Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems I

Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems II

SOLID MECHANICS AND ITS APPLICATIONS Volume 156

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A.F. Vakakis • O.V. Gendelman • L.A. Bergman • D.M. McFarland • G. Kerschen • Y.S. Lee

Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems I

A.F. Vakakis • O.V. Gendelman • L.A. Bergman • D.M. McFarland • G. Kerschen • Y.S. Lee

Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems II

Alexander F. Vakakis Department of Mechanical Science and Engineering University of Illinois Urbana, Illinois, USA and Mechanics Division National Technical University of Athens Athens, Greece Lawrence A. Bergman Department of Aerospace Engineering University of Illinois Urbana, Illinois, USA

Oleg V. Gendelman Faculty of Mechanical Engineering Technion – Israel Institute of Technology Haifa, Israel D. Michael McFarland Department of Aerospace Engineering University of Illinois at Urbana-Champaign Urbana, Illinois, USA Young Sup Lee Department of Mechanical and Aerospace Engineering New Mexico State University Las Cruces, New Mexico, USA

Gaëtan Kerschen Department of Aerospace and Mechanical Engineering University of Liège Liège, Belgium

ISBN-13: 978-1-4020-9125-4

e-ISBN-13: 978-1-4020-9130-8

Library of Congress Control Number: 2008940435 © 2008 Springer Science+Business Media, B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 987654321 springer.com

Contents Volume 1

Preface

vii

Abbreviations

xi

1 Introduction

1

2 Preliminary Concepts, Methodologies and Techniques

15

2.1 2.2 2.3 2.4 2.5

Nonlinear Normal Modes (NNMs) Energy Localization in Nonlinear Systems Internal Resonances, Transient and Sustained Resonance Captures Averaging, Multiple Scales and Complexification Methods of Advanced Signal Processing 2.5.1 Numerical Wavelet Transforms 2.5.2 Empirical Mode Decompositions and Hilbert Transforms 2.6 Perspectives on Hardware Development and Experiments

16 28 38 54 70 71 77 81

3 Nonlinear Targeted Energy Transfer in Discrete Linear Oscillators with Single-DOF Nonlinear Energy Sinks

93

3.1 Configurations of Single-DOF NESs 3.2 Numerical Evidence of TET in a SDOF Linear Oscillator with a SDOF NES 3.3 SDOF Linear Oscillators with SDOF NESs: Dynamics of the Underlying Hamiltonian Systems 3.3.1 Numerical Study of Periodic Orbits (NNMs) 3.3.2 Analytic Study of Periodic Orbits (NNMs) 3.3.3 Numerical Study of Periodic Impulsive Orbits (IOs) 3.3.4 Analytic Study of Periodic and Quasi-Periodic IOs 3.3.5 Topological Features of the Hamiltonian Dynamics 3.4 SDOF Linear Oscillators with SDOF NESs: Transient Dynamics of the Damped Systems 3.4.1 Nonlinear Damped Transitions Represented in the FEP 3.4.2 Dynamics of TET in the Damped System

93 98 108 108 124 135 137 157 165 166 171

v

vi

3.5 Multi-DOF (MDOF) Linear Oscillators with SDOF NESs: Resonance Capture Cascades and Multi-frequency TET 3.5.1 Two-DOF Linear Oscillator with a SDOF NES 3.5.2 Semi-Infinite Chain of Linear Oscillators with an End SDOF NES 4 Targeted Energy Transfer in Discrete Linear Oscillators with Multi-DOF NESs

Contents

233 237 269 303

4.1 Multi-Degree-of-Freedom (MDOF) NESs 4.1.1 An Alternative Way for Passive Multi-frequency Nonlinear Energy Transfers 4.1.2 Numerical Evidence of TET in MDOF NESs 4.2 The Dynamics of the Underlying Hamiltonian System 4.2.1 System I: NES with O(1) Mass 4.2.2 System II: NES with O(ε) Mass 4.2.3 Asymptotic Analysis of Nonlinear Resonant Orbits 4.2.4 Analysis of Resonant Periodic Orbits 4.3 TRCs and TET in the Damped and Forced System 4.3.1 Numerical Wavelet Transforms 4.3.2 Damped Transitions on the Hamiltonian FEP 4.4 Concluding Remarks

303

Index

369

304 309 317 320 325 328 336 347 347 352 365

Contents Volume 2

5 Targeted Energy Transfer in Linear Continuous Systems with Singleand Multi-DOF NESs

1

5.1 Beam of Finite Length with SDOF NES 5.1.1 Formulation of the Problem and Computational Procedure 5.1.2 Parametric Study of TET 5.2 Rod of Finite Length with SDOF NES 5.2.1 Formulation of the Problem, Computational Procedure and Post-Processing Algorithms 5.2.2 Computational Study of TET 5.2.3 Damped Transitions on the Hamiltonian FEP 5.3 Rod of Semi-Infinite Length with SDOF NES 5.3.1 Reduction to Integro-differential Form 5.3.2 Numerical Study of Damped Transitions 5.3.3 Analytical Study 5.4 Rod of Finite Length with MDOF NES 5.4.1 Formulation of the Problem and FEPs 5.4.2 Computational Study of TET 5.4.3 Multi-Modal Damped Transitions and Multi-Scale Analysis 5.5 Plate with SDOF and MDOF NESs 5.5.1 Case of a SDOF NES 5.5.2 Case of Multiple SDOF NESs 5.5.3 Case of a MDOF NES 5.5.4 Comparative Study with Linear Tuned Mass Damper

1 1 6 12 13 18 39 66 67 75 86 99 99 109 117 132 142 147 150 153

6 Targeted Energy Transfer in Systems with Periodic Excitations

161

6.1 Steady State Responses and Generic Bifurcations 6.1.1 Analysis of Steady State Motions 6.1.2 Numerical Verification of the Analytical Results 6.2 Strongly Modulated Responses (SMRs) 6.2.1 General Formulation and Invariant Manifold Approach 6.2.2 Reduction to One-Dimensional Maps and Existence Conditions for SMRs 6.2.3 Numerical Simulations 6.3 NESs as Strongly Nonlinear Absorbers for Vibration Isolation 6.3.1 Co-existent Response Regimes

162 162 175 177 177 187 194 202 202 v

vi

Contents

6.3.2 Efficiency and Broadband Features of the Vibration Isolation 6.3.3 Passive Self-tuning Capacity of the NES

206 213

7 NESs with Non-Smooth Stiffness Characteristics

229

7.1 System with Multiple NESs Possessing Clearance Nonlinearities 7.1.1 Problem Description 7.1.2 Numerical Results 7.2 Vibro-Impact (VI) NESs as Shock Absorbers 7.2.1 Passive TET to VI NESs 7.2.2 Shock Isolation 7.3 SDOF Linear Oscillator with a VI NES 7.3.1 Periodic Orbits for Elastic Vibro-Impacts Represented on the FEP 7.3.2 Vibro-Impact Transitions in the Dissipative Case: VI TET

229 230 233 241 242 251 259

8 Experimental Verification of Targeted Energy Transfer

311

8.1 TET to Ungrounded SDOF NES (Configuration II) 8.1.1 System Identification 8.1.2 Experimental Measurements 8.2 TET to Grounded SDOF NES (Configuration I) 8.2.1 Experimental Fixture 8.2.2 Results and Discussion 8.3 Experimental Demonstration of 1:1 TRCs Leading to TET 8.3.1 Experimental Fixture 8.3.2 Experimental TRCs 8.4 Steady State TET under Harmonic Excitation 8.4.1 System Configuration and Theoretical Analysis 8.4.2 Experimental Study

311 312 314 320 322 324 330 331 333 342 344 347

9 Suppression of Aeroelastic Instabilities through Passive Targeted Energy Transfer

353

9.1 Suppression of Limit-Cycle Oscillations in the van der Pol Oscillator 9.1.1 VDP Oscillator and NES Configurations 9.1.2 Transient Dynamics 9.1.3 Steady State Dynamics and Bifurcation Analysis 9.1.4 Summary of Results 9.2 Triggering Mechanism for Aeroelastic Instability of an In-Flow Wing 9.2.1 The Two-DOF Aeroelastic Model 9.2.2 Slow Flow Dynamics 9.2.3 LCO Triggering Mechanism 9.2.4 Concluding Remarks 9.3 Suppressing Aeroelastic Instability of an In-Flow Wing Using a SDOF NES 9.3.1 Preliminary Numerical Study 9.3.2 Study of LCO Suppression Mechanisms

260 279

353 355 359 381 394 397 399 402 428 447 453 453 463

Contents

vii

9.3.3 Robustness of LCO Suppression 9.3.4 Concluding Remarks 9.4 Experimental Validation of TET-Based, Passive LCO Suppression 9.4.1 Experimental Apparatus and Procedures 9.4.2 Results and Discussion 9.5 Suppressing Aeroelastic Instability of an In-Flow Wing Using a MDOF NES 9.5.1 Revisiting the SDOF NES Design 9.5.2 Configuration of a Wing with an Attached MDOF NES 9.5.3 Robustness of LCO Suppression – Bifurcation Analysis 9.6 Preliminary Results on LCO Suppression in a Wing in Unsteady Flow

487 501 502 502 508 520 520 532 536 559

10 Seismic Mitigation by Targeted Energy Transfer

571

10.1 The Two-DOF Linear Primary System with VI NES 10.1.1 System Description 10.1.2 Simulation and Optimization 10.1.3 Computational Results 10.2 Scaled Three-Story Steel Frame Structure with NESs 10.2.1 Characterization of the Three-Story Linear Frame Structure 10.2.2 Simulation and Optimization of the Frame-Single VI NES System 10.2.3 Simulation and Optimization of the Frame-VI NES-Smooth NES System 10.3 Experimental Verification 10.3.1 System Incorporating the Single-VI NES 10.3.2 System Incorporating Both VI and Smooth NESs 10.4 Observations, Summary and Conclusions

572 572 574 576 581 582 585 595 603 606 609 615

11 Suppression of Instabilities in Drilling Operations through Targeted Energy Transfer

619

11.1 Problem Description 11.2 Instability in the Drill-String Model 11.3 Suppression of Friction-Induced Limit Cycles by TET 11.3.1 Addition of an NES to the Drill-String System 11.3.2 Parametric Study for Determining the NES Parameters 11.4 Detailed Analysis of the Drill-String with NES Attached 11.4.1 NES Efficacy 11.4.2 Robustness of LCO Suppression 11.4.3 Transient Resonance Captures 11.5 Concluding Remarks

620 625 627 629 630 633 634 637 639 640

12 Postscript

645

Index

649

Preface

This monograph evolved over a period of nine years from a series of papers and presentations addressing the subject of passive vibration control of mechanical systems subjected to broadband, transient inputs. The unifying theme is Targeted Energy Transfer – TET, which represents a new and unique approach to the passive control problem, in which a strongly nonlinear, fully passive, local attachment, the Nonlinear Energy Sink – NES, is employed to drastically alter the dynamics of the primary system to which it is attached. The intrinsic capacity of the properly designed NES to promote rapid localization of externally applied (narrowband) vibration or (broadband) shock energy to itself, where it can be captured and dissipated, provides a powerful strategy for vibration control and the opens the possibility for a wide range of applications of TET, such as, vibration and shock isolation, passive energy harvesting, aeroelastic instability (flutter) suppression, seismic mitigation, vortex shedding control, enhanced reliability designs (for example in power grids) and others. The monograph is intended to provide a thorough explanation of the analytical, computational and experimental methods needed to formulate and study TET in mechanical and structural systems. Several practical engineering applications are examined in detail, and experimental verification and validation of the theoretical predictions are provided as well. The authors also suggest a number of possible future applications where application of TET seems promising. The authors are indebted to a number of sponsoring agencies. The Office of Naval Research – ONR (AFV, LAB), the National Science Foundation – NSF (AFV, DMM, LAB), the Air Force Office of Scientific Research – AFOSR (AFV, DMM, YSL, LAB), the Fund for Basic Research of the National Technical University of Athens (AFV), the Hellenic Secretariat for Research and Development (AFV), the Horev Fellowship Trust, the Israel Science Foundation (OG), the Belgian National Fund for Scientific Research (GK), and the Mavis Memorial Fund Fellowship of the College of Engineering of the University of Illinois (YSL) provided financial support for this work which enabled the realization of this long-standing, multinational collaborative research effort. In addition, the authors are greatly appreciative

vii

viii

Preface

of the long-standing interest and support of Mr. Jim Lally of PCB Piezotronics, Inc., whose generosity through equipment support has enabled much of the experimental work reported in this monograph. The authors would like to express their gratitude to Professor Leonid Manevitch of the sem*nov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russia, for his many shared insights which have proved to be critical to many of the developments included in this monograph; the authors greatly benefited from many stimulating discussions with him over the years in Moscow, Athens and Urbana. In addition, AFV will be always indebted to his late advisor and teacher, Professor Thomas K. Caughey, who through his teaching of asymptotic techniques, nonlinear dynamics and mechanical vibrations influenced much of the analytical work included in this work. Many colleagues and former or current graduate and undergraduate students of the authors have, through their contributions, influenced various parts of this research effort. The authors are indeed indebted to (in no particular order) Dr. S. Tsakirtzis, Dr. F. Georgiadis, Dr. P. Panagopoulos, Dr. F. Nucera, Prof. Yu.V. Mikhlin, Prof. D. Quinn, Prof. R. H. Rand, Prof. T. Strganac, Prof. P. Cizmas, Prof. T. Kalmar-Nagy, Prof. V. Rothos, Prof. V. Pilipchuck, Dr. D. Gorlov, Dr. A. Musienko, Prof. X. Ma, Dr. X. Jiang, Mr. T. Sapsis, Mr. Yu. Starosvetsky, Mr. I. Karayannis, Mr. W.J. Hill, Mr. C. Nichkawde, Mr. R. Viguie, Prof. J.C. Golinval, Prof. A. Santini, Prof. D. Wang, Prof. S.C. Hong, Dr. A. Musienko, Mr. P. Kourdis, Mr. R. Viguie, Mr. J. Kowtko, Dr. Y. Wang, Mr. S. Hubbard, Ms. N. Fanouraki, Mr. C. Dumcum, Ms. C. Tripepi, Mr. F. Lo Iacono, Mr. G. Barone, Ms. M. Wise, and Ms. I. Rizou. The Departments of Aerospace Engineering and Mechanical Science and Engineering at the University of Illinois at Urbana- Champaign, the College of Engineering of the University of Illinois at Urbana-Champaign, and the Mechanics Division of the Department of Applied Sciences of the National Technical University of Athens generously provided space, resources, travel accommodation and hospitality during this nine year period, for which the authors are indeed grateful. In addition, the authors would like to acknowledge the use of materials from their papers in archival journals and conference proceedings published by Springer Verlag, the American Society of Mechanical Engineers – ASME, Elsevier, World Scientific, the Society of Industrial and Applied Mathematics – SIAM, the American Institute of Aeronautics and Astronautics – AIAA, the American Institute of Physics – AIP, Sage Journals, and John Wiley & Sons. The original sources of these materials are referenced throughout this monograph. Moreover, the authors are appreciative of the efforts of the editorial staff at Springer Verlag, particularly of Ms. Nathalie Jacobs and Ms. Anneke Pot of Springer NL, and of the careful job of typesetting performed by Karada Publishing Services, particularly Ms. Jolanda Karada. Of course, any errors that remain are solely the responsibility of the authors. Last, but not least, the authors would like to express their gratitude to their family members for their continued and unconditional support over the course of this long effort: Fotis Sr., Anneta, Elpida, Brian, Elias, Vasiliki, Sotiria, Marianna

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

ix

and Fotis Jr. (AFV), Anechka, Miriam, Sheina and Hava (OG), Jane (LAB), Karen (DMM), Carine, Julie and Maxime (GK), Ju Eun, Rose and Erica (YSL). This monograph is dedicated to them. A.F. Vakakis O. Gendelman L.A. Bergman D.M. McFarland G. Kerschen Y.S. Lee

Abbreviations

AZ BHA BPC CEP CX-A DOF EDM EMD ETM FE FEP FFT FI FRET HF IFC IMF IO LCO LF LO LPC LSR MDOF MF NATA NES NLBVP NLS eq. NNM NP

Attenuation zone Bottom-hole-assembly Branch point of cycles Critical energy peak Complexification-averaging Degree-of-freedom Energy dissipation measure Empirical mode decomposition Energy transaction measure Finite element Frequency-energy plot Fast Fourier Transform Frequency index Fluorescence resonance energy transfer High frequency Impulsive forcing condition Intrinsic mode function Impulsive orbit Limit cycle oscillation Low frequency Linear oscillator Limit point cycle Lyapunov–Schmidt reduction Multi-degree-of-freedom Middle frequency Nonlinear aeroelastic test apparatus Nonlinear energy sink Nonlinear boundary value problem Nonlinear Schrödinger equation Nonlinear normal mode North pole

xi

xii

NS bifurcation NS NES POD POM PZ RCC r.m.s. response SDOF SIM SMR SN bifurcation SP SRC TET TMD VDP oscillator VI WT

Abbreviations

Neimark–Sacker bifurcation Non-smooth NES Proper orthogonal decomposition Proper orthogonal mode Propagation zone Resonance capture cascade Root mean square response Single-degree-of-freedom Slow invariant manifold Strongly modulated response Saddle-node bifurcation South pole Sustained Resonance Capture Targeted energy transfer Tuned mass damper Van der Pol oscillator Vibro-impact Wavelet transform

Chapter 1

Introduction

Any process in nature involves to a certain extent some type of energy transfer. From an engineering point of view, certain processes of energy transfer are undesired but still inevitable, as, for instance, energy dissipation in electromechanical systems; whereas other processes are desired and highly beneficial to the design objectives, the classical example from mechanical engineering being the addition of a vibration absorber to a machine for eliminating unwanted disturbances. Targeted energy transfers (TETs), where energy of some form is directed from a source (donor) to a receiver (recipient) in a one-way irreversible fashion, govern a broad range of physical phenomena. One basic example of TET in nature, is resonance-driven solar energy harvesting governing photosythesis (Jenkins et al., 2004), where energy from the Sun is captured by photobiological antenna chromophores and is then transferred to reaction centers through a series of interactions between chromophore units (van Amerongen et al., 2000; Renger et al., 2001). In addition, basic problems in biopolymers concern energy self-focusing, localization and transport (Kopidakis et al., 2001), with applications in photosynthesis (Hu et al., 1998) and bioenergetic processes (Julicher et al., 1997). From the engineering point of view, the scaling down of engineering applications from macro- to micro- and nano-scales dictates an understanding of the mechanisms governing TET and energy exchanges between components possessing different characteristic lengths with dynamics governed by different time-scales. For example, as pointed out by Wang et al. (2007) in applications such as molecular electronic devices where the scales of the dynamics are at the level of individual molecules, classical concepts of heat transport do not apply, and heat is transported by energy transfer through discrete molecular vibration excitations. Hence, understanding and analyzing energy transfer mechanisms in molecular dynamics (such as, resonance energy transfer) is key in conceiving devices or studying processes for specific macromolecular applications, such as, for example, in the area of photophysics (Andrews, 2000; Jenkins and Andrews, 2002, 2003). Moreover, molecular dynamic simulations of energy transfers (for example, through solitonic waves) in mechanistic molecular or atomistic models have been used to study thermodynamic processes, such as, melting of polymer crystals and phase transitions in polymer-

1

2

1 Introduction

clay nanocomposites (Ginzburg and Manevitch, 1991; Berlin et al., 1999; Ginzburg et al., 2001; Berlin et al., 2002; Gendelman et al., 2003). In Musumeci et al. (2003) issues related to nonlinear mechanisms for energy transfer and localization in biological macromolecules and related applications to biology are discussed. Moreover applications of nonlinear energy transfer in a broad area of applications ranging from cancer detection (Meessen, 2000; Vedruccio and Meessen, 2004) to wireless power transfer (Kurs et al., 2007) have been reported in the recent literature. Therefore, it is not surprising that TET phenomena have received much attention in applications from diverse fields of applied mathematics, applied physics, and engineering. Representative examples are the works by Aubry and co-workers on passive targeted energy transfer (TET) between nonlinear oscillators and/or discrete breathers (Kopidakis et al., 2001; Aubry et al., 2001; Maniadis et al., 2004; Memboeuf and Aubry, 2005; Maniadis and Aubry, 2005), on breather-phonon resonances (Morgante et al., 2002), and on quantum TET between nonlinear oscillators (Maniadis et al., 2004). The dynamical mechanisms considered in these works were based on imposing conditions of nonlinear resonance between interacting dynamical systems in order to achieve TET from one to the other, and then ‘breaking” this condition at the end of the energy transfer to make it irreversible. A mechanism of TET along a line or surface by means of coherent traveling solitary waves is examined in Nistazakis et al. (2002); specifically, the transfer of a solitary wave to a targeted position was studied in the nonlinear Schrödinger (NLS) equation, the underlying nonlinear dynamical mechanism being resonance energy transfer from an ac drive to the solitary wave. Applications of energy localization and TET in diverse applications, such as, biological macromolecules – proteins and DNA, arrays of Josephson junctions in superconductivity applications, and molecular crystals are given in Dauxois et al. (2004), including analytical, computational and experimental results. In other complex phenomena, such as turbulence and chaotic dynamics, multiscale energy transfers between different spatial and temporal scales govern the dynamics. Perhaps the best known example is turbulence, where mechanical energy is supplied to a fluid system at relatively large length scales, peculiar spatiotemporal coherent structures are formed at intermediate scales, and dissipation of energy occurs at short scales (Bohr et al., 1998). Hence, energy transfer between these scales is what makes turbulence possible. Examples of works on multi-scale energy transfers in fluids are the works by Kim et al. (1996) and Tran (2004) who studied nonlinear energy transfers in fully developed turbulence, and by Brink et al. (2004) who studied nonlinear interactions and multi-scale energy transfers among inertial modes of a rotating fluid, modeling it as a network of coupled oscillators. All nonlinear energy transfers involve to a certain extent some type of nonlinear resonance between a donor and a receptor. Resonance energy transfer has been identified as an important mechanism for energy and electronic transports in the area of photophysics of macromolecules (Jenkins and Andrews, 2002, 2003; Andrews and Bradshaw, 2004), and has been recognized as the principal mechanism for electronic energy transport in molecular chains following initial excitations (Daniels et al., 2003). Esser and Henning (1991) analyzed energy transfer and bi-

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

3

furcations in a condensed molecular system. Fluorescence resonance energy transfer (FRET) where fluorescent energy from an excited fluorophore is transferred to lightabsorbing molecules lying in close proximity, has been well studied; FRET has been applied as an optical microscopy technique for developing biosensors and examining physiological processes with highly temporal and spatial resolution (Cardullo and Parpura, 2003; Berland et al., 2005). Additional applications of FRET range from in vivo medical diagnosis of infections (Hwang et al., 2006), to detection of targeted DNA sequences (Xu et al., 2005), and development of biosensors and medical probes (Yesilkaya et al., 2006). Dodaro and Herman (1998) studied analytically energy transfers in liquids through resonant vibration interactions, using a molecular dynamics approach to study the probability of vibration energy transfer between atoms. An example of a study of laser-assisted resonance energy transfer is provided in Allco*ck et al. (1999), and application of TET in the field of molecular motors of cochlear cells was considered in Spector (2005). An important additional application of TET is in the area of energy harvesting, that is, of the development of efficient and reliable energy harvesters capable of efficiently capturing ambient energy from a variety of media. For example, photosynthetic organisms have developed efficient sunlight harvesting apparatus to fuel their metabolisms (Hu et al., 1998); also, dendrimeric polymers are being considered as energy harversters in nanodevices (Andrews and Bradshaw, 2004). In engineering applications energy harvesters were studied for converting ambient vibrations into usable electrical energy (Glynne-Jones et al., 2004; Lesieutre et al., 2004; Cornwell et al., 2005; Roundy, 2005; Kim et al., 2005; Stephen, 2006) but their performance is limited by the fact that ambient vibration is often low-level and broadband, occurring in random bursts. The passive nonlinear TET designs discussed in this work, result in broadband energy transfer between structural components, in some cases even at low energy levels; hence, they hold promise towards alleviating the current restrictions of current energy mechanical harvesters of ambient vibration. Additional studies of nonlinear energy transfer in lattice models have been performed to model heat flux and test the validity of the classical Fourier law of heat conduction (Gendelman and Savin, 2004; Balakrishnan and Van den Broeck, 2005). Wang (1973) analyzed TET between nonlinearly interacting waves, and Kevrekidis et al. (2004) studied localization and resonance-induced energy transfer in mechanical lattices with geometric nonlinearity. Spire and Leon (2004) studied energy absorption due to resonance of impeding waves by discrete molecular chains, resulting in generation of solitons in these chains; it was found that both nonlinearity and discreteness effects are prerequisites for this type of nonlinear energy absorption. If we restrict our focus to purely mechanical systems possessing no dissipation and executing vibrations, still one can point out a variety of dynamic phenomena involving strong nonlinear energy transfers. Often the process of passive nonlinear vibration energy exchange is described in terms of nonlinear interaction between different structural modes with either close or well-separated frequencies. Such exchange is not possible in linear dynamical systems since, except for the case of modes with closely spaced frequencies (giving rise to the classical beat phenom-

4

1 Introduction

enon), modes in these systems are uncoupled and can not exchange energy between them in a passive way. In the presence of nonlinearity, however, nonlinear energy interactions can occur due to internal resonances, even between structural modes with widely spaced frequencies (Guckenheimer and Holmes, 1983; Wiggins, 1990; Nayfeh and Mook, 1995; Nayfeh, 2000). In nonlinear Hamiltonian systems irreversible transfer of energy is generally precluded due to conservation of the phase volume and by virtue of the Poincaré recurrence theorem; however, in certain cases the Hamiltonian dynamics can be trapped in bounded regions of the state space for relatively long time, with subsequent release (Zaslavskii, 2005). In addition, there are special cases where complete and irreversible (targeted) transfer of energy occurs between coupled nonlinear oscillators [(Nayfeh and Mook, 1995; see also the discussion of Fermi Targeted Energy Transfer in Maniadis and Aubry (2005)]; such irreversible nonlinear energy transfers occur on heteroclinic orbits of appropriately defined slow flows of the dynamics, they occur asymptotically as time tends to infinity, and are not robust as they are realized only at specific energy levels (in fact, perturbations of these orbits destroys the irreversibility of energy transfer, and lead to excitations of quasi-periodic orbits in the slow flows). In general, nonlinear energy transfers may be realized due to symmetry-breaking as nonlinear mode bifurcations, or through spatial energy localization phenomena from the formation of localized nonlinear normal modes (NNMs) (King and Vakakis, 1995; Boivin et al., 1995; Vakakis et al., 1996; Vakakis et al., 2002; Lacarbonara et al., 2003; Jiang et al., 2005). In the works by Nayfeh and Nayfeh (1994), Nayfeh and Mook (1995), Oh and Nayfeh (1998), Nayfeh (2000) and Malatkar and Nayfeh (2003) a new form of nonlinear energy transfer between widely spaced modes in harmonically forced structures is analyzed; this mechanism of passive energy transfer is caused by resonance interaction of the slow modulation of a higher mode (generated from a Hopf bifurcation) with a lower one. This type of energy transfer is peculiar, in the sense that the interacting modes need not satisfy conditions of internal resonance. Kerschen et al. (2008) discuss an alternative form of nonlinear modal interaction between highly energetic NNMs. Indeed, at low energies these modes may possess incommensurate linearized natural frequencies so they do not satisfy internal resonance conditions. Due to the energy dependence of their frequencies, however, at higher energies the same NNMs may become internally resonant, as their energy-dependent frequencies may become commensurate resulting in strong nonlinear modal interactions. This underlines the fact that important, essentially nonlinear phenomena (such as this one) may be missed when resorting to perturbation techniques based on linear (harmonic) generating functions, whose range of validity is restricted to small-amplitude motions and/or weak nonlinearities [but for a perturbation technique based on strongly nonlinear yet simple (non-smooth) generating functions, valid in strongly nonlinear regimes (but not in weakly nonlinear ones!) refer to Pilipchuk (1985, 1988, 1996), Pilipchuk et al. (1997) and Pilipchuk and Vakakis (1998)]. This monograph is devoted to the study of targeted energy transfer (TET) phenomena in dissipative mechanical and structural systems possessing essentially non-

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

5

linear local attachments. We will show that the addition to a linear system of a local attachment possessing essential (nonlinearizable) stiffness nonlinearity, may significantly alter the global dynamics of the resulting integrated system. The reason lies in the lack of a preferential resonance frequency of the attachment, which, in principle, enables it to engage in nonlinear resonance with any mode of the linear system, at arbitrary frequency ranges (provided, of course, that no mode has a node in the neighborhood of the point of attachment). The actual scenario of single-mode or multi-mode nonlinear resonance interaction of the attachment with the linear system will depend on the level and spatial distribution of the instantaneous vibration energy of the integrated system. We will show that under certain conditions, passive TET from the linear system to the NES occurs, i.e., a one-way and irreversible (on the average) flow of energy from the linear system to the attachment, which acts, in effect, as a nonlinear energy sink – NES. Moreover, in contrast to the classical linear vibration absorber whose action is narrowband, we will show that under certain conditions the NES can resonantly interact with the linear system in a broadband fashion, and engage in a resonance capture cascade with a set of structural modes over a broad frequency range; then the NES, acts in essence, as a passive, adaptive, broadband boundary controller. Hence, viewed in the context of vibration theory, the NES can be regarded as a generalization of the concept of the classical linear vibration absorber (or tuned mass damper – TMD). Viewed in the context of the theory of dynamical systems, however, the addition of the essentially nonlinear NES introduces degeneracies in the free and forced dynamics of the integrated system, opening the possibility of higher co-dimensional bifurcations and complex dynamical phenomena, certain of which might be compatible to the design objectives of the specific engineering application considered. As a preliminary illustrative example of TET, we consider a two degree-offreedom (DOF) dissipative unforced system described by the following equations: y¨1 + λ1 y˙1 + y1 + λ2 (y˙1 − y˙2 ) + k(y1 − y2 )3 = 0 εy¨2 + λ2 (y˙2 − y˙1 ) + k(y2 − y1 )3 = 0.

(1.1)

Physically, these equations describe a damped linear oscillator (LO) with mass and natural frequency normalized to unity, and viscous damping coefficient λ1 ; and an essentially nonlinear attachment with normalized mass ε, normalized nonlinear stiffness coefficient k, and viscous damping coefficient λ2 . Note that system (1.1) cannot be regarded as a small perturbation of a linear system due to the strongly nonlinear coupling terms. The detailed study of this type of dynamical systems is postponed until Chapter 3, and here we will only provide a brief numerical demonstration of TET by studying its transient dynamics. To this end, we simulate numerically system (1.1) for parameter values ε = 0.1, k = 0.1, λ1 = 0.01 and λ2 = 0.01. The selected initial conditions correspond to an impulse F = Aδ(t) imposed to the linear oscillator [where δ(t) is Dirac’s delta function – this impulsive forcing is equivalent to imposing the initial velocity y˙1 (0+) = A] with the system being initially at rest, i.e., y1 (0) = y2 (0) = y˙2 (0) = 0

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1 Introduction

Fig. 1.1 Evolution of the energy ratio κ for impulse strength A = 0.5.

and y˙1 (0+) = A. Hence, the initial energy is stored only in the LO. The instantaneous transfer of energy from the LO to the nonlinear attachment can be monitored by computing the non-dimensional energy ratio κ, which denotes the portion of instantaneous total energy stored in the nonlinear attachment, κ=

E2 , E1 + E2

E1 =

1 2 (y + y˙12 ), 2 1

E2 =

ε k y˙2 + (y1 − y2 )4 , 2 4

(1.2)

where E1 and E2 are the instantaneous energies of the LO and the attachment, respectively. Of course, all quantities in relations (1.2) are time dependent. In Figures 1.1 and 1.2 we depict the evolution of the energy ratio κ for impulse strengths A = 0.5 and A = 0.7, respectively. From Figure 1.1 it is clear that only a small amount of energy (of the order of 7%) is transferred from the LO to the nonlinear attachment. However, for a slightly higher impulse the energy transferred climbs to almost 95% (see Figure 1.2), within a rather short time (up to t = 15, which is much less than the characteristic time of viscous energy dissipation in the LO). In this case, almost the entire impulsive energy is passively transferred from the LO to the nonlinear attachment, which acts as nonlinear energy sink. It should be mentioned that the mass of the attachment in this particular example is just 10% of the mass of the LO (and it will be shown that this mass can be reduced even further with similar TET results). From this example, it appears that passive TET from the directly excited LO to the essentially nonlinear attachment in (1.1) is realized when the energy exceeds a certain critical threshold. The mathematical description of the TET process poses distinct challenges, since this phenomenon is transient (instead of steady state), and

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7

Fig. 1.2 Evolution of the energy ratio κ for impulse strength A = 0.7.

essentially nonlinear (instead of weakly nonlinear). Traditionally, what one does when dealing with systems of coupled oscillators like (1.1) is to consider the structure of periodic or quasi-periodic orbits of the corresponding undamped, Hamiltonian system; however, given that TET is a strongly nonlinear transient phenomenon that occurs in the dissipative system, at this point it remains unclear what its relation is to the dynamics of the underlying Hamiltonian system. In addition, it is not obvious how to analytically study the TET phenomenon, as this occurs in the strongly damped transient dynamics, where the majority of current techniques from nonlinear dynamics are inapplicable; yet an analytical study of TET is required in order for one to gain an understanding of the underlying dynamics, and apply it to practical engineering designs. Some additional obvious open questions that arise from this preliminary example concern the time scale of TET compared to the characteristic time scales of the dynamics of the LO and the NES; the realization and robustness of TET subject to other initial conditions or external excitations (such as, for example, time periodic ones); and the possible extension of TET to multi-degreeof-freedom linear oscillators or flexible structures with local essentially nonlinear attachments. These are some of the problems that we will be concerned with in this monograph. According to the commonly accepted and perhaps correct point of view, the historical development of mechanics and, in particular, dynamics since Newton became possible because the observed motion of celestial bodies was modeled by almost conservative and nearly integrable mathematical dynamical models. Thus, the statement that the orbits of planets are ellipses allowed Newton to discover the classical laws of gravitation and motion. The solution of this problem initiated a breath of important developments in applied mathematics, which eventually grew to the gen-

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1 Introduction

eral theory of integrable systems (Arnold, 1980). This type of systems possesses as many independent first integrals of motion as their number of degrees of freedom, so their n-dimensional dynamics may be reduced (at least in principle) to singledegree-of-freedom (SDOF) dynamical systems; alternatively put, the n-dimensional phase spaces of these systems are foliated by families of n-dimensional tori, so their motions may be reduced to periodic or quasi-periodic rotations on the surfaces of multi-dimensional tori. For such integrable dynamical systems no energy exchanges can occur between different modes of rotation and, of course, a process like TET is not possible at all. Later observations demonstrated that both assertions mentioned above concerning the dynamics of celestial bodies (that is conservativity and integrability) are not exact. Indeed, these systems are not exactly conservative, primarily due to tidal phenomena; the famous manifestation of these effects is the one sidedness of the Moon. Celestial systems are also not exactly integrable, as gravitational multi-body interactions spoil the integrability; this led to the study of the celebrated threebody problem, whose proof of non-integrability led Poincaré to the development of modern geometrical dynamical systems theory and chaotic dynamics (Poincaré, 1899; Barrow-Green, 1996). Indeed, despite numerous attempts of more than two centuries, the three-body problem, that is, the dynamics of three bodies interacting via gravitational forces could not be analytically solved, until it was proven by Poincaré to be non-integrable. Until the time of Poincaré common wisdom was the Lagrangian view, that once a dynamical system is modeled by a set of differential equations its analytical solution is a matter of developing the necessary mathematical techniques; Poincaré proved that there are dynamical systems – even of simple configuration – for which no analytical solutions can exist (hence, for example, the impossibility of long-term weather prediction). Despite their non-integrability, dynamical systems in celestial mechanics are often close to integrable ones, with the characteristic value of the perturbations from integrability being of the order of about 10−3 or less. This is the reason that problems of celestial mechanics provided also a major thrust to the development of regular and singular perturbation techniques in applied mathematics. With the help of these techniques, the dynamics of integrable Hamiltonian systems perturbed by small Hamiltonian perturbations were analyzed and understood rather well in the framework of the celebrated KAM (Kolmogorov–Arnold–Moser) theory (Arnold, 1963a, 1963b, 1964). If the perturbation is small enough, then for the majority of initial conditions quasi-periodic motions of the perturbed Hamiltonian system persist under the perturbation (on ‘sufficiently irrational’ multi-dimensional tori), whereas for special values of initial conditions (corresponding to countably infinite internal resonances of the perturbed Hamiltonian system) invariant tori are destroyed and replaced by thin layers of chaotic motions. These chaotic layers prevent the existence of a sufficient number of independent analytic independent first integrals of motion, leading to non-integrability of the perturbed Hamiltonian system. Much less is known about the effects of non-Hamiltonian perturbations, and so in this area theoretical developments concern mainly low-dimensional systems. The main effects known include scattering by resonance and capture into the resonance

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(Armold, 1988). The former effect occurs when the orbit of the perturbed system is slightly modified due to passage through resonance [by a perturbation of O(ε1/2 ), where ε is the characteristic strength of the perturbation]. Capture into the resonance occurs for a small subset of initial conditions, but the resulting variation of the perturbed trajectory is of O(1). Still, many interesting and important dynamical systems of practical significance can not be described as small perturbations of integrable ones. For this more general class of systems there are no rigorous analytical methods of solution. One approach for analyzing this class of problems is to try to apply perturbation techniques far beyond the formal boundaries of their applicability (sometimes such an approach can bring about success, for example, see the method discussed in Section 2.4). Another approach is to seek some important partial solution of the problem, for example, with the help of methods such as, harmonic balance, multiple scales, nonlinear normal modes (NNMs) (Nayfeh and Mook, 1995; Vakakis et al., 1996; Verhulst, 2005). The systems under consideration in this work, exhibiting passive TET phenomena, belong to this latter category. Indeed, considering the dynamical system of the preliminary example (1.1), it is non-integrable, non-Hamiltonian, and besides some special cases cannot be expressed in the form of a perturbed integrable dynamical system. As mentioned previously, added challenges arise due to the type of responses that we will be interested in, namely, damped transient motions instead of steady state ones. It follows that standard perturbation techniques from the theory of dynamical systems dealing with periodic or nearly periodic motions are generally inapplicable in the problem of TET in systems of coupled discrete or continuous oscillators. Moreover, it is not clear for what types of dynamical systems is TET possible at all, and, even if it is possible, under what conditions it can be realized. Hence, in our study of TET in mechanical and structural systems certain important issues need to be addressed, including: • The type of structural modification needed for realization of passive TET in a dynamical system, and the class of dynamical systems capable of TET. • The robustness of the TET phenomenon to changes (and uncertainties) in system parameters, initial conditions and external excitations. • The physical understanding of the dynamical mechanisms governing the TET, and the mathematical analysis of TET. • The ways to enhance and optimize TET in a system according to a specific set of design objectives, and in the framework of practical applications where TET is useful. • The comparison of TET-based designs to alternative current linear or nonlinear, passive or active designs. • Provided that TET is theoretically proven to be beneficial according to a specific design objective, the practical implementation of these designs in engineering applications. In this monograph we will attempt to address some of these issues, but, of course, no complete answers to all questions can be provided at this point. Instead, this work can be regarded as a first attempt towards addressing some of the above issues, and,

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1 Introduction

thus, as a demonstration that the nonlinear phenomenon of passive TET, under certain conditions, can prove to be beneficial to a broad range of practical engineering applications. This monograph is intended for the reader who has a general acquaintance with analytical dynamics and the basic theory of ordinary differential equations. Some more advanced issues, not normally taught in standard engineering curricula and necessary for understanding the forthcoming material, are reviewed in Chapter 2. These include the issue of localization in mechanical systems, the concepts of nonlinear normal modes and resonance capture, as well as a survey of the complexification-averaging (CxA) technique which will be frequently employed in this work. In the same chapter perspectives on the experimental fixtures developed for studying TET will be discussed. Chapter 3 is central to our discussion, as it provides the theoretical basis of our study of TET. This includes the analysis of the main mechanisms for TET in the simplest possible system that can exhibit this phenomenon, namely, a singledegree-of-freedom (SDOF) linear oscillator with a SDOF essentially nonlinear attachment; some additional theoretical results on conditions for optimal TET and on TET in multi-degree-of-freedom (MDOF) linear oscillators with SDOF essentially nonlinear attachments are also included in this chapter. Chapter 4 analyzes discrete linear oscillators with MDOF essentially nonlinear attachments, and demonstrates enhanced and more complex forms of TET for this type of dynamical systems. In Chapter 5 we extend our theoretical analysis of TET to flexible structures, by considering beams, rods and plates with essentially nonlinear SDOF and MDOF attachments; we show that TET can be beneficial for shock isolation of this class of systems. Chapter 6 treats TET in discrete oscillators under periodic external excitations; it turns out that such systems can possess rather unusual response regimes which can be related to different regimes of periodic or quasi-periodic TET, some of which turn out to be favorable in the context of vibration isolation. The analysis in Chapter 7 concerns TET in systems with attachments that possess non-smooth nonlinearities; special attention is given to the analysis of TET to attachments with vibro-impact nonlinearities, since these systems are proved to be especially suitable for applications where shock isolation at a fast time scale is required. Experimental studies that validate the nonlinear TET phenomenon are reviewed in Chapter 8, which also includes a discussion of issues related to practical implementation of TET in engineering applications. In Chapters 9 to 11 implementation of TET to practical problems is discussed. These include, passive suppression of aeroelastic instabilities by means of TET to SDOF and MDOF lightweight essentially nonlinear attachments (Chapter 9); application of TET to seismic mitigation problems, considering attachments with smooth, as well as vibro-impact nonlinearities (Chapter 10); and passive suppression of drillstring instabilities in oil drilling applications by means of TET (Chapter 11). These applications demonstrate the potential of TET-based passive designs as efficient solutions to a broad range of important problems encountered in engineering practice. Our discussion of TET is concluded in Chapter 12 with some perspectives on the

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passive designs discussed in this work, and on potential future extensions of this work.

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Chapter 2

Preliminary Concepts, Methodologies and Techniques

As mentioned in the Introduction (Chapter 1), the study of targeted energy transfer (TET) in strongly nonlinear and non-conservative oscillators poses some distinct technical challenges, and dictates the use of concepts, formulations, analytical methodologies and computational techniques from different fields of applied mathematics and engineering, such as dynamical systems and bifurcation theory, theory of asymptotic approximations, numerical signal processing, and experimental dynamics. Therefore, before we initiate our study of the nonlinear dynamics of TET, it is appropriate to provide first some background information related to certain key concepts and methodologies that will be applied in the work that follows. Specifically, we will briefly discuss the concepts of nonlinear normal mode (NNM) and nonlinear mode localization in discrete and continuous oscillators, and the occurence of nonlinear internal resonances, transient resonance captures (TRCs) and sustained resonance captures (SRCs) in undamped or damped, forced or unforced systems of coupled oscillators. These concepts will provide us with the necessary theoretical framework to base our theoretical study of the dynamics of TET; moreover, using these concepts we will be able to identify, interprete, and place into the right context complex nonlinear dynamical phenomena related to TET. Then, we will outline the basic elements of a special perturbation technique, namely, the complexification-averaging (CX-A) technique which will be one of the basic mathematical tools employed for performing the analytical derivations required for our theoretical studies. This will be followed by discussion of some selected advanced signal processing techniques, namely, wavelet transforms – WTs, empirical mode decomposition – EMD, and Hilbert transforms, which will be especially suitable for post-processing the computational nonlinear dynamical responses related to TET, and for identifying the corresponding underlying nonlinear modal interactions that govern TET or influence its effectiveness. In essence, we will work towards the formulation of an integrated post-processing methodology for analyzing strongly nonlinear transient (or steady state) modal interactions in systems with strong nonlinearities. We will end this chapter by providing some preliminary re-

15

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2 Preliminary Concepts, Methodologies and Techniques

marks on the development of the necessary hardware required for our experimental work, undertaken to validate and confirm the theoretical results related to TET. We start our discussion by considering the concept of nonlinear normal mode (NNM), which will be central in our theoretical investigation of the dynamics of TET in systems of coupled oscillators.

2.1 Nonlinear Normal Modes (NNMs) Engineers and physicists traditionally associate the concept of normal mode with linear vibration theory and regard it as closely related to the principle of linear superposition. Indeed, a classical result of linear vibration theory is that the normal modes of vibration of a multi-degree-of-freedom (MDOF) discrete system can be employed to decouple the equations of motion through an appropriate coordinate (modal) transformation, and to express its free or forced oscillations as superpositions of modal responses. Another result of classical linear theory is that the number of normal modes of vibration cannot exceed the number of degrees of freedom (DOF) of a discrete system, and that any forced resonances of the system under external harmonic excitation always occur in neighborhoods of frequencies of normal modes. Although in nonlinear systems the principle of superposition does not (generally) hold, nevertheless the concept of the normal mode can still be employed. Rosenberg (1966) defined a nonlinear normal mode (NNM) of an undamped discrete MDOF system as a synchronous periodic oscillation where all material points of the system reach their extreme values or pass through zero simultaneously; hence, the NNM oscillation is represented by either a straight modal line (similar NNM) or a modal curve (non-similar NNM) in the configuration space of the system. NNMs are generically non-smimilar, since similarity (which is always the case in linear theory) can only be realized when special symmetries exist (Vakakis et al., 1996). Lyapunov (1947) proved the existence of n synchronous periodic solutions (NNMs) in neighborhoods of stable equilibria of n-DOF Hamiltonian systems with no internal resonances, and Weinstein (1973) and Moser (1976) extended Lyapunov’s result to MDOF Hamiltonian systems with internal resonances. As discussed below, an important feature that distinguishes NNMs from linear normal modes is that they can exceed in number the degrees of freedom of an oscillator; in cases where this occurs, essentially nonlinear modes (having no analogs in linear theory) are generated through NNM bifurcations, breaking the symmetry of the dynamics and resulting in nonlinear energy localization (motion confinement) phenomena. Similar NNMs are analogous to linear normal modes, in the sense that their modal lines do not depend on the energy of the free oscillation and space-time separation of the governing equations of motion can still be performed; however, as mentioned previously, this type of NNMs is realized only when special symmetries occur, and are not typical (generic) in nonlinear systems. More generic are non-similar NNMs, whose modal curves do depend on energy; this energy depen-

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dence prevents the direct separation of space and time in the governing equations of motion by means of non-similar NNMs, which complicates their analytical computation (Kauderer, 1958, Manevitch and Mikhlin, 1972; Vakakis et al., 1996). In this work, we will adopt a more extended definition of NNMs, defining an NNM as a (not necessarily synchronous) time-periodic oscillation of a nondissipative nonlinear dynamical system. This enables us to extend the NNM definition to cases of systems in internal resonance, where the resulting strongly nonlinear modal interactions render the free oscillation non-synchronous (King and Vakakis, 1996). Viewed in a different context, whereas in the absence of internal resonance a NNM can be represented by a modal line or curve in the configuration space of the system – so that functional relations of the form yi = yˆi (y1 ), y1 ≡ yˆ1 (y1 ), i = 1, . . . , n can be established between the coordinates yj (hence, Rosenberg’s original NNM definition), no such functional relations hold when internal resonances occur. Still, our extended NNM definition applies to this later case as well. The extension of the concept of NNM to non-conservative systems with damping was studied by Shaw and Pierre (1991, 1993), who introduced the concept of damped NNM invariant manifold to account for the fact that the free oscillation of a damped nonlinear system is a non-synchronous, decaying motion. This NNM invariant manifold formulation is based on ideas developed by Fenichel (1971) regarding persistence and smoothness of invariant manifolds in dynamical systems, and computes damped NNM invariant manifolds of the damped dynamical flow by parametrizing the damped NNM response in terms of a reference displacement and a reference velocity. For sufficiently weak damping, the damped NNM invariant manifold can be viewed as perturbation (and analytic continuation) of the NNM of the corresponding undamped Hamiltonian system. When a motion is initiated on a damped NNM invariant manifold of a MDOF system, the response of each coordinate is in the form of a decaying oscillation with non-trivial phase difference with regard to the other coordinates. A computationally efficient extension of the invariant manifold methodology was proposed by Nayfeh and Nayfeh (1993) who reformulated the NNM invariant manifold method in a complex framework. When no resonances exist, the NNM invariant manifolds of a MDOF discrete oscillator are two-dimensional, and the NNMs are uncoupled from each other. When internal resonances exist, there occur strongly nonlinear interactions between NNMs which couple them; this causes an increase of the dimensionality of the corresponding NNM invariant manifold. NNMs and NNM damped invariant manifolds will play a central role in our discussion of TET and related strongly nonlinear transient dynamical phenomena. Moreover, our study will indicate that a prerequisite for realization of TET from linear systems to strongly nonlinear boundary attachments is the existence of some form of energy dissipation in the system; although in this study the main energy dissipation mechanism considered is (weak) viscous damping, other forms of energy dissipation may also qualify for TET, such as, for example, energy transmission to the far field of unbounded media by traveling waves (see Section 3.5.2). A paradoxical fact, however, is that although TET is realized only in the (weakly) dissipative system, in essence its dynamics is governed by the dynamics, and, especially, the

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NNMs of the underlying non-dissipative system. Indeed, it will be shown, that the properties and bifurcations of NNMs of the non-dissipative system determine the conditions (i.e., the ranges of system parameters, external excitations and initial conditions) for the realization of TET in the dissipative system. The topological structure and bifurcations of the NNMs of the underlying non-dissipative systems will be carefully studied in this work – especially in the frequency-energy domain, since the energy dependencies of NNMs (and damped NNM invariant manifolds) play a key role regarding TET; this holds especially for NNMs whose spatial distributions change from non-localized to localized with decreasing energy. But there are additional benefits to be gained by adopting a NNM-based framework for our study. It will be shown that NNM bifurcations govern complex nonlinear transitions occurring in the damped dynamics with decreasing energy. This becomes clear when one considers that in a weakly damped system the NNMs and NNM bifurcations are preserved as weakly damped NNM invariant manifolds and as bifurcations of these manifolds, respectively, which lie in neighborhoods of the corresponding undamped NNMs. It follows that the weakly damped, transient, nonlinear dynamics follow approximately paths along NNM invariant manifolds, and that bifurcations of NNM invariant manifolds appear as sudden transitions (jumps) in the damped transient dynamics. These may lead to complex, multi-modal and multi-frequency complex transitions in the dynamics, which, however, may be fully interpreted, modeled and analytically studied by adopting a theoretical framework based on NNMs and damped NNM invariant manifolds. More importantly, using such a framework TET can be analyzed and optimized according to a set of design criteria, which is needed for the implementation of TET in practical applications. Indeed a NNM-based approach seems to be natural for the study of TET and the associated strongly nonlinear phenomena discussed in this work. Returning now to our brief review of NNM-related works, constructive methods for computing NNMs in discrete oscillators with no internal resonances have been developed (see, for example, Rand, 1971, 1974; Manevitch and Miklhin, 1972; Mikhlin, 1985; Bellizzi and Bouc, 2005), and NNMs in systems with internal resonances (where strong nonlinear modal interactions take place) have also been studied (see, for example, Boivin et al., 1995; King and Vakakis, 1996; Nayfeh et al., 1996; Jiang et al., 2005a). In an additional series of works (King and Vakakis, 1993, 1994, 1995a; Vakakis and King, 1995; Andrianov, 2008) methodologies for analysing the NNMs (and their bifurcations) of nonlinear elastic and continuous systems have been developed. In King and Vakakis (1994) stationary and traveling solitary waves (breathers) in a class of nonlinear partial differential equations are regarded as localized NNMs over domains of infinite spatial extent and are studied analytically. These methods and some additional ones for analyzing and computing NNMs in discrete and continuous oscillators are reviewed in Manevitch et al. (1989), Vakakis et al. (1996), Vakakis (1996, 1997, 2002), Pierre et al. (2006), Kerschen et al. (2008a) and Peeters et al. (2008). An additional interesting feature of NNMs, which clearly distinguishes them from classical linear normal modes, is that they can exceed in number the degrees of freedom of a dynamical system. This is due to NNM bifurcations which may also

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Fig. 2.1 NNMs of system (2.1); —: stable, - - -: unstable NNMs.

lead to NNM instability (a feature which, again, is distinct from what predicted by linear theory). We illustrate this by a simple example. To this end, we consider the following two-DOF Hamiltonian system with cubic stiffness nonlinearities (Vakakis and Rand, 1992a, 1992b): y¨1 + y1 + y13 + K(y1 − y2 )3 = 0 y¨2 + y2 + y23 + K(y2 − y1 )3 = 0

(2.1)

Due to its symmetry, this system possesses only similar NNMs which are computed by imposing the following functional relationship: y2 = yˆ2 (y1 ) ≡ c y1

(2.2)

where c ∈ R is a real modal constant. Substituting (2.2) into (2.1), we derive the following algebraic equation satisfied by the modal constant: K(1 + c)(c − 1)3 = c(1 − c2 )

(2.3)

In Figure 2.1 the real values of the modal constant c are depicted for varying coupling stiffness coefficient K, from which we infer that a pitchfork bifurcation (Wiggins, 1990) of NNMs occurs in the Hamiltonian system. This type of bifurcation is realized due to the symmetry of system (2.1) and is expected to ‘break’ into saddle node (SN) bifurcation(s) when this symmetry is perturbed. Referring to Figure 2.1, we note that system (2.1) always possesses the NNMs y2 = ±y1 corresponding to solutions c = ±1 of (2.3), irrespectively of the coupling strength K; these correspond to in-phase and out-of-phase similar NNMs, respectively, which

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can be regarded as continuations of the two normal modes of the corresponding linear system. However, as noted from the bifurcation diagram of Figure 2.1, the nonlinear system possesses two additional NNMs which bifurcate from the outof-phase NNM at K = 1/4. The bifurcating NNMs are out-of-phase, essentially nonlinear, time-periodic motions of system (2.1) having no analogs in linear theory; both of these NNMs become strongly localized as K → 0 (i.e., for sufficiently weak coupling) to one of the two SDOF oscillators of system (2.1). Hence, in the limit of weak coupling nonlinear mode localization occurs in the symmetric system. In the next section nonlinear localization in dynamical systems is discussed in more detail. This simple example demonstrates that the NNMs of a dynamical system may exceed in number its degrees of freedom. In this particular case, the NNM bifurcation is due to 1:1 internal resonance between the two SDOF nonlinear oscillators of system (2.1). An additional interesting conclusion drawn from this specific example is that NNM bifurcations may result in mode instability; indeed, for K < 1/4 the outof-phase NNM x2 = −x1 becomes unstable (Vakakis et al., 1996), a result which, as shown below, has implications on the global Hamiltonian dynamics of system (2.1). We mention that the instability of the out-of-phase NNM is manifested in the form of modulated (instead of a periodic) oscillation, and not in the form of an exponentially growing motion; in other words, in system (2.1) only orbital stability (Nayfeh and Mook, 1995) has meaning, as Lyapunov asymptotic stability is not possible in the nonlinear Hamiltonian oscillator (2.1) due to the dependence of the frequency of oscillation on the energy. To show this, we construct numerical Poincaré maps of the global dynamics. First, we reduce the dynamical flow of system (2.1) on its three-dimensional isoenergetic manifold, defined by the relation H (y1 , y˙1 , y2 , y˙2 ) ≡

y 2 + y22 y14 + y24 + K(y1 − y2 )4 y˙12 + y˙22 + 1 + =h 2 2 4

(2.4)

where h is the (conserved) level of energy. Then we intersect the isoenergetic flow by the two-dimensional cut section = {y1 = 0, y˙1 > 0} ∩ {H = h}

(2.5)

which is everywhere transverse to the flow. Moreover, the resulting two-dimensional Poincaré map is orientation-preserving due to the restriction imposed on the sign of the velocity y˙1 at the cut section. In Figure 2.2 we depict the Poincaré maps of system (2.1) for the low energy level h = 0.4, and two values of K corresponding to relatively strong (Figure 2.2a) and weak (Figure 2.2b) coupling. We note that the in-phase NNM normal mode (appearing as the upper equilibrium point in both maps) is orbitally stable, as it appears as a center surrounded by closed orbits (which are intersections of invariant tori of the Hamiltonian with the cut section ). Considering the out-of-phase NNM, above the bifurcation it is stable, whereas below it is unstable and possesses a double hom*oclinic loop, as inferred from Figure 2.2b. The (seemingly smooth) hom*oclinic orbits (loops) are formed by the coalescence of the stable and unstable invariant manifolds of the unstable out-

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Fig. 2.2 Poincaré maps of the global dynamics of system (2.1) for low energy h = 0.4: (a) K = 0.4 > 1/4, (b) K = 0.1 < 1/4.

of-phase NNM, and represent the boundaries between trajectories that encircle only one of the bifurcating NNMs and those that enclose both. The Poincaré plots of Figure 2.2 (which correspond to a relatively low value of energy) are rather deceiving, however, since they give the impression that the global dynamics of the oscillator (2.1) is regular and completely predictable [in fact, for low energies the dynamics can be asymptotically approximated by the method of multiple scales (Vakakis and Rand, 1992a)]. In fact, since the oscillator (2.1) is non-integrable, ‘rational’ and some ‘irrational’ invariant tori of the flow are expected to ‘break’ according to the KAM theorem (MacKay and Meiss, 1987), giving rise to random-like chaotic motions; local (small-scale) chaos then results in layers of stochasticity surrounding count-

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able infinities of stable subharmonic periodic orbits (Guckenheimer and Holmes, 1983; Veerman and Holmes, 1985; Wiggins, 1990) that result from the ‘breakdown’ of invariant tori. This local chaos is due to exponentially small splittings of the stable and unstable manifolds of unstable subharmonic orbits, producing transverse intersections of these manifolds close to resonance bands of the dynamics (Veerman and Holmes, 1986). Apart from causing global qualitative changes in the dynamics, the NNM bifurcation depicted in Figure 2.1 gives rise to global (large-scale) chaos in the Hamiltonian system (2.1), and, hence, to large-scale instability. This is a consequence of the splitting and transverse intersections of the stable and unstable invariant manifolds that form the seemingly smooth (at low energies) hom*oclinic loops of the unstable NNM in the Poincaré map of Figure 2.2b; this results in large-scale hom*oclinic tangles and chaotic Smale horseshoe maps (Wiggins, 1990) leading to largescale chaos in system (2.1). This is demonstrated in the Poincaré maps of Figure 2.3, corresponding to relatively high energy levels h = 50.0 and h = 150.0, and weak coupling (i.e., after the NNM bifurcation has taken place). We note that there is a large region [a sea of stochasticity (Lichtenberg and Lieberrman, 1983)] in each of these maps, inside which the orbits of the oscillator seem to wander in an erratic fashion. These regions contain chaotic motions, i.e., motions with extreme sensitivity on initial conditions. In each Poincaré map the region of large-scale chaos occupies a neighborhood of the unstable out-of-phase NNM and the domain where transverse intersections of the invariant manifolds of that NNM occur. The occurrence of large-scale chaotic motions in the Hamiltonian system (2.1) is a direct consequence of the pitchfork bifurcation of NNMs, since they appear only after the NNM bifurcation has occurred (i.e., only for K < 1/4). Therefore, a necessary condition for large-scale chaos in system (2.1) is the orbital instability of the out-of-phase NNM (since only then can large-scale transverse hom*oclinic intersections of invariant manifolds occur). As a result, in this case the bifurcation of NNMs increases the complexity of the global dynamics and adds global instability into the system. This is a first indication of the global effects on the dynamics that a NNM bifurcation can introduce. In the course of this work we will show that NNM bifurcations can affect in a critical way the dynamics of TET, and that they play an important role when optimizing for robust, fast-scale and strong passive TET from a directly forced linear system to an essentially nonlinear boundary attachment. To illustrate the frequency-energy dependence and some additional interesting features of NNMs we consider another example of a two-DOF system, consisting of a nonlinear oscillator linearly coupled to a linear one (Kerschen et al., 2008a): y¨1 + 2y1 − y2 + 0.5y13 = 0 y¨2 + 2y2 − y1 = 0

(2.6)

In contrast to (2.1) this system is not symmetric so it can possess only non-similar NNMs. As mentioned previously, this is the generic type of NNMs encountered in dynamical systems, so this example aims to demonstrate certain features of nonsimilar NNMs that are typical for a broad class of nonlinear coupled oscillators.

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Fig. 2.3 Poincaré maps of the global dynamics of system (2.1) for high energies and K = 0.1 < 1/4: (a) h = 50.0, (b) h = 150.0.

The non-similar NNMs of this system are approximately computed by the method of harmonic balance (Nayfeh and Mook, 1995), i.e., by seeking timeperiodic responses in the form y1 (t) ≈ A cos ωt,

y2 (t) ≈ B cos ωt

(2.7)

Note that the computation of non-similar NNMs is approximate, in contrast to the exact expressions derived for the similar NNMs in the previous example. When the ansatz (2.7) is substituted into (2.6), and a matching of coefficients of the various harmonic functions is performed, we obtain the following expressions for the amplitudes:

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2 Preliminary Concepts, Methodologies and Techniques

Fig. 2.4 NNMs of system (2.6) depicted in a frequency-energy plot (FEP); the corresponding modal curves in the configuration plane are inset, horizontal and vertical axes in these plots depict the displacements of the nonlinear and linear oscillators, respectively.

8(ω2 − ω22 )(ω2 − ω12 ) A=± 3(ω2 − 2

1/2 ,B =

A 2 − ω2

(2.8)

√ with the natural frequencies of the linearized system given by ω1 = 1 and ω2 = 3. This result demonstrates the frequency dependence of the amplitudes of the NNMs of system (2.6). The appropriate graphical depiction of NNMs is key to their exploitation. In this work extensive use will be made of frequency-energy plots (FEPs) where the amplitude of a NNM is plotted as function of its (conserved) energy. The NNMs of system (2.6) were computed numerically (Peeters et al., 2008) and are depicted in Figure 2.4. There exist two main backbone branches of NNMs, an in-phase branch, S11+, originating (for low energies) from the first linearized natural frequency and an out-of-phase one, S11−, originating from the second linearized natural frequency. The notation ‘S’ used for these NNMs refer to the symmetric character of these solutions [i.e., both oscillators of (2.6) execute synchronous motions], whereas the indices indicate that the two oscillators of system (2.6) vibrate with the same dominant frequency. A detailed discussion of FEPs and the corresponding notations of branches of NNMs depicted on them, is postponed until Section 3.3. The FEP of Figure 2.4 clearly shows that the nonlinear modal parameters have a strong dependence on the (conserved) energy of the oscillation. Specifically, the frequencies of the in-phase and out-of-phase NNMs increase with energy, which reveals the hardening characteristic of system (2.6). Moreover, the modal curves

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change with increasing energy, since the in-phase NNM tends to localize to the linear oscillator (i.e., its modal curve tends to become vertical in the corresponding configuration plane with increasing energy), whereas the out-of-phase NNM tends to localize to the nonlinear oscillator (its modal curve tends to become horizontal with increasing energy). This tendency of NNMs to localize with varying energy is key for the realization of TET in the corresponding weakly dissipative system, as discussed in Chapter 3 [see also Pilipchuck (2008) for an additional study of nonlinear mode localization and TET due to the dependence of the shapes of NNMs on energy]. In this work we will make extensive use of the FEP, and show that it is a valuable tool not only for examining NNMs of Hamiltonian systems, but also for investigating nonlinear transitions leading to TET in weakly dissipative ones. Another salient feature of NNMs is that they may nonlinearly interact without their linearized natural frequencies necessarily satisfying conditions of internal resonance. These strongly nonlinear modal interactions [which differ from nonlinear modal interactions considered in the current literature, see (Nayfeh, 2000) for example] occur at relatively high energy levels (so that nonlinear effects are dominant in the motion), and can be clearly studied by representing the NNMs in the FEP. Such internally resonant NNMs have no counterparts in linear theory and are generated through NNM bifurcations. The FEP of system (2.6) depicts internally resonant NNMs at high energies (see Figure 2.4). In particular, we note an additional branch of NNMs lying on a subharmonic tongue emanating from the in-phase backbone branch S11+. This tongue is denoted by S31, since it corresponds to a 3:1 internal resonance of the in-phase and out-of-phase NNMs at those energy levels. Surpris√ ingly, the ratio of the linearized natural frequencies of system (2.6) is equal to 3, but due to the energy dependence of the frequencies of the NNMs, a 3:1 ratio between the two frequencies of the NNMs can still be realized; hence, conditions of 3:1 internal resonance are realized at high energies, although no such conditions are possible at lower energies. This result clearly demonstrates that NNMs can be internally resonant without necessarily having commensurate linearised natural frequencies, a feature that is rarely discussed in the literature. This also underlines that important features of nonlinear dynamics can be missed when resorting exclusively to perturbation techniques based on linearized generating solutions, and, thus, being limited to small-amplitude motions (Kerschen et al., 2008a). To better illustrate this interesting high-energy nonlinear resonance mechanism, the branch S11− is represented by dashed line as S33− in the FEP of Figure 2.4, at a third of its frequency. This is permissible, because a periodic solution of period T is also periodic with period 3T , so the branch S33− can be considered as identical to S11−. Using this notation it is clear that at the energy range of 3:1 internal resonance there occurs a smooth transition from branch S11− to branch S33− through the subharmonic tongue S31. In Figure 2.5 we present a closeup of the FEP in the energy range of existence of the 3:1 internally resonance NNMs; the subharmonic tongue is more clearly depicted, and the stability of the various branches of internally resonance is examined. This discussion indicates that additional nonlinear resonance scenarios are realized when we further increase the energy of the seemingly simple system (2.6).

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Fig. 2.5 Energy range of the FEP of existence of 3:1 internally resonant NNMs in system (2.6), –•–•–•– unstable NNMs; the corresponding modal representations in the configuration plane are depicted at selected energies.

This is supported by the fact that for increasing energy the frequencies of the outof-phase NNMs on branch S11− increase steadily, whereas the√frequencies of the in-phase NNMs on S11+ tend to the asymptotic limit ω2 = 3. Following this reasoning, we expect the existence of a countable infinity of internal resonances between the in-phase and out-of-phase NNMs at specific higher energy ranges. This is confirmed by the numerical results presented in Kerschen et al. (2008a). In this work we will investigate in detail FEPs similar to those depicted in Figures 2.4 and 2.5, and show that the energy dependencies of the NNM backbone branches and sub-

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harmonic tongues of NNMs dictate the different forms of possible targeted energy transfers in coupled oscillators with essentially nonlinear attachments. The previous examples highlight the advantages of adopting nonlinear theoretical frameworks (instead of linearized ones) for analyzing nonlinear dynamical responses. As shown previously, there are cases where a nonlinear dynamical system may possess essentially nonlinear modes or can exhibit essentially nonlinear dynamical responses that have no counterparts in linear theory. Applying linear concepts such as modal analysis and frequency response plots to such a nonlinear system may model only partially the dynamics, so alternative approaches that take into full account the effects of the nonlinearity must be applied instead. In that context, the concepts of NNM and damped NNM invariant manifold provide a solid theoretical framework for analyzing, interpreting and modeling strongly nonlinear responses of dynamical systems. An additional important characteristic of NNMs relates to forced resonances, since in analogy to linear theory, forced resonances of nonlinear systems excited by periodic excitations occur in neighborhoods of NNMs (Mikhlin, 1974) [this may lead to quite complex structures of forced resonances as discussed by King and Vakakis (1995b)]. Hence, knowledge of the structure of NNMs of a nonlinear oscillator can provide valuable insight on its fundamental or secondary (subharmonic, superharmonic or combinantion) resonances (Nayfeh and Mook, 1995), a feature which is of considerable engineering importance. The structure of forced resonances of nonlinear oscillators is determined, in essence, from the structure and bifurcations of their NNMs, so performing forced response analysis based on linear eigenspaces and not taking into account the possibility that essentially nonlinear modes might exist, may lead to inadequate modeling of the dynamics. Moreover, it was shown in recent studies (Pesheck, 2000; Pesheck et al., 2002; Jiang et al., 2005b; Touzé et al., 2004, 2007a, 2007b; Touzé and Amabili, 2006) that NNMs can provide effective bases for constructing reduced-order models of the dynamics of discrete and continuous nonlinear oscillators. Indeed, NNM-based Galerkin projections for discretizing the dynamics were proven to be more accurate in predicting the nonlinear dynamics of these systems compared to linear modebased Galerkin projections. These results demonstrate one additional application of NNMs; that is, even though NNMs do not satisfy orthogonality properties (as classical linear normal modes do) they can still be used as bases for accurate, low-order Galerkin projections of the dynamics of discrete and continuous weakly or strongly nonlinear oscillators. The resulting low-order reduced models are expected to be much more accurate compared to linear mode-based ones (especially in systems with strong or even nonlinearizable nonlinearities). The reason for the enhanced accuracy of NNM-based reduced-order models lies on the invariance properties of NNMs, and on the fact that they represent exact solutions of the free or forced nonlinear dynamics of the oscillators considered. Hence, free or forced oscillations of a nonlinear structure in the neighborhoods of NNMs can be accurately captured by either isolated NNMs (in the absence of multi-modal nonlinear interactions), or by a small subset of NNMs (when internal resonances between NNMs occur). Hence, NNMs hold promise as bases for efficient and accurate low-order reduction of the

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dynamics of systems with many degrees-of-freedom, for example, of finite-element (FE) computational models; this holds, in spite the fact that NNMs do not satisfy any form of orthogonality conditions. NNMs can be applied in additional areas of vibration theory, as in the area of modal analysis and system identification. Traditional techniques for modeling the dynamics of nonlinear structures are based on the assumptions of weak nonlinearities and of a nonlinear modal structure similar to that of the underlying linearised system. As shown in the previous examples even a simple two-DOF system can have more normal modes than its degrees of freedom; hence, in performing nonlinear modal analysis one should consider the possibility that certain of the modes might be essentially nonlinear, having no counterparts in linear theory. The bifurcating NNMs of the previous examples represent precisely this type of essentially nonlinear modes; they change qualitatively the modal structure of the dynamical system by adding essentially nonlinear components that do not exist in the context of linear theory. It follows that the concept of the NNM can provide the necessary framework for developing nonlinear modal analysis techniques, capable of modeling essentially nonlinear dynamics (Kerschen et al., 2005). It was mentioned previously that the bifurcating similar NNMs of the symmetric system (2.1) become localized to either one of the two oscillators of the system as coupling between them becomes weak, so that nonlinear mode localization occurs in the weakly coupled system. Moreover, we showed that the non-similar NNMs of the non-symmetric system (2.6) become localized with varying energy, even in the absence of NNM bifurcations. Hence, it becomes clear that nonlinear localization is an important feature of the dynamics of coupled nonlinear dynamical systems. Nonlinear mode localization and its applications are discussed in the next section.

2.2 Energy Localization in Nonlinear Systems One of the most interesting features of NNMs is that they may induce nonlinear mode localization in dynamical systems, i.e., a subset of NNMs may be spatially localized to subcomponents of dynamical systems. Mode localization may occur also in linear systems composed of multiple coupled subsystems (Anderson, 1958; Pierre and Dowell, 1987; Hodges, 1982), however, it only results due to the interplay between break of symmetry (structural disorder) and weak coupling between subsystems. In nonlinear systems, structural disorder is not a prerequisite for mode localization, since the dependence of the frequency of oscillation on the amplitude (energy) provides an ‘effective disorder’ (or ‘mistuning’) in the dynamics (Vakakis et al., 1993; Vakakis, 1994; King et al., 1995; Vakakis et al., 1996). Nonlinear mode localization was realized in both examples of systems of unforced coupled oscillators examined in Section 2.1, either due to a bifurcation of similar NNMs in the symmetric system (2.1), or due to the energy dependence of the nonlinear mode shape of non-similar NNMs in system (2.6). Moreover, forced nonlinear localization in systems under harmonic excitation has been studied (Vakakis,

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1992; Vakakis et al., 1994), and nonlinear mode localization in flexible systems with smooth (Vakakis, 1994; Aubrecht and Vakakis, 1996; Aubrecht et al., 1996) and non-smooth nonlinearities (Emaci et al., 1997) has been investigated. A review of mode localization in systems governed by nonlinear partial differential equations was given in Vakakis (1996). We will demonstrate some aspects of nonlinear mode localization in coupled oscillators by considering two examples, one involving a low-dimensional cyclic system, and a second one with a nonlinear medium of infinite spatial extent. The latter example will underline the theoretical link between NNMs and solitary waves. We start by considering a cyclic assembly of coupled oscillators, governed by the following set of ordinary differential equations (Vakakis et al., 1993): y¨i + yi + εµyi3 + εk(yi − yi+1 ) + εk(yi − yi−1 ) = 0, y0 ≡ yN ,

yN+1 ≡ y1

i = 0, . . . , N (2.9)

This symmetric system possesses similar NNMs, which can be approximately computed by the method of multiple scales (Nayfeh and Mook, 1995) as follows: ⎫ y1 (t) = a1 cos[(1 + εα)t + β1 ] + O(ε) ⎪ ⎪ ⎪ y2 (t) = yN−1 (t) = −a2 cos[(1 + εα)t + β1 ] + O( ) ⎬ (N = 2p + 1) y3 (t) = yN−2 (t) = −a3 cos[(1 + εα)t + β1 ] + O( ) ⎪ ⎪ ⎪ ⎭ • • • or ⎫ y1 (t) = a1 cos[(1 + εα)t + β1 ] + O(ε) ⎪ ⎪ ⎪ ⎪ y2 (t) = yN−1 (t) = −a2 cos[(1 + εα)t + β1 ] + O(ε) ⎪ ⎪ ⎪ ⎬ y3 (t) = yN−2 (t) = a3 cos[(1 + εα)t + β1 ] + O(ε) (N = 2p + 1) ⎪ • • • ⎪ ⎪ ⎪ yp−1 (t) = yp+1 (t) = (−1)p ap−1 cos[(1 + εα)t + β1 ] + O(ε) ⎪ ⎪ ⎪ ⎭ yp (t) = (−1)p+1ap cos[(1 + εα)t + β1 ] + O(ε) (2.10) where p is an integer and ak ≥ 0; the phase β1 depends on the initial conditions, and εα = εα(a1 ) is the small amplitude-dependent nonlinear correction to the frequency of the NNM. The ratios (an /am ) in (2.10) were determined in (Vakakis et al., 1993) for systems with even and odd degrees of freedom. In Figure 2.6 we present a subset of NNMs for systems (2.9) with N = 4 and N = 5 degrees of freedom. It can be shown that for fixed (conserved) energy level, the parameter that conrols nonlinear mode localization is the ratio (k/µ), i.e., the relative magnitude of coupling with respect to stiffness nonlinearity. For low values of this ratio the NNMs depicted in Figures 2.6a, b become localized to the first oscillator, i.e., the amplitude a1 becomes much larger than the corresponding amplitudes of the other oscillators; this occurs in spite of direct coupling between oscillators. As the coupling to nonlinearity ratio (k/µ) increases from relatively small

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2 Preliminary Concepts, Methodologies and Techniques

Fig. 2.6 Nonlinear localization in the cyclic system (2.9) with, (a) N = 4, and (b) N = 5 degrees of freedom; —: stable NNMs, — — —: unstable NNMs; - - - -: asymptotic approximations.

to relatively large values, there occur two distinct scenarios of delocalization, as the energy of the NNM gradually becomes spatially extended. Specifically, for the system with even DOF (N = 4, see Figure 2.6a), the localized NNM branches become delocalized through a bifurcation with the out-of-phase (spatially extended) NNM a1 = a2 = a3 = a4 ; this bifurcation signifies the end of localization in this system. A different scenario of delocalization occurs in the system with odd DOF (N = 5, see Figure 2.6b), since as the ratio (k/µ) increases the localized branches of NNMs become delocalized through smooth transitions to spatially extended NNMs; the ab-

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sence of NNM bifurcations in this case is a reflection of the symmetry group of this system which differs from that of the system with even DOF (for group theoretic approaches to problems in dynamical systems, see Manevitch and Pinsky, 1972a, 1972b, and also Manevitch et al., 1970; Vakakis et al., 1996). We note that although the previous results prove that for weak coupling to nonlinearity ratios (strong) localization of motion occurs in the first oscillator of system (2.9), due to cyclic symmetry this result can be extended to each of the other oscillators. Hence, we can prove that system (2.9) possesses N (strongly) localized NNMs with the vibration being passively confined mainly to one of the oscillators. Moreover, these localized NNMs are stable, and, hence, physically realizable (Vakakis et al., 1993); in the same reference, it is proven that additional (weakly) localized NNMs occur, with motion passively confined mainly to a subset of oscillators. Nonlinear localization can greatly infuence the transient structural response since it can lead to passive motion confnement of disturbances generated by external forces. When localized NNMs of such structures are excited by external impulsive forces, the oscillations remain passively confined close to the point where they are initially generated instead of ‘spreading’ through the entire structure. Such passive confinement can also occur in linear systems but only in the presence of disorder and weak substructure coupling (Pierre and Dowell, 1987). To demonstrate the passive nonlinear motion confinement phenomenon, we consider the impulsive response of the cyclic system (2.9). As discussed previously, as k/µ → 0+ branches of similar NNMs localize (strongly) to a single oscillator. A system with N = 50 oscillators is considered in the numerical simulations and the numerical results are obtained by finite element (FE) computations (Vakakis et al., 1993). A force with unit magnitude and duration t = 0.2 is applied to the first oscillator, and the transient response of the system is depicted in Figure 2.7 for parameters εµ = 0.3, εk = 0.05 and k/µ = 0.166; at this energy level and for the chosen system parameters the cyclic system possesses strongly localized NNMs, so passive nonlinear motion confinement of the impulsive response is expected. Indeed, as shown in Figure 2.7 the nonlinear response remains confined to the directly forced oscillator, instead of ‘leaking’ to the entire system. For comparison purposes the responses of the corresponding linear system with εµ = 0 are also shown in the plots of Figure 2.7, from which we conclude that in the linear case there is a gradual ‘spreading’ of the impulsive energy to all oscillators; moreover, the spreading of energy in the linear system becomes increasingly more profound as time increases. The motion confinement of disturbances in the nonlinear system can only be attributed to the excitation of strongly localized NNMs by the external impulse, yielding motion confinement due to their invariance properties. In fact, it is the invariance property of the stable strongly localized NNM that yields transient motion confinement of disturbances in the system under consideration. The second example will demonstrate that there is a theoretical link between NNMs and spatially localized solitary waves in nonlinear media of infinite spatial extent. For this we consider spatially localized NNMs in the following nonlinear partial differential equation (King and Vakakis, 1994):

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2 Preliminary Concepts, Methodologies and Techniques

Fig. 2.7 Passive confinement of the impulsive response of the cyclic system with N = 50 DOF for a single impulse applied to the first oscillator; normalized displacements are depicted at different snapshots, —: nonlinear system, - - -: linear system.

ut t + u + ελuxx + εαu3 = 0,

−∞ < x < ∞

(2.11)

where x and t are the spatial and temporal independent variables, respectively, λ, α > 0, 0 < ε 1, and the short-hand notation for partial differentiation has been adopted. This equation represents the ‘continuum limit’ approximation of weakly modulated out-of-phase oscillations of an infinite chain of coupled oscillators with

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cubic stiffness nonlinearities. For additional works on localized NNMs in nonlinear chains we refer to Manevitch (2001) and Manevitch and Pervouchine (2003). A first integral of motion of (2.11) is given by

2 εα 4 1 +∞ ∂u 2 ∂u 2 (2.12) u dx + u − ελ + H = 2 −∞ ∂t ∂x 2 provided that H < ∞ (this holds for stationary localized wave solutions of the type considered in our analysis). We seek stationary, spatially confined and time-periodic solutions of the nonlinear medium (2.11) in the form of NNMs, by expressing the response of an arbitrary point of the medium in terms of the response of a reference point x = x0 , u(x, t) = U [x, u0 (t)], u0 (t) ≡ u(x0 , t) (2.13) satisfying the compatibility condition U [x0 , u0 (t)] ≡ u0 (t), and the additional conditions: lim u(x, t) = 0, u(x, t) = u(x, t + T ), t ∈ R (2.14a) x→±∞

By the first of the above relations we require spatial localization of the envelope, and by the second time-periodicity. Moreover, due to the odd stiffness nonlinearity of (2.11), the following additional symmetry of the envelope is satisfied: U [−x, −u0(t)] = −U [x, u0 (t)],

x∈R

(2.14b)

It follows that we only need to confine our analysis to x ≥ 0 and u0 (t) ≥ 0. In essence, expression (2.13) represents a modal function for the sought NNM oscillation in functional space, and, viewed in that context, it may be regarded as an infinite-dimensional extension of modal relations satisfied by NNMs of finitedimensional oscillators. Note that by (2.13) we assume that the sought NNM is nonsimilar, so it is anticipated that the modal function will depend on the energy of the oscillation (or, equivalently, on the amplitude of the NNM oscillation). In addition, by (2.13) we make the assumption that the NNM is a synchronous oscillation where all points of the medium vibrate in-unison, so that the response of each point may be parametrized in terms of the reference response u0 (t) of the reference point x = x0 . Combining relations (2.12) and (2.13) we derive the following functional equation governing the modal function U (King and Vakakis, 1994), +∞ 2 2 2 2 −∞ {[U (x, A) − U (x, u0 )] − ελ[Ux (x, A) − Ux (x, u0 )]}dx +∞ 2 −∞ Uu0 (x, u0 )dx (εα/2)[U 4 (x, A) − U 4 (x, u0 )] Uu 0 u 0 + +∞ 2 −∞ Uu0 (x, u0 )dx +{−U (x0, u0 ) − ελUxx (x0 , u0 ) − εαU 3 (x0 , u0 )}Uu0 (x, u0 = = −U (x, u0 ) − ελUxx (x, u0 ) − εαU 3 (x, u0

(2.15)

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2 Preliminary Concepts, Methodologies and Techniques

where again the short-hand notation for partial differentiation is used. The amplitude A > 0 of the NNM is the maximum amplitude attained by the response u0 (t) of the reference point, when the system reaches the position of maximum potential energy. Equation (2.15) has to be solved simultaneously with the following two additional conditions: lim U [x, u0 (t)] = 0 (2.16) x→±∞

{−U (x0 , A) − ελUxx (x0 , A) − εαU 3 (x0 , A)}Uu0 (x, A) = −U (x, A) − ελUxx (x, A) − εαU 3 (x, A)

(2.17)

Condition (2.16) is self-explanatory [it corresponds to the first of relations (2.14a)], whereas condition (2.17) needs further justification. A careful examination of the functional relation (2.15) reveals that it becomes singular when the system reaches the position of maximum potential energy u0 = A; indeed, the coefficient of the highest-order partial derivative Uu0 u0 becomes zero when u0 = A, so this represents a regular singular point of the mathematical problem. Therefore, the solution for U (x, u0 ) must be, (i) first asymptotically approximated in semi-open intervals 0 ≤ u0 (t) < A, and then, (ii) analytically continued up to the maximum potential energy level u0 = A; this analytical continuation is accomplished by imposing the condition (2.17) which guarantees that the solution of the functional equation (2.15) is extended up to the point of maximum potential energy. The non-similar NNM governed by relations (2.15)–(2.17) was solved asymptotically in King and Vakakis (1994), leading to the following analytical approximation for the modal function U [x, u0 (t)] = [a1(0) (x) + εa1(1) (x) + O(ε2 )]u0 (t) + [εa3(1) (x) + O(ε3 )]u30 (t) + O[εu50 (t)]

(2.18)

where a1(0) (x) = sec hz, a1(1) (x) = [(1/24)(αA2 + K1 + AK2 )z cosh z − (αA2 /48) sinh z] tanh z sec h2 z, (1)

a3 (x) = −(α/8)(1 − sec h2 z) sec hz, z = A(3α/8λ)1/2 (x − x0 ),

+∞ +∞ (0) 2 (0)2 λ a1 (x) dx / a1 (x) dx , K1 = − −∞

K2 = −

+∞

−∞

−∞

(0) (1) (0)4 2a1 (x)a3 (x) + (α/2)a1 (x)

dx /

+∞ −∞

(0)2 a1 (x)dx

and prime denotes differentiation with respect to x. Note that although no space-time separation is possible for this problem (since the sought NNM is non-similar), the

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

35

derived asymptotic approximation is based on solving an hierarchy of subproblems at increasing orders of ε which are separable, so they can be solved analytically (King and Vakakis, 1994). After computing the approximation for the modal function (2.18), the reference response u0 (t) is computed by substituting (2.18) into (2.11) and evaluating the resulting expression at the reference point x = x0 . This results in the following nonlinear modal oscillator: (0)

u¨ 0 (t) + [1 + ελa1

(1)

+ [εα + ε2 λa3

(1)

(x0 ) + ε2 λa1

(x0 )]u0 (t)

(x0 )]u30 (t) + O[ε 2u50 (t), ε 3 ] = 0

(2.19)

For specific initial conditions the response u0 (t) of the modal oscillator can be computed in closed form in terms of Jacobian elliptic functions. This computes also the frequency of the oscillation of the non-similar NNM, and reveals its dependence on energy (the initial condition). For example, for initial conditions u0 (0) = A, u˙ 0 (0) = 0 (i.e., for initiation of the NNM oscillation at the point of maximum potential energy) the solution of (2.19) is expressed as u0 (t) = A cn (pt, k 2 ), (0)

p = {1 + ελa1

(1)

(x0 ) + ε2 λa1

(1)

(x0 ) + [εα + ε2 λa3

(x0 )]A2 }1/2 (2.20)

(1)

where k 2 = [εα + ε 2 λa3 (x0 )]A/2p2 is the elliptic modulus (Byrd and Friedman, 1954). The frequency of the NNM coincides with the frequency of the periodic response (2.20), πp ω = ω(A) = (2.21) 2K(k) where K(•) is the complete elliptic integral of the first kind (Byrd and Friedman, 1954). This completes the analytic approximation of the NNM of system (2.11). The solution u(x, t) = U [x, u0 (t)] given by expressions (2.18)–(2.21) represents a stationary, spatially localized, time-periodic response of the nonlinear medium, i.e., a stationary breather or stationary solitary wave. Since this stationary wave represents synchronous (in-unison) oscillations of all points of the nonlinear medium, it can be regarded as a localized NNM of the medium of infinite spatial extent. Hence, the previous results provide a theoretical link between NNMs and stationary solitary waves (breathers) in nonlinear partial differential equations. In Figure 2.8 we depict snapshots of the stationary breather for parameters α = 1.2, λ = 0.9, A = 0.25, x0 = 0 and ε = 0.01. As discussed in King and Vakakis (1994), based on the stationary solution (2.18)–(2.21) a family of travelling breathers of system (2.11) can be computed by imposing the following Lorentz coordinate transformation: √ √ t − vx/ ελ x + vt ελ (2.22) √ , u0 √ u(x, ˜ t) = U 1 + v2 1 + v2

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2 Preliminary Concepts, Methodologies and Techniques

Fig. 2.8 Localized NNM of system (2.11) – stationary breather.

The traveling wave velocity (group velocity) v is related to the frequency ω by modifying the frequency-energy relation as follows: √ πp 1 + v 2 ω = ω(v, A) = (2.23) 2K(k) We note that the NNM (2.18) can be considered as special case of the traveling breather solution (2.22) with zero group velocity, v = 0. From a practical point of view, nonlinear mode localisation phenomena can be implemented in active or passive vibration isolation designs, where unwanted disturbances generated by external forces are initially spatially confined to predetermined, specially designed subcomponents of the structure, and then passively or

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actively dissipated locally. Indeed, inducing localized NNMs in flexible structures of large spatial extent is expected to enhance the controllability of these structures, since in designing for active control one would need to consider only local structural components where the unwanted disturbances are to be confined, instead of considering the structures in their entireties; of course, issues of observability, controllability, spill-over effects, and possible instabilities by excitation of unwanted or unmodelled modes should be addressed in these control designs. In addition, the study of motion confinement phenomena due to nonlinear effects can prove to be beneficial in applications where such localization phenomena are unwanted. For example, localization of vibration energy in rotating turbine blade assemblies can be catastrophic since it may lead to failure of high-speed rotating blades. Understanding the interplay between (and effects of) structural disorders, coupling forces and stiffness or damping nonlinearities on localization can prevent such failures and prolong the operational life of mechanical or structural components. We end this section by providing a remark concerning the relation between nonlinear mode localization and nonlinear targeted energy transfer (TET) phenomena considered in this work. Simply stated, nonlinear mode localization can be regarded as a static way of passive energy confinement: in structures with localized NNMs, energy confinement can be achieved only as long as stable localized NNMs are excited either by the external excitations and/or the initial conditions of the problem; it follows that energy confinement through nonlinear mode localization relies mainly on passive confinement of disturbances at the points of their generation through direct excitation of stable localized NNMs. It follows that no passive energy transfer is possible in this case. On the other hand, TET can be regarded as a dynamic way of passive energy confinement: indeed, TET relies of the passive, directed transfer of unwanted vibration energy from the point of its generation to isolated or sets of nonlinear energy sinks (NESs) where this energy is confined and dissipated locally; moreover, we will show that TET can be realized for a broad range of external excitations and/or initial conditions, and can result in broadband energy transfer between different parts of a system. As mentioned in the previous section (and as discussed in detail in the following chapters), nonlinear mode localization plays a key role in TET, as TET critically depends on the variation of the shapes of excited NNMs, from being non-localized to becoming localized with varying energy. In the next section we continue our discussion of introductory concepts by discussing the nonlinear phenomena or internal resonances, transient resonance captures (TRCs) and sustained resonance captures (SRCs) in nonlinear dynamical systems.

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2 Preliminary Concepts, Methodologies and Techniques

2.3 Internal Resonances, Transient and Sustained Resonance Captures A general n-DOF time-invariant, linear Hamiltonian vibrating system with n distinct natural frequencies possesses n linear normal modes which form a complete orthogonal basis in R n ; if a natural frequency has multiplicity p – for example, due to special symmetries of the system – the set of (n − p + 1) independent normal modes can be complemented by (p − 1) generalized modes (Meirovitch, 1980) to form again a complete and orthogonal basis in R n . This can be extended to infinite dimensions in the case of bounded, time-invariant, unforced linear continuous systems [since unbounded elastic media possess continuous spectra of eignevalues and support waves instead of vibration modes (Courant and Hilbert, 1989)]. Viewed from a geometric perspective, the 2n-dimensional phase space of the nDOF linear time-invariant Hamiltonian system is foliated by an infinite family of invariant n-tori, parametrized by the Hamiltonian (which in most cases coincides with the total conserved energy of the motion); this is due to the fact that linear systems are always integrable. To give an example, consider the following two-DOF linear system of coupled oscillators: y¨1 + y1 + K(y1 − y2 ) = 0 y¨2 + y2 + K(y2 − y1 ) = 0

(2.24)

This system possesses an in-phase mode with√natural frequency ω1 = 1, and an outof-phase mode with natural frequency ω2 = 1 + 2K. To get a geometric picture of the dynamics in phase space, we introduce the action-angle variable transformation, (y1 , y˙1 , y2 , y˙2 ) ∈ R 4 → (I1 , I2 , φ1 , φ2 ) ∈ (R + × R + × S 1 × S 1 ), which can be regarded as a form of nonlinear polar transformation (Persival and Richards, 1982), and defined by the relations y1 = 2I1 /ω1 sin φ1 , y˙1 = 2I1 ω1 cos φ1 y2 = 2I2 /ω2 sin φ2 , y˙2 = 2I2 ω2 cos φ2 (2.25) In terms of the new variables, the system of coupled oscillators (2.24) can be expressed as follows: I˙1 = 0 ⇒ I1 = I10 I˙2 = 0 ⇒ I2 = I20 φ˙ 1 = ω1 φ˙ 2 = ω2

(2.26)

The leading two equations in (2.26) are trivially solved, and represent conservation of energy for each of the two normal modes of system (2.24); actually, these

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Fig. 2.9 Foliation of the phase space of two-DOF linear Hamiltonian system (2.24) by an infinite family of invariant two-tori parametrized by energy.

additional first integrals of motion render the two-DOF linear system (2.24) fully integrable (for linear systems the integrability property can be extended to R n ). It follows that at a given energy level the dynamics of (2.24) is reduced to the dynamics of the angles φ1 and φ2 on an isoenergetic two-torus T 2 , with the resulting motion being either periodic (if the frequency ratio ω1 /ω2 is a rational number) or quasi-periodic (if ω1 /ω2 is irrational). By varying the energy of the motion (through changes in initial conditions) the dynamics in phase space takes place on different isoenergetic two-tori, so the entire phase space of system (2.24) is foliated by an infinite family of invariant two-tori parametrized by energy. In Figure 2.9 we present a schematic depiction of this family of isoenergetic two-tori which are invariant for the dynamical flow of (2.24). We note that the limiting cases where only one of the two modes of the system is excited by the initial conditions (i.e., I1 = 0 or I2 = 0) correspond to degeneracies of the family of tori and are represented by one-dimensional manifolds (lines) as shown in Figure 2.9. Returning to our discussion of the general n-DOF time-invariant linear Hamiltonian system, the energy imparted at t = 0 in the system by the initial conditions is partitioned among the linear modes (i.e., the motion takes place on a specific n-torus T n in phase space), and no further energy exchanges between modes is possible for t > 0. Each linear mode conserves its own energy and participates accordingly in the (periodic or quasi-periodic) response of the system through linear superposition with the responses of the other modes. This nice structure of the linear phase space in terms of the foliation by the infinite family of invariant tori is not expected to be preserved when the Hamiltonian system is perturbed by nonlinear terms. For example, considering perturbations of the integrable Hamiltonian system (2.24) by nonlinear non-Hamiltonian perturbations, no tori survive the perturbation. For Hamiltonian perturbations, however, the KAM (Kolmogorov–Arnold–Moser) theorem (MacKay and Meiss, 1987) guarentees that ‘sufficiently irrational’ tori (i.e., those for which the ratio ω1 /ω2 is ‘poorly’ approximated by rational numbers in a number-theoretic setting) are pre-

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served, filled with quasi-periodic orbits. Generically, the remaining invariant n-tori of the infinite foliation ‘break-up’ by the perturbation, leading to an infinite number of stable-unstable pairs of periodic orbits of arbitrarily large periods and to chaotic trajectories located in local chaotic layers; this renders the perturbed nonlinear system non-integrable (this scenario was demonstrated in the example with the twoDOF Hamiltonian system and the corresponding Poincaré maps of Figure 2.3 in Section 2.1). There are, however, cases of integrable nonlinear Hamiltonian systems where the foliation of phase space by invariant tori is still preserved (in similarity to the linear case) (Moser, 2003); it is conjectured, however, that full intergability is not a generic property of nonlinear Hamiltonian systems. The previously described scenario of ‘break-up’ of rational and ‘insufficiently irrational’ tori in nonlinear Hamiltonian systems underlines a nonlinear dynamical mechanism that enables energy exchanges between modes, even if they are well separated in frequency (clearly, this would not be possible in linear theory). This mechanism is the phenomenon of internal resonance which results in nonlinear coupling between modes, and gives rise to mode bifurcations and nonlinear beat phenomena during which strong energy exchanges between modes occur. This is not possible in linear theory, since, as discussed above, there is no mechanism for exchanging energy between well separated modes (although, it is well-known that beat phenomena can occur when linear modes are closely spaced in frequency). Internal resonances in nonlinear Hamiltonian systems are associated with the failure of the averaging theorem with respect to certain ‘slow angles’ of the problem in neighborhoods of resonance manifolds. We show this in the following brief exposition which follows Arnold (1988), Lochak and Meunier (1988) and Verhulst (2005). Consider the following 2n-dimensional nonlinear Hamiltonian system in action-angle variables, I˙ = εF (φ, I ) (I, φ) ∈ (R +n × T n ) (2.27) φ˙ = ω(I ) + εG(φ, I ) which, for |ε| 1 is a perturbation of the 2n-dimensional integrable Hamiltonian system, I˙ = 0, φ˙ = (I ). We consider the general case where the n frequencies ω = [ω1 . . . ωn ]T depend on the n-vector of actions I [this is typical in nonlinear Hamiltonian systems, but it does not hold for the linear system (2.24)–(2.26)]. In (2.27) we assume that F and G are sufficiently smooth functions which are 2πperiodic in the n-vector of angles φ = [φ1 , . . . , φn ]T ; moreover, as in previous examples, by T n we denote the n-torus. It follows that the n-vector of functions F can be expanded in complex Fourier series in terms of the n angles as follows: F (φ, I ) =

+∞

ck1 ...kn (I )ej (k1 φ1 +...+kn φn )

(2.28)

k1 ,...,kn =−∞

where j = (−1)1/2 and ck1 ...kn (I ) is an n-vector of complex coefficients of the harmonic characterized by the indices (k1 , . . . , kn ) ∈ Z n . A resonance manifold of

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

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the dynamics of (2.28) is defined by the relation, ˆ ˆ ˆ kˆ1 ω1 (I ) + kˆ2 ω1 (I ) + . . . + kˆn ωn (I ) = 0 ⇒ I = I (k1 :k2 :...:kn )

(2.29)

for some (kˆ1 , kˆ2 , . . . , kˆn ) ∈ Z n , provided that the corresponding vector of Fourier ˆ ˆ ˆ coefficients in (2.28) does not vanish, ckˆ1 kˆ2 ...kˆn (I (k1 :k2 :...:kn ) ) = 0. If the resonance n manifold is a low-dimensional submanifold of R , in its neighborhood we can average out the angles that do not participate in the internal resonance (these angles possess time-like behavior and are regarded as ‘fast’ angles), and reduce accordingly the dimensionality of the dynamics. This is performed by defining appropriate ‘slow’ angles (which are not time-like and cannot be averaged out of the dynamics) as combinations of the angles that participate in the resonance condition (2.29). In essence, the internal resonance provides nonlinear coupling between all participating modes, and results in energy exchanges between these modes. In the absence of internal resonance, all angles in (2.27) possess time-like behavior (and, hence, are ‘fast’ angles) so they can be averaged out of the problem to reduce it to the following n-dimensional averaged dynamical system, I˙a = εc0...0 (Ia )

(2.30)

i.e., in terms of the vector of coefficients of the Fourier term in (2.28) not depending on φ. Given an initial condition I (0) = Ia (0), it can be proven that I (t) − Ia (t) = O(ε) on the timescale 1/ε (Verhulst, 2005). In the absence of internal resonances no nonlinear modal interactions occur, and each mode retains its energy, in similarity to the linear case [at least correct to O(1) – small modal energy exchanges occur at higher orders of ε so they are insignificant]. It follows that no significant energy exchanges between modes can occur in the absence of internal resonances. The effect of an internal resonance on the dynamics of a nonlinear system is illustrated by the following example. We consider a two-DOF system composed of a linear oscillator weakly coupled to a strongly nonlinear attachment (Vakakis and Gendelman, 2001), y¨1 + Cy13 + ε(y1 − y2 ) = 0 y¨2 + ω22 y2 + ε(y2 − y1 ) = 0

(2.31)

where the stiffness characteristic of the weak coupling, 0 < ε 1, is the small parameter of the problem. For ε = 0 the two oscillators become uncoupled, and the nonlinear system is integrable. We wish to study the effects of internal resonance on the dynamics of this system when we perturb it by weak coupling terms. In terms of the terminology introduced in Chapter 3, this system represents a linear oscillator (LO) with an attached grounded nonlinear energy sink (NES) (see Section 3.1). First, we bring this system in the form (2.27) by transforming in terms of the action-angle variables (I1 , I2 , φ1 , φ2 ) ∈ (R + ×R + ×T 2 ) of the unperturbed system, 1/3

y1 = I1

cn [2K(1/2)φ1/π, 1/2]

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2 Preliminary Concepts, Methodologies and Techniques 1/3

y˙1 = −[I1 ω1 (I1 )2K(1/2)/π] sn [2K(1/2)φ1/π, 1/2] × dn [2K(1/2)φ1/π, 1/2] y2 = 2I2 /ω2 sin φ2 y˙2 = 2I2 ω2 cos φ2

(2.32)

1/3

where ω1 (I1 ) = I1 is the frequency of oscillation of the uncoupled nonlinear oscillator, K(1/2) is the complete elliptic integral of the first kind (Byrd and Friedman, 1954), and = (4C)−1/6 [3π/K(1/2)]1/3, = [3π 4 C/8K 4 ]1/3 . Introducing these transformations into the perturbed system (2.31), we express it in the form (2.27): I˙1 = εF1 (I1 , I2 , φ1 , φ2 ) I˙2 = εF2 (I1 , I2 , φ1 , φ2 ) φ˙ 1 = ω1 (I1 ) + εG1 (I1 , I2 , φ1 , φ2 ) φ˙ 2 = ω2 + εG2 (I1 , I2 , φ1 , φ2 )

(2.33)

By construction, the functions F1 , F2 , G1 and G2 are 2π-periodic in φ1 and φ2 and are listed explicitly later in this section, and also in Vakakis and Gendelman (2001). Equations (2.33) represent a two-frequency dynamical system in (R + × R + × 2 T ), and are in a form directly amenable to two-frequency averaging (Lochak and Meunier, 1988). Indeed, by applying straightforward averaging with respect to the two angles φ1 and φ2 we obtain the following simplified averaged system: I˙1a = ε(1/4π 2 ) I˙2a = ε(1/4π 2 )

2π 0

2π

F1 (I1a , I2a , φ1 , φ2 )dφ1 dφ2 = 0

2π 0

2π

F2 (I1a , I2a , φ1 , φ2 )dφ1 dφ2 = 0

(2.34)

which is of the general form (2.30). Hence, in the averaged system the two oscillators [conserve to O(1)] their initial energies, inspite of the weak coupling. Clearly, this is will not be case when internal resonances occur in the dynamics. The condition under which the dynamics of the averaged system (2.33) accurately describes the dynamics of the full system (2.34) has been addressed in previous works (Neishtadt, 1975; Morozov and Shilnikov, 1984; Arnold, 1988). Arnold’s theorem (1988) answers this question. If the condition

d ω1 (I1 ) = 0 dt ω2 is satisfied along the trajectories of the dynamical flow of (2.33), then the full dynamics is close to the averaged dynamics up to time of O(1/ε). That is, if √ I (0) = Ia (0), then I (t) − Ia (t) ≤ κ ε for 0 < t < 1/ε.

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

43

The condition of the theorem precludes any trajectory of (2.33) from being captured on a resonance manifold. According to our previous discussion, the conditions for the existence of an (m : n) resonance manifold of (2.33) are as follows: mω1 (I1 ) − nω2 = 0, 2π 2π Fp (I1 , I2 , φ1 , φ2 )e−j (mφ1 −nφ2 ) dφ1 dφ2 = 0, 0

p = 1, 2 (2.35)

where m and n are integers. In what follows we study in detail the 1:1 internal resonance in the dynamics of the Hamiltonian system (2.31) or (2.33), corresponding to the following level of the action of the nonlinear oscillator: ω1 (I1 ) − ω2 = 0 ⇒ I1 ≡ I1(1:1) = (ω2 /)3 (2.36) √ We restrict our analysis to an O( ε) boundary layer of the 1:1 resonance manifold by defining the ‘slow’ angle ψ = φ1 − φ2 , and introducing the angle transformation √ (1:1) (φ1 , φ2 ) → (ψ, φ2 ) and the action transformation I1 = I1 + εξ . Introducing these transformations into the last of equations (2.33), we express the independent variable as t = (φ2 /ω2 ) + O(ε), which shows that φ2 is time-like, and, hence, a ‘fast’ angle. It follows that we can replace t by φ2 as the independent variable of the remaining three equations of (2.33), and obtain the following reduced local dynamical system in the neighborhood of the (1:1) resonance manifold, ξ =

√

∂ F˜1 (1:1) (1:1) εω2−1 F˜1 (I1 , I2 , ψ, φ2 ) + εω2−1 (I , I2 , ψ, φ2 ) + O(ε3/2 ) ∂I1 1

I2 = εω2−1 F˜2 (I1(1:1) , I2 , ψ, φ2 ) + O(ε3/2 ) √ ψ = εω1 (I1(1:1) )ω2−1 ξ + εω2−1 [ω1 (I1(1:1) )ξ 2 /2 ˜ 1 (I +G 1

(1:1)

˜ 2 (I , I2 , ψ, φ2 ) − G 1

(1:1)

, I2 , ψ, φ2 )] + O(ε 3/2 )

(2.37)

where primes denote differentiation with respect to φ2 , and the notation Fp (I1 , I2 , φ1 = ψ + φ2 , φ2 ) ≡ F˜p (I1 , I2 , ψ, φ2 ), p = 1, 2, and a similar nota˜ p , p = 1, 2 are adopted. We emphasize that due to 1:1 internal resonance tion for G only one ‘fast’ (time-like) angle remains in the dynamics of the reduced averaged system (the angle φ2 ), and the new ‘slow’ angle ψ appears. Moreover, although averaging with respect to the ‘fast’ angle φ2 can still be performed, this cannot be done with respect to the ‘slow’ angle ψ since the conditions of the averaging theorem do not apply with respect to that angle; hence, 1:1 internal resonance is associated with failure of the averaging theorem in the neighborhood of the corresponding resonance manifold. The dynamics of the local model (2.37) describes the nonlinear interac√which tion between the two oscillators in the O ε neighborhood of the 1:1 resonance manifold can be analyzed by asymptotic techniques such as the method of multiple

44

2 Preliminary Concepts, Methodologies and Techniques

Fig. 2.10 Phase portrait (slow flow dynamics) for µ = 1.0 of the leading-order approximation of the slow angle for 1:1 internal resonance of the Hamiltonian system (2.31).

scales. This was performed in Vakakis and Gendelman (2001) were the following asymptotic solutions of (2.37) were derived using the method of multiple scales (Nayfeh and Mook, 1995), √ √ εω2 I1 (φ2 ) = I1(1:1) + C ( εφ2 ) + O(ε) (1:1) ω1 (I1 ) √ I2 (φ2 ) = I20 + O ε √ √ ψ(φ2 ) = C( εφ2 ) + O( ε) (2.38) where prime denotes differentiation with respect to the argument √ of the function, and the leading-order approximation of the slow angle C(ζ ), ζ = εφ2 , is governed by the following equation, (2.39) C (ζ ) + µ cos C(ζ ) = 0 √ 3/2 where µ = 0.9897 I20 /[ω2 K(1/2)]. In (2.38) I20 is a real constant determined by the initial conditions, and we recall that φ2 = ω2 t + O(ε) is the only ‘fast’ (time-like) angle of the problem. The phase portrait of the slow angle is presented in Figure 2.10 for µ = 1.0; due to the cyclicity of the slow angle the dynamics is restricted in the strip −3π/2 < C(ζ ) ≤ π/2. We note that there exists a stable equilibrium at (C, C ) = (−π/2, 0) and an unstable equilibrium at (C, C ) = (π/2, 0). These correspond to a stable inphase NNM, and an unstable out-of-phase NNM of system (2.31), respectively. In these NNMs both oscillators vibrate with identical frequencies satisfying the condition of 1:1 internal resonance; in either of these NNMs the nonlinear oscillator agjusts its frequency (through its amplitude) to be equal to the frequency of the linear oscillator. The stable in-phase NNM is surrounded by a family of quasi-periodic orbits, corresponding to continuous energy exchanges between the in-phase and out-

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

45

of-phase NNMs. The limit of this family of quasi-periodic orbits is a hom*oclinic loop (which appears as heteroclinic loop in the plot of Figure 2.10, but, in actuality it connects the unstable NNM with itself). On this hom*oclinic orbit there is gradual (asymptotic) transfer of energy from the in-phase NNM to the unstable out-of-phase NNM as t → ±∞; however, this orbit is only realized at a specific energy level and, due to its degeneracy, is sensitive to perturbations in initial conditions. Outside the hom*oclinic loop there exist mixed-mode librations of the dynamics corresponding to smaller energy interactions between the two NNMs of the system. There exist two additional localized NNMs in system (2.31) with both oscillators possessing identical dominant frequencies, but these are not gereneared due to 1:1 internal resonance, and, hence, are not captured by the previous singular perturbation analysis [actually, both these localized modes can be approximated by regular perturbation analysis of (2.31)]. Indeed, there exists an additional out-of-phase stable NNM localized to the linear oscillator, with negligible amplitude of the nonlinear oscillator; clearly, this linearized mode can not be captured by the previous analysis where nonlinear effects and 1:1 internal resonance play the central role. Moreover, away from the 1:1 resonance manifold defined by (2.36) there exists an additional branch of stable out-of-phase NNMs where both oscillators possess equal dominant frequencies, but not equal to ω2 ; this NNM is localized to the nonlinear oscillator (Vakakis and Gendelman, 2001; Vakakis et al., 2003). Internal resonances represent a fundamental mechanism for nonlinear dynamic interactions in nature, and their affects are evident in a broad range of complex phenomena, ranging from resonance interactions of asteroids with Jupiter (Dermott and Murray, 1983) and cardiac arrythmias (Guevara et al., 1981), to nonlinear resonances in plasmas and fluids (Gildenburg et al., 2001; Mendonça et al., 2003) and nonlinear modal interactions in unforced and forced flexible engineering structures (Nayfeh and Mook, 1995). Moreover, internal resonances give rise to essentially nonlinear dynamic behavior, such as bifurcations and chaotic motions, and prevent local linearization of dynamical systems by smooth coordinate transformations in neighborhoods of equilibrium points [see theory of normal forms (Guckenheimer and Holmes, 1983; Wiggins, 1990)]. But more importantly for our discussion, internal resonances provide a fundamental nonlinear mechanism for energy transfer between interacting systems or modes, thus paving the way for realization of directed or targeted energy transfers from a component of a dynamical system to another. In view of the fact that the majority of systems considered in this work will possess some form of energy dissipation, we need to extend the notion of internal resonance to dissipative (non-conservative) dynamical systems and, hence, introduce the concept of resonance capture. Resonance capture (or capture/entrapment into a resonance manifold) can be regarded as a form of transient internal resonance, whereby an orbit of the dynamical system is captured in the neighborhood of a resonance manifold in phase space, triggering vigorous energy exchanges between different components of the system. Moreover, in similarity to internal resonances, resonance captures prevent the direct application of the averaging principle, particularly in systems with multiple frequencies (Arnold, 1988; Sanders and Verhulst, 1985); on the other hand, reso-

46

2 Preliminary Concepts, Methodologies and Techniques

nance captures lead to interesting energy exchanges and dynamic interactions in celestial mechanics, orbital mechanics, and even in particle dynamics (Koon et al., 2001; Belokonov and Zabolotnov, 2002; Itin et al., 2000). Resonance captures play an important role in targeted energy transfer in dissipative systems, and provide the necessary conditions for irreversible and one-way, passive energy transfer from a component of a dynamical system to a different one, which the later component acting, in essence, as a nonlinear energy sink. In the following exposition we provide some definitions that will help us classify the different types of resonance captures that will be encountered in this work. Consider the following general nonlinear non-conservative (dissipative) system in polar form with multiple phase angles (Sanders and Verhulst, 1985): I = εR(φ, I ) φ = ω(I )

(2.40)

where I ∈ R +p , φ ∈ T q (generally, q ≤ p), and ω(I ) = [ω1 (I ), ω2 (I ), . . . , ωq (I )]T . The dimension of the p-vector I may be even greater than that of the original dynamical system depending on the required number of fast-frequency frequency decompositions (this is indeed the case in this work, see Chapter 9). In (2.40) the p-vector I represents energy-like amplitudes (like the actions in the previous example), whereas φ is a q-vector of angles. The set of points in D ⊂ R p where ωi (I ) ≡ 0, i = 1, 2, . . . , q defines a resonance manifold. This resonance condition is necessary but not sufficient, since as shown previously, if ωi (I ) = 0, i = 1, 2, . . . , q, the internal resonance manifold ˆ ω(I ) = 0, kˆ ∈ Z q }, where the corresponding is defined as the set {I ∈ R p : k, Fourier coefficients from R(φ, I ) are not identically zero (and the notation •, • denotes internal product between two vectors). Assume that the dynamical flow of system (2.40) intersects transversely the resonant manifold. In similarity to internal resonance realized in Hamiltonian or conservative systems, capture into resonance may occur for some phase relations satisfying the condition that an orbit of the dynamical system reaching the neighborhood of a resonant manifold continues in such a way that the commensurable frequency relation is approximately preserved; in this situation not all phase angles are fast (timelike) variables, so classical averaging cannot be performed with respect to these angles. As a result, over the time scale 1/ε the exact and averaged solutions diverge up to O(1) (Arnold, 1988). This is similar to what holds for internal resonance. There are no commonly accepted definitions for transient or sustained resonance capture in the literature. For example, according to the definition provided by Bosley and Kevorkian (1992), if an internal resonance occurs at a time instant t = t0 , with the non-trivial frequency combination σ = k1 ω1 + k2 ω2 + · · · + kq ωq , where ki ∈ N, i = 1, . . . , q, vanishing at that time instant, then, sustained resonance capture (SRC) is defined to occur when the condition σ ≈ 0 persists for times

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

47

t − t0 = O(1). On the other hand, transient resonance capture (TRC) refers to the case when σ makes a single slow passage through zero. Quinn (1997a) provides a slightly different definition of resonance capture, as follows. The possible behavior of trajectories near the resonance manifold on the time scale 1/ε is described according to the following three scenarios: (i) Capture, where solutions are unbounded in backward time, however, captured trajectories remain bounded for forward times of O(1/ε), i.e., a sustained resonance capture occurs in forward time; (ii) Escape, where solutions grow unbounded in forward time, however, in backward time, solutions remain bounded for times of O(1/ε), i.e., a sustained resonance exists in backward time; and (iii) Pass-through, where solutions do not remain in the neighborhood of the resonance manifold in either forward or backward time, and no sustained resonance occurs. The definitions that will be adopted in this work differ slightly from the ones provided by Bosley and Kevorkian (1992) and Quinn (1997a) and follow more closely those provided by Burns and Jones (1993). These definitions are especially suitable for analyzing resonance captures in multi-phase dynamical systems, i.e., in systems possessing multiple phase angles, some of which become slowly-varying in neighborhoods of resonance manifolds. Consider an unforced n-DOF system and denote its linearized natural frequencies by ωk , k = 1, . . . , n. We define the following conditions: (i) Internal Resonance, for motions for which there exist ki ∈ Z, i = 1, 2, . . . , n, such that k1 ω1 + · · · + kn ωn ≈ 0, i.e., some combination of linear natural frequencies satisfy commensurability; (ii) Transient Resonance Capture (TRC), as capture into a resonance manifold which occurs and continues for a certain period of time (for example, on the time scale 1/ε), followed by a transition to escape from capture; this includes the phenomenon of pass-through-resonance as defined by Quinn (1997a); and, (iii) Sustained Resonance Capture (SRC) (denoted also as permanent resonance capture by Burns and Jones, 1993), defined as a resonance capture that will never escape with increasing time. SRCs are quite likely for pendulum-like equations (or called pendulum normal forms) obtained by partial averaging in the neighborhood of a given resonance of the dynamics. An unstable equilibrium point of the corresponding unperturbed pendulum system should be non-degenerate by Neishtadt’s Condition B (Arnold, 1988), which is another weaker transversality condition. For example, a SDOF pendulum equation possesses an unstable equilibrium point (i.e., a saddle point) when the mass is vertically upward, and a hom*oclinic orbit originating from the saddle point and enclosing the stable equilibrium indicating the vertically-downward position. Accordingly, SRCs were formulated by two theorems (Burns and Jones, 1993), one regarding existence of an attractor near the resonance manifold, and the other regarding its domain of attraction and hence the likelihood of resonance captures tending asymptotically to the resonant attractor. A mechanism for resonance capture in perturbed two-frequency Hamiltonian systems was studied by Burns and Jones (1993) where it was shown that the most probable mechanism for resonance capture involves the interaction between the asymptotic structures of the averaged system and the resonance. It was shown that, if

48

2 Preliminary Concepts, Methodologies and Techniques

the system satisfies a less restrictive condition (referred to as Condition N in Lochak and Meunier, 1988) regarding transversal intersection of the averaged orbits of the resonance manifold, resonance capture can be viewed as an event with low probability, and passage through resonance is the typical behavior on the time scale 1/ε. Necessary conditions were proved in Kath (1983a) both for entrainment to TRC and for its continuation (and thus the possible indication of unlocking or escape from resonance capture after a finite time) by successive near-identity transformations; a sufficient condition was also derived for the continuation of transient resonance by means of matched asymptotic expansions (Kath, 1983b). On the other hand, transition to escape from resonance capture was studied by Quinn (1997b) in a coupled Hamiltonian system consisting of two identical oscillators, with each possessing a hom*oclinic orbit when uncoupled. Focusing on intermediate energy levels at which transient resonant motion occurs, Quinn analyzed the existence and behavior of those motions in equipotential surfaces whose trajectories are shown to remain in the transiently stochastic region for long times, and, finally, to escape from (or leak out of) the opening in the equipotential curves and proceeding to infinity. Regarding passage through resonance, one may refer to, for example, Neishtadt (1975). The phenomenon of passage-through resonance is sometimes referred to as non-stationary resonance caused by excitations having time-dependent frequencies and amplitudes (Nayfeh and Mook, 1995). For more details on resonance captures in multi-frequency systems, one can also refer to Bakhtin (1986), Lochak and Meunier (1988), Dodson et al. (1989) and Neishtadt (1997, 1999). Additional works on resonance captures in undamped and damped oscillators are discussed in Section 3.4. For a demonstrative example of transient resonance capture, we consider again the system of coupled oscillators (2.31) but now with weak dissipative terms: y¨1 + ελy˙1 + Cy13 + ε(y1 − y2 ) = 0 y¨2 + ελy˙2 + ω22 y2 + ε(y2 − y1 ) = 0

(2.41)

We wish to examine the perturbation of the 1:1 internal resonance discussed previously by the weak dissipative terms. Transforming into action-angle variables as previously, we reduce the system into a form similar to (2.33), but now with weak dissipative terms added (Vakakis and Gendelman, 2001), I˙1 = ε ×

1/3

3I1 π 2K(1/2)( cn4 + 2 sn2 dn2 ) 1/3

−2λI1 ω1 (I1 )K(1/2) 2 2 1/3 sn dn + I1 cn sn dn − π

≡ εF1 (I1 , I2 , φ1 , φ2 ) I˙2 = −ε 2λI2 cos φ2 + 2

2I2 cos φ2 ω2

2I2 sin φ2 sn dn ω2

2I2 1/3 sin φ2 − I1 cn ω2

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

49

≡ εF2 (I1 , I2 , φ1 , φ2 ) 2/3 −1 4K 2 (1/2)I1 4 2 2 φ˙ 1 = ω1 (I1 ) + ε cn + 2 sn dn π2 2K(1/2) 2I2 1/3 1/3 2 cn sn dn + I1 cn − × −λI1 ω1 (I1 ) sin φ2 cn π ω2 ≡ ω1 (I1 ) + εG1 (I1 , I2 , φ1 , φ2 ) sin φ 2I 2 2 1/3 λ 2I2 ω2 cos φ2 + sin φ2 − I1 cn φ˙ 2 = ω2 + ε √ ω2 2I2 ω2 ≡ ω2 + εG2 (I1 , I2 , φ1 , φ2 )

(2.42)

where the arguments of the Jacobi elliptic functions cn, sn, dn (Byrd and Friedman, 1954) are given by [2K(1/2)φ1/π, 1/2], and the previous notations and parameter definitions hold. We note that the right-hand side exprssions of (2.42) are 2π-periodic in the angles φ1 and φ2 , and that for λ = 0 they provide the explicit expressions of the right-hand sides of system (2.33). In the absence of resonance capture both φ1 and φ2 are ‘fast’ angles, so straightforward two-phase averaging may be applied to (2.42), yielding the following averaged system: I˙1a = −ελI1a I˙2a = −ελI2a

(2.43)

The averaged dynamics predict exponential decays for both actions of the system, as no O(1) nonlinear interactions occur; in that cese, each oscillator vibrates (approximately) independently from the other, resembling a damped SDOF system. In this case condition A of Arnold’s (1988) averaging theorem holds. However, when trajectories of the dissipative system are captured on the 1:1 resonance manifold of the system (2.35) the averaging theorm fails, as only one of the angles remains ‘fast’, and the ‘slow’ phase√ ψ = φ1 − φ2 enters into the asymptotic analysis. Restricting our attention to an O( ε) boundary layer of the 1:1 resonance manifold and working similarly to the Hamiltonian case, we obtain the following local dynamical system (since it is valid only in the neighborhood of the 1:1 resonance manifold), which is identical in form to (2.37): ξ =

√

εω2−1 F˜1 (I1

(1:1)

, I2 , ψ, φ2 ) + εω2−1

∂ F˜1 (1:1) (I , I2 , ψ, φ2 ) + O(ε3/2 ) ∂I1 1

I2 = εω2−1 F˜2 (I1(1:1) , I2 , ψ, φ2 ) + O(ε3/2 ) √ ψ = εω1 (I1(1:1) )ω2−1 ξ + εω2−1 [ω1 (I1(1:1) )ξ 2 /2

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2 Preliminary Concepts, Methodologies and Techniques

˜ 1 (I (1:1) , I2 , ψ, φ2 ) − G ˜ 2 (I (1:1) , I2 , ψ, φ2 )] + O(ε 3/2 ) +G 1 1

(2.44)

In the above equations primes denote differentiation with respect to the remaining ‘fast’ angle φ2 , the notation Fp (I1 , I2 , φ1 = ψ + φ2 , φ2 ) ≡ F˜p (I1 , I2 , ψ, φ2 ), p = ˜ p , p = 1, 2 is imposed. 1, 2, is adopted, and a similar notation for G The solution of the local system (2.44) can be approximated by the method of multiple scales as follows (Vakakis and Gendelman, 2001): √ √ εω2 (1:1) + C ( εφ2 ) I1 (φ2 ) = I1 (1:1) ω1 (I1 ) √ + ε{ω2−1 Fˆ10 (I1(1:1) , I20 , C( εφ2 ), φ2 ) √εφ2 + [T1 (ζ )D(ζ ) + q1 (ζ )]dζ } + O(ε 3/2 ) √ √ I2 (φ2 ) = I20 + εI21 ( εφ2 ) + O(ε) √ √ √ ψ(φ2 ) = C( εφ2 ) + εD( εφ2 ) + O(ε)

(2.45)

φ2 √ √ Fˆ1 (I1(1:1) , I20 , C( εφ2 ), δ)dδ, and Fˆ1 where Fˆ10 (I1(1:1) , I20 , C( εφ2 ), φ2 ) = denotes the zero-mean component of function F˜1 (i.e., the function itself minus its average with respect to φ2 over one period, 0 < φ2 ≤ 2π); I20 is a real constant determined by the initial conditions, and the slowly varying terms are explicitly computed as follows: I21 (ζ ) = (2πω2 )

−1

2π 0

F˜2 (I1(1:1) , I20 , C(ζ ), δ)dδ

∂ F˜1 (1:1) (I , I20 , C(ζ ), δ)dδ ∂ψ 1 0 2π ∂ Fˆ10 (1:1) −1 (I − , I20 , C(ζ ), δ)C (ζ ) q1 (ζ ) = (2πω2 ) ∂C(ζ ) 1 0

T1 (ζ ) = (2πω2 )−1

+ +

2π

∂ F˜1 (1:1) (I , I20 , C(ζ ), δ)I21 (ζ ) ∂I2 1

∂ F˜1 (1:1) ω2 C (ζ ) dδ (I1 , I20 , C(ζ ), δ) (1:1) ∂I1 ω (I )

√ ζ = εφ2

1

1

(2.46) √ Hence, the entire solution relies on the computation of the O(1) and O( ε) approximations for, C and D, of the ‘slow’ angle ψ, which we now proceed to discuss. We start by considering the O(1) approximation. It can be shown (Vakakis and Gendelman, 2001) that the leading-order approximation for the slow phase angle

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

51

Fig. 2.11 Phase portrait (slow flow dynamics) for µ = 1.0 of the leading-order approximation of the slow angle for 1:1 internal resonance of the dissipative system (2.41): (a) weak damping, ν = 0.5; (b) strong damping, ν = 1.2.

√ C(ζ ), ζ = εφ2 , is governed by a perturbation of equation (2.39) of the Hamiltonian system, (2.47) C (ζ ) + µ cos C(ζ ) = −(λω2 /3) ≡ ν where µ was defined in (2.39). Hence, to O(1) the weak dissipation introduces a non-hom*ogeneous term in the pendulum-type equation decribing the slow evolution of the slow angle. In Figure 2.11 we provide the phase portraits of the slow flow (2.47) for µ = 1.0 and ν = 0.5, 1.2. Depending on the relative values of µ and ν, the phase portrait of the slow flow dynamics, either possess (for sufficiently weak damping, if µ > ν – see Figure 2.11a), or not (for relatively strong damping, if µ < ν – see Figure 2.11b) a stable/unstable pair of equilibrium points and a closed hom*oclinic loop. Hence, we conclude that by increasing the dissipation above a critical threshold no resonance captures can be realized in the system (2.41), as evidenced by the lack of equilibrium points in the corresponding slow flow.

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2 Preliminary Concepts, Methodologies and Techniques

Fig. 2.12 Type A and B orbits during resonance capture on a resonance manifold.

Resonance capture in system (2.41) can occur only if dissipation remains below the critical threshold (i.e., for µ > ν – see Figure 2.11a), in which case, to leading order, there are two types of slow flow orbits in the neighborhood of the resonance manifold: closed periodic orbits inside the hom*oclinic loop of the unstable equilibrium (such as the type-A orbit of Figure 2.12) corresponding to sustained resonance capture of the dynamics on the resonance manifold; and open orbits outside the hom*oclinic loop (depicted as type-B orbits in Figure 2.12) corresponding to passage through resonance, and transient resonance capture according to our previous definition. By taking into account higher-order terms in the asymptotic analysis we can show that no sustained resonance captures (SRC) can occur in the dissipative system under consideration. Indeed, the stable and unstable equilibrium points of the O(1) slow flow are the first-order approximations of in-phase and out-of-phase slowly decaying orbits, respectively, satisfying conditions of approximate 1:1 internal resonance (according to the definitions of Section 2.1 these decaying motions take place on damped NNM invariant manifolds). It follows that when higher-order terms are taken into account in the asymptotic analysis the equilibrium points in the slow phase plot of Figure 2.11a are replaced by slowly decaying orbits, with the decay occurring at time scale εt. In addition, the degenerate hom*oclinic loop of the O(1) slow flow (which defines the domain of transient resonance capture) ‘breaks up’ under the perturbation by higher-order (slower) terms, and the stable equilibrium becomes an attractor. To show the effects √ on the slow flow dynamics of higher-order (slower) terms, we consider the O( ε) correction to the ‘slow’ angle governed by the following quasi-linear equation: D (ζ ) − T2 T1 (ζ )D(ζ ) = T2 q1 (ζ ) + q2 (ζ ) where ζ =

√ εφ2 and

(2.48)

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

T2 = ω2−1 ω1 (I1

(1:1)

)

q2 (ζ ) = (2πω22 )−1 ω1 (I1

(1:1)

+ (2πω2 )−1

53

2π

) 0

2π

(1:1) Fˆ10 (I1 , I20 , C(ζ ), δ)dδ

˜1 −G ˜2 G

+ (2ω2 )−1 ω1 (I1(1:1) )

(1:1)

(I1

ω2 C (ζ ) ω1 (I1

(1:1)

,I20 ,C(ζ ),δ)

dδ

2

)

The solution of (2.48) was explicitly computed in√Vakakis and Gendelman (2001) and is not repeated here. When the correction D( εφ2 ) is taken into account, the slow flow phase portrait depicted in Figure 2.11a is perturbed in the following way. The hom*oclinic loop ‘breaks up’ (this is to be expected as it represents a highly degenerate structure of the slow flow dynamics) and replaced by the independent stable and unstable manifolds of the unstable equilibrium point (the out-of-phase√ NNM), whereas the stable center (the in-phase NNM) becomes an attractor. To O( ε) the two components of the stable invariant manifold of the unstable equilibrium √ point define the domain of attraction of 1:1 resonance capture. We note that at O( ε) the two equilibrium points of the slow flow phase portrait of Figure 2.11a still appear as equilibrium points, and only when O(ε) terms (that is, higher-order terms) are taken into account in the perturbation analysis they become slowly decaying orbits on the corresponding damped NNM invariant manifolds; this indicates that √ the envelopes of these orbits decay at a slower time scale compared to the O( ε) slow flow. In Figure 2.13 we present a computation of the ‘break up’ of the hom*oclinic loop carried out by Panagopoulos et al. (2004) for a different system, however, the qualitative features of the slow flow hold for the present system as well. The plot in this figure presents a projection √ of the extended phase space (ψ, ψ , t) of the slow flow dynamics [note that the O( ε) equation (2.48) for the ‘slow’ angle possesses a slowly-varying non-hom*ogeneous term] onto the plane (ψ, ψ ), and provides an √ O( ε) approximation of the domain of attraction of 1:1 resonance capture. We end this section by noting that TRCs will play a central role in our discussion of targeted energy transfer phenomena in coupled oscillators with strongly nonlinear attachments, so this subject will be revisited in the coming chapters. In the next section we discuss an analytical methodology that will be employed throughout this work to theoretically analyze, understand and predict TET phenomena in strongly nonlinear transient dynamics, as well as the associated transient and sustained resonance captures that govern these phenomena. Moreover, this technique is especially suitable for analyzing and identifying strongly nonlinear modal interactions occurring during the damped transitions considered in this work.

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2 Preliminary Concepts, Methodologies and Techniques

√ Fig. 2.13 Perturbation of the hom*oclinic loop of the phase portrait of the ‘slow’ angle by O( ε) terms, showing the domain of attraction of 1:1 resonance capture; the unperturbed hom*oclinic loop is indicated by dashed lines (Panagopoulos et al., 2004).

2.4 Averaging, Multiple Scales and Complexification Different versions and combinations of multiple scales and averaging techniques are widely used for analyzing the responses of nonlinear dynamical systems (Kevorkian and Cole, 1996), and a general review of asymptotic methods in mechanics is provided in Andrianov et al. (2003). In this work we will make extensive use of a special technique, the so called complexification-averaging (CX-A) technique, based on complexification of the dynamics and then averaging over ‘fast’ time-scales. We will show that the CX-A technique is especially suitable for analyzing strongly nonlinear transient responses of the type that we will be concerned with in our study of TET. Indeed, the employment of this special technique is dictated by the fact that the majority of TET problems considered in this work will be formulated in the transient domain (although in Chapter 6 we will examine steady state TET as well), so conventional perturbation or asymptotic techniques such as the methods of averaging, multiple scales and Lindtstead–Poincaré which are more suitable for

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analyzing steady state motions (such as periodic orbits) are not directly applicable in the majority of problems that we will be concerned with in this work (however, these methods will be employed after application of CX-A in our further analysis of the resulting slow flows). Moreover, the systems considered in this work possess strong (and even nonlinearizable) nonlinearities, so perturbation techniques based on linear generating functions and based on the assumption of weak nonlinearity, again are not directly applicable for the TET-related problems examined herein. The complex representation of a nonlinear oscillatory system was initially considered as a phenomenological model that provides enhanced possibility for analyzing nonlinear effects (Scott et al., 1985; Kosevitch and Kovalyov, 1989). Moreover, the use of complexification techniques is widely used in the applied physics literature for studying nonlinear dynamics and wave phenomena. It has been shown recently (Manevitch, 1999, 2001) that this type of complexified models can be formally obtained for anharmonic oscillators and nonlinear oscillatory chains, through the CX-A technique, in order to replace the classical equations of motion by a set of first-order complex (modulation) equations. The method is based on an initial transformation of real coordinates to complex ones, and subsequent use of averaging or multiple scale expansions with further selection of resonance terms for obtaining the main nonlinear approximations (Manevitch 1999, 2001; Gendelman and Manevitch, 2003). We illustrate the different approaches for applying the CX-A method by means of two examples. The first deals with a common and simple model widely used in the nonlinear dynamics literature, namely, the weakly nonlinear Duffing oscillator forced by weak harmonic excitation. The second example concerns the application of the CX-A method to the analysis of the strongly nonlinear transient response of the system of damped oscillators (2.41) undergoing 1:1 transient resonance capture; in this way, we will demonstrate an alternative approach for analyzing transient resonance captures in that system. The first example provides a formal introduction to the CX-A method considering a system that has been analyzed in the literature with a breath of analytical techniques; hence, it will help us relate and compare the application of the CX-A technique with other analytical methods of nonlinear dynamics. The second example demonstrates the application of the CX-A technique to a more complicated, strongly nonlinear transient problem, which lies beyond the formal range of validity of weakly nonlinear conventional methods. Additional and more complicated applications of the CX-A technique will be presented throughout this work to a variety of problems; indeed, this technique will serve as our main theoretical tool for obtaining analytic approximations of the transient damped dynamics of coupled, essentially nonlinear oscillators leading to TET. We start by analyzing the dynamics of the following harmonically forced oscillator, (2.49) y¨ + y + ε(8y 3 − 2 cos t) = 0 where 0 < ε 1 is a dimensionless formal small parameter.

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The first step for applying the method is complexification of the dynamics, which is performed by introducing the new complex variable, ψ = y˙ + jy

(2.50)

˙ → ψ corresponds physwhere j = (−1)1/2 . The transformation of variables (y, y) ically to studying the dynamics from a fixed to a rotating coordinate frame. Transforming the original equation (2.49) in terms of the complex variable (2.50), and recognizing that cos t = (ej t + e−j t )/2, we obtain the following alternative complex differential equation of motion: ψ˙ − j ψ + ε[j (ψ − ψ ∗ )3 − (ej t + e−j t )] = 0

(2.51)

where ∗ denotes complex conjugate. This equation is exact, as it is derived from the original real equation of motion without omitting any terms in the process. At this point we make an assumption regarding the dynamics. In particular, we aim to study the dynamics of (2.51) under the assumption of 1:1 (fundamental) resonance, i.e., under condition that the response of the oscillator has a dominant harmonic component with frequency equal to the frequency of the harmonic excitation. Hence, we express the complex variable in the following polar form: ψ(t) = ϕ(t)ej t

(2.52)

As shown in later chapters, under proper modifications – i.e., multi-fast frequency partitions – the CX-A method can be extended to systems whose responses possess more than one dominant (fast) frequency components. Substituting the representation (2.52) into (2.51) yields the following alternative (still exact) equation of motion: ϕ˙ + j ε[ϕ 3 exp(2j t) − 3|ϕ|2 ϕ + 3|ϕ|2 ϕ ∗ exp(−2j t) − ϕ ∗3 exp(−4j t)] − ε[1 + exp(−2j t)] = 0

(2.53)

At this point there are two ways of proceeding with the analysis, both of which involve approximations of a certain extent. The first way to proceed is to apply a multiple scales analysis to system (2.53) and reduce the problem to solving an hierarchy of linear subproblems at orders of increasing powers of the formal small parameter, εk , k = 0, 1, 2, . . . . In what follows we will demonstrate the application of this approach by analyzing (2.53). The second way of approximately analyzing (2.53) is to average out terms possessing (fast) frequencies higher than unity; this amounts to assuming that (2.52) represents a slow-fast partition of the dynamics (note that no such slow-fast partition is imposed in the first way to solving the problem), with ej ωt representing the fast oscillation, and ϕ(t) its (complex) slow modulation. This alternative approach, which is especially suitable in neighborhoods of resonances of strongly nonlinear transient problems [where the nonlinear terms are not scaled by a formal small parameter as in (2.53)], will be demonstrated in the second example considered later in this section.

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Since no formal assumption regarding the fast frequencies of the system (2.53) was imposed, the multiple scales singlular perturbation technique is applied to analyze its dynamics. To this end, the following asymptotic decomposition of the dependent variable, and the corresponding transformation of the independent variable are introduced: ϕ(t) = ϕ0 (τ0 , τ1 , . . .) + εϕ1 (τ0 , τ1 , . . .) + ε2 ϕ2 (τ0 , τ1 , . . .) + O(ε 3 ) ∂ ∂ d = + ε[1 + εf2 (τ1 )] + O(ε2 ) dt ∂τ0 ∂τ1

(2.54)

where τ0 = t is the fast time scale, and τ1 = εt is the leading-order slow time scale; the higher-order, slower time scales τk , k = 2, 3, . . . are obtained by proper inversion of the second of equations (2.54) once the slow functions f2 (τ1 ), . . . are determined (see discussion below). We emphasize the point that the second of expansions (2.54) is slightly different than those used in conventional multiple scales expansions, and the necessity for introducing slow multiplicative factors such as f2 (τ1 ) in the O(ε2 ) terms will be explained below. Apparently this type of decomposition has been used for the first time by Lighthill (1960), but in the rather different context of problems in aerodynamics. Transforming the slow flow (2.53) by (2.54), we obtain the following hierarchy of linear subproblems at different orders of approximation. The subproblem at O(1) yields the following solution: ∂ϕ0 = 0 ⇒ ϕ0 = ϕ0 (τ1 ) ∂τ0

(2.55)

which indicates that the main approximation for ϕ is slowly-varying (at time scale τ1 = εt); this indicates that under the assumptions of this analysis the ansatz (2.52) indeed represents a slow-fast partition of the dynamics (although this was not assumed a priori). Proceeding to the next order of approximation, we obtain the following linear subproblem governing ϕ1 : ∂ϕ0 ∂ϕ1 + + j [ϕ03 exp(2j τ0 ) − 3|ϕ0 |2 ϕ0 + 3|ϕ0 |2 ϕ0∗ exp(−2j τ0 ) ∂τ1 ∂τ0 − ϕ0∗3 exp(−4j τ0)] − 1 − exp(−2j τ0 ) = 0

(2.56)

This equation represents the O(ε) approximation of the slow flow dynamics of the system, i.e., it governs approximately the slow evolution of the complex amplitude ϕ with time. In order to avoid the secular growth of ϕ1 with respect to the fast time scale, i.e., to avoid a response that will not be uniformly valid with increasing time, we need to eliminate from (2.56) non-oscillating terms. Hence, the following condition must be imposed: ∂ϕ0 − 3j |ϕ0 |2 ϕ0 − 1 = 0 ∂τ1

(2.57)

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Equation (2.57) is integrable, yielding the following first integral of motion for the O(ε) approximation [but not for the original equation of motion (2.49) or (2.53)]: h=

3j |ϕ0 |4 + ϕ0∗ − ϕ0 2

(2.58)

This means that the O(ε) approximation can be analytically computed in closed form. It should be mentioned that the appearance of a first integral of motion is a common feature of CX-A calculations for Hamiltonian systems. Indeed, the exact system (2.49) has a time-dependent Hamiltonian, and by applying averaging, it can be shown that (2.58) is a first integral of the corresponding slow flow. After introducing a polar decomposition of ϕ0 in terms of a real amplitude and a real phase, ϕ0 (τ1 ) = N(τ1 ) exp[j δ(τ1 )], equations (2.57) and (2.58) are rewritten as: ∂N = cos δ, ∂τ1 h=

∂δ 1 = 3N 2 − sin δ ∂τ1 N

3 4 N − 2N sin δ = const 2

(2.59)

Introducing the notation Z = N 2 , combining the first of equations (2.59) with the first integral of motion h into a single equation in terms of Z, and integrating it by quadratures we obtain the following explicit solution for the amplitude N: ⎧ √ √ 2 ⎫1/2 ⎪ ⎪ ⎨ aq sn2 32 pqτ1 , k + bp 1 + cn 32 pqτ1 , k ⎬ N(τ1 ) = − 2 ⎪ ⎪ √ √ ⎩ ⎭ q sn2 32 pqτ1 , k + p 1 + cn 32 pqτ1 , k

(2.60)

where a and b are the two real roots of the algebraic equation 2 4Z − (3/2)Z 2 − h = 0 (with the other two roots being complex and expressed as m±j n) and the remaining parameters are defined according to: 1 −(p − q)2 + (a − b)2 p = (m − a)2 + n2 , q = (m − b)2 + n2 , k = 2 pq In the above expressions k is the modulus of the Jacobi elliptic functions sn(•) and cn(•). From (2.60) the real phase δ(τ1 ) is evaluated directly from the first of equations (2.59). It should be mentioned that the expression for the O(1) approximation ϕ0 has been computed by considering O(ε) terms and applying the method of multiple scales. The same result could be obtained without formal resort to the method of multiple scales by merely omitting all non-resonant terms from the initial (exact)

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equation (2.53) – i.e., by performing ‘naive averaging’ with respect to the fast time scale τ0 (actually this approach will be used in the second example of CX-A technique that follows). This observation means that the seemingly voluntary trick of omitting non-resonant terms from the orgininal exact equation (2.53) may be substantiated by formal use of multiple scales, and thus the efficiency of the CX-A approach may be explained formally, at least for the case of weak nonlinearity. The computation of the next approximation constitutes a somewhat non-trivial problem. To this end, the explicit expression for the first approximation is obtained by solving equation (2.56) after eliminating secular terms through (2.57), 1 3 ϕ1 (τ0 , τ1 ) = − ϕ03 exp(2j τ0 ) + |ϕ0 |2 ϕ0∗ exp(−2j τ0 ) 2 2 j 1 − ϕ0∗3 exp(−4j τ0 ) + exp(−2j τ0 ) + C1 (τ1 ) 4 2

(2.61)

where the slow-varying function C1 (τ1 ) is a constant of integration with respect to the fast time scale τ0 , and is computed by considering the equation governing the O(ε 2 ) approximation: ∂ϕ2 ∂ϕ0 3j 5 ϕ exp(4j τ0 ) + f2 (τ1 ) − ∂τ0 ∂τ1 4 0 + − (15j/2)|ϕ0|2 ϕ03 + 3j ϕ02 C1 (τ1 ) − 3ϕ02 exp(2j τ0 ) 51j 3 ∂C1 |ϕ0 |4 ϕ0 + 3|ϕ0|2 − ϕ02 − 3j [2|ϕ0|2 C1 (τ1 ) + ϕ02 C1∗ (τ1 )] + + ∂τ1 4 2 + − (69j/4) |ϕ0 |4 ϕ0∗ + 3j ϕ02 C1 (τ1 ) + 6j |ϕ0|2 C1∗ (τ1 ) + 6|ϕ0 |2 exp(−2j τ0) + (21j/4)|ϕ0|2 ϕ0∗3 − (9/4)ϕ0∗2 − 3j ϕ0∗2 C1∗ (τ1 ) exp(−4j τ0 ) + (3j/4)ϕ0∗5 exp(−6j τ0 ) = 0

(2.62)

Secular terms in (2.62) are eliminated by imposing the following condition: f2 (τ1 )

∂ϕ0 dC1 (τ1 ) 51j 3 |ϕ0 |4 ϕ0 + 3|ϕ0|2 − ϕ02 + + ∂τ1 dτ1 4 2 − 3j 2|ϕ0|2 C1 (τ1 ) + ϕ02 C1∗ (τ1 ) = 0

(2.63)

Now it is possible to demonstrate that the term containing the unknown function f2 (τ1 ) is unavoidable, so it is necessary to be included in the initial multiple scale expansions (2.54). Indeed, if we set f2 ≡ 0 equation (2.63) has the following solution,

17 τ1 17 19j C1 (τ1 ) = |ϕ0 |2 ϕ0 − + D− |ϕ0 (u)|2 du (3j |ϕ0|2 ϕ0 + 1) 24 72 6 0

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where D is a real constant of integration. The integral term leads to global divergence of the solution, although at time scales of order higher than 1/ε. It should be mentioned that normal averaging procedures guarantee the accuracy at similar time scale, but the approach developed here enables the extension of the analytical solution to even larger time scales. In other words, in order to avoid weak secularity of C1 (τ1 ) we need to introduce an additional function f2 (τ1 ) through the definition (2.54). The first way to compute f2 (τ1 ) is to set the function C1 (τ1 ) equal to zero, and to compensate for the secular terms in (2.63) by appropriate selection of f2 (τ1 ), as follows: (51j/4)|ϕ0|4 ϕ0 + 3|ϕ0 |2 − (3/2)ϕ02 , C1 ≡ 0 (2.64) f2 = − 3j |ϕ0 |2 ϕ0 + 1 The approximate solution for this choice of f2 is computed by combining the previous results (2.59)–(2.63). The dependence of the slow time scale on the original temporal variable is obtained by appropriate inversion of (2.54) with account of the explicit expression (2.64). These expressions may be trivially computed but are not presented here due to their awkwardness. This way of computing f2 (τ1 ) and C1 (τ1 ) has two shortcomings. First, it is inapplicable in the vicinity of stationary points of equation (2.57) because of divergence of f2 there. Second, the slow time variable becomes complex, and additional divergence problems may occur in neighborhoods of the poles of the elliptic functions in (2.60). Despite these shortcomings, the previously outlined procedure may be performed at any order of approximation. However, it is possible to derive an analytic approximation free from the above shortcomings. To this end, one can demonstrate that the requirements of non-diverging C1 (τ1 ), and of real and non-diverging f2 (τ1 ) may be satisfied by a unique choice of these functions as follows: f2 =

17 |ϕ0 |2 , 6

C1 =

17 19j |ϕ0 |2 ϕ0 − 24 72

(2.65)

Then, the corresponding approximation for the solution is given by

1 3 ψ = ϕ0 exp(j t) + ε − ϕ03 exp(3j t) + |ϕ0 |2 ϕ0∗ exp(−j t) 2 2

j 19j 1 ∗3 17 2 |ϕ0 | ϕ0 − exp(j t) − ϕ0 exp(−3j t) + exp(−j t) + 4 2 24 72 (2.66) where ϕ0 = N(εt) exp[j δ(εt)]. The shortcoming of this approach is that in order to compute the second-order approximation we have to solve the equation that eliminates secular terms of the equation at the third degree of the small parameter, which is a rather cumbersome task. We now compare the results obtained by CX-A approach with direct numerical simulations of equation (2.49) for different values of the small parameter and various initial conditions. The numerical parameters used for these simulations are

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Fig. 2.14 CX-A solution of (2.49) for initial conditions y(0) = y(0) ˙ = 0: (a) ε = 0.065, (b) ε = 0.13, (c) ε = 0.03; exact solution is represented by crosses (+ + +), the analytical approximation based on (2.64) by a solid line (—), and the analytical approximation based on (2.65) by diamonds (♦♦♦).

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Fig. 2.15 CX-A solution of (2.49) for initial conditions y(0) = 0.7, y(0) ˙ = 0 (close to fundamental resonance) and ε = 0.5; exact solution is represented by crosses (+ + +), the analytical approximation based on (2.64) by a solid line (—), and the analytical approximation based on (2.65) by diamonds (♦♦♦).

listed in the corresponding figure captions. The results depicted in Figure 2.14 indicate that the analytical approximation including terms up to O(ε 2 ) and based on (2.65) provides a better approximation to the solution, compared to the corresponding analytical approximation based on (2.64) with C1 ≡ 0. Besides, the accuracy of the analytical approximation decreases with increasing values of the small parameter ε, at least in the range considered in the simulations. It should be stressed that large values of ε do not necessarily imply that the derived analytical approximations will be poor. The numerical simulation depicted in Figure 2.15 demonstrates that close to fundamental resonance the analytical solution is close to the exact solution despite the relatively large value of ε used in this particular simulation. Both analytical approximations based on (2.64) and (2.65) provide good approximations to the exact solution, even at relatively large times. The results presented in Figures 2.16–2.18 provide comparisons of exact solutions with the analytical approximation (2.66) based on conditions (2.65), for various values of ε and initial conditions. We note that the accuracy of the analytical approximation decreases with increasing ε. In general, for these simulations the analytical approximation based on conditions (2.64) provides accuracy comparable to the othe analytical approximations depicted in these figures. From the analysis of the dynamical system (2.49) we conclude that the CX-A technique, when applied together with a modified multiple scales procedure, provides good analytical approximations for the forced nonlinear response. Moreover, in regions of resonance the CX-A approach provides good approximations even for relatively large values of the small parameter of the problem (i.e., beyond the formal range of applicability of the multiple-scales approach). Two different approaches were proposed for computing the higher-order approximations, both providing rather reliable predictions in their corresponding regions of applicability. It

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Fig. 2.16 CX-A solution of (2.49) for initial conditions y(0) = 0, y(0) ˙ = 0, (a) ε = 0.03, (b) ε = 0.1; exact solution is represented by a solid line (—), and the analytical approximation based on (2.65) by diamonds (♦♦♦).

should be mentioned that the dimensionless formal parameter ε in (2.49), commonly regarded as the small parameter in conventional asymptotic analyses of this problem, turns out not to be a ‘true’ perturbation parameter. Specifically, the analysis of the previous example demonstrates that the accuracy of the derived asymptotic approximations depends on the relationship between the frequency of the slow modulation ϕ and the (fast) frequency of the main (fundamental) resonance of the problem; however, it is not yet clear how one could select appropriate perturbation parameters to scale this relationship in the analysis, so this issues remains an open problem.

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Fig. 2.17 CX-A solution of (2.49) for initial conditions y(0) = 1, y(0) ˙ = 0 (close to fundamental resonance) and ε = 0.1; exact solution is represented by solid line (—), and the analytical approximation based on (2.65) by diamonds (♦♦♦).

Another possibility for accurate asymptotic expansions using the CX-A technique arises in cases when the initial conditions of the response are in the neighborhood of the stationary point of equation (2.57) (or, in other terms, close to the regime of fundamental resonance of the problem). The small parameter in this case measures the deviation of the response from the stationary point. This case is of major importance in applications of the CX-A technique when systems with strong nonlinearity are studied (where the nonlinear terms are not scaled by a formal small parameter – this is the case in the next example of application of the CX-A technique). Thus, it is justified to apply the CX-A technique even in dynamical systems that do not formally satisfy the conditions of the averaging theorem (see, for example, Kevorkian and Cole, 1996), but only in response regimes that are either close to exact resonance, or in the domains of attraction of the corresponding resonance manifolds. This observation paves the way for the application of the CX-A technique to TET-related problems, where transient or sustained resonance captures on fundamental or subharmonic resonance manifolds are dominant in the corresponding damped, nonlinear transient resposes. It should be mentioned that the very presentation of the equations of motion in complex form [i.e., equation (2.53)], and the elimination of non-resonant terms for the modulation equations much resembles the well-known method of normal forms (Guckenheimer and Holmes, 1983; Wiggins, 1990; Nayfeh, 1993; Kahn and Zarmi, 1997). Still, the CX-A technique outlined above is based on different ideas of multiple scales and averaging and seems to lead to essential simplifications of these well-known methods. In the second example considered in this section we demonstrate the application of the CX-A technique to a strongly nonlinear transient problem, and show that the method is capable of analytically modeling the regime of 1:1 transient resonance

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Fig. 2.18 CX-A solution of (2.49) for initial conditions y(0) = 0.7, y(0) ˙ = 0 (close to fundamental resonance), (a) ε = 1.0, (b) ε = 1.3; exact solution is represented by a solid line (—), and the analytical approximation based on (2.65) by diamonds (♦♦♦).

capture (i.e., of 1:1 transient resonance) in a system of coupled oscillators, in accordance with the previous discussion. To this end, we reconsider the two-DOF system of coupled damped oscillators (2.41) examined in the previous section, and apply the CX-A technique to study the regime of 1:1 TRC (Vakakis and Gendelman, 2001); this response regime was studied in the previous section using an alternative methodology, i.e., by resorting to action-angle transformations and analyzing the corresponding local model in the neighborhood of the 1:1 resonance manifold by the method of multiple scales. Rewriting the system (2.41) in the form

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y¨1 + ελy˙1 + Cy13 + ε(y1 − y2 ) = 0 y¨2 + ελy˙2 + ω2 y2 − εy1 = 0

(2.67)

where ω2 = ω22 + ε, and introducing the new complex variables, ψ1 = y˙1 + j ωy1 ,

ψ2 = y˙2 + j ωy2

(2.68)

we express (2.67) as the following set of first-order complex differential equations: (j ω + ελ) jε ψ˙ 1 − (ψ1 + ψ1∗ ) − (ψ1 − ψ1∗ ) 2 2ω jε jC (ψ2 − ψ2∗ ) = 0 + 3 (ψ1 + ψ1∗ )3 + 8ω 2ω ελ jε (ψ1 − ψ1∗ ) = 0 ψ˙ 2 − j ωψ2 + (ψ2 + ψ2∗ ) + 2 2ω

(2.69)

The set of equations (2.69) is exact, and, in contrast to the previous example, it represents a strongly nonlinear system since the nonlinear terms are not scaled by a small parameter and the initial conditions are assumed to be O(1) quantities. We now seek an approximate solution of (2.69) based on the assumption of 1:1 resonance, i.e., by assuming that both oscillators execute slowly-modulated oscillations with identical ‘fast’ frequencies equal to ω: ψ1 = ϕ1 ej ωt ,

ψ2 = ϕ2 ej ωt

(2.70)

In essence, in the regime of 1:1 TRC we partition the dynamics in terms of the ‘slow’ complex amplitudes ϕi , i = 1, 2 modulating the ‘fast’ oscillatory terms ej ωt . Hence, in contrast to the previous example, and in the absence of a formal small parameter scaling the nonlinear terms, we make the basic assumption that there exists a single fast frequency ω in the dynamics as a means of simplifying the analysis. This is needed in view of the fact that formal application of the method of multiple scales [at least with linear trigonometric generating functions – but see Belhaq and Lakrad (2000) and Lakrad and Belhaq (2002) for extensions of the multiple scales method with Jacobi elliptic functions and Yang et al. (2004) and Chen and Cheung (1996) for extension of averaging and other perturbation schemes based on elliptic generating functions] is not justified in this strongly nonlinear problem. Substituting (2.70) into (2.69) and averging out terms that contain fast frequencies higher than ω (such as terms multipled by e2j ωt , e3j ωt , . . .) we obtain the following approximate slow flow valid in the regime of 1:1 TRC:

ω ελ j 3j C jε ω− + ϕ1 − ϕ2 = 0 |ϕ1 |2 ϕ1 + ϕ˙1 + 3 2 2 2 8ω 2ω ϕ˙2 +

ελ jε ϕ2 + ϕ1 = 0 2 2ω

(2.71)

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The fact that (2.71) is an averaged system, among other approximations, poses certain restrictions concerning the time domain of its validity. As mentioned earlier, when first-order averaging is performed in systems in standard form containing a small parameter ε [for example, see relations (2.33) and (2.42)], the validity of the results is only up to times of O(1/ε). In the CX-A approach described above there is no formal small parameter to describe the slowly-varying character of the complex modulations ϕ1 and ϕ2 , so we cannot provide a formal result regarding its range of validity. In this regard, we can only state that the averaged slow flow (2.71) is valid only up to finite times, as long as the basic assumptions outlined above (regarding the slow-fast partition and the existence of a single ‘fast’ frequency in the dynamics) are satisfied. Returning to the analysis of the slow flow (2.71), in order to account for the amplitude decays of the two oscillators due to damping dissipation we introduce the new variables, σ1 and σ2 defined by the relations, ϕi = σi e−ελt /2, i = 1, 2, and express the averaged slow flow in the following form: ω j 3j Ce−ελt jε ω− σ1 − σ2 = 0 |σ1 |2 σ1 + 3 2 2 8ω 2ω jε σ1 = 0 σ˙ 2 + 2ω σ˙ 1 +

(2.72)

We now show that the above dynamical system is fully integrable. To this end, we multiply the first of equations (2.72) by the complex conjugate σ1∗ ; then we take the complex conjugate of the same first equation and multiply it by σ1 . We perform similar operations on the second of equations (2.72), i.e., we first multiply it by σ2∗ and then multiply its complex conjugate by σ2 . By adding the so derived four complex expressions we show that the averaged system (2.72) possesses the following first integral of motion: σ˙ 1 σ1∗ + σ˙ 1∗ σ1 + σ˙ 2 σ2∗ + σ˙ 2∗ σ2 = 0 ⇒ d(|σ1 |2 + |σ2 |2 = 0 ⇒ |σ1 |2 + |σ2 |2 = ρ 2 dt

(2.73)

This first integral is a conservation-of-energy-like integral of the averaged system when expressed in terms of the σ -variables. This enables us to express the complex amplitudes in the following polar representations: σ1 = ρ sin θ ej δ1 ,

σ2 = ρ cos θ ej δ2

(2.74)

which, when substituted into (2.72) and following certain algebraic manipulations, reduce the isoenergetic averaged dynamics (i.e., for ρ = const) to the following dynamical system on the two-torus (δ, θ ) ∈ T 2 : ε ω 3Cρe−ελt sin2 θ + cot 2θ cos δ = 0 δ˙ + − 2 8ω3 ω

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Fig. 2.19 Phase plots of the reduced slow flow (2.75): (a) case of no resonance capture, ρ = 7.84; and cases of 1:1 resonance capture, (b) ρ = 16.0, (c) ρ = 100.0 and (d) ρ = 225.0.

θ˙ +

ε sin δ = 0 2ω

(2.75)

In (2.75) we introduced the phase difference δ = δ1 − δ2 , which denotes the relative phase between the two oscillators during 1:1 resonance, and the angle θ which determines their corresponding amplitudes (θ ≈ 0 denotes localization of the oscillation to the linear oscillator, whereas θ ≈ π/2 denotes localization to the nonlinear oscillator). Moreover, in the averaged slow flow there occurs a slow ‘drift’ of the ‘instantaneous equilibrium points’ of the reduced flow (2.75) due to the previously introduced exponentially decaying coordinate transformation that relates the complex amplitudes ϕi and σi . Hence, the present analysis accurately captures the O(ε) slow ‘drift’ of the equilibrium points of the slow flow [as discussed in the analysis of system (2.41) in Section 2.3]. The numerical integrations of system (2.75) for varying values of the initial first integral ρ reveal clearly the 1:1 resonance capture in the system. These results are presented in the (δ, θ ) phase plots of Figure 2.19 for parameters ω = 1.0, C = 2.0, ε = 0.1, λ = 1.0 and initial conditions δ(0) = 0.0 and θ (0) = 0.01 (i.e., for motion initially localized to the linear oscillator). For ρ = 7.84 (see Figure 2.19a) the initial energy localized in the linear oscillator remains confined to that oscillator (indicated by the fact that θ is in the neighborhood of zero for the entire duration of

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Fig. 2.20 Transient response of the nonlinear oscillator of system (2.67) for, (a) h = 0.5 (no resonance capture), (b) h = 0.8 and (c) h = 1.125 (cases of 1:1 TRC); — exact numerical simulation, ♦♦♦ CX-A analysis.

the motion). At higher values of ρ (see Figures 2.19b–d) we note targeted energy transfer from the linear to the nonlinear oscillator; indeed, orbits that start initially with θ ≈ 0, after some transients settle to damped oscillations with θ ≈ π/2, i.e., localize to the nonlinear oscillator. Of particular interest is the fact that the analytical results capture accurately not only the 1:1 resonance capture of the dynamics, but also the transition to resonance capture as the dynamics is attracted towards the neighborhood of the 1:1 resonance manifold. To assess the accuracy of the analytical predictions obtained by the CX-A technique, in Figure 2.20 we compare the theoretically predicted response y1 (t) of the nonlinear oscillator through application of the previous CX-A technique, to the corresponding numerical response derived by direct numerical simulation of the original equations of motion (2.67). For these results we used the system parameters ω22 = 0.9, C = 5.0, ε = 0.1 and λ = 0.5, and set all initial conditions to zero √ except for the initial velocity of the linear oscillator, y˙2 (0) = 2h. In Figure 2.20a

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we depict the low-energy damped response for h = 0.5; in this case no resonance capture occurs in the dynamics, and there is poor agreement between the analytical and numerical results. This is justified by the fact that the basic assumption of the CX-A analysis (i.e., that both oscillators possess a single dominant fast frequency nearly equal to ω2 ) does not hold in this low-energy regime. In Figures 2.20b, c where 1:1 TRC (and TET) takes place there is satisfactory agreement between the predicted and numerical transient responses, although some overshooting or undershooting can be noted in certain time intervals. These errors can be attributed to the averaging approximations introduced in the CX-A analysis, and to the strong nonlinearities of the system considered. These results demonstrate the potential of the CX-A technique to accurately model strongly nonlinear transient responses under conditions of resonance capture. In this work the CX-A technique will be applied to various problems involving TRCs in coupled oscillators whose responses possess single or multiple fast frequencies. It will be shown that this method is a valuable analytical tool for studying strongly nonlinear damped responses resulting in single- or multi-frequency TET. Moreover, coupled with advanced signal processing algorithms, the CX-A technique can be also applied to studies of identification of strongly nonlinear modal interactions governing TET in practical applications, such as aeroelastic instability suppression, shock isolation of flexible structures, and passive seismic mitigation. In the next section we provide a brief discussion of some advanced signal processing techniques that will be used throughout this work to analyze strongly nonlinear transient responses related to TET.

2.5 Methods of Advanced Signal Processing The strongly nonlinear dynamics governing TET require the use of special techniques for their analysis and post-processing. In this work we will make extensive use of advanced signal processing techniques that are especially suited for postprocessing non-stationary nonlinear time series. In this section we provide a brief introduction to these techniques. Specifically, one way to carry out the study of strongly nonlinear weakly dampled dynamics coupled oscillators considered in this work will be to superimpose the wavelet transform (WT) spectra of the transient responses in frequency-energy plots (FEPs) of the corresponding Hamiltonian dynamics, as discussed in Section 2.1. In performing this procedure we recognize that the effect of weak damping on the transient dynamics is rather parasitic (in the sense that it does not generate ‘new dynamics,’ but rather acts as perturbation of the underlying Hamiltonian responses), so that the damped transient responses are expected to occur in neighborhoods of periodic (or quasi-periodic) Hamiltonian motions. Once this is recognized, the interpretation of the damped nonlinear dynamics and the full understanding of the associated multi-frequency modal interactions become possible. In addition, analysis of strongly nonlinear damped transitions will be performed by applying Empirical Mode Decomposition (EMD) to the measured time series,

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Hilbert-transforming the resulting Intrinsic Model Functions (IMFs) and then comparing the results to Wavelet Transform spectra. We will show that this process can help us identify and classify the different strongly nonlinear transitions that take place in multi-frequency transient data of the type considered in TET applications. Since these methodologies will be applied throughout this work, in what follows we give a brief exposition of their basic elements and provide some preliminary examples of their applications to nonlinear time series analysis.

2.5.1 Numerical Wavelet Transforms The WT can be viewed as a basis for functional representation, but is at the same time a relevant technique for time-frequency analysis. In contrast to the Fast Fourier Transform (FFT) which assumes signal stationarity, the WT involves a windowing technique with variable-sized regions. Small time intervals are considered for high frequency components, whereas the size of the interval is increased for lower frequency components thereby providing better time and frequency resolutions than the corresponding FFTs. Hence, the Wavelet Transform (WT) can be viewed as the ‘dynamic’ extension of the ‘static’ Fourier Transform (FT), in the sense that instead of decomposing a time series (signal) in the frequency domain using the cosine and sine trigonometric functions (as in the FT), in the WT alternative families of orthogonal functions are employed which are localized in frequency and time. These families of orthogonal functions, the so-called wavelets can be adapted in time and frequency to provide details of the frequency components of the signal during the time interval analyzed. These wavelets result from a mother wavelet function through successive iterations. As a result, the WT provides the transient evolution of the main frequency components of the time series, in contrast to the FT that provides a ‘static’ description of the frequency of the signal. In this work, the results of applying the numerical WT are presented in terms of WT spectra. These contour plots depict the amplitude of the WT as a function of frequency (vertical axis) and time (horizontal axis). Heavy shaded regions correspond to regions where the amplitude of the WT is high, whereas lightly shaded ones correspond to low amplitudes. Such plots enable one to deduce the temporal evolutions of the dominant frequency components of the signals analyzed. The Matlab program used for the WT computations reported in this work was developed at the University of Liège by Dr. V. Lenaerts in collaboration with Dr. P. Argoul from the Ecole Nationale des Ponts et Chaussées (Paris, France). Two types of mother wavelets ψM (t) are considered: (a) The Morlet wavelet which is a Gaussian-windowed complex si2 nusoid of frequency ω0 , ψM (t) = e−t /2 ej ω0 t ; and (b) the Cauchy wavelet of order n, ψM (t) = [j/(t + j )]n+1 , where j = (−1)1/2 . The frequency ω0 for the Morlet WT and the order n for the Cauchy WT are user-specified parameters which allow one to tune the frequency and time resolutions of the results. It should be noted that these two mother wavelets provide similar results when applied to the signals con-

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sidered in the present work. In recent works by Argoul and co-workers (Argoul and Le, 2003; Le and Argoul, 2004; Yin et al., 2004; Erlicher and Argoul, 2007), the continuous Cauchy Wavelet transform was applied to system identification of linear dynamical systems. We demonstrate the application of the numerical WT by an example taken from the dissertation thesis by Tsakirtzis (2006). Specifically, we consider a two-DOF linear system weakly coupled to a three-DOF attachment composed of strongly nonlinear coupled oscillators (this system will be studied in detail in Chapter 4, Section 4.1.2, where TET from linear systems to strongly nonlinear MDOF attachments will be analyzed): u¨ 1 + (ω02 + α)u1 − αu2 + ελu˙ 1 = F1 (t) u¨ 2 + (ω02 + α + ε)u2 − αu1 − εv1 + ελu˙ 2 = F2 (t) µv¨1 + C1 (v1 − v2 )3 + ε(v1 − u2 ) + ελ(v˙1 − v˙2 ) = 0 µv¨2 + C1 (v2 − v1 )3 + C2 (v2 − v3 )3 + ελ(2v˙2 − v˙1 − v˙3 ) = 0 µv¨3 + C2 (v3 − v2 )3 + ελ(v˙3 − v˙2 ) = 0

(2.76)

In Figures 2.21–2.23 we present the WT spectra of the relative responses v2 − v1 and v3 − v2 of the strongly nonlinear attachment for parameters ε = 0.2, α = 1.0, C1 = 4.0, C2 = 0.05, ελ = 0.01, µ = 0.08, and ω02 = 1.0; in the simulations out-of-phase impulsive excitations are considered, F1 (t) = −F2 (t) = Y δ(t) with zero initial conditions. First, we consider the WT spectra of the weakly forced responses depicted in Figure 2.21. In this case there occurs strong targeted energy transfer (TET) from the directly forced linear system to the nonlinear attachement (amounting to nearly 90% of input energy transferred and dissipated by the attachment). Examination of the WT spectra reveals certain interesting features of the dynamics. Indeed, we note that there occurs a transient resonance capture (TRC) of the dynamics of the relative response v1 − v2 by a strongly nonlinear mode whose frequency varies in time and lies in between the two natural frequencies of the uncoupled and undamped linear system; that this is a strongly nonlinear mode is signified by the fact that it does not lie close to either one of the linear natural frequencies of the system, which implies that this mode localizes predominantly to the nonlinear attachment. The strong nonlinearity of the response is further signified by the occurrence of an initial multi-frequency beat oscillation (subharmonic or quasi-periodic), as evidenced by the existence of an initial high frequency component in the spectrum of v1 − v2 . In addition, the second nonlinear stiffness-damper pair of the attachment (corresponding to the relative response v2 − v3 ) absorbs (and dissipates) broadband energy from the both modes of the linear system; this is evidenced by the fact that the corresponding WT spectrum of Figure 2.21b possesses a broad range of frequency components that includes both natural frequencies of the linear system. These results indicate that strong TET in this case is associated with TRCs of the dynamics of the nonlinear attachment by strongly nonlinear modes that predom-

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Fig. 2.21 WT spectra of the relative responses, (a) ν1 −ν2 , and (b) ν2 −ν3 of system (2.76) for outof-phase impulse excitation of magnitude Y = 0.1; the linear natural frequencies of the uncoupled and undamped linear system (ε = 0) are indicated by dashed lines.

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inantly localize to the attachment; moreover these TRCs take place over a broad frequency range, resulting in broadband TET. Hence, it becomes clear that the numerical WT spectra provide important information not only regarding the frequency contents of the nonlinear responses, but also on the temporal evolution of each individual frequency component as the interaction between the linear and nonlinear subsystems progresses in time. This underlines the usefulness of the WT as a tool to analyze essentially nonlinear dynamical interactions of the type considered in this work. By increasing the magnitude of the impulse to Y = 1.0, there occurs a marked deterioration of TET from the linear system to the nonlinear attachment. In Figures 2.22a, b we depict the corresponding WT spectra of the relative responses of the nonlinear attachment in this case, which reveal the reason for poor TET. Indeed, the dynamics of the nonlinear attachment appears to engage in sustained resonance capture (SRC) predominantly with two weakly nonlinear modes lying in the corresponding neighborhoods of the in-phase and out-of-phase modes of the unforced and undamped linear system. Moreover, the fact that the weakly nonlinear in-phase and out-of-phase modes localize predominantly to the linear system, prevents significant localization of the vibration to the NES, a feature that contributes to weaker TET. We conclude that weak TET in this case is associated with SRC of the dynamics with weakly nonlinear modes that are predominantly localized to the linear subsystem. Finally, in Figures 2.23a, b we depict the corresponding WT spectra for the system with stong out-of-phase excitation Y = 1.5. Similarly to the case depicted in Figures 2.21a, b, we note the occurrence of a strong TRC of the dynamics on a strongly nonlinear mode localized predominantly to the nonlinear attachment; this TRC leads to strong TET from the linear system to the attachment. Comparing the WT spectra of Figures 2.23a, b to those of the case of weak TET (depicted in Figures 2.22a, b), we note that in the later case the transient responses are dominated by sustained frequency components (i.e., by SRCs), indicating excitation of weakly nonlinear modes which are mere analytic continuations of linearized modes of the system. On the contrary, in cases where strong TET occurs, the frequencies of the nonlinear modes involved in the TRCs are not close to linearized natural frequencies, indicating that these are strongly nonlinear modes having no linear analogs; as a result, these modes localize predominantly to the NES. A general conclusion drawn from the examination of these WT spectra is that the TET efficiency of system (2.76) may be explained by the examination of the resonance captures depicted in the WTs of the transient responses. Indeed, strong TET in the system is associated with TRCs of the dynamics with essentially (strongly) nonlinear modes localized predominantly to the nonlinear attachment; whereas weak TET involves SRCs, i.e., sustained excitation of weakly nonlinear modes (i.e., modes that are analytic continuations of linearized modes of the system) localized predominantly to the linear system. This application demonstrates clearly the potential of the numerical WT as a tool for analyzing and interpreting strongly nonlinear transient dynamics in terms of transient or sustained resonance captures. Moreover, when combined with Em-

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Fig. 2.22 WT spectra of the relative responses, (a) ν1 −ν2 , and (b) ν2 −ν3 of system (2.76) for outof-phase impulse excitation of magnitude Y = 1.0; the linear natural frequencies of the uncoupled and undamped linear system (ε = 0) are indicated by dashed lines.

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Fig. 2.23 WT spectra of the relative responses, (a)ν1 − ν2 , and (b) ν2 − ν3 of system (2.76) for outof-phase impulse excitation of magnitude Y = 1.5; the linear natural frequencies of the uncoupled and undamped linear system (ε = 0) are indicated by dashed lines.

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pirical Mode Decomposition and the Hilbert transform it can form the basis of an integrated nonlinear approach for identifying the transient dynamics as well as the modal interactions that occur in the dynamics of systems with strongly nonlinear substructures.

2.5.2 Empirical Mode Decompositions and Hilbert Transforms The Empirical Mode Decomposition (EMD) is a technique for decomposing a signal in terms of intrinsic oscillatory modes that are termed intrinsic mode functions (IMFs). The IMFs satisfy the following three main conditions, which are imposed in an ad hoc fashion: (a) For the duration of the entire time series, the number of extrema and of zero crossings of each IMF should either be equal or differ at most by one; (b) at any given time instant, the mean value (moving average) of the local envelopes of the IMFs defined by their local maxima and minima should be zero; and (c) the linear superposition of all IMFs should reconstruct the original time series. The EMD algorithm for computing the intrinsic mode functions (IMFs) of a signal (time series), say x(t), is called sifting process and involves the following steps (Huang et al., 1998a, 1998b, 2003): (a) Consider separately the envelopes defined by the local maxima and minima of x(t), and interpolate the locus of all local maxima of x(t) through a spline 1 (t); simapproximation, thus constructing an upper envelope of the signal emax ilarly interpolate the locus of all local minima of x(t) thus creating a lower 1 (t). envelope of the signal, emin (b) Compute the moving average R1 (t) between the lower and the upper envelopes, and define the modified, zero-mean signal h1 (t) = x(t) − R1 (t). (c) Repeat this procedure k times starting from h1 (t) until the signal computed at the k-th iteration, say h1k (t) ≡ c1 (t), satisfies the properties of an IMF therefore one stop criterion must be applied. The stop criteria of the repeatable procedure can be various; one of them is being applied in each case. In our applications we use either the standard deviation between the (k − 1)-th and k-th steps or the number of successive repetitions of the sifting process. This process yields the first IMF of the signal x(t), namely, c1 (t). (d) The second-order remainder of the signal, x2 (t), is defined by the relation x2 (t) = x(t) − c1 (t), on which the previous procedure is repeated to extract the second IMF, c2 (t). (e) The outlined procedure is repeated until the n-th order remainder, xn (t), becomes a monotonic function of time. As discussed above, one can employ alternative convergence criteria for completing the outlined iterative algorithm; two of them are extracted directly from the afore-mentioned properties of the IMFs. The first convergence criterion determines convergence when the following standard deviation between the (k − 1)-th and k-th steps,

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Fig. 2.24 Schematic presentation of application of the empirical mode decomposition to the signal x(t) = sin ω0 t + sin 3ω0 t where ω0 = 2π.

T |h1(k−1)(t) − h1(k) (t)|2 SD = h21(k−1)(t) t =0 is reduced below a preset tolerance, and T is the signal duration; in this work this tolerance was chosen in the range [0.2, 0.3]. Practically, this criterion implies that the k-th iteration h1k (t) ≡ c1 (t) is approximately (within the specified tolerance) zero-mean. A second convergence criterion consistent with the properties of the IMFs, is to determine convergence by computing the successive repetitions of the sifting process, and determining if the number of zero crossings and the number of extrema are equal or differ by one for S repetitions; in this work S was chosen to be equal to either 2 or 3. In this study, we utilize Matlab codes developed by Rilling et al. (2003) to perform numerical EMD. Figure 2.24 depicts schematically the extraction of IMFs from the signal x(t) = sin 2πt + sin 6πt. Since there is no control of the sifting process, end effects appear in the results. Following the previous notation the two IMFs of this signal are computed as c1 (t) ≈ sin 6πt (i.e., the high-frequency component is extracted first), and c2 (t) = x(t) − c1 (t) ≈ sin 2πt. By the construction algorithm outlined above, the lowest-order IMFs contain the oscillatory components (IMFs) of the signal with the highest frequency components. As the order of the IMFs increases, their corresponding frequency contents decrease accordingly. Hence, EMD analysis extracts oscillating modulations or modes imbedded in the data, which could be regarded as the ‘oscillatory building blocks’

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of the signal. It follows that the essence of the EMD method is to empirically identify the intrinsic oscillatory modes in the data (time series), and to categorize them in terms of their characteristic time scales, by considering the successive extreme values of the signal. Hence, the result of the analysis is a multi-scale separation of the time series in terms of its oscillating components, with the different time scales being extracted automatically by the algorithm itself. As discussed below, the EMD algorithm, when combined with the Hilbert transform can provide further insightful information on the decomposition of the signal. After applying the EMD analysis to the time series, the extracted IMFs are Hilbert-transformed in order to compute their approximate transient amplitudes and phases. The Hilbert transform H [c(t)] ≡ c(t) ˆ of a signal (time series) c(t) is defined as follows: +∞ 1 c(τ ) 1 dτ ≡ ∗ c(t) (2.77) c(t) ˆ = π t − τ πt −∞ where (*) denotes the convolution operator: t f (τ )g(t − τ )dτ = f (t) ∗ g(t) = −∞

t −∞

f (t − τ )g(τ )dτ

Hence, the Hilbert transform does not change the domain of the signal, as it transforms the signal from the time domain to the time domain. In the context of the following analysis, the Hilbert transform of the signal c(t) can be regarded as the ‘imaginary’ part of the signal, enabling one to perform a complexification of that signal. Indeed, defining the complexified analytical signal ψ(t) = c(t) + j c(t) ˆ

(2.78)

where j = (−1)1/2 , we compute its amplitude A(t) and phase ϕ(t) by expressing the complexification in polar form: ψ(t) = A(t)ej ϕ(t ) = A(t) cos ϕ(t) + j A(t) sin ϕ(t)

(2.79)

It follows that the signal can be represented in the form c(t) = A(t) cos ϕ(t)

(2.80)

with amplitude and phase given by A(t) =

c(t)2 + c(t) ˆ 2,

ϕ(t) = tan−1

c(t) ˆ c(t)

(2.81)

These decompositions enable one to compute the instantaneous frequency of the signal c(t) according to the following definition: f (t) =

˙ˆ − c(t) ϕ(t) ˙ c(t)c(t) ˆ c(t) ˙ = 2 2π 2π[c(t) + c(t) ˆ 2]

(2.82)

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Therefore, by applying the Hilbert transform to each IMF component resulting from EMD of a signal, we can determine the variation of the instantaneous frequency of each IMF; this, in turn, enables us to get valuable insight into the dominant frequency components that are contained in each IMF and to study resonant modal interactions between IMFs of responses of different components of a system. It is precisely these results that make the combined EMD-Hilbert transform useful for the TET problems considered in this work. Indeed, the decomposition of the transient responses of different components of a system in terms of their oscillatory components (IMFs), and the subsequent computation of the instantaneous frequencies of these IMFs, provides a useful tool for studying nonlinear resonant interactions between these components. To this end, we say that a (k:m) transient resonance capture (TRC) occurs between two IMFs c1 (t) and c2 (t) with phases ϕ1 (t) and ϕ2 (t), respectively, whenever their instantaneous frequencies satisfy the following approximate relation, kϕ1 (t) − mϕ2 (t) ≈ const ⇒ k ϕ˙ 1 (t) ≈ mϕ˙2 (t),

t ∈ [T1 , T2 ]

(2.83)

The time interval [T1 , T2 ] defines the duration of the TRC between the two IMFs. A more complete picture for the TRC between two IMFs can be gained by constructing appropriate phase plots of the dynamics of the phase difference ϕ12 (t) = ϕ1 (t) − ϕ2 (t). More specifically, a resonance capture is signified by the existence of a loop in the phase plot of ϕ12 (t) when plotted against ϕ˙12 (t), whereas absence of (or escape from) TRC is signified by time-like (that is, monotonically varying) behavior of ϕ12 (t) and ϕ˙ 12(t). In addition, the ratio of instantaneous frequencies of the IMFs, ϕ˙1 (t)/ϕ˙2 (t), provides an estimate of the order of the resonance capture. Ending this brief exposition we mention that the dominant (see discussion below) IMFs of a signal have usually a physical interpretation as far as their characteristic scales are concerned; indeed, certain IMFs may possess instantaneous frequencies that are nearly identical to resonance frequencies of components of the system examined, but this need not always be the case. This implies that certain IMFs may represent artificial (non-physical) oscillating modes of the data. As shown in Kerschen et al. (2006, 2008b), the leading-order (dominant) IMFs coincide with the responses of the slow flow generated by the set of modulation equations of the system; this interesting observation, paves the way for a physics-based interpretation of the IMFs, in terms of the slow flow dynamics (which represent the ‘essential’ dynamics of the system). EMD when combined with the WT enables one to determine the dominant IMFs of a nonlinear time series. This is achieved by superimposing the plots of instantaneous frequencies of the IMFs to the corresponding WT spectra of the time series. The instantaneous frequencies of the dominant IMFs should coincide with the main (dominant) harmonic components of the corresponding WT spectra in the corresponding time windows of the response. It follows that by combining EMD and the WT one is able to determine the main dominant oscillating components in a measured time series and, hence, to perform order reduction and low-order modeling of measured transient signals. In this work EMDs and numerical WTs are imple-

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mented in Matlab . Focusing in the specific applications examined in this work, this integrated approach provides the characteristic time scales of the dominant nonlinear dynamics and the modal interactions occurring between components of a system. Moreover, by adopting this analysis one can identify and analyze the most important nonlinear resonance interactions that are responsible for nonlinear energy exchanges and TET between these components.

2.6 Perspectives on Hardware Development and Experiments We conclude this chapter by discussing certain issues related to the experimental validation of the theoretical results related to TET derived in this work. Experimental studies of TET will be performed by considering SDOF nonlinear oscillators attached to SDOF or MDOF linear systems. As discussed in the theoretical derivations of Chapter 3, important prerequisites for the realization of passive TET in these systems is that the nonlinear attachments possess essential (nonlinearizable) stiffness nonlinearities, and that there exists weak damping dissipation in the integrated linear system – nonlinear attachment configuration. The later is easily implementable, since to a certain extent all practical experimental fixtures possess some degree of damping (inherent damping, or damping added at joints or supports); so the main concern in the experiments is with regard to the accurate measurement and estimation of damping in the exterimental fixtures. The former requirement of essential stiffness nonlinearity, however, is more difficult to implement, so in the experimental work special care was paid towards the design and practical implementation of essentially nonlinear stiffness elements and the accurate measurement of their stiffness characteristics. Passive stiffness nonlinearity in practical settings can be implemented by taking advantage of geometric nonlinearity realized during oscillations of elastic elements. Following this approach, recent works employed different linear spring combinantions to develop geometrically nonlinear stiffness designs. Virgin et al. (2007) considered absorbers with geometrically nonlinear stiffnesses and studied their vibration isolation capacities. Carella et al. (2007a, 2007b) considered vertical linear springs acting in parallel with oblique linear springs, and showed that this configuration could be designed to possess zero dynamic stiffness at their static equilibrium positions. DeSalvo (2007) combined horizontal and vertical linear springs in an arrangement yielding a geometrically nonlinear overall stiffness characteristic, and applied this design to the problem of passive seismic mitigation. Lee et al. (2007) designed spring mechanisms with ‘negative stiffness in the large’ and applied them to vehicle suspension designs; their approach was based on the largeamplitude post-buckling behavior of elastic ‘springing’ (thin shell) elements. In our approach essential (nonlinearizable) stiffness nonlinearity of the third degree was realized experimentally by adopting the simple configuration of Figure 2.25. A thin rod (piano wire) with no pretension was clamped at both ends, and was restricted to perform transverse vibrations at its center. Assuming that the

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Fig. 2.25 Realization of essential stiffness nonlinearity of the third degree.

wire is composed of linearly elastic material, a static force F will cause a transverse displacement x, which from geometry can be expressed as: F = kx[1 − L(L2 + x 2 )−1/2 ]

(2.84)

The stiffness characteristic k = 2EA/L represents the stiffness constant of the wire in axial displacement, E and A are the modulus of elasticity and cross sectional area of the wire, respectively, and L the half-length of the wire. The nonlinear forcedisplacement relationship (2.84) is a consequence of the geometric nonlinearity of this system, eventhough the wire itself is linearly elastic. For small displacements x we Taylor-expand the expression in the bracket of (2.84) about x = 0, yielding (L2 + x 2 )−1/2 =

x2 1 3x 4 − + + O(x 6 ) L 2L3 8L5

(2.85)

so that the force displacement relation (2.84) is approximated as follows: F =

k 3 EA x + O(x 5 ) = 3 x 3 + O(x 5 ) 2L2 L

(2.86)

Hence, the geometric nonlinearity of the system considered produces, to the leading order of approximation, a cubic stiffness nonlinearity with coefficient C = EA/L3 . Moreover, the corrective terms for increasing displacement are of higher order in x, and do not add a linear term in the stiffness characteristic (2.86). If, however, the thin wire is preloaded, a highly undesirable linear term, proportional to the initial

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Fig. 2.26 Experimental realization of Configuration I of nonlinear attachment (grounded attachment with essential cubic stiffness nonlinearity): (a) experimental fixture, (b) schematic describing the various components of the fixture.

preload tension, appears, and the resulting stiffness becomes linearizable. Hence, special care in the experimental setups was given to minimize pretension in the wire; in practical realizations of (2.86) a small linear term (due to unavoidable small pretension) always appears, however, this does not affect the TET results. In the experiments three different configurations of essentially nonlinear attachments were considered. The first configuration (labeled Configuration I) consists of a grounded, essentially nonlinear attachment (termed nonlinear energy sink – NES, see Chapter 3), and its practical implementation is depicted in the experimental fixture of Figure 2.26. The fixture consists of two single-degree-of-freedom oscillators connected by means of a linear coupling stiffness. The left oscillator (the linear system) is grounded by means of a linear spring, whereas the right one (the NES) is grounded by means of a nonlinear spring with essential cubic nonlinearity (the

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Fig. 2.27 Experimental realization of Configuration II of nonlinear attachment (ungrounded attachment with essential cubic stiffness nonlinearity): (a) experimental fixture, (b) schematic describing the various components of the fixture, (c) schematic indicating the NES portioned from the linear oscillator.

clamped wire design presented in Figure 2.25); an additionalviscous damper exists in the NES. The second configuration of essentially nonlinear attachment (NES) (labeled Configuration II) consists of an ungrounded nonlinear attachment, that is coupled to the linear system through an essential stiffness element. In Figure 2.27 we depict this Configuration. The advantage of this design compared to Configuration I is its versatility, since it can be connected to ungrounded structures (such as moving ones); moreover, it will be shown that even lightweight ungrounded NESs can be effective passive absorbers and local energy dissipators, making them primary candidates for realizing TET in practical applications. Experimental results with fixtures implementing Configurations I and II will be reported in Chapters 3 and 8 of this work (for example, an experimental fixture depicting an ungrounded NES

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Fig. 2.28 Experimental realization of a vibro-impact attachment: (a) experimental fixture, (b) detail of VI NES.

configuration attached to a two-DOF linear system of coupled oscillators is depicted in Figure 3.96). A third experimental configuration with a vibro-impact attachment will be considered in our study of passive seismic mitigation by means of TET. The vibro-

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impact configuration is depicted in Figure 2.28. In this design, the essential stiffness nonlinearity of the attachment is realized by vibro-impacts, which, as argued in Chapter 7, can be viewed as a limiting case of a family of ‘smooth’ essentially nonlinear stiffnesses; in that context, the vibro-impact nonlinearity can be regarded as the ‘strongest possible’ stiffness nonlinearity of this family of essentially nonlinear stiffnesses. In the experimental fixture considered in this work, the vibroimpact nonlinearity of the attachment is realized by imposing rigid restrictors to the free motion of the mass of the attachment (see Figure 2.28b). We will demonstrate that, apart from their relative simplicity, properly designed vibro-impact attachments can act as strong passive absorbers and energy dissipators of broadband vibrations from the structures to which they are attached. Vibro-impact TET will concern us in Chapters 7 and 10.

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Chapter 3

Nonlinear Targeted Energy Transfer in Discrete Linear Oscillators with Single-DOF Nonlinear Energy Sinks

In this chapter we initiate our study of passive nonlinear targeted energy transfer – TET (or, so-called nonlinear energy pumping) by considering discrete systems consisting of linear coupled oscillators (refered to from now on as ‘primary systems’) with single-DOF (SDOF) essentially nonlinear attachments. In later chapters we will extend this study to discrete and elastic continuous systems with SDOF or MDOF nonlinear attachments. We aim to show that under certain conditions, the nonlinear attachments are capable of passively absorbing and locally dissipating significant portions of vibration energy of the primary systems to which they are attached. Moreover, this passive targeted energy transfer will be shown to occur over broad frequency ranges, due to the capacity of the nonlinear attachments will be capable to engage in transient resonance (i.e., in transient resonance captures) with linear modes of the primary systems at arbitrary frequency ranges. Then, in essence, these essentially nonlinear attachments will act as nonlinear energy sinks (NESs). By applying analytic methodologies especially developed for studying strongly nonlinear transient regimes (such as the CX-A method introduced in Section 2.4), performing numerical simulations, and post-processing the results by means of the signal analysis techniques discussed in Section 2.5, we will be able to study, model and understand the dynamical mechanisms governing passive nonlinear TET in the systems under consideration. Moreover, we will formulate appropriate measures for assessing the TET efficiency of different configurations of NESs, which, ultimately, will enable us to establish conditions for optimal TET in the systems considered. At the end of this chapter we will extend the study of TET to infinite-DOF chains with SDOF essentially nonlinear attachments and investigate TET generated by impeding elastic waves to boundary NESs.

3.1 Configurations of Single-DOF NESs The realization of nonlinear targeted energy transfer – TET (or nonlinear energy pumping) was first observed by Gendelman (2001) who studied the transient dy-

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namics of a two-DOF system consisting of a damped linear oscillator (LO) (designated as ‘primary system’) that was weakly coupled to an essentially (strongly) nonlinear, damped attachment, i.e., an oscillator with zero linearized stiffness. The need for essentially nonlinearity was emphasized, since linear or near-integrable nonlinear systems have essentially constant modal distributions of energy that preclude the possibility of energy transfers from one mode to another; moreover, such essentially nonlinear oscillators do not have preferential resonant frequencies of oscillation, which enables them to resonantly interact with modes of the primary system at arbitrary frequency ranges. Returning to the work by Gendelman (2001), he showed that, whereas input energy is imparted initially to the LO, a nonlinear normal mode (NNM) localized to the nonlinear attachment can be excited provided that the imparted energy is above a critical threshold. As a result, TET occurs and a significant portion of the imparted energy to the LO gets passively absorbed and locally dissipated by the essentially nonlinear attachment, which acts, in essence, as nonlinear energy sink (NES). This result was extended in other works. A slightly different nonlinear attachment was considered in Gendelman et al. (2001) and Vakakis and Gendelman (2001). In these papers (some results of which are reviewed in Section 2.3), the nonlinear oscillator (the NES) was connected to ground using an essential nonlinearity. This configuration (refered to as ‘Configuration I’ in Section 2.6) is depicted in Figure 3.1. TET was then defined as the one-way (irreversible on the average) channeling of vibrational energy from the directly excited linear primary structure to the attached NES. The underlying dynamical mechanism governing TET was found to be a transient resonance capture (TRC) (Arnold, 1988) of the dynamics of the nonlinear attachment on a 1:1 resonance manifold (see Section 2.3 for related definitions). An interesting feature of the dynamics discussed in these works is that a prerequisite for TET is damping dissipation; indeed, in the absence of damping, typically, the integrated system can only exhibit nonlinear beat phenomena (caused by internal resonances, see Section 2.3), whereby (the conserved) energy gets continuously exchanged between the linear primary system and the nonlinear attachment, but no TET can occur. Nonlinear TET in two-DOF systems was further investigated in several recent studies. In Vakakis (2001), the onset of nonlinear energy pumping was related to the zero crossing of a frequency of envelope modulation, and a criterion (critical threshold) for inducing nonlinear energy pumping was formulated. The degenerate bifurcation structure of the NNMs, which reflects the high degeneracy of the underlying nonlinear Hamiltonian system composed of the undamped LO coupled to an undamped attachment with pure cubic stiffness nonlinearity, was explored by Gendelman et al. (2003). Vakakis and Rand (2004) discussed the resonant dynamics of the same undamped system under condition of 1:1 internal resonance and showed the existence of synchronous (NNMs) and asynchronous (elliptic orbits) periodic motions; the influence of damping on the resonant dynamics and TET phenomena in the damped system was studied in the same work. The structure and bifurcations of NNMs of the mentioned two-DOF system with pure cubic stiffness nonlinearity were analyzed in Mikhlin and Reshetnikova (2005).

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Fig. 3.1 Impulsively loaded primary structure weakly coupled to a grounded NES (referred to as ‘Configuration I’ in Section 2.6).

Kerschen et al. (2005) showed that the superposition of a frequency-energy plot (FEP) depicting the periodic orbits of the underlying Hamiltonian system, to the wavelet transform (WT) spectra of the corresponding weakly damped responses represents a suitable tool for analyzing energy exchanges and transfers taking place in the damped system. Goyal and Whalen (2005) considered a nonlinear energy sink design for mitigating vibrations of an air spring supported slab; the NES used in that work is similar to the grounded version of essentially nonlinear attachment (NES Configuration I) considered in this chapter. A procedure for designing passive nonlinear energy pumping devices was developed in Musienko et al. (2006), and the robustness of energy pumping in the presence of uncertain parameters was assessed in Gourdon and Lamarque (2006). Koz’min et al. (2007) performed studies of optimal transfer of energy from a linear oscillator to a weakly coupled grounded nonlinear attachment, employing global optimization techniques. Additional theoretical, numerical and experimental results on nonlinear TET were reported in recent works by Gourdon and Lamarque (2005) and Gourdon et al. (2007). The first experimental evidence of nonlinear energy pumping was provided by McFarland et al. (2005a). TRCs leading to TET were further analyzed experimentally in Kerschen et al. (2007), whereas application of nonlinear energy pumping to problems in acoustics, was demonstrated experimentally by Cochelin et al. (2006). In most of the above-mentioned studies, grounded and relatively heavy nonlinear attachments (NESs) were considered (i.e., Configuration I NESs – see Section 2.6), which clearly limits their applicability to practical applications. Gendelman et al. (2005) introduced a lightweight and ungrounded NES configuration (refered to as ‘Configuration II’ in Section 2.6) which led to efficient nonlinear energy pumping from the LO to which it was attached. This alternative configuration is depicted in Figure 3.2. Although there is no complete equivalence between the grounded and

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Fig. 3.2 Impulsively loaded primary structure connected to an ungrounded and lightweight NES (referred to as ‘Configuration II’ in Section 2.6).

ungrounded NES configurations depicted in Figures 3.1 and 3.2, it can be shown that, through a suitable change of variables the governing equations (and dynamics) of these two NES configurations may be related (Kerschen et al., 2005). To show this, we consider the simplest possible system with an NES of Configuration II, namely a SDOF LO with a SDOF ungrounded nonlinear attachment, x¨ + x + C(x − ν)3 = 0 εν¨ + C(ν − x)3 = 0

(Config. II NES)

and show that through a series of coordinate transformations it can be cast into a form that nearly resembles a primary system with an attached grounded NES of Configuration I. In the above system the lightweightness of the NES is ensured by requiring that 0 < ε 1; all other variables are treated as O(1) quantities. Through the change of variables, x = ε(z − w),

ν = εz + w

the above system is expressed as ε(1 + ε)¨z + ε(z − w) = 0 (1 + ε)w¨ + ε(w − z) +

C(1 + ε)4 3 w =0 ε

(Config. I NES).

These equations correspond to a linear primary system (composed of a mass with no grounding stiffness) linearly coupled to an NES of Configuration I. Moreover a comparison between these two systems shows that an ungrounded NES (Config. II) with small mass ratio ε with respect to the mass of the primary system and coupled through essential nonlinearity to a LO (the primary system), is equivalent to a grounded NES (Config. I) with large mass ratio (1 + ε)/ε and stiff grounding

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nonlinearity, that is weakly coupled to an ungrounded mass (the primary system) by means of a weak linear coupling stiffness. This result provides an initial hint on the fact that the mass of the NES affects differently the dynamics of the two considered configurations. Indeed, it will be shown in this work that NES Configuration II is most effective for relatively light mass (which makes it an excellent candidate as a practical vibration absorption device), whereas, on the contrary, NES Configuration I is effective for relatively heavier mass. The dynamics of a two-DOF system composed of a linear primary oscillator coupled to an ungrounded and light-weight NES was analyzed in a series of recent papers. Lee et al. (2005) focused on the dynamics of the underlying Hamiltonian system. The different families of periodic orbits of the strongly nonlinear system were depicted in a frequency-energy plot (FEP) (see Section 2.1 for the appropriate definition), which was shown to possess: (i) a backbone curve with periodic orbits satisfying the condition of fundamental (1:1) internal resonance; and (ii) a countable infinity of subharmonic branches, with each branch corresponding to a different realization of an subharmonic resonance between the LO and the NES. In Kerschen et al. (2006a), the energy exchanges in the damped system were interpreted based on the topological structure and bifurcations of the periodic solutions of the underlying undamped system. It was observed that TET can be realized through two distinct mechanisms, namely fundamental and subharmonic TET. It was also noted that a third mechanism, which relies on the excitation of so-called impulsive periodic and quasi-periodic orbits, is necessary to initiate either one of the TET mechanisms through nonlinear beating phenomena. These impulsive orbits were studied using different analytic methods in Kerschen et al. (2008). These theoretical findings were validated experimentally in McFarland et al. (2005b). Gendelman (2004) provided a different perspective of TET dynamics by computing the damped NNMs of a LO coupled to an NES using the invariant manifold approach. He showed that the rate of energy dissipation in this system is closely related to the bifurcations of the NNM invariant manifold. To complement this approach, Panagopoulos et al. (2007) analyzed how initial conditions determine the specific equilibrium point of the slow flow dynamics that is eventually reached by the trajectories of the system. Manevitch et al. (2007a, 2007b), Quinn et al. (2008) and Koz’min et al. (2008) discussed the conditions that should be satisfied by the system and forcing parameters for optimal TET to occur (i.e., so that the maximum portion of the vibration energy of the LO gets passively transferred and locally dissipated by the NES in the least possible time). We conclude this bibliographical review on the dynamics of linear oscillators coupled to NESs by mentioning that alternative designs for SDOF NES have also been proposed. In (Georgiades et al., 2005) and (Karayannis et al., 2007), TET at a fast time-scale was achieved using NESs with non-smooth stiffness characteristics (clearances and impacts); NESs with non-smooth stiffness characteristics will be considered in detail in Chapter 7 of this work. In (Gendelman and Lamarque, 2005) and (Avramov and Mikhlin, 2006) an NES characterized by multiple states of equilibrium positions was considered (this was achieved through a snap-through stiffness element). Moreover, as reported in Laxalde et al. (2007), nonlinear energy

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Fig. 3.3 The two-DOF system with essential stiffness nonlinearity.

pumping can also be realized using an NES with hysteretic nonlinearity. Finally, multi-degree-of-freedom (MDOF) NESs were first introduced in Tsakirtzis et al. (2005) and will be discussed in detail in Chapter 4 of this work.

3.2 Numerical Evidence of TET in a SDOF Linear Oscillator with a SDOF NES In this section we demonstrate TET by considering the simplest possible system of coupled oscillators capable of exhibiting this phenomenon. The system is depicted in Figure 3.3 and consists of a damped SDOF linear oscillator (LO), which acts as the primary system, coupled to an ungrounded attachment (NES Configuration II) through a pure cubic stiffness which lies in parallel to a viscous damper. The seemingly simple configuration of this two-DOF system is quite deceptive, since, as shown below, its dynamics possesses rich and complex structure, including capacity for TET. We mention at this point that the requirement of essential stiffness nonlinearity of the NES plays a key role in the realization of TET, since it precludes the existence of a preferential resonance frequency for the NES. This follows from the fact that an essentially nonlinear NES is not a priori tuned to any specific frequency, unlike the classical tuned mass damper (TMD) (Frahm, 1911; Den Hartog, 1947). Hence, depending on its instantaneous energy the NES is capable of oscillating over a broad frequency range. The equations of motion of the system of Figure 3.3 are given by m1 x¨ + c1 x˙ + c2 (x˙ − ν) ˙ + k1 x + k2 (x − ν)3 = 0 m2 ν¨ + c2 (˙ν − x) ˙ + k2 (ν − x)3 = 0

(3.1)

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where x(t) and ν(t) refer to the displacement of the LO and of the NES, respectively. Equations (3.1) are rescaled according to the mass m1 of the LO and assume the form x¨ + λ1 x˙ + λ2 (x˙ − ν˙ ) + ω02 x + C(x − ν)3 = 0 ˙ + C(ν − x)3 = 0 εν¨ + λ2 (˙ν − x) where ε=

m2 , m1

ω02 =

k1 , m1

C=

k2 , m1

λ1 =

(3.2) c1 , m1

λ2 =

c2 m1

(3.3)

We will be mainly concerned with systems possessing lightweight Configuration II NESs, i.e., systems (3.3) with large mass asymmetry. Hence, we will assume throughout that 0 < ε 1, so that ε can be regarded as the small parameter of the perturbation, asymptotic and averaging analyzes that follow. The assumption of lightweight NESs is of practical significance, as the considered NES designs can be realized with minimal mass modifications of the mechanical or structural systems to which they are attached. The system (3.2) is assumed to be initially at rest, with an impulse of magnitude X applied to the LO; this is equivalent to initiating the system with initial conditions x(0) ˙ = X, x(0) = ν(0) = ν˙ (0) = 0 and no external forcing. Numerical integration of system (3.2) is carried out for varying values of the impulse X and fixed parameters ε = 0.05, ω02 = C = 1, λ1 = λ2 = 0.002. Note that weak damping is chosen in order to better highlight the different dynamical phenomena that occur in this system. A quantitative measure of the capacity of the NES to passively absorb and locally dissipate impulsive energy from the LO can be obtained by computing the following energy dissipation measures (EDMs): t λ2 0 [ν˙ (τ ) − x(τ ˙ )]2 dτ × 100, ENES,t 1 = lim ENES (t) (3.4) ENES (t) = 2 t 1 (X /2) The EDM ENES (t) represents the percentage of impulsive energy that is absorbed and dissipated by the NES up to time instant t, whereas ENES,t 1, the percentage of impulsive energy that is eventually dissipated by the NES up to the end of damped motion (i.e., ENES,t 1 is the asymptotic limit that ENES (t) reaches with increasing time in this passive system). The dependence of ENES,t 1 on the impulse magnitude X is depicted in Figure 3.4. This figure shows that the NES is most effective at intermediate levels of energy, where it dissipates as much as 94% of the input impulsive energy. In addition, there exists a well-defined threshold of input energy below which no significant energy dissipation by the NES can be achieved. This critical energy level represents a lower input energy bound below which TET is insignificant and the NES is ineffective. The existence of this critical energy level becomes apparent when we consider the transient dynamics of the coupled system. In Figure 3.5 we present the transient damped dynamics at a low-energy level; specifically we consider the impulse

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Fig. 3.4 Percentage of impulsive energy eventually dissipated in the NES as a function of the magnitude of the impulse; symbols refer to the simulations of Figures 3.5–3.8.

Fig. 3.5 Transient dynamics of the two-DOF system (low energy level; X = 0.05): (a) LO displacement; (b) NES displacement and (c) percentage of instantaneous total energy in the NES.

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Fig. 3.6 Transient dynamics of the two-DOF system (intermediate energy level; X = 0.12): (a) LO displacement; (b) NES displacement; (c) percentage of instantaneous total energy in the NES and (d) close-up of the NES response.

magnitude X = 0.05 corresponding to point A of Figure 3.4. Clearly, oscillation of the LO possesses much higher amplitude than that of the NES. The NES undergoes small oscillations and most of the impulsive energy remains localized to the LO. This becomes apparent when we compute the percentage of instantaneous total energy stored in the NES, D(t) =

ε ν˙ 2 (t) + (C/2)[x(t) − ν(t)]4 × 100 x˙ 2 (t) + ω02 x 2 (t) + εν˙ 2 (t) + (C/2)[x(t) − ν(t)]4

(3.5)

which is depicted in Figure 3.5c. It follows that no interesting energy transfer occurs from the LO to the NES in this case, and the response remains localized mainly to the LO. This explains why a relatively small portion of the input energy (44%) is dissipated by the NES at this low-energy regime. Moving now to the intermediate energy regime, X = 0.12 (point B in Figure 3.4), a completely different dynamical behavior is realized (see Figure 3.6). The motion is now strongly localized to the NES, as evidenced by its higher amplitude compared to that of the LO. A substantial variation in the NES frequency is also observed,

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which is the indication of the strongly nonlinear nature of its oscillation. Inspite of the fact that initially the energy is entirely stored in the LO, it quickly flows back and forth between the two oscillators. After t = 15 s, 87% of the instantaneous total energy is stored in the NES, but this number drops down to 3% immediately thereafter. Throughout this nonlinear beating phenomenon, a reversible energy transfer occurs, which, however, results in near optimal energy dissipation. At this intermediate energy regime, as much as 94.4% of the total input energy is dissipated by the damper of the NES. Another evidence of the nonlinear beating in this case is that the system performs fast oscillations with frequency close to 1 rad/s, modulated by a slowlyvarying envelope (see Figure 3.6d for a close-up of the NES response); this is due to the fact that a 1:1 transient resonance capture (TRC) between the LO and the NES takes place. It is interesting to note that no a priori tuning of the NES parameters was necessary in order to achieve this result. It is the variation of the frequency of the NES due to damping dissipation that plays the role of ‘tuning’ (but also of ‘detuning’ at later times) for the realization of, and escape from TRC. This is markedly different from ‘classical’ nonlinear beat phenomena caused by internal resonances in Hamiltonian coupled oscillators with linearizable nonlinear stiffnesses, where the ratio of the linearized eigenfrequencies dictates the type of internal resonance between modes that is realized [see, for example, spring-pendulum systems in Nayfeh and Mook (1995)], and no escape from internal resonance is possible once it is initiated. When the magnitude of the applied impulse is further increased (X = 0.2, Figure 3.7 and point C in Figure 3.4), the motion still localizes to the NES, but a different type of energy exchange is encountered. Indeed, during the initial stage of the motion (until approximately t = 15 s), a nonlinear beating phenomenon takes place as in the previous case. However, after this continuous energy exchange between the two oscillators takes place, an irreversible energy flow from the LO to the NES occurs, nonlinear energy pumping is triggered, and TET is realized. Figure 3.7d, which superposes both responses, illustrates that these are completely synchronized during this latter regime. In other words, they vibrate in an in-phase fashion with the same apparent frequency. The underlying dynamical phenomenon causing nonlinear TET can therefore be related to capture in the neighborhood of a 1:1 resonant manifold of the dynamics. The transient nature of the resonance capture is evident in Figure 3.7c, since energy is released back to the LO around t = 300 s. However, when this occurs, the total remaining energy level of the system is small compared to its initial value. Another manifestation of TET is that the envelopes of the displacements decrease monotonically (i.e., no modulation is observed in contrast to the beating regime), with the envelope of the LO decreasing faster than that of the NES. Overall, 87.6% of the total input energy is dissipated by the NES in this case. At a higher-energy regime (X = 0.5, Figure 3.8 and point D in Figure 3.4), no further qualitative change appears in the dynamics. The nonlinear beating phenomenon dominates the early regime of the motion. A weaker but faster energy exchange is observed, since 32% of the total energy is transferred to the NES after t = 4 s. The triggering of TET still occurs, but the irreversible energy transfer from the LO to the NES is slower compared to the previous simulation. This is why energy dis-

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Fig. 3.7 Transient dynamics of the two-DOF system (moderate-energy level; X = 0.2): (a) LO displacement; (b) NES displacement; (c) percentage of instantaneous total energy in the NES and (d) superposition of both displacements during nonlinear TET.

sipation is less efficient than at the moderate-energy regime, and only 50% of the total input energy is dissipated by the NES. Another interesting simulation is shown in Figure 3.9 for parameters ε = 0.05, ω02 = C = 1, λ1 = 0, λ2 = 0.002 and X = 0.1039. As in previous simulations, after an initial nonlinear beating (until approximately t = 150 s), a distinct regime of the transient dynamics is realized. As Figure 3.9d shows, the transient dynamics in the second regime is captured on a 1:3 resonant manifold of the dynamics, with the LO oscillating three times faster than the NES; it is noteworthy that the NES envelope grows during a few cycles, indicating that the NES is extracting energy from the LO. This simulation is further evidence that the NES has no preferential resonant frequency, and it can engage in (fundamental and subharmonic) resonance with the LO at multiple frequency ranges. To highlight the fundamental difference between the SDOF NES and the classical linear TMD we compare their capacities for vibration absorption by performing an additional series of simulations. To this end, the two configurations depicted in Figure 3.10 are used in a parametric study where we vary, (i) the spring constant k1 of the LO (and therefore the natural frequency of the LO, ω0 ), and (ii) the magnitude X of the impulse applied to the LO. The three-dimensional plots in Figures 3.11 and

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Fig. 3.8 Transient dynamics of the two-DOF system (high-energy level; X = 0.5): (a) LO displacement; (b) NES displacement and (c) percentage of instantaneous total energy in the NES.

3.12 display the energy dissipated by the TMD and the SDOF NES, respectively, as functions of parameters ω0 and X. Due to the linear superposition principle, the normalized energy dissipated by the TMD does not depend on the impulse magnitude. There is a specific value of ω0 for which the energy dissipation in the TMD is maximum (95.38% of the total input energy). Any deviation in the frequency content of the LO response from this regime decreases the TMD performance, signifying the well-known result that the TMD is only effective when it is tuned to the natural frequency of the LO. Unlike the TMD, the NES performance depends critically on the impulse magnitude, which is an intrinsic limitation of this type of nonlinear absorbers. This is confirmed in Figure 3.12. This figure indicates that the effectiveness of the NES is not significantly influenced by changes in the natural frequency of the LO. More precisely, for values of ω0 beyond a critical threshold there exists impulse magnitudes for which the NES dissipates a significant amount of the total input energy. Moreover, there are alternative mechanisms by which the NES can induce TET in this system. For example, for (ω0 , X) = (2.3, 0.31) the response (depicted in Figure 3.13) reveals that the LO vibrates three times faster than the NES, i.e., similarly to what was observed in Figure 3.9. Hence, a 1:3 resonance between the LO and the NES seems to be the mechanism responsible for the sudden increase in

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Fig. 3.9 Transient dynamics of the two-DOF system (X = 0.1039, λ1 = 0): (a) LO displacement; (b) NES displacement; (c) percentage of instantaneous total energy in the NES and (d) superposition of both displacements during nonlinear energy pumping.

performance in this region. Such subharmonic TET is realized in the narrow zones of increased energy dissipation of the plot of Figure 3.12, which are quite distinct from the main region of increased energy dissipation where fundamental TET is realized through 1:1 TRCs. Therefore, it appears that the NES can dissipate a substantial amount of the total input energy through fundamental (1:1) as well as subharmonic (m:n) resonances. In conclusion, even though the performance of the TMD is not affected by the level of total energy in the system, it is limited by its inherent sensitivity to uncertainties in the natural frequency of the primary system. In contrast, provided that the energy is above a critical threshold, the SDOF NES is capable of robustly absorbing transient disturbances over a broad range of frequencies. Hence, the NES may be regarded as an efficient passive absorber, possessing adaptivity to the frequency content of the vibrations of the primary system. This is due to its essential stiffness nonlinearity, which precludes the existence of any preferential resonance frequency. It is also shown in this section, that a seemingly simple system comprising of a damped LO and an essentially nonlinear attachment may exhibit complicated dynamics and transitions, including fundamental and subharmonic resonances, nonlinear beating phenomena, multi-frequency responses and strong motion localization to

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Fig. 3.10 Comparison of the linear and nonlinear energy absorbing devices: (a) TMD coupled to a LO; (b) NES coupled to a LO.

either oscillator. The most interesting feature of this system is arguably its capability to realize passive and irreversible energy transfer phenomena from the impulsively loaded LO to the NES, in spite of the relative lightness of the NES compared to the mass of the LO. The complexity of the problem dictates a systematic study of the damped and undamped dynamics of the integrated discrete system composed of the linear oscillator with an essentially nonlinear attachment (the NES). In the following sections of this chapter we will employ a combination of numerical and analytical techniques, including direct numerical simulations; special analytical methodologies capable of modeling both qualitatively and quantitatively transient, strongly nonlinear damped transitions; and advanced signal processing techniques to analyze the resulting nonlinear and non-stationary signals. First, we will consider the system without damping. Even though damping is a prerequisite for TET (as the numerical results discussed in this section indicate), we will show that for sufficiently weak damping the dynamics of TET is governed, in essence, by the underlying Hamiltonian dynamics; the weak damping dissipation then controls the transient damped transitions along branches of NNMs of the Hamiltonian system. After studying the complex dynamics of the Hamiltonian system we will examine damped transient responses,

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Fig. 3.11 TMD performance.

Fig. 3.12 NES performance.

modal energy exchanges and TET between the linear primary system and the attached NES. Experimental verification of the theoretical results will be presented later in this chapter.

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Fig. 3.13 Two-DOF system with LO coupled to a NES (ω0 = 2.3 N/m and X = 0.31 m/s): (a) system response and (b) close-up.

3.3 SDOF Linear Oscillators with SDOF NESs: Dynamics of the Underlying Hamiltonian Systems In this section we consider the undamped version of the system depicted in Figure 3.3, by setting c1 = c2 = 0 and eliminating all external forces. We will perform separate studies of (i) the periodic orbits (NNMs), (ii) the quasi-periodic orbits, and (iii) the so-called impulsive orbits of the Hamiltonian system; in addition, we will provide a geometric interpretation of the Hamiltonian dynamics in terms of slow manifolds of the dynamics that will help us understand the mechanisms generating TET in the damped system.

3.3.1 Numerical Study of Periodic Orbits (NNMs) In the absence of damping and external forcing terms, the equations of motion (3.2) become, x¨ + ω02 x + C(x − ν)3 = 0 ε ν¨ + C(ν − x)3 = 0

(3.6)

where the small parameter is used to denote the lightweightness of the NES; in the following analytical studies ε will be considered as one of a perturbation parameter of the problem. Before proceeding with the study of the periodic orbits, we make a note regarding the degeneracies of system (3.6). For this, we recast the equations of motion in terms of phase variables:

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

⎧ ⎫ ⎡ 0 ⎪ ⎪ ⎪ x˙ ⎪ ⎬ ⎢ ⎨ 2 z˙ −ω 0 =⎢ ⎣ ν˙ ⎪ 0 ⎪ ⎪ ⎭ ⎩ ⎪ w˙ 0

1 0 0 0

0 0 0 0

0 0 1 0

⎧ ⎫ ⎤⎧ ⎫ ⎪ 0 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎨ ⎬ ⎨ ⎥ z −C(x − ν) ⎬ ⎥ + ⎦⎪ν ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ w ⎩ ⎭ 3 (−C/ε)(ν − x)

109

(3.7)

This representation shows that the linear part of system (3.7) possesses two zero eigenvalues and a pair of purely imaginary eigenvalues; in addition, the system possesses cubic stiffness nonlinearities. This indicates that the system may undergo co-dimension three bifurcations for changes of its parameters, and, in fact, its entire four-dimensional phase space coincides with its Center Manifold (Guckenheimer and Holmes, 1983; Wiggins, 1990). This highly degenerate structure is responsible for the complex dynamics of the system (3.2), despite its seemingly simple configuration. We also mention that although the full unfolding of the dynamics and the study of the bifurcation structure of the dynamical system (3.7) is a formidable task (and well beyond the scope of this work), it is still possible to analytically study its dynamics related to the TET phenomena under interest.

3.3.1.1 The Numerical Algorithm Returning to system (3.6), the periodic orbits (or NNMs, see Section 2.1) for a given period T are computed using the method of non-smooth transformations (Pilipchuk, 1985; Pilipchuk et al., 1997). This method formulates the problem of computing the periodic solutions in terms of a nonlinear boundary value problem (NLBVP) over a bounded domain. This NLBVP is solved using a shooting method. Because a nonlinear system is considered, multiple periodic solutions (NNMs) may coexist for a fixed period T . Once a periodic solution is computed for a specific period T , the procedure is restarted for another value of T , say T + T . To this end, different strategies may be used; the sequential continuation method is considered herein. The continuation of the periodic orbits is carried out until the entire frequency range of interest is investigated, and eventually, a branch of periodic solutions is numerically computed. The stability of the computed periodic orbits is determined numerically by application of Floquet theory. Necessary (but not sufficient) conditions for bifurcation and stability-instability exchanges are satisfied when two Floquet multipliers of the corresponding variational problems coincide at +1 or −1. Since the periodic orbits of a two-DOF Hamiltonian system are considered in this study, two Floquet multipliers of the variational problem are always equal to +1, whereas the other two form a reciprocal pair. The stability results are verified using direct numerical simulations of the equations of motion. The complete procedure (i.e., NLBVP formulation, shooting method, sequential continuation and stability analysis) is applied to system (3.6) within the range of energies of interest.

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Fig. 3.14 The non-smooth functions τ (u) and e(u).

The first step of the numerical algorithm for computing the periodic orbits is to formulate an equivalent two-point NLBVP over a finite domain. This is performed using the method of non-smooth transformations first introduced by Pilichuck (1985). This method can be applied to the numerical and analytical study of the periodic orbits (and their bifurcations) of strongly nonlinear dynamical systems. To apply the method we express the sought periodic solutions in terms of two nonsmooth variables, τ (•) and e(•), as follows: ν(t) = e(t/α)y1 (τ (/α)),

x(t) = e(t/α)y2 (τ (t/α))

(3.8)

where α = T /4 represents the (yet unknown) quarter period. The non-smooth functions τ (•) and e(•) are defined according to the expressions τ (u) =

π 2 sin−1 sin u , π 2

e(u) = τ (u)

(3.9)

and replace the independent time variable from the equations of motion (prime denotes differentiation with respect to the argument throughout this derivation, whereas dot is used to denote differentiation with respect to time); their graphic depiction is given in Figure 3.14. Substituting (3.8) into (3.6), we impose ‘smoothening conditions’ (Pilipchuk et al., 1997) to eliminate singular terms from the resulting equations, such as terms proportional to

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e (u) = τ (u) = 2

∞

111

[δ(u + 1 − 4k) − δ(u − 1 − 4k)].

k=−∞

where by δ(•) we denote Dirac’s function. Setting equal to zero the component of the transformed equations that is multiplied by the non-smooth variable e, we formulate the following two-point NLBVP in terms of the non-smooth variable τ in the interval −1 ≤ τ ≤ 1, y1 = y3 y2 = y4 C y3 = − α 2 (y1 − y2 )3 ε y4 = −ω02 α 2 y2 − Cα 2 (y2 − y1 )3 y1 (−1) = y1 (1) = 0 y2 (−1) = y2 (1) = 0

(3.10)

where primes denote differentiation with respect to the non-smooth variable τ , and a state space formulation was utilized. The boundary conditions above result from the afore-mentioned smoothening conditions. The NLBVP (3.10) was solved using a shooting method, by matching at τ = 0 the two solutions shot from the left and right boundary points τ = ±1. The numerical algorithm is similar to the one used in Pilipchuck et al. (1997). Hence, the problem of computing the periodic solutions of the undamped system (3.6) is reduced to solving the NLBVP (3.10) formulated in terms of the bounded independent variable τ ∈ [−1, 1], with the quarter-period α playing the role of the nonlinear eigenvalue. It is noted that the solutions of the NLBVP can be approximated analytically through regular perturbation series (Pilipchuk et al., 1997), however, this will not be attempted herein where only numerical solutions will be considered. We just mention here that (3.10) is amenable to direct analytical study in terms of simple mathematical functions. We note that the NLBVP (3.10) provides the solution only in the normalized halfperiod −1 ≤ t/α ≤ 1 ⇒ −1 ≤ τ ≤ 1. To extend the result over a full normalized period equal to 4 one needs to add the component of the solution in the interval 1 ≤ t/α ≤ 3; to perform this one takes into account the symmetry properties of the nonsmooth variables τ and e by adding the antisymmetric image of the solution about the point (yi , t/α) = (0.1), as shown in Figure 3.15. By construction, it follows that the computed periodic solutions satisfy the initial conditions, x(−α) = ν(−α) = 0 ˙ = y2 (−1)/α. We note at this point that since and ν˙ (−α) = y1 (−1)/α, x(−α) (3.6) is an autonomous dynamical system these initial conditions can be shifted arbitrarily in time; for example, they can be applied to the initial time t = 0 instead of t = −a = −T /4. However, in what follows we will respect the formulation of the NLBVP (3.10), and retain the initial conditions at t = −T /4.

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Fig. 3.15 Construction of the periodic solutions v(t) = e(t/α)y1 (τ (t/α)) and x(t) = e(t/α)y2 (τ (t/α)) over an entire normalized period −1 ≤ t/α ≤ 3 from the solutions yi (τ (t/α)), i = 1, 2 of the NLBVP (3.10), computed over the half normalized period −1 ≤ t/α ≤ 1.

Referring to the general form of the periodic orbit depicted in Figure 3.15, we introduce the following classification: (i)

Symmetric periodic orbits (NNMs) Snm± correspond to orbits that satisfy the conditions, ν˙ (−T /4) = ±˙ν (+T /4) ⇒ y1 (−1) = ±y1 (+1) and x(−T ˙ /4) = ±x(+T ˙ )/4) ⇒ y2 (−1) = ±y2 (+1) with n being the number of half-waves in y1 (and ν), and m the number of half-waves in y2 (and x) in the half-period interval −T /4 ≤ t ≤ +T /4 ⇔ −1 ≤ τ ≤ +1. Hence, the periodic solutions on the branch of NNMs Smn+ (Smn−) pass through the origin of the configuration plane (x, ν) with positive (negative) slope. The ratio (m:n) indicates the order of the internal resonance realized during the given periodic motion. For instance, a 1:1 internal resonance is realized on both branches S11±, which means that both the LO and the NES vibrate with the same dominant frequency. Since the periodic orbits considered are nolinear, they will possess additional harmonics at multiples of this dominant frequency, but the amplitudes of these harmonics are expected to be small. Similarly, on branches S13±, a 1:3 internal resonance is realized between the two oscillators (with the LO oscillating 3 times faster than the NES), and there are two dominant harmonic components in the responses, one around

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1 rad/s and one around 1/3 rad/s; as shown later, the amplitudes of these two harmonic components may vary along the branches S13±. Also, the (+) and (-) signs in the notations of these branches indicate whether the corresponding periodic solutions pass through the origin of the configuration plane with positive or negative slopes, respectively. For instance, an in-phase (out-of-phase) motion of the system is realized on S11+ (S11−). (ii) Unsymmetric periodic orbits (NNMs) Upq are orbits that do not satisfy the conditions of the symmetric orbits. In particular, orbits U (m + 1)m bifurcate from the branch of symmetric NNMs S11− at T /4 ≈ mπ/2, and exist approximately within the intervals mπ/2 < T /4 < (m + 1)π/2, m = 1, 2 . . . . For example on branches U 21±, a 2:1 internal resonance is realized between the two oscillators (with the NES oscillating two times faster than the LO), and there are two dominant harmonic components, one around 2 rad/s and one around 1 rad/s; we note that the magnitudes of these two harmonic components may vary along branches U 21±. As mentioned previously, the numerical solution of the two-point NLBVP (3.10) is constructed utilizing a shooting method, details of which can be found in Lee et al. (2006). In brief terms, the NLBVP is solved as follows. For a given nonlinear eigenvalue a (the quarter period of the NNM) the solutions of the NLBVP are computed at different energy levels; it is expected that at a given energy level there might co-exist multiple nonlinear periodic solutions sharing the same minimal period. Periodic orbits that correspond to synchronous motions of the two oscillators of the system, and pass through the origin of the configuration plane are termed nonlinear normal modes (NNMs) in Vakakis et al. (1996), but a more extended definition of NNMs is adopted in this work (see Section 2.1) to include all periodic motions (and not just synchronous ones). The different families of computed periodic solutions are depicted in three types of plots. In the first two types of plots, we assume zero initial displacements x(−T /4) = ν(−T /4) = 0, and depict the initial velocities ν˙ (−T /4) = y1 (−1)/α and x(−T ˙ /4) = y2 (−1)/α of the periodic orbits as functions of the quarter-period α = T /4 of the (conserved) energy of that orbit: h = (1/2)[ε ν˙ 2 (−T /4) + x˙ 2 (−T /4)] = (1/2α 2 )[εy12(−1) + y22 (−1)]. In the third type of plots, we depict the frequencies of the periodic orbits as functions of their energies h. These plots clarify the bifurcations that connect, generate or eliminate the different branches (families) of periodic solutions (NNMs). As mentioned previously, the stability of the computed periodic orbits is determined by Floquet analysis and by performing direct numerical simulations of the equations of motion (3.6). The numerical results correspond to system (3.6) with parameters and ε = 0.05, ω0 = 1.0 and C = 1.0, in the energy range 0 < h < 1. In Figures 3.16 and 3.17 we depict the bifurcation diagrams of the initial velocities ν˙ (−T /4) and x(−T ˙ /4) of the computed periodic orbits for varying quarter-period α and energy h. Since the dynamical behavior of the system on the various branches of NNMs

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will be discussed in detail in the following sections, we make only some general and preliminary observations at this point. To illustrate the computational results, in Figure 3.18 we present time series of representative periodic motions on branches S11+, S13+ and U 21+, together with the corresponding motions in the configuration plane of the system, (x, ν). Figure 3.19 depicts the Fourier transforms of the time series to illustrate the frequency content of these periodic motions. Considering the bifurcation diagrams of Figures 3.16 and 3.17 we make the following remarks. The NNM branches Snn− exist in the quarter-period intervals 0 < α < nπ/2, and their initial conditions satisfy the following limiting relationships (see Figure 3.16): ˙ =∞ lim {|˙ν (−α)|, |x(−α)|}

α→0

and

lim {|˙ν (−α)|, |x(−α)|} ˙ = 0.

α→nπ/2

In the energy domain, these symmetric branches exist over the entire range 0 < h < 1. We note that branches Snn− are, in essence, identical to the branch S11−, since they are identical to it over the domain of their common minimal period (actually, the branches Snn− are derived by branch S11− by repeating it n times); similar remarks can be made regarding the branches S(kn)km±, k integer, which are identified with Snm±. Considering the neighborhoods of branches S11± and referring to Figure 3.16, the branches S11+ and U 21 bifurcate out at point α = π/2 where S11− disappears (similar behavior is exhibited by the branches S31, S21, . . .). For π/2 ≤ α ≤ π a bifurcation from S11+ to S13+ takes place without change of phase; similar bifurcations take place at higher values of α for branches S15+, S17+, . . . . For α ≈ 3π/2 the branches S13+ and S13− coalesce with branch S11−, with similar coalescences with branch S11− taking place at higher values of α for the pairs of branches S15±, S17±, . . . . The unsymmetric NNM branches U (m + 1)m bifurcate from the symmetric branches S(m + 1)(m + 1) – at quarter-periods α = mπ/2. It turns out that certain periodic orbits on these branches, termed impulsive orbits – IOs, are of particular importance concerning TET in the damped system. The IOs satisfy the additional initial condition y1 (−1) ≡ ν˙ (−α) = 0, and correspond to zero crossings of the branches U (m + 1)m in the bifurcation diagram of Figure 3.17a. Taking into account the formulation of the NLBVP (3.10), it follows that IOs satisfy initial conditions ν(−T /4) = ν˙ (−T /4) = x(−T /4) = 0, and x(−T ˙ /4) = 0, which happen to correspond to the initial state of the undamped system (3.6) (being initially at rest) after application of an impulse of magnitude x(−T ˙ /4) = y2 (−1)/α to the linear oscillator. This implies that if the LO of the system (being initially at rest) is forced impulsively and one of the stable IOs is excited, a portion of the imparted energy is transferred directly to the invariant manifold corresponding to that IO, and, as a result energy is passively transferred from the LO to the NES during the initial cycle of the motion; in subsequent cycles of the response energy gets continuously transferred back and forward between the NES and the LO, and a nonlinear beat phenomenon is formed. We will show that the excitation of IOs provides one of the possible mechanisms for triggering TET in the damped system. A detailed analy-

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Fig. 3.16 Normalized initial velocities of periodic orbits as functions of the quarter-period α; solid (dashed) lines correspond to positive (negative) initial velocities: (a) |y1 (−1)| vs. α, (b) |y2 (−1)| vs. α (S11: ◦, S13: , S15: , S31: , S21: ♦ with in-phase as filled-in, and branches U without symbol).

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Fig. 3.17 Initial velocities of the periodic solutions as functions of energy; solid (dashed) lines correspond to positive (negative) initial velocities; unstable solutions are denoted by crosses: (a) |v(−T ˙ /4)| vs. α, (b) |x(−T ˙ /4)| vs. α.

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Fig. 3.18 Periodic motion on (a) S11+; (b) S13+ and (c) U 21+ (ε = 0.05, C = 1); NES response - - -, LO response —.

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Fig. 3.19 Power spectral density of the periodic motion on (a) S11+; (b) S13+ and (c) U 21+ (ε = 0.05, C = 1); left plots correspond to the LO response, and right plots to the NES response.

sis and discussion of the role of IOs on TET will be carried out in the following sections. Similar classes of IOs can be realized also in a subclass of S- branches. In particular, this type of orbits can be realized on NNM branches S(2k +1)(2p+1)±, k = p, but not on periodic orbits that do not pass through the origin of the configuration plane (such as S21, S12, . . . ). NNM Branch S11− is a particular case, where the

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IO is realized only asymptotically, as the energy tends to zero, and the motion is localized completely in the linear oscillator.

3.3.1.2 Frequency-Energy Plots (FEPs) A more suitable representation of the computed NNMs is to depict their frequency indices (FIs) as functions of their energies h in a frequency-energy plot (FEP). A first introduction of this type of plots was made in Section 2.1, where it was shown that they clearly depict and clarify the bifurcations that generate or eliminate the different branches of periodic solutions (NNMs) of a Hamiltonian system. To construct the FEP of the Hamiltonian system (3.6), the FI of a NNM on branches Snm± and U nm± is defined as the ratio of its two indices multiplied by the driving frequency ωf of the system on the branch, i.e., F I = nωf /m; the driving frequency is the frequency of the harmonic component closest to the natural frequency of the LO, ω0 = 1 rad/s, and slightly varies from one branch to another and even along the same branch. For instance, S21± is characterized by the frequency index F I = (2/1) × 0.97 = 1.94, as is U 21±, and S13± is characterized by F I = (1/3) × 1.05 = 0.35. This rule holds for every branch, the only exception being the two branches S11±, which form the main backbones of the FEP. For these two backbone branches we utilize as FI the common dominant frequency of oscillation of the LO and the NES (as the condition of 1:1 resonance is satisfied pointwise on these branches). In Figure 3.20 we depict the FEP of the Hamiltonian system (3.6) for parameters ε = 0.05, ω02 = C = 1. A periodic orbit (NNM) is represented by a point in the plot. A NNM branch, represented by a solid line, is a collection of periodic orbits possessing the same qualitative features. Bifurcation points are also indicated in that plot, with (+) and (o) used to indicate changes of stability. We note that, if the system was linear (i.e., if the essential cubic nonlinearity was replaced by a linear stiffness), the FEP would merely consist of two horizontal lines appearing passing through two natural frequencies of the corresponding two-DOF system. A consequence of the frequency convention (FI) adopted is that smooth transitions between certain branches translate to ‘jumps’ in the FEP (see for instance the dashed line between S15± and S11− in Figure 3.20). The complexity of the FEP is an indication of the complexity and richness of the nonlinear dynamics of this two-DOF Hamiltonian system. As discussed previously, this is a consequence of the high degeneracy of the dynamical system (3.6). To understand the different types of periodic motions realized in this system, close-ups of several branches are provided in Figure 3.21. The corresponding periodic orbits are represented in the configuration plane (ν, x) of the system. The aspect ratio is set so that increments on the horizontal and vertical axes are equal in size, enabling one to directly deduce whether the motion is localized to the LO (vertical line) or to the NES (horizontal line). Although a systematic analytic study of the various types of periodic solutions of the FEP is postponed until in the next section, some preliminary remarks are due at this point. The backbones of the FEP are formed by NNM branches on which the

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.20 Frequency-energy plot (FEP) depicting the periodic orbits of the Hamiltonian system (3.6); impulsive orbits (IOs) are denoted by bullets (•); bifurcation points are denoted by (+) when four Floquet multipliers are equal to +1, and (◦) when two Floquet multipliers are equal to +1 and two to −1.

system response is nearly monochromatic (see, Figure 3.18a). Specifically, in- and out-of-phase synchronous vibrations of the two particles are realized on S11+ and S11−, respectively. These NNMs are strongly nonlinear analogs (continuations) of the in-phase and out-of-phase linear normal modes of the corresponding two-DOF linear system with all stifnesses being linear. However, unlike the classical linear normal modes, the shapes and frequencies of the NNMs are energy dependent. The natural frequency of the LO (ω0 = 1 rad/s, identified by a frequency index equal to unity) divides naturally the NNMs into higher- and lower-frequency nonlinear modes. Figure 3.21a depicts the NNMs on the higher-frequency out-of-phase branch S11−. Due to their energy dependence, they become localized to the LO or to the NES as ω → 1+ or ω 1, respectively. Two saddle-node bifurcations can also be observed on this branch. In Figure 3.21b, the NNMs on the lower-frequency

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Fig. 3.21 Close-ups of specific branches of the FEP: (a) S11−; (b) S11+; (c) S12±; (d) S13±; (e) S21±; (f) U 21; (g) U 43; (h) U 65; (i) U 12; stability-instability boundaries are represented as in Figure 3.20 and IOs are indicated by triple asterisks; the plots for U 43 and U 65 consist of two nearly spaced branches, but only one of these is presented for clarity; since the motion is nearly identical on the two branches composing S12, S21, U 12 and U 21 (c, e, i, f), only the oscillations on one of the there branches are depicted in the configuration plane.

in-phase branch S11+ are depicted; these motions localize to the nonlinear attachment as the total energy in the system decreases. For further energy decrease, S11+ ceases to exist and is continued by S13±, S15±, etc., as shown in Figure 3.20. There is a sequence of higher- and lower-frequency branches of subharmonic periodic motions Snm± and U nm± with m = n. These NNM branches are termed subharmonic tongues, and they bifurcate out from the backbone branches S11±. Unlike the NNMs on the backbones, the tongues consist of multi-frequency periodic solutions (see, i.e., Figures 3.19b, c). Specifically, each tongue occurs in the neighborhood of an internal resonance between the LO and the NES. Due to the essential nonlinearity of the system (3.6), there exists a countable infinity of tongues Snm± and U nm± in the FEP. This means that the NES is capable of engaging in every possible n:m internal resonance with the LO, with n and m being relative prime integers; clearly, only a subset of these tongues can be represented in Figure 3.20. We mention at this point that the existence of a countable infinity of periodic orbits for

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.21 Continued.

this system can be proved rigorously by applying subharmonic Melnikov analysis (Guckenheimer and Holmes, 1983; Wiggins, 1990). As discussed in Veerman and Holmes (1985, 1986) the generation of infinitely countable subharmonic orbits is related to the non-integrability (Lichtenberg and Lieberman, 1983) of the Hamiltonian system (3.6). Specifically, these countable infinities of subharmonic motions are generated from the breakdown of invariant KAM tori, and they give rise to low-

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scale chaotic layers close to the corresponding resonance bands of these motions. The generation and stability of subharmonic motions in non-integrable Hamiltonian systems can be studied through the use of averaging methodologies (Holmes and Marsden, 1982; Greenspan and Holmes, 1983; Veerman and Holmes, 1986; Wiggins, 1990). A few symmetric tongues are now described in more detail. The periodic motions on these branches are also considered as NNMs according to our extended definition introduced in Section 2.1 (for a study of non-synchronous NNMs in systems with internal resonances in appropriately defined modal spaces, see Vakakis et al., 1996). •

• •

•

The family S1(2k + 1)±, k = 1, 2, etc., exists in neighborhoods of frequency indices F I = 1/(2k + 1). Each family is composed of two in-phase and outof-phase branches. For fixed k each of the two branches S1(2k + 1)± is linked through a smooth transition with its neighboring branches S1(2k − 1)± or S1(2k + 3)±, and exists only for a finite energy interval. The pair S1(2k + 1)± is eliminated through a saddle node bifurcation at a higher energy level, as illustrated in Figure 3.21d for branches S13±. The pairs of branches S1(2k)±, k = 1, 2, etc. bifurcate from the branches S1(2k + 1)±. For instance, the coalescence of the pair of branches S12± with the branch S13+ for decreasing energy is depicted in Figure 3.21c. The families Sn1±, n = 2, 3, etc. appear in neighborhoods of frequency indices F I = n, i.e., at progressively higher frequencies with increasing n. These tongues emanate from S11− and coalesce with S11+ at higher energies. These coalescences seem to occur through jumps represented by dashed lines in the FEP of Figure 3.20, but as explained previously this is an artifact of the frequency convention (frequency indexing) adopted in the FEP. Focusing now on the unsymmetric tongues, the family U (m + 1)m bifurcates from branch S11−. At a higher energy level, the two branches composing the tongues are eliminated through saddle-node bifurcations. An additional family of unsymmetric solutions is U m(m + 1) and in Figure 3.20 this family is depicted only for frequency indices F I < 1. The shapes of these orbits in the configuration plane are similar to those of U (m + 1)m, but rotated by π/2, as illustrated in Figures 3.21f, i. Periodic motions on the unsymmetric tongues are not NNMs because there exist non-trivial phases between the two oscillators, so they correspond to Lissajous curves in the configuration plane.

As mentioned previously, there exist special periodic orbits on the tongues that satisfy the conditions x(−T ˙ /4) = 0 and ν(−T /4) = ν˙ (−T /4) = x(−T /4) = 0. These orbits, termed impulsive orbits (IOs), have important practical significance, since they correspond to impulsive forcing of the LO of system (3.6). These orbits are indicated by bullets in Figure 3.20 and triple asterisks in Figure 3.21. In principle, IOs can be realized on any subharmonic tongue, with the exception of tongues on which the periodic orbits do not pass through the origin of the configuration plane (for example, S12±). As far as the backbone curves are concerned, an IO on the outof-phase branch S11− is realized only asymptotically as the energy tends to zero,

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and the motion is completely localized to the LO; similarly, there is no finite-energy IO on branch S11+.

3.3.2 Analytic Study of Periodic Orbits (NNMs) In an effort to better understand the dynamics and localization phenomena that occur in different frequency/energy ranges of system (3.6), we proceed to the analytical study of the periodic solutions shown in the FEP of Figure 3.20. Representative examples of this analysis will be given for periodic orbits on the backbone branches S11± and on selected subharmonic tongues, namely S13± and U 21±. Without loss of generality we assume that ω02 = 1 and express system (3.6) in the form: x¨ + x + C(x − ν)3 = 0 ε ν¨ + C(ν − x)3 = 0

(3.11)

The analysis will be based on the complexification-averaging method (CX-A) first introduced by Manevitch (1999) and briefly outlined in Section 2.4. This technique will also be applied in later sections to analyze the strongly nonlinear transient dynamics of the damped version of system (3.6).

3.3.2.1 Backbone Branches S11± The backbone branches S11± correspond to motions where the two oscillators of the system possess identical dominant frequency components. The analytical study is performed by applying the CX-A methodology through a slow-fast partition of the dynamics. Following the method, we introduce the new complex variables: ψ1 = x˙ + j ωx

and ψ2 = ν˙ + j ων

(3.12)

where ω is the dominant (fast) frequency of oscillation and j = (−1)1/2. Expressing the displacements and accelerations of the linear and nonlinear oscillators of the system in terms of the new complex variables, we obtain, x=

ψ1 − ψ1∗ , 2j ω

x¨ = ψ˙ 1 −

ν=

ψ2 − ψ2∗ , 2j ω

jω ν¨ = ψ˙ 2 − (ψ2 + ψ2∗ ) 2

jω (ψ1 + ψ1∗ ) 2 (3.13)

where the asterisk denotes complex conjugate. Since nearly monochromatic (at fast frequency ω) periodic solutions of the equations of motion are sought, and since we make the assumption that the two oscil-

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lators vibrate with the same fast frequency, the previous complex variables are approximately expressed in terms of fast oscillations of frequency ω, ej ωt , modulated by slowly varying (complex) amplitudes φi (t), i = 1, 2: ψ1 (t) = φ1 (t)ej ωt

and ψ2 (t) = φ2 (t)ej ωt

(3.14)

This amounts to partitioning the dynamics into slow- and fast-varying components, with the modulations of the (approximately harmonic) fast oscillations ej ωt providing the essential slow flow dynamics of the system. Hence, the study of periodic orbits on NNM branches S11± is reduced to studying the slow flow dynamics. This pattern of reducing the problem to the slow flow dynamics by means of CX-A analysis will be used throughout this work, as a means to separate the essential (slow flow) from the unessential (fast-flow) dynamics of the problem. Note that no a priori restriction was imposed on the frequency ω of the fast oscillation, which allows us to develop asymptotic approximations of the NNM branches over their entire domains of existence (since the fast frequency may vary within a branch of NNMs). At the same time, by the ansatz (3.14) we assume that the periodic motion is dominated by a single fast frequency harmonic (which holds for periodic motions on both backbone braches S11±). The analysis of more complex periodic motions (for example, NNMs on subharmonic tongues) dictates more complicated assumptions than (3.14); examples of such more involved analyses are provided in the following sections. Substituting expressions (3.14) and (3.13) into (3.11) yields the following alternative expressions for the equations of motion, which are exact up to this point: φ1 ej ωt − φ1∗ e−j ωt jω (φ1 ej ωt + φ1∗ e−j ωt ) + 2 2j ω 3 φ1 ej ωt − φ1∗ e−j ωt − φ2 ej ωt + φ2∗ e−j ωt +C =0 2j ω

jω j ωt j ωt j ωt ∗ −j ωt ˙ (φ2 e + φ2 e ε φ2 e + φ2 j ωe − ) 2 3 φ1 ej ωt − φ1∗ e−j ωt − φ2 ej ωt + φ2∗ e−j ωt =0 (3.15) −C 2j ω φ˙ 1 ej ωt + φ1 j ωej ωt −

The basic approximation related to the CX-A technique is that we perform averaging of equations (3.15) with respect to the fast frequency ω, after which only terms containing the fast frequency remain (to a first approximation). This leads to the following set of complex modulation equations, which constitute the approximate slow flow reduction of the dynamics: φ˙1 + (j ω/2)φ1 − (j/2ω)φ1 + (j C/8ω3 ) × (−3|φ1|2 φ1 + 3φ12 φ2∗ − 3φ22 φ1∗ + 3|φ2 |2 φ2 + 6|φ1 |2 φ2 − 6|φ2 |2 φ1 ) = 0

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ε[φ˙ 2 + (j ω/2)φ2 ] − (j C/8ω3 ) × (−3|φ1|2 φ1 + 3φ12 φ2∗ − 3φ22 φ1∗ + 3|φ2 |2 φ2 + 6|φ1 |2 φ2 − 6|φ2 |2 φ1 ) = 0 (3.16) Introducing the polar representations φ1 = Aej α and φ2 = Bejβ in equation (3.16), where A, B are real amplitudes and α, β real phases, and setting separately the real and imaginary parts of the resulting equations equal to zero, the following set of real modulation equations is obtained governing the slow evolution of the real amplitudes and phases of the modulations φi , i = 1, 2: BC A˙ + [(3A2 + 3B 2 ) sin(α − β) + 3AB sin(2β − 2α)] = 0 8ω3 A 3CA3 6AB 2 C ωA − − − 2 2ω 8ω3 8ω3 BC − [(−9A2 − 3B 2 ) cos(α − β) + 3AB cos(2β − 2α)] = 0 8ω3 AC [(3B 2 + 3A2 ) sin(α − β) + 3AB sin(2β − 2α)] = 0 ε B˙ − 8ω3 Aα˙ +

3B 3 C 6A2 BC εωB − − 3 2 8ω 8ω3 AC − [(−9B 2 − 3A2 ) cos(α − β) + 3AB cos(2β − 2α)] = 0 8ω3

εB β˙ +

(3.17)

The first and third of equations (3.17) that describe the evolutions of the two real amplitude modulations, can be combined to yield εB B˙ A˙ + = 0 ⇒ A2 + εB 2 = N 2 A

(3.18)

where N is a constant of integration. Clearly, (3.18) is a conservation-of-energy-like relation for the slow flow, as it is directly linked to conservation of total energy in the undamped system (3.11) during free oscillation. It follows that the modulation equations (3.17) can be reduced by one, with the addition of the algebraic relation (3.18). The periodic solutions on the backbone branches S11± are computed by setting the derivatives with respect to time in (3.17) equal to zero, i.e., by imposing stationarity conditions on the modulation equations. The resulting first and third equations are trivially satisfied if we assume identity of phases, α = β, whereas the second and fourth equations become: ωA A 3CA3 6AB 2 C BC − − − − [−9A2 − 3B 2 + 3AB] 3 3 2 2ω 8ω 8ω 8ω3

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=

127

A 3C ωA − − (A − B)3 = 0 2 2ω 8ω3

3B 3 C 6A2 BC AC εωB − − − [−9B 2 − 3A2 + 3AB] 3 3 2 8ω 8ω 8ω3 3C εωB + (A − B)3 = 0 = 2 8ω3

(3.19)

The amplitudes A and B can be estimated by combining these equations, which leads to the following analytic expressions for the periodic motions (NNMs) on the backbone branches S11±: ψ1 − ψ1∗ = (A/ω) cos ωt 2j ω

1/2 −εω2 4ω2 ε(ω2 − 1)3 = cos ωt ω2 − 1 3C[(1 + ε)ω2 − 1]3

x(t) ≈ X cos ωt =

ψ2 − ψ2∗ = (B/ω) cos ωt 2j ω 1/2 4ω2 ε(ω2 − 1)3 = cos ωt 3C[(1 + ε)ω2 − 1]3

ν(t) ≈ V cos ωt =

(3.20)

Since a single fast frequency was assumed in the slow-fast partitions (3.14), and only terms containing this fast frequency were retained after averaging the complex equations (3.15), the analytical expressions (3.20) are only approximations of the original dynamics of (3.11). It is interesting to note that the ratios of the amplitudes of the linear and nonlinear oscillators on both branches S11± are given approximately by the following simple form: X −εω2 = 2 (3.21) V ω −1 This relation shows that if the mass ε of the NES is small (as assumed in this study), and the frequency is not close to unity, the motion is always localized to the NES. Indeed, as one would expect intuitively, the oscillation localizes to the LO sufficiently close to its resonant frequency ω = ω0 = 1. This result is compatible to the fact that NNM branches of the FEP with large curvatures represent strongly nonlinear oscillations, as they correspond to strong dependence of frequency on energy. √ There is a region in the frequency domain, 1/(1 + ε) < ω < 1, where the coefficients X and V are imaginary, indicating that no in-phase or out-of-phase NNMs can √ occur there. Indeed, the in-phase backbone branch S11+ exists only for ω ≤ 1/(1 + ε), whereas the out-of-phase branch S11− for ω ≥ 1. Moreover, the analytical approximations of branches in the FEP are computed by noting that the conserved energy of the system is given by

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Fig. 3.22 Analytic approximation of the backbone branches S11± in the FEP.

E=

(V − X)4 X2 +C 2 4

(3.22)

Taking into account expressions (3.20), we construct the analytic FEP depicted in Figure 3.22 for parameters ε = 0.05 and C = 1. The analytical approximations are in close agreement with the exact numerical backbone of the FEP depicted in Figure 3.20. However, the simple ansatz (3.14) restricts the validity of the plot to regions where subharmonic tongues are encountered, since in these regions more than one dominant fast harmonic components are present in the response. For example, it is not possible to predict in the plot of Figure 3.22 that S11+ ceases to exist for decreasing energy, where it is continued by S13±, since during that transition two dominant fast frequency components, namely, ω and 3ω are present in the responses. To model this transition, the monochromatic ansatz (3.14) needs to be modified, as performed in the next section.

3.3.2.2 Symmetric Tongues S13± The in-phase and out-of-phase subharmonic periodic motions on branches S13± are analyzed in this section. Along these tongues, the LO vibrates three times faster than the NES, and two fast frequencies, ω and 3ω, are necessary for accurately modeling the periodic orbits. To this end, the CX-A technique is modified by introducing four new complex variables, ψ1 , . . . , ψ4 , defined as follows: ψ1 = x˙1 + j ωx1

and ψ3 = x˙2 + 3j ωx2

ψ2 = ν˙ 1 + j ων1

and ψ4 = ν˙2 + 3j ων2

(3.23)

These lead to the following analytic approximations for the responses of the two oscillators: x(t) ≡ X1 cos ωt + X2 cos 3ωt ≡ x1 (t) + x2 (t)

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

ν(t) ≡ V1 cos ωt + V2 cos 3ωt ≡ ν1 (t) + ν2 (t)

129

(3.24)

Following the procedure outlined in the previous section, the complex variables (3.23) are partitioned in terms of slow and fast components as follows: ψ1,2 (t) = φ1,2 (t)ej ωt

and ψ3,4 (t) = φ3,4 (t)e3j ωt

(3.25)

which when substituted into the equations of motion (3.11) yield the following complex equations of motion: φ˙ 1 ej ωt + φ1 j ωej ωt + φ˙ 3 e3j ωt + φ3 3j ωe3j ωt − (j ω/2)(φ1 ej ωt + φ1∗ e−j ωt ) − (3j ω/2)(φ3e3j ωt + φ3∗ e−3j ωt ) + (2j ω)−1 (φ1 ej ωt − φ1∗ e−j ωt ) + (6j ω)−1 (φ3 e3j ωt − φ3∗ e−3j ωt ) + (C/2j ω)[(φ1 ej ωt − φ1∗ e−j ωt ) − (φ2 ej ωt − φ2∗ e−j ωt ) + 1/3(φ3 e3j ωt − φ3∗ e−3j ωt ) − 1/3(φ4 e3j ωt − φ4∗ e−3j ωt ]3 = 0 ε[φ˙ 2 ej ωt + φ2 j ωej ωt + φ˙4 e3j ωt + φ4 3j ωe3j ωt − (j ω/2)(φ2 ej ωt + φ2∗ e−j ωt ) − (3j ω/2)(φ4e3j ωt + φ4∗ e−3j ωt )] − (C/2j ω)[(φ1 ej ωt − φ1∗ e−j ωt ) − (φ2 ej ωt − φ2∗ e−j ωt ) + (1/3)(φ3 e3j ωt − φ3∗ e−3j ωt ) − (1/3)(φ4 e3j ωt − φ4∗ e−3j ωt )]3 = 0

(3.26)

Averaging independently over each of the two fast frequencies ω and 3ω, we derive a set of four complex differential equations governing the time evolutions of the slow modulations φ1 , . . . , φ4 . Then, introducing the polar transformations φ1 = Aej α , φ2 = Bejβ , φ3 = Dej γ and φ4 = Gej δ and separating the real and imaginary parts, we obtain a set of eight real modulation equations governing the slow evolutions of the amplitudes and phases of the four complex modulations. The next step is to consider identical phase angles, α = β = γ = δ (in the absence of dissipative terms this does not restrict the generality of the analysis), and then set the derivatives of the real amplitudes equal to zero. This leads to the following set of algebraic equations, which compute the amplitudes of the harmonic components of the responses at frequencies ω and 3ω, A=

εω2 B, 1 − ω2

D=

9εω2 G, 1 − 9ω2

3B 2 CGZ2 Z12 + 9CB 3 Z13 + 2CBG2 Z1 Z22 + 12ω4 εB = 0 9CB 3 Z13 + 108CB 2GZ2 Z12 + CG3 Z23 + 18ω4 εG = 0 where

(3.27)

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3 Nonlinear TET in Discrete Linear Oscillators

εω2 9εω2 − 1, Z = −1 (3.28) 2 1 − ω2 1 − 9ω2 These coefficients are related to the amplitudes X1 , V1 , X2 and V2 of approximations (3.24) by Z1 =

X1 = A/ω,

V1 = B/ω,

X2 = D/3ω,

V2 = G/3ω

(3.29)

Figure 3.21 depicts these harmonic components along the NNM branch S13+ for varying frequency ω. Starting with frequency ω ≈ 0.6, the third harmonic components X2 and V2 are small, meaning that the corresponding oscillations are nearly monochromatic, i.e. x(t) ≈ X1 cos ωt, ν(t) ≈ cos ωt. When frequency decreases the amplitudes of the basic harmonic components X1 and V1 also decrease, with X1 decreasing nearly quadratically and V1 approximately linearly. These results indicate that the subharmonic motions on S13+ become increasingly localized to the nonlinear oscillator (the NES) as branch S11+ makes a smooth transition to S13+. At the same time, the LO starts developing the third harmonic component with amplitude X2 and frequency 3ω, which is responsible for the cubic shape of the subharmonic motion in the configuration plane. As the branch S13+ approaches the triple coalescence point G3 , the components X1 and V1 further decrease, whereas X2 and V2 undergo a sudden increase (in absolute value) and the third harmonic components become dominant in the motions of both oscillators. Eventually the responses become again nearly monochromatic (but now at fast frequency 3ω, where ω ≈ 1/3), and the responses are approximated as x(t) ≈ X2 cos 3ωt, ν(t) ≈ V2 cos 3ωt with X2 and V2 having opposite signs (i.e., the linear and nonlinear oscillators are now moving in out-of-phase fashion). A transition to the out-of-phase branch S33− (or equivalently S11−) is therefore realized as we approach the triple coalescence point G3 . The FEP computed by the analytic approximations of the CX-A method is presented in Figure 3.21d for parameters ε = 0.05, C = 1 and frequencies in the neighborhood of ω = 1/3. This plot highlights the triple coalescence of the two branches S13± with branch S11− (not shown in the plot) at point G3 . The evolutions of the NNMs of system (3.11) along the branch S13− present an interesting, though paradoxical feature of the dynamics. Indeed, as point G1 in Figure 3.21d is reached, the depiction of the NNMs in the configuration plane indicates localization to the NES, i.e., that ν(t) x(t). Under this condition, an inspection of the equations of motion (3.11) reveals that the nonlinear attachment vibrates nearly independently, in essence driving the LO. However, as we increase energy and move toward point G2 , it can be shown that the force generated by the linear spring tends to overcome that of the nonlinear spring, which means that the motion of the LO becomes less influenced by the motion of the nonlinear attachment. Once point G2 is reached, both the LO and the nonlinear attachment approximately vibrate as a set of uncoupled linear oscillators with natural frequencies fixed in the ratio 1/3. In other words, in the neighborhood of point G2 the strongly nonlinear dynamical system (3.11) oscillates approximately as the following system of uncoupled linear oscillators: ν¨ + (1/9)ν = 0, x¨ + x = 0 (3.30)

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Fig. 3.23 Frequency dependence of the amplitudes X1 , V1 , X2 , V2 on S13+; G3 is the point of triple coalescence of branches S13± with S11− (see also Figure 3.21d).

When we increase the energy even further and reach the triple coalescence bifurcation point G3 , the force generated by the nonlinear spring is now negligible compared to that generated by the linear spring; then, the LO vibrates nearly independently and drives the nonlinear attachment. This behavior explains why subharmonic tongues S13+ (as well as other U − and S− tongues) appear as nearhorizontal segments in the FEP of Figure 3.20: the reason is that on these tongues the strongly nonlinear system behaves nearly as a system of uncoupled linear oscillators, and the frequency of oscillation becomes nearly independent from energy. The smooth transition from S13− to S15− and the triple coalescence of S15± with S11− follow a process similar to what was just described. This also holds for the other lower-frequency branches S12±, S14±, S16±, S17±, etc. Regarding the higher-frequency branches S21±, S31±, etc., the only difference is that they emanate from S11− and coalesce into S11+.

3.3.2.3 Unsymmetric Tongues U 21± A similar methodology applies when studying U − tongues. As a representative example, the dynamics of system (3.11) on the two branches U 21± is now examined. Periodic oscillations on these branches carry two dominant harmonic components with frequencies ω and 2ω, resulting in a 2:1 internal resonance. Accordingly, the following complex variables:

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3 Nonlinear TET in Discrete Linear Oscillators

ψ1 = x˙1 + j ωx1

and ψ3 = x˙2 + 2j ωx2

ψ2 = ν˙ 1 + j ων1

and ψ4 = ν˙2 + 2j ων2

(3.31)

are introduced, and represented in terms of slow-fast components as follows: ψ1,2 = φ1,2 ej ωt

and ψ3,4 = φ3,4 e2j ωt

(3.32)

The resulting transformed equations are averaged over the two fast frequencies ω and 2ω, and the additional polar transformations φ1 = Aej α , φ2 = Bejβ , φ3 = Dej γ and φ4 = Gej δ are introduced. As shown in Figure 3.21f, periodic motions on U 21± are represented by closed loops (i.e., Lissajous curves) in the configuration plane. The expressions x(t) = (A/ω) sin ωt + (D/2ω) sin 2ωt ≡ x1 (t) + x2 (t), v(t) = (B/ω) sin ωt + (G/2ω) sin 2ωt ≡ v1 (t) + v2 (t)

(3.33)

provide an appropriate ansatz for modeling this type of motions, and the phase angles can be assigned the values α = β = γ = δ = 0 without loss of generality. If expressions (3.33) were defined using cosine functions, one would model open loops in the configuration plane, so, for example, this would apply for analyzing branches on the subharmonic tongues S21±. Imposing stationarity conditions on the equations of the slow flow leads to the determination of the real amplitudes of the harmonic components A=

εω2 B, 1 − ω2

D=

4εω2 G, 1 − 4ω2

6CB 3 Z13 + 3CBG2 Z1 Z22 + 8ω4 εB = 0, 24CB 2 GZ2 Z12 + 3CG3 Z23 + 64ω4 εG = 0

(3.34)

where

εω2 4εω2 − 1, Z = −1 2 1 − ω2 1 − 4ω2 Equations (3.34–3.35) can be solved exactly yielding 4εω4 (Z2 − 8Z1 ) 32εω4 (2Z1 − Z2 ) , G=± B=± 3 9CZ1 Z2 9CZ23 Z1 Z1 =

(3.35)

(3.36)

with the remaining two amplitudes being computed by the first two of equations (3.34). The presence of the ± signs shows that up to four solutions can coexist for a fixed value of the frequency ω. However, only two of these solutions represent distinct periodic motions and generate the two branches U 21±. Figure 3.24 depicts the variation with frequency of the coefficients X1 = A/ω, V1 = B/ω, X2 = D/2ω and V2 = G/2ω, for the two subharmonic tongues (cor-

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Fig. 3.24 Frequency dependence of the amplitudes X1 , V1 , X2 and V2 on U 21+.

responding to parameters ε = 0.05 and C = 1). Starting from lower frequencies, U 21± originates from S11+, since X2 and V2 are nearly equal to zero and X1 and V1 have identical signs. With increasing frequency both oscillators start developing a significant harmonic component with frequency 2ω. Around ω = 0.97, X1 and V1 decrease rapidly, whereas X2 and V2 undergo a sudden increase (in absolute values). Eventually, the branch S11− is reached by both branches U 21± through a triple coalescence point (S22− with U 21±). This is similar to what observed for the subharmonic tongues S13±. The previous analysis can also be used to analytically compute the impulsive orbit (IO) on U 21±. This orbit corresponds to all initial conditions zero except for x(0) ˙ = 0, which yields √ √ 3 3 (3.37) B = −2G ⇒ Z1 = (Z2 /12) 2 + 2 10 − 100 Taking into account expressions (3.35), we derive an analytical estimate for the (U 21) of the IO on the tongues U 21±: frequency ωIO ⎡ (U 21) ωIO

=⎣

(25 + 2−1/3 552/3 − 5101/3 ) + ε(2 − 4101/3 + 2102/3 ) (1 + ε)(40 − 8101/3 + 4102/3 )

(3.38)

⎤1/2 (540 − 270101/3 + 216102/3 ) + ε(216102/3 − 2160) + ε 2 (−624 + 96101/3 + 96102/3 ) ⎦ + 2(1 + ε)(40 − 8101/3 + 4102/3 )

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3 Nonlinear TET in Discrete Linear Oscillators

A better estimate can be obtained if an additional third harmonic component is included in the ansatz (3.31–3.33). From (3.38) we conclude that the frequency of the IO depends essentially on the mass ratio ε; moreover, it can be proven that it does not depend on the coefficient of the nonlinearity of the attachment. As ε → 0 (U 21) → 1, and the frequency of the IO tends to the natural it can be shown that ωIO frequency of the linear oscillator. The degree of localization of the IO can be estimated by considering the ratio V /Y of the maximum amplitudes attained by the nonlinear attachment and the linear oscillator during one period of the motion. It can be shown that this ratio is independent of the coefficient of nonlinearity of the attachment, and the stiffness of the linear oscillator, but depends only on the mass ratio ε. Moreover, stronger localization to the nonlinear attachment occurs for small mass ratios, with V /Y → 1.65 as ε → 0. It is interesting to note that this localization limit appears to be independent of the actual parameters of the system, and depends only on its configuration. These results show that best localization results for the IO are realized for light attachments, and that the degree of localization obtained in the limit of small mass ratios reaches a parameter-independent limit. As mentioned in the previous section, if a stable, localized impulsive orbit is excited by external forcing or by the initial conditions of the system, then during the first cycle of the motion energy is rapidly transferred from the directly excited LO to the nonlinear attachment, and from there on a continuous exchange of energy between the two oscillators occurs in the form of a nonlinear beat phenomenon; as shown in the next section, the excitation of such nonlinear beats provides conditions for the realization of efficient TET in the damped, impulsively forced system. This issue will be studied in detail in the following exposition. Similar analysis can be performed to model the dynamics of nonlinear beat phenomena on the other unsymmetric branches U m(m + 1) and U (m + 1)m. We note that due to the essential nonlinearity of the system considered, the nonlinear beat phenomena on the U -branches do not require any a priori ‘tuning’ of the nonlinear attachment, since at specific frequency-energy ranges the nonlinear attachment passively ‘tunes itself’ in an internal resonance with the linear oscillator. This represents a significant departure from the ‘classical’ nonlinear beat phenomena observed in coupled oscillators with linearizable nonlinear stiffnesses [for example, in a spring-pendulum system (Nayfeh and Mook, 1985)], where the ratio of the linearized natural frequencies of the components dictates the possible types of internal resonances that can be realized. This observation further highlights the enhanced versatility of the NES as vibration absorber due to its essential stiffness nonlinearity. A systematic analytical study of IOs of the Hamiltonian system (3.11) is postponed until Section 3.3.4, whereas a numerical study of these special orbits is performed in the next section.

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Fig. 3.25 Manifold of impulsive orbits (IOs) represented in the FEP; periodic impulsive orbits are denoted by bullets (•).

3.3.3 Numerical Study of Periodic Impulsive Orbits (IOs) In Section 3.3.1.2 we discussed the existence of periodic and quasi-periodic IOs, which correspond to non-zero initial velocity of the LO with all other initial conditions zero. Since these are the exact orbits that are directly excited after the application of an impulsive excitation to the LO, they have an important significance in practical applications of TET. An extensive series of computations of IOs was carried out employing the numerical algorithm described in Section 3.3.1.1 with the additional restriction that v(0) ˙ = 0. The results are presented in Figure 3.25. Because the NES is capable of engaging in a countable infinity of n : m internal resonances with the LO, with n and m being relative prime integers, there exists a countable infinity of periodic IOs, which are aligned along a smooth curve in the FEP. In addition, one can reason-

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.26 Time series of representative periodic IOs for ε = 0.05, C = 1: (a) low-energy IO U 14; (b) moderate-energy IO U 54; (c) high-energy IO S31.

ably assume that there exists an uncountable infinity of quasi-periodic IOs, which correspond to irrational ratios of frequencies of oscillation of the LO and the NES. The periodic and quasi-periodic IOs form a smooth manifold of solutions in the FEP, which is of significant practical importance. This is due to the fact that this manifold provides the impulse magnitude needed to excite an IO for a specified frequency. As shown below a subset of periodic IOs represents stable oscillations of the system that strongly localize to the NES. It follows that if such a stable periodic IO

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137

is excited by an external shock, strong energy transfer from the directly excited LO to the NES takes place over a period of the oscillation. In the Hamiltonian system such an IO is repeated as time progresses, and energy gets continuously exchanged between the LO and the NES; however, in the weakly damped system, an initial excitation of a stable, periodic IO that is strongly localized to the NES leads to strong TET, and, in fact, as shown in the following sections this represents one of three possible mechanisms for generating TET in the damped system. The time series of three representative periodic IOs are depicted in Figure 3.26. Comparing the relative magnitudes attained by the linear and nonlinear oscillators in each of the IOs depicted in that Figure, we note the following. The low-energy periodic IO U 14, which corresponds to a 1:4 internal resonance between the two oscillators, is localized to the LO (see Figure 3.26a). If this orbit is excited by an external shock, a very small fraction of the input energy is transferred to the NES during the nonlinear beating phenomenon. The moderate-energy IO U 54 (see Figure 3.26b) is strongly localized to the NES. The excitation of this orbit channels a major portion of the induced energy from the directly excited LO to the nonlinear attachment during a period of the oscillation. Regarding the high-energy periodic IO S31 (see Figure 3.26c), the NES still undergoes a motion with a larger amplitude than that of the LO, but localization to the NES is less pronounced compared to the IO U 54. In Figure 3.27 representative periodic IOs are depicted in the configuration plane (v, x) of the Hamiltonian system. By construction, these IOs have a common feature: each orbit passes with vertical slope through the origin of the configuration plane. These plots indicate that low-energy periodic IOs with X ≤ 0.078 are localized to the LO (where x(0) ˙ = X is the only non-zero initial condition of the IO). In contrast, moderate-energy periodic IOs in the range X ∈ [0.104, 0.158] are localized to the NES. As far as the high-energy periodic IOs with X ≥ 0.58 are concerned, energy is shared between the two oscillators. Due to the significance of IOs as a basic underlying mechanism for realizing TET in the damped system, in the following section we provide an extensive analytical study of the manifold of IOs in the FEP of system (3.11). Due to the complexity of the problem, it turns out that we need to perform three separate analytical studies of IOs, in the high-, moderate- and low-energy regimes, respectively.

3.3.4 Analytic Study of Periodic and Quasi-Periodic IOs Motivated by the numerical results of the previous section, low-energy (i.e., S1m and U 1m, m > 1), moderate-energy (i.e., U (k + 1)k, k > 1) and high-energy (i.e., Sn1 and U n1, n > 1) impulsive orbits will be analyzed separately. To this end, we reconsider the undamped Hamiltonian system, x¨ + x + C(x − ν)3 = 0

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.27 Representative periodic IOs in the configuration plane (for ε = 0.05, C = 1): (a) low-; (b) moderate- and (c) high-energy orbits; the horizontal and vertical axes represent the NES and LO displacements, respectively, and their aspect ratio is set so that increments on the horizontal and vertical axes are equal in size, enabling one to directly deduce whether the motion is localized to the LO (near vertical) or to the NES (near horizontal).

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

ε ν¨ + C(ν − x)3 = 0

139

(3.39)

where, as previously, we assume that 0 < ε 1, indicating a lightweight nonlinear attachment. We recall that an IO of the dynamical system (3.39) is defined as the orbit corresponding to initial conditions ν(0) = ν(0) ˙ = x(0) = 0 and x(0) ˙ = 0. The singularity in the second of equations (3.39) (since as εto0 the highest derivative is eliminated) can be removed by introducing the following rescalings: x → (8ε/C)1/2 x,

ν → (8ε/C)1/2 ν

(3.40)

so that (3.39) can be transformed into the form x¨ + x + 8ε(x − ν)3 = 0 ν¨ + 8(ν − x)3 = 0

(3.41)

subject to initial conditions ν(0) = ν˙ (0) = x(0) = 0 and x(0) ˙ = X. The additional coordinate transformation, y1 = x + εν,

y2 = x − ν

(3.42)

renders the dynamical system into the following final form, y1 + εy2 =0 1+ε y1 + εy2 + 8(1 + ε)3 y23 = 0 y¨2 + 1+ε

y¨1 +

(3.43)

subject to initial conditions: y˙1 = y˙2 = Y = 0,

y1 = y2 = 0

(3.44)

Note that for notational consistency we have replaced in (3.44) the initial condition X by Y . In physical terms, the new ccordinate y1 denotes the motion of the center of mass of the two oscillators, whereas coordinate y2 their relative response. These new coordinates are natural for describing and studying TET phenomena in the corresponding weakly damped system, since the capacity of the nonlinear attachment to passively absorb and locally dissipate energy from the LO depends on the relative displacement y2 and its derivative, rather on the absolute response ν. The dynamical system (3.43–3.44) is equivalent to systems (3.39) and (3.41), and has the advantage that the small parameter does not multiply any of the time derivatives of the dependent variables. Hence, system (3.43–3.44) is considered in the following analytical study of IOs. Examining (3.44) we deduce that, correct to first order, the center of mass of the system undergoes a linear oscillation of unit frequency, whereas the relative motion between the LO and the nonlinear attachment is governed by a strongly nonlinear ordinary differential equation with cubic nonlinearity. This O(1) partition of the linear and nonlinear dynamics is one addi-

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.28 Graphic computation of periodic IOs (•) from the bifurcation diagram of periodic orbits: (a) k = 4, ε = 0.05; (b) k = 6, ε = 0.05.

tional advantage for considering the transformed dynamical system (3.43–3.44) in the following analysis. Introducing the rescaled time τ = ωt, where ω is a characteristic frequency of the motion, solving the first of equations (3.43) and substituting into the second, the dynamical system is reduced to the following form: y1 (τ ) = (1 + ε)1/2 Y sin kτ + O(ε) y2 (τ ) + 8(1 + ε)2 k 2 y23 (τ ) = −k 2 (1 + ε)1/2 Y sin kτ + O(ε)

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

y2 (0) = 0,

y2 (0) = Y/ω,

k = ω−1 (1 + ε)−1/2

141

(3.45)

where primes denote differentiation with respect to τ . We note that in terms of the normalized time the LO performs approximate harmonic oscillations with normalized frequency k, and the problem of computing the IOs of system (3.39) is reduced to solving the second of equations (3.45). We note at this point that the reduced system approximates well the original system (3.39) only at moderate or large energies of the motion, i.e., in response regimes where the O(1) approximations dominate over the (omitted) O(ε) corrections. At low energies, however, O(ε) terms are expected to play a dominant role in the response, so the reduced system (3.45) may not be used to approximate the response in these regimes. Approximations to the periodic IOs are computed by imposing on (3.45) the periodicity condition y2 (τ ) = y2 (τ + 2π), ∀τ ∈ R + , and the additional initial conditions y2 (0) = 0 and y2 (0) = Y/ω. Note that by imposing the 2π-periodicity condition on y2 (τ ) we impose the additional restriction of integer values for k ∈ N + . In Figure 3.28 we present the graphic computation of periodic IOs. The bifurcation diagrams in these plots depict the 2π-periodic solutions of y2 (τ ) that satisfy only the initial condition y2 (0) = 0, ε = 0.05 and k = 4, 6; the corresponding normalized initial conditions (π/2)y2 (0) are depicted versus the initial condition Y . The periodic IOs are then computed as intersections of the plots in these bifurcation diagrams with the lines (π/2)y2 (0) = Y/ω = k(1 + ε1/2 )Y since at these intersections the second initial condition in (3.45) is satisfied as well. The classification of the impulsive periodic orbits follows the notation introduced in Section 3.3.1.2 for symmetric and unsymmetric periodic orbits (S− or U − orbits, respectively). In Figure 3.29 we depict some representative periodic IOs reconstructed from the approximations y1 (τ ) and y2 (τ ) of the reduced system (3.45), and compare them to the exact IOs computed numerically from the original equations (3.39). It can be observed that the reduced system approximates well the original system at moderate and high energies, but not at small ones. Of particular interest is the impulsive orbit U 54 depicted in Figure 3.29b, which is in the form of a modulated signal or beat (that is, a ‘fast’ oscillation modulated by a ‘slow’ envelope). This orbit occurs at a moderate energy level, and its fast frequency is close to the eigenfrequency of the linear oscillator, so that near 1:1 internal resonance between the linear oscillator and the nonlinear attachment occurs. In the next section it will be shown that such IOs possess two close, rationally related frequency components, which when superimposed produce the observed beating behavior. It follows that for such moderate-energy impulsive orbits one can approximately partition the dynamics into slow and fast components and employ averaging arguments. No such slow-fast partition, however, of the dynamics is possible for the other types of impulsive periodic orbits depicted in Figures 3.29a, c.

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.29 Periodic IOs for ε = 0.05, C = 1: (a) low-energy orbit U 14; (b) moderate-energy orbit U 54; (c) high-energy orbit S31; — exact; - - - reconstruction based on the reduced system (3.45).

3.3.4.1 IOs at Moderate Energy Levels The moderate-energy impulsive orbits U (k + 1)k are first analyzed. An IO representative of this family, U 54, is shown in Figure 3.29b. Motivated by this result, we seek a solution of the system of equations (3.45) (with general k) in the form of a beat, i.e., of a fast oscillation with normalized frequency k (the frequency of the LO response) modulated by a slowly varying envelope. Because such a solution

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may be modeled as a superposition of two harmonics with closely spaced frequencies, a condition of near 1:1 internal resonance is realized for this moderate-energy IO. It follows that in this case the dynamics can be partitioned into slow and fast components, so we introduce the new complex variable ψ(τ ) = w (τ ) + j kw(τ )

(3.46)

which is further expressed as ψ(τ ) =

'

ϕ(τ ) exp(j kτ ) () * ' () *

Slow component

(3.47)

Fast component

Substituting (3.47) and (3.48) into the second of equations (3.45), and averaging out fast components with frequencies at multiples of k, the following modulation (slow flow) equation in complex form is obtained: ϕ (τ ) +

3(1 + ε)2 j jk k 2 Y (1 + ε)1/2 j ϕ(τ ) − |ϕ(τ )|2ϕ(τ ) = 2 k 2

(3.48)

This complex equation governs the slow temporal evolution of the magnitude and phase of the envelope of the response y2 (τ ) on the moderate energy IO. It turns out that the modulation equation (3.48) is exactly integrable, with first integral of motion given by H (ϕ) = j α|ϕ|2 − (jβ/2)|ϕ|4 − jρϕ ∗ − jρϕ = const

(3.49)

where asterisk denotes complex conjugate, and the coefficients in (3.49) are given by α = k/2, β = 3(1 + ε)2 /k and ρ = k 2 Y (1 + ε)1/2 /2. Hence, the solutions of the averaged system (3.48) can be derived in closed form. Introducing the final polar transformation, ϕ(τ ) = N(τ ) exp[j δ(τ )] (3.50) and taking into account the first integral (3.49), the expressions for the amplitude and phase of the envelope of y2 (τ ) are obtained in real form: 2 4 + Y Y Y − β N4 − + 4ρ (4ρN) (3.51a) cos δ = 2α N 2 − ω ω ω dN 2 (τ ) = (3.51b) dτ , 1 ± 16ρ 2 N 2 (τ ) − {2α[N 2 (τ ) − (Y/ω)2 ] − β[N 4 (τ ) − (Y/ω)4 ] + 4ρ(Y/ω)}2 2 These systems are complemented by the initial conditions δ(0) = 0,

N 2 (0) = (Y/ω)2

(3.51c)

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3 Nonlinear TET in Discrete Linear Oscillators

When integrated by quadratures subject to the initial condition (3.51c) equation (3.51b) can be recast into the following form:

N2 2 Y ω

u−

Y 2 1/2 ω

du [I3 + I2 u + I1

u2

− u3 ]1/2

τ

=± 0

βdξ 2

(3.52)

where the coefficients of the denominator of the integrand of the left-hand side are defined as follows: I1 = −(Y/ω)2 + (4α/β),

I2 = −(4α 2 /β 2 ) + (Y/βω)[8ρ + β(Y/ω)3 ]

I3 = [(4ρ/β) − (2αY/βω) + (Y/ω)3 ]2

(3.53)

The definite integral (3.52) can be expressed in terms of elliptic functions (Gradshteyn and Ryzhik, 1980): y du = g cn−1 (cos φ, m) = g F (φ, m) 1/2 1/2 (u − b) [(u − c)(u − c∗ )]1/2 b (a − u) (3.54) with b < y ≤ a,

c = b1 + j a1 ,

g = (AB)−1/2 ,

c ∗ = b1 − j a1

m = [(a − b)2 − (A − B)2 ]/4AB

A2 = (a − b1 )2 + a12 , B 2 = (b − b1 )2 + a12

−1 (a − y)B − (y − b)A φ = cos (a − y)(B + (y − b)A

(3.55)

In the above expressions cn−1 (•,•) is the inverse Jacobi elliptic cosine, F (•,•) the incomplete elliptic function of the second kind, and m the modulus. Expression (3.54) can be applied to solve (3.52) by assigning the parameter value b = (Y/ω)2 , and computing a and (c, c∗ ) as the (single) real and complex pair of roots of the equation I3 + I2 u + I1 u2 − u3 = 0, respectively. As a result, the solution of (3.52) is given by:

(a − N 2 )B − (N 2 − b)A βτ βτ ,m ⇒ = cn ,m cos φ = cn 2g (a − N 2 )B + (N 2 − b)A 2g (aB + bA) − (aB − bA) cn βτ , m 2g ⇒ N 2 (τ ) = (3.56) βτ (B + A) + (A − B)cn 2g , m This expression computes the amplitude squared of the slow modulation, N 2 , as a function of the normalized time τ for the moderate-energy IO. It can be easily verified that the above expression satisfies the initial condition (3.51c), i.e.,

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Fig. 3.30 Slow evolutions of (a) the amplitude squared N 2 (τ ), and (b) phase δ(τ ) of the modulation (envelope) of y2 (τ ) for a moderate-energy IO under conditions of 1:1 internal resonance.

N 2 (0) = b = (Y/ω)2 . Once N 2 (τ ) is approximated through (3.56), the phase δ(τ ) of the modulation is computed through (3.51a). Schematics of the evolutions of N 2 (τ ) and δ(τ ) over one cycle of the IO are depicted in Figure 3.30. The analytic approximation of the response y2 (τ ) is then computed by combining the expressions (3.46), (3.47) and (3.50), y2 (τ ) ≈ −

j N(τ ) j [kτ +δ(τ )] e + cc = −(j/2k)N(τ )ej δ(τ ) ' ' () * 2k Slow component

=

N(τ ) sin[kτ + δ(τ )] k

j kτ e()

* +cc

Fast component

(3.57)

with cc denoting the complex conjugate, and the normalized time defined according ˜ to τ = [k(1 + ε)1/2 ]−1 t. The solution (3.57) has normalized frequency (k) ≈ k + δ (τ ); since δ (τ ) is a slowly varying quantity, the normalized frequency can ˜ be approximated further as (k) ≈ k + δ (τ )τ , where •τ denotes average with respect to normalized time τ .

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3 Nonlinear TET in Discrete Linear Oscillators

The analytic expression (3.57) approximates moderate-energy IOs of system (3.45) under condition of 1:1 internal resonance. It is interesting to note that this expression is valid for periodic as well as quasi-periodic IOs, since no periodicity condition has yet been imposed on the solution (as the initial condition Y is yet undetermined and k is assumed to be real but not necessarily integer). To compute periodic IOs in the region of 1:1 internal resonance, we must impose additional 2π-periodicity conditions on y1 (τ ) and y2 (τ ). Considering the first of expressions (3.45), 2π-periodicity of y1 (τ ) implies that k must be a positive integer; this, however, does not imply necessarily that 2π is the minimal period of y1 (τ ). Considering the approximation (3.57) for y2 (τ ), a minimal 2π-normalized period is imposed on the amplitude N 2 (τ ) and phase δ(τ ); from (3.51a) and (3.56) this implies that cn(βτ/2g, m) must be 2π-periodic. Combining all previous arguments, we conclude that periodic moderate-energy periodic IOs are obtained provided that the following conditions are enforced: k ∈ N+

and 4K(m)

2g = 2π β

(Periodic IOs)

(3.58)

where K(m) is the complete elliptic integral of the first kind (see Figure 3.30b). We note that by the conditions (3.58) y2 (τ ) has a minimal normalized period ˜ equal to 2π/(k) ≈ 2π/(k + δ (τ )τ ) = 2π/(k + 1), and y1 (τ ) a minimal normalized period equal to 2π/k. Hence, a (k + 1) : k internal resonance occurs between the LO and the NES for the computed moderate-energy IO, which, for large values of k, satisfies the initial assumption of near 1:1 internal resonance. It follows that for sufficiently large integers k, the two oscillators of system (3.41) [and of the original dynamical system (3.39) with appropriate rescalings] execute oscillations, x(τ ) = [y1 (τ ) + εy2 (τ )]/(1 + ε) and ν(τ ) = [y1 (τ ) − y2 (τ )]/(1 + ε); these are indeed in the form of beats, since they represent the superposition of two signals with near identical normalized frequencies equal to k and k + 1. Moreover, by the above construction of the IOs, the higher the positive integer k is, the closer the IO satisfies the condition of 1:1 internal resonance, and the more valid the beat assumption (and the slow-fast partition) for the IO becomes. Summarizing, the procedure for computing an analytic approximation for a moderate-energy periodic IO is outlined below: • • • • • • •

Select the order of the internal resonance k Determine the coefficients α, β and ρ in (3.53) Consider a specific initial condition Y Compute the denominator of the integrand (3.52) using expressions (3.53) Compute the roots a, b, c and c∗ of the denominator of the integrand of (3.54) by solving the algebraic equation I3 + I2 u + I1 u2 − u3 = 0 Compute the coefficients g and m, hence compute the coefficient 4K(m)(2g/β) If 4K(m)(2g/β) is equal to 2π, the periodicity condition for y2 (τ ) is satisfied, and the periodic IO U (k + 1)k is realized. If not, modify Y and return to step 4

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Table 3.1 Initial conditions for moderate-energy periodic IOs. Periodic IO U 21 U 32 U 43 U 54 U 65 U 76 U 87 U 98 U 10–9

• •

(k (k (k (k (k (k (k (k (k

= 1) = 2) = 3) = 4) = 5) = 6) = 7) = 8) = 9)

x(0) ˙ (exact)

x(0) ˙ (analytic)

0.5794 0.2398 0.1581 0.1263 0.1115 0.1039 0.1000 0.0977 0.0965

0.2697 0.1675 0.1288 0.1099 0.0999 0.0944 0.0914 0.0898 0.0889

Using (3.51a) and (3.56), compute the amplitude N(τ ) and the phase δ(τ ) of the envelope of the IO From (3.45) and (3.57), compute y1 (τ ) and y2 (τ ), and transform them back to the original variables x(t) and ν(t), taking into account the rescalings (3.40).

In Table 3.1 we present a comparison between the exact and analytically predicted initial conditions for certain moderate-energy unsymmetric periodic IOs, for the system (3.39) with ε = 0.05, C = 1. Apart from U 21, satisfactory agreement between theory and numerics is obtained, which confirms that the previous analysis is valid near the region of 1:1 internal resonance; indeed, as predicted, the accuracy of the analytical predictions for the family of IOs U (k + 1)k is expected to improve with increasing k. In Figure 3.31 we present comparisons between analytical and numerical time series of the responses x(t) and ν(t) for the periodic IOs U 43 and U 65. The analytical approximations were computed based on the previous analysis, whereas the numerical simulations by directly integrating the governing equations (3.39). The analytical periodic IOs can also be represented in the FEP of the Hamiltonian system, when noting that the (conserved) energy of each IO is given by Y 2 /2, and the corresponding frequency index by ω=

˜ 1 1 (k) ≈ + 1/2 1/2 k(1 + ε) (1 + ε) k(1 + ε)1/2

(3.59)

The analytically predicted IOs are shown in Region I of the plot of Figure 3.32a, which when compared to the exact result of Figure 3.32b, validates the previous analytical methodology. We end this section with some remarks. First, the periodicity conditions (3.58) can be generalized by substituting the second of these relations with the more general relation 4K(m)(2g/β)p = 2π, p ∈ N + , which amounts to p waveforms for y2 (τ ) in the normalized interval τ ∈ [0, 2π]; however, in order to ensure that the modulations N 2 (τ ) and δ(τ ) are still slow compared to the fast oscillation exp(j kτ ), we must require that k p. The second remark concerns the fact that IOs not satisfying the periodicity conditions (3.58) are quasi-periodic beats that can still be partitioned in terms of slow-fast components. Indeed, by varying k one obtains a

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.31 Comparisons between analytical approximations (dashed lines) and direct numerical simulations (solid lines) of moderate-energy periodic IOs: (a) U 43; (b) U 65.

one-dimensional manifold possessing an uncountable infinity of quasi-periodic impulsive orbits, and a countable infinity of periodic impulsive orbits imbedded onto it. In this case, the quantity 4K(m)(2g/β) with k non-integer defines the (slow) frequency of the envelope modulation of the quasi-periodic response y2 (τ ), which is a function of the initial condition Y . As a final remark we note that relations (3.56) may be used to estimate the maximum amplitude attained by the slow envelope, Nmax = a 1/2 , where a was defined previously as one of the real roots of the integrand in (3.54). This measure (which is valid for periodic as well as quasi-periodic IOs) is directly related to the energy passively transferred from the LO to the nonlinear attachment during a cycle of the nonlinear beat. Moreover, although during the nonlinear beat (i.e., the moderateenergy IO) energy is continuously exchanged between the LO and the nonlinear attachment, when damping is added to the Hamiltonian system (3.39) this energy exchange is replaced by targeted energy transfer (TET) to the attachment (Kerschen et al., 2005). Hence, the maximum amplitude Nmax of the slow envelope directly affects the effectiveness of TET in the system under consideration. It can be shown that Nmax increases with increasing k, as the 1:1 resonance region is approached

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Fig. 3.32 IOs represented in the FEP of the system: (a) analytic predictions; (b) exact results; regions I, II and III correspond to moderate-, high- and low-energies, respectively.

from higher frequencies, though this increase reaches a definite limit (Lee et al., 2005). The relation between moderate-energy IOs of the Hamiltonian system and TET in the weakly damped one will be discussed in detail in later sections.

3.3.4.2 IOs at High-Energy Levels We now proceed to analyze high-energy IOs of the general form Sn1 and U n1. Judging from the results depicted in Figure 3.26, high-energy IOs have distinctly different waveforms than moderate-energy ones, since they do not appear in the form of beats. Hence, the analytical methodology of the previous section cannot be applied for analyzing this class of IOs, and a separate analysis must be developed.

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.33 Response y2 (t) for the high-energy IOs, (a) S71; (b) S91.

To this end, the approximate dynamical system (3.45) is expressed in terms of the original time variable t, yielding: y1 (t) = (1 + ε)1/2 Y sin[(1 + ε)−1/2 t] + O(ε) y¨2 (t) + 8(1 + ε)y23 (t) + (1 + ε)−1 y1 (t) = 0 + O(ε), y2 (0) = 0,

y˙2 (0) = Y

(3.60)

At sufficiently high energy levels, the essentially nonlinear coupling stiffness behaves almost as a rigid connection. It is therefore reasonable to assume that x(t) ≈ ν(t) ⇒ |y1 (t)| |y2 (t)| in this regime. Then, the relative displacement is expressed as a superposition of slow and fast components y2 (t) ≈

'

s(t) ()

*

Slow component

+

'

f (t) ()

*

(3.61)

Fast component

where for high-energy IOs it is natural to assume that |f (t)| |s(t)|. This is illustrated in Figure 3.33 for the high-energy IOs S71 and S91. Substituting (3.61) into the second of equations (3.60) and the accompanying initial conditions, yields the following differential equation possessing slow and fast varying parts: f¨ + s¨ + 8(1 + ε)(f 3 + 3f 2 s + 3f s 2 + s 3 )

Y t =− sin (1 + ε)1/2 (1 + ε)1/2 s(0) + f (0) = 0 ⇒ s(0) = f (0) = 0, s˙ (0) + f˙(0) = Y ⇒ s˙ (0) = 0,

f˙(0) = Y

(3.62)

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Setting separately equal to zero the fast and slow components of (3.62), and taking into account that , |f (t)| |s(t)| we find that the fast dynamics is governed by an unforced oscillator of Duffing-type, f¨ + 8(1 + ε)f 3 = 0,

f (0) = 0,

f˙(0) = Y

the solution of which is readily obtained in closed form,

K(1/2) 1 f (t) = −A cn η t + , η 2

(3.63)

(3.64)

where A = 2−1 (1 + ε)1/2 Y and η = A[8(1 + ε)]1/2 . The expressions cn(•,•) and K(1/2) in (3.64) denote the Jacobi elliptic cosine function and the complete elliptic integral of the first kind, respectively. Substituting (3.64) into (3.62) and averaging out the fast dynamics we obtain the following approximate dynamical system governing the slow dynamics s¨ + 8(1 + ε)[3f 2 T s + s 3 ] = −(1 + ε)1/2 Y sin[(1 + ε)−1/2 t],

s(0) = s˙ (0) = 0 (3.65) where the average of the fast oscillation f 2 T can be explicitly computed according to (Gradshteyn and Ryzhik, 1980): f 2 T =

1 T

T

A2 cn2 (ηt, 1/2)dt =

A2 [E(π, 1/2) − 2K(1/2)] K(1/2)

(3.66)

where T = 4K(1/2)/2η, and E(•,•) is the incomplete elliptic function of the second kind. Because |s(t)| 1, to a first approximation the cubic term can be neglected in (3.65), so the slow flow dynamical system can be reduced approximately to the following linear system that can be solved explicitly: s(t) ≈

−Y (1 + ε)1/2 sin[(1 + ε)−1/2 t] 24(1 + ε)2 f 2 T − 1 +

Y sin[24(1 + ε)f 2 T t] [24(1 + ε)2 f 2 T − 1][24(1 + ε)f 2 T ]

(3.67)

Combining the solutions (3.64) and (3.67), the relative displacement y2 (t) for the high-energy IO can be approximated by the analytical expression:

1 K(1/2) , (3.68) y2 (t) ≈ −A cn η t + η 2 ' () * Fast component

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3 Nonlinear TET in Discrete Linear Oscillators Table 3.2 Initial conditions for high-energy periodic IOs.

+

Periodic IO

x(0) ˙ (exact)

x(0) ˙ (analytic)

U 21 (n = 2) S31 (n = 3) S51 (n = 3) S71 (n = 7) U 81 (n = 8) S91 (n = 9)

0.58 1.59 4.85 9.75 12.80 16.30

0.82 1.84 5.12 10.03 13.10 16.58

−Y (1 + ε)1/2 Y sin[24(1 + ε)f 2 T t] sin[(1 + ε)−1/2 t] + 2 2 24(1 + ε) f T − 1 [24(1 + ε)2 f 2 T − 1][24(1 + ε)f 2 T ] ' () * Slow component

Then, the IO in terms of the original variables can be evaluated by combining the first of expressions (3.60) and (3.68), and inversing the coordinate transformations y1 = x + εν, y2 = x − ν. To compute the initial condition Y corresponding to a specific high-energy periodic IO, a periodicity condition similar to that for the moderate-energy case should be imposed. This periodicity condition is formulated as follows: n

4K(1/2) = 2π(1 + ε)1/2 , η

n ∈ N+

(Periodic IOs)

(3.69)

and amounts to a n : 1 internal resonance between the LO and the NES. This condition requires that the period of the slow component s(t) is n times the period of the fast component f (t), with the overall (not necessarily) minimal period of y2 (t) being equal to 2π(1 + ε)1/2 [i.e., equal to the period of y1 (t)]. From (3.69) the corresponding initial condition for Y is computed: Y (n) =

K 2 (1/2)n2 π 2 (1 + ε)3/2

8ε C

1/2 (3.70)

where the rescaling (3.40) is taken into account. Table 3.2 presents the comparison between the predicted and exact initial conditions for a few symmetric and unsymmetric high-energy IOs for a system with ε = 0.05 and C = 1. Good agreement between theory and numerics is noted. In Figure 3.34 we depict the analytical time series for the IOs S71 and S91 and compare them to the corresponding exact solutions derived by direct integrations of the equations of motion (3.39). Overall, satisfactory agreement is obtained, particularly when the order n of the internal resonance is increased. The total energy of the IO is computed as E = Y 2 /2, whereas the frequency index of an orbit is given by ω ≈ n. Employing (3.70), an analytic expression for the locus of high-energy IOs in the FEP can be derived as 4εK 4 (1/2)ω4 E= (3.71) Cπ 4 (1 + ε)3

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Fig. 3.34 Comparisons between analytical approximations (dashed lines) and direct numerical simulations (solid lines) of high-energy periodic IOs: (a)S71; (b) S91.

This approximation is presented in Region II of Figure 3.32a and compares well with the exact high-energy IO manifold of Figure 3.32b.

3.3.4.3 IOs at Low-Energy Levels The low-energy periodic IOs S1m and U 1m are finally analyzed. As mentioned previously, at low energies, O(ε) terms are expected to play a dominant role in the response, so the reduced system (3.45) may not be used to approximate the IOs in this case. Instead the rescaled dynamical system (3.41) is reconsidered, x¨ + x + 8ε(x − ν)3 = 0 ν¨ + 8(ν − x)3 = 0 x(0) ˙ = Y,

x(0) = ν(0) = ν˙ (0) = 0,

0<ε 1

(3.72)

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3 Nonlinear TET in Discrete Linear Oscillators

where for coherence with the previous two sections, the initial condition is denoted by Y . Figure 3.29a illustrates that low-energy IOs are characterized by (i) motions of the two oscillators with very small amplitudes, and (ii) a much larger amplitude of oscillation of the LO; motivated by these numerical results we assume that in low-energy IOs it holds that |ν(t)| |x(t)| 1. Taking into account this assumption it appears that an appropriate ansatz for the low-energy IOs is x(t) = Y sin t + · · · ,

ν(t) = ' B () sin t * + ' Fast component

s(t) ()

*

+···

(3.73)

Slow component

with |B| |Y | 1 and |s(t)| |Y | 1. In contrast to the analysis of the previous section, the component of the NES response with frequency close to unity is regarded as the fast component, whereas the second component s(t) is regarded as the slow component of the solution. Substituting (3.73) into the second of equations (3.72) yields the following differential equation: −B sin t + s¨ (t) + 8[(B − Y )3 sin3 t + 3(B − Y )2 s(t) sin2 t + 3(B − Y )s 2 (t) sin t + s 3 (t)] = 0

(3.74)

Setting separately equal to zero the slow and fast components of (3.74), we partition the dynamics into the following slow and fast components: −B sin t + 8(B − Y )3 sin3 t + 24(B − Y )s 2 (t) sin t = 0 s¨(t) + 24(B − Y )2 s(t) sin2 t + 8s 3 (t) = 0

(3.75)

The method of harmonic balance is applied to the first of equations (3.75), i.e., to the fast component of the dynamics, leading to the relation: −B + 6(B − Y )3 + 24(B − Y )f 2 (t) = 0 ⇒ −B + 6(−Y )3 ≈ 0 ⇒ B ≈ −6Y 3

(3.76)

Focusing now in the slow component of the dynamics [the second of equations (3.75)], the fast term sin2 t is averaged out to yield the following averaged slow flow dynamical system: (3.77) s¨(t) + 12(B − Y )2 s(t) + 8s 3 (t) ≈ 0 π since sin2 tT = 1/π 0 sin2 tdt = 1/2. In view of the fact that |B| |Y | and |s(t)| |Y | 1, expression (3.77) may be approximated by the simplified linear equation (3.78) s¨(t) + 12Y 2 s(t) ≈ 0 which is readily solved, by imposing the initial conditions for the impulsive orbit:

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Table 3.3 Initial conditions for low-energy periodic IOs. Periodic IO U 1–22 S19 U 2–15 U 16 S13 U 12 U 34

(m = 22) (m = 9) (m = 15/2) (m = 6) (m = 3) (m = 2) (m = 4/3)

x(0) ˙ (exact)

x(0) ˙ (analytic)

0.0083 0.0201 0.0241 0.0299 0.0555 0.0781 0.0942

0.0083 0.0203 0.0243 0.0304 0.0609 0.0913 0.1369

√ 6Y 2 s(t) ≈ √ sin 12 Y t 12

(3.79)

Combining the previous results, the low-energy IOs of system (3.72) are analytically approximated as follows: x(t) ≈ Y sin t,

√ 6Y 2 ν(t) ≈ −6Y 3 sin t + √ sin 12 Y t 12

(3.80)

Depending on the non-zero initial condition Y , relations (3.80) describe either periodic or quasi-periodic low-energy IOs. As in the analytical derivations of the previous two sections, the periodicity of the solution (3.80) is ensured by applying a periodicity condition, i.e., by imposing a 1 : m internal resonance between the LO and the nonlinear attachment: √ 1 12 Y = , m

m ∈ N+

(Periodic IO)

(3.81)

Because of the slow-fast partition in the ansatz (3.73), the analytic approximation (3.80) is expected to be in better agreement with the exact solution for large integers m, that is, for sufficiently small energies. Taking the rescaling (3.40) into account an approximation of the low-energy periodic IO of the original dynamical system (3.39) is obtained in the following form: √ √ 8ε 8ε x(t) ≈ √ [m sin(t/m) − sin t] (3.82) sin t, ν(t) ≈ √ 2 3C m 4 3Cm3 Table 3.3 presents a comparison between predicted and exact low-energy periodic IOs for the system with ε = 0.05 and C = 1. Again, good agreement between the analytical and exact values is observed. Figure 3.32 depicts the analytical and exact time series for the IOs U 1–22 and S13, from which good agreement is noted. The total energy of a low-energy IO is equal to E = Y 2 /2, whereas its frequency index is ω ≈ 1/m. Employing the resonance condition (3.81), a surprisingly simple but accurate analytic approximation of the locus of low-energy IOs in the FEP is obtained:

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Fig. 3.35 Comparisons between analytical approximations (dashed lines) and direct numerical simulations (solid lines) of low-energy periodic IOs: (a) U 1–22; (b) S13.

εω2 (3.83) 3C The locus of IOs is depicted in Region III of Figure 3.32a. Overall, good agreement is obtained between the predictions and the exact results, which demonstrates the accuracy of the analysis. In summary, we studied the periodic and quasi-periodic IOs of the strongly nonlinear Hamiltonian system (3.39). These are responses of the system initially at rest and excited by an impulsive force applied to the linear oscillator. As shown in later sections IOs directly affect the TET capacity of the damped system, i.e., the capacity of the nonlinear attachment to passively absorb broadband energy from the linear oscillator in a one-way, irreversible fashion. The manifold of quasi-periodic and periodic IOs in the FEP was analytically studied by considering separately the high-, moderate- and low-energy regimes. Different analytical methods were applied to analyze the IOs in these regimes. Of particular interest are moderate-energy IOs in the neighborhood of 1:1 internal resonance of the system which are in the forms of nonlinear beats, with the motion localized mainly to the nonlinear oscillator. As shown in a later section the excitation of an IO in the 1:1 internal resonance regime E=

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represents a very effective dynamical mechanism for strong passive TET from the linear oscillator to the nonlinear attachment.

3.3.5 Topological Features of the Hamiltonian Dynamics In this section we focus in the intermediate-energy region, and provide some remarks on the topological features of the dynamics in phase space under conditions of 1:1 internal resonance. Our aim is to relate solutions, such as NNMs on branches S11± and IOs, to certain global topological features of the Hamiltonian dynamics of system (3.6). Through a suitable change of variables we will reduce the isoenergetic dynamics to a three-dimensional sphere, and discuss how the critical energy threshold required for TET in the damped system (discussed in Section 3.2) can be directly related to a similar critical energy threshold in the Hamiltonian system, above which the IOs are in the form of nonlinear beats with strong energy exchanges between the LO and the nonlinear attachment. Finally, we will discuss how the topology of the phase space close or away from a hom*oclinic connection of the slow flow dynamics affects the qualitative features of the IOs discussed in Sections 3.3.3 and 3.3.4. The following exposition follows closely (Quinn et al., 2008). Considering again the two-DOF Hamiltonian system (3.6) and setting (without loss of generality) ω0 = 1, x¨ + x + C(x − ν)3 = 0 ε ν¨ + C(ν − x)3 = 0

(3.84)

we recall from Section 3.3.2.1, that solutions in the neighborhoods of the two backbone branches S11± of the FEP can be analytically modeled using the CX-A technique. Indeed, assuming the following ansatz for these solutions: x(t) ≈

A(t) cos[ωt + α(t)], ω

ν(t) ≈

B(t) cos[ωt + β(t)] ω

(3.85)

we obtain the set of four modulation equations (3.17) that govern the slow evolution of the amplitudes A(t), B(t) and phases α(t), β(t) of the two oscillators. Note that the ansatz (3.85) indicates that conditions of 1:1 internal resonance are realized in the dynamics, so that the harmonic components of frequency ω in the response of the two oscillators. dominate over all other higher harmonics (this would not occur, for example, in neighborhoods of, or on subharmonic and superarmonic tongues, see Sections 3.3.1 and 3.3.2). Introducing the phase difference φ = α − β, the slow flow equations (3.17) can be reduced to the following three-dimensional autonomous dynamical system on the cylinder (R + × R + × S 1 ), a˙ 1 =

−3a2C sin φ[(a12 + a22) − 2a1 a2 cos φ] 8

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3 Nonlinear TET in Discrete Linear Oscillators

3a1 C sin φ[(a12 + a22 ) − 2a1 a2 cos φ] 8ε 1 3C 2 [(a1 + a22 ) − 2a1a2 cos φ] φ˙ = − 2 8

a1 a2 1 1− × cos φ − 1 − cos φ ε a2 a1

a˙ 2 =

(3.86)

where the notation a1 = A, a2 = B was utilized. In Section 3.3.2.1, the analytic modeling of periodic orbits that satisfy the exact 1:1 internal resonance condition was considered; moreover, since we were interested on steady state solutions, we imposed stationarity conditions to the derived modulation equations (i.e., the terms containing derivatives with respect to time were set equal to zero). In this section, a more general analysis is carried out in the sense that fast oscillations with frequencies ω ≈ 1 and modulated by slowly-varying envelopes are sought. In other words, we are primarily interested in the dynamics near the region of 1:1 internal resonance, which corresponds to the intermediate-energy regime of the FEP in the notation of the previous sections. It turns out that the autonomous dynamical system (3.86) is fully integrable, as it possesses the following two independent first integrals of motion: √ a12 + ( εa2 )2 ≡ r 2 a 2 3C 2 a12 +ε 2 + (a + a22 − 2a1 a2 cos φ)2 ≡ h 2 4 32 1

(3.87)

The first equation is a consequence of energy conservation in (3.84), and enables us to introduce a second angle ψ into the problem, defined by

π π π a1 ψ tan + =√ (3.88) , ψ∈ − , 2 4 2 2 εa2 Taking into account the first integrals of (3.87) and introducing the new angle into the problem, the slow flow dynamical system (3.86) can be further reduced to system on a three-dimensional sphere, r˙ = 0 ψ˙ =

√ −3Cr 2 [(1 + ε) − (1 − ε) sin ψ − 2 ε cos ψ cos φ] sin φ 3/2 8ε

√ 1 3Cr 2 − [(1 + ε) − (1 − ε) sin ψ − 2 ε cos ψ cos φ] 2 2 16ε

sin ψ cos φ × (1 − ε) − 2ε1/2 cos ψ

φ˙ =

(3.89)

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Fig. 3.36 Topology of the reduced phase space: (a) three-dimensional sphere (r, φ, ψ) ∈ (R + × S 1 × S 1 ), (b) projection of the reduced dynamics onto the unit disk.

where (r, φ, ψ) ∈ (R + × S 1 × S 1 ) (see Figure 3.36). Then, the second of the first integrals of motion (3.87) can be expressed in the form 3Cr 2 r2 1/2 2 3 + sin ψ + =h [(1 + ε) − (1 − ε) sin ψ − 2ε cos ψ cos φ] 8 16ε2 (3.90) Considering the isoenergetic dynamical flow corresponding to r = const, the orbits of the system lie an a topological two-sphere, and follow the level sets of the first integral of motion (3.90). Projections of the isoenergetic reduced dynamics onto the unit disk at different energy levels are depicted in Figure 3.37. The north pole (NP) at ψ = π/2 lies at the center of the disk, while the south pole (SP) ψ = −π/2 is mapped onto the entire unit circle. In this projection, trajectories that pass through the SP approach the unit circle at φ = π/2 and are continued at φ = −π/2. If the response is localized to the LO, so that a2 a1 , the phase variable ψ lies close to +π/2. In contrast, a localized response in the nonlinear attachment (i.e., a1 a2 ) implies that ψ ≈ −π/2. Before we examine the dynamics near the region of 1:1 internal resonance, we reconsider the periodic motions on branches S11±, corresponding to the equilibrium points of the slow flow (3.89). These equilibrium points are explicitly evaluated by the following expressions: ψ˙ = 0 ⇒ sin φeq = 0 ⇒ φeq = 0, π φ˙ = 0 ⇒ cos ψeq −

with

3Cr 2 (1 + ε)2 [1 − sin(ψeq + γeq )] cos(ψeq + γeq ) = 0 8ε (3.91)

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Fig. 3.37 Projection of the dynamics of the isoenergetic manifold onto the unit disk at different energy levels (ε = 0.1, C = 2/15); (a) r = 1.00, (b) r = 0.375, (c) r = 0.25.

√ 2 ε cos φeq (3.92) tan γeq = 1−ε Equilibrium points satisfying φeq = 0 correspond to in-phase periodic motions and generate the backbone branch S11+ for varying frequency and energy; those corresponding to φeq = π, represent out-of-phase periodic motions and generate the other backbone S11−. In the projections of the phase space shown in Figure 3.37, periodic motions (NNMs) on S11+ appear as equilibrium points that lie on the horizontal axis to the right of the origin, whereas periodic motions on S11− as equilibrium points that lie on the horizontal axis to the left of the origin. With increasing energy, i.e., as r → ∞, both equilibrium points approach the value

1−ε lim ψeq = arctan √ (3.93) r→∞ 2 ε cos φeq so that, for 0 < ε 1 and in the limit of high energies we have that ψeq,S11+ > 0 and ψeq,S11− < 0. With increasing energy the in-phase NNMs on S11+ localize to the LO, while the out-of-phase NNMs on S11− localize to the nonlinear attachment (the NES). The degree of localization is controlled only by the mass ratio ε, and for small but finite values of this ratio the high-energy localization is incomplete, as the limiting values of ψeq,S11+ and ψeq,S11− do not attain π/2 in magnitude. Considering now the low-energy limit, it is easily shown that for sufficiently small values of r the equilibrium equation for ψeq degenerates to the simple limiting relation cos ψeq → 0. Therefore, we conclude that as r → 0+, the following values are attained by the equilibrium value for ψ: lim ψeq,S11+ = −π/2 and

r→0+

lim ψeq,S11− = +π/2

r→0+

(3.94)

It follows that in the limit of small energies the in-phase NNM on S11+ localizes to the nonlinear oscillator, while the out-of-phase NNM on S11− to the LO. However,

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Fig. 3.38 Topology of the branch S11− for varying ε and C = 2/15.

unlike the high-energy limits (3.93), as r → 0 localization is complete to either the LO or the nonlinear attachment. In the transition from high to low energies, the branch of out-of-phase NNMs S11− undergoes two saddle-node bifurcations. In the first bifurcation, a new pair of stable-unstable equilibrium points is generated near ψ = +π/2. As energy decreases a second (inverse) saddle-node bifurcation occurs that anhiliates the unstable equilibrium generated by the first bifurcation, together with the stable branch of S11− that existed for higher energy values. It should be noted, however, that these bifurcations occur only below a certain critical mass ratio ε, i.e., only for sufficiently light attachments. This is demonstrated in Figure 3.38, which depicts the variation of the out-of-phase branch S11− in the (ψeq , r) plane for three values of the mass ratio ε; note that no bifurcations occur for the higher value of for ε. Figures 3.37a, b, c depict the above-mentioned bifurcations in projections of the phase space of the isoenergetic dynamics. Projections of the topological structure of the phase space of the system before the first (higher energy) bifurcation, in between the two bifurcations, and below the second (lower energy) bifurcation are depicted in Figures 3.37a, b and c, respectively. An alternative representation of these bifurcations in the FEP was depicted in Figures 3.20 and 3.21a for branch S11−. We now focus on the topology of the impulsive orbits (IOs) in the neighborhood of ω = 1, under conditions of 1:1 internal resonance. From the discussion of Sections 3.3.3 and 3.3.4, it is clear that an IO corresponds to the initial condition a2 (0) = 0 ⇒ ψ(0) = π/2. In terms of the spherical topology of the isoenergetic flow, an IO is therefore coincident with a trajectory passing through the NP, which renders this graphical representation particularly attractive. The IO computed from

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Fig. 3.39 IOs passing through the NP (the origin of the projection), and orbits passing through the SP for ε = 0.1 and C = 2/15: (a) r = 0.25, (b) r = 0.36, (c) r = 0.37, (d) r = 0.386, (e) r = 0.387, (f) r = 0.40, (g) r = 0.44, (h) r = 0.46, (i) r = 0.50; the shift of the IO from the left to the right between (g) and (h) is an artifact of the projection.

the slow flow (3.89), together with the trajectory passing through the SP (corresponding to the orbit having as only non-zero initial condition the velocity of the LO) are shown in Figure 3.39 for varying values of the energy-like parameter r (on different isoenergetic manifolds). We note that the depicted IOs may be either periodic or quasi-periodic. In Figures 3.39c, d a third isolated trajectory is seen which lies on the same energy level as the trajectory passing through the NP. Starting from the low-energy isoenergetic manifold of Figure 3.39a, we note that the IO makes a small excursion in the spherical phase space, and remains localized close to ψ = +π/2; it follows that in this case, the energy exchange between the LO to the nonlinear attachment is insignificant, and the oscillation remains confined predominantly to the LO. The same qualitative behavior is preserved until the critical energy r = rcr = 0.3865 (occurring between Figures 3.39d and 3.39e), for which

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the IO coincides with two hom*oclinic loops in phase space; these turn out to be the hom*oclinic loops of the unstable hyperbolic equilibrium (NNM) on S11− which exists between the two saddle-node bifurcations discussed previously. For r > rcr the topology of the IO changes drastically, as it makes much larger excursions in phase space; this means continuous, strong energy exchange between the LO and the nonlinear attachment in the form of nonlinear beats. At an even higher value of energy, r ≈ 0.4495, the IO passes through both the NP and SP (this occurs between Figures 3.39g, h), and 100% of the energy is transferred back and forth between the LO and the nonlinear attachment during the occurring nonlinear beats. We conclude that for fixed mass ratio ε and nonlinear coefficient C, the geometries of the IOs undergo significant changes for varying energy: for low energies, the IOs are localized to the LO, whereas above a critical energy threshold the IOs appear as nonlinear beats, whereby energy gets continuously exchanged between the LO and the nonlinear attachment. Moreover, at specific energy levels almost the entire (conserved) energy of the motion gets transferred back and forth between the linear and nonlinear oscillators. It turns out that the critical value of the energy-like variable, rcr , can be directly related to the energy threshold required for TET in the weakly damped system. Indeed, as we recall from the numerical results of Section 3.2, strong TET phenomena in the damped system (3.2) occur only when the external impulsive excitation applied to the LO (i.e., the initial energy of the system) exceeds a certain critical value. The threshold for TET in the damped system can be directly related to the existence of a critical energy level (signified by rcr ) in the underlying Hamiltonian system, above which the IO makes large excursions in phase space and nonlinear beats corresponding to strong energy exchanges between the LO and the nonlinear attachment are initiated. Moreover, conditions for optimal TET in the damped system can be formulated by studying the topology of the IOs in the neighborhood of the hom*oclinic loops in the slow flow of the Hamiltonian system. These remarks provide a first indication of the intricate relation between IOs and TET, and of the importance of understanding the Hamiltonian dynamics in order to correctly interpret strongly nonlinear transitions and TET in the weakly dissipative system. A systematic study of the dynamics of the damped system will carried out starting from the next section. Figure 3.40 depicts the maximum excursion attained by an IO from the NP (i.e., the measure ||ψNP || = |π/2 − ψNP |), as function of r and different values of the mass ratio ε; as discussed above this measure provides a good picture of the energy exchange that occurs between the linear and nonlinear oscillators. Considering the results of Figure 3.40 there are two interesting findings. First, below a critical mass ratio there occurs a discontinuity in this energy exchange. For instance, for ε = 0.25, the variation of ||ψNP || is continuous with r (Figure 3.40d); the reason is that the branch S11− does not undergo any saddle-node bifurcations for this mass ratio (see Figure 3.38), so no hom*oclinic loops exist (and, hence, no significant topological change in the shape of the IOs occurs) as r varies. On the contrary, for smaller mass ratios, the IOs undergo significant topological changes as r varies (see Figure 3.39), which leads to the discontinuities in energy exchanges noted in Figures 3.40a–c.

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Fig. 3.40 Amplitude of the IO as function of r for C = 2/15 and: (a) ε = 0.01, (b) ε = 0.05, (c) ε = 0.10, (d) ε = 0.25.

The second interesting finding is that the mass ratio has a critical influence on the capacity of the nonlinear attachment to passively absorb energy from the LO during a cycle of the motion. Specifically, we note that for ε = 0.01, only a small amount of energy is transferred from the LO to the nonlinear attachment, as evidenced by the small value of ||ψNP || in Figure 3.40a. However, for ε = 0.1 and ε = 0.25, complete energy exchange between the two oscillators takes place (i.e., the upper bound ||ψNP || = π is reached for a specific value of r) during a cycle of the motion (see Figures 3.40c, d). The energy level r = rcomplete for which complete energy exchange occurs between the LO and the nonlinear attachment during the beating phenomenon is related to the energy of the impulsive orbit, hNP =

3Cr 4 r2 + 2 32

(3.95)

and to the energy of the trajectory passing through the SP: hSP =

3Cr 4 r2 + 4 32ε2

(3.96)

Equating these two energies, we ensure that an orbit initiated from the NP (i.e., an IO) passes also from the SP, signifying that there occurs complete energy transfer from the LO to the nonlinear attachment during a cycle of the ensuing nonlinear beat. This provides the sought after critical value for rcomplete as follows:

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hNP = hSP ⇒ rcomplete =

8ε2 3C(1 − ε2 )

165

1/2 (3.97)

According to this expression, for ε = 0.1 and C = 2/15, there is complete energy exchange between the two oscillators when r = rcomplete = 0.4495, which is in agreement with the results depicted in Figure 3.40. Because no complete energy exchange can be achieved for small mass ratios, expression (3.97) only holds for sufficiently large values of ε. These results conclude our numerical and analytical study of the dynamics of the Hamiltonian system (3.6). In the next section we start our systematic study of the dynamics of the weakly dissipative system, which will include a detailed discussion of damped transitions and of targeted energy transfer (TET) phenomena. We will show that for sufficiently weak damping (which is a reasonable and practical assumption for typical mechanical systems and structural components) the underlying Hamiltonian dynamics govern, in essence the damped responses, with damping playing a rather parasitic role, in the sense that it does not ‘produce’ to any new dynamics; this observation, however, is not intended to diminish the important role that damping plays on TET phenomena, as discussed below. Viewed in this context, we will then argue that the excitation of stable IOs giving rise to strong energy exchanges between the LO and the nonlinear attachment, provides an important mechanism for strong TET in the weakly damped system. Moreover, conditions for optimal TET will be closely related to the topology of orbits of the underlying Hamiltonian system, and especially to the topology of the manifold of IOs. Hence, the response of the Hamiltonian system and the analysis presented in the previous sections provide the necessary framework for understanding and analyzing the responses of the weakly damped system, for interpreting complex nonlinear modal interactions and transitions, and, more importantly, for designing NESs with optimal TET capacities.

3.4 SDOF Linear Oscillators with SDOF NESs: Transient Dynamics of the Damped Systems Based on our knowledge of the Hamiltonian dynamics, we initiate our study of the transient dynamics of the weakly damped system (3.2), which is reproduced here for convenience: x¨ + λ1 x˙ + λ2 (x˙ − ν˙ ) + ω02 x + C(x − ν)3 = 0 ˙ + C(ν − x)3 = 0 εν¨ + λ2 (˙ν − x)

(3.98)

Again we will assume that the nonlinear attachment is lightweight, 0 < ε 1. In an initial series of numerical simulations we demonstrate the intricate relation between the weakly dissipative and Hamiltonian dynamics.

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3.4.1 Nonlinear Damped Transitions Represented in the FEP The aim of this section is to show that the previously studied structure of the underlying Hamiltonian dynamics of (3.98) greatly influences the transient dynamics of the weakly damped system. When viewed from this perspective, one can systematically interpret complex multi-frequency transitions between different nonlinear normal modes (NNMs) in the damped dynamics, by relating them to transitions between different branches of NNMs in the FEP of Figure 3.20. Unless otherwise noted, in the following simulations of this section we consider system (3.98) with parameters ε = 0.05, ω0 = 1.0, C = 1.0, and weak damping, λ1 = 0, λ2 = 0.0015. In the first numerical simulation (see Figure 3.41) we initiate the motion on the high-energy unstable IO on branch U 21 corresponding to initial conditions ν(−T /4) = ν(−T ˙ /4) = x(−T /4) = 0 and x(−T ˙ /4) = X = −0.579. Even though the excited IO is unstable, there is strong targeted energy transfer (TET) from the (directly excited) LO to the NES, as evidenced by the rapid and strong build-up of the oscillation amplitude of the NES (note that the NES is initially at rest). Moreover, due to the instability of the excited IO the motion escapes immediately from branch U 21 to land on S11+ through a frequency transition (jump). As energy further decreases due to viscous dissipation the motion follows a multi-mode transition visiting the branches S13+, S13−, S15−, S15+, . . . , i.e., it follows the basic backbone curve of the frequency-energy plot (FEP) of Figure 3.20a. This is shown in Figure 3.41c where the wavelet transform (WT) spectrum of the relative displacement (ν − x) is superimposed to the FEP of the underlying Hamiltonian system. Although this plot provides a purely phenomenological interpretation of the damped transitions in terms of the undamped Hamiltonian dynamics, it validates our previous assertion regarding the parasitic role of weak damping in the transient dynamics. Indeed, damping does not generate any new dynamics, but merely influences the damped transitions (jumps) between different branches of NNMs of the Hamiltonian system. Clearly, by depicting the damped dynamics on the FEP, we are able to interpret complex multi-frequency transitions such as the ones shown in Figures 3.41a, b, involving the participation of multiple nonlinear modes in the transient response. A more detailed consideration of this nonlinear damped transition can be found in Lee et al. (2006). In the second simulation we initiate the motion on the moderate-energy stable IO on branch U 76 (corresponding to the non-zero initial condition X = −0.1039). In Figures 3.42a, b we depict the transient responses of the LO and the NES, indicating that there occurs stronger TET to the NES in this case. Moreover, since the initially excited special orbit on U 76 is stable, there occurs a prolonged initial oscillation of the system on that branch at the early stage of the motion (see Figure 3.42c). As energy decreases due to damping dissipation there occurs a transition (jump) to the stable branch S13−, where the NES engages into a transient 1:3 internal resonance with the LO; this is referred to as a 1:3 transient resonance capture (TRC) (Arnold, 1988; Quinn, 1997 – see also Section 2.3). As energy decreases even further due to viscous dissipation there occurs escape from 1:3 TRC, and the motion evolves along branches S15, S17, . . .. as in the previous simulation.

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Fig. 3.41 Damped transition initiated on the unstable IO on branch U 21: transient responses of (a) the LO and (b) the nonlinear oscillator (NES); (c) WT spectrum of (v − x) superimposed to the Hamiltonian FEP.

This second simulation provides the first numerical evidence that the excitation of a stable IO close to the 1:1 resonance manifold represents one of the mechanisms for strong TET in system (3.98). Lee et al. (2006) showed that the strongly nonlinear damped transitions depicted in Figures 3.41 and 3.42, are sensitive to damping, since for small damping variation a qualitatively different series of multi-modal transitions may result. An additional observation drawn from these numerical simulations is that the excitation of a stable IO prolongs the initial phase of nonlinear beats between the LO and the NES, resulting in strong TET to the NES. Indeed, by comparing the time series of Figures 3.41a, b and 3.42a, b we conclude that when an unstable IO is initially excited (so that no significant initial beating occurs), TET from the LO to the NES is weaker.

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Fig. 3.42 Damped transition initiated on the stable IO on branch U 76: transient responses of (a) the LO and (b) the nonlinear oscillator (NES); (c) WT spectrum of (v − x) superimposed to the Hamiltonian FEP.

In the third series of damped transitions depicted in Figure 3.43 we study damped transitions initiated by exciting low-, moderate- and high-energy IOs of the system with λ1 = λ2 = 0.005. The qualitative differences between these transitions are evident, indicating the sensitivity of the dynamics of system (3.98) on the initial conditions (or, equivalently, on the initial energy of the motion). For initial condition X = 0.05 (corresponding to a low-energy IO, Figure 3.43a) the response possesses a frequency component around ω = 0.2 rad/s during the initial stage of the motion, which indicates excitation of the low-energy IO. As discussed in Sections 3.3.3 and 3.3.4, such an IO is localized to the LO, and this is why a transition to S11− is observed after a short multi-frequency initial transient. Eventually, only

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Fig. 3.43 WT spectra of the transient damped response (v − x) of the two-DOF system (3.98) interpreted in the FEP for excitation of: (a) a low-energy IO, X = 0.05; (b) a moderate-energy IO, X = 0.12; (c) a moderate-energy IO, X = 0.2, and (d) a high-energy IO, X = 0.5.

a small portion of vibration energy is transferred to, and dissipated by the NES in this case, a result which is compatible with the fact that passive TET is ‘triggered’ only above a critical energy threshold (Section 3.2). Figure 3.43a also illustrates that the dynamics is weakly nonlinear at this low-energy level, since after the initial transients the dominant frequency component of the damped motion is near the linearized frequency ω0 = 1, and the response is narrowband. Qualitatively different transient dynamics is encountered for initial condition X = 0.12 and excitation of a moderate-energy IO (see Figure 3.43b). Strong and sustained harmonic components appear in this case, and the damped motion never fully enters into the domain of attraction of the 1:1 resonant manifold; instead, the damped response is in the form of a prolonged nonlinear beat, which results in strong TET from the LO to the NES. This regime of motion is strongly nonlinear, as revealed by the appearance of multiple strong sustained harmonics over a relatively broadband frequency range. Increasing further the initial condition to X = 0.2 (and exciting still a moderateenergy IO, see Figure 3.43c), gives rise to a different damped transition scenario. Specifically, there occurs a rapid transition of the damped dynamics from the IO to

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Fig. 3.44 EDM when an IO is excited, as function of the non-zero initial condition of that IO.

branch S11+, where sustained 1:1 TRC is initiated. This transition is similar to that encountered in the numerical simulation of Figure 3.41, and results in moderate TET from the LO to the NES. A similar transition is noted for the initial condition X = 0.5 corresponding to excitation of a high-energy IO, and shown in Figure 3.43d. In summary, different transition scenarios are realized in the damped dynamics depending on the energy of the IOs that are initially excited. These different transitions may result in enhanced (or weaker) TET from the LO to the NES, depending on the excitation (or lack of) of nonlinear beat pheneomena leading to strong localization of the motion to the NES. To further emphasize this point, in Figure 3.44 we depict the energy dissipation measure (EDM) (i.e., the percentage of input energy dissipated by the NES) when an IO is excited, as function of the non-zero initial condition X of that IO; the system parameters for these simulations are selected as ε = 0.5, ω02 = 1, C = 1, λ1 = λ2 = 0.01. The positions of some representative (stable and unstable) IOs are indicated in that plot as well. Low-energy impulsive orbits are located below the critical energy threshold, and their excitation results in weak TET. Optimal TET is associated with the excitation of moderate-energy IOs, located just above the energy threshold and satisfying conditions of near 1:1 internal resonance between the LO and the NES (i.e., U 54, U 43, . . . ). By further increasing the initial condition of the IO we get deterioration of TET, as we leave the regime of 1:1 internal resonance so that less pronounced nonlinear beats are realized when an IO is excited (see Figure 3.26 and the analysis of Section 3.3.4).

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The results of this section show the clear relation between TET and the strongly nonlinear multi-mode (and multi-frequency) transitions that take place in the FEP. This naturally leads to a detailed discussion of the alternative mechanisms for the realization of TET in system (3.98), a task addressed in the next section.

3.4.2 Dynamics of TET in the Damped System We now study the capacity for targeted energy transfer (TET) of the lightweight ungrounded NES considered in the previous sections; that is, its capacity to passively absorb and locally dissipate vibration energy from the SDOF linear oscillator (LO), without ‘spreading back’ the absorbed energy. We will show that key to understanding TET in the weakly damped system is our knowledge of the topological structure of the orbits of the underlying Hamiltonian system, as it is the undamped dynamics that influences in a essential way the weakly damped transitions and the resulting strongly nonlinear modal interactions. The first mechanism for TET, fundamental TET or fundamental energy pumping, is due to 1:1 transient resonance capture (TRC) of the dynamics, and is realized when the damped motion traces approximately the in-phase backbone curve S11+ of the FEP of Figure 3.20, at relatively low frequencies ω < ω0 . The second mechanism, subharmonic TET, resembles the first, but is realized when the motion takes place along a lower frequency subharmonic tongue Snm, n < m of the FEP; it is due to n : m TRC, and is less efficient than fundamental TET. The third mechanism, TET through nonlinear beats, is the most powerful mechanism for TET, as it involves the initial excitation of an IO at a higher frequency tongue, at frequencies ω > ω0 . In the following sections we will discuss each TET mechanism separately through numerical simulations and analysis.

3.4.2.1 TET through Fundamental Transient Resonance Capture (TRC) The first mechanism for TET involves excitation of the branch of in-phase NNMs S11+, where the LO and the NES oscillate with identical frequencies in the neighborhood of the fundamental frequency ω0 . In Figure 3.21b we depict a detailed plot of branch S11+ of the Hamiltonian system i.e., the set (3.98) with λ1 = λ2 = 0], and note that at higher energies the in-phase synchronous periodic oscillations (NNMs) are spatially extended (involving finite-amplitude oscillations of both the LO and the nonlinear attachment). However, since the nonlinear mode shapes of NNMs on S11+ strongly depend on the level of energy, and as energy decreases they become localized to the nonlinear attachment. This low-energy localization is a basic characteristic of the two-dimensional NNM invariant manifold corresponding to S11+; moreover, this localization property is preserved in the weakly damped system, where the motion takes place on a two-dimensional damped NNM invariant manifold (Shaw and Pierre, 1991, 1993).

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This means that when the initial conditions of the damped system place the motion on the damped NNM invariant manifold corresponding to S11+, for decreasing energy the mode shape of the resulting oscillation makes a transition from being initially spatially extended to being localized to the NES. This, in turn, leads to passive transfer of energy from the LO to the NES. As shown below, the underlying dynamical phenomenon governing fundamental TET is TRC on a 1:1 resonance manifold of the damped system. As discussed in Section 2.3, TRC is a form of transient nonlinear resonance between two modes of a system, followed by escape from the capture regime. TRCs and sustained resonance captures (SRCs) have been studied extensively in weakly varying Hamiltonian systems and in non-conservative oscillators [(Kevorkian, 1971, 1974; Gautesen, 1974; Neishtadt, 1975, 1986, 1987, 1997, 1999; Haberman, 1983; Kath, 1983; Arnold, 1988; Bosley and Kevorkian, 1992; Quinn et al., 1995; Bosley, 1996; Quinn, 1997; Vakakis and Gendelman, 2001; Vainchtein et al., 2004); see also the discussion in Wiggins (1990) on the interaction of resonance bands in weakly damped oscillators using geometrical methods]. Regarding the study of energy exchanges and nonlinear dynamical interactions caused by TRCs, we mention the work by Neishtadt (1975) on the transition of a Hamiltonian system across a separatrix (separatrix crossing) caused by periodic parametric excitation due to a slowly varying frequency; the work by Friedland (1997) on trapping into resonance in adiabatically varying systems driven by externally launched pump waves; on continuous resonant growth of induced nonlinear waves (Aranson et al., 1992); on the excitation of an oscillatory nonlinear system to high energy by weak chirped frequency forcing (Marcus et al., 2004); and on a method based on resonance capture to control transitions between different regimes of Hamiltonian systems (Vainchtein and Mezic, 2004). However, with the exception of the paper by Quinn et al. (1995) these works deal with systems without damping; on the other hand, in contrast to the results reported in this work Quinn et al. (1995) did not consider strong inertial asymmetry, which as shown below is a necessary condition (along with weak dissipation) for realizing TET through TRC. We note that in the absence of damping, no TET, i.e., irreversible energy transfer, can occur on motions initiated on branch S11+. The reason is that in the absence of energy dissipation the distribution of energy between the linear and nonlinear components is ‘locked’ (due to the invariance of the NNM manifold S11+), so no localization can occur to either one of these system components. In addition, unlike the phenomenon of internal resonance encountered in conservative oscillators, during TRC the frequency of oscillation of the NES varies with time, depending on the amount of energy transferred from the LO; therefore, it is indeed possible to escape from the fundamental resonance capture regime if the frequency of the NES departs away from the neighborhood of the natural frequency of the LO, ω0 . Finally, we note that although the NES has no preferential resonant frequency (as it possesses nonlinearizable stiffness nonlinearity), it may synchronize with the LO along S11+ due to the invariance properties of the damped NNM manifold, and this occurs passively, without the need to ‘tune’ the NES parameters. This demonstrates

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the enhanced versatility of the systems with essential nonlinearities considered in this work. Numerical evidence of fundamental TET in the damped system (3.98) is presented in Figure 3.45 for ω0 = 1, C = 1, ε = 0.05, λ1 = λ2 = 0.002. Weak damping is considered in order to better highlight the TET phenomenon, and the motion is initiated on a NNM on S11+ corresponding to initial conditions x(0) = ν(0) = 0, x(0) ˙ = 0.175, ν˙ (0) = 0.386. Considering the transient responses depicted in Figures 3.45a, b, we note that the envelope of the response of the LO decays more rapidly than that of the NES. The detail of the response presented in Figure 3.45c indicates that motion along S11+ corresponds to in-phase vibration of the two masses with identical fast frequency, confirming that the transient dynamics is locked into 1:1 transient resonance capture (TRC). The percentage of instantaneous energy stored in the NES is presented in Figure 3.45d, confirming that as the damped motion follows branch S11+ with decaying energy, an irreversible and complete energy transfer takes place from the LO to the NES, at least until escape from resonance capture occurs around t ≈ 300 s. We commend that the reversal in instantaneous energy suffered by the NES for t > 300 s occurs at the very late stage of the response where the energy of the system has almost completely been dissipated by damping. Finally, in Figure 3.45e, the Morlet WT spectrum of the relative response between ν(t)−x(t) is superposed to the backbone of the Hamiltonian FEP, confirming that the in-phase branch S11+ is approximately traced by the damped transient response. This validates our previous conjecture that the TET dynamics in the damped system is mainly governed by the topological structure and bifurcations of the periodic (and quasi-periodic) motions of the underlying Hamiltonian system. We now proceed to analytically study the fundamental TET mechanism by analyzing system (3.98) through the complexification-averaging (CX-A) technique discussed in Sections 2.4 and 3.3.2. Even though (3.98) is a strongly nonlinear system of coupled oscillators, analytical modeling of its transient dynamics leading to TET can still be performed. Indeed, motivated by the time series of the transient responses of Figures 3.45a, b we will partition the transient dynamics into slow and fast components, and then reduce our study to the investigation the corresponding slow flow dynamics of the system. The slow flow governs the essential (important) dynamics of the weakly damped system, as well as the nonlinear modal interactions that occur between the LO and the NES and lead to fundamental TET. As discussed in Sections 2.4 and 3.3.2 the CX-A technique is especially suited for studying TET, as it can be applied to the analysis of transient, strongly nonlinear responses that possess multiple distinct fast frequencies, yielding the reduced slow flow dynamics that govern the slow modulations of these fast components (namely, their amplitudes and phases). Clearly, the CX-A approach provides a good approximation of the exact dynamics only as long as the corresponding assumptions of the analysis are satisfied, and within the time domain of validity of the associated averaging operations [see (Sanders and Verhulst, 1985) and the discussion in Section 2.4]. There are important motivations for reducing the dynamics of (3.98) to the slow flow. First, as mentioned above, the slow flow-dynamics can be regarded as the im-

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Fig. 3.45 Fundamental TET (ω0 = 1, C = 1, ε = 0.05, λ1 = λ2 = 0.002): (a) LO displacement; (b) NES displacement; (c) superposition of system displacements (solid line: LO; dashed line: NES); (d) percentage of instantaneous total energy in the NES; and (e) WT spectrum of the relative response (v − x) superposed to the backbone of FEP of the underlying Hamiltonian system.

portant (essential) dynamics of the system (after the non-essential fast dynamics has been factored out of the analysis), since it determines the long-term behavior of the response. In addition, being in the form of a set of first-order ordinary differen-

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tial equations, the reduced slow flow dynamical system, although still nonlinear, is generally easier to analyze than the original strongly nonlinear equations of motion. Finally, the derived slowly-varying amplitudes and phases represent meaningful features of the transient responses and offer a sharper and clearer characterization of the system dynamics than the original time series. Focusing on fundamental TET, the following new complex variables are introduced in the system of equations (3.98), ψ1 (t) = x(t) ˙ + j x(t) ψ2 (t) = ν(t) ˙ + j ν(t)

(3.99)

where j = (−1)1/2 . Since fundamental TET corresponds to 1:1 transient resonance capture (TRC), we will assume that the transient dynamics possess a single dominant fast frequency ω ≈ ω0 = 1, and introduce the following slow-fast partitions of the new complex variables, ψ1 (t) = ϕ1 (t)ej t ψ2 (t) = ϕ2 (t)ej t

(3.100)

where ϕi (t), i = 1, 2, are slowly varying complex modulations of the fast components. It should be clear that by the ansatz (3.100) the validity of the following analysis is only valid in the neighborhood of ω0 = 1 of the FEP. Expressing the system responses in terms of the new complex variables, x = (ψ1 − ψ1∗ )/2j,

ν = (ψ2 − ψ2∗ )/2j

(3.101)

where asterisk ∗ denotes complex conjugate, substituting into (3.98), and performing averaging with respect to the fast frequency (i.e., omitting terms with fast frequencies greater than or equal to unity), the following set of approximate, slow modulation equations governing the (slow) evolutions of the complex modulations is derived: ϕ˙1 − (ελ/2)(ϕ2 − ϕ1 ) − (3j C/8)|ϕ1 − ϕ2 |2 (ϕ1 − ϕ2 ) + (ελ/2)ϕ1 = 0 ϕ˙2 + (j/2)ϕ2 + (λ/2)(ϕ2 − ϕ1 ) − (3j C/8ε)|ϕ2 − ϕ1 |2 (ϕ2 − ϕ1 ) = 0 (3.102) For the sake of simplicity, from now on we will assume that λ1 = λ2 = λ in (3.102), without restricting the generality of the analysis. To obtain a set of real modulation equations, we express the complex amplitudes in polar forms, ϕi (t) = ai (t)ejβi (t ), i = 1, 2, substitute these into (3.102), and set separately equal to zero the real and imaginary parts of the resulting expressions. By introducing the phase difference φ(t) = β1 (t) − β2 (t), the final set of real modulation equations can be cast in the form of an autonomous dynamical system: a˙ 1 − (ελ/2)a2 cos φ + ελa1 + (3C/8)(a12 + a22 − 2a1 a2 cos φ)a2 sin φ = 0

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a˙ 2 + (λ/2)a2 − (λ/2)a1 cos φ − (3C/8ε)(a12 + a22 − 2a1a2 cos φ)a1 sin φ = 0 φ˙ + (λ/2)[(εa2 /a1 ) + (a1 /a2 )] sin φ − 1/2 + (3C/8)(a12 + a22 − 2a1 a2 cos φ) × {(1/ε)[1 − (a2 /a1 ) cos φ] − [1 − (a1 /a2 ) cos φ]} = 0

(3.103)

The variables a1 and a2 represent the (real) amplitudes of the slowly-varying envelopes of the linear and nonlinear responses, respectively, whereas φ(t) the phase difference of the evolutions of these envelopes. The reduced dynamical system (3.103) governs the slow flow dynamics of the fundamental TET. In particular, 1:1 TRC, the underlying dynamical mechanism of TET, is associated with non-time-like evolution of the phase angle φ or, equivalently, failure of the averaging theorem with respect to that angle (Sanders and Verhulst, 1985; Verhulst, 2005). Indeed, in case that φ would exhibit time-like behavior, we could regard it as a fast angle and apply the averaging theorem over φ to prove that the amplitudes a1 and a2 decay exponentially with time, nearly independently from each other (see also the discussion in Section 2.4). Then, no significant energy exchanges between the linear and nonlinear oscillators would take place, and no TET would be possible. ˙ φ) Figure 3.46a depicts the dynamics of 1:1 TRC in the slow flow phase plane (φ, for system (3.103) with ε = 0.05, λ = 0.01, C = 1, ω0 = 1 and initial conditions a1 (0) = 0.24, a2(0) = 0.01, φ(0) = 0. The oscillatory behavior of the phase variable in the neighborhood of the in-phase limit φ = 0+ confirms the occurrence of 1:1 TRC in the neighborhood of the in-phase NNM branch S11+. As evidenced by the build-up of amplitude a2 of the envelope of the NES depicted in Figures 3.46b, d, this leads to fundamental TET from the LO to the NES. Escape from the 1:1 TRC is associated with time-like behavior of φ and rapid decrease of the amplitudes a1 and a2 , as predicted by applying averaging in (3.103). A comparison of the analytical approximations (3.101–3.103) with direct numerical simulation of (3.98) subject to the previous initial conditions is presented in Figure 3.4c confirming the accuracy of the analysis. The discrepancy between analysis and numerical simulation noted for T > 50S is attributed to the escape of the dynamics from the regime of 1:1 TRC, where the assumptions of the analysis are not valid any more. Moreover, due to the averaging operations associated with the CX-A technique, the resulting analytical approximation is not expected to be valid for relatively large times (see the discussion on the relation between averaged and exact dynamics in Section 2.4).

3.4.2.2 TET through Subharmonic TRC Subharmonic TET involves excitation of a low-frequency subharmonic S-tongue of NNMs for frequencies ω < ω0 . As mentioned in Section 3.3.1.2, by low-frequency tongues we mean families of NNMs of the underlying Hamiltonian system with the nonlinear attachment engaging in m:n internal resonance with the LO (where m, n are integers with m < n). Another feature of a low-frequency tongue Smn, m < n is that it is represented by a nearly horizontal line in the FEP, since on the tongue

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Fig. 3.46 Dynamics of fundamental TET: (a) 1:1 TRC in the slow flow; (b) nornalized amplitude modulations; (c) comparison between analytical approximation (dashed line) and direct numerical simulation (solid line) of NES response (vt ); (d) system responses, [dashed line x(t), solid line v(t)].

the strongly nonlinear response resembles that of a linear system with the NES oscillating slower than the LO and the ratio of their frequencies being approximately equal to m/n < 1 (see the discussion about oscillations on tongues S13± in Section 3.3.2.2). Moreover, to each rational number m/n, m > n there corresponds a pair of closely spaced tongues, composed of in-phase (Smn+) and an out-of-phase (Smn−) periodic motions, respectively; finally, these tongues exist over finite energy ranges. Hence, a countable infinity of low-frequency subharmonic tongues exists over finite energy ranges of the Hamiltonian system corresponding to

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Fig. 3.47 Subharmonic TET (ω0 = 1, C = 1, ε = 0.05, λ1 = λ2 = 0.001): (a) LO displacement; (b) NES displacement; (c) superposition of system displacements (solid line: LO; dashed line: NES); (d) percentage of instantaneous total energy in the NES; and (e) WT spectrum of the relative response (v − x) superposed to the backbone of FEP of the underlying Hamiltonian system.

λ1 = λ2 = 0 in (3.98). As mentioned in Section 3.3.1.2 this is a direct sequence of the non-integrability of this strongly nonlinear Hamiltonian system under examination. To explain subharmonic TET in the damped system (3.98), we focus in the particular pair of lower tongues S13±, and refer to Figure 3.21d. As discussed in Sec-

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tion 3.3.2.2, at the extremity of this tongue (i.e., at the maximum energy of the tongue), the oscillation is localized to the LO. However, as in the case of fundamental TET, the reduction of energy by damping dissipation leads to gradual delocalization of the motion from the LO and localization to the NES; as a result, passive energy transfer from the LO to the NES, i.e., subharmonic TET, takes place. It follows that, as in the case of fundamental TET, it is the change of shape of NNMs on S13± that eventually leads to subharmonic TET in the damped system. Again, one can invoke arguments of invariance and persistence of the damped NNM manifold resulting from the perturbation due to weak damping of the corresponding NNM invariant manifolds S13± of the underlying Hamiltonian system. In this case, the underlying dynamics causing TET is an m:n TRC that occurs in the neighborhood of an m:n resonance manifold of the dynamics, as discussed later in this section. The transient dynamics for motion initiated on the stable branch S13− (with initial conditions x(0) = ν(0) = 0, x(0) ˙ = −0.0497, ν˙ (0) = 0.0296) is displayed in Figure 3.47 for ω0 = 1, C = 1, ε = 0.05, and λ1 = λ2 = 0.001. Despite the presence of viscous dissipation, the NES response grows continuously as it passively absorbs and locally dissipates vibration energy from the LO whose amplitude rapidly decreases. Figure 3.47d shows that subharmonic TET takes place until approximately t = 900 s, during which almost complete energy transfer from the LO to the NES is realized. The WT spectrum of Figure 3.47e demonstrates clearly that the damped response traces approximately the subharmonic tongue S13− until it reaches the backbone curve of the FEP, after which it traces that branch. This provides further evidence of the close relation of the weakly damped and Hamiltonian dynamics, and highlights the mechanism governing TET in this case. It is interesting to note that for the specific 1:3 subharmonic TET shown in Figure 3.47, the LO oscillates with a frequency approximately three times that of the NES. Moreover, due to the stability properties of the tongues S13±, subharmonic TET can only take place for out-of-phase relative motions between the LO and the NES (i.e., for excitation of the stable out-of-phase NNMs on tongue S13−), and not for in-phase ones, since the in-phase tongue S13+ is unstable (see Figure 3.21d). To demonstrate the analysis of the dynamics governing subharmonic TET, we focus on 1:3 TRC in the neighborhood of tongue S13−. However, similar analysis can be applied to other cases of subharmonic resonance captures leading to TET. Due to the fact that motion in the neighborhood of S13− possesses two main harmonic components with frequencies ω and ω/3, the transient damped responses of system (3.98) are expressed as x(t) = x1 (t) + x1/3 (t),

ν(t) = ν1 (t) + ν1/3 (t)

(3.104)

where the indices 1 and 1/3 indicate that the respective terms possess dominant frequencies equal to ω and ω/3, respectively. As in the case of fundamental TET, we introduce the following new complex variables: ψ1 (t) = x˙1 (t) + j ωx1 (t) ≡ ϕ1 (t)ej ωt , ψ3 (t) = x˙1/3 (t) + j (ω/3)x1/3 (t) ≡ ϕ3 (t)ej (ω/3)t

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ψ2 (t) = ν˙ 1 (t) + j ων1 (t) ≡ ϕ2 (t)ej ωt , ψ4 (t) = ν˙ 1/3 (t) + j (ω/3)ν1/3 (t) ≡ ϕ4 (t)ej (ω/3)t

(3.105)

Again slow-fast partitions of the dynamics are introduced, but this is performed in a different way than in the case the fundamental TET case, to reflect the existence of two fast frequencies ω and ω/3 in the responses during 1:3 TRC. Although ω ≈ 1 during 1:3 TRC in the neighborhood of tongue S13−, we opt to keep ω as a yet undetermined frequency parameter for the time being. In (3.105) the variables ϕi (t), i = 1, . . . , 4 represent slowly varying complex modulations of the fast oscillations with frequencies ω and ω/3. Expressing the responses x and ν and their time derivatives in terms of the new complex variables, i.e., x=

ψ1 − ψ1∗ 3(ψ3 − ψ3∗ ) + , 2j ω 2j ω

ν=

ψ2 − ψ2∗ 3(ψ4 − ψ4∗ ) + 2j ω 2j ω

(3.106)

and substituting the resulting expressions into (3.98), we perform averaging over each of the two fast frequencies ω and ω/3, and derive the following set of complex coupled differential equations governing the slow evolutions of the four complex modulations, ϕ˙1 + (j ω/2 − j/2ω)ϕ1 + (ελ/2)(2ϕ1 − ϕ2 ) + (j C/8ω3 ){3[9ϕ33 − 27ϕ32ϕ4 − 9ϕ43 − (ϕ1 − ϕ2 )|ϕ1 − ϕ2 |2 + 27ϕ3ϕ42 − 18(ϕ1 − ϕ2 )|ϕ3 − ϕ4 |2 ]} = 0 ϕ˙3 + (j ω/6 − 3j/2ω)ϕ3 + (ελ/2)(2ϕ3 − ϕ4 ) + (j C/8ω3 ){−9[ϕ1(2(ϕ3 − ϕ4 )(ϕ1∗ − ϕ2 ) − 3(ϕ3∗ − ϕ4∗ )2 ) + ϕ2 (2(ϕ4 − ϕ3 )(ϕ1∗ − ϕ2 ) + 3(ϕ3∗ − ϕ4∗ )2 ) + 9(ϕ3 − ϕ4 )|ϕ3 − ϕ4 |2 ]} = 0 ϕ˙2 + (j ω/2)ϕ2 + (λ/2)(ϕ2 − ϕ1 ) − (j C/ε8ω3 ){3[9ϕ33 − 27ϕ32ϕ4 − 9ϕ43 − (ϕ1 − ϕ2 )|ϕ1 − ϕ2 |2 + 27ϕ3ϕ42 − 18(ϕ1 − ϕ2 )|ϕ3 − ϕ4 |2 ]} = 0 ϕ˙4 + (j ω/6)ϕ4 + (λ/2)(ϕ4 − ϕ3 ) − (j C/ε8ω3 ){−9[ϕ1(2(ϕ3 − ϕ4 )(ϕ1∗ − ϕ2 ) − 3(ϕ3∗ − ϕ4∗ )2 ) + ϕ2 (2(ϕ4 − ϕ3 )(ϕ1∗ − ϕ2 ) + 3(ϕ3∗ − ϕ4∗ )2 ) + 9(ϕ3 − ϕ4 )|ϕ3 − ϕ4 |2 ]} = 0

(3.107)

where it is assumed that λ1 = λ2 = λ. The complex amplitudes are expressed in polar form, ϕi (t) = ai (t)ejβi (t ), i = 1, . . . , 4, which when substituted into (3.107) and upon separation of real and imaginary parts lead to an autonomous set of seven slow flow real modulation equations in terms of the amplitudes ai = |ϕi |, i = 1, . . . , 4, and three phase differences

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defined as φ12 = β1 − β2 , φ13 = β1 − 3β3 , and φ14 = β1 − 3β4 . Due to its complexity, the autonomous system that governs the slow flow of 1:3 TRC is not reproduced in its entirety here, but is only expressed in the following compact form: a˙ 1 + (ελ/2)(2a1 − a2 ) + g1 (a, φ) = 0 a˙ 3 + (ελ/2)(2a3 − a4 ) + g3 (a, φ) = 0 a˙ 2 + (λ/2)(a2 − a1 ) + g2 (a, φ)/ε = 0 a˙ 4 + (λ/2)(a4 − a3 ) + g4 (a, φ)/ε = 0 φ˙ 12 + f12 (a) + g12 (a, φ; ε) = 0 φ˙ 13 + f13 (a) + g13 (a, φ) = 0 φ˙ 14 + f14 (a) + g14 (a, φ; ε) = 0

(3.108)

In the system above, gi and gij are 2π-periodic functions in terms of the phase angles φ = (φ12 φ13 φ14 )T , and a is the (4 × 1) vector of amplitudes, a = [a1 a2 a3 a4 ]T . As in the case of fundamental TET, strong energy exchanges between the LO and the NES can occur only if a subset of phase angles φij does not exhibit time-like behavior, that is, when some phase angles possess non-monotonic behavior with respect to time. This can be deduced from the structure of the slow flow (3.108), where it is clear that if all phase angles exhibit time-like behavior and functions gi are small, averaging over these phase angles (which could then be regarded as fast angles) would lead to decaying amplitudes. In that case no significant energy exchanges between the LO and the NES could take place. As a result, 1:3 subharmonic TET is associated with non-time-like behavior of (at least) a subset of the slow phase angles φij in (3.108). Figure 3.48 depicts the results of the numerical simulation of the slow flow (3.107) for ε = 0.05, λ = 0.03, C = 1 and ω0 = 1. The motion is initiated on branch S13− with initial conditions ν(0) = x(0) = 0, ν˙ (0) = 0.01499, and x(0) ˙ = −0.059443. The issue of computing the corresponding initial conditions for the slow flow (3.107) is non-trivial and indeterminate, as this system possesses more dimensions than the exact problem. The discussion of this issue is postponed until Section 9.2.2.2 in Chapter 9, and here it suffices to state that the initial conditions for the complex amplitudes and the value of the frequency of the slow flow model (3.107) are computed by minimizing the difference between the analytical and numerical responses of the system in the interval t ∈ [0, 100]: ϕ1 (0) = −0.0577, ϕ4 (0) = 0.0134,

ϕ2 (0) = 0.0016, ω = 1.0073

ϕ3 (0) = −0.0017 (3.109)

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Fig. 3.48 Dynamics of subharmonic 1:3 TET: (a, b) amplitude modulations; (c–e) phase modulations.

This result proves that indeed frequency ω is close to unity, in accordance to our previous discussion. Before proceeding with discussing the numerical results, we mention that the initial conditions required for the solution of the set modulations (3.107) exceeds in number the available initial conditions of the original problem (3.98); the reason, of course, is that, due to decompositions (3.104, 3.105) we are in need to define initial conditions separately for each of the harmonic components at frequencies

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ω and ω/3. The method of defining the initial conditions adopted above, although not conceptually elegant and non-unique, nevertheless provides satisfactory initial conditions for the slow flow as judged by the following numerical results. The initial conditions (3.109) indicate that the energy at t = 0 is almost entirely stored in the fundamental frequency component of the LO. Figures 3.48a, b depict the slow evolutions of the amplitudes ai . As judged from the build-up of amplitude a4 and the corresponding decay of a1 , it becomes evident that 1:3 subharmonic TET involves primarily energy transfer from the fundamental component of the LO to the 1/3 subharmonic component of the NES. Considering the evolution of the amplitude a2 , we conclude that a smaller amount of energy is transferred from the fundamental component of the LO to the fundamental component of the NES. These conclusions are supported by the plots of Figures 3.48c–e, where the temporal evolutions of the phase differences φ12 = β1 − β2 , φ13 = β1 − 3β3 , and φ14 = β1 −3β4 are presented. Absence of strong energy exchanges between the fundamental and 1/3 subharmonic components of the LO response is associated with the time-like behavior of the corresponding phase difference φ13 , whereas strong energy transfer from the fundamental component of the LO response to both fundamental and 1/3 subharmonic components of the NES response, is associated with early-time oscillatory (i.e., non-time-like) behavior of the corresponding phase differences φ12 and φ14 . Oscillatory behaviors of φ12 and φ14 signify 1:1 and 1:3 TRCs, respectively, between the fundamental component of the LO response and the fundamental and 1/3 subharmonic components of the NES response. With progressing time, the phase variables become eventually time-like, signifying escapes from the corresponding TRCs. We note that the oscillations of φ12 and φ14 take place in the neighborhood of π, which confirms that, in this particular example, 1:3 subharmonic TET involves out-of-phase relative motions between the LO and the NES (since they take place in the neighborhood of tongue S13−). The predictive capacity of the analytical slow flow model (3.107, 3.108) in the regime of 1:3 subharmonic TET is demonstrated by the result depicted in Figure 3.49. It can be observed that the analytically predicted NES response is in satisfactory agreement with the exact response obtained by direct simulation of equations (3.98); this, in spite of the fact that transient and strongly nonlinear dynamics is considered. However, the analytic model fails to accurately model the response in the later regime, where escape from 1:3 TRC occurs. This occurs because during this regime the damped response leaves the neighborhood of tongue S13− and approximately evolves along the backbone curve of the FEP. Eventually, the next tongue S15 is reached, and at that point the motion cannot be described by the ansatz (3.104, 3.105) anymore, since the 1/3 subharmonic component gradually diminishes becoming unimportant and a new 1/5 subharmonic component enters into the dynamics. As a result, the considered analytical model looses validity.

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Fig. 3.49 Transient damped response of the NES during 1:3 subharmonic TET: comparison between analytical slow-flow approximation (dashed line) and direct numerical simulation (solid line).

3.4.2.3 TET through Nonlinear Beats The previous two TET mechanisms cannot be ‘triggered’ with the NES being initially at rest, since both require non-zero initial velocity for the NES, i.e., ν˙ (0) = 0. This means that neither fundamental nor subharmonic TET can occur immediately after the application of an impulsive excitation to the LO. An alternative TET mechanism, however, TET through nonlinear beats, not only surpasses this limitation, but proves to be the most powerful TET mechanism since it is capable of initiating stronger energy transfers from the LO to the NES compared to the above-mentioned two TET mechanisms. This TET mechanism is based on the initial excitation of IOs (especially, moderate-energy ones, close to the 1:1 resonance manifold) which have been discussed in detail in Sections 3.3.3 and 3.3.4. As mentioned previously, the excitation of stable localized IOs in the regime of 1:1 internal resonance of the Hamiltonian system (with the system being initially at rest subject to impulsive excitations of the LO – equivalently, with initial conditions x(0) ˙ = 0 and ν(0) = ν˙ (0) = x(0) = 0), leads to rapid transfer of energy from the LO to the NES during a cycle of the motion. This transfer is realized through nonlinear beats. We will show that in the weakly damped system, such IOs play the role of transient bridging orbits that direct the damped motion into the domain of attraction of a resonant manifold, which eventually leads to (triggers) either fundamental or subharmonic TET. Recalling the analysis of Section 3.3.4.1, the class of moderate-energy IOs occurs only above a critical energy threshold. It follows, that the corresponding triggering mechanism for TET is effective only for input energies above this critical threshold. Indeed, as shown in Section 3.3.3, low-energy (or equivalently low-frequency) IOs transfer a small fraction of the input energy from the LO to the NES, so they

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cannot induce TET. It should also be noted that, due to the essential (nonlinearizable) nonlinearity of the NES the considered nonlinear beating phenomena do not require any a priori tuning of the nonlinear attachment: at a specific frequencyenergy range corresponding to n:m resonance capture, the essential nonlinearity of the NES passively adjusts the amplitude to fulfill the required resonance conditions. This represents a significant departure from classical nonlinear beat phenomena observed in coupled oscillators with linearizable nonlinear stiffnesses where the ratio of the linearized natural frequencies of the components dictates the type of internal resonance that can be realized. To validate our conjecture, we perform a numerical simulation where system (3.98) is initiated at the IO on U 21 (corresponding to initial conditions x(0) = ν(0) = ν˙ (0) = 0, and x(0) ˙ = 0.5794, for system parameters ω0 = 1, C = 1, ε = 0.05, λ1 = λ2 = 0.005). As evidenced in the instantaneous energy plot of Figure 3.50c, a nonlinear beating phenomenon takes place in the initial stage of the motion until approximately T = 50 s; this corresponds to the initial excitation of the damped analogue of the IO on U 21. During the nonlinear beat phenomenon, the relative displacement ν(t) − y(t) possesses two main frequency components (around 1 and 2 rad/s), but the higher harmonic is barely visible in the WT spectrum plot of Figure 3.50d. After this initial nonlinear energy exchange between the two oscillators, the dynamics makes a transition to the damped in-phase NNM manifold S11+, and the dynamics is captured into the domain of attraction of the 1:1 resonant manifold. Eventually, fundamental TET takes place. We note that TET through nonlinear beats also occurs in the numerical simulation depicted in Figure 3.42; in that case, however, the initial beats due to excitation of the IO on U 76 lead, first to a transition to small duration fundamental TET, and then to a second transition to a more prolonged 1:3 subharmonic TET. This underlines the fact that although damping cannot generate new dynamics in the system, it critically influences the damped transitions between branches of solutions of the underlying Hamiltonian system. Finally, we note that TET through nonlinear beats proves to be the most efficient TET mechanism. Further discussion of this TET mechanism is postponed until Section 3.4.2.4 where conditions for optimal TET are discussed. In the next section we discuss TET from the alternative view of damped NNM manifolds, which highlights more clearly the role of damping on TET.

3.4.2.4 Damped NNM Manifolds and Fundamental TET In this section we wish to further demonstrate the important role of damping on fundamental TET. Although the analysis will be carried out under the assumption of 1:1 resonance capture leading to fundamental TET, it can be extended to the more complicated case of m:n subharmonic TET, with appropriate modifications. Reconsidering equations (3.98), which describe the two-DOF damped dynamics of an essentially nonlinear system, it is clear that they cannot be solved exactly (i.e., in explicit analytic form). However, as shown in Section 3.4.2.1 fundamental TET can be approximately analyzed by performing averaging in the vicinity of the 1:1 reso-

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Fig. 3.50 TET through nonlinear beats, excitation of IO U 21 (ω0 = 1, C = 1, ε = 0.05, λ1 = λ2 = 0.005): (a) LO displacement; (b) NES displacement; (c) percentage of instantaneous total energy in the NES; and (d) WT spectrum of the relative response (v − x) superposed to the backbone of FEP of the underlying Hamiltonian system.

nance manifold (or for the more complicated case of subharmonic TET, by multiphase averaging in the neighborhoods of the corresponding resonance manifolds – see Section 3.4.2.2). We note that even the resulting reduced averaged system (3.102–3.103) is still too complicated to be solved analytically, although its state space may be reduced to three dimensions, unlike the exact system (3.98). Approximate solutions of the averaged system governing fundamental TET may be computed based on two different approximations, each of which is now discussed. In the following analysis we will relax the condition λ1 = λ2 = λ enforced in (3.102), and instead adopt independent values for both damping constants. The first option to analyze the averaged system in the regime of 1:1 resonance capture is to suppose that the damping coefficient λ is small; it follows that the zerothorder approximation to solving (3.102) is the undamped system which is completely integrable as discussed previously. The effect of non-zero damping may then be described by application of appropriate asymptotic procedures. Such an approach, however, does not seem meaningful for studying TET, since as shown below TET strongly depends on the value of damping, so that the mentioned low-order perturbation scheme cannot be expected to describe the details of this strong dependence.

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The second perturbation approach for analyzing the averaged dynamics, is based on the assumption of strong mass asymmetry between the LO and the NES, as described by the small parameter ε in (3.98); this means that we will focus on linear oscillators with lightweight NESs. This approach does not necessarily assume small damping, and instead relies on perturbation analysis considering the NES mass ε as the small parameter. This approach is considered in this section, for a system with parameters, λ1 = 0, λ2 = ελ, C = 4ε/3 and ω0 = 1. The two latter conventions do not affect the generality of the analysis, since they may be satisfied by appropriate rescalings of the dependent and independent variables of the averaged system. We start our analysis of fundamental TET by considering the system of averaged (complex modulation) equations (3.102). Introducing the following change of complex variables, ϕ1 + εϕ2 1+ε χ2 = ϕ 1 − ϕ 2

χ1 =

the modulation equations (3.102) take the form: χ˙ 1 +

jε (χ1 − χ2 ) = 0 2(1 + ε)

χ˙ 2 +

j λ(1 + ε) j (1 + ε) (χ2 − χ1 ) + χ2 − |χ2 |2 χ2 = 0 (3.110) 2(1 + ε) 2 2

We recall that the slow flow system (3.102), and, hence (3.110) was derived under the assumption of 1:1 resonance between the LO and the NES, and so this model is valid only in the neighborhood of the 1:1 resonance manifold of the underlying Hamiltonian system. As in (3.102) the complex coordinates χ1 and χ2 describe the oscillations of the center of mass of, and the relative displacement between the LO and the NES, respectively. By successive differentiation and simple algebra, the above averaged system may be reduced to the following single modulation equation governing the slow flow of 1:1 resonance capture in the damped dynamics:

d j λ(1 + ε) j (1 + ε) d 2 χ2 2 χ χ |χ + + − | χ 2 2 2 2 dt 2 2 2 dt 2 +

jε (λχ2 − j |χ2 |2 χ2 ) = 0 4

(3.111)

This equation is integrable for λ = 0, but here we are interested in the damped case λ > 0. More precisely, we assume that λ ε, so we treat λ as an O(1) quantity. Equation (3.111) may be analyzed by the multiple scales approach (Nayfeh and Mook, 1995). To this end, we introduce the new time scales, τi = εi t, i = 0, 1 . . ., which are treated as distinct independent variables in the following analysis. Expressing the time derivatives in (3.111) as

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∂ ∂ d = +ε + O(ε2 ), dt ∂τ0 ∂τ1

d2 ∂2 ∂2 = + 2ε + O(ε2 ) ∂τ0 ∂τ1 dt 2 ∂τ02

(3.112)

substituting (3.112) into (3.111), and retaining only O(1) terms we derive the following first-order modulation equation,

∂ 2 χ2 ∂ j λ j 2 χ χ |χ (3.113) + + − | χ 2 2 2 2 =0 ∂τ0 2 2 2 ∂τ02 which possesses the following exact first integral of motion:

λ j ∂χ2 j 2 χ2 + χ2 − |χ2 | χ2 = M(τ1 , τ2 , . . .) + ∂τ0 2 2 2

(3.114)

In expressing the constant of integration M as function of the slow-scales τ1 , τ2 , . . ., we recognize that the first integral of motion (3.114) refers only to the first-order dynamics, i.e., it is only constant correct to O(1); mathematically, the slow variation of the first integral (3.114) is justified by the fact that the multiple scales of the problem are considered to be distinct and independent from each other. Hence, by (3.114) we allow slow variation of the dynamics, but at higher-order (slower) time scales. By the same reasoning, the equilibrium points, (τ1 , τ2 , . . .) of the first-order system (3.113) may be constant with respect to the first-order time scale, but may slowly vary with respect to higher-order (superslow) time scales; hence, the equilibrium points may depend on the higher-order superslow time scales τ1 , τ2 , . . . . These equilibrium points of the slow flow are computed by solving the following algebraic equation: j λ j + − ||2 = M(τ1 , τ2 , . . .) 2 2 2

(3.115)

Clearly, if an equilibrium is stable it holds that (τ1 , τ2 , . . .) =

lim χ2 (τ0 , τ1 , τ2 , . . .) < ∞

τ0 →+∞

whereas it holds that (τ1 , τ2 , . . .) =

lim

τ0 →−∞

χ2 (τ0 , τ1 , τ2 , . . .) < ∞

if that equilibrium is unstable. One can show that the first-order dynamical system (3.113) does not possess any limit sets besides equilibrium points [for instance by applying Bendixon’s criterion (Guckenheimer and Holmes, 1982; Wiggins, 1990)]. Since we will carry the analysis only up to O(ε), we omit from here on slow time scales of order higher than one and express the solution of (3.115) in the following polar form: (3.116) (τ1 ) = N(τ1 ) exp(j γ (τ1 ))

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Upon substituting into (3.115) and separating real and imaginary terms, we reduce the computation of the equilibrium points of the slow flow to λ2 Z(τ1 ) + Z(τ1 )[1 − Z(τ1 )]2 = 4|M(τ1)|2

(3.117)

where Z(τ1 ) ≡ N 2 (τ1 ). The number of solutions of equation (3.117) depends on |M(τ1 )| and λ. The function on the left-hand side can be either monotonous, or can have a maximum and a minimum. In the former case the change of |M(τ1 )| has no effect on the number of solutions and equation (3.117) provides a single positive solution. In the latter case, however, the change of |M(τ1 )| brings about a pair of saddle-node bifurcations, and hence multiple solutions. In order to distinguish between the different cases, we check the roots of the derivative with respect to Z(τ1 ) of the left-hand side of (3.117): + 1 + λ2 − 4Z + 3Z 2 = 0 ⇒ Z1,2 = [2 ± 1 − 3λ2 ] 3 (3.118) √ It follows that for λ < 1/ 3 there exist two additional real roots and √ a pair of saddle-node bifurcations, whereas at the critical damping value λ = 1/ 3 the two saddle-node bifurcation points coalesce forming the typical structure of a cusp. Extending these results to equation (3.117), if a single equilibrium exists, this equilibrium is stable with respect to the time scale τ0 . If three equilibrium points exist, two of them are stable nodes, and the third is an unstable saddle with respect to the time scale τ0 . Therefore, the O(1) dynamics is attracted always to a stable node. The characteristic rate of attraction of the dynamics near a node may be evaluated by linearizing equation (3.114), and considering the following perturbation of the dynamics near an equilibrium point: χ2 (τ0 , τ1 ) = (τ1 ) + δ(τ0 ),

|δ| ||

(3.119)

Upon substitution of (3.119) into (3.114) yields the following linearized equation,

λ j ∂δ j δ + δ − j ||2 δ − 2 δ ∗ = 0 (3.120) + ∂τ0 2 2 2 where asterisk denotes complex conjugate. Rewriting equation (3.120) as

λ j j ∂ 2 + + − j || δ = 2 δ ∗ ∂τ0 2 2 2

(3.121)

taking its complex conjugate and combining the two equations, we derive an expression that explicitly computes the evolution of the perturbation δ(τ0 ) (note that depends only on τ1 and not on the time scale τ0 ), ∂ 1 ∂2 2 2 +λ + (1 + λ − 4Z + 3Z ) δ = 0 ⇒ ∂τ0 4 ∂τ02 δ = δ0 exp[(−λ ± j ω)t/2]

(3.122)

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√ where ω = 3Z 2 − 4Z + 1. Solution (3.122) reveals that the linearized dynamics in the vicinity of the equilibrium points depends on λ and Z. The following possible alternatives are now√described. For relatively large values of damping above the critical value, λ > 1/ 3, there exists a single stable node in the O(1) dynamics. For Z > 1 or Z < 1/3 the attraction of the dynamics to that node is through oscillations [i.e., ω is real-underdamped cases], whereas for 1 > Z > 1/3 the attraction is through a decaying motion [i.e., ω is imaginary – overdamped case). √ For relatively small damping values, λ < 1/ 3, the situation is more complex, since there exist two additional real equilibrium points given by (3.118). For Z > 1 or Z < 1/3 the attraction of the dynamics to the stable node is oscillatory (underdamped cases), whereas for 1 > Z > Z1 or Z2 > Z > 1/3 the attraction is through a decaying motion (overdamped cases). For Z1 > Z > Z2 we obtain an unstable equilibrium, and the linearized model predicts exponential growth in the dynamics. In summary, as Z slowly decreases due to its dependency on the slow-time scale τ1 , and depending on the damping value λ, the O(1) √ dynamics undergoes qualitative changes (bifurcations). In particular, if λ > 1/ 3 we anticipate the dynamics to remain always stable, since in that case there exists a single slowly-varying at√ tracting manifold of the O(1) averaged flow. However, if λ < 1/ 3 the dynamics becomes unstable, in which case we expect that the O(1) averaged flow will make a sudden transition from one attracting manifold to another for slowly decreasing Z. In order to study this complicated damped transition, one should investigate the slow evolution of the equilibrium of the O(1) averaged flow (τ1 ). To this end, we consider the O(ε) terms in the multiple-scale expansion (3.111– 3.112):

∂ 2 χ2 ∂ j λ j 2 2 χ2 + χ2 − |χ2 | χ2 + ∂τ0 ∂τ1 ∂τ1 2 2 2

j ∂ λ j χ2 − |χ2 |2 χ2 + [λχ2 − j |χ2 |2 χ2 ] = 0 + (3.123) ∂τ0 2 2 4 We are interested in the behavior of the solution of the O(ε) averaged flow in the neighborhood of a stable equilibrium point, or equivalently, in the neighborhood of the damped NNM invariant manifold (τ1 ) = limτ0 →+∞ χ2 (τ0 , τ1 ). Therefore, by taking the limit τ0 → +∞ in equation (3.123) we obtain the following equation which describes the evolution of the dynamics at the slower time scale τ1 :

λ j j ∂ j + − ||2 + (λ − j ||2 ) = 0 (3.124) ∂τ1 2 2 2 4 In deriving this equation we take into account that on the slowly-varying, stable invariant manifold there is no dependence of the dynamics on τ0 , since (τ1 ) was defined previously as the equilibrium point of the O(1) averaged flow (3.113–3.114). Hence, the differential equation (3.124) describes the slow evolution of the stable equilibrium points of equation (3.113) (these are equilibrium points with respect to

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the fast time scale τ0 , but not with respect to the slow time scale τ1 and to slow time scales of higher orders, which, however are omitted from the present analysis). The slowly varying equilibrium (τ1 ) provides an O(ε) approximation to the damped NNM manifold of the dynamics of the system (3.98); this is an invariant manifold of the damped dynamics and can be regarded as the analytical continuation for weak damping of the corresponding NNM of the underlying Hamiltonian system (Shaw and Pierre, 1991, 1993). Rearranging equation (3.124) in the form

λ j ∂∗ j 2 ∂ + − j || − j 2 = − (λ − j ||2 ) (3.125) 2 2 ∂τ1 ∂τ1 4 and adding to it its complex conjugate, we obtain the following explicit expression for the slowly varying derivative of the equilibrium point of the O(1) slow flow: −λ + j ||2 − 3 ||4 − λ2 ∂ = (3.126) ∂τ1 2 1 + λ2 − 4 ||2 + 3 ||4 Using the polar representation, (τ1 ) = N(τ1 ) exp(iγ (τ1 )), and separating real and imaginary parts, equation (3.126) yields the following set of real differential equations governing the slow evolution of the magnitude and phase of the stable equilibrium points of the O(1) averaged flow (i.e., of the stable damped NNM manifolds), −λN ∂N = 2 ∂τ1 2 1 + λ − 4Z + 3Z 2 (Z − 3Z 2 − λ2 ) ∂γ = ∂τ1 2 1 + λ2 − 4Z + 3Z 2

(3.127)

where Z(τ1 ) ≡ N 2 (τ1 ). The first of equations (3.127) can be integrated exactly by quadratures to yield (1 + λ2 ) ln Z(τ1 ) − 4Z(τ1 ) + (3/2)Z 2 (τ1 ) = K − λτ1

(3.128)

where K is a constant of integration [it actually depends on the higher-order time scales τ2 , τ3 , . . ., but these are nor considered here as the analysis is restricted to O(ε)]. Expression (3.128) implicitly determines the evolution of Z(τ1 ) and, consequently, of N(τ1 ). The slow evolution of the phase γ (τ1 ) is described by the second of equations (3.127), and may be computed by direct integration once Z(τ1 ) is known; due to the implicit form of (3.128), however, this task cannot be performed analytically and requires a numerical solution. Essential information concerning the qualitative behavior of the solution may be extracted from relation (3.127) even √ without explicitly solving it. Indeed, for sufficiently strong damping, λ > 1/ 3, the denominator on the right-hand side

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Fig. 3.51 Response of the averaged system (3.111) in the regime of 1:1 resonance capture, for ε = 0.05, λ = 0.2 < 1/3, and initial conditions given by χ1 (0) = 0.7 + 0j , χ2 (0) = 0.7 + 0j .

terms is always positive, and the first equation describes a monotonous decrease of Z(τ1 ) towards zero with increasing τ1 . In other words, we conjecture that the slowly varying dynamics remains always on the in-phase√damped NNM manifold S11+. By contrast, for relatively weak damping, λ < 1/ 3, the velocity ∂Z/∂τ1 is a negative quantity for Z > Z1 , but becomes divergent as the limit Z → Z1 is approached from above. We cannot proceed to any statement regarding the sign of the velocity when the amplitude is in the range Z2 > Z > Z1 , as the equilibrium point is unstable there; therefore, we infer that as Z decreases below the critical amplitude Z1 the damped dynamics should be attracted to a NNM damped manifold distinct from S11+. This distinct manifold is a weakly nonlinear (linearized) branch of the damped NNM invariant manifold S11−. Of course, this conclusion is valid only for the averaged system (3.110–3.111), which was derived under the condition of 1:1 resonance capture. In the original system (3.98) attraction of the dynamics to other (i.e., different from 1:1) subharmonic or superharmonic resonance manifolds may take place, depending on the initial conditions and the system parameters. Similar averaging arguments could be used to study such more complex damped transitions. The previous analytical findings are illustrated by performing numerical simulations of the averaged system (3.111) for parameters ε = 0.05, λ = 0.2, and initial conditions χ1 (0) = 0.7 + 0j, χ2 (0) = 0.7 + 0j . The time evolution of the square of the modulation of the envelope of the NES response, |χ2 |2 , is depicted in Figure 3.51. Clearly, both the magnitude and frequency of the envelope modulation of the NES response tend to zero as the trajectory approaches the critical value

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Fig. 3.52 Real and imaginary parts of the complex modulation χ2 of the NES plotted against time, in the regime of 1:1 resonance capture for ε = 0.05, λ = 0.2, and initial conditions χ1 (0) = 0.7 + 0j , χ2 (0) = 0.7 + 0j .

Z1 =0.979. In the vicinity of this value, the trajectory jumps to the alternative stable attractor S11−. This point may be further illustrated using the three-dimensional plot depicted in Figure 3.52, where the real and imaginary parts of the complex envelope modulation of the NES, χ2 , are plotted in a parametric plot for increasing time. The damped trajectory of the envelope modulation of the NES starts from zero, gets attracted initially by the stable damped NNM manifold S11+, and then makes a transition (jump) to the weakly nonlinear, low-energy stable NNM manifold S11−. In order to check the validity of the asymptotic approximations, we performed direct simulations of the original set (3.98) (i.e., of the exact system before averaging) with the same initial conditions used for the plots of Figures 3.51 and 3.52; the result is presented in Figure 3.53. It is clear from this figure that the damped dynamics is initially attracted by the damped NNM manifold S11+, as evidenced by the in-phase 1:1 resonant oscillations of the NES and the LO, with nearly unit frequency. With diminishing amplitude of the NES, the critical amplitude is reached close to t ∼ 50 s, and a transition of the damped dynamics to a the out-of-phase linearized low-energy regime S11− takes place, with the motion localizing to the LO. This is in accordance with the predictions of the averaging analysis. The next simulation illustrates the dynamics of the averaged system (3.110) for the case of low damping (see Figure 3.54). The system parameters are chosen as ε = 0.05, λ = 0.03, and the initial conditions as χ1 (0) = 0.9 + 0j and χ2 (0) = 0.9 + 0j . Despite the low damping value, the qualitative behavior of the dynamics is similar to the previous case, although it takes much more time for the dynamics to escape away from the damped NNM invariant manifold S11+. It should be mentioned that, technically, the multiple-scale analysis developed above is not formally valid in this case, because the damping coefficient is of O(ε) and

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Fig. 3.53 Direct numerical simulation of the damped system (3.98) for parameters ε = 0.05, λ1 = 0, λ2 = 0.01, and initial conditions x(0) = v(0) = x(0) ˙ = 0 and x(0) ˙ = 0.7; the dynamics correspond to the analytical results of Figures 3.50 and 3.51.

Fig. 3.54 Response of the averaged system (3.111) in the regime of 1:1 resonance capture, for ε = 0.05, λ = 0.03, and initial conditions given by χ1 (0) = 0.9 + 0j and χ2 (0) = 0.9 + 0j .

not of O(1) as assumed in the analysis. To check, however, the applicability of the approximation in this case, the original system (3.98) was again simulated for parameters and initial conditions corresponding to the ones of the averaged model. The result is presented in Figure 3.55. It is difficult to judge whether any real transition (jump) occurs at t ∼ 480 s, but a gradual change of the NES frequency starts at this

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Fig. 3.55 Direct numerical simulation of the damped system (3.98) for parameters ε = 0.05, λ1 = 0, λ2 = 0.0015, and initial conditions x(0) = v(0) = x(0) ˙ = 0 and x(0) ˙ = 0.9; the dynamics correspond to the analytical result of Figure 3.54.

Fig. 3.56 Exact solution of the NES oscillation v(t) – solid line, superimposed to the analytically predicted envelope modulation [computed from (3.128)] – dotted line, up to the point of transition away from the damped NNM manifold S11+.

time instant and reveals escape from the regime of 1:1 resonance capture, thereby confirming the analytic findings. It is instructive to compare the result of the direct numerical simulation with the analytic expression (3.128) that computes approximately the modulation of the envelope of the response of the NES. The result of this comparison is presented in Figure 3.56. Expression (3.128) provides an accurate prediction for the modulation of the envelope of the NES response as long as the damped dynamics is in the

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1:1 resonance capture regime, i.e., before the escape from the damped NNM manifold S11+. Still, the description of the response is not complete since the initial conditions should also be taken into account. The averaging approach successfully describes the process up to times of O(1/ε), but is not suitable for later times since the limit τ0 → +∞ is irrelevant in this case, as the dynamics makes a transition away from the manifold S11+ at finite time. Fortunately, this latter problem is even easier to address. Indeed, if one is interested only in the behavior of the system up to a time scale of O(1), i.e., only during the initial transient regime of the motion, then to a first approximation it is possible to neglect all terms of O(ε) from the problem; this is shown below. To this end, we reconsider the damped two-DOF system (3.98) in the form x¨ + λ1 x˙ + λ2 (x˙ − v) ˙ + ω02 x + C(x − v)3 = 0 ˙ + C(v − x)3 = 0 ε v¨ + λ2 (v˙ − x)

(3.129)

with λ1 = 0, ω02 = 1, λ2 = ελ and C = ε, 0 < ε << 1. We introduce the change of variables, y1 = x + εv, y2 = x − v, where y1 describes the motion of the center of mass of the system, and y2 the relative motion between the LO and the NES. System (3.129) is then transformed into the following form: y1 + εy2 =0 1+ε y1 + εy2 + (1 + ε)λy˙2 + (1 + ε)y23 = 0 y¨2 + 1+ε

y¨1 +

(3.130)

The important advantage of system (3.130) compared to (3.129) is that the highest derivatives are now multiplied by unity, and the perturbation parameter is shifted to the remaining terms. This permits the application of standard perturbation techniques (such as the methods of multiple scales or averaging) to the analysis of the dynamics. To a first approximation, we retain only terms of O(1) in (3.130), rendering the resulting analytical transient approximations valid only up to times of O(1), i.e., only in the initial, strongly nonlinear regime of the motion: y¨1 + y1 = 0 (Early-time approximation)) (3.131) y¨2 + λy˙2 + y23 = −y1 More accurate approximation to the dynamics may be obtained by carrying the analysis beyond the O(1) approximation, for example, by analyzing the transformed system (3.130) by the method of multiple scales or averaging. This, however, would recover the averaging results of the previous analysis which are valid up to times of O(1/ε), so this option is not pursued further here. We note that the damping term in the second of equations (3.131) appears now as an O(1) quantity, so the approximation is justified only if λ = O(1); in the following simulations this condition is satisfied. Besides, the implicit assumption is that O(y1 ) = O(y2), i.e., that the amplitude of the oscillation of the center of

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mass is comparable to the amplitude of the relative oscillation between the LO and the NES. This assumption is correct only during the initial regime of the motion, as further evolution of the variables brings about diffentiation of the relative scaling between amplitudes, and the coupling term of order ε in the equation for v is not negligible anymore. Furthermore, equations (3.131) do not conserve energy in the absence of damping, which means that they are not suitable for describing the global dynamics of the system (3.129). We wish to develop analytical approximations of the early-time transient responses modeled by the dynamical system (3.131), subject to the general initial conditions y1 (0) = Y1 ,

y˙1 (0) = V1

y2 (0) = Y1 − Y2 ,

y˙2 (0) = V1 − V2 )

(3.132)

where Y1 and V1 are the initial displacement and velocity of the LO, respectively, and Y2 and V2 the corresponding initial conditions for the NES. Hence, correct up to a time scale of O(1), the system decomposes approximately to an unforced, undamped LO and a strongly damped and strongly nonlinear oscillator forced by the linear one; in essence, the approximately linear oscillation of the center of mass drives the strongly nonlinear relative oscillation between the LO and the NES. As in the previous analysis carried out in this section we focus only on the early-time response under the condition of 1:1 transient resonance capture (TRC). In terms of the approximate system (3.131), this means that the relative displacement y2 is assumed to perform fast oscillations with frequency nearly equal to unity, possibly modulated by a slowly-varying envelope. This paves the way for a slow-fast partition of the early-time dynamics. Solving the first of equations (3.131), we may reduce the approximate system to a single nonlinear differential equation: y1 = Y1 cos t + V1 sin t y¨2 + λy˙2 + y23 = −Y1 cos t − V1 sin t, y2 (0) = Y1 − Y2 ,

y˙2 (0) = V1 − V2

(3.133)

Restricting the analysis to the subset of initial conditions that correspond to the domain of attraction of the 1:1 resonance manifold (i.e., that provide the conditions for 1:1 TRC), we introduce the following slow flow partition of the dynamics: ψ(t) ≡ y2 (t) + j y˙2 (t) = ϕ(t) ej t

(3.134)

where ej t represents the fast oscillation of the system and ϕ(t) the corresponding slow modulation. Clearly, the original variables can be recovered using the relations y2 (t) = [ψ(t) − ψ ∗ (t)]/2j

and y˙2 (t) = [ψ(t) + ψ ∗ (t)]/2

(3.135)

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where the asterisk denotes complex conjugate. Introducing the expressions (3.134) and (3.135) into (3.133) we obtain y1 = Y1 cos t + V1 sin t ϕe ˙ j t + j ϕej t + = −Y1

(λ − 1) j (ϕej t + ϕ ∗ e−j t ) + (ϕej t − ϕ ∗ e−j t )3 2 8

ej t + e−j t ej t − e−j t − V1 2 2j

(3.136)

with initial condition ϕ(0) = (V1 − V2 ) + j (Y1 − Y2 ). To explore the slow flow dynamics (3.136) we perform time averaging with respect to the fast frequency, and obtain the following reduced, early-time slow flow system: ϕ˙ +

(λ + j ) 3j Y1 V1 |ϕ|2 ϕ = − − ϕ− , 2 8 2 2j

ϕ(0) = (V1 −V2 )+j (Y1 −Y2 ) (3.137)

This complex modulation equation governs approximately the slow dynamics of the early-time dynamics in the neighborhood of the 1:1 resonance manifold. To derive a set of real modulation equations, we employ the polar form representation, ϕ(t) = N(t) ej δ(t ) , and set separately real and imaginary parts equal to zero to derive a set of two real modulation equations governing the amplitude and the phase: N˙ + (λ/2)N = −(Y1 /2) cos δ + (V1 /2) sin δ N δ˙ + (N/2) − (3N 3 /8) = (Y1 /2) sin δ + (V1 /2) cos δ N(0) = (Y1 − Y2 )2 + (V1 − V2 )2 , tan δ(0) = (Y1 − Y2 )/(V1 − V2 )

(3.138)

From the physical viewpoint, the amplitude N(t) may be associated with a characteristic amplitude of the early-time nonlinear oscillations. The 1:1 damped invariant NNM manifold of the early-time dynamics corresponds to the set of equilibrium points of the slow flow (3.138) up to time scale of O(1). In order to determine this set we impose stationarity conditions N˙ = δ˙ = 0 yielding the following relations: N 6 − (8/3)N 4 + (16/9)(1 + λ2 )N 2 − (16/9)(Y12 + V12 ) = 0 cos δ = [V1 N(1 − 3N 2 /4) − λY1 N ]/(Y12 + V12 ) sin δ = [Y1 N(1 − 3N 2 /4) + λV1 N]/(Y12 + V12 )

(3.139)

The stability of an equilibrium point is specified by the nature of the eigenvalues of the Jacobian matrix of the linearization of system (3.138) evaluated at that equilibrium point:

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Fig. 3.57 Early-time dynamics: surface of equilibrium points A = f (N, λ) as solutions of the first of equations (3.139).

, µ1,2 = (1/2) −λ ± 4(9N/8 − 1/2N)(N/2 − 3N 3 /8)

(3.140)

Bifurcations of equilibrium points can be studied by considering the topology of the two-dimensional surface A = f (N, λ), where A = Y12 + V12 ; this is depicted in Figure 3.57. This surface is defined by the first of equations (3.139), and all equilibrium points lie on it. The folding lines on this surface form the boundaries separating the parameter regions where one or three equilibrium points exist. These are defined by the following equation: ∂ 6 N − (8/3)N 4 + (16/9)(1 + λ2 )N 2 − (16A/9) = 0 (Folding curves) ∂N (3.141) A projection of the fold to the plane (λ, A) is obtained by eliminating N between equation (3.141) and the first of equations (3.139). The point on the plane (λ, A) where the two folding curves intersect is computed as (λdeg = 1/31/2 , Adeg = 0.39506), and can be regarded as the most degenerate point of the surface of equilibrium points. In Figure 3.58 we depict the two folding curves projected onto the plane (λ, A) with the degenerate point of intersection also indicated. In the region between the two folding curves, the early-time, slow flow dynamical system (3.138) possesses three equilibrium points, whereas in the complementary region only one. Qualitative changes in the dynamics are anticipated as the folding curves are crossed transversely. It should be mentioned that the equilibrium points discussed above are the only limit sets of the equation (3.137). This fact may be rigorously proved with the help of Bendixon’s criterion (Guckenheimer and Holmes, 1982). That is why the classification of phase trajectories on the basis of the equilibrium points to which they are attracted is justified.

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Fig. 3.58 Early-time dynamics: projection of the fold of the surface A = f (N, λ) onto the plane (λ, A).

In order to study the evolution of the trajectories of the dynamical system in phase space, we consider again the energy-like quantity N(t), and provide the following alternative expression related to the responses of the original system (3.129): , 2 + (x(t) − v(t))2 (3.142) ˙ − v(t)) ˙ N(t) = (x(t) The same quantity was defined previously by the polar transformation of the slow complex amplitude, φ(t) = N(t) ej δ(t ) (3.143) and its temporal (slow) evolution is governed by the first of equations (3.138). Hence, it is possible to compare directly the dynamics of the exact system (3.129) and the averaged dynamics governed by the early-time slow modulation equations (3.137) or (3.138). To study quantitative changes in the dynamics of the exact system associated with bifurcations of equilibrium points of the reduced early-time dynamical system (3.138), we consider two case studies corresponding to different values of damping and initial conditions. First, we consider system (3.129) with damping λ2 = λ = 0.1, and initial conditions corresponding to A = 0.1. The additional damping coefficient is chosen to be zero in the following computations, i.e., λ1 = 0. This corresponds to a point inside the area defined by the folding curves in the (λ, A) plane (see Figure 3.58), which means that the reduced early-time system (3.138) possesses three equilibrium positions. These are computed as: (δ, N) = (0.109375, 0.345185) (Lower Focus) (δ, N) = (0.306884, 0.955293) (Middle Saddle) (δ, N) = (2.755332, 1.278643) (Upper Focus) Taking into account the previously introduced coordinate transformations, this specific case corresponds to the following two-parameter set of initial conditions of the original dynamical system (3.129)

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Fig. 3.59 Domains of attraction of the early-time averaged system (3.138) with λ = 0.1 and A = 0.1, superimposed to a grid of initial conditions (δ(0), N(0)) with each symbol indicating the level of efficiency of TET realized in the exact damped system (3.129): (+) EDM > 70%, (◦) 50% < EDM < 70%, () 30% < EDM < 50%, (*) 10% < EDM < 30%.

Y1 = 0.0,

V1 = 0.316228,

Y2 = Y1 − N(0) cos δ(0),

V2 = V1 − N(0) cos δ(0)

(3.144)

with parameters δ(0) and N(0). The exact system (3.129) was integrated for ε = 0.01 using each time a different initial point ( δ(0), N(0)) on the plane (δ, N). For each simulation we computed the corresponding energy dissipation measure – EDM, i.e., the percentage of initial energy of the system that is eventually dissipated by the damper of the NES damper. Our effort was to relate the effectiveness of TET in the original system (3.129), to the domains of attraction of the stable equilibrium points of the early-time averaged system (3.138). In Figure 3.59 we depict the domains of attraction of the upper and lower foci of the reduced system, superimposed to a grid of initial points ( δ(0), N(0)). The different symbols of the grid points are related to the percentage of total initial energy eventually dissipated by the NES for the corresponding initial condition. The picture clearly demonstrates that most efficient TET in the exact system (3.129) occurs if the dynamics of the early-time reduced system (3.138) is initiated inside or below the basin of attraction of the upper focus. A worthwhile caution is that the depicted basin of attraction is only approximate since it is computed only up to a time scale of O(1), and, hence, is valid only in the initial high-energy regime of the dynamics [since in the transformed system (3.130) only O(1) terms were retained in the analysis]. The following numerical simulations support the abovementioned conclusion.

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Fig. 3.60 Transient response of the exact system (3.129) for initial conditions, Y1 = 0.0, V1 = 0.316228, Y2 = −0.698246, V2 = 0.266712: (a) LO and NES displacements; (b) evolution of N(t) from the early-time averaged system (3.138).

In Figure 3.60a we depict the exact responses of the LO and the NES for an initial condition inside the basin of attraction of the upper focus of the averaged system ( N(0) = 0.7, δ(0) = 1.5), whereas in Figure 3.60b we depict the corresponding temporal evolution N(t) computed by integrating the reduced system (3.138). The results indicate that around t = 20 s the dynamics is in the domain of attraction of

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the upper focus. At t = 50–60 s the trajectory escapes this regime, and this coincides with a rather abrupt decrease of the amplitudes of oscillation. It is interesting to note that the LO is continuously oscillating with unit frequency, whereas the NES is oscillating with a lower frequency than that. This result may be understood in terms of the previously discussed invariant manifold approach; that is, an abrupt change of the dynamical regime is caused by the breakdown of the invariant manifold by the saddle-node bifurcation described previously. In Figure 3.60 one can also clearly distinguish the crossover between the initial transient and the slow evolution of the invariant manifold. Up to t = 50 s the oscillations of the dynamical flow around the upper focus are clear, whereas afterwards a rapid escape of the dynamics away from the domain of attraction of the upper focus takes place. In Figures 3.61a, b the corresponding plots for an initial condition inside the basin of attraction of the lower focus ( N(0) = 0.5, δ(0) = 0.4) are depicted. Around t = 50 s the dynamics is attracted by the lower focus, and the two oscillators are oscillating with approximately unit frequency. In this case, the amount of energy transferred from the LO to the NES is small. From the viewpoint of invariant manifolds, this case corresponds to the situation without bifurcation. One cannot expect efficient TET in this case, since the NES is almost not excited. The case when the NES is initially at rest, which is important from a practical viewpoint, is now considered. This corresponds to excitation of an impulsive orbit (IO). In order to study the EDM (the percentage of energy eventually dissipated by the NES) for given initial velocities of the LO and for varying damping values λ, we performed an additional series of numerical simulations. The results are depicted in Figure 3.62, superimposed to the folding boundary curves of Figure 3.58. The plot shows that most efficient TET occurs when the dynamics is initiated above and close to the upper folding boundary curve. This result can be related to previous results based on the damped NNM manifold approach, and demonstrates that the most efficient TET is realized when the dynamics is attracted to the stable damped NNM manifold, close to the point of bifurcation of that manifold. Otherwise, if the dynamics is attracted relatively far from the bifurcation point, it undergoes a few cycles of oscillation around the stable focus before breaking down (these cycles are, in fact, nonlinear beats). Therefore we conclude that with the NES initially at rest, TET is most efficient in the region close (but above) the upper folding boundary curve in the (λ, A) plane. This conclusion is supported by the simulations of Figures 3.63a, b depicting the transient √ responses of exact system (3.129) for initial conditions (Y1 = Y2 = V2 = 0, V1 = 0.25), and parameters λ = 0.25 and ε = 0.01. In this case, the EDM is over 50%. The plots demonstrate that around t = 35 s the dynamics of the system is attracted by the stable focus of the averaged system (corresponding to δ = 2.427 and N = 1.3105), with the amplitude of LO decreasing smoothly up to t = 100 s. The amplitude of the NES increases up to t = 45 s and then decreases abruptly. Figure 3.64 depicts√the response of system (3.129) for initial conditions Y1 = Y2 = V2 = 0, V1 = 0.065, and λ = 0.3, ε = 0.01. The selected initial conditions and damping value correspond to an initial point lying below the lower folding boundary curve of the (λ, A) plane. The EDM is below 20% in this case. The plots

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.61 Transient response of the exact system (3.129) for initial conditions, Y1 = 0.0, V1 = 0.316228, Y2 = −0.194709, V2 = −0.144303: (a) LO and NES displacements; (b) evolution of N(t) from the early-time averaged system (3.138).

demonstrate that around t = 10 s the dynamics is attracted by the basin of attraction of the focus corresponding to δ = 0.9055 and N = 0.2556. The amplitude of the LO decreases smoothly, while the amplitude of the NES remains almost constant. It is interesting to note that both oscillators are oscillating with unit frequency, implying the continuity (i.e., lack of bifurcation) of the invariant manifold.

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Fig. 3.62 Efficiency of TET as expressed by EDM (%), for varying damping values and initial conditions, with the NES being initially at rest; the fold predicted by the early-time averaging analysis is also shown.

The results presented above demonstrate that the problem of identifying appropriate initial conditions for enhanced TET may be reduced to the problem of predicting of the domains of attraction for a limited number of equilibrium points of the early-time averaged slow flow. This latter approximation does not coincide with the damped NNM manifold approach discussed in the beginning of this section, and the resulting two-dimensional slow phase plane (N, δ) does not coincide with that of the damped NNM manifold at later stages of the dynamical process. Nevertheless, equilibrium points in this slow phase plane obviously correspond to damped NNM manifolds, which provide a direct connection between these two approaches. In other words, the approach developed above enables the determination of the specific equilibrium point eventually reached by the dynamics of the system for the majority of initial conditions. Once this question is answered, the dynamics and efficiency of TET may be assessed using the damped NNM manifold framework. Consequently, the combination of the two methods leads to the analytical modeling of TET dynamics over its entire time span, and answers the question of robustness of TET to changes in initial conditions. The numerical results presented in this section clearly support the predictions of these analytical methodologies. In summary, in this section we analyzed the damped dynamics of the essentially nonlinear two-DOF system (3.98) or (3.129) under conditions of 1:1 resonance capture. The resulting fundamental TET was studied by considering the damped NNM

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. √ 3.63 Transient response of the exact system (3.129) for initial conditions, Y1 = 0.0, V1 = 0.25, Y2 = 0.0, V2 = 0.0: (a) LO and NES displacements; (b) evolution of N(t) from the early-time averaged system (3.138).

manifolds of the slow flow, and by analyzing the attraction of the dynamics on these manifolds, as well as by studying damped transitions between damped NNM manifolds. More importantly, we demonstrated that the rate of energy dissipation by the NES, i.e., TET efficiency, is closely related to the bifurcation structure of the NNM

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Fig. √ 3.64 Transient response of the exact system (3.129) for initial conditions, Y1 = 0.0, V1 = 0.065, Y2 = 0.0, V2 = 0.0: (a) LO and NES displacements; (b) evolution of N(t) from the early-time averaged system (3.138).

invariant manifolds. Indeed, it was found numerically that with the NES initially at rest (i.e., when an impulsive orbit is excited), optimal TET is realized when the damped dynamics is attracted by a stable in-phase damped NNM invariant manifold, close to the point of bifurcation of that manifold, or equivalently, close (but above) the upper folding boundary curve in the (λ, A) plane depicted in Figure 3.62; this

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folding curve was computed by performing an analysis of the early-time dynamics. This naturally leads us to the more detailed study of the conditions for optimal fundamental TET in system (3.98), which is performed in the next section.

3.4.2.5 Conditions for Optimal Fundamental TET In Section 3.3.5 we discussed some topological features of the Hamiltonian dynamics of the two-DOF system (3.98) with no damping terms. Focusing in the intermediate-energy region close to the 1:1 resonance manifold of the Hamiltonian system we studied the topological changes of intermediate-energy impulsive orbits (IOs) for varying energy (see Figure 3.39). Specifically, we found that above the critical value of energy-like variable r = rcr (see Section 3.3.5) the topology of intermediate-energy IOs changes drastically, as these make much larger excursions into phase space, resulting in continuous strong energy exchanges between the LO and the nonlinear attachment in the form of strong nonlinear beats. We also mentioned in Section 3.3.5 that this critical energy of the Hamiltonian system may be directly related to the energy threshold required for TET in the weakly damped system (as discussed in Section 3.2 and Figure 3.4). In this section we study the intermediate-energy dynamics of the weakly damped system (3.98), x¨ + λ1 x˙ + λ2 (x˙ − v) ˙ + ω02 x + C(x − v)3 = 0 ˙ + C(v − x)3 = 0 ε v¨ + λ2 (v˙ − x)

(3.98)

in an effort to formulate conditions for optimal fundamental TET; as usual we assume that 0 < ε 1. It follows that our study will be necessarily restricted to the neighborhood of the 1:1 resonance manifold of the underlying Hamiltonian system, and the damped dynamics will be studied under the condition of 1:1 resonance capture. However, the ideas and techniques presented here can be extended to study optimal conditions for the more general case of m:n subharmonic TET. To initiate our analysis, we set ω02 = 1 in (3.98), and consider the following ansatz for the damped responses close to the 1:1 resonance manifold of the Hamiltonian system (i.e., for ω ≈ 1): x(t) ≈

a1 (t) cos [ωt + α(t)] , ω

v(t) ≈

a2 (t) cos [ωt + β(t)] ω

(3.145)

Substituting (3.145) into (3.98) and averaging out all frequency components with frequencies higher than ω, we derive a system of four modulation equations governing the slow evolution of the amplitudes a1 (t), a2 (t) and phases α(t), β(t) of the two oscillators; this defines the slow flow of system (3.98) in the neighborhood of the 1:1 resonance manifold. In Section 3.3.5 we found that the slow flow of the corresponding undamped system is fully integrable and can be reduced to the sphere (R + ×S 1 ×S 1 ). Motivated

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by these results, we√introduce the phase difference φ = α − β, the energy-like 2 variable r 2 = a12 + ( εa √2 ) , and the angle ψ ∈ [−π/2, π/2] defined by the relation tan[ψ/2 + π/4] = a1 / εa2 . Enforcing the condition of weak damping by rescaling the damping coefficients according to λ1 → ελ1 , λ2 → ελ2 , and expressing the slow flow equations in terms of the new variables, we reduce the slow flow of the damped dynamics to the sphere (r, φ, ψ) ∈ (R + × S 1 × S 1 ): . rr˙ = − ελ1 (1 + sin ψ) + ελ2 (1 + ε) − (1 − ε) sin ψ − 2ε1/2 cos ψ cos ϕ 2 ψ˙ =

ϕ˙ =

√ −3Cr 2 (1 + ε) − (1 − ε) sin ψ − 2 ε cos ψ cos ϕ sin ϕ 8ε3/2 λ2 ελ1 cos ψ + (1 − ε) cos ψ − 2ε1/2 sin ψ cos ϕ − 2 2 √ 1 3Cr 2 − (1 + ε) − (1 − ε) sin ψ − 2 ε cos ψ cos ϕ 2 16ε2

sin ϕ 1/2 sin ψ cos ϕ − ε1/2 λ2 × (1 − ε) − 2ε cos ψ cos ψ

(3.146)

We note that when λ1 = λ2 = 0 the slow flow reduces to the integrable system (3.89) on a two-torus possessing the first integral (3.90). For non-zero damping, however, the slow flow dynamics is non-integrable and the dimensionality of the system (3.146) cannot be further reduced. In Figure 3.65 projections of damped IOs to the three-dimensional space (r, φ, ψ) ∈ (R + ×S 1 ×S 1 ) are depicted for three different initial energy levels; these results were obtained by direct numerical simulations of the damped system (3.98) subject to initial conditions corresponding to IOs, and can be directly compared to the plots of Figure 3.37 which depict isoenergetic projections of the underlying Hamiltonian dynamics. In the damped case, however, instead of the equilibrium points corresponding to NNMs on branches S11± we get in-phase and out-of-phase damped NNM invariant manifolds (Shaw and Pierre, 1991, 1993). For the case of large initial energy there is an initial transient phase (denoted as Stage I in Figure 3.65b) as the orbit gets attracted to the damped NNM manifold S11+; this is followed by the slow evolution of the damped motion along S11+ as energy decreases due to damping dissipation, with the motion predominantly localized to the NES as evidenced by the fact that ψ(t) ≈ −π/2 (Stage II in Figure 3.65b). Finally, the damped NNM S11+ becomes unstable, and the dynamics makes a final transition to the weakly nonlinear (linearized) NNM manifold S11−; the resulting out-of-phase oscillations are localized predominantly to the LO, as evidenced by the fact that limt →∞ ψ(t) = π/2 (Stage III in Figure 3.65b). TET in this case occurs predominantly during Stage I (TET through nonlinear beat) and Stage II (fundamental TET). For lower initial energy (i.e., in the intermediate energy level), the initial transients of the dynamics during the attraction to S11+ possess larger amplitudes

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.65 Phase space projection of damped IOs for ε = 0.1, C = 2/15, and λ1 = λ2 = 0.1: (a) projection definition, (b) r(0) = 2.0, (c) r(0) = 1.0, (d) r(0) = 0.5.

(Stage I, Figure 3.65c), leading to an increase of the resulting TET due to nonlinear beats; in later times, Stages II and III of the dynamics are similar to the corresponding ones of the higher-energy case. Compared to the previous case, TET is enhanced, especially during the initial transients of the motion where the LO and the NES undergo larger-amplitude nonlinear beats. Qualitatively different dynamics is observed when the initial energy is further decreased; this can be noted from the projection of Figure 3.65d, where the low-energy motion rapidly localizes to the LO as the dynamics gets directly attracted by the weakly nonlinear branch of the damped NNM manifold S11−, and, as a result, TET drastically diminishes. In essence, for this low energy value only Stage III of the dynamics is realized. An analytical study of the stability of the damped NNM manifolds S11±, which, as we showed, affects the damped transitions of system (3.98) and the resulting TET, is carried out in Quinn et al. (2008). In that work a detailed study of TET efficiency as judged by the time required by the NES to passively absorb and dissipate a significant amount of initial energy of the LO is performed as well. A representative

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Fig. 3.66 Damped IO simulations at various initial energy levels: projections of the damped motions onto the unit disk for (a) r(0) = 0.45, (b) r(0) = 0.46, (c) r(0) = 0.47, (d) r(0) = 0.50, (e) r(0) = 0.75; (f) time T required for decay of r(0) by a factor of e−1 as a function of r(0), circles refer to the projections (a–e).

result of this study is presented in Figure 3.66, depicting the time T required for the initial value of the energy-like variable, r(0), to decay by a factor of e, when the motion is initiated on an IO: r(T ) = r(0) e−1 (3.147) We note that for a classical viscously damped SDOF linear oscillator with damping constant ελ, the corresponding time interval T would be equal to ελ/2. The numerical results depicted in Figure 3.66 were derived for parameters ε = 0.05, λ1 = 0, λ2 = 0.2 and C = 2/15, and the damped IOs in Figures 3.66a–e are only depicted in the time interval 0 < t < T . We note that as we increase r(0) from 0.46 to 0.47 there is a drastic reduction in T , signifying drastic enhancement of TET efficiency. This is associated with a sudden ‘excursion’ of the damped IO in the projection of the phase space, as the dynamics makes a transition from a motion that is predominantly localized to the LO (Figures 3.66a, b) to a motion where large relative motion

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.67 Percentage of energy dissipated in system (3.148) when intermediate-energy damped IOs are excited (ε = 0.05, C = 1 and ελ = 0.005): solid lines correspond to excitation of specific periodic IOs, and the dashed line indicates the energy remaining in the system at t = 25 s.

between the LO and the NES takes place (see Figure 3.66c); this, in turn leads to enhanced TET through nonlinear beats. It is interesting to note that the sudden jump in TET efficiency in Figure 3.66 occurs in the intermediate-energy regime, in the neighborhood of the 1:1 resonance manifold of the Hamiltonian system. In this regime of the dynamics the slow flow model (3.146) is valid, so it can be used to study the conditions for optimal TET efficiency. An analytic study of the conditions for optimal TET through the excitation of intermediate-energy IOs is carried out in Sapsis et al. (2008), and elements of this study will be reproduced here. Although the study is carried out under the assumption of 1:1 resonance capture, and is based on CX-A, the analysis of the resulting averaged slow flow is different than that carried out for the underlying Hamiltonian system. Hence, we reconsider the two-DOF system (3.98) with λ1 = λ2 = ελ and ω02 = 1, x¨ + ελx˙ + ελ(x˙ − v) ˙ + x + C(x − v)3 = 0 εv¨ + ελ(v˙ − x) ˙ + C(v − x)3 = 0

(3.148)

with initial conditions corresponding to excitation of an impulsive orbit (IO), v (0) = v˙ (0) = x (0) = 0 and x(0) ˙ = X, and 0 < ε 1. In Figure 3.67 we depict the dissipation of instantaneous energy in this system with ε = 0.05, C = 1

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and ελ = 0.005 (these parameter values will be used in the remainder of this section, unless stated otherwise), when damped IOs are excited. In accordance with previous findings of this Chapter, we find that strong energy dissipation, i.e., strong TET, is realized in the intermediate energy region and more specifically in the neighborhood of the 1:1 resonance manifold of the underlying Hamiltonian system (we note that the FEP of the corresponding undamped system with the positions of periodic IOs indicated, is depicted at Figure 3.20). Moreover, optimal TET, as judged by the strongest energy dissipation in the least possible time in the plot of Figure 3.67, is realized for initial impulses X (i.e., initial energies) in the range between the periodic IOs U 65 and U 76; from the FEP of Figure 3.20, we note that these periodic IOs are close to the energy level of a saddle-node bifurcation of the linearized and strongly nonlinear components of the backbone branch S11−. At this energy level, an unstable hyperbolic periodic orbit is generated on the strongly nonlinear component of S11−. As shown below, it is the hom*oclinic orbit of this hyperbolic periodic orbit that affects the topology of nearby IOs and defines conditions for optimal TET in the weakly damped system. This observation is in accordance with the discussion of Section 3.3.5 and the results depicted in Figures 3.39 and 3.66, indicating that above a critical energy level the topology of the IOs changes drastically, with IOs making large excursions in phase space (actually, this critical energy level in the Hamiltonian system may be defined as the energy where with the IO coincides with the hom*oclinic orbit – Figure 3.39d). Finally, we note that these observations are also in accordance with the findings of the approach based on damped NNM invariant manifolds (see Section 3.4.2.4), where it was noted that the most efficient TET is realized when the damped dynamics is attracted to a stable damped NNM manifold, close to the point of bifurcation of that manifold. The analytical study of conditions for optimal fundamental TET is carried out by applying the CX-A technique to system (3.148) under condition of 1:1 internal resonance between the LO and the NES. Moreover, only intermediate-energy IOs are considered, focusing to those lying close to the 1:1 resonance manifold with dominant (fast) frequency ω ≈ 1 (see the FEP of Figure 3.20). Applying the usual complexification, ψ1 (t) = v(t) ˙ + j v(t) ≡ φ1 (t) ej t ,

ψ2 (t) = x(t) ˙ + j x(t) ≡ φ2 (t) ej t

and performing averaging with respect to the fast term ej t , we derive the following set of complex modulation equations, φ˙ 1 + (j/2)φ1 + (λ/2) (φ1 − φ2 ) − (3j C/8ε) |φ1 − φ2 |2 (φ1 − φ2 ) = 0 φ˙ 2 + (ελ/2) (2φ2 − φ1 ) + (3j C/8) |φ1 − φ2 |2 (φ1 − φ2 ) = 0

(3.149)

with initial conditions φ1 (0) = 0 and φ2 (0) = X. Introducing the new complex variables, u = φ 1 − φ2 φ1 = (u + w)/(1 + ε) (3.150) ⇔ φ2 = (w − εu)/(1 + ε) w = εφ1 + φ2

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we express system (3.149) as u˙ +

3(1 + ε)j C 2 w − εu (1 + ε)λ u+w |u| u + j u− − ελ =0 2 8ε 2(1 + ε) 2(1 + ε)

w˙ + j ε

u+w w − εu + ελ =0 2(1 + ε) 2(1 + ε)

(3.151)

with initial conditions u (0) = −X and w (0) = X. Hence, we have reduced the problem of studying intermediate-energy damped IOs of the initial system of coupled oscillators (3.148) to the above system of first-order complex modulation equations governing the slow flow close to the 1:1 resonance manifold. These equations are valid only for small- and moderate-energy IOs, i.e., for initial conditions X < 0.5 (see Figure 3.67), since above this level the fast frequency of the response depends significantly on the energy level and the assumption ω ≈ 1 is violated. Since we are interested in the study of optimal energy dissipation by the NES, we shall now derive expressions for the various energy quantities in terms of the complex modulations u and w. These expressions will be further exploited in an effort to study conditions on u and w that optimize TET. Thus, for computing the instantaneous total energy stored in the LO we derive the expression: EL (t) ≡ ≈

1 2 [x (t) + x˙ 2 (t)] 2 |w − εu|2 1 1 (Im[φ2 ej t ])2 + (Re[φ2 ej t ])2 = |φ2 |2 = 2 2 2 (1 + ε)2 (3.152)

The instantaneous energy stored in the NES is approximately evaluated as: 1 C ENL (t) = εv˙ 2 (t) + [x(t) − v(t)]4 2 2 1 C ε(Im[φ1 ej t ])2 + (Re[vej t ])4 ≈ 2 2

1 u + w jt 2 C jt 4 (3.153) ε Im e = + (Re[ue ]) 2 1+ε 2 Finally, the most important energy measure as far as our analysis is concerned will be the energy dissipated by the damper of the NES, approximated as: t EDISS (t) =

t ελ [x˙ (t) − v˙ (t)] dt ≈ ελ

(Re[uej t ])2 dt

2

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

= ελ

215

t . (Re [u])2 cos2 t + (Im [u])2 sin2 t − Re [u] Im [u] sin 2t dt 0

t = ελ

1 + cos 2t 1 − cos 2t + (Im [u])2 − Re [u] Im [u] sin 2t dt 2 2

(Re [u])2 0

(3.154)

Omitting terms with fast frequencies greater than unity from the integrand (this is consistent with our analysis based on averaging with respect to the fast frequency equal to unity), the above integral can be approximated by the following simple expression: ελ EDISS (t) ≈ 2

t

/

ελ (Re [u]) + (Im [u]) dt = 2 2

2

t |u (t)|2 dt

(3.155)

Hence, within the approximations of the analysis, the energy dissipated by the NES is directly related to the modulus of u (t) which characterizes the relative response between the LO and the NES. It follows, that enhanced TET in system (3.148) is associated with the modulus |u (t)| attaining large amplitudes, especially during the initial phase of motion where the energy is at its highest. Returning to the slow flow (3.151), the second modulation equation can be solved explicitly as follows:

t ε ε (j + λ) t ε(ελ − j ) w(t) = X exp − + exp − (j + λ)(t − τ ) u (τ ) dτ 2 2 (1 + ε) 2 0

(3.156) which, upon substitution into the first modulation equation yields: j + λ ε2 + (1 + ε)2 3j C(1 + ε) 2 |u| u + u˙ − u 8ε 2(1 + ε)

ε (j + λ) t ελ − j X exp − = 2(1 + ε) 2

(ελ − j ) +ε 2 (1 + ε)

2 t 0

ε exp − (j + λ)(t − τ ) u (τ ) dτ, 2

u(0) = −X (3.157)

This complex integro-differential equation governs the slow flow of a damped IO in the intermediate-energy regime, as it is equivalent to system (3.151). It follows that the above dynamical system provides information on the slow evolution of the damped dynamics close to the 1:1 resonance manifold.

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Fig. 3.68 Slow flow (3.151) of a damped IO in the intermediate-energy regime of Figure 3.67.

In Figure 3.68 we present a typical solution of (3.151) depicting the slow flow of a damped IO in the upper intermediate-energy regime of Figure 3.67. The initial ‘wiggles’ in the slow flow represent the initial attraction of the IO dynamics by the damped NNM manifold S11+, and correspond to initial nonlinear beats in the full response. Although short in duration, the energy dissipated by the NES in the initial regime of nonlinear beats can be quite significant as discussed below. In Figure 3.69 we examine the dynamics of the averaged system (3.151) [or equivalently (3.157)] over the entire intermediate-energy regime of damped IOs. Starting from relatively high energies (i.e., the highest value of impulsive magnitude X, Figure 3.69a), the initial regime of nonlinear beats (corresponding to the attraction of the dynamics to the stable damped NNM invariant manifold S11+) leads to strong energy exchanges between the LO and the NES; as the dynamics settles to S11+ the energy exchanges diminish and slow energy dissipation is noted in both oscillators; finally the dynamics makes the transition to the linearized damped NNM submanifold S11− at the later stage where nearly the entire energy of the system has been dissipated. We conclude that in the upper region of the intermediate-energy regime TET is relatively weak as the impulsively excited LO retains most of its energy throughout the oscillation. As the impulsive energy decreases (see Figures 3.69b, c) the initial regime of nonlinear beats expands and stronger energy exchanges between the impulsively forced LO and NES take place; moreover, the dynamics instead of settling to S11+, proceeds to make a transition to the weakly nonlinear branch of S11−. These features of the slow dynamics enhance TET in the system, as judged by the efficient dissipation of energy in both oscillators. Overall, optimal energy dissipation, and hence optimal TET, is realized

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Fig. 3.69 Slow flow (3.151) or (3.157) of damped IOs in the intermediate-energy regime: (a) X = 0.30 (upper regime).

in Figure 3.69d, where the initial regime of beats is replaced by a (super)slow oscillation during which the entire energy of the LO gets transferred to the NES over a single half-cycle; some of this energy gets ‘backscattered’ to the LO at a later stage of the motion, during some low-amplitude nonlinear beats, but the major amount of energy gets dissipated during the initial half-cycle energy transfer where the energy of the system is at its highest; this provides the condition for optimal TET in this system, and corresponds to the ‘ridge’ in Figure 3.67 at X ≈ 0.11. A slight decrease of the impulsive magnitude X changes qualitatively the slow dynamics, as both oscillators now settle into linearized responses and negligible TET takes place; in this case the slow dynamics gets directly attracted to the weakly nonlinear branch of S11−. Hence, the slow dynamics of the damped IOs in the intermediate-energy regime is quite complex. Indeed, based on the qualitative features of the damped IO dynamics we may divide the intermediate-energy regime of Figure 3.67 into three distinct subregimes; these can be distinguished by the features of the slow flow dynamics (3.157) during the initial, highly energetic stage of the impulsive motion where most TET is realized. In the upper subregime corresponding to higher impulsive magni-

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Fig. 3.69 Slow flow (3.151) or (3.157) of damped IOs in the intermediate-energy regime: (b) X = 0.19.

tudes (see Figures 3.69a–c) TET through nonlinear beats takes place. The middle subregime (see Figure 3.69d) is the regime of optimal TET, and is governed by the most complex dynamics, since the initial slow flow dynamics consists of a single ‘super-slow’ half-cycle during which the entire energy of the LO gets transferred to the NES. Hence, it appears that the initial nonlinear beats realized in the upper subregime degenerate to a single ‘super-slow’ half-cycle of the slow flow as the middle subregime is approached. As shown in the following analysis, the dynamical mechanism that leads to this ‘super-slow’ degeneration of the slow dynamics in Figure 3.69d is the hom*oclinic orbit of the unstable damped NNM on S11− that is generated by the saddle-node bifurcation at the critical energy level between the periodic IOs U 65 and U 76 in the FEP of Figure 3.20. Finally, the lower subregime is characterized by linearized motion predominantly localized to the LO, with complete absence of nonlinear beats and negligible TET. We note that this disussion can be directly related to the analysis presented in Section 2.3 where the dynamics of a two-DOF system of a different configuration (with a grounded NES) was studied asymptotically in the neighborhood of the 1:1 resonance manifold of the dynamics. Indeed, the hom*oclinic orbit of the unstable

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219

Fig. 3.69 Slow flow (3.151) or (3.157) of damped IOs in the intermediate-energy regime: (c) X = 0.12.

undamped NNM on S11− of the Hamiltonian system (3.148) studied in the present section, is similar to the hom*oclinic loop appearing in Figure 2.10 of the Hamiltonian system (2.31). As shown in Section 2.3, when sufficiently weak damping is added to the system [refer to condition µ > ν in equation (2.47)] the Hamiltonian hom*oclinic loop is perturbed (to first order as in Figure 2.11a, and to second order as schematically shown in Figure 2.13). Hence, following a similar reasoning, we can relate the results regarding TET efficiency of this section to the damped system (2.41) with grounded NES, by relating the dynamics in the neighborhood of the perturbed hom*oclinic orbit of that system to TET efficiency. The previous discussion and results provide ample motivation for focusing in the initial, highly energetic regime of the slow flow dynamics (3.151) [or equivalently (3.157)], as this represents the most critical stage for TET. Hence, we consider the modulation equation (3.157) and restrict the analysis to the initial stage of the dynamics. Mathematically, we will be interested in the dynamics up to times of O(1/ε1/2 ), and for initial conditions (impulses) of order X = O(ε1/2 ). Under these assumptions we consider the integral term on the right-hand side of (3.157)

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Fig. 3.69 Slow flow (3.151) or (3.157) of damped IOs in the intermediate-energy regime: (d) X = 0.11 (optimal TET).

and express it as follows:

I ≡ε

(ελ − j ) 2 (1 + ε)

1 =ε 2(1 + ε)

2 t

ε exp − (j + λ) (t − τ ) u (τ ) dτ 2

2 t

ε exp − (j + λ)(t − τ ) u (τ ) dτ + O(ε2 ) 2

When t = O(ε−1/2), we have also that (τ − t) = O(ε−1/2); it follows that by expanding the exponential in the integrand in Taylor series in terms of ε, the integral I can be approximated as

1 I ≈ε 2 (1 + ε)

2 t u (τ ) dτ + O(ε3/2 ) 0

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221

Fig. 3.69 Slow flow (3.151) or (3.157) of damped IOs in the intermediate-energy regime: (e) X = 0.09 (lower regime).

or, by invoking the mean value theorem of integral calculus, as I ≈ 2−2 (1 + ε)−2 εt u (t0 ) for some t0 in the interval 0 < t0 < t. Given that t = O(ε−1/2 ) and u (t0 ) = O(X) = O(ε1/2 ), we prove that for times smaller than O(ε−1/2 ), the integral is ordered as I = O (ε), and hence is a small quantity. Taking this result into account, and introducing the variable transformations u = ε 1/2 z

and X = ε 1/2 Z

to account for the scaling of the initial condition (impulse) X = O(ε1/2 ), we express the modulation equation (3.157) in the form z˙ −

jZ j +λ 3j C 2 |z| z+ z=− +O(ε, ε 1/2 λ), 8 2 2

z(0) = −Z,

t up to O(ε−1/2 ) (3.158)

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3 Nonlinear TET in Discrete Linear Oscillators

where the variable z and initial condition Z are assumed to O(1) quantities, unless otherwise noted. Finally, introducing the rescalings z→ the new notation,

4 3C

1/2

z,

w→

3C B =− 4

4 3C

1/2 w

(3.159a)

1/2 (3.159b)

Z

ˆ the system is and the additional scaling for the damping coefficient, λ = ε 1/2 λ, brought into the following final form, z˙ −

jB j + ε1/2 λˆ j 2 |z| z+ z= +O(ε), 2 2 2

z(0) = B,

t up to O(ε−1/2) (3.160)

and all quantities other than the small parameter ε are assumed to be O(1) quantities. The complex modulation equation (3.160) provides an approximation to the initial slow flow dynamics, and is valid formally only up to times of O(ε−1/2). In Figure 3.70 we compare the initial approximation of the slow flow (3.158) or (3.160) and the full slow flow (3.151) or (3.157), by computing the predicted energy dissipated in the intermediate-energy regime of damped IOs by the two approximations. This comparison clearly validates the slow flow approximation (3.160) in the intermediate-energy level of interest in this study. Introducing the polar transformation, z = Nej δ , substituting into (3.160) and separating real and imaginary parts, this system can be expressed in terms of the following two real modulation equations: B ε1/2 λˆ N = sin δ + O(ε), N(0) = B N˙ + 2 2 1 1 B cos δ + O(ε), δ(0) = 0 δ˙ + − N 2 = 2 2 2N

(3.161)

These equations govern the slow evolutions of the amplitude N and phase δ of the complex modulation z of the IO, during the initial (high-energy) regime of the dynamics. In Figure 3.71 we depict the initial regime of slow flow dynamics (3.160–3.161) for ε = 0.05, λˆ = 0.4472 and three different normalized impulses (initial conditions) B. For B above the critical level Bcr (λˆ = 0.4472) ≈ 0.3814, the slow flow model (3.160) predicts large excursion of the damped IO in phase space. In fact, after executing relatively large-amplitude transients, the orbit is being ultimately attracted by the stable in-phase damped NNM S11+ (which, within the order of approximation of the present analysis, appears as a fixed point, although as shown in previous sections in actuality it ‘drifts’ slowly, i.e., it depends on higher order time scales); these initial transients correspond to the nonlinear beats (the ‘wiggles’) observed in the initial stage of the full slow flow model (3.157) in the upper subregime

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223

Fig. 3.70 Percentage of energy dissipated when intermediate-energy damped IOs are excited (ε = 0.05, C = 1 and λε 1/2 λˆ = 0.1): (a) full slow slow (3.151) or (3.157), (b) approximation of the slow flow in the initial stage of the dynamics, (3.158) or (3.160).

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3 Nonlinear TET in Discrete Linear Oscillators

of the intermediate-energy regime (see Figures 3.68 and 3.69a–c). Note, that since the model (3.160–3.161) is valid only for the initial stage of the slow flow dynamics, it cannot predict the eventual transition of the dynamics from S11+ to S11− in the later, low-energy (linearized) stage of the oscillation (nor the slow ‘drift’ of the dynamics on these damped invariant manifolds). ˆ there is a significant qualitative change For B below the critical level Bcr (λ), in the dynamics as the IO executes small-amplitude oscillations, and the dynamics is being attracted to the weakly nonlinear out-of-phase damped NNM S11−; this corresponds to the weakly nonlinear dynamics realized in the lower subregime of the intermediate-energy range (see Figure 3.69e). It follows, that the critical orbit that separates these two qualitatively different regimes of the dynamics is a ‘perturbed hom*oclinic orbit’ realized for B = Bcr (λˆ ). This special orbit is formed by one of the branches resulting from the ‘break-up’ of the Hamiltonian hom*oclinic loop when weak damping is added to the system. We recall that the hom*oclinic loop of the unstable undamped NNM S11− is generated due to a saddle-node bifurcation that occurs at an energy level between the periodic IOs on branches U 65 and U 76 in the FEP of Figure 3.20. The damped perturbed hom*oclinic orbit appears as the initial ‘super-slow’ half-cycle in the plot of Figure 3.69d, and corresponds to the case of optimal TET in the system. In Figure 3.71 we depict the portion of this damped hom*oclinic orbit corresponding to the solution of the slow flow dynamical systems (3.160–3.161) for the given initial condition z(0) = B; we note that these are peculiar forms of dynamical systems, as the initial conditions appear also as excitation terms on their right-hand sides. In what follows, the damped perturbed hom*oclinic orbit will be analytically studied, in an effort to analytically model the optimal TET regime depicted in Figure 3.69d. This analysis is analogous to, but different from the analytical study performed in Section 2.3 concerning the ‘breakup’ of the hom*oclinic orbit (depicted in Figure 2.10) of system (2.31) with grounded NES when damping was added. Reconsidering system (3.160–3.161), we seek its solution in the following regular perturbation series form: z(t) = z0 (t) + ε1/2 λˆ z1 (t) + O (ε) ,

B = B0 + ε1/2 λˆ B1 + O (ε)

(3.162)

Substituting into (3.160) and considering only O(1) terms we derive the following system at the first order of approximation, z˙ 0 −

j j j B0 |z0 |2 z0 + z0 = , z0 (0) = B0 2 2 2

(3.163a)

or in terms of the polar transformation z0 = N0 ej δ0 , B0 sin δ0 , N0 (0) = B0 N˙ 0 = 2 1 1 B0 cos δ0 , δ0 (0) = 0 δ˙0 + − N02 = 2 2 2N0

(3.163b)

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Fig. 3.71 Parametric plots for λˆ = 0.4472 Im[z] against Re[z] with t being the parametrizing variable: initial regime of slow flow dynamics of intermediate-energy damped IOs for different normalized impulses B [slow flow (3.160) or (3.161)].

We note that there exist no damping terms in this first order of approximation, as these terms enter into the problem at the next order of approximation. It can be proved that the undamped slow flow possess the following Hamiltonian (first integral of the motion): j j j B0 ∗ j B0 |z0 |2 − |z0 |4 − z − z0 = h 2 4 2 0 2

(3.164)

where the asterisk denotes complex conjugate. This relation reduces (3.163b) to the following one-dimensional slow flow:

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.72 Roots of f (a, B0 ) = 0 (the additional real root for a > 1 is not shown).

2a˙ = f (a; B0 ) sin δ0 =

1/2

√ 2 d a , B0 dt

,

2 1 1 f (a; B0 ) ≡ 4B0 a − a − a 2 + B04 + B02 2 2 a(0) = B02 ,

δ0 (0) = 0

(3.165)

where we introduced the notation a(t) ≡ N02 (t). The roots of the polynomial f (a; B0) depend on the parameter B0 (see Figure 3.72). For B0 > B0 cr ≈ 0.36727 the polynomial f (a; B0 ) possesses two real distinct roots for a, whereas for B0 < B0 cr four distinct real roots. For B0 = B0 cr two of the real roots coincide, so f (a; B0) possesses only three distinct real roots then: a1 = B0 cr < a2 = a3 = 0.4563 < a4 = 2.9525,

B = B0 cr ≈ 0.36727

In this case, it will be proven that the system (3.165) possesses a hom*oclinic orbit, which we now proceed to compute explicitly. Indeed, for B0 = B0,cr the reduced slow flow dynamical system can be integrated by quadratures as a˙ =

1 1/2 (a − B02 cr ) (a2 − a)2 (a4 − a) 4 a dυ ⇒t=4 (a2 − υ) (υ − B02 cr ) (a4 − υ) 2

1/2

(3.166)

B0 cr

where the initial condition a(0) = B02 cr was taken into account, and it was recognized that a(t) ≥ B02 cr for t ≥ 0. We note that (3.166) provides the unique solution of the problem (3.165). The definite integral in the expression above can be explicitly evaluated (Gradshteyn and Ryzhik, 1980) to yield the following analytical hom*oclinic orbit of the first-order system (3.163):

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems (−)

γ1 γ2

√ √ γ γ γ1 γ2 1 2 t + γ2 cosh2 t γ1 sinh2 8 8 , ⎡ ⎤ (−) d a (t) 2 h (−) ⎦ δ0 (t) = δ0h (t) = sin−1 ⎣ B0 cr dt

N02 (t) ≡ ah (t) = a2 −

227

(3.167a)

(3.167b)

where γ1 = a2 − B02 cr , γ2 = a4 − a2 , and only the branch of the solution corresponding to t ≥ 0 is taken. The solution (3.167) assumes the limiting values, √ N0 (0) = B0 cr and limt →+∞ N0 (t) = a2 . Of course, the solution (3.167) can be extended for t < 0, but the resulting branch of the hom*oclinic orbit is not a solution √ of the problem (3.163), and satisfies the limiting relation limt →−∞ N0 (t) = a2 . We mention that system (3.166) provides an additional hom*oclinic loop for (3.163) [which, however, does not satisfy the initial condition a(0) = B02 cr ]: γ1 γ2

√ √ γ1 γ2 γ1 γ2 2 2 t + γ2 sinh t γ1 cosh 8 8 , ⎡ ⎤ (+) d a (t) 2 h (+) ⎦ δ0 (t) = δ0h (t) = sin−1 ⎣ B0 cr dt

N02 (t) ≡ ah(+) (t) = a2 +

(3.168a)

(3.168b)

√ a4 and This hom*oclinic loop assumes the limiting values, N0 (0) = √ limt →±∞ N0 (t) = a2 . In Figure 3.73 the two hom*oclinic loops corresponding to (3.167a, b) and (3.168a, b) are depicted. These loops are shown by dashed lines for the full range −∞ < t < +∞, with the branch (3.167a) of the hom*oclinic solution of problem (3.165) being identified by solid line. This completes the solution of the O(1) approximation of the hom*oclinic solution of (3.160–3.161), and we now proceed to consider the O(ε) problem, which takes into account (to the first order) the effects of damping. We will be especially interested in studying the perturbation of the hom*oclinic solution (3.167a, b) of the O(1) problem when weak damping [of O(ε1/2 )] is added. The O(ε 1/2 ) analysis will also provide a correction due to damping of the critical value of the implulse (initial condition) corresponding to the hom*oclinic solution (see Figure 3.71). Substituting (3.162) in (3.160) and considering O(ε1/2 ) terms, we derive the following problem at the next order of approximation: z˙ 1 −

j ∗ 2 j 1 j B1 (z1 z0 + 2 |z0 |2 z1 ) + z1 = − z0 + , 2 2 2 2

z1 (0) = B1

(3.169)

This is a complex quasi-linear ordinary differential equation with a nonhom*ogeneous term. Although the following analysis applies for the general class of solutions of (3.169), from hereon we will focus only in the solution corresponding to the perturbation of the hom*oclinic orbit (3.167a, b) of the O(1) problem. The perturbed hom*oclinic solution z1h (t) of (3.169) is written as

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.73 hom*oclinic orbits (3.167a, b) and (3.168a, b): (a)ah(±) (t), (b) parametric plot of Im[z] against Re[z] with t being the parametrizing variable; the solid line represents the hom*oclinic solution of the slow-flow problem (3.165).

z1h (t) = z1HS (t) + z1P I (t)

(3.170)

i.e., it is expressed as a superposition of the general hom*ogeneous solution z1HS (t) and of a particular integral z1P I (t). Key in solving the problem, is the computation of two linearly independent hom*ogeneous solutions of (3.169), since then, a particular integral may be systematically computed by either solving the differential

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229

equation satisfied by the Wronskian of the hom*ogeneous solutions, or through the method of variation of parameters. We can easily prove (by simple substitution into the complex hom*ogeneous equation) that one hom*ogeneous solution of (3.169) can be computed in terms of the time (1) derivative of the O(1) hom*oclinic solution as z1HS (t) = z˙ 0h (t), ∈ R. At this point we decompose the complex solution into real and imaginary parts: z1h (t) = x1h (t) + jy1h (t),

z0h (t) = x0h (t) + jy0h (t)

(3.171)

Then the first hom*ogeneous solution of (3.169) is expressed as ⎫ 2 x˙0h (t) ⎪ ⎪ ⎬ B0 cr ⎪ 2 ⎪ (1) (t) = y˙0h (t) ⎭ y1HS B0 cr

(1) x1HS (t) =

(1) ⇒ z1HS =

2 [x˙0h (t) + j y˙0h (t)] B0 cr

(First hom*ogeneous solution) (3.172)

where the real constant was selected so that the first hom*ogeneous solution (1) (1) (1) satisfies the initial conditions x1HS (0) = 0, y1HS (0) = +1 ⇒ z1HS (0) = j . In addition, the hom*ogeneous solution (3.173) satisfies the limiting conditions (1) (1) (t) = 0 and limt →+∞ y1HS (t) = 0. limt →+∞ x1HS To compute a second linearly independent hom*ogeneous solution of (3.169) it is convenient to carry the entire analysis to the real domain, by decomposing (3.169) into the following set of two real quasi-linear coupled ordinary differential equations with non-hom*ogeneous terms: 2 + 3y 2 − 1)/2 x x˙1h (x0h x0h y0h 1h 0h + 2 + y 2 − 1)/2 y˙1h y −(3x0h −x y 1h 0h 0h 0h =

−x0h /2 (B1 cr − y0h ) /2

(3.173)

Note that problem (3.173) governs the O(ε1/2 ) perturbation of the O(1) hom*oclinic solution (3.167a, b), and the real constant B1 cr on the right-hand side denotes the O(ε 1/2 ) correction to B0 cr in (3.162). We seek a second hom*ogeneous solution of (2) (2) (3.173) satisfying the initial conditions, x1HS (0) = −1, y1HS (0) = 0. Accordingly, we consider the following relation satisfied by the Wronskian of (3.173): (1)

(2)

(2)

(1)

W (t) = x1HS (t)y1HS (t) − x1HS (t)y1HS (t)

(3.174a)

From the theory of ordinary differential equations the Wroskian then satisfies the following relation:

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3 Nonlinear TET in Discrete Linear Oscillators

W˙ (t) = 0 ⇒ W (t) = W (0) = 1

(3.174b)

which provides a means for computing the second hom*ogeneous through the relation, (1)

(1) (2) (2) (1) (2) x1HS (t)y1HS (t) − x1HS (t)y1HS (t) = 1 ⇒ x1HS (t) =

(2)

x1HS(t)y1HS (t) − 1 (1)

(3.175)

y1HS (t)

When this expression is substituted into the second of equations (3.173) with the non-hom*ogeneous term dropped, yields the following first-order quasi-linear differ(2) ential equation governing y1HS , (2) y˙1HS

+ a21

(1) x1HS (1) y1HS

(2)

+ a22 y1HS =

a21 (1) y1HS

,

(2)

y1HS (0) = 0

(3.176)

2 + 3y 2 − 1)/2, a 2 2 with a11 = x0h y0h , a12 = (x0h 21 = −(3x0h + y0h − 1)/2, and 0h a22 = −x0h y0h . The solution of (3.176) provides the second linearly independent hom*ogeneous solution of (3.169), which is computed explicitly as follows: ⎫ (1) (2) ⎪ x1HS (t)y1HS (t) − 1 ⎪ (2) ⎪ ⎪ x1HS (t) = ⎪ (1) ⎪ ⎬ y1HS (t) ⎧ ⎫ t t (1) ⎨ ⎬ ⎪ x1HS (s) a21(τ ) ⎪ (2) ⎪ ⎪ exp − + a a y1HS (t) = (s) (s) ds dτ 21 22 ⎪ (1) (1) ⎪ ⎩ ⎭ ⎭ y1HS (τ ) y1HS (s) τ

(2)

(2)

(2)

⇒ z1HS = x1HS (t) + jy1HS(t)

(Second hom*ogeneous solution) (3.177)

As mentioned previously, the second hom*ogeneous solution satisfies the initial (2) (2) (2) conditions x1HS (0) = −1, y1HS (0) = 0 ⇒ z1HS (0) = −1, and, contrary to (2) (3.172) it diverges with time, since it holds that limt →+∞ x1HS (t) = +∞ and (2) limt →+∞ y1HS (t) = +∞. Making use of the two linearly independent hom*ogeneous (3.172) and (3.177) we may compute a first particular integral by the method of variation of parameters. Indeed, by expressing the real and imaginary parts of the particular integral z1P I (t) = x1P I (t) + jy1P I (t), in the form (1) (2) x1P I (t) x1HS(t) x1HS (t) + c2 (t) (3.178) = c1 (t) (1) (2) y1P I (t) y (t) y (t) 1HS

1HS

and evaluating the real coefficients c1 (t) and c2 (t) by substituting into (3.173), we obtain the following explicit solution of problem (3.169) which provides the O(ε1/2 ) perturbation of the hom*oclinic orbit:

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

x1h (t) y1h (t) ⎡

231

⎤

t (1) (t) x1HS x (τ ) − y (τ ) B 0h 1cr 0h (2) (2) y1HS (τ ) − x1HS (τ ) dτ ⎦ − = ⎣1 + (1) 2 2 y1HS (t) ⎡ + ⎣2 +

t 0

⎤

(2) x0h (τ ) (1) x1HS (t) B1cr − y0h (τ ) (1) y1HS (τ ) + x1HS (τ ) dτ ⎦ (2) 2 2 y1HS (t) (3.179)

This analytical expression contains three yet undetermined real constants, namely, the coefficients 1 , 2 , and the correction to the initial condition for motion on the hom*oclinic orbit, B1 cr . By imposing the initial condition of (3.169), z1h (0) = B1 cr ⇒ x1h (0) = B1 cr , y1h (0) = 0, we compute the two coefficients as follows: 1 = 0 and 2 = −B1 cr (3.180) Then, taking into account that the components of the second hom*ogeneous solution (2) (2) (t) and y1HS (t) in the second additive term of (3.179) diverge as t → +∞, and x1HS in order to obtain bounded solutions for x1h (t) and y1h (t) as t → +∞, we require that +∞ −B1 cr +

x0h (τ ) (1) B1 cr − y0h (τ ) (1) y1HS (τ ) + x1HS (τ ) dτ = 0 2 2

(3.181a)

This evaluates B1 cr according to the following expression: +∞ (1) (1) x0h (τ )y1HS (τ ) − y0h (τ )x1HS (τ ) dτ 0 B1 cr = +∞ (1) 2− x1HS (τ ) dτ

(3.181b)

This completes the solution of the problem (3.169) and computes the perturbation of the hom*oclinic orbit in the damped system (3.160–3.161) with O(ε1/2 ) damping. In summary, the analytic approximation of the perturbed hom*oclinic orbit is given by zh (t) = z0h (t) + ε1/2 λˆ z1h (t) + O (ε) ,

ˆ = B0 cr + ε1/2 λB ˆ 1 cr + O (ε) Bcr (λ) (3.182)

, where z0h (t) = ah(−) (t) exp[δh(−) (t)] and ah(−) (t), δh(−) (t) are computed by (3.167a, b); z1h (t) = x1h (t) + jy1h (t), where x1h (t) and y1h (t) are computed by (3.179), (3.180) and (3.182b); B0 cr ≈ 0.36727; and B1 cr is computed by (3.181b).

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.74 Slow flow response in the regime of optimal TET (‘super-slow’ half-cycle of TET), for ε = 0.05, C = 1 and λ = ε1/2 λˆ = 0.1; comparison of full slow slow (3.151) or (3.157), of the approximation of the slow flow in the initial stage of the dynamics (3.158) or (3.160), and of the asymptotic solution (3.182).

For ε = 0.05 and λˆ = 0.4472 we estimate the initial condition as Bcr (λˆ = 0.4472) ≈ 0.3806, which compares to the numerical value of 0.3814 derived from the numerical integration of the initial approximation of the slow flow (3.160–3.161) (see Figure 3.71). Taking into account the previous coordinate transformations and rescalings for B, the previous analytical result leads to an estimated initial condition (impulse) of X = 0.0983 for optimal TET (i.e., for the excitation of the damped hom*oclinic orbit), compared to the numerical result of X = 0.1099 derived from the full averaged slow flow (3.157) (see Figure 3.69d); we note that the error is of O(ε = 0.05) and compatible to our previous asymptotic derivations. In Figure 3.74 we provide a comparison of the three approximate models for the slow flow dynamics in the regime of optimal TET; the asymptotic analysis correctly predicts the half-cycle ‘super-slow’ transfer of energy from the LO to the NES in the initial regime of the motion, although it underestimates the maximum amplitude of the response during this half-cycle; this can be explained by the fact that the slow flow approximation (3.158) or (3.160) is only valid in the initial regime of the motion. This completes the analytical study of the regime of optimal TET in system (3.148) when intermediate-energy damped IOs are excited. In summary, in the weakly damped system, optimal TET is realized for initial energies where the excited damped IOs are in the neighborhood of the hom*oclinic orbit of the unstable

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out-of-phase damped NNM S11−; in the underlying Hamiltonian system this unstable NNM is generated at a critical energy through a saddle-node bifurcation. We studied analytically the perturbation of the hom*oclinic orbit in the weakly damped system, which introduces an additional ‘super slow’ time scale in the averaged dynamics and leads to optimal TET from the LO to the NES in a single ‘super-slow’ half cycle. At higher energies, this ‘super-slow’ half cycle is replaced by strong nonlinear beats (these are generated due to the attraction of the dynamics to the stable in-phase damped NNM S11+), which yield significant but non-optimal TET through nonlinear beats. At lower energies than the one corresponding to the optimal TET regime, the dynamics is attracted by the stable, weakly nonlinear (linearized), out-of-phase damped NNM S11− and TET is negligible. The above-mentioned conclusions are valid for the weakly damped system (3.148), under the assumption of sufficiently small ε, i.e., of for lightweight NESs and systems with strong mass asymmetries. In Figures 3.75–3.77 we study TET in system (3.148) for excitation of intermediate-energy damped IOs over a wider range of mass asymmetry ε and damping ελ; these plots were derived by direct numerical integrations of the differential equations of motion, and monitoring the instantaneous energy of the system versus time. Numerical results indicate that, by increasing ε (i.e., by decreasing the mass asymmetry) and the damping coefficient ελ, the capacity of the NES for optimal TET deteriorates. This is due to the fact that by increasing the inertia of the NES the amplitude of the relative response between the LO and the NES decreases, which hinters the capacity of the damper of the NES to effectively dissipate energy. Moreover, by increasing damping in the system, the damper of the LO dissipates an increasingly higher portion of the vibration energy which leads to deterioration of TET; this markedly slows energy dissipation in the system, as judged by comparing the time intervals required for energy dissipation in the plots of Figure 3.77 and the corresponding time intervals in the regimes of optimal TET in the plots of Figures 3.75 and 3.76.

3.5 Multi-DOF (MDOF) Linear Oscillators with SDOF NESs: Resonance Capture Cascades and Multi-frequency TET Up to now we examined TET in a two-DOF system consisting of SDOF damped linear oscillator (LO) coupled to an essentially nonlinear attachment, acting, in essence, as nonlinear energy sink (NES). In this section, we extend the analysis to MDOF LOs with SDOF essentially nonlinear boundary attachments. The main result reported in this section is that the SDOF NES can interact with (and extract energy from) multiple linear modes of the linear system to which it is attached, due to resonance capture cascades (RCCs). Indeed, we will show that through RCCs the NES can passively extract broadband vibration energy from the linear system (i.e., over wide frequency ranges), through multi-frequency TET. What enables a SDOF NES to interact with multiple linear modes over arbitrary frequency ranges is its essential stiffness nonlinearity, which enables it to engage in

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.75 Energy dissipation in system (3.148) when damped IOs are excited for mass assymetry ε = 0.03: (a) ελ = 0.015, (b) ελ = 0.003, (c) ελ = 0.006.

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235

Fig. 3.76 Energy dissipation in system (3.148) when damped IOs are excited for mass assymetry ε = 0.1: (a) ελ = 0.005, (b) ελ = 0.01, (c) ελ = 0.02.

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.77 Energy dissipation in system (3.148) when damped IOs are excited for mass assymetry ε = 0.2: (a) ελ = 0.01, (b) ελ = 0.02, (c) ελ = 0.04.

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transient resonance capture (TRC) with any highly energetic linear mode irrespective of its frequency, provided, of course, that this mode has no node at the point of attachment of the NES. Then, the NES extracts energy from each specific mode, before escaping from TRC and engaging in transient resonance the next one. In the passive system considered, what controls the order with which modes participate in these RCCs is the initial state of the system, the external excitation (being narrowband or broadband), and the actual rate of energy dissipation due to damping (since the instantaneous energy level of the NES passively ‘tunes’ its instantaneous frequency). These concepts are discussed in the following sections and demonstrated for the case of a two-DOF linear LO with a SDOF NES attachment. Then, we consider a semi-infinite chain of LOs with a single NES attached to its end, as a first attempt to extend the concept of passive TET to linear waveguides with local essentially nonlinear attachments.

3.5.1 Two-DOF Linear Oscillator with a SDOF NES The system considered is depicted in Figure 3.78, and consists of a two-DOF damped LO (designated as the primary system) coupled to a SDOF NES. The equations of motion are given by: m1 x¨1 + c1 x˙1 + k1 x1 + k12 (x1 − x2 ) = 0 m2 x¨2 + c2 x˙2 + cv (x˙2 − v) ˙ + k2 x2 + k12 (x2 − x1 ) + C(x2 − v)3 = 0 ε v¨ + cv (v˙ − x˙ 2 ) + C(v − x2 )3 = 0

(3.183)

The variables x1 (t) and x2 (t) refer to the displacements of the oscillators of the (primary) linear system, whereas v(t) refers to the displacement of the NES. As in the previous sections, a lightweight NES is considered by requiring that ε m1 , m2 , with 0 < ε 1 being a small parameter characterizing the strong mass asymmetry of the system. As in the analysis for the two-DOF system considered in the previous sections, first we discuss the dynamics of the underlying Hamiltonian system obtained by setting all damping terms equal to zero; then we analyze the nonlinear transitions in the weakly damped system and relate these transitions to the Hamiltonian dynamics.

3.5.1.1 Frequency-Energy Plot (FEP) of the Underlying Hamiltonian System It is not necessary to perform an exhaustive calculation of the periodic orbits of underlying Hamiltonian system of (3.183), since the dynamics governing TET can be studied by considering the following two subsets of orbits in the Hamiltonian FEP: (i) the backbone branches of periodic orbits under conditions of 1:1:1 internal resonance, and (ii) the manifolds of impulsive orbits (IOs). Note that since in this

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Fig. 3.78 The three-DOF system consisting of a two-DOF primary LO with an essentially nonlinear, lightweight NES.

case the primary linear system possesses two degrees of freedom there exist multiple backbone sub-branches (depending on the relative phases between the three oscillators of the system during 1:1:1 internal resonance), and multiple manifolds of IOs in the FEP. An analytic approximation of the backbone branches of the Hamiltonian system can be derived by applying the complexification-averaging (CX-A) technique. To this end, the following complex variables are introduced, ψ1 = x˙1 + j ωx1 ,

ψ2 = x˙2 + j ωx2 ,

ψ3 = v˙ + j ωv

(3.184)

where ω is the common dominant frequency of oscillation during 1:1:1 internal resonance. Following the CX-A procedure as discussed in the previous sections (i.e., averaging over the fast frequency ω, expressing the resulting complex modulations in polar form, and imposing stationarity conditions for the resulting real amplitudes and phases) the following analytical approximation for the NNMs on the backbone branches of the Hamiltonian system is obtained, x1 (t) ≈ A sin ωt,

x2 (t) ≈ B sin ωt,

v(t) ≈ D sin ωt

where

4εω2 c2 A= 3C(c2 − c1 )3

1/2 ,

D = c2 A, B = c1 A,

(3.185)

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c1 = k1 + k12 − ω2 m1 /k12 ,

239

c2 = −k12 − c1 (ω2 m2 − k2 − k12) /εω2

The backbone branches can be constructed by varying the frequency ω and calculating the corresponding total energies of the NNMs. Figure 3.79 depicts the backbone branches, denoted by S111, of the system with parameters m1 = m2 = k1 = k2 = k12 = C = 1 and ε = 0.05. NNMs depicted as projections of the threedimensional configuration space (v, x1 , x2 ) of the system are inset. When the projections of the NNMs are close to horizontal (vertical) lines, the motion is localized to the NES (primary system). Four characteristic frequencies, f1L , f2L , f1H and f2H are defined in this FEP. At high energy levels and finite frequencies, the essential nonlinearity behaves as a rigid link, and the system is reduced to the following system of two linear coupled oscillators: m1 x¨1 + k1 x1 + k12 (x1 − x2 ) = 0 (ε + m2 )x¨2 + k2 x2 + k12 (x2 − x1 ) = 0

(3.186)

For the above parameters the natural frequencies of this system are given by f1H = 0.9876 rad/s and f2H = 1.7116 rad/s. At low energy levels, the stiffness of the essential nonlinearity tends to zero, and the system is again reduced to the primary two-DOF √ LO, the natural frequencies of which are given by f1L = 1 rad/s and f2L = 3 rad/s. From Figure 3.79, we note that the two frequencies f1L and f2L divide the FEP into three distinct regions. The first region defined by ω ≥ f2L , consists of the backbone sub-branch S111+−+, where the (+) and (−) signs characterize the relative phases between the three masses of the system, and indicate whether the extremum of the amplitude of the corresponding oscillator during the synchronous 1:1:1 periodic motion (NNM) is positive or negative, respectively. On this sub-branch, the primary LO vibrates in an out-of-phase fashion, and the motion becomes increasingly localized to the LO or the NES as ω → f2L or ω → ∞, respectively. The second region defined by f1L ≤ ω ≤ f2H , consists of two distict sub-branches, namely S111+−− and S111++−. These branches coalesce at point S111+0− (depicted as the grey dot in Figure 3.79), where the initial velocity of the mass m2 is zero. On S111+−− the LO vibrates in an out-of-phase fashion, and the motion localizes to the NES as the frequency leaves the neighborhood of f2H . On S111++− the LO oscillates in in-phase fashion, and the vibration localizes to the LO as ω → f1L . The third region corresponding to ω ≤ f1H , consists of the sub-branch S111+++, where the LO vibrates in in-phase fashion, and the motion localizes to the NES as the frequency tends away from f1H . Due to the energy dependence of the NNMs along the sub-branches of S111, interesting and strong energy exchanges may occur between the primary LO and the NES when weak damping is introduced in the system. Indeed, the weakly damped system possesses damped NNM manifolds which can be considered as analytic continuations for weak damping of the NNMs of the Hamiltonian system. Since, these manifolds are invariant for the dynamical flow, when a damped response is initiated

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Fig. 3.79 Analytic approximation of the backbone branch of (3.183): NNMs depicted as projections of the three-dimensional configuration space of the system are superposed; the horizontal and vertical axes in these plots are the responses of the nonlinear and primary systems, respectively [top plot (v, x1 ), bottom plot (v, x2 )].

on a damped NNM manifold, it stays on it for the entire duration of the decaying oscillation. Two specific sub-branches, namely S111+−− and S111+++, play an important role for the realization of fundamental TET in system (3.183). Due to the dependence of the frequency of the damped oscillation on the instantaneous energy, irreversible channeling of vibration energy from the LO to the NES takes place as the damped continuations of the NNMs S111+−− and S111+++ are traced from high to low frequencies (since the shapes of the corresponding NNMs localize from the LO to the NES as frequency decreases – see Figure 3.79). Hence, both in-phase

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and out-of-phase fundamental TET can be realized in this system, corresponding to in-phase or out-of-phase motions of the oscillators of the primary LO, respectively; this shows the adaptivity of the NES to different initial conditions and represents a generalization of the concept of fundamental TET discussed in Section 3.4.2.1 for the two-DOF system. A detailed stability analysis of S111+−− and S111+++ was not performed, but the following numerical simulations and experimental results show that these are stable oscillations, at least for the parameter values considered in this work. The backbone of the FEP of the Hamiltonian system can also be computed numerically. Assuming that a NNM is realized for the initial velocity vector ˙ and zero initial displacements, this vector together with the [x˙1 (0) x˙2 (0) v(0)] period of the motion, T , are computed by satisfying the following periodicity condition: T ˙ ) x1 (T ) x2 (T ) v(T ) x˙1 (T ) x˙2 (T ) v(T T T ˙ − 0 0 0 x˙1 (0) x˙ 2 (0) v(0) = 000000 (3.187) The numerical computation was carried out in Matlab using optimization techniques. For a given value of the period T the objective function to be minimized is the norm of the left-hand side of equation (3.187), and the optimization variables are the non-zero initial conditions. By varying the period, the backbone branch represented in Figure 3.80 is obtained; a small subset of subharmonic tongues (see Sections 3.3.2.2 and 3.3.2.3) has also been identified using this algorithm. We note the close agreement between the backbones computed numerically and analytically (compare Figures 3.79 and 3.80). Another important feature of the FEP concerns the manifolds of IOs. The essential role of IOs for TET has been discussed extensively in Section 3.4; the periodic IOs of system (3.183) correspond to the special initial conditions, x˙1 (0) = ˙ = 0 (or, equivalently, to two im0, x˙2 (0) = 0 and x1 (0) = x2 (0) = v(0) = v(0) pulses applied to the LO with the system initially at rest). Contrary to the two-DOF examined in Sections 3.3.3 and 3.3.4, two distinct families of IOs are realized in the three-DOF under consideration: in-phase IOs correspond to two in-phase impulses of identical magnitudes applied to the two masses of the LO at t = 0, corresponding to initial conditions, x˙ 1 (0) = x˙2 (0) = 0; out-of-phase IOs correspond to two out-of-phase impulses of equal magnitude applied to the two masses of the LO, and initial conditions x˙1 (0) = −x˙2 (0) = 0. Contrary to the two-DOF system examined previously, no IOs can be realized in the three-DOF system by applying a single impulse to either one of the masses of the LO. Similarly, however, to the two-DOF system, the excitation of stable IOs localized to the NES, leads to rapid and significant energy transfer from the LO to the NES during a cycle of the oscillation of the three-DOF system; when damping is introduced this leads to effective, fast scale TET from the LO to the NES. It follows that for system (3.183) there exist two distinct IO manifolds, consisting of periodic and quasi-periodic in-phase and out-of-phase IOs, respectively. The computations depicted in Figure 3.80 were re-

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Fig. 3.80 Numerical computation of the FEP (backbone branches and periodic IOs) of system (3.183) for m1 = m2 = k1 = k2 = k12 = C = 1 and ε = 0.05; black dots and squares denote out-of-phase and in-phase IOs, respectively; unstable NNMs are denoted by (×); IOs 1–6 refer to Figures 3.81 and 3.82.

stricted to periodic IOs corresponding to low-order internal resonances between the LO and the NES. As mentioned above, no periodic orbits corresponding to impulsive excitation of only one of the masses of the primary system were detected. However, we conjecture that for this type of impulsive excitation quasi-periodic impulsive orbits could still exist, and are such that the NES resonates with a mode of the primary system only above a certain energy threshold. Moreover, it was observed that strong nonlinear interaction of the NES with the in-phase mode of the LO is triggered at lower energy levels compared to the out-of-phase mode.

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Fig. 3.81 Representative in-phase IOs: (a) IO 1, (b) IO 2, (c) IO 3 (see Figure 3.80); left column: time series; - -— x1 (t), - -O- - x2 (t), - -- - v(t); right column: two-dimensional projections of IOs and instantaneous percentage of total energy carried by the NES during a cycle of the IO.

The in-phase manifold of IOs consists of in-phase impulsive orbits (++0) located on in-phase subharmonic tongues, with the masses of the primary linear system oscillating an in-phase fashion. This manifold is depicted as a smooth curve in the FEP. Representative in-phase IOs labeled as IO 1, IO 2 and IO 3 in Figure 3.80 are illustrated in Figure 3.81. When the phase differences between the masses of

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Fig. 3.82 Representative out-of-phase IOs: (a) IO 4, (b) IO 5, (c) IO 6 (see Figure 3.80); left column: time series; - -- - x1 (t), - -O- - x2 (t), - -- - v(t); right column: two-dimensional projections of IOs and instantaneous percentage of total energy carried by the NES during a cycle of the IO.

the system are trivial, the motion of the NNM in the configuration space (x1 , x2 , v) takes the form of a simple curve; in the case of non-trivial phase differences the motion corresponds to a Lissajous curve. For IO 1, the oscillations of the two masses of the linear primary system are almost identical and nearly monochromatic; the corresponding oscillation of the NES

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Fig. 3.83 Maximum percentage of energy transferred from the LO to the NES during a cycle of the IO (dashed line: in-phase IOs; dotted line: out-of-phase IOs); the FEP of Figure 3.80 is superimposed to this plot, unstable NNMs are denoted by (×).

has two dominant harmonic components, one equal to the dominant frequency of oscillation of the primary system, and the other equal to one-third of that frequency. Hence, a 1:1:3 internal resonance (IR) between the two masses of the primary system and the NES is realized. The nonlinear beat resulting due to this internal resonance is clearly deduced in the plots of Figure 3.81. For IO 1, the energy exchange between the LO and the NES is insignificant, as the maximum percentage of total energy transferred from the LO to the NES during a cycle is just 0.17%. For IOs 2 and 3, however, which correspond to 3:3:2 and 5:5:2 internal resonances, respectively, energy transfer from the LO to the NES during a cycle of the nonlinear beat is much stronger, reaching levels of 35% and 15%, respectively (the notation p:p:q internal resonance implies that the frequencies of oscillation of the first mass of the LO, the second mass of the LO and the NES are in ratios equal to p:p:q). The out-of-phase manifold of IOs consists of out-of-phase impulsive orbits (+−0) located on out-of-phase subharmonic tongues. This manifold is also represented by a smooth curve in the FEP. Representative out-of-phase IOs (labeled as IO 4–6 in Figure 3.80 and corresponding to 1:1:3, 2:2:3 and 6:6:5 internal resonances, respectively) are shown in Figure 3.82. In Figure 3.83 we present a study of maximum energy transferred from the LO to the NES during a cycle of the nonlinear beat resulting from excitation of in-phase or out-of-phase IOs. Superimposed to the plot of maximum energy transferred is the FEP of Figure 3.80, indicating the backbone branch and the two manifolds of

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IOs. What is evident from this plot is that there exist two critical energy thresholds, one for each of the in-phase and out-of-phase IOs, above which the IOs transfer a significant amount of energy from the LO to the NES during a cycle of the nonlinear beat; moreover, the energy threshold for out-of-phase IOs, hc2 , is higher than the corresponding one for in-phase IOs, hc1 . For instance, for the out-of-phase IO 4 located below the energy threshold hc2 , the maximum energy transferred to the NES during a cycle of the nonlinear beat is approximately 0.15% of the energy of the LO, whereas for the out-of-phase IO 6 located above that threshold the corresponding percentage of energy transferred is nearly 60%. It is interesting to note that the in-phase and out-of-phase thresholds hc1 and hc2 are located close to the corresponding energies where the saddle node bifurcations, I for in-phase NNMs and II for out-of-phase NNMs, take place; these bifurcations generate unstable branches of in-phase and out-of-phase NNMs as shown in Figure 3.83. From the discussion of Sections 3.3.5 and 3.4.2.5 we recall that in the two-DOF system a similar bifurcation exists in the corresponding FEP. In that system the hom*oclinic loops of the unstable NNM generated from the saddle-node bifurcation affect drastically the topologies of nearby IOs, since IOs lying inside the hom*oclinic loops are localized in phase space and the corresponding motions of the system are predominantly localized to the LO; on the contrary, IOs lying outside the hom*oclinic loops and being close to the 1:1:1 resonance manifold of the Hamiltonian dynamics make large excursions in phase space and correspond to strong nonlinear beats where significant energy is being exchanged between the linear and nonlinear oscillators. It appears that similar dynamics take place in the three-DOF considered here: the strong energy exchanges for in-phase or out-of-phase IOs in the neighborhoods of the saddle node NNM bifurcations I or II (actually, IOs having energies slightly higher that the energies of these saddle-node bifurcations), are affected by their proximities to hom*oclinic loops of unstable in-phase or out-ofphase NNMs, respectively, and to the corresponding 1:1:1 resonance manifolds at frequencies f1L and f2L , respectively. Based on the discussion and results of Section 3.4.2.5 we may deduce that the excitation of the damped analogs of these IOs lead to optimal in-phase and out-of-phase fundamental TET in the weakly damped three-DOF system. Another similarity to the dynamics of the two-DOF system is that, the two manifolds of in-phase and out-of-phase IOs of the three-DOF system play important roles regarding fundamental and subharmonic TET in the weakly damped system. However, a distinct feature of the dynamics of the weakly damped three-DOF system is the occurrence of resonance capture cascades (RCCs). This is a new feature of TET dynamics, whereby the NES passively extracts energy from both modes of the primary LO, as it engages sequentially in transient nonlinear resonance with both of them. This is discussed in the next section.

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3.5.1.2 Dynamics of the Damped System: Resonance Capture Cascades We now consider the dynamics of the weakly damped system (3.183). As in the case of the two-DOF system considered in Section 3.4, the underlying Hamiltonian dynamics determine, in essence, the weakly damped transitions and the energy exchanges between the LO and the NES. The first series of numerical simulations verifies that both in-phase and out-ofphase fundamental TET can occur in the weakly damped system, corresponding to in-phase or out-of-phase relative motions of the two masses of the LO. The simulations were carried out for the following specific system: x¨1 + 0.005x˙1 + x1 + (x1 − x2 ) = 0 x¨2 + 0.005x˙2 + 0.002(x˙2 − v) ˙ + x2 + (x2 − x1 ) + (x2 − v)3 = 0 0.05v¨ + 0.002(v˙ − x˙2 ) + (v − x2 )3 = 0

(3.188)

so that the small parameter of the problem is given by ε = 0.05; moreover, the assumption of weak damping is satisfied. The motion is first initiated on a NNM on the backbone branch S111+++, and the resulting motion involves in-phase oscillations of all three masses of the system with the same apparent frequency, as shown in Figures 3.84a, b. The temporal evolution of the instantaneous frequencies of the responses can be followed by superimposing their wavelet transform (WT) spectra to the FEP (as performed in Section 3.4 for the two-DOF system). In Figure 3.84c, the WT spectrum of the relative response v(t) − x2 (t) is superposed to the backbone of the FEP (represented by a solid line). As mentioned in previous sections this representation is purely schematic since it superposes a damped WT spectrum to the undamped FEP; nevertheless, this representation helps us deduce the essential influence of the underlying Hamiltonian dynamics on the weakly damped transitions, and is only used for purely descriptive purposes. The plot of Figure 3.84c clearly illustrates that as the total energy in the system decreases due to viscous dissipation, the response closely follows the backbone branch S111+++; in actuality, the response takes place on the damped NNM invariant manifold which results as perturbation of S111+++ when weak damping is added to the system. The dynamical flow is captured in the neighborhood of a 1:1:1 resonance manifold leading to prolonged 1:1:1 TRC. Figure 3.84d depicts the trajectories of the phase difference 1 (t) ≡ φv (t) − φx1 (t) between v(t) and x1 (t), and the phase difference 2 (t) ≡ φv (t)−φx2 (t) between v(t) and x2 (t); these phase variables are computed directly from the transient responses v(t), x1 (t) and x2 (t) by applying the Hilbert transform. A non-time-like behavior of the two phase differences is noted, which provides further evidence of the occurrence of 1:1:1 TRC. Figure 3.84e confirms that in-phase fundamental TET, i.e., passive and irreversible (on the average) energy transfer from the LO to the NES, takes place. In the second simulation the motion is initiated on S111+−−. In the initial stage of the motion ( 0 < t < 100 s) out-of-phase fundamental TET is realized, with

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Fig. 3.84 In-phase fundamental TET on the damped NNM invariant manifold S111+++: (a) transient responses; (b) close-up of the time series, - -- - x1 (t), - -•- - x2 (t), - -- - v(t); (c) WT spectrum of v(t) − x2 (t) superposed to the FEP; (d) trajectories of phase differences; (e) percentage of instantaneous total energy in the NES.

the two masses of the LO oscillating in an out-of-phase fashion (see Figures 3.85a, b). During this initial regime of the motion, the envelopes of all responses decrease monotonically, but the envelope of the NES seems to decrease more slowly than those of the masses of the linear primary system; TET to the NES occurs during this stage of the motion (see Figure 3.85e). Around t = 80 s, the displacement of the second mass of the primary system, x2 (t), becomes very small, and a transition from the out-of-phase damped NNM S111+−− to the in-phase damped NNM

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Fig. 3.85 Initial out-of-phase fundamental TET on the damped NNM invariant manifold S111+−−, followed by 1:3:3 subharmonic TET: (a) transient responses; (b, c) close-ups of the time series during fundamental and subharmonic TET, - -- - x1 (t), - -•- - x2 (t), - -- - v(t); (d) WT spectrum of v(t) − x2 (t) superposed to the FEP; (e) trajectories of phase differences; (f) percentage of instantaneous total energy in the NES.

S111++− occurs. When the end of S111++− is traced by the damped dynamics (close to the point of saddle-node bifurcation that eliminates the stable/unstable pair of NNMs in the FEP), escape from 1:1:1 TRC occurs, which results in time-like behavior of the phase differences in Figure 3.85e. The plots of Figures 3.85c, d, f show that this is soon followed by 1:3:3 subharmonic TRC leading to subharmonic TET as the damped motion traces the damped analogue of the in-phase tongue S113. Con-

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sidering the notation used for the subharmonic tongues, we generalize the notation introduced for the subharmonic tongues of the two-DOF, in Section 3.3.1.2: a subharmonic tongue Sppq contains periodic motions with two dominant frequencies, namely ω and pω/q. We conclude that the NES extracts vibration energy from the LO through two distinct TET mechanisms, that is, initial out-of-phase fundamental TET, followed by subharmonic TET. We now proceed to verify the existence of energy thresholds above which excitations of IOs can trigger in-phase or out-of-phase fundamental TET. In the results depicted in Figure 3.86, the damped motion is initiated by exciting the in-phase IOs 1 and 2, located below and above the energy threshold hc1 of Figure 3.83, respectively; we recall that the in-phase IOs are generated by applying two in-phase identical impulses to the two masses of the LO at t = 0. By noting the resulting responses we conclude that the dynamics is markedly different in the two cases. Indeed, when IO 1 is excited, the NES does not extract a significant amount of energy from the LO, as the damped motion is nearly linear and remains localized predominantly to the LO; this is due to the fact that the damped dynamics traces the weakly nonlinear (linearized) branch S111++− with decreasing energy (see Figure 3.86c). When IO 2 is excited, however, qualitatively different dynamics takes place, since in the initial stage of the response strong nonlinear beats take place leading to TET; during this phase significant energy is dissipated by the damper of the NES. With decreasing energy (and frequency) of the NES escape from the regime of nonlinear beats occurs, and the dynamics makes a transition to the damped NNM S111+++, at which point significant in-phase fundamental TET is realized. Overall, multifrequency TET from the LO to the NES takes place in this case, underscoring the adaptivity of the NES to initial conditions; indeed, depending on the specific initial conditions of the system, the NES passively ‘tunes itself’ and transiently resonates with different modes of the primary system, absorbing and dissipating vibration energy from the LO. Likewise, if the damped oscillation is initiated by exciting an out-of-phase IO (i.e., by applying two out-of-phase but equal in magnitude impulses to the two masses of the LO at t = 0) located below the energy threshold hc2 (IO 4 in Figure 3.83), the response traces the linearized damped NNM S111+−+, on which the motion localizes predominantly to the LO throughout. However, if an out-ofphase IO above the energy threshold is excited (IO 6 in Figure 3.83), after an initial regime of nonlinear beats the damped motion makes a transition to the damped NNM S111+−− with decreasing energy, and out-of-phase fundamental TET takes place. Another case of practical importance is when a single impulse is applied to one of the masses of the LO (the primary system). We recall that for single applied impulses to the LO no periodic IOs were detected, but instead both the in-phase and out-of-phase modes of the LO participate in the damped response; hence, a multimodal response is anticipated in this case, which opens the possibility of interesting multi-frequency nonlinear transitions and energy exchanges in the system. In the following simulations we consider a slightly modified system (3.183), in the sense that no grounded stiffness for mass m2 exists (i.e., k2 = 0), and an addi-

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Fig. 3.86 Excitation of IO 1 (a, c, e) and IO 2 (b, d, f): (a) absence of TET on the linearized branch S111++−; (b) initial TET through nonlinear beats, followed by inphase TET on S111+++; (c, d) WT spectra of v(t) − x2 (t) superposed to the FEP; (e, f) percentages of instantaneous total energy in the NES.

tional dashpot of constant c12 is placed between the two masses m1 and m2 of the LO. The numerical values of the system parameters were selected to be identical to the ones of an experimental fixture (discussed in Section 3.5.1.3), and are listed in Table 3.4. These parameters were identified using experimental modal analysis and the restoring-force technique (see Section 3.5.1.3). An impulsive force in the form of a half-sine pulse of duration 0.01 s is applied to mass m1 of the LO; the

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3 Nonlinear TET in Discrete Linear Oscillators Table 3.4 System parameters of the experimental three-DOF system (Figure 3.96). Parameter

Value

m1 m2 ε k1 k2 k12 C c1 c2 c12 cv

0.6285 kg 1.213 kg 0.161 kg 420 N/m 0 N/m 427 N/m 4.97 × 106 N/m3 0.05 to 0.1 Ns/m 0.5 to 0.9 Ns/m 0.2 to 0.5 Ns/m 0.3 to 0.35 Ns/m

Fig. 3.87 Damped response for single half-sine force at mass 1 m , with peak 1 N and duration 0.01 s: (a) WT spectrum of v(t) − x2 (t) superposed to the FEP, (b) percentage of instantaneous total energy in the NES.

peak amplitude of the applied impulse was selected in the range 1–40 N to highlight the qualitatively different damped transitions and energy exchanges taking place at different energy levels. In Figure 3.87 we depict the damped responses for excitation of mass m1 with a half-sine force with peak equal to 1 N. Although both linear modes participate (at least initially) in the response, the contribution of the in-phase linear mode is dominant and more persistent (see Figure 3.87a). It is clear that the weakly nonlinear damped NNM S111++− is mainly excited in this case, so the response remains localized to the LO and not more than 0.3% of the instantaneous total energy is transferred to the NES at any given time. As a result, negligible TET takes place in this case. Note that there is also a small contribution from the higher weakly nonlinear damped NNM S111+−+ but this does not affect significantly TET in this case.

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Fig. 3.88 Damped response for single half-sine force at mass m1 , with peak 15 N and duration of 0.01 s: (a–c) transient responses, (d) WT spectrum of v(t) − x2 (t) x t - superposed to the FEP, (e) percentage of instantaneous total energy in the NES.

By increasing the forcing peak to 15 N (see Figure 3.88), the initial energy of the system exceeds the critical threshold for in-phase TET (see the FEP of Figure 3.80). The branch S111+++ is excited in this case, and the instantaneous total energy in the NES remains below 40% of the total energy of the system at any given instant of the motion. After t = 5.5 s, the participation of the in-phase mode in the system

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Fig. 3.89 Damped response for single half-sine force at mass m1 , with peak 27 N and duration 0.01 s: percentage of instantaneous total energy in the NES.

response is negligible, a sign that a significant portion of the energy contained in this mode has been transferred to and dissipated by the NES. Higher-frequency components are present in the relative displacement across the nonlinear spring v(t)−x2 (t) (see Figure 3.88c), but these are mainly non-dominant harmonics of the damped response and do not correspond to a nonlinear resonance interaction of the NES with the out-of-phase linear mode. Hence, the energy initially imparted to the out-ofphase linear mode remains in that mode, and is dissipated by the dampers of the LO; this explains the relatively weak TET evidenced for this force level. The damped dynamics remains qualitatively unchanged until the force peak reaches 27 N, where the percentage of instantaneous energy transferred to the NES reaches levels of up to 70% (Figure 3.89). A qualitatively different picture of the damped dynamics, however, occurs when the force peak increases to 28 N (see Figure 3.90). This is due to the fact that for this level of impulsive force the initial energy of the system exceeds the threshold for occurrence of out-of-phase TET in the system (see Figure 3.80). From the numerical results of Figure 3.90 it is clear that in this case the damped dynamics possess two distinct regimes. In the initial regime of the motion (0 < t < 2 s) the NES engages in 1:1:1 TRC with the high-frequency out-of-phase linear mode (see Figure 3.90d) as it traces the NNM branches S111+−+ and S111+−−. During this initial stage of the dynamics there occurs strong out-of-phase TET at a fast time scale, so that at t ≈ 1 s, the NES carries 89% of the instantaneous total energy, and the participation of the out-of-phase linear mode in the damped response drastically decreases with time as it looses energy to the NES. We conclude that in the initial stage of the motion the NES extracts energy from the out-of-phase linear mode and locally dissipates it without ‘spreading it back’ to the LO. In terms of the previously introduced notation out-of-phase fundamental TET takes place during this initial stage of the damped dynamics, and the motion resembles that depicted in Figure 3.85. In that case, however, at the later stage of the motion the response underwent a transition to a low-frequency subharmonic tongue, whereas in the present case a different damped transition follows after the initial excitation of S111+−−.

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Fig. 3.90 Damped response for single half-sine force at mass m1 , with peak 28 N and duration 0.01 s : (a–c) transient responses, (d) a resonance capture cascade (RCC) in the WT spectrum of v(t) − x2 (t) superposed to the FEP, (e) percentage of instantaneous total energy in the NES.

Indeed, for t > 2 s there occurs a damped transition to the damped NNM S111+++, as the NES escapes TRC with the out-of-phase linear mode and engages in TRC with the in-phase linear mode of the LO; as a result, starting from t = 2 s the NES starts extracting energy from the in-phase linear mode and, from t = 3.5 s strong in-phase TET to the NES occurs, with the instantaneous total energy in the NES reaching levels of 90% of total instantaneous energy of the system. This is an example of occurrence of a resonance capture cascade (RCC), i.e., of a sequential transient resonance interaction of the NES with both modes of the

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Fig. 3.91 Damped response for single half-sine force at mass m1 , with peak 40 N and duration 0.01 s: (a–c) transient responses, (d) a resonance capture cascade WT spectrum of v(t) − x2 (t) superposed to the FEP, (e) percentage of instantaneous total energy in the NES.

primary system. The NES first extracts and dissipates almost the entire energy of the out-of-phase linear mode, before engaging in resonance and extracting energy from the in-phase mode linear mode. What triggers the RCC is the dependence of the instantaneous frequency of the NES on its energy, and, more importantly, the lack of a preferential resonance frequency of the NES due to its essential stiffness nonlinearity. It follows that depending on its instantaneous energy, the NES is capable of resonantly interacting with both linear modes, extracting energy from the

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Fig. 3.92 Damped response for single half-sine force at mass m1 , with peak 6 N and duration 0.25 s: WT spectrum of v(t) − x2 (t) superposed to the FEP.

higher-frequency mode before engaging the lower frequency one. What is especially notable is that the process of RCC is adaptive and purely passive, as the NES ‘tunes itself’ with the most highly energetic linear modes irrespective of their frequencies, before making a transition to modes with lower energies. RCC gives rise to multi-frequency TET from the LO to the NES, which becomes increasingly more broadband as the number of linear modes participating in the RCC increases (see Figure 3.90c). We emphasize the capacity of the NES to engage in TRC with modes of the primary system at arbitrary frequency ranges (provided, of course, that these modes do not posses nodes close to the point of attachment of the NES), as this underlines the broadband feature of nonlinear TET; this is qualitative different from the narrowband action of the classical linear vibration absorber, and is a feature of the NES that renders it especially suitable for practical applications. Moreover, the phenomenon of RCC is a distinct feature of MDOF LOs with attached NESs, as it cannot be realized in the two-DOF system examined in previous sections. Figure 3.91 proves that RCCs are robust and persist for higher peak force amplitudes. For the increased impulsive level of 40 N considered in that Figure, the initial TRC of the NES dynamics with the out-of-phase linear mode (resulting in suppression of the out-of-phase linear mode during the first few cycles of the damped response), and the subsequent damped transition to TRC with the in-phase linear mode are even more evident. Finally, in Figure 3.92 we compare the damped transitions of the previous case (force peak of 40 N and duration 0.01 s) to the ones occurring for an impulsive force of longer duration (0.25 s) but smaller peak (6 N) so that the total initial energy imparted to the system by the impulse remains constant. We note that due to the increased peak duration, the participation in the damped response of the out-ofphase linear mode drastically decreases (compare Figures 3.91 and 3.90d), so that in-phase TET occurs from the beginning of the motion and no RCC occurs. This case is similar to the case presented in Figure 3.84, where direct excitation of the backbone branch S111+++ was considered (the only difference being the stronger higher harmonics that occur in the present case).

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From the previous results we conclude that the duration and amplitude of the applied half-sine pulse have an important influence on the damped dynamics and TET in the system. Depending on these parameters (but also on damping), different branches of the FEP may be excited or traced during the damped nonlinear transitions, affecting the strength of TET. To provide an additional example of the complex, multi-frequency transitions that can take place in coupled oscillators with essentially nonlinear local attachments, we consider the following alternative three-DOF system (Kerschen et al., 2006a): x¨2 + ω02 x2 + λ2 x˙2 + d(x2 − x1 ) = 0 ˙ + d(x1 − x2 ) + C(x1 − v)3 = 0 x¨1 + ω02 x1 + λ1 x˙1 + λ3 (x˙1 − v) εv¨ + λ3 (v˙ − x˙1 ) + C(v − x1 )3 = 0

(3.189)

with parameters ω02 = 136.9, λ1 = λ2 = 0.155, λ3 = 0.544, d = 1.2 × 103 , ε = 1.8, and C = 1.63×107, corresponding the linearized natural frequencies ω1 ≡ 2πf1 = 11.68 rad/s and ω2 ≡ 2πf2 = 50.14 rad/s. In Figure 3.93a we present the relative response v(t) − x1 (t) of the system for initial displacements x1 (0) = 0.01, x2 (0) = v(0) = −0.01 and zero initial velocities. The multi-frequency content of the transient response is evident, and is quantified in Figure 3.93b, where the instantaneous frequency of the time series is computed by applying the numerical Hilbert transform (Huang et al., 1998). As energy decreases due to damping dissipation, an interesting RCC takes place, involving as many as eight TRCs. The complexity of the RCC is evidenced by the fact that of these eight TRCs only two (labeled IV and VII in Figure 3.93b) involve the linearized in-phase and out-of-phase modes of the linear oscillator, while the remaining ones correspond to essentially nonlinear interactions of the NES with a number of low- and high-frequency nonlinear modes of the system (which apparently have no analogues in the linearized dynamics). During each TRC there occur energy exchanges between the NES and the the nonlinear mode involved in the resonance capture, after which escape from TRC occurs and the NES engages in transient resonance with the next mode of the series. Clearly, the main ‘tuning’ parameter that controls this purely passive RCC is the instantaneous energy of the system and its rate of decrease due to damping dissipation. In essence, the NES acts as a passive, broadband boundary controller, absorbing, confining and eliminating vibration energy from the linear oscillator. In the two additional applications that follow, we demonstrate the occurrence of RCCs in coupled MDOF oscillators with essentially nonlinear attachments. In the first application we consider the six-DOF system x¨1 + 0.014x˙1 + 2x1 − x2 = 0 x¨2 + 0.014x˙2 + 2x2 − x1 − x3 = 0 x¨3 + 0.014x˙3 + 2x3 − x2 − x4 = 0

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Fig. 3.93 Resonance capture cascade (RCC) in the damped transient dynamics of system (3.189): (a) relative response v(t) − x1 (t), (b) instantaneous frequency of v(t) − x1 (t) computed by the Hilbert transform (eight TRCs indicated).

x¨4 + 0.014x˙4 + 2x4 − x3 − x5 = 0 x¨5 + 0.0141x˙5 − 0.0001v˙ + 2x5 − x4 + (x5 − v)3 = 0 0.05v¨ + 0.0001(v˙ − x˙5 ) + (v − x5 )3 = 0

(3.190)

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Fig. 3.94 RCC in the damped dynamics of the six-DOF system (3.190) following direct excitation of the fourth linear mode: (a) relative response v(t) − x5 (t), (b) WT spectrum of v(t) − x5 (t) superposed to the FEP.

with initial excitation of only the fourth mode of the linear primary system. In Figure 3.94 we depict the relative response v(t) − x5 (t), along with its WT spectrum superimposed to the FEP of the underlying Hamiltonian system of (3.190); for clarity, only the first four linear modes are depicted in the FEP. We note that a RCC occurs in this case, leading to multi-frequency TET from the primary system to the NES. After an initial TRC of the NES dynamics with the fourth linear mode (labeled TRC 1 in Figure 3.94), a damped transition occurs after which the NES engages in TRC with the second linear mode (TRC 2). At a later stage of the dynamics a second damped transition occurs leading to final TRC of the NES dynamics with the first linear mode (TRC 3). This application illustrates clearly the usefulness of the utilization of combined WT spectra and FEPs as tools for interpreting useful nonlinear transitions. In the next application we consider an (N + 1)-DOF linear chain of coupled oscillators (the primary system) with a grounded NES (Configuration I – see Section 3.1) attached to its end (Vakakis et al., 2003). Each linear oscillator of the chain possesses unit mass and grounding stiffness ω02 , and is coupled to its neighboring oscillators by linear stiffnesses of characteristic d. The primary system possesses (i) T ] and (N+1) correspond(N +1) mass-normalized eigenvectors φ (i) = [φ0(i) . . . φN ing distinct eigenfrequencies ωi , i = 0, 1, . . . , N. The responses of the oscillators of the primary system are denoted by x0 (t), . . . , xN (t), where x0 (t) is the response of the point of attachment to the NES. These responses are then expressed in modal series: N φi(k) ak (t), i = 0, 1, . . . , N xi (t) = k=0

We express the equations of motion of the system using modal coordinates for the primary system

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Fig. 3.95 RCC in the damped dynamics of the 11-DOF system (3.191) with N = 9: (a) multifrequency response of the NES, (b) instantaneous frequency (t) of the NES versus time (the 10 linear modes of the chain are denoted by dashed lines).

v(t) ¨ + Cv (t) + ελv(t) ˙ + ε v(t) − 3

N

φ0(k) ak (t)

=0

k=0

a¨ m (t)

2 + ωm am (t)

+ ελa˙ m (t) + ε

N

φ0(k) φ0(m) ak (t) −

φ0(m) v(t)

=0

k=0

(3.191) with m = 0, 1, . . . , N. For the numerical simulation we considered a chain of ten linear oscillators (N = 9) with parameters ω02 = 0.4, d = 3.5, C = 5.0, λ = 0.5, ε = 0.1 and initial conditions v(0) = v(0) ˙ = 0, xm (0) = 0, m = 0, 1, . . . , 9 and x˙m (0) = 0, , m = 0, 1, . . . , 8, x˙ 9 (0) = 70. This corresponds to an impulsive excitation being applied at t = 0 to the oscillator of the chain most distant from the NES. In Figure 3.95 we

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present the transient response of the attachment v(t), together with its instantaneous frequency of oscillation (t) versus time. Note the strong RCC taking place in the damped dynamics involving as many as six of the linearized modes of the chain, including both modes located at the boundaries of the frequency spectrum of the chain. This application further demonstrates the capacity of the NES for broadband TET from the primary system. The final series of numerical simulations demonstrates the superior performance of an essentially nonlinear attachment (NES) as passive absorber of shock energy of a linear MDOF system of coupled oscillators, when compared to the classical linear absorber (or tuned mass damper – TMD). To this end, we consider the following eleven-DOF system with a strongly nonlinear end attachment (Ma et al., 2008): ε v¨ + ελ(v˙ − x˙0 ) + C(v − x0 )3 = 0 x¨0 + ελx˙0 + ω02 x0 − ελ(v˙ − x˙ 0 ) − C(v − x0 )3 + d(x0 − x1 ) = 0 x¨j + ελx˙j + ω02 xj + d(2xj − xj −1 − xj +1 ) = 0, x¨9 + ελx˙9 + ω02 x9 + d(x9 − x8 ) = 0

j = 1, . . . , 8 (3.192)

In this example we consider an ungrounded lightweight NES (of Configuration II – see Section 3.1) by assuming that 0 < ε 1. We assume that the system is initially at rest, and an impulse of magnitude X is applied at t = 0 to the left boundary of the linear chain, corresponding to initial conditions, v(0) = v(0) ˙ = 0; xp (0) = 0, p = 0, . . . , 9; x˙9 (0+) = X; and x˙k (0) = 0, k = 0, . . . , 8. To study TET efficiency, i.e., the capacity of the NES to passively absorb and locally dissipate impulsive energy from the linear chain, we employ the instantaneous and asymptotic energy dissipation measures (EDMs) defined by relations (3.4), suitably modified for system (3.192): t λ2 [v(τ ˙ ) − x˙0 (τ )]2 dτ 0 ENES (t) = × 100, ENES,t 1 = lim ENES (t) t 1 X2 /2 In Figure 3.96a we present the plot of the EDM ENES,t 1 as function of the stiffness characteristic C of the NES, for impulse strength X = 4.3, system parameters ω02 = 1.0, d = 2.0, and two values of damping, namely, ελ = 0.0125, and 0.025 (Ma et al., 2008). For comparison, we also depict the corresponding EDMs for a chain with a linear TMD attached at its end, with identical parameter values. Clearly, the TMD proves to be effective only in a narrow band of small stiffness values, i.e., in the neighborhood of resonance with the chain. On the contrary, the NES proves to be more effective than the TMD, since it is capable of passively absorbing a significant portion of the impulsive energy of the chain over a wide range of values of C; this is due to the capacity of the NES to engage in resonance capture and passively absorb energy from any of the modes of the chain, irrespective of their actual natural frequencies. We note that as much as 37% of input energy is passively

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Fig. 3.96 Comparison of the TET for the case of linear (TMD) and strongly nonlinear NES) attachments: (a) EDM ENES,t1 for varying stiffness C, and two damping values; (b) EDM ENES (t) for specific values of C and fixed damping ελ = 0.0125.

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absorbed and eventually dissipated by the NES, and that, even away from the region of optimal TET, the NES is capable of significant TET. In Figure 3.96b we depict the instantaneous EDM ENES (t) for the case of optimal TET and ελ = 0.0125, and compare it to the corresponding EDM for the linear TMD with optimal parameters. It is interesting to note that the sequence of early-time arrivals to the attachments of reflected wavepackets from the boundaries of the chain are associated with sudden increases of the rates of energy dissipation. In Ma et al. (2008) the capacity for TET of the system (3.192) is related to the shapes and energies of the underlying proper orthogonal modes (POMs) of the transient dynamics (Cusumano et al., 1994; Georgiou et al., 1999; Azeez and Vakakis, 2001; Ma and Vakakis; 1999). It is shown that enhanced TET is related to excitation of dominant highly energetic POMs that localize to the NES. This observation is then used for constructing accurate low-dimensional reduced-order models for the TET dynamics.

3.5.1.3 Experimental Demonstration of Multi-Frequency TET Although experimental TET results will be presented in detail in Chapter 8, in this section we provide some preliminary experimental evidence in support of the previous theoretical findings. The experimental measurements reported here were performed using the fixture depicted in Figure 3.97, composed of a two-DOF linear oscillator (the primary system) coupled to an essentially nonlinear ungrounded SDOF attachment (an NES of Configuration II). The primary system consists of two cars made of aluminum angle stock which are supported on a straight air track (that reduces friction forces during the oscillation). The NES consists of a shaft supported by two linear bearings; steel plates on the shaft clamp two steel wires configured with practically no pretension, realizing the essential cubic stiffness nonlinearity C (see Section 2.6 for a discussion on the practical realization of essential cubic stiffness nonlinearity, and also Chapter 8). The wires are connected to the primary system through clamps at their outer ends. A short half-sine force pulse representative of a broadband input is applied to the left car (of mass m1 ) of the primary system (see Figure 3.97), and the damped responses of the three oscillators are measured using accelerometers. Estimates of velocities and displacements are obtained by numerically integrating the measured acceleration time series, and the resulting signals are high-pass filtered to remove spurious components introduced by the integration procedure. The parameters of the experimental fixture were measured before the experimental tests. Prior to system identification, the cars of the primary system and the NES were weighed as m1 = 0.6285 kg, m2 = 1.213 and ε = 0.161 kg, respectively, which implies a low mass ratio equal to 8.7%. Experimental modal analysis was then carried out to measure the stiffness and damping parameters of the integrated three-DOF experimental system. First, the primary system was disconnected from the NES, and experimental modal analysis was performed using the stochastic subspace identification method (Van Overschee and De Moor, 1996) to provide the

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Fig. 3.97 Experimental fixture: (a) NES, (b) schematic of the two-DOF primary system and the SDOF NES.

two natural frequencies estimates of 1.95 Hz and 6.25 Hz, respectively. Because the masses of the primary system were known, the stiffness and damping parameters k1 , k12 , c1 , c2 , and c12 could be deduced from this experimental modal analysis, and are listed in Table 3.4. In the second step of modal analysis the primary system was clamped, an impulsive force was applied to the NES using a modal hammer, and the NES acceleration and applied force were measured. The restoring force surface method (Masri and Caughey, 1979) was then used to estimate the coefficient of the essential nonlinearity C and the damping coefficient cv of the NES. For further details about the procedure, the reader is referred to Chapter 8. The identified system parameters of the experimental fixture are listed in Table 3.4. Damping estimation is a difficult problem in this fixture due to the presence of several ball joints and bearings, and of

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Fig. 3.98 WT spectra of the experimental relative responses v(t) − x2 (t) superposed to the FEP for impulsive forces of duration approximately 0.01 s : (a–d) Cases I–IV.

the air track. It was found that damping was rather sensitive to the force level, which is the reason why ranges rather than fixed values are given in Table 3.4. Even though the air-track greatly reduced friction forces in the system, at low forcing amplitudes friction seemed to intervene significantly with the experimental measurements. In the first series of experimental tests, the mass m1 was impulsively excited by impulsive forces of durations around 0.01 s and varying magnitudes. Four cases of (gradually increasing) input energy were considered, labeled as I (0.0103 J), II (0.0258 J), III (0.0296 J) and IV (0.0615 J), respectively. The superposition of the WT spectrum of the relative response across the nonlinear spring of the NES to the FEP for each case is depicted in Figure 3.98. Starting with the case of lowest impulsive energy (Case I, Figure 3.98a) the damped NNM S111+++ is excited from the beginning of the motion. This means that the input energy is already above the critical energy threshold for in-phase fundamental TET, but below the energy threshold for resonance with the out-of-phase linear mode; this case resembles the low-energy numerical case depicted in Figure 3.87a.

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Moving to Case II (Figure 3.98b) S111+++ is again excited, but higher harmonic components are now present, and the dynamics resembles the corresponding numerical simulation presented in Figure 3.88d. By slightly increasing the input impulsive energy (Case III, Figure 3.98c), the energy threshold for resonance interaction of the NES dynamics with the out-of-phase linear mode is exceeded; as a result the damped NNM S111+−− is initially excited leading to initial out-of-phase fundamental TET, followed by a damped transition a to S111+++ and in-phase fundamental TET. Hence, in this case there is experimental confirmation of a resonance capture cascade (RCC) that occurs in the transient dynamics of the system. The series of TRCs that occur in this RCC resembles the numerical simulation of Figure 3.90d. Finally, in the experimental measurement corresponding to the highest energy input, Case IV (Figure 3.98d), a stronger RCC similar to the one occurring in Case III takes place resembling the corresponding highest-energy simulation of Figure 3.91d. Overall, the experimental findings are in accordance with, and validate the numerical results discussed in Section 3.5.1.2. Further results for the RCC taking place in Case IV are displayed in Figure 3.99. During the first few cycles of the damped response, the NES clearly resonates with the out-of-phase mode and strong out-of-phase TET is realized, with as much as 87% of the instantaneous total energy being passively captured by the NES around t ≈ 2 s; after this initial regime of the motion the participation of the out-of-phase mode in the dynamics drastically reduces. After t = 2 s a damped nonlinear transition in the dynamics takes place and the NES engages in TRC with the in-phase linear mode extracting energy from it in in-phase fundamental TET. The comparison of the experimentally measured results of Figures 3.99c, e with the corresponding theoretically predicted ones depicted in Figures 3.99d, f, shows close agreement between experiment and theory in the initial highly energetic phase of the motion 0 < t < 4 s. Specifically, the sequential interaction of the NES with both linear modes during the RCC is accurately reproduced by the numerical model. The observed discrepancies between experimental and theoretical results that occur in the later, low-energy regime of the damped motion, may be attributed to the sensitivity of the low-energy dynamics of the system on unmodeled friction forces in the bearings and the air-track of the experimental fixture. We note, however, that since in the later stage of the motion the energy level of the system is small, no significant qualitative features of the dynamics are missed due to dry friction effects. No attempt was made to optimize TET in the experimental fixture, i.e., to maximize energy dissipation by the NES, since the purpose of the experimental tests was to confirm the numerical predictions and, especially the occurrence of RCCs in the three-DOF system. Some additional experimental results are presented in Figure 3.100, to show that RCCs and multi-modal (multi-frequency) TET can occur for even smaller mass ratios. For this particular experimental series we considered two mass ratios equal, to 6% (corresponding to m1 = 1.1295 kg, m2 = 1.553 kg and ε = 0.161 kg – Figure 3.100a) and 8.7% (Figure 3.100b). The applied impulsive force to mass m1 was kept fixed, with duration equal to 0.15 s. A clear RCC is observed for the system with reduced mass ratio (Figure 3.100a); as pre-

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Fig. 3.99 Experimental RCC, Case IV: (a–c) measured responses; (d) theoretically predicted NES response; (e, f) measured and theoretically predicted percentage of instantaneous total energy in the NES.

dicted by the numerical simulations, the out-of-phase damped NNM S111+−− is excited in the initial stage of the motion (leading to out-of-phase fundamental TET), followed by a transition of the dynamics to TRC with the in-phase damped NNM S111+++ (and in-phase fundamental TET). For the system with increased mass ratio (Figure 3.100b) a similar, albeit weaker RCC is observed in the experimental measurements.

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Fig. 3.100 RCCs in the WT spectra of experimental relative responses v(t) − x2 (t) superimposed to the FEP: (a) 6%; (b) 8.7% mass ratio, peak duration of 0.15 s.

3.5.2 Semi-Infinite Chain of Linear Oscillators with an End SDOF NES In this final section we study the dynamics of a semi-infinite linear chain of coupled oscillators with an essentially nonlinear attachment (NES) at its boundary. Considering first the undamped system we analyze families of localized nonlinear standing waves situated inside the lower or upper attenuation zones of the dynamics of the linear chain, with energy being predominantly confined to the NES. In addition, we estimate the energy radiated from the NES back to the chain, when the NES is excited under non-resonant conditions by wavepackets with dominant frequencies inside the propagation zone of the dynamics of the chain. We show that in this system TET from the semi-infinite chain to the NES is possible even in the absence of damping. The TET dynamics, however, is qualitatively different in this case: instead of TET through TRCs as in the case of finite-DOF weakly damped oscillators considered previously, TET in the undamped infinite-DOF system relies on the excitation of in-phase standing waves localized to the NES. Passive TET from the semi-infinite linear chain to the NES is confirmed numerically. The analysis of the undamped system follows closely the work by Manevitch et al. (2003). Then we analyze the weakly damped semi-infinite linear chain with a weakly damped essentially nonlinear oscillator attached (Vakakis, 2001). Using a reduction approach, we reduce the dynamics to a complex integro-differential equation and then analyze TET using the complexification-averaging approach (CX-A). We show that TET in the weakly damped system is generated by TRCs as in the case of finitedimensional discrete oscillators discussed in previous sections.

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3.5.2.1 Dynamics of the Chain-NES Interaction The dynamics of linear or nonlinear periodic chains with local attachments (or ‘defects’) is a research area with many interesting applications, such as in the areas of optical and magneto-optical waveguide periodic arrays, semiconductor superlattices, layered composite media, micro- or nano-lattices used as thermal barriers, in photonic band-gap materials (photonic crystals), and bio-molecular engines (see, for example, the works by Chen and Mills, 1987; Eggleton et al., 1996; Akozbek and John, 1998). Gendelman and Manevitch (2000) examined the dynamics of a semi-infinite string with a strongly nonlinear oscillator attached to its end, and studied energy transfer from the string to the attached oscillator through impeding short rectangular pulses. They found that excitation of vibrations in the oscillator was possible through this nonlinear interaction. Lazarov and Jensen (2007) studied the influence of stiffness nonlinearities on the filtering properties (i.e., the low-frequency bands) of infinite linear chains with attached nonlinear oscillators; they found that the position of low-frequency bands in these systems depended on the form of the nonlinearity and the level of energy of the motion. Goodman et al. (2004) analyzed the dynamic interaction of a nonlinear Schrödinger soliton with a local defect and proposed a mechanism of resonance energy transfer from the impeding soliton to a nonlinear standing wave localized at the defect. Additional works (Kivshar et al., 1990; Forinash et al., 1994; Goodman et al., 2002a, b) examined nonlinear interactions of standing or traveling waves in infinite nonlinear media with local defects. The system under consideration is a semi-infinite chain of coupled linear oscillators, whose free end is weakly coupled to an essentially nonlinear attachment. We wish to study the possibility of passive TET from the chain to the nonlinear attachment, which then acts, in essence, as an NES. Each oscillator of the chain is grounded and possesses only next-neighbor interactions. Assuming no damping in the system, the set of equations governing the dynamics is given by x¨k + c2 (2xk − xk−1 − xk+1 ) + ω02 xk = 0,

k<0

x¨0 + c2 (x0 − x−1 ) + ε(x0 − v) + ω02 x0 = 0 v¨ + 8av3 − ε(x0 − v) = 0

(3.193)

where xk denotes the response of the k-th oscillator of the linear chain, v the response of the NES, c2 the linear coupling stiffness between adjacent oscillators, and ω02 the linear grounding stiffness of each oscillator. The dimensionless perturbation parameter 0 < ε 1 scales the weak coupling between the linear chain and the NES, and the parameter a denotes the strength of the essential (nonlinearizable) stiffness nonlinearity of the attachment. Note that in this case we consider a grounded form of NES (of the type presented in Figure 3.1 – Configuration I), and the mass of the NES is not assumed to be small (as in previous sections). Instead, in the following analysis the small parameter characterizes the weak coupling between the semi-infinite chain and the NES.

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Before discussing the chain-NES dynamic interaction, we examine briefly the dynamics of the infinite chain with no boundaries and no nonlinear attachment. The dispersion relation of the infinite linear chain is composed of two attenuation zones (AZs) and a single propagation zone (PZ) in the frequency domain (Brillouin, 1953; Mead, 1975). In the AZs the chain supports two families of standing waves with exponentially decaying envelopes, which represent near field solutions of the infinite chain. The lower AZ is in the frequency range ω ∈ , [0, ω0 ), whereas the upper

AZ extends up to arbitrarily large frequencies, ω ∈ ( ω02 + 4c2 , ∞). In the PZ, , ω ∈ (ω0 , ω02 + 4c2 ), the infinite chain supports two families of traveling waves that propagate unattenuated in opposite directions of the chain. It is well known that energy through the chain can only propagate by means of traveling waves, i.e., only with frequencies inside the PZ. The bounding frequencies ωb1 = ω0 and ωb2 = ,

ω02 + 4c2 that separate the two AZs from the PZ correspond to in-phase and outof-phase normal mode oscillations (i.e., synchronous non-decaying standing waves) of the infinite chain (Mead, 1975). Now suppose that the integrated semi-infinite chain-NES system is initially at rest, and at t = 0 an impulse F δ(t) is applied to an oscillator of the chain. Then, the motion of the system at t = 0+ is a conservative free oscillation and the energy transfer through the chain and to the nonlinear oscillator may be approximately analyzed with the help of linear theory. The first basic problem of the chain-NES dynamics is to establish the type of excitation of the NES by the chain. Clearly, the excitation of the nonlinear oscillator is caused by an initial right-going traveling wave propagating through the chain; depending on the form of this wave, the initial chain-NES dynamic interaction may occur under resonance or non-resonance conditions. Resonance interaction is most probable if the time and distance needed for the wave to travel through the chain and impede to the NES suffice for the formation of a wave packet with primary frequency > ω0 . Then the excitation of the NES occurs approximately under condition of 1:1 resonance. In that case, the chain may be approximately simulated as a single particle acting on the nonlinear oscillator with prescribed force, i.e., possessing certain amplitude and frequency, and applied during a known time interval. All these parameters may be obtained simply by solving the linear problem for the chain. On the other hand, non-resonant interaction between the chain and the NES corresponds to the situation when the wave packet disturbance in the chain does not have sufficient time and space to form into a cohesive wave form, and, as a result the force that excites the nonlinear oscillator is non-harmonic. The next basic problem of the chain-NES dynamics focuses on the radiation (backscattering) of energy from the NES back to the chain after the initial wave has impeded to it. This process is the most interesting from an analytical point of view, and as shown below, can be divided into two essentially different parts: (a) the transient radiation of excess energy from the NES back to the chain through traveling

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3 Nonlinear TET in Discrete Linear Oscillators

or near-field waves; and (b) the formation at the NES of a localized standing wave mode. In the following analysis we discuss these issues separately. We first consider radiation (backscattering) of energy from the NES to the semiinfinite chain through traveling waves, i.e., waves with frequencies in the PZ. Specifically, we consider the state of the system after the main impulsive excitation of the chain commences. The NES is excited with a wave packet with predominant frequency > ω0 (i.e., in the PZ on the linear chain), as these are the only waves that can travel from the source of the excitation through the chain and impede to the NES; moreover, this frequency most probably belongs to the zone of moderate wavenumbers corresponding to the maximum of group velocity. Therefore, under conditions of 1:1 resonance the energy of the NES is radiated back to the chain in the diapason of moderate wavenumbers, and for qualitative purposes the energy radiation may be studied in the continuum approximation. These assumptions regarding the radiation process will be proved and validated a posteriori by the derived results. In this case we propose the following ordering of the variables of system (3.193), v = O(1), xk = O(ε), k = 0, −1, −2, . . . . Hence, to a first approximation we consider the following continuum approximation of equations (3.193): 2 ∂ 2 x(s, t) 2 2 ∂ x(s, t) − c r + ω02 x(s, t) ≈ 0, 0 ∂t 2 ∂s 2

c2 r0

s≤0

∂x(0, t) ≈ εv(t) ∂s

v (t) + 8αv 3 (t) ≈ 0

(3.194)

where r0 is the distance between oscillators, so that the k-th oscillator corresponds to the position s = kr0 , k = 0, −1, −2, . . . of the one-dimensional continuum, and primes denote differentiation with respect to s. In deriving (3.194) we replaced the infinite set of variables xk (t), k ≤ 0 by the continuous variable x(s, t), s ≤ 0, and the semi-infinite set of ordinary differential equations of (3.193) by a single partial differential equation [the first of relations (3.194)]. The last equation in (3.194) describes (to a first approximation) a vibration of the nonlinear oscillator with constant amplitude and frequency. In fact, this is only an approximation since in actuality the amplitude and frequency of the nonlinear attachment varies slowly with time due to energy loss by energy radiation to the chain. However, it will be shown that this radiation effect is of order O(ε2 ), and, therefore, the variations of the amplitude and the frequency of the nonlinear oscillator are nearly adiabatic up to O(ε2 ). The flow of energy through the chain in the continuum limit may be estimated by recalling that the energy stored in the spatial interval a < s < b of the chain is computed as 2 2 1 b ∂x 2 2 ∂x 2 2 (3.195) + c r0 + ω0 x ds Eab (t) = 2 a ∂t ∂s

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so the flow of energy in the chain is approximated as dEab = dt

b

(x˙ x¨ + c2 r02 x x˙ + ω02 x ) ds

a

b =

x˙ x¨ + c2 r02 x x˙ + (c2 r02 x − x) ¨ x˙ ds

a

b =

c2 r02

1 1 (x x) ˙ ds =c2 r02 x x˙ 1x=b − x x˙ 1x=a

(3.196)

a

Therefore, the rate of total energy radiated from the nonlinear oscillator back to the chain is estimated by setting a = −∞, b = 0 in (3.196), and taking into account that (due to causality) the chain is motionless in the far field s → −∞: dENES dEchain = c2 r02 x (0) x(0) ˙ =− dt dt

(3.197)

The rate of energy loss of the NES due to radiation is the negative of the corresponding energy gain by the chain, which is a consequence of the lack of damping dissipation in system (3.193). We note that the energy contained in the NES depends only on its instantaneous frequency of oscillation, and this fact is crucial in our discussion. Indeed, considering the dominant harmonic component at frequency ω of the (approximately) periodic response v(t) of the NES, the outgoing radiated harmonic traveling wave in the chain may be expressed as xω (s, t) ≈ Aω ej (ωt +βs),

β = (cr0 )−1 (ω2 − ω02 )1/2 ,

ω ≥ ω0 ,

where j = (−1)1/2 . Due to the fact that this is a traveling wave emanating from the NES due to energy backscattering, it propagates in the direction of decreasing negative s, i.e., away from the NES and towards the far field s → −∞. The amplitude of this wave may be computed from the second of equations (3.194), −j ε Zω Aω ≈ , c ω2 − ω02

(3.198)

where Zω is the amplitude of the harmonic of v(t). Substituting this result into (3.197), and averaging over the period T = 2π/ω, we derive the following approximate expression for the rate of energy radiation at frequency ω in the PZ of the linear chain: ε2 ω |Zω |2 dEchain ≈ , , ω ≥ ω0 (3.199) dt 2c ω2 − ω2 0

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3 Nonlinear TET in Discrete Linear Oscillators

Hence, energy radiation is indeed of O(ε 2 ) which validates our previous assertions and assumptions. In actuality, the energy of the oscillator decreases slowly due to energy radiation back to the chain, and so does its instantaneous frequency of oscillation until it approaches the neighborhood of the lower bounding frequency ωb1 = ω0 . Clearly, expression (3.199) is not valid in the neighborhood of this bounding frequency, since the assumed scaling of v and xk does not hold there; this means that as ω slowly decreases towards ωb1 the traveling wave ansatz becomes invalid since the dynamics of the system approach the qualitatively different state of 1:1 resonance, which should be considered separately. The previous scenario is supported by the findings reported in Vakakis (2001) (and in Section 3.5.2.3), where numerical simulations of the dynamic interaction of a damped NES with a damped linear chain of coupled oscillators are presented. It is numerically shown (and analytically proven), that after some initial irregular transients (corresponding to the energy radiation phase described previously), 1:1 TRC between the in-phase normal mode of the chain (at frequency ω0 ) and the NES takes place. During this TRC strong energy exchanges between the two systems occur. The results (3.195–3.199) concerning monochromatic energy radiation from the nonlinear oscillator to the chain can be extended to the case of transient energy radiation. To show this, we Laplace-transform the first two linear equations of (3.194), assuming that the chain is initially at rest, and imposing the far field condition lims→−∞ x(s, t) = 0. This leads to the following expression for the Laplace transform U (0, p) = L[u(0, t)], where p is the Laplace transform variable, εV (p) 1/2 ε + c p2 + ω02 ε 1 2 + O(ε ) = εV (p) 1/2 − 2 2 c p + ω02 c p2 + ω02

U (0, p) =

(3.200)

where V (p) = L[v(t)]. Inverse Laplace-transforming the above expression and substituting the result into the last of equations (3.194) we obtain the following nonlinear integro-differential equation governing the transient energy radiation from the NES back to the chain: v(t) ¨ + 8αv 3 (t) =

ε t − ε v(t) − v(τ )J0 [ω0 (t − τ )] dτ c 0 t ε2 3 + 2 v(τ ) sin ω0 (t − τ )dτ + O(ε ) c ω0 0

(3.201)

In agreement with the previous simplifying analysis, the integral terms on the righthand side that govern energy radiation to the chain are of O(ε 2 ). In Section 3.5.2.3 we discuss in detail the solution of this integro-differential equation.

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From the above discussion we conclude that after a wavepacket impedes to the NES, its energy is slowly radiated back to the chain in an O(ε2 ) nonlinear dynamic interaction, until the dynamics approaches a regime of 1:1 resonance close to the lower bounding frequency ωb1 = ω0 . The dynamics of this resonance interaction is studied in the next section.

3.5.2.2 Nonlinear Resonance Interactions and TET We now focus in nonlinear resonance interactions occurring between the NES and the semi-infinite chain in the neighborhood of the lower bounding frequency of the PZ of the infinite chain. Later we will extend the analysis to resonance interactions occurring the the neighborhood of the upper bounding frequency. We commence right from the beginning that the problem of resonance in the system under consideration is by no means trivial, as we deal with a problem possessing an infinite number of DOFs and a local essential (strong) nonlinearity; moreover, the transient nature of the examined dynamical interactions complicates even further the analysis. Since no common ways exist to proceed with this problem, we need to apply some simplifying propositions that will enable us to analytically approximate in a self-consistent way the dynamic phenomena under investigation. We follow closely the analysis by Manevitch et al. (2003). However, as in the previous section, the validity of the assumptions made has to be checked a posteriori when the analytical results are derived. First, we assume that 1:1 resonance between the semi-infinite chain and the NES occurs at a frequency smaller than the lower bounding frequency ωb1 = ω0 , i.e., inside the lower AZ of the dispersion relation of the linear chain. It follows that the amplitudes of the responses of the oscillators of the chain decay exponentially with increasing distance from the NES. This basic simplifying assumption will be checked (and validated) through numerical simulations later. An additional simplification is achieved by supposing that the shape of this exponential amplitude decay is fairly approximated by a single exponent which is consistent with the dispersion relation of the linear chain, xj ≈ x0 eκj ,

j ≤ 0,

ω02 − 2 ≈ 2c2 (cosh κ − 1)

(3.202)

where denotes the fast frequency of oscillation of the oscillators of the chain [as explained below in relation (3.205)], and κ the frequency-dependent rate of exponential decay. It follows that, in contrast to the analysis of the previous section, we now seek standing-wave solutions localized to the NES. The assumption (3.202) introduces an approximation in the analysis, since it omits nonlinear effects in the decay rate which are present in the system; for an asymptotic study of near field solutions in nonlinear layered media we refer to Vakakis and King (1995). Substituting (3.202) into (3.193) we reduce approximately the dynamics to a system of two coupled ordinary differential equations:

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3 Nonlinear TET in Discrete Linear Oscillators

x¨0 + x0 [c2(1 − e−κ ) + ω02 ] + ε(x0 − v) = 0 v¨ + 8αv 3 − ε(x0 − v) = 0

(3.203)

This indicates that the problem of studying the resonance interaction of the NES with the semi-infinite chain can be reduced approximately to the simpler problem of resonance interaction between the NES and the nearest to it oscillator of the chain. Clearly, the biggest advantage gained by the above reduction is that the study of the resonance interaction may be performed by applying the CX-A method introduced in previous sections. To this end, we,introduce the complex variables ψ1 = x˙0 + j ωx0 and ψ2 = v˙ + j ωv with ω = c2 (1 − e−κ ) + ω02 + ε, which reduces (3.203) to the following set of first-order complex modulation equations: ψ˙ 1 − (j ω/2)(ψ1 − ψ1∗ ) − (j ω/2)(ψ1 − ψ1∗ ) + (j ε/2ω)(ψ2 − ψ2∗ ) = 0 ψ˙ 2 − (j ω/2)(ψ2 − ψ2∗ ) + (j a/ω3)(ψ2 − ψ2∗ )3 + (j ε/2ω)(−ψ2 + ψ2∗ + ψ1 − ψ1∗ ) = 0

(3.204)

where asterisk denotes complex conjugate. We now introduce the following approximate slow-fast partition of the dynamics, implying that in the studied resonance interactions there exists a single dominant fast frequency : ψk = ϕk (t) ej t ,

k = 1, 2

(3.205)

The fast frequency is assumed to be in the neighborhood of ωb1 = ω0 , and the complex amplitudes φk (t) to be slowly-varying; this implies that φ˙ k (t) = O(ε) or smaller. Substituting (3.205) into (3.204) and averaging over the fast frequency we obtain the following set of modulation equations governing the evolutions of the slow-varying complex amplitudes, ϕ˙1 − j µ1 ϕ1 + (j ε/2ω) ϕ2 = 0 ϕ˙2 + j µ2 ϕ2 − (3j a/ω3) |ϕ2 |2 ϕ2 + (j ε/2ω)ϕ1 = 0

(3.206)

where µ1 = ω − and µ2 = − (ω/2) + (ε/2ω). There are two different ways to proceed with the analysis of (3.206), both of which are equivalent. In the first approach we express the complex variables in in polar form, φk (t) = ak (t) ejβk (t ) , k = 1, 2, substitute into (3.206) and set the real and imaginary parts separately equal to zero. Then the following system of real modulation equations results: a˙ 1 − (ε/2ω) a2 sin(β2 − β1 ) = 0 ⇒ a12 + a22 = ρ 2 (3.207a) a˙ 2 + (ε/2ω) a1 sin(β2 − β1 ) = 0 a1 β˙1 − µ1 a1 + (ε/2ω) a2 cos(β2 − β1 ) = 0

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

a2 β˙2 + µ2 a2 − (3 a a23 /ω3 ) + (ε/2ω) a1 cos(β2 − β1 ) = 0

277

(3.207b)

Provided that a1 a2 = 0, we define the new phase difference variable θ = β2 − β1 and combine equations (3.207b) to get: θ˙ + µ2 + µ1 −

3αa22 ε + ω3 2ω

a2 a1 − a2 a1

=0

(3.207c)

Equations (3.207a, c) form an autonomous set of nonlinear evolution equations. The integral relation between the two amplitudes in (3.207a) is an energy-like expression indicating conservation of the total energy of the undamped system during the motion. Indeed, we note that for the type of localized standing waves considered here the total energy of the integrated semi-infinite chain-NES is finite and conserved. The stationary solutions of (3.207a, c) correspond to (approximately) timeperiodic localized standing waves of the integrated system. These are computed by solving the following set of nonlinear algebraic equations: a12 + a22 = ρ 2 , θ = 0

3αa22 ε ε a1 a2 ω + − =0 (3.208) + − 2 2ω 2ω a2 a1 ω3 , where we recall that ω = c2 (1 − e−κ ) + ω02 + ε and that the exponential decay factor κ is expressed in terms of the fast frequency by the second of relations (3.202), i.e., the linear dispersion relation of the infinite chain. Combining all these results we derive the following expression relating the fast frequency of oscillation to the decay factor κ (through the frequency ω): / 01/2 ω = ω02 + ε + (1/2)(2 − ω02 ) + (1/2) (ω02 − 2 )1/2 (ω02 − 2 + 4c2)1/2 (3.209) Since we are interested in localized standing waves with frequencies close to the lower bounding frequency ωb1 = ω0 but inside the lower AZ, we introduce at this point a frequency detuning parameter δω defined by the relation: 2 = ω02 − ε2 δω2 This leads to the following algebraic relations governing the amplitudes and decay factors of the nonlinear standing wave motions: ω = ω0 + [ε(1 + cδω)/2ω0 ] + O(ε2 ),

a12 + a22 = ρ 2

1 [ω0 + (ε/2ω0 )(1 + cδω)] + (ε/2ω0 ) − 3aa22[ω0−3 − (3ε/2ω05)(1 + cδω)] 2

a2 a1 + O(ε2 ) = 0 + (ε/2ω0 ) − (3.210) a2 a1

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3 Nonlinear TET in Discrete Linear Oscillators

The frequency of the slow modulation corresponding to the stationary solution is obtained by considering the phase relations (3.207b) and taking into account that θ = 0 ⇒ β1 = β2 : β˙1 = β˙2 = O(ε) = ω − − (ε/2ω) (a2 /a1 ) = (ε/2ω) 1 + cδω − (a2 /a1 ) + O(ε 2 )

(3.211)

This result is consistent with our assumption of slowly-varying phases. For fixed energy ρ and detuning frequency δω the set (3.210) is solved numerically for the amplitudes a1 and a2 . Then the corresponding phases are computed by means of (3.211). The localized standing wave solutions with frequency close to the lower bounding frequency of the chain are then approximated as follows: x0 (t) ≈ (a1 /ω) sin [t + β1 (t)] , x˙ 0 (t) ≈ a1 cos [t + β1 (t)] ,

xp (t) ≈ x0 (t) eκp , p ≤ 0

v(t) ≈ (a2 /ω) sin [t + β2 (t)] , v(t) ˙ ≈ a2 cos [t + β2 (t)]

(3.212)

This is a synchronous oscillation with constant amplitude, fast frequency = (ω02 − ε2 δω2 )1/2 , and effective frequency ωeffective = + β˙1 = ω0 + of (ε/2ω0 )[1 + cδω − (a2 /a1 )] + O(ε 2 ). In order to comply 1 the 1 assumptions 1 1 with the analysis these quantities should satisfy the relations, 1β˙1 1 = 1β˙2 1 (as this separates the slow and fast dynamics), and ωeffective = + β˙1 < ω0 (since this satisfies the condition that the frequency of the standing waves lies inside the lower AZ of the chain). In Figure 3.101 we depict the energy dependence of ωeffective for parameters c2 = 1, ω02 = 0.4, ε = 0.1, α = 5/8 and varying frequency detuning δω. These solutions correspond to a1 > 0 and a2 > 0, i.e., to in-phase motions between the NES and the adjacent oscillator of the chain, localized to the NES. Hence, the 1:1 resonance interaction between the NES and the chain close to the lower bounding frequency ωb1 = ω0 gives rise to a continuous family of localized, slowly modulated standing waves that lie inside the lower AZ of the chain; the decay rates of these waves increase as the frequency detuning δω is increased, further inside the lower AZ. We now discuss a second approach for analyzing the averaged set of complex slow modulations (3.206) that takes in account the integrability features of this set of equations. We start by noting that the set (3.206) is completely integrable, since it possesses the following two first integrals of motion: ρ 2 = |ϕ1 |2 + |ϕ2 |2 H = −j µ1 |ϕ1 |2 − j µ2 |ϕ2 |2 − (3j α/2ω3 ) |ϕ2 |4 + j λ(ϕ1∗ ϕ2 + ϕ2∗ ϕ1 ) (3.213)

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Fig. 3.101 Energy dependence of the effective frequencies of in-phase localized standing waves inside the lower AZ, for varying frequency detuning δω.

where λ = ε/2ω, µ1 = ω − and µ2 = − (ω/2) + (ε/2ω). We recall that ω is the reference frequency related to the rate of exponential decay κ, whereas is the (single) dominant fast frequency of the 1:1 resonance interaction. Taking into account the first integral of motion, we express the slowly-varying complex amplitudes as ϕ1 = ρ cos φ ej δ1 and ϕ2 = ρ sin φ ej δ2 , where ϕ and δk , k = 1, 2 are time-dependent angle variables. Employing the second integral of motion the set modulation equations (3.206) is transformed as follows: ϕ˙ = λ sin δ δ˙1 = µ1 − λ cos δ tan ϕ δ˙ = (µ1 + µ2 ) − (3αρ 2 /ω3 ) sin2 ϕ − λ cos δ (tan ϕ − cot ϕ) C = (µ1 + µ2 ) sin2 ϕ − 3αρ 2 /ω3 sin4 ϕ + 2λ sin ϕ cos ϕ cos δ (3.214) where C is a first integral of the motion, and δ = δ1 − δ2 . Setting Z ≡ sin2 ϕ we can solve exactly this first-order slow flow approximation. Indeed, the following analytic solution of (3.214) can be derived: / 01/2 Z˙ = 4λ2 Z(1 − Z) − [C − Z(µ1 − µ2 ) + (3αρ 2 /2ω3 )Z 2 ]2 / 2 0−1/2 ⇒ 4λ Z(1 − Z) − [C − Z(µ1 − µ2 ) + (3αρ 2 /2ω3 )Z 2 ]2 dZ = t + S (3.215) where S is a constant of integration, and the integral in (3.215) can be explicitly expressed in terms of elliptic integrals. Returning to the slow flow (3.214), it is of interest to study the case when the effective frequency of oscillation ωeffective is exactly equal to the prescribed fast frequency of the resonance, i.e., ωeffective = . In that case there is no slow frequency

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3 Nonlinear TET in Discrete Linear Oscillators

modulation of the fast oscillation (since then it holds that the slow-phases are stationary, i.e., β˙1 = β˙2 = 0), and the integrated chain-NES system executes purely time-periodic oscillations at the fast frequency . This special (pure fast frequency) solution is computed by solving the following (extended) set of stationary equations: 0 = sin δ ⇒ δ = 0, π ⇒ µ1 ∓ λ tan ϕ = 0 (µ1 + µ2 ) − (3αρ 2 /2ω3 ) sin2 φ ± λ(tan φ − cot φ) = 0

(3.216)

which leads to the following amplitude-frequency relation for this special, purely fast-frequency solution: ρ = ρ() =

1/2

2 µ1 () + λ2 () 3 λ2 () − µ2 () 3α µ1 () µ21 ()

(3.217)

This solution exists only in the finite interval ωmin < ω < ω0 inside the lower AZ, where ωmin is the solution of the equation µ1 ()µ2 () − λ2 () = 0. A typical plot depicting this solution is presented in Figure 3.102a. Note the breakdown of the analytical approximation in the neighborhood of the lower bounding frequency ωb1 = ω0 = 1. The corresponding physical energy of the oscillation is given by

ρ 2 () λ2 2 + µ (3.218) E() = 1 2(λ2 + µ21 ) 1 − exp(−2κ) where κ is the exponential decay rate of the localized standing wave. The corresponding plot is presented at Figure 3.102b. Note the abrupt energy increase as the PZ is approached, a feature consistent with the fact that inside the PZ the standing wave solution is transformed to a traveling wave propagating in the semi-infinite chain and corresponding to unbounded energy. In summary, we proved the existence of a family of nonlinear standing wave solutions localized to the NES, and possessing effective frequencies situated inside the lower AZ of the dispersion relation of the linear chain. Physically, during these motions the chain executes synchronous in-phase oscillations, which are also inphase with the NES responses. In the following analysis we prove the existence of a similar family of localized standing waves with effective frequencies situated in the upper AZ of the linear chain, corresponding to out-of-phase oscillations of adjacent pairs of oscillators. Hence, we , consider localized standing waves of (3.193) with frequencies in the range ωb2 = ω02 + 4c2, +∞ . Following the procedure outlined previously, we introduce the following assumption of exponential decay for the amplitudes of the oscillators of the chain, xk = (−1)k x0 eνk , k ≤ 0,

ω02 − 2 = 2c2(cosh κ − 1),

κ = pπ + ν, p ∈ Z (3.219)

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Fig. 3.102 Purely fast localized standing waves for parameters ω0 = 1, a = 5/8, c = 1 and ε = 0.1: (a) dependence of the energy-like variable ρ on the effective frequency ωeffectiveective = , (b) dependence of energy on frequency.

where denotes again the (common) fast frequency of the linear oscillators and the NES, and out-of-phase motions are assumed. Substituting the ansatz (3.219) into (3.193) we reduce the problem of computing localized standing waves inside the upper AZ to the following system of coupled oscillators:

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3 Nonlinear TET in Discrete Linear Oscillators

x¨0 + x0 [c2 (1 − e−ν ) + ω02 + ε] − εv = 0 v¨ + 8av3 − ε(x0 − v) = 0

(3.220)

Introducing the reference frequency ω = [c2 (1 − e−ν ) + ω02 + ε]1/2 the analysis of system (3.220) follows the steps outlined above for the reduced system (3.203), but for an important modification. This is dictated by the fact that, in contrast to the family of in-phase localized standing waves considered previously, out-of-phase standing waves can exist only above a certain energy threshold since their frequencies must exceed the upper bound ωb2 . Moreover since the oscillation is expected to be strongly localized to the NES it is logical to impose the additional requirement that |a2 | |a1 |. This amounts to rescaling the amplitudes of the slow flow in terms of the small parameter of the problem according to, a1 = εb1 , a2 = b2 and b1 , b2 = O(1). Taking into account these assumptions, and performing a similar analysis to that adopted for the in-phase localized standing waves, we derive the following stationary solutions corresponding to time-periodic, out-of-phase, localized standing waves of the chain-NES system: b22 = ρ 2 + O(ε2 ),

θ ≡ β1 − β2 = 0,

b1 = ρ[ω2 − (6ρ 2 /ω2 ) + ε]−1 + O(ε2 ) β˙1 = β˙2 = ω − − (b2 /2ωb1 ) 2 = ω02 + 4c2 + ε2 δω ω = (ω02 + 2c2 )1/2 + (ε/2)(ω02 + 2c2 )−1/2 [1 − 2−1/2c δω] + O(ε2 ) (3.221) This set of stationary conditions is similar to the set (3.208, 3.211) for in-phase, localized standing waves. Moreover, the solution (3.221) is valid only when the conditions |β˙1 | and + β˙1 > (ω02 + 4c2)1/2 hold. In Figure 3.103 we depict the dependence of the effective frequency ωeffective = + β˙1 with respect to the energy-like quantity ρ = (a12 + a22 )1/2 for the family of out-phase localized standing waves. These computations were performed for c2 = 1, ω02 = 0.4, ε = 0.1, α = 5/8 and varying frequency detuning parameter δω. We note that close to the upper bounding frequency there exists an approximately linear dependence of the effective frequency on energy. The localized solution corresponds to a1 < 0, a2 > 0, a2 |a1 |, i.e., to out-of-phase oscillations between the NES and the nearest to it linear oscillator of the chain. An analytical estimate for the energy threshold for the family of out-of-phase localized standing waves is now derived. To this end, we express the energy-like quantity ρ as ρ = ρ (0) + εr, where ρ (0) is a constant and r is the variation of the energy in the neighborhood of the upper bounding frequency ωb1 . Substituting this expression into the third of equations (3.221) provides a way for determining the constant ρ (0) ; indeed, ρ (0) is chosen so to eliminate the O(1) term from the

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

283

Fig. 3.103 Energy dependence of the effective frequencies of out-of-phase localized standing waves inside the upper AZ, for varying frequency detuning δω.

frequency of the slow variation thus rendering the analytical solution consistent with the assumptions made. This leads to the following estimate: ρ (0) = (ω02 + 2c2 )3/2 /3α (ω02 + 4c2)1/2 − (1/2)(ω02 + 2c2 )1/2 (3.222) As a result, the frequency of the slow modulation becomes an O(ε) quantity, given by the following expression: 1

ρ (0) r β˙1 = β˙2 = ε − (0) − 2 2ρ (ω02 + 2c2 )1/2 (ω0 + 2c2 ) (0)2 6αρ × (ω02 + 2c2) − 2 (ω0 + 2c2 ) 2 ⎫ ⎬ (0)2 6αρ B 2 2 (ω + 2c ) − + O(ε2 ) + 0 2ρ (0) (ω02 + 2c2 )1/2 (ω02 + 2c2 ) ⎭ (3.223) where

= (δω) = (1/2)(ω02 + 2c2)−1/2 (1 − 2−1/2cδω)

and

B = B(r, δω) = −ρ ×

(ω02

(0)

1 + 2 (ω02

6αρ (0)2 + 2c ) − 2 (ω0 + 2c2) 2

2 1/2

+ 2c )

−2

+r

(ω02

−

12αρ (0) r (ω02 + 2c2 )

+

12αρ (0)2

(ω02 + 2c2 )3/2

6αρ (0)2 + 2c ) − 2 (ω0 + 2c2) 2

−1

284

3 Nonlinear TET in Discrete Linear Oscillators

Finally, the amplitudes of oscillation of the problem are approximated as follows: a1 = ερ

(0)

6αρ (0)2 ω02 + 2c2 − 2 ω0 + 2c2

−1 + ε2 B(r, δω) + O(ε3 )

a2 = ρ (0) + εr + O(ε2 )

(3.224)

This solution indicates that close to the upper bounding frequency the effective frequency ωeffective = + β˙1 of the out-of-phase localized standing waves vary linearly with increasing energy, a result which is consistent with the numerical result of Figure 3.102. The energy threshold for the existence of this family is given by ρcr (δω) = ρ (0) + ε rcr (δω) + O ε2 , and is approximated by the requirement that ˙ on the threshold it must be satisfied that ωeffective = ωb2 , or, + β1 = ωb2 ⇒ β˙1 = 0 + O ε2 . This leads to the following algebraic expression for determining rcr (δω): (0)2 1 6αρ

(δω)ρ (0) rcr

(δω) − (0) ω02 + 2c2 − 2 1/2 − 2 2ρ ω0 + 2c2 ω0 + 2c2 ω2 + 2c2 0

2 6αρ (0)2 B(rcr , δω) 2 2 =0 + 1/2 ω0 + 2c − 2 ω0 + 2c2 2ρ (0) ω02 + 2c2

(3.225)

This completes the analytical study of the out-of-phase localized standing waves in the system (3.193). In the remainder of this section we perform a series of numerical simulations in order to highlight the role that the computed families of localized standing waves play on TET from the chain to the NES. We note that, in contrast to our previous studies of TET in weakly damped finite-DOF coupled oscillators, TET in the present problem takes place even in the absence of damping. This is due to the fact that the energy radiation from the NES to the far-field of the semi-infinite chain [i.e., as s → −∞ in the continuum approximation (3.194)] has an equivalent effect to damping dissipation in finite-DOF discrete oscillators, and, hence, induces the necessary frequency variation of the NES response required for TET. We performed a series of numerical simulations with a chain composed of 200 oscillators with an essentially nonlinear oscillator (the NES) attached to its right end. In the first series of simulations the initial conditions of all oscillators are set equal to zero, except for x˙−3 (0) = X = 0; in essence, this simulates an initial impulse of magnitude X applied to the fourth oscillator from the NES. The total instantaneous energy of the system was monitored to verify energy conservation and ensure accuracy of the numerical simulations. In addition, care was taken to select the time window of the simulations small enough to avoid the interference due to reflected waves from the left free end of the chain in the measurements. For a small enough impulse neither in-phase nor out-of-phase localized standing waves (modes) are excited (see Figure 3.104). For a sufficiently strong impulsive magnitude, however, excitation of the in-phase localized standing wave occurs. This

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Fig. 3.104 Numerical simulation of the chain-NES interaction for weak impulse excitation of the fourth oscillator of the chain: Response of the NES.

is shown the the numerical simulations depicted in Figure 3.105 for impulsive magnitude X = 50, and system parameters ω0 = 1.5, c2 = 2.0, 8α = 0.5 and ε = 0.3. In Figure 3.105a we depict the transient responses of the NES and its neighboring oscillator of the chain, whereas in Figure 3.105b we depict the temporal evolution of the instantaneous frequency ωNL (t) of the NES. In the plot of Figure 3.105a we note an initial regime of strong dynamic interaction between the chain and the NES, after which the system settles into a time-periodic localized standing wave motion, with energy predominantly confined to the NES. This time-periodic solution is the theoretically predicted localized in-phase standing wave inside the lower AZ of the infinite chain. This is confirmed by the fact that its frequency (i.e., the asymptotic value reached by ωNL (t) in Figure 3.105b) is equal to 1.497 < ωb1 = ω0 ; by the near-exponential decay of the amplitudes of the oscillators (see Figure 3.105c – the small discrepancies noted for distant oscillators is due to the fact that they have not reached a complete steady state motion at the time of the measurement); and by the near in-phase oscillations of the chain and the NES. In Figure 3.105d we depict the instantaneous fraction of initial energy contained in the leading 26 oscillators of the chain and the NES; as time increases this energy reaches an asymptotic value that represents the fraction of total initial energy transferred to the localized standing wave. Hence, passive TET from the undamped semi-infinite chain to the undamped NES occurs through the excitation of the in-phase standing wave localized to the NES. This is qualitatively different compared to the mechanisms of TET for finite-DOF, weakly damped oscillators, which relied either on fundamental and subharmonic TRCs or on the excitation of nonlinear beats. In the absence of damping in the infinite-dimensional system, radiation to the far field provides an energy dissipation mechanism similar to damping, which drives the dynamics to the domain of attraction of the localized in-phase standing wave, and, hence, generates TET.

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.105 Numerical simulation of the chain-NES interaction for strong impulse excitation: (a) — v(t), - - - x0 (t); (b) instantaneous frequency ωNL (t) of the NES.

In the second series of numerical simulations we study the excitation of the localized out-of-phase standing wave inside the upper AZ of the linear chain. In our simulations we could not establish the occurrence of TET in the impulsively excited chain through excitation of the out-of-phase family of localized standing waves, i.e., we could not reproduce the scenario for TET discussed above, which relied on the excitation of the in-phase family of standing waves. As an alternative, we wish to numerically demonstrate the existence of the out-of-phase family of localized waves. To this end, we initiate the system by exponentially decaying out-of-phase initial conditions for the 25 leading oscillators of the chain, and observe an initial regime of chain – NES dynamic interaction, after which a time-periodic localized

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Fig. 3.105 Numerical simulation of the chain-NES interaction for strong impulse excitation: (c) near exponential decay of the amplitudes of the oscillators when the localized in-phase standing wave is excited; (d) evolution of instantaneous normalized energy of the leading 26 oscillators and the NES.

out-of-phase standing wave is formed, with energy predominantly confined to the NES. In Figure 3.106a we depict the corresponding evolution of the instantaneous frequency ωNL (t) of the NES, which eventually enters into the higher AZ, above the upper bounding frequency ωb2 = 3.2015. In Figure 3.106b we depict the instantaneous normalized energy of the leading 18 oscillators of the chain and the NES, representing the portion of the total energy ‘trapped’ in the localized standing wave.

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.106 Numerical simulation of the chain-NES interaction for out-of-phase exponentially decaying initial excitation of the leading 25 oscillators: (a) instantaneous frequency ωNL (t) of the NES; (b) evolution of instantaneous normalized energy of the leading 18 oscillators and the NES.

The main conclusion drawn from the analytical and numerical results of this section is that passive TET can occur in the undamped semi-infinite chain of linear oscillators with a weakly coupled, essentially nonlinear end attachment; that is, impulsive energy from the chain can be transferred irreversibly to the nonlinear oscillator (which acts as an NES) under conditions of nonlinear 1:1 resonance. The only scenario for TET established by the numerical simulations is through the exci-

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tation of families of in-phase standing waves (nonlinear modes) situated inside the lower AZ of the linear chain, and localized to the NES. Based on the previous theoretical and numerical results we can formulate the following scenario for passive TET from the semi-infinite chain to the nonlinear oscillator. An initial impulsive excitation of the chain causes energy to propagate towards the NES (and also away from the NES to the far field) through traveling wavepackets with predominant frequencies inside the PZ ω ∈ (ωb1 , ωb2 ) of the chain (actually, the only way to transfer energy through the linear chain is by exciting traveling waves). After these traveling wavepackets impede to the nonlinear oscillator they excite it initially with frequencies inside the PZ of the chain, under non-resonant conditions (this is confirmed by the numerical result of Figure 3.105b). These initial non-resonant interactions cause initial near-adiabatic radiation of energy from the nonlinear oscillator back to the chain, a process that reduces its instantaneous frequency; indeed the radiation of energy from the nonlinear oscillator back to the chain has the same effect as energy dissipation due to damping in finite-DOF discrete coupled oscillators. After sufficient radiation of energy, the instantaneous frequency of the nonlinear oscillator reaches from above the lower bounding frequency ωb1 = ω0 of the chain, where conditions for 1:1 resonance between the chain and the nonlinear oscillator are established. This eventually leads to excitation of an inphase localized standing wave (mode) of the integrated chain-attachment system. Once this localized mode is excited, energy is ‘trapped’ in the nonlinear oscillator, and no further energy radiation back to the chain is possible afterwards, since the motion takes place on an invariant nonlinear normal mode manifold localized to the nonlinear oscillator. As a result, there occurs confinement of energy to the NES and passive TET. An interesting feature of this TET scenario is that it is realized in the absence of damping. This contrasts to our studies of finite-DOF systems of coupled oscillators, where TET occurred only in the presence of damping dissipation, through TRCs in neighborhoods of the corresponding resonant manifolds. In the infinite-DOF undamped system considered in this section the far field acts as an effective energy dissipater, ‘absorbing’ irreversibly energy in the form of traveling waves propagating away from the nonlinear oscillator. Hence, in the scenario outlined above for the undamped infinite-DOF system TET is realized through the eventual excitation of a standing wave localized to the NES rather than through TRCs. Due to the invariance property of the family of localized standing waves, once such a standing wave is excited the motion remains confined to the NES and no energy radiation to the semi-infinite chain is possible afterwards. In the next section we formulate an alternative analytical methodology for studying TET in the corresponding weakly damped system, and examine the mechanisms for TET in that case.

3.5.2.3 Integro-Differential Formulation To study TET in the weakly damped, semi-infinite chain with the nonlinear end attachment we adopt a different methodology by reducing the dynamics to a single

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3 Nonlinear TET in Discrete Linear Oscillators

integro-differential equation in terms of the NES response v(t) (Vakakis, 2001). The system considered is the weakly damped variant of (3.193), x¨k + c2 (2xk − xk−1 − xk+1 ) + ελx˙k + ω02 xk = 0,

k>0

x¨0 + c2 (x0 − x1 ) + ε(x0 − v) + ελx˙0 + ω02 x0 = 0 v¨ + Cav3 + ελv˙ − ε(x0 − v) = 0

(3.226)

with initial conditions, xi (0) = x˙i (0) = 0, i = p; xp (0) = 0, x˙p (0) = X; and v(0) = v(0) ˙ = 0. In addition, compared to (3.193), in (3.226) we change the indexing of the oscillators of the chain from negative to positive The initial conditions correspond to impulsive excitation of the (p + 1)-th oscillator of the chain, with the system being initially at rest. Before proceeding, however, with the analysis we present numerical evidence of TET from the semi-infinite chain of weakly damped oscillators to the damped NES. The numerical simulations were carried out by numerically integrating a model of 101 oscillators. Careful monitoring of the transient wave propagation in the model assured that no unwanted reflexions of waves due to the finiteness of the numerical chain occurred, so that the analytical condition of semi-infinite chain was accurately simulated in the temporal window of the results presented herein. In the first simulation we consider a chain with parameters ε = 0.1, λ = 0.5, C = 5.0, c2 = 1.5, ω02 = 0.9, p = 2 and X = 4. In Figure 3.107 we depict the transient responses of the nonlinear attachment and the adjacent linear oscillator, from which it is concluded that no TET from the chain to the NES occurs in the system in this case. Note that in the absence of TET the linear oscillator executes a nearly monochromatic (single-frequency) fast oscillation with a slowly decaying envelope, whereas the NES executes a multi-frequency oscillation with no discernable dominant harmonic component. Next, we consider a system with parameters ε = 0.1, λ = 0.5, C = 5.0, c2 = 1.5, ω02 = 0.4 and initial conditions as previously. In Figure 3.108 we depict the transient responses of the NES and its adjacent linear oscillator, from which TET from the chain to the NES is noted. Considering the transient response of the NES we note that after an initial regime of multi-frequency transients, the response appears to settle to a single-frequency fast oscillation modulated by a slow varying envelope. Moreover, TET appears to occur predominantly in the regime of single-frequency fast oscillation. Relating these results to the TET scenario outlined in the previous section for the undamped system, we deduce that in the initial multi-frequency regime the NES radiates (backscatters) energy to the chain as its frequency decreases inside the PZ of the chain. After sufficient energy radiation to the far field, the instantaneous frequency of the NES approaches from above the lower bounding frequency ωb1 = ω0 of the PZ, and TRC takes place at frequency ωb1 = ω0 . At this point fundamental TET from the chain to the NES takes place in similar way to the two-DOF system (see Section 3.4.2.1); in that context TET in the damped system is qualitatively different than TET in the undamped system which is due to excitation of an in-phase standing wave localized to the NES. Hence,

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Fig. 3.107 Case of absence of TET from the chain to the NES: (a) response of the NES, (b) response of the neighboring to the NES linear oscillator.

the numerical results of Figure 3.108 appear to confirm in general terms the TET scenario formulated in the previous section for the undamped chain-NES system. Similar TET results were obtained for the system with parameters ε = 0.1, λ = 0.5, C = 5.0, c2 = 3.5, ω02 = 0.4 and initial conditions as previously (Vakakis, 2001). The numerical results indicate that depending on the system parameters and the level of impulsive excitation, TET in the system is realized. The fact that TET appears to be coincidental with the settlement of the NES response to a regime of a single-frequency fast oscillation modulated by a slowly varying envelope, provides strong motivation to apply slow-fast partition of the dynamics in the regime of TET and apply once again the CX-A methodology. Before we proceed to studying this partition, however, it is necessary to reduce the dynamics of the chain-NES interaction by taking into account the linear structure of the chain dynamics. Indeed, we will show that the dynamics of the infinite system (3.226) can be reduced (with

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3 Nonlinear TET in Discrete Linear Oscillators

Fig. 3.108 Case of TET from the chain to the NES: (a) response of the NES, (b) response of the neighboring to the NES linear oscillator.

no approximation) to a single integro-differential equation. To perform this task we make use of the analytical results of Lee (1972) and Wang and Lee (1973) who, in essence, derived the Green’s functions of the free and forced damped chain of linear oscillators in explicit form. To this end, the response of the k-th oscillator of the chain (3.226) can be symbolically expressed as follows: xk (t) = X[Gk−p (t) + Gk+p−3 (t)] + ε[v(t) − x0 (t)] ∗ [Gk (t) + Gk+1 (t)],

k≥0 (3.227)

where (∗) denotes the convolution operation. The kernel Gm (t) = G−m (t) is defined as

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

Gm (t) = e

−ελt /2

t

293

J0 (ω02 − ε2 λ2 /4)1/2 (t 2 − τ 2 )1/2 J2m (2cτ )dτ

≡ e−ελt /2Hm (t)

(3.228)

where J2m (•) denotes the Bessel function of the first kind of order 2m. Using (3.227) we express the response x0 (t) of the linear oscillator adjacent to the NES in the following integro-differential form: / 0 x0 (t) = X[Gp (t) + Gp+1 (t)] + ε v(t) − X[Gp (t) + Gp+1 (t)] ∗ [G0 (t) + G1 (t)] + O(ε 2 )

(3.229)

with p ≥ 0. Substituting (3.229) into the last of equations (3.226) we obtain the following reduced dynamical system, in the form of a single integro-differential equation governing the motion of the NES: v¨ + ελv˙ + Cv3 + εv =

0 / εX[Gp (t) + Gp+1 (t)] + ε2 v(t) − X[Gp (t) + Gp+1 (t)] ∗ [G0 (t) + G1 (t)] + O(ε3 )

(3.230)

This equation is supplemented by the initial conditions v(0) = v(0) ˙ = 0. It follows that the problem of studying the dynamics of TET in system (3.226) is reduced to the equivalent problem of studying the dynamics of the integrodifferential equation (3.230) with zero initial conditions. Clearly, direct application of the CX-A technique developed in the previous section is not possible at this point, due to the apparent lack of a single ‘fast’ frequency in the non-hom*ogeneous term on the right-hand side of (3.230). Hence, before proceeding with the analysis of this equation it is necessary to examine carefully the frequency content of the nonhom*ogeneous term; if this term can be approximated by a slowly modulated fast monochromatic oscillation, it will render the integro-differential equation (3.230) amenable to direct CX-A analysis. Since the quantity Gm−1 (t) + Gm (t) ≡ e−ελt /2[Hm−1 (t) + Hm (t)] appears repeatedly in (3.230) we start our analysis by studying the spectral content of this quantity. As shown by Wang and Lee (1973), Hm (t) can be expressed in the following alternative form (which highlights its spectral content): π cos mθ j ω(θ)t −1 Hm (t) = π e − e−j ω(θ)t dt, 2j ω(θ ) 0 1/2 ω(θ ) = ω02 + 4c2 sin2 (θ/2) (3.231)

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3 Nonlinear TET in Discrete Linear Oscillators

By (3.231) Hm (t) is expressed as a superposition of a continuum of harmonics with frequencies in the range [ω0 , (ω02 + 4c2)1/2 ], which is coincident with the PZ of the infinite undamped linear chain, where time-harmonic traveling waves can propagate unattenuated upstream or downstream through the chain. Outside this frequency range (in the two AZs) the chain acts as a filter, exponentially attenuating harmonic signals and producing merely near-field solutions. Hence, a first conclusion is that the reduced system (3.230) highlights the fact that the NES is forced by a continuum of impeding harmonics in the range of the PZ of the linear chain. We now asymptotically analyse (3.232) in order to show that after some initial multi-frequency transients, Hm (t) performs oscillations dominated by the single ‘fast’ frequency ωb1 = ω0 , which is the lower bounding frequency of the dispersion relation of the chain; this finding will pave the way for applying the CX-A methodology to the reduced system. Considering the time dependence of the integral (3.231) we note that for t 1 the harmonic terms in the integrand perform fast oscillations; it follows that for sufficiently long times we can apply the method of stationary phase (Bleistein and Handelsman, 1986) to asymptotically approximate Hm (t) + Hm−1 (t) as follows:

1/2 ej (ω0 t +π/4) 2 = (t 1) 2j πc2 ω0 t

ej (ω0 t +3π/4) 4ω0 3/2 c2 − + 2[m2 + (m + 1)2 ] − 1 32j π 1/2ω0 c2 t ω02

Hm (t) + Hm−1 (t)

+ O(t −5/2 ) + cc,

t 1

(3.232)

where ‘cc’ denotes complex conjugate and m = 0, 1, 2, . . . . We note that at sufficiently long times (i.e., after the multi-frequency early transients have died out) the quantity Hm (t) + Hm−1 (t) settles approximately to a fast oscillation with frequency ω0 modulated by an algebraically decaying ‘slow’ envelope. Similar algebraic time decay rates for anharmonic chains were derived by Sen et al. (1996). A short time analytic approximation for Hm (t) + Hm−1 (t) is derived by Taylorexpanding the exponentials in (3.232) close to t = 0, and performing successive integrations with respect to θ of the resulting coefficients of powers of t,

Hm (t) + Hm−1 (t)

(t 1)

≈ π −1

i=1,3,5,...

ti [Ii (m) + Ii (m − 1)] , i!

t 1 (3.233)

where

π

Ii (m) =

ω(i−1) (θ ) cos mθ dθ,

i = 1, 3, 5, . . .

An interesting observation is that for fixed i the quantity Ii (m) becomes zero for m ≥ (i + 1)/2. It follows that as the order m increases we must consider higher orders of t in the early time expansion (3.233) to obtain accurate approximations. This observation is consistent with the existence of exceedingly larger initial ‘silent’

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regions in Gp (t) + Gp+1 (t) in the non-hom*ogeneous term of the reduced integrodifferential equation with increasing p (i.e., as the impulse is shifted further downstream away from the NES). The point of matching of the short- and long-time approximations can be computed by imposing an appropriate criterion, for example, by minimizing in time the error quantity 01/2 / Er(t) = [g(t 1)(t) − g(t 1)(t)]2 + [g˙(t 1)(t) − g˙(t 1) (t)]2 g(t) ≡ [Hm (t) + Hm−1 (t)] This quantitative criterion provides the time interval [0, t ∗ ] of validity of the Taylorseries based approximation, and the beginning of the range of validity of the longterm asymptotic approximation (3.232). The error at the point of matching, Er(t ∗ ), can be made arbitrarily small by including a sufficient number of terms in the two approximations. Similar matching techniques of short- and long-time local solutions have been introduced in previous works (for example, Salenger et al., 1999) to construct global analytical approximations of strongly nonlinear responses of coupled oscillators. In Figure 3.109a we depict a comparison of the short and long time approximations with the (exact) numerical simulation for the quantity Hp (t) + Hp+1 (t) for the chain whose responses are shown in Figure 3.108. Since the impulse is applied in the fourth particle of the we have that p = 2; the short time approxima system tion was derived up to O t 3 , whereas the long time asymptotic approximation up to O t −3/2 . In the same figure we depict the error Er(t) versus time from where the instant of transition t ∗ is determined. Better approximations can be obtained by improving the accuracy of the long time asymptotic approximation. The previous discussion proves that, in the TET regime and after certain initial multi-frequency transients the non-hom*ogeneous term of the reduced equation (3.230) possesses a dominant harmonic with fast frequency ω0 . This finding enables us to apply the CX-A method to analyze TET from the chain to the NES. The solution of the reduced system is developed in two steps. For t ∈ [0, t ∗ ) the short-term solution of the reduced system is expressed in Taylor series whose coefficients are computed by matching respective powers of t on the left- and right-hand sides. For t ≥ t ∗ we express the quantities Gp (t) + Gp+1 (t) and [G0 (t) + G1 (t)] on the right-hand side of (3.230) using the long-time asymptotic approximation (3.232). We then apply the CX-A method by partitioning the dynamics into fast and slowcomponents using as initial condition the state of the system at t ∗ (as computed by the Taylor series expansions of the previous step). Elaborating further on the second step, to approximate v(t) we introduce the complex variable ψ(t) = v(t) ˙ + j ω0 v(t), and express ψ(t) in polar form, ψ(t) = ϕ(t)ej ω0 t , where φ(t) represents the slowly varying modulation of the fast oscillation ej ω0 t . Moreover, it is of help to introduce the complex amplitude σ (t) defined by φ(t) = σ (t)e−ελt /2. Finally, we use the following compact notation for the longtime asymptotic solution (3.232),

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Fig. 3.109 Matching the local approximations (3.232) and (3.233): (a) short and long time approximations for [Hp (t) + Hp+1 (t)], p = 2, compared to the exact numerical simulation for the response depicted in Figure 3.107; (b) error function Er(t) for the same system determining the transition point t ∗ .

Hm−1 (t) + Hm (t)

(t 1)

≡ h(t; m) ej ω0 t + O(t −5/2 ) + cc

where

1/2

ej π/4 ej 3π/4 2 4ω0 3/2 h(t; m) = − 2j πc2 ω0 t 32π 1/2 ω0 j c2 t c2 2 2 × + 2[m + (m + 1) ] − 1 ω02

(3.234)

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Introducing the new variables, t˜ = t − t ∗ , σ (t) = σ (t˜ + t ∗ ) ≡ σ˜ (t˜), h(t; p) = ˜ t˜; p), t˜ ≥ 0, and omitting the tildes from the resulting expressions h(t˜ + t ∗ ; p) ≡ h( we derive the following slow flow approximation governing the dynamics of the complex modulation ∗ j ω02 − ε 3j Ce−ελt e−ελt |σ |2 σ σ˙ + σ− 2ω0 8ω03 ε2 = εX h(t; p + 1) + 2j ω0

t σ (τ ) h(t − τ ; p + 1) dτ 0

t −ε X

h(τ ; p + 1) h(t − τ ; 1) dτ + O(ε3 )

2

(3.235)

The initial condition σ (0) is determined by computing the Taylor series solution at the transition point t = t ∗ ⇒ t˜ = 0. We note that due to the approximations involved, the solution of (3.235) is expected to be valid only up to times of O(1/ε2 ). Hence, the problem of studying TET in the weakly damped system (3.226) is reduced approximately to the analyis of the dynamics of the complex modulation equation (3.235). This analysis is similar to the ones performed in previous sections for studying the slow flow of the two-DOF system, and is not carried out further. We state, however, that the reduction of the dynamics to (3.235) indicates that TET in the weakly damped system (3.225) is due to TRC of the NES dynamics in the neighborhood of a 1:1 resonance manifold at frequency ω0 ; in that sense, TET in the weakly damped system can be regarded as qualitatively different from the TET mechanism in the corresponding undamped system which was due to excitation of an in-phase family of standing waves localized to the NES. Viewed in a different context, however, the TET dynamics in the damped and undamped systems possess a similarity. Indeed the spectral study of the non-hom*ogeneous term of the reduced system carried out in this section confirms the TET scenario of the previous section, namely, that TET from the semi-infinite chain to the NES occurs when the frequency of the NES approches from above the lower bound of the PZ of the chain. Similar results were obtained in Dumcum (2007) where the analysis was extended to semiinfinite linear chains with lightweight ungrounded NESs (of Configuration II – see Section 3.1). A final note concerns the initial multi-frequency transients that occur after a traveling wavepacket propagating in the semi-infinite chain impedes on the NES (see Figure 3.108). In this regime the NES interacts with traveling waves possessing frequencies inside the PZ of the chain, and radiates energy to the far field of the chain. Traveling waves, however, can be regarded as the continuum limit of the closely packed resonances of a chain composed of a large (but finite) number of coupled oscillators, as this number tends to infinity. Viewed in that context, the initial multifrequency transients resulting from the dynamic interaction of the NES with im-

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peding traveling waves propagating in the semi-infinite chain, can be viewed as the continuum limit of resonance capture cascades (RCCs) occurring between subsets of linear modes of the finite but high-DOF chain and the NES, as the number of DOF of the chain tends to infinity. This provides an interesting physical background to the complex traveling wave-NES dynamic interaction that occurs in the initial stage of the NES dynamics.

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Chapter 4

Targeted Energy Transfer in Discrete Linear Oscillators with Multi-DOF NESs

4.1 Multi-Degree-of-Freedom (MDOF) NESs In the previous chapter we considered targeted energy transfer (TET) from linear discrete primary systems to single-degree-of-freedom (SDOF) essentially nonlinear attachments (or nonlinear energy sinks – NESs). In this chapter we extend our discussion of nonlinear TET to multi-DOF essentially nonlinear NESs. The reason for doing so is twofold. First, we aim to show that through the use of MDOF NESs it is possible to passively extract vibration energy simultaneously from multiple linear modes of primary systems. This feature normally does not appear in the case of SDOF NESs, since as shown in Chapter 3, in such attachments multi-frequency TET (involving resonance interactions of the NESs with multiple linear modes) can only occur through resonance capture cascades (RCCs); i.e., through sequential transient resonance captures (TRCs) involving only one linear mode at a time. Second, we wish to show that by using MDOF NESs we can improve the efficiency and robustness of TET, even at small energy levels. This represents a qualitatively new feature in TET dynamics, since as we discussed in Chapter 3, strong TET from primary discrete systems to SDOF NESs can be realized only when the energy exceeds a well-defined critical threshold (e.g., see Figure 3.4). The general study of the nonlinear dynamical interactions of linear primary systems with MDOF essentially nonlinear NESs is a formidable problem from an analytical point of view, due to the high-order degeneracies of the governing dynamics that lead to high-co-dimension bifurcations (Guckenheimer and Holmes, 1983; Wiggins, 1990). However, we will show in this chapter that if the aim of the analysis is narrowed to focus on TET dynamics, asymptotic analysis can still be applied to study analytically certain aspects of the problem. The following exposition draws results from the thesis by Tsakirtzis (2006). Additional works on MDOF NESs were performed by Gourdon et al. (2005, 2007) and Gourdon and Lamarque (2005), whereas Musienko et al. (2006) studied nonlinear energy transfers from a linear oscillator to a system of two attached SDOF NESs. Ma et al. (2008) studied TET from a chain of particles to a two-DOF essentially nonlinear attachment at its end by ap-

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plying proper orthogonal decomposition (POD); they related TET in that system to the localization properties of proper orthogonal modes and to the energy distribution among them. POD-based reduced-order modeling of the TET dynamics was also discussed in that work.

4.1.1 An Alternative Way for Passive Multi-frequency Nonlinear Energy Transfers It was shown in Chapter 3 that SDOF essentially nonlinear attachments (NESs) are capable of passively absorbing energy from multiple linear modes of primary systems through resonance capture cascades (RCCs). The resulting multi-frequency TET occurs through sequential transient resonance captures (TRCs), as the nonlinear attachment engages in resonance capture with each linear mode involved in the RCC in the neighborhood of its own natural frequency (e.g., close to the corresponding resonance manifold), before escaping TRC and engaging in resonance with the next linear mode of the sequence at a different frequency. Moreover, it was shown that RCCs lead to sequential, multi-frequency energy transfer from all participating linear modes to the nonlinear attachment, which then acts, in essence, as broadband NES. As an example we consider a system consisting of a two-DOF primary system that is weakly coupled to a SDOF essentially nonlinear NES with governing equations of motion given by Tsakitzis et al. (2005): u¨ 1 + u1 (ω02 + 2α) − α u2 = 0 u¨ 2 + u2 (ω02 + α + ε) − α u1 − ε v = 0 v¨ + C v3 + ε β v˙ + ε(v − u¨ 2 ) = 0

(4.1)

The parameters used in the following simulation are assigned the numerical values, α = 1, ω0 = 1, β = 2, C = 3, and ε = 0.1, with all initial conditions being assumed zero, except for the initial velocity u˙ 1 (0) = 25.0. In the plot of Figure 4.1 we depict the numerical wavelet transform (WT) spectrum of the transient response of the NES, from which the occurrence of an RCC is deduced. Indeed, in the initial phase of the motion the NES resonates (or engages in TRC) with the higher outof-phase linear mode, resulting in passive energy absorption from that mode in the neighborhood of the higher natural frequency (Vakakis et al., 2003; Panagopoulos et al., 2004). As energy decreases due to damping dissipation, an escape of the dynamics from this initial TRC occurs, and the NES engages in transient resonance with the lower in-phase linear mode; in turn, this results in TET from the lower mode to the NES in the neighborhood of the lower natural frequency. In the final phase of the motion the dynamics escapes from this second TRC as well, and settles into a linearized regime as the motion decays to zero due to damping dissipation; due to

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Fig. 4.1 Wavelet analysis of the transient response of an SDOF NES engaged in an RCC (frequencies in Hz).

Fig. 4.2 Primary system with MDOF NES.

the low amplitude of the motion, the nonlinear effects (and, hence, the effect of the NES) are negligible at this late stage of the dynamics. What was described above constitutes a RCC, leading to multi-frequency TET from both modes of the primary system to the SDOF NES. However, since this energy transfer takes place in a sequential manner, the SDOF NES does not engage in simultaneous resonance with both linear modes of the primary system. In an attempt to device an NES capable of extracting simultaneously energy from multiple linear modes of the primary system to which it is attached, we consider an alternative design by adding to the NES more degrees of freedom. As another motivational example, we consider the system depicted in Figure 4.2, composed of a two-DOF linear primary system weakly coupled to a three-DOF essentially nonlinear attachment. We aim to study the capacity of the MDOF NES to passively absorb and locally dissipate vibration energy initially induced to the primary system. Assuming that the two modes of the uncoupled primary system (i.e., for ε = 0) possess natural frequencies ω1 and ω2 , the equations of motion are given by x ε 2 x1 − ε + v1 = 0 x¨1 + ελx˙1 + ω12 + 2 2

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x ε 1 x2 − ε − v1 = 0 x¨2 + ελx˙2 + ω22 + 2 2

x2 − x1 + C1 (v1 − v2 )3 = 0 µv¨1 + ελ(v˙1 − v˙2 ) + ε v1 + 2 µv¨2 + ελ(2v˙2 − v˙1 − v˙3 ) + C1 (v2 − v1 )3 + C2 (v2 − v3 )3 = 0 µv¨3 + ελ(v˙3 − v˙2 ) + C2 (v3 − v2 )3 = 0

(4.2)

where the variables x1 and x2 are the linear modal co-ordinates (that is, they describe the amplitudes of the linear in-phase and out-of-phase modes of the primary system), and vi , i = 1, 2, 3 are the absolute displacements of the particles of the NES. In the following numerical simulations an initial excitation of the primary system is considered, with the NES being initially at rest. Before considering energy transactions in the coupled system, it is instructive to discuss the dynamics of the two degenerate systems resulting in the limit of zero coupling, i.e., as ε → 0. The degenerate nonlinear attachment possesses three nonlinear normal modes (NNMs); as discussed in Section 2.1 these are synchronous free periodic motions where all coordinates of the system vibrate in-unison, in similarity to the modes of classical linear vibration theory (Vakakis et al., 1996). The first NNM of the decoupled NES possesses zero frequency and corresponds to a rigidbody mode of the decoupled NES. In addition, an in-phase NNM exists satisfying the relation [v2 (t) − v3 (t)] = [v1 (t) − v2 (t)], and an out-of-phase one satisfying the relation [v2 (t) − v3 (t)] = −[v1 (t) − v2 (t)]. Based on these observations, we introduce at this point the nonlinear modal coordinates z1 (t), z2 (t) and z3 (t), defined as z3 (t) = [v2 (t) − v3 (t)] + [v1 (t) − v2 (t)] z2 (t) = [v2 (t) − v3 (t)] − [v1 (t) − v2 (t)] z1 (t) = v1 (t) + v2 (t) + v3 (t)

(4.3)

representing the coordinates of the three NNMs of the decoupled NES. The corresponding backbone curves (i.e. the frequency-energy dependences) of the linear and nonlinear modes of the decoupled primary system and the decoupled NES √ are depicted in Figure 4.3 for C1 = C2 = 0.15, µ = 0.33, ω1 = 1.0, ω2 = 3 and ε = 0. At crossing points between different backbone curves (i.e., at points A, B and C) internal resonances may occur, since at these points the frequency of a NNM coincides to the natural frequency of one of a linear mode of the primary system. It follows that in the proposed design there exists the possibility of simultaneous resonance captures between multiple NNMs of the NES with the two linear modes of the primary system. When non-zero but weak coupling is introduced (0 < ε 1), system (4.2) is expected to possess NNMs that are perturbations of the aforementioned modes of the two decoupled linear and nonlinear subsystems. Moreover, the resulting dynamics are expected to exhibit added complexity due to the multi-modal dynamical

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Fig. 4.3 Frequencies of the linear and nonlinear modes of the uncoupled primary system and the NES (corresponding to ): — linear, - - - nonlinear modes.

interaction of the linear and essentially nonlinear subsystems; this is especially true close to points of internal resonances where bifurcations of NNMs (Vakakis et al., 2003) are expected to occur that further complicate the dynamics. In Figure 4.4 the transient responses of the coupled system of are depicted for ε = 0.25 and initial conditions u˙ 1 (0) = 5.0, u˙ 2 (0) = −5.0, with all other initial conditions zero. This corresponds to initial excitation of the anti-phase linear mode of the decoupled primary system. Comparing the responses of the linear and nonlinear subsystems we clearly deduce that the NES passively absorbs vibration energy from the primary system. Moreover, this energy is absorbed in multiple frequencies, which is an indication of the occurring complex dynamical interactions. In Figures 4.4c and 4.4d the Fast Fourier Transforms (FFTs) of the nonlinear modal responses z2 (t) and z3 (t) are depicted, respectively, whereas in Figure 4.5 the corresponding wavelet spectra of these responses are presented. As discussed in Section 2.5.1, the WT spectra reveal not only the frequency contents of the nonlinear modal responses, but also the temporal evolution of each individual frequency component; this is key to understanding the transient nonlinear interactions that occur between the primary system and the NES. Indeed, the WT spectra depicted in Figure 4.5 reveal that a series of transient resonance captures (TRCs) occurs, which we now proceed to discuss. Specifically, the out-of-phase NNM of the NES [corresponding to z3 (t)] absorbs energy at three main frequencies, two of which are close to the natural frequencies of the linear in-phase and out-of-phase linear modes, and one is lower than these. Hence, the MDOF NES appears to resonate simultaneously with both linear modes, extracting energy simultaneously from both. The additional lower frequency component indicates the presence of an essentially nonlinear mode that exists in the coupled system; as shown in Lee et al. (2005) in systems of this type (composed of weakly coupled linear and nonlinear components), there can exist numerous branches of stable and unstable NNMs resulting from bifurcations under conditions of internal resonance. The in-phase NNM [corresponding to z2 (t)] exhibits similar

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Fig. 4.4 Transient responses of system (4.2): (a) x1 (t) —, z2 (t) - - - - - -, (b) x2 (t) —, z3 (t) - - -, (c) FFT of z2 (t), (d) FFT of z3 (t).

behavior, though its interaction with the out-of-phase linear mode takes place after some initial time delay. This mode also absorbs energy in a multi-frequency fashion, and resonates with both linear modes of the primary system; the presence of the lower NNM is again noted in the in-phase nonlinear modal response. The results presented in this section provide a numerical demonstration that, indeed, MDOF NESs can act as passive energy absorbers of vibration energy over wide frequency ranges. This is due to the occurrence of simultaneous TRCs at different frequency ranges, resulting from resonance interactions of multiple NNMs of the NES with multiple linear modes of the primary system. Motivated by this preliminary numerical evidence, we now proceed to a more systematic numerical study of targeted energy transfer (TET) phenomena from linear oscillators to attached MDOF NESs.

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Fig. 4.5 Wavelet transform spectra of the transient responses of the MDOF NES (frequencies in Hz): (a) z3 (t), (b) z2 (t).

4.1.2 Numerical Evidence of TET in MDOF NESs In this section we will study systematically the efficiency of passive TET in the system depicted in Figure 4.2. The study follows closely Tsakirtzis (2006) and Tsakirtzis et al. (2007). This system consists of a two-DOF primary linear oscillator connected through a weak linear stiffness of constant ε (which is the small parameter of the problem, i.e., 0 < ε 1) to a three-DOF NES with essential stiffness nonlinearities. Each mass of the primary system is normalized to unity, and the stiffnesses of the NES possess pure cubic characteristics with constants C1 and C2 . Each mass of the nonlinear attachment is equal to µ, and both linear and nonlinear subsystems possess linear viscous dampers with small constants ελ. Assuming that impulsive excitations F1 (t) and F2 (t) are applied to the primary system and that no direct forcing excites the nonlinear attachment, the equations of motion are given by

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u¨ 1 + (ω02 + α)u1 − αu2 + ελ1 u˙ 1 = F1 (t) u¨ 2 + (ω02 + α + ε)u2 − αu1 − εv1 + ελ1 u˙ 2 = F2 (t) µv¨1 + C1 (v1 − v2 )3 + ε(v1 − u2 ) + ελ2 (v˙1 − v˙2 ) = 0 µv¨2 + C1 (v2 − v1 )3 + C2 (v2 − v3 )3 + ελ2 (2v˙2 − v˙1 − v˙3 ) = 0 µv¨3 + C2 (v3 − v2 )3 + ελ2 (v˙3 − v˙2 ) = 0

(4.4)

As mentioned in the previous section, in the limit ε → 0 system (4.4) is decomposed into two uncoupled , oscillators: a two-DOF linear primary system with natural fre-

quencies ω1 = ω02 + 2α and ω2 = ω0 < ω1 corresponding to the out-of-phase and in-phase linear modes, respectively; and a three-DOF NES with a rigid body mode, and two flexible nonlinear normal modes – NNMs (Tsakirtzis et al., 2005). Our first aim is to study the dynamics of system (4.4), and, in particular, the efficiency (strength) of TET from the forced primary system to the NES. In this section the study of the damped dynamics is performed through direct numerical simulations of the equations of motion and post-processing of the transient results. We do this in order to establish the ranges of parameters for which efficient targeted energy transfer from the primary system to the NES takes place. In later sections we will study TET in (4.4) using analytic techniques. An extensive series of numerical simulations is performed over different regions of the parameter space of the system, in order to establish the system parameters for which optimal passive TET from the primary system to the NES occurs. Moreover, by varying the linear coupling stiffness α of the primary system, we study the influence of the spacing of the two eigenfrequencies ω1 , and ω2 on TET. The numerical simulations are carried out by assigning different sets of initial conditions of the primary system, with the NES always being initially at rest. To assess the strength of passive TET from the primary system to the NES, the following energy dissipation measure (EDM) is numerically computed: ελ2 t E(t) = (v˙1 (τ ) − v˙2 (τ ))2 + (v˙2 (τ ) − v˙3 (τ ))2 dτ (4.5) Ein 0 where Ein is the input energy provided to the system by the initial conditions. This non-dimensional EDM represents the instantaneous portion of input energy dissipated by the NES up to time instant t; it follows that by means of (4.5) we can obtain a qualitative measure of the effectiveness of the MDOF NES to passively absorb and locally dissipate vibration energy from the primary system. Clearly, due to the fact the system examined is purely passive (with energy being continuously lost due to damping dissipation) the instantaneous EDM should reach a definite asymptotic limit which is symbolically denoted as ENES = lim E(t) t 1

(4.6)

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Fig. 4.6 EDM for varying values of single impulse Y (impulsive forcing condition I1) and coupling stiffness a of the primary system.

This asymptotic EDM represents the portion of input energy that is eventually dissipated by the NES. In the following exposition, the asymptotic evaluation (4.6) is used as a measure of the efficiency of TET from the primary system to the MDOF NES. We note, however, that the EDMs (4.5) and (4.6) can not describe the time scale of TET, i.e., how rapidly energy gets transferred and dissipated by the NES; clearly, in certain applications the time scale of energy transfer is an important factor for assessing NES efficiency but this issue will not be pursued further in this section (however, it will be revisited in later sections and chapters). It suffices to state that the use of NESs with non-smooth nonlinearities drastically decreases the time scale of energy dissipation (Georgiadis et al., 2005); in addition, as shown in Section 3.4.2.5 the excitation of impulsive orbits affects the time-scale of TET dynamics. As shown below, for weak coupling between the primary system and the NES, efficient passive TET from the primary system to the NES can be achieved for small values of the mass parameter µ and nonlinear characteristic C2 of the NES with all other parameters being quantities of O(1). This combination of system parameters leads to large relative displacements between the particles of the NES, which, in turn, leads to large energy dissipation by the dampers of the NES. Hence, a basic conclusion drawn from the numerical study is that lightweight MDOF NESs with weak nonlinear stiffnesses C2 are effective energy absorbers and dissipators; this is an interesting conclusion from the practical point of view, since it renders such lightweight NESs applicable for a diverse set of engineering applications. The numerical simulations were performed for the following system parameters: ε = 0.2,

α = 1.0,

µ → ε2 µ = 0.08,

C1 = 4.0, ω02 = 1.0

C2 = 0.05,

ελ1 = ελ2 = ελ = 0.01,

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Fig. 4.7 EDM for varying values of the in-phase impulses Y (impulsive forcing condition I2) and coupling stiffness a of the primary system.

Fig. 4.8 EDM for varying values of the out-of-phase impulses Y (impulsive forcing condition I3), and coupling stiffness a of the primary system; symbols A and B at the plot corresponding to α = 1 refer to the results depicted in Figures 4.11 and 4.12, respectively.

and three types of impulsive forcing conditions – IFCs (or, equivalently, initial conditions – velocities) for the primary system: (i) single IFC designated by I1, corresponds to F1 (t) = Y δ(t) (or, equivalently, u˙ 1 (0) = Y ), and all other initial conditions zero; (ii) in-phase IFC designated by I2, with F1 (t) = F2 (t) = Y δ(t) and all other initial conditions zero; and (iii) out-of-phase IFC I3, with F1 (t) = −F2 (t) = Y δ(t) and all other initial conditions zero. In Figures 4.6–4.8 we depict the asymptotic EDM ENES (e.g., the portion of input energy eventually dissipated by the NES) as function of the magnitude of the

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Fig. 4.9 The system with SDOF NES attachment whose dynamics is compared to the system depicted in Figure 4.2.

impulse Y and the linear coupling stiffness of the primary system α, for the above three types of IFCs. In all cases, a significant portion (reaching as high as 86% for IFC I1; 92% for IFC I2; and 90% for IFC I3) of the input energy gets passively absorbed and dissipated by the MDOF NES. This significant passive TET occurs in spite of the fact that the (directly forced) primary linear system and the NES have identical dashpots. Moreover, the energy transfer is broadband, since the vibration energy absorption takes place over wide frequency ranges. Whereas the portion of energy eventually dissipated at the NES depends on the level of energy input and the closeness of the natural frequencies of the primary system (as expected, since the system considered is nonlinear), this dependence is less pronounced compared to the case of the SDOF NES. This is concluded when comparing the performance of the MDOF NES to that of the SDOF NES depicted in Figure 4.9 (Vakakis et al., 2004) – this is performed in the comparative plot of Figure 4.10 for a system with α = 0.2, and IFCs I1-I3 – and also by considering the results reported in Chapter 3. The system with SDOF NES whose response is depicted in Figure 4.10 is identical to that of Figure 4.2, but with the MDOF NES being replaced by a single mass of magnitude 3µ grounded by means of an essential cubic stiffness nonlinearity with characteristic C = 1.0 and weak viscous damper ελ. So it is clear that a significant improvement of efficiency of TET is achieved by using the multi-DOF NES; in addition, TET for the case of the MDOF NES is more robust to variations of the input force compared to the SDOF case. Particularly notable is the capacity of the MDOF NES to absorb a significant portion of the input energy even for low applied impulses. Such low-energy targeted energy transfer is markedly different from the performance of SDOF NESs, where, as reported in previous works (Vakakis et al., 2004; McFarland et al., 2004) and in Chapter 3 of this work, TET is ‘activated’ only when the magnitude of input energy exceeds a certain critical threshold. For the case of the MDOF NES such a critical energy threshold can only be detected in the energy plot for α = 4 of Figure 4.8, e.g., only in the case when the primary system possesses well separated natural

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Fig. 4.10 Comparisons of EDMs for primary systems attached to SDOF and MDOF NESs, and impulsive forcing conditions: (a) I1, (b) I2, and (c) I3.

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frequencies and is excited by out-of-phase initial conditions. In all other cases (Figures 4.6–4.8) no such critical input energy threshold is identified. This interesting dynamical feature of the MDOF NES will be reconsidered in more detail in a later section; here it suffices to state that the capacity of MDOF NES for low-energy TET is enabled by the rich structure of periodic orbits (NNMs) of the underlying Hamiltonian system, a subset of which localize to the NES with decreasing energy due to damping dissipation. Of particular interest is the plot of ENES depicted in Figure 4.8 corresponding to α = 1 (for the case when the natural frequencies of the uncoupled primary system are equal to ω1 = 1.7321, ω2 = 1.0 rad/sec) and out-of-phase impulse excitations. In that plot we note that for sufficiently small impulse magnitudes, the portion of energy dissipated by the MDOF NES develops an initial local minimum before reaching higher values. To gain insight into the dynamics of targeted energy transfer in that region, in Figures 4.11 and 4.12 the numerical spectra of Cauchy wavelet transforms (WTs) of the internal relative NES displacements [v2 (t) − v1 (t)] and [v3 (t) − v2 (t)] at points labeled A and B of Figure 4.8 are depicted. Point A corresponds to the case of relatively weak TET from the primary system to the MDOF NES, whereas, point B to a case where nearly 90% of the input energy gets absorbed and eventually dissipated by the NES. The WT spectra depict the amplitude of the WT as function of frequency (vertical axis) and time (horizontal axis). Heavy shaded areas correspond to regions where the amplitude of the WT is high whereas lightly shaded regions correspond to low amplitudes. Such plots enable one to deduce the temporal evolutions of the dominant frequency components of the signals analyzed. Comparing the two responses of Figures 4.11 (point A) and 4.12 (point B), it is clear that the enhanced TET noted in the later case is due mainly to the largeamplitude transient relative response [v3 (t) − v2 (t)]. Moreover, judging from the corresponding WT spectrum, this time series consists of a ‘fast’ oscillation with frequency close to ω1 , that is modulated by a large-amplitude ‘slow’ envelope. Additionally, one notes that this modulated response is not sustained over time, but takes place only in the initial phase of the motion and escapes from this regime of the motion at approximately t = 50. Similar behavior is noted for the time series of the other relative response, [v2 (t) − v1 (t)] depicted in Figure 4.12. It is well established (Vakakis et al., 2004; Panagopoulos et al., 2004; McFarland et al., 2004) that this represents a TRC of the NES dynamics on a resonance manifold near the out-ofphase linear mode of the uncoupled primary system, which results in enhanced and irreversible energy transfer from the primary system to the NES. Comparing the responses of Figures 4.12 and 4.11, it is clear that in the later case (where weaker TET occurs) the transient responses are dominated by sustained frequency components indicating excitation of NNMs, rather than occurrence of TRCs. The frequencies of some of the excited NNMs differ from the linearized natural frequencies ω1 and ω2 , indicating the presence of essentially nonlinear modes in the response, having no linear analogs. From the above discussion it is clear that the transient dynamics of the dissipative system of Figure 4.2 is rather complex. Moreover, the numerical results depicted in

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Fig. 4.11 Internal NES relative displacements for out-of-phase impulses (IFC I3) with Y = 1 and α = 1 (point A in Figure 4.8): (a) Time series, (b) Cauchy wavelet transforms; the linear natural frequencies of the uncoupled (ε = 0) primary system are indicated by dashed lines.

Figures 4.6–4.12 indicate that the MDOF NES leads to enhanced TET compared to the SDOF NES, a conclusion that provides ample motivation for a systematic and detailed study of the corresponding transient dynamics. This is performed in the following sections. We start our study by considering the underlying Hamiltonian system (i.e., the corresponding system with no dissipation), and show that the Hamiltonian dynamics influences drastically the weakly damped responses and, hence, controls TET.

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Fig. 4.12 Internal NES relative displacements for out-of-phase implulses (IFC I3) with Y = 1.5 and α = 1 (point B in Figure 4.8): (a) Time series, (b) Cauchy wavelet transforms; the linear natural frequencies of the uncoupled (ε = 0) primary system are indicated by dashed lines.

4.2 The Dynamics of the Underlying Hamiltonian System The results reported in the previous sections provide ample motivation to study the dynamics of the system depicted in Figure 4.2. Our aim is to better understand the different regimes of the motion, and the dynamic mechanisms that govern passive TET from the directly excited primary system to the MDOF NES (Tsakirtzis, 2006; Tsakirtzis et al., 2007). A first step towards analyzing the dynamics of system (4.4) is to study the structure of the periodic orbits of the corresponding Hamiltonian system (with no damping terms, ελ = 0). Then, to show that passive TET as well as other type of complicated transient dynamics of the weakly damped system (4.4)

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can be explained and interpreted in terms of transitions between different branches of periodic orbits of the Hamiltonian system in an appropriate frequency-energy plot (FEP). The reasoning behind this plan has to do with the intricate relationship between the damped and weakly undamped systems, and on the paradoxical fact that the weakly damped dynamics is mainly determined by the underlying Hamiltonian dynamics (Lee et al., 2005; Kerschen et al., 2006). Indeed, the effect of damping in the transient dynamics is parasitic, as it does not generate new dynamics but only invokes transitions between different branches of solutions (NNMs) of the underlying Hamiltonian system. It follows that although damping is prerequisite for TET, the dynamics of TET is mainly determined by the underlying Hamiltonian structure of the dynamics. We will employ both analytical and numerical techniques to show that the undamped (Hamiltonian) system possesses a surprisingly complicated structure of periodic orbits that give rise to complicated phenomena and damped transitions. This result should not be unexpected given the high degeneracy of the linear structure of the dynamical system (4.4), which is expected to lead to complicated, highcodimension bifurcations on the corresponding high-dimensional center manifold. Although such a general bifurcation study is beyond the scope of this work, we will show that the underlying Hamiltonian dynamics influence the weakly damped transient dynamics of Figure 4.2, and, in essence, governs TET. To provide an indication of the degeneracy of the system with an attached MDOF NES, we reconsider equations (4.4) and set the damping parameters and forcing terms equal to zero. Changing into modal coordinates of the primary (linear) system, w1 = u1 +u2 , w2 = u1 −u2 , the equations of motion can be placed in the following form: w¨ 1 + ω02 w1 + (ε/2)(w1 − w2 ) − εv1 = 0 w¨ 2 + (ω02 + 2α)w2 − (ε/2)(w1 − w2 ) + εv1 = 0 µv¨1 + C1 (v1 − v2 )3 + ε[v1 − (1/2)(w1 − w2 )] = 0 µv¨2 + C1 (v2 − v1 )3 + C2 (v2 − v3 )3 = 0 µv¨3 + C2 (v3 − v2 )3 = 0 Placing these equations into state form we obtain:

(4.7)

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(4.8) with κ = C2 /C1 . In the limit of zero coupling between the primary system and the MDOF NES, ε → 0, the combined system degenerates into a system with two pairs of imaginary eigenvalues and three double zero eigenvalues. Clearly, this is a highly degenerate dynamical system, with a ten-dimensional center manifold (which coincides with the entire phase space of the system). Such degenerate dynamical systems possess highly co-dimensional bifurcation structures, which give rise to complicated regular and chaotic dynamics (Guckenheimer and Holmes, 1983; Wiggins, 1990), and their study is beyond the current state-of-the-art. However, by narrowing our aim to the study of TET, it is possible to apply analytical techniques to the study of the dynamics of this highly degenerate system. Hence, we reconsider the dynamics of the five-DOF essentially nonlinear Hamiltonian system which is derived by removing the damping terms from equations (4.4). There are various numerical algorithms that compute the periodic orbits of this system, and in this work the numerical algorithm described in Tsakirtzis et al. (2005) is followed. To compute the periodic orbits of this system, first it is assumed that a periodic orbit of the Hamiltonian system is realized for the initial velocity vector [u˙ 1 (0) u˙ 2 (0) v˙1 (0) v˙2 (0) v˙3 (0)] with zero initial displacements; then, the algorithm computes this initial condition vector together with the period T , for which the following periodicity condition is satisfied: [u1 (T ) u2 (T ) v1 (T ) v2 (T ) v3 (T ) u˙ 1 (T ) u˙ 2 (T ) v˙1 (T ) v˙2 (T ) v˙3 (T )]T − [0 0 0 0 0 u˙ 1 0 u˙ 2 0 v˙1 0 v˙2 0 v˙3 0]T = 0

(4.9)

The algorithm has been implemented in Matlab using optimization techniques. For a given value of the period T , the objective function to minimize is the norm of the

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left-hand side of equation (4.9), with the optimization variables being the five nonzero initial velocities. By varying the period, a frequency-energy plot (FEP) can be drawn, depicting the dominant frequency of a periodic motion (NNM) as function of the corresponding (conserved) energy of the Hamiltonian system. When more than one dominant frequencies exist (for example, when two coordinates have different dominant frequency components), the lowest of these dominant frequencies is depicted in the FEP. In the following sections we consider two configurations of MDOF NESs, principally distinguished by the order of magnitude of their masses. The aim of the study is to assess the influence of the masses of the NESs on TET.

4.2.1 System I: NES with O(1) Mass The first system configuration considered (referred to from now on as ‘System I’) consists of a relatively heavy nonlinear attachment, and system parameters: µ = 1.0, ω02 = 1.0, α = 1.0, ε = 0.1, C1 = 2.0, C2 = ε2 (System I) The small value of the nonlinear characteristic C2 was dictated by the numerical results of the previous section, where it was found that for small values of C2 enhanced TET from the primary system to the NES was realized. First, we discuss certain features of the dynamics of this system in the frequency-energy plot (FEP). A first observation related to the system of equations (4.4), is that for no damping and forcing, and in the limit of small energy and finite frequencies the dynamics of the system is approximately governed by the following linear subsystem of equations (4.4): u¨ 1 + (ω02 + α)u1 − αu2 = 0 u¨ 2 + (ω02 + α + ε)u2 − αu1 − εv1 = 0 (Limit of low energies finite frequencies) µv¨1 + ε(v1 − u2 ) = 0

(4.10)

In that case the periodic orbits of the full undamped and unforced nonlinear system tend to the three eigenmodes of the linear subsystem (4.10), with corresponding eigenfrequencies, f1 = 1.7473, f2 = 1.0265, and f3 = 0.3054 rad/s. A second observation is that in the limit of high energies and finite frequencies the essentially nonlinear stiffnesses of system (4.4) behave approximately as massless rigid links, resulting in the following alternative approximate linear subsystem: u¨ 1 + (ω02 + α)u1 − αu2 = 0 u¨ 2 + (ω02 + α + ε)u2 − αu1 − εv1 = 0 (Limit of high energies finite frequencies) 3µv¨1 + ε(v1 − u2 ) = 0

(4.11)

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Fig. 4.13 Frequency-energy plot (FEP)of the Hamiltonian dynamics of System I; symbols A, B and C are the initial conditions for the damped transitions depicted in Figures 4.26, 4.27 and 4.28, respectively.

Then the periodic motions of the Hamiltonian system tend asymptotically to the linear eigenfrequencies, fˆ1 , fˆ2 and fˆ3 of subsystem (4.11). For System I these frequencies are equal to fˆ1 = 1.766, fˆ2 = 1.0248, and fˆ3 = 0.1766 rad/s. These observations are important in order to understand the complicated structure of periodic orbits of the Hamiltonian System I in the FEP. This will lead also to clear interpretations of multi-frequency damped transitions, as sudden jumps between distinct branches of solutions in the FEP. The FEP for the periodic orbits of System I is depicted in Figure 4.13, together with two enlarged regions Z1, and Z2 showing in detail certain domains of the plot (see Figures 4.14, 4.15). Indicated also in the plot are the natural frequencies fi , fˆi of the limiting linear systems (4.10) and (4.11). Unless in the neighborhood of one of the six natural frequencies fi , fˆi , i = 1, 2, 3, the response of the primary subsystem is small, and the motion is localized to the nonlinear attachment.

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Fig. 4.14 Enlarged region Z1.

Fig. 4.15 Enlarged region Z2.

Regarding the general features of the FEP, we note that it contains two basic types of branches: backbone (global) branches consisting of multi-frequency periodic motions defined over extended frequency and energy ranges; and local branches termed subharmonic tongues consisting of multi-frequency periodic motions, with frequencies defined only in neighborhoods of certain basic frequencies. Each tongue is defined over a finite energy range, and consists of two subharmonic branches of periodic solutions (NNMs), which at a critical energy value coalesce in a bifurcation that

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Fig. 4.16 Time series of the periodic motions of System I corresponding to the indices indicated at the FEP of Figures 4.13 and 4.14.

signifies the end of that particular tongue and the elimination of the corresponding subharmonic motion. Moreover, there exists a regular backbone branch where the last mass of the nonlinear attachment (the NES) has nearly zero amplitude (e.g., v3 ≈ 0). Periodic motions (NNMs) on this regular backbone are approximately monochromatic, that is, all coordinates of System I vibrate approximately in-unison with identical dominant frequencies; NNMs on that regular backbone branch correspond to either in-phase or out-of-phase relative motions of the particles of the system. On this branch, the motion is always localized to the first two masses of the nonlinear attachment, except in the vicinity of the natural frequencies of the low-energy limiting linear subsystem (4.10), and at the extremities of the two lower tongues observed in Figure 4.13; one of these tongues occurs at f1 /3 = 0.58 rad/s, and the other at f2 /3 = 0.34 rad/s. A countable infinity of additional subharmonic tongues occurs in the neighborhoods of frequencies that are in rational relationships to the basic frequencies f1 , f2 and f3 of subsystem (4.10), but these are not represented in the FEP of Figure 4.13. The time histories depicted in Figures 4.16a, b (points 1 and 2) show that the mo-

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tion on the regular backbone branch is mainly monochromatic, and that, indeed, the MDOF NES vibrates with the same frequency as the primary system. At point 1 the displacements of the two masses of the primary system oscillate in out-of-phase fashion, whereas at point 2 in in-phase fashion. Another interesting feature of the FEP of System I is that, besides the regular backbone branch, there exist additional singular backbone branches at higher values of the energy (the term ‘singular’ is justified by the analysis of the next section). Each of the singular backbone branches may also carry tongues of subharmonic periodic motions. For instance, a lower tongue appears around f2 /3 = 0.34 rad/s for each of the three singular backbone branches depicted in Figure 4.14. There are basic qualitative differences between the additional singular backbone branches and the main backbone branch: first, the amplitude of oscillation of the last mass of the NES takes finite values at the singular backbone branches; second, for periodic motions on the singular backbone branches the particles of the system oscillate with differing dominant frequency components (this contrasts to the regular backbone branch where all particles oscillate with identical dominant frequency components). Indeed, the singular backbone branches consist of subharmonic motions that are defined over wide frequency ranges of the FEP, in contrast to subharmonic motions on the tongues that are localized to frequencies rationally related to fi and fˆi . It is interesting that the additional family of backbone curves of System I is not limited to the three singular branches depicted in Figure 4.13. Indeed, as shown later a countable infinity of singular backbone branches exists in the FEP, a result substantiated by numerical evidence. In particular, an extended computation of the periodic orbits of System I performed at a fixed dominant frequency ω = 1.5 rad/s, yielded as many as eleven distinct periodic orbits (NNMs) distinguished by their energy and frequency contents (i.e., they possess different composition of harmonics); however, some of these orbits are unstable. The computed initial conditions of these orbits are listed in Table 4.1, together with their corresponding energies. All these periodic orbits (NNMs) on the singular backbones have two common features: first, the motion of System I is always strongly localized to the nonlinear attachment; second, they all correspond to approximately the same motion of the primary system, since the linear out-of-phase mode is predominantly excited at this particular frequency. The difference between these periodic solutions becomes clear when the Fast Fourier Transforms (FFTs) of the corresponding time series are considered. Whereas the relative displacement [v2 (t) − v3 (t)] contains always the dominant component at ω = 1.5 rad/s, the dominant harmonic component of [v1 (t) − v2 (t)] varies depending on the specific orbit considered. This enables us to label the singular backbone branches with the notation S1jp. The first index refers to the dominant frequency of the primary system (in this case ω = 1.5 rad/s), whereas the second indicates that the dominant frequency of [v1 (t) − v2 (t)] is j times the dominant frequency of the primary system; the third index indicates that the dominant frequency of [v2 (t) − v3 (t)] is p times of that of the primary system. Following this notation, the regular backbone branch of the FEP of Figure 4.13 is labeled as S111 (since it is approximately monochromatic, i.e., all particles oscillate with identical domi-

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Table 4.1 Initial conditions and energies of the periodic orbits of System I for ω = 1.5 rad/s. Solutions 1 2 3 4 5 6 7 8 9 10 11

Feature 1:1 1:1 1:3 1:3 1:3 1:4 1:4 1:5 1:5 1:6 1:6

u˙ 1 (0) 0.9388 0.9649 0.7854 0.7855 0.7851 0.7333 0.7278 0.7300 0.7105 0.7083 0.7081

u˙ 2 (0) –0.2456 –0.2484 –0.2296 –0.2269 –0.2240 –0.2291 –0.1484 –0.1582 –0.2134 –0.2091 –0.1551

v˙1 (0) –7.4273 –8.1159 –2.4476 –3.0519 –3.6277 3.8334 –14.3179 –19.7692 8.1081 13.9267 –25.9220

v˙2 (0) –6.2384 –5.5059 –10.9023 –10.5010 –10.0961 –15.0532 1.5783 7.2405 –19.7099 –25.9124 14.0430

v˙3 (0) 13.2846 13.2263 13.0534 13.2539 13.4223 10.9568 12.4052 12.1996 11.3402 11.7220 11.5617

Energy 2.1327 2.1337 2.1701 2.1701 2.1701 2.2576 2.2576 2.4718 2.4649 2.7004 2.7004

nant frequency), and the additional backbones as S131, S141, S151, . . . . Generally speaking, the higher the dominant harmonic of [v1 (t) − v2 (t)] is, the higher is the energy of the corresponding periodic orbit. Starting from ω ≈ 0.22 rad/s coalescences between different backbone branches occur sequentially as shown in Figure 4.15; these are saddle-node (SN) bifurcations. Coalescences occur between two branches with similar motion, labeled by (a) and (b) (for instance, S151a coalesces with S151b). At the coalescence points, the motion is identical to that on the regular backbone branch, meaning that the coalescing branches meet the regular backbone branch at the coalescence points. Hence, with diminishing frequency the different families of singular backbone branches eventually disappear through coalescences, and a single low frequency singular backbone branch eventually emerges, termed lower singular backbone branch. On this branch, the last mass of the NES has very small displacement but the overall motion of System I is still localized to the first two masses of the nonlinear attachment; this is confirmed by the simulations of Figures 4.16c, d corresponding to points 3 and 4 on the lower singular backbone branch. Summarizing, the most interesting feature of the frequency-energy plot (FEP) of System I is the existence of a countable infinity of closely spaced singular backbone branches that extend over wide ranges of frequencies and energies. This feature of the dynamics is novel, and differs from the FEPs discussed in Chapter 3 corresponding to SDOF NESs. In the following section we consider the same primary system – MDOF NES configuration but with O(ε) masses, in order to assess the effect on the dynamics of a reduction of the NES masses.

4.2.2 System II: NES with O(ε) Mass We now reconsider the system depicted in Figure 4.2 with weak nonlinear stiffness C2 and small NES masses; this system we label as ‘System II’. It is shown that by reducing the masses of the NES the complexity of the dynamics increases, and the

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capacity of the system for TET is significantly enhanced. Hence, the unforced and undamped system (4.4) is considered again with parameters: ε = 0.2,

α = 1.0,

µ → ε2 µ = 0.08,

C1 = 4.0,

C2 → ε2 C2 = 0.05,

ω02 = 1.0 (System II)

We are interested to study the effect on the dynamics of a reduction of the masses of the nonlinear attachments, and to relate TET to the topological structure of periodic orbits of the FEP of the underlying Hamiltonian system. Moreover, we wish to compare the FEP of this system to that of System I. The underlying Hamiltonian system in this case takes the form: u¨ 1 + (ω02 + α)u1 − αu2 = 0 u¨ 2 + (ω02 + α + ε)u2 − αu1 − εv1 = 0 ε2 µv¨1 + C1 (v1 − v2 )3 + ε(v1 − u2 ) = 0 ε 2 µv¨2 + C1 (v2 − v1 )3 + ε2 C2 (v2 − v3 )3 = 0 ε2 µv¨3 + ε2 C2 (v3 − v2 )3 = 0

(4.12)

Regarding ε as a perturbation parameter of the problem, system (4.12) is expected to possess complicated dynamics as ε → 0 since it is essentially (strongly) nonlinear, high-dimensional, and singular [in three of equations (4.12) the highest derivatives are multiplied by the perturbation parameter squared]. The periodic orbits of System II were computed utilizing the numerical algorithm described in the previous section for System I. In Figure 4.17 the periodic orbits of (4.12) are presented in a FEP, and in Figure 4.18 some representative orbits are presented. Since the numerical algorithm could not reliably capture the lowest frequency branch, this was analytically computed (as discussed later) and superimposed to the numerical results. These results provide an indication of the complexity of the dynamics. As for the case of System I, the FEP contains both a regular backbone and a family of singular backbones. In this case, however, the singular backbone branches are not densely packed as in System I. Moreover, for System II the backbone branches of periodic orbits (NNMs) are defined over wider frequency and energy ranges compared to System I, and no subharmonic tongues were revealed. Hence, it appears that by reducing the masses of the NES the local subharmonic tongues are eliminated; that is, there are no subharmonic motions at frequencies rationally related to the natural frequencies f1 , f2 , f3 of the linear subsystem (4.10) (for System II these frequencies assume the values f1 = 1.8529, f2 = 1.5259, f3 = 0.9685 rad/s). As for the case of System I, in the limit of high energies and moderate frequencies, System II reaches the linear limiting system (4.11), with corresponding limiting natural frequencies given by fˆ1 = 1.7734, fˆ2 = 1.1200, and fˆ3 = 0.7960 rad/s.

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Fig. 4.17 FEP of the periodic orbits of System II; indices refer to the time series depicted in Figure 4.18.

It is interesting to consider the dynamics of System II at points A, B and C of the plot of Figure 4.17, i.e., at points where the regular backbone branch crosses the natural frequencies of the low-energy limiting linear system (4.10). At these points it holds approximately that, v1 ≈ v2 and v2 ≈ −v3 , so the system may be approximately decomposed into two subsystems: the subsystem (4.10) (the limiting linear system for low energies and finite frequency), and a strongly nonlinear system composed of the first two masses of the NES with their center of mass being approximately motionless. At points A, B and C the linear subsystem vibrates on one of its linear modes at frequencies f1 , f2 or f3 , whereas the nonlinear attachment adjusts its energy to oscillate with the same frequency. Hence, the energy of the nonlinear subsystem (together with the energy of the linear subsystem) determines the points of crossing A,B and C of the regular backbone curve with each of the natural frequencies of the linear limiting subsystem (4.10). An additional remark is that the reduction of the masses of the NES causes a ‘spreading out’ of the closely spaced members of the family of singular backbones of System I. As a result, multiple subharmonic periodic orbits coexist over wider energy ranges compared to System I (though some of these orbits are unstable and, hence, not physically realizable). The elimination of the subharmonic tongues and the spreading of the family of singular backbone curves imply that in System II

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Fig. 4.18 Periodic orbits at specific points (indicated by numbers) in the FEP of Figure 4.17 (System II).

subharmonic motions are realized only on the singular backbone curves (instead of tongues as in System I) that extend over wide regions of the FEP. These features of the FEP will have profound effects on the transient responses of the weakly damped System II, which are examined in the next section. Moreover, it will be shown that System II possesses enhanced TET properties compared to System I.

4.2.3 Asymptotic Analysis of Nonlinear Resonant Orbits In this section we initiate the analytical study of the Hamiltonian dynamics of system (4.4) (with zero damping and forcing terms). Specifically, we mathematically study certain aspects of NNMs on the regular and singular backbone curves, and explain analytically the multiplicity (fine structure) of the family of singular backbone

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

Fig. 4.18 Continued.

329

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branches in the FEP. In the following analysis we consider in detail only the undamped and unforced System I, and show that the results can be extended to System II by an appropriate time transformation. To this end we consider the system of coupled oscillators (Tsakirtzis, 2006; Tsakirtzis et al., 2007), u¨ 1 + (ω02 + α)u1 − αu2 = 0 u¨ 2 + (ω02 + α + ε)u2 − αu1 − εv1 = 0 µv¨1 + C1 (v1 − v2 )3 + ε(v1 − u2 ) = 0 µv¨2 + C1 (v2 − v1 )3 + ε2 C2 (v2 − v3 )3 = 0 µv¨3 + ε2 C2 (v3 − v2 )3 = 0

(4.13)

and assume that all parameters other than ε are O(1) scalars. The main goal of the analysis is to study the periodic motions (NNMs) of this system that possess a dominant frequency ω away from the natural frequencies of the limiting linear system that results as ε → 0. First, only non-resonant motions are considered. Under the condition of absence of linear resonances, and assuming that the system executes a periodic oscillation with frequency ω, the approximations u¨ 1 ≈ −ω2 u1 and u¨ 2 ≈ −ω2 u2 are introduced, which approximately reduce the two leading differential equations of (4.13) to the following algebraic relations (taking α = ω02 = 1 for simplicity): u1 ≈

εv1 2 (1 − ω )(3 − ω2 ) + ε(2 − ω2 )

= O(ε) (ω away from roots of denominator)

εv1 (2 − ω2 ) u2 ≈ = O(ε) 2 (1 − ω )(3 − ω2 ) + ε(2 − ω2 )

(4.14)

These approximate algebraic relations replace (and thus simplify) two of the ordinary differential equations of system (4.13). The rationale behind this approximation is that away from their resonances the two linear oscillators vibrate approximately in a harmonic fashion with common frequency ω. It follows that in the absence of resonance the Hamiltonian dynamics is governed mainly by the MDOF NES, as the response of the linear system is approximately computed by (4.14). Moreover, for frequencies ω away from the roots of the denominator of (4.14) (i.e., the linearized natural frequencies of the limiting system as ε → 0), the periodic orbits of (4.13) are mainly localized to the MDOF NES, and governed approximately by the following reduced system: v¨1 + C1 (v1 − v2 )3 + εv1 = 0 v¨2 + C1 (v2 − v1 )3 + ε2 C2 (v2 − v3 )3 = 0 v¨3 + ε2 C2 (v3 − v2 )3 = 0

(4.15)

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√ where the rescaling of time t → µt was introduced. Finally, with the change of variables, 3z = v1 + v2 + v3 , q1 = v1 − v2 and q2 = v2 − v3 , the reduced system (4.15) is expressed as: z¨ + (ε/3)[z + (2q1 /3) + (q2 /3)] = 0 q¨1 + (ε/3)[z + (2q1 /3) + (q2 /3)] + 2C1 q13 − ε2 C2 q23 = 0 q¨2 + 2C2 ε2 q23 − C1 q13 = 0

(4.16)

The variable z describes the (slow time scale) oscillation of the center of mass of the MDOF NES, whereas the variables q1 and q2 are the relative oscillations between the NES masses (which occur at a faster time scale). As a result, the reduced system can be further decomposed into a ‘slowly varying’ component, i.e., the z-oscillator, and two ‘fast varying’ components, namely the coupled oscillators governing q1 and q2 . The reduced system (4.16) is the starting point for the pertubation analysis that follows. Before proceeding further we show that the dynamics of System II (possessing small NES masses) can be reduced also to the form (4.16) by a transformation of the time variable. Indeed, considering the undamped and unforced System II √ – equations (4.12) – the time transformation τ = ε µ is introduced. Assuming that the dominant frequency ω of the periodic orbit is away from the linear resonances, it can be shown that the responses of the linear subsystem can be expressed approximately as, u1 ≈ u2 /(2ε2 µ − ω2 ) and u2 ≈ εv1 [ε + 2ε 2 µ − ω2 − 2/(2ε2 µ − ω2 )], so the system reduces again to system (4.16). Hence, the following analytical results √ derived for System I also apply to System II for the rescaled time variable τ = ε µ.

4.2.3.1 The Low-Frequency Limit √ Assuming that ω ε/3, i.e., that the dominant frequency of the response is much less than the linearized natural frequency of the first equation of the set (4.16), we may approximately neglect the second derivative z¨ from the first equation, and derive the following approximate algebraic expression for z: z ≈ −(2q1/3) − (q2 /3)

(4.17)

This approximation is valid only in the low-frequency limit, since only for sufficiently small frequencies the inertia term in the linear oscillator in (4.16) is of much smaller magnitude that the stiffness terms. Hence, we can reduce further system (4.16) to a system of two essentially nonlinear coupled oscillators: q¨1 + 2C1 q13 − ε2 C2 q23 = 0 q¨2 + 2C2 ε2 q23 − C1 q13 = 0

(Low frequency limit)

(4.18)

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This is a symmetric system in the terminology of Rosenberg (1966), and its periodic solutions are similar nonlinear normal modes (NNMs) satisfying linear modal relationships of the form q2 = kq1 , where k is the modal constant (see Section 2.1). In addition, these are synchronous periodic motions of system (4.18) where both coordinates oscillate in-unison, reaching their extreme values of the same instant of time, so that the resulting motion is represented by a straight line in the configuration plane (q1 , q2 ) of the reduced system. Substituting the relation q2 = kq1 into (4.18), and imposing the requirement that both equations produce identical periodic solutions, we derive the following equation for determining the modal constant k, possessing two real roots: ε2 (C2 /C1 ) k 4 + 2 (C2 /C1 ) ε2 k 3 − 2k − 1 = 0 ⇒ 1 3ε 2 C2 + O(ε2 ) k1 = − − 2 32C1

2C1 2/3 −2/3 1 k2 = ε − + O(ε2/3 ) C2 2

(Regular root) (Singular root) (4.19)

The characterization of the two roots as ‘regular’ and ‘singular’ is related to the analysis that follows below. Summarizing, at the low frequency (and low energy) limit the system possesses two branches of periodic solutions. These are precisely the two low regular and low singular backbones shown in the FEP of Figure 4.13 of System I and of Figure 4.17 for System II. The periodic solutions (NNMs) of the system on these low frequency branches are computed through integration by quadratures of either one of equations (4.18) after the modal relation q2 = kq1 is imposed: q¨1 + (2C1 − ε2 C2 k1,2 )q13 = 0 q2 = k1,2 q1 ,

z ≈ −(2q1 /3) − (q2 /3),

(Low frequency limit)

(4.20)

The solutions of the reduced system (4.20) can expressed analytically in terms of elliptic functions. These periodic solutions represent the low-frequency/low-energy asymptotic limits of the branches of NNMs of System I (and also of System II through the time transformation discussed previously).

4.2.3.2 The Case of Finite O(1) Frequencies The other limiting case is when the basic frequency ω of the periodic orbit is of O(1), but away from the linear resonances. In this case the term εz/3 in the first equation of system (4.16) is small compared to the second derivative z¨ , so we may neglect it and express approximately the (slow) oscillation of the center of mass of the MDOF NES as follows: z ≈ (ε/3ω2 )[(2q1/3) + (q2 /3)] + O(ε2 ) = (ε/9ω2 )(2q1 + q2 ) + O(ε2 ) (4.21)

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It follows that in this case we may reduce the system (4.16) to a two-DOF system, similarly to the low-frequency case [see equation (4.18)]:

1 ε 1 + 2 (2q1 + q2 ) + 2C1 q13 − ε2 C2 q23 = 0 (O(1) frequency) q¨1 + 9 ω q¨2 + 2C2 ε2 q23 − C1 q13 = 0

(4.22)

System (4.22) represents a perturbed system of two coupled nonlinear oscillators, with ε being the perturbation parameter. This may be regarded as a non-symmetric perturbation of the symmetric system (4.18) derived for the low-frequency limit; however the added non-symmetric term (ε/9)(1 + 1/ω2 )(2q1 + q2 ) may produce non-trivial perturbations to the dynamics (since it represents a perturbation of an already degenerate-symmetric system), and requires careful consideration in the asymptotic analysis. Indeed, in what follows we prove that there exist two general classes of periodic solutions of (4.22): regular solutions based on regular perturbation analysis of the reduced set; and singular solutions based on singular asymptotic expansions of that set. These two types of asymptotic solutions correspond to the two types of backbone curves identified in previous sections in the FEP of System I, namely, regular and singular backbone branches with each type possessing distinct topological features and dynamical characteristics. Starting with regular perturbation analysis, and omitting terms that depend on ε from (4.22) the following generating symmetric system is obtained: 3 q¨10 + 2C1 q10 =0 3 =0 q¨20 − C1 q10

(4.23)

whose periodic solutions may be exactly computed by quadratures (and expressed in terms of elliptic functions). The solutions are similar NNMs in the terminology of Rosenberg (1966), since they satisfy the linear modal relationship q2 = (−1/2)q1. Recalling the previous analysis of the lower limiting case, we infer that solutions of (4.22) that are expressed as perturbations of the generating solutions obtained from system (4.23) can be regarded as finite-frequency analogs of the ones lying on the low regular backbone branch corresponding to the regular root k1 = −1/2 + O(ε2 ) in relations (4.19). The perturbed solutions for q1 (t) and q2 (t) are expressed as regular perturbations of the generating solutions of (4.23): q1 (t) = q10 (t) + εq11 (t) + ε2 q12 (t) + · · · q2 (t) = q20 (t) + εq21 (t) + ε2 q22 (t) + · · ·

(4.24)

Substituting (4.24) into (4.22) an hierarchy of problems is derived (in increasing powers of ε) that govern the higher-order corrections to the high-frequency periodic solutions. These regular perturbation solutions lie on a single backbone branch of the FEP of System I, which is the high-frequency (and high-energy) limit of the

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regular backbone curve. Based on this approximation it is confirmed that the regular backbone in System I consists of a single branch and does not possess the fine structure of the family of singular backbone branches (see Figures 4.13–4.15). Moreover, this analytic approximation leads to the following estimate for the oscillation of the right end mass of the NES, v3 ≈

εq1 + O(ε2 ), 6ω2

ω = O(1)

(4.25)

which in the high-frequency limit is negligible. This fully confirms the numerical results reported in Section 4.2.1. We now consider periodic solutions of (4.22) that may be regarded as finite frequency continuations of periodic solutions on the low singular backbone, and correspond to the singular root k2 = (2C1 /C2 )2/3 ε−2/3 in (4.19). Based on the numerical findings of Figures 4.13–4.15, we deduce that for increasing frequency (and energy) there occurs a series of bifurcations giving rise to additional singular backbones containing solutions of increasingly higher frequency content. For finite [i.e., O(1)] frequencies we obtain an entire family of singular backbone branches that is densely packed in energy. The following analysis aims to analytically study this family of singular backbones for O(1) frequencies (but away from linear resonances). In this case we approximate the solutions through singular asymptotic analysis, and introduce the transformations (q1 , q2 ) → (Q2 = εq2 , η = q2 − k2 q1 ). Substituting these transformations into (4.22) the following rescaled equations are obtained, which govern periodic solutions (NNMs) on the family of singular backbone branches: ¨ 2 + (3/2)C2Q3 + (3ε/2)C2 Q22 η = 0 Q 2 1/3

C22 1 1 2C1 1/3 1 + 2 Q2 Q22 η = ε2/3 η¨ + 6 2C1 9 C2 ω

(4.26)

The first equation represents an O(ε) parametric perturbation of a strongly nonlinear oscillator. The second equation is singular, as noted from the small coefficient of the derivative term. It is a quasi-linear equation with combined parametric and external excitations. It is well known that this type of excitation produces families of periodic solutions of increasingly higher frequency content (in the case of pure parametric excitation these periodic solutions lie on stability-instability boundaries according to Floquet theory). Hence, from the model (4.26) we may indirectly infer the existence of countable infinities of periodic solutions (due to combined parametric/external resonances) with increasingly higher frequency contents. These correspond to the family of periodic solutions realized on the family of singular backbone curves in the FEP; moreover, the previous analytical arguments indicate that the numerically observed fine structure of singular backbones of Figure 4.13 consists of a countable infinity of distinct branches. Apart from the common basic frequency ω, different members of the family of singular backbones possess increasingly higher harmonics

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at frequencies nω, n = 2,3,. . . , which are generated by the previously described combined parametric and external resonances in the second of equations (4.26). The fine structure of the family of singular backbone branches (interestingly enough, it resembles quantization at closely spaced discrete values of energy) is analytically studied by defining the averaged energy of oscillation, E, of a periodic orbit (NNM): 3 2 v˙32 v˙22 εv12 C1 q14 ε 2 C2 q24 v˙12 Et = + + + + + = (4.27) 2 2 2 2 4 4 t 2 3

ε C1 q14 ε 2 C2 q24 2q1 q2 3˙z2 1 2 2 = + q˙ + q˙1 q˙2 + q˙2 + z+ + + + 2 3 1 2 3 3 4 4 t

We claim that the averaged value of the potential energy between NNMs on distinct singular branches is almost unaffected by the perturbation due to the fine structure of the family. This claim is based in the following reasoning. As mentioned previously, the fine structure is formed due to parametric and external resonances in the second of equations (4.26), which, in addition to oscillations at the basic frequency ω, produce high-frequency harmonics in η possessing similar amplitudes but increasingly higher frequency components nω, n = 2, 3, . . . . Actually, it holds that |η| ∼ |q2|/k2 ∼ ε2/3 |q2 |. Hence, the corrections to the potential energies due to singular perturbations will be insignificant, and, as a result, the fine structure of the family of singular backbones will be determined mainly by fluctuations of the averaged kinetic energy T . The fluctuations of the kinetic energy between different branches of the family of singular backbones is evaluated as follows: 4 2 5 1 2 3˙z 2 T t = + q˙ + q˙1 q˙2 + q˙2 2 3 1 t 5 4 2 1 −2 3˙z −1 2 2 + k2 (q˙2 − η) ∼ ˙ + q˙2 k2 (q˙2 − η) ˙ + q˙2 2 3 t 3 2 η˙ 2 1 1 1 3˙z2 q˙22 + 1+ ∼ + 2 + 2 ∼ T0 + 2 η˙ 2 t (4.28) 2 3 k2 k2 3k2 t 3k2 where T0 is the average value of the kinetic energy, and we have taken into account that since q˙2 and η˙ have different dominant frequencies they average out from the final expression in (4.28). Now, taking into account that at high frequencies it holds that, Q2 ∼ ω ⇒ q2 ∼ ε−1 ω, and that |η| ∼ |q2 |/k2 ∼ ε2/3 |q2 | ∼ ε −1/3 ω, it follows that η˙ 2 ∼ n2 ω2 |η|2 ∼ ε −2/3 n2 ω4 . From (4.28) it is concluded that in the high frequency limit the averaged kinetic energy behaves according to T t = T0 +

ε2/3 C0 ω4 n2 ⇒ 3

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Et = E0 +

ε2/3 C0 ω4 n2 ∼ ω4 (ε−2 + D0 n2 ω2 ε2/3 ) 3

(4.29)

since E0 ∼ ε−2 ω4 . Hence, the splitting distances between members of the family of singular backbones of System I is of O(n2 ω2 ε2/3 ), where n is the order of parametric resonance of the periodic solution for η (or equivalently, the high-frequency harmonic component in η). On the logarithmic scale used to depict energy in the numerical FEP of Figure 4.13, the splitting distance is scaled according to ln(Et ) ∼ ln(ε −2 + D0 n2 ω2 ε2/3 ) ∼ n2 ω2 ε8/3 . This analytical result is in agreement with the numerical results depicted in the FEP. The previous analysis directly applies also to System II. Indeed, taking into account the previous rescaling of time that relates Systems I and II, we only need to √ apply the frequency rescaling, ω → ω/(ε µ), to extend the previous analytical findings to System II. The resulting scaling in the frequency-energy plot of System II is ln(Et ) ∼ n2 ω2 ε2/3 , correctly predicting the ‘spreading out’ of the fine structure of the family of singular backbone branches.

4.2.4 Analysis of Resonant Periodic Orbits We now consider resonant nonlinear responses and transient resonance captures (TRCs) of system (4.4) by reducing the dynamics of system to a single integrodifferential equation; we then discuss methodologies for the analytical treatment of the reduced system. First, we focus on the resonant motions of system (4.4). Specifically, we study the nonlinear undamped and damped dynamics in the neighborhoods of the linear natural frequencies of the system, and discuss methods to analyze the resonant nonlinear interactions between the linear primary system and the MDOF NES. Contrary to the non-resonant analysis of Section 4.2.3, during resonance the components of the linear subsystem oscillate with finite amplitudes, and strong energy exchanges with the NES take place. It is precisely these motions close to resonances that lead to TET phenomena when damping is introduced. First, we study analytically the periodic orbits (NNMs) of the undamped and unforced system (4.4) that result from resonance interactions, i.e., that possess dominant frequency components close to the O(1) natural frequencies of the linear limiting system (4.10). To this end, we introduce again the coordinate transformation R=

v1 + v2 + v3 , 3

X1 = v2 − v1 ,

X2 = v3 − v2

(4.30)

where X1 , X2 and R denote the two relative displacements, and the displacement of the center of mass of the MDOF NES, respectively. Substituting (4.30) into (4.4), and omitting damping terms for the moment, the undamped equations of motion take the form:

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u¨ 1 + (ω02 + α)u1 − αu2 = 0 ε u¨ 2 + (ω02 + α + ε)u2 − αu1 − εR = − (2X1 + X2 ) 3 ε 3µR¨ + εR − εu2 = (2X1 + X2 ) 3 (2X1 + X2 ) − u2 ) = 0 µX¨ 1 + 2C1 X13 − C2 X23 − ε(R − 3 µX¨ 2 + 2C2 X23 − C1 X13 = 0

(4.31)

In the following analysis, unless explicitly noted, the system parameters are assumed to be O(1) quantities. Considering the transformed set of equations (4.31), it is noted that the motion of the center of mass of the NES also executes a linear (but slow) motion which results as weak perturbation of the rigid body mode R¨ = 0. The next step of the analysis involves a linear coordinate transformation that brings the leading three linear equations of system (4.31) into Jordan canonical form (note that the last two equations are perturbations of essentially nonlinear, i.e., nonlinearizable, equations). To this end, we introduce the linear modal transformation ⎧ ⎫ ⎡ ⎫ ⎤⎡ ⎤⎧ 10 0 T1,1 T2,1 T3,1 ⎪ ⎪ ⎨ u1 ⎪ ⎬ ⎬ ⎨ Q1 ⎪ ⎢ ⎥⎢ ⎥ 0 ⎦ ⎣ T1,2 T2,2 T3,2 ⎦ Q2 u2 = ⎣ 0 1 (4.32) ⎪ ⎪ ⎪ +√ ⎩ ⎪ ⎭ ⎭ ⎩ 0 0 1 3µ R T1,3 T2,3 T3,3 Q3 where Tij denotes the j -th component of the i-th eigenvector of the following symmetric matrix: ⎤ ⎡ 2 −α 0 (ω0 + α) +√ ⎥ ⎢ (4.33) = ⎣ −α (ω02 + α + ε) −ε 3µ ⎦ +√ + 0 −ε 3µ ε 3µ Substituting the transformation (4.32) into (4.31), the following alternative set of equations of motion is obtained: ε Q¨ 1 + ωˆ 12 Q1 = (2X1 + X2 )(T1,3 / 3µ − T1,2 3 ε Q¨ 2 + ωˆ 22 Q2 = (2X1 + X2 )(T2,3 / 3µ − T2,2 3 ε Q¨ 3 + ωˆ 32 Q3 = (2X1 + X2 )(T3,3 / 3µ − T3,2 3 (2X1 + X2 ) µX¨ 1 + 2C1 X13 − C2 X23 + ε 3 = ε (T1,3 / 3µ − T1,2 )Q1 + (T2,3 / 3µ − T2,2 )Q2 + (T3,3 / 3µ − T3,2 )Q3 µX¨ 2 + 2C2 X23 − C1 X13 = 0

(4.34)

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where the linearized natural frequencies are defined as follows: ωˆ 12 = ω12 + ε/2 + O(ε2 ), ωˆ 22 = ω22 + ε/2 + O(ε2 ), ωˆ 32 = ε/3µ + O(ε2 ) (4.35) and ω1 > ω2 are the two natural frequencies of the uncoupled linear system corresponding to ε = 0. The elements Tij in (4.34) are defined as follows: √ √ T1,1 = −1/ 2 + O(ε), T1,2 = +1/ 2 + O(ε), T1,3 = 0 + O(ε), √ √ T2,1 = +1/ 2 + O(ε), T2,2 = +1/ 2 + O(ε), T2,3 = 0 + O(ε), T3,1 = 0 + O(ε),

T3,2 = 0 + O(ε),

T3,3 = 1 + O(ε)

Physically, the variables Q1 (t) and Q2 (t) are modal coordinates of the out-ofphase and the in-phase modes, respectively, of the uncoupled linear primary system; whereas Q3 (t) is the coordinate describing the (slow) motion of the center of mass of the MDOF NES. It is noted that the following relations hold between the linearized frequencies ωˆ i and the natural frequencies fi of the linear limiting (4.10) (these are defined in Section 4.2.1): √ √ ωˆ 1 = f1 + O(ε), ωˆ 2 = f2 + O(ε), ωˆ 3 = f3 + O( ε) = O( ε) Considering the system of equations (4.34), we partition it into two subsets: a set of three linear uncoupled oscillators that are weakly ‘forced’ by terms that depend linearly on the NES relative displacements; and a set of two coupled, essentially nonlinear oscillators that govern the relative displacements within the MDOF NES. This partition is very useful in the following analysis in order to perform a reduction of the dynamics to a single integro-differential equation. Finally, motivated again by the numerical results of the previous section, we introduce the additional assumption that the stiffness characteristic C2 of the NES is small; this is imposed by introducing the rescaling C2 → ε 2 C2 = O(ε 2 ). Under these assumptions, and assuming that 0 < ε 1, the first subset of three uncoupled linear equations of the system (4.34) can be solved explicitly as follows: Q˙ 1 (0) sin ωˆ 1 t ωˆ 1 √ t ε −T1,2 + T1,3 / 3µ [2X1 (τ ) + X2 (τ )] sin ωˆ 1 (t − τ )dτ + 3ωˆ 1 0 Q˙ 2 (0) sin ωˆ 2 t Q2 (t) = Q2 (0) cos ωˆ 2 t + ωˆ 2 √ t ε −T2,2 + T2,3 / 3µ [2X1 (τ ) + X2 (τ )] sin ωˆ 2 (t − τ )dτ + 3ωˆ 2 0 Q˙ 3 (0) sin ωˆ 3 t Q3 (t) = Q3 (0) cos ωˆ 3 t + ωˆ 3

Q1 (t) = Q1 (0) cos ωˆ 1 t +

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√ t ε −T3,2 + T3,3 / 3µ + [2X1 (τ ) + X2 (τ )] sin ωˆ 3 (t − τ )dτ 3 0 (4.36) Hence, the modal coordinates of the linear subsystem and the displacement of the center of mass of the MDOF NES are expressed (in exact form) in terms of the relative displacements X1 (t) √ and X2 (t) between the particles of the NES. Note, however, that since ωˆ 3 = O( ε), the center of mass of the NES executes a slow oscillation; this was anticipated previously by the observation that this motion is the ¨ 3 = 0. weak perturbation of the rigid body motion Q Considering now the last of equations (4.34), and taking into account the previous rescaling C2 → ε2 C2 = O(ε 2 ), the following analytic approximation for the variable X2 (t) is obtained: µX¨ 2 = C1 X13 − 2ε2 C2 X23 ⇒ t τ C1 X13 (s)dsdτ + O(ε2 ) X2 (t) = µ−1 0

(4.37)

where we have taken into account that the MDOF NES is initially at rest [so that the initial conditions X2 (0) = X˙ 2 (0) = 0 were imposed in (4.37)]. As a result, the relative displacement X2 (t) is approximately expressed in terms of the relative displacement X1 (t). Finally, substituting the previous results into the fourth of equations (4.34) the full dynamics is approximately reduced to a single, essentially nonlinear integro-differential equation in terms of the dependent variable X1 (t): X¨ 1 + (2C1 /µ)X13 + (2ε/3µ)X1

t τ t = −(ε/3µ2 C1 X13 (s)ds dτ + ε2 Cˆ 2 µ−1 0

τ 0

3 C1 X13 (s)ds dτ

Q˙ 1 (0) + (ε/µ) (T1,3 / 3µ − T1,2 ) Q1 (0)cosωˆ 1 t + sin ωˆ 1 t ωˆ 1 √ ε(−T1,2 + T1,3 / 3µ) t + 3ωˆ 1 0

τ w −1 3 × 2X1 (τ ) + µ C1 X1 (s)ds dw sin ωˆ 1 (t − τ )dτ 0

Q˙ 2 (0) sin ωˆ 2 t + (T2,3 / 3µ − T2,2 ) Q2 (0) cos ωˆ 2 t + ωˆ 2 √ ε(−T2,2 + T2,3 / 3µ t + 3ωˆ 2 0

τ w × 2X1 (τ ) + µ−1 C1 X13 (s)ds dw sin ωˆ 2 (t − τ )dτ

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Q˙ 3 (0) + (T3,3 / 3µ − T3,2 ) Q3 (0)cos ωˆ 3 t + sin ωˆ 3 t ωˆ 3 √ ε(−T3,2 + T3,3 / 3µ) t + 3 0

τ w −1 3 × 2X1 (τ ) + µ C1 X1 (s)ds dw sin ωˆ 3 (t − τ )dτ + O(ε3 ) 0

≡ εf1 (X1 ; ε) + ε2 f2 (X1 ; ε) + O(ε3 )

(4.38)

Strongly nonlinear dynamical systems with similar structure to (4.38) were analyzed asymptotically in Vakakis et al. (2004) and Panagopoulos et al. (2004). Solutions that possess a dominant (fast frequency) harmonic component, may be portioned into slow and fast components by imposing the following ansatz: X1 (t) ≈ A(t)cosθ (t)

(4.39)

where A(t) and θ (t) represent the slowly-varying amplitude and phase of the response, respectively. Hence, by expressing the solution of (4.38) in the form (4.39) the solution is expressed as a fast oscillation modulated by slowly varying envelope. Clearly the (slow) variation of the envelope represents the important (essential) dynamics that govern the resonance interactions between the primary system and the MDOF NES. Substituting (4.39) into (4.38) we obtain the following approximate modulation equations that govern the slow evolution of the amplitude and phase, dA(t) ≈ εg1 (A(t), θ (t), ε 1/2 t, ωˆ 1 t, ωˆ 2 t) + ε2 g2 (A(t), θ (t), ε 1/2 t, ωˆ 1 t, ωˆ 2 t) dt dθ (t) ≈ (t) + εh1 (A(t), θ (t), ε 1/2 t, ωˆ 1 t, ωˆ 2 t) dt + ε2 h2 (A(t), θ (t), ε 1/2 t, ωˆ 1 t, ωˆ 2 t) √ πA(t) 2C1 /µ (t) = √ 2K(1/ 2)

(4.40)

where√(t) = O(1) is the instantaneous frequency of the fast oscillation, and K(1/ 2) is the complete elliptic integral of the first kind. The functions gi and hi , i = 1, 2 in (4.40) are 2π-periodic in the slow angle θ and the slow time ε1/2 t, but their dependences on the other time scales ωˆ 1 t and ωˆ 2 t depend on the specific values of the linearized natural frequencies ωˆ 1 and ωˆ 2 . This means that the terms on the right-hand sides of relations (4.40) might be either periodic or quasi-periodic functions in terms of the fast time t, depending on if the frequency ratio ωˆ 1 /ωˆ 2 is a rational or irrational number, respectively. We note that these terms also depend on the slow time ε1/2 t. Equations (4.40) are modulation equations and apply for arbitrary values of the basic fast frequency of the solution. For further analysis we need to impose addi-

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tional restrictions on the fast frequency (t), and confine the analysis locally in frequency; this will introduce an additional slow independent variable in the modulation equations that will enable us to analyze resonant periodic orbits of the Hamiltonian system with frequencies close to the natural frequencies of f1 and f2 of the linear subsystem (4.10) [or equivalently – correct to O(ε) – to the frequencies ωˆ 1 and ωˆ 2 ]. To provide an example of such a local analysis we restrict the fast frequency (t) to be approximately equal to ωˆ 1 , and aim to study resonant periodic motions of the Hamiltonian system with dominant frequencies close to the higher natural frequency of the linear subsystem. To this end, we define the amplitude of oscillation, R, by the following frequency relation: √ πR 2C1 /µ ωˆ 1 = √ (4.41) 2K(1/ 2) and introduce two new variables, namely, a slow angle variable χ(t) and an amplitude perturbation α(t): √ χ(t) = θ (t) − ωˆ 1 t, A(t) = R + εα(t) (4.42) By √ considering the relations (4.42) into (4.40) we study periodic motions in an O( ε)-neighborhood of the 1-1 resonance manifold in the phase space of the system, defined by the resonance condition (4.41). Hence, we aim√to reduce the general modulation equation (4.40) to a local system valid in the O( ε)-neighborhood of this 1-1 resonance manifold. Substituting (4.42) into the general modulation equations (4.40) the following local modulation equations are obtained: dα(t) ≈ ε1/2 G α(t), χ(t) + ωˆ 1 t, ωˆ 1 t, ωˆ 2 t, ε 1/2 t; ε (4.43) dt √ dχ(t) πα(t) 2C1 /µ ≈ ε1/2 √ + εH α(t), χ(t) + ωˆ 1 t, ωˆ 1 t, ωˆ 2 t, ε 1/2 t; ε dt 2K(1/ 2) where G and H represent appropriately defined functions with the arguments shown above. Further analysis of the reduced modulation equations (4.43) can be performed by applying perturbation techniques, for example by applying the method of averaging [indeed, equations (4.43) are in standard form for applying averaging over the ‘fast’ time variable t] or the method of multiple scales [as performed in Panagopoulos et al. (2004)]. The analysis will yield approximate asymptotic expressions for the periodic orbits and their frequencies. In addition, the dynamical flow in the approximate slow phase plane of the modulation equations (4.43) can be derived. It is clear that the analysis (and the dynamics of the local model) will depend among other factors on the nature of the ratio of the linearized natural frequencies ωˆ 1 /ωˆ 2 . For example, if this ratio is rational the functions ε 1/2 G and εH in (4.43) become periodic functions in the fast time t (so, for example, simple averaging can be applied with respect to the fast time scale in order to analyze the

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local dynamics); whereas, if the frequency ratio is irrational the same functions become quasi-periodic in the fast time scale. These observations will dictate the type of asymptotic analysis that should be applied to study the dynamics of the local undamped system (4.43). We note that the above reduction into modulation equations governing the slow flow dynamics can be applied also to analytically study transient resonance captures (TRCs) in the weakly damped dynamics (for example, the dynamics depicted in Figure 4.12). Indeed, considering the weakly damped system (4.4), and applying the previous reduction process, the five equations of motion can be reduced to the following single reduced integro-differential equation: X¨ 1 + (2C1 /µ)X13 + ε λˆ X˙ 1 + (2ε/3µ)X1 = εfˆ1 (X1 ) + ε2 fˆ2 (X1 ) + O(ε 3 ) (4.44) where ε λˆ denotes a weak damping coefficient, and ε fˆ1 , ε2 fˆ2 are integro-differential operators analogous to the operators εf1 , ε2 f2 in (4.38), respectively, but modified to account for the additional weak damping terms. The analysis follows the general steps discussed previously, and can be applied to study local TRCs in neighborhoods of resonance manifolds defined by frequency relations similar to (4.41) (Panagopoulos et al., 2004). Perhaps a disadvantage of the described approach for studying resonant motions is that the resulting integro-differential equations are quite complicated, which makes their analytical treatment cumbersome. To address this limitation, in the remainder of this section we formulate an alternative approach for analyzing the global structure of the resonant periodic orbits (NNMs) of the undamped and unforced system (4.4), based on complexification and averaging (CX-A). This approach is similar to the analytical approach introduced in Chapter 3, and in the context of the present analysis, it is applied only to study the resonant periodic orbits that are connected to the regular backbone branch [where all particles of system (4.4) oscillate with identical dominant frequencies]; however, similar analysis can be applied to develop analytic approximations for solutions on the family of singular backbone branches and on the local subharmonic tongues. This can be performed by selecting in each case the appropriate ansatz to replace the one utilized in the following analysis. The alternative method for analyzing resonant motions in system (4.4) relies on complexification of the dynamics, followed by slow / fast partition of the response (see Section 2.4). The analysis is performed under the assumption that the resonant response possesses a single ‘fast’ frequency (satisfying a rational relation with a linear eigenvalues of the primary system), that is modulated by a ‘slowly’ varying envelop containing the important (essential) dynamics that we wish to study. The following procedure outlines the formulation of a slow flow problem, e.g., the derivation of the set of slow modulation equations governing the essential dynamics. As discussed in Lee et al. (2006) and demonstrated in Section 3.3.2 this procedure can be extended to study periodic or quasi-periodic motions possessing more than one ‘fast’ frequencies.

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The first step of the alternative analytical method based on CX-A is to introduce the following set of complex dependent variables, each of which contains as real part the velocity of a particle of the system and as imaginary part the corresponding displacement multiplied by the (single) fast frequency: ψ1 = u˙ 1 + j ωu1 , ψ2 = u˙ 2 + j ωu2 , ψ3 = v˙1 + j ωv1 , ψ4 = v˙2 + j ωv2 and ψ5 = v˙3 + j ωv3 where j = (−1)1/2 , and ω is the dominant (fast) frequency of the periodic resonant motion that we wish to study. Then, the displacements and accelerations can be expressed in terms of the new complex variables and their complex conjugates; for example, considering the velocity and acceleration of the first mass of the primary system we obtain, u˙ 1 = (ψ1 − ψ1∗ )/2j ω and u¨ 1 = ψ˙ 1 − (j ω/2)(ψ1 + ψ1∗ ), where (∗ ) denotes complex conjugate. Moreover, since we seek approximately monochromatic periodic solutions in the fast time scale (i.e., solutions that possess a single common fast frequency), the previous complex variables may be expressed in polar form as ψ1 = φ1 ej ωt , ψ2 = φ2 ej ωt , ψ3 = φ3 ej ωt , ψ4 = φ4 ej ωt , ψ5 = φ5 ej ωt

(4.45)

where the complex, time-varying amplitudes φi (t), i = 1, . . . , 5, are slowly-varying amplitude modulations of the ‘fast’ oscillations ej ωt . Employing the ansatz (4.45) it is possible to perform a partition of the resonant response of the system into slow and fast components, and to derive the approximate set of modulation equations governing the slow flow dynamics. This is performed by expressing the undamped and unforced equations (4.4) in terms of the complex variables (ψi and then) φi , and averaging the transformed equations over the fast variable ωt to retain only terms of fast frequency ω. In essence, this averaging process amounts to disregarding terms in the nonlinear equations of motion that possess fast components possessing frequencies higher than ω; the resulting approximate set of averaged equations is expected to be valid only in neighborhoods of the FEP close to the fast frequency ω. Adopting the previously described averaging procedure we derive the following approximate set of first-order complex equations governing the amplitudes φi : jα j ω j ω02 − − φ˙1 + φ1 (φ1 − φ2 ) = 0 2 2ω 2ω jα jε j ω j ω02 ˙ − − φ 2 + φ2 (φ2 − φ1 ) − (φ2 − φ3 ) = 0 2 2ω 2ω 2ω

jε jω j C1 ˙ φ3 − µ φ3 + − 3 |φ3 − φ4 |2 (φ3 − φ4 ) = 0 (φ3 − φ2 ) + 2 2ω 8ω3

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j C1 jω ˙ + − 3 |φ4 − φ3 |2 (φ4 − φ3 ) µ φ 4 + φ4 2 8ω3 j C2 − 3 |φ4 − φ5 |2 (φ4 − φ5 ) = 0 8ω3

j C2 jω ˙ + µ φ 5 + φ5 − 3 |φ5 − φ4 |2 (φ5 − φ4 ) = 0 2 8ω3 +

(4.46)

This represents the (approximate) slow flow of the undamped and unforced dynamical system (4.4) under the specific assumptions made. In a final step we introduce the following polar transformations: φ1 = A 1 e j a 1 ,

φ2 = A 2 e j a 2 ,

φ3 = A 3 e j a 3 ,

φ4 = A 4 e j a 4 ,

φ5 = A 5 e j a 5

which, when substituted into (4.46), and upon setting separately the real and imaginary parts equal to zero, yield the following set of ten real modulation equations governing the (real) amplitudes Ai and phases ai : α sin(a2 − a1 ) = 0 A˙ 1 − A2 2ω α ω ω02 − − (A1 − A2 cos(a2 − a1 )) = 0 A1 a˙ 1 + A1 2 2 2ω α ε sin(a2 − a1 ) − A3 sin(a3 − a2 ) = 0 A˙ 2 − A1 2ω 2ω α ω ω02 − − (A1 cos(a2 − a1 ) − A2 ) A2 a˙ 2 + A2 2 2 2ω −

ε (A2 − A3 cos(a3 − a2 )) = 0 2ω

(A2 + A24 )C1 εA2 sin(a3 − a2 ) − A4 3 sin(a4 − a3 ) = 0 µA˙ 3 − 2ω 8ω3 ε ω (A3 − A2 cos(a3 − a2 )) µA3 a˙ 3 + µA3 − 2 2ω −3

(A23 + A24 )C1 (A3 − A4 cos(a4 − a3 )) = 0 8ω3

µA˙ 4 + 3

(A23 + A24 )C1 A3 (A24 + A25 )C2 A5 sin(a − a ) − 3 sin(a5 − a4 ) = 0 4 3 8ω3 8ω3

µA4 a˙ 4 + µA4 −3

(A2 + A23 )C1 ω −3 4 (A4 − A3 cos(a4 − a3 )) 2 8ω3

(A24 + A25 )C2 (A4 − A5 cos(a5 − a4 )) = 0 8ω3

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(A25 + A24 )A4 sin(a5 − a4 ) = 0 8ω3

µA5 a˙ 4 + µA5

(A2 + A24 )C2 ω −3 5 (A5 − A4 cos(a5 − a4 )) = 0 2 8ω3

(4.47)

An inspection of (4.47) verifies that the steady state amplitudes satisfy the algebraic relationship A21 + A22 + µ(A23 + A24 + A25 ) = N 2 , which may be regarded as an energy-like expression indicating conservation of total energy of the resonant periodic motion of the unforced and undamped system (4.4). Alternatively, this represents a first integral of the slow flow (4.47). To compute periodic resonant solutions of the system, we impose two stationarity requirements in (4.47), namely, that, (i) the phase differences are trivial, a1 = a2 = a3 = a4 = a5 = a, where a is arbitrary; and (ii) the derivatives of the amplitudes are equal to zero, A˙ i = 0. The first condition can hold since the system is undamped. By imposing these stationarity conditions we obtain the following set of nonlinear algebraic equations: ω02 α ω − − A1 (A1 − A2 ) = 0 2 2ω 2ω ω02 α ε ω − − A2 (A2 − A1 ) − (A2 − A3 ) = 0 2 2ω 2ω 2ω ε 3C1 ω − (A3 − A2 ) − (A3 − A4 )3 = 0 2 2ω 8ω3 3C2 ω 3C1 µA4 − (A4 − A3 )3 − (A4 − A5 )3 = 0 3 2 8ω 8ω3 ω 3C2 µA5 − (A5 − A4 )3 = 0 2 8ω3 µA3

(4.48)

which governs the steady state amplitude of the resonant motions with fast frequency ω. By numerically solving it for varying frequency ω we obtain an approximation for the main backbone branch of the system (based on the assumption that the averaging operation is valid). Once the state amplitudes are numerically computed, the analytical approximation for the corresponding periodic orbit (NNM) of the system is given by A1 A2 sin(ωt + a), u2 = sin(ωt + a) ω ω A3 A4 sin(ωt + a), w2 = sin(ωt + a), w1 = ω ω u1 =

w3 =

A5 sin(ωt + a) ω (4.49)

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(a)

(b)

Fig. 4.19 Approximate regular backbone branches obtained from equations (4.48): (a) System I, (b) System II.

In Figure 4.19 we depict the approximate main backbone branches in the FEP for Systems I and II, resulting from the numerical solution of the set of steady state equations (4.48). The analytical results are in agreement with the numerical FEPs depicted in Figures 4.13 and 4.17; this validates the outlined analytical complexification/averaging method. As mentioned previously, by modifying appropriately the ansatz (4.45) the previous analysis can be extended to approximate other types of periodic solutions in the FEPs of Systems I and II. Depending on the dominant fast frequencies of the motions of the particles of the system, one should define appropriate complex variables ψi , i = 1, . . . , 5, and select suitable slow/fast partitions of the dynamics. Moreover, the complexification / averaging analysis can be applied to study damped transient responses of the full system (4.4), in similarity to the analysis performed in Chapter 3. These results conclude the study of the FEP of periodic orbits of the underlying Hamiltonian system which results by neglecting the damping and forcing terms from (4.4). In the following section we present a study of damped transitions and TET in system (4.4) by adding weak damping and considering impluses applied to the linear primary system. We will show that the weakly damped transitions (and TET) of the impulsively forced system can be studied in terms of the underlying Hamiltonian dynamics.

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4.3 TRCs and TET in the Damped and Forced System The topological portraits of the FEPs of the Hamiltonian Systems I and II provide a clear indication of the complex topology of the periodic orbits of the undamped and unforced dynamics. In this section we show that this rich topological structure of periodic orbits of the underlying Hamiltonian systems leads to complicated transient dynamics of the forced and damped systems, including multi-frequency transitions between different branches of solutions, isolated TRCs and resonance capture cascades. The study of transitions in the damped dynamics is performed by superimposing the wavelet transform (WT) spectra of the transient responses to the FEPs of the underlying Hamiltonian systems (Tsakirtzis, 2006; Tsakirtzis et al., 2007). In that way, and while supposing that the effect of weak damping is purely parasitic (as it cannot generate ‘new dynamics,’ but rather acts as perturbation of the underlying Hamiltonian response), the transient responses occur in neighborhoods of branches of periodic (or quasi-periodic) solutions of the corresponding Hamiltonian systems. Once this is recognized, the interpretation of the damped dynamics is possible, and an understanding of the resulting multi-frequency transitions can be gained.

4.3.1 Numerical Wavelet Transforms The transient dynamics of the damped and forced system is processed by numerical wavelet transforms (WTs). The results are presented in terms of WT spectra, which are contour plots depicting the amplitude of the WT as function of frequency (vertical axis) and time (horizontal axis). Heavy shaded areas correspond to regions where the amplitude of the WT is high whereas lightly shaded regions correspond to low amplitudes. Both Morlet and Cauchy WTs were considered, but these two mother wavelets provided similar results when applied to the signals considered herein. Representative WT spectra of the transient nonlinear responses of system (4.4) are presented in Figures 4.20–4.25. Specifically, we reconsider the responses of System II for α = 1.0 and impulsive forcing condition (IFC) I3, studied previously in Figures 4.8, 4.11 and 4.12. Referring to the plot depicted in Figure 4.8 (with α = 1.0), a peculiar behavior of the efficiency of targeted energy transfer (TET) from the primary linear system to the MDOF NES was noted. In particular, when the primary system was excited by a pair of out-of-phase impulses of magnitude Y , strong TET to the NES occurs at low energy levels (i.e., for weak applied impulses), with values of EDM reaching levels of 90% for Y = 0.1 (point C in Figure 4.8). By increasing the magnitude of the applied impulse the eventual energy transfer to the NES first decreases (with EDM reaching nearly 50% for Y = 1.0 – point A in Figure 4.8), before increasing again to higher levels (with EDM being nearly equal to 90% for Y = 1.5 – point B in Figure 4.8); further increase of Y decreases the portion of input energy eventually dissipated by the NES.

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Fig. 4.20 WT spectrum of the relative NES displacement (v1 − v2 ) of System II for out-of-phase impulse magnitude ; the linear natural frequencies of the uncoupled (ε = 0) primary system are indicated by dashed lines.

Fig. 4.21 WT spectrum of the relative NES displacement (v2 − v3 ) of System II for out-of-phase impulse magnitude ; the linear natural frequencies of the uncoupled (ε = 0) primary system are indicated by dashed lines.

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In an attempt to understand the reason for this peculiar behavior of TET in this system, we computed the WT spectra of the relative NES displacements [v1 (t) − v2 (t)] and [v2 (t) − v3 (t)]. In Figures 4.20 and 4.21 the WT spectra of System II for out-of-phase impulsive excitation of magnitude Y = 0.1 are depicted. In this case there occurs strong TET from the primary system to the NES (amounting to nearly 90% of input energy). Examination of the corresponding WT spectra reveals the following features of the dynamics: (i)

There occurs a strong TRC of the dynamics of the relative displacement [v1 (t) − v2 (t)] with an essentially nonlinear mode (i.e., with no counterpart in the linearized system) whose frequency varies in time and lies between the two linearized natural frequencies of the primary system. The fact that this is an essentially (strongly) NNM is signified by the fact that its frequency does not lie close to either one of the linearized natural frequencies of the system; this implies that this mode localizes predominantly to the NES. The strong nonlinearity of the response of the NES is further signified by the occurrence of an initial multi-frequency beat oscillation (subharmonic or quasi-periodic), as evidenced by the existence of an initial high frequency component in the spectrum of [v1 (t) − v2 (t)]. (ii) The second nonlinear stiffness-damper pair of the MDOF NES (corresponding to the relative displacement [v2 (t)−v3 (t)]) absorbs (and dissipates) broadband energy from the primary system; this is evidenced by the fact that the WT spectrum of [v2 (t) − v3 (t)] exhibits a wide range of frequency components, which includes the linearized natural frequencies of the primary system. These results indicate that strong TET in this case is associated with TRCs of the dynamics by strongly nonlinear modes that predominantly localize to the NES; moreover these TRCs take place over a wide frequency range, resulting in broadband TET from the primary structure to the NES. These results underline the validity of the numerical WT, which in this complicated dynamical problem provides important information not only on the frequency contents of the nonlinear responses, but also on the temporal evolution of each individual frequency component as the strongly nonlinear interaction between the linear and nonlinear subsystems progresses in time. By increasing the magnitude of the impulse to Y = 1.0, we note a marked deterioration of TET from the primary system to the NES. In Figures 4.22 and 4.23 we depict the WT spectra for [v1 (t) − v2 (t)] and [v2 (t) − v3 (t)] for this case, revealing the reason for poor TET: namely, the dynamics of the MDOF NES appears to engage in sustained resonance capture (SRC) predominantly with two weakly nonlinear modes lying in neighborhoods of the linearized in-phase and out-of-phase modes of the primary system. The fact that both of these weakly nonlinear modes localize predominantly to the primary system, prevents significant localization of the vibration to the NES, and, hence, leads to weaker TET. This prevents strong broadband TET from the primary system to the NES. We conclude that weak TET in this case is associated with SRC of the NES dynamics with weakly nonlinear modes which are predominantly localized to the primary system.

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Fig. 4.22 WT spectrum of the relative NES displacement (v1 − v2 ) of System II for out-of-phase impulse magnitude Y ; the linear natural frequencies of the uncoupled (ε = 0) primary system are indicated by dashed lines.

Fig. 4.23 WT spectrum of the relative NES displacement (v2 − v3 ) of System II for out-of-phase impulse magnitude Y = 1.0; the linear natural frequencies of the uncoupled (ε = 0) primary system are indicated by dashed lines.

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Fig. 4.24 WT spectrum of the relative NES displacement (v1 − v2 ) of System II for out-of-phase impulse magnitude Y = 1.5; the linear natural frequencies of the uncoupled (ε = 0) primary system are indicated by dashed lines.

Fig. 4.25 WT spectrum of the relative NES displacement (v2 − v3 ) of System II for out-of-phase impulse magnitude Y = 1.5; the linear natural frequencies of the uncoupled (ε = 0) primary system are indicated by dashed lines.

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Finally, in Figures 4.24 and 4.25 we depict the corresponding WT spectra for Y = 1.5. Similarly to the case for Y = 0.5 (see Figures 4.20 and 4.21), we note the occurrence of strong TRC of the dynamics of the NES with a strongly nonlinear mode localized predominantly to the NES; this TRC leads to strong TET from the primary system to the NES. Comparing the WT spectra of Figures 4.22 and 4.23 to those corresponding to weak TET (Figures 4.22 and 4.23), we note that in the later case the transient responses are dominated by sustained frequency components (i.e., by SRCs), indicating excitation of weakly nonlinear modes that are mere analytic continuations of linearized modes of System II. On the contrary, in cases where strong TET is realized, the frequencies of the nonlinear modes involved in the corresponding TRCs are not close to the linearized natural frequencies ω1 and ω2 , indicating the presence in the response of strongly nonlinear modes with no linear counterparts; these modes localize predominantly to the NES.

4.3.2 Damped Transitions on the Hamiltonian FEP Starting with System I, we perform a series of numerical simulations to study the transient dynamics of system (4.4) with weak damping, in an effort to demonstrate that complicated transitions in the dynamics of the weakly damped system closely follow branches of the underlying Hamiltonian system. We aim to show that, for sufficiently weak damping, damped transitions can be interpreted as jumps between different branches of periodic solutions of the FEP of Figure 4.14. Hence, we aim to show that TET in the system of Figure 4.2 [or in system (4.4)] is governed, in essence, by the topological structure of the NNMs of the underlying Hamiltonian system; this, occurs in spite the fact that, as discussed in Chapter 3, damping is a prerequisite for TET for the systems considered. In the following simulations the motion of the system is initiated with different initial conditions, and there is no external forcing; the system parameters for System I were defined in Section 4.2.1, and the damping coefficients in (4.4) were assigned the (small) values ελ1 = 8 × 10−3 and ελ2 = 1.6 × 10−3 . Hence, in what follows only weakly damped nonlinear transitions are examined. First, the motion is initiated at point A of a lower subharmonic tongue emanating from the main backbone curve of the FEP of the system in Figure 4.13, and the resulting damped transient responses are depicted in Figure 4.26. It is noted that although the MDOF NES starts with almost no energy, after t = 1500 s it passively absorbs nearly all of the energy of the (initially excited) linear primary system in an irreversible fashion. Moreover, TET from the linear primary system to the NES coincides with the transition from a subharmonic tongue to the main backbone curve with decreasing energy (due to damping dissipation) as evidenced from the plots of Figure 4.26c; these plots depict the superposition of the FEP of Figure 4.13 to the WT spectra of the transient responses [v1 (t) − v2 (t)] and [v2 (t) − v3 (t)]. These plots should be viewed from a purely phenomenological point of view, as they superpose weakly damped (the WT spectra) to undamped (the branches of periodic orbits on the FEP)

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Fig. 4.26 Transient response of the weakly damped System I for initial conditions at point A of the FEP of Figure 4.13: (a) Time series, (b) partition of instantaneous energy of the system, and (c) WT spectra depicted in the FEP of the underlying Hamiltonian System I.

responses, and they should be used only for descriptive purposes. Nevertheless this type of superpositions help us interpret transitions that occur in the damped responses in terms of the topological portrait of the periodic orbits of the underlying Hamiltonian system; in this particular case, the only transition in the dynamics takes place from the subharmonic branch where the motion is initiated, to the main backbone branch, and there are no other transitions or jumps between branches of solutions (i.e., the transition is smooth with decreasing energy – see Figure 4.26c). Concerning the damped responses of Figure 4.26a, we note the nearly complete absence of motion of the third particle of the MDOF NES, in accordance to our previous discussion regarding the periodic motions (NNMs) on the regular backbone branch of System I.

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Fig. 4.27 Transient response of the weakly damped System I for initial conditions at point B of the FEP of Figure 4.13: (a) Time series, (b) partition of instantaneous energy of the system, and (c) WT spectra depicted in the FEP of the underlying Hamiltonian System I.

Next, the motion is initiated at point B on a branch of the family of singular backbones of the FEP of Figure 4.13, namely, on branch S161. The results of this simulation are depicted in Figure 4.27, and some major qualitative differences are observed compared to the previous simulation. In this case the last mass of the NES executes large-amplitude oscillations, and the dominant frequency components of the WT spectra of [v1 (t) − v2 (t)] and [v2 (t) − v3 (t)] differ (in contrast to motions on the regular backbone curve that are nearly monochromatic); finally, the motion is nearly localized to the MDOF NES. Indeed, the WT spectrum of the relative displacement [v2 (t)−v3 (t)] follows a singular backbone branch, engaging at t ≈ 550 s

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Fig. 4.28 Transient response of the weakly damped System I for initial conditions at point C of the FEP of Figure 4.13: (a) Time series, (b) partition of instantaneous energy of the system, and (c) WT spectra superimposed to the FEP of the underlying Hamiltonian System I.

in 1:1 TRC with the in-phase linearized mode at the natural frequency f2 of system (4.10). On the other hand, the WT spectrum of [v1 (t)−v2 (t)] does not generally follow the same singular backbone branch since its dominant harmonic component is six times the dominant harmonic component of [v2 (t) − v3 (t)]. When the dominant frequency of [v1 (t) − v2 (t)] gets close to the neighborhood of the regular backbone branch, it is possible that TRCs occur involving the regular backbone S111 and the singular backbone S161. An additional simulation is depicted in Figure 4.28, with the motion initiated on point C of S131c (see Figure 4.13) not far from the coalescence point of this branch with S131d (see Figure 4.15). Once the motion reaches the coalescence point for diminishing energy, a bifurcation occurs, which is clearly evidenced by the envelopes of the relative displacements of the NES. In addition, we note the occurrence of an

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interesting resonance capture at the final stage of the motion when the dominant harmonic component of the relative displacement [v1 (t) − v2 (t)] (which is three times the dominant harmonic component of [v2 (t) − v3 (t)]) appears to engage in resonance capture with one of the lower tongues emanating from the regular backbone curve. This is precisely the type of resonance capture conjectured previously, leading to strong energy exchanges between the particles of the NES. As in the previous simulation, throughout the motion almost all of the energy of vibration is localized to the MDOF NES. It is interesting to note that in general weak TET occurs in System I. This is concluded by performing a series of numerical simulations with initial forcing conditions similar to those considered in Section 4.1.2 (with IFCs I1-I3), and computing the portion of total impulsive energy eventually dissipated by the MDOF NES. In all cases it was found that only a small portion of input energy is eventually transferred to (and locally dissipated by) the NES. A representative result of weak TET is depicted in Figure 4.29 for the case of single impulsive excitation with magnitute Y = 1.5 applied to the left mass of the linear subsystem (corresponding to impulsive forcing condition I1). We now consider the transient damped dynamics leading to TET in System II, corresponding to weak nonlinear stiffness C2 and small NES masses. We will show that by decreasing the masses of the MDOF NES the complexity of the dynamics increases, and the capacity for TET significantly improves compared to System I. In the following simulations the motion of the system is initiated with different initial conditions, and no external forcing is considered; the system parameters for System II were defined in Section 4.2.2, and the damping coefficients in (4.4) were assigned the values ελ1 = ελ2 = ελ = 0.01. So, again, only weakly damped nonlinear transitions are considered in what follows. Revisiting an earlier result, we wish to reconsider and study in more detail the damped transitions associated with the peculiar behavior of the TET plot of System II for α = 1.0 and IFC I3 depicted in Figure 4.8. More specifically, in Section 4.1.2 it was numerically shown that when the linear system is excited by a pair of out-ofphase impulses of magnitude Y, strong TET from the linear primary system to the NES occurs even at low values of the impulse (with EDM as high as 90% for Y = 0.1); by increasing the magnitude of the impulse, initially TET deteriorates (with EDM reaching nearly 50% for Y = 1.0), before improving back to high levels (with EDM increasing up to nearly 90% for Y = 1.5). Further increase of Y decreases the portion of input energy that is eventually dissipated by the NES, so that TET deteriorates. The WT spectra of the responses of the particles of the NES for System II were depicted in Figures 4.20–4.25, and it was postulated that strong TET is associated with transient resonance captures (TRCs) of the transient dynamics by strongly nonlinear modes predominantly localized to the NES; whereas, weak TET is associated with sustained resonance captures (SRCs) of the dynamics by weakly nonlinear modes predominantly localized to the linear system. We wish to confirm these results by studying the WT spectra of the NES responses superimposed to the FEP of System II (depicted in Figure 4.17); by doing so we wish to observe directly

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Fig. 4.29 Weak TET in System I for IFC I1 of magnitude Y = 1.5: (a) Time series, (b) partition of instantaneous energy of the system, and (c) WR spectra superimposed to the FEP of the underlying Hamiltonian System I.

the resulting TRCs and transitions between branches of periodic solutions. The WT spectra superimposed to the FEP for System II are depicted in Figures 4.30–4.32. In Figure 4.30 the damped responses corresponding to IFC I3 and Y = 0.1 [i.e., impulses F1 (t) = −F2 (t) = Y δ(t) and zero ICs in system (4.4)] are presented. These responses correspond to point C of the TET diagram of Figure 4.8. In this case both relative displacements [v1 (t) − v2 (t)] and [v2 (t) − v3 (t)] of the NES follow regular backbone branches in the FEP as energy decreases due to damping dissipation. The relative displacement [v1 (t)−v2 (t)] has a dominant frequency component which approaches the linearized natural frequency f2 of the limiting system (4.10) with decreasing energy; in contrast, [v2 (t) − v3 (t)] has two strong harmonic components that approach the linearized natural frequencies f2 and f3 with de-

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Fig. 4.30 Damped responses of System II for IFC I3 with Y = 0.1: (a, b) Cauchy WT spectra of the relative displacements [v1 (t) − v2 (t)] and [v2 (t) − v3 (t)] superimposed to the FEP of the Hamiltonian System II; these responses correspond to point C of the TET diagram of Figure 4.8.

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Fig. 4.30 Damped responses of System II for IFC I3 with Y = 0.1: (c) partition of instantaneous energy of the system; these responses correspond to point C of the TET diagram of Figure 4.8.

creasing energy; this indicates that TET occurs simultaneously with two modes of the linear limiting system (4.10). Moreover, the same regular backbone branches are tracked by the response throughout the motion, and strong TET occurs right from the early stage of the dynamics. This explains the high value of EDM (˜90%) that is realized even for this low level of impulsive excitation; clearly, this can not be realized through the use of SDOF NESs, as TET to this type of attachments takes place (is ‘activated’) only above a certain critical energy level. Hence, the described low-energy TET is a unique feature of the MDOF NES configuration. By increasing the magnitude of the impulse to Y = 1.0 TET from the primary system to the MDOF NES significantly decreases. The damped response of System II in this case is depicted in Figure 4.31. Some major qualitative differences are observed compared to the lower-impulse simulation of Figure 4.30. Judging from the partition of the instantaneous energy among the linear and nonlinear systems, it is concluded that targeted energy transfer is significantly delayed, and, hence, occurs at lower energy levels; this explains the weak TET to the NES (EDM˜50% in this case). This delay is explained when one studies the WT spectra of the NES relative responses superimposed to the FEP of Figure 4.31a. Noting that in the initial stage of the motion the dominant WT components of the NES relative displacements occur close to the linearized frequency f1 , we conclude that in the initial (high energy)

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Fig. 4.31 Damped responses of System II for IFC I3 with Y = 1.0: (a, b) Cauchy WT spectra of the relative displacements [v1 (t) − v2 (t)] and [v2 (t) − v3 (t)] superimposed to the FEP of the Hamiltonian System II; these responses correspond to point A of the TET diagram of Figure 4.8.

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Fig. 4.31 Damped responses of System II for IFC I3 with Y = 1.0: (c) partition of instantaneous energy of the system; these responses correspond to point A of the TET diagram of Figure 4.8.

stage of the motion there occurs strong resonance capture of the damped motion by the linearized out-of-phase mode of the limiting system (4.10). This yields a motion mainly localized to the (directly excited) primary linear system, with only a small portion of energy ‘spreading out’ to the NES. As energy decreases due to damping dissipation, the damped motion ‘escapes’ from the initial out-of-phase resonance capture, and follows regular backbone branches; this results in TET (as in the simulations of Figure 4.30), which, however, occurs with a delay, at a stage where the energy of the system is small due to damping dissipation. Hence, no significant TET from the primary system to the NES takes place in this case. By increasing the magnitude of the impulse to Y = 1.5, the dynamics escape from the strong initial out-of-phase resonance capture, yielding once again strong TET. This is depicted in Figure 4.32, showing that the NES relative responses possess multiple strong frequency components, indicating that strong TET takes place over multiple frequencies. Note in this case the early strong TET from the primary system to the NES, resulting in EDM of nearly 90%. These results are in agreement with the conclusions drawn from the study of the WT spectra of the NES relative responses of the same system (see Figures 4.20–4.25 in Section 4.3.1). The superposition of the WT spectra to the FEP of the underlying Hamiltonian System II provides additional valuable insight to the sequences of resonance captures (transient or sustained) that facilitate or hinter TET from the primary system to the NES. This confirms the value of the FEP as a tool for interpreting the transient dynamics of the strongly nonlinear systems considered herein.

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4 Targeted Energy Transfer in Discrete Linear Oscillators with Multi-DOF NESs

Fig. 4.32 Damped responses of System II for IFC I3 with Y = 1.5: (a, b) Cauchy WT spectra of the relative displacements [v1 (t) − v2 (t)] and [v2 (t) − v3 (t)] superimposed to the FEP of the Hamiltonian System II; these responses correspond to point B of the TET diagram of Figure 4.8.

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Fig. 4.32 Damped responses of System II for IFC I3 with Y = 1.5: (c) partition of instantaneous energy of the system; these responses correspond to point B of the TET diagram of Figure 4.8.

Similar results were obtained for alternative forcing excitations of the linear primary system, confirming the strong TET capacity of the NES in System II. A last example of strong TET is depicted in Figure 4.33, for the case of single impulse excitation of magnitude Y = 1.5 (IFC I1 – Figure 4.6, case α = 1.0). Notice the strong multi-frequency content of the WT spectra of the internal displacements of the MDOF NES, proving that TET from the primary system to the NES takes place in a broadband fashion [i.e., simultaneously from the three linearized modes of the limiting subsystem (4.10)]; this results in nearly 85% of input energy being eventually transferred to, and dissipated by the MDOF NES. Compare this picture to the corresponding plot of Figure 4.29c for System I, where the NES dynamics is narrowband and weak TET occurs.

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Fig. 4.33 Damped responses of System II for IFC I1 with Y = 1.5: (a, b) Cauchy WT spectra of the relative displacements [v1 (t) − v2 (t)] and [v2 (t) − v3 (t)] superimposed to the FEP of the Hamiltonian System II.

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4.4 Concluding Remarks The results presented in this chapter demonstrate that MDOF essentially nonlinear attachments (MDOF NESs) can be designed to be efficient and robust passive broadband absorbers of vibration or shock energy from the primary systems to which they are attached. Moreover, the extraction of vibration energy occurs in a multifrequency fashion, through simultaneous dynamic interactions of multiple modes of the nonlinear attachments with multiple modes of the primary systems. This form of multi-frequency energy exchange is different than the resonance capture cascades encountered in the previous chapter, where TET to SDOF NESs occurs in a sequential manner, i.e., through resonance capture cascades. The dynamical systems considered in this work possess complicated dynamics due to their degenerate structures. The considered MDOF NES has strong passive TET capacity, extracting in some cases as much as 90% of the vibration energy of the primary system to which it is attached. The capacity of the MDOF essentially nonlinear attachment to absorb broadband vibration energy was demonstrated numerically in this section, but it can also be analytically studied by a reduction process of the governing system of ordinary differential equations, and local slow/fast partition of the damped dynamics. It was shown that MDOF essentially nonlinear attachments may be more efficient energy absorbers compared to SDOF ones, since they are capable of absorbing energy simultaneously from multiple structural modes, over wider frequency and energy ranges. Passive TET by the MDOF NES can be related to transient resonance captures (TRCs) of the damped dynamics, whereby orbits of the system in phase space are transiently captured in neighborhoods of resonance manifolds. An interesting dynamical feature of the considered MDOF NES configurations is the existence of two classes of backbone branches in their frequency-energy planes: isolated regular backbone branch containing NNMs where all particles of the primary system and the NES oscillate with identical dominant frequencies; and additional families of densely packed singular backbone branches containing NNMs where particles oscillate with differing dominant harmonic components. It was proved that these families of singular backbones contain countable infinities of backbone branches, which are mainly generated by combined parametric and external resonances between the two relative displacements of the particles of the NES. It is conjectured that this interesting energy ‘quantization’ of the families of singular backbone branches may represent different modes of nonlinear interaction and energy exchange between the particles of the essentially nonlinear, MDOF attachment. Finally, it was shown that complex transitions in the damped dynamics of the system with attached MDOF NES may be related to transitions or jumps between branches of NNMs of the underlying Hamiltonian system. In that context, TRCs leading to TET may be related to damped motions in neighborhoods of certain invariant manifolds of the underlying Hamiltonian system. The methodologies and results presented in this chapter pave the way for applying lightweight MDOF essentially nonlinear attachments as shock and vibration

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absorbers of unwanted disturbances of structures. The proposed designs are modular and can be designed to be lightweight; hence they can be conveniently attached to existing structures with minimal structural modifications. Application of MDOF NESs for shock isolation of elastic continua is considered in the next chapter.

References Georgiadis, F., Vakakis, A.F., McFarland, M., Bergman, L.A., Shock isolation through passive energy pumping caused by non-smooth nonlinearities, Int. J. Bif. Chaos (Special Issue on ‘Non-Smooth Dynamical Systems: Recent Trends and Perspectives’), 15(6), 1–13, 2005. Gourdon, E., Lamarque, C.H., Energy pumping for a larger span of energy, J. Sound Vib. 285, 711–720, 2005. Gourdon, E., Coutel, S., Lamarque, C.H., Pernot, S. , Nonlinear energy pumping with strongly nonlinear coupling: Identification of resonance captures in numerical and experimental results, in Proceedings of the 20th ASME Biennial Conference on Mechanical Vibration and Noise, Long Beach, California, September 24–28, 2005. Gourdon, E., Pernot, S, Lamarque, C.H., Energy pumping with multiple passive nonlinear absorbers, in Proceedings of EUROMECH Colloquium 483 on Geometrically Nonlinear Vibrations of Structures, FEUP, Porto, Portugal, July 9–11, 2007. Guckenheimer, J., Holmes, P., Nonlinear Oscillations, Dynamical System, and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983. Kerschen, G., Lee, Y.S., Vakakis, A.F., McFarland, D.M., Bergman, L.A., Irreversible passive energy transfer in coupled oscillators with essential nonlinearity, SIAM J. Appl. Math. 66, 648– 679, 2006. Lee, Y.S., Passive Broadband Targeted Energy Transfers and Control of Self-Excited Vibrations, PhD Thesis, Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 2006. Lee, Y.S., Kerschen, G., Vakakis, A.F., Panagopoulos, P.N., Bergman, L.A., McFarland, D.M., Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment, Physica D 204 (1–2), 41–69, 2005. Ma, X., Vakakis, A.F., Bergman, L.A., Karhunen–Loeve analysis and order reduction of the transient dynamics of linear coupled oscillators with strongly nonlinear end attachments, J. Sound Vib. 309, 569–587, 2008. McFarland, D.M., Bergman, L.A., Vakakis, A.F., Experimental study of nonlinear energy pumping occurring at a single fast frequency, Int. J. Nonlinear Mech. 40, 891–899, 2004. Musienko, A.I., Lamarque, C.H., Manevitch, L.I., Design of mechanical energy pumping devices, J. Vib. Control 12(4), 355–371, 2006. Panagopoulos, P.N., Vakakis, A.F., Tsakirtzis, S., Transient resonant interactions of linear chains with essentially nonlinear end attachments leading to passive energy pumping, Int. J. Solids Struct. 41(22–23), 6505–6528, 2004. Rosenberg, R., On nonlinear vibrations of systems with many degrees of freedom, Adv. Appl. Mech. 9, 155–242, 1966. Tsakirtzis, S., Passive Targeted Energy Transfers From Elastic Continua to Essentially Nonlinear Attachments for Suppressing Dynamical Disturbances, PhD Thesis, National Technical University of Athens, Athens, Greece, 2006. Tsakirtzis, S., Kerschen, G., Panagopoulos, P.N., Vakakis, A.F., Multi-frequency nonlinear energy transfer from linear oscillators to MDOF essentially nonlinear attachments, J. Sound Vib. 285, 483–490, 2005.

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Tsakirtzis, S., Panagopoulos, P.N., Kerschen, G., Gendelman, O., Vakakis, A.F., Bergman, L.A., Complex dynamics and targeted energy transfer in systems of linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments, Nonl. Dyn. 48, 285–318, 2007. Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A., Normal Modes and Localization in Nonlinear Systems, Wiley Interscience, New York, 1996. Vakakis, A.F., Manevitch, L.I., Gendelman, O., Bergman, L.A., Dynamics of linear discrete systems connected to local essentially nonlinear attachments, J. Sound Vib. 264, 559–577, 2003. Vakakis, A.F., McFarland, D.M., Bergman, L.A., Manevitch, L.I., Gendelman, O., Isolated resonance captures and resonance capture cascades leading to single- or multi-mode passive energy pumping in damped coupled oscillators, J. Vib. Acoust. 126 (2), 235–244, 2004. Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.

Chapter 5

Targeted Energy Transfer in Linear Continuous Systems with Single- and Multi-DOF NESs

Up to now we have considered passive targeted energy transfer from linear discrete systems of coupled oscillators to attached SDOF and MDOF NESs. In this chapter we extend the study of TET dynamics to linear elastic continua possessing attached NESs attached to their boundaries. Our study builds on the formulations, methodologies and results discussed in previous chapters, in an effort to demonstrate that appropriately designed and placed essentially nonlinear local attachments may affect the global dynamics of elastic systems to which they are attached. More importantly, we show that such nonlinear attachments can passively absorb and locally dissipate significant portions of shock-induced energy inputs from directly excited linear continua. This paves the way for practical implementation of TET and the concept of NES to flexible systems encountered in engineering practice.

5.1 Beam of Finite Length with SDOF NES The first class of elastic systems considered is composed of linear beams with attached NESs, with general configuration depicted in Figure 5.1 (Georgiades, 2006; Georgiades et al., 2007). Specifically, we consider an impulsively forced, simply supported, damped linear beam, with an attached essentially nonlinear, damped SDOF oscillator (the NES). As in the case of discrete oscillators considered in Chapters 3 and 4, we will show that the NES can passively and irreversibly absorb a major portion of the impulsive energy of the beam. Moreover, TET from the linear beam to the NES can be optimized by appropriate design and placement of the attachment.

5.1.1 Formulation of the Problem and Computational Procedure Assuming that the beam dynamics is governed by linear Bernoulli–Euler theory, the equations of motion of the integrated system are given by

1

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.1 Linear beam with attached SDOF NES.

EIyxxxx (x, t) + ελyt (x, t) + myt t (x, t) + {C[y(d, t) − v(t)]3 + ελ[yt (d, t) − v(t)]}δ(x ˙ − d) = F (t)δ(x − a) εv(t) ¨ + C[(v(t) − y(d, t)]3 + ελ[v(t) ˙ − yt (d, t)] = 0

(5.1)

with zero initial conditions. In (5.1), E is the Young’s modulus, I the moment of inertia of the cross section, and m the mass per unit length of the beam; moreover, proportional distributed viscous damping is assumed for the beam, and the short-hand notation for partial differentiation is enforced, e.g., (·)xx ≡ ∂ 2 (·)/∂x 2 , (·)t ≡ ∂(·)/∂t, . . . . By adopting the usual assumption 0 < ε 1, the system is assumed to possess weak viscous damping, and the NES is assumed to be lightweight compared to the mass of the beam. Clearly, this last assumption is important for the practical implementation of this design, since in practical engineering applications one requires that the NES does not add significant new weight or modify considerably the overall structural configuration. In addition, we assume that the attachment possesses essential cubic stiffness nonlinearity, which, together with viscous damping dissipation are prerequisites for the realization of TET. We now discuss certain aspects of the computational study of the transient dynamics of the essentially nonlinear damped system (5.1). First, we consider the set of linear normal modes of the simply supported beam with no damping, external forcing, or NES attached. This is given by φr (x) = (2/mL)1/2 sin(rπx/L), ωr = (rπ)2 (EI /mL4 )1/2 ,

r = 1, 2, . . .

(5.2)

where φr (x) and ωr are the mode shape and natural frequency of the mode, respectively. Since these modes are solutions of a Sturm–Liouville eigenvalue problem

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they satisfy the following orthonormality relations:

L

mφr φs dx = δrs

L

and

EI 0

∂ 2 φr ∂ 2 φs dx = ωr2 δrs , ∂x 2 ∂x 2

r, s = 1, 2, . . .

Considering now the integrated damped beam-NES system, its transient response is numerically computed by projecting the dynamics of the partial differential equation (with attached NES) in the functional space defined by the complete and orthonormal base of normal modes (5.2). To this end, we non-dimensionalize the system (5.1) by introducing the following new normalized parameters and variables: τ =t

EI , m

ε1 =

ε , m

c=

C , EI

λ1 =

ελ , m

Q(x, τ ) =

F (x, τ ) EI m

which bring (5.1) into the following non-dimensional form: yxxxx (x, τ ) + λ1 yτ (x, τ ) + yτ τ (x, τ ) + {c[y(d, τ ) − v(τ )]3 + λ1 [yτ (d, τ ) − v(τ ˙ )]}δ(x − d) = Q(τ )δ(x − a) ε1 v(τ ¨ ) + c[(v(τ ) − y(d, τ )]3 + λ1 [v(τ ˙ ) − yτ (d, τ )] = 0

(5.3)

In (5.3) dots denote differentiation with respect to the scaled time variable τ . To project the dynamics of (5.3) in the infinite-dimensional base of orthonormal normal modes (5.2) we express the transverse displacement field y(x, t) in the series form ∞ y(x, τ ) = ar (τ )φr (x) (5.4) r=1

Substituting (5.4) into (5.3), leads to ∞ r=1

ar (τ )

∞

∞

r=1

r=1

d 4 φr (x) + λ1 a˙ r (τ )φr (x) + a¨ r (τ )φr (x) 4 dx

⎧ ∞ 3 ⎫ ∞ ⎨ ⎬ ar (τ )φr (d) − v(τ ) + λ1 a˙ r (τ )φr (d) − v(τ ˙ ) + c δ(x − d) ⎩ ⎭ r=1

r=1

= F (τ )δ(x − a) ∞ ∞ 3 ¨ −c ar (τ )φr (d) − v(τ ) − λ1 a˙ r (τ )φr (d) − v(τ ˙ ) = 0 (5.5) ε1 v(t) r=1

r=1

By multiplying (5.5) by an arbitrary modeshape φp (x), integrating with respect to x from 0 to L, and enforcing the orthonormality conditions satisfied by the normal modes, yields the following set of coupled nonlinear ordinary differential equations governing the modal amplitudes ap (τ ), p = 1, 2, . . . :

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

a¨ p (τ ) + ωp2 ap (τ ) + ε1 a˙ p (τ ) ⎧ ∞ 3 ⎫ ∞ ⎨ ⎬ + c ar (τ )φr (d) − v(t) + λ1 a˙ r (τ )φr (d) − v(τ ˙ ) φp (d) ⎩ ⎭ r=1

= q(τ )φp (a) ¨ ) + c v(τ ) − ε1 v(τ

r=1

∞

3 ar (τ )φr (d)

˙ )− + λ1 v(τ

r=1

∞

a˙ r (τ )φr (d) = 0

r=1

(5.6) We note that the essential nonlinearity and the damping term of the NES couples all modes through the infinite summation terms, whereas, the linear part of the system decouples completely by the projection to the orthonormal basis of normal modes of the uncoupled linear beam. It follows that although the NES is local and lightweight it introduces global effects in the dynamics of the integrated system. This is due, of course, to the essential (strong) stiffness nonlinearity of the system. As in Chapters 3 and 4, a quantitative measure of TET (that is, a measure of the effectiveness of the NES to passively absorb and locally dissipate energy from the directly forced beam) is given by the energy dissipation measure (EDM) which quantifies the instantaneous portion of impulsive energy of the beam that is dissipated by the damper of the NES: ENES (τ ) ≡

τ

˙ − λ1 v(u)

T

F (τ ) 0

∞

r=1 ∞

2 a˙ r (u)φr (d)

du (5.7)

a˙ r (τ )φr (a)dτ

r=1

In this passive system, the EDM (5.7) reaches an asymptotic limit denoted by ENES,τ 1 ≡ limτ 1 ENES (τ ), which quantifies the portion of impulsive energy that is eventually dissipated by the NES over the entire duration of the decaying motion. In the numerical simulations the infinite series (5.4) was truncated to include only a finite number of modes; this is equivalent to performing an approximate projection of the dynamics to a finite-dimensional basis of orthonormal normal modes. The dimensionality of the truncated space that is required for accurate numerical simulations is determined by performing a convergence study. It was found that N = 5 modes were sufficient for accurately computing the transient dynamics. Representative examples of typical convergence results are depicted in Figures 5.2a, b, were we depict the EDM ENES,τ 1 ≈ ENES (τ = 150) as function of the nonlinear coefficient c and the position d of the NES, respectively. For these simulations the impulsive force was selected as a half sine pulse,

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

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Fig. 5.2 Convergence study of the EDM ENES,τ 1 for the truncated system with N = 1, 2 and 5 modes as function of (a) NES stiffness C and (b) NES position d.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

q(t) =

A sin(2πτ/T ), 0 ≤ τ ≤ T /2 0, t > T /2

(5.8)

with A = 10.0 and T = 0.4/π. Moreover, the system parameters were assigned the numerical values: EI = 1.0, m = 1.0, L = 1.0, ε1 = 0.1, a = 0.3, λ1 = 0.05, with d = 0.65 for the convergence plot of Figure 5.2a, and c = 1.322 × 103 for that depicted in Figure 5.2b. Studying the plots of Figures 5.2a, b we note convergence of the results for N = 5 modes, justifying the mode truncation implemented in the following results. Considering the dependence of the EDM ENES,τ 1 , on the nonlinear coefficient c of the NES (see Figure 5.2a), we note that for c of O(103) the EDM is significant, reaching values above 80%; this indicates that strong TET is realized in this case. Considering the dependence of ENES,τ 1 on the NES position (Figure 5.2b), we note two regions of strong TET (with the EDM reaching values of the order of 80%), corresponding to placement of the NES between the boundaries and the center of the beam. On the contrary, significantly weaker TET is realized when the NES is placed near the center of the beam (where the second and fourth normal modes of the uncoupled beam possess nodes), or near the boundaries of the beam where the response of the beam is small. These results indicate that an appropriately designed and placed NES can passively absorb and locally dissipate a major portion of the energy induced to the beam by the external shock; moreover, this passive energy absorption is broadband and irreversible (on the average), as verified by the significant levels of energy eventually dissipated by the damper of the NES. In the following section we present the results of a parametric study of TET in the system, in an effort to optimize TET from the beam to the NES. Although an optimization study of TET should address not only maximization of the EDM ENES,τ 1 , but also the issue of the time scale governing the energy transfer, in this section we only focus on the former issue, leaving the discussion of the later issue (i.e., of the time scale of TET) for Chapters 7, 9 and 10.

5.1.2 Parametric Study of TET The following simulations are performed for the half-sine shock excitation (5.8) with A = 10.0, T = 0.4/π and system parameters, EI = 1.0, m = 1.0, L = 1.0, ε1 = 0.1, a = 0.3 and λ1 = 0.05. In addition, by the results of the convergence study we truncate the discretized set of equations (5.6) to N = 5 terms, which corresponds to a strongly nonlinear set of five coupled modal oscillators. In the first parametric study we keep the (light) mass of the NES fixed and compute the asymptotic EDM ENES,τ 1 as function of the nonlinear coefficient c and the position d of the NES. Viewed in context, the plots of Figure 5.2 for N = 5 modes can be regarded as different ‘slices’ of the three-dimensional plot of Figure 5.3a.

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Fig. 5.3 EDM ENES,τ 1 as function of the NES c and the position d: (a) three-dimensional plot, (b) contour projection in the (c, d) plane.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.4 Case of strong TET to the NES, transient responses of the system with and without NES attached: (a) NES response, (b) beam response at the point of attachment.

There are two regions of strong TET in the (c, d) plane realized for c = 1.32 × 103 and d = 0.348, with optimal NES energy dissipation reaching the level of 83.3%. Moreover, for c of O(103) the effectiveness of the NES appears to be robust in variations of the nonlinear coefficient c; this is indicated by the two distinct ‘strips’ of sustained high values of energy dissipation in the plots of Figure 5.3. Moreover, the plots of Figure 5.3 reveal a strong dependence of TET on NES position d. This should be expected, given that by placing the NES closer to the center

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Fig. 5.4 Case of strong TET to the NES, transient responses of the system with and without NES attached: (c) beam response at x = 0.65.

of the beam, the NES is hindered from interacting with half of the beam modes that possess nodes near that position. A general conclusion drawn from the plots in Figure 5.3 is that a lightweight, essentially nonlinear NES can be designed and appropriately placed to passively absorb a major portion (of the order of 80%) of shock energy induced in the beam. The described energy absorption is broadband (as it involves shock excitation and multi-modal beam response), and is realized over a wide frequency range. This observation highlights the advantage of the proposed NES design compared to existing designs based on linear vibration absorbers: that is, the capacity of the NES to absorb effectively broadband energy over wide ranges of frequencies and system parameters, in contrast to the linear absorber whose action is narrowband. As we discussed in Chapters 3 and 4, the main reason behind the capacity of the NES for broadband passive vibration absorption is its essential nonlinearity (and the corresponding absence of preferential resonance frequencies), which enables it to engage in transient resonance captures (TRCs) or resonance capture cascades (RCCs) with isolated or sets of structural modes on arbitrary frequency ranges. To demonstrate the significant reduction in amplitude of the beam vibrations achieved due to passive TET, in Figure 5.4 we depict the transient responses of the NES, of the point of attachment of the beam and of another point of the beam at position x = 0.8. The system parameters for these simulations were fixed to the values c = 1.322 × 103 and d = 0.65; by the results depicted in Figure 5.3 this corresponds to a case of strong TET with ENES,τ 1 ≈ 83%; for comparison purposes we also depict the corresponding responses of the beam with no NES attached. We

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.5 Case of weak TET to the NES, transient responses of the system with and without NES attached: (a) NES response, (b) beam response at the point of attachment.

note the drastic reduction in the envelope of oscillation of the transient responses of the beam when the NES is attached, due to the rapid absorption and local dissipation of impulsive energy by the NES. The multi-frequency content clearly evidenced in

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Fig. 5.5 Case of weak TET to the NES, transient responses of the system with and without NES attached: (c) beam response at x = 0.65.

the NES transient response indicates broadband absorption of vibration energy by the NES from different structural modes of the beam, and demonstrates clearly the capacity of the NES to absorb and dissipate broadband energy from the beam. We note, especially, the early high frequency transients resembling a nonlinear beat (i.e., a transient ‘bridging’ orbit) followed by the transition of the dynamics towards lower frequencies – and resonance capture – as time progresses. In the plots in Figure 5.5 we depict the transient responses of the system for parameters c = 2 and d = 0.65 where TET is much weaker, corresponding to EDM ENES,τ 1 ≈ 58%. In this case, we note a much smaller reduction of the envelope of the beam responses when the NES is attached; moreover, the reduction of the envelope occurs over a longer time scale compared to the case of efficient TET of Figure 5.4. In addition, in this case the response of NES is of much smaller amplitude than the beam response, which indicates the inability of the NES to absorb and dissipate a major part of the impulsive energy of the beam. These results highlight the usefulness of the parametric plots of Figure 5.3. Indeed, using such plots one can determine optimal NES parameters for which strong and robust TET from the beam to the NES takes place. Moreover, these plots can form the basis for practical NES designs, capable of significantly reducing the level of unwanted vibration of flexible structural components forced by external shocks. To be able to better study the robustness of TET, one needs to extend the present parametric study to include changes in initial conditions and system parameters of the linear structure (in this case the beam). This addresses the need of studying how

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

the effectiveness of the NES is affected by changes due to structural degradation, and the initial state of the system. The results of this section provide a first evidence of TET from a linear elastic continuum to an attached NES. In contrast to our study of TET in discrete systems carried out thus far, to address passive energy transfers in the beam-NES configuration we needed to carefully study the effect that the location of the NES had on TET efficiency. This is due to the fact that if the NES is attached close to a node of a structural mode of the flexible system, its capacity to passively absorb and dissipate energy from that mode is drastically diminished, so TET deteriorates. In this respect, the performance of parametric studies similar to the one presented in this section can help us determine optimal placements of NESs to structural assemblies.

5.2 Rod of Finite Length with SDOF NES In an effort to extend our study of TET with linear elastic continua in attached NESs we now consider an impulsively forced (dispersive) rod of finite length that rests on a linear elastic foundation and possesses a SDOF NES attached to its end. The results reported in this section are drawn from the works by Georgiades (2006), Tsakirtzis (2006), Tsakirtzis et al. (2007a) and Panagopoulos et al. (2007), which should be consulted for more details. We mention that a new feature of the analysis presented in this section is the detailed post-processsing of the time series of the nonlinear dynamical interaction between the NES and the elastic continuum through a combination of numerical wavelet transforms (WTs) and Empirical Mode Decompositions (EMDs). It will be shown that the combination of these numerical transforms will enable us to study in detail the mechanisms governing the strongly nonlinear dynamical interactions and energy exchanges between the rod and the NES. This task will be performed by computating the evolutions of the dominant harmonic components of the corresponding time series, ultimately yielding multiscale analysis of the transient nonlinear dynamics, and identification of the principal resonance modal interactions that occur between the continuum and the NES that are responsible for TET (or lack of it). In the following exposition we systematically study passive broadband TET from the linear dispersive rod to the attached ungrounded, strongly (essentially) nonlinear SDOF NES. What distinguishes (and complicates) the present study compared to our studies of discrete oscillators discussed in Chapters 3 and 4, is the fact that due to the essentially nonlinear coupling between the continuum and the NES there occurs simultaneous nonlinear coupling between the NES and the infinity of modes of the rod, opening the possibility for transient nonlinear modal interactions of increased complexity. It is precisely this type of compicated nonlinear modal interactions, however, that gives rise to TET in the system under consideration. In the previous section we considered a beam with an attached NES, and demonstrated that strong TET was possible in that system. Moreover, in Vakakis et al. (2004a) the different types of dynamic interactions of a semi-infinite dispersive rod

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with a grounded essentially nonlinear attachment were analytically and numerically studied, but no attempt to systematically study TET was undertaken. In that work (which will be reviewed in Section 5.3) it was shown that the attachment initially engages in nonlinear resonance with incoming traveling elastic waves; as the energy of the attachment decreases due to damping and radiation of energy back to the rod, the attachment engages in 1:1 transient resonance capture (TRC) with the in-phase mode of the dispersive rod, leading to TET from the rod to the attachment; further decrease of energy of the NES leads to escape of the dynamics from TRC and appearance of the linearized regime of the motion. No study of the efficiency of TET, however, was undertaken in that work. Nevertheless, the computational and analytical results reported in Vakakis et al. (2004a) reveal that resonant interactions of elastic continua with local essentially nonlinear attachments can give rise to complex resonant phenomena; this provides further evidence that local essentially nonlinear attachments may introduce global changes in the dynamics of the elastic continua to which they are attached.

5.2.1 Formulation of the Problem, Computational Procedure and Post-Processing Algorithms The system under consideration consists of an linear elastic rod of mass distribution M and length L resting on a linear elastic foundation with distributed stiffness k and distributed viscous damping δ. At its right boundary the rod is coupled to an ungrounded, lightweight end attachment of mass m M by means of an essentially nonlinear cubic stiffness of constant C in parallel to a viscous damper ελ (see Figure 5.6). The elastic foundation renders the dynamics of the rod dispersive and introduces a cut-off frequency; in the frequency spectrum of the corresponding rod of infinite length this frequency separates the domains traveling and attenuating waves. As the constant of the elastic foundation tends to zero this cut-off frequency also tends to zero, and the dynamics of the rod become non-dispersive. It is assumed that the left boundary of the rod is clamped, that an impulsive force (shock) F (t) is applied at position x = d (where x is measured from the left clamped end of the rod), and that the entire system is initially at rest. Under these assumptions, the governing equations of motion of the system are expressed as follows: ∂ 2 u(x, t) ∂u(x, t) + F (t)δ(x − d) − ku(x, t) − δ 2 ∂t ∂x

∂u(L, t) − C[u(L, t) − v(t)]3 δ(x − L) − ελ − v(t) ˙ δ(x − L) ∂t

EA

=M

∂ 2 u(x, t) , ∂t 2

0≤x≤L

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.6 Linear dispersive rod with attached SDOF NES. Table 5.1 Leading eigenfrequencies of the uncoupled undamped rod (k = 1). Normal Mode Eigenfrequency (Hz)

1 0.29

2 0.77

3 1.26

4 1.76

u(0, t) = 0, u(x, 0) = 0,

C[u(L, t) − v(t)]3 + ελ ∂u(x, 0) = 0, ∂t

5 2.26

6 2.75

7 3.25

8 3.75

∂u(L, t) − v(t) ˙ = mv(t) ¨ ∂t

v(0) = 0,

v(0) ˙ =0

(5.9)

This is an initial value (Cauchy) problem governed by a set of coupled partial and ordinary differential equations, with essential nonlinearities. Clearly, (5.9) represents a well-posed mathematical problem, but no analytic solution for the transient dynamics is possible; hence, one must resort to numerical methods for its solution. It is emphasized that of interest in this work is the study of the transient nonlinear dynamics of the system, especially at the initial stage of the motion (i.e., immediately after the imposition of the external shock) where the energy of the system is at its highest and strong nonlinear dynamical interactions between the rod and the NES are anticipated. It is precisely these strongly nonlinear dynamical interactions that we aim to analyze in detail in this section. In the following numerical simulations the transient dynamics was computed by performing a finite element (FE) discretization of the equations of motion (5.9). The methodology was developed in the thesis by Georgiades (2006). First we mention that the eigenfrequencies of the linear rod on an elastic foundation with no damping and forcing terms and no NES attached, are given by (in rad/s): ωq =

(2q − 1)2

k π 2 EA + , 2 M 4L M

q = 1, 2, . . .

(5.10)

In Table 5.1 we present the leading eigenfrequencies of the uncoupled and undamped rod [with NES detached – expression (5.10)] for parameters L = 1, EA = 1.0, M = 1.0, δ = 0.05, m = 0.1, ε = 0.1, λ = 0.5, and elastic foundation equal to k = 1.

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The set of discrete equations resulting from application of the FE method were integrated using the adaptive Newmark Algorithm (Geradin and Rixen, 1997). As an excitation of the rod, the following impulsive half-sine pulse is considered: A sin(2πτ/T ), 0 ≤ τ ≤ T /2 F (t) = (5.11) 0, t > T /2 with varying amplitude A and period equal to T = 0.1T1, where T1 is the period of the first mode of the linear rod with no NES attached; this assures that the impulsive excitation is of sufficiently small duration compared to the characteristic time scale of the system T1 . The shock is applied at position d = 0.3 on the rod (see Figure 5.6), and the system parameters were assigned the numerical values, L = 1,

EA = 1.0,

M = 1.0,

δ = 0.05,

m = 0.1,

ε = 0.1,

λ = 0.5

In the performed simulations the results are post-processed by computing a set of energy measures, in order to study the energy absorbed and dissipated in the NES attachment, as well as the energy exchanges between the NES and the rod during their transient nonlinear interaction. Post-processing of the numerically computed time series of the rod and the NES is performed in two different ways. First, we perform a spectral analysis of the computed time series by employing numerical Wavelet Transforms (WTs), and constructing numerical WT spectra of the responses. As explained in Section 2.5, WT spectra enable one to determine accurately the dominant frequency components in the transient responses, and, in addition, to study the evolutions of these dominant harmonic components in time. Such wavelet spectra enable one to better understand the ‘slow flow’ dynamics of the studied rod-NES interactions. In an alternative approach, the time series are analyzed by Empirical Mode Decomposition (EMD) (see Section 2.5). Through this numerical algorithm one decomposes the computed time series (signals) in terms of intrinsic mode functions (IMFs) which can be regarded as oscillatory modes embedded in the signal. By construction, the superposition of all IMFs regenerates the signal. Further analyzing the IMFs by means of the Hilbert Transform one determines the dominant frequency components of the IMFs, which, when compared to the corresponding WT spectra, enables one to examine in detail the resonant dynamic interactions that occur between the rod and the NES. The nonlinearity and the non-stationarity of the computed transient signals force us to combine both mentioned techniques for postprocessing of the data. The first step of the post-processing analysis that supercedes the application of the previous methods, however, is to introduce certain energy measures. These were first introduced by Georgiades (2006). Certain of these measures help us assess the accuracy of the numerical simulations: indeed, the total energy of the system – including the energy dissipated by the dampers – should be approximately preserved, not only at each time step of the numerical integration, but also for the entire time window of the numerical simulation. Additional energy measures enable us to study

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

carefully the energy exchanges that occur between the rod and the NES; as shown below, by constructing energy transaction histories between the rod and the NES, we are able to identify the distinct dynamical mechanisms that govern TET (i.e., nonlinear beats, irreversible, one-way TET, or a combination of both). It is interesting to note that the aforementioned energy exchange mechanisms can be eventually related to the results of WT and EMD of the computed time series of the rod and NES responses. To this end, the normalized energy dissipated by the damper of the NES up to time t, is computed by the following EDM: t ελ [u(L, ˙ τ ) − v(τ ˙ )]2 dτ 0 ENES(t ) = × 100 (5.12) t 1 F (t) u(d, ˙ t)dt 2 0

i.e., the percentage of impulsive energy applied to the rod that is dissipated by the damper of the NES up to time t. For the passive system considered here this EDM reaches a definitive asymptotic limit with increasing time, ENES,t 1 = limt 1 ENES (t). This represents the percentage of impulsive energy that is eventually dissipated by the NES during the entire duration of the motion. In addition, for each simulation we estimate the normalized total dissipated energy over the entire duration of the motion, which, to ensure accuracy of the FE simulation, should be approximately equal to unity: Edamp,NES(t) + Edamp,rod (t) ≈1 t 1 Ein

ηtotal (t) ≡ lim

(5.13)

In (5.13), Edamp,NES(t) and Edamp,rod (t) are the energies dissipated by the NES and the rod up to time t, respectively, whereas Ein is the input impulsive energy. By ensuring that this ratio assumes numerical values close to unity, we also ensure that the FE simulation is performed for a sufficiently long time interval, so that no essential transient dynamics is missed outside the time window of the numerical study. Additional energy measures utilized to check the accuracy of the simulations are discussed in Georgiades (1996). An additional important energy measure computes the instantaneous transction of energy between the rod and the NES. Assuming that the NES is an open system that exchanges continuously energy with the rod (through energy absorption from incoming waves or energy backscattering to the rod), one defines the following Energy Transaction Measure (ETM), ETrans , between the NES and the rod: ETrans = Ek,NES + Ed,NES + Edamp,NES

(5.14)

In the above relation denotes the corresponding energy difference between two subsequent time steps; Ek,NES (t) = (1/2)mv˙ 2 (t) is the instantaneous kinetic energy and Ed,NES(t) = (1/4)C[u(L, t) − v(t)]4 the instantaneous potential energy of the NES; whereas

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

t

Edamp,NES(t) = ελ

17

[u(L, ˙ τ ) − v(τ ˙ )]2 dτ

is the energy dissipated by the damper of the NES up to the time instant t. The ETM is an important energy measure from a physical point of view, since it helps one identify instantaneous inflow or outflow of energy between the rod and the NES; in particular, when there is inflow of energy from the rod to the NES, it holds that ETrans > 0, whereas, negative values of the ETM (ETrans < 0) correspond to backscattering of energy from the NES to the rod. Moreover, in the limit when the time step t of the numerical simulation tends to zero, the ratio ETrans / t represents the instantaneous power inflow to or outflow between the rod and the NES. Clearly, efficient TET from the rod to the NES is signified by strong positive energy transactions (ETrans > 0) throughout the transient response of the system, but especially in the initial regime of the motion (i.e., immediately after the application of the external shock), when the energy of the system is at its highest level. In cases where there are only positive spikes of the ETM, there occurs irreversible energy transfer from the rod to the NES, that is, energy is continuously transferred from the rod to the NES where it is eventually dissipated by viscous damping. This scenario for TET will be designated as irreversible TET. As shown below, there are alternative TET scenarios corresponding to different types of energy transactions between the rod and the NES. For example, it is possible to obtain TET through nonlinear beats, corresponding to alternating series of positive and negative spikes in the energy transaction between the rod and the NES. This indicates that energy flows back and forth from the rod to the NES, with the overall average of the ETM ETrans being positive; in this case the NES backscatters significant portions of energy back to the rod, but, on the average, it absorbs and dissipates a certain portion of the input energy of the rod. Finally, it is possible to obtain a combination of the aforementioned energy transaction scenarios, i.e., an initial stage of irreversible TET, followed by a regime of TET through nonlinear beats. Finally, we make some remarks concerning the post-processing of the numerical results through WT and EMD. In the following numerical simulations we will be interested mainly in the high-energy transient dynamics at the early stage of the response, i.e., immediately after the application of the external shock. Therefore, in certain cases we will need to divide the time series into early and late parts, since, as the amplitudes of the responses get smaller due to damping dissipation the corresponding numerical wavelet traces are too light to be tracable. As in Chapters 3 and 4, the WT spectra will be employed to study the temporal evolutions of the dominant frequency components of the time series, as well as the nonlinear modal interactions occurring between the rod and the NES. One disadvantage of the WT compared to the EMD is its ineffectiveness to detect complex details of the time series, such as, intrawaves in the nonlinear signals, i.e., oscillatory components of the time series possessing frequency components that vary rapidly within a characteristic period. This is one of the reasons that EMD is employed as an alternative tool for the post-processing the numerical time series.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

As discussed in Section 2.5, EMD provides the characteristic time scales of the dominant nonlinear dynamics of the rod-NES interaction. Moreover, by adopting this analysis one can identify and analyze the most important nonlinear resonance interactions between the rod and the NES which are responsible for the nonlinear energy exchanges between these two subsystems. To this end, we say that a k:m transient resonance capture (TRC) occurs between the IMF c1 (t) of the rod, and the IMF c2 (t) of the NES (with corresponding phases ϕ1 (t) and ϕ2 (t), respectively), whenever their instantaneous frequencies satisfy the following approximate relation: k ϕ˙ 1 (t) − mϕ˙ 2 (t) ≈ const,

for t ∈ [T1 , T2 ]

The time interval [T1 , T2 ] defines the duration of the said TRC. A more complete picture of the TRC between the two mentioned IMFs can be gained by constructing appropriate phase plots that involve the phase difference ϕ12 (t) ≈ ϕ1 (t) − ϕ2 (t) and its time derivative. More specifically, a TRC is signified by the existence of a small loop in the phase plot of ϕ12 (t) versus ϕ˙ 12(t); whereas absence of (or escape from) TRC is signified by time-like (that is, monotonically varying) behavior of ϕ12(t) and ϕ˙ 12(t). In addition, the ratio of instantaneous frequencies of the IMFs, ϕ˙1 (t)/ϕ˙2 (t), provides a confirmation of the order of the k:m TRC. It is precisely these features that make EMD useful for studying the strongly nonlinear transient problem considered in this section. Indeed, the decomposition of the rod and NES transient responses in terms of their oscillatory components (the IMFs), and the subsequent computation of their instantaneous frequencies, provides a useful tool for studying nonlinear resonant interactions between the NES and the modes of the rod. In what follows we provide results of this analysis.

5.2.2 Computational Study of TET In Georgiades (2006) four main sets of FE simulations were performed for the system parameters and the half-sine applied external shock defined in the previous section. What distinguished the first and second sets of FE simulations was the different parameter values for the elastic foundation of the rod, k, the NES stiffness, C, and the magnitude of the applied shock, A. Specifically, the first set of simulations was performed for a dispersive rod with fixed distributed elastic foundation k = 1.0 , for 22 values of the nonlinear characteristic in the range C ∈ [0.001, 20], and 15 values of the shock amplitude in the range A ∈ [0, 500]. This gave a total of 22 × 15 = 330 possible pairs (C, A), all of which were simulated in the first series. Similarly, the second set of numerical simulations involved the same 330 numerical simulations but for an elastic foundation with k = 0, corresponding to a non-dispersive rod. Each of the transient simulations of the first two series was performed for a sufficiently large time interval, so that at the end of the simulation at least 99% of the input shock energy was damped by the distributed viscous damping of the rod and the discrete viscous damper of the NES.

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Fig. 5.7 Contour plot of the percentage of shock energy eventually dissipated by the NES, ENES,t1 , as function of the nonlinear stiffness, C, and the shock amplitude, A, for a dispersive rod (k = 1).

The numerical study has multiple objectives. The first objective is to study the ranges of parameters for which the NES is capable of passively absorbing and locally dissipating a significant portion of the shock energy applied to the rod, and, in addition, to investigate robustness of the NES performance to certain parameter variations. The second objective is to study the dynamical mechanisms that influence TET from the rod to the NES, and in the way, to determine the most favorable conditions for the realization of strong TET. An additional objective is to analyze in detail the TRCs between the NES and the rod responsible for TET (and the characteristic time scales of these interactions) though the use of WTs and EMD. In Figure 5.7 we depict the contour plot of the EDM ENESt 1 , as function of the parameters C and A for the FE simulations corresponding to the dispersive rod, k = 1 (the results corresponding to the non-dispersive rod k = 0 can be found in Georgiades (2006) and Georgiades et al. (2007). In the remainder of this work, wherever we mention ‘the EDM’, we will be referring to the asymptotic value ENES,t 1 . Regions of the plot where the EDM is relatively large correspond to strong TET from the rod to the NES, indicating that a significant percentage of the shock energy of the rod is eventually absorbed and dissipated by the NES. These numerical results reveal that when strong shocks are applied, enhanced TET occurs (with more

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.8 Contour plots of the percentage of shock energy eventually dissipated by the NES, ENES,t1 , as function of the NES mass m, and the shock amplitude A for four different values of the nonlinear stiffness C, and a dispersive rod (k = 1).

than 75% of shock energy eventually dissipated by the NES) when the essential stiffness nonlinearity is relatively weak. By contrast, when smaller shocks are applied, strong TET occurs (corresponding to ENES,t 1 > 75%) over a wide range of values of the essentially nonlinear stiffness of the NES. This should be expected, since when the energy is high, a stiff essential nonlinearity amounts to a near-rigid connection between the rod and the NES, yielding small relative velocities across the NES damper, and, hence, small energy dissipation by the NES. In an additional set of FE simulations four distinct values of the nonlinear stiffness of the NES are considered, namely, C = 0.004, 0.01, 2.0, 10.0, for varying mass of the NES in the range m ∈ [0.01, 0.1] (for a total of 11 values), and shock amplitude in the range A ∈ [1, 420] (for a total of 13 values). Therefore for each value of C there were 11 × 13 = 143 possible pairs (m, A), all of which were realized in the numerical simulations. Again, to ensure that the numerical integration was of sufficient duration, an additional requirement was imposed, namely that at least 99% of the shock energy should be dissipated at the end of each of FE simulation. In Figure 5.8 we depict the EDM as function of the NES mass m and the shock amplitude A for four chosen values of the nonlinear stiffness characteristic (C = 0.004, 0.01, 2.0, 10.0), and a dispersive rod (k = 1). As in Figure 5.7, we deduce that there are parameter regions where strong TET from the rod to the NES

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Table 5.2 FE simulations of the system whose TET plot appears in Figure 5.7. FE Simulation – Application No.

Phenomena

Group

C

A

ENES,t1 (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

B B-I B B B B B I B B B B B B B B B-I B B-I I B

b b a a a c c a b b b b c b b c a c a a c

0.02 0.004 6 9 2 5 5 0.01 0.01 0.005 0.02 0.06 0.08 0.09 0.1 0.2 0.8 0.8 5 0.2 20

180 180 10 10 20 50 100 20 260 400 200 100 420 100 60 460 10 180 3 5 500

76 72 76 74 75 60 51 67 76 75 75 76 60 74 74 52 71 56 69 67 21

is realized. Moreover, as the value of the nonlinear stiffness characteristic increases the region of strong TET shifts to smaller shock amplitudes and becomes narrower. In addition, in parameter ranges where relatively strong TET occurs there appears to be nearly negligible dependence of the EDM on the NES mass for values of m > 0.02. These results indicate that the NES can be designed to passively absorb and locally dissipate a significant portion of the applied (broadband) shock energy of the rod. Moreover, the NES can be designed so that the passive TET from the rod to the NES is both strong and robust to small changes in the impulsive energy and the system parameters. These results demonstrate the efficacy of using lightweight essentially nonlinear local attachments as passive absorbers and local energy dissipaters of broadband energy from elastic continua. This result extends the results reported in previous chapters where discrete linear oscillators with local essentially nonlinear attachments were studied. We now proceed to a detailed analysis of the dynamics governing TET from the rod to the NES in the system of Figure 5.6. Considering the dispersive rod with k = 1, 21 FE simulations [termed from now on ‘Applications’ (Georgiades, 2006)] were considered for the system whose TET plot is depicted in Figure 5.7. In Table 5.2 we present the system parameters used for each application, together with the corresponding EDMs and the characterization of the corresponding dynamical phenomena. ‘B’ indicates the occurrence of nonlinear beat phenomena in the tran-

22

5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.9 Relative responses between the end of the rod and the NES for Applications 1, 17, 20 (see Table 5.2).

sient responses of the rod and the NES. ‘I’ indicates irreversible (one-way) energy transfer from the rod to the NES; we note, however, that even in these cases there exists an initial region, albeit small, where very early nonlinear beat phenomena occur, so we may designate the phenomenon as being predominantly irreversible energy transfer. Finally, the designation ‘B-I’ indicates early nonlinear beat phenomena in the transient dynamics, followed by irreversible energy transfer from the rod to the NES. These designations refer to the previous discussion regarding energy transactions between the linear and nonlinear components of the system considered. A simple comparison of the different applications listed in Table 5.2 reveals that, with the exception of Applications 7, 16, and 21, all applications correspond to rather strong TET, since a major part of the input (broadband) vibration energy in the rod is passively absorbed and dissipated by the NES. This observation is in itself interesting since it shows that strong TET in the system under consideration occurs over wide combinations of input energy and system parameters. It follows that a study of TET efficiency in different applications can only be performed on a relative (i.e. comparative) basis, and in that context the EDM can only be considered as a relative indicator of TET efficiency. Specific examples for all three types of the afore-mentioned dynamical mechanisms (‘B’, ‘B-I’ and ‘I’) are discussed below. The Applications listed in Table 5.2 are partitioned into three main groups. Group (a) consists of Applications 3, 4, 5, 8, 17, 19 and 20 with relatively strong TET from the rod to the NES, corresponding to relatively small input energies (shocks). All three dynamical mechanisms (B, I, and B-I) are realized in the Applications of

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Fig. 5.10 Relative responses between the end of the rod and the NES for Applications 2, 7, 14, 19 (see Table 5.2).

Group (a). The second group of Applications 1, 2, 9, 10, 11, 12, 14 and 15 [labeled as Group (b)] is again characterized by relatively strong TET, but corresponds to higher levels of input energy; these Applications involve the dynamical mechanisms B and B-I. Finally, Group (c) consists of Applications 6, 7, 13, 16, 18 and 21 with relatively weak energy transfers, and higher levels of input energy; all Applications in this group are characterized by persistent nonlinear beat phenomena (mechanism B), involving continuous energy exchanges between the rod and the NES. Typical transient relative displacements of the NES with respect to the edge of the rod are presented in Figures 5.9 and 5.10. In each of these plots (as in the ones that follow), each Application is characterized by its group and the governing dynamical mechanism; for example, in Figure 5.9 Application 1 is labeled by (b, B), and so on. The measure of relative displacement between the NES and the edge of the rod affects directly the efficiency of TET, since the capacity of the NES to dissipate energy transferred from the rod is directly related to the relative velocity across its viscous damper. It follows that enhanced energy dissipation by the NES is realized when this relative displacement (and its time derivative) attains large magnitudes, especially in the critical initial regime of the motion where the energy is still relatively large (and energy dissipation due to damping in the rod is still small). Examples of cases where large, early relative displacements between the rod and the NES occur are Applications 1 (Figure 5.9) and 2 (see Figure 5.10) with corre-

24

5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

sponding dissipation measures ENES,t 1 = 76% and 72%, respectively; whereas an example where small relative displacement occurs is Application 7 (Figure 5.10) corresponding to ENES,t 1 = 51%. An interesting feature of the NES is its capacity to interact with more than one structural modes of the rod (this is done sequentially through resonance capture cascades – RCCs, see below). Indeed, due to the essential coupling nonlinearity, the NES is simultaneously ‘coupled’ to all modes of the rod [as can be realized from the differential equations (5.9)], so it has the capacity to nonlinearly resonate with structural modes over wide frequency ranges, provided, of course, that the initial conditions are appropriate. Such multi-modal and multi-frequency interactions of the NES with the rod may lead to multi-frequency targeted energy transfer and complex dynamic phenomena, such as, abrupt transitions between different dynamical regimes. These interactions become apparent in the WT spectra of the dynamics, although in some cases they may be visible in the time series themselves. For instance, in Figure 5.9 – Application 1 the frequency content of the NES response is rich, and the RCC is evident; this is also the case in Figure 5.10 – Application 14. A useful computational tool for studying the nonlinear dynamic interaction between the rod and the NES is the study of the transient energy transaction history between these two subsystems. In Figures 5.11a, b we depict the energy transaction histories between the NES and the rod for Applications 1 and 17, where strong TET from the rod to the NES occurs (76% of shock energy dissipated by the NES in Application 1, and 71% in Application 17) (Georgiades, 2006). In these plots we note the strong positive spikes of energy transmission from the rod to the NES and the small negative spikes of energy backscattered from the NES to the rod; this explains the relatively high values of the EDM realized in these applications. In addition, in both applications there is a positive net balance of energy transferred from the rod to the NES during the critical early regime of the response where the overall energy of the motion is relatively high. In Figures 5.12a, b we depict the energy transaction histories for two Applications (7 and 21) corresponding to relatively weak TET (51% of shock energy eventually dissipated by the NES in Application 7, and 21% in Application 21); in these simulations we note that strong backscattering of energy from the NES to the rod occurs, which explains the corresponding weaker energy transfers. An alternating series of positive and negative spikes of energy transfers is an indication that nonlinear beat phenomena between the rod and the NES occur (dynamical mechanism ‘B’ in Table 5.2). This is especially evident in the energy transaction history of Application 1 (see Figure 5.11a), where nonlinear beat phenomena with strong positive spikes are clearly detected. In Application 17 (see Figure 5.11b) the series of strong initial nonlinear beats is followed by irreversible (one-way) energy transfer (dynamical mechanism I in Table 5.2) from the rod to the NES, as evidenced by the late series of positive – only energy spikes. Similar persisting nonlinear beats are observed in the energy transaction histories depicted in Figures 5.12a, b where applications with relatively weaker TET are depicted. The distinctive feature of the beats in these cases is that the negative and positive energy spikes are of comparable magnitudes, preventing strong ‘flow of energy’ from the rod to the NES. In

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Fig. 5.11 Case of strong TET, ETMs between the rod and the NES for (a) Application 1 (case ‘B’, nonlinear beats); (b) Application 17 (case ‘B-I’, initial nonlinear beats followed by irreversible energy transfer).

Figure 5.13 we depict the energy transaction for Application 20 where irreversible energy transfer from the rod to the NES occurs right from the beginning of the dynamics, and nonlinear beat phenomena are completely absent; indeed, in Appli-

26

5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.12 Case of weak TET, ETMs between the rod and the NES for (a) Application 7 (case ‘B’, nonlinear beats); (b) Application 21 (case ‘B’, nonlinear beats).

cations 8 and 20 there is only irreversible ‘flow of energy’ from the rod to the NES, where the energy is localized to the NES and dissipated by the NES damper. The

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Fig. 5.13 Case of strong TET, ETMs between the rod and the NES for Application 21 (case ‘I’, irreversible energy transfer).

resulting TET is relatively strong in this case, and comparable to the strong TET realized in applications governed by the dynamical mechanisms B and B-I. In all considered simulations, the energy exchanges between the rod and the NES are realized in the form of spikes, which reflects the fact that the external excitation itself is in the form of a spike (short pulse); this generates forward- and backwardpropagating pulses in the rod which are either reflected at the left (clamped) boundary of the rod, or are partially reflected and transmitted into the NES at its right boundary. Numerical plots such as the ones depicted in Figures 5.11–5.13 enable one to study in detail the transient energy exchanges between the rod and the NES, and, more importantly, to determine the dynamical mechanisms that govern these energy exchanges. In addition, it is possible to deduce the precise time windows of the dynamics where, either strong TET to the rod, or backscattering of energy from the NES back to the rod take place. In the following study we relate the previous energy transaction histories to the TRCs that take place due to nonlinear modal interactions between the rod and the NES. In Figure 5.14 we depict the WT spectra of the relative transient responses between the edge of the rod (from now referred to as ‘the rod’) and the NES, for four cases where either strong TET occurs [cases (a, B-I) – Application 17; (a, B) – Application 1; and (a, I) – Application 20] or weaker TET is realized [case (c, B) – Application 7]. The WT spectra reveal the dominant frequency components of the corresponding responses, as well as their temporal evolutions with decreasing energy due to damping dissipation. Considering Application 17 [case (a, B-I)] where

28

5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.14 WT spectrum of the relative motion of the NES with respect to the edge of the rod: (a) Application 1 – case ‘B’; (b) Application 7 – case ‘B’.

strong TET from the rod to the NES occurs (see Figure 5.14c), we observe early (i.e., high energy) transient resonant interactions of the NES with predominantly the first and second modes of the rod, as well as a weaker early NES resonant interaction

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Fig. 5.14 WT spectrum of the relative motion of the NES with respect to the edge of the rod: (c) Application 17 – case ‘B-I’; (d) Application 20 – case ‘I’; the first three eigenfrequencies of the uncoupled linear rod are indicated.

30

5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

with the third mode of the rod; all these early interactions are realized in the form of nonlinear beats. Moreover, we observe a nonlinear transient capture of the dominant frequency component of the dynamics by a nonlinear mode whose frequency shifts below the first linearized mode of the rod. During this low-frequency transition the dynamics localizes gradually to the NES with decreasing energy; similar transitions were detected in previous chapters (see also Lee et al., 2005; Kerschen et al., 2005) in the dynamics of discrete linear oscillators coupled to NESs. The aforementioned early resonant interactions explain the nonlinear beats observed in the early response regime (mechanism ‘B’), whereas the low frequency transition of the dominant harmonic yields one-way irreversible energy transfers from the rod to the NES (mechanism ‘I’) in this Application. Similar transient capture of the dynamics by a nonlinear mode is deduced in the WT spectrum of Figure 5.14a [Application 1 – (a, B)], however, in this case the frequency variation of the nonlinear mode (dominant harmonic) takes place in between the first and second eigenfrequencies of the rod. Similarly to Application 17 (Figure 5.14c) this transition yields strong TET from the rod to the NES. Additional early beats between the NES and the second and third modes of the rod take place (mechanism ‘B’, as in Application 17); more importantly, however, there occurs a secondary late transition of the dynamics from the nonlinear mode to the first rod mode, after which additional persistent beats between the NES and the first rod mode are realized (mechanism ‘B’). This late transition is qualitatively different from the dynamics depicted in Figure 5.14c. No such low frequency transitions occur in the WT spectra of the relative transient responses of Applications 7 [case (c, B) – weaker TET from the rod to the NES], and 20 [case (a, I) – strong TET], that are presented in Figures 5.14b, d, respectively. In the case of weaker TET (Figure 5.14b) we observe strong and persistent resonance locking of the relative response at the frequency of the second linearized mode of the rod, with persistent nonlinear beats observed in the transient response. It is interesting to note that in this case there is complete absence of resonance interactions between the NES and the first mode of the rod. In the case of stronger TET in Application 20 (see Figure 5.14d) there is similar resonance locking of the relative response at the first linearized mode of the rod, which, however, is not as persistent as in the WT spectrum of Figure 5.14b. In both cases, there is the absence of transient capture of the early (high energy) relative motion by a nonlinear mode localized at the NES (as in Figures 5.14a, c). Finally, we note clearly the multi-modal content of the dynamics of the rod-NES interaction, reaffirming our previous comment with regard to the capacity of the NES to resonantly interact with a set of linearized modes of the rod. In general, such multi-modal resonant interactions enhance the effectiveness of nonlinear TET in the system, and lead to complex dynamical phenomena such as resonance capture cascades (RCCs). The WT spectra, when combined with empirical mode decomposition (EMD) of the transient responses of the rod and the NES form a powerful computational tool that can be utilized to reveal additional dynamical features of the resonance interactions occurring in the system. This is discussed in what follows.

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Fig. 5.15 EMD analysis of Application 17 – case ‘B-I’: (a) IMF-based reconstructed transient response of the edge of the rod and the NES compared to numerical simulations.

For a more detailed study of the nonlinear resonance interactions between the rod and the NES two representative cases of EMD analysis concerning Applications 7 and 17 are considered (Georgiades et al., 2007). To increase the accuracy of the analysis, the early and late transient responses of Application 7 are analyzed separately, whereas, no such separation was deemed necessary for Application 17. In each case we analyze through EMD the transient responses of the edge of the rod and of the NES. Examination of the IMFs of these transient responses and their instantaneous frequencies provides insightful information concerning the resonance interactions that occur between the rod and the NES. Indeed, the computation of the instantaneous frequencies of the IMFs, combined with the previous WT spectra provide us with the opportunity to interpret the WT results in terms of resonance interactions between specific IMFs of the rod and the NES. In what follows we will apply this methodology to examine in detail resonance interactions in Applications 17 [case (a, B-I)] and 7 [case (c, B)]. In Figure 5.15a we present IMF-based reconstructions of the transient responses of the edge of the rod and the NES for Application 17; complete agreement between numerical simulation and IMF-based reconstruction is observed, proving the validity of the EMD analysis for decomposing the transient nonlinear responses through IMFs. Representative IMFs are depicted in Figure 5.15b. Next, decompositions of the IMFs in terms of their instantaneous amplitudes and phases were performed in order to examine their individual frequency contents. This information should be analyzed together with the corresponding WT spectrum of the relative transient response between the edge of the rod and the NES (see Figure 5.14a); from that plot it is clearly observed that in this case strong nonlinear TET is associated with low

32

5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.15 EMD analysis of Application 17 – case ‘B-I’: (b) IMFs of the transient response of the edge of the rod and the NES.

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frequency ‘locking’ of the dynamics to a nonlinear mode below the first eigenfrequency of the rod (at 0.29 Hz). In Figure 5.16a we depict the instantaneous frequencies of the second IMF of the NES, and the fifth and ninth IMFs of the rod end response; in the same Figure we superimpose these instantaneous frequencies to the wavelet spectra of the corresponding simulated transient responses. We note that these IMFs are dominant since they coincide with dominant frequency components of the wavelet spectra at different time windows of the dynamics. The following conclusions are drawn from these results. It is clear that the 2nd IMF of the NES and the 9th IMF of the rod possess nearly constant instantaneous frequencies precisely at the low frequency range of the nonlinear mode of the WT spectrum of Figure 5.14a; hence, these IMFs engage in 1:1 TRC in the initial (high energy) stage of the transient dynamics. This 1:1 TRC becomes apparent by considering the corresponding phase plot of the phase difference φ2NES(t) − φ9Rod (t) in the early time window where the 1:1 TRC occurs (see Figure 5.16b). Indeed, resonance capture between two IMFs is indicated by the non-time-like, ‘slow’ evolution of the difference between their corresponding phase difference, so that the averaging theorem cannot be applied with respect to that phase difference, and preventing averaging out that phase from the dynamics. It is precisely such resonance captures that lead to passive TET from the rod to the NES, as quantified by the EDM. Moreover, the fact that the mentioned 1:1 TRC takes place in the early stage of the dynamics where the energy of the system is at its highest level, explains the strong TET observed in this application. In this resonance capture regime, the 2nd (dominant) IMF of the rod coincides in frequency with the dominant harmonic component of the transient response of the NES, whereas the 9th IMF of the rod coincides with the lowest of the dominant harmonic components of the transient response of the edge of the rod. These results (together the ones presented below) demonstrate the capacity of the combined EMD-WT analysis to accurately identify the oscillatory components of the rod and NES time series that engage in TRC, and, are ultimately responsible for passive TET phenomena from the rod to the NES. In Figure 5.17a we depict the exact and IMF-based reconstructed responses for Application 7 [case (c, B) – weaker TET], from which again complete agreement between simulations and IMF reconstructions is observed. Representative IMFs of the early (high energy) responses of the edge of the rod and the NES are depicted in Figure 5.17b. Consideration of the resonance interactions between the IMFs of the rod and the NES reveals the reason that weak TET is realized in this application. Referring to the WT spectrum of the relative response between the edge of the rod and the NES for this application (see Figure 5.14c), we established ‘locking’ of the dynamics in the vicinity of the second linearized eigenfrequency of the rod (close to 0.77 Hz). Examining the temporal evolutions of the instantaneous frequencies of the IMFs of the early transient responses of the edge of the rod and the NES (see Figure 5.18a), we note that the 1st IMF of the NES and the 5th IMF of the rod develop delayed frequency ‘plateaus’ close to 0.77 Hz for t > 12. Moreover, examining the phase plot of the phase difference φ1NES (t) − φ5Rod (t) over the time window where the frequency plateaus are realized, we note the characteristic loops that are indica-

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.16 Nonlinear 1:1 TRC in Application 17 between the 2nd IMF of the NES and the 9th IMF of the edge of the rod: (a) instantaneous frequencies of the two IMFs; (b) phase plot of the phase difference indicating the 1:1 TRC.

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Fig. 5.17 EMD analysis of Application 17 – case ‘B’: (a) IMF-based reconstructed transient responses of the edge of the rod and the NES versus numerical simulations – early and late responses are treated separately.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.17 EMD analysis of Application 17 – case ‘B’: (b) IMFs of the early transient responses of the edge of the rod and the NES.

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tive of 1:1 resonance capture between these two IMFs (see Figure 5.18b). However, since this TRC occurs at a late stage of the response (i.e., at the stage where a significant portion of the initial energy of the system has already been dissipated due to damping), the resulting TET from the rod to the NES is not as strong as in the previously discussed Application 17, where the corresponding TRC takes place at the early highly eneregetic stage of the dynamics. In Figure 5.18 we also show that in Application 7 there occurs an additional ‘delayed’ 1:1 TRC between the 2nd IMF of the NES and the 6th IMF of the edge of the rod at a frequency near the first eigenfrequency of the rod (0.29 Hz), which, however, does not lead to significant energy transfer from the rod to the NES. Finally, from the plots of Figure 5.18a we note that, by superimposing the instantaneous frequencies of the IMFs to the WT spectra of the respective numerical time series, we infer that the 1st and 2nd IMFs of the NES coincide with the higher and lower dominant harmonics, respectively, of the time series of the NES, but only during the later, low-energy stage of the motion. Similar conclusions can be drawn with regard to the 5th and 6th IMFs of the rod. Summarizing, it appears that strong TET in the system under consideration is associated with TRCs between IMFs of the NES and rod responses at specific frequency ranges and during the critical early stage of the motion where the energy of the system is at high level; delayed TRCs between IMFs of the rod and the NES that occur at diminished energy level result in weaker TET from the rod to the NES. In terms of the corresponding WT spectra, strong energy exchanges and early (highenergy regime) TRCs between IMFs are associated with ‘locking’ of the dynamics with nonlinear normal modes that localize to the NES as the energy of the system diminishes due to damping dissipation. The results of this section demonstrate the efficacy of using lightweight essentially nonlinear attachments – NESs as passive absorbers of broadband (shock) energy from elastic structures. The resulting irreversible TET of shock energy to the NESs, eliminate in an effective way unwanted structural disturbances. Hence, the proposed design can be regarded as a new paradigm for passive shock isolation of elastic structures. An interesting (and appealing) feature of the NES concept is that, although an NES represents only a local alteration of the physical configuration of a structure, it can affect the global structural dynamics. The reason behind this seemingly paradoxical finding (and also being the basic feature that distinguishes the NES from previous absorber designs mentioned in the literature), is the essential stiffness nonlinearity of the NES, which enables it to resonantly interact (i.e., to engage in resonance captures) with structural modes at arbitrary frequency ranges, provided, of course, that its point of attachment is not close to nodes of the structural modes of interest. A new feature of the study of TET carried out in this section is the use of combined Wavelet Transforms (WTs) and Empirical Mode Decomposition (EMD) as a tool for identifying the specific TRCs responsible for nonlinear modal interactions between the NES and the structure to which it is attached. It was found that there exist at least three distinct dynamical mechanisms governing the NES-rod nonlinear resonance interactions; namely, nonlinear beat phenomena (mechanism ‘B’), direct one-way irreversible energy transfers from the rod to the NES (mechanism ‘I’), or

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.18 Nonlinear 1:1 TRCs in Application 7 between the 1st IMF of the NES and the 5th IMF of the edge of the rod, and the 2nd IMF of the NES and the 6th IMF of the rod: (a) instantaneous frequencies of the IMFs of the NES and the edge of the rod; (b) phase plots of the phase differences indicating the two 1:1 TRCs.

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a combination of the two (mechanism ‘B-I’). Although no direct association of any one of these three mechanisms to the strength of TET to the NES can be discerned based on the results presented herein, some interesting observations based on the previous computational findings can still be made. Indeed, relatively strong TET is associated with the occurrence of early nonlinear beats in the response (cases ‘B’ and ‘B-I’); this is not to say, however, that nonlinear beats always lead to relatively strong TET (counterexamples are Applications 7, 16, 18 and 21 in Table 5.2). These observations regarding early nonlinear beats are consistent with results reported in Sections 3.3 and 3.4 (see also Kerschen et al., 2005), where it was found that the most efficient mechanism for TET in the twoDOF system considered there was the excitation of early nonlinear beats (or, of stable impulsive IOs close to the 1:1 resonance manifold of the dynamics). Returning to the results reported in this section and motivated by the previous discussion, we conjecture that strong TET in the rod-NES configuration is similarly ‘triggered’ by early nonlinear beat phenomena occurring in the neighborhood of the 1:1 resonance manifold of the frequency-energy plot (FEP) of the underlying Hamiltonian system (i.e., of the undamped rod with undamped attached NES). To prove this conjecture one needs to follow a methodology similar to the one developed in Sections 3.3 and 3.4 for the two-DOF system. First, we need to construct the nonlinear FEP of the periodic (and quasi-periodic) orbits of the underlying Hamiltonian system (a challenging task in itself). Then, we need to compute periodic and quasi-periodic orbits with initial conditions that ‘trigger’ strong TET; finally, by superimposing the computed FEP to WT spectra of the numerical transient responses of the damped system we wish to prove that transient responses producing strong TET are ‘triggered’ by periodic or quasi-periodic nonlinear beats in the FEP. In the following section we provide some preliminary results towards interpreting damped transitions of the finite rod-NES system in terms of the FEP of the underlying Hamiltonian system.

5.2.3 Damped Transitions on the Hamiltonian FEP In this section we follow an alternative approach in our study of multi-frequency transitions in the transient dynamics of the viscously damped dispersive finite rod with the NES (see Figure 5.6). First, we will compute the periodic orbits of the underlying Hamiltonian system with no damping and external forcing and depict them in a frequency-energy plot (FEP); this will be similar to the plots constructed for the Hamiltonian dynamics of the discrete systems examined in the previous chapters. As shown in Section 3.3 this representation enables one to clearly distinguish between the different types of periodic motions in terms of backbone curves, subharmonic tongues and manifolds of impulsive orbits (IOs). Then, the dynamics of the damped and forced system will be considered and the corresponding WT spectra will be depicted in the FEP in an effort to interpret complex damped multi-frequency responses in terms of transitions between different

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

branches of periodic solutions on the FEP. Finally, the damped dynamics will be decomposed by EMD, that is, the computed time series will be decomposed in terms of intrinsic mode functions (IMFs) at different characteristic time (or frequency) scales. Comparisons of the evolutions of the instantaneous frequencies of the IMFs with the WT spectra of the corresponding time series, will enable us to identify the dominant IMFs of the signals and the time scales at which the dominant dynamics evolve at different time windows of the responses. Moreover, by superimposing the WT spectra of the damped responses to the FEP of the underlying Hamiltonian system, will be able to clearly relate multi-scaled transitions occurring in the transient damped dynamics, to transitions between different solution branches in the FEP. As a result, we aim to develop a physics-based, multi-scaled approach and provide the necessary computational tools for multi-scaled analysis of complex multi-frequency transitions occurring in the dynamics of essentially nonlinear dynamical systems (Tsakirtzis, 2006; Panagopoulos et al., 2007). The first step in our computational approach is to study the Hamiltonian system derived by omitting damping and forcing terms from the equations of motion (5.1). The reason for studying the Hamiltonian dynamics, is that, as shown in Chapters 3 and 4, for sufficiently weak damping the transient damped dynamics of system (5.1) is expected to approximately trace the branches of periodic or quasi-periodic solutions of the corresponding Hamiltonian system. To this end, we rewrite the equations of motion (5.9) in normalized form, omitting the forcing terms and adding general initial conditions for the rod and the NES: −

∂ 2 u(x, t) ∂u(x, t) ∂ 2 u(x, t) + + ω02 u(x, t) + ελ1 = 0, 2 ∂x ∂t ∂t 2

∂u(L, t) = −ε v(t), ¨ ∂x

u(0, t) = 0

C[u(L, t) − v(t)] + ελ2 3

u(x, 0) = r(x),

0≤x≤L

∂u(L, t) − v(t) ˙ = ε v(t) ¨ ∂t

∂u(x, 0) = s(x), ∂t

v(0) = v0 ,

v(0) ˙ = v˙0

(5.15)

In addition, we explicitly denote the lightweightness of the NES by the small parameter 0 < ε 1, and allow for different damping coefficients for the rod and the NES. Setting λ1 = λ2 = 0 we derive the following Hamiltonian system: ∂ 2 u(x, t) ∂ 2 u(x, t) 2 + ω u(x, t) − = 0, 0 ∂t 2 ∂x 2 ε v(t) ¨ + C[v(t) − u(L, t)]3 = 0,

0≤x≤L

u(0, t) = 0,

∂u(L, t) = −ε v(t) ¨ (5.16) ∂x

Initial conditions are omitted from (5.16) since the problem of computing the undamped periodic orbits of the Hamiltonian system constitutes a nonlinear boundary

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41

value problem (NLBVP); this is in contrast to problems (5.9) and (5.15) which are formulated as Cauchy (initial value) problems. To compute T -periodic solutions of system (5.16) the displacements of the rod and the NES are expressed in the following series forms (Panagopoulos et al., 2007): u(x, t) =

∞

Ck (x) cos[(2k − 1)t] +

k=1

v(t) =

∞

Vc,k cos[(2k − 1)t] +

k=1

∞

Sk (x) sin[(2k − 1)t]

k=1 ∞

Vs,k sin[(2k − 1)t]

(5.17)

k=1

where by = 2π/T we denote the basic frequency of the time-periodic motion. We note that the above infinite series expressions represent exact periodic solutions of the NLBVP (5.16), as they are, in essence, the Fourier series expansions of the sought solutions. Approximations in the computations will be made when the infinite series are truncated for computational purposes. Substituting (5.17) into the (linear) partial differential equation in (5.16) and taking into account the imposed boundary conditions, the following series of linear boundary value problems (BVPs) are obtained, governing the evolutions in space of the distributions Ck (x) and Sk (x), k = 1, 2, . . . : d 2 Ck (x) + [(2k − 1)2 2 − ω02 ]Ck (x) = 0 dx 2 d 2 Sk (x) + [(2k − 1)2 2 − ω02 ]Sk (x) = 0 dx 2 dCk (L) = ε(2k − 1)2 2 Vc,k , Ck (0) = Sk (0) = 0, dx dSk (L) = ε(2k − 1)2 2 Vs,k dx

(5.18)

The general solutions of the first two linear ordinary differential equations in (5.18) are expressed as , Ck (x) = Cˆ k ∈ x (2k − 1)2 2 − ω02 , Sk (x) = Sˆk ∈ x (2k − 1)2 2 − ω02 where Cˆ k = ,

ε(2k − 1)2 2 Vc,k (2k − 1)2 2 − ω02 cos L [(2k − 1)2 2 − ω2 ]

(5.19)

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Sˆk = ,

ε(2k − 1)2 2 Vs,k (2k − 1)2 2 − ω02 cos L [(2k − 1)2 2 − ω2 ]

The expressions (5.19) are valid over the entire frequency range ∈ [0, ∞), i.e., for harmonics with frequencies in both the propagation zone (PZ) and attenuation zone (AZ) of the uncoupled linear rod of infinite length. However, depending on the value of the frequency , the solutions (5.19) may change qualitatively assuming the form of traveling waves or attenuating standing waves. Indeed, for values of the fundamental frequency satisfying (2k −1)22 −ω02 < 0 for some k ∈ N + (inside the AZ of the dispersive rod), the following well-known relations can be employed: √ sin(j α) = j sinh(α), cos(j α) = cosh(α) with j = −1 Then expressions (5.19) yield time-periodic standing waves with attenuating spatial envelopes. On the contrary, time-periodic solutions satisfying the condition (2k − 1)2 2 − ω02 > 0 for some k ∈ N + (inside the PZ of the rod), correspond to time-periodic traveling waves of constant amplitude that propagate freely in the rod until they reach either one of its boundaries where they scatter. We note that resonances (standing waves) in the rod can only occur inside the PZ of the corresponding infinite rod, as they result from positive interference of left- and right-going traveling waves. Expressions (5.19) are derived in terms of the amplitudes Vs,k and Vc,k of the harmonics of the NES. These are computed by substituting (5.17) and (5.19) into the nonlinear ordinary differential equation in (5.16), yielding the following algebraic expression in terms of an infinite series with respect to the index k (Panagopoulos et al., 2007): −ε

∞

(2k − 1) Vc,k cos[(2k − 1)t] + Vs,k sin[(2k − 1)t] 2

2

k=1

+C

∞

1 − ε(2k − 1) [(2k − 1) 2

2

2

2

− ω02 ]−1/2

,

2 2 2 tan L (2k − 1) − ω0

k=1

× Vc,k cos[(2k − 1)t] + Vs,k sin[(2k − 1)t]

3 =0

(5.20)

Expanding the cubic power in (5.20), and setting the resulting coefficients of the trigonometric functions cos[(2k−1)t] and sin[(2k−1)t], k = 1, 2, . . . separately equal to zero, one derives an infinite set of nonlinear algebraic equations for the amplitudes Vs,k and Vc,k , whose solution completely determines the time-periodic solutions of the Hamiltonian system (5.16). In the numerical computations the infinite set of nonlinear algebraic equations resulting from (5.20) was truncated by considering terms only up to the fifth harmonic (i.e., k = 1, 2, 3), and omitting higher harmonics. The resulting truncated set

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43

Fig. 5.19 FEP of the Hamiltonian system (5.16) based on the truncated system (5.19, 5.20) with k = 1, 2, 3: (a) backbone branches of periodic motions and tonges of subharmonic motions.

of six nonlinear algebraic equations was then numerically solved for the amplitudes Vs,k and Vc,k , which also determined approximately the time-periodic response of the rod through relations (5.17) and (5.19). In Figure 5.19 we depict the approximate branches of time-periodic solutions of the Hamiltonian system in the FEP; specifically we employ the previously derived truncated system to compute the approximate amplitude of the relative displacement [v(t) − u(L, t)] of the truncated system (for k = 1, 2, 3) for varying values of the fundamental frequency and fixed parameters ε = 0.05, ω0 = 1.0, C = 1.0, L = 1.0 and λ1 = λ2 = 0. Only the frequency range covering the two leading modes

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.19 FEP of the Hamiltonian system (5.16) based on the truncated system (5.19, 5.20) with k = 1, 2, 3: (b) details of regions I and II; numbers () correspond to the periodic orbits depicted in Figures 5.20 and 5.21, and letters (•) to the numerical simulations of damped transitions.

of the uncoupled linear rod is considered in the FEP, which depicts the logarithm of the energy of a periodic orbit, log10 (E), as function of the fundamental frequency (in rad/s) of that orbit. The (conserved) energy E of the periodic orbit is computed by the following expression: E=

1 2

L ∂u(x, t) 2

∂t

dx +

1 2

L ∂u(x, t) 2

∂x

1 dx + ω02 2

L 0

u2 (x, t)dx

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45

Fig. 5.19 FEP of the Hamiltonian system (5.16) based on the truncated system (5.19, 5.20) with k = 1, 2, 3: (c) details of regions I and II; numbers () correspond to the periodic orbits depicted in Figures 5.20 and 5.21, and letters (•) to the numerical simulations of damped transitions.

1 1 + εv 2 (t) + C[v(t) − u(L, t)]4 2 4

(5.21)

Considering the FEP of Figure 5.19, we discern the existence of two lowfrequency asymptotes. These correspond to the two leading modes of the linear uncoupled rod, ωn =

ω02 +

(2n − 1)2 π 2 , 4L2

n = 1, 2

(5.22)

where for the chosen parameters these are given by ω1 = 1.8621 rad/s and ω2 = 4.8173 rad/s. In addition, there exist two high-frequency asymptotes at frequencies ωˆ 1 and ωˆ 2 . Noting that at high energies and finite frequencies the essentially nonlinear stiffness of system (5.16) behaves as a massless rigid link, the high-frequency asymptotes are computed as the eigenfrequencies of the following alternative limiting linear system: ∂ 2 u(x, t) ∂ 2 u(x, t) + ω02 u(x, t) − = 0, 2 ∂t ∂x 2

0≤x≤L

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

u(0, t) = 0,

∂ 2 u(L, t) ∂u(L, t) = −ε ∂x ∂t 2

(5.23)

i.e., of the dispersive rod with a mass ε attached to its right end. The eigenfrequencies of this limiting system are computed by solving the following transcendental equation, , , ωˆ 2 − ω02 2 2 (5.24) tan L ωˆ − ω0 = ε ωˆ 2 which for the chosen parameters are given by ωˆ 1 = 1.7728 rad/s and ωˆ 2 = 4.5916 rad/s. Since the principal aim for constructing the FEP is to interpret the (weakly) damped dynamics of system (5.15) in terms of the topological structure of the periodic solutions of the underlying Hamiltonian system, it is necessary to discuss certain qualitative features of this plot. A first observation concerns the complexity of the FEP. For comparison purposes, we note that for the system where the attachment is connected by means of a linear stiffness, the FEP consists of straight horizontal lines corresponding to the countable infinity of linear vibration modes whose mode shapes and frequencies do not depend on the energy of the vibration. It follows that all curves in the FEP deviating from the horizontal direction represent essentially nonlinear periodic motions of the nonlinear system, having no counterparts in linear theory and localizing mainly to the nonlinear attachment. By the same token, branches of solutions that are nearly horizontal represent weakly nonlinear motions, as they can be regarded as mere perturbations of linearized vibration modes; these solutions are mainly confined to the elastic rod. That the addition of a single, lightweight essentially nonlinear NES introduces such drastic, strongly nonlinear effects in the FEP (occurring over wide frequency and energy ranges), proves that the addition of the local NES induces global effects on the dynamics of the system. This is caused by the fact that, due to its essential nonlinearity, the NES is capable of interacting with any of the modes of the rod over arbitrary frequency ranges. Proceeding to discuss the specific details of the FEP of Figure 5.19, there exist two types of branches of periodic motions, namely backbone (global) branches and subharmonic tongues (local) tongues. These are similar to the corresponding branches and tongues of the FEP of the two-DOF discussed in Section 3.3.1.2 (see Figure 3.20). Backbone branches consist of nearly monochromatic time-periodic solutions possessing dominant harmonic components and higher harmonics at integer multiples of the dominant harmonics. These branches are defined over extended frequency and energy ranges, and typically are composed of strongly nonlinear periodic solutions that are mainly localized to the nonlinear attachment. Exceptions are in neighborhoods of the linearized eigenfrequencies of the rod, ω1 , ω2 , . . . , where the spatial distributions of the periodic motions resemble those of the corresponding rod mode shapes and are localized to the rod; and in neighborhoods of the highenergy asymptotes ωˆ 1 , ωˆ 2 , . . . , where the relative displacements between the nonlinear attachment and the rod end tend to zero (i.e., the nonlinear coupling stiffness is nearly unstretched) and, as a result, the nonlinear effects are nearly negligible. At

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Fig. 5.20 Periodic orbits on backbone branches of the FEP: (1) = 0.6 rad/s, (2) = 1.3 rad/s, (3) = 1.75 rad/s, (4) = 2.3 rad/s, (5) = 4.5 rad/s, (6) = 1.87 rad/s, (7) = 4.83 rad/s; — rod end, - - - NES.

these energy ranges the corresponding segments of the backbone branch in the FEP appear as nearly horizontal segments. In the plots depicted in Figure 5.20 some representative periodic motions on the backbone branch are depicted. These solutions are regarded as analytically predicted time-periodic solutions of the system, since their initial conditions are determined by solving the truncated system (5.20) with the index assuming the values

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

k = 1, 2, 3. The accuracy of these solutions is confirmed by comparing them to direct numerical simulations of the equations of motion (Tsakirtzis, 2006; Panagopoulos et al., 2007). An additional set of periodic solutions of the FEP of Figure 5.19 are realized on subharmonic tongues (local branches); these are multi-frequency time-periodic motions, with frequencies being approximately equal to rational multiples of the eigenfrequencies ωn of the uncoupled rod. Each tongue is defined over a finite energy range and is composed of two distinct branches of subharmonic solutions. At a critical energy level the two branches coalesce in a bifurcation that signifies the end of that particular tongue and the elimination of the corresponding subharmonic motions at higher energy values. It can be proven that there exists a countable infinity of tongues emanating from the backbone branches at frequencies being in rational relation with respect to the eigenfrequencies of the uncoupled linear rod, ωn . On a given subharmonic tongue the responses at any point of the rod and of the attachment resemble those of two linear oscillators, albeit possessing different (but rationally related) eigenfrequencies. Hence, the interesting (and paradoxical) observation can be drawn, that on the essentially nonlinear subharmonic tongues (they are characterized as such since they exist due to the strong stiffness nonlinearity of the system) the rod-attachment system behaves nearly as an equivalent two-frequency linear system. This observation, which is similar to that concerning the subharmonic orbits of the two-DOF system studied in Section 3.3.1.2 [also see Kerschen et al. (2005)] provides a hint on the rich and complex dynamics of the system considered herein. In the FEP of Figure 5.19, we depict only a subset of leading subharmonic tongues. For example, the tongue depicted in Region I (see Figure 5.20b) is in the vicinity of ω4 /3; it follows that subharmonic motions on this tongue correspond to responses where the nonlinear attachment possesses a dominant harmonic with frequency ω4 /3 (and a minor harmonic at ω4 ), whereas the response of the rod end possesses a dominant harmonic at frequency ω4 (and a minor harmonic at ω4 /3). (n) In the following exposition a tongue labeled as Tp/q will denote the branch of subharmonic motions where the frequency of the dominant harmonic component of the nonlinear attachment is nearly equal to (p/q)ωn , whereas that of the rod end equals ωn . It follows that the relative displacement [v(t) − u(L, t)] during a sub(n) harmonic motion on tongue Tp/q possesses two main harmonics at frequencies ωn and (p/q)ωn . Using this notation, the subharmonic tongue depicted in Figure 5.20b (4) (4) is labeled as T1/3 . In Figure 5.21 three subharmonic orbits on the tongue T1/3 of the FEP are depicted in the neighborhood of frequency ω4 /3. We mention that although all these subharmonic orbits coexist, i.e., they possess the same fundamental frequency , they correspond to qualitatively different dynamics. We now focus on the damped dynamics of system (5.15). This study was performed through direct simulations of the governing equations of motion and post-processing of the computed time series. The transient responses of the rodattachment system with viscous dissipation are computed by a finite element code developed for Matlab. This code is different from the FE code discussed in Section 5.2.1 and will be employed also in Section 5.3 to model a rod or infinite length;

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(4)

49

Fig. 5.21 Periodic orbits on the subharmonic tongue T1/3 (Region I, Figure 5.19b): (8) = 3.72 rad/s, log(Energy) = 0.645; (9) = 3.72 rad/s, log(Energy) = 1.015; (10) = 3.72 rad/s, log(Energy) = 0.48; — rod end, - - - NES.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

comparison of the results given by this code with the results obtained by the FE code described in Section 5.2.1 was also performed in order to ensure convergence and accuracy of the results. For these computations the rod was discretized into 200 finite elements, which ensured a five-digit convergence regarding the computation of the three leading modes of the rod. This was deemed to be sufficient for the computations presented herein, as we will be interested only in a frequency range encompassing the first three linearized modes of the rod. Regarding the numerical integrations of the equations of motion (5.15), the Newmark algorithm was utilized with parameters chosen to ensure unconditional stability of the numerical algorithm. The sampling frequency was such that the eigenfrequencies of the leading three modes of the rod were less than 6% of the sampling frequency. Regarding viscous dissipation, proportional damping in the rod was assumed, by expressing the damping matrix in the form D = a1 M + a2 K, where M and K are the mass and stiffness matrices of the rod. The parameters used for the FE computations were chosen as ε = 0.05, C = 1.0, L = 1.0, ω0 = 1.0, λ2 = 0.02, α1 = 0.001, and α2 = 0.0, and the damped responses were initiated with different sets of initial conditions of the rod and the NES. The first numerical simulation is performed with initial conditions corresponding to point A on the main backbone branch of the FEP at frequency = 0.6 rad/s (see Figure 5.19a). These initial conditions for the rod and the nonlinear attachment are approximately computed as follows: v(0) ≈ {Vc,1 cos(ωt) + Vc,2 cos(3ωt) + Vc,3 cos(5ωt)}t =0 ⇒ v(0) ≈ −0.1650 , , ω2 − ω02 cos(ωt) + Cˆ 2 sin x 9ω2 − ω02 cos(3ωt) u(x, 0) ≈ Cˆ 1 sin . , + Cˆ 3 sin x 25ω2 − ω02 cos(5ωt) u(0, 0) ≈ −0.0052

t =0

⇒ (5.25)

with Vc1 = −0.1597, Vc2 = −0.054, Vc3 = 0.0001, and Cˆ 1 = 0.0027, Cˆ 2 = −0.0079 and Cˆ 3 = −0.00002. In the undamped system these initial conditions correspond to a periodic motion that is predominantly localized to the nonlinear attachment (the NES). In Figure 5.22 we depict the damped responses of the NES and the point of its connection to the rod, together with the wavelet transform spectrum of the damped relative motion [v(t) − u(L, t)] superimposed to the FEP of the Hamiltonian system. We observe that as energy decreases due to damping dissipation the motion appears to trace closely the lower, in-phase backbone branch of the corresponding Hamiltonian system. This observation confirms that for sufficiently weak damping the damped response lies close to the dynamics of the underlying Hamiltonian system (in fact, as discussed in Chapters 3 and 4 the damped motion takes place on the damped invariant NNM manifold lying on the neighborhood of the corresponding NNM manifold of the Hamiltonian system). The nonlinear dynamic interaction between the rod and the NES during this damped transition is now examined in more detail.

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Fig. 5.22 Damped response initiated at point A of the FEP of Figure 5.19a: (a) transient responses v(t) and u(L, t); (b) WT spectrum of the relative response [v(t) − u(L, t)] superimposed to the FEP of the Hamiltonian system.

In this particular application the damped motion is initiated close to the subhar(1) monic tongue T1/3 (see Figure 5.22b), so a weak 1:3 TRC occurs at least in the beginning of the motion; indeed, in that early response regime both the rod end and

52

5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.23 ETM between the rod and the NES for damped responses depicted in Figure 5.22.

the NES execute in-phase oscillations, with the rod oscillating nearly three times faster than the NES. Since there is a countable infinity of low subharmonic orbits emanating from the lower in-phase backbone branch of the FEP with decreasing energy, the damped dynamics passes through a sequence of TRCs of increasing order; hence, as energy decreases the rod oscillates increasingly faster compared to the NES. Moreover, since the lower in-phase backbone branch of the FEP does not undergo any major topological changes with decreasing energy – apart from the minor topological changes related to the bifurcations that generate the countable infinities of low-energy subharmonic tongues – the damped transition is also smooth and does not undergo major sudden transitions in frequency. The transient energy transaction measure (ETM) [defined by relation (5.14)] between the rod and the NES is depicted in Figure 5.23. It indicates the presence of (weak) nonlinear beat phenomena between the rod and the NES, with continuous energy being exchanged between them. We note that in the damped transition of Figure 5.22 the motion is predominantly localized to the nonlinear attachment throughout the motion, so that only weak energy exchanges occur between the two subsystems. As shown in Section 5.2.2 [but also in Georgiades et al. (2007)] for different sets of initial conditions stronger energy exchanges may occur, resulting in strong TET from the rod to the NES. There we showed that TET may be realized through either nonlinear beats, one-way energy transfers from the rod to the

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53

Fig. 5.24 Damped response initiated at point B of the FEP of Figure 5.19a. Transient responses v(t) and u(L, t).

attachment (evidenced by a series of positive-only spikes in the ETM plot), or a combination of both these mechanisms.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.25 Damped response initiated at point B of the FEP of Figure 5.19a. WT spectrum of the relative response [v(t) − u(L, t)] superimposed to the FEP of the Hamiltonian system.

Summarizing, with decreasing energy the damped transition traces the lower inphase backbone branch of the FEP. Since the damped orbit is initiated in the neigh(1) , the rod and the NES are initially locked borhood of the subharmonic tongue T1/3 into 1:3 TRC (with the rod oscillating nearly three times as fast as the NES, albeit with much smaller amplitude). As energy decreases due to damping dissipation the oscillation of the rod becomes increasingly faster than that of the NES (with ever decreasing amplitude), as the damped dynamics visits neighborhoods of higher-order (1) tongues T(1/n) , n > 3 lying along the lower in-phase backbone branch (see Figure 5.22b). As a result, the dynamics engages in increasingly higher-order in-phase TRCs, which, however, are realized at increasingly smaller time intervals. Since the lower backbone branch of the FEP does not undergo any major topological changes, no major (abrupt) transitions occur in the damped dynamics for this particular simulation. An interesting series of nonlinear transitions is observed in the second numerical simulation of the damped dynamics depicted in Figures 5.24–5.26, and correspond(4) ing to initial condition of the system at point B on the subharmonic tongue T1/5 of the FEP. That is, the motion is initiated on an undamped subharmonic orbit with dominant frequencies = 2.214 rad/s ≈ ω4 /5 and ω4 , see Figure 5.19a. Transitions in the damped dynamics are clearly evidenced by the irregular amplitude modulations of the time series (especially the one corresponding to the nonlinear attachment), or equivalently, by their multi- frequency contents. A better representation of the transitions in the damped dynamics is achieved by superimposing the

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

55

Fig. 5.26 ETM between the rod and the NES for the damped responses depicted in Figures 5.24 and 5.25.

WT spectrum of the relative motion [v(t)−u(L, t)] to the FEP of the undamped system (see Figure 5.25). The following transitions are then discerned with decreasing energy of the motion: (4)

I. Initial high energy transition from the subharmonic tongue T1/5 (where the mo(1) ; two dominant harmonics appear at frequencies tion is initiated) to tongue T2/3 ω4 and ω4 /5 during this early stage of the response; (1) II. Subharmonic transient resonance capture (TRC) on T2/3 with the nonlinear attachment possessing a nearly constant dominant harmonic component at frequency 2ω1 /3 and a minor harmonic at frequency ω1 ; (1) (1) (1) III. Transition from tongue T2/3 to tongue T1/3 and subharmonic TRC on T1/3 ; this secondary TRC is signified by the strong harmonic at frequency ω1 /3 and the weaker harmonic at frequency ω1 ; IV. Final low-energy transition to the linearized (low-amplitude) state, where the response of the nonlinear attachment is nearly zero and the dynamics is dominated by the response of the linear rod; the motion ends up being confined predominantly to the linear rod as its response decays to zero.

These complex transitions are caused by the fact that the essentially nonlinear attachment lacks a preferential frequency of oscillation (since it possesses zero linearized stiffness), which enables it to engage in fundamental or subharmonic TRCs with different modes of the linear rod over broad frequency ranges. Equivalently, the essential stiffness nonlinearity of the attachment generates a series of resonance capture cascades (RCCs) between the NES and the rod. As discussed in Section 3.5 such RCCs may lead to strong, multi-frequency TET.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.27 EMD analysis of the NES response at Stage 1 of the damped transition of Figure 5.25: (a) instantaneous frequency of the 1st IMF superimposed to the WT spectrum of the transient response; (b) reconstruction of the transient response using the 1st IMF.

The WT spectra of the relative responses superimposed to the FEP provide a clear picture of the TRCs occurring in the damped transitions (see Figures 5.22b and 5.25). Even in cases were complex, multi-scale transitions take place, the depiction of WT spectra against appropriate FEPs provides a clear explanation and interpretation of the damped transitions. Hence, the methodology followed in this

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Fig. 5.28 EMD analysis of the rod end response during Stage I of the damped transition of Figure 5.25: (a, b) instantaneous frequencies of the 1st and 2nd IMFs superimposed to the WT spectrum of the corresponding transient response.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.28 EMD analysis of the rod end response during Stage I of the damped transition of Figure 5.25: (c) reconstruction of the transient response using the 1st and 2nd IMF.

work can be extended to other applications were post-processing analysis of complex multi-frequency signals is performed. We now proceed to the multi-scale analysis of the damped transition initiated on (4) the subharmonic tongue T1/5 (depicted in Figure 5.24) by applying EMD. Each of the four transitions I–IV identified in Figure 5.25 will be examined separately, with the aim to model the dynamics during each transition and determine the characteristic time scales where the nonlinear resonance interactions between the rod and the nonlinear attachment (or NES) take place. (4) Starting with the initial high-energy transition from tongue T1/5 (where the mo(1) (Stage I, 0 < t < 160 s), EMD analysis indicates tion is initiated) to tongue T2/3 that the NES response is dominated by its 1st IMF (see Figure 5.27), whereas, the rod end response is approximately modeled by two dominant IMFs, namely, its 1st and 2nd IMFs (see Figure 5.28). Proceeding to the analysis of the instantaneous frequencies of the dominant IMFs of the NES and rod end responses, we notice that these coincide with dominant harmonic components of the corresponding transient responses; hence, one concludes that the nonlinear dynamics of the rod-NES transient interaction during Stage I of the damped transition is low-dimensional, with the dynamics of the NES resembling the response of a single-DOF oscillator with frequency being approximately equal to ω4 /5 ≈ 2.214 rad/s, and the dynamics of the rod end resembling the superposition of two single-DOF oscillators with frequencies ω4 and ω4 /5, respectively.

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Fig. 5.29 EMD analysis of the NES response in Stage II of the damped transition of Figure 5.25: (a, b) instantaneous frequences on the 1st and 2nd IMFS superimposed to the WT spectrum of the corresponding transient response.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.29 EMD analysis of the NES response in Stage II of the damped transition of Figure 5.25: (c) reconstruction of the transient response using the 1st and 2nd IMF.

Hence, the EMD analysis indicates there is one dominant time scale in the transient dynamics of the NES and two dominant time scales in the dynamics of the rod end. These results are confirmed by the time series reconstructions depicted in Figures 5.27b and 5.28c, which prove the low-dimensionality of the NES-rod end nonlinear interaction during this initial (and highly energetic) stage of the motion. Moreover, during Stage I it is observed that the 1st IMF of the NES response is in near 1:5 resonance with the 1st IMF of the rod end response, and in near 1:1 resonance with 2nd IMF of the rod end response. These IMF TRCs are responsible for the energy exchanges that occur between the rod and the NES during Stage I of the transition. Proceeding now to the more complicated damped transition occurring during (1) Stage II (160 < t < 420 s – where the dynamics is captured on tongue T2/3 ), the NES response appears to be dominated (and modeled) by its two leading IMFs (see Figure 5.29), which indicates that in this case the NES responds like a two-DOF oscillator. Considering the rod end response one establishes the existence of three dominant IMFs (the leading three IMFs depicted in Figure 5.30), with the instantaneous frequency of the 1st IMF executing modulated oscillations, and that of the 2nd IMF suffering sudden transitions (jumps) with increasing time. This type of complex behavior of the IMFs is distinctly different from what was observed in Stage I and is characteristic of intrawaves in the time series. The existence of intrawaves in oscillatory modes (IMFs) is one of the nonlinear effects detected in typical nonlinear systems, such as the forced Duffing oscillator, the Lorenz system and the Rossler

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chaotic attractor, with the WT spectra not being able to detect them; as mentioned in Huang et al. (1998), ‘. . . in fact such an instantaneous frequency presentation actually reveals more details of the system: it reveals the variation of the frequency within one period, a view never seen before . . . ’. The time series reconstructions depicted in Figures 5.29b and 5.30b confirm that the superposition of the dominant IMFs accurately models the damped transition during this Stage of the motion. Note that the higher dimensionality of the NES and rod end responses observed in this case, signifies that the complexity of the dynamics increases compared to Stage I. Considering the resonance interactions between the IMFs of the NES and the rod end responses during Stage II of the damped response, the 1st IMF of the NES is in near 2:3 internal resonance with the 2nd IMF of the rod end in the time interval 160 < t < 250 s, and with the 3rd IMF of the rod end in the interval 250 < t < 350 s. Moreover, there appears to be 1:1 internal resonance between the 1st IMF of the NES and the 3rd IMF of the rod end in the time interval 160 < t < 250 s. (1) (1) to tongue T1/3 is signified by the The transition of the dynamics from tongue T2/3 decrease of the instantaneous frequency of the 1st IMF of the NES in the interval t < 350 s. The 1st IMF of the rod end possesses an oscillatory instantaneous frequency close to ω4 in the interval 160 < t < 300 s, and between ω3 and ω4 in the interval 300 < t < 420 s (due to intrawaves, as discussed above). This EMD result agrees qualitatively with the late excitation of the 3rd linear mode of the rod, as indicated by the WT spectrum of the time series. The 2nd IMF of the rod end possesses an instantaneous frequency that is approximately equal to ω1 for 160 < t < 250 s, and is oscillatory about ω2 for 250 < t < 350 s; note that the WT spectrum of the time series of the rod end response does not indicate any excitation of the second mode of the rod during Stage II (which demonstrates the clear advantage of using EMD when analyzing complex signals, compared to the WT). In Figures 5.31–5.33 the results of the EMD analysis of the damped response in Stages III and IV (t > 420 s) are depicted. In this case the NES response possesses three dominant IMFs, whereas the response of the rod end possesses four. (1) Resonance capture of the dynamics on tongue T1/3 is signified by the fact that the instantaneous frequency of the 3rd IMF of the NES response (which is dominant) is approximately equal to ω1 /3 in the time interval 420 < t < 820 s (with the exception of a ‘high frequency burst’ in the neighborhood of t = 500 s, which, however is of no practical significance as it corresponds to small amplitude of the IMF and is (1) noise dominated); whereas, the transition from T1/3 to the linearized regime is signified by the decrease of the instantaneous frequency of the same IMF for t > 820 s. It is interesting to note that in the time interval where the ‘high frequency burst’ of the 3rd IMF of the NES occurs, the 4th IMF of the NES ‘locks’ to the value ω1 /3, and, hence, through superposition provides the necessary correction in the reconstruction of the overall time series in that time interval. Moreover, by studying the waveform of the 5th IMF of the NES one notes that this IMF dominates the tran(1) to the linearized regime occurring for t > 800 s. Considering the sition from T1/3 IMFs of the rod end response, one notes intrawaves centered at the linearized eigenfrequencies of the rod, ω1 , . . . , ω4 , similarly to those observed in the EMD analysis of the response in Stage II.

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Fig. 5.30 EMD analysis of the rod end response in Stage II of the damped transition of Figure 5.25: (a, b) instantaneous frequencies of the 1st and 2nd IMFs superimposed to the WT spectrum of the corresponding transient response.

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Fig. 5.30 EMD analysis of the rod end response in Stage II of the damped transition of Figure 5.25: (c) instantaneous frequencies of the 3rd IMFs superimposed to the WT spectrum of the corresponding transient response; (d) reconstruction of the transient response using the 1st, 2nd and 3rd IMF.

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Fig. 5.31 EMD analysis of the NES response in Stages III–IV of the damped transition of Figure 5.25: instantaneous frequencies (superimposed on the wavelet transform of the response), and time series of the (dominant) 3rd, 4th and 5th IMFs.

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Fig. 5.32 EMD analysis of the NES response in Stages III–IV of the damped transition of Figure 5.25. Reconstruction of the response by superposing the three dominant IMFs.

These results demonstrate the usefulness of EMD as a computational tool for post-processing transient nonlinear responses that involve multiple resonance captures and escapes. In fact, the previous results indicate that the EMD can capture delicate features of the dynamics (such as intrawave effects or participation of multiple modes in different time windows of the response) that are not evident in the corresponding WT spectra. Nevertheless, the presented computational analysis shows that the combination of EMD and WT forms a powerful computational methodology for post-processing and modeling of complex nonlinear transient responses of practical structural systems. In summary, damped nonlinear transitions of system (5.15) can be analyzed by a combination of numerical WT and EMD. These post-processing algorithms are capable of analyzing even complex nonlinear transitions, by providing the dominant frequency components (or equivalently the time scales) were the nonlinear phenomena take place. In addition, the EMD can detect delicate features of the dynamics, such as intrawaves – i.e., IMFs with modulated instantaneous frequencies, which the WTs cannot accurately sense. More importantly, the superposition of the dominant IMFs of the signal accurately reconstructs the signal, and, hence, these dominant IMFs may be interpreted in terms of outputs of intrinsic modal oscillators. It follows, that the determination of the dominant IMFs of a complex nonlinear signal, paves the way for modeling this signal as a superposition of the responses of intrinsic modal oscillators, for determining the dimensionality of the governing dynamics, and for ultimately performing multi-scaled system identification of the underlying dynamics of the system.

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Fig. 5.33 EMD analysis of the rod end response in Stages III–IV of the damped transition of Figure 5.25. Instantaneous frequencies (superimposed on the WT spectrum of the response) of the (dominant) 1st, 2nd, 3rd and 4th IMF.

5.3 Rod of Semi-Infinite Length with SDOF NES We now extend our study of strongly nonlinear dynamic interactions and TET in elastic continua with strongly nonlinear end attachments, by analyzing the damped dynamics of a semi-infinite linear dispersive rod possessing a local essentially nonlinear end attachment. We study resonant interactions of the attachment with incident traveling waves from the rod, as well as 1:1 TRCs of the nonlinear attachment with an in-phase mode at the bounding frequency between the PZ and AZ of the rod. This study can be considered as extension of the study of finite rod-NES dynamics carried out in Section 5.2, and of the analysis of semi-infinite linear chain-NES dynamic interaction studied in Section 3.5.2. As pointed out by Goodman et al. (2002), the interaction of incident traveling solitary waves with a local defect can lead to various dynamic phenomena, such as, speed up or slow down of the traveling wave; scattering of the wave to multiple independent wavepackets; or even trapping of the wave at the point of defect in the form of a localized mode (standing wave). Goodman et al. (2002) investigated in detail the complicated dynamics resulting from soliton – local impurity interaction for the case of the sine – Gordon equation. If the local nonlinear attachment considered in

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this section is regarded as a type of ‘defect’, the dynamical phenomena considered are similar, but in the context of linear wave-guide/local nonlinear defect interaction. Additional studies on soliton-defect dynamic interactions were performed in Cao and Malomed (1995), Zei et al. (1992), Goodman and Haberman (2004) and in references therein. In other related works, localized modes in a multi-coupled periodic system of infinite extent with a single nonlinear disorder were analyzed by Cai et al. (2000); symmetric, anti-symmetric and asymmetric localized modes were computed, and their stability was analyzed in that work. Komech (1995) studied the dynamics of an infinite string with an attached nonlinear oscillator and showed that each finiteenergy solution of the integrated system tends to a stationary solution as t → ±∞. Trapped (localized) modes in stop bands (AZs) of two-dimensional waveguides with obstacles were discussed in Linton et al. (2002) and in a series of works referenced therein. In a more applied work (Qu, 2002), order reduction techniques for engineering systems with local nonlinearities were discussed. In related works, Kotousov (1996) studied wave propagation in elastic continua with local nonlinearities; ElKhatib et al. (2005) studied suppression of bending waves in a beam by means of a tuned vibration absorber; and Komech and Komech (2006) studied long-term asymptotics of finite-energy solutions of a Klein–Gordon equation with a local oscillator attachment.

5.3.1 Reduction to Integro-differential Form We consider a general linear undamped elastic waveguide (designated as the primary system), coupled to an essentially nonlinear attachment (the NES) by means of a weak linear stiffness. The local attachment is grounded, and possesses unit mass, viscous damping and nonlinearizable stiffness nonlinearity of the third degree. Denoting by v(t) the displacement of the NES, and by u(x O , t) the displacement of the primary system at the point of attachment O in the direction of v(t), we obtain the following governing differential equation for motion of the attachment (see Figure 5.34): (5.26) v(t) ¨ + λv(t) ˙ + εv(t) + Cv3 (t) = εu(x O , t) In (5.26) the small parameter 0 < ε 1 scales the weak coupling, λ denotes the viscous damping coefficient, and C the coefficient of the stiffness nonlinearity; the spatial coordinate x parametrizes the undeformed configuration of the primary system in its configuration space. Assuming that the primary system is initially at rest and that an external force F (x A , t) is applied at point A at t = 0 (see Figure 5.34), we express its response at the point of attachment O, u(x O , t), in terms of its corresponding Green’s functions gOO and gOA : t t F (x A , τ )gOA (t − τ )dτ − ε[u(x O , τ ) − v(τ )]gOO (t − τ )dτ u(x O , t) = −∞

−∞

(5.27)

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.34 Elastic wave guide with a weakly coupled grounded NES.

The Green’s function gOO denotes the displacement at point O of the primary system in the direction of v(t) due to a unit impulse applied at the same point and the same direction; whereas gOA denotes the displacement at point O of the primary system in the direction of v(t), due to a unit impulse applied at point A in the direction of the external force. Substituting (5.27) into (5.26) and iterating repeatedly the previous procedure in order to express u(x O , t) on the right-hand side in terms of the afore-mentioned Green’s functions and the NES displacement v(t), we obtain the following general integro-differential equation governing the oscillation of the nonlinear attachment v(t) ¨ + λv(t) ˙ + εv(t) + Cv3 (t) = ε[F (x A , t) ∗ gOA (t)] + ε2 [−F (x A , t) ∗ gOA (t) ∗ gOO (t) + v(t) ∗ gOO (t)] + · · · + εn (−1)n−1 F (x A , t) ∗ gOA (t) ∗ gOO (t) ∗ · · · ∗ gOO (t) ' () * + (−1)n v(t) ∗ gOO (t) ∗ · · · ∗ gOO (t) ' () *

(n−1) terms

+ ···

(5.28)

(n−1) terms

where (∗) denotes the convolution operator. What makes possible the reduction of the governing equations of motion to integro-differential form is the assumption of linearity of the primary system (i.e., the elastic waveguide). Terms on the right-hand side of (5.28) containing only the external excitation F (x A , t) represent non-hom*ogeneous ‘forcing’ terms of the above dynamical system, and govern, in

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essence, the dynamics of the attachment predominantly influenced by the external excitation. Similarly, terms on the right-hand side that contain integrals in terms of v(t) represent the dynamics of the attachment predominantly influenced by its complex nonlinear dynamic interaction with the primary system, including scattering of waves from the NES back to the waveguide and targeted energy transfer effects. We conclude that the representation (5.28) leads to a natural partition of the dynamics of the attachment. In the remainder of this section we employ the general expression (5.28) to study the nonlinear dynamic interaction of a dispersive linear rod of infinite spatial extent resting on a continuous elastic foundation (the primary system), with an essentially nonlinear grounded attachment that is weakly coupled to its right boundary. This system can be regarded as the semi-infinite extension as L → ∞ of the rod-NES system depicted in Figure 5.6 (but with a grounded instead of an ungrounded NES). The analysis follows closely (Vakakis et al., 2004). Depending on the specific initial conditions and the external forces considered, we distinguish between two systems, and label them as Systems I and II. System I is forced by an impulsive excitation applied to a single point of the semi-infinite rod, with all initial conditions of the primary system and the local attachment being assumed as zero. System II is unforced with the excitation being provided by an initial displacement of the rod in the form of a finite unit step, with all other initial conditions being set equal to zero. After providing detailed mathematical descriptions of the two systems, a study of the different regimes of the rod-attachment interaction is carried out using computational and analytical tools.

5.3.1.1 System I: Impulsive Excitation Assuming for the moment that the primary system is an undamped linearly elastic rod of finite length L resting on a continuous linear elastic foundation, the dynamics become one-dimensional with governing equations of motion given by −

∂ 2 u(x, t) ∂ 2 u(x, t) 2 + ω u(x, t) + = F δ(x + e)δ(t), 0 ∂x 2 ∂t 2

−L ≤ x ≤ 0

v(t) ¨ + λv(t) ˙ + Cv3 (t) + ε[v(t) − u(0, t)] = 0 ∂u(0, t) + ε[v(t) − u(0, t)] = 0, ∂x u(x, 0) =

u(−L, t) = 0

∂u(x, 0) = v(0) = v(0) ˙ =0 ∂t

(5.29)

The point of attachment is situated at x = 0, normalized material properties for the rod are used, and the normalized stiffness of the elastic foundation is denoted by ω02 ; in addition, all geometric and material properties of the rod are assumed to be uniform. It is assumed that an impulse of magnitude F is applied at x = −e > −L at t = 0.

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Fig. 5.35 Rod on elastic foundation with weakly coupled grounded NES.

Taking the limit L = ∞ one obtains a semi-infinite, impulsively loaded dispersive rod. The rod of infinite length possesses a PZ corresponding to ω > ω0 , where traveling wave solutions exist, and an AZ for 0 < ω < ω0 where localized standing waves (near-field) solutions are realized. At the bounding frequency ωb = ω0 (separating the AZ and the PZ) the rod oscillates in an ‘in-phase’ normal mode with all points executing an identical synchronous time-periodic oscillation of constant amplitude. The Green’s function of the rod of infinite length describing the response of the rod at position x and time t due to a unit impulse applied at point x¯ and time instant t¯ is given by ¯ (5.30) ¯ t − t¯) = J0 ω0 (t − t¯)2 − (x − x) ¯ 2 H (t − t¯ − x + x) g1 (x − x, where J0 (·) denotes the Bessel function of zero-th order and first kind, and H (·) Heaviside’s function. Then, in terms of the general expression (5.28), the two Green’s functions gOO and gOA are expressed as follows: gOO (t) = g1 (0, t) = J0 (ω0 t)H (t) gOA (t) = g1 (0 + e, t) = J0 ω0 t 2 − (0 + e)2 H (t − 0 − e)

(5.31)

Substituting (5.31) into (5.28), we obtain the following governing integrodifferential equation for the nonlinear attachment for System I (Vakakis et al., 2004): t v(t) ¨ + λv(t) ˙ + Cv3 (t) + εv(t) − ε2 v(τ )J0 [ω0 (t − τ )]dτ

= εF J0 ω0 t 2 − e2 H (t − e)

t

−ε F 2

J0 ω2 τ 2 − e2 H (τ − e)J0 [ω0 (t − τ )]dτ + O(ε3 )

≡ εF1 (t) + ε2 F2 (t) + O(ε3 )

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v(0) = v(0) ˙ = 0,

L→∞

(System I)

71

(5.32)

Hence, System I is an impulsively loaded semi-infinite dispersive rod with an essentially nonlinear end attachment. The two expressions on the right-hand side are pure non-hom*ogeneous terms and represent the leading-order ‘direct forcing’ of the nonlinear attachment due to the impulsive excitation. The integral term on the lefthand side models the leading-order interaction of the NES with the dispersive rod, including energy radiation from the NES back to the rod and energy entrapment by the NES in the form of localized vibrations. These effects will be studied in more detail in the following exposition. Note, however, that the system (5.32) provides only an approximation to the dynamics since it omits O(ε 3 ) and higher-order terms; it follows, that the derived results can only be asymptotically valid in the limit of weak coupling as ε → 0. For the case of finite rod the above dynamical system must be modified to account for wave effects due to reflections at the boundaries of the rod. The modifications of (5.32) due to finiteness of the dispersive rod can be analytically studied by applying Laplace transform with respect to the temporal variable directly to the system of equations (5.29) for 0 > x > −L > −∞. To this end, we Laplace-transform the corresponding equations of motion (5.29) and solve the first equation to obtain Y (x, s) − (s 2 + ω02 )Y (x, s) = F δ(x + e) ⇒ Y (x, s) = A cosh[(s 2 + ω02 )1/2 (L + x)] + B sinh[(s 2 + ω02 )1/2 (L + x)] −L + F δ(ξ + e)h(x − ξ, s)dξ (5.33) 0

where Y (x, s) = L[u(x, t)] is the Laplace transform of u(x, t), s the Laplace vari,

able, and h(x, s) = α−1 sinh αx, α = α(9s) = s 2 + ω02 . The unknowns A and B in (5.33) are computed by imposing the transformed boundary conditions [the third of relations (5.29)]. Evaluating the resulting expression at x = 0 we obtain the following expression for the Laplace-transformed displacement of the point of connection of the rod with the nonlinear attachment, Y (0, x) = (s) cosh αL + [α coth αL + ε]−1 {εV (s) − ε(s) coth αL − (s)α sinh αL} (5.34) where V (s) is the Laplace transform of v(t), and (s) is computed as (s) =

0 −L

F δ(x + e)h(−L − x, s)dx = h(−L + e, s)

For ε 1 we expand (5.34) in ascending powers of the small parameter to obtain the following final approximate expression for Y (0, s), Y (0, s) = F α −1 (− tanh αL cosh αe + sinh αe)[1 − εα−1 tanh αL]

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+ εα −1 V (s) tanh αL + O(ε2 ) (5.35) , with a = a(s) = s 2 + ω02 , and x = −e being the point of application of the impulsive load. The Laplace inversion of relation (5.35) can be expressed in terms of left- and right-going waves propagating in the finite rod (Caughey, 1987). Since this approach can add to our physical insight of the early-time nonlinear dynamic interactions occurring in System I we proceed to discuss it briefly. To this end, we express the hyperbolic functions in (5.35) in terms of exponentials as follows: tanh αL = 1 − 2e−2αL + 2e−4αL − 2e−6αL + · · · sinh αe =

eαe − e−αe , 2

cosh αe =

eαe + e−αe 2

Substituting these expressions into (5.35) we obtain the following approximate expression approximating the early-time dynamics of the point of connection of the rod to the NES: Y (0, s) = F α −1 [−e−αe + eα(−2L+e) + eα(−2L−e) + · · ·] − εF α −2 [−e−αe + eα(−2L+e) + eα(−2L−e) + · · ·] + εα −1 V (s)[1 − 2e−2αL + · · ·] + O(ε2 )

(5.36)

The exponentials in (5.36) represent arrivals of individual longitudinal waves at the point of attachment O of the rod, caused either due to the impulsive excitation or due to reflections from the boundaries of the finite rod. The advantage of considering the Laplace-transformed response in the form (5.36) instead of (5.35) is that the former can be directly inverted to yield an approximation of the early-time transient dynamics of the system. Indeed, applying inverse Laplace transform to (5.36) we obtain the following early-time approximation of the dynamics of the connecting point, u(0, t) = L−1 [Y (0, s)], in the form of left- and right-going traveling waves: u(0, t) = . - − F J0 ω0 t 2 − e2 H (t − e) − J0 ω0 t 2 − (2L − e)2 H (t − (2L − e)) + · · ·

t

+ εF

J0 [ω0 (t − τ )]J0 ω0 τ 2 − e2 H (τ − e)dτ

t

−

J0 [ω0 (t − τ )]J0 ω0 τ 2 − (2L − e)2 H (τ − (2L − e))dτ − · · ·

+ε

t

+ O(ε ) 2

v(τ )J0 [ω0 (t − τ )]dτ − 2

t

v(t − τ )J0 ω0 τ 2 − (2L)2 H (τ − 2L)dτ + · · ·

(5.37)

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Terms in the above expression multiplied by the Heaviside function H (t − e) are waves arriving at the nonlinear attachment after propagating through a length equal to e, i.e., they originate at the point of forcing directly after application of the impulsive load; terms multiplied by H (t − 2L) are waves arriving at the nonlinear attachment after traveling through the entire length of the rod and after being reflected from the fixed boundary at x = −L, and so on. Hence, expression (5.37) enables us to study in detail the early-time dynamic interaction of the nonlinear attachment with individual incoming wavepackets propagating through the dispersive rod. Substituting (5.37) into the second of relations (5.29) we obtain a model for the early time dynamics of the nonlinear attachment; this model is in the form of incident and reflected traveling waves. Hence, one is able to study the early- time nonlinear dynamic interaction of the nonlinear attachment with the leading incoming wavepackets from the rod generated by the impulse. In the limit of the semi-infinite rod, L = ∞ , we recover System I [equation (5.32)]. A similar wave-based earlytime analysis can be applied to the dynamics of System II, which we now proceed to examine.

5.3.1.2 System II: Initial Step Displacement Distribution Considering again the finite rod-NES configuration, we assume that there is no external excitation, and that a finite-step initial displacement distribution is imposed. This leads to the following system of governing equations: ∂ 2 u(x, t) ∂ 2 u(x, t) 2 + ω u(x, t) − = 0, 0 ∂t 2 ∂x 2

−L < x < 0

v(t) ¨ + λv(t) ˙ + Cv3 (t) + ε[v(t) − u(0, t)] = 0 ∂u(0, t) + ε[v(t) − u(0, t)] = 0, ∂x

u(−L, t) = 0

u(x, 0) = D[H (x + d1 ) − H (x + d2 )], ∂u(x, 0) = v(0) = v(0) ˙ =0 ∂t

−L ≤ −d2 < −d1 ≤ 0 (5.38)

where d2 − d1 = d > 0, D denotes the magnitude of the step of the initial displacement, and L the length of the rod. In this case, we need to modify the previous Green’s function formulation in order to compute the transient response. Indeed, an initial rod displacement u(x0 , t0 ) produces the equivalent force distribution u(x0 , t0 )δ (t0 ) (Morse and Feshbach, 1953), where prime denotes generalized differentiation (Richtmyer, 1985) of the delta function with respect to its argument. Following the methodology of Section 5.3.1.1 and letting L → ∞ we obtain the following approximate integro-differential equation governing the dynamics of the nonlinear attachment:

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v(t) ¨ + λv(t) ˙ + εv(t) + Cv3 (t) t 0 = ε u(ξ, 0)δ (τ )g1 (x − ξ, t − τ )dξ dτ 0

−ε

2

+ε

−∞

t

0 t

2

τ

0 −∞

u(ξ, 0)δ (τ )g1 (x − ξ, t − λ)dξ dλ g1 (0, t − τ )dτ

v(t − τ )g1 (0, τ )dτ

+ O(ε3 )

(5.39)

x=0

Performing manipulations on the right-hand side of the above equation we derive the following integro-differential equation governing the nonlinear dynamics of System II:

t

v(t) ¨ + λv(t) ˙ + Cv 3 (t) + εv(t) − ε 2

v(τ )J0 [ω0 (t − τ )]dτ

= εD H (t − d1 ) − H (t − d2 ) − tω0

−ε D 2

t

−d1

−d2

, H (t + ξ ) J1 ω0 t 2 − ξ 2 dξ t2 − ξ2

H (τ − d1 ) − H (τ − d2 ) − τ ω0

−d1

−d2

, H (τ + ξ ) 2 2 J1 ω0 τ − ξ dξ τ2 − ξ2

× J0 [ω0 (t − τ )]H (t − τ )dτ + O(ε3 ) ≡ εF1 (t) + ε2 F2 (t) + O(ε 3 ) v(0) = v(0) ˙ = 0,

L→∞

(System II)

(5.40)

As for System I, this asymptotic model is approximate [since terms of O(ε3 ) or of higher order are omitted], and converges to the exact system in the limit of weak coupling ε → 0. In summary, System II models a semi-infinite, unforced dispersive rod with a finite-step initial displacement distribution, zero initial velocity, and an essentially nonlinear end attachment to its free end. Similarly to System I the two integrals on the right-hand side of equation (5.40) are pure non-hom*ogeneous terms representing the leading-order ‘direct forcing’ of the nonlinear attachment due to the initial step displacement distribution of the rod. The integral term on the left-hand side models the leading-order interaction between the attachment and the dispersive rod and is identical to the corresponding term for System I. Moreover, one can develop expressions analogous to (5.36) and (5.37) to study the early time dynamic interaction of the nonlinear attachment with incoming waves propagating through the dispersive finite rod. In the following section we perform computational simulations of Systems I and II to study the nonlinear dynamical interaction of the semi-infinite rod with the nonlinear attachment (the NES). Two computational models will be considered. The

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first model utilizes Neumann expansions to replace the integrals on the left-hand sides of Systems I and II by an infinite set of first-order ordinary differential equations; the second model is based on finite element simulations of the original equations of motion.

5.3.2 Numerical Study of Damped Transitions To study damped transitions in Systems I and II it is necessary to numerically integrate the corresponding governing equations of motion (5.32) and (5.40). Both systems can be reduced to the following compact form: t 3 2 v(τ )J0 [ω0 (t − τ )]dτ v(t) ¨ + λv(t) ˙ + Cv (t) + εv(t) − ε 0

= εF1 (t) + ε F2 (t) + O(ε ) 2

3

v(0) = v(0) ˙ =0

(5.41)

which, as we proceed to show can be expressed as an infinite set of ordinary differential equations. To perform this operation we take into account the property of the Bessel function of zero-th order (Watson, 1980), J0 [ω0 (t − τ )] =

∞

Jk (ω0 t)Jk (ω0 τ )

(5.42)

k=−∞

which upon substitution into (5.41) leads to the following alternative representation of Systems I and II in terms of infinite sets of ordinary differential equations: ⎤ ⎡ Neumann Series Expansion ) *' ( ⎥ ⎢ ∞ ⎥ ⎢ ⎥ v(t) ¨ + λv(t) ˙ + Cv 3 (t) + εv(t) − ε2 ⎢ (ω t)ϕ (t) + 2 J (ω t)ϕ (t) J k 0 k ⎥ ⎢ 0 0 0 ⎦ ⎣ k=1 = εF1 (t) + ε2 F2 (t) + O(ε3 ) ϕ˙ k (t) = Jk (ω0 t)v(t), v(0) = v(0) ˙ = 0,

k = 0, 1, 2, . . .

ϕk (0) = 0,

k = 0, 1, 2, . . .

(5.43a)

In essence, the O(ε2 ) integral term in relation (5.41) was expressed as a Neumann series expansion. It is interesting to note that the set (5.43a) presents a clear representation of the effects of dispersion of the linear medium on the dynamics; indeed, in the limit ω0 → 0 (i.e., in the limit of no elastic foundation and a nondispersive semi-infinite rod) only the zero-th amplitude ϕ0 (t) survives in (5.43a),

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and the infinite set degenerates to the following set of two ordinary differential equations: vt) ¨ + λv(t) ˙ + Cv3 + εv(t) − ε2 ϕ0 (t) = εF1 (t) + ε2 F2 (t) + O(ε3 ) ϕ˙ 0 (t) = v(t) v(0) = v(0) ˙ = 0,

ϕ0 (0) = 0

(Non-dispersive limit)

(5.43b)

It follows that non-zero amplitudes ϕk (t), k = 1, 2, . . . represent the dispersive effects on the dynamics. From a mathematical point of view the amplitudes ϕk (t) in (5.43a) are the coefficients of the Neumann expansion (Watson, 1980) of the integral term t v(τ )J0 [ω0 (t − τ )]dτ 0

in (5.41), following the Neumann expansion of the Bessel function in expression (5.42). As mentioned previously, this integral term models the leading-order dynamical interaction of the nonlinear attachment with the rod, including energy exchanges between these two subsystems. It will be shown below that a disadvantage of the described Neumann seriesbased model (5.43a) is that it fails to converge for t 1, since high-order terms of the infinite summation on √ the left-hand side grow to finite values as time increases, and Jk (u) ≈ O(1/ u) as u 1 independently of the order k. No such convergence problem is encountered, however, in the simpler non-dispersive model (5.43b). Nevertheless, as we shall see, the representation (5.43a) is still valid for early time prediction of the transient interaction between the rod and the nonlinear attachment. In Figure 5.36 we present numerical simulations for System I [equations (5.43a)] √ with parameters ε = 0.1, ω0 = 0.9, C = 5.0, F = −10, λ = 0.5, e = 1 and 11 amplitudes, ϕ0 (t), . . . , ϕ10 (t), being taken into account. In Figures 5.54a, b we depict the forcing functions εF1 (t) and ε 2 F2 (t) as defined by equations (5.32), and in Figure 5.36c we present the response v(t) of the nonlinear attachment computed using the Neumann series-based model (5.43a). The response of the same system computed by the finite element (FE) approach is presented in Figure 5.36d. For the FE computations, we consider directly the original System I, relations (5.29), with the length of the rod being chosen sufficiently long (L = 400) to avoid numerical pollution of the results by reflected waves originating from the free right boundary. The number of elements used in the simulations ensured a five-digit convergence of the leading modes of the rod. In the case of impulse excitation (System I), the delta function in equation (5.29) was modeled using a half-sine pulse whose area was equal to the amplitude F of the delta function. The frequency of the pulse was set to 10 Hz (higher frequency pulses were also considered but it was found that above 10 Hz, the response was no longer influenced by the pulse frequency). Regarding the FE numerical integration of the equations of motion (5.29), the Newmark algorithm (Geradin and Rixen, 1994) was considered with parameters chosen to ensure unconditional stability of the algorithm (the same

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77

Fig. 5.36 Damped response of System I with forcing F = −10: (a) εF1 (t); (b) ε 2 F2 (t); (c) NES response v(t), model (5.43a) based on Neumann series expansions; (d) NES response v(t) based on FE computations; (e) leading amplitudes ϕk (t), k = 0, 1, . . . , 10; (f) instantaneous NES frequency of v(t).

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

FE model was employed for the simulations carried out in Section 5.2.3 to discretize the finite rod). In some cases, slight numerical damping was added to ensure stability of the numerical results. Finally, in the FE simulations the sampling frequency was such that the distance travelled by the waves in one time step never exceeded the distance between two successive nodes (so the Courant condition was satisfied). Comparing the Neumann series-based and FE simulations (Figures 5.36c and 5.36d), good agreement is obtained in the early time (highly nonlinear) phase of the motion, approximately up to t = 20 s. After that early time regime the predictive capability of the Neumann representation deteriorates, and there is disagreement between the two computations. The reason for the lack of convergence of the Neumann series as time increases can be understood by examining the behavior of the time series of the amplitudes ϕk (t), as depicted in Figure 5.36e. We conclude that the participation of the high-order amplitudes is no longer negligible with increasing time. As a result, the Neumann expansion, ∞ 2 ε J0 (ω0 t)ϕ0 (t) + 2 Jk (ω0 t)ϕk (t) k=1

[which replaces the integral in the integro-differential equation (5.41)] no longer converges with increasing time as more terms are added to the summation. We note at this point that the Bessel functions of the first kind behave asymp−1/2 totically as follows, Jk (ω0 t) ∼ O(ω0 t −1/2 ), t 1 irrespective of the order k = 0, 1, . . . . It is concluded, therefore, that the model (5.43a) based on Neumann series expansions is valid only in the early-time dynamics [this does not apply, however, for the non-dispersive model (5.43b) as discussed previously]. The Neumann series-based model, however, has the advantage not to possess any integral term and to directly depict the effects of dispersion through the amplitudes ϕp (t), p = 1, 2, .... In Figure 5.36f we present the time evolution of the instantaneous frequency (t) of the nonlinear attachment, computed by applying a numerical Hilbert transform to the exact (FE) time series depicted in Figure 5.36d [but see also the work by Chandre et al. (2003) for alternative methods of frequency extraction from a time series]. Such instantaneous frequency plots will be useful in what follows, in our study of transitions of the damped dynamics between different dynamical regimes. From Figure 5.36f we conclude that the transient response takes place in the neighborhood of the cut-off frequency ω = ω0 = 1 that separates the attenuation and propagation zones of the dispersive rod of infinite spatial extent. To demonstrate that the described rod-attachment dynamics is caused by the essential stiffness nonlinearity of the attachment, in Figure 5.37 we depict the response of System I with the essential nonlinearity replaced by a linear stiffness of constant C = 5.0, and all other parameters being left unchanged. We note the low retention of energy by the linear attachment and the nearly negligible amplitudes ϕk (t), k = 0, 1, . . . , 10 in this case. We now proceed to study the different regimes of the rod-nonlinear attachment dynamic interaction through FE computations. This investigation reveals the main

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79

Fig. 5.37 System I with linear attachment and forcing F = −10: (a) attachment response v(t) based on the model (5.43a) with Neumann series expansion; (b) leading amplitudes ϕk (t), k = 0, 1, . . . , 10.

regimes of the transient dynamics, and the mechanisms that govern the energy exchanges between the rod and the NES during different stages of the damped transient motion. To study the different regimes of the motion, we analyze the effect that the variation of the magnitude of the excitation has on the rod-nonlinear attachment dynamics. Specifically, the response v(t) and the instantaneous frequency (t) of the nonlinear attachment are √ computed using FE computations for System II with parameters, ε = 0.1, ω0 = 0.9, C = 5.0, λ = 0.05, d1 = −6.0, d2 = −8.0, d = 2.0 and varying magnitude D. In the following discussion we only consider FE computations, although as mentioned previously the Neumann series-based models (5.43a, b) can also be used to study the early-time nonlinear response. Different amplitudes D will be considered and we start our study by examining the case D = 4.5. In Figure 5.38 we depict the transient responses of System II, from which we deduce the presence of three different regimes of motion labeled as Regimes 1 (0–100 s), 2 (100–300 s) and 3 (470–800 s). Moreover, the Regimes 2 and 3 are separated by a relatively large transition period (300–470 s). We make the following remarks concerning these regimes of the motion.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.38 FE simulations for System II with D = 4.5: (a) NES response v(t); (b) NES instantaneous frequency (t); (c) forces in the linear and nonlinear springs; (d) responses v(t) — and u(0, t) - - - for t 1 (Regime 3a).

In the early-time, high-frequency Regime 1 (see Figures 5.38b, c) the nonlinear attachment (NES) interacts with incoming travelling wavepackets possessing frequencies inside the PZ of the dispersive rod (i.e., with ω > ω0 ). As a result, the instantaneous frequency of the attachment also is situated inside the PZ, (t) > ω0 . Considering the transitions of the damped dynamics of the nonlinear attachment, after an early amplitude build up to a maximum level, the dynamics makes a transition to a weakly modulated oscillation (Regime 2 in Figures 5.38b, c) caused by wave radiation from the nonlinear attachment back to the rod. This regime of weakly modulated, nearly time-periodic oscillation of the NES possesses a (fast) frequency nearly equal to ω0 , which is the bounding frequency between the AZ and PZ of the rod; this frequency is also the frequency of the in-phase mode of the rod. Such a weakly modulated motion of the NES possessing a single fast-frequency is typical in fundamental TET regimes (see for example, the discussion in Section 3.4.2.1). Hence, during Regime 2, the NES engages in 1:1 TRC with the in-phase normal mode of the dispersive rod at frequency ω0 , which yields nonlinear passive extraction of energy from that mode.

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81

Fig. 5.39 System II, shape of the rod for t 1: (a) case D = 4.5 (Regime 3a); (b) case D = 10.0 (Regime 3b).

As the energy of the NES decreases due to energy radiation and damping dissipation the dynamics can no longer sustain the 1:1 TRC, so escape from TRC follows; hence, energy is radiated back to the rod and the instantaneous frequency of the NES decreases until it reaches the low level (t) = O(ε1/2 ) [a discussion regarding the linearized motion occurring at this frequency is given below (Panagopoulos et al., 2004)]. At the end of this transition the dynamics becomes almost linear (see Figures 3.39b, c, d) and Regime 3 of the motion is reached. In actuality there are two alternative possibilities for the evolution of the dynamics of the integrated rod-NES system after escape from TRC (Regime 2); these will be denoted as Regimes 3a and 3b from here on. Both Regimes correspond to low-amplitude linearized oscillations of the system. Regime 3a (see Figures 3.39b, c, d) consists of weakly modulated periodic motions in the neighborhood of (t) ≈ ω0 . The in-phase mode of the rod is excited, and the NES vibrates in an out-of-phase fashion and with much smaller amplitude than the rod. These assertions can be proved analytically in this case, due to the nearly linear nature of the dynamics (Panagopoulos et al., 2004). Following the analysis in that work, for sufficiently small amplitudes of oscillation of the NES and

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

ignoring damping for the moment, it can be shown that the response of System II in Regime 3a can be approximated as εω02 x + 1 Y ej ω0 t + cc, x ≤ 0 u(x, t) ≈ ε − ω02 v(t) ≈ V ej ω0 t + cc =

εY ej ω0 t ε − ω02

+ cc

(5.44)

where ‘cc’ denotes the complex conjugate. Taking into account the actual numerical values of the parameters used for the simulations, we obtain the analytical estimate V /Y = −0.125, which is in satisfactory agreement with the FE numerical simulation of Figure 3.39d. The mode shape of the rod in Regime 3a computed through FE computations is depicted in Figure 3.40a; it is noted that it is not exactly a straight line [as predicted by the analytical expression (5.44)], a discrepancy attributed to the finiteness of the rod in the FE simulations and to higher-order terms that were neglected from the above simplified linearized approximation. This type of linearized motion with approximate frequency ω0 is not the only possible long-time settling response of System II. The numerical confirmation of this assertion is given in Figure 5.40 which corresponds to amplitude D = 10.0 with parameters as defined above and damping added to the rod. We note that until the long-time Regime of the motion is reached, the dynamics is qualitatively similar to the undamped case presented in Figure 5.38. However, after escape from 1:1 TRC the NES settles into Regime 3b, consisting of low-frequency, weakly modulated oscillations of the system well inside the AZ of the dispersive rod; moreover, from the mode shape of the assembly depicted in Figure 5.39b, we conclude that this low-frequency motion is strongly localized to the NES, with the rod undergoing small amplitude, near-field oscillations close to the point of connection. This type of localized motion is similar to the localized modes studied inside AZs of discrete linear chains with essentially nonlinear attachments [we recall the results reported in Section 3.5.2 and in Manevitch et al. (2003)]. The frequency of this localized motion into which the system settles was analytically approximated in Panagopoulos et al. (2004) for √ the case of a discrete linear chain with a nonlinear end attachment, as ωesc = O( ε). In addition, as shown in Figure 3.41c, in Regime 3b the attachment and the point of connection of the rod execute nearly in-phase motions. Following Panagopoulos et al. (2004), the response of the system in this linearized regime is approximated as follows: u(x, t) ≈ Y (x)ej ωt + cc

, , 2 2 2 2 ej ωt + cc, ω < ω0 , x ≤ 0 = A exp − x ω0 − ω + B exp x ω0 − ω v(t) ≈ V ej ωt + cc with the frequency ω computed by solving

(5.45a)

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83

Fig. 5.40 FE simulations for System II with D = 10.0; (a) NES response v(t); (b) NES instantaneous frequency (t); (c) response v(t) — and u(0, t) - - - for t 1 (Regime 3b).

εω2 − ε − ω2

,

, , εω2 ω02 − ω2 exp −2L ω02 − ω2 = − − ω02 − ω2 ε − ω2 (5.45b)

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.41 FE simulations for System II with (a) D = 2.0 and (b) D = 8.0.

For ε = 0.1, ω02 = 0.9 and L = 400 √ we compute the frequency of the localized mode as ω = 0.3 rad/s = O( ε), which agrees with the corresponding values derived from the numerical simulations of Figure 5.40. Moreover, the same approximate analysis estimates the steady state localized√mode shape as V /Y (0) ≈ [1 − (ω2 /ε)]−1 , which confirms that when ω = O( ε) it holds that V Y (0). Hence, the linearized motion is localized to the nonlinear attachment, which oscillates in an in-phase fashion with respect to the rod end. In Figure 5.41 we examine the dynamics of System II for amplitudes D = 2 and D = 8. For the weakest excitation (Figure 5.41a), Regime 1 is of short duration, whereas Regime 2 (1:1 TRC) cannot be realized since the excitation is not sufficiently strong; as a result, the entire motion takes place entirely in Regime 3a, i.e., it is nearly linear. Since no 1:1 TRC takes place in the neighborhood of the frequency ω0 , the strength of TET is minimal in this case. For the case of strongest excitation (Figure 5.41b), Regimes 1 and 2 can clearly be deduced, whereas, the eventual transition to Regime 3a is not depicted in the time window considered for the numerical simulations. Similar dynamics is noted in the FE numerical simulations of System I. In Figure 5.42 we depict the response of√System I with parameters identical to System II and F = −10, ε = 0.1, ω0 = 0.9, C = 5.0, λ = 0.05. The response of the

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85

Fig. 5.42 FE simulations for System I with F = −10.0: (a) NES response v(t); (b) NES instantaneous frequency (t).

nonlinear attachment v(t) is presented in Figure 5.42a, whereas the corresponding instantaneous frequency (t) is shown in Figure 5.42b. After early time transients due to dynamic interaction of the nonlinear attachment with incoming travelling waves (Regime 1), the response settles into a weakly modulated periodic oscillation with frequency near ω0 (Regime 2). For t > 200 s, escape from TRC occurs, and from t > 400 s, the nearly linear Regime 3a is reached. By adding damping to the rod, the localized mode can also be excited (i.e., Regime 3b) as in the case of System II. Hence, the dynamics is qualitatively similar to what is depicted in Figure 5.38. The previous numerical results are in agreement with the TET scenario outlined in Section 3.5.2 for the semi-infinite chain with an essentially nonlinear end attachment. That is, an initial dynamic interaction during which wave radiation from the nonlinear attachment to the semi-infinite chain is realized. This is followed by TET due to 1:1 TRC of the NES with the in-phase mode of the chain at the lower bounding frequency ωb1 = ω0 , followed by escape from TRC, and eventual transition to nearly linear motion. A similar scenario is realized in the continuous system examined in this section, but now the 1:1 TRC occurs between the normal mode of the rod at the bounding frequency ωb = ω0 of the rod of infinite extent (note that in this case a single bounding frequency exists, whereas in the semi-infinite chain

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

considered in Section 3.5.2 there were two such bounding frequencies). This shows the robustness of the TET phenomenon, as it is realized in the two semi-infinite systems with different configurations. In the following section we perform an analytical study of the different regimes of the transient response of System I, in order to gain more insight into the underlying nonlinear dynamical mechanisms governing the nonlinear attachment-rod interaction. The analysis can be extended also for System II and for more general classes of transient excitations.

5.3.3 Analytical Study In this section we will analyze dynamic interactions of the NES with impeding traveling waves possessing frequencies inside the PZ of the rod of infinite length (that is, nonlinear interactions in Regime 1 of the dynamics). In addition, we will study analytically weakly modulated responses of the NES possessing fast frequencies close to the bounding frequency ωb = ω0 , under conditions of 1:1 TRC; this will correspond to Regime 2 of the dynamics (Vakakis et al., 2004). We initiate our analytical study by examining the dynamic interaction of the essentially nonlinear attachment with incoming traveling waves with frequencies inside the PZ (ω > ω0 ) of the dispersive rod of infinite spatial extent. To this end, we analyze the dynamics of the NES forced by a single monochromatic incident wave, Aej (ωt −kx ), with j = (−1)1/2 and k being the wavenumber. If, to a first approximation, we neglect higher-frequency components in the reflected wave (that are generated by the essential stiffness nonlinearity of the NES), and consider only wave components at the basic frequency of the incident wave, we can express the rod response as follows: u(x, t) = Aej (ωt −kx) + Bej (ωt +kx) + cc

(5.46a)

that is, as a superposition of the incident and reflected waves. In (5.46a), B is the amplitude of the reflected wave, A the (prescribed) amplitude of the incident wave, and ‘cc’ denotes complex conjugate. Substituting (5.46a) into the governing linear partial differential equation of the rod with no NES attached, we compute the wavenumber k by the following dispersion relation (which also defines the PZ of the rod of infinite length): k = (ω2 − ω02 )1/2 ,

ω ≤ ω0

(5.46b)

We now make the basic assumption that the NES engages in resonance with the incoming wave. This resonance interaction may be regarded as analog of the resonance interaction of the NES with nonlinear normal modes (NNMs) of the discrete chain of particles, inside the PZ of that system (Vakakis et al., 2003). Assuming that the nonlinear attachment possesses no damping (λ = 0), and that it (approximately) oscillates with frequency ω (i.e., the frequency of the incident wave), we express its response as:

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

v(t) = Zej ωt + cc

87

(5.46c)

Clearly, this ansatz is only an approximation, as it omits higher harmonics generated by the essential nonlinearity. Substituting (5.46c) together with (5.46a, b) into the governing differential equations (5.29) (with F = 0), we obtain the following approximate algebraic relationships for the complex amplitudes A, B and Z, (−j k + ε)B ≈ εZ − (j k + ε)A −ω2 Z + 3CZ 2 Z ∗ − ε(A + B − Z) ≈ 0

(5.47)

where (*) denotes complex conjugate. Combining these two equations, expressing the complex amplitude of the attachment in polar form, Z = |Z|ej φ , and setting separately equal to zero the real and imaginary parts of the resulting complex equation we obtain the following relations in terms of real variables: (−m − ω02 + 3C|Z|2 + ε)|Z| cos φ + ε2 m−1/2 |Z| sin φ = 2εA (−m − ω02 + 3C|Z|2 + ε)|Z| sin φ − ε2 m−1/2|Z| cos φ = −2ε2 Am−1/2 (5.48) where m = ω2 − ω02 > 0, and without loss of generality we assume that the prescribed amplitude A is a real number. Eliminating φ from this set of equations we derive the following frequency-amplitude relationship between |Z| and m, that computes the approximate steady state response of the nonlinear attachment caused by the incident monochromatic traveling wave of amplitude A (Vakakis et al., 2004), m3 + 2(ω02 − 3C|Z|2 − ε)m2 + [(ω02 − 3C|Z|2 − ε)2 − 4ε2 (A/|Z|)2 ]m + [ε4 − 4ε4 (A/|Z|)2 ] ≈ 0

(5.49)

with m assumed to be an O(1) quantity. Once a solution m = m(|Z|) is computed, the corresponding phase φ is obtained by either one of equations (5.48), as follows: tan(φ/2) ≈ (1/2)(m + ω02 − 3C|Z|2 − ε − 2εA/|Z|)−1 × − 2ε2 m−1/2 ± 4ε4 m−1 − 4(m + ω02 − 3C|Z|2 − ε − 2εA/|Z|) . 1/2 (5.50) × (−m − ω02 + 3C|Z|2 + ε − 2εA/|Z|) For the considered system parameters equation (5.49) possesses always two or three real roots for m as functions of |Z|. However, if we take into account that due to our previous assumptions we seek solutions only inside the PZ of the rod we must pose the additional inequality constraints, m > 0 and ω02 − 3C|Z|2 < 0, which restrict the solution to a single branch m = m(|Z|). This branch is depicted in Figure 5.43a for system parameters ω02 = 1.0, C = 3.0, ε = 0.1, A = 5.0; the corresponding phase φ is presented in Figure 5.43b. We note that in the neighborhood of the bounding frequency ωb = ω0 the incoming traveling wave degenerates into a standing wave, i.e., the in-phase normal mode, and so the previous analysis is not

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

valid in that region. The inapplicability of the presented analysis close to ωb = ω0 is manifested in Figure 5.43a by the elimination of the single-validness of the solution branch m = m(|Z|) in the neighborhood of point O; this is an indication that bifurcations take place as the bounding frequency is approached from above, but these cannot be analytically studied by the simplified analysis considered herein. In fact, this series of bifurcations can be studied analytically by considering the solutions of equation (5.49) in the neighborhood of the bounding frequency ωb = ω0 , through appropriate rescaling of the frequency variable and modification of the ansatz for the sought analytical solution. Regarding the plot of the phase depicted in Figure 5.43b, we note that as the amplitude |Z| of the NES increases the phase φ reaches the limit π/2. Moreover, we make the observation that in the limit A → 0, i.e., for small-amplitude incoming waves, the point of crossing of solutions E tends to O and there is single-validness of the solution m = m(|Z|) over the entire permissible region of the plot of Figure 5.43a. In Figure 5.44 we verify the analytically predicted dynamic interactions in the PZ of the dispersive rod by performing direct numerical simulations for System I with parameters ω02 = 1.0, C = 3.0, ε = 0.1, λ = 0, F = −40, e = 1.0 and 11 terms considered in the Neumann expansion [expressions (5.32) and (5.43a)]. In this case the undamped nonlinear attachment undergoes a steady state periodic oscillation with amplitude approximately equal to 1.5 and frequency equal to 2 rad/s; clearly, this represents a resonance of the nonlinear attachment inside the PZ of the dispersive rod, which, although differing from the case of single incident wave (since in the numerical simulation the excitation of the attachment is in the form of an incident wave packet), nevertheless it reveals a frequency-amplitude dependence that agrees with the resonance plot of Figure 5.43a. The nonlinear dynamic interactions of the NES with incident traveling waves are responsible for the built up of the NES response during Regime 1 of the motion, as depicted in the FE simulations of Figures 5.38 and 5.40–5.42. As the frequency of the NES decreases due to damping dissipation and energy radiation (backscattering) to the rod, the dynamics of the NES enters into the regime of 1:1 TRC with the in-phase normal mode of the rod at the bounding frequency ωb = ω0 (that is, Regime 2 of the motion). Then the NES executes slowly modulated oscillations with fast frequency which can be analytically studied by applying the complexificationaveraging (CX-A) technique (Manevitch, 2001) discussed in Section 2.4 and applied in Chapter 3. To study the transition of the damped dynamics from Regime 1 to Regime 2 of the motion, we will apply an order reduction methodology based on CX-A by assuming that the dynamics possesses a single fast frequency in the neighborhood of the bounding frequency ωb . To this end, we reconsider System I with e = 1, ω0 = 1 and weak viscous damping: ∞ 3 2 v(t) ¨ + ελv(t) ˙ + Cv (t) + εv(t) − ε J0 (t)ϕ0 (t) + 2 Jk (t)ϕk (t) k=1

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89

Fig. 5.43 Resonance interaction of the NES traveling with amplitude A = 5.0: (a) NES amplitude |Z| as function of the frequency variable m; (b) NES phase φ as function of amplitude |Z|; shaded regions denote inadmissible ranges of solutions.

= εF J0

t 2 − e2 H (t − e)

t

− ε2 F

J0

τ 2 − e2 H (τ − e)J0 (t − τ )dτ + O(ε3 )

≡ εF1 (t) + ε2 F2 (t) + O(ε3 ) ϕ˙ k (t) = Jk (t)v(t), v(0) = v(0) ˙ = 0,

k = 0, 1, 2, . . . ϕk (0) = 0,

k = 0, 1, 2, . . .

(5.51)

Since we aim to study the transient dynamics of this system under condition of 1:1 TRC, we express the NES response v(t) in the form of a weakly modulated fast oscillation with frequency ωb = ω0 , and the amplitudes ϕk (t) as slowly varying functions. Without loss of generality we assume from this point on that ω0 = 1.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.44 Resonance interaction of the NES in the PZ of the dispersive rod: (a) NES response v(t); (b) instananeous NES frequency (t).

Hence, we will introduce a slow-fast partition of the dynamics of (5.51) that will enable us to focus on the slow dynamics of the system, and thus study the transition of the damped dynamics towards 1:1 TRC. To perform this task we introduce the new complex variables, zk (t) = ϕ˙k (t) + j ϕk (t),

k = 0, 1, . . .

ψ(t) = v(t) ˙ + j v(t)

(5.52)

where j = (−1)1/2 , and express (5.51) as the following set of complex ordinary differential equations, j ελ ˙ ψ(t) − [ψ(t) + ψ ∗ (t)] + [ψ(t) + ψ ∗ (t)] 2 2 jC j [ψ(t) − ψ ∗ (t)]3 + ε − [ψ(t) − ψ ∗ (t)] + 8 2

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems ∞

+

91

εj J0 (t)[z0 (t) − z0∗ (t)] + εj Jk (t)[zk (t) − zk∗ (t)] + O(ε 2 ) 2 k=1

≡ εF1 (t) + ε2 F2 (t) + O(ε3 ) zk (t) + zk∗ (t) = −j Jk (t)[ψ(t) − ψ ∗ (t)],

k = 0, 1, 2, . . .

(5.53)

where (∗) denotes the complex conjugate. These equations are exact, since they involve no simplifications compared to the original System I [relation (5.51)]. We now introduce the slow-fast partition of the dynamics by adopting the following representations for the dependent complex variables in (5.53), zk (t) ≈ Bk (t), ψ(t) ≈ A(t)e

k = 0, 1, 2, . . .

jt

(5.54)

where A(t) and B(t) are slowly varying complex amplitudes. By these representations we approximate ψ(t) as a slowly modulated time-periodic oscillation with fast frequency ω0 = 1, and zk (t) as slowly varying complex functions. These expressions are expected to be valid only in the regime of 1:1 TRC, i.e., in Regime 2. Before we substitute the approximations (5.54) into (5.53), we need to represent the Bessel functions and the forcing functions εF1 (t) and ε 2 F2 (t) in terms of complex slow-fast partitions, in a way compatible to our CX-A analysis. One way to perform this task is by means of two-point quasi-fractional approximants as shown in Guerrero and Martin (1988), Martin and Baker (1991) and Chalbaud and Martin (1992). Indeed, employing the results of Guerrero and Martin (1988), we can approximately partition the leading-order Bessel functions of the first kind in terms of ‘slow’ and ‘fast’ components as follows: Jn (t) ≈ (1 + t)−1/2 wn (t)ej t + cc ≡ On (t)ej t + cc, n = 0, 1 6N 6N i pi t i 1 i=0 Pi t , vn (t) ≈ 6i=0 (5.55) wn (t) = [un (t) − j vn (t)], un (t) ≈ 6N N i i 2 i=0 qi t i=0 qi t where ‘cc’ denotes complex conjugate. Hence, the two leading-order Bessel functions are expressed in terms of a fast oscillation ej t modulated by the slowly varying functions On (t). Choosing q0 = 1, the remaining (3N + 2) parameters are determined by solving the equations N ∞ N ∞ 2k s k i k t qs t ak t = Pi t (−1) (2k)! s=0

+

N i=0

k=0

pi t i

∞ k=0

i=0

(−1)k

t 2k+1 (2k + 1)!

k=0

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

N

qN−s t −s

s=0 N

∞

Bk t −k

k=0

qN−s t −s

∞

s=0

=

PN−i t −i

i=0

bk t −k

N

=

k=0

N

pN−i t −i

(5.56a)

i=0

where ak = 2−n

(k/2) p=0

n + 1/2 k − 2p

(−1)p , + n + 1)

22p p!(p

Bk = 2Re (βk ),

bk = −2Im (βk ), ⎧ ⎫ ⎬

p k ⎨ n p 7 1 (−j ) (−j ) 1/2 1/2 βk = √ s(s − 1) + − n2 + , k − p 2p p! ⎭ 4 π (1 + j ) ⎩ k p=1

s=1

j = (−1)1/2

(5.56b)

In Guerrero and Martin (1988) the numerical values for the coefficients qi , pi and Pi in the above expressions are provided for N ≤ 5. Our numerical computations showed that the quasi-fractional approximations (5.55) and (5.56) provide approximations to the leading-order Bessel functions of the first kind, J0 (t) and J1 (t), that are virtually identical to the exact values of these functions. Moreover, we may use these relations to approximate higher-order Bessel functions of the first kind using the following recursive formula: Jp−1 (t) + Jp+1 (t) = Jp+1 (t) =

2p JP (t) ⇒ t

2p Jp (t) − Jp−1 (t), t

p = 1, 2, . . .

(5.57)

Our numerical computations indicate that as the order of the Bessel function increases we need to consider increasingly more terms in the quasi-fractional approximations (5.55) (i.e., we must increase N) in order to achieve good agreement with the exact solutions as t → 0. However, except for a small neighborhood of t = 0, there is complete agreement of the quasi-fractional approximations with the exact solutions when N ≤ 5. Considering now the forcing functions εF1 (t) and ε2 F2 (t), we will apply the slow-fast partition (5.55) to represent these functions in terms of slow and fast complex components. Considering first the forcing function εF1 (t), we can express it in the following form: √ 2 u2 + 2eu ej ( u +2eu−u e−j e H (u) ej t +cc εF1 (t) ≈ εF O0 () * ' () * ' Slow component

Fast component

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

≡ εWi (u)H (u)ej t + cc,

u=t −e

93

(5.58)

where W1 (u) denotes the ‘slow’ modulation of the ‘fast’ oscillation with frequency ω0 = 1. Similarly, the second-order forcing function ε2 F2 (t) can be approximately partitioned in terms of slow and fast dynamics, by introducing the new variable s = τ − e: ε2 F2 (t) ≈ −ε F H (u)e ' 2

−j e

u 0

O0 (u − s)O0 ()

√2 s 2 + 2es ej ( s +2es−s) ds * '

Fast component

Slow component

≡ ε2 W2 (u)H (u)ej t + cc,

ej t +cc () *

u=t −e

(5.59)

Returning now to the equations of motion (5.53), we substitute into them the slow-fast partitions (5.54), (5.55), (5.58) and (5.59), and retain only fast terms of frequency ω0 = 1 (or equivalently, we average out harmonic components with fast frequencies higher than unity). This yields the following set of approximate modulation equations governing the slow evolutions of the complex amplitudes A(t) and Bk (t), k = 0, 1, . . . :

ελ 3j C 2 j ˙ + A+ |A| A A+ 2 2 8 ∞ jε j ∗ ∗ 2 + ε − A + O0 (B0 − B0 ) + j ε Ok (Bk − Bk ) + O(ε ) 2 2 k=1

= εW1 (t − e)H (t − e) + ε2 W2 (t − e)H (t − e) + O(ε3 ) Bk = −j AOk∗ ,

k = 0, 1, 2, . . .

(5.60)

Substituting the infinite set of algebraic equations governing the slowly varying functions Bk (t) into the first of differential equations (5.60), this set can be reduced to a single complex differential equation governing the slow evolution of the amplitude A(t):

ελ 3j C 2 jε j j ˙ + A+ |A| A + ε − A + O0 [−j AO0∗ − (−j AO0∗)∗ ] A+ 2 2 8 2 2 ∞ + jε Ok [−j AOk∗ − (−j AOk∗ )∗ ] + O(ε2 ) k=1

= εW1 (t − e)H (t − e) + ε2 W2 (t − e)H (t − e) + O(ε3 )

(5.61)

Hence, the slow flow nonlinear dynamic interaction between the dispersive rod and the nonlinear attachment in the 1:1 TRC regime (Regime 2) is approximately gov-

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

erned by the reduced complex modulation equation (5.61). This means that the slow flow dynamics can be reduced to a set of two first-order real amplitude and phase modulations for the motion of the attachment. These are determined by expressing the complex amplitudes in polar form: Ok = γk ej δk ,

A = aej b , Wi = ζi ej σi ,

k = 0, 1, 2, . . .

i = 1, 2

(5.62)

which when substituted into (5.61) and upon separation of real and imaginary parts, leads to the following set of real (slow flow) modulation equations: a(t) ˙ + (ελ/2)a(t) +ε

2

(1/2)a(t)γ02 (t) sin 2[b(t)

− δ0 (t)] +

∞

a(t)γk2 (t) sin 2[b(t)

− δk (t)] + O(ε)

k=1

. = εζ1 (t − e) cos[σ1 (t − e) − b(t)] + ε2 ζ2 (t − e) cos[σ2 (t − e) − b(t)] + O(ε3 ) H (t − e) ˙ + (1/2) − (3C/8)a 2 (t) b(t) + ε (−1/2) + εγ02 (t) sin2 [b(t) − δ0 (t)] + 2ε = ε[ζ1 (t − e)/a(t)] sin[σ1 (t − e) − b(t)]

∞

γk2 (t) sin2 [b(t) − δk (t)] + O(ε 2 )

k=1

. + ε2 [ζ2 (t − e)/a(t)] sin[σ2 (t − e) − b(t)] + O(ε3 ) H (t − e)

(5.63)

An inspection of the reduced slow flow (5.63) indicates that the condition of slow amplitude modulation is always satisfied, since a(t) ˙ = O(ε). In order to get a similar condition for the slow modulation for the phase as well, we impose the following additional restriction: ˙ = O(ε) ⇒ (1/2) − (3C/8)a 2(t) = O(ε) b(t)

(5.64)

Provided that this condition is satisfied, the solution of (5.63) provides the following analytic approximation for the NES-rod nonlinear interaction in Regime 2 of the motion v(t) =

ψ(t) − ψ ∗ (t) ≈ a(t) sin[t + b(t)] 2j

v(t) ˙ =

ψ(t) + ψ ∗ (t) ≈ a(t) cos[t + b(t)] 2

(5.65)

with the frequency of oscillation of the NES given approximately by (t) ≈ 1 + ˙ = 1 + O(ε). b(t)

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

95

Numerical integrations of system (5.63) were performed and compared to exact solutions derived by direct numerical simulations of System I [relations (5.51)]. Provided that the assumptions of the analysis were satisfied, satisfactory agreement between analysis and numerics was noted; a representative result is depicted in Figure 5.45 for System I with parameters ε = 0.1, ω0 = 1.0, C = 5.0, F = −15.0, e = 1.0, λ = 3, and 11 terms taken into account in the Neumann expansion. We ˙ is note that, except for the early stage t < 10 where the frequency correction b(t) not of O(ε) , the analytical approximation for v(t) is close to the exact numerical simulation. In the early regime t < 10 the amplitude modulation is not small, and hence it may not be studied by the analytical model (5.63); this regime of the motion (Regime 1 in the computational simulations of Section 5.3.2) represents interaction of the NES with impeding traveling waves from the rod, and, hence, is away from the 1:1 resonance manifold of the system (and so conditions for 1:1 TRC are not met). It follows that the NES response in Regime 1 cannot be studied by the simple ansatz (5.52–5.54) (but refer to the previous analysis in this section leading to expressions (5.49) and (5.50)]. From Figure 5.45a we note that the analytical approximation predicts accurately the slow amplitude decrease of the oscillation of the NES due to damping dissipation in Regime 2 of the motion. Since the low-order analytical model (5.63–5.64) results from the Neumann series-based model (5.43a), its validity is restricted only to the early-time response of the system, i.e., during the transition from Regime 1 to the regime of 1:1 TRC, that is, Regime 2. The analytical model, however, is not valid in the regime of escape from TRC when the transition of the dynamics to the linearized Regimes 3a or 3b is realized (this is discussed in Section 5.3.2). This becomes clear when we consider the FE simulation of the dynamics of System I with system parameters as set above (see Figure 5.45d), where divergence from the Neumann series-based numerical solution is noted with progressing time. However, the derived low-order analytical model accurately models the dynamics in the transition towards, and during the Regime 2, i.e., at least up to t = 40 s. The analytical approach presented can be used to analyze alternative rod-NES configurations. For example, one can prove that the unforced and undamped nondispersive rod-attachment system with ω0 = 0 cannot sustain 1:1 TRC and, hence, no Regime 2 can occur in its transient dynamics. This should be expected as in the non-dispersive case the bounding frequency is zero (ωb = 0) and the rod of infinite extent does not possess an AZ; in this case the rod of infinite extent supports traveling waves with every possible frequency. In this case the dynamics is governed by the set of equations (5.43b) with no forcing and damping terms, v(t) ¨ + Cv3 (t) + ε[v(t) − εϕ0 (t) + O(ε2 )] = 0, ϕ˙0 (t) = v(t),

ϕ(0) = 0

v(0) = 0,

v(0) ˙ =0 (5.66)

where an initial displacement for the nonlinear attachment is assumed, and all other initial conditions are set to zero. Introducing the variables z0 (t) = ϕ˙0 (t) + j ϕ0 (t), and ψ(t) = v(t) ˙ + j v(t), and expressing these into the polar forms, ψ(t) ≈ a(t)ej b(t )ej ωt and z0 (t) ≈ B0 (t), we derive the following set of amplitude and

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.45 Transition from Regime 1 to 2, System I: (a, b) evolutions of the amplitude a(t) and ˙ of the frequency correction b(t); (c) analytical - - - and (Neumann series-based) numerical — solutions for v(t); (d) FE numerical solution for v(t).

phase modulation equations [that are analogous to relations (5.63)]: a(t) ˙ + ε2 a(t)[1 − cos 2b(t)] + O(ε3 ) = 0 ˙ + [ω − (1/2) − (3C/8)a 2(t)] − ε/2 + (ε2 /2) sin 2b(t) + O(ε3 ) = 0 (5.67) b(t) with ω being an arbitrary reference frequency. Due to the lack of a bounding frequency (in view of the non-dispersiveness of the rod), by varying the reference frequency ω the analytical model (5.67) is valid during the entire decaying motion of the attachment. Indeed, imposing the condition that the quantity

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97

Fig. 5.46 Non-dispersive semi-infinite rod with nonlinear attachment: (a) Neumann series-based response of the attachment v(t); (b) FE response v(t); (c) amplitude ϕ0 (t); (d) instantaneous frequency of the attachment (t).

ω − 1/2 − (3C/8)a 2(t) in the second of equations (5.67) is a quantity of O(ε), ˙ is also of O(ε), and the relations one guarantees that the frequency correction b(t) (5.67) describe slow-varying modulations. As a result, the motion of the nonlinear attachment consists of a single regime, that is, a decaying oscillation with energy being continuously radiated to the rod in the form of traveling waves. This analytical prediction is confirmed by the numerical simulation of equations (5.66) depicted in Figure 5.46; this numerical simulation is performed for system parameters ε = 0.1, ω0 = 0, C = 5.0, λ = 0, and initial condition v(0) = 0.7. We note that in agreement with our previous discussion, the response of the attachment

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

v(t) is composed of a single regime, that is, of a continuously decaying oscillation. The decay of the NES oscillation is due to radiation of energy to the rod during the entire regime of the motion; this is confirmed by the nearly-constant oscillation of the amplitude ϕ0 (t), which decays only after the motion of the NES reaches a sufficiently low level and the level of radiated energy from the attachment to the rod also diminishes. This result indicates that the dispersiveness of the rod dynamics influences in an essential way the qualitative dynamics of the rod-nonlinear attachment interaction. Moreover, in contrast to the dispersive case we note good agreement between the Neumann series-based and FE simulations (Figures 5.46a, b) of the transient dynamics. This should be expected, since, as discussed previously, the cause of non-convergence of the Neumann series-based numerical solution is the non-negligible contributions to the response from high-order terms of the Neumann series expansion in (5.43a) with increasing time. These non-converging terms, however, are completely missing in the non-dispersive case since only the leading amplitude ϕ0 (t) survives from the infinite series of amplitudes ϕi (t), see expression (5.43b). This concludes our examination of the nonlinear dynamics of the semi-infinite dispersive rod possessing an end nonlinear attachment (NES). The lack of a linear part in the NES stiffness nonlinearity enables the NES to engage in resonance interactions not only with incident traveling waves from the rod, but also with the in-phase standing wave (normal mode) of the semi-infinite rod at the bounding frequency separating its propagation and attenuation zones. Relating the results of this section to the previous results of this work, resonance interactions of the NES with traveling waves in the PZ of the linear continuous medium can be considered as the ‘continuum limit’ of resonance capture cascades (RCCs) occurring between normal modes of finite-DOF discrete oscillators with attached NESs (Panagopoulos et al., 2004). Viewed in that context, the complicated resonance interactions occurring in Regime 1 of the NES response can be viewed as resonance interactions of the NES with traveling waves in the continuous spectrum of frequencies in the PZ of the linear elastic medium. As the energy of the NES decreases due to damping dissipation and energy radiation back to the rod in the form of traveling waves is realized, the instantaneous frequency of the NES continuously decreases and approaches the bounding frequency ωb = ω0 from above. Then, the nonlinear attachment engages in 1:1 TRC with the in-phase mode of the rod, in similarity to TRCs studied in previous sections. This TRC can only occur due to the dispersion property of the linear medium [hence, characterized by Manevitch (2003) as apotheosis of dispersion!], and provides conditions for the realization of passive TET from the linear medium to the NES.

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99

5.4 Rod of Finite Length with MDOF NES In this section we reconsider the finite dispersive rod and study its complex nonlinear dynamic interactions with a multi-DOF essentially nonlinear end attachment (a MDOF NES). This can be considered as extension of our studies of the discrete linear oscillator with an attached MDOF NES of Chapter 4, and of the finite rod with an end SDOF NES of Section 5.2. We will make use of frequency energyplots (FEPs) for depicting and interpreting essentially nonlinear damped transitions in terms of the undamped dynamics, and, additionally, of Empirical Mode Decomposition (EMD) for decomposing the transient dynamics in terms of multi-scaled intrinsic mode functions (IMFs). This will enable us to perform multi-scale identification of the dominant nonlinear resonant interactions that occur between the rod and the MDOF NES, and to formulate an integrated physics-based, multi-scale method for analyzing and modeling strongly nonlinear, complex dynamical interactions. The analysis of this section closely follows the works by Tsakirtzis (2006) and Tsakirtzis et al. (2007a), which should be consulted for further details.

5.4.1 Formulation of the Problem and FEPs We consider a finite, dispersive linear rod on an elastic foundation clamped at its left end, and coupled at its right end to an essentially nonlinear MDOF ungrounded attachment (the NES). The MDOF NES possesses three small masses coupled by means of essentially nonlinear (nonlinearizable) stiffnesses situated in parallel to weak viscous dampers (see Figure 5.47). Moreover, the masses of the NES are assumed to be small, so that their summation is equal to the mass of the SDOF NES considered in Section 5.2, i.e., m1 + m2 + m3 = 0.1. This will enable us to make direct comparisons of the performances of the SDOF and MDOF NESs without introducing potential added mass effects in the dynamics. In addition, viscous damping in the system is assumed to be weak by setting λ 1. Assuming unidirectional vibration of the system, and denoting by v1 (t), v2 (t) and v3 (t) the displacements of the three masses of the NES, and by u(x, t) the distributed displacement of the rod at position x, we obtain the following governing differential equations for the rod: ∂u(x, t) ∂ 2 u(x, t) ∂ 2 u(x, t) 2 − + ω u(x, t) + λ = F (t)δ(x − d), 1 0 ∂t ∂t 2 ∂x 2 u(0, t) = 0,

∂u(L, t) = ε[v1 (t) − u(L, t)], ∂x

u(x, 0) = 0,

0≤x≤L

∂u(x, 0) =0 ∂x (5.68a)

and the MDOF NES: m1 v¨1 (t) + ε[v1 (t) − u(L, t)] + C1 [v1 (t) − v2 (t)]3 + λ[v˙1 (t) − v˙2 (t)] = 0

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.47 Linear dispersive elastic rod with an attached MDOF NES.

v1 (0) = v˙1 (0) = 0 m2 v¨2 (t) + C1 [v2 (t) − v1 (t)]3 + C2 [v2 (t) − v3 (t)]3 + λ[v˙2 (t) − v˙1 (t)] + λ[v˙2 (t) − v˙3 (t)] = 0 v2 (0) = v˙2 (0) = 0 m3 v¨3 (t) + C2 [v3 (t) − v2 (t)]3 + λ[v˙3 (t) − v˙2 (t)] = 0 v3 (0) = v˙3 (0) = 0

(5.68b)

Hence, we assume that the system is initially at rest, and that a shock is applied at position x = d of the rod. In the above equations ε is the constant of the linear coupling stiffness between the rod and the MDOF NES, and, depending on its value there is either weak or strong coupling between the rod and the NES; in fact, one of the aims of the following computational study is to investigate the effect of the coupling term on the TET performance. In (5.68) λ1 and λ denote the viscous damping coefficients of the rod and the NES, respectively, and C1 , C2 the coefficients of the essential stiffness nonlinearities of the MDOF NES (see Figure 5.47). Moreover, in the following analysis the length of the rod is normalized to L = 1 . The frequency ω0 is the non-dimensional distributed elastic support of the rod and introduces dispersive effects in its dynamics; as discussed in previous sections this frequency represents the cut-off frequency in the spectrum of the dynamics of the uncoupled, infinite dispersive rod; that is, the bounding frequency separating the attenuation (0 < ω < ω0 ) and propagation zones (ω > ω0 ) of the rod on the elastic foundation. For prescribed excitation the equations of motion (5.68a, 5.68b) were solved numerically using the Matlab FE code described in Section 5.2.1, employing an implicit time integration scheme based on the adapted Newmark algorithm (Gerandin and Rixen, 1997). At each time step of the numerical integration the total energy balance was computed in order to ensure that the relative energy error between subsequent steps of the computation was kept less than 0.001%, and that the to-

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101

Table 5.3 The leading eigenfrequencies of the uncoupled dispersive rod (ω0 = 1, L = 1.0). Normal Mode

1

2

3

4

5

6

7

8

9

10

Eigenfrequency 1.8621 4.8178 7.9194 11.046 14.184 17.329 20.48 23.638 26.802 29.973 (rad/sec)

tal accumulative energy error throughout the entire computation was kept less than 1%. Strong coupling between the clamped rod and the NES is a prerequisite for the occurrence of strong nonlinear modal interactions between the linear and nonlinear subsystems. The reason is that weak coupling would not excite sufficiently the NES, so insignificant nonlinear effects in the damped responses would be realized. We point out that this holds due to the clamped condition at the left boundary of the rod, which restricts the rod response to low amplitudes. We note, however, that for different boundary conditions (e.g., free left boundary) the rod response might attain higher amplitudes under shock excitation, so that strongly nonlinear modal interactions might be realized even for weak coupling with the NES (see, for example, the discrete system of Figure 4.2). In Table 5.3 we present the leading eigenfrequencies of the uncoupled clamped rod (with MDOF NES detached) on an elastic foundation with ω0 = 1. The first step of our study is to construct the FEP of the corresponding undamped and unforced Hamiltonian system with λ = λ1 = 0 and F (t) = 0 in relations (5.68a, 5.68b). Then, as shown in our previous studies, the FEP can help us understand and interpret damped transitions involving strongly nonlinear modal interactions between the rod and the NES. To this end, we consider the following Hamiltonian system: ∂ 2 u(x, t) ∂ 2 u(x, t) 2 + ω u(x, t) − = 0, 0 ∂t 2 ∂x 2 u(0, t) = 0,

0≤x≤L=1

∂u(L, t) = ε[v1 (t) − u(L, t)] ∂x

m1 v¨1 (t) + ε[v1 (t) − u(L, t)] + C1 [v1 (t) − v2 (t)]3 = 0 m2 v¨2 (t) + C1 [v2 (t) − v1 (t)]3 + C2 [v2 (t) − v3 (t)]3 = 0 m3 v¨3 (t) + C2 [v3 (t) − v2 (t)]3 = 0

(5.69)

We omit initial conditions at this point since we will examine the nonlinear boundary value problem (NLBVP) governing the periodic orbits of this system; in contrast, the original problem (5.68a, 5.68b) is formulated as an initial value (Cauchy) problem. Analytical approximations of the T -periodic orbits are sought in the form of the following Fourier series:

102

5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs ∞

u(x, t) =

Ck (x) cos[(2k − 1)t],

v1 (t) =

k=1 ∞

v2 (t) =

∞

V1,k cos[(2k − 1)t]

k=1

V2,k cos[(2k − 1)t],

v3 (t) =

k=1

∞

V3,k cos[(2k − 1)t] (5.70)

k=1

where = 2π/T denotes the basic frequency of the periodic motion. Substituting (5.70) into the differential equations (5.69) and taking account the imposed boundary conditions for the rod, we obtain the following series of linear BVPs governing the spatial distributions Ck (x) of the rod: −

∞

[(2k − 1)]2 Ck (x) cos[(2k − 1)t]

k=1

+ ω02

∞

Ck (x) cos[(2k − 1)t] −

k=1

∞

Ck (x) cos[(2k − 1)t] = 0

k=1

⇒ −Ck (x) + {ω02 − [(2k − 1)]2 }Ck (x) = 0 Ck (0) = 0,

dCk (L) = ε[v1k − Ck (L)] dx

(5.71)

An explicit solution of (5.71) provides the following analytical expression for Ck (x), k = 1, 2, 3, . . . , in terms of the corresponding coefficients V1k of the NES:

, Ck (x) = Cˆ k sin x (2k − 1)2 2 − ω02 (5.72) Cˆ k = , (2k

− 1)2 2

− ω02

εV1k

,

, 2 cos L (2k − 1) 2 − ω02 + ε sin L (2k − 1)2 2 − ω02

For example, taking into account only the three leading terms in the series of u(x, t), we derive the following expression for the displacement at the end of the rod during the time-periodic motion: u(L, t) ≈ ,

εV11 cos t

, 2 − ω02 cot L 2 − ω02 + ε

+,

εV13 cos 3t

, 2 2 2 2 9 − ω0 cot L 9 − ω0 + ε

+,

εV15 cos 5t

, 2 2 2 2 25 − ω0 cot L 25 − ω0 + ε

Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems

103

Clearly, this expression holds only as long as (2k − 1)2 2 − ω02 ≥ 0 ⇒ 2 ≥ ω02 (2k − 1)−2 ,

k = 1, 2, 3, . . .

i.e., only when lies in the PZ of the k-th harmonic; this requirement is satisfied for all harmonics if 2 ≥ ω02 , i.e., for periodic orbits with basic frequency in the PZ of the rod of infinite extent). In that case the solution of the rod response is spatially extended (non-localized) in the form of traveling waves, whose positive interference produces the vibration modes evidenced in the transient dynamics. ˜ then, for k ≥ k˜ the trigonometric However, if (2k˜ − 1)2 2 − ω02 < 0 for some k, functions in expressions (5.71) and (5.72) should be replaced by hyperbolic ones, ˜ and higher harmonics of the rod response and the spatial distributions of the k-th u(x, t) become spatially localized (representing near-field solutions with exponentially decaying envelopes). In that case the corresponding time-periodic motion of ˜ harmonic and higher) the rod possesses a set of harmonics (starting from the k-th in the form of spatially decaying standing waves, or near-field solutions localized close to the boundaries of the rod. The qualitative changes in the time-periodic motion of the rod (that is, from spatially extended harmonics to spatially decaying ones) due to changes in the frequency of oscillation, are caused by the dispersion effects introduced by its elastic foundation. Assuming that 2 ≥ ω02 , i.e., that the basic harmonic of the response is situated inside the PZ of the rod (a similar procedure holds if a harmonic lies inside the AZ), the corresponding amplitudes, V1,k , V2,k and V3,k , k = 1, 2, 3, . . . , of the corresponding harmonics of the MDOF NES are computed by substituting the relations (5.72) into the last three nonlinear differential equations of the set (5.69). Expanding the powers of the resulting trigonometric expressions, and setting the coefficients of the resulting trigonometric functions cos[(2k − 1)t], k = 1, 2, . . . , equal to zero, we derive an infinite set of coupled nonlinear algebraic relations in terms of the amplitudes V1,k , V2,k and V3,k governing the time-periodic response of the MDOF NES with basic frequency . For computational reasons, this infinite set of algebraic equations must be truncated by considering terms only up to the fifth harmonic (i.e., k = 1, 2, 3 only), and omitting higher harmonics. The resulting truncated set of nine nonlinear algebraic equations is numerically solved for the amplitudes V1,k , V2,k and V3,k , which completes the analytic approximation of the periodic motion of the system through relations (5.70) and (5.72). The set of nine equations is too lengthy to be reproduced here and can be found in the thesis by Tsakirtzis (2006). In the following numerical results we consider two configurations of MDOF NESs, which are principally distinguished by the strength of the coupling stiffness ε, and the magnitudes of the nonlinear coefficients C1 and C2 . Indeed, our aim is to study the influence of the coupling stiffness and the coefficients of the essential nonlinear stiffnesses of the MDOF NES on TET. The first configuration considered (referred to from now on as ‘System I’) consists of a highly asymmetric MDOF NES, in the sense that it possesses strongly dissimilar nonlinear stiffness coefficients. The parameters of System I are listed below:

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

C1 = 1.0, L = 1.0,

C2 = 0.001, ω0 = 1.0,

ε = 6.6,

λ = λ1 = 0,

m1 = m2 = m3 = 0.1/3 (System I)

In Figure 5.48 we depict the FEP of System I, computed by the previously outlined analytical approximation. The FEP of Figure 5.48 depicts the dependence of the basic frequency in rad/s of the time-periodic oscillation on the (conserved) logarithm of the energy of this oscillation, log10 (E). Only the frequency range covering the two leading modes of the uncoupled linear rod is considered in the FEP of Figure 5.48. The energy E of the periodic orbit is computed by the expression E=

1 2

L ∂u(x, t) 2

∂t

dx +

1 2

L ∂u(x, t) 2

∂x

1 dx + ω02 2

L

u2 (x, t)dx

1 1 1 1 + ε[v1 (t) − u(L, t)]2 + m1 v12 (t) + m2 v22 (t) + m3 v32 (t) 4 2 2 2 1 1 + C1 [v2 (t) − v1 (t)]4 + C2 [v3 (t) − v2 (t)]4 4 4

(5.73)

In Figure 5.49 some representative periodic orbits on different branches of the FEP are depicted. Comparisons of these results with direct FE simulations of the equations of motion (5.69) (computed for the initial conditions predicted by the analytical model) confirmed the accuracy of the analytical computations (Tsakirtzis, 2006). Considering the FEP depicted in Figure 5.48 one discerns two low-frequency asymptotes, which correspond to the two leading modes of the linear uncoupled rod with eigenfrequencies given by: (2n − 1)2 π 2 ωn = ω02 + , n = 1, 2 (Low-energy asymptotes) (5.74) 4L2 For the parameters corresponding to System I these are computed as, ω1 = 1.8621 rad/s and ω2 = 4.8173 rad/s. In the limit of high energies there exist additional frequency asymptotes, denoted by ωˆ i , i = 1, 2, . . . , corresponding to the eigenfrequencies of the system with rigid connections between the rod and the MDOF NES. High-energy periodic orbits close to these asymptotes are weakly nonlinear motions that predominantly localize to the rod. These high-frequency asymptotes are computed as the eigenfrequencies of the dispersive rod with a mass equal to m1 + m2 + m3 = 0.1 attached to its right end. As in the FEPs considered in Sections 3.3 and 4.2, the FEP of Figure 5.48 possesses (global) backbone branches of periodic orbits and (local) subharmonic tongues. Backbone branches consist of nearly monochromatic periodic solutions possessing a dominant harmonic component and higher harmonics at integer multiples of the dominant harmonic; these branches are defined over extended frequency and energy ranges and are composed of periodic solutions mainly localized to the MDOF NES, except in neighborhoods of the linearized eigenfrequencies of the rod

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Fig. 5.48 FEP of System I based on the truncated system (5.70) and (5.72) with k = 1, 2, 3: digits () correspond to the periodic orbits depicted in Figure 5.49; the low- and high-energy asymptotes close to the first mode of the rod are shown in dashed lines; point A () refers to the numerical simulations, and WT analysis of Section 5.4.3.

(see Figure 5.49). Subharmonic tongues are composed of multi-frequency periodic motions with frequencies at rational multiples of the eigenfrequencies ωn of the uncoupled rod. Each tongue is defined over a finite energy range, and is composed of two distinct branches of subharmonic solutions, which, at a critical energy level, coalesce in a bifurcation that signifies the end of that particular tongue and the elimination of the corresponding subharmonic motions for higher energies. In this non-integrable dynamical system there exist countable infinite subharmonic tonques emanating from backbone branches at frequencies in rational multiples of the eigenfrequencies of the uncoupled linear rod. To study the effect on the FEP of strong coupling between the rod and the MDOF NES and of stronger essential nonlinearity C2 , we consider a second set of parameters and label the corresponding system as ‘System II’: C1 = 1.0,

C2 = 0.01,

ε = 9.0,

λ = λ1 = 0,

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.49 Representative periodic orbits of the FEP of System I.

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Fig. 5.50 FEP of System II based on the truncated system (5.70) and (5.72) with k = 1, 2, 3; digits () correspond to the periodic orbits depicted in Figure 5.51; the low- and high-energy asymptotes close to the first mode of the rod are shown in dashed lines; point B () refers to the numerical simulations, WTs and EMD analysis of Section 5.4.3.

L = 1.0,

ω0 = 1.0,

m1 = m2 = m3 = 0.1/3 (System II)

The FEP of System II is depicted in Figure 5.50. Except in the neighborhoods of the low- and high-energy asymptotes ωi and ωˆ i , i = 1, 2, . . . , the branches of periodic solutions are essentially nonlinear, as indicated by their high curvatures and strong dependencies on energy. The fact that all subharmonic tongues in the FEP are nearly horizontal does not mean that the dynamics are weakly nonlinear; on the contrary, the dynamics on the subharmonic tongues is essentially nonlinear. As explained in Section 3.3 and in Lee et al. (2005), on a subharmonic tongue, the strongly nonlinear system (5.69) oscillates approximately as a system of uncoupled linear oscillators, albeit with different frequencies, say ωp and (m/n)ωp , where ωp is the p-th eigenfrequency of the uncoupled rod and (m/n) is rational; as a result, the strongly nonlinear regimes on the subharmonic tongues resemble the dynamics of coupled linear oscillators with rationally related frequencies.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.51 Representative periodic orbits of the FEP of System II.

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The high-energy asymptotes of the FEP of System II (in similarity to System I), indicate – as expected – that at sufficiently high energies System II resembles a rod with a concentrated end mass equal to m1 + m2 + m3 = 0.1. This means that the dynamics of System II close to these high-energy asymptotes is weakly nonlinear, with the corresponding oscillations being predominantly localized to the rod. Hence, high-energy, weakly nonlinear dynamics may occur in System II (as in System I). Moreover, there is a region of the FEP (labeled as ‘Region I’ in Figure 5.50), where the responses of all NES masses and the rod end possess nearly identical amplitudes. In Figure 5.51 representative periodic motions lying on backbone branches of the FEP of System II are depicted.

5.4.2 Computational Study of TET We now study passive TET from the finite rod to the MDOF NES, by studying numerically the damped dynamics of system (5.68a, 5.68b). Indeed, we will examine the efficacy of using a MDOF NES as passive absorber and efficient dissipater of broadband energy from the elastic rod under shock excitation. Given that the examined NESs are lightweight, local and in modular form (i.e., they can be attached to existing elastic structures with minimal structural modifications and added mass), our TET study can pave the way for applying the concept of NES to shock isolation of practical flexible systems. An additional aim of the following study is to show that weakly damped, nonlinear transitions in the system examined can be interpreted and understood by means of the FEP of the underlying Hamiltonian system; in that context, complex, multi-frequency dynamical transitions of the weakly damped system may be interpreted as transitions between branches of periodic solutions (NNMs) of the FEP. The same conclusion was drawn from our previous studies of damped transitions and TET in Sections 3.4, 4.3.2, and 5.2.3. Finally, we will perform multiscale analysis of the damped responses by the combined WT-EMD methodology discussed in Section 5.2.3; this will enable us to study the strongly nonlinear modal interactions that occur between the rod and the MDOF NES and give rise to TET. We study TET in the system depicted in Figure 5.47 by computing the asymptotic values of the corresponding energy dissipation measures (EDMs), i.e., of the percentages of shock energy of the rod that are (eventually) dissipated by the dampers of the MDOF-NES, as system parameters vary. The following parametric study is performed by numerically integrating the governing equations of motion (5.68a, 5.68b) using the previously described FE discretization. The numerical simulations are performed for a shock of the form A sin(2πt/T ), 0 < t ≤ T /2 F (t) = (5.75) 0, t > T /2 where T = 0.1T1 , where T1 is the period of the first mode of the linear rod. Moreover, we assume that the shock is applied at position d = 0.2 from the clamped (left)

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Table 5.4 The leading modal critical viscous damping rations of the uncoupled dispersive rod (ω0 = 1, L = 1.0). Normal Mode

1

2

3

4

5

6

7

8

9

10

Modal critical viscous ratio 0.2685 0.1038 0.0631 0.0453 0.0352 0.0289 0.0244 0.0211 0.0187 0.0166

end of the rod. Although the form of the applied shock is kept fixed throughout the following parametric study of TET, the shock magnitude A is changed to investigate the effect of the level of shock energy input on TET (Tsakirtzis, 2006; Tsakirtzis et al., 2007a). The finite rod with L = 1.0 and ω0 = 1 was discretized into 200 finite elements, which ensured a five-digit convergence of the eigenfrequencies and shapes of its leading modes. In the simulations weak damping for the rod was assumed, modeled by a damping matrix which was expressed as linear superposition of mass and stiffness matrices, i.e., D = a1 M + a2 K with a1 = 0 and a2 = 0.01. The leading eigenfrequencies of the uncoupled rod (with the MDOF NES detached) are listed in Table 5.3, whereas the corresponding modal critical viscous damping ratios are presented in Table 5.4. The FE model was integrated by the Newmark algorithm. Finally, the sampling frequency was chosen as less than 6% of the eigenfrequencies of the excited modes (i.e., the leading three modes) of the rod. In the following parametric study (Tsakirtzis, 2006; Tsakirtzis et al., 2007a) we vary the coupling stiffness ε (which will be proven to be a critical parameter for TET efficiency), and the magnitude A of the applied shock, for five different sets of nonlinear coefficients C1 and C2 . Moreover, we wish to study the effect of NES asymmetry on TET, that is, the effect of asymmetric nonlinear oscillator pairs on the capacity of the MDOF NES to passively absorb and dissipate shock energy from the rod. To this end, we consider the following five pairs of nonlinear stiffnesses for the MDOF NES: Application I: (C1 , C2 ) = (1.0, 0.001) Application II:

(C1 , C2 ) = (1.0, 0.01)

Application III:

(C1 , C2 ) = (1.0, 0.1)

Application IV:

(C1 , C2 ) = (1.0, 1.0)

Application V:

(C1 , C2 ) = (1.0, 10.0)

In Application I the essential stiffness nonlinearity of the pair of NES oscillators that lies the furthest from the rod was chosen to be much weaker than the corresponding nonlinearity of the pair that is directly connected to the rod through the coupling linear stiffness ε (see Figure 5.47). As we proceed from Application I to III this asymmetry decreases, until it is completely eliminated in Application III (which corresponds to a ‘symmetric’ NES), and reversed in Applications IV and V. The rationale for studying this asymmetry is that TET efficiency (i.e., the capacity of the MDOF NES to passively absorb and locally dissipate shock energy from the

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rod) depends, in essence, on the capacity of the pairs of NES oscillators (or, at least one of these pairs) to execute large-amplitude relative (internal) oscillations, since only then the dampers of the NES can dissipate major portions of the shock energy transferred from the rod. Hence, we wish to examine if a relatively weak essential nonlinearity in at least one pair of the NES oscillators affects the capacity of the NES to execute large-amplitude relative motions and yield effective TET. On the other hand, it is clear that in the other extreme, where both essential stiffnesses of the NES are weak, we should expect deterioration of TET, as this would hinter the capacity of the MDOF NES to engage in simultaneous multi-modal nonlinear interactions with the rod (see for example the results of Chapter 4). Hence, it is necessary to carefully examine how the asymmetry of the MDOF NES, and, specifically, weak essential nonlinearity in a pair of NES oscillators affects TET in this system. In each of the above five applications the FE simulations are performed for parameters L = 1.0, λ = 0.01/2, m1 = m2 = m3 = 0.1/3, ω0 = 1.0 and nonlinear stiffnesses as listed above. Zero initial conditions for the system are applied. In the series of numerical simulations performed for each application we consider coupling stiffness values in the range ε ∈ [0.1, 10] with a step of ε = 0.1 for a total of 100 values; in addition, we consider amplitudes of the applied shock in the range A ∈ [10, 200] with a step A = 10 for a total of 20 values. Hence, to each application corresponds a total of 20 × 100 = 2000 pairs (ε, A) all of which are realized in the parametric study. The computational procedure for studying TET efficiency in each of the five applications is as follows. For each pair (ε, A) we integrate numerically the FE model of the system (5.68a, 5.68b) for a sufficiently large time interval so that at least 99.5% of the input shock energy is eventually damped during the simulation; this ensures that no essential dynamics is missed in the transient simulations due to inappropriate selection of the time interval of integration. Then, we assess TET efficiency from the rod to the MDOF NES by computing the following EDM: t ελ{[v˙2 (τ ) − v˙1 (τ )]2 + [v˙3 (τ ) − v˙2 (τ )]2 }dτ × 100 (5.76) ENES,t 1 = lim 0 T t 1 ∂u(d, τ ) dτ F (τ ) dτ 0 i.e., the percentage of shock energy of the rod that is eventually dissipated by the MDOF NES; high values of ENES,t 1 indicates strong TET. As mentioned previously, the EDM (5.76) does not provide any information regarding the time scale of the TET dynamics, i.e., how fast energy fom the rod is passively absorbed and dissipated by the MDOF NES. Instead, we will focus only on the percentage of input energy dissipated by the NES, and postpone the issue of the time scale of TET until Chapters 8, 9 and 10. In Figure 5.52 we depict contour plots of the EDM ENES,t 1 as function of the parameters ε and A for Application I, i.e., for the highly asymmetric MDOF NES. We note that there is a wide region of strong TET corresponding to relatively strong coupling (ε > 4) and moderate to large amplitudes of input shock (A > 70); in this

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.52 Application I – contour plots of EDM ENES,t1 , as a function of linear coupling stiffness ε, and the shock amplitude A; Cases 1 and 2 () refer to the simulations depicted in Figures 5.53 and 5.54.

region the MDOF NES is highly efficient and dissipates a major portion of input shock (ENES,t 1 > 75%); what is even more significant from a practical point of view is that this high TET efficiency is robust to variations in the system parameters considered. However, we should note that these results correspond to zero initial conditions of the system, so there can be no assurance regarding robustness of NES efficiency with respect to different sets of initial conditions. In summary, for strong linear coupling there occurs strong TET from the rod to the MDOF NES (when it is initially at rest) over a wide range of initial shocks. Moreover, the weaker the applied shock is, the stronger the coupling stiffness should be for strong TET to occur. An additional conclusion drawn from the plot of Figure 5.52 is that, compared to the SDOF NES examined in previous chapters, strong TET in the MDOF NES occurs over wider ranges of input energies (shocks). Indeed, in Chapter 3 where SDOF NESs were considered, it was found that TET was sensitive to the level input energy, in the sense that optimal TET was achieved for specific levels of input energy and away from these levels TET deteriorated markedly (see for example the results depicted Figures 3.4 and 3.44). On the contrary, the results depicted in Figure 5.52 indicate that the MDOF NES provides better and more robust TET performance,

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Fig. 5.53 Application I – Case 1; (a, b) NES displacements [v2 (t) − v1 (t)] and [v3 (t) − v2 (t)] superimposed to the response of the right end of the rod; (c) transient energy measures.

since strong TET is maintained over wider ranges of input energy. This result is important from a practical point of view, since in engineering applications requiring effective shock absorption is achieved when strong TET performance over a wide range of shock energies. In order to study in more detail the damped dynamics governing TET from the rod to the MDOF NES we examined in detail two specific cases, labeled as Cases 1 and 2 in the contour plot of Figure 5.52. Case 1 (see Figure 5.53) corresponds to strong coupling, moderate applied shock, (ε, A) = (6, 110), and strong TET, ENES,t 1 = 81.15%; Case 2 (see Figure 5.54) corresponds to weak coupling, moderate applied shock, (ε, A) = (0.2, 110), and weak TET ENES,t 1 = 11.89%. Hence, we aim to relate the damped dynamics to the strong or weak NES efficiency of the MDOF NES for these two cases. It should be clear that the enhanced performance of the MDOF NES in Case 1 is mainly due to the large-amplitude relative displacement [v3 (t) − v2 (t)], which exceeds that of the rod end especially in the early stage of the dynamics (i.e., in the most highly energetic regime of the dynamics); this, in turn, leads to a large-amplitude relative velocity [v˙3 (t) − v˙2 (t)] and to

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.54 Application I – Case 2; (a, b) NES displacements [v2 (t) − v1 (t)] and [v3 (t) − v2 (t)] superimposed to the response of the right end of the rod; (c) transient energy measures.

strong shock energy dissipation by the damper of the second pair of NES oscillators. This is reflected in the plots of transient damped energies (see Figure 5.53c), where it is deduced that the NES dampens a significant portion of input energy during the early (highly energetic) stage of the response. The large amplitude of the relative displacement [v3 (t) − v2 (t)] in this case (which is mainly due to the weak nonlinear coupling stiffness C2 ) explains the large amount of energy damped by the viscous damper of the second pair of oscillators of the MDOF NES. It follows that the high asymmetry of the NES in Application I proves to be beneficial for TET. An investigation of the nonlinear modal (resonance) interactions giving rise to strong TET in cases like this one will be carried out in Section 5.4.3. In Figure 5.55 we present the TET efficiency plot for Application II, i.e., for reduced NES asymmetry compared to Application I. The computational procedure outlined for Application I was also applied to Application II, that is, we varied the linear coupling stiffness in the range ε ∈ [0.1, 10] for a total of 100 values, and the shock amplitude in the range A ∈ [10, 200] for a total of 20 values. This gave a total of 20 × 100 = 2000 possible pairs (ε, A), all of which where realized for constructing the NES efficiency plot of Figure 5.55. Similarly to Application I, we ensured that each of the numerical simulations was performed for a sufficiently long

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Fig. 5.55 Application II – contour plot of EDM ENES,t1 as a function of linear coupling stiffness ε, and the shock amplitude A; Case 3 () refers to the simulation depicted in Figure 5.56.

time interval, so that at least 99.5% of the input shock energy was damped in the time window considered in the simulations. In Application II (as in Application I) there is a wide region of the plot where strong TET occurs and the EDM exceeds 75%. However, we note that the region of strong TET is slightly diminished compared to Application I (see Figure 5.52). This implies that reducing NES asymmetry, reduces (even slightly) the capacity of the NES to dissipate a significant portion of the shock energy of the rod. As in Application I strong TET occurs for stiff coupling and moderate to large amplitudes of applied shock (i.e., at moderate to high energy levels). In addition, there are small regions in the range A ∈ [100, 120] and ε ∈ [8, 9] where strong TET from the rod to the NES (ENES,t 1 > 80%) take place; in one of these regions we have the global optimal value ENES,t 1 ≈ 84.11%. In Figure 5.56 we consider the simulations corresponding to ε = 8.6 and A = 100 (labeled as Case 3). We note the rapid dissipation of shock energy by the NES in the early (highly energetic) regime of the motion; this is primarily due to the high-amplitudes of the relative displacement [v3 (t) − v2 (t)]. Moreover, judging from the waveforms of the rod end response and the relative displacements of the MDOF NES we infer that the efficient dissipation of energy by the NES is caused by a series of transient resonance captures (TRCs) occurring in the transient dynamics.

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.56 Application II – Case 3; (a, b) NES displacements [v2 (t) − v1 (t)] and [v3 (t) − v2 (t)] superimposed to the response of the right end of the rod; (c) transient energy measures.

Three additional series of numerical simulations corresponding to Applications III–V are depicted in Figures 5.57–5.59. As the NES asymmetry reverses, the region of efficient dissipation of energy by the NES also diminishes. This means that NES asymmetry by itself is insufficient to improve NES efficiency: for effective TET to occur the NES asymmetry must be related to strong nonlinear characteristic C1 and weak nonlinear characteristic C2 . This conclusion is interesting from a practical point of view, i.e. for the design of MDOF NESs as passive shock absorbers. Summarizing these observations, we conclude that strong and robust TET in the system of Figure 5.47 is realized for strong linear coupling between the rod and the MDOF NES, weak coupling in the second oscillator pair of the NES (composed of the coupled masses m2 and m3 ), and strong coupling in the first oscillator pair (composed of the coupled masses m1 and m2 ). In addition, strong TET is realized for strong to moderate amplitudes of the applied shock. It appears that strong coupling between the rod and the NES and strong nonlinear stiffness C1 yield strong transfer of shock energy from the rod to the MDOF NES; whereas, weak nonlinear stiffness C2 yields effective dissipation of the transferred shock energy as it leads to largeamplitude relative response [v3 (t) − v2 (t)]. Moreover, in all cases considered the

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Fig. 5.57 Application III – contour plot of EDM ENES,t1 , as a function of linear coupling stiffness ε, and the shock amplitude A.

passive absorption of energy by the MDOF NES is broadband, contrary to conventional linear designs (based on linear vibration absorbers) where energy absorption is narrowband. This feature makes the proposed design novel and applicable to a diverse range of practical applications. To better understand the nonlinear modal interactions between the rod and the MDOF NES and the associated TRCs leading to TET, in the next section we analyze two representative numerical simulations by combined numerical wavelet transforms (WTs) and empirical mode decomposition (EMD). We show that by superimposing the WT spectra of the responses to the corresponding Hamiltonian FEPs, and studying TRCs between individual IMFs of the rod and NES responses, we can study the nonlinear modal interactions occurring in the transient nonlinear dynamics of the system under consideration.

5.4.3 Multi-Modal Damped Transitions and Multi-Scale Analysis The aim of this section is to study multi-modal interactions in the transient damped dynamics of the rod-MDOF NES system. This is performed through the use of numerical WTs and EMDs, which yields the identification of the dominant TRCs in

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

Fig. 5.58 Application IV – contour plot of EDM ENES,t1 , as a function of linear coupling stiffness ε, and the shock amplitude A.

the rod-MDOF nonlinear dynamic interaction, and paves the way for multi-scale analysis of the transient dynamics. The numerical simulations considered are computed utilizing the FE model described in the previous sections, but with no applied shock excitation. Instead, each of the examined damped motions is initialized with initial conditions corresponding to a specific point of the backbone branch of the corresponding FEP of the Hamiltonian system (studied in Section 5.4.1). We then wavelet-transform each of the relative transient responses [v1 (t) − u(L, t)], [v1 (t) − v2 (t)] and [v2 (t) − v3 (t)], superimpose the resulting WT spectra to the Hamiltonian FEP, and, finally, analyze the time series by EMD. This post-processing program helps us to clearly identify the entire sequence of multi-frequency, multimodal nonlinear transitions that occur in the damped nonlinear dynamics. The first numerical simulation is performed for System I, i.e., for the system with parameters, C1 = 1.0, C2 = 0.001, ε = 6.6, λ = 0.01/2, L = 1.0, ω0 = 1.0, m1 = m2 = m3 = 0.1/3, and initial conditions corresponding to point A on the backbone branch of the FEP of Figure 5.48. In the undamped system, this initial condition corresponds to a periodic motion (NNM) that is predominantly localized to the NES, with both the rod and the NES performing oscillations with an identical basic frequencies equal to = 3.4 rad/s. The specific initial conditions for the rod and the nonlinear attachment are approximately computed using the analytical method of Section 4.3 (with three terms in the truncated series) as follows:

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Fig. 5.59 Application V – contour plot of EDM ENES,t1 , as a function of linear coupling stiffness ε, and the shock amplitude A.

, U (x, 0) = α1 sin x 2 − ω02

,

, + α3 sin x 92 − ω02 + α5 sin x 252 − ω02 v1 (0) =

∞

V1,k ,

v2 (0) =

k=1

∞

V2,k ,

k=1

v3 (0) =

∞

V3,k

k=1

with α1 = −0.107074,

α3 = −0.0430936,

α5 = −0.00285114

V11 = 0.0639564,

V13 = 0.0781734,

V15 = 0.00494063

V21 = 0.835016,

V23 = 0.0162039,

V31 = −0.00116088,

V25 = −0.00341636

V33 = −0.0000465294,

V35 = −5.29816 × 10−7

In Figures 5.60a–d we depict the relative responses of the system, together with their WT spectra superimposed to the FEP of Figure 5.48. As energy decreases due to damping dissipation the motion makes a damped transition that traces closely the main backbone branch of the FEP. This observation confirms once again that for

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5 TET in Linear Continuous Systems with Single- and Multi-DOF NESs

sufficiently weak damping the damped response is dominated by the dynamics of the underlying Hamiltonian system. The nonlinear dynamic interaction between the rod and the NES during this damped transition is now examined in more detail. In the following exposition we adopt the notation regarding subharmonic tongues first (n) introduced in Section 5.2.3; namely, a subharmonic tongue labeled as Tp/q denotes the branch of subharmonic motions where the frequency of the dominant harmonic component of the NES response is nearly equal to (p/q)ωn , whereas that of the rod end is equal to ωn (the n-th linearized eigenfrequency of the rod). It follows that (n) for a subharmonic motion initiated on tongue Tp/q , the relative displacement between the rod and an NES mass, or the relative displacements between NES masses possess two main harmonics at frequencies ωn and (p/q)ωn . In the present simulation, the motion starts at the point of main backbone with frequency = 3.4 rad/s. From Figures 5.60b–d we deduce that the WT spectra of the relative displacements initially trace the main backbone, as energy decreases. Then, the dynamics makes (4) (2) (3) (1) transitions along a series of subharmonic tongues, such as T1/5 , T2/3 , T1/3 , T2/3 , and (1) (1) (1) T1/2 (the tongues T2/3 and T1/2 are not depicted in the FEP of Figure 5.48). Specifically, considering the response [v2 (t) − v3 (t)] (see Figure 5.60b), following its initialization at point A of the backbone curve, its frequency content is broadband, as there is (weak) excitatio