Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems (Solid Mechanics and Its Applications)

Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems I Nonlinear Targeted Energy Transfer in Mecha...

Author: Alexander F. Vakakis | Oleg V. Gendelman | Lawrence A. Bergman | D. Michael McFarland | Gaëtan Kerschen | Young Sup Lee

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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

Series Editor:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For other titles published in this series, go to www.springer.com/series/6557

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 Semenov 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-


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 Allcock 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-


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

6

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


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-

8

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


9

(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,

10

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


11

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-

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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-


17

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 = yî (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

18


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


19

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

20


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 homoclinic loop, as inferred from Figure 2.2b. The (seemingly smooth) homoclinic orbits (loops) are formed by the coalescence of the stable and unstable invariant manifolds of the unstable out-


21

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-

22


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) homoclinic loops of the unstable NNM in the Poincaré map of Figure 2.2b; this results in large-scale homoclinic 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 homoclinic 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:

24


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


25

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-


27

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

28


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,


29

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): yï + 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

30


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-


31

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):

32


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


33

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)

34


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


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

36


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


37

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.

38


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


39

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-

40


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


41

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]

42

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

0

2π 0

2π

F2 (I1a , I2a , φ1 , φ2 )dφ1 dφ2 = 0

(2.34)

0

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.


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)

0

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


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-


45

of-phase NNMs. The limit of this family of quasi-periodic orbits is a homoclinic 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 homoclinic 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 homoclinic 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


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


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 homoclinic 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


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 homoclinic 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


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

50


˜ 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


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-homogeneous 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 homoclinic 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.

52


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 homoclinic 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 homoclinic 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 homoclinic 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)


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

0

+ (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 homoclinic 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 homoclinic 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-homogeneous 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.

54


√ Fig. 2.13 Perturbation of the homoclinic 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 homoclinic 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


55

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.

56


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.


57

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)

58


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)


59

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

60


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


61

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 (♦♦♦).

62


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


63

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.

64


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


65

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

66


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)


67

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

68


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


69

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

70


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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3 Nonlinear TET in Discrete Linear Oscillators

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 ε ν¨ + 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:


⎧ ⎫ ⎡ 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)

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(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


e (u) = τ (u) = 2

∞

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[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


113

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-


115

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).

116


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.


117

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

118


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


119

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

120


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


121

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

122


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-


123

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,

124


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-


125

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

126


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


=

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

128


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)


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)

130


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)


131

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:

132


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-


133

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 )

134


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.


135

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-

136


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


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

138


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).


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-

140


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)


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.

142


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


143

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)

144


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.,


145

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 .

146


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


147

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

148


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


149

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.

150


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)


151

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 =

0

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

152

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


153

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 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,


161

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

162


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


163

the IO coincides with two homoclinic loops in phase space; these turn out to be the homoclinic 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 homoclinic 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 homoclinic 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.

164


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:


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.

166


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

178


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)

182


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.

184


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-

186


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.


187

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

188


∂ ∂ 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 ))


189

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)

190


√ 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


191

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

192


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


193

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

194


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


195

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

196


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 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.

202


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


203

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

204


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.


205

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

206


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


207

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

208


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


209

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

210


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


211

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

212


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 ε v¨ + 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


213

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 homoclinic 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 homoclinic 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

214


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

0

0


= 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

/

0

ελ (Re [u]) + (Im [u]) dt = 2 2

2

0

t |u (t)|2 dt

(3.155)

0

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.

216


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


217

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-

218


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 homoclinic 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 homoclinic orbit of the unstable


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 homoclinic 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 homoclinic 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 homoclinic 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)

220


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

0

2 t

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

0

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


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)

222


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


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).

224


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 homoclinic orbit’ realized for B = Bcr (λˆ ). This special orbit is formed by one of the branches resulting from the ‘break-up’ of the Hamiltonian homoclinic loop when weak damping is added to the system. We recall that the homoclinic 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 homoclinic 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 homoclinic 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 homoclinic 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 homoclinic 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)


225

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:

226


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 homoclinic 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 homoclinic 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 homoclinic 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 homoclinic 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 homoclinic loop assumes the limiting values, N0 (0) = √ limt →±∞ N0 (t) = a2 . In Figure 3.73 the two homoclinic 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 homoclinic solution of problem (3.165) being identified by solid line. This completes the solution of the O(1) approximation of the homoclinic 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 homoclinic 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 homoclinic 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 nonhomogeneous 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 homoclinic orbit (3.167a, b) of the O(1) problem. The perturbed homoclinic solution z1h (t) of (3.169) is written as

228


Fig. 3.73 Homoclinic 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 homoclinic 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 homogeneous solution z1HS (t) and of a particular integral z1P I (t). Key in solving the problem, is the computation of two linearly independent homogeneous solutions of (3.169), since then, a particular integral may be systematically computed by either solving the differential


229

equation satisfied by the Wronskian of the homogeneous solutions, or through the method of variation of parameters. We can easily prove (by simple substitution into the complex homogeneous equation) that one homogeneous solution of (3.169) can be computed in terms of the time (1) derivative of the O(1) homoclinic 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 homogeneous 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 homogeneous solution) (3.172)

where the real constant was selected so that the first homogeneous solution (1) (1) (1) satisfies the initial conditions x1HS (0) = 0, y1HS (0) = +1 ⇒ z1HS (0) = j . In addition, the homogeneous solution (3.173) satisfies the limiting conditions (1) (1) (t) = 0 and limt →+∞ y1HS (t) = 0. limt →+∞ x1HS To compute a second linearly independent homogeneous 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-homogeneous 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) homoclinic 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 homogeneous 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:

230


W˙ (t) = 0 ⇒ W (t) = W (0) = 1

(3.174b)

which provides a means for computing the second homogeneous 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-homogeneous 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 homogeneous 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

0

(2)

(2)

(2)

⇒ z1HS = x1HS (t) + jy1HS(t)

(Second homogeneous solution) (3.177)

As mentioned previously, the second homogeneous 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 homogeneous (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 homoclinic orbit:


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 +

0

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 homoclinic 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 homogeneous 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)

0

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)

0

This completes the solution of the problem (3.169) and computes the perturbation of the homoclinic orbit in the damped system (3.160–3.161) with O(ε1/2 ) damping. In summary, the analytic approximation of the perturbed homoclinic 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).

232


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 homoclinic 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 homoclinic 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 homoclinic 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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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.


235


236




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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)


c1 = k1 + k12 − ω2 m1 /k12 ,

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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.


243

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

246


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 homoclinic loops of the unstable NNM generated from the saddle-node bifurcation affect drastically the topologies of nearby IOs, since IOs lying inside the homoclinic 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 homoclinic 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 homoclinic 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

248


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


249

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-


251

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

252

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.


253

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

254


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+−−.


255

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

256


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


257

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).

258


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


259

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)

260


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


261

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

262


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


263

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.

264


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


265

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

266


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.


267

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-

268


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.


269

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.

270


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 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

272


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


273

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

274


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.


275

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:

276


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


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

278


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)


279

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

280


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)


281

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:

282


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


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

+ 2c )

2 1/2

−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


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


285

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.

286


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


287

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.

288


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-


289

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

290


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,


291

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

292


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


Gm (t) = e

−ελt /2

t

293

J0 (ω02 − ε2 λ2 /4)1/2 (t 2 − τ 2 )1/2 J2m (2cτ )dτ

0

≡ 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-homogeneous 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 nonhomogeneous 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)

294


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, . . .

0

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’


295

regions in Gp (t) + Gp+1 (t) in the non-homogeneous 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-homogeneous 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),

296


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)


297

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)

0

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-homogeneous 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-

298


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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4 Targeted Energy Transfer in Discrete Linear Oscillators with Multi-DOF NESs

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


305

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

306


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


307

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

308


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.


309

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

310


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)


311

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,

312


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


313

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

314


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.


315

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

316


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.


317

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)

318


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)


319

(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

320


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)


321

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 , fî of the limiting linear systems (4.10) and (4.11). Unless in the neighborhood of one of the six natural frequencies fi , fî , i = 1, 2, 3, the response of the primary subsystem is small, and the motion is localized to the nonlinear attachment.

322


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


323

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-

324


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 fî . 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-


325

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

326


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.


327

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

328


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



329

330


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)


331

√ 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)


333

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

334


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


335

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

336


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:


337

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)

338


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 +


339

√ 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)

0

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

0

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

0

0

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

340


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

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


µA˙ 5 + 3

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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.


351



352


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


357

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-

358


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.


359

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)

360


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.


361

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.

362


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.


363

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.

364


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.


365

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

366


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

2

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) ε v(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


3

they satisfy the following orthonormality relations:

L

mφr φs dx = δrs

L

and

0

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, . . . :

4


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)

0

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,


5

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.

6


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.


7

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.

8


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


9

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

10


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


11

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

12


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


13

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

14


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.


15

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

16


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


t

Edamp,NES(t) = ελ

17

[u(L, ˙ τ ) − v(τ ˙ )]2 dτ

0

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.

18


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.


19

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

20


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


21

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


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


23

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


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


25

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


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


27

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


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


29

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


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.


31

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


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.


33

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-

34


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.


35

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.

36


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.


37

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

38


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.


39

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

40


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


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)

42


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


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

44


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

0

∂t

dx +

1 2

L ∂u(x, t) 2

0

∂x

1 dx + ω02 2

L 0

u2 (x, t)dx


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

46


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


47

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

48


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;


(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.

50


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.


51

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


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


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.

54


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


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.

56


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


57

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.

58


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.


59

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.

60


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


61

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.

62


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.


63

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.

64


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.


65

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.

66


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


67

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)

68


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-homogeneous ‘forcing’ terms of the above dynamical system, and govern, in


69

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.

70


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

0

= 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 )

0

≡ εF1 (t) + ε2 F2 (t) + O(ε3 )


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-homogeneous 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]

72


+ 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

0

t

−

J0 [ω0 (t − τ )]J0 ω0 τ 2 − (2L − e)2 H (τ − (2L − e))dτ − · · ·

0

+ε

t

0

+ 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 + · · ·

0

(5.37)


73

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:

74


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

0

+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 )

0

(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

0

= 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

0

−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-homogeneous 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


75

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),

76


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


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).

78


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


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.

80


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.


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

82


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)


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)

84


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


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

86


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:


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

88


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


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 )

0

≡ 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.

90


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

92


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


≡ 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-

94


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)


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

96


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


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

98


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.


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

100


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-


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


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:

104


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

0

∂t

dx +

1 2

L ∂u(x, t) 2

0

∂x

1 dx + ω02 2

L

u2 (x, t)dx

0

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


105

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,

106


Fig. 5.49 Representative periodic orbits of the FEP of System I.


107

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.

108


Fig. 5.51 Representative periodic orbits of the FEP of System II.


109

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)

110


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


111

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

112


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

114


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


115

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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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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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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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) excitation of tongues in the range 2–4.5 rad/s, and (1) (1) (1) , T1/2 and T1/5 . At the late, low-energy also of the lower-frequency tongues T2/3 regime of the motion the response possesses low-frequency content. Considering next the response [v1 (t) − v2 (t)] (see Figure 5.60c), we note that stronger frequency components exist compared to the response [v2 (t) − v3 (t)]. Indeed the frequency content is broadband in the range 2–5 rad/s, and after the initiation of the motion on (2) (3) (4) the Hamiltonian backbone branch, subharmonic tongues, such as T2/3 , T1/3 , T1/5 , (1) (1) and T1/2 , are traced, which indicates that there occur simultaneous TRCs of the T2/3 MDOF NES with the 1st, 2nd, 3rd and 4th modes of the rod. This can be explained by observing that the relative response [v1 (t) − v2 (t)] corresponds to the deformation of the (stiffer) nonlinear stiffness C1 which belongs to the part of the MDOF NES that interacts directly with the rod through the (stiff) coupling stiffness ε. At the later (low-energy) stage of the motion the response [(v1 (t)−v2 (t)] makes a final transition back to the backbone curve, tracing the weakly nonlinear mode close to the first eigenfrequency of the rod, ω1 ; this contrasts to the low-frequency content of the relative response [v2 (t) − v3 (t)] in the low-energy regime. Hence, we observe the strongly broadband frequency content of the relative response [v1 (t) − v2 (t)] reflecting the simultaneous transient resonance interaction (TRCs) of the MDOF NES with as many as four modes of the rod. Similar broadband content is observed in the WT spectrum of the relative response [v1 (t) − u(L, t)] (see Figure 5.60d), which corresponds to the deformation of the linear coupling stiffness ε. However, in this case the frequency content of the signal is in the range 2.5–5 rad/s, with additional excitation of the lower-frequency (4) subharmonic tongue T1/5 . There is weak excitation of the low-frequency tongues (1) (1) and T1/2 , and in the late, low-energy stage of the motion the response posT2/3


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Fig. 5.60 Relative responses of System I for initial condition at point A of the FEP of Figure 5.48 ( = 3.4 rad/s): (a) time series.

122


Fig. 5.60 Relative responses of System I for initial condition at point A of the FEP of Figure 5.48 ( = 3.4 rad/s): (b, c) WT spectra superimposed to the FEP.


123

Fig. 5.60 Relative responses of System I for initial condition at point A of the FEP of Figure 5.48 ( = 3.4 rad/s): (d) WT spectra superimposed to the FEP.

sesses higher frequency content compared to the relative displacements of the NES masses. Summarizing, the nonlinear dynamical interaction between the rod and the MDOF NES is broadband, with as many as four modes of the rod participating simultaneously in these resonance interactions. Moreover, the dynamics seem to occur in the neighborhoods of subharmonic tongues of the FEP. A second numerical simulation was performed for System II, with parameters C1 = 1.0, C2 = 0.1, ε = 9.0, λ = 0.01/2, L = 1.0, ω0 = 1.0, m1 = m2 = m3 = 0.1/3, and initial condition at point B on the backbone branch of the FEP of Figure 5.50. In the undamped system this initial condition gives rise to a periodic oscillation (NNM) predominantly localized to the NES, with both the rod and the NES performing oscillations with identical basic frequencies equal to = 4.4 rad/s. The specific initial conditions for the rod and the nonlinear attachment are approximately computed as follows:

,

, 2 2 2 2 U (x, 0) = α1 sin x − ω0 + α3 sin x 9 − ω0

, 2 2 + α5 sin x 25 − ω0 v1 (0) =

∞ k=1

V1,k ,

v2 (0) =

∞ k=1

V2,k ,

v3 (0) =

∞ k=1

V3,k

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Fig. 5.61 Relative responses of System II for initial condition at point A of the FEP of Figure 5.50 ( = 4.4 rad/s): (a) time series.


125

Fig. 5.61 Relative responses of System II for initial condition at point A of the FEP of Figure 5.50 ( = 4.4 rad/s): (b, c) WT spectra superimposed to the FEP.

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Fig. 5.61 Relative responses of System II for initial condition at point A of the FEP of Figure 5.50 ( = 4.4 rad/s): (d) WT spectra superimposed to the FEP.

with α1 = −0.30063,

α3 = 0.06949,

V11 = 0.3329,

V13 = 0.115,

V21 = 1.1067,

V23 = 0.0331,

V31 = −0.6119,

α5 = −0.00229

V15 = 0.005565 V25 = −0.001618

V33 = −0.0263,

V35 = −0.00082

In Figures 5.61a–d we depict the three transient relative displacements of this system, together with the WT spectra of these responses superimposed to the FEP of Figure 5.50. As in the previous numerical simulation, there occurs a series of nonlinear multi-frequency transitions in the dynamics. In this case, the motion is initiated at point B of the main backbone (with frequency = 4.4 rad/s); during the initial regime of the motion the damped dynamics traces the main backbone of the FEP for decreasing energy. Then, there occurs a series of TRCs involving a set of modes of the rod over broad frequency ranges. Specifically, a study of the WT spectrum of the relative response [(v2 (t) − v3 (t)] (see Figure 5.61b) indicates that there occur broadband resonance interactions between the second pair of oscillators of the MDOF NES and the 1st, 2nd, 3rd and 5th modes (but not with the 4th mode) (5) (3) (2) (2) , T1/3 , T3/4 and T2/3 are traced by the of the rod, as the subharmonic tongues T1/4 damped dynamics for decreasing energy. Stronger modal interactions are noted in the spectrum of [v1 (t) − v2 (t)] in the same broadband range above ω1 ; this was ex-


127

pected, since this response represents the deformation of the stiffer nonlinear spring C1 which belongs to the pair of oscillators of the NES that is directly connected to the rod. Moreover, in the late, low-energy regime of the motion the dynamics traces the weakly localized nonlinear mode on the backbone curve close to the first rod eigenfrequency, ω1 . Higher frequency components are noted in the spectrum of the relative response [v1 (t) − u(L, t)], with its broadband content being in the range above 3 rad/s. It is interesting to note that these findings are confirmed by the EMD analysis performed below. Compared to the previous numerical simulations of System I (see Figure 5.60), we conclude that in System II there occur stronger nonlinear multi-modal interactions between the rod and the MDOF NES (since it possesses stiffer rod-NES coupling), over higher and broader frequency ranges. This should be expected, since the strong coupling between the rod and the MDOF NES excites higher frequency modes of the rod, compared to System I, and yields higher frequency nonlinear modal interactions between the rod and the MDOF NES. We now analyze the numerical simulations of Figure 5.61 of System II by EMD, in our effort to identify the dominant resonance interactions that occur between the rod and the MDOF NES, as well as the time scales (or frequencies) at which these resonance interactions are realized. A more detailed presentation of this analysis, together with the full EMD study of the numerical simulations of Figure 5.60 for System I can be found in Tsakirtzis (2006) and Tsakirtzis et al. (2007a). As in Section 5.2.3 the dominant IMFs of a time series are designated as the IMFs whose instantaneous frequencies coincide with dominant harmonics of the corresponding WT spectrum. Hence, the EMD is a multi-scale analysis with the potential to identify the ‘essential’ dynamics (resonance interactions) embedded in the measured responses of coupled systems. The results of EMD analysis of the transient responses of System II are depicted in Figures 5.62–5.64. The EMD of the relative response [v2 (t) − v3 (t)] provided six IMFs, of [v1 (t) − v2 (t)] eight IMFs, of [v1 (t) − u(L, t)] nine IMFs, and of the rodend response nine IMFs. As mentioned previously, the IMFs are oscillatory components possessing different time scales (or frequencies) embedded in the time series, and their superposition reconstructs the measured time series. However, since the IMFs are constructed in an ad hoc fashion (see Section 2.5), only a subset of IMFs are expected to be dominant, i.e., to capture the essential dynamics embedded of the signals. Moreover, the dominant IMFs have usually a physical interpretation in terms of the characteristic time scales of the signal, whereas the ‘artificial’ (unimportant) IMFs represent artificial (non-physical) oscillating modes in the data. Comparisons of the instantaneous frequencies of the IMFs to the corresponding WT spectra indicates that the relative response [v2 (t) − v3 (t)] is dominated by its 1st and 2nd IMFs (see Figures 5.62c, d); the relative response [v1 (t) − v2 (t)] by its 1st, 2nd and 3rd IMFs (see Figures 5.63d–f); and the rod end response u(L, t) by its 1st, 2nd and 3rd IMFs (see Figures 5.64d–f). These results are confirmed by the signal reconstructions depicted in Figures 5.62e, 5.63g and 5.64g which prove that the nonlinear interaction between the rod and MDOF NES is low dimensional, occurring, at most, over four time (frequency) scales.

128


Fig. 5.62 Relative response [v2 (t) − v3 (t)] of System II initiated at point B of the FEP of Figure 5.51 ( = 4.4 rad/s: (a, b) time series of 1st and 2nd (dominant) IMFs superimposed to the transient response.

By studying the instantaneous frequencies of the dominant IMFs of these responses we infer that the damped transitions possess four dominant time scales, corresponding to the 1st, 2nd, 3rd and 5th linearized eigenfrequencies of the rod. Hence, we confirm the results of the previous section, namely, that the MDOF NES engages in nonlinear interactions [or transient resonance captures (TRCs)] with four modes of the rod, so that energy exchanges between the two subsystems occur at


129

Fig. 5.62 Relative response [v2 (t) − v3 (t)] of System II initiated at point B of the FEP of Figure 5.51 ( = 4.4 rad/s: (c, d) instantaneous frequencies of the dominant IMFs superimposed to the WT spectrum of [v2 (t) − v3 (t)] (early time zoomed plots are also included). (e) signal reconstruction of the response using the superposition of the two dominant IMFs.

130


Fig. 5.63 Relative response [v1 (t) − v2 (t)] of System II initiated at point B of the FEP of Figure 5.50 ( = 4.4 rad/s): (a, b, c) time series of 1st, 2nd and 3rd (dominant) IMFs superimposed to the transient response; (d, e, f) instantaneous frequencies of the dominant IFMs superimposed to the wavelet spectrum of [v1 (t) − v2 (t)] (early time zoomed plots are also included); (g) signal reconstruction of the response using the superposition of the three dominant IMFs.

four dominant time scales. It is interesting to note that no nonlinear modal interactions between the 4th mode of the rod and the NES were observed, even though the NES does interact with the 3rd and 5th rod modes; this result is in agreement with the WT analysis of the same simulation discussed previously. The specific TRCs that occur between the IMFs of the relative displacements of the NES masses and the rod end response can be studied in detail by considering the plots of instan-


131

Fig. 5.63 Continued. Table 5.5 Main TRCs and multimodal interactions in the transient responses of System II (see Figures 5.62–5.64).

taneous IMF frequencies in Figures 5.62c, d, 5.63d–f and 5.64d–f. A synopsis of these TRCs is given in Table 5.5. The numerous TRCs listed in Table 5.5 provide an indication of the complexity of the broadband nonlinear modal interactions oc-

132



curring between the rod and the MDOF NES; these TRCs can be clearly detected and systematically categorized by analyzing the corresponding time series by EMD. Judging from the complex nonlinear TRCs occurring between the IMFs of the rod end response and the IMFs of the relative responses of the MDOF NES, we conclude that the increase of the DOF of the NES enhances its capacity to resonantly interact with multiple modes of the linear continuum, compared to the SDOF NES case. This yields a wider range of modal interactions between the rod and the MDOF NES, and enhanced broadband TET.

5.5 Plate with SDOF and MDOF NESs In our final study of TET in elastic systems we consider a more complicated system, namely, a thin plate lying on an elastic foundation with SDOF or MDOF NESs attached to it. This study is discussed in more detail in the thesis by Georgiades (2006) and in the paper by Georgiades and Vakakis (2008), so here we will only present a


133


synopsis of the main results. In addition to broadband TET, we study the strongly nonlinear modal interactions in the damped dynamics by employing the combined WT-EMD post-processing technique outlined previously. The following NES configurations will be considered in this section: (i) a single ungrounded, essentially nonlinear SDOF NES; (ii) a set of two SDOF NESs attached at different points of the plate; and (iii) a single MDOF NES with multiple essential stiffness nonlinearities. We will examine in detail the underlying dynamical mechanisms governing TET in these cases by detecting the dominant TRCs that occur between the plate modes and the NESs. Moreover, we will perform comparative studies of the performance of the three considered NES configurations with the case of single or multiple linear Tuned-Mass-Dampers (TMDs).

134


Fig. 5.64 Rod end response u(L, t) of System II initiated at point B of the FEP of Figure 5.50 ( = 4.4 rad/s): (a, b, c) time series of 1st, 2nd and 3rd (dominant) IMFs superimposed to the transient response; (d, e, f) instantaneous frequencies of the dominant IMFs superimposed to the wavelet spectrum of u(L, t) (early time zoomed plots are also included); (g) signal reconstruction of the response using the superposition of the three dominant IMFs.

The plate on the linear elastic foundation is depicted in Figure 5.65. It consists of a linear isotropic elastic plate with mass distribution per unit area equal to M, width W , length L, thickness h, and distributed proportional viscous damping per unit area equal to d. The plate is clamped on one edge only, with all other edges remaining traction-free, and is resting on a distributed elastic foundation with stiffness per unit area equal to k. The plate is assumed to be sufficiently thin, so that to a


135

Fig. 5.64 Continued

first approximation its shear deformation may be neglected (the so-called Kirchhoff assumptions). Hence, the governing partial differential equation of motion and the associated boundary conditions are given by (assuming that the plate is initially at rest) ∂ 2w ∂w + kw = F (t)δ(x − bx , y − by ) +d 2 ∂t ∂t ∂w(x, y, 0) ∂w(x, 0, t) = 0, w(x, y, 0) = 0, =0 w(x, 0, t) = 0, ∂x ∂t D∇ 4 w + M

My (0, y) = My (W, y) = My (x, L) = 0 Qy (0, y) = Qy (W, y) = Qy (x, L) = 0

(5.77)

136


Fig. 5.64 Continued

where F (t) is the applied external excitation, δ(·, ·) is Dirac’s generalized function, and the differential operator ∇ applies to both x and y directions. In (5.77) the variables My (·, ·) and Qy (·, ·) denote the internal bending moments about the yaxis and the shear forces along the y-axis of the plate, respectively (see Figure 5.65). Moreover, the flexibility D in the equation of the plate is defined as D=

Eh3 12(1 − ν 2 )

where E is the modulus of elasticity and ν Poisson’s ratio for the material of the plate (Leissa, 1993). The problem (5.77) is non-dimensionalized, with the following numerical values assigned for the plate parameters: W = L = 1,

h = 0.01,

M = 1,

D = 1,

ν = 0.3,

k = 100

(5.78)

which are in accordance to the assumptions of thin plate theory; in the following simulations the damping coefficient is assigned the value d = 10 for the case of a


137

Fig. 5.64 Continued

single force (shock) applied to the plate, and d = 15 for the case of multiple forces applied to the plate. We assume that at t = 0 a single or multiple transient forces (shocks) are applied to the plate. Each shock has the form of a half sine pulse: A sin(2πt/T ), 0 ≤ t ≤ T /2 F (t) = (5.79) 0, t > T /2 In the case of a single applied shock, its position on the plate is given by, (bx , by ) = (1, 1); whereas, in the case of multiple applied shocks, their positions on the plate are given by (bx1 , by1) = (0.6, 0.5) (for shock 1), (bx2 , by2) = (0.5, 0.5) (for shock 2) and (bx3 , by3 ) = (0.4, 0.5) (for shock 3). Unless otherwise stated, in single shock simulations the shock amplitude is taken as A = 100 and its period as T = 0.1T5 where T5 is the period of the fifth mode of the linear plate no attachment. This requirement ensures that the applied shock has sufficiently small duration to directly excite at least the leading five modes of the plate; this enables us to study the capacity of the NES(s) to passively absorb broadband vibration energy from multiple plate modes. In multiple shock simulations, the shock amplitudes are taken as A1 = 25, A2 = −100 and A3 = 25, and the shocks are assumed to possess a common period equal to T = 0.1T5. The partial differential equation in (5.77) was discretized using 4-node quadrilateral elements, and non-conforming shape functions with corner nodes (with 12

138


Fig. 5.65 Linear cantilever plate on elastic foundation.

degrees of freedom) (Zienkiewicz and Taylor, 2000; Liu and Quek, 2003). In each node, the transverse displacement s and the rotations about the x- and y-axes were considered. For this specific finite element, the explicit forms of the matrix of shape functions was derived by Melosh (1963), and expressed simply in terms of local normalized coordinates at each node. The FE discretization is described in detail in the thesis by Georgiades (2006), based on the energy approach, i.e., on the estimation of the energies of a single finite element from Kirchoff’s plate theory and based on this estimate on the derivation of the corresponding FE mass matrices and FE displacements. Using a connection matrix that indicates which nodes correspond to adjacent elements, the full structural matrices of the plate were constructed (Georgiades, 2006). Then, the discrete system was solved numerically using the Newmark Algorithm [for details, see (Geradin and Rixen, 1997)]. In Table 5.6 the leading natural frequencies and corresponding mode shapes of the unforced and undamped plate estimated using the FE simulation, are presented. A sensitivity analysis was performed to find the required number of discrete elements for convergence of the results; it was determined that a total of 10 × 10 elements (10 in each direction) were sufficient for accurate transient numerical simulations. The verification of the accuracy of the natural frequencies was performed for the case of the plate with no


139

Table 5.6 FE computation of the leading modes of the plate on elastic foundation (k = 100) with no NES attached.

elastic foundation (k = 0) using a model in ANSYS, and, also by comparing with the results reported in Leissa (1993). We now assume that a single ungrounded NES is attached at position (x, y) = (dx , dy ) of the plate. The NES is assumed to be lightweight – of mass ε 1 – and to possesses an essentially nonlinear (nonlinearizable) cubic stiffness with characteristic C, in parallel to a viscous damper λ. With the addition of the NES the modified equations of motion together with the boundary and initial conditions take the form: D∇ 4 w + M

∂ 2w ∂w + kw +d 2 ∂t ∂t

/ 0 + C[w(dx , dy ) − v(t)]3 + λ[w(d ˙ x , dy ) − v(t)] ˙ δ(x − dx , y − dy ) = F (t)δ(x − bx , y − by ) ˙ − w(d ˙ x , dy )] = 0 ε v(t) ¨ + C[v(t) − w(dx , dy )]3 + λ[v(t) w(x, 0, t) = 0,

∂w(x, 0, t) =0 ∂x

My (0, y) = My (W, y) = My (x, L) = 0 Qy (0, y) = Qy (W, y) = Qy (x, L) = 0 w(x, y, 0) = 0,

∂w(x, y, 0) = 0, ∂t

v(0) = 0,

v(0) ˙ =0

(5.80)

140


The dynamics of the NES is incorporated into the discretized FE equations of motion by expanding accordingly the system matrices, and adding a nonlinear stiffness component (due to the essential cubic nonlinearity) (Georgiades, 2006). The TET capacity of the NES is studied by estimating the following energy dissipation measure (EDM): t λ [w(d ˙ x , dy , τ ) − v(τ ˙ )]2 dτ 0 × 100 (5.81) ENES (t) = t F (τ )w(b ˙ x , by , τ )dτ 0

This represents the percentage of shock energy of the plate dissipated by the damper of the NES up to time t. It is clear that with increasing time the EDM reaches an asymptotic limit, ENES,t 1 = limt 1 ENES (t), which represents the portion of the shock energy of the plate that is eventually dissipated by the NES by the end of the oscillation. The percentage of input shock energy dissipated by the distributed viscous damping of the plate up to time instant t is computed by:

1 L W t ∂w(x, τ ) dτ dxdy d 2 0 0 ∂t 0 Eplate(t) = × 100 (5.82) t F (τ )w(b ˙ x , by , τ )dτ 0

Combining (5.81) and (5.82), the percentage of input shock energy dissipated by the plate-NES system up to time instant t is computed as follows: Etotal(t) = ENES (t) + Eplate(t)

(5.83)

Similar formulations hold when multiple SDOF NESs, MDOF NESs or linear tuned mass dampers (TDMs) are attached to the plate. We only mention that for the case of single MDOF NES attached to the plate the equations of motion are given by: ∂ 2w ∂w + kw +d 2 ∂t ∂t / 0 + C0 [w(dx , dy ) − v(t)] δ(x − dx , y − dy ) = F (t)δ(x − bx , y − by )

D∇ 4 w + M

¨ + C0 [v(t) − w(dx , dy )] + λ[v(t) ˙ − u(t)] ˙ + C1 [v(t) − u(t)]3 = 0 m1 v(t) ¨ + λ[u(t) ˙ − v(t)] ˙ + C1 [u(t) − v(t)]3 m2 u(t) + λ[u(t) ˙ − s˙ (t)] + C2 [u(t) − s(t)]3 = 0 m3 s¨(t) + λ[˙s (t) − u(t)] ˙ + C2 [s(t) − u(t)]3 = 0 w(x, 0, t) = 0,

∂w(x, 0, t) =0 ∂x


141

My (0, y) = My (W, y) = My (x, L) = 0 Qy (0, y) = Qy (W, y) = Qy (x, L) = 0 w(x, y, 0) = 0,

∂w(x, y, 0) =0 ∂t

v(0) = v(0) ˙ = u(0) = u(0) ˙ = s(0) = s˙ (0) = 0

(5.84)

Details of the FE formulation and the corresponding structural matrices of the integrated system can be found in Georgiades (2006). The effectiveness of the NES to passively absorb and locally dissipate the shock energy of the plate (i.e., the TET efficiency) is studied by computing the following energy dissipation measures (EMDs): t λ [u(τ ˙ ) − v(τ ˙ )]2 dτ ENES 1 (t) = t 0 × 100 F (τ )w(b ˙ x , by , τ )dτ 0

t λ [˙s (τ ) − u(τ ˙ )]2 dτ 0 ENES 2 (t) = t × 100 F (τ )w(b ˙ x , by , τ )dτ

(5.85)

0

i.e., the percentage of shock energy dissipated by each of the two dampers of the MDOF NES up to time t. The summation of these two EMDs provides a measure of the TET efficiency of the MDOF NES. It should be clear that the two EMDs reach definite asymptotic limits ENES 1,2,t 1 = limt 1 ENES 1,2 (t) as time progresses. The results of the simulations are post-processed using numerical WTs and EMD. In addition, we examine the portion of shock energy dissipated by each of the two subsystems, i.e., the plate and the NES(s); moreover, in certain simulations we examine the energy transaction measure (ETM) ETrans, as defined by relation (5.14) in Section 5.2.1. The ETM is an important energy measure from a physics point of view, since it helps us identify inflow or outflow of energy from the plate to the NES or vice versa; in particular, when there is inflow from the plate to the NES it holds that ETrans > 0, whereas, negative values of the ETM correspond to backscattering of energy from the NES to the plate. We recall that the ratio ETrans/ t where

t tends to zero represents the power flow from the plate to the attachment and vice versa. In the following sections all simulations are performed for a sufficiently long time window so that at least 96.5% of the energy of the system is been eventually dissipated by damping. This ensures that no essential dynamics is missed in the transient simulations due to insufficient time of numerical integration. All sets of simulations with SDOF nonlinear attachments are performed with using a (10 × 10) FE mesh for the plate. To better depict the results we divide the plate into ‘yslices’ corresponding to fixed values of y. The parametric studies of the plate with

142


an attached MDOF NES are performed with the NES located at all x-positions on three ‘y-slices’, namely, y = 1.0, 0.7 and 0.3, again using a (10 × 10) FE mesh.

5.5.1 Case of a SDOF NES In the first parametric study we examine TET in a plate forced by single or multiple shocks, possessing a single SDOF NES attachment. We perform four main sets of FE simulations, and examine the influence of the variation of the NES parameters and input energy on TET from the plate to the NES, using as criterion the portion of total energy eventually dissipated by the NES, ENES,t 1 = limt 1 ENES (t). Unless otherwise stated, the mass of the NES is taken as ε = 0.05, the nonlinear stiffness characteristic as C = 103 , and the damping coefficient as λ = 0.1. In the first set of simulations we examined the influence of the nonlinear stiffness coefficient C and the NES position on TET. In Figure 5.66 we depict the asymptotic limit of EMD ENES,t 1 , as function of C and the x-position of the NES on the ‘slice’ y = 1. In Figure 5.66a (corresponding to C in the range [100, 3000]) we depict the results for single shock excitation, whereas in Figure 5.66b (with C ∈ [100, 1000]) we depict the corresponding results for multi-shock excitation; for the simulations depicted in Figure 5.66b the mass of NES is fixed to the smaller value ε = 0.005. We conclude that for a fixed x-position of the NES, TET is robust for variations of C when C is in the range O(102) − O(103). Moreover, the variation of the xposition of the NES on a fixed ‘y-slice’ affects strongly TET; this sensitivity can be explained by the fact that certain locations of the NES may be close to nodal curves of the different modes of the plate, where the capacity of the NES to passively absorb and dissipate energy from these particular modes is hintered. This becomes clear when we depict TET efficiency as function of the position of the NES for fixed values of C (see Figure 5.67). The maximum efficiency of the NES occurs when it is positioned at the corners of the plate, with maximum values of the EDM ENES,t 1 reaching levels of 87.72% and 89.28%, for NES positions (x, y) = (0, 1) and (1, 1), respectively. The interpretation of the results depicted in Figure 5.67 (corresponding to single shock excitation) must be performed in conjunction with the results of Table 5.6, which depicts the nodal curves of the five leading modes of the linear cantilever plate (with no NES attached). The strips close to the ‘y-slices’ y = 0.8 and y = 0.9 are close to the free edge of the plate and the nodal lines of the 3rd and 5th plate modes; as a result, the NES efficiency is low in these regions. Similarly, in strips close to x = 0.2, 0.3, 0.7 and 0.8, the NES efficiency is relatively low (of the order of 40%); again, this can be interpreted by the fact that these strips are in neighborhoods of nodal curves of the 4th mode of the plate. Finally, there is a strip in the middle of the plate, x = 0.5, where the lowest value of TET is noted; this is due to the fact that this region is in the neighborhood of nodes of the 2nd and 5th modes of the plate. Moreover, as depicted in Figure 5.67 the efficiency of


143

Fig. 5.66 Parametric study of TET as function of NES stiffness C, and x-position on the plate for the ‘y-slice’ y = 1: (a) single shock excitation with NES mass ε = 0.05, and (b) multiple shock excitation.

144


Fig. 5.67 Parametric study for TET efficiency as function of NES position on the plate for C = 103 and NES mass ε = 0.005; the clamped edge is at y = 0.

the NES decreases when the NES is located closer to the clamped end where the displacements of the plate are reduced and the nonlinear effects are negligible. In the second set of numerical simulations, we consider single shock excitation and examine the influence of the damping coefficient λ on TET, with the NES attached at every possible position on the plate. In Figure 5.68 we depict the asymptotic EDM ENES,t 1 , as function of the NES damping coefficient λ in the range λ ∈ [0.01, 0.5], and the x-position of the NES for a representative ‘y-slice’, y = 1. We note a deterioration of NES efficiency with decreasing damping coefficient. This trend, however, does not necessarily mean that by indefinitely increasing NES damping we will achieve continuous enhancement of NES efficiency. Indeed, for sufficiently large values of NES damping the relative motion between the NES and the plate is expected to significantly decrease (as the connection between the plate and the NES becomes more rigid); this, in turn, will result to decrease of the relative motion across the damper of the NES, and, hence, to the deteoration of the capacity of the NES damper to dissipate shock energy. In the final set of simulations of this series, we examine the influence of the input energy and the nonlinear stiffness characteristic on TET, when the NES is located at position (x, y) = (0, 1). We assume that a single shock is applied to the plate, and examine shock amplitudes in the range A ∈ [0.001, 100]. In Figure 5.69 we depict the results of this parametric study. We note the strong dependence of TET efficiency on the magnitude of the shock: for low shock magnitudes and irrespec-


145

Fig. 5.68 Parametric study of TET as function of the damping coefficient λ and the x-position of the NES on the plate, for fixed y = 1.

Fig. 5.69 Parametric study of TET as a function of the shock magnitude (input energy) and stiffness C, for NES position at (x, y) = (0, 1).

tive of the values of C TET is of the order of 87%; however, for increasing shock magnitudes and nonlinear stiffness characteristics, TET efficiency decreases within a small range, of the order of 7%. A conclusion from this first parametric study is that strong TET can be realized from the plate to a SDOF NES, especially when the NES is located at points of the plate corresponding to antinodes of energetically high plate modes. Indeed, it is possible to passively transfer from the plate and locally dissipate at the NES as much as 87% of the shock energy of the plate. Moreover, the integrated plate-NES system can be designed for robust TET.

146


Fig. 5.70 Dominant IMFs of the response of the SDOF NES and of the plate at the point of attachment, superimposed to the corresponding time series.


147

To study in more detail the nonlinear modal interactions that give rise to TET in the plate-SDOF NES system, we isolate a specific case corresponding to a plate with parameters as specified previously, and the SDOF NES attached at (x, y) = (0, 0.5), with parameters C = 3 × 103 , λ = 0.1 and ε = 0.05. We assume that there is a single applied shock in the form of a half-sine applied at (x, y) = (1, 1), i.e., at one of the free corners of the plate. In this specific case 64.35% of the shock energy of the plate gets eventually transferred to and locally dissipated by the NES. We examine the transient nonlinear resonance interactions (TRCs) occurring between the plate response at the point of attachment to the NES, and the NES response, and focus mainly in the highly energetic, early stage of the response where the nonlinear effects are expected to be more profound. Specifically we wish to analyze nonlinear modal interactions and TRCs in the dynamics in the early time interval 0 < t < 5, where more than 75% of the shock energy is been dissipated. Following the postprocessing methodology outlined in previous sections, the damped responses of the plate and the NES are analyzed by numerical WTs and then decomposed in terms of Intrinsic Mode Functions (IMFs) using EMD (Georgiades, 2006). By superimposing the instantaneous frequencies of the IMFs to the corresponding WT spectra of the transient responses we can deduce the dominant IMFs of the responses, as depicted in Figures 5.70a–e. Following our post-processing methodology, the determination of the dominant IMFs of the plate and NES responses enables us to detect the dominant TRCs that govern TET in this case. Considering the plate response, the 5th and 6th IMFs are dominant (see Figures 5.71a, b), whereas the 1st, 2nd and 3rd IMF of the NES response are also dominant (see Figures 5.71c–e). By computing the ratios of the instantaneous frequencies of the dominant IMFs of the plate and the NES responses we can identify the possible types of k:m TRCs that occur in the transient dynamics, as well as the corresponding time intervals where these TRCs occur. Considering the instantaneous frequency plots depicted in Figure 5.71 we deduce 1:1 TRCs between (a) the 5th IMF of the plate and the 3rd IMF of the NES at frequency close to 1.9 Hz in the time interval 0.25 < t < 3.25 (TRC I); (b) the 5th IMF of the plate and the 1st IMF of the NES close to 1.9 Hz in the time interval 6 < t < 12 (TRC II); and (c) the 6th IMF of the plate and the 2nd IMF of the NES close to 1.9 Hz in the time interval 8 < t < 14 (TRC III). These TRCs are responsible for passive TET in this specific case, and this post-processing analysis enables us to identify the corresponding nonlinear resonance interactions between the embedded oscillatory modes in the plate and NES responses.

5.5.2 Case of Multiple SDOF NESs The parametric study of TET efficiency in terms of NES location carried out in the previous section revealed locations where TET is weak, and alternative locations where TET is strong (with corresponding asymptotic EDM of the order of more than 70%). The computational study carried out in this section aims to investigate possi-

148


Fig. 5.71 Instantaneous frequencies of the dominant IMFs superimposed to the WT spectra of the corresponding responses: (a, b) 6th and 7th IMF of the plate response; (c, d, e, f) 1st, 2nd, 3rd and 4th IMF of the SDOF NES response; dominant 1:1 TRCs between IMFs are indicated by I, II and III.


149

Fig. 5.72 Comparative study of TET efficiency when using two single NESs, and a set of two NESs; single shock excitation is considered.

ble enhancement of TET through the use of multiple SDOF NESs. The parameters used for the plate and the applied shock in the following numerical simulations are identical to those employed in the simulations of Section 5.5.1. In this case, however, we consider two SDOF NES attached to the plate, with each NES possessing mass ε = 0.005 (i.e., 0.5% of the total mass of the plate), stiffness C = 103 , and damping coefficient λ = 0.1. A single shock excitation is applied to the plate, of the same form and position as in the previous section. In the following numerical simulations we examine seven specific cases (designated as Cases A–G) where two SDOF NESs are located at various positions on the plate. Of specific interest are cases where the NESs are located at nodal curves of plate modes. In Figure 5.72 we depict a bar diagram depicting NES efficiencies for all seven cases considered. In that diagram we compare the efficiencies of single NESs attached separately to either one of the two locations occupied by the set of two NESs, to the efficiency of the set of two NESs when they are both simultaneously attached to the plate. In each case we indicate the position of the two NESs. For case A TET corresponding to attaching separately the two NESs is 36.13% and 21.58%, respectively, but this number increases to 61.02% when both NESs are attached simultaneously to the same locations; we note that the later number exceeds

150


the sum of the previous two, i.e., the synergetic TET achieved by the set of two NESs is enhanced compared to the sum of TETs when each of the two NESs is applied in isolation. This demonstrates a positive synergy effect of the set of two NESs, which, however, is not expected to persist in the other cases where NES locations more favourable to TET are considered. In case B, the first NES is located at position (x, y) = (0.2, 0, 8), which is a crossing point of the nodal curves of the 3rd and 4th plate modes, and the second NES at (x, y) = (0.5, 1.0), which is the location corresponding to weakest TET when a single NES is used. Again, in this case synergetic TET for the set of two NESs increases significantly to 60.4%, which exceeds the sum of the individual TETs achieved when the NESs are applied separately at the same locations. Hence, in this case TET again benefits from positive synergy of the set of NESs. To further investigate synergetic NES effects on TET, in cases C, D, E and F we consider the set of NESs at positions on the plate where high individual TETs are achieved for a single SDOF NES (between 60–70% – such locations are at the edges of the plate, x = 0, 1). For cases C, D, E and F we place the NESs at the edges of the plate x = 0, 1, with y = 0.6, 0.7, 0.8 and 0.9, respectively. For these cases we note a slight improvement (about 10%) of the synergetic efficiency of the set of two NESs (see Figure 5.72). Finally, there are two locations where the efficiency of the single SDOF NES is very high (more than 80%); these are the two free corners of the plate. The numerical simulations indicate that by attaching the two NESs simultaneously to these locations (case G), we obtain a combined TET efficiency of 89.9%, which can be considered as the optimal synergetic TET efficiency that can be achieved by the set of two NESs on the plate. We note, however, only marginal improvement in TET efficiency compared to using the two NESs in isolation at the same locations. In conclusion, the use of the set of two NESs improves TET efficiency in regions where the use of single (isolated) NESs leads to poor TET performance. In such regions there occur positive synergisms between the two NESs of the set, which leads to TET efficiencies that exceed the sum of the efficiencies of single NESs when these are applied in isolation. The use of multiple NESs, however, improves only marginally TET efficiency in locations where the isolated NESs already yield good TET performance.

5.5.3 Case of a MDOF NES We now consider TET from the plate to a single MDOF NES. The equations of motion of this system are given by (5.84), and TET efficiency is judged by the energy dissipation measures (5.85) and their long-time asymptotic values. The first set of numerical simulations of this series was performed in order to examine the effect of the linear coupling stiffness on TET. The plate parameters are identical to the ones used for the case of SDOF NES attachment, and the applied shock is the half-sine excitation used in previous simulations, defined by relation (5.79); the duration of the applied shock was selected sufficiently small to directly excite at


151

Fig. 5.73 Parametric study of TET for a MDOF NES located at every possible x-position on the ‘slice’ y = 1, for coupling stiffness C0 = 300.

least the leading five modes of the plate – however, additional plate modes may be indirectly excited through nonlinear coupling provided by the MDOF NES. The three masses of the MDOF NES are assumed to be small, m1 = m2 = m3 = 0.005/3, with the total mass of the MDOF NES being equal to the smallest mass of the SDOF NES used in the parametric study of the previous section. In this way we wish to perform a comparative study of the SDOF and MDOF NESs, without any effects due to added mass. The two nonlinear stiffnesses of the NES are selected as C1 = 5.0 and C2 = 0.1, whereas the two dampers of the NES possess identical damping coefficients λ = 0.1. In Figure 5.73 we depict the long-time asymptotic value of the combined EDM ENES 1,t 1 + ENES 2t 1 = limt 1 [ENES 1 (t) + ENES 2 (t)], which represents the portion of shock energy of the plate eventually dissipated by the two dampers of the MDOF NES, as function of the coupling stiffness C0 (in the range [10, 1000]), and the x-position of the NES for the representative ‘y-slice’ y = 1. A conclusion from these numerical results is that TET appears to be robust for varying C0 provided that this is above the threshold C0 = 100, and for every possible x-position of the NES (as depicted in Figure 5.73). Moreover, strong TET from the plate to the MDOF NES is realized for relatively stiff coupling stiffnesses. The levels of optimal TET values attained by using MDOF NESs reaches levels of 80%, which are comparable to the ones attained by using SDOF NESs. Similarly to the case of the SDOF NES (see Section 5.5.1), the variation of the position of the MDOF NES appears to strongly

152


Fig. 5.74 Parametric study of TET for a MDOF NES located at every possible position on the plate, for coupling stiffness C0 = 300.

affect TET, as depicted in Figure 5.73; however, this will be more evident in the second set of simulations that we now proceed to discuss. In the second set of simulations we examine the influence of the position of the MDOF NES on TET; for this, we examine MDOF NES placement at every possible position on the plate, and apply the same NES parameters as in the previous set of simulations. In addition, we fix the coupling stiffness to C0 = 300. In Figure 5.74 we depict TET as function of the NES position. Predictably, the highest values of TET are obtained when the MDOF NES is situated at the free corners of the plate, reaching 85.85% for NES position at (x, y) = (1, 1), and 76.67% at (0, 1). At these positions the MDOF NES can interact with all plate modes, as no nodal curves of low-order modes are located nearby. Similarly to the case of the SDOF NES, the interpretation of the results depicted in Figure 5.74 should be carried out in conjunction with Table 5.6, which depicts the nodal curves of the leading five modes of the uncoupled plate with no elastic support (labeled from hereon as the ‘plate modes’). From Figure 5.74 we deduce that the efficiency of the NES decreases when it is located closer to the clamped end, where the displacements of the plate are small and the nonlinear effects are less profound. Since passive TET is the result of nonlinear resonance interactions (TRCs) between the plate and the NES, it is reasonable to expect that in low-amplitude regimes the effectiveness of the MDOF should de-


153

teriorate. As demonstrated, however, in Section 4.1.2 and in Tsakirtzis (2006) and Tsakirtzis et al. (2007b) it is possible (under certain forcing conditions and at definite ranges of NES parameters) to achieve efficient TET from a directly forced linear system to a MDOF NES, even at low amplitude regimes; such a case, however, was not realized in the simulations considered herein. An additional set of numerical simulations examining the influence on TET efficiency of the magnitude of the applied shock and the (linear) coupling stiffness C0 is reported in Georgiades (2006) and Georgiades and Vakakis (2008). It was found that TET does not depend significantly on the variation of shock input. The analysis of the complex nonlinear modal interactions and the corresponding TRCs that govern TET in this system can be performed by applying the combined Wavelet/EMD post-processing methodology. As the analysis becomes quite involved (since each of the transient responses of all three masses of the NES that must be considered in this case) we do not present any results here, and refer the reader to the thesis by Georgiades (2006) for a detailed presentation. In the next section we conclude our study of TET in the plate-NES system by performing a comapartive study of energy absorption performance with the linear vibration absorber (or tuned mass damper – TMD); this study will demonstrate the qualitative differences in the dynamics and superior shock absorption performance of the nonlinear designs compared to conventional linear ones.

5.5.4 Comparative Study with Linear Tuned Mass Damper We now present the results of a parametric study of the plate with a linear TMD attached. We will assess the capacity of the TMD to absorb and locally dissipate shock energy from the plate by varying the TMD parameters and its position on the plate. Then we will compare the results with the corresponding ones derived when applying SDOF or MDOF NESs to the plate. We perform a series of simulations considering single and multiple shock excitations applied to the plate. In each of these sets, the efficiency of the TMD to passively absorb and locally dissipate shock energy from the plate is estimated by the following asymptotic limit: t λ [w(d ˙ x , dy , τ ) − v(τ ˙ )]2 dτ 0 ETMD,t 1 = lim ETMD (t), ETMD (t) = × 100 t t 1 F (τ )w(b ˙ x , by , τ )dτ 0

(5.86) where v(t) is the response of the TMD. This represents the portion of the shock energy of the plate that is eventually dissipated by the damper of the TMD, and is similar to the EDMs defined in previous sections to assess the TET capacity of SDOF and MDOF NESs. In the first set of these simulations we examine the potential of the TMD to absorb and dissipate shock energy by varying its stiffness and its location on the

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Fig. 5.75 Efficiency ETMD,t1 of the linear TMD as a function of its stiffness and its x-position on the plate for the ‘y-slice’ y = 1, for single shock excitation.

plate, for single shock excitation. The system parameters of the plate were defined in Section 5.5.1, the TMD mass is selected as ε = 0.05, and its damping coefficient as λ = 0.1; these are identical to the parameters used for the simulations with the SDOF NES attachment in Section 5.5.1, so the two sets of simulations can be directly compared. In Figure 5.75 we depict the efficiency of the TMD, ETMD,t 1 , as function of its stiffness and x-position on the plate, for the fixed ‘y-slice’ y = 1 (i.e., at the free edge of the cantilever plate). This result should be compared to the plot of Figure 5.66a for the SDOF NES. We note that the variation of the location of the TMD strongly affects its efficiency, in a manner similar to the SDOF and MDOF NES attachments examined previously. Indeed, when the TMD is located at positions close to nodal lines of the plate, the TMD can not interact with the corresponding plate modes, and therefore the absorption of shock energy from the plate deteriorates. Moreover, when the TMD is ‘tuned’ to the i-th plate mode, i.e., when its stiffness is equal to kln = ωi2 ε, where ωi is the i-th natural frequency of the (uncoupled and linear) plate, its efficiency in extracting energy from that mode is high. However, for relatively high stiffness values of the TMD, i.e., when it is ‘detuned’ from the leading plate modes, its efficiency deteriorates, as expected. Comparing to the plot of Figure 5.66a we note that the TET efficiency for the case of SDOF NES does not show such dependence on stiffness, and hence, its performance is more robust to stiffness variations.


155

Fig. 5.76 Efficiency ETMD,t1 of the linear TMD as a function of its stiffness and its x-position on the plate for the ‘y-slice’ y = 1, for multiple shock excitation.

In the second set of simulations we examine the effectiveness of the TMD for variation of its stiffness and its location, for the case of multiple applied shocks. The parameters used are identical to the first set of simulations presented above, but the damping coefficient of the plate was selected as d = 15, and the TMD mass as ε = 0.005; these system parameters are identical to the ones used in Section 5.5.1, where the efficiency of the SDOF NES for multiple shock excitations was examined. In Figure 5.76 we present the results for this set of simulations. Again, the stiffness and position of the TMD appear to strongly affect its efficiency. Comparing these results to the plot of Figure 5.66b we note again the insensitivity of the performance of the SDOF NES to stiffness variations for the case of multiple shock excitation. In order to compare the relative performance of the various nonlinear and linear absorber configurations considered, we performed an additional set of simulations for fixed attachment placement at (x, y) = (0, 1), away from the source of the single applied shock at (bx , by ) = (1, 1). For this set of simulations the plate parameters are defined as in Section 5.5.1, and all SDOF attachments considered possess mass ε = 0.005 (or 0.005% of the plate mass), and viscous damper coefficients λ = 0.1. For the simulations corresponding to a MDOF NES, each of the three NES masses was chosen equal to 0.005/3, and the two viscous damping coefficients where set to λ = 0.1; hence no added mass effects were introduced. For simulations where the coupling stiffness C0 varies the two nonlinear stiffness coefficients of the NES are selected as C1 = 5.0 and C2 = 0.1; when the stiffness C1 varies the other stiffness

156


Fig. 5.77 Comparison of TET efficiency of the different nonlinear and linear attachments for varying coupling stiffness.

coefficients are selected as C0 = 103 and C2 = 0.1; whereas when C2 varies the other stiffness coefficients are given by C0 = C1 = 103. In Figure 5.77 we depict the results for this set of simulations, from which we deduce that the strongest and most robust TET is achieved for the case of SDOF NES. As expected, the TMD is effective only when it is ‘tuned’ to energetically strong plate modes, and its performance rapidly deteriorates when ‘detuning’ occurs. In contrast, the essentially nonlinear SDOF and MDOF NESs possess no such detuning limitations as they lack preferential ‘tuning’ frequencies; hence, these NESs are capable of engaging in transient resonance (TRC) with plate modes at arbitrary frequency ranges, with the only controlling parameter determining the resulting sequence of TRCs being their instantaneous energies which ‘tune’ accordingly their instantaneous frequencies. Finally, we examine in more detail the performances of the optimal configurations of the SDOF NES and the TMD, in order to compare their corresponding rates of TET from the plate. For this final set of simulations we considered multiple shock excitations with plate parameters defined in Section 5.5.1 and distributed damping coefficient d = 15. The attachments are located at one of the free corners of the plate, (x, y) = (0, 1), and their parameters are chosen as ε = 0.005, λ = 0.1, C = 100 for the SDOF NES, and kln for the TMD (i.e., it is tuned to the 4th natural frequency of the uncoupled linear plate). For the specific optimal configurations considered, the corresponding TET efficiencies are 88.94% for the SDOF NES, and 82.24% for the TMD. Although the percentages of shock energy eventually dissipated by these two configurations are comparable, the corresponding rates of TET


157

Fig. 5.78 Transient energy dissipated for single shock excitation for a plate with or without attachments; the optimal configurations for the SDOF NES and the TMD are considered.

differ drastically. The rates of shock energy dissipation can be deduced from the plots of Figure 5.78, where the total energy dissipation measure Etotal(t) [see relation (5.83)] is depicted as a function of time; for comparison purposes the rate of energy dissipation in the plate with no attachment is also depicted. We note that for multi-shock excitation the required time for the integrated plate-TMD system to dissipate 90% of the applied shock energy is approximately t ≈ 10.5, whereas the corresponding time for the integrated plate-SDOF NES system is less than t = 3 (these results, of course, hold only for this multiple excitation case where three linear modes are excited with almost equal energy). Hence, nonlinear dissipation of shock energy occurs on a faster time scale, a result which is in agreement with the findings of previous works; actually, as shown in the thesis by Georgiades (2006), the rate of nonlinear energy dissipation can be further increased by employing NESs with non-smooth stiffness characteristics, a feature which has already been explored in seismic mitigation designs (Nucera et al., 2007). The issue of vibro-impact NESs will be studied in detail in Chapter 7, whereas their application to seismic mitigation will be studied in Chapter 10. In conclusion, apart from the lack of robustness of TET to changes of parameters, the rate of shock energy absorption and dissipation in the linear design is smaller compared to the nonlinear design. Summarizing, this comparative study demonstrates the improved robustness of the considered NES designs, as well as the faster rate of nonlinear shock energy dissipation when essentially nonlinear attachments are used. This is an expected

158


finding, given that the NESs do not possess the single-tuning-frequency limitation of the TMD; instead, since they possess no preferential set of resonance frequencies they are capable of engaging in TRC with any plate mode (provided that the mode has no node close to the point of attachment to the NES), over broad frequency ranges. It is this capacity for broadband energy absorption that renders the NES an efficient and adaptive passive boundary controller. From a practical point of view, it is often encountered in engineering practice the situation that due to fatigue or joint degradation the natural frequencies of a structure may gradually change, detuning attached TMDs and rendering them inefficient; in such situations NESbased designs should be able to still remain ‘tuned’ to structural modes and thus maintain the efficiency of passive shock mitigation, with no further structural or design modifications required.

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Chapter 6

Targeted Energy Transfer in Systems with Periodic Excitations

Previous chapters demonstrated that the addition of a relatively lightweight strongly nonlinear attachment to a primary (discrete or continuous) linear structure under shock excitation can drastically modify its transient dynamic response and bring about the TET phenomenon. Hence, it is not unreasonable to expect that similar salient dynamical behavior will be revealed also for the case of external periodic excitation. The transition from shock (broadband) to periodic (narrowband) excitation, however, is not trivial, and the application of nonlinear energy sinks (NESs) to structures under narrowband excitation deserves special consideration. For, example, it is not obvious that the capacity for TET of an NES under conditions of shock excitation of a primary structure can be extended to the case of periodic excitation. This chapter treats exactly this problem. We aim to demonstrate that the steady state response of a primary system under harmonic excitation with an attached NES exhibits not only common steady state and weakly modulated responses, but also a very special type of responses characterized by large modulations of the resulting oscillations; this response type is referred to as Strongly Modulated Response (SMR), and may be regarded as the extension of the TET phenomenon to structures under periodic (narrowband) excitation. Moreover, we demonstrate that SMRs are related to relaxation oscillations of the corresponding averaged dynamical flows (the slow flows of the dynamics), and in fact, one can regard SMRs as a form of repetitive TETs under the action of persistent periodic forcing. The possible application of NESs as strongly nonlinear vibration absorbers for vibration isolation of harmonically forced single- and multi-DOF primary subsystems is then discussed, and it is shown that under certain conditions, the efficiency of the NESs as vibration isolators can far exceed that of properly tuned linear absorbers (or tuned mass dampers – TMDs).

161

162


6.1 Steady State Responses and Generic Bifurcations 6.1.1 Analysis of Steady State Motions We initiate our study by considering a primary SDOF linear oscillator under harmonic external excitation with an ungrounded, lightweight and essentially nonlinear NES attached (Gendelman et al., 2006; Gendelman and Starosvetsky, 2007; Gendelman et al., 2008; Starosvetsky and Gendelman, 2008a, b). This system is described by the following set of equations, 4 y¨1 + ελ(y˙1 − y˙2 ) + (1 + εσ )y1 + ε(y1 − y2 )3 = εA cos t 3 4 3 εy¨2 + ελ(y˙2 − y˙1 ) + ε(y2 − y1 ) = 0 3

(6.1)

where y1 and y2 are the displacements of the linear oscillator and the attachment respectively, ελ the damping coefficient, εA the amplitude of the external force, and εσ a frequency detuning parameter. The parameter ε 1 is the small parameter of the problem which scales the coupling between the two oscillators, the damping forces, the amplitude of the external force, the detuning parameter, and the mass of the NES. The coefficients A, λ, σ are adopted to be of O(1). The coefficient of the nonlinear term can be modified by proper rescaling of the dependent variables and the forcing amplitude; the value (4ε/3) is chosen for the sake of convenience. We will analyze the steady state periodic responses of (6.1) by the complexification-averaging (CX-A) methodology introduced in previous chapters. To this end, we apply the following coordinate transformations, denoting the centerof-mass and relative displacements of the system, v = y1 + εy2 w = y1 − y2

(6.2)

and then switch the analysis to complex variables: ϕ1 exp(j t) = v˙ + j v ϕ2 exp(j t) = w˙ + j w

(6.3)

By (6.3) we partition the dynamics into slow and fast components, and make the additional ansatz that the sought steady state responses are in the form of fast oscillations exp(j t) modulated by slowly-varying complex amplitudes ϕi (t). Moreover, it is clear that we seek periodic solutions of (6.1) with dominant frequencies identical to the frequency of the external periodic force, and approximately equal to the eigenfrequency of the linear oscillator (that is, the frequency detuning εσ provides a slight frequency mismatch). Hence, we will be interested in fundamental nonlinear resonances of system (6.1).


163

After substitution of (6.2) and (6.3) into (6.1) and subsequent averaging over the fast oscillations of frequency unity, we obtain the following slow flow (complex modulation) equations: ϕ˙ +

εA j εσ (ϕ1 + εϕ2 ) jε (ϕ1 − ϕ2 ) − = 2(1 + ε) 2(1 + ε) 2

ϕ˙ 2 + λ(1 + ε) −

j ϕ2 + (ϕ2 − ϕ1 ) 2 2(1 + ε)

εA j εσ (ϕ1 + εϕ2 ) j (1 + ε) − |ϕ2 |2 ϕ2 = 2(1 + ε) 2 2

(6.4)

The system of equations (6.4) has a complicated structure and cannot be solved analytically. The first step towards analyzing its steady state solutions is to perform local analysis of its equilibria (fixed points). Such analysis is of significant physical interest, since these points correspond to periodic responses of the system described by equation (6.1). To find the fixed points we equate the time derivatives of (6.4) to zero (ϕ˙ 1 = ϕ˙ 2 = 0) thus obtaining the following complex algebraic relations: εA j εσ (ϕ10 + εϕ20 ) jε (ϕ10 − ϕ20 ) − = 2(1 + ε) 2(1 + ε) 2 λ(1 + ε) −

j ϕ20 + (ϕ20 − ϕ10 ) 2 2(1 + ε)

εA j εσ (ϕ10 + εϕ20 ) j (1 + ε) − |ϕ20|2 ϕ20 = 2(1 + ε) 2 2

(6.5)

By simple algebraic manipulations system (6.5) may be reduced to the following more convenient form:

σ2 2σ A2 2 2 4 6 λ + |ϕ |ϕ | + | + |ϕ | = 20 20 20 1−σ (1 − σ )2 (1 − σ )2 |ϕ20 |2 σ −1 + θ20 = sgn(1 − σ ) tan λ(1 − σ ) λ

(1 + εσ )ϕ2 − j (1 + ε)A 1 + εσ |ϕ20 | cos θ20 ϕ10 = ⇒ |ϕ10 | = 1−σ 1−σ cos θ10

(1 + ε)A (6.6) θ10 = sgn(1 − σ ) tan−1 tan θ20 − (1 + εσ ) |ϕ20 | cos θ20 The polynomial in the first of equations (6.6) can be brought into the following compact form, α1 Z + α2 Z 2 + α3 Z 3 + α4 = 0 (6.7) where

164


|ϕ20 |2 = Z, α2 =

α1 = λ2 +

2σ α3 = 1, (1 − σ )

σ2 , (1 − σ )2

α4 =

−A2 (1 − σ )2

(6.8)

Depending on the system parameters and the coefficients (6.8), the generic polynomial (6.7) can have one or three positive (real) solutions. Therefore, due to continuity one expects generically that at certain special critical values of the parameters A, λ and σ two of these solutions will coalesce, yielding bifurcations of steady state periodic solutions; generically, these will be saddle-node (SN) bifurcations (although, as we will see later, generic Hopf bifurcations can also be realized). At points of bifurcation, both the polynomial (6.7) and its derivative with respect to Z should be equal to zero: 3α3 Z 2 + 2α2 Z + α1 = 0

(6.9)

It follows that to compute the bifurcation points we need to satisfy simultaneously the set of equations (6.7) and (6.9); this yields the bifurcation curve in parameter space (A, λ, σ ) where SN bifurcations occur. Quite remarkably, the positions of fixed points of the slow flow do not depend on the value of the NES mass ε, as indicated by the following relation which yields the SN bifurcation points in parameter space: 3α3 (α1 α2 − 9α3 α4 )2 + 2α2 (α1 α2 − 9α3 α4 )(6α1 α3 − 2α22 ) + α2 (6α1 α3 − 2α22 )2 = 0

(6.10)

The projections of the solutions of (6.10) to the two-dimensional plane of parameters (A, λ) for σ = 3 are presented in Figure 6.1. Additional projections for various positive and negative detuning values σ are presented in Figures 6.2 and 6.3. The three-dimensional surface of the SN bifurcation boundary is presented in Figures 6.4 and 6.5. The plot depicted in Figure 6.4 is related to positive values of the detuning parameter, whereas the plot of Figure 6.5 corresponds to negative values of the detuning parameter. In addition to SN bifurcations, where a stable steady state solution of the slow flow simply disappears when it coalesces with an unstable one, there exists one additional generic bifurcation scenario for loss of stability, namely, the realization of Hopf bifurcations (Guckenheimer and Holmes, 1983; Wiggins, 1990). In order to study this type of bifurcations of the slow flow (6.4), we should explore the conditions for stability of the steady state solutions. To this end, we reconsider the equations of the slow flow (6.4) and introduce the small (complex) perturbations δi (t), i = 1, 2 of the fixed points in the following form: ϕ1 = ϕ10 + δ1 ,

|δ1 | 1

ϕ2 = ϕ20 + δ2 ,

|δ2 | 1

(6.11)


165

Fig. 6.1 Curves of SN bifurcations for σ = 3.

Fig. 6.2 Curves of SN bifurcations for fixed positive detuning σ .

Substituting (6.11) into (6.4) and keeping only linear terms with respect to δi , i = 1, 2 in the resulting variational equations we obtains the following linearized system of equations in terms of the perturbations, δ˙1 = −

jε j εσ (δ1 + εδ2 ) (δ1 − δ2 ) + 2(1 + ε) 2(1 + ε)

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Fig. 6.3 Curves of SN bifurcations for fixed negative detuning σ .

Fig. 6.4 Surface of the SN bifurcations for positive values of detuning σ .

δ˙1∗ =

j εσ (δ1∗ + εδ2∗ ) jε (δ1∗ − δ2∗ ) − 2(1 + ε) 2(1 + ε)

δ˙2 =

j (δ2 − δ1 ) j εσ (δ1 + εδ2 ) −λ(1 + ε)δ2 − + 2 2(1 + ε) 2(1 + ε) + j (1 + ε)|ϕ20|2 δ2 +

j (1 + ε) 2 ∗ ϕ20 δ2 2


167

Fig. 6.5 Surface of the SN bifurcations for negative values of detuning σ .

δ˙2∗ =

j (δ2∗ − δ1∗ ) j εσ (δ1∗ + εδ2∗ ) −λ(1 + ε)δ2∗ + − 2 2(1 + ε) 2(1 + ε) − j (1 + ε)|ϕ20|2 δ2∗ −

j (1 + ε) ∗2 ϕ20 δ2 2

(6.12)

where asterisk denotes complex conjugate. The characteristic polynomial of the linearized system (6.12) is given by µ4 + γ 1 µ 3 + γ 2 µ 2 + γ 3 µ + γ 4 = 0

(6.13)

with |ϕ20| = N20 , γ1 = λ(1 + ε), γ3 = λε(εσ 2 + 1)/4,

(ε + 1)2 λ2 + (ε2 σ 2 + 1) 3ε 3 3ε 3 4 2 + + N20 , + (ε2 σ − 1)N20 + γ2 = 2 4 4 4 γ4 =

3ε 2 (1 − σ )2 4 ε2 σ (1 − σ ) 2 ε2 [(1 − σ )2 λ2 + σ 2 ] N20 + N20 + 16 4 16

(6.14)

We note that the amplitude N20 provides the first-order approximation for the amplitude of steady state periodic oscillation of the relative response w = y1 − y2 [i.e., the displacement between the linear oscillator and the NES – see equations (6.1)]. This amplitude is directly related to the efficiency of steady state TET in the system considered, since as discussed in previous chapters the capacity of the NES to passively absorb and locally dissipate a significant portion of the energy of the linear oscillator is directly tied to the relative response w attaining large amplitudes. Indeed, large amplitudes of the relative response w signifies resonance interaction of the NES with the linear oscillator (which is a prerequisite for TET), and, in addition, it guarantees that the damping element coupling the NES to the linear oscillator strongly dissipates vibration energy at steady state.

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Returning now to the variational system (6.12), a generic Hopf bifurcation in the slow flow (6.4) occurs at points where the characteristic polynomial possesses a pair of pure imaginary roots, µ = ±j (6.15) where is scalar denoting approximately the characteristic frequency of the periodic orbit that is generated from the bifurcating fixed point. Clearly, this periodic orbit of the slow flow modulations (6.4) corresponds to a torus of the full dynamical system (6.1). Before proceeding to study the occurrence of Hopf bifurcations in the slow flow we mention that the linearized variational equations (6.12–6.14) can also be employed to recover the previous boundaries for SN bifurcations in parameter space. Indeed, SN bifurcations of fixed points of the slow flow (6.4) correspond to points where roots of the characteristic polynomial (6.13) are real and change sign from negative to positive as the system parameters change. This provides an alternative way for studying SN bifurcations in the steady state dynamics. Returning now to our study of Hopf bifurcations, substituting (6.15) into the characteristic polynomial (6.13) and splitting the resulting expression into real and imaginary parts, we obtain expressions for the boundary of Hopf bifurcation in parameter space and for estimating the characteristic frequency : γ32 − γ2 γ3 γ1 + γ4 γ12 = 0

1/2 γ3 1 ε2 σ 2 + ε 2 ⇒ =± = γ1 2 1+ε

(6.16)

Additional algebraic manipulations reduce the first of expressions (6.16) in the fol2 , lowing simpler form, in terms of the amplitudeZ = N20 ν1 Z 2 + v2 Z + v3 = 0 where the coefficients in (6.17) are given by 3 3 3 3 3 v1 = − ε 4 σ 2 λ2 − ε 2 σ λ2 − ε 3 σ λ2 − λ2 ε5 σ 2 − λ2 ε3 8 8 4 16 16 3 3 3 3 − ε 4 σ λ2 − λ2 ε − λ2 ε2 − ε3 σ 2 λ2 8 16 8 16 1 2 1 2 4 3 1 4 2 2 1 2 2 1 3 2 v2 = λ ε − λ ε σ − ε σ λ + ε σ λ + ε σ λ 4 4 4 4 4 1 2 5 3 1 2 2 1 3 2 2 − λ ε σ + λ ε − ε σ λ 4 4 4 1 1 1 1 1 v3 = ε 3 σ 2 λ2 − λ2 ε − λ4 ε4 σ − λ4 ε3 − λ4 ε 8 16 8 16 16 1 1 1 1 1 − λ4 ε2 − λ2 ε5 σ 4 − λ4 ε4 σ 2 − λ4 ε5 σ 2 − λ4 ε3 σ 2 8 16 8 16 16

(6.17)


169

Fig. 6.6 Curves of Hopf bifurcations for σ = 0.5 and ε = 0.05 (no SN bifurcations exist for the parameter values chosen); points I and II refer to the bifurcation points of Figure 6.7.

1 1 − λ4 ε2 σ − λ4 ε3 σ 8 4 In addition to (6.17), the amplitude Z should also satisfy (6.7). Eliminating Z from these two equations, we obtain the following Hopf bifurcation boundaries in parameter space, which also provide the boundaries of stability exchange of the fixed points of the slow flow (6.4): α1 Z1 + α2 Z12 + α3 Z13 + α4 = 0 α1 Z2 + α2 Z22 + α3 Z23 + α4 = 0

(6.18)

At points where Hopf bifurcations are realized the bifurcating fixed points are computed by the expression: , (−v2 ∓ v22 − 4v3 v1 ) Z1,2 = (6.19) 2v1 The region in parameter space of unstable fixed points of the slow flow (6.4) is bounded by the two boundaries given by (6.18). In Figure 6.6 we depict the projection of the stability boundary (or boundary of Hopf bifurcations) to the (λ, A) parameter plane for fixed values of σ andε. By now we have established the boundaries in three-dimensional parameter space (A, λ, σ ) where SN and Hopf bifurcations of the slow flow (6.4) are realized. However in cases where more than one periodic solutions co-exist for the same set of system parameters some uncertainty remains, as to which of these solutions under-

170


Fig. 6.7 Hopf bifurcations for the case of a single periodic solution of (6.4), for λ = 0.2,σ = 0.5, ε = 0.05; no SN bifurcations exist for these parameter values.

goes the bifurcation. In order to address this issue and thus complete the analysis, we need to combine the previously computed bifurcation results in a single plot. One way to present these√ results is the plot of Figure 6.7, where the amplitude of the periodic solution N20 = Z is depicted as function of the amplitude of the external harmonic excitation A for fixed values of the detuning σ , the damping λ and the NES mass ε. An alternative way is the combined bifurcation diagram of Figure 6.8 [which also depicts the boundaries of stability of the steady state periodic solution of (6.1)] where the SN and Hopf bifurcation curves are presented in the parameter plane (A, λ) for fixed values of the detuning σ and the NES mass ε. Additional bifurcation results for several parameter values are depicted in Figures 6.8–6.11. The bifurcation results presented at Figures 6.6–6.11 require some further discussion. Considering the bifurcation diagram of Figure 6.6, the region within the boundaries of Hopf bifurcations relates to the unstable periodic solutions of the slow flow (6.4), whereas the region outside these boundaries relates to stable ones. The bifurcation diagram presented in Figure 6.7 depicts Hopf bifurcations for the case where there exists a single branch of periodic solutions of the slow flow (6.4), and corresponds to the ‘slice’ for λ = 0.2 of the diagram of Figure 6.6. Considering now the boundaries for Hopf bifurcation depicted in Figure 6.8, we note that they lie inside the region where three periodic solutions exist due to SN bifurcations. In order to determine which of these periodic solutions exhibits Hopf bifurcations we construct the bifurcation diagram of Figure 6.9, from which it becomes clear that it is the low-amplitude branch of the periodic solutions. Indeed, the resulting unstable region of the low branch of solutions refers to the internal region of the Hopf bifurcation boundary presented in Figure 6.8 for damping value λ = 0.2. The interior instability region between two SN bifurcation boundaries of Figure 6.8


171

Fig. 6.8 Curves of Hopf and SN bifurcations on the plane (λ, A) for σ = 1.2 and ε = 0.05; points I–IV refer to the bifurcation points of Figure 6.9.

Fig. 6.9 Hopf and SN bifurcations for the case of co-existing periodic solutions of (6.4), for λ = 0.2, σ = 1.2, ε = 0.05.

corresponds to the instability region of the middle-amplitude solution branch in Figure 6.9 for the same value of the damping. It should be mentioned that for λ = 0.2 the large-amplitude branch of periodic solutions remains always stable for the parameter ranges considered in the plot.

172


Fig. 6.10 Curves of Hopf and SN bifurcations on the plane (λ, A) for σ = 5 and ε = 0.05; points I–IV refer to the bifurcation points of Figure 6.11.

Fig. 6.11 Hopf and SN bifurcations for the case of co-existing periodic solutions of (6.4), for λ = 0.1, σ = 5, ε = 0.05.

The last set of bifurcation diagrams (Figures 6.10 and 6.11) for σ = 0.5 is qualitatively different from the previous cases, since in this case both the low- and middle-amplitude branches of periodic solutions undergo Hopf bifuractions. Indeed the Hopf bifurcations in this case bring a qualitatively new kind of loss of stability as predicted by the stability boundaries of Figure 6.10. Thus, the lower boundary


173

Fig. 6.12 Frequency response diagram (fundamental resonance plot) for A = 0.4, λ = 0.2 and ε = 0.01; bold lines refer to unstable regions of periodic solutions, and thin lines to stable regions (SN and Hopf bifurcations are also noted).

of Hopf bifurcations in Figure 6.10 corresponds to Hopf bifurcations of the middle branch of periodic solutions of the slow flow, whereas the upper boundary in Figure 6.10 is related to bifurcations of the lower branch of periodic solutions. It is essential to note that the both branches coalesce at the fold point (A ≈ 2.18 for σ = 0.5). Additional important information concerning the local bifurcations of the periodic solutions of the slow flow (6.4) may be obtained √by constructing frequency response diagrams; these depict the amplitude N20 = Z of the steady state periodic solution as function of the detuning parameter σ , for fixed values of the amplitude of external forcing A, damping λ, and NES mass ε. In Figure 6.12 we present a representative frequency response diagram, with bifurcation points and stability types of branches of solutions also marked in that diagram. Recalling the assumptions of the analysis, the depicted frequency response provides an approximate fundamental resonance plot of system (6.1). Although plots of this type do not convey much new information compared to the previously considered bifurcation diagrams, they are directly applicable to the problem of vibration isolation since they depict the amplitudes of steady state responses in the frequency domain. We postpone discussion of the issue of vibration isolation until Section 6.3. Regarding the frequency response of Figure 6.12 we only mention at this point that there exists an upper stable branch of steady state periodic solutions, corresponding to large-amplitude stable periodic oscillations of the NES relative to the linear oscillator; this branch co-exists with a stable low-amplitude branch of periodic responses corresponding to low-amplitude relative oscillations. Additional examples of frequency response diagrams are presented in later sections.

174


Fig. 6.13 Relative response y1 (t) − y2 (t) of system (6.1) for A = 0.225, λ = 0.2, ε = 0.05 and σ = 0; initial conditions are y1 (0) = 0.29, y˙1 (0) = 0.25, y2 (0) = 0, and y˙2 (0) = −0.15.

Summarizing, the presented analysis indicates that there exist four types of bifurcations of equilibrium points of the slow flow (6.4); namely, SN bifurcations in the domain of existence of three equilibrium points; Hopf bifurcations in the domain of existence of a single equilibrium point; Hopf bifurcation of one equilibrium point in the domain of existence of three equilibrium points; and Hopf bifurcations of two equilibrium points in the domain of existence of three equilibrium points. Of course, all these scenarios are different combinations of the two generic co-dimension 1 bifurcations, namely SN and Hopf bifurcations. We note that equilibrium points of the slow flow (6.4) correspond to periodic solutions of the original dynamical system (6.1), whereas periodic orbits generated (or eliminated) by Hopf bifurcations of equilibrium points of the slow flow correspond to periodic or quasi-periodic oscillations on two-tori of the original system. A periodic or quasi-periodic oscillation on a bifurcating two-torus corresponds to rational or irrational ratio, respectively, of the frequency of the bifurcating periodic solution of the slow flow and the basic ‘fast’ frequency – taken as unity, see ansatz (6.3) – of the dynamics.


175

Fig. 6.14 Relative response y1 (t) − y2 (t) of system (6.1) for A = 0.24, λ = 0.2, ε = 0.05 and σ = 0;. initial conditions are y1 (0) = 0.29, y˙1 (0) = 0.25, y2 (0) = 0, and y˙2 (0) = −0.15.

6.1.2 Numerical Verification of the Analytical Results As explained in detail in Section 2.4, the use of the CX-A approach for the analytical treatment of essentially nonlinear systems assumes that the approximation (which is formally justified only for weakly nonlinear systems), will remain correct in the limit when the small parameter of the problem becomes of order unity. This assumption requires additional verification. In order to achieve this goal, we perform direct numerical simulations of the initial system (6.1) and verify independently the predictions of the analysis. The response regimes for certain sets of parameters are presented in Figures 6.13 and 6.14. For the parameter values used for the numerical simulations of Figure 6.13, equa3 2 tion (6.8) reduces to the √ simple form 0.04Z + Z = 0.224 , which yields the single real solution N20 = Z = 0.577. This value is in agreement with the amplitude of the numerical solution depicted in Figure 6.13. Moreover, the conditions (6.17) suggest that the combination of the parameters used for this particular simulation corresponds to a stable periodic solution [i.e., to a stable fixed point of the slow flow equations (6.4)], which again is in agreement with the findings of the numerical simulation. The parameters for the simulation depicted in Figure 6.14 were selected in order to study the response in the zone where the analysis predicts that the periodic so-

176


Fig. 6.15 Fast Fourier Transform of the response depicted in Figure 6.14.

lution is unstable. Indeed, the numerical solution is in the form of a quasi-periodic oscillation, as evidenced by the slowly modulated fast oscillation of Figure 6.14. For this particular simulation the corresponding point in parameter space is rather close to the boundary of stability; therefore it is reasonable to suggest that the modulation frequency should be close to the characteristic frequency at the point of Hopf bifurcation. According to the second of relations (6.16), this value is predicted as = 0.109. In order to measure this frequency numerically, we perform Fast Fourier Transform (FFT) of the response presented in Figure 6.14 and obtain the frequency spectrum depicted in Figure 6.15. We deduce that, in addition to the main peak corresponding to the response at the excitation frequency (i.e., the ‘fast’ oscillation equal to unity), the FFT spectrum possesses a pair of secondary peaks which are symmetric with respect to the main peak. The distance between each of the two secondary peaks and the main peak corresponds to the modulation frequency. Direct measurement yields the value 0.11, which is in near agreement with the theoretical prediction. We conclude that the analytic approach presented above yields reliable predictions of the behavior of original forced system (6.1). Moreover, the approach is rather sensitive, since the only difference between the plots of Figures 6.13 and 6.14 is a 4% difference in the forcing amplitude A. Still, this difference brings about qualitatively different responses and the analytic approach succeeds to capture this fact. So, the results of the CX-A technique are reliable and valid, at least in the


177

regime of fundamental nonlinear resonances, and as long as the slow-fast partition of the dynamics (6.3) (which was a basic assumption of the analysis) is justified. Still, we should pay attention to the special sets of initial conditions used for producing the responses of Figures 6.13 and 6.14. The reason is that the response under consideration should be close enough to the fixed point of the slow flow, since the analysis presented above is only local. It follows that if the initial conditions are not specially tailored, the flow could be well attracted to alternative response regimes which do not satisfy the assumptions of the analysis, and hence, are not predicted by it. Indeed, it is a well known feature of (forced or unforced) nonlinear dynamical systems that they may possess qualitatively different co-existing solutions; in fact Guckenheimer and Holmes (1983) present examples where chaotic solutions co-exist with arbitrarily many stable or unstable periodic solutions [see discussion on Newhouse sinks and also refer to (Newhouse, 1974)]. It is true that in many harmonically excited systems (especially weakly nonlinear ones) steady state responses such as the ones discussed above (that is, either stable or Hopf-modulated) may be the only types of steady state motions that can be possibly realized by these systems. For the essentially nonlinear system (6.1), however, this is not the case as we proceed to discuss below.

6.2 Strongly Modulated Responses (SMRs) 6.2.1 General Formulation and Invariant Manifold Approach The last claim made in the previous section is substantiated by performing numerical simulations of the original system (6.1) for the same parameter values used to generate the responses depicted in Figures 6.13 and 6.14, but now with zero initial conditions. The results are presented in Figures 6.16 and 6.17, respectively. In both plots we can see a qualitatively new type of response regime involving a strongly modulated, nearly periodic oscillation. In the beginning of each cycle the amplitudes of both responses v = y1 + εy2 and w = y1 − y2 grow slowly. Then, after a certain amplitude threshold is reached, the amplitude of the motion of the center of mass v abruptly decreases, whereas the relative response between the NES and the linear oscillator, w, is excited with subsequent characteristic decay. This process appears to be similar to transient TET realized in an impulsively forced oscillator coupled to an NES, as discussed in Chapter 3; indeed, one form of TET in this type of impulsively forced oscillators was realized through modulated fast oscillations of the relative response between the NES and the linear oscillator. In this case, however, we deal with steady state (periodic) TET caused by periodic external forcing applied to the linear oscillator. Qualitatively, the regimes of the strongly modulated responses of Figures 6.16 and 6.17 appear to be similar. Still, quantitative differences regarding the envelopes of the responses and the frequencies of the modulations can be discerned, despite the

178


Fig. 6.16 Strongly modulated responses (SMRs) of system (6.1) for A = 0.225, λ = 0.2, ε = 0.05, σ = 0 and zero initial conditions.

minor differences between the system parameters of the two simulations. Moreover, in the simulations used for computations of responses at Figures 6.13 and 6.14 the detuning parameter σ was chosen to be zero. From (6.7) it immediately follows that the averaged flow has only one fixed point in both cases, and the responses presented at Figures 6.13 and 6.14 correspond to exactly these fixed points. The results depicted in Figures 6.16 and 6.17 are very different indeed. This means that the system can exhibit steady state response regimes (such as the presented strongly modulated ones) which in principle can not be captured by local analysis of the fixed points of the averaged flow. In order to distinguish this type of steady state response from those derived by the local analysis of the previous sections and related to the fixed (equilibrium) points of the slow flow (6.4), we denote it as a Strongly Modulated Response (SMR). The


179

Fig. 6.17 Strongly modulated responses of system (6.1) for A = 0.24, λ = 0.2, ε = 0.05, σ = 0 and zero initial conditions.

width of the amplitude modulation is equal to the response amplitude, and the analytical treatment of such responses poses distinct challenges. Indeed, as discussed above in order to analyze SMRs local analysis of the slow flow equations (6.4) is insufficient, and, rather global analysis of the dynamics is required. In general, such a challenging analytical problem is hardly solvable, since the slow flow (6.4) is essentially nonlinear and evolves in four-dimensional phase space. Still, assuming that the mass ε [which can also be regarded as a mass ratio in the normalized equations (6.1)] is sufficiently small, it may be used as a small parameter for performing singular perturbation analysis. Then, the invariant manifold approach may be applied to analyze SMRs, rather similarly to the procedure outlined in Section 3.4.2 for the unforced case. It should be mentioned that in the local analysis of the previous sections the smallness of ε was not required and not assumed.

180


We begin the analysis of SMRs by combining the two first-order equations of the slow flow system (6.4) through simple manipulations, and reducing the slow flow to the following single second-order complex ordinary differential equation:

d j (1 + ε) j ε(1 − σ ) d 2 ϕ2 2 |ϕ2 | ϕ2 + αϕ2 − ϕ2 + dt 2 2(1 + ε) dt 2

j εβ (1 + εσ ) j ε (1 − σ ) j (1 + ε) εA εAβ 2 |ϕ2 | ϕ2 − αϕ2 − − ϕ2 = + 2(1 + ε) 2 2 2(1 + ε) 2 (6.20) where α=

λ(1 + ε)2 + j (1 − ε2 σ ) 2(1 + ε)

β=

j (1 + εσ ) 2(1 + ε)

In the sequence, we perform a multiple scales analysis of the response of (6.20) by introducing the new independent time scales τk = ε k t, k = 0, 1, . . . (where τ0 is the leading-order time scale, and τ1 , τ2 , . . . are slow scales of increasing order), and expressing the response and the time derivatives in (6.20) as follows: ϕ2 = ϕ2 (τ0 , τ1 , . . .) ∂ ∂ d = +ε + ···, dt ∂τ0 ∂τ1

d2 ∂2 ∂2 = + 2ε + ··· dt 2 ∂τ0 ∂τ1 ∂τ02

(6.21)

Substituting (6.21) into (6.20) and setting equal to zero the coefficients of powers of ε, we derive the following hierarchy of problems at successive orders of approximation:

j ϕ2 j ∂ 2 ϕ2 ∂ λϕ2 2 |ϕ | + − =0 (6.22a) + ϕ O(ε 0 ) : 2 2 ∂τ0 2 2 2 ∂τ02

j ϕ2 j (1 − σ ) ∂ 2 ϕ2 ∂ λϕ2 1 2 |ϕ2 |2 ϕ2 O(ε ) : 2 + − |ϕ2 | ϕ2 + + ∂τ0 ∂τ1 ∂τ1 2 2 2 4

j (1 − σ )ϕ2 j j λ(1 − σ ) ∂ λϕ2 jA σ 2 + − |ϕ2 | ϕ2 + + ϕ2 − =0 + ∂τ0 2 2 2 4 4 4 •

•

•

(6.22b)

Equation (6.22a) describes the leading-order approximation of the evolution of the slow flow (averaged) dynamics. This equation can be trivially integrated,

∂ λ j j ϕ2 + ϕ2 − |ϕ2 |2 ϕ2 = C(τ1 , τ2 , . . .) ϕ2 + (6.23) ∂τ0 2 2 2


181

where C(τ1 , τ2 , . . .) is an arbitrary function of higher-order time scales. Higherorder time scales are not considered in the current analysis, since as shown below the dynamical phenomena in question are captured by the leading-order approximations (6.22a,b); hence, for the sake of brevity only the dependences on the time scales τ0 and τ1 will be denoted explicitly in the following analysis. It follows that the fixed (equilibrium) points of (6.23) with respect to the time scale τ0 are denoted by and obey the following algebraic equation: λ j j (τ1 ) + (τ1 ) − |(τ1 )|2 (τ1 ) = C(τ1 ) 2 2 2

(6.24)

Equation (6.24) can be solved in closed form by expressing the fixed points in polar form, i.e., (τ1 ) = N(τ1 ) exp[j θ (τ1 )], and substituting in (6.24) to derive the following real equation for the amplitude of the fixed point: λ2 N 4 (τ1 ) + [N 2 (τ1 ) − N 4 (τ1 )]2 = 4|C(τ1 )|2 N 2 (τ1 )

(6.25a)

Introducing the new variable Z(τ1 ) = N 2 (τ1 ), (6.25a) is expressed in the form λ2 Z(τ1 ) + Z(τ1 )[1 − Z(τ1 )]2 = 4|C(τ1 )|2

(6.25b)

which is a cubic polynomial amenable to explicit solutions. Once Z(τ1 ) is computed, the corresponding phase θ (τ1 ) of the fixed point is evaluated by the following expression:

−1 1 − Z(τ1 ) (6.26) θ (τ1 ) = arg C(τ1 ) − tan λ The number of real and positive solutions of the cubic equation (6.25b) depends on the values of the ‘parameters’ |C(τ1 )| and λ. The homogeneous part of equation (6.25b) can be either monotonous or it may possess a maximum and a minimum. In the former case the variation of |C(τ1 )| has no effect on the number of solutions, and equation (6.25b) possesses a single positive solution. In the latter case, however, the variation of |C(τ1 )| will generate a saddle-node (SN) bifurcation of fixed points, where a new stable-unstable pair of positive fixed points is generated. In order to distinguish between these two cases we should check whether the derivative of the homogeneous part of (6.25b) with respect of Z has any real roots: √ 2 ∓ 1 − 3λ2 ∂ 2 2 λ Z + Z [1 − Z] = 0 ⇒ Z1,2 = (6.27) ∂Z 3 √ From this result it follows that for λ < 1/ 3 (i.e., for relatively weak damping) two roots of the homogeneous √ problem are generated through a SN bifurcation; at for λ > the√critical value λ = 1/ 3 these roots coalesce, and are non-existent √ 1/ 3. This is the typical structure of a cusp, with the value λ = 1/ 3 representing the critical damping value. Of course, the results of the present analysis at O(1) are similar to those reported in Section 3.4.2.4 – the only difference being that the external forcing terms appear at higher orders of approximation.

182


Fig. 6.18 Projection of the slow invariant manifold of the system (SIM) for λ = 0.2; the unstable branch is denoted by dashed line, and arrows denote hypothetic transitions (jumps) in the regime of relaxation oscillations.

It is easy to show through equation (6.23) that if a single fixed point of (6.25b) exists, it is stable (in the form of a node) with respect to the leading-order time scale τ0 . If there are three fixed points, two of them are stable (nodes) and one is unstable (a saddle point). Therefore at the leading-order approximation governed by the time scale τ0 the dynamics of the slow flow (6.4) will be attracted to either one of these nodes. In fact, equation (6.24) defines a two-dimensional slow √ invariant manifold(SIM) of the dynamics. In the case λ < 1/ 3 the fold lines L1,2 = {N(τ1 ) = Z1,2 , θ (τ1 ) ∈ [0, 2π)} divide the stable and unstable branches of the SIM. In Figure 6.18 we depict the projection of the two-dimensional SIM on the plane (N, C); the fold lines correspond to the local maximum and minimum points of the SIM. It is well-known (Arnold et al., 1994; Guckenheimer et al., 2005; Guckenheimer et al., 2006) that such a folding structure of the SIM may give rise to relaxationtype oscillations, characterized by sudden transitions (jumps) of the response during each cycle (the hypothetic sudden transitions between the two stable branches are denoted by arrows at Figure 6.18). We conjecture that such relaxation oscillations occur in the SMRs described above. Still, such motions may be possible only if the dynamical flow can reach the fold lines L1,2 , while following the two branches of the SIM with respect to the slow time scale τ1 . In order to assess this possibility one should investigate the behavior of the flow on the SIM given by (τ1 ). To this end, we consider the O(ε) suproblem (6.22b) derived by the multiple scales expansion. In particular, we are interested in the be-


183

havior of the solution on the stable branches of the SIM in the limit at the fast time scale τ0 tends to infinity, i.e., the limit (τ1 ) = lim ϕ2 (τ0 , τ1 ). Considering τ0 →+∞

equation (6.22b) in the limit τ0 → +∞, and taking into account the asymptotic stability of points on the stable branches with respect to time scale τ0 , we obtain the following equation for motion on the SIM governed by the slow time scale τ1 :

j λ(1 − σ ) jA (1 − σ ) ∂ λ i j σ 2 2 || + + − || + + − =0 ∂τ1 2 2 2 4 4 4 4 (6.28) Equation (6.28) can be written in compact form as follows:

j ∂ 2 λ j 2 ∂ + − j || − =G 2 2 ∂τ1 2 ∂τ1

j λ(1 − σ ) jA 1−σ σ 2 || − + + (6.29) G=− 4 4 4 4 Adding to (6.29) its complex conjugate and performing the necessary algebraic manipulations it is possible to extract the following closed form expression for the evolution of the slow dynamics on the SIM with respect to the slow time scale τ1 : (. / ) 2 λ − j + 2j ||2 G + j 2 G∗ ∂ = (6.30) ∂τ1 λ2 + 1 − 4 ||2 + 3 ||4 We note that for the particular case of no applied external harmonic force, A = 0, the above expression corresponds to TET in the corresponding impulsively forced or unforced system studied by the method of invariant manifolds (up to an insignificant frequency shift – see Section 3.4.2.4). This observation provides further evidence of the relationship between SMRs in the harmonically forced system and TET in the impulsively forced or unforced one. Returning now to relation (6.30) for the harmonically forced system, it is possible to reduce it to the following form: ( ) −λ + j (1 − 4σ ) ||2 + σ − λ2 (1 − σ ) − 3(1 − σ ) ||4 ∂ = / (6.31) . ∂τ1 2 λ2 + 1 − 4 ||2 + 3 ||4 Expressing this complex relation in terms of its modulus and phase through the polar transformation (τ1 ) = N(τ1 ) exp[j θ (τ1)], we obtain the following dynamical system on the cylinder (N, θ ) ∈ (R + × S 1 ) governing the slow evolution on the SIM at time scale τ1 : −λN ∂N / = . 2 ∂τ1 2 λ + 1 − 4N 2 + 3N 4 (1 − 4σ )N 2 + σ − λ2 (1 − σ ) − 3(1 − σ )N 4 ∂θ / . = ∂τ1 2 λ2 + 1 − 4N 2 + 3N 4

(6.32)

184


Fig. 6.19 Phase portrait of the slow evolution of the SIM for the case of no external harmonic excitation, A = 0.

The phase portrait of system (6.32) is presented at Figure 6.19. It is clear from the first equation of (6.32) that the phase trajectories on the upper stable branch slowly evolve directed towards the fold line L2 , whereas the trajectories on the lower stable branch are not able to reach the fold line L1 . This means that although the dynamics can make a sudden transition (jump) from the upper stable branch of the SIM to the lower one, it cannot make a similar transition back. Actually, this ends up as a rather trivial observation since in the absence of external harmonic forcing the dynamics cannot reach a non-trivial steady state regime, as it is damped out by dissipation towards the state of trivial (zero) equilibrium. In order to allow for jumps from the lower stable branch of the SIM back to the upper one (and, therefore, to provide the necessary condition for the occurrence of relaxation oscillations) the slow flow in the vicinity of L1 should undergo bifurcation. That is, at some subset of L1 the orbits of the slow flow lines on the lower branch of the SIM should become tangent to L1 . Such points correspond to fixed points of a desingularized slow flow (Guckenheimer et al., 2006), where the numerator of equation (6.30) vanishes. In order to investigate these special points, one should compute the fixed points of the slow flow equation (6.30) for arbitrary amplitudes of the external harmonic function A. The appropriate condition reads (λ − j + 2j ||2 )G + j 2 G∗ = 0

(6.33)

and possesses two sets of solutions (fixed points). The first set is trivial and is computed by setting G = 0; this solution corresponds to fixed points of the initial equa-


185

tion (6.20), i.e., to fixed points of the global flow that (quite naturally) lie on the SIM. The other set of solutions of (6.33) satisfies the following conditions: 3||4 − 4||2 + 1 + λ2 = 0 exp(2j arg G) =

−j 2 λ + j (2||2 − 1)

(6.34)

The first equation in (6.34) coincides with the equation for the fold lines; therefore, as expected, the solutions of this type describe the folded singularities of the slow flow. Bifurcations of such singularities correspond to violations of the transversality condition and, therefore, yield qualitative changes of the flow in the vicinity of the fold line. Specifically, these bifurcations result in a switch of the directions of the flow lines and therefore provide the necessary conditions for relaxation oscillations. What is even more interesting is that equations (6.34) may be solved in closed form. Indeed, introducing again the polar transformation (τ1 ) = N(τ1 ) exp[j θ (τ1)], (6.34) yields the following solutions for the positions of the singularities on the fold lines L1 and L2 : ⎡ ⎤ λN1 1/2 ⎦ L1 : N1 = Z1 , θ = 1,2 ≡ γ01 ± cos−1 ⎣ , 2 2 2 A 1 − N1 + λ ⎡ ⎤ λN 2 1/2 ⎦ L2 : N2 = Z2 , θ = 3,4 ≡ γ02 ± cos−1 ⎣ , 2 2 2 A 1 − N2 + λ N1,2 = (4/3) ± [(4/3)2 − 4(1 + λ2 )/3]1/2 ⎡ ⎤ λ ⎦ , k = 1, 2 γ0k = sin−1 ⎣ , (1 − Nk2 )2 + λ2

(6.35)

We conclude that for sufficiently weak external harmonic excitations, that is, for amplitudes of the harmonic excitation below the first critical threshold, A < A1 crit = ,

λN1

(6.36a)

(1 − N12 )2 + λ2

no bifurcation close to the lower fold line L1 can occur. Then, the slow flow in the vicinity of both fold lines of the SIM remains qualitatively similar to that depicted in Figure 6.19, providing no possibility for the occurrence of relaxation oscillations (and thus of SMRs) in the slow flow (6.4). As the forcing amplitude approaches the value A → A1 crit from below, a SN bifurcation occurs at L1 , as θ → γ01 , and a pair of singularities is formed; in the interval between these points, the flow in the

186


Fig. 6.20 Phase portrait of the slow evolution of the SIM for the case when A1 crit < A < A2 crit and σ = 0.5; only the stable branches of the SIM are depicted.

vicinity of L1 reverses direction. A similar SN bifurcation occurs in the neighborhood of the higher fold line L2 as the amplitude of the external force reaches the second (higher) critical threshold, A → A2 crit = ,

λN2

(6.36b)

(1 − N22 )2 + λ2

again from below. The representative phase portrait describing the evolution of the slow flow on the SIM for the amplitude of the external harmonic force in the range A1 crit < A < A2 crit is presented in Figure 6.20. In Figure 6.21 we present the corresponding phase portrait for the case A > A2 crit. From the plots of Figures 6.20 and 6.21 we deduce that after the occurrence of the SN bifurcations close to the fold lines L1 and L2 , there exists a subset of orbits on the SIM that carry the flow to L1 , thus providing the possibility for a jump to the upper stable branch of the SIM, and, hence, to relaxation oscillations. Indeed, the flow can reach the fold L2 and then jump down again to L1 , thus closing the loop of the relaxation oscillation and giving rise to the SMRs of the slow flow depicted in Figures 6.16 and 6.17. It is interesting to note that the previously derived critical thresholds for the amplitude of the harmonic excitation, A1 crit and A2 crit, do not depend on the detuning parameter σ . Still one cannot conclude that the SMRs are robust to changes in the detuning parameter, since the condition A > A1 crit is necessary, but by no means sufficient for the occurrence of SMRs in the slow flow (6.40). In other terms, if this condition is valid then sudden transitions (jumps) between the stable branches of


187

Fig. 6.21 Phase portrait of the slow evolution of the SIM for the case when A > A2 crit and σ = 0.5; only the stable branches of the SIM are depicted.

the SIM may occur, but there is no guarantee that the series of these transitions will accumulate to stable attractors of the slow dynamics in the form of SMRs. In order to obtain the missing sufficient conditions for the occurrence of SMRs one should investigate more delicate aspects of the slow flow dynamics (6.4). This is performed in the next section.

6.2.2 Reduction to One-Dimensional Maps and Existence Conditions for SMRs Studying carefully the phase portrait depicted in Figure 6.20 we deduce that there exists an interval of θ , namely, 1 < θ < 2 , where all orbits on the SIM arrive to the fold line L1 and then depart from it. In the regime of relaxation oscillations, an orbit in the previously constructed phase cylinder (N, θ ) ∈ R + × S 1 initially jumps from a point of this interval on L1 to the upper branch of the SIM; then it slowly evolves following an orbit of the slow flow towards the upper fold line L2 , before jumping back to the lower stable branch of the SIM; following an orbit of the slow flow it moves towards the lower fold line L1 reaches it in one of the points of the interval θ ∈ [ 1 , 2 ]; following this the orbit jumps up to the upper branch of the SIM and the cycle of the relaxation oscillation (SMR) continues indefinitely. Therefore, it is natural to consider this relaxation regime in terms of a onedimensional Poincaré map P of the interval [ 1 , 2 ] of the fold line L1 into itself:

188


P :[

1,

2]

P

−→ [

1,

2] , θ

→ P (θ )

In the regime of relaxation oscillations (SMRs) this map takes a point on the fold line L1 of the cylinder (N, θ ) ∈ R + × S 1 and maps it into L1 under the action of the slow flow (6.30). Clearly, a stable SMR will correspond to an attractor of this map (for example, a period-k fixed point), so the conditions for existence of this attractor will provide the necessary and sufficient conditions for existence of the corresponding SMR in the slow flow (6.4), and, hence, also in the original dynamical system (6.1) for NES mass ε sufficiently small. In order to construct the one-dimensional map P , we should consider separately its ‘slow’ and ‘fast’ components during a cycle of the relaxation oscillation. As far as the slow components of the map are concerned, these correspond to the parts of the relaxation cycle spent on the lower and the upper stable branches of the SIM. Hence, we may use equations (6.30) and directly connect the ‘exit’ and ‘landing’ points on the fold lines L1 and L2 . Due to complexity of the associated expressions, in the following developments the ‘slow’ components of the map P are evaluated numerically. As for the ‘fast components of the map, it is clear that the function ϕ2 should be continuous at the points of transition between the ‘fast’ and the ‘slow’ components. Therefore, to model the jumps which provide the ‘fast’ components of the map we should define appropriately the complex invariant C(τ1 ) defined by equation (6.24). If the value of C(τ1 ) is known at the point of start of the jump (the ‘exit’ point) between the two fold lines, it is possible to compute the amplitude N and phase θ corresponding to the point of ‘landing’ of the jump unambiguously, and thus to complete the definition of the map P . The procedure of numerical integration should be performed twice, however, one for each of the two stable (upper and lower) branches of the SIM; hence, two values for the invariant C(τ1 ) should be computed for each of the two ‘fast’ components of the map in order to determine the ‘landing’ points of the jumps in L1 and L2 . These two fast components correspond to the two jumps between fold lines during each cycle of the relaxation oscillation. Fortunately, these fast elements of the mapping cycle can be written down in closed form. For example, if one knows the values of N and θ at the ‘exit’ point of the jump on the fold line L1 , say (N1 , θ01 ), and denotes the ‘landing’ point on the upper stable branch of the SIM by (Nu , θu ), one may compute the value of Nu from the polynomial (6.25b) by exploiting the invariance of C(τ1 ) on the fast component of the jump, λ2 Z1 + Z1 (1 − Z1 )2 = λ2 Zu + Zu (1 − Zu )2 = 2λ2 2 1 + 1 − 3λ2 + 3 − 1 − 3λ2 ⇒ = 27 9 2 1 + 1 − 3λ2 Zu = Nu2 = 3

(6.37)


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where Z1 = N12 . Then, combining (6.24) and (6.37), one obtains the following explicit expression for the phase variable at the ‘landing’ point of the jump: 2 Nu − N12 λ −1 + θ01 θu = tan λ2 − 1 − N12 Nu2 − 1 √ 9λ 1 − 3λ2 −1 (6.38a) = θ01 + tan √ −1 + 15λ2 − 1 − 3λ2 Hence, the part of map which corresponds to the jump from the lower fold line L1 to the upper stable branch of the SIM, is very simple – the amplitude switches to Nu and the phase rotates by a constant angle. Similarly, the jump from an ‘exit’ point (N2 , θ02 ) on the upper fold line L2 to the point (Nd , θd ) on the lower stable branch of the SIM is described by the following map: 8 2 N2 → Nd = 1 − 1 − 3λ2 3

θ02

Nd2 − N22 λ + θ02 → θd = tan λ2 − 1 − Nd2 N22 − 1 √ 9λ 1 − 3λ2 −1 = θ02 − tan √ −1 + 15λ2 + 1 − 3λ2

−1

(6.38b)

It should be stressed that for each point of the interval [ 1, 2 ] only one computation is required for a single cycle of the map. The outlined construction of the one-dimensional Poincaré map P is somewhat similar to the procedure developed in Guckenheimer et al. (2006) for analyzing chaotic attractors in regimes of relaxation oscillations occurring in low-dimensional phase spaces. Clearly, not every orbit which starts from the lower fold line L1 of the SIM will land within the interval [ 1 , 2 ], since it may be attracted to alternative attractors lying either on the upper or lower stable branches of the SIM. Of course, only those points which are mapped into this interval can carry sustained relaxation oscillations and yield SMRs. Representative examples of return maps are illustrated in Figures 6.22 and 6.23. The map depicted in Figure 6.22 is defined for all points of the interval θ ∈ [ 1 , 2 ], since all of these points are mapped into the same interval under the action of the map, which is clearly contracting. Therefore applying the contracting map theorem one proves the existence of a stable attractor of the map in the interval [ 1 , 2 ], which corresponds to a sustained regime of relaxation oscillations and thus to an SMR of the slow flow (6.4). In this case, the attractor is the stable period-one fixed point θe ≈ 0.51. By increasing the detuning parameter value (with the values of the forcing amplitude and the damping parameters remaining unchanged) we notice qualitative changes in the return map P (see Figure 6.23). As it becomes clear from the plot of

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Fig. 6.22 One-dimensional map P for A = 0.6, λ = 0.2 and σ = 1; the stable attractor of the map is denoted by dashed line.

Fig. 6.23 One-dimensional map P for A = 0.6, λ = 0.2 and σ = 2.9.

Figure 6.23 all orbits of the map are inclined and not all points originating from the interval [ 1 , 2 ] on the lower fold line L1 land inside the corresponding interval on L2 . Indeed, there is a region of the interval on L1 which results in unsustainable cycles of relaxation oscillations, since orbits in the phase cylinder originating in that region are getting attracted by a stable attractor of the SIM before they can reach L2 .


191

Fig. 6.24 Sequence of maps in the region 1 < θ < 1 close to the upper critical detuning σ = σR = 2.69; the positions of the stable and unstable period-one fixed points (SMRs) are marked by bold dashed and solid lines, respectively.

In addition, it is clear that in this case the map possesses no period−k fixed points, so that no SMRs can exist. Indeed, every orbit initiated in the interval [ 1 , 2 ] on the lower fold line L1 escapes this interval after a sufficiently large number of cycles of relaxation oscillations and settles into one of the regimes of stable steady state motions studied by the local analysis of Section 6.1.1. It follows that in this case the system can exhibit transient relaxation oscillations but not sustained ones, so no SMRs are possible. From the discussion above we conclude that for higher values of the detuning parameter σ the stable period-one fixed point of the map corresponding to an SMR of the slow flow disappears. By varying the detuning parameter and by carefully studying the structure of the map, we may determine the value of σ for which the periodone attractor of the map disappears, and thus investigate the dynamical mechanism responsible for its appearance. Hence, we obtain an analytical tool for determining the frequency region of existence of SMRs. For the system considered with A = 0.6 and λ = 0.2, the boundaries of the detuning parameter within which the SMR exists are determined as σR = 2.69 > σ > σL = −2.0546. Considering the transformations relating the slow flow to the exact equations of motion (6.1), we conclude that SMRs in system (6.1) exists in an O(ε) neighborhood of the exact resonance. Our next goal is to investigate the mechanism that generates the limit cycles related to SMRs when the detuning parameter passes into the range of existence σ ∈ (σL , σR ) of these motions. In Figure 6.24 we depict a sequence of maps close to the upper critical value of the detuning parameter σ = σR = 2.69. At the criti-

192


Fig. 6.25 Projection of the stable LCO corresponding to a SMR for A = 0.3, λ = 0.2 and σ = 0.5: dashed lines refer to ‘fast’ jumps between the two stable branches of the SIM; solid lines refer to ‘slow’ evolutions on the stable branches of the SIM.

cal value σ = σR a SN bifurcation occurs, and for decreasing values of detuning a stable-unstable pair of period-one fixed points is generated. Interpreting this result in terms of the averaged problem (6.4) in four-dimensional phase space, this bifurcation generates a stable-unstable pair of limit cycle oscillations (LCOs); moving one step further, these LCOs correspond to motions on stable and unstable two-tori, i.e., SMRs, in the dynamics of the original system (6.1). We note that depending on the rotation numbers (Guckenheimer and Holmes, 1983; Wiggins, 1990) of the orbits on these two-tori, these can be either periodic (for rational rotation numbers) or quasi-periodic (for irrational rotation numbers); in turn, these lead to periodic or quasi-periodic SMRs in the original system (6.1). This global bifurcation is not related to the behavior of fixed points or of homoclinic orbits of the slow flow (6.4) (the latter are absent in this generic case), and may not be addressed by local analysis. Still, the presence of the small parameter ε (that characterizes the smallness of the NES mass compared to the mass of the linear oscillator) allows us to reduce the global flow to the one-dimensional nonlinear map P and thus to demonstrate this global bifurcation of the slow flow in terms of a local bifurcation of the map. The previous global analysis of the dynamics identified the mechanism of creation and annihilation of the stable and unstable periodic orbits (LCOs) of the slow flow in the neighborhood of the upper boundary σ = σR of the frequency detuning range of existence of SMRs. A projectionof a representative stable LCO (SMR) on the phase cylinder (N, θ ) ∈ R + × S 1 is presented at Figure 6.25. This orbit clearly depicts the slow evolution of the dynamics on the upper and lower stable branches of the SIM (denoted by solid lines), and the fast transitions (jumps – denoted by dashed lines) when the orbit reaches the fold lines L1 and L2 .


193

Fig. 6.26 A stable period-two periodic orbit of the one-dimensional map P .

Once the dynamics is reduced to the one-dimensional map P , one expects that the system will exhibit generic bifurcations that occur in general classes of this type of dynamical systems (Guckenheimer and Holmes, 1983). Indeed, period-doubling bifurcations of the map P are expected to occur at certain parameter values. For example, for A = 1.0, λ = 0.05 and σ = 0 the map possesses a stable periodtwo fixed point [or a period-two periodic orbit of the averaged slow flow (6.4)], as depicted in Figure 6.27. No additional period doubling bifurcations (i.e. from period-two to fixed points of higher order) were observed in the map, however, so that period doublings in this map appear to be rather ubiquitous. In Figure 6.2.2 the zones where period-two fixed points exist are depicted in the (λ, A) plane for σ = 0. It should be mentioned that the analytical approach developed in this section is valid only in the limit ε → 0, i.e., only for the case of lightweight NESs. The study of SMRs in the slow flow for finite values of ε requires the computation of higherorder expansions for equation (6.20) and the study of higher-order subproblems in the hierarchy (6.22). Clearly, this is a rather cumbersome task. Moreover, the value of such an endeavor is questionable anyway, since the obtained refinement in the analytical results will be of the order of the error introduced by the averaging procedure. Hence, in the following section we content ourselves to comparing the derived analytical predictions with direct numerical simulations of the original system (6.1) and of the averaged slow flow (6.4).

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Fig. 6.27 Zones of stable period-two fixed points of the map P (for σ = 0).

Fig. 6.28 Comparison between analytical prediction (thick line) and numerical simulation of (6.4) (thin line) for ε = 0.01, σ = 0; initial condition is denoted by (•), and the ‘fast’ components of the analytical map are denoted by dashed lines.

6.2.3 Numerical Simulations Our next goal is to verify numerically the analytical prediction of the existence of the SMR attractors described in the previous section. In Figures 6.28–6.30 the analytical predictions obtained by employing the one-dimensional map reduction is


195



compared to direct numerical solutions of the slow flow (6.4) for identical initial conditions and varying small parameter ε. ‘Fast’ components of the analytical solutions (based on the one-dimensional map) are computed from the invariant (6.24) and, therefore, are not related to the corresponding numerical orbits (which spiral around the slow manifold due to the pair of complex conjugate eigenvalues of the

196


Fig. 6.31 SMR computed by direct numerical simulation of (6.1) (black line) compared to the analytical prediction based on the one-dimensional map (grey line), for A = 0.6, λ = 0.2, σ = 1.0 and ε = 0.01.

corresponding Jacobian). So, only the ‘slow’ components of the analytical solutions should be compared with the numerical simulations. It is clear from these plots that the actual orbits of the slow flow (6.4) only slightly deviate from the analytically predicted orbits; however, deviations between analytical predictions and numerical simulations increase with increasing ε, as expected. Indeed, as discussed above for moderate values of ε analytical results may not be used for quantitative predictions. To get an estimate of the accuracy of the analytical solution based on the onedimensional map reduction, in Figure 6.31 we compare the analytical prediction with direct numerical simulation of the original system (6.1) with initial conditions constrained to be on the SIM and for identical parameters to the ones used for constructing the plot of Figure 6.22. The range of frequency detuning for existence of the SMR attractor is computed by the analytical solution as −0.9 < σ < 1; whereas the corresponding interval computed from direct numerical simulations of system (6.1) for ε = 0.01 is given by −0.9 < σ < 0.9. It is essential to note that we expect the accuracy of the analytical prediction to improve with decreasing ε, in agreement with the asymptotic analysis of the previous section. From this result and the comparison depicted in Figure 6.31 we conclude that the analytical predictions are in agreement with the direct numerical simulations of the SMRs. An additional use of the one-dimensional mapping technique discussed in the previous section has to do with its capacity to predict peculiar transient behavior in the response. Indeed, we showed analytically in the previous section that the slow flow dynamics may exhibit a few cycles of relaxation oscillations before the dynamics is eventually attracted to a stable (unmodulated) periodic response. In order to verify this analytical prediction we consider a case where no stable SMR attractor


197

Fig. 6.32 The analytical map diagram in the absence of a stable SMR.

exists in the slow flow (6.4); hence, we consider a system with parameters A = 0.2, λ = 0.2 and σ = 0.35. By picking the initial value of phase angle as θ (0) = 0 on the lower fold line L1 and plotting the map, we obtain the response depicted in Figure 6.32; in this case the number of cycles of the map is equal to two. In order to check this analytical prediction, in Figure 6.33 we depict the corresponding direct numerical simulation of the original system (6.1) with identical initial conditions. From the transient response of Figure 6.33 it is clear that before the dynamics settles into the stable periodic response, it exhibits two cycles of relaxation oscillations, exactly as predicted by the analytical model. It should be mentioned, however, that in order to get such coincidence, the parameter ε needed to be picked as small. The latter requirement reflects the fact that the regime of transient relaxation oscillations occurs in close proximity to the bifurcation of the stable–unstable pair of LCOs that generates the regime of SMRs in the system; hence, the structure of the dynamical flow structure of this particular system is expected to be sensitive to changes of parameters. The next direct numerical simulation is related to the analytical prediction of period doubling bifurcation of SMRs (see the plot of Figure 6.2.2). To this end, we performed simulations of system (6.1) for parameters in the analytically predicted zone of period doubling bifurcations, e.g., A = 0.8 and λ = 0.053 (Figure 6.34), as well as in the zone where a stable period-one solution of the analytical map is predicted to exist very close to the point of the period doubling bifurcation, e.g., A = 0.7 and λ = 0.065 (Figure 6.35). For these simulations the remaining parameters of the problem are chosen as ε = 0.005 and σ = 0. The numerical results are presented in two forms: (i) in terms of direct time series, and, (ii) in terms of !

two-dimensional Poincaré maps !: −→ , (w, w) ˙ → ! (w, w) ˙ on the two-

198


Fig. 6.33 Transient response of system (6.1) with parameters and initial conditions identical to those of the one-dimensional map of Figure 6.32), A = 0.3, λ = 0.2, σ = 1.1 and ε = 0.0005.

dimensional ‘cut section’ = {(v, w, v, ˙ w) ˙ ∈ R 4 , v = 0, v˙ ≥ 0} [here we employ the transformed variables (6.2)]. It is interesting to note that the numerical simulation of Figure 6.34 reveals clearly the period doubling bifurcation of the SMR. As far as the computed Poincaré maps ! are concerned, they demonstrate that in actuality both responses are chaotic-like at a time scale of O(1/ε 2 ). Of course, the analytical treatment presented in the previous section can not be used to make any statement regarding the dynamics of the system governed by this time scale [as it is restricted to O(ε) terms]. It is instructive at this point to analyze the frequency contents of a typical SMR. The analysis is performed with the help of Hilbert transform and Hilbert Vibration Decomposition (Feldman, 2006), i.e., by EMD. The harmonic components of the SMR for a case of low-amplitude harmonic excitation are presented in Figure 6.36. As expected, during most of the SMR cycle the instantaneous frequency of the dominant harmonic component of the response (i.e., of the dominant IMF) is identical to the forcing frequency, whereas the instantaneous frequency of the secondary harmonic component (the secondary IMF) is a three times multiple of that of the dominant component; this is clearly due to the essential cubic stiffness nonlinearity of system (6.1). Still, it should be mentioned that in the regions of fast transitions from high to low frequencies, the frequency of the dominant harmonic component decreases in a rather essential manner and the 1:3 TRC with the secondary component is also destroyed. This means that in these regions the dynamics escapes 1:1 fundamental TRC with the external force and is recaptured again into resonance after dissipation of energy has occurred. A similar phenomenon of breakdown of 1:1 fundamental resonance during ‘jump down’ fast-scale frequency transitions is revealed in the strongly modulated quasi-


199

!

Fig. 6.34 Poincaré map !: −→ , (w, w) ˙ → !(w, w), ˙ and corresponding time series of (6.1) in the analytically predicted zone of SMR period doubling.

!

Fig. 6.35 Poincaré map !: −→ , (w, w) ˙ → ! (w, w), ˙ and corresponding time series of (6.1) for an SMR close to the point of period doubling bifurcation.

periodic response (SMR) corresponding to large-amplitude external harmonic excitation (see Figure 6.37). In this case the frequency of the dominant harmonic component overshoots above unity when the fast ‘jump down’ takes place. The secondary component behaves in an even more complicated manner, as its instan-

200


Fig. 6.36 Hilbert decomposition of the w(t) component of the SMR: instantaneous frequency and amplitudes of the dominant and secondary harmonic components (IMFs) of the response for A = 0.2, λ = 0.2 and ε = 0.05; dominant IMF — solid line, secondary IMF - - - dashed line.

taneous frequency varies essentially when the frequency of the first component is almost constant; hence, no clear TRC occurs in this case. One may conjecture that the secondary harmonic component passes through a number of resonances with the dominant one, but the simulation can not clearly confirm this conjecture. Based on the above theoretical consideration and analysis of the frequency content of SMRs, the relationship between SMRs in the periodically forced system and TET in the impulsively forced one is revealed. Viewed in the context of TRCgoverned TET discussed in previous chapters, SMRs are governed by a similar underlying mechanism, namely, repeated (periodic or quasi-periodic) series of TRCs of the steady state dynamics during the ‘slow’ components of the SMRs, with subsequent escapes during the ‘fast’ components of the SMRs, followed by recaptures. The series of TRCs are related to increased TET from the linear oscillator (LO) to the NES and dissipation of vibration energy by the NES damper. It was shown in this section that for the case of small NES mass ε, both TET in impulsively forced oscillators and SMRs in periodically forced ones, may be successfully treated with the help of asymptotic approaches based on singular perturbations and invariant manifold considerations. The only difference between these two cases is that in the case of TET there is a single fast transition (jump) of the transient


201

Fig. 6.37 Hilbert decomposition of the w(t) component of the SMR: instantaneous frequencies and amplitudes of the dominant and secondary harmonic components (IMFs) of the response for A = 1.4, λ = 0.2 and ε = 0.05; dominant IMF — solid line, secondary IMF - - - dashed line.

dynamics from the breakdown point of the SIM; whereas for SMRs such jumps occur periodically or quasi-periodically. In addition, there occur reverse jumps during which fast frequency increases take place in the steady state dynamics. These reverse jumps occur due to energy inputs provided by the exciting force, and are consistent with the fact that during an SMR there occurs an energy balance between the input energy provided by the external harmonic excitation and the energy dissipated by the dampers of the NES and the LO. Thus, the relationship between transient TET and steady state SMRs is clear, both from the mathematical and physical points of view: the SMRs can be viewed, in essence, as periodic or quasi-periodic versions of transient TET. Still, the fact that the periodic (or quasi-periodic) sequence of TETs noted in the SMRs occurs under the action of constant-amplitude harmonic excitations, is still somewhat puzzling. One may conjecture that an SMR represents a non-trivial consequence of the interaction of external (fundamental) and subharmonic TRCs occurring in the essentially nonlinear system (6.1). In the following section we examine the use of NESs as vibration absorbers of steady state motions of harmonically forced oscillators.

202


6.3 NESs as Strongly Nonlinear Absorbers for Vibration Isolation The most popular current solution in passive vibration isolation designs is the linear vibration absorber or tuned mass damper (TMD), where an additional linear SDOF oscillator is added to an existing linear or weakly nonlinear structure for the purpose of attenuating vibration over a narrow frequency range centered at the natural frequency of the absorber (Frahm, 1911; Ormondroyd and Den Hartog, 1928). The effective bandwidth of the TMD is governed by the damping of the absorber, and a trade-off exists between attenuation efficiency and bandwidth (Ormondroyd and Den Hartog, 1928; Bykhovsky, 1980). However the use of a linear absorber poses distinct problems when the excitation frequency is not fixed, and the frequency response in the neighborhood of structural resonances outside the narrowband action of the TMD can be adversely affected, to the extent that resonant peaks can become very steep (Roberson, 1952). To achieve linear broadband vibration absorption, multi-absorber designs have been considered both theoretically (Carcaterra et al., 2005, 2006; Koç et al., 2005; Carcaterra and Akay, 2007) and experimentally (Akay et al., 2005). In addition, adaptive tuned mass dampers have also been considered towards this goal (Bonello et al., 2005; Brennan, 2006). Alternative designs employed vibration absorbers with nonlinear stiffness elements; Carella et al. (2007a, 2007b), Virgin and Davies (2003) and Virgin et al. (2007) considered buckled struts and absorbers with geometrically nonlinear stiffnesses and studied their vibration isolation capacities; their approaches, however, differ from the approach considered in this work, as no targeted energy transfer was considered. In a separate series of studies, nonlinear vibration absorbers with linearized stiffness characteristics were studied (Shaw et al., 1989; Natsiavas, 1992, 1993a,b,c; Rice and McCraith, 1987), but the case of the strong (essential) stiffness nonlinearity received less attention. The methods and results presented in the previous sections of this chapter enable one to investigate the variety of response regimes exhibited by the NES attached to a primary SDOF linear oscillator under external harmonic excitation, and on the condition of 1:1 fundamental resonance. Therefore, it is possible to employ the previously developed techniques and the derived results in order to assess the performance of the NES as strongly nonlinear vibration absorber. Moreover, the followed approach can be generalized to the more general class of discrete or continuous primary linear systems with more degrees of freedom possessing SDOF or MDOF essentially nonlinear attachments.

6.3.1 Co-existent Response Regimes The treatment presented in the previous sections allowed us to figure out two main types of the steady state regular response regimes of system (6.1); namely, time-


203

periodic or weakly modulated steady state responses [corresponding to equilibrium points or LCOs generated through SN or Hopf local bifurcations of the slow flow (6.4)], and strongly modulated responses (SMRs) [due to relaxation-type oscillations generated from global bifurcations in the slow flow (6.4)]. Moroever, for this essentially nonlinear system one can expect ever more complicated irregular steady state responses, involving high-order resonances or even chaotic orbits, although such responses are typically realized for relatively large values of forcing amplitudes A. In this section we will restrict our consideration only to cases where 1:1 fundamental resonances occur, i.e., when the primary system and the NES oscillate with frequencies that are nearly identical to the forcing frequency. Consequently, the two types of regular steady state responses mentioned above are the only possible responses for the system. In order to assess the efficiency of the NES as nonlinear vibration absorber, one should in principle describe all possible responses realized at the ranges of system parameters of interest, since by omitting any co-existing stable steady state regimes we may jeopardize the efficiency and robustness of the our vibration isolation designs. Then, it is possible to decide whether in the worst case scenario we get satisfactory conditions of vibration isolation, excluding the possibility of transitions to other unfavorable response regimes for small changes of the system parameters or initial conditions. So, it is of considerable importance to be able to predict the co-existence of the possible response regimes over the frequency and parameter ranges considered in the vibration isolation design. Not less important task is a prediction of domains of attraction for every response regime that can be realized in the forced system. This problem is of special interest due to the multiplicity of possible steady state responses. Even if some of these responses are not favorable and compatible to the vibration isolation objective, and one cannot eliminate these regimes by appropriate choice of parameters, it may be still possible to reduce their domains of attraction in the space of initial conditions. It is clear that every practical system is expected to work only in certain finite range of possible initial conditions and the designer should be able to estimate this region. If one can design the system in a way that these realistic initial conditions will never lead the dynamics to undesired response regimes, then the problem of vibration isolation can still be solved despite theoretical possibility of problematic responses. In this section, we present certain examples of co-existence of different timeperiodic or weakly modulated steady state responses and SMRs. For the sake of convenience, we make use of the frequency response diagrams described in Section 6.1, and for each diagram we indicate the zone of existence of SMRs as well. Representative frequency response diagrams corresponding to different topologies of branches of responses are depicted in Figures 6.38–6.40. As as will be discussed below, co-existing regimes of stable SMRs can also provide favorable conditions for steady state TET. This should not be surprising given that SMRs can be viewed as periodic or quasi-periodic versions of TET realized in transiently forced oscillators (see our previous discussion in Section 6.2). These results suggest a satisfactory agreement between the analytically predicted and numerically obtained periodic steady state response amplitudes. In order to ver-

204


Fig. 6.38 Numerical verification of the frequency response diagram for A = 0.4, λ = 0.2 and ε = 0.01: (O) numerical integration; bold (thin) line indicates unstable (stable) periodic response.


ify numerically the co-existence of the various system regimes predicted in the previous section, we have picked the most interesting cases which reveal the existence of three distinct regimes for the same set of parameters but for different initial conditions. The selected cases relate to the frequency response diagrams presented in


205


Fig. 6.41 Co-existence of three response regimes corresponding to different initial conditions for the system with parameters A = 0.4, λ = 0.2, σ = 1.5 and ε = 0.001.

Figures 6.38–6.40, and a representative result of three co-existent steady state responses is depicted in Figure 6.41.

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6.3.2 Efficiency and Broadband Features of the Vibration Isolation The first task that needs to be addressed in order to use the NES as a vibration absorber is its tuning, so we initiate our study of nonlinear vibration isolation from this task. To this end, we reconsider the harmonically forced system (6.1) and expressed it in the following slightly modified form: y¨1 + ελ(y˙1 − y˙2 ) + y1 + εk (y1 − y2 )3 = εA cos ωt εy¨2 + ελ(y˙2 − y˙1 ) + εk (y2 − y1 )3 = 0

(6.39)

We note that only the primary LO possesses damping, whereas the NES is considered to be undamped. Since we consider the regime of fundamental resonance of this system we suppose that the forcing frequency is close to unity, ω ≈ 1, and introduce the parameter k of the NES which needs to be tuned for optimal vibration isolation performance (as defined by the energy criteria defined below). For the case of the classical linear vibration absorber (i.e., with linear coupling stiffness between the LO and the secondary system), the tuning of parameter k is performed by equating the frequency of the absorber to the resonance frequency of the LO. In the essentially nonlinear case considered herein, however, the tuning procedure is more complicated. Direct application of the analytical procedure outlined in previous sections enables one to estimate the co-existing steady state responses of system (6.39) over the frequency range of interest, and, therefore, to optimize the target function selected as criterion for the vibration isolation. We select two such possible energy criteria. First, the total energy of system (6.39) defined as Etot =

y˙ 2 y2 y˙12 (y1 − y2 )4 + ε 2 + 1 + εk 2 2 2 4

(6.40a)

and, second, the kinetic energy of the primary LO to be isolated: Ekin =

y˙12 2

(6.40b)

Moreover, since the above energy criteria are time dependent due to the time-varying external forcing and damping terms, the corresponding averages of these quantities are considered instead. A typical frequency response plot depicting the dependence of the average of the total energy of the system Etot on frequency is depicted in Figure 6.42, for A = 0.3, λ = 0.4 and ε = 0.1, and varying values of the tuning parameter k. From this plot we conclude that the average total energy in the system can be efficiently reduced by varying the parameter k, while keeping all other parameters fixed; moreover, for relatively high values of k this reduction is made robust over a wide frequency range. It should be mentioned that further increase of k above the values considered in Figure 6.42, causes the appearance of a co-existing large-amplitude steady state


207

Fig. 6.42 Averaged total system energy frequency response in the regime of 1:1 fundamental resonance.

response regime (similar to one presented in Figure 6.41), and, following the reasoning of the previous section, this leads to deterioration of vibration isolation. In addition, the non-monotonic dependence of the averaged total energy on frequency is attributed to SMRs that exist precisely at the regimes of non-monotonicity. So, the enhanced energy reduction and absorption for the system with high value of k is related to this specific response regime. At this stage it is important to emphasize that there is definitely the possibility for additional co-existing stable steady state regimes at certain frequency detuning ranges, σ > 0. However in the vicinity of the 1:1 fundamental resonance there is a region where a single stable periodic attractor and a stable strongly quasi-periodic (SMR) attractor co-exist. In this co-existence region the optimization carried out in this section is valid. Additional co-existing steady state attractors are in the form of periodic lower-amplitude regimes, which may improve even more the efficiency of vibration isolation compared to the regime of SMRs. The previous discussion dictates the establishment of a performance criterion according to which the vibration isolation capacity of the optimally tuned nonlinear absorber (NES) considered herein can be compared to that of the corresponding optimally tuned classical linear vibration absorber (TMD) with identical parameters A, λ and ε. This will enable us to judge the improvement achieved in the proposed vibration isolation design based on the use of essential stiffness nonlinearities. Before we proceed with a particular tuning, some classical results concerning the tuning of the damped, forced linear absorber (Den Hartog, 1956) are reviewed at this point. Here we consider only the main results that will assist us in our discussion on

208


tuning that follows. To this end, we write the system treated by Den Hartog (1956) as follows: M y¨1 + c(y˙1 − y˙2 ) + y1 + kLIN (y1 − y2 ) = P0 cos(ωt) my¨2 + c(y˙2 − y˙1 ) + kLIN (y2 − y1 ) = 0 The main goal of the tuning process outlined by Den Hartog was to reduce the displacement of the primary LO, based on the steady state action of the attached damped linear absorber. The following normalized quantities are introduced: mass ratio, µ = m/M; natural frequency of the absorber, ωa2 = kLIN /m; natural frequency of the primary LO, 2n = K/M; ratio of natural frequencies, f = ωa /n ; and forced frequency ratio, g = ω/n . A frequency response plot depicting the steady state response |y1 | of the primary LO for varying damping parameters is depicted in Figure 6.43. All plots pass through the fixed points of intersection P and Q, which, therefore, are invariant to variations of the damping parameter. By varying the ratio of natural frequencies f these fixed points can be shifted up and down. Thus the tuning process followed by Den Hartog was based on the requirement that P and Q lie at equal heights, and then, the adjustment of the damping parameter to render the frequency response curve to pass through these fixed points with horizontal (or near horizontal) tangent. Equal-height fixed points P and Q correspond to Den Hartog (1956): f =

1 1+µ

(6.41)

Then, the adjustment of the damping parameter was performed according to an analytic expression developed by Den Hartog (1956) which is not reproduced here. To translate these tuning results in terms of the system parameters of (6.37), we set M = 1, c = ελ, P0 = εA and m = ε. Then, according to the previous results we obtain the following expression for optimal tuning of the linear vibration absorber: kLIN =

ε (1 + ε)2

(6.42)

It can be proved (Den Hartog, 1956) that due to the linearity of the problem, similar tuning holds for the optimization of the total energy stored in the system. The criterion for comparing the vibration isolation performances of the NES and the tuned linear absorber is now formulated. Specifically, defining as optimization criterion for our vibration isolation study the minimization of the highest peak of the averaged total system energy (labeled as critical energy peak – CEP) over the frequency range of interest, the performance of the linear and nonlinear designs will be compared by comparing the corresponding CEPs. The rules according to which these two absorbers are to be compared are formulated as follows: (i) the mass, damping, external forcing amplitude and frequency of the primary LO are to remain fixed for a particular study; and (ii) the linear and nonlinear vibration absorbers will possess identical masses but their stiffness parameters


209

Fig. 6.43 Linear vibration absorber, frequency response plot of the primary LO for varying damping values; the points of intersection P,Q are denoted.

k and kLIN will be optimized (tuned) independently. The averaged total energy criterion will be used for the comparison, since, as discussed in previous sections there exists the possibility that the steady state responses of the nonlinear system will be strongly amplitude-modulated (of the beating type); in that case the averaged criterion can be applied for assessing the efficacy and effectiveness of the nonlinear vibration isolation design over a period of the strong modulation (beating). Of course, this doesn’t rule out the possibility that undesired isolated amplitude peaks might occur within a cycle of the response, but the alternative criterion of reducing the maximum absolute peak amplitude will not be considered herein. A typical comparison study of the performances of the linear and nonlinear absorbers under the previously specified conditions is presented in Figure 6.44, for A = 0.2, λ = 0.1 and ε = 0.1. Clearly, the optimally tuned nonlinear absorber is more effective and its performance is more robust to frequency variations compared the optimally-tuned linear absorber for the considered set of system parameters. Although the linear absorber provides better isolation in the narrowband region close to its tuning frequency, the NES provides consistent reduction of the averaged energy peaks over the entire frequency range of interest, and, hence, better performance over a wider frequency range compared to the linear absorber. The frequency response plot at Figure 6.44 is somewhat unusual – the response curve exhibits a number of discontinuities (range of non-monotonicity) in the zone where the SMR regime exists, and it appears to be rather smooth outside this region. This result has been obtained as a result of numerical simulations, and one may conjecture that for different values of the frequency the dynamical flow in that zone

210


Fig. 6.44 Averaged total system energy frequency responses for linear and nonlinear vibration absorbers; the zone of existence of SMRs is indicated.

might be attracted to either one of multiple co-existing stable steady state regimes; that is, either to a stationary (or weakly quasi-periodic) response, or, alternatively, to an SMR. It follows that when designing the essentially nonlinear vibration absorber for optimal vibration isolation, one should care about the regimes to which the flow is attracted, since some of these might be rather undesirable. Therefore, a study of the effect of the initial conditions on nonlinear vibration isolation is required. From the persistence point of view, such an obstacle could undermine the entire endeavor, since rather often the domains of attraction of different response regimes are mixed, and, in addition, in some cases the boundaries of domains of attraction might even be of fractal nature (Jackson, 1991). Still, as mentioned previously, if one can demonstrate that at certain ranges of initial conditions the dynamics of the system is not attracted to problematic attractors, then the design of the NES as nonlinear absorber can still be deemed as satisfactory. In order to illustrate this point, we performed Monte Carlo simulations of the steady state dynamics for different values of randomly picked initial conditions. The parameters used are identical to those of Figure 6.38, but with ε = 0.001. It should be mentioned that the shape of the frequency response plot does not depend on the value of ε in the framework of the approximations used in the previous analysis. We explore two different values of the detuning parameter, namely, σ = 1.5 (a regime of co-existence of desirable SMRs with periodic or weakly modulated stationary solutions) and σ = 2.0 (a regime where no SMRs exist). For both selected values of the detuning parameter there exist two stationary responses, namely, a (desirable) low-amplitude solution, and an (undesirable) high-amplitude one.


211

Fig. 6.45 Monte Carlo simulations of steady state attractors of the dynamics for randomly varying initial conditions, A = 0.4, λ = 0.2, σ = 1.5, ε = 0.001 and y20 = 0; the SMR attractor is denoted by (×), the low-amplitude stationary attractor by (◦), and the high-amplitude (undesirable) stationary attractor by (•).


The results of the Monte Carlo simulations are presented at Figures 6.45 and 6.46. One can see that for σ = 1.5 no trajectory is attracted to the undesirable high-

212



amplitude stationary response which is unfavorable for vibration isolation; instead, trajectories are attracted by, either the low-amplitude stationary steady state solution or the SMR. Both of these stable solutions are favorable to vibration isolation, with the SMR solution being preferable since, as discussed previously, it represents a version of steady state TET. On the contrary, by increasing the detuning parameter to σ = 2.0 we reach a regime where no SMRs exist, and, as a result, a subset of initial conditions leads to attraction of the dynamics by the undesirable high-amplitude stationary stable response. From these simulations it appears that the regime of SMRs represents a rather strong attractor, since it prevents the dynamical flow from being attracted by the undesirable high-amplitude stationary response. This result, however, is only based on this specific series of Monte Carlo simulations and is not based on any rigorous analytical proof. Moreover, the above example may be regarded as of little practical significance, since it corresponds to a very small (and hence impractical) value of the mass ratio ε; indeed, for this value of the mass ratio the previously developed theoretical asymptotic analysis is expected to yield satisfactory quantitative predictions. However, qualitatively similar results can be obtained for cases of more practical significance, which, however, are beyond the range of validity of our previous analytical approximations. This is demonstrated in Figures 6.47–6.49, which correspond to increased mass ratio ε = 0.05 and varying detuning parameter σ . In all three cases shown and for all values of randomly chosen initial conditions considered, the absorber dynamics are not attracted by an undesirable high-amplitude stationary response. Rather its steady state response is either an SMR or a low-amplitude stationary solution,


213


which ensures robust vibration isolation for all initial conditions considered in the Monte-Carlo simulation. Although a more detailed investigation may be required to proceed into final and analytically rigorous conclusions, the presented results indicate robustness of nonlinear vibration isolation by the considered NES design. The nonlinear absorber has an additional feature which justifies its use: it possesses a self-tuning capacity to variations of the external frequency, and thus can be applied for the case of linear primary systems with many degrees of freedom. The reason behind this self-tuning capacity of the NES is its essential nonlinearity which, in contrast to the linear vibration absorber, prevents the existence of a preferential resonance frequency. This extends the capacity of the NES to engage in resonance interaction with multiple modes of the linear primary system to which it is attached (and, hence, to provide multi-modal passive vibration isolation). The next section provides a demonstrative example of this feature of the NES.

6.3.3 Passive Self-tuning Capacity of the NES To study the vibration isolation properties of the NES when applied to a MDOF system, we consider a harmonically excited two-DOF system of linear coupled oscillators (the primary linear system) with a nonlinear energy sink (NES) attached to it. The masses of the linear oscillators are assumed to be identical (and taken as equal to unity without loss of generality). The system is described by the following

214



equations: y¨2 + k2 y2 + k1 (y2 − y1 ) = εF2 cos ωt y¨1 + k2 y1 + k1 (y1 − y2 ) + εkv (y1 − v)3 + ελ(y˙1 − v) ˙ = εF1 cos ωt εv¨ + εkv (v − y1 )3 + ελ(v˙ − y˙1 ) = 0

(6.43)

where y1, y2 and v are the displacements of the linear oscillators and the NES, respectively; ελ is the damping coefficient; and εFi , i = 1, 2 are the amplitudes of the weak excitations applied to each of the linear oscillators. The system (6.43) may be rescaled by introducing the following normalized variables and parameters: t=

k1 τ,

k2 k˜2 = , k1

λ λ˜ = √ , k1

F1 F˜1 = , k1

F2 F˜2 = , k1

kv k˜v = (6.44) k1

Substitution of (6.44) into (6.43) yields the following rescaled system: y2

+ (k˜2 + 1)y2 − y1 = εF˜2 cos ωτ ˜ 1 − v ) y1

+ (k˜2 + 1)y1 − y2 = εF˜1 cos ωτ − ε k˜v (y1 − v)3 − ε λ(y ˜ − y1 ) = 0 εv

+ εk˜v (v − y1 )3 + ε λ(v

(6.45)

where primes denote differentiation with respect to the normalized independent variable τ . The two natural frequencies of the primary linear system are assumed to be


215

of the same order of magnitude, incommensurate and well separated [i.e., their difference is of O(1)]. It follows that the normalized coupling stiffness k˜2 is selected in a way to provide two distinct incommensurate natural frequencies√for the primary linear system; e.g., k˜2 = 1 yields the natural frequencies ω2 = 3 and ω1 = 1. Although the assignment of numerical values may seem rather restrictive, it will become clear from the following development that the particular values of the natural frequencies are of no significance for the analysis, as long, of course, as the previous assumptions are enforced. Indeed, the only significant restriction is the absence of beats in the linear primary system, that is, the requirement of well separated natural frequencies. For cases where the natural frequencies of the primary system are closely spaced a separate asymptotic analysis must be performed. At this point we transform the dynamical system (6.45) in terms of modal coordinates of the primary system, √ y1 = (x1 + x2 ) / 2 √ y2 = (x1 − x2 ) / 2 (6.46) yielding the following transformed system of equations:

3 x1 + x2 ε A˜ 2 ε λ˜ εk˜v (x1 + x2 ) + 3x2 = − √ cos ωτ − √ √ √ −v − √ −v 2 2 2 2 2

3

εA˜ 1 ε λ˜ (x1 + x2 ) ε k˜v (x1 + x2 ) x1 + x1 = √ cos ωτ − √ √ √ −v − √ − v 2 2 2 2 2

3 (x + x ) (x1 + x2 ) εv + εk˜v v − + ε λ˜ v − 1√ 2 = 0 (6.47) √ 2 2 x2

Note that the introduced modal transformation decouples the left-hand-sides of the linear oscillators of the primary system but there is still O(ε) coupling between the two linear modes through the nonlinear and damping terms on the right-hand-sides. In (6.47), we have introduced the notations A˜ 1 = F˜1 + F˜2 and A˜ 2 = F˜1 − F˜2 , which represent the modal harmonic excitations of the two linear modes. The additional rescalings, x1 x2 x˜ 1 = √ ; x˜2 = √ ; 2 2 bring the system to the following simpler form:

ε˜ =

ε 2

(6.48)

x˜2 + 3x˜2 = −˜ε A˜ 2 cos ωτ − ε˜ k˜v (x˜1 + x˜2 − v)3 − ε˜ λ˜ (x˜1 + x˜2 − v ) ˜ x˜1 + x˜ 2 − v ) x˜1 + x˜1 = ε˜ A˜ 1 cos ωτ − ε˜ k˜v (x˜ 1 + x˜2 − v)3 − ε˜ λ( ˜ − x˜1 + x˜2 ) = 0 ε˜ v + ε˜ k˜v (v − x˜1 + x˜2 )3 + ε˜ λ(v

(6.49)

216


For the sake of notational convenience the tildes in the variables will be omitted from now on in the analysis. From the rescaled dynamical system (6.49) we note that due to its essential stiffness nonlinearity the NES is directly coupled to both modes of the primary system, so it may engage in resonance interactions with both. The steady state solutions of system (6.49) will be analyzed using the CX-A method. Before we perform the direct calculation of the periodic steady state responses it is convenient to perform a final coordinate transformation that will bring the rescaled dynamical system (6.49) into its final form. To this end, leaving the first two equations of (6.49) unchanged we replace the third equation (governing the response of the NES) by adding the first two equations of (6.49) and then subtracting from the sum the third equation. Defining the new variable w according to, w ≡ x1 + x 2 − v

(6.50)

system (6.49) is rewritten in the following final form: x2 + 3x2 = −εA2 cos ωτ − εkv w3 − ελw x1 + x1 = εA1 cos ωτ − εkv w3 − ελw

(6.51)

w + 3x2 + x1 = ε (A1 − A2 ) cos ωτ − (1 + 2ε)kv w3 − (1 + 2ε)λw which will be the basis of the following analysis. We begin our treatment by considering harmonic excitations with frequencies close to the first natural frequency of the primary system; hence, we consider direct harmonic excitation of the lower linear mode only. We will refer to this resonance as in-phase fundamental resonance. To this end, we introduce the weak frequency detuning εσ defined as ω = 1 + εσ (6.52) which measures the closeness of the excitation frequency ω to the first natural frequency. To apply the CX-A methodology we introduce the following new complex variables, x1 + ix1 = ϕ1 exp[j (1 + εσ )τ ] x2 + ix2 = ϕ2 exp[j (1 + εσ )τ ] w + iw = ϕw exp[j (1 + εσ )τ ]

(6.53)

where ϕ1 , ϕ2 , ϕw are assumed to be slowly evolving complex modulations of a fast oscillation at the frequency of the external excitation. By the slow-fast partition in the ansatz (6.53) we seek steady state solutions possessing a single fast frequency equal to the first natural frequency of the primary system. Introducing (6.53) into (6.51) and averaging over one period of the external excitation we obtain the following slow flow dynamical system, which governs the slow evolutions of the modulations ϕ1 , ϕ2 and ϕw in the neighborhood of the first linear mode:


217

3j εkv −εA2 ελ |ϕw |2 ϕw − ϕw + 2 8 2 3j εkv εA1 ελ |ϕw |2 ϕw − ϕw + ϕ˙ 1 + j εσ ϕ1 = 2 8 2

3j j 1 + εσ ϕw − ϕ2 − ϕ1 ϕ˙ w + j 2 2 2 ϕ˙ 2 + j (εσ − 1)ϕ2 =

=

ε (A1 − A2 ) 3j (1 + 2ε)kv λ(1 + 2ε) |ϕw |2 ϕw − + ϕw 2 8 2

(6.54)

In order to analytically estimate the periodic responses of the system we consider the fixed points of (6.54) by setting ϕ˙1 = ϕ˙ 2 = ϕ˙ w = 0. This yields the following system of nonlinear algebraic equations, −εA2 3εj kv ελ |ϕw0 |2 ϕw0 − ϕw0 + 2 8 2 3εj kv εA1 ελ |ϕw0 |2 ϕw0 − ϕw0 + j εσ ϕ10 = 2 8 2

3j j 1 + εσ ϕw0 − ϕ20 − ϕ10 j 2 2 2

j (εσ − 1)ϕ20 =

=

ε(A1 − A2 ) 3(1 + 2ε)j kv λ |ϕw0 |2 ϕw0 − (1 + 2ε) ϕw0 + 2 8 2

(6.55)

where subscript ‘0’ denotes the value of the corresponding modulation at the fixed point. The solutions of (6.55) can be calculated by simple algebraic manipulations. Indeed, we may reduce the computation to a single third-order inhomogeneous polynomial in terms of the modulus of the complex amplitude ϕw0 , by recognizing that the system (6.55) can be manipulated to yield, α2 α¯ 2 |ϕw0 |6 + (α2 α¯ 1 + α1 α¯ 2 ) |ϕw0 |4 + α1 α¯ 1 |ϕw0 |2 = α3 α¯ 3 where

(6.56)

3ελ λ λ(1 + 2ε) 1 α1 = j εσ + + + + 2 4(εσ − 1) 4σ 2

3j kv 3j kv (1 + 2ε) 9j εkv + + α2 = − 16(εσ − 1) 16σ 8

α3 =

A1 ε(A1 − A2 ) 3εA2 − − 4(εσ − 1) 4σ 2

and overbar denotes complex conjugate. The phase of ϕw0 and the remaining two amplitudes are then computed through the relations,

218


Fig. 6.50 Steady state frequency response for in-phase fundamental resonance (for ω ≈ 1.0); dashed lines refer to unstable periodic solutions and solid lines to stable ones.

ϕw0 =

−α3 α1 + α2 |ϕw0 |2

ϕ20 =

3εkv j εA2 j ελ |ϕw0 |2 ϕw0 + + ϕw0 2(εσ − 1) 8(εσ − 1) 2(εσ − 1)

ϕ10 = −

j A1 3kv jλ |ϕw0 |2 ϕw0 + + ϕw0 2σ 8σ 2σ

(6.57)

In Figure 6.50 we depict the frequency response plot for this type of excitation. SN and Hopf bifurcations can be analytically studied by performing a stability analysis of the computed fixed points, in accordance to the procedure developed in Section 6.1. Points of bifurcation are indicated in the plot of Figure 6.45. In similarity to previous sections, large values of |ϕw0 | indicate strong steady state TET from the forced primary system to the NES, and, hence, enhanced vibration isolation. Considering now the out-of-phase fundamental resonance of the system by assuming an excitation with frequency close to the second eigenfrequency of the primary system, √ ω = 3 + εσ (6.58) we follow a similar CX-A procedure to derive the approximate frequency response plot of Figure 6.51. In similarity to the plot depicted in Figure 6.50, this frequency response plot is based on the analysis of the slow flow dynamics.


219

Fig. 6.51 Steady state frequency response plot for out-of-phase fundamental resonance (for ω ≈ 1.732); dashed lines refer to unstable periodic solutions and solid lines to stable ones.

It is rather reasonable to expect that SMRs will exist for system (6.49) as well. As in a case of periodic or weakly modulated steady state regimes (generated by Hopf bifurcations), analytical treatment of SMRs must be carried out separately for the cases of in-phase and out-of-phase fundamental resonances. Such reduction is possible in the case of sufficiently small NES mass ε. To this end, we reconsider the original equations (6.49) in slightly modified form, for excitation frequencies close to the first natural frequency of the linear primary system (that is, for the case of in-phase fundamental resonance): x2 + 3x2 = −εA2 cos [(1 + εσ )τ ] − εkv (x1 + εx2 − v)3 − ελ(x1 + εx2 − v ) x1 + x1 = εA1 cos [(1 + εσ )τ ] − εkv (x1 + εx2 − v)3 − ελ(x1 + εx2 − v ) εv + kv (v − x1 − εx2 )3 + λ(v − x1 − εx2 ) = 0

(6.59)

In this case the ordering of terms with respect to the small parameter ε can only be balanced if we assume that x1 ∼ O(1), v ∼ O(1) (under condition of 1:1 resonance) and x2 ∼ O(ε). If this is the case, then terms related to x2 in the last two equations of (6.59) bring about corrections of at most O(ε 2 ) and can be neglected in lower-order approximations. Following this argument the three-DOF system (6.59) can be trivially reduced to a two-DOF system similar to (6.1) with appropriate notational accommodation. Thus, the treatment of SMRs in this case may be accomplished similarly to the process described in Section 6.2 and does not

220


need any further analytical consideration. We note, however, that this reduction can only be made due to our assumption of well-separated linear modes of the primary system. When primary systems with closely-spaced modes are considered then coupling (beat phenomena) between linear modes occur and the above reduction to a two-DOF can not be accomplished. In that case a global analysis of the slow flow dynamics of the original, full three-DOF system (6.45) must be performed to study the regimes of SMRs (this raises interesting questions concerning the form of the necessary Poincaré map reduction that needs to be developed to study relaxation oscillations – SMRs in the corresponding higher-dimensional slow flow, and the corresponding bifurcations of that map). The case of out-of-phase fundamental resonances of (6.59), i.e., of SMRs in the vicinity of the second natural frequency can be treated similarly, and the associated problem can be reduced to a two-DOF problem as well. It should be mentioned that SMRs are expected to exist in an O(ε)-neighborhood of the exact resonance. As mentioned above, since the two natural frequencies of the primary system of (6.59) are well separated, no overlap of in-phase and out-of-phase SMRs is expected in the system under consideration. The open question, of course, that merits further investigation is the interaction of these two types of SMRs when the primary system has closely spaced modes; however, this interesting question is not dealt with in this work. Motivated by the previously outlined tuning procedure for the case of SDOF linear primary system (see Section 6.3.2), the principal aim of the tuning procedure for selecting the coupling stiffness kv and the coupling damper λ in the three-DOF system (6.45), should be to allow excitement of in-phase or out-of-phase SMRs for fixed NES mass. We will demonstrate this tuning procedure by means of a demonstrative example. Specifically, we consider the following values for the amplitudes of the excitation and the NES mass, A1 = 1, A2 = 3, and ε = 0.01. We need to select the NES parameters (kv , λ) in such a way as to allow excitation of SMRs in the neighborhood of each mode of the linear primary system. In order to determine these values, in the neighborhood of each excited mode we vary the NES parameters (kv , λ) with a small step, and for each pair we construct the corresponding onedimensional Poincaré maps that we introduced in Section 6.2 for studying relaxation oscillations. Then, we obtain the frequency ranges of existence of in-phase and outof-phase SMRs (as the stable fixed points of the corresponding one-dimensional maps) in the neighborhood of each excited mode. By ensuring that for the selected pair (kv , λ) there exist both stable in-phase and out-of-phase SMRs in the frequency ranges of interest (in the neighborhoods of the two linear modes) we satisfy the basic requirement of the tuning procedure. Suppose that the described procedure yields a pair of NES parameters kv = 1, λ = 0.2 that allows SMRs in the neighborhood of each of the modes of the primary system. For this NES parameter pair we plot the corresponding in-phase and out-of-phase fundamental frequency response diagrams as in Figure 6.52; these diagrams possess branches of stable periodic oscillations that co-exist with regimes of stable SMRs (computed as outlined above). Despite the excitation of SMRs in the vicinity of each mode we note that an undesirable situation has developed in the


221

Fig. 6.52 Fundamental resonance plots: (a) in-phase, (b) out-of-phase; dashed lines refer to unstable periodic solutions and solid lines to stable ones; the domains of existence of SMRs are indicated; in the shaded region undesirable conditions for vibration isolation develop.

neighborhood of the left bound of the domain of existence of SMRs for the case of in-phase fundamental resonances (see Figure 6.52a). Indeed, observing the diagram of Figure 6.52b we can see that the lower-frequency Hopf bifurcation (point A) occurs on the lower stable branch of periodic solutions slightly before the lower (left) boundary of the domain of existence of SMRs. This bifurcation causes an undesirable effect on the system response since the loss of stability of the lower branch may be accompanied by the jump of the dynamics to the upper stable branch which yields large-amplitude vibrations; clearly this is an unfavorable scenario for vibration isolation. In order to avoid this effect our design should be to translate the lower Hopf bifurcation of the lower branch into the domain of existence of SMRs, in order to ensure the transition from the lower stable branch of periodic solutions into the regime of SMRs with increasing frequency detuning, thus avoiding potential undesirable jumps to the larger-amplitude stable branch of solutions. This can be achieved easily by appropriate design of the system parameters, e.g., by increasing the damping parameter from λ = 0.2 to λ = 0.4, as depicted in Figure 6.53 where the corresponding frequency response plots of the redesigned system are depicted. The frequency response diagrams presented in Figure 6.53 suggest that by increasing the frequency detuning there is the possibility of transitions from the lower stable branch of periodic solutions to the regime of SMRs for both the in-phase and out-of-phase modes. The following direct numerical simulations carried out for

222


Fig. 6.53 Transition of the Hopf bifurcation point on the lower stable branches of periodic solutions of fundamental, (a) in-phase and (b) out-of-phase resonance plots by increasing the damping λ; dashed lines refer to unstable periodic solutions and solid lines to stable ones; the domains of existence of SMRs are indicated.

this system in the entire frequency range under study (including both modes) and zero initial conditions, reveal that no undesirable transitions (jumps) to the upper branches of periodic solutions are realized. It is essential to note that there still exists the possibility of transitions of the dynamics to the upper stable branches for different (non-zero sets of initial conditions. These regimes may be excited, for example, by relatively large initial displacements; e.g., this is the case when strong non-harmonic or even non-periodic but repeated excitations – in the form of impulses or trains of pulses – are applied, which are capable of inducing transitions of the dynamics to the high-amplitude stable solution branches, and thus affect inadversely the vibration isolation design. Such excitations are not considered in this work, however, and investigation of their effects on the steady state dynamics of systems with attached NES is left as an open problem. Summarizing the results of the example provided above, we have seen that it is not enough to excite in-phase or out-of-phase SMRs for achieving satisfactory vibration isolation performance; what is also important is to design the system so that low-frequency Hopf bifurcation points on the lower stable branches of periodic solutions occur well inside the domain of existence of SMRs. Indeed, by shifting these bifurcation points on the lower branches of periodic responses into the domain of existence of SMRs, we ensure effective and robust vibration isolation by application of the NES.


223

Fig. 6.54 Numerical frequency responses: (a) max(y12 + y22 ); (b) max(y1 ); (c) max(y2 ); (d) mean amplitude of y1 (t); (e) mean amplitude of y2 (t); superimposed to each plot is the corresponding frequency response of the optimally tuned linear absorber (dotted lines).

In order to study more systematically the vibration mitigation performance of the NES, we construct numerical frequency response plots for zero initial conditions over the entire frequency range of interest that include both natural frequencies of the linear primary system. Referring to system (6.43), in the considered frequency response plots we depict the variations of the following quantities for varying excitation frequency ω: (i) max(y12 + y22 ) (see Figure 6.54a); (ii) max(y1 ) and max(y2 ) (see Figures 6.54b, c); and (iii) mean amplitudes of y1 (t) and y2 (t) (see Figures 6.54d, e). For comparison, on the same plots we also depict the corresponding frequency responses of the optimally tuned linear absorber; the linear absorber was tuned numerically by minimizing the summation of the two resonant peaks of the corresponding frequency response curve. For these simulations the parameters of the problem were selected as A1 = 1, A2 = 3, kv = 1, λ = 0.4 and ε = 0.01. We now perform a numerical parametric study of the amplitudes of the external excitations for which SMRs in the nonlinear design provide better vibration isolation performance than the optimally tuned absorber in the linear design. It should be noted that both linear and nonlinear vibration absorbers are tuned according to the same criteria discussed above (that is, minimal sum of resonant peaks of the max-

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Fig. 6.55 Comparisons of linear and nonlinear designs in the (A1 , A2 ) plane; dots indicate better performance of the SMR-based nonlinear design.

imal response deflection). In Figure 6.55 we depict schematically the zone on the (A1 , A2 ) plane for which the SMR-based nonlinear vibration isolation is preferable. From the results presented in Figure 6.55 we note that better performance of SMR-based nonlinear vibration isolation is realized for relatively high amplitudes of external excitations. This result is not surprising since the analytical model predicts that the upper branch of the SIM does not depend on the amplitude of excitation; it follows that for some low excitation amplitudes SMRs may be already excited, however, the system response will be rather high compared to the one realized when a linear absorber is applied. In the case of high amplitude excitation, the response of the system with linear absorber attached will overcome the SMRs. This is due to the fact that SMRs are weakly affected by the increase of the excitation amplitudes; this is contrary to the case when the linear absorber is attached. In order to demonstrate the robustness of the nonlinear vibration isolation achieved by excitation of either the lower branch of stable periodic responses or the regime of SMRs, we have performed a series of direct numerical integrations of the original dynamical system (6.47) for random sets of initial conditions. Specifically, we randomly picked 300 triplets of initial displacements in the ranges −0.5 ≤ xi (0) ≤ 0.5, i = 1, 2 and −0.5 ≤ v(0) ≤ 0.5, and zero initial velocities, under conditions of in-phase or out-of-phase fundamental resonances. Two frequency detuning values were selected for each type of fundamental resonance. Referring to the previously discussed frequency response diagrams of the system with NES attached [see Figures 6.50–6.53], the first value of frequency detuning


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Fig. 6.56 Attractors of the dynamics for random sets of the initial conditions: (a) in-phase fundamental resonance, σ = −1; (b) in-phase fundamental resonance, σ = −0.8; (c) out-of-phase fundamental resonance, σ = −1.5, (d) out-of-phase fundamental resonance, σ = −1.2; dots denote the lower stable branch of periodic orbits, and diamonds denote the regime of stable SMRs.

refers to the region of two co-existing stable periodic regimes, and is out of the domain of existence of stable SMRs; while for the second detuning value there is co-existence of the regime of stable SMRs and the upper branch of stable periodic responses. Hence, the frequencies of external excitation are defined as follows: for the in-phase fundamental resonances, ω = 1√+ εσ , with σ = −0.8, −1.0; for the out-of-phase fundamental resonances, ω = 3 + εσ , with σ = −1.2, −1.5. The system parameters are selected as A1 = 1, A2 = 3, ε = 0.01, λ = 0.4 and kv = 1. The results of these simulations are depicted in Figure 6.56. The attractor reached in each of the numerical simulations (corresponding to a triplet of initial conditions chosen from the random set) is marked by a diamond when the attractor is a stable SMR, or by a circle when the attractor is on the upper branch of stable periodic solutions. It is clear from these results that for values of the frequency detuning in the regime of co-existence of two branches of stable periodic motions, all simulations are attracted by the periodic solution on the lower branch. For values of frequency detuning in the regime of co-existence of SMRs and stable periodic responses, all numerical simulations are attracted by a stable SMR. These results provide additional confirmation of the robustness of the steady state regimes related to

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the lower stable branch of periodic solutions and the regime of stable SMRs. Both these regimes are favorable to the objective of vibration isolation. It is essential to note that the considered initial displacements possess much larger magnitudes than the amplitudes of the external excitations. The results presented in this chapter indicate that there are some major differences between commonly used designs based on linear vibration absorbers (TMDs) and the essentially nonlinear absorber (NES) designs proposed herein. First, the performance of the NES-based design depends on the amplitude of the external excitation, i.e., it works properly only for specific ranges of forcing amplitudes. The TMD, at least in theory, is free of this drawback as the dynamics is purely linear and hence energy-independent; however, in practice too high amplitudes of the external forcing are also problematic in the linear design, as they yield prohibitively large displacements of the damper which either leads to degeneration of the system due to fatigue or drives the dynamics out of the linear regime. On the other side, if the excitation is small enough, no protection is required in the first place. From this viewpoint, the NES may be plausible despite its limitations concerning the permissible range of external excitation amplitudes. The nonlinear absorber has two main advantages compared to the TMD. The first is based on the fact that, simply, it is more effective – it yields better vibration isolation performance. However, with this comes the possible drawback that alternative undesirable response regimes might co-exist with the ones that are favorable for vibration isolation. Still, as demonstrated in the previous study, proper design of the NES will guarantee that for certain ranges of initial conditions these problematic responses are not excited at all. The regime of SMRs greatly facilitates robust vibration absorption, since through its extended domain of attraction it is capable of deferring the dynamical flow from the undesired dynamic attractors. Consequently, the regime of SMRs is a rather desirable response regime for effective vibration mitigation. This conclusion is a bit unexpected in the sense that normally, quasi-periodic responses are rather undesirable in engineering systems. Yet, this counter-intuitive conclusion is perhaps in line with the overall theme of this work, which is the consideration of essentially nonlinear designs in engineering practice. This contrasts with the view that nonlinearities in engineering are counterproductive and hence should be avoided at all cost. The second major advantage of the proposed essentially nonlinear absorber is its capability to work over broad frequency bands. While connected to a two-DOF primary system, the NES demonstrates much better performance than a properly tuned TMD. To achieve comparable broadband performance by means of linear vibration absorbers, one should design them to possess more degrees of freedom. These advantageous properties of the strongly nonlinear vibration absorbers were revealed by means of theoretical analysis and numeric simulation; experimental verifications on the subject are few and incomplete. Hence, there is the need for further experimental study and validation of the proposed NES designs. Moreover, the study of applying SDOF or MDOF NESs for vibration isolation of periodically forced continuous elastic systems can be carried out by extending the methodologies and optimization procedures developed in this chapter. In the context of applying such


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designs to practical engineering problems, a systematic and rigorous study of the issue of robustness of nonlinear vibration isolation in the presence of co-existing unfavorable response regimes deserves further investigation.

References Akay, A., Xu, Z., Experiments on vibration absorption using energy sinks, J. Acoust. Soc. Am. 118(5), 3043–3049, 2005. Arnold, V.I., Afrajmovich, V.S., Il’yashenko, Yu.S., Shil’nikov, L.P., Dynamical Systems V. Encyclopedia of Mathematical Sciences, Springer-Verlag, Berlin/New York, 1994. Bonello, P., Brennan, M.J., Elliot, S.J., Vincent, J.F.V., Jeronimidis, G., Designs for an adaptive tuned vibration absorber with variable shape stiffness element, Proc. Roy. Soc. A 461 (2064), 3955–3976, 2005. Brennan, M.J., Some recent developments in adaptive tuned vibration absorbers/neutralizers, Shock Vib. 13(4–5), 531–543, 2006. Bykhovsky, I.I., Fundamentals of Vibration Engineering, Robert E. Krieger, New York, 1980. Carcaterra, A., Akay, A., Koç, I.M., Near-irreversibility in a conservative linear structure with singularity points in its modal density, J. Acoust. Soc. Am. 119(4), 2141–2149, 2006. Carcaterra, A., Akay, A., Koç, I.M., Theoretical foundations of apparent-damping phenomena and nearly irreversible energy exchange in linear conservative systems, J. Acoust. Soc. Am. 121(4), 1971–1982, 2007. Carrella, A., Brennan, M.J., Waters, T.P., Static analysis of a passive vibration isolator with quasizero-stiffness characteristic, J. Sound. Vib. 301(3–5), 678–689, 2007a. Carrella, A., Brennan, M.J., Waters, T.P., Optimization of a quasi-zero-stiffness isolator, J. Mech. Sc. Tech. 21(6), 946–949, 2007b. Den Hartog, J.P., Mechanical Vibrations, McGraw-Hill, New York, 1956. Feldman, M., Time-varying vibration decomposition and analysis based on the Hilbert transform, J. Sound Vib. 295, 518–530, 2006. Frahm, H., Device for damping vibrations of bodies, U.S. Patent 989,958, 1911. Gendelman, O.V., Starosvetsky, Y., Quasi-periodic response regimes of linear oscillator coupled to nonlinear energy sink under periodic forcing, J. Appl. Mech. 74, 325–331, 2007. Gendelman, O.V., Gourdon, E., Lamarque, C.-H., Quasi-periodic energy pumping in coupled oscillators under periodic forcing, J. Sound Vib. 294, 651–662, 2006. Gendelman, O.V., Starosvetsky, Y., Feldman, M., Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: Description of response regimes, Nonl. Dyn. 51, 31–46, 2008. Guckenheimer, J., Holmes, P., Nonlinear Oscillations, Dynamical System, and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983. Guckenheimer, J., Hoffman, K., Weckesser, W., Bifurcations of relaxation oscillations near folded saddles, Int. J. Bif. Chaos 15, 3411–3421, 2005. Guckenheimer, J., Wechselberger, M., Young, L.-S., Chaotic attractors of relaxation oscillators, Nonlinearity 19, 701–720, 2006. Jackson, E.A., Perspectives of Nonlinear Dynamics, Vol. 1, Cambridge University Press, 1991. Koç, I.M., Carcaterra, A., Xu Z., Akay, A., Energy sinks: Vibration absorption by an optimal set of undamped oscillators, J. Acoust. Soc. Am. 118(5), 3031–2042, 2005. Natsiavas, S., Steady state oscillations and stability of non-linear dynamic vibration absorbers, J. Sound Vib. 156, 227–245, 1992. Natsiavas, S., Dynamics of multiple degree-of-freedom oscillators with colliding components, J. Sound Vib. 165, 439–453, 1993a.

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Natsiavas, S., Vibration absorbers for a class of self-excited mechanical systems, J. Appl. Mech. 60, 382–387, 1993b. Natsiavas, S., Mode localization and frequency veering in a damped system with dissimilar components, J. Sound Vib. 165, 137–147, 1993c. Newhouse, S.E., Diffeomorphisms with infinitely many sinks, Topology 13, 9–18, 1974. Ormondroyd, J., Den Hartog, J.P., The theory of the dynamic vibration absorber, Trans. ASME 50, A9–A22, 1928. Rice, H.J., McCraith, J.R., Practical non-linear vibration absorber design, J. Sound Vib. 116, 545– 559, 1987. Roberson, R.E., Synthesis of a nonlinear dynamic vibration absorber, J. Franklin Inst. 254, 205– 220, 1952. Shaw, J., Shaw, S.W., Haddow, A.G., On the response of the nonlinear vibration absorber, Int. J. Nonlinear Mech. 24, 281–293, 1989. Starosvetsky, Y., Gendelman, O.V., Attractors of harmonically forced linear oscillator with attached nonlinear energy sink II: Optimization of a nonlinear vibration absorber, Nonl. Dyn. 51, 47–57, 2008a. Starosvetsky, Y., Gendelman, O.V., Strongly modulated response in forced 2DOF oscillatory system with essential mass and potential asymmetry, Physica D 237, 1719–1733, 2008b. Szmolyan, P., Wechselberger, M., Relaxation oscillations in R3, J. Diff. Eq. 200, 69–104, 2004. Virgin, L.N., Davis, R.B., Vibration isolation using buckled struts, J. Sound Vib. 260, 965–973, 2003. Virgin, L.N., Santillan, S.T., Plaut, R.H., Vibration isolation using extreme geometric nonlinearity, in Proceedings Euromech Colloquium 483 on Geometrically Nonlinear Vibrations of Structures, July 9–11, FEUP, Porto, Portugal, 2007. Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.

Chapter 7

NESs with Non-Smooth Stiffness Characteristics

In this chapter we consider NESs with non-smooth stiffness nonlinearities, such as, clearances and vibro-impacts. Apart from their interesting nonlinear dynamics, the additional motivation for studying this class of nonlinear attachments is their capacity to absorb shock energy at fast time scales. The consequence of this capacity for rapid passive energy absorption is that this type of NESs are good candidates for applications where the targeted energy transfer (TET) from the directly forced primary structure to the NES(s) must be accomplished at the very early stage of the motion, that is, immediately after the application of the external shock where the energy is at its highest level; examples are, structures under seismic excitation or cars during collision. We will provide a theoretical basis for assessing the capacity of NESs with nonsmooth nonlinearities for TET at fast time scales, and postpone until Chapter 10 the discussion of the application of NESs with vibro-impact nonlinearities to the important problem of passive seismic mitigation of structures. For works on the mechanics of systems with non-smooth stiffness or damping nonlinearities we refer to the monographs by Babitsky (1998), Persson (1998), Brogliato (1999), Wiercigroch and de Kraker (2000), Babitsky and Krupenin (2001), Glocker (2001), Awrejcewicz and Lamarque (2003) and references therein. In the first two sections we provide numerical evidence of the capacity for shock isolation of NESs with non-smooth stiffnesses. In the following sections we will be focusing on systems with NESs possessing clearance or vibro-impact nonlinearities, in an effort to study certain aspects of the complex dynamics of these systems and related them to TET.

7.1 System with Multiple NESs Possessing Clearance Nonlinearities We initiate the study of TET in coupled mechanical oscillators with NESs possessing non-smooth stiffness characteristics (referred to from now on as non-smooth NESs – NS NESs) by studying the shock isolation properties of a system of two 229

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coupled non-conservative linear oscillators with NESs possessing clearance nonlinearities (Georgiades, 2006; Georgiades et al., 2005). Apart from the fact that such non-smooth stiffness elements introduce strong nonlinearities to the system, they are rather easy to implement in practical settings since they can be realized by means of assemblies of linear stiffnesses. This renders the proposed designs practical in their implementation, and convenient to study experimentally under realistic forcing conditions (e.g., refer to the experimental work reported in Chapter 10).

7.1.1 Problem Description The system considered in this section is depicted in Figure 7.1. The primary system is composed of two weakly coupled, viscously damped linear oscillators (LOs – referred to as LO 1 and 2), where the small parameter of the problem, 0 < ε 1, scales the weak coupling. To each of the two LOs there is an attached NS NES (labeled as NES 1 or 2 – see Figure 7.1) through a weak linear stiffness; each NES possesses viscous damping and an internal restoring linear spring which acts in parallel to a linear stiffness with clearance. We assume that an external shock F (t) is applied to LO 1, and wish to examine the capacity of the two NS NESs to passively and rapidly extract shock energy from (and thus isolate from shock) the primary system. That is, we aim to study the capacity of the NS NESs to absorb shock energy from the primary system at a sufficiently fast time scale and reduce the level of vibration of the primary system at the initial stage of the motion where the energy is at its highest level. Note that this objective is more demanding than similar objectives for TET posed in previous Chapters, where effectiveness of TET was judged primarily based on asymptotic energy dissipation measures (EDMs), i.e., by studying the capacity of a (‘smooth’) NES to absorb significant portions of the energy induced in a primary system, without paying much attention on the time scale of TET (i.e., on how rapid TET is). In addition, contrary to our studies of NESs with smooth stiffness characteristics carried out in previous chapters, in the present study we do not make any assumption regarding the smallness of the NES masses m1 , . . . , m4 , allowing them to be O(1) quantities. Hence, we aim to show that appropriately designed NS NESs can rapidly absorb significant fractions of the broadband shock energy of the primary system through passive TET, and locally dissipate this energy without ‘spreading it back’ to the primary system. If this type of TET can occur at a sufficiently fast time scale, then this should result in drastic reduction of the level of vibration of the primary system at the critical initial stage of the motion (that is, immediately after the shock has been applied), and, hence, to effective passive shock isolation. As shown below, the non-smooth stiffness characteristics of the NESs can, indeed, yield broadband TET on a fast time scale. Returning to the system of Figure 7.1, the non-smooth stiffnesses of the two NS NESs are piecewise linear, and expressed as


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Fig. 7.1 Primary system with multiple NES with clearance nonlinearities.

⎧ ⎪ ⎨ k12 v1 + k11(v1 + e1 ), v1 ≤ −e1 −e1 < v1 < e1 G1 (v1 ) = k12 v1 , ⎪ ⎩ k v + k (v − e ), v ≥ e 12 1 11 1 1 1 1 and

⎧ ⎪ ⎨ k22 v2 + k21(v2 + e2 ), v2 ≤ −e2 −e2 < v2 < e2 G2 (v2 ) = k22 v2 , ⎪ ⎩ k v + k (v − e ), v ≥ e 22 2 21 2 2 2 2

(7.1a)

(7.1b)

where v1 = x2 − x1 , v2 = x4 − x3 , and e1 , e2 denote the clearances at the upper and lower NESs, respectively. We note that in the limits k11 , k21 → ∞, the clearance nonlinearities approach vibro-impact (VI) limits, so the NS NES design considered herein can be extended to the VI case as well. As mentioned above, the considered non-smooth stiffness nonlinearities represent strong forms of stiffness nonlinearities. This can be easily inferred from the VI limit, where each vibro-impact represents an impulsive (pseudo) excitation of the system, and, as a result, excites modes of the system over a broad frequency range. Therefore, we anticipate that the dynamic interactions in the system of Figure 7.1 will be broadband. We introduce at this point the following coordinate transformations: u1 = (x1 + x2 )/2,

v1 = x2 − x1 ,

u2 = (x3 + x4 )/2,

v2 = x4 − x3

(7.2)

In physical terms, variables v1 and v2 are the relative displacements of the internal masses m2 and m4 of the two NESs with respect to their frames, whereas, u1 and u2 are related to the motions of their centers of mass. Using these new coordinates the

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equations of motion of the system are expressed as follows: u¨ 1 + εα1 (v1 /2 + u1 − y1 )3 = 0 v¨1 + 2εα1 (v1 /2 + u1 − y1 )3 + 4G1 (v1 ) + 4λv˙1 = 0 y¨1 + λy˙1 + ω12 y1 + εα1 (−v1 /2 − u1 + y1 )3 + ε(y1 − y2 ) = F (t) y¨2 + λy˙2 + ω22 y2 + εα2 (−v2 /2 − u2 + y2 )3 + ε(y2 − y1 ) = 0 u¨ 2 + εα2 (v2 /2 + u2 − y2 )3 = 0 v¨2 + 2εα2 (v2 /2 + u2 − y2 )3 + 4G2 (v2 ) + 4λv˙2 = 0

(7.3)

with zero initial conditions. Unless otherwise noted, in the following exposition the masses of the system are assigned the values M1 = M2 = 1 and m1 = m2 = m3 = m4 = 1/2. The system of equations (7.3) is solved numerically in Matlab (Georgiades, 2006; Georgiades et al., 2005). The accuracy of the numerical computations is ensured by computing the overall instantaneous energy of the system at each step of the computation, as well as the amount of energy dissipated up to that time instant by the dampers of the system; it is checked that the summation of these two energy measures is identical (within a small numerical round-off error) to the energy input to the system provided by the external shock. A first quantitative assessment of TET efficiency in the system is performed by defining appropriate energy dissipation measures (EDMs), i.e., by computing the percentage of shock energy dissipated by each NES up to a specified time instant (as discussed in previous chapters). To this end the following two EDMs are introduced, t λ v˙i2 (ξ )dξ 0 ENES i (t) = T × 100, i = 1, 2 (7.4) F (τ )y˙1 (τ )dτ 0

where T is the duration of the external shock. An additional quantitative measure to study the efficacy of utilizing the nonsmooth NESs for shock isolation is the computation of nonlinear shock spectra; in this study these will be plots of the maximum level of vibration attained by the LO 2 for varying grounding stiffness ω12 (see Figure 7.1) and fixed shock and fixed system parameters. The use of shock spectra is a standard technique for designing linear shock isolation systems. In the nonlinear case the use of such spectra is limited by their dependence on the magnitude of the excitation (or the energy of the motion). Nevertheless, as reported below they can provide useful information for judging the effectiveness of the NESs when the relative duration of the shock with regard to a characteristic time scale of the problem varies; such a characteristic time scale can be chosen as being equal to either one of the characteristic periods of the linear primary system, Ti = 2π/ωi , i = 1, 2. Moreover, one can compare the nonlinear


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spectra with reference linear spectra corresponding to systems of similar designs but with no NESs attached in order to assess and compare the effect of TET on shock isolation. Finally, we note that since the system of Figure 7.1 possesses a symmetrical design, it is capable of dual-mode shock isolation, i.e., of preventing transient disturbances of either one of the LOs of the primary system from being transmitted to the other. Clearly, the underlying dynamical mechanism governing this type of shock isolation is passive nonlinear TET.

7.1.2 Numerical Results In the following series of numerical simulations we consider finite half-sine external excitations of the form: , 0≤t≤T A sin 2πt T (7.5) F (t) = 0, t >0 In Figure 7.2 we depict the portion of shock energy dissipated by the dampers of the two NESs for varying clearance e1 of NES 1. For this series of simulations the characteristics of the force were set equal to A = 104 and T = 10−4 , whereas the other parameters of the system were assigned the following values: ε = 0.27, k11 = 102,

λ = 0.1, k21 = 10,

ω12 = ω22 = 0.5,

e2 = 0.05,

k12 = k22 = 0,

α1 = α2 = 85

To obtain these results we integrated the equations of motion and computed the transient EDMs (7.4). The plot of Figure 7.2 depicts the asymptotic limit ENES,t 1 ≡ limt 1 {ENES 1 (t) + ENES 2 (t)}, which represents the percentage of shock energy that is eventually dissipated by the two NESs during the entire duration of the motion. From the plot of Figure 7.2 we conclude that the maximum value of ENES,t 1 is approximately 75% and occurs for e1 ≈ 0.23. For clearances greater that this optimal value the internal mass of NES 1 does not possess enough amplitude to overcome the clearance, and the nonlinear effects in the dynamics of NES 1 are eliminated. Note that even above this optimal clearance the damper of NES 1 still dissipates about 42% of shock energy, but this amount is not affected by the clearance e1 anymore. For e1 > 0.23, however, there are still nonlinear clearance effects in the dynamics of NES 2, as this NES possesses sufficiently large relative amplitude |v2 | to overcome the clearance e2 . Referring to the plot of Figure 7.2, we note that in the range of small clearances e1 there is an almost linear increase of ENES,t 1 as function of clearance. The ineffectiveness of NES 1 for small clearances can be explained by noting that the internal restoring stiffness k11 used in these particular simulations is rather strong; it follows that at low clearances the motion x1 of the internal mass m2 of NES 1

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Fig. 7.2 Percentage of shock energy, ENES,t1 , eventually dissipated by the two NESs as a function of clearance e1 , for A = 104 and T = 10−4 ; points A and B correspond to the plots of Figures 7.4 and 7.5, respectively.

Fig. 7.3 Percentage of shock energy, ENES,t1 , eventually dissipated by the two NESs as a function of clearance e1 , for A = 105 and T = 10−4 .


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Fig. 7.4 Transient responses corresponding to point A of Figure 7.2: (a) instantaneous normalized energy in the primary system; (b) transient responses of LO 1 and NES 1.

is heavily constrained by its restoring stiffness k11 , yielding small relative displacement v1 . As a result, only a small fraction of the total external energy ends up being dissipated by the internal damper of NES 1. With increasing clearance, the displacement of the internal mass of NES 1 also increases, which is reflected on the corresponding increase of ENES,t 1 in Fig-

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Fig. 7.4 Transient responses corresponding to point A of Figure 7.2: (c) transient responses of LO 2 and NES 2.

ure 7.2. With increasing clearance e1 one expects the percentage of shock energy dissipated by NES 1 to reach a maximum before settling to a linear motion when the clearance is large enough to eliminate the nonlinear effects in that NES. A similar phenomenon was observed in the dynamics of a vibro-impacting beam (Azeez and Vakakis, 2001) where proper orthogonal decomposition was used to quantatively identify regions of maximum nonlinear vibro-impact interaction between the beam and the rigid contstraints which caused the vibro-impacts. The effect of increasing the shock amplitude on ENES,t 1 is investigated in the plot of Figure 7.3, corresponding to force characteristics A = 105 and T = 10−4 , and all other parameters held fixed. We note that since the system under consideration is nonlinear one expects that the dynamics will change qualitatively (and obviously quantitatively) with increasing energy input. In this case the maximum portion of energy dissipated by the two NESs again reaches levels up to nearly 75% (yielding an optimal clearance of e1 ≈ 1.83). For sufficiently low clearances, again there is a nearly linear increase of ENES,t 1 as function of e1 ; however, as the clearance increases a saturation-like effect is noted regarding TET efficiency. To study the time scale of TET in this system we study the diminishing of the energy stored in the primary system in time. In Figures 7.4 and 7.5 we depict representative plots of the instantaneous normalized energy (with respect to the shock energy) of the primary linear system for simulations corresponding to points A (e1 = 0.15) and B (e1 = 0.23) of the plot of Figure 7.2. For comparison purposes the corresponding plots for systems with no NESs attached are also shown. Note the significant reduction of the normalized energy when the NESs are attached


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Fig. 7.5 Transient responses corresponding to point B of Figure 7.2: (a) instantaneous normalized energy in the primary system; (b) transient responses of LO 1 and NES 1.

(see Figures 7.4a and 7.5a); also note the significant reduction (nearly 40%) of the corresponding maximum amplitude of y2 (t) (see Figures 7.4c and 7.5c). Clearly, the energy absorbed and dissipated by the NESs results in enhanced shock isolation of the primary system. A special note is appropriate at this point, with regard to the fast time scale of energy dissipation in the system when NESs are attached. This yields considerable

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Fig. 7.5 Transient responses corresponding to point B of Figure 7.2: (c) transient responses of LO 2 and NES 2.

energy transfer from the primary system to the NESs right from the beginning of the motion (i.e., during the energetically high regime of the dynamics). The capacity of the NS NESs to rapidly absorb shock energy in the initial highly energetic regime is critical to their role as shock isolators, and is evidenced by the drastic reduction of the initial peaks in the transient responses of the LOs 1 and 2, occurring immediately after the application of the external shock (see Figures 7.4b, c and 7.5b, c). We now construct nonlinear shock spectra of the response of the system of Figure 7.1. For fixed external shock, these spectra depict the maximum amplitude reached by the LO 2 of the primary system as function of the grounding stiffness ω12 . In essence, we investigate the maximum amplitude of y2 (t) by varying the duration T of the shock with respect to the period of free oscillation of the LO 1 of the primary system. For comparison purposes we also compute the linear shock spectra of the following two reference systems: (i) the linear primary system with no NESs attached, and, (ii) the primary system with no NESs attached, but possessing masses M1 = M2 = 2 (that is, when the mass of each of the removed NESs is integrated into the corresponding mass of the LO to which it was originally attached). The second reference system is considered in order to cancel any missing mass effects from the comparisons of the shock spectra of the linear and nonlinear systems. In Figure 7.6 we depict the nonlinear spectrum for the system with parameters ε = 0.27,

λ = 0.1,

k11 = k21 = 5 × 102 ,

ω22 = 5.0,

e2 = 0,

α1 = α2 = 85

e1 = 0.9,

k12 = k22 = 0,


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Fig. 7.6 Nonlinear shock spectra of the system with NESs attached, and the two linear reference systems, for A = 105 and T = 10−4 .

and shock characteristics A = 105 and T = 10−4 . Referring to the nonlinear spectrum we note that by adding the NESs we are able to reduce the maximum of y2 (t) by as much as 80% (for ω12 ≈ 5); in addition, we are able to nearly eliminate the resonance of the linear spectra in the neighborhood of ω12 = 5. These results demonstrate the drastic effect of broadband TET from the primary system to the NESs on the shock isolation of the primary system. However, since the system under consideration is nonlinear the depicted nonlinear spectrum is energy dependent, and as such, it will vary when the forcing level varies. Similar conclusions can be drawn by examining the nonlinear spectrum of Figure 7.7 corresponding to e1 = 0.51, force characteristics A = 5 × 104 and T = 10−4 , and all other parameters held fixed. For this weaker shock excitation, there is again significant reduction of the amplitude of the nonlinear system compared to the two linear reference systems. Again, we note the near complete elimination of the linear resonance close to ω12 = 5 from the nonlinear spectrum. Additional results reported in Georgiades (2006) demonstrate the insensitivity of the EDM with respect to variations of the clearance stiffnesses k11 and k21 , indicating that the capacities for TET of the NESs do not change significantly in the vibro-impact limit, i.e., for k11, k21 → ∞. The results reported in this section provide numerical evidence of the capacity of NS NESs for TET at a fast scale, and demonstrate the efficacy of using NS NESs in shock isolation designs. The significance of using NS NESs lies in the fact that they can rapidly absorb a major portion of the shock energy of the primary system, during

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Fig. 7.7 Nonlinear shock spectra of the system with NESs attached, and the two linear reference systems, for A = 5 × 104 and T = 10−4 .

the critical initial (highly energetic) stage of the dynamics, that is, immediately after the external shock is applied. Moreover, the clearance nonlinearities utilized herein are easily implementable in practical settings, since they are realized by means of assemblies of linear stiffness elements. In the limits of large clearance stiffnesses the nonlinearities approach VI limits. The numerical simulations performed in the VI limits indicate that the capacities for TET of the NESs are not significantly altered. Indeed, results similar to those reported in this section hold in the VI case as well, provided, of course, that the vibro-impacts occur elastically, i.e., that they do not introduce additional sources of dissipation in the system [for a discussion of the effects of inelastic impacts on vibro-impact dynamics of elastic systems, the reader is referred to Emaci et al. (1997)]. Hence, the afore-mentioned conclusions are expected to hold also for the VI case. Finally, we point out that the attached NS NESs affect the global dynamics of the integrated system in a time-varying fashion. This is evidenced by the capacity of the NS NESs to passively absorb shock energy from the primary system early on, that is, right at the initiation of the motion when the energy is at its highest (and damage due to shock is expected typically to occur). As the overall energy level decreases due to damping dissipation, the nonlinear effects of the NS NESs on the dynamics gradually decrease, and the system settles into linearized motion regimes. It follows that the effects of the NS NESs on the system dynamics are expected to be significant only in the highly energetic initial stages of the shock-induced motion, i.e., at precisely the regime where fast and efficient energy dissipation should be achieved in order to effectively isolate the primary system.


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7.2 Vibro-Impact (VI) NESs as Shock Absorbers We now provide a second application of a non-smooth NES, by studying a SDOF LO (the primary system) with an ungrounded vibro-impact (VI) NES. We aim to study the use of the VI NES as shock absorber. Then, by considering different configurations of primary LOs with VI attachments we will investigate the capacity of this type of attachments to rapidly absorb and dissipate significant portions of shock energy applied to the primary systems. In addition, we will perform parametric studies to determine the dependence of energy dissipation by the VI attachment on the system parameters. To perform these tasks we will employ nonlinear shock spectra similar to the ones considered in the previous section in order to demonstrate that a properly designed VI attachment can significantly reduce the maximum level of shock response of primary systems (over wide frequency ranges) through fast-time broadband TET. This is in contrast to classical linear vibration absorbers, whose actions are restricted to be narrowband. As in the previous section we will show that VI attachments can significantly reduce or even completely eliminate resonances from the corresponding shock spectra, thus providing strong, robust and broadband shock protection to the primary structures to which they are attached. The regular and chaotic dynamics and bifurcations of vibro-impact (VI) oscillators have been studied extensively in the literature (Shaw and Rand, 1989; Babitsky, 1998; Brogliato, 1999; Kryzhevich and Pliss, 2005; Thota et al., 2006; Thota and Dankowicz, 2006). In an additional series of papers (Masri and Caughey, 1966; Masri and Ibrahim, 1973; Filippov, 1988; Pfeiffer and Glocker, 2000; BlazejczykOkolewska, 2001; Leine and Nijmeijer, 2004; Peterka and Blazejczyk-Okolewska, 2005; Sun and Luo, 2006), VI dampers were considered for reducing the vibration levels of structures under periodic or stochastic excitation. Shaw and Holmes (1982, 1983), Shaw (1985) and Shaw and Shaw (1989) applied methods from the geometrical theory of nonlinear dynamics to analyze the dynamics of free and forced dynamics of systems with piece-wise nonlinearities, including systems undergoing vibro-impacts. In additional recent works, Gorelyshev and Neishtadt (2006) discussed the extension of adiabatic perturbation theory to VI systems; Mikhlin et al. (1998), Leine et al. (2000), Czolczynski and Kapitaniak (2004), Wen et al. (2004), Kryzhevich and Pliss (2005), Dupac and Marghitu (2006), Leine (2006), Lin et al. (2006), Halse et al. (2007), Luo et al. (2007) studied periodic orbits, bifurcations and chaos in discrete and continuous oscillators with clearance or vibro-impact nonlinearities; Ivanov (1993, 2003, 2004) analyzed singularities of the dynamics of systems with bilateral and unilateral constraints and discussed properties of the solutions of systems with Coulomb friction; Zhuravlev (1976, 1977) investigated vibro-impact oscillations using non-smooth coordinate transformations [for an additional application of this method, see also Azeez et al. (1999)], and Pilipchuk (1985, 1988, 2001, 2002) extended this approach by considering non-smooth transformations of the dependent (temporal) variable of the problem; in a recent work Thomsen and Fidlin (2008) developed an analytical methodology for analyzing the dynamics of systems undergoing near-elastic vibro-impacts, by extending the method of discontinuous

242


transformations in conjunction with an extended averaging principle; Pinnington (2003) analyzed energy exchange and dissipation due to collisions in a line of coupled oscillators; (Valente et al., 2003) provided a geometrical analysis of the dynamics of a vibro-impacting two-DOF system; Salapaka et al. (2001) studied the dynamics of a linear oscillator impacting with a vibrating platform; Quinn (2005) investigated the oscillations of two parametrically excited pendula undergoing vibroimpacts; Li and Darby (2006) reported experimental work on the effect of an impact damper on a MDOF system; Zhao and Dankowicz (2006) analyzed degenerate grazing dynamics of impact microactuators using analytical and numerical techniques; Lancioni and Lenci (2007) studied the forced dynamics of a semi-infinite beam on unilateral elastic springs, and Murphy and Morrison (2002) studied computationally and experimentally instabilities and bifurcations of a vibro-impacting string; Hu and Schiehlen (2003) discussed multi-scale simulation of impact responses with applications ranging from wave propagation to rigid body dynamics; Sampaio and Soize (2007) formulated measures that quantify nonlinear effects for uncertain systems, whereas Azeez and Vakakis (2001) approached the issue of quantification of nonlinear effects in the dynamics of vibro-impacting systems by means of the method of proper orthogonal decomposition. Vedenova and Manevitch (1981), Vedenova et al. (1985), Gendelman (2006) and (Meimukhlin and Gendelman, 2007) examined modeling inelastic impacts with smooth, essentially nonlinear stiffness functions of high degree; Nayeri et al. (2007) investigated the action of multi-unit impact dampers in systems under stochastic excitation, and Namachchivaya and Park (2005) developed an analytical approach based on averaging for studying the dynamics of VI systems under stochastic excitation; Wagg (2007) used energy balance analysis to examine multi-modal systems undergoing vibro-impacts, and studied effective restitution coefficients; Shaw and Pierre (2006) applied tuned impact dampers to rotating structural components and assessed their performance; and Veprik and Babitsky (2001) investigated vibration isolation of a SDOF linear oscillator with a dynamic vibration absorber with motionlimiting constraints.

7.2.1 Passive TET to VI NESs Following Karayannis et al. (2008), we consider the system depicted in Figure 7.8 consisting of a SDOF LO with mass m coupled to an attachment with vibro-impact (VI) nonlinearity. Apart from the weak restoring stiffness k1 , it is assumed that the nonlinear attachment undergoes two-sided inelastic impacts when it reaches the left and right limits of the clearance 2e. We assume that the LO is forced by a half-sine shock F (t) of magnitude A and duration T , T A sin 2πt T , 0≤t ≤ 2 (7.6) F (t) = 0, t ≥ T2


243

Fig. 7.8 SDOF LO with vibro-impact (VI) NES.

and that the system has zero initial conditions. The system possesses two viscous dampers with characteristics c and c1 , with additional passive energy dissipation resulting from the inelastic collisions of the attachment mass µ. We model inelastic impacts by introducing the coefficient of restitution rc, defined by the relation rc = −

x˙1+ − x˙2+

x˙1− − x˙2−

(7.7)

where the superscripts (∓) refer to velocities before and after impacts, respectively. Clearly, the coefficient of restitution assumes values in the range 0 < rc < 1, with unity corresponding to perfectly elastic, and zero to purely plastic impacts. Although the coefficient of restitution depends on the composition of the impacting bodies and surfaces, and on the magnitudes of the velocities of the bodies during impact, in this study it is assumed to remain constant during each individual numerical simulation. For sufficiently large clearances no vibro-impacts occur, and the system becomes linear (in effect we obtain a LO with a classical linear vibration absorber attached to it). It follows that by increasing the clearance to sufficiently large values we will be able to compare the dynamics of the linear and VI-systems, and, hence, assess the effects of vibro-impacts on the transient dynamics and on shock isolation. We wish to study passive TETs from the directly forced LO to the VI NES, leading to passive shock isolation of the LO. We note that even though the VI system under consideration is strongly nonlinear, in intervals between impacts it behaves in a purely linear fashion. Hence, in time windows in between vibro-impacts the nondimensionalized equations of motion are expressed as the following linear set: x¨1 + ωn2 x1 + λx˙1 − λ1 (x˙2 − x˙1 ) − 2 (x2 − x1 ) = N(t)

244


εx¨2 + 2 (x2 − x1 ) + λ1 (x˙2 − x˙1 ) = 0

(7.8)

where ωn2 = k/m, 2 = k1 /m, λ = c/m, λ1 = c1 /m, ε = µ/m, N(t) = F (t)/m. This linear system of equations is numerically integrated until the condition of impact |x2 − x1 | = e is realized. At that time instant the two conditions of continuity of displacements and discontinuity of (jump in) velocities are imposed, according to the following formulas: x˙1+ =

x˙ 1− (1 − ε rc) + εx˙ 2− (rc + 1) , 1+ε

x˙2+ =

x˙1− (1 + rc) + x˙2− (ε − rc) 1+ε

(7.9)

Then, the system (7.8) is numerically integrated subject to the new initial conditions following the impact, until the impact condition is met again and the previous procedure is repeated. It follows that a precise computation of the time instants of vibro-impacts is crucial for the correct numerical integration of the transient dynamics of the considered two-DOF system. The transient VI dynamics is simulated utilizing a Matlab code which computes precisely the time instants where impacts occur. This class of VI dynamical problems is especially challenging from the analysis and computational point of view, since the time instants where the vibro-impacts occur are determined by the solution itself, so they cannot be determined a priori, i.e., before solving the problem. Moreover, the essential (strong) nonlinearity of the system is generated precisely at these time instants (due to the impulsive excitations applied to the integrated system); in addition, at exactly these time instants significant portions of energy are dissipated due to inelastic collisions between the attachment mass and the mass of the linear oscillator (due to the fact that rc < 1). It follows that in order to computationally model this class of problems correctly one must pay special attention to the accurate computation of the time instants of vibro-impacts, as well as the energy dissipated during each impact. In the numerical code the accuracy of the transient computation was evaluated by checking that at each time instant of the simulation the total initial energy of the system (provided by the external shock) equaled the sum of instantaneous kinetic and potential energies of the system, as well as the energy dissipated by the two viscous dampers and the inelastic vibro-impacts up to that time instant. The efficiency of the VI NES to passively absorb and locally dissipate shock energy from the LO is evaluated by computing the instantaneous energy dissipation measure (EDM) and its asymptotic limit, i.e., of the percentage of shock energy that is eventually dissipated by the NES damper and by the inelastic vibro-impacts: EVI NES,t 1 = lim EVI NES (t), t 1

EVI NES (t) =

t

c1 [x˙1 (τ ) − x˙2 (τ )]2 dτ +

0

Pt p=1

T 0

f (τ )x˙1 (τ )dτ

p × 100

(7.10)


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The EDM EVI NES (t) represents the percentage of shock energy dissipated up to time instant t; p , p = 1, . . . , Pt , the amount of energy dissipated during the p-th vibro-impact; and Pt the number of vibro-impacts that occur during the decaying motion up to time instant t. We perform a series of parametric studies for the following system parameters, m = 1, ωn2 = 1, λ = λ1 = 0.1, A = 10, T = 0.1(2π/ωn ), rc = 0.6; in the simulations we varied the mass ratio of the VI NES ε, the coupling frequency squared 2 , and the clearance e. In Figure 7.9 we depict contour plots of the asymptotic EDM EVI NES,t 1 for fixed clearance and varying mass ratio and coupling frequency squared; whereas in Figure 7.10 we present the corresponding contour plot for the linear system with relatively large clearance (e ≈ 2.0) so that no vibroimpacts occur during the entire decaying motion. Considering first the linear case (see Figure 7.10), we note that the asymptotic EDM assumes high values (reaching a maximum of 73.7%) on a narrow zone corresponding to linear resonance of the attachment with the primary system. Away from this zone the EDM deteriorates to less than 50%. This is expected, as the effectiveness of the linear attachment as shock absorber is narrowband and, as a result, its performance deteriorates away from the condition of linear resonance with the primary system. By contrast, the performance of the VI attachment is broadband, so its effectiveness to passively absorb and dissipate shock energy is expected to extend over wider parameter ranges. Regarding the performance of the VI NES (see Figure 7.9), we note the existence of two regions in each of the EDM contour plots, corresponding to weak and stiff coupling frequency , respectively, compared to the grounding frequency ωn of the primary system. In the region of weak coupling frequency, an increase of the clearance up to the value of 1.25 results in an increase of the EDM and, hence, enhancement of TET; above this clearance value, however, we note deterioration of the EDM. In contrast, in the region of stiff coupling frequency, increasing the clearance results in an increase of the EDM, and, hence, enhances the performance of the VI NES as shock absorber. As expected, for relatively large values of the clearance, the EDM contour plot of the VI system approaches the linear plot of Figure 7.10, as fewer or no vibro-impacts occur during the decaying motion. Clearly, the number and timings of vibro-impacts are expected to affect significantly TET from the primary LO to the NES, and, hence, the effectivess of the VI NES as shock absorber. To study in more detail the dependence of TET on the occurring vibro-impacts, we examined the differences in the asymptotic EDM contour plots between the VI and linear systems, and attempted to relate these differences to the number of vibro-impacts occurring in the corresponding nonlinear responses. In Figure 7.11 we depict representative results that address this issue; positive (negative) differences in the asymptotic EDM correspond to surplus (deficit) of energy dissipated by the VI NES when compared to the linear vibration absorber. From these plots we infer that the number of vibro-impacts plays an important role in the enhancement of the asymptotic EDM in the VI case. By increasing the clearance, the number of vibro-impacts diminishes (as expected), but this does not necessarily imply that the effectiveness of the VI NES as shock isolator deteriorates. For example, for clearance e = 0.25 (not shown in Figure 7.11) although the num-

246


Fig. 7.9 Contour plots of the energy dissipation measure EVI NES,t1 for varying 2 and mass ration ε: (a) e = 0.25; (b) e = 1.00; (c) e = 1.25; (d) e = 1.50.


247

Fig. 7.10 Contour plots of the energy dissipation measure EVI NES,t1 for the linear system (no vibro-impacts, e ≈ 2.0), for varying 2 and mass ratio ε.

ber of vibro-impacts is of the order of 6000, the performance of the VI NES is not significantly enhanced compared to cases with larger clearances (see Figure 7.10); on the other hand, for e = 1.50 there occur at most 3 vibro-impacts (and in most of the contour plot of Figure 7.11b no vibro-impacts occur at all), yet the VI NES shows as much as 18% better EDM efficiency compared to the linear absorber in this case. This can be explained by noting that for small clearances and mass ratios the relative displacements and velocities across the viscous damper c1 of the VI NES are expected to be small, so that relatively small amounts of shock energy are dissipated by that damper (these results agree with the findings reported in the previous section). In addition, if vibro-impacts start immediately after or even during the application of the external shock, for small mass ratios the attachment is prevented from attaining sufficient acceleration and velocity, and as a result, the exchange of momentum between the LO and the NES and the amount of shock energy dissipated due to vibro-impacts may be small even if a large total number of vibro-impacts occurs. From the plots of Figure 7.11a we note that by increasing the mass ratio, we can achieve enhanced EDM; moreover, the regions of intense vibro-impacts coincide with the regions of enhanced nonlinear shock absorption efficiency. Of course, in regions where no vibro-impacts occur the performance of the VI attachment converges to that of the linear absorber. It follows that efficient shock absorption of the vibro-impact NES can be achieved even for small number of occurring vibro-impacts. These findings should be considered in conjunction with the fact that in the two limits e → 0 and e ≈ 2 the two-DOF system of Figure 7.8 degenerates into two different linear systems: either a SDOF linear oscillator with mass equal to (m + µ),

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Fig. 7.11 Difference of asymptotic EDM contour plots between the VI and linear systems and corresponding number of vibro-impacts: (a) e = 0.50; positive (negative) values correspond to surplus (deficit) of energy dissipated by the VI NES.

or a linear oscillator of mass m with an attached linear vibration absorber of mass µ. For intermediate values of e and small values of 2 , the response of the system is strongly nonlinear, and there exist extensive regions in the parameter plane (ε, 2 )


249

Fig. 7.11 Difference of asymptotic EDM contour plots between the VI and linear systems and corresponding number of vibro-impacts: (b) e = 1.00; positive (negative) values correspond to surplus (deficit) of energy dissipated by the VI NES.

where the efficiency of the VI NES exceeds that of the linear absorber by as much as 29% (for a clearance e ≈ 1). In terms of the asymptotic EDM, for mass ratios in the range 0.2 < ε < 0.5 the VI NES eventually absorbs and locally dissipates 60–80%

250


Fig. 7.11 Difference of asymptotic EDM contour plots between the VI and linear systems and corresponding number of vibro-impacts: (c) e = 1.50; positive (negative) values correspond to surplus (deficit) of energy dissipated by the VI NES.

of the shock energy, even for a relatively small number of total vibro-impacts (from 2 to 6). In Figure 7.12 we depict the asymptotic EDM EVI NES,t 1 as function of clearance e for a VI NES with 2 = 0.005 and mass ratio ε = 0.3. For this small value


251

Fig. 7.12 Asymptotic EDM EVI NES,t1 as a function of clearance e for 2 = 0.005 and ε = 0.3.

of coupling frequency, there is a nearly linear variation of EVI NES,t 1 for small clearances. However, for increasing clearance we note saturation in the plot of the asymptotic EDM yielding to an optimal value of this measure. Further increase of results in deterioration of the asymptotic EDM, until the linear regime is reached, where no vibro-impacts occur and the EDM becomes independent of clearance [the constant level of EDM in that regime is due to the percentage of shock energy dissipated by the coupling viscous damper c1 (see Figure 7.8)]. In conclusion, superior shock absorption by the VI NES compared to the linear absorber is attained for intermediate clearances (i.e., away from the two limiting linear systems corresponding to zero of large clearances), small coupling frequencies 2 , and large mass ratios ε. Moreover, high shock absorption efficiencies may be attained for even small total number of vibro-impacts, provided that conditions for sufficient momentum and energy exchanges between the primary system and the NES during vibro-impacts are realized. In the next section we examine the shock isolation properties of VI NES-based designs by constructing nonlinear shock spectra and comparing these to reference linear spectra.

7.2.2 Shock Isolation To study the capacity for shock isolation of the VI NES we consider the system of Figure 7.13a consisting of a two-DOF linear primary system with an attached VI

252


Fig. 7.13 Primary structure on foundation with VI NES attached: (a) system configuration; (b) spectra depicting maximum normalized force transmitted to ground as a function of normalized frequency ratio.

NES. As with the study of the previous section, the following exposition follows closely (Karayannis et al., 2008). The mass m represents the primary structure to be isolated, resting on an elastic foundation of mass m3 . An external half-sine shock (7.6) is applied to the primary structure, and our aim is to assess the capacity for shock isolation of a VI NES-based design. The computational study of this system is performed for the following parameter values: m = 1,

m3 /m = 5,

ω32 = k3 /m3 ,

ωn2 = k/m = 5,

λ = c/m = 0.1,


253

Fig. 7.14 Transient response x3 of the foundation under resonance, ω3 /ωn = 1.

λ1 = c1 /m = 0.1, rc = 0.6,

λ3 = c3 /m3 = 0.1ω3 ,

µ = 0.35,

e = 0.6,

A = 10,

k1 = 0.005,

T = 0.2π

In Figure 7.13b we depict the nonlinear shock spectrum for the vibro-impacting system. For comparison purposes, we present also two additional reference shock spectra of the following two limiting linear systems: the system obtained in the limit of large clearances when no vibro-impacts occur (i.e., when the system degenerates to a primary system attached to a linear vibration absorber); and the system obtained in the limit of zero NES mass (that is, the primary system with no NES attached). In each shock spectrum we depict the maximum value of normalized forced transmitted to ground, FˆTrans = max{c3 x˙3 + m3 ω32 x3 }/(J ω3 ), as a function of the non-dimensional frequency ratio (ω3 /ωn ). The parameter J in the denominator of the normalized transmitted force FˆTrans represents the shock impulse defined by

T

J =

F (t)dt

(7.11)

0

From the results depicted in Figure 7.13b we infer that due to TET of shock energy from the primary structure to the VI NES there is significant reduction of the normalized transmitted force; in the resonance region ω3 ≈ ωn , FˆTrans reaches levels which are 40% less compared to the reference linear system with no attachment, and 32% less compared to the reference system with linear absorber attached. Moreover, in the VI system complete elimination of the resonance in the shock spectrum is realized, so effective shock isolation is provided over a broad frequency range. Hence, in contrast to the narrowband action of the linear vibration absorber, the VI

254


Fig. 7.15 Primary structure with sensitive component and VI NES: (a) system configuration; (b) spectra of maximum normalized acceleration of the sensitive component as function of nondimensional frequency ratio.

NES provides effective broadband shock isolation. In Figure 7.14 we present the response x3 of the foundation under resonance, ω3 /ωn = 1, for the nonlinear and reference linear configurations discussed previously. Note the significant suppression of the maximum response of the foundation in the case of VI NES, signified by the rapid decay of the corresponding transient motion. The second system considered in our shock isolation study is depicted in Figure 7.15a (Karayannis et al., 2008). We wish to study the transmission of shockinduced vibrations from the directly forced primary structure to a sensitive component (denoted by the mass m3 ) that is connected to the mass of the primary system m. A vibro-impact NES is attached to the primary system, which itself is excited by


255

Fig. 7.15 Primary structure with sensitive component and VI NES: (c) transient response x3 of the sensitive component under linear resonance, ω3 /ωn = 1.

the half-sine shock (7.6). This shock isolation study is performed for the following parameter values: m = 1,

m3 /m = 0.01,

λ1 = c1 /m = 0.1, rc = 0.6,

ω32 = k3 /m3 ,

ωn2 = k/m = 5,

λ3 = c3 /m3 = 0.1ω3 ,

µ = 0.35,

e = 0.6,

λ = c/m = 0.1,

A = 10,

k1 = 0.005,

T = 0.2π

In Figure 7.15b we depict the nonlinear shock spectrum of this system, and compare it to two reference spectra, corresponding to linear systems with, either no VI attachment, or a linear vibration absorber attached instead of an VI NES. These spectra are computed by depicting the maximum normalized acceleration of the sensitive component, G = max{x¨3 /(u˙ m ω3 )}, as a function of the non-dimensional frequency ratio (ω3 /ωn (. The velocity u˙ m in the denominator of the normalized acceleration G is computed by the following relation: u˙ m =

J , m

T

J =

F (t)dt

(7.12)

0

From these results we deduce that the VI attachment provides effective shock isolation of the sensitive component over a broad range of frequency ratios, by significantly reducing the maximum normalized acceleration G. Moreover, in the critical resonance region ω3 ≈ ωn the reduction in acceleration due to TET of shock energy to the VI NES is significant compared to the reference linear systems; in

256


fact, in the nonlinear spectrum the linear resonance close to ω3 /ωn = 1 is almost completely suppressed. In Figure 7.15c we depict the transient response x3 of the sensitive component for the nonlinear and reference linear systems under resonance, ω3 /ωn = 1. We note the drastic reduction of the level vibration due to TET of shock energy to the VI NES. To study the robustness of the shock isolation performance of the VI NES we performed a final series of numerical simulations, where for fixed impulsive excitation (A = 10, T = 0.2π) and linear system parameters as defined above, we varied the parameters of the VI NES and computed shock spectra similar to that depicted in Figure 7.15b; we then compared these nonlinear spectra to the ones of the corresponding linear systems with infinite clearance (i.e., primary systems with linear absorbers attached) in order to judge the effect that vibro-impacts have on the shock isolation. In each of these plots, the system parameters that are not varied are assigned the numerical values defined above. The results are depicted in Figure 7.16. In Figure 7.16a we depict the comparison between the nonlinear and linear spectra for five values of VI NES mass, µ = 0.1, 0.2, 0.35, 0.4 and 0.5, keeping all other NES parameters fixed. We note that the larger the VI NES mass is, the better the relative performance of the VI NES becomes with respect to the linear absorber design; therefore, we conclude that the mass of the VI NES is an important parameter for shock isolation (this was to be expected, since as mentioned previously, the mass of the VI NES affects its momentum exchange with the linear oscillator during vibroimpacts. In Figure 7.16b we examine the effect of varying the clearance e on shock isolation, by considering nonlinear spectra for four different clearances, e = 0.1, 0.2, 0.4 and 0.6 and all other NES parameters being kept fixed. We note that for decreasing clearance the linear and nonlinear spectra converge, and that better relative shock isolation performance of the VI NES is achieved for larger clearances. In Figure 7.16c we examine the effect of varying the viscous damping coefficient of the VI NES, by considering five different damping values, namely, λ1 = 0.01, 0.05, 0.2, 0.3 and 0.4. From these results we conclude that the variation of viscous damping of the VI NES does not affect significantly its effectiveness. Similarly, the variation of the coefficient of restitution of the vibro-impacts from the value rc = 0.6 (this series of plots is not shown here) has a marginal effect on the performance of the VI NES. Finally, in Figure 7.16d we depict the effect on the nonlinear shock spectra of the variation of the restoring linear stiffness k1 of the VI NES. In that study we considered five different values of linear stiffness, namely, k1 = 5 × 10−3 , 1 × 10−2 , 0.1, 0.5 and 1.0; by comparing the nonlinear shock spectra with the corresponding linear ones, we note that there is robustness of shock elimination within the examined range of stiffness values. By further increasing the stiffness (k > 5.0), no vibro-impacts occur and the response of the system becomes purely linear. The results presented in Figure 7.16 demonstrate robustness of the shock isolation provided by the VI NES for changes of its parameters. Summarizing, we studied alternative shock isolation designs of primary structures with attached VI NESs. We showed that VI NESs can be designed as effective shock isolators, providing significant reduction of maximum responses of the primary systems over broad frequency ranges. Hence, appropriately designed VI


257

Fig. 7.16 Parametric study of shock spectra of maximum normalized acceleration of the sensitive component as function of ω3 /ωn and (a) the VI NES mass µ; (b) the clearance e; for comparison, the corresponding linear spectra of systems with infinite clearances (e.g., with linear vibration absorbers attached) are also depicted.

NESs can act as broadband, passive shock isolators, which may significantly reduce or even completely eliminate resonances from shock spectra. The results presented

258


Fig. 7.16 Parametric study of shock spectra of maximum normalized acceleration of the sensitive component as function of ω3 /ωn and (c) the VI NES viscous damping λ1 ; and (d) the restoring stiffness k1 ; for comparison, the corresponding linear spectra of systems with infinite clearances (e.g., with linear vibration absorbers attached) are also depicted.

indicate that, in designing VI NESs as shock isolators, important design parameters that should be taken into account are the clearances, the coupling stiffnesses and the NES masses. The parametric studies performed in this section indicate that


259

depending on the application and the types of applied shocks, there is a range of clearances for which effective shock energy absorption and dissipation occurs; outside this range, smaller or larger clearances result in deterioration of shock absorption, as the vibro-impacting systems approach linear limiting systems. Moreover, better vibro-impact shock absorption is anticipated for weak coupling stiffness between the primary systems and the attached NESs, and relatively large values of NES masses compared to the masses of the corresponding primary systems. These results are in agreement with findings of Section 7.1.

7.3 SDOF Linear Oscillator with a VI NES In Section 7.2 we provided numerical evidence of the potential of vibro-impact (VI) attachments as passive shock absorbers. In this section we study in more detail the unforced dynamics of a SDOF linear primary system with a VI attachment, in an effort to explore the richness and complexity of the vibro-impact dynamics and relate it to TET. Our methodology will be similar to that followed in Sections 3.3 and 3.4 for the analogous system with ungrounded NES with smooth nonlinear stiffness characteristics. First, we will explore the rich structure of periodic orbits (and impulsive orbits – IOs) of the VI dynamics in a frequency-energy plot (FEP) of the Hamiltonian system undergoing purely elastic vibro-impacts; our aim will be to study if the FEP of the Hamiltonian VI system possesses a structure of backbone curves, subharmonic tongues and impulsive orbits, similar to the one of the system with ‘smooth’ NES. Then, we will study damped transitions in the dynamics of the VI system undergoing inelastic vibro-impacts, by superimposing the wavelet transform (WT) spectra of transient damped responses to the Hamiltonian FEP; our effort will be to demonstrate schematically that such representations contribute to the identification and interpretation of complex resonance captures and multi-modal transitions occurring in these damped transitions. Finally, we will proceed to make some preliminary remarks and conjectures regarding the mechanisms for TET in the VI system. Overall, our aim will be to demonstrate that vibro-impacts in oscillators with even small clearances can introduce great complexity in the unforced dynamics of the two-DOF system considered. Our exposition will follow the results of Nucera et al. (2007) and Lee et al. (2008), which should be consulted for additional results on this topic. Application of VI NESs to the problem of seismic mitigation was studied by Nucera et al. (2008), and the results of these works will be discussed in Chapter 10.

260


Fig. 7.17 SDOF LO with VI NES.

7.3.1 Periodic Orbits for Elastic Vibro-Impacts Represented on the FEP To study the underlying dynamical mechanisms and associated TRCs that generate passive TET in systems with attached VI NESs, and in order to demonstrate the complexity that a single VI NES (with even small clearance) can introduce to the dynamics, we consider the simplest possible primary system – VI NES configuration, namely a SDOF LO coupled to a VI NES (see Figure 7.17). As in the case of the NES with smooth essential nonlinearities we will show that a clear interpretation of damped VI transitions governing TET in the shock-excited system may be gained by depicting the WT spectra of these motions in a FEP of the underlying VI Hamiltonian system (i.e., for the system with purely elastic impacts and no viscous damping elements). As in in the case of the NES with smooth nonlinearities we will demonstrate that for sufficiently weak dissipation, damped VI transitions take place near branches of periodic or quasi-periodic motions of the corresponding undamped system. Hence, by studying the structure of periodic orbits of the Hamiltonian system, we should be able to gain an understanding of the governing dynamics of the weakly damped dynamics, and to be able to clearly identify complex multi-frequency transitions and resonance captures producing strong energy exchanges and TET in the weakly damped VI system. We initiate our study by constructing the FEP of the Hamiltonian VI system with no viscous damping dissipation and purely elastic vibro-impacts. We consider the VI model introduced in Section 7.2.1. In time intervals between vibro-impacts the equations of motion are purely linear and given by (with an alternative notation being adopted for the system parameters compared to Section 7.2.1), m1 x¨1 + k1 x1 + c1 x˙1 + k2 (x1 − x2 ) = 0 m2 x¨2 + k2 (x2 − x1 ) = 0 or in non-dimensional form:

(7.13)


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x1 + x1 + λx1 + σ (x1 − x2 ) = 0 εx2 + σ (x2 − x1 ) = 0

(7.14)

In the normalized equations (7.14) primes denote differentiation with respect to the transformed temporal variable ξ = (k1 /m1 )1/2 t, and λ = c1 /(k1 m1 )1/2 , ε = m2 /m1 , σ = k2 /k1 are dimensionless damping, mass and stiffness parameters, respectively. Moreover, in (7.14) we consider normalized displacements defined by the rescalings xi → xi /e, i = 1, 2, so in terms of these normalized displacements vibro-impacts occur when |x1 − x2 | = 1. As in Section 7.2 the mass m2 of the NES is not necessarily small compared to the mass m1 of the LO, so the ratio ε is assumed to be an O(1) quantity. Assuming zero viscous damping, λ = 0, and considering purely elastic impacts (corresponding to restitution coefficient ρ = 1), the VI system becomes Hamiltonian. The velocities of the LO and the NES after an impact (denoted by superscripts +) are computed in terms of the corresponding velocities before impact (denoted by superscripts −) as follows: x1+ =

x1− (1 − ε) + 2εx2− , 1+ε

x2+ =

2x1− + (ε − 1)x2− 1+ε

(7.15)

In the numerical algorithm used for our computational study the linear equations (7.14) are integrated as long as the no-impact condition |x1 − x2 | < 1 is satisfied. When |x1 − x2 | = 1 a (purely elastic) impact occurs and dicontinuities in the velocities take place, whereas the displacements remain continuous through the vibro-impact process; the velocities immediately after the impact are computed by relations (7.15), and the numerical integration of the linear system (7.14) resumes with the new initial conditions until the next vibro-impact occurs and the outlined procure is iterated. As discussed in Section 7.2.1 and Nucera et al. (2007), the precise computation of the time instants of vibro-impacts is essential for the accuracy of the numerical simulations. This accuracy was checked by recording the total energy of the VI motion, ensuring its conservation throught the entire interval of the simulation. The total (conserved) energy of the normalized Hamiltonian system H (x1, x1 , x2 , x2 ) is computed in terms of the initial conditions of the normalized system (7.14) as follows: H (x1 , x1 , x2 , x2 ) =

εx22 (0) + x12 (0) σ [x2 (0) − x1 (0)]2 + x12 (0) + = h (7.16) 2 2

Assuming zero initial velocities, the critical threshold below which no vibro-impacts can occur is computed as hcrit = σ/2. Throughout this section the normalized mass and stiffness parameters are taken as ε = 0.1 and σ = 0.1, resepctivelly. The periodic solutions of the Hamiltonian vibro-impacting system were computed by employing the method of non-smooth transformations first introduced by Pilipchuk (1985) and Pilipchuk et al. (1997), and applied in Section 3.3.1.1 for computing the periodic orbits (NNMs) of the corresponding system with smooth essen-

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tial nonlinearity (i.e., with a ‘smooth’ NES). To this end, we introduce the following coordinate transformations: x1 (ξ ) = e(ξ/α)y1 (τ (ξ/α)),

x2 (ξ ) = e(ξ/α)y2 (τ (ξ/α))

(7.17)

where α = T /4 represents the quarter period of the periodic motion, and the nonsmooth functions τ (·) and e(·) are defined according to expressions (3.9). Then, we obtain the following NLBVP in terms of the variables y1 , y2 , y3 = y1 and y4 = y2 , in analogy to system (3.10) of Section 3.3.1.1: y1 = y3 y2 = y4 y3 = −α 2 y1 − σ α 2 (y1 − y2 ) y4 = −(σ α 2 /ε)(y2 − y2 ) y1 (±1) = 0,

y2 (±1) = 0

(7.18)

In system (7.18) primes denote differentiations with respect to the non-smooth variable τ , and the periodic orbits are computed subject to zero initial velocities. Vibro-impacts occur when |y1 − y2 | = 1, where discontinuity conditions in the velocities are imposed, in similarity to conditions (7.15). The solution of the NLBVP (7.18) computes the VI periodic orbit over half its period T = 4α (i.e., for −1 ≤ τ ≤ 1 ⇒ −α ≤ ξ ≤ α); to extend this orbit over the entire period of the oscillation we take into account the form of the non-smooth transformations (7.17) (see also related discussion in Section 3.3.1.1). Moreover, taking into account the transformations (7.17), the conserved energy of the VI periodic orbit is expressed as h=

1 [εy 2 (−1) + y12 (−1)] 2α 2 2

(7.19)

The NLBVP (7.18) is solved by a shooting algrorithm in the bounded domain −1 ≤ τ ≤ 1. However, unlike the shooting method employed for the case of the NES with smooth nonlinearity (Lee et al., 2005), matching at τ = 0 of the two solutions initiated from the left and right boundaries (τ = ±1) is not helpful in the present VI problem. This is due to the fact that symmetric VI periodic orbits are expected to exhibit vibro-impacts at τ = 0, so matching solutions at this point is meaningless. Therefore, for the VI problem the following matching procedure is adopted (Lee et al., 2008): for fixed quarter period α the set of equations of the NLBVP is solved as an initial value problem with initial conditions defined at the left boundary, yi (−1) = 0, yi (−1) = 0, i = 1, 2; then we perform matching of the solution of the initial value problem at the right boundary through the inequalities, |yi (+1)| < tol 1, i = 1, 2, where the tolerance is taken as tol ∼ O(10−5 )−O(10−6 ). This procedure ensures that the NLBVP (7.18) is approximately solved (that is, within the presecribed numerical tolerance).


263

Fig. 7.18 Vibro-impact nonlinearity as limiting case of a family of smooth, essentially nonlinear stiffnesses.

It is anticipated that the seemingly simple VI system of Figure 7.17 will possess a very complicated structure of periodic orbits in the FEP. This expectation is justified when considering that vibro-impact nonlinearity represents a very strong (and degenerate) form of nonlinearity. A way to view this is by considering the following family of odd essentially nonlinear stiffnesses, fn (u) = kn u2n+1 ,

n = 0, 1, 2, . . .

(7.20)

where the coefficient kn depends on the exponent and is selected so that the normalization condition fn (±1) = 1, n = 0, 1, 2, . . . is satisfied. Then, the vibro-impact nonlinearity corresponding to purely elastic impacts is obtained as the (degenerate) discontinuous limit f∞ (u) = limn→∞ fn (u) (Vedenova et al., 1985; Manevitch et al., 1989). Viewed in this context, vibro-impact nonlinearity can be considered as the ‘strongest possible’ stiffness nonlinearity of the family (7.20). Following this reasoning, Pilipchuk (1985, 1988, 1996) developed an asymptotic methodology based on non-smooth transformations and non-smooth generating functions, that is applicable to strongly nonlinear regimes (Pilipchuk et al., 1997, 1998; Salenger and Vakakis, 1998). In fact, it is interesting to note that Pilipchuk’s technique is not applicable to weakly nonlinear regimes, where conventional perturbation methods of the dynamical systems theory based on harmonic generating functions are applicable! For more information of this method we refer to the above-mentioned works by Pilipchuk and to (Vakakis et al., 1996). The anticipated high complexity of the structure VI periodic orbits dictates the use of careful notation for their representation in the FEP (Lee et al., 2008). In principle, the basic notation introduced in Section 3.3.1.1 for the FEP of the dynamics of the Hamiltonian system with ‘smooth’ NES is followed, with an additional index being introduced characterizing the pattern of the occurring vibro-impacts. To this end, we employ the following notation for depicting the various types of VI periodic orbits in the FEP.

264


Symmetric VI periodic orbits are denoted as SmnE(O)pp±, and satisfy the conditions xi (ξ ) = ±xi (ξ + T /2), ∀ξ ∈ R, i = 1, 2, where T is the period of the motion. Similarly to the case of ‘smooth’ NES, symmetric VI orbits correspond to synchronous oscillations of the LO and the VI NES, and typically are represented by curves in the configuration plane (x1 , x2 ). Unsymmetric VI periodic orbits labeled as U mnpq±, do not satisfy the conditions of the symmetric ones (for notational simplicity, whenever p = q we will adopt the convention, U mnpp± ≡ U mn±). These orbits correspond to asynchronous motions of the two oscillators and are represented by open or closed (Lissajous) curves in the configuration plane (x1 , x2 ). The integer index m denotes the number of half-waves in the VI NES response within a half-period, whereas the integer index n denotes the corresponding number of half-waves in the LO response; clearly, the ratio (m:n) indicates the order of nonlinear resonance that occurs between the VI NES and the LO on the given VI periodic orbit. The index E or O denotes the symmetry pattern of the vibroimpacts, and has meaning only for symmetric VI orbits: E(O) denotes an even (odd) symmetry of occurring vibro-impacts within a half-period; it follows that the notation E(O) implies that a vibro-impact occurs (does not occur) at quarter-period ξ = α ≡ T /4. The integer indices p and q denote the number of vibro-impacts that occur in the first and second quarter-period, respectively, of a given VI periodic orbit; it follows that for symmetric orbits it holds that p = q for symmetric orbits. Finally, the (+) sign corresponds to in-phase VI periodic motions, where, for zero initial displacements the initial velocities of the LO and the NES possess identical signs at the beginning of both the first and second half-periods of the periodic motion; otherwise, the VI periodic motion is deemed to be out-of-phase and the (−) sign is used. Finally, the two in-phase and out-of-phase linear modes of the system are denoted by Lmm±, and are, in fact, equivalent to L11±. The FEP of the Hamiltonian VI system for ε = 0.1 and σ = 0.1 is depicted in Figure 7.19, with some representative VI periodic orbits presented in Figure 7.20. The complexity of the bifurcations that generate the VI periodic orbits can be inferred from the bifurcation diagrams of Figure 7.21 where the initial velocities of the LO and the VI NES (for zero initial displacements) are depicted as functions of the total energy h. This complexity was anticipated in view of the degenerate vibro-impact nonlinearity of this system. Note that the FEP in Figure 7.19 is obtained for system (7.14) with all displacements being normalized with respect to the clearance e, so that vibro-impacts occur whenever the absolute value of the relative displacement between the two particles becomes equal to unity in magnitude. Considering the original system (7.13) with clearance e, its Hamiltonian is expressed as, Hˆ = e2 H where H is the normalized Hamiltonian defined by (7.16). This implies that, for fixed system parameters, ε = 0.1 and σ = 0.1, the Hamiltonian structure of the original (non-normalized) system will be identical to that of Figure 7.19; it follows that for a larger (smaller) clearance, the entire structure of VI periodic orbits will be preserved by just shifted towards higher (lower) energy regimes. So, the introduced normalization allows us to study all possible VI responses of the original system by considering a single ‘normalized’ FEP for fixed mass and coupling stiffness ratios. It is interesting to


265

Fig. 7.19 FEp of periodic orbits of the Hamiltonian system with VI NES for ε = σ = 0.1; the manifold of VI IOs is indicated by the dashed line (- - -), whereas the thresholds for vibro-impacts for the in-phase and out-of-phase modes are denoted by bullets (•); unstable branches are denoted by crosses, and energy regimes I–IV referred to in Section 7.3.2.2 are indicated.

note that this normalization does not hold for the system with smooth stiffness nonlinearity (i.e., the ‘smooth’ NES); this can be easily deduced when noting that the introduced normalizations change the form of the system with smooth nonlinearities. Indeed, as we recall from the discussion of Chapter 3, the topological structure of the FEP of the system with ‘smooth’ NES (and the corresponding bifurcation structure of the Hamiltonian periodic orbits) is affected by both the mass ratio and the essentially stiffness nonlinearity of the NES. We now make some comments and remarks regarding the ‘normalized’ FEP of Figure 7.19. First, we note that the two dots indicate the critical energy thresholds below which oscillations without vibro-impacts occur, and the dynamics of the two-DOF system is purely linear. Clearly, only the in-phase and out-of-phase linear modes L11± exist below these energy thresholds. As we increase the energy the motion above these energy thresholds vibro-impacts start occurring, giving rise to two main branches of symmetric periodic VI NNMs: the branch of out-of-phase symmetric VI NNMs S11O00−, which bifurcates from the out-of-phase linear mode L11−, after which this mode becomes unstable; and the branch of symmetric inphase VI NNMs S13O00+ which bifurcates out of the in-phase linear mode L11+, after which this linear mode also becomes unstable. For convenience, from hereon the shortened notations, S11O00− = S22E11− = S33O22− = . . . ≡ S11−

266


Fig. 7.20 Solutions of the NLBVP (7.18) for ε = σ = 0.1: (a) symmetric VI periodic orbit on the backbone S13O00+ ≡ S13+, α = 1.7, h = 0.58875; (b) symmetric VI periodic orbit S73O33−, α = 1.6459, h = 7.3348; (c) unsymmetric VI periodic orbit U 8353−, α = 4.9065, h = 1.9458.

and S13O00+ = S26E11+ = . . . ≡ S13+, will be adopted for these two main backbone branches, which will be referred to as backbone (global) branches of the FEP. Both backbone branches exist over broad frequency and energy ranges, and, except for the neighborhoods of the bifurcation points with L11±, they correspond to oscillations that are mainly localized to the VI NES. A basic bifurcation in the VI FEP is the saddle node (SN) bifurcation of the two backbone branches S11− (at


267

Fig. 7.21 Bifurcation diagrams (incomplete) depicting the initial velocities as functions of the total energy h for ε = σ = 0.1: (a) VI NES, (b) LO; unstable branches are marked by crosses.

268


h ≈ 0.06), which signifies the elimination of the unstable branch S11− that bifurcates out of L11−; the stable branch S11− that is generated after this SN bifurcation maintains its stability for increasing energies. As shown below, this SN bifurcation of the backbone branch S11− affects the capacity of the VI NES for TET. The additional in-phase backbone branch S13+ that bifurcates out of the in-phase linear mode L11+ is stable until high energies where zones of instability appear. In Figure 7.20a we depict a typical stable VI orbit on the in-phase backbone branch S13+. We note that in the corresponding FEP of the system with ‘smooth’ NES studied in Section 3.3 (see Figure 3.20), there exist two backbone branches S11±. As indicated by the time series of Figure 7.20a, however, on the in-phase VI backbone branch S13+ three sign changes for the LO velocity within half a period are realized, compared to only one for the NES velocity (Lee et al., 2008); this highfrequency component becomes more prominent at higher energies (in addition, as shown in the next section, a 3:1 TRC occurs in the neighborhood of this branch when weak damping is added to the system). A different class of VI periodic solutions of the FEP lies on subharmonic tongues (local branches); these are multi-frequency periodic motions, possessing frequencies that are rational multiples of one of the linearized eigenfrequencies of the system. In similarity to the FEP of the system with ‘smooth’ NES (see Figure 3.20), each subharmonic tongue is defined over a finite energy range, and is composed of a pair of branches of in- and out-of-phase subharmonic oscillations. Depending on the behavior of these VI subharmonic tongues with varying energy, the FEP is portioned into four main Regimes, which are labeled as Regimes I–IV in Figure 7.19. In the highest energy Regime IV, out-of-phase VI subharmonic orbits (both symmetric and unsymmetric) bifurcate out of the backbone branch S11−. With increasing energy they form subharmonic tongues of out-of-phase motions with almost constant frequencies, until they reach the manifold of VI IOs (see discussion below) after which they change to in-phase motions until they coalesce with the inphase backbone branch S13+ at specific energy levels; this signifies the end of these tongues and the elimination of the corresponding subharmonic motions for higher energy values. This is similar to what was observed in the FEP of the system with ‘smooth’ NES; however, the unsymmetric subharmonic tongues of the system with NS NES do not fold on themselves to reach back the out-of-phase backbone branch S11−. In Regime III of the FEP the bifurcation behavior of subharmonic tongues is similar to Regime IV. The apparent difference is that the manifold of VI IOs undergoes a discontinuous transition on branch S31O11−, caused by the two bifurcations of that branch with the unsymmetric subharmonic branches U (15)578− and U 8353− in that region (see Detail I in Figure 7.19). The subharmonic orbits in Regime II exhibit different bifurcation behavior than in Regimes IV and III. In fact, there appear to be no subharmonic tongues bifurcating from S11−; instead, small subharmonic tongues appear to lie along the manifold of VI IOs, and eventually merge with the in-phase backbone branch S13+ with decreasing energy. For example, the in-phase unsymmetric branch U 21+ bifurcates from S13+ and turns into the out-of-phase unsymmetric branch U 21−


269

after it crosses the manifold of VI IOs. In addition, for frequencies in between the two linearized frequencies ω1 and ω2 there exist multiple subhharmonic branches bifurcating in a degenerate (higher co-dimensional) bifurcation from the in-phase linear mode L11+ (see Detail II in the FEP of Figure 7.19). These subharmonic branches coexist with the in-phase backbone branch S13+, which is unstable in most of Regime II. Examples of this type of subharmonic branches are U 2201+, U 5511+, U 4421+, . . . in the FEP of Figure 7.19. Finally, the lowest energy Regime I of the FEP is defined for energies below the bifurcation point of the linear mode L11−. The manifold of VI IOs meets the stable out-of-phase linear mode L11− at a bifurcation point that coincides with the critical energy level hcrit = σ/2; we recall that for h < hcrit no vibro-impacts are possible and the dynamics of the system is completely linear. As in the case of the system with ‘smooth’ NES there exists a countable infinity of subharmonic tongues, corresponding to symmetric or unsymmetric VI subharmonic motions with different patterns of vibro-impacts during a cycle of the oscillation. As in the case of smooth nonlinearity (see Figure 3.20), unsymmetric VI periodic orbits are represented by closed (Lissajous) curves in the configuration plane of the system. In Figures 7.20b, c we depict two representative symmetric and unsymmetric VI orbits on two subharmonic branches of the FEP. Finally, there exists a third class of VI motions in the FEP, which are denoted as VI impulsive orbits (VI IOs). These are VI periodic solutions corresponding to zero initial conditions of the system, except for the initial velocity of the LO. As discussed in Section 3.3 a VI IO represents, in essence, the response of the system being initially at rest and forced by a single impulse applied to the LO at time t = 0+. Apart from the clear similarity of a VI IO to the Green’s function defined for the corresponding linear system, the importance of studying this class of orbits stems from their essential role regarding passive TET from the linear oscillator to the NES [see Section 3.4.2 and Lee et al. (2005), Kerschen et al. (2005, 2006, 2008a)]. Indeed, in the case of the NES with smooth nonlinearities, IOs (which, under some conditions, are in the forms of nonlinear beats) play the role of bridging orbits occurring in the initial phase of TET and ‘channeling’ a significant portion of the applied impulsive energy from the linear primary system (in this case the LO) to the NES at a relatively fast time scale; as discussed in Section 3.4.2 this represents the most efficient scenario for passive TET (i.e., TET through nonlinear beats – see Sections 3.4.2.4 and 3.4.2.5). Although the aforementioned results refer to damped impulsive orbits, the dynamics of the underlying Hamiltonian system determines, in essence, the dynamics of the damped system as well, provided that damping is sufficiently small. It follows that the IOs of the VI Hamiltonian system govern, in essence, the initial phase of TET from the LO to the NES. The numerical results show that VI periodic and quasi-periodic IOs form a manifold in the FEP, containing a countable infinity of periodic IOs, and an uncountable infinity of quasi-periodic IOs. For the system under consideration the approximation of the manifold of VI IOs was computed numerically, and in Figure 7.19 is superimposed to the FEP; in general, the manifold resembles a smooth curve, with the exception of a number of outliers, which are due to the adopted convention for the frequency index in the

270


Fig. 7.22 Two representative VI IOs for ε = σ = 0.1 (left U 5231−, right U 4122−): 9a) unnormalized responses x1 (t) (LO) and x2 (t) (VI NES); (b) relative responses x2 (t) − x − 1(t); (c) representation in the configuration plane (x1 , x2 ).

FEP (Lee et al., 2008). On a VI subharmonic tongue, a VI IO is realized whenever the relative motion between that LO and the VI NES changes from in-phase to out-of-phase. Representative VI IOs are depicted in Figure 7.22. In general, the IOs become increasingly more localized to the VI NES as their energy decreases, a result which is in agreement with previous results for NESs with smooth essential nonlinearities (Kerschen et al., 2008a). As energy increases, the VI IOs tend


271

towards the in-phase mode, i.e., their plot in the configuration plane (x2 , x1 ) tends to a straight line of slope 0.084π [since the eigenvector for L11+ on the (x2 , x1 ) plane is equal to (3.702, 1)]. Moreover, we note there is no critical energy threshold for the appearance of VI IOs since there are no low- energy VI motions (the system is linear for low energy levels), and that the dominant frequency of a VI IO depends on the clearance e (i.e., on the energy regime where the VI periodic orbits exist). Due to the degenerate VI nonlinearity of this system, it is expected that higherco-dimensional bifurcations will occur in the dynamics. One case of indication of such degenerate bifurcations in presented in the Detail II of the FEP of Figure 7.19, where multiple branches of symmetric and unsymmetric VI periodic orbits (branches S95E22−, U 5421−, U 4520−, S55O11−, U 3201−, . . . ) are noted to bifurcate from the in-phase linear mode L11+ at the point of generation of the in-phase backbone branch S13+. In addition, VI periodic orbits below the branch S73E33− appear to lie along the VI IO manifold, for example, tongues U 21±, S95E22−, U 5421−, U 4520−, S55O11−, U 3401− and U 44− in the FEP of Figure 7.19. It is interesting to note that the complexity of the FEP is solely due to the clearance e that gives rise to vibro-impacts. Indeed, in the limit of no clearance, e → 0, the entire structure of VI orbits depicted in the FEP of Figure 7.19 collapses to two horizontal lines corresponding to the linear modes L11±. We conclude that, due to the degeneracy of the VI dynamics, even a small clearance can generate significant complexity, including chaotic orbits, as discussed below. The global dynamics of the Hamiltonian VI system (7.14–7.15) was studied by constructing Poincaré maps resulting from the projection of the isoenergetic dynamics (i.e., of the dynamical flow corresponding to fixed value of h) on the twodimensional ‘cut section’, = {(x1 , x1 , x2 , x2 ) ∈ R 4 /H (x1 , x1 , x2 , x2 ) = h, x1 = 0, x˙1 > 0} which is transverse to the flow except at points where x˙1 = 0. Similar Poincaré map constructions for vibro-impact dynamics were considered in Mikhlin et al. (1998), and for the VI system under consideration are depicted in Figure 7.23. Below the energy level h = hcrit = σ/2 = 0.05 no vibro-impacts occur (see Figure 7.23a), and the only possible periodic solutions are the linear stable modes L11+. At energy levels above this critical threshold (see Figures 7.23b–h), vibroimpacts occur (at |x2 −x1 | = 1), and generate a countable infinity of subharmonic VI orbits that replace the two linear modes L11±; this complex structure of orbits is a direct consequence of the non-integrability of the Hamiltonian VI dynamics. When vibro-impacts occur, the sections of the Poincaré maps corresponding to |x2 | > 1 are cut-off from the Poincaré maps, and the last bounding points that are included in the map are those for which the conditions {x1 = 0 and |x2 | = 1} hold. For increasing energies, the ‘stochastic seas’ (i.e., the regions of chaotic motions in the Poincaré maps) diminish, and the domains of regular motion expand. An additional use of the Poincaré map is that it can help us identify or infer the existence of global features of the dynamics, such as homoclinic and heteroclinic

272


Fig. 7.23 Poincaré maps of VI dynamics for σ = ε = 1 and varying energy.


273

Fig. 7.24 Damped transition for no viscous dissipation and weakly inelastic impacts: (a) WT spectrum of the relative response x2 − x1 superimposed to the FEP; (b) logarithm of instantaneous energy versus time.

loops. For example, at the energy level h = 0.06 (see Figure 7.23c) we identify stable and unstable VI periodic orbits U 44− in the neighborhood of the linear mode L11− and the unstable NNM S11− (see also the location of this branch in the FEP of Figure 7.19). This infers the existence of a homoclinic loop that connects the unstable periodic orbit S11−. The topologies of VI IOs on branches such as U 44− (which lie in the neighborhood of the SN bifurcation of S11−) are greatly influenced by the family of homoclinic orbits of the unstable branch S11− and affect significantly the efficiency of TET from the LO to the NES. This is similar to what occurs for the case of smooth nonlinearity (see discussion in Section 3.4.2.5), where it was found that close to this family of homoclinic orbits conditions for optimal TET are realized. Indeed, as shown in the next section, excitation of stable VI IOs in the neighborhood of the family of homoclinic orbits of S11− provides conditions for optimal VI TET, since large-amplitude relative displacements between the LO

274


Fig. 7.25 Damped transition for ρ = 0.7, λ = 0: (a) time series; (b) WT spectrum of the relative response x2 − x1 .

and the VI NES are realized in that region and the time scale of the resulting TET is affected as well. Apart from the compact representation of VI periodic motions, the FEP is a valuable tool for understanding the nonlinear resonant interactions (transient resonance captures – TRCs) that govern energy exchanges and TET during damped transitions in the weakly dissipative VI system. This is due to the fact that for sufficiently weak dissipation (caused by inelastic vibro-impacts, viscous damping or both), the damped VI dynamics is expected to be realized in neighborhoods of branches in the FEP of the underlying VI Hamiltonian system. This is demonstrated in Figure 7.24


275

Fig. 7.25 Damped transition for ρ = 0.7, λ = 0: (c) percentages of total energy dissipated by the LO and the VI NES; (d) WT spectrum of the relative response x2 − x1 superimposed to the FEP.

where we depict the wavelet transform (WT) spectra of the damped response of the system with ε = σ = 0.1, λ = 0 and ρ = 0.95 superimposed to the Hamiltonian FEP [in this particular simulation the unnormalized responses of the system are depicted (Nucera et al., 2007)]. The motion is initiated on a subharmonic tongue, and we may distinguish three distinct stages in the resulting damped VI transition. In an initial stage, the motion remains in the neighborhood of the subharmonic tongue were it is generated, yielding an initial persistent subharmonic TRC. As a result, subharmonic VI TET takes place from the LO to the NES, and efficient energy dis-

276


Fig. 7.26 Damped transition for ρ = 1.0, λ = 0.005λcrit : (a) time series; (b) WT spectrum of the relative response x2 − x1 .

sipation occurs, as evidenced by the energy dissipation plot of Figure 7.24b. In the second stage of the damped motion the dynamics makes a transition to a different lower energy subharmonic tongue, which signifies the occurrence of a different subharmonic TRC (and subharmonic TET) in the damped dynamics. Escape from this second TRC regimes leads to a transition of the dynamics to the manifold of VI IOs during the third stage of the motion, before the dynamics becomes linear, and undergoes a final transition to the linear mode L11+ (the final stage of the response). It will be shown in the next section that the transition of the damped dynamics along the manifold of VI IOs during the third stage of the motion is associated with a complex series of multiple TRCs with subharmonic tongues existing in the vicinity


277

Fig. 7.26 Damped transition for ρ = 1.0, λ = 0.005λcrit : (c) percentages of total energy dissipated by the LO and the VI NES; (d) WT spectrum of the relative response x2 − x1 superimposed to the FEP.

of this manifold. It follows, that by studying VI transitions in the FEP and relating them to rates of energy dissipation by the VI NES, we should be able to identify the most effective damped transitions from a TET point of view. In a more general context, in the next section we will perform a systematic study of the dynamics of TET in the two-DOF system of Figure 7.17 by assuming inelastic impacts and viscous dissipation in the LO, and analyzing the resulting transient responses by numerical WTs. Then, we will superimpose the resulting WT spectra to the FEP of Figure 7.19

278


Fig. 7.27 Damped transition for ρ = 0.7, λ = 0.05λcrit : (a) time series; (b) WT spectrum of the relative response x2 − x1 .

(similarly to Figure 7.24), in an effort to interprete the damped transitions in terms of the underlying Hamiltonian dynamics, and to identify the governing dynamical mechanisms for VI TET.


279

Fig. 7.27 Damped transition for ρ = 0.7, λ = 0.05λcrit : (c) percentages of total energy dissipated by the LO and the VI NES; (d) WT spectrum of the relative response x2 − x1 superimposed to the FEP.

7.3.2 Vibro-Impact Transitions in the Dissipative Case: VI TET We now consider the weakly dissipative normalized system (7.14) with λ = 0 and inelastic impacts. Then, the relations (7.15) computing the normalized velocities of the LO and the VI NES immediately after an impact in terms of the corresponding velocities before impact, are replaced by the following expressions:

280

x1+ =


(1 − ερ)x1− + ε(1 + ρ)x2− , 1+ε

x2+ =

(1 + ρ)x1− + (ε − ρ)x˙2− 1+ε

(7.21)

where 0 < ρ ≤ 1 is the coefficient of restitution. Throught the numerical simulations of this section, and unless otherwise noted, we assume that ε = 0.1, σ = 0.1, ρ = 0.7 and λ = 0.005λcrit, where λcrit = 2 is the value of critical viscous damping for the LO (hence, weak viscous damping is assumed) (Lee et al., 2008). Before we proceed to analyze damped transitions, we investigate the competion between the two energy dissipation sources present in the system, namely, viscous dissipation in the LO and inelastic impacts in the VI NES. For this purpose, the damped motion was initiated on the stable VI IO lying on the subharmonic tongue U 8344−. In Figures 7.25–7.27 we depict the damped responses for the following cases: (a) for no viscous dissipation in the LO and inelastic impacts (ρ = 0.7, λ = 0 – Figure 7.25); (b) for viscous dissipation λ = 0.005λcrit and purely elastic impacts (ρ = 1.0, λ = 0.005λcrit – Figure 7.26); and (c) for a combination of viscous dissipation and inelastic impacts (ρ = 0.7, λ = 0.005λcrit – Figure 7.27). Comparing the WT spectra of Figures 7.25b, d and 7.26b, d, we note distinct patterns of energy exchange and dissipation in the damped transient dynamics. For purely elastic impacts the response is linear and the WT spectra lie along the two linear modes L11±; when inelastic impacts occur, there occurs a strongly nonlinear transition of the VI dynamics along strongly nonlinear subharmonic tongues and the in-phase backbone branch S13+, until, at the later stage of the response, the dynamics settles into linearized motion along the modes L11±. A similar, albeit weaker, nonlinear transition is noted for the case of combined inelastic impacts and viscous dissipation (see Figures 7.27b, d), where the damped dynamics traces, primarily the backbone branch S13+ (i.e., there occurs an immediate 1:1 TRC of the dynamics of the NES and the in-phase mode L11+ right from the beginning of the motion), and, secondarily, higher frequency subharmonic tongues. We conclude that the addition of weak viscous dissipation in the LO does not affect significantly the VI damped transitions. We focus now in the study of the mechanisms that govern VI TET by fixing ρ = 0.7 and λ = 0.005λcrit, in order to compare the dynamical mechanisms for VI TET to the corresponding mechanisms for the case of ‘smooth’ NES discussed in Chapter 3. As discussed in Section 3.4.2, in the smooth case the following three mechanisms for TET were established: (a) fundamental TET, where the damped in-phase NNM invariant manifold S11+ is excited; (b) subharmonic TET, where a low-frequency subharmonic tongue is excited; and (c) TET through nonlinear beats, where an IO close to the 1:1 resonance manifold of the dynamics is excited. Our study of TET in the VI case will follow similar lines, by considering energy exchanges between the LO and the NES for alternative types of initial excitation of the system. In particular, we will study VI TET when in-phase or out-of-phase periodic orbits lying on backbone and subharmonic tongues are excited, as well as, when the damped motion is initiated by exciting VI IOs at various energy levels. In what follows we examine each of these cases separately.


281

Fig. 7.28 Damped transition initiated on S13+: (a) time series; (b) percentages of total instantaneous energy in the LO and the VI NES; Stages A–D of the damped transition are indicated.

7.3.2.1 Fundamental VI TET In Figure 7.28 we present the damped response of the system for initial conditions on the in-phase backbone branch S13+ and initial normalized energy h ≈ 10.0. There are four distinct stages in the damped response, which are denoted as Stages A–D in Figure 7.28. In the highly energetic initial Stage A there occurs a 1:1 TRC in the dynamics, with the response possessing a strong harmonic at the frequency of the in-phase linear mode and a weaker harmonic at 3ω1 . This is evident by examining the detailed plot depicted in Figures 7.29a, where it is clear that the relative transient

282


Fig. 7.28 Damped transition initiated on S13+: (c) percentages of total energy dissipated by the LO and the VI NES; (d) WT spectrum of the relative response x2 − x1 ; Stages A–D of the damped transition are indicated.

response x2 −x1 in Stage A possesses a third harmonic component; moreover, in the detailed plot of Figure 7.30a it is noted that the WT spectrum of the relative response possesses two clear components, a main one at frequency ω1 , and a secondary one at 3ω1 on the subharmonic branch S31O11−. This leads to fundamental VI TET from the LO to the VI NES, with almost 85% of the initial energy (nearly 45% by the LO and 40% by the VI NES) being dissipated during this initial stage of the motion. The nonlinear modal interactions that lead to fundamental VI TET will be examined in more detail later by EMD, where the governing 1:1 TRC will be more clearly identified.


283

Fig. 7.28 Damped transition initiated on S13+: (e) WT spectrum of the relative response x2 − x1 superimposed to the FEP; Stages A–D of the damped transition are indicated.

Fig. 7.29 Damped transition initiated on S13+, time series of responses: (a) Stage A, (b) Stage B, (c) Stage C.

284


Fig. 7.30 Damped transition initiated on S13+, WT spectrum of the relative response x2 − x1 superimposed to the FEP: (a) Stage A, (b) Stage B.

Stage B (see Figures 7.29b and 7.30b) corresponds to a regime of pure 1:1 TRC as the third harmonic component is nearly eliminated, and the LO and the VI NES execute in-phase oscillations with frequencies approximately equal to ω1 . It is clear that the weakly damped dynamics follows approximately the in-phase backbone branch S13+ until this branch becomes unstable (i.e., at the bifurcation point where the subharmonic branches U 11+ and U 21± bifurcate out of this branch – see Fig-


285

Fig. 7.30 Damped transition initiated on S13+, WT spectrum of the relative response x2 − x1 superimposed to the FEP: (c) Stage C.

ure 7.19). This provides further evidence of the connection between the weakly damped dynamics and the dynamics of the underlying Hamiltonian system. During Stage C of the damped motion there occurs a series of TRCs along subharmonic tongues on the FEP, whereas in the low-energy Stage D vibro-impacts cease to occur, the motion is purely linear, and energy dissipation is solely due to viscous damping of the LO. As expected, the linear dynamics consist of a combination of the damped analogs of the linear in-phase and out-of-phase modes L11±, with mode L11+ being more dominant in the response. We conclude that in this numerical simulation there occurs fundamental VI TET due to 1:1 TRC of the dynamics of the VI NES at frequency ω1 . Recalling that ω1 is the natural frequency of the in-phase linear mode L11+, we conclude that during fundamental VI TET the LO and the VI NES engage in in-phase 1:1 resonance capture. This VI TET mechanism is analogous to fundamental TET discussed in the case of smooth nonlinearity (see Section 3.4.2.1). The next simulation examines the damped response of the system for initial conditions on the out-of-phase backbone branch S11− at h ≈ 0.8 (see Figure 7.31). There is insignificant TET from the LO to the NES in this case, since (as in the case of smooth nonlinearity) the initial energy of the motion localizes predominantly to the VI NES right from the beginning of the motion; then, localization to the VI NES is maintained throughout, as the damped VI motion approximately traces the backbone branch S11−. In fact, in this case vibro-impacts occur only during a short initial stage of the motion (i.e., for ξ < 10 – see Figure 7.31a), where almost 90% of total energy is dissipated. In the purely linear regime where no vibro-impacts occur (for ξ > 10) the response is mainly composed of the damped analogue of the out-

286


Fig. 7.31 Damped transition initiated on S11−: (a) time series; (b) percentages of total instantaneous energy in the LO and the VI Nes.

of-phase linear mode L11−, with a weaker participation of the in-phase damped mode L11+. The participation of these closely spaced modes in the linear response produces a beat phenomenon, which is evidenced by the strong energy exchange noted in the plot of Figure 7.31b. We conclude that there is immediate escape of the transient damped dynamics from the initially excited out-of-phase backbone branch S11−, followed by settlement of the response in alternative response regimes. This is a general conclusion drawn from the performed numerical simulations (Lee et al., 2008), and holds for motions that are initiated on all branches and tongues of the FEP other than the in-


287

Fig. 7.31 Damped transition initiated on S11−: (c) percentage of total energy dissipated by the LO and the VI NES; (d) WT spectrum of the relative response x2 − x1 .

phase backbone branch (we recall that this was also the case for the case of smooth nonlinearity, see Section 3.4).

7.3.2.2 VI TET through Excitation of VI IOs Having established the mechanism of fundamental TET in the system with VI NES, we now consider the possibility of alternative mechanisms for VI TET based on the excitation of VI IOs. As shown in Section 3.4.2 excitation of IOs on certain energy

288


Fig. 7.31 Damped transition initiated on S11−: (e) WT spectrum of the relative response x2 − x1 superimposed to the FEP.

ranges provides the mechanism for most efficient TET in the system with NES with smooth nonlinearity. In the following study we study efficiency of TET from the LO to the NES when VI IOs are excited in the four previously defined Regimes I–IV of the FEP. We recall from the case of NES with smooth nonlinearity that IOs play an important role as far as TET is concerned; this holds especially for IOs lying in the neighborhood of the family of homoclinic orbits of the unstable in-phase damped NNMs S11+, close to the 1:1 resonance manifold of the damped dynamics (see discussion in Section 3.4.2.5). In Figure 7.32 we depict the damped response of the system when a VI IO in Regime I is excited. Since the initial energy of the motion is relatively low, vibroimpacts occur only during the short-duration initial stage of the dynamics, and afterwards the dynamics become completely linear, involving continuous energy exchanges between the two linear modes of the system (with no vibro-impacts) with frequencies ω1 and ω2 . Due to the closely spaced linear natural frequencies, a linear beat develops and energy is predominantly dissipated by viscous dissipation in the LO. In this case insignificant TET from the LO to the VI NES occurs. The damped responses for initial excitation of the VI IO on the subharmonic tongue S95E22− in Regime II of the FEP are depicted in Figure 7.33. In this case the dynamics cannot exhibit a 1:1 TRC, since the in-phase backbone branch S13+ is unstable at the specific initial energy level considered in this simulation. As a result, the damped dynamics may be divided into four distinct stages, labeled A–D in Figure 7.33. Stages A–C are strongly nonlinear, whereas, the low-energy Stage D is linear with no vibro-impacts occurring in that late stage of the response. In Stage


289

Fig. 7.32 Damped transition initiated on a VI IO in regime I: (a) time series; (b) percentages of total instantaneous energy in the LO and the VI NES.

A the damped dynamics follows approximately the tongue S95E22− (where the motion is initiated) with decreasing energy. Nearly 50% of the total initial energy is dissipated during this stage of the response, with 33% of total energy being dissipated due to TET from the LO to the VI NES. The damped dynamics in Stages B and C is complex, as it undergoes transitions along subharmonic tongues such as U 5421−, U 4520−, S55O11− and U 3201− lying close to the manifold of IO. Finally, when sufficient energy is dissipated and no additional vibro-impacts can occur the dynamics settles into the linear Stage D, where predominant contribution of mode L11+ is realized.

290


Fig. 7.32 Damped transition initiated on a VI IO in regime I: (c) percentages of total energy dissipated by the LO and the VI NES; (d) WT spectrum of the relative response x2 − x1 .

In Figure 7.34 the damped dynamics for excitation of the VI IO on the subharmonic tongue S32O11 in Regime III of the FEP is presented. For the selected initial energy level for this simulation 1:1 TRC is possible (since the backbone branch S13+ is stable at the initial energy level considered), and five distinct stages of the damped motion (labeled as A–E in Figure 7.34) are inferred. In Stage A the damped motion follows the subharmonic tongue S31O11− which acts as bridging orbit for the dynamics to make the transition for its initial state to 1:1 TRC (which is realized in Stage B). In Stage B there occurs a 1:1 TRC as the stable in-phase backbone branch S13+ is excited; as a result, fundamental TET from the LO to the VI NES is realized, so that nearly 50% of the total initial energy is dissipated by the VI NES by


291

Fig. 7.32 Damped transition initiated on a VI IO in regime I: (e) WT spectrum of the relative response superimposed to the FEP.

the end of this stage of the motion. As energy decreases due to viscous dissipation and inelastic impacts, the in-phase backbone branch S13+ becomes unstable and the damped dynamics makes a transition to Stages C and D; these stages are similar to those occuring in Regime II of the FEP, and the dynamics follows complex transitions along subhramonic tongues, similar to the ones depicted in Figure 7.33. At the later, low-energy Stage E the dynamics is linear and dominated by mode L11+. We conclude that by exciting VI IOs lying in Regime III of the FEP the ‘bridging orbit’ scenario is realized (as in the case of the system with ‘smooth’ NES), leading eventually to fundamental VI TET. This scenario yiels efficient TET from the LO to the VI NES. Finally, when IOs in the high-energy Regime IV of the FEP are excited (see Figure 7.35) the damped transitions are similar to those realized in Regime III, with TET efficiency at the end of fundamental VI TET reaching levels of nearly 55%. In conclusion, we identify two mechanisms for VI TET, namely, fundamental VI TET due to 1:1 TRC, and VI TET through excitation of a VI IO leading eventually to fundamental VI TET. These are similar to the corresponding TET mechanisms for the case of ‘smooth’ NES. No subharmonic VI TET (caused by TRC of the dynamics on an isolated VI subharmonic tongue) could be realized in the numerical simulations of the dynamics of the VI system under consideration, as the VI dynamics seem to engage in series of TRCs involving multiple subharmonic tongues (instead of an isolated one) lying close to the manifold of VI IOs. However, as shown in the simulations of Figure 7.24, subharmonic VI TET is indeed possible in the VI system of Figure 7.17. We conjecture, therefore, that, probably, subharmonic VI TET is a mechanism for TET in systems with very weak viscous damping and

292


Fig. 7.33 Damped transition initiated on a VI IO in regime II: (a) time series; (b) percentages of total instantaneous energy in the LO and the VI NES.

weakly inelastic impacts. A final conclusion drawn from the previous simulations is that lack of fundamental TET in Regimes I and II of the FEP can be attributed to the instability of the in-phase backbone branch S13+ in the corresponding energy ranges. This is an additional indication of the strong relation that exists between the Hamiltonian dynamics and the weakly damped transitions. The specific nonlinear resonance interactions leading to TET in the VI system can be further analyzed by post-processing the corresponding time series by Empirical Mode Decomposition (EMD). We demonstrate this by analyzing in detail the mechanism for fundamental VI TET and showing that it corresponds to a 1:1 TRC.


293

Fig. 7.33 Damped transition initiated on a VI IO in regime II: (c) percentages of total energy dissipated by the LO and the VI NES; (d) WT spectrum of the relative response x2 − x1 .

To this end, we analyze the high-energy damped transition depicted in Figure 7.28 for motion initiated on the in-phase backbone branch . As shown in previous chapters, decomposition of nonlinear damped transitions by EMD leads to multi-scale nonlinear system identification of the governing dynamics, and provides the means for identifying nonlinear resonance (modal) interactions between substructures, as well as the time (of frequency) scales where these modal interactions occur. In Figure 7.36 we depict the results of EMD analysis, from which we conclude that the response of the VI NES possesses a single dominant IMF [the leading one – denoted by c1 (NES)], whereas, the LO response possesses two dominant IMFs [the

294


Fig. 7.33 Damped transition initiated on a VI IO in regime II: (e) WT spectrum of the relative response superimposed to the FEP.

first and second ones – denoted by c1 (LO) and c2 (LO)]. Denoting by φ11 the instantaneous phase of the dominant IMF c1 (NES) computed by the numerical Hilbert transform, and by φ21 and φ22 the corresponding phases of c1 (LO) and c2 (LO), and computing the phase differences θ1 = φ11 −φ21 and θ2 = φ11 −φ22, TRCs occurring between IMFs can be studied in detail. Indeed, as discussed in previous Chapters, when a specific phase difference exhibits time- like (i.e., monotonic with time) behavior over a specified time interval, it may be regarded as a ‘fast’ angle, and, hence, may be averaged out of the dynamics; this eliminates the possibility of resonance interaction between the corresponding IMFs. On the contrary, non-time-like behavior of a phase difference precludes the direct application of the averaging theorem with respect to that angle, and the possibility for resonance interaction between the corresponding IMFs. In Figure 7.36d we note non-time-like behavior of the phase difference θ1 during Stage A of the damped motion, which indicates that a 1:1 TRC between the dominant IMF of the VI NES and the first dominant IMF of the LO is realized (denoted as RC1 in Figure 7.36d). Escape from this resonance capture is signified by the time-like behavior of θ1 in Stage B, followed by an additional 1:1 TRC (RC3) in the weakly nonlinear Stage C, and by the purely linear Stage D. This delayed lowenergy TRC signifies the excitation of the in-phase linear mode L11+ towards the end of the damped transition. An additional 1:1 TRC (RC2) between the dominant IMF of the VI NES and the second dominant IMF of the LO is revealed by the nontime-like behavior of the phase difference θ2 in stage B of the decaying response. As discussed previously, this corresponds to the regime of pure 1:1 TRC between the VI NES and the LO, which leads to fundamental TET.


295

Fig. 7.34 Damped transition initiated on a VI IO in regime III: (a) time series; (b) percentages of total instantaneous energy in the LO and the VI NES.

It is interesting to note that the aforementioned partitioning of the phase evolution plot of Figure 7.36d in terms of distinct stages, coincides with the corresponding partitioning introduced previously in the plots of Figure 7.28 (see, especially the energy exchange plot of Figure 7.28b). This demonstrates that the EMD technique although applied (by construction) in an ad hoc manner, can still lead to physically relevant results. Efforts towards a rigorous physical interpretation of EMD results in terms of the slow flow dynamics of a system, and application of EMD in the context of nonlinear, non-parametric system identification are undertaken currently (Kerschen et al., 2008b).

296


Fig. 7.34 Damped transition initiated on a VI IO in regime III: (c) percentages of total energy dissipated by the LO and the VI NES; (d) WT spectrum of the relative response x2 − x1 .

We end this section by mentioning that the presented VI TET results are by no means optimized, that is, higher TET efficiencies may be achieved when alternative sets of initial conditions are considered. This leads us naturally to the discussion of TET efficiency in the system with VI NES carried out in the next section.


297

Fig. 7.34 Damped transition initiated on a VI IO in regime III: (e) WT spectrum of the relative response superimposed to the FEP.

7.3.2.3 Efficiency of VI TET We aim to study the efficiency of VI TET in the system of Figure 7.17 by introducing certain definitions related to the capacity of the VI NES to passively absorb and dissipate vibration energy from the LO, as well as the time required for this VI dissipation to occur. Specifically, we denote by ξLI the normalized time instant when the last vibro-impact in a given simulation occurs (that is, for ξ > ξLI the transient response is purely linear); by ξ95% the time required for 95% of the initial energy of the system to get dissipated by viscous dissipation and inelastic vibro-impacts; and by EVI NES the percentage of initial energy that is eventually dissipated by the VI NES due to inelastic vibro-impacts (i.e., during the entire duration of the damped motion). In this context, the ratio EVI NES /ξLI represents the average measure of the percentage of energy dissipated by the VI NES per unit time, whereas, the ratio 95/ξ95% the average percentage of energy dissipated per unit time until 95% of total energy is dissipated. The ratio EVI NES /ξLI provides a measure of VI TET efficiency per unit time as long as vibro-impacts occur (i.e., for ξ ≤ ξLI ), and is used as a means of judging the rate (time scale) of energy dissipation (efficiency) by the VI NES only; on the contrary, the measure 95/ξ95% is used to study the overall rate of energy dissipation in the system (including the combined the effects of inelastic vibro- impacts and viscous damping dissipation). Clearly, higher values of the energy measure EVI NES , and/or lower values of the time measure ξ95% , result in more efficient VI TET in the system under consideration. Moreover, if there are no other sources of dissipation, higher values of the average rate EVI NES /ξLI indicate high VI TET efficiency in the nonlinear regime of

298


Fig. 7.35 Damped transition initiated on a VI IO in regime IV: (a) time series; (b) percentages of total instantaneous energy in the LO and the VI NES.

the damped response, i.e., in the regime where vibro-impacts are realized. However, in the presence of additional viscous damping dissipation in the LO, the average rate 95/ξ95% provides a better indicator of the overall efficiency of TET to dissipate a significant prortion of the total initial energy of the system throughout the damped motion. In Figure 7.37 we depict the measures ξLI , ξ95% , EVI NES and the average rates EVI NES /ξLI , 95/ξ95% as functions of initial energy, for damped VI responses initiated on VI IOs over a wide energy range of the FEP. The system parameters used for these simulations are ρ = 0.7, λ = 0.005λcrit and ε = σ = 0.1. As expected,


299

Fig. 7.35 Damped transition initiated on a VI IO in regime IV: (c) percentages of total energy dissipated by the LO and the VI NES; (d) WT spectrum of the relative response x2 − x1 .

with increasing energy more vibro-impacts occur, as indicated by the increase of the normalized time measure ξLI with increasing energy in Figure 7.37a. Judging from the dependence of the energy measure EVI NES on energy (see Figure 7.37a), we conclude that most efficient VI TET is realized when VI IOs are excited in Regimes III and IV of the FEP (the highest VI TET efficiency is above 65% for this series of simulations). Moreover, VI TET in Regimes III and IV occurs at a relatively fast time scale, as indicated by the relatively small values of the normalized time measure ξ95% in the corresponding energy ranges (see Figure 7.37a).

300


Fig. 7.35 Damped transition initiated on a VI IO in regime IV: (e) WT spectrum of the relative response superimposed to the FEP.

Focusing now on the average rates depicted in Figure 7.37b, we deduce again that the most efficient rates of the overall energy dissipation measure 95/ξ95%, are realized in Regimes III and IV, although the highest rates of energy dissipated by the VI NES during vibro-impacts, EVI NES /ξLI , are realized in the lower energy Regimes I and II. We conclude that, although in these Regimes there occurs strong TET from the LO to the VI NES as long as vibro-impacts last (i.e., there is more efficient energy dissipation per vibro-impact), the overall duration of vibro-impacts is small (due to the small level of overall energy), as reflected by the relatively small values of the corresponding overall TET efficiency rates 95/ξ95%. In an additional series of numerical simulations we computed the previous measures for simulations corresponding to excitations of VI IOs at varying energies and restitution coefficients , and the fixed viscous damping coefficient λ = 0.005λcrit = 0.01. This study identified the regimes of efficient VI TET when both the energy of the excited VI IO and the coefficient of restitution of vibro-impacts are varied. The results are presented in Figures 7.38 and 7.39, from which we conclude that the most efficient TET takes place when highly-energetic VI IOs are excited (in Regimes III and IV of the FEP), and for smaller restitution coefficients, i.e., for highly inelastic vibro-impacts. This last result is not as obvious as it might seem from a first reading; indeed, although it is clear that the average rate EVI NES /ξLI is favored when the restitution coefficient increases (as this results in increased dissipated energy per vibro-impact), this does not necessarily imply that the overall TET efficiency as measured by the average rate 95/ξ95% is also favored (for example, refer to the average rates depicted in the plot of Figure 7.37b).


301

Fig. 7.36 Results of EMD analysis of the damped transition of Figure 7.28: (a) IMFs of the VI NES response (maximum amplitudes indicated); (b) IMFs of the LO response.

In conclusion, we showed that a SDOF primary LO with an attached VI NES possesses very complicated dynamics. In the absence of energy dissipation, vibroimpacts give rise to a variety of periodic (and quasi-periodic) motions, which when represented in the FEP yield a quite complex topology of periodic and quasiperiodic orbits. In the limit of zero clearance the entire complex VI FEP degenerates to just two linear (in- and out-of-phase) modes. By superimposing WT spectra of weakly damped responses to the Hamiltonian FEP we were able to study complicated transitions, and deduce the different mechanisms for passive TET from the LO to the VI NES. As in the case of smooth stiffness nonlinearity (the ‘smooth’ NES), both fundamental and subharmonic TET can be realized by the VI NES. The most

302


Fig. 7.36 Results of EMD analysis of the damped transition of Figure 7.28: (c) WT spectra of the dominant IMFs; (d) evolutions of the phase differences θ1 and θ2 indicating the occurrence of TRCs in the dynamics.

efficient mechanism for TET, however, is through the excitation of highly-energetic VI IOs, in similarity to the case of smooth nonlinearity. In contrast to NESs with smooth essential nonlinearities, however, VI NESs are capable of passively absorbing and locally dissipating significant portions of the energy of the primary systems to which they are attached, at sufficiently fast time scales; this renders them especially suitable for applications where rapid energy dissipation is rquired, such as, passive seismic mitigation, where shock elimination in the early, highly energetic regime of the structural response is a critical requirement. The application of


303

Fig. 7.37 Study of efficiency of TET in the VI system for excitation of VI IOs: (a) measures ξLI , ξ95% and EVI NES as functions of total energy; (b) average rates EVI NES /ξLI and 95/ξ95% as functions of total energy.

304


Fig. 7.38 Efficiency of TET in the VI system for excitation of VI IOs: left column, average rates EVI NES , ξLI and ξ95% as functions of total energy and restitution coefficient for λ = 0.01; right column, corresponding projections in the total energy-restitition coefficient plane.


305

Fig. 7.39 Efficiency of TET in the VI system for excitation of VI IOs: left column, average rates EVI NES /ξLI and 95/ξ95% as functions of total energy and restitution coefficient for λ = 0.01; right column, corresponding projections in the total energy-restitition coefficient plane.

combinations of NESs with smooth and non-smooth stiffness characteristics to the problem of seismic mitigation will be examined in detail in Chapter 10.

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Chapter 8

Experimental Verification of Targeted Energy Transfer

Experimental verification of TET will be carried out in this chapter. Some preliminary experimental results concerning TET from a two-DOF linear primary system to an attached ungrounded SDOF NES (of Configuration II – see Section 3.1) were already presented in Section 3.5.1.3, where experimental verification of resonance capture cascades (RCCs) was also performed. In this chapter we focus on some basic experimental tests that confirm in a systematic way certain of the previous theoretical predictions related to TET from linear primary systems to grounded or ungrounded NESs (i.e., NES Configurations I and II – see Sections 2.6 and 3.1). All experimental results were carried out in the Linear and Nonlinear Dynamics and Vibrations Laboratory of the University of Illinois, Urbana–Champaign. Further experimental studies of TET will be presented in Chapter 9, in our study of aeroelastic instability suppression of in-flow wings through the use of SDOF NESs with essential cubic nonlinearities; and in Chapter 10, in our study of TET-based passive seismic mitigation designs for seismically forced structures with essentially nonlinear and/or vibro-impact NESs.

8.1 TET to Ungrounded SDOF NES (Configuration II) The first experimental study of TET is carried out with the fixture depicted in Figure 2.27, consisting of a SDOF linear oscillator (LO) with an attached ungrounded NES of Configuration II (according to the notation introduced in Section 3.1 – refer to the NES configuration depicted in Figure 3.2). The experimental results of this work are fully discussed in McFarland et al. (2005b), and here only a summary of the main findings is provided. Assuming that the LO is forced by an external excitation, the experimental fixture is modeled by the following two-DOF system: M x¨ + ελ1 x˙ + ελ2 (x˙ − v) ˙ + kx + C(x − v)3 = p(t)

311

312


Fig. 8.1 Experimental impulsive force.

εv¨ + ελ2 (v˙ − x) ˙ + C(v − x)3 = 0

(8.1)

From Section 2.6 we recall that the essential cubic nonlinearity of the NES is realized by means of wires with nearly no pretension that are connected to the LO using clamps. In the described experiment, the wire span was adjusted to 12 in, and further details about the construction of the essential nonlinearity can be found in McFarland et al. (2005a, b). A long-stroke shaker provided a controlled (and repeatable) short impulse to the LO; a representative input (broadband) force is depicted in Figure 8.1. The response of both oscillators was then measured using accelerometers. Estimates of the corresponding velocity and displacement time series were obtained by numerically integrating the measured acceleration time series. The resulting signals were then high-pass filtered to remove the spurious components introduced by the numerical integration procedure.

8.1.1 System Identification The first stage of the experimental study concerns the system identification of the various parameters of the model. This is needed in order to perform comparisons of the experimental results with theoretical predictions (using discrete models), and thus identify the experimentally measured nonlinear dynamics. The goal of system identification is to exploit input and output measurements performed on the structure using vibration sensing devices in order to estimate all the parameters governing the equations of motion. Prior to system identification, the LO and the NES were weighed as M = 1.266 kg and ε = 0.140 kg, respectively, which implies a mass ratio ε/M = 0.11.


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Table 8.1 Parameters of model (8.1) identified through experimental modal analysis, and the restoring force surface method. Parameter

Estimated Value

M ε k ελ1 ελ2 C α

1.266 kg 0.140 kg 1143 N/m 0.155 Ns/m 0.4 Ns/m 1.85 × 106 N/m2.8 2.8

System identification was carried out in two separate steps. First, the LO was disconnected from the NES, and modal analysis was performed on the disconnected LO employing the stochastic subspace identification method (Van Overschee and De Moor, 1996). This yielded natural frequency and the critical viscous damping ratio estimates equaling 4.78 Hz and 0.2%, respectively. Because the mass of the LO was known, the stiffness and the damping parameters were easily deduced from this modal analysis, and their values are listed in Table 8.1. In the second step of the system identification, the LO was clamped, and an impulsive force was applied to the NES using an instrumented modal hammer. Both the NES acceleration and applied force were measured, and the restoring force surface method (Masri and Caughey, 1979) was employed to estimate the nonlinearity C and the damping coefficient ελ2 from these measurements. In essence, Newton’s second law is applied to estimate the nonlinear stiffness force by the following relation: ˙ x, x) ˙ = p − ε v¨ (8.2) fNL (v, v, ˙ x, x) ˙ is the restoring force and p the external force [for simplicity, where fNL (v, v, the temporal dependences are omitted from (8.2)]. Hence, the time history of the restoring force can be estimated directly from the measurement of the acceleration and the external force and from the knowledge of the mass ε. This is illustrated for the 21 N force level in Figure 8.2a. The representation of the restoring force in terms of the relative displacement (v-x) is depicted in Figure 8.2b, and demonstrates that indeed the linear component of the nonlinear stiffness is negligible; in other words, the experimental wire assembly behaved as an essential nonlinearity (as predicted). The model, fNL (v, v, ˙ x, x) ˙ = ελ2 (v˙ − x) ˙ + C(v − x)3 (8.3) could then be fitted to the measured estimate of the restoring force, and least-squares parameter estimation could be used to obtain the values of coefficients C and ελ2 . For greater flexibility, the functional form of the nonlinear stiffness was replaced by the more general expression, ˙ x, x) ˙ = ελ2 (v˙ − x) ˙ + C |v − x|α sign(v − x) fNL (v, v,

(8.4)

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Fig. 8.2 Measured restoring force represented as a function of (a) time and (b) relative displacement (v − x).

The three unknown parameters in (8.4), namely the nonlinear coefficient, the exponent of the nonlinearity and the dashpot constant, were estimated by following the same procedure as in Kerschen et al. (2001); i.e., one seeks the minimum of the normalized mean-square error between the measured and predicted restoring forces as a function of the exponent of the nonlinearity. The resulting parameters are listed in Table 8.1. The best results were obtained using an exponent equal to 2.8, which is not far from the theoretical value of 3.

8.1.2 Experimental Measurements Two series of experimental tests were performed in both of which the LO was impulsively loaded. In the first series of tests, the damping in the NES was kept relatively weak in order to highlight the different mechanisms for targeted energy transfer (TET). Additional tests were performed to investigate whether TET could still be realized when the viscous damping of the NES was increased.

8.1.2.1 Weak Damping Case In the weak damping case, several force levels ranging from 21–55 N were considered, but for conciseness, only the results for the lowest and highest force levels are displayed in Figures 8.3 and 8.4. At the 21 N forcing level, the acceleration and displacement of the NES are higher than those of the LO, which indicates targeted transfer of vibration energy to the NES. The percentage of instantaneous total energy carried by the NES depicted in Figure 8.3c illustrates that strong energy exchanges between the LO and the NES take place, and that this channeling of energy to the NES is not irreversible. Indeed, after 0.23 s, as much as 88% of the


315

Fig. 8.3 Experimental results for low damping (21 N force level): (a) measured accelerations (solid line: LO; dashed line: NES); (b) measured displacements (solid line: LO; dashed line: NES); (c) percentage of instantaneous total energy in the NES, and (d) LO displacement (solid line: with NES attached; dashed line: with only grounded dashpot).

total instantaneous energy is present in the NES, but this level drops down to 1.5% immediately thereafter. Hence, in this case, energy rapidly flows back and forth between the two oscillators, which is characteristic of a nonlinear beat phenomenon. Another indication that nonlinear beating occurs is that the envelope of the NES response undergoes large modulations in this case. At the 55 N forcing level, the nonlinear beat phenomenon still dominates the early regime of the motion. A less vigorous but faster energy exchange is now observed as 63% of the total energy is transferred to the NES after 0.12 s. These quantities also hold for the intermediate force levels listed in Table 8.2. These observations are in close agreement with the numerical studies in Section 3.2; indeed, in this case due to the excitation of IOs there is smaller transfer of energy from the LO to the NES, but in a faster fashion as the force level increases. The main qualitative difference from the case of the lowest force level is that there exists now a second regime of motion. Indeed, after approximately t = 2.5 s, the motion is captured in the domain of attraction of the 1:1 resonant manifold, as

316


Fig. 8.4 Experimental results for low damping (55 N force level): (a) measured accelerations (solid line: LO; dashed line: NES); (b) measured displacements (solid line: LO; dashed line: NES); (c) percentage of instantaneous total energy in the NES, and (d) LO displacement (solid line: with NES attached; dashed line: with only grounded dashpot). Table 8.2 Nonlinear beating phenomenon: energy transferred from the LO to the NES, and corresponding transfer time. Excitation level (N) 21 29 34 45 55

Energy transferrred to the NES (%) 88 72 67 64 63

Transfer time (s) 0.23 0.20 0.19 0.13 0.12

clearly evidenced in Figure 8.4c. This graph also demonstrates the irreversibility of the energy transfer in this case, at least until escape from resonance capture occurs around t = 6.2 s. Another manifestation of 1:1 resonance capture is that the envelopes of the displacement and acceleration time series decrease almost monotonically during this regime, and no envelope modulation is observed. The system is capable of sustaining the resonance capture during a significant interval of the mo-


317

Fig. 8.5 Experimental results for high damping (31 N force level): (a) measured accelerations (solid line: LO; dashed line: NES); (b) measured displacements (solid line: LO; dashed line: NES); (c) percentage of instantaneous total energy in the NES; (d) energy dissipated in the NES (solid line: measurements; dashed line: simulations), and (e) restoring coupling force.

tion (i.e., in the interval 2.5 < t < 6.2 s). Hence, experimental confirmation of TET from the LO to the NES is established in this case. A qualitative way to assess the efficiency of energy dissipation by the NES is to compare the responses of the LO in the following two cases: (a) when the NES

318


is attached to the LO (which corresponds to the presented results); and (b) when the NES is disconnected and replaced by a dashpot installed between the LO and ground. This later configuration corresponds to a SDOF LO with just added damping. Case (b) was not realized in the laboratory, but the system dynamics was studied using numerical simulation. Figures 8.3d and 8.4d compare the corresponding displacements of the LO for the aforementioned two different system configurations. It can be observed that the NES performs much better than the grounded dashpot for the 21 N forcing level, but this is less obvious for the 55 N forcing level. This might mean that for weak damping (when the nonlinear beat phenomenon is capable of transferring a significant portion of the total energy to the NES), nonlinear beating might be a more useful mechanism for energy dissipation compared to resonance capture leading to irreversible energy transfer.

8.1.2.2 Strong Damping Case We now consider the experimental response of a system with strong damping. Several force levels in the range 31–75 N are considered in this case, but only the results for the 31 N level are presented herein. The damping constant λ2 in the coupling element was identified as 1.48 Ns/m, which means that the damping element that couples the two oscillators (and is situated in parallel to the essential stiffness nonlinearity) can no longer be considered as being of order ε. The increase in damping is also reflected in the measured restoring coupling force depicted in Figure 8.5e. The experimental system response shown in Figures 8.5a, b is almost entirely damped out after about 5 to 6 cycles of oscillation. The NES acceleration and displacement are still higher than the corresponding responses of the LO, which means that TET may also occur in the presence of stronger damping. The percentage of instantaneous total energy in the NES in Figure 8.5c never reaches values close to 100% as in the previous case, but we conjecture that this is due to the faster decay of the response due to strong damping; as soon as energy is transferred to the NES, it is almost immediately dissipated by the dashpot, and irreversible TET takes place. This is fully compatible with the theoretical results. A quantitative measure of energy dissipation is available through the computation of the energy dissipation measure (EDM), i.e., of the energy dissipated in the NES normalized by the total input energy: t ελ2 [v(τ ˙ ) − x(τ ˙ )]2 dτ Ediss,NES,%(t) = 100 (8.5) 0 t p(τ )x(τ ˙ )dτ 0

Experimental and simulated estimates of the EDM are depicted in Figure 8.5d. This demonstrates that as much as 96% of the total input energy is dissipated by the NES, which indicates high TET efficiency by the NES. Moreover, there is satisfactory agreement between theoretical predictions and experimental measurements,


319

Fig. 8.6 Superposition of the WT spectrum of the relative displacement across the nonlinearity and the backbone of the frequency energy plot: (a) forcing amplitude 55 N, system with low damping; (b) forcing amplitude 31 N, system with high damping.

validating the mathematical analysis and the corresponding models developed in Chapter 3.

8.1.2.3 Post Processing Results and Additional Experiments To study further the nonlinear mechanisms governing energy transfers in the experimental system, we applied the numerical wavelet transform (WT) to selected time series for systems with weak and strong damping. In Figure 8.6, we depict the Morlet WT spectra of the relative displacements v(t)−x(t) of the systems with (a) weak damping and forcing amplitude 55 N (Figure 8.6a) and (b) strong damping and forcing amplitude 31 N (Figure 8.6b). Superimposed to the WT spectra are the backbone curves of the frequency energy plot (FEP) of the model (8.1), represented by solid line. We recall that the shading of the WT spectra denotes the relative amplitude of the dominant harmonic components of the corresponding damped motions, as computed through the WT. Based on the experimental WT spectra depicted in Figure 8.6 we deduce that the dynamics of the system is indeed nonlinear. A strong indication of this fact is that the predominant frequency component of the NES varies with energy; moreover, there appear strong harmonic components during the initial nonlinear beating phenomena in these responses. Once these harmonic components disappear, fundamental TET due to 1:1 transient resonance capture is triggered, as the transient dynamics traces approximately the in-phase backbone branch S11+. Hence, the experimental results confirm the theoretical finding that the formation of initial nonlinear beats provides the ‘bridging dynamics’ for triggering fundamental (or subharmonic) TET in the two-DOF system (8.1).

320


Another remark regarding the experimental results concerns the fact that, in the weakly damped system (see Figure 8.6a) the predominant harmonic component of the NES traces the backbone branch for most of the duration of the time series. This is not so apparent in the heavily damped response of Figure 8.6b, and validates our theoretical finding that the weakly damped, transient dynamics can be interpreted and understood in terms of the topological structure of periodic orbits (NNMs) of the underlying Hamiltonian system in the FEP. Additional measurements were performed for a system with stiffer stiffness nonlinearity by shortening the span of the wire from 12 in to 10 in and increasing the wire diameter from 0.010 in to 0.020 in; this yields a nonlinear coefficient of C = 1.65 × 107 N/m3 . Inspection of the accelerations and displacements shown in Figures 8.7a, b reveals that the NES is no longer vibrating symmetrically with respect to its equilibrium position, particularly in the time interval 1 < t < 3 s. Interestingly enough, as can be judged from the results depicted in Figures 8.7c, d, there is an almost pointwise agreement between the experimental accelerations and those predicted by the theoretical model (8.1). A better understanding of this particular regime of the motion can be gained from a snapshot of the configuration space depicted in Figure 8.7e. Apparently, the transient dynamics is captured in the domain of attraction of a subharmonic tongue in the FEP, with the corresponding motion not being symmetric with respect to the origin of the configuration space. Indeed, it turns out that in this case the motion takes the form of a closed loop (Lissajous curve), which might mean that an unsymmetric U −branch is excited. However, due to the existence of a countable infinity of tongues, and due to the presence of damping, it is difficult to identify with certainty the specific subharmonic tongue that is excited in this particular experimental run. Nonetheless, this result provides further evidence of the complex dynamical behavior that this two-DOF system with essential nonlinearity can possess, in full agreement with the theoretical predictions of Chapter 3; moreover, it provides further experimental evidence of the intricate relationship between the Hamiltonian and weakly damped dynamics. Finally, Figure 8.7f illustrates that a nonlinear beating occurs during this particular regime, with the resulting energy exchanges being not so vigorous, since less than 40% of vibration energy gets transferred at any given time of the beating cycle from the LO to the NES.

8.2 TET to Grounded SDOF NES (Configuration I) We now proceed to discuss a second series of experimental results regarding TET from a linear primary system to a grounded NES of Configuration I according to the notation introduced in Section 3.1 (with configuration depicted in Figure 3.1). Our exposition follows closely (McFarland et al., 2005a).


321

Fig. 8.7 Case of increased stiffness nonlinearity: (a) measured accelerations (solid line: LO; dashed line: NES); (b) measured displacements (solid line: LO; dashed line: NES); comparison between predicted (dashed line) and measured (solid line) accelerations: (c) LO; (d) NES; (e) motion in the configuration plane, and (f) percentage of instantaneous total energy contained in the NES.

322


8.2.1 Experimental Fixture The experimental fixture employed for this study is depicted in Figure 2.26. It consists of two SDOF oscillators (the linear primary system and the NES) connected by means of a linear coupling stiffness. The left oscillator (the linear subsystem) is grounded by means of a linear spring, whereas the right one (the nonlinear energy sink – NES) is grounded by means of a nonlinear spring with essential cubic nonlinearity. To dissipate the transferred energy, a viscous damper exists connecting the NES to ground. Transient (shock) excitation of the linear primary system is provided by means of a rod that impacts elastically with the left mass. More details regarding the experimental fixture and the realization of the essential cubic stiffness nonlinearity by means of wires with no pretension can be found in Section 2.6. The two-DOF system model of the experimental fixture is governed by the equations of motion M y¨ + Ky + ελy˙ + ε(y − v) = F (t) mv¨ + εcv˙ + Cv 3 − ε(y − v) = 0

(8.6)

for zero initial conditions. Weak coupling between the linear primary system and the NES is assumed by requiring that ε K. Assuming that F (t) is an impulsive (broadband) excitation of finite duration, we aim to show experimentally that broadband energy initially imparted to the linear subsystem is passively transferred to the NES in an irreversible (on the average) way, where this energy is confined and dissipated without ‘spreading back’ to the linear primary system. The eigenfrequency of the linear subsystem, and the viscous damping factors of both subsystems, were estimated by performing experimental modal analysis tests using an instrumented modal hammer manufactured by PCB Piezotronics, Inc. to provide the necessary excitation. Data analysis was performed using the Diamond modal analysis package developed at Los Alamos National Laboratory. The modal parameters of the linear primary system were estimated to be M = 0.834 kg,

K = 993 N/m,

ελ = 0.129 Ns/m

corresponding to the eigenfrequency ω0 = 35.63 rad/s, and the viscous damping ratio ζ = 2.3 × 10−3 ; hence, the linear oscillator is lightly damped. The mass and damping parameters of the NES, and the linear coupling stiffness connecting the primary system and the NES, were estimated by performing experimental modal analysis of the linear SDOF oscillator which results when the coupling stiffness is grounded and the nonlinear spring is disconnected. The system parameters were estimated as m = 0.393 kg,

ε = 114 N/m,

εc = 0.454 Ns/m

The stiffness characteristic of the nonlinear spring of the NES was identified by performing a series of static tests, wherein known displacements were imposed


323

Fig. 8.8 Identified stiffness characteristic of the essentially nonlinear spring of the NES: • experimental measurement, — fitted cubic polynomial.

to the NES mass and the corresponding restoring forces created by the wire were measured. A cubic polynomial was then fitted to the experimental measurements, yielding the following nonlinear force-displacement relationship: f (x) = 166x + 1.36 × 107 x 3

(8.7)

Hence, a small (undesirable) linear term is identified in the stiffness of the NES which, however, is negligible for the characteristic displacements of interest in our experiment. In Figure 8.8, the fitted model of the essential stiffness nonlinearity of the NES is compared to experimental measurements. Experimental data collected included the input (broadband) force, the acceleration of each mass, and the force associated with the nonlinear stiffness. These signals were digitized using two Siglab Model 20-42, 4-channel spectrum analyzers. One of these ran at a sampling rate which allowed it to store 8 s of time-history data, which was deemed adequate to capture all of the significant TET phenomena realized in the experimental fixture. The other analyzer acquired input force and the acceleration of the linear subsystem at a much higher rate, allowing the reconstruction of these short signals in adequate detail. The discrete signals thus acquired were first transferred to Matlab for post-processing, including correcting biases and synchronization of the time data. All signals were shifted to start at time t = 0, and the processed time histories were imported to Mathematica for all further manipulations, during which the experimental signals were commonly interpolated by cubic spline approximations for simplicity.

324


Fig. 8.9 Strong experimental impulsive force for inducing nonlinear TET in the experimental fixture.

8.2.2 Results and Discussion Two series of experimental tests were performed. In the first series of tests, consisting of eight trials (labeled ep1 through ep8), the input force was strong enough to produce significant nonlinear effects, including TET. Four additional trials (labeled NOEP1 through NOEP4) were performed with the input force reduced to where the system response was practically linear and no TET occurred. The aforementioned sets of experiments were dictated by the theoretical prediction (see Chapter 3), that passive TET in the considered system of coupled oscillators can only be realized if the imparted energy gets above a critical threshold. As shown below, the experimental results fully validate this theoretical prediction. In what follows, typical results from each of the two trial sets are provided. First, experimental results for the case EP2 (when TET is realized) will be presented, corresponding to the experimentally produced impulsive force depicted in Figure 8.9. This force is applied to the linear primary system with the entire system being initially at rest, and is approximately 6.25 ms in duration; this forcing level is typical of the strong excitation required for inducing nonlinear passive TET in the experimental fixture. To perform comparisons between measured results and theoretical predictions, we carried out an additional independent numerical simulation of the transient response of system (8.6) subjected to the experimentally measured force and zero initial conditions. In Figure 8.10 a comparison between the experimental and theoretical acceleration time series of the two subsystems is depicted, from which very good agreement is noted. Such agreement is typical of what was observed in all experimental trials. Nonlinear TET is noted, especially at the early stage of the dynamics (0 < t < 4 s) when the energy of the system is relatively high and the nonlinear effects are more pronounced. Moreover, by studying the experimental time series of Figure 8.10, one notes that during this early TET regime the NES oscillates with a dominant ‘fast’ frequency that is approximately equal to the eigenfrequency of the linear subsystem, i.e., a 1:1 TRC takes place.


325

Fig. 8.10 Case when passive TET from the linear primary system to the NES occurs: experimental — and numerical – – – acceleration time series of (a) the linear (directly excited) primary system, and (b) the NES.

A quantitative measure of passive TET in the system is performed by computing the energy dissipation measure (EDM), t εc ENES (t) = v˙ 2 (τ )dτ (8.8) Ei (T ) 0 representing the instantaneous portion of the input energy Ei (t) dissipated by the viscous damper of the NES, where T is the duration of the external impulse. The input energy measure is computed from the following expression: t F (τ )y(τ ˙ )dτ 00

For 0 < ε 1, the equilibrium point (0,0) is always an unstable spiral. There exists a stable ω-limit set (or limit cycle) on which the solution can be approximated by averaging in the form, x(t) = 2 cos ωt + O(ε), where the frequency is given by ω = 1 − ε2 /16 + O(ε3 ) (Guckenheimer and Holmes, 1983). Figure 9.1 depicts the LCO of the VDP oscillator in the phase plane when ε = 0.1, and shows that there is small deviation between the original and averaged solutions [of O(ε)]. We wish to study the suppression of the LCO of the VDP oscillator by adding a lightweight NES to it. Throughout this study we will assume that ε is the small parameter of the problems considered, and assign it the value ε = 0.01 1. We will consider two alternative NES attachments, namely, grounded and ungrounded NESs, as depicted in Figure 9.2. We designate the corresponding configurations of VDP oscillators with grounded and ungrounded NESs as VDPNES1 and VDPNES2 , respectively. The grounded NES (labeled as NES1 ) is linearly coupled to the VDP oscillator, whereas the ungrounded NES (NES2 ) is coupled to the VDP

356


Fig. 9.2 System configuration with (a) grounded (VDPNES1 ) and (b) ungronded NES (VDPNES2 ).

oscillator through essentially nonlinear stiffness in parallel to a viscous damping element. The equations of motion for configuration VDPNES1 (Figure 9.2a – Configuration I in the notation of Section 3.1) are expressed as x¨ − εx(1 ˙ − x 2 ) + x + εC (x − y) = 0 εmy¨ + ελy˙ + κy 3 + εC (y − x) = 0

(9.1)

where x and y are the displacements of the VDP and NES masses, respectively; εm is the ratio of the NES and VDP masses, ελ the damping coefficient, εC, the linear coupling stiffness, and κ the nonlinear stiffness coefficient. The equations of motion for configuration VDPNES2 (Figure 9.2b – Configuration II in the notation of Section 3.1) are given by x¨ − εx(1 ˙ − x 2 ) + x + ελ(x˙ − y) ˙ + κ(x − y)3 = 0 εmy¨ + ελ(y˙ − x) ˙ + κ(y − x)3 = 0

(9.2)

where similar definitions for the parameters hold. The underlying Hamiltonian dynamics, as well as the transient damped dynamics of NESs connected to singledegree-of-freedom linear oscillators were studied in Vakakis and Gendelman (2001) and Vakakis et al. (2003) for NES1 , and in Kerschen et al. (2006a) and Lee et al. (2005) for NES2 . Moreover, the dynamics of these NES configurations have been studied extensively in previous chapters of this monograph.


357

First, we perform computational parametric studies of systems (9.1) and (9.2) to identify parameter subsets where the LCO of the VDP oscillator can be suppressed or even completely eliminated. For this purpose, we restrict the NES parameters to the ranges 0.01 ≤ εm ≤ 0.5, 0.01 ≤ κ ≤ 1.0, and 0.01 ≤ ελ ≤ 0.3 for NES1 and 0.01 ≤ ελ ≤ 0.1 for NES2 . For configuration VDPNES1 , two cases of linear coupling stiffness are considered: εC = 1.0 (weaker) and εC = 10.0 (stronger). We select initial conditions for the numerical simulations outside the LCO of the uncoupled VDP oscillator [specifically, (x(0), y(0), x(0), ˙ y(0)) ˙ = (3.5, 0, 0, 0)], and integrate the equations of motion (9.1a) and (9.1b) for sufficiently long time to ensure that the initial transients die out. Then we compute the root-mean-square (r.m.s.) amplitude of the corresponding steady state response. Since the LCO√response of the uncoupled VDP oscillator possesses an r.m.s. √ amplitude equal to 2, steady state responses with r.m.s. amplitudes less than 2 imply suppression of the LCO. Figures 9.3 and 9.4 depict contour plots of VDP r.m.s. amplitudes on the minimum parameter planes, and on the sections corresponding to mass ratio of 5% and nonlinear stiffness of 0.2 for configurations VDPNES1 and VDPNES2 , respectively. In both cases, we observe that very small damping (e.g., 1%) causes complicated (or non-uniform) steady state amplitudes of the VDP oscillator. For the grounded NES configuration (Figure 9.3), the steady state amplitudes of the VDP oscillator vary relatively uniformly on the parameter plane whether the linear coupling is weak or strong. For example, on the plane corresponding to εm = 5% (κ = 0.2), the steady state amplitudes become smaller as the damping increases; that is, they seem to be independent of the nonlinear stiffness (the mass ratio). It is generally observed that strong damping leads to better suppression results for the grounded NES. On the other hand, we do not observe uniform behavior of steady state r.m.s. amplitudes for the ungrounded NES (Figure 9.4). Instead, a larger mass ratio, stronger damping and sufficiently weaker nonlinear coupling tend to produce a larger degree of suppression. A general conclusion drawn from these plots, however, is that the addition of light passive (but essentially nonlinear) NESs may completely eliminate the LCOs of the VDP oscillator over relatively wide parameter ranges. More systematic studies of the effects of the parameters will be provided later utilizing the method of numerical continuation. In Figure 9.5, we show that an NES with an even small mass ratio can passively eliminate the LCO of the VDP oscillator. Evidently, the grounded NES1 appears to be more effective in eliminating the LCO in a shorter period compared to the ungrounded NES2 ; however, such isolated simulations cannot be used to draw general conclusions regarding the relative effectiveness of the two NES configurations in suppressing LCOs. Indeed, this relative performance ought to be evaluated by considering aspects such as robustness of LCO suppression over wide NES parameter ranges, as well as issues of applicability of these NES configurations to practical designs.

358


Fig. 9.3 Contour plots of steady state r.m.s. amplitudes of (grounded) configuration VDPNES1 with respect to mass, damping, and nonlinear stiffness coefficient of the NES: (a) weaker linear coupling (εC = 1.0); (b) stronger linear coupling (εC = 10.0).


359

Fig. 9.4 Contour plots of steady state r.m.s. amplitudes of (ungrounded) configuration VDPNES2 with respect to mass, damping, and nonlinear stiffness coefficient of the NES.

9.1.2 Transient Dynamics In this section, we construct slow flow dynamical models in order to analytically study the transient and steady state motions of the VDP oscillator with attached NES. This is achieved by applying the complexification-averaging technique (CXA) of Manevitch (2001). The analytical study of the transient dynamics will reveal the dynamical mechanism responsible for LCO suppression, whereas the study of the steady state dynamics will address issues of robustness of LCO suppression.

9.1.2.1 Slow Flow Model Following the CX-A technique, we introduce at this point the new complex variables ψ1 = y˙ + jy and ψ2 = x˙ + j x where j 2 = −1, and substitute them into the equations of motion (9.1a, b) through the relations y=

1 (ψ1 − ψ1∗ ), 2j

y˙ =

1 (ψ1 + ψ1∗ ), 2

y¨ = ψ˙ 1 −

j (ψ1 + ψ1∗ ) 2

x=

1 (ψ2 − ψ2∗ ), 2j

x˙ =

1 (ψ2 + ψ2∗ ), 2

x¨ = ψ˙ 2 −

j (ψ2 + ψ2∗ ) 2

(9.3)

360


Fig. 9.5 Transient responses depicting LCO elimination for initial conditions (x(0), y(0), x˙ (0), y(0)) ˙ = (3.5, 0, 0, 0): (a) configuration VDPNES1 , εm = 5%, ελ = 15%, κ = 0.2, εC = 1.0; (b) configuration VPDNES2 , εm = 5%, ελ = 5%, κ = 0.2.


361

where, as usual, the asterisk denotes complex conjugate. Then, we express the complex variables in polar form: ψ1 (t) = ϕ1 (t)ej t ,

ψ2 (t) = ϕ2 (t)ej t

(9.4)

where ϕi (t), i = 1, 2, represent the slowly-varying complex amplitudes, and ej t the fast-varying components at the (linearized) natural frequency of the VDP oscillator. By expressions (9.3) we partition the transient dynamics into slow and fast components, that is, we make the assumption that the transient responses of systems (9.1) and (9.2) are composed of fast oscillations that are modulated by slowly varying envelopes. Averaging out all fast-frequency components with frequencies higher than unity, we obtain a set of two complex-valued modulation equations that govern the slow flow dynamics. For configuration VDPNES1 , the slow flow dynamical model is expressed as

C 1 λ C 3κ +j 1− ϕ1 − j ϕ2 + j ϕ1 |ϕ1 |2 ϕ˙ 1 = − 2 m m 2m 8εm ϕ˙ 2 = −j

εC 1 1 ϕ1 + (ε + j εC) ϕ2 − ε ϕ2 |ϕ2 |2 2 2 8

(9.5)

whereas for VDPNES2 it reads

λ 1 λ 3κ j+ ϕ1 + ϕ2 + j ϕ˙ 1 = (ϕ1 − ϕ2 ) |ϕ1 − ϕ2 |2 2 m 2m 8εm ϕ˙ 2 =

ελ ε ε 3κ ϕ1 + (1 − λ) ϕ2 − ϕ2 |ϕ2 |2 − j (ϕ1 − ϕ2 ) |ϕ1 − ϕ2 |2 (9.6) 2 2 8 8

These represent the averaged slow flows of the corresponding VDP-NES configurations. In Figures 9.6 and 9.7, we examine the validities of the averaged systems by comparing the averaged responses to the (numerically) exact solutions. Validation is demonstrated both in cases when the LCO survives the addition of the NES, and also in cases when it is suppressed. Although the systems under consideration do not necessarily possess weak nonlinear terms, we may recall from the classical averaging theorem that an O(ε) approximation for stable dynamics of vector fields is guaranteed up to a time scale of O(1/ε) (Sanders and Verhulst, 1985). From the comparisons depicted in Figures 9.6 and 9.7 we verify that the averaged systems (9.5) and (9.6) provide satisfactory approximations to the original dynamics. The only small deviation can be found in the case when the LCO is sustained in configuration VDPNES1 (Figure 9.6a). In this case the exact NES response possesses an additional fast, high-frequency component, e3j t , which is filtered out in our averaging process. For a more precise approximation we need to include this high-frequency term, yielding a two-frequency averaged slow dynamical system (see, for example, the CX-A analysis carried out in Sections 3.3.2.2 and 3.3.2.3); however, this will not

362


Fig. 9.6 Validity of the averaged slow dynamics model for configuration VPDNES1 , (a) when the LCO survives (εm = 5%, κ = 1.0, ελ = 1%, εC = 0.1), and (b) when the LCO is eliminated (εm = 5%, κ = 1.0, ελ = 30%, εC = 1.0); initial conditions are given by (x(0), y(0), x˙ (0), y(0)) ˙ = (3.5, 0.1, 0, 0).


363

Fig. 9.7 Validity of the averaged slow dynamics model for configuration VPDNES2 , (a) when the LCO survives (εm = 5%, κ = 1.0, ελ = 1%), and (b) when the LCO is eliminated (εm = 5%, κ = 1.0, ελ = 10%); initial conditions are given by (x(0), y(0), x˙ (0), y(0)) ˙ = (2.5, 0.01, 0, 0).

be considered in this study. Instead, we will examine this high-frequency component via numerical wavelet transformations later in this section.

364


Returning now to the complex modulation equations (9.5) and (9.6), we express the slowly-varying complex amplitudes in polar form ϕi (t) = αi (t)ejβi (t ) ,

αi , βi ∈ R,

i = 1, 2

(9.7)

where αi (t) and βi (t) represent the slowly-varying amplitude and phase of each degree of freedom, respectively. Substituting into the averaged systems (9.5) and (9.6), and setting separately equal to zero the real and imaginary parts, we obtain two sets of real modulation equations. For configuration VDPNES1 , these are written as λ C α1 − α1 sin φ 2m 2m ε εC α1 sin φ α˙ 2 = (4 − α22 )α2 + 8 2

3κ 1 C ˙ − 1 − εC α1 α2 + (εmα12 − α22 ) cos φ α1 α2 φ = 2 m 8εm

α˙ 1 = −

(9.8)

whereas, for VDPNES2 , they are given by λ λ 3C 3C α1 + α2 cos φ − α2 (α12 + α22 ) sin φ + α1 α22 sin 2φ 2m 2m 8εm 8εm ελ ε ε α1 cos φ + (1 − λ)α2 − α23 α˙ 2 = 2 2 8 3C 3C 2 α1 (α12 + α22 ) sin φ − α α2 sin 2φ + 8 8 1 3C 3C 1 α1 α2 (α12 + 2α22 ) − α1 α2 (2α12 + α22 ) α1 α2 φ˙ = − α1 α2 + 2 8εm 8 3C λ (εmα12 + α22 ) sin φ + εmα14 − α24 − 3(1 − εm)α12 α22 cos φ − 2m 8εm 3C α1 α2 (εmα12 − α22 ) cos 2φ (9.9) − 8εm α˙ 1 = −

In the relations above, α1 and α2 are the envelopes of the amplitude of the NES and VDP oscillator, respectively, and φ ≡ β1 − β2 , the phase difference between the oscillations of the VDP oscillator and the NES. The slow dynamical models (9.8) and (9.9) approximate the transient dynamics of the two NES configurations considered, and can be used to study the LCO suppression mechanism based on passive TET from the VDP oscillator to the NES.


365

9.1.2.2 LCO Suppression Mechanism The LCO suppression mechanism is now explored by directly studying the transient dynamics. We consider two cases: (i) when the LCO either survives the action of the NES or is not effectively suppressed, and (ii) when the LCO is completely eliminated through TET from the VDP oscillator to the NES. As a first step, the transient responses of the system for initial conditions chosen outside or inside the LCO of the uncoupled VDP oscillator are plotted, and their amplitude and frequency contents are compared. In addition, by studying the dynamics of the slow flows (9.8) or (9.9) with prescribed initial conditions, we can examine the behavior of orbits in the slow flow phase space (α1 , α2 , φ), and examine the possible existence of transient or permanent resonance captures (RCs) in the dynamics. If a resonance capture occurs, the phase variable φ exhibits non-time-like (but rather oscillatory) behavior (as discussed in Section 9.3, and also in Bosley and Kevorkian, 1992), and the correspond˙ plane appears in the form of a spiral. Specifically, when ing trajectory in the (φ, φ) a 1:1 resonance capture occurs at a specific time instant t = t0 , the slow phase difference φ becomes small, and the frequencies of the VDP oscillator and the NES become ‘locked’ in an approximately 1:1 relationship up to times t − t0 = O(1). As discussed in Section 2.3, we distinguish between two types of resonance capture. Transient resonance capture (TRC) refers to the case when the resonance capture persists for a certain period of time [i.e., on the time scale O(1/ε)], followed by an escape from the resonance capture regime and subsequent time-like behavior of the slow phase difference φ. By contrast, sustained resonance capture (SRC) occurs when the resonance capture is permanent. We note that, since the slow flow models (9.8) and (9.9) are based on the assumption of a single fast frequency (identical for both the VDP oscillator and the NES), only 1:1 resonance captures can be modeled by the outlined slow flow analysis (higher-order RCs can be studied by including multiple fast frequency components in the derived slow flow models). As shown in previous sections, another way for examining the transient dynamics is through application of numerical wavelet transforms (WTs). Moreover, WT spectra can be superimposed to appropriately defined frequency-energy plots (FEPs) to provide a clear interpretation of the damped dynamics in terms of the underlying Hamiltonian ones. As discussed previously, the purpose of superimposing the transient damped dynamics on FEPs is to show that for weak damping (negative for the VDP oscillator and positive for the NES) the evolutions of the transient responses of the damped systems in the frequency-energy plane follow closely branches of periodic orbits of the underlying Hamiltonian systems. To this end we need to evaluate the total instantaneous energy of both considered configurations, as outlined below. The initial total energy for configuration VDPNES1 is given by 1 1 εC κ (x(0) ˙ 2 + εmy(0) (x(0) − y(0))2 + y(0)4 ≡ E0 ˙ 2 ) + x(0)2 + 2 2 2 4 (9.10) whereas for VDPNES2 it is given by E(0) =

366


E(0) =

1 1 κ (x(0) ˙ 2 + εmy(0) ˙ 2 ) + x(0)2 + (x(0) − y(0))4 ≡ E0 2 2 4

(9.11)

For both NES configurations, we can numerically compute the energy generated or dissipated by the nonlinear damper in the VDP oscillator as t VDP Ed (t) = ε x(τ ˙ )2 (x(τ )2 − 1)dτ (9.12) 0

The energy dissipated by the NES can be expressed, for VDPNES1 , as t y(τ ˙ )2 dτ EdNES (t) = ελ

(9.13)

0

and for VDPNES2 as EdNES (t)

t

= ελ

˙ ) − y(τ ˙ ))2 dτ (x(τ

(9.14)

0

Thus, we express the total energy in the following way: E total(t) = E(0) − EdVDP (t) − EdNES (t)

(9.15)

for both configurations. We then consider the energy exchange between the VDP oscillator and the NES, which is basically a competition between the nonlinear damping in the VDP oscillator (which either dissipates energy or feeds energy to the system) and the positive damping in the NES (which only dissipates energy). For this purpose, we define the energy components E VDP (t) and E NES (t) for each subsystem as E VDP (t) ≡ E0VDP − EdVDP (t),

E NES (t) ≡ E0NES − EdNES (t)

(9.16)

where E0VDP or E0NES represents the initial energy imposed by the initial conditions, with the initial potential energy stored in the coupling stiffness shared equally by both degrees of freedom. Figure 9.8 depicts the transient dynamics of configuration VDPNES1 (with parameters εm = 5%, κ = 1.0, ελ = 1%, εC = 0.1) for initial conditions outside the LCO of the uncoupled VDP oscillator given by (x(0), y(0), x(0), ˙ y(0)) ˙ = (3.5, 0.01, 0, 0). All results are exact, resulting from numerical integration of system (9.1), with the exception of the plot of Figure 9.8b which depicts the dynamics of the averaged system (9.8). The time series depicted in Figure 9.8a indicate that the LCO of the VDP oscillator survives the presence of the NES. From the phase ˙ depicted in Figure 9.8b, computed from the averaged system (9.8) portrait (φ, φ) with identical initial conditions, we can predict that a 1:1 SRC occurs when the LCO is retained. However, the WT analysis of the response (Figure 9.8c) indicates that the steady state LCO dynamics is captured into a 3:1 SRC. In order to study such subharmonic resonance captures, we would need to employ at least a two-fastfrequency slow dynamics model; however, this will not be considered in this study.


367

Fig. 9.8 Transient dynamics of configuration VDPNES1 when the LCO survives, case of 3:1 SRC ˙ for the averaged system as total energy decreases: (a) time series; (b) motion on the plane (φ, φ) (9.8); (c) instantaneous frequencies; (d) WT spectrum on the FEP; (e) energy exchange measures.

368


Fig. 9.8 Transient dynamics of configuration VDPNES1 when the LCO survives, case of 3:1 SRC as total energy decreases: (c) instantaneous frequencies; (d) WT spectrum on the FEP.


369

Fig. 9.8 Transient dynamics of configuration VDPNES1 when the LCO survives, case of 3:1 SRC as total energy decreases: (e) energy exchange measures.

The 3:1 SRC becomes clearer if we superimpose the WT spectrum of the transient response on the FEP of the underlying Hamiltonian system which is derived by eliminating the damping terms from system (9.1); this is performed in Figure 9.8d where the WT spectrum of the relative displacement x(t)−y(t) is plotted against the total instantaneous energy of the system. The initial state of the system is away from any periodic solutions of the underlying Hamiltonian system, but the response approaches the vicinity of and is finally captured in a 3:1 internal resonance manifold as the total energy decreases to reach the steady state LCO. This proves conclusively that the surviving LCO is the product of a 3:1 SRC. We then compare the energy exchanges between the components of the system in Figure 9.8e. Since the motion is initiated outside the LCO of the uncoupled VDP oscillator, the total energy decreases until it reaches the steady state value corresponding to the surviving LCO. The energy dissipation by the NES is monotonically increasing with time. On the other hand, the nonlinear damper of the VDP oscillator initially dissipates energy up to time 20T , where T = 2π is the linear natural period of the VDP oscillator; afterwards it feeds energy into the system, decreasing monotonically. As a result, the energy fed by the VDP oscillator and the energy dissipated by the NES reach a balance to produce the surviving LCO. This type of sustained LCO is similar to vortex-induced resonant vibrations of a circular cylinder, where the fluid-structure interaction yielding LCOs of the cylinder immersed in the fluid flow exhibits synchronization and lock-in between the vortex and vibration frequencies (Skop and Griffin, 1973; Griffin and Skop, 1973).

370


Fig. 9.9 Transient dynamics of configuration VDPNES1 when the LCO survives, case of 3:1 SRC as total energy increases: (a) time series; (b) WT spectrum on the FEP.


371

Figure 9.9 depicts another possible case of the dynamics of VDPNES1 with the LCO surviving the action of the NES, for motion initiated inside the LCO of the uncoupled VDP oscillator; in this case, only the time series and FEP are considered, for system parameters identical to the previous case. We note that the dynamics is captured by the 3:1 resonance manifold as the total energy increases. We now examine the transient dynamics of configuration VDPNES1 when the LCO is completely eliminated by the action of the NES. The only difference in parameters from the ones used for the simulations of Figure 9.8 is the damping of the NES, which in this case is taken equal to ελ = 0.1. In Figure 9.10 we depict the exact transient dynamics of the system resulting from numerical integration of system (9.1), except for Figure 9.10b where the averaged dynamics [system (9.8)] is considered. The time series of Figure 9.10a indicates an initial 3:1 transient resonance capture (the NES possesses two frequency components, equal to unity and three), which becomes a 1:1 resonance capture at the late stage of LCO suppression. This behavior is clearer when one inspects the WT analysis results depicted in Figure 9.10c. The slow flow model (9.8) predicts a 1:1 TRC as time increases (see Figure 9.10b), and, as discussed previously, it is not capable of modeling the initial 3:1 TRC. By studying the transient response superimposed on the FEP (Figure 9.10d) the TRCs and the associated transitions in the dynamics become clear. Initially the dynamics is captured close to the 3:1 resonance manifold of the underlying Hamiltonian system, and the resulting TRC is sustained for a certain period of time as the total energy decreases. As the energy reaches the value where the S31− branch meets the S11− branch, the dynamics escapes from 3:1 TRC and engages in 1:1 TRC. As energy decreases further and the LCO is completely eliminated, the dynamics escapes from 1:1 resonance capture at the later stage of the motion. Examining the corresponding energy exchanges of Figure 9.10e, we clearly identify the reasons for LCO elimination. Contrary to the case when the LCO survives (Figure 9.8e), the energy dissipation by the NES counterbalances the energy supply provided by the nonlinear (negative) damping of the VDP oscillator. It is interesting to note that the NES passively adjusts the rate of energy dissipation so to precisely counterbalance the energy input fed by the VDP oscillator. Moreover, in this case the rates of energy dissipation or generation asymptotically reach steady state values, whereas the corresponding ones of Figure 9.8e are almost linearly increasing or decreasing as the LCO is retained. Therefore, for LCO elimination it is necessary that TET from the VDP oscillator to the NES should occur at a sufficiently fast time scale and be strong enough to overcome the energy fed by the nonlinear damper of the VDP oscillator. Similar studies were performed for configuration VDPNES2 with a mass ratio εm = 0.05 and coefficient of essentially nonlinear coupling stiffness equal to κ = 1.0. The damping values ελ = 1% and ελ = 10% were considered for the cases when the LCO survives or is eliminated by the action of the NES. Initial conditions were equal to (x(0), y(0), x(0), ˙ y(0)) ˙ = (3.5, 0.1, 0, 0) for motions initiated outside the unperturbed LCO, and (1.0, 1.0, 0, 0) for those initiated inside the unperturbed LCO. Contrary to the configuration with grounded NES, in this case the underlying Hamiltonian system possesses a very complicated structure of periodic

372


Fig. 9.10 Transient dynamics of configuration VDPNES1 when the LCO is eliminated, case of initial 3:1 TRC and subsequent 1:1 TRC as total energy decreases: (a) time series; (b) motion on ˙ of the averaged system (9.8). the plane (φ, φ)


373

Fig. 9.10 Transient dynamics of configuration VDPNES1 when the LCO is eliminated, case of initial 3:1 TRC and subsequent 1:1 TRC as total energy decreases: (c) instantaneous frequencies; (d) WT spectrum on the FEP.

374


Fig. 9.10 Transient dynamics of configuration VDPNES1 when the LCO is eliminated, case of initial 3:1 TRC and subsequent 1:1 TRC as total energy decreases: (e) energy exchange measures.

solution branches (as shown in the corresponding FEP), which suggests a higher possibility for strong resonance interactions between the VDP oscillator and the NES. Mostly, similar conclusions as for VDPNES1 can be drawn, except for the following. Regardless of whether the motion is initiated outside or inside the unperturbed LCO, when the LCO survives, the resulting TRCs on the FEP of the relative displacement x(t) − y(t) [describing the structure of periodic orbits of the Hamiltonian system resulting by eliminating damping terms from system (9.2)] are generally more complex – although the dominant frequency contents lock into a 1:1 frequency ratio; the complex, underlying Hamiltonian dynamics may be responsible for this. Also, the initial energy exchanges between the VDP oscillator and the NES now occur on a fast scale (i.e., within a few linearized natural periods). A general conclusion of the previous transient simulations is that, in order to achieve complete elimination of the LCO, the NES must be designed to cause an overall continuous reduction of the instantaneous total energy of the system; that is, it must stimulate energy flow in the system towards a path from superharmonic to subharmonic resonance captures on the FEP. This goal does not merely imply an increase in damping (although this is definitely beneficial in certain cases), but rather, a careful selection of the system parameters (such as mass ratio and coupling stiffness) that influence the underlying Hamiltonian dynamics. In accordance to results reported in previous chapters, the topology of the branches of periodic solutions of the underlying Hamiltonian influences to a great extent the TRCs and the sudden


375

Fig. 9.11 Transient dynamics of configuration VDPNES2 when the LCO survives (εm = 5%, κ = 1.0, ελ = 1%), for initial conditions (x(0), y(0), x˙ (0), y(0)) ˙ = (3.5, 0.1, 0, 0): (a) time ˙ of the averaged system (9.9). series; (b) motion on the plane (φ, φ)

376


Fig. 9.11 Transient dynamics of configuration VDPNES2 when the LCO survives (εm = 5%, κ = 1.0, ελ = 1%), for initial conditions (x(0), y(0), x˙ (0), y(0)) ˙ = (3.5, 0.1, 0, 0): (c) instantaneous frequencies; (d) WT spectrum on the FEP.


377

Fig. 9.11 Transient dynamics of configuration VDPNES2 when the LCO survives (εm = 5%, κ = 1.0, ελ = 1%), for initial conditions (x(0), y(0), x˙ (0), y(0)) ˙ = (3.5, 0.1, 0, 0): (e) energy exchange measures.

transitions (jumps) in the transient damped dynamics. For example, reducing (increasing) the mass ratio for configuration VDPNES2 , shifts the entire FEP towards lower (higher) energies. Hence, an optimization study is required to achieve robust passive LCO suppression, a topic that is addressed in the next section.

378


Fig. 9.12 Transient dynamics of configuration VDPNES2 when the LCO survives (εm = 5%, κ = 1.0, ελ = 1%), for initial conditions (x(0), y(0), x˙ (0), y(0)) ˙ = (1.0, 1.0, 0, 0): (a) time series; (b) WT spectrum on the FEP.


379

Fig. 9.13 Transient dynamics of configuration VDPNES2 when the LCO survives (εm = 5%, κ = 1.0, ελ = 10%), for initial conditions (x(0), y(0), x˙ (0), y(0)) ˙ = (3.5, 0.1, 0, 0): (a) time ˙ of the averaged system (9.9). series; (b) motion on the plane (φ, φ)

380


Fig. 9.13 Transient dynamics of configuration VDPNES2 when the LCO survives (εm = 5%, κ = 1.0, ελ = 10%), for initial conditions (x(0), y(0), x˙ (0), y(0)) ˙ = (3.5, 0.1, 0, 0): (c) instantaneous frequencies; (d) WT spectrum on the FEP.


381

Fig. 9.13 Transient dynamics of configuration VDPNES2 when the LCO survives (εm = 5%, κ = 1.0, ελ = 10%), for initial conditions (x(0), y(0), x˙ (0), y(0)) ˙ = (3.5, 0.1, 0, 0): (e) energy exchange measures.

9.1.3 Steady State Dynamics and Bifurcation Analysis In this section, we will explore the steady state dynamics of the system by utilizing the method of numerical continuation, and relate LCO suppression to the bifurcation structure of the steady state solutions. The results will provide optimal design parameters for the NES, guaranteeing robustness of LCO elimination.

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9.1.3.1 Study of Robustness of LCO Elimination We now show numerically that LCO elimination due to the action of the NES is the result of a Hopf bifurcation. The steady state solutions of the full equations of motion (9.1) and (9.2) are computed through numerical continuation performed using the package MATCONT in Matlab (Dhooge et al., 2003). We consider the configuration VDPNES1 first, with the linear coupling stiffness εC regarded as a control parameter, and fixing the other NES parameters to the values κ = 1.0, ελ = 3%. We consider two mass ratios, namely, εm = 10% where supercritical LCOs exist, and εm = 50% where only subcritical LCOs exist. We remark that the Hopf bifurcation in the steady state dynamics proves to be subcritical, i.e., to involve a subcritical LCO near the bifurcation point. By subcritical we denote that the LCO occurs below the critical parameter value. In Figure 9.14 we present the bifurcation structure of steady state amplitudes of configuration VDPNES1 , governed by system (9.1). At the bifurcation plots, the notation BPC denotes a branch point of cycles, a point LPC, a limit point cycle, and a point NS, a Neimark–Sacker bifurcation (Golubitsky and Schaeffer, 1985). From these results it is clear that the LCOs are eliminated through a Hopf bifurcation that occurs at εC = 0.0415 for εm = 10%, and εC = 0.2579 for εm = 50%. We note that all non-trivial curves in the bifurcation diagram represent periodic oscillations of the system, that is, LCOs. The first case (εm = 10%) is regarded as a case of ‘malign’ system nonlinearities (as far as LCO suppression is concerned) because, although the trivial equilibrium is stable after the bifurcation point, the solution may jump onto coexisting stable LCOs if sufficiently strong perturbations are imposed; hence, the LCO suppression cannot be characterized as robust in this case. On the other hand, the latter case (εm = 50%) provides robust elimination of LCOs, since the stable trivial solution that results after the Hopf bifurcation is a global attractor of the dynamics (this is a case of ‘benign’ system nonlinearities). We also remark that, when the system exhibits malign nonlinearities, it possesses a complicated bifurcation structure. For example, considering the bifurcation diagram of Figure 9.14a, there exist LPC bifurcations (or equivalently, saddle-node bifurcations of equilibria), BPC bifurcations (or pitchfork bifurcations of equilibria), and NS bifurcations (or Hopf bifurcation of equilibrium positions) (Golubitsky and Schaeffer, 1985); it follows that the steady state dynamics of the system can assume complex forms such as higher-dimensional periodic or quasi-periodic orbits on two-tori generated at points of NS bifurcations. Similar bifurcation studies were performed for the configuration VDPNES2 governed by system (9.2), with mass ratio εm as the control parameter. Similar results were obtained, such as sub- and supercritical LCOs; however, the system with an ungrounded NES does not possess a complex bifurcation structure even in the case of malign system nonlinearities (there simply exist two LPC points in the bifurcation diagram). Although the detailed results are not discussed at this point, we will reexamine these behaviors utilizing the averaged system later in this section. Finally, we remark that the existence of a Hopf bifurcation in the steady state dynamics can be proven theoretically, employing the method of Lyapunov–Schmidt


383

Fig. 9.14 Bifurcation diagrams of steady state amplitudes of configuration VDPNES1 (κ = 1.0, ελ = 3%), stable (unstable) branches are marked by solid (dotted) lines: (a) case when supercritical LCOs exist (malign nonlinearity), εm = 10%; (b) case when subcritical LCOs exist (benign nonlinearity), εm = 50%.

reduction (LSR) (Golubitsky and Schaeffer, 1985). By this methodology, singularities are removed by projecting the complement of the null space of the linearized system operator and the null space itself, onto the range and its complement space, respectively. Then, the generalized implicit function theorem is invoked. Hence, LSR can be directly applied to the full equations of motion (9.1) and (9.2); alternatively, LSR can be applied to the slow flow models (9.8) and (9.9), to show that their

384


equilibrium positions exhibit pitchfork bifurcations. More details are not included in this study, but are left as future work.

9.1.3.2 Analytical Study We now examine analytically the steady state dynamics of both configurations VDPNES1 and VDPNES2 , utilizing the slow flow dynamics models (9.8) and (9.9). In so doing we demonstrate the capacity of these models to accurately capture the steady state dynamics and their bifurcations of systems (9.1) and (9.2), respectively. We will analyze each NES configuration separately. Starting from Configuration VDPNES1 , we express the slow flow model (9.8) in the following compact form: α˙ 1 = f1 (α1 , α2 , φ) α˙ 2 = f2 (α1 , α2 , φ) α1 α2 φ˙ = g(α1 , α2 , φ)

(9.17)

Then, the steady state amplitudes α1 , α2 and the phase difference φ are obtained by solving the set of homogeneous nonlinear algebraic equations f1 (α1 , α2 , φ) = f2 (α1 , α2 , φ) = g(α1 , α2 , φ) = 0 where trivial solutions are excluded. It turns out that in this case we can find the analytical solutions in explicit form. Indeed, taking into account the analytical expression for f1 , we can solve explicitly for amplitude α1 C f1 (α1 , α2 , φ) = 0 ⇒ α1 = − α2 sin φ λ

(9.18)

which, when plugged into the relation f2 (α1 , α2 , φ) = 0, yields

C2 2 sin φ α22 = 4 1 − λ

(9.19)

Since α22 > 0, the parenthesis on the right-hand side must be positive, so that λ > C 2 sin2 φ ⇒ sin2 φ < and thus

λ ≡ crit , C2

or | sin φ|
0, the solution is unstable, which implies divergent response of the wing. For comparison, we depict the results predicted from both steady aerodynamics and quasi-steady aerodynamics. The model based of steady aerody-

402


namics (S) predicts that the typical section model is neutrally stable for < SF (that is, that all eigenvalues are purely imaginary); when = SF , the bending (heave) and torsion (pitch) frequencies merge in a phenomenon called coalescence or merging frequency flutter (Dowell et al., 1995). If > SF , the dynamics becomes unstable, and the responses diverge. On the other hand, the model based on quasi-steady aerodynamics (QS) predicts flutter at a lower value of the reduced ve< (S) locity, (QS) F F , where the real parts of the complex pair of eigenvalues change sign from negative to positive (see Figure 9.24). We note that in the quasi-steady case there exists a tendency for the frequencies to merge but complete merging does not occur (Dowell et al., 1995). For the above numerical parameter values, we obtain a flutter speed equal to (QS) ≡ F = 0.87. Note that steady aerodynamics predicts a higher flutter speed F than quasi-steady theory (i.e., (S) F = 1.03), and also a higher coalescence frequency at the flutter speed for the steady flow condition. Now we include nonlinear stiffnesses in both degrees of freedom of the model. Clearly, stability behavior of the trivial solution y = α ≡ 0 will follow the linear analysis since the trivial solution is a hyperbolic equilibrium point, so we can invoke the Hartman–Grobman Theorem and claim topological conjugacy between the linear and nonlinear local vector fields sufficiently close to the hyperbolic equilibrium (Guckenheimer and Holmes, 1983). Also, due to the hardening nature of the nonlinearities of the system, which are expected to limit the amplitudes of the responses, the nonlinear system may possess LCOs at supercritical speeds (i.e., for > F ). In Figure 9.25 we depict typical responses at subcritical and supercritical speeds of the linearized and nonlinear systems, respectively. We see that the linearized system predicts divergent responses, which, clearly, are not realistic, whereas, in actuality the nonlinearities restrict the growth of the diverging wing responses so that LCOs are developed instead. It follows that the development of LCOs is a direct product of aeroelastic instability in the system.

9.2.2 Slow Flow Dynamics We now construct a slow flow model of the dynamics of system (9.37), based on slow-fast partition of the dynamics, and valid for response regimes both before and after flutter instability has occurred. For the construction of the model we separate the important (slow flow) from the secondary or non-essential (fast flow) dynamics and utilize ideas from the CX-A methodology discussed in previous chapters.

9.2.2.1 Dominant (Fast) Frequencies in the Responses To establish an accurate slow flow dynamical model capturing reliably and robustly the full nonlinear response of system (9.37), we need to consider the dominant fre-


403

Fig. 9.25 Time responses at (a) subcritical ( = 0.5) and (b) supercritical ( = 0.95) reduced velocities in the linear and nonlinear models; initial conditions are given by (y(0), α(0), y (0), α (0)) = (0.01, 0, 0, 0).

404


quency components of the dynamics in the different stages of the motion. These will be regarded as the ‘fast’ frequencies in the dynamics, in terms of which ‘slow’ modulation equations will be developed. This procedure will establish also the dimensionality of the slow flow of the transient nonlinear dynamics. First, we examine the dominant frequency components in the transient responses for varying reduced velocity . Figure 9.26 depicts power (FFT) spectra of the responses of the heave and pitch modes normalized with respect to their respective maxima at each value of the reduced velocity. For subcritical reduced speeds (i.e., < F ), there are only two dominant frequency components, ωheave and ωpitch , related to the two linearized natural frequencies for the heave and pitch (in terms of the non-dimensional frequencies defined previously the two dominant components are and unity, respectively). When the reduced velocity exceeds the flutter speed (i.e., > F ), there appear two dominant frequency components, ωpitch and 3ωpitch, for the heave mode, and one dominant frequency component, ωpitch , for the pitch mode. This clearly shows that both below and above the flutter speed, the aeroelastic response of the wing contains at most three dominant frequency components, related to the two linear natural frequencies of the linear flutter model. That is, the lowest component corresponds to the heave mode ωheave ≈ = 0.5, the middle one to the pitch mode, ωpitch ≈ 1, and the highest one to approximately three times that of the pitch mode, 3ωpitch ≈ 3. In the following exposition we will refer to these three dominant frequencies by LF, MF, and HF (low, middle, and high frequency), respectively. The FFT analysis provides an averaged (static) view of the frequency content of the transient nonlinear signals. Since the phenomena studied in this work are essentially nonlinear and transient, we resort instead to frequency decompositions based on numerical wavelet transforms (WTs), which provide information on the temporal evolutions of the dominant harmonic components of the subcritical or supercritical transient responses of the wing. This will enable us to clearly establish and study dynamical transitions occurring between different regimes of the transient motions. We perform WT analysis for two specific reduced velocities, namely, = 0.5 and = 0.95, corresponding to subcritical and supercritical wing responses, respectively, and depicted in Figures 9.27 and 9.28. However, the results presented herein can be similarly extended to other subcritical or supercritical reduced speeds. Considering the plots of the WT spectra depicted in Figures 9.27 and 9.28, we note that when the flow speed is less than the flutter speed (in a subcritical regime – see Figure 9.27) the linearized natural frequency of the heave mode appears as the main frequency component in heave response, and that of the pitch mode as the minor. On the other hand, the pitching response possesses both heave and pitch harmonics with the pitch harmonic being the dominant one. Apparently, there exists a frequency relation of ωpitch ≈ 2ωheave as we may expect from the relation satisfied by the two linearized natural frequencies, i.e., a 1:2 internal resonance occurs in the transient dynamics of the wing when a flow speed is less than the flutter speed. In addition, we can deduce the existence of a non-negligible frequency component at the linearized heave natural frequency ( = 0.5) in the pitch mode, so that the lowest likewise frequency components both in heave and pitch modes appear to


405

Fig. 9.26 Normalized power spectrum with respect to the reduced velocity; (a) heave mode, (b) pitch mode; heave ωheave and pitch ωpitch denote frequency components close to the linear natural frequencies of heave and pitch, respectively (ωheave ≈ = 0.5 and ωpitch ≈ 1); initial conditions are (y(0), α(0), y (0), α (0)) = (0.01, 0, 0, 0).

406


Fig. 9.27 WT spectra of transient responses at a subcritical speed ( = 0.5); (a) heave mode, (b) pitch mode; initial conditions are given by (y(0), α(0), y (0), α (0)) = (0.01, 0, 0, 0).


407

Fig. 9.28 WT spectra of transient responses at a supercritical speed ( = 0.95); (a) heave mode, (b) pitch mode; initial conditions are given by (y(0), α(0), y (0), α (0)) = (0.01, 0, 0, 0).

408


interact with each other up to τ ≈ 80. Later we will show numerically that this transient dynamics is also captured in a 1:1 resonance manifold (Quinn, 1997a; Vakakis and Gendelman, 2001), so that a 1:1 TRC occurs in the dynamics as well. Clearly, below the critical flutter velocity the energy extracted from the flow is being channeled and then exchanged between the nonlinear modes through resonance captures (as discussed below). In the supercritical regime, however, a qualitative change in the dynamics occurs, since a sudden transition (jump) between frequency components takes place (see Figure 9.28). To understand this transition we will need to partition the dynamics into three separate phases: (i) an initial transient period, where likewise frequency components are likely to match each other leading to 1:1 TRC; (ii) a transition or escape from TRC into new frequency or resonance relations, where the basic heave harmonic gradually triggers the pitch mode at its dominant harmonic and then dies out, whereas at the same time a higher frequency component develops nearly at three times the linearized frequency of the pitch mode (i.e., a 3:1 superharmonic component); and finally, (iii) the generation of an LCO as a steady state response develops, where the resulting dominant harmonics are a 3:1 superharmonic component in heave mode, and a component at the pitch linearized frequency in pitch mode, resulting in a 3:1 sustained resonance capture (SRC). We will explore each one of these phases of the dynamics in detail, since they constitute the triggering mechanism for LCOs in the in-flow rigid wing.

9.2.2.2 Multi-Phase Averaging Before proceeding to analyzing the different regimes of the previously outlined LCO triggering mechanism, it will be necessary to develop a slow flow model of the dynamics through multi-phase averaging, taking into account the previous WT-based frequency analysis. To this end, we reconsider the equations of motion (9.37), and through a linear coordinate transformation express them in the following inertially decoupled form: y + ζ1 y + k11 y + k12 α + n11 y 3 + n12 α 3 = 0 α + ζ2 y + k21y + k22α + n21 y 3 + n22 α 3 = 0

(9.42)

where the coefficients are defined as follows: ζ1 ≡ µCL,α (rα2 + γ xα )/D, k11 ≡ rα2 2 /D, k21 ≡ −xα 2 /D, n11 ≡ rα2 ξy /D,

ζ2 ≡ −µCL,α (γ + xα )/D

k12 ≡ {µCL,α

2

(rα2 + γ xα ) − rα2 xα }/D

k22 ≡ {rα2 − µCL,α n12 ≡ −xα ξα /D,

2

(γ + xα )}/D

n21 ≡ −xα ξy /D,

n22 ≡ ξα /D (9.43)


409

and it holds that D = rα2 − xα2 > 0 for any mass distribution of the rigid wing. Note that only the coefficients ζ1 , ζ2 , k12 , k22 are functions of the reduced velocity . Moreover, the condition k12 ≡ 0 that results from the previous linearized eigenvalue analysis provides the following analytical (linearized) approximation for the flutter speed: F

≡

rα2 xα µCL,α (rα2 + γ xα )

(9.44)

It follows that k12 < 0 implies that < F , so that no flutter occurs. Motivated by our previous numerical analysis and the fact that there exist at most three dominant (fast) frequency components in the subcritical and supercritical inflow wing responses, we decompose the heave and pitch transient responses in terms of the following three dominant frequency components: y (τ ) = y1 (τ ) + y2 (τ ) + y3 (τ ) α (τ ) = α1 (τ ) + α2 (τ ) + α3 (τ )

(9.45)

The subscripts 1, 2, and 3 denote terms possessing three distinct dominant (fast) frequencies, proportional to ej τ , ej τ , and e3j τ , respectively. This representation is similar to our previous theoretical analyses of transient responses possessing multiple fast frequency components (for example, see Sections 3.3 and 3.4). Now, following the CX-A methodology (Manevitch, 2001) we introduce the following new complex variables: ψ1 = y1 + j y1,

ψ3 = y2 + jy2 ,

ψ2 = α1 + j α1 ,

ψ4 = α2 + j α2 ,

ψ5 = y3 + 3jy3 ψ6 = α3 + 3j α3

(9.46)

where j 2 = −1. Then, we may express the original real variables in (9.37) in terms of the new complex ones as follows: y=

1 1 1 (ψ1 − ψ1∗ ) + (ψ3 − ψ3∗ ) + (ψ5 − ψ5∗ ) 2j 2j 6j

α=

1 1 1 (ψ2 − ψ2∗ ) + (ψ4 − ψ4∗ ) + (ψ6 − ψ6∗ ) 2j 2j 6j

1 (ψ1 + ψ1∗ + ψ3 + ψ3∗ + ψ5 + ψ5∗ ) 2 1 α = (ψ2 + ψ2∗ + ψ4 + ψ4∗ + ψ6 + ψ6∗ ) 2 j j 3j (ψ1 + ψ1∗ ) − (ψ3 + ψ3∗ ) − (ψ5 + ψ5∗ ) y = ψ1 + ψ3 + ψ5 − 2 2 2 j j 3j (ψ2 + ψ2∗ ) − (ψ4 + ψ4∗ ) − (ψ6 + ψ6∗ ) (9.47) α = ψ2 + ψ4 + ψ6 − 2 2 2 y =

410


At this point we introduce a slow-fast partition of the dynamics, by partitioning the complex responses into slow and fast parts, ψ1 (τ ) = ϕ1 (τ ) ej τ , ψ3 (τ ) = ϕ3 (τ ) ej τ , ψ5 (τ ) = ϕ5 (τ ) e3j τ ψ2 (τ ) = ϕ2 (τ ) ej τ , ψ4 (τ ) = ϕ4 (τ ) ej τ , ψ6 (τ ) = ϕ6 (τ ) e3j τ

(9.48)

where ϕk (τ ), k = 1, 2, . . . , 6 represent slowly-varying, complex-valued amplitude modulations. In expressing the variables according to (9.48) we assume that the transient responses are composed of ‘fast’ oscillations [represented by the complex exponentials in (9.48)] modulated by ‘slow’ envelopes [represented by the complex amplitudes ϕk (τ )]. These partitions are fully compatible with the results of numerical simulations. It should be clear that the (slow) temporal evolutions of the modulations ϕk (τ ) govern the important (essential) dynamics of system (9.37) in an appropriately defined slow flow phase space. Interestingly enough, the dimensionality of the slow flow phase space – in this case this space is 12-dimensional – exceeds the dimensionality of the phase space of the original system (9.37) – which is four-dimensional; this is due to the fact that the dimensionality of the slow flow phase space depends on the number of dominant harmonics that govern the transient dynamics – in this case three. Substituting (9.47) and (9.48) into (9.37) and applying multi-phase averaging (Lochak and Meunier, 1988) over the frequency components ej τ , ej τ , and e3j τ we obtain six complex-valued, ordinary differential equations governing the slow evolutions of the modulations, in the compact form ϕ + F (ϕ; ) = 0,

F, ϕ ∈ C 6 ,

∈R

(9.49)

where the reduced velocity is regarded an the independent parameter of the problem. The detailed form of the slow flow (9.49) is given below:

1 k11 j k12 3j ϕ1 + ζ1 + j − ϕ1 − ϕ2 − (n11 ϕ1 |ϕ1 |2 + n12 ϕ2 |ϕ2 |2 ) 2 2 83 j 3j (n11 ϕ1 |ϕ3 |2 + n12 ϕ2 |ϕ4 |2 ) − (n11 ϕ1 |ϕ5 |2 + n12 ϕ2 |ϕ6 |2 ) = 0 4 12

k22 1 k21 j 3j ζ2 − j ϕ1 + − ϕ2 − ϕ2 + (n21 ϕ1 |ϕ1 |2 + n22 ϕ2 |ϕ2 |2 ) 2 2 83 −

j 3j (n21 ϕ1 |ϕ3 |2 + n22 ϕ2 |ϕ4 |2 ) − (n21 ϕ1 |ϕ5 |2 + n22 ϕ2 |ϕ6 |2 ) = 0 4 12 1 j k 3j 12 ϕ4 − (n11 ϕ3 |ϕ1 |2 + n12 ϕ4 |ϕ2 |2 ) ϕ3 + [ζ1 + j (1 − k11)]ϕ3 − 2 2 42 j 3j − (n11 ϕ3 |ϕ3 |2 + n12 ϕ4 |ϕ4 |2 ) − (n11 ϕ3 |ϕ5 |2 + n12 ϕ4 |ϕ6 |2 ) 8 12 j + (n11 ϕ3∗2 ϕ5 + n12 ϕ4∗2 ϕ6 ) = 0 8 −


411

1 j 3j ϕ4 + (ζ2 − j k21)ϕ3 + (1 − k22 )ϕ4 − (n21 ϕ3 |ϕ1 |2 + n22 ϕ4 |ϕ2 |2 ) 2 2 42 j 3j − (n21 ϕ3 |ϕ3 |2 + n22 ϕ4 |ϕ4 |2 ) − (n21 ϕ3 |ϕ5 |2 + n22 ϕ4 |ϕ6 |2 ) 8 12 j + (n21 ϕ3∗2 ϕ5 + n22 ϕ4∗2 ϕ6 ) = 0 8

1 k11 j k12 j ζ1 + j 3 − ϕ5 − ϕ6 − ϕ5 + (n11 ϕ5 |ϕ1 |2 + n12 ϕ6 |ϕ2 |2 ) 2 3 6 42 j j (n11 ϕ5 |ϕ3 |2 + n12 ϕ6 |ϕ4 |2 ) − (n11 ϕ5 |ϕ5 |2 + n12 ϕ6 |ϕ6 |2 ) 4 72 j + (n11 ϕ33 + n12 ϕ43 ) = 0 8

k22 1 k21 j j ζ2 − j ϕ5 + 3− ϕ6 − ϕ6 + (n21 ϕ5 |ϕ1 |2 + n22 ϕ6 |ϕ2 |2 ) 2 3 2 3 42 −

j j (n21 ϕ5 |ϕ3 |2 + n22 ϕ6 |ϕ4 |2 ) − (n21 ϕ5 |ϕ5 |2 + n22 ϕ6 |ϕ6 |2 ) 4 72 j + (n21 ϕ33 + n22 ϕ43 ) = 0 8

−

(9.50)

The plots presented in Figures 9.29 and 9.30 demonstrate the validity of the averaged system with ‘optimally’ determined initial conditions (see below) compared to the (numerically) exact solutions for both subcritical and supercritical reduced velocities. Our careful numerical study (not fully shown herein) indicates that the three-harmonic slow flow model (9.49) approximates well the original dynamics at the entire range of reduced speeds, i.e., the averaged system is valid for accurately modeling the nonlinear dynamics over the entire subsonic fluid-structure interaction regime. We note that each of the dominant harmonic components in the model (9.45) can be recovered from the averaged system (9.49), according to the following expressions: y1 (τ ) =

1 Im[ϕ1 (τ )ej τ ],

y2 (τ ) = Im[ϕ3 (τ )ej τ ], y3 (τ ) =

1 Im[ϕ5 (τ )e3j τ ], 3

α1 (τ ) =

1 Im[ϕ2 (τ )ej τ ]

α2 (τ ) = Im[ϕ4 (τ )ej τ ] α3 (τ ) =

1 Im[ϕ6 (τ )e3j τ ] 3

(9.51)

so we may reconstruct the heave and pitch responses directly from the decompositions (9.45). An interesting point now discussed concerns the choice of initial conditions of the averaged set of equations (9.50). Because we need twelve initial conditions for the averaged system, and we possess only four available initial conditions for the full

412


Fig. 9.29 Validation of the averaged system (9.50) for a subcritical reduced velocity ( with initial conditions (y(0), α(0), y (0), α (0)) = 10−3 , 10−3 , 0, 0).

= 0.5),

system (9.37), the problem of determining the appropriate initial conditions of the slow flow model becomes indeterminate. This problem also arose in Section 3.4.2.2, where the analytical study of subharmonic TET in a two-DOF system was carried out.


Fig. 9.30 Validation of the averaged system (9.50) for a supercritical reduced velocity ( with initial conditions (y(0), α(0), y (0), α (0)) = 10−3 , 10−3 , 0, 0).

413

= 0.95),

Returning to the problem of determining the initial conditions of the slow flow (9.50), we may express the initial conditions directly from the decomposition (9.45), y (0) = y1 (0) + y2 (0) + y3 (0) ,

α (0) = α1 (0) + α2 (0) + α3 (0)

y (0) = y1 (0) + y2 (0) + y3 (0) ,

α (0) = α1 (0) + α2 (0) + α3 (0) (9.52)

414


which leads to the expressions ϕ1 (0) = y1 (0) + j y1 (0) , ϕ3 (0) = y2 (0) + jy2 (0) , ϕ5 (0) = y3 (0) + 3jy3 (0) , under the restrictions

ϕ2 (0) = α1 (0) + j α1 (0) ϕ4 (0) = α2 (0) + j α2 (0) ϕ6 (0) = α3 (0) + 3j α3 (0)

(9.53)

1 1 Im ϕ1 (0) + Im ϕ5 (0) Im ϕ3 (0) = y (0) − 3

1 1 Im ϕ2 (0) + Im ϕ6 (0) Im ϕ4 (0) = α (0) − 3 Re ϕ3 (0) = y (0) − [Re ϕ1 (0) + Re ϕ5 (0)] Re ϕ4 (0) = α (0) − [Re ϕ2 (0) + Re ϕ6 (0)]

(9.54)

The determination of the initial conditions for the averaged system (9.50) is thus converted to the optimization problem of computing the minimum of the normalized mean square error, + E x − xa 2 E x − E[x] 2 in the time interval 0 < τ < τˆ for some τˆ , where the quantity to be minimized is regarded as function of the six sought initial conditions {ϕ1 (0), . . . , ϕ6 (0)}; E[·] denotes the mean value, and · the norm based on the standard inner product; moreover, x(τ ) ≡ (y(τ ), α(τ ))T and xa (τ ) ≡ (ya (τ ), αa (τ ))T are the response vectors of the exact and averaged systems, respectively. Note that the solution of this optimization problem may not be unique, since it depends on the topological properties and singularities of the solution manifold in the corresponding space (for example, the solution manifold may have several local minima so that the ‘optimal’ solution can be computed as any one of them). We may avoid this lack of uniqueness by expressing the solutions of the optimization problem in Taylor series ϕi (τ ) =

N

ϕij τ j + O(τ N+1 )

(9.55)

j =0

where ϕij ∈ C, i = 1, . . . , 6 as τ → 0 (hence, we assume that |τˆ | 1), and matching the series with the exact solutions at a specified matching time instant to determine uniquely each of the Taylor coefficients. Then we can construct the normal equation and find the so-called Moore-Penrose least squares solution which should be unique in terms of the Fredholm Alternative Theorem (i.e., see Keener, 2000). However, this kind of matching – in spite of uniqueness – may not provide good long-term results, particularly for higher-order approximations or multi-phase averaging. For example, Keener (1977) studied the validity of the two-timing method


415

(method of multiple scales, or, equivalently, first-order averaging method) for limit cycles for large times; he showed that the approximate solution, which is pointwise valid only for times of order O(1/ε), is orbitally valid for large times in the sense that the approximate solution (although not necessarily pointwise valid for all times) approaches a valid approximation of a stable limit cycle. Therefore, in this work we consider approximate solutions in the sense of orbital validity by way of the ‘optimal’ initial conditions, instead of pointwise accuracy which is guaranteed only up to a small time scale and may provide inaccurate results in the long run. Considering the analysis of the slow flow (9.49–9.50), two different formulations for analyzing the slow flow dynamics can be followed, by expressing the complex quantities in Cartesian or polar coordinates. Let us first consider the slow flow equations in Cartesian coordinates. Expressing ϕk (τ ) = z2k−1 (τ ) + j z2k (τ ), k = 1, . . . , 6, where zi ∈ R, ∀i into (9.50), we obtain twelve (real-valued) slow flow modulation equations: Z = G(Z; ),

Z = (z1 , . . . , z2 ) ∈ R 12 ,

∈R

(Cartesian coordinates) (9.56) Alternatively, expressing the complex quantities in polar form, we express ϕk (τ ) = ak (τ ) ejβk (τ ) , ak ∈ R + , βk ∈ S 1 , k = 1, . . . , 6, which when substituted into (9.50), and upon separation of real and imaginary parts, leads again to a set of 12 slow flow equations in the form ak = f˜k (a, β; ) and ak βk = g˜k (a, β; )

(9.57)

The exact form of these equations is given below:

ζ1 a2 3n12 2 3n12 2 n12 2 a1 + a1 − sin (β1 − β2 ) k12 + a + a =0 a + 2 2 42 2 2 4 6 6 ζ2 a1 cos (β1 − β2 ) 2

a1 3n21 2 3n21 2 n21 2 sin (β1 − β2 ) k21 + a + a =0 + a + 2 2 3 6 5 42 1

ζ1 a4 3n12 2 3n12 2 n12 2 sin (β3 − β4 ) k12 + a + a a3 + a3 − a + 2 2 22 2 4 4 6 6 n11 2 n12 2 a3 a5 sin (3β3 − β5 ) + a a6 sin (β3 + 2β4 − β6 ) = 0 + 8 8 4

ζ2 3n21 2 3n21 2 n21 2 a3 sin (β3 − β4 ) k21 + a + a a4 + a3 cos (β3 − β4 ) + a + 2 2 22 1 4 3 6 5 n22 2 n21 2 a4 a6 sin (3β4 − β6 ) + a a5 sin (2β3 + β4 − β5 ) = 0 + 8 8 3 a2 +

416


ζ1 a6 3n12 2 3n12 2 n12 2 sin (β5 − β6 ) k12 + a + a + a5 − a + 2 6 22 2 2 4 12 6 n11 3 n12 3 a3 sin (3β3 − β5 ) − a sin (3β4 − β5 ) = 0 − 8 8 4

ζ2 3n21 2 3n21 2 n21 2 a5 sin (β5 − β6 ) k21 + a a a6 + a5 cos (β5 − β6 ) + a + + 2 6 2 3 12 5 22 1 n22 3 n21 3 a sin (3β4 − β6 ) − a sin (3β3 − β6 ) = 0 − 8 4 8 3

a1 3n11 2 3n11 2 n11 2 2 − k11 − a − a a 1 β1 + a − 2 2 3 6 5 42 1

a2 3n12 2 3n12 2 n12 2 cos (β1 − β2 ) k12 + a + a =0 − a + 2 42 2 2 4 6 6

ζ1 a2 3n22 2 3n22 2 n22 2 2 − k22 − a4 − a6 + a1 sin (β1 − β2 ) a 2 β2 + a2 − 2 2 4 2 6 2

3n21 2 3n21 2 n21 2 a1 cos (β1 − β2 ) k21 + a + a =0 a + − 2 2 3 6 5 42 1

a3 3n11 2 3n11 2 n11 2 1 − k11 − a − a a 3 β3 + a − 2 22 1 4 3 6 5

a4 3n12 2 3n12 2 n12 2 cos (β3 − β4 ) k12 + a + a − a + 2 22 2 4 4 6 6 n11 2 n12 2 a3 a5 cos (3β3 − β5 ) + a a6 cos (β3 + 2β4 − β6 ) = 0 +, 8 8 4

ζ2 a4 3n22 2 3n22 2 n22 2 1 − k22 − a a + a3 sin (β3 − β4 ) a4 β4 + a − − 2 4 4 6 6 2 22 2

3n21 2 3n21 2 n21 2 a3 cos (β3 − β4 ) k21 + a a a + + − 2 4 3 6 5 22 1 n22 2 n21 2 a a6 cos (3β4 − β6 ) + a a5 cos (2β3 + β4 − β5 ) = 0 + 8 4 8 3

a5 3n11 2 3n11 2 n11 2 9 − k11 − a − a a 5 β5 + a − 6 22 1 2 3 12 5

a6 3n12 2 3n12 2 n12 2 cos (β5 − β6 ) k12 + a + a − a + 6 22 2 2 4 12 6 n11 3 n12 3 a3 cos (3β3 − β5 ) + a cos (3β4 − β5 ) = 0 + 8 8 4

ζ2 a6 3n22 2 3n22 2 n22 2 9 − k22 − a − a + a5 sin (β5 − β6 ) a 6 β6 + a − 6 2 4 12 6 2 22 2 a5


a5 3n21 2 3n21 2 n21 2 cos (β5 − β6 ) k21 + a + a a + 6 22 1 2 3 12 5 n22 3 n21 3 a4 cos (3β4 − β6 ) + a cos (3β3 − β6 ) = 0 + 8 8 3

417

−

(9.58)

By combining these equations we derive the following alternative autonomous set of slow flow modulation equations: ak = fk (a, φ; ),

, ai ap φip = gn (a, φ; ) (polar coordinates)

(9.59)

where k = 1, . . . , 6, φip ≡ βi − βp and (n; i, p) = (1; 1, 2), (2; 3, 5), (3; 3, 6), (4; 4, 5), (5; 4, 6). There are only five independent phase relations in (9.60) representing five phase differences between components of the solution with the following physical meanings: φ12 represents the interaction between LF heave and LF pitch; φ35 represents the interaction between MF heave and HF heave; φ36 represents the interaction between MF heave and HF pitch; φ45 represents the interaction between MF pitch and HF heave; and φ46 represents the interaction between MF pitch and HF pitch. It can be shown that all other possible phase differences arising in (9.60) can be expressed in terms of these five independent phase variables; for example, the phase interaction between MF heave and MF pitch can be expressed as φ34 =

1 (φ35 − φ45 ) 3

or

1 (φ36 − φ46 ) . 3

Although the modulation sets (9.57) and (9.58) are equivalent, in the following analysis we will be using the modulation equations in the polar form, equations (9.58), or in autonomous form, equations (9.60), since they provide direct information for the amplitudes of the components of the solution, as well as for the phases representing the nonlinear interactions between these components. A well recognized mathematical deficiency, however, of the equations in polar form relates to the mathematical singularity of the polar transformation at the origin, which renders the set (9.58) invalid for analyzing the dynamics when some of the components have zero (or nearly zero) amplitudes. In that case the modulation equations in Cartesian form, (9.57), should be used instead.

9.2.2.3 Bifurcation Analysis of Steady State Dynamics Before we employ the slow flow model to study the LCO triggering mechanism, we perform a steady state bifurcation analysis of the dynamics utilizing MATCONT in Matlab (Dhooge et al., 2003), in conjunction with the algorithm introduced in Kubíˇcek (1976) utilizing parameterization with respect to the arc length of equilibrium loci (see Section 9.1.3). To this end, we consider the original equations of motion (9.42), and express them in the following first-order form:

418


x = X(x; ) where x = (y, α, y , α )T ,

∈R

(9.60)

Direct application of MATCONT on these first-order differential equations provides bifurcation diagrams that yield information on the global dynamics of the full system at steady state, as depicted in Figure 9.31. As we discussed in Section 9.2.1, the (stable) trivial equilibrium x = 0 undergoes a Hopf bifurcation at the flutter speed F = 0.87, and changes its stability with simultaneous generation of a stable LCO. When the reduced velocity reaches the divergence flutter speed D = 1.767, two unstable non-trivial equilibrium points are computed. The solution curve for heave appears to be almost vertical at D , while that for pitch this does not hold. The physical interpretation is that, for reduced velocities higher than the divergence flutter speed, almost every heave position can be an equilibrium position whereas the pitch mode attains a specific equilibrium position. We note that, if we zoom out the vertical axis in Figure 9.31a to the same order as in Figure 9.31b, then the heave equilibrium curve also looks like a parabola; but this understanding may not be physically meaningful. In any case, these nontrivial equilibrium points are unstable so that they are not physically realizable. The divergence due to flutter represents a ‘static’ instability from a dynamics point of view (Blevins, 1990) and the corresponding reduced speed can be computed from static balance as , 2 (9.61) D ≡ rα /(γ µCL,α ) Since the above results provide only global information regarding to where and what type of bifurcations occur, and how large the LCO amplitudes are, they will not help us understand the nonlinear modal interactions that generate the fluid-structure instabilities that eventually act as LCO triggering mechanisms. To address this issue it is necessary to perform bifurcation analysis of the averaged system (9.57) for the trivial equilibrium via MATCONT, and for (9.60) for the nontrivial LCOs utilizing Kubíˇcek’s method (1976). The reason for using two different approaches is dictated by the possible singularities built in (9.60) when one of the amplitudes becomes zero; then, the set degenerates to a set of differential-algebraic equations and becomes unsolvable using MATCONT since this package only solves differential equations of the standard form x = X(x; σ ) where x ∈ R n , σ ∈ R k . Figure 9.32 depicts the numerical continuation results for steady state amplitudes and phase differences for the multi-phase averaged system (9.60). We note that after the speed exceeds the critical value of flutter the HF heave and MF pitch components are dominant at steady state, a result that is consistent with numerical simulations. These steady state results will be revisited in a later section where analytical study of the LCO triggering mechanism is carried out. Because our wing model assumes small oscillations, |α| < 10◦ ≈ 0.1745 rad, the numerical solutions at higher supercritical speeds may deviate from physical observations. In addition, we may not observe the secondary Hopf bifurcation by our slow flow analysis, since at least five dominant harmonics are required for its computation (Liu and Dowell, 2004). Moreover, only supercritical LCOs will be obtained due to the specific parameter choices used in our numerical study.


419

Fig. 9.31 Bifurcation analysis of the steady state dynamics of system (9.42): ‘H’ and ‘BP’ stand for Hopf bifurcation point and branching point, respectively; stable (unstable) motion is represented by a solid (dashed) line.

We now examine the possible existence of other equilibrium solutions of the slow flow equations (9.60). In fact, there exist many other non-trivial but degenerate equilibrium solutions; Figure 9.33 presents one of these computed by numerical

420


Fig. 9.32 Steady state LCO amplitudes and phase differences from the averaged system (9.60); (a) heave amplitudes, (b) pitch amplitudes.


421

Fig. 9.32 Steady state LCO amplitudes and phase differences from the averaged system (9.60); (c, d) phase differences.

422


continuation of equilibrium positions, corresponding to near-trivial values for all amplitudes except for the LF pitch a2 . The LF pitch amplitude a2 in Figure 9.33a is estimated as a2 = a4 = 0.5a4 where a4 is the MF pitch amplitude in Figure 9.32b. Although this relation can be obtained analytically as shown below, we may intuitively guess it by examining the expression of the amplitudes in (9.51) and noting that at a specific reduced speed all amplitudes except LF or MF pitch are almost trivial, and that the contribution of the heave mode to the total energy at steady state is rather negligible. Then, α ≈ a4 when only MF pitch is dominant, and if only LF pitch is dominant, then we may approximately compute α ≈ a2 / = a4 by (9.51). Figures 9.34 and 9.35 depict heave and pitch responses corresponding to the steady state motions for = 0.95 in the plots of Figures 9.32 and 9.33, respectively. Both cases correspond to identical initial conditions, (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0). However, the corresponding initial conditions used for integrating the slow flow equations (9.60) are different. For example, the initial conditions used for performing the numerical simulations depicted in Figure 9.34 are selected to be ‘optimal’ in the sense discussed in Section 9.2.2.2 (where the overdeterminacy of the problem of selecting the initial conditions of the slow flow was discussed), in order to accurately approximate the exact solutions which will be used later in the study of LCO triggering mechanism; whereas, the initial conditions utilized for the numerical simulations of Figure 9.35 are slightly different. Since numerical studies can be made only for stable motions, it turns out that the prediction depicted in Figure 9.35 is not meaningful in our study of the LCO triggering mechanism because it is based on a degenerate equilibrium of the slow flow modulation equations, and does not provide any information on the HF heave component which is observable in actuality. It turns out that the degenerate equilibrium solutions presented in Figure 9.33 can be derived analytically from a subsystem of the slow flow model (9.60), namely, from the multi-phase averaged system with two dominant frequencies corresponding to LF and MF components; this subsystem will be called the two-frequency averaged system hereafter. For this subsystem, we neglect HF terms y3 (τ ) and α3 (τ ) in (9.45), or, equivalently, the corresponding complex variables ψ5 and ψ6 , so the resulting complex-valued slow flow equations (9.49) contain only the complex modulations ϕ1 , . . . , ϕ4 ∈ C 4 . Then, we obtain the following reduced two-frequency averaged system in polar form:

ζ1 a2 3n12 2 3n12 2 a + =0 a1 − sin φ12 k12 + a 2 2 2 4 42 2

ζ2 a1 3n21 2 3n21 2 a2 + a1 cos φ12 + a + =0 sin φ12 k21 + a 2 2 42 1 2 3

ζ1 a4 3n12 2 3n12 2 a3 + a3 − a + =0 sin φ34 k12 + a 2 2 22 2 4 4

ζ2 a3 3n21 2 3n21 2 sin φ34 k21 + a3 = 0 a4 + a3 cos φ34 + a1 + 2 2 2 2 4

a1 +


423

Fig. 9.33 Additional steady state LCO solution, amplitudes and phase differences predicted by the averaged system (9.60); (a) amplitudes, (b) phase differences (mod 2π was applied for the plot of φ36 ).

424


Fig. 9.34 Total and component responses for steady state solutions corresponding to amplitudes and phases depicted in Figure 9.32 for = 0.95; (a) heave mode response and (b) pitch mode response.


425

Fig. 9.35 Total and component responses corresponding to steady state solutions of the averaged system depicted in Figure 9.33 for = 0.95; (a) heave, (b) pitch.

426


Fig. 9.35 Total and component responses corresponding to steady state solutions of the averaged system depicted in Figure 9.33 for = 0.95; (c, d) phase differences.


427

Fig. 9.36 Analytical steady state pitch amplitudes and their stability when (φ12 , φ34 ) = (π/2, π/2), computed from the two-frequency averaged system (9.63).

3n11 2 3n11 2 3n22 2 3n22 2 ζ1 a − + a + − a12 sin φ12 a a 42 1 2 3 42 2 2 4 2

1 3n12 2 3n12 2 3n21 2 3n21 2 2 − a + + a + − a k =0 cos φ12 a22 k12 + a a 21 1 2 42 2 2 4 42 1 2 3

a3 a4 3n11 2 3n11 2 3n22 2 3n22 2 ζ2 k22 − k11 − a a a3 a4 φ34 + a − + a + − a32 sin φ34 2 22 1 4 3 22 2 4 4 2

1 3n12 2 3n12 2 3n21 2 3n21 2 2 − cos φ34 a42 k12 + a + + a + − a k =0 a a 21 3 2 4 4 4 3 22 2 22 1

a1 a2 φ12 +

a1 a2 2

k22 − k11 −

(9.62) In this case there appear only two phase expressions, related to phase interactions between the LF heave and LF pitch modes, φ12 = β1 − β2 , and also between MF heave and MF pitch modes, φ34 = β3 − β4 . = φ = 0 in (9.62), and For steady state solutions, we set a1 = · · · = a4 = φ12 34 obtain a set of six algebraic equations, from which the equilibrium solutions can be computed. First we consider the case where (φ12 , φ34 ) = (mπ, nπ) and m, n ∈ Z, and substitute sin φ12 = sin φ34 ≡ 0, cos φ12 = cos φ34 ≡ ±1 into (9.62) to obtain a1 = a3 = 0, i.e., only trivial solutions for the heave mode. Then we obtain the following two relations satisfied by the corresponding non-trivial amplitudes:

3n12 2 3n12 2 2 a =0 (9.63) a + a2 k12 + 42 2 4 4

428


a42

3n12 2 3n12 2 a =0 k12 + a + 22 2 4 4

(9.64)

The solutions of (9.63) and (9.64) yield the following four cases: (i) a2 = a4 ≡ 0: trivial solutions (ii) a4 ≡ 0buta2 = 0: non-trivial LF pitch mode, a2 = 2 −k12 /(3n12 ) if k12 /n12 < 0 (iii) a2 ≡ 0buta4 = 0: non-trivial MF pitch mode, a4 = 2 −k12 /(3n12 ) if k12 /n12 < 0 (or

>

F)

(iv) a2 = 0anda4 = 0: non-trivial LF and MF pitch modes, a2 =

2 −k12/n12 , 3

a4 =

2 −k12 /n12 3

if k12 /n12 < 0

(9.65)

Similarly, for the following combinations of phase differences, (φ12 , φ34 ) = (mπ, (2n + 1)π/2), ((2m + 1)π/2, nπ), ((2m + 1)π/2, (2n + 1)π/2), where m, n ∈ N, we can also compute the same equilibrium solutions as in (9.67). Furthermore, we can evaluate their stability analytically (Lee et al., 2005b). Figure 9.37 depicts one specific set of steady state amplitudes corresponding to (φ12 , φ34 ) = (π/2, π/2); these steady state solutions for the two-frequency averaged system can be regarded as a degenerate subset of the three-frequency averaged system (9.49). Using the slow flow models developed in this section we are in the position to study the dynamical mechanism that ‘triggers’ aeroelastic instabilities (LCOs) in the two-DOF in-flow wing (9.37) or (9.42). We will prove that these instabilities are caused by series of transient and sustained resonance captures which ultimately direct energy from the flow to the pitch mode. The identification of the LCO triggering mechanism will be key to the passive LCO mitigation designs and strategies developed in later sections.

9.2.3 LCO Triggering Mechanism In this section, we show numerically and analytically that series resonance captures are responsible for the triggering and development of LCOs in the in-flow rigid wing. We start by presenting some motivating numerical results.

9.2.3.1 Numerical Results Considering the slow flow equations (9.49) with ‘optimal’ initial conditions (see Section 9.2.2.2), we examine first the dynamics of the system at subcritical reduced speeds. Figure 9.37a depicts the heave and pitch responses at a subcritical reduced


429

Fig. 9.37 Response of the slow flow (9.49) at a subcritical speed ( = 0.5): (a) normalized heave and pitch responses; (b) energy variations with respect to time; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).

430


Fig. 9.37 Response of the slow flow (9.49) at a subcritical speed ( = 0.5): (c) phase differences in time; (d) phase differences in their phase planes; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).


431

Fig. 9.37 Response of the slow flow (9.49) at a subcritical speed ( = 0.5): (e, f) phase interactions between different frequency components in time and phase plane; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).

432


Fig. 9.37 Response of the slow flow (9.49) at a subcritical speed ( = 0.5): (g) instantaneous frequencies; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).

speed ( = 0.5), normalized by their respective maximum amplitudes in order to compare their frequency contents. The existence of a 1:2 internal resonance between the heave and pitch modes (i.e., ωpitch ≈ 2ωheave ) was already mentioned in Section 9.2.2.1 from the wavelet transform analysis (Figure 9.27). In addition, the possible occurrence of 1:1 TRCs was also suggested. One may deduce the occurrence of these two types of resonance interactions directly from the plot of Figure 9.37a. Indeed, considering the dominant frequencies in each response up to τ ≈ 100, we approximately compute the frequency ratio of heave to pitch modes as 1:2; moreover, at later times the frequency ratio of these two modal responses becomes 1:1, and the modal responses become out-of-phase as shown in the zoomed plot. The existence of a 1:1 TRC in the transient responses can be verified by the phase analysis depicted in Figures 9.37c–f. We note the lack of phase interactions involving the HF components, which underlines the fact that no such resonance interactions involving these components occur at subcritical speeds (this is in accordance to the WT results of Figure 9.27 which indicate that there are no HF components in the modal responses). Note the wandering behavior of the phase difference φ56 in Figure 9.37c, and the time-like behaviors of the phase interactions of the MF components with HF components in Figures 9.37e–f. If we examine the phase interaction φ12 between the LF heave and LF pitch components, and the phase interaction φ34 between the MF heave and MF pitch components, we clearly establish their nontime-like behaviors (see Figure 9.37c); this is also revealed in the form of spirals


433

in the appropriate phase planes (see Figure 9.37d). We conclude that these phase differences cannot be regarded as ‘fast’ angles so they may not be averaged out of the dynamics, which indicates that the corresponding components are involved in internal resonance or resonance capture. The fact that the LF heave and LF pitch components are involved in 1:1 TRC (instead of 1:1 internal resonance) is indicated by the clear escape from the resonance capture regime, as evidenced from the plot of the phase difference φ12 at τ ≈ 150. The utilization of these phase interactions as evidence for internal resonances or resonance captures is confirmed by the instantaneous frequencies depicted in Figure 9.37g, computed by the following relations [recall the decompositions (9.45– 9.48) together with the polar coordinate transformations ϕk (τ ) = ak (τ )ejβk (τ ) ], ω1h (τ ) = = β1 (τ ),

ω2h (τ ) = 1 + β3 (τ ),

ω3h (τ ) = 3 + β5 (τ )

ω1α (τ ) = = β2 (τ ),

ω2α (τ ) = 1 + β4 (τ ),

ω3α (τ ) = 3 + β6 (τ ) (9.66)

where βk = tan−1 (Im ϕk /Re ϕk ), k = 1, . . . , 6, and their derivatives are regarded as slow frequency corrections to the fast dominant values (Zniber and Quinn, 2003). For the 1:2 internal resonance between the heave and pitch modes, the frequency ω2α (τ ) is compared to 2ω1h (τ ). Then, we check that the frequency relation, ω2α − 2ω1h ≈ 0, persists in the entire time interval, which clearly implies the occurrence of 1:2 internal resonance throughout the transient response. We remark that this internal resonance is possible only because of our choice of the specific ratio between natural frequencies, i.e., = 0.5. On the other hand, comparing ω1h (τ ) and ω1α (τ ), and ω2h (τ ) and ω2α (τ ), we clearly verify that 1:1 TRC and escape from resonance capture between likewise LF components occur in the response. Then, it is natural to expect the occurrence of energy exchanges between the heave and pitch modes, similar to those occurring between modes in internal resonance in systems with dissipation [see Figure 9.37b, (Greenlee and Snow, 1975)]. Next, we apply similar arguments to explore the triggering mechanism that gives rise to LCOs at supercritical flow speeds. Basically, the LCO triggering mechanism is composed of three main stages as discussed in the wavelet transform (WT) analysis of Section 9.2.2.1. This classification is established by studying the corresponding energy exchanges between the heave and pitch modes (see Figure 9.38b), the phase interactions between dominant frequency components of these modes (see Figure 9.38c-f), and the corresponding instantaneous modal frequencies (see Figure 9.38g). We refer to these three stages as Stages I, II, and III, with main corresponding features, 1:1 TRC, escape from 1:1 TRC, and finally 3:1 SRC, respectively. Each of these regimes occurring for a supercritical reduced speed = 0.95 is considered in detail below. Starting from Stage I, initial transients (up to τ ≈ 20) involve a 1:2 internal resonance (see Figure 9.38g) which may initially cause strong energy exchanges between the heave and pitch modes (as shown in Figure 9.38b). Then, a 1:1 TRC is realized in the dynamics. Indeed, comparing the time responses of Figure 9.38a, we find that the amplitudes of both modes (and thus their respective energies) increase

434


Fig. 9.38 Response of the slow flow (9.48) at a supercritical speed ( = 0.95): (a) normalized heave and pitch responses; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).


435

Fig. 9.38 Response of the slow flow (9.48) at a supercritical speed ( = 0.95): (b) energy variations with respect to time; (c) phase differences in time; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).

436


Fig. 9.38 Response of the slow flow (9.48) at a supercritical speed ( = 0.95): (d) phase differences in their phase planes; (e) phase interactions between different frequency components in time and phase plane; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).


437

Fig. 9.38 Response of the slow flow (9.48) at a supercritical speed ( = 0.95): (f) phase interactions between different frequency components in time and phase plane; (g) instantaneous frequencies; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).

438


during this initial 1:1 TRC, until the energy of the heave mode reaches its maximum at the end of the Stage I. In Chapter 3 we found that in-phase 1:1 TRC is the underlying mechanism for fundamental TET from a SDOF LO to an essentially nonlinear SDOF attachment (the NES). In analogy, in the case of the in-flow rigid wing, the pitch mode can be regarded as the primary system, and the heave mode as the NES to which vibration energy is irreversibly transferred during Stage I. Our aeroelastic model possesses a positive damping component in the heave mode and a negative damping component in the pitch mode. Thus, unlike the usual nonlinear TET phenomenon, energies in both heave and pitch modes increase during 1:1 TRCs as energy from the flow feeds directly to both modes. Besides, energy dissipation by the heave mode (as for the NES) plays no significant role in the competition between the two damping mechanisms. This also explains the viewpoint that the initial excitation of the heave mode acts or triggers initiation and development of the pitch mode. It is remarkable that the nonlinear beating phenomenon which is caused by 1:2 internal resonance between heave and pitch was reported in Section 3.4.2 as the most efficient mechanism to transfer or initiate TET (corresponding to TET through nonlinear beats – see Section 3.4.2.3). Hence, it seems that energy exchange between heave and pitch during Stage I occurs in the most efficient way from a TET point of view. That is why the initial short occurrence of 1:2 internal resonance makes possible maximum energy transfer. The occurrence of 1:1 TRCs are verified by the non-time-like behaviors of the phase differences between likewise frequency components (see Figure 9.38c), and by spirals formed in the corresponding projections of the phase space (see Figure 9.38d). In particular, the instantaneous frequencies shown in Figure 9.38g indicate the occurrence of 1:1 TRCs, since the frequencies of likewise frequency components lie, on average, very close to each other, following straight lines, whereas in later times some of the components show frequency shifting (Zniber and Quinn, 2003) with increasing energy. We now consider Stage II of the transient dynamics. Once the heave mode reaches its maximum amplitude, escapes from 1:1 TRCs occur. Superimposed time responses show the corresponding transitions in the dynamics (refer to Zoom A in Figure 9.38a); the in-phase 1:1 TRCs of Stage I turn to 3:1 TRCs gradually as the heave amplitude decreases and pitch amplitude increases. Thus the energy exchanges between the two modes (see Figure 9.38b) follow the typical behavior of escape from resonance capture (Kerschen et al., 2005, 2006a); in particular, timelike behaviors of likewise phase differences (with φ56 being the most prominent – see Figure 9.38c), and escapes from spirals in projected phase planes (see Figure 9.38d) confirm escape from the in-phase 1:1 TRCs of Stage I. In the meanwhile, the phase differences between MF heave and HF heave and between MF pitch and HF heave do not exhibit time-like behaviors anymore, which are precursors for the occurrence of 3:1 SRCs (Figure 9.38e). These observations are confirmed also in terms of the plots of instantaneous frequencies depicted in Figure 9.38g. Finally, we discuss Stage III, during which the LCO fully develops and the steady state aeroelastic instability is reached. As a result of the escapes from TRCs in Stage II, the steady state dynamics finally settles into a series of 3:1 SRCs. Examining the


439

zoomed-in plots of the responses depicted in Figure 9.38a, we can establish the frequency relation between the two modes as nearly 3:1, and the occurrence of inphase modal oscillations. The energy exchanges between heave and pitch modes become balanced, on average, with most energy being imparted to the pitch mode (see Figure 9.38b). The occurrence of SRCs can be verified in a way similar to Stages I and II, i.e., by the occurrence of non-time-like phase behaviors and of spirals in the corresponding phase planes, or, more directly, in terms of plots of instantaneous frequencies. In particular, comparing the instantaneous frequencies 3ω2α (τ ) and ω3h (τ ) in this stage, we can find a good alignment between them on average, i.e., 3ω2α − ω3h ≈ 0 (see Figure 9.38g). An interesting note concerns the fact that the HF heave (or the steady state resonance frequency component) undergoes upward frequency shift so that 3:1 SRC to MF pitch is made possible, as energy is continuously fed from the flow into the system. It is also remarkable that the likewise phase differences in Figures 9.38c, d in this stage imply the existence of 1:1 SRCs.

9.2.3.2 Analytical Proof of the LCO Triggering Mechanism In the previous section we analyzed numerical simulations to study the LCO triggering mechanism. The numerical findings can be confirmed analytically employing partial averaging, which is a local analysis. Specifically, we will perform averaging only for non-resonant (fast) phase angles possessing time-like behavior in order to remove the non-essential dynamics and derive a reduced-order slow flow model (Zniber and Quinn, 2003). In an effort to confirm that an internal resonance and a series of resonance captures are responsible for the LCO triggering mechanism, we study the resonance captures that occur in the slow flow at each stage of the response. Then, the order of approximation and its validity on the corresponding time scales can be verified when escapes from resonance captures occur. For example, in order to prove the existence of SRCs in Stage III of the LCO triggering mechanism, the existence of steady state equilibrium points of the slow flow model will serve as the necessary (but not sufficient) condition (Quinn, 1997; Zniber and Quinn, 2003). Since the averaged system possesses sensitive dependence on initial conditions (due to the fact that the problem of assigning initial conditions to the slow flow is indeterminate – see discussion in Section 9.2.2.2), ‘optimal’ initial conditions for the slow flow modulation equations in polar form (9.60) will be utilized in the following analysis. As a result, we may deduce different phase behaviors and different steady states (see Figures 9.39c,d) than the ones in the corresponding plots of Figure 9.38 which were computed using the ‘optimal’ initial conditions for the complex-valued modulation equations (9.49). Figures 9.39a, b depict envelopes of the responses of the heave and pitch components, respectively. We may expect that the dominant contribution to the initial triggering of LCOs comes from the MF heave component, so that it develops the likewise counterpart, MF pitch in Stage I. Then, from Stage II until the dynamics

440


Fig. 9.39 Amplitude and phase responses computed with the averaged slow flow (9.60) for a supercritical reduced velocity ( = 0.95); initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).



441

442





443

444


reaches its steady state, MF pitch together with MF heave act as the driving mechanism to further raise the amplitude of the HF heave response. This intuition is visualized in the phase responses. The initial triggering of the LCO corresponds to non-time-like behaviors of the likewise phase differences (and thus by the existence of 1:1 TRCs) in Stage I (see Figure 9.39c). On the other hand, time-like responses of the other phase differences imply that no other effective triggering mechanisms exist during this initial stage. In Stage II, however, escape from 1:1 TRCs of likewise HF components occurs while the other likewise frequency components remain still ‘locked’ in the regime of 1:1 TRCs. Non-time-like behaviors of the phase interactions between MF heave and HF heave (corresponding to the phase difference φ35 ) and between MF pitch and HF heave (phase difference φ45 ) support the previous argument. At steady state when the fully developed LCO is realized during Stage III, only the phase difference between LF heave and LF pitch indicates capture of these components into 1:1 SRC, while the other phase differences escape from 1:1 TRCs. Indeed only the phase interaction between MF pitch and HF heave exhibits non-time-like behavior, which coincides with the numerical simulations of the previous sections [where the original dynamical system (9.37) or (9.42) was used, before averaging is applied]. In Figures 9.39f, h the instantaneous frequency of each harmonic component is plotted by means of relations (9.66); however, this time the slowly varying frequency corrections βk , k = 1, . . . , 6, are directly computed from the slow flow equations (9.60). Frequency locking between the LF components persists almost during the entire time interval, and, in addition, the steady state resonance frequencies show upward frequency shift along with increasing energy input from the flow. The frequencies of the MF components are kept locked in 1:1 resonance capture, on average, and become unlocked only after Stage II, whereas those of the HF components are unlocked just at the end of Stage I. In particular, the frequency shift in HF heave engages into another locking at three times the frequency of the MF pitch, as we already observed in the previous section. Examining the slow dynamics in each stage separately, in Stage I we perform partial averaging over all time-like phase variables except the phases ψI = (φ12 , φ34 , φ56 ), and construct the following reduced-order model describing 1:1 TRCs between surviving components in the form: a = fI (a; ψI ),

ψI = gI (a; ψI )

(9.67)

where a ∈ R 6 and ψI ∈ T 3 = S 1 × S 1 × S 1 . It is sufficient to show that there occur escapes from 1:1 TRCs after some time interval of frequency locking. When solving the reduced-order model (9.67), we use ‘optimal’ initial conditions which are the same to those used for the slow flow equations (9.60). This leads to the phase interactions depicted in Figures 9.40a, b where τA and τB refer to the approximate time instants when the dynamics is captured into, and escape from resonance capture, respectively. Hence, we analytically verify that there exist 1:1 TRCs during Stage I. When transition to escape occurs at time instant τB , the partially-averaged system (9.67) loses validity since the assumptions upon which it is derived are no


445

Fig. 9.40 Analytical study of 1:1 TRCs at Stage I for = 0.95: (a) phase interactions; (b) frequency shiftings; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).

longer valid. This becomes clear when one compares the exact and averaged HF heave responses in Figure 9.40d.

446


Fig. 9.40 Analytical study of 1:1 TRCs at Stage I for = 0.95: (c) heave amplitude components; (d) pitch amplitude components; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).


447

For Stage II we perform a similar analysis as in Stage I. Taking into account the resonance interactions between heave and pitch components in this stage of the dynamics, we define the non-time-like variables of the problem as ψI I = (φ12 , φ34 , φ35 , φ45 , φ345 ) ∈ T 5 and construct a slow flow model in the form: a = fI I (a; ψI I ),

ψI I = gI I (a; ψI I )

(9.68)

We note that in this case there exist only three phases as independent variables. As for the initial conditions for (9.68), we use the state of the dynamics at instant τB when the system enters Stage II, starting from the ‘optimal’ initial conditions for (9.60). From the results depicted in Figure 9.41 it is evident that the dynamics is settling down into a steady state motion at time instant τC . In particular, the (resonant) frequencies of the HF components shift upward to their original values close to a value equal to three, so that the system can sustain 3:1 resonance captures. Note that the high-frequency modulations disappear through the previous partial averaging. Finally, in Stage III we derive a slow flow model with non-time-like phase variables ψI I I = (φ12 , φ45 , φ346 ), yielding: a = fI I I (a; ψI I I ),

ψI I I = gI I I (a; ψI I I )

(9.69)

When solving this reduced-order model, we perform a similar analysis with Stage II, and find that there exist equilibrium points in the slow flow, which is proof of existence of SRCs. Recall that in this case the time derivatives of the phase variables, βk , k = 1, . . . , 6 in (9.27) act as effective detuning frequency components. Therefore, instead of finding equilibrium conditions for detuning parameters as in Zniber , i = 1, 3, 5 and Quinn (2003), we focus on finding such conditions that βi ≈ βi+1 for 1:1 resonance captures, and 3β4 ≈ β5 for 3:1 resonance captures. In this way, we may define the intervals where frequency lockings occur as evidence for resonance captures. In Figure 9.42 we depict the steady state response of the slow flow (9.69) that prove the occurrence of 3:1 SRCs, or equivalently the development of the LCO instability. Since it is not feasible to explicitly compute the equilibrium points of the slow flow (9.69), we may instead verify implicitly their existence, for example, by examining the plots of Figures 9.42a–d we regard to convergence of the slow dynamics at steady state values as proof of existence of the corresponding equilibrium points. Besides, we have already computed some of these steady state solutions by way of numerical continuation in Section 9.2.2.3. Hence, we may conclude that in Stage III 3:1 resonance captures are sustained permanently in time, i.e., that 3:1 SRCs are realized in the dynamics.

9.2.4 Concluding Remarks We investigated the LCO triggering mechanisms in a two-DOF in-flow rigid wing model in subsonic flow by employing quasi-steady aerodynamics. Reviewing fun-

448


Fig. 9.41 Analytical study of the transient dynamics at Stage II for tions; (b) frequency shiftings.

= 0.95: (a) phase interac-


Fig. 9.41 Analytical study of the transient dynamics at Stage II for components; (d) pitch amplitude components.

449

= 0.95: (c) heave amplitude

450


Fig. 9.42 Analytical study of 3:1 SRCs at Stage III for = 0.95: (a) phase interactions; (b) frequency shiftings; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).


451

Fig. 9.42 Analytical study of 3:1 SRCs at Stage III for = 0.95: (c) heave amplitude components; (d) pitch amplitude components; initial conditions correspond to (y(0), α(0), y (0), α (0)) = (10−3 , 10−3 , 0, 0).

452


damental aspects of linear flutter theory, we established a slow flow system based on multi-phase averaging which exhibits good agreement with the original dynamics. Through the method of numerical continuation we analyzed the steady state dynamics of the slow flow. The slow flow system showed sensitive dependence on initial conditions due to the resulting complexity of the dynamics in phase space and the indeterminacy of the problem of computing initial conditions from the exact system. Hence, the initial conditions for the slow flow were determined through an optimization process. It is interesting to note that even at subcritical speeds 1:1 transient resonance captures occur between heave and pitch harmonic components, which are responsible for strong energy exchanges between aeroelastic modes. We also found that the LCO triggering mechanism is composed of a series of dynamical phenomena, starting with a 1:1 transient resonance capture, followed by escape from resonance capture, and then a final 3:1 sustained resonance capture between the heave and pitch modes. After exploring the triggering mechanism numerically at each of the aforementioned stages of the dynamics, we proved our claims analytically by means of partial averaging and by studying the resulting slow flow models. The identification of the dynamical mechanism generating LCOs in system (9.37) or (9.42) provides the necessary framework for developing strategies and techniques for their passive suppression. Indeed, based on a purely phenomenological view of the problem of LCO formation, one might arrive at the conclusion that the problem of LCO suppression can be formulated as a problem of suppressing a steady state oscillation. Instead, as shown in this section, the problem of LCO triggering and development is a problem involving series of transient and sustained resonance captures between heave and pitch harmonic components, i.e., it is a problem of transient modal interactions. The premise adopted in this work, therefore, is that by eliminating the triggering mechanism for LCOs, one would be to eliminate the aeroelastic instability at the early stage of its development, before it reaches steady state. Hence, the problem of LCO suppression is converted to the problem of suppressing the transient triggering mechanism, i.e., to a problem formulated in the transient domain. Definitely, dealing with a strongly nonlinear problem in the transient (instead of steady state) domain is a challenging undertaking, yet the TET-based designs based on transient directed energy transfers developed in this work offer a solution with the potential to address the present problem. This is due to the fact that, as shown in previous chapters, essentially nonlinear attachments of the type considered in this work are capable of passively absorbing and locally dissipating broadband vibration energy from primary structures in a one way irreversible fashion; this renders such attachments suitable candidates for suppressing aeroleastic instabilities of the inflow rigid wing. This issue is addressed in the following sections.


453

9.3 Suppressing Aeroelastic Instability of an In-Flow Wing Using a SDOF NES The triggering mechanism of limit cycle oscillation (LCO) of a wing due to aeroelastic instability was studied in the previous section, where it was shown that a cascade of resonance captures constitutes the LCO triggering mechanism in the form of (i) initial attraction into 1:1 TRC, (ii) escape from this capture and finally, and (iii) entrapment of the dynamics in 3:1 SRC leading to full development of the LCO. Alternatively, an initial excitation by the flow of the heave mode acts as the triggering mechanism for the eventual activation of the pitch mode through transient nonlinear modal interactions involving the aforementioned resonance captures and escapes; the eventual excitation of the pitch mode signifies the appearance of an LCO of the in-flow wing in flow. In general, efforts have been made to control LCOs by means of active control schemes (Ko et al., 1997; Friedman et al., 1997) or by inducing autoparametric resonances (Fatimah and Verhulst, 2003; Tondl et al., 2000). In this section we study the efficacy of passively suppressing aeroelastic instabilities (LCOs) in the two-DOF in-flow rigid wing model (9.37) or (9.42) by attaching a SDOF NES. In the process we will provide an application of TET to an engineering problem of practical significance. First, we perform computational parametric studies that clearly demonstrate (at least) three fundamental mechanisms of LCO suppression by means of TET from the in-flow wing to the NES. Then, we further investigate the numerically detected LCO suppression mechanisms by performing numerically time-frequency analysis by means of WTs and EMD, and applying the CX-A technique performing multiphase averaging. By computing appropriately defined measures of energy dissipated by the NES, we explore the energetic transactions associated with each LCO suppression mechanism in terms of transient modal interactions. Furthermore, we prove that LCO suppression is due to TRCs between heave and pitch harmonic components and the NES, during which strong TET from the wing to the NES take place. Then, we address the issue of robustness of LCO suppression by performing bifurcation analysis in order to detect stable co-existing stable attractors in the steady state dynamics which can have either a beneficial or an inadverse effect regarding LCO suppression. In the process we describe how the three LCO suppression mechanisms are related to the bifurcation picture of the steady state dynamics. The following exposition follows the work reported in Lee et al. (2007a).

9.3.1 Preliminary Numerical Study We examine the two-DOF rigid in-flow wing model of Figure 9.23 with an attached SDOF NES (see Figure 9.43). Assuming small motions and using the principle of virtual work (Dowell et al., 1995), we derive the equations of motion of the wing-

454


NES assembly as follows: ˙ ) mh¨ + Sα α¨ + Kh (h + c1 h3 ) + qSCL,α (α + h/U + cs (h˙ − d α˙ − z˙ ) + ks (h − dα − z)3 = 0 ˙ ) Iα α¨ + Sα h¨ + Kα (α + c2 α 3 ) − qeSCL,α (α + h/U ˙ + dks (dα + z − h)3 = 0 + dcs (d α˙ + z˙ − h) ˙ + ks (z + dα − h)3 = 0 ms z¨ + cs (˙z + d α˙ − h)

(9.70)

or in non-dimensional form y + xα α + 2 y + ξy y 3 + µCL,α (y +

α)

+ ελ(y − δα − v ) + C(y − δα − v)3 = 0 rα2 α + xα y + rα2 α + ξα α 3 − γ µCL,α (y +

α)

+ δελ(δα + v − y ) + δC(δα + v − y)3 = 0 εv + ελ(v + δα − y ) + C(v + δα − y)3 = 0

(9.71)

where the notation of the previous section applies for (9.73), and the normalized parameters in (9.74) are defined as follows: , y = h/b, v = z/b, C = b2ks /mωa2 , δ = d/b, γ = e/b, ra = Ia /(mb2), xa = xcg /b, τ = ωa t, ε = ms /m, λ = cs /ms ωa , ξy = c1 b2 2 , ξa = c2 ra2 , = ωh /ωa , ωh = Kh /m, ωa = Ka /Ia The quantities ωh and ωa are the linearized frequencies of the heave and pitch modes, respectively, primes denote differentiation with respect to the normalized time τ , µ = ρ∞ bS/2m is the density ratio and = U/(bωa ) the reduced fluid velocity. Note that through the offset d (or δ) from the elastic axis, the NES (with corresponding response z or v) interacts with both the heave (response h or y) and pitch (angle α) modes. We now perform computational parametric studies of the dynamics of system (9.75) to identify parameter subsets where LCOs of the wing can be suppressed or even completely eliminated. Initial conditions close to the trivial equilibrium position are considered; i.e., we set all initial conditions equal to zero except for the initial velocity y (0) = 0.01. Regarding the wing parameters, we take these to be identical to the ones utilized in Section 9.2 where the LCO triggering mechanism was studied, xα = 0.2,

rα = 0.5,

µ = (10π)−1 ,

y = 0.4,

CL,α = 2π,

= 0.5,

ξy = ξα = 1

(9.72)


455

Fig. 9.43 Two-DOF rigid wing model with an attached SDOF NES.

which gives a flutter speed equal to F = 0.87 (see Figure 9.24). There are four control parameters for the NES, namely, the mass ratio of the NES and the wing, ε; the normalized damping coefficient, λ; the normalized coefficient of the essentially-nonlinear stiffness, C; and the normalized offset attachment, δ. In the numerical study, we will consider parameter variations in the ranges 0.01 ≤ ε ≤ 0.1 (i.e., for practical reasons we consider as small as possible mass ratios), 0.1 ≤ λ ≤ 1, 1 ≤ C ≤ 20, and −1 ≤ δ ≤ 1. Our methodology for performing the computational parametric study is as follows. Using the aforementioned initial conditions and parameter sets, we integrate the equations of motion (9.75) for sufficiently long time to assure that initial transients die out. Then we compute the root-mean-square (r.m.s.) amplitude of the resulting steady state response. Comparing the steady state pitch (or heave) amplitudes in r.m.s. with and without NES attached, we may infer partial or complete LCO suppression. Partial LCO suppression can be inferred by computing the amplitude ratio (in the pitch mode) for systems with and without NES attached, which should be less than unity. Complete LCO suppression is inferred when this amplitude ratio tends to zero as the steady state of the system is approached. More specifically, we introduce the following definition for the amplitude ratio in the pitch mode,

α =

Steady state pitch amplitude (rms) with NES × 100% Steady state pitch amplitude (rms) without NES

and a similar expression for the heave mode y .

(9.73)

456


(a)

= 0.9, ε = 0.01

(b)

= 0.9, ε = 0.02

Fig. 9.44 Steady state amplitude ratio in the pitch mode, α for various mass ratios and reduced speeds; the areas enclosed by the thick curves indicate parameter domains where α < 60%; the r.m.s. steady state pitch amplitude for the system with no NES is 0.11 rad for = 0.9, and 0.17 rad for = 0.95; initial conditions are all zero except for y (0) = 0.01.

Figure 9.44 depicts the steady state amplitude ratio in the pitch mode a for mass ratios equal to ε = 0.01 and 0.02, and reduced speeds = 0.9 and 0.95. Here we only consider the reduction in the amplitude of the pitch mode because this is the dominant mode during the aeroelastic instability (actually, the overall amplitude


(c)

= 0.95, ε = 0.01

(d)

= 0.95, ε = 0.02

Fig. 9.44 Continued

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Fig. 9.45 The first LCO suppression mechanism for = 0.9, δ = 90%, ε = 1%, λ = 0.1, and C = 10; zero initial conditions are used except for y (0) = 0.01.

of the NES is similar to that of the pitch mode). From these numerical computations we conclude that LCO suppression is more probable when the NES is attached far from the elastic axis of the wing; a possible reason for this is that, for relatively large offsets, the NES interacts efficiently with both heave and pitch modes, which permits nonlinear energy exchanges between both modes and the NES. Moreover, it appears that by attaching the NES aft of the elastic axis (i.e., for δ < 0) more effective LCO suppression is realized. This can be inferred heuristically by examining the system (9.71); indeed energy dissipation by the NES due to the damping term ελ(y − δα − v ) is maximized if the NES interacts with both heave and pitch modes under condition of 1:1 resonance capture, and δ is negative. This will be revisited in a later section where bifurcation analysis of the steady state dynamics will be performed. In addition, we note that if the aim is to achieve good suppression results for increased flow speeds, we might need to consider higher mass ratios of the NES; for example, most of the regions where instability is suppressed by about 40% for ε = 0.01 and = 0.9 (the area enclosed by thick curves in Figure 9.44a) disappear when the same mass ratio is used but the reduced speed is increased to = 0.95 (see Figure 9.44c). Finally, we remark that this computational parametric study provides simple comparison of r.m.s. steady state amplitude reduction (more precisely, of average power reduction) under a specific set of initial conditions. Hence, the results depicted in Figure 9.44 can be only indicative of the capacity of NES for


459

Fig. 9.46 The second LCO suppression mechanism for = 0.9, δ = 90%, ε = 1%, λ = 0.2, and C = 20; zero initial conditions are used except for y (0) = 0.01.

LCO suppression, and in order to draw conclusions of a more general nature a more detailed study should be performed addressing the issue of possible co-existing attractors of the steady state dynamics; this is performed in a later section. From this preliminary computational parametric study, we deduce the existence of three dynamical mechanisms for LCO suppression, as depicted in Figures 9.45– 9.47; moreover, Figure 9.48 presents the case where the NES fails to suppress the aeroelastic instability. By studying these preliminary numerical results, some general conclusions can be drawn regarding the nonlinear dynamics of LCO suppression by the NES. In the first suppression mechanism (Figure 9.45) the action of the NES yields repeated burst-outs and eliminations of aeroelastic instabilities, eventually leading to complete LCO suppression; the second mechanism (Figure 9.46) results again in partial LCO suppression, but with no instability burst-outs; whereas, the third mechanism (Figure 9.47) results in complete LCO suppression. Apart from the phenomenological differences in the action of the NES in the three mentioned LCO suppression mechanisms, there appear to be some common features in the dynamics. First, the nonlinear dynamics of the NES-wing interaction involves broadband energy exchanges in the transient (as opposed to steady state) domain; second, it appears that the essential stiffness nonlinearity of the NES initially prevents the aeroelastic instability from growing above a certain amplitude; at a later phase of the response, conditions for resonance capture between the aeroelas-

460


Fig. 9.47 The third LCO suppression mechanism for = 0.9, δ = 90%, ε = 1%, λ = 0.4, and C = 40; zero initial conditions are used except for y (0) = 0.01.

tic modes of the wing and the NES are fulfilled and passive targeted energy transfers (TETs) from the wing to the NES take place leading to further LCO suppression. Below we discuss separately the aforementioned three LCO suppression mechanisms, by post-processing the corresponding numerical time series, focusing only on the main dynamical features, and leaving the discussion of further details to a following section.

9.3.1.1 The First LCO Suppression Mechanism This mechanism is characterized by a recurrent series of suppressed burst-outs of the heave and pitch modes of the wing, leading eventually to complete suppression of the aeroelastic instabilities. In the initial phase of transient burst-outs, a series of developing instabilities of predominantly the heave mode is effectively suppressed by proper transient ‘activation’ of the NES, which passively tunes itself1 to the fast frequency of the developing aeroelastic instability; as a result, the NES engages 1 Because an NES possesses no preferential resonance frequency (due to its essential stiffness nonlinearity), the NES can resonantly interact with any mode of the primary system to which it is attached. By locally dissipating the absorbed energy by means of its viscous damper, it can exhibit an escape from one resonance capture to another, engaging in resonance capture cascades (see Section 3.5 and Kerschen et al., 2006a).


461

Fig. 9.48 Case of no LCO suppression for = 0.9, δ = 90%, ε = 2%, λ = 0.4, and C = 40; zero initial conditions are used except for y (0) = 0.01.

in 1:1 TRC with the heave mode, passively absorbing broadband energy from the wing, thus eliminating the burst-out. In the latter phase of the dynamics, the energy fed by the flow does not appear to directly excite the heave and pitch modes of the wing, but, instead, it seems to get transferred directly to the NES until the wing is entirely at rest and complete LCO suppression is achieved. At the initial stage of the recurrent burst-outs, at time instants when the pitching LCO is nearly eliminated, most of the energy induced by the flow to the wing is absorbed directly by the NES with only a small amount of energy being transferred to the heave mode, so that both the NES and the heave mode reach their maximum amplitude modulations. This is followed by suppression of the burst-out, and this process is repeated until at a later stage of the dynamics complete suppression of the aeroelastic instability is reached. It will be shown in a later section that the beating-like (quasi-periodic) modal interactions observed during the recurrent burstouts turn out to be associated with Neimark–Sacker bifurcations (Kuznetsov, 1995) of a periodic solution of system (9.71), and to be critical for determining domains of robust LCO suppression.

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9.3.1.2 The Second LCO Suppression Mechanism This mechanism is characterized by intermediate or partial suppression of LCOs. The initial action of the NES is the same as in the first suppression mechanism. Targeted energy transfer to the NES then follows under conditions of 1:1 TRC, followed by conditions of 1:1 SRC where both heave and pitch modes attain constant (but non-zero) steady state amplitudes. We note that the heave mode response can grow larger than in the corresponding system with no NES attached (exhibiting an LCO), at the expense of suppressing the pitch mode. We also note that, in contrast to the first suppression mechanism, the action of the NES is non-recurring in this case, as the NES acts only during the early phase of the motion stabilizing the wing and suppressing the developing LCO.

9.3.1.3 The Third LCO Suppression Mechanism This mechanism is the most effective for suppressing the aeroelastic instabilities, as it results in complete and permanent elimination of LCOs. The dynamics of the wing-NES interaction resembles the second suppression mechanism (at least in a phenomenological way), leading, however, to asymptotic suppression of the LCO and possessing radically different underlying dynamics (as discussed later). Again, transient TET from the aeroelastic modes of the wing to the NES is caused by nonlinear modal interactions during 1:1 TRCs. Both heave and pitch modes, as well as the NES, exhibit exponentially decaying responses resulting in attraction of the dynamics to a trivial attractor. In general, for increasing reduced speeds larger NES masses are required for the realization of this suppression mechanism and complete elimination of LCOs.

9.3.1.4 Case of No LCO Suppression The action of the NES is not always beneficial to the objective of LCO suppression. Indeed, for certain combinations of system parameters, initial conditions and reduced flow speeds the transient dynamics of system (9.71) may be attracted to nontrivial attractors that enhance the aeroelastic instability. This underlines the need for a robustness study of the proposed NES designs. It follows that the NES might have an inadverse effect on the dynamics when it does not act as an efficient energy absorber of the oscillating wing undergoing aeroelastic excitation. In some cases, as the representative result of Figure 9.48 shows, the steady state amplitudes of LCOs may grow even larger compared to the corresponding system with no NES attached. Depending on the particulars of the cases examined (system parameters, initial conditions, flow speeds) the steady state dynamics of system (9.71) may result in superharmonic frequency relations between aeroelastic modes. Then, similarly to the behavior observed in the LCO triggering mechanism in the in-flow wing with no NES attached (see Section 9.2),


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the dynamics might undergo a transition from 1:1 to 3:1 locking of frequency ratios between the heave and pitch modes; this implies the possibility of occurrence of an initial 1:1 TRC followed by a transition to a 3:1 SRC between the heave and pitch modes of the wing, and the generation of aeroelastic instability. This observation suggests that in order to effectively suppress the aeroelastic instabilities, the NES must interact with both heave and pitch modes in a way as to prevent direct energy transfer from the flow to the wing modes through subharmonic and superharmonic resonance captures (similar conclusions were drawn in Section 9.1 where LCO suppression in the van der Pol oscillator was studied).

9.3.2 Study of LCO Suppression Mechanisms In this section, we investigate numerically and analytically the three LCO suppression mechanisms mentioned previously. First, we numerically post-process the transient responses of the wing-NES system (9.71) in order to determine the dominant harmonic components and the underlying nonlinear resonant interactions that give rise to TET and result in aeroelastic instability suppression. The post-processing techniques that are employed include numerical wavelet transforms (WTs) and empirical mode decompositions (EMDs) combined with Hilbert transforms (see Section 2.5). Based on these numerical post-processing results, in the following section we analytically study the corresponding nonlinear modal interactions by performing fast-slow partitions of the transient dynamics employing complexification and multi-frequency averaging; the resulting reduced-order slow flow model fully capture the wing-NES nonlinear interactions, and provide a full understanding and modeling of the three instability suppression mechanisms. Through this systematic plan of study we aim to formulate a new paradigm for passive TET-based LCO mitigation.

9.3.2.1 Numerical Study We first explore the nonlinear dynamics and energetic interactions governing each suppression mechanism by numerically computing the instantaneous energy exchanges between the aeroelastic modes and the attached NES of the self-excited system (9.71). To this end, we need to define certain energy measures. The instantaneous total energy of the wing-NES system can be expressed as a sum of the instantaneous kinetic and potential energies of the wing and the NES as follows:

1 2 1 2 1 2 2 Total E y (τ ) + rα α (τ ) + xα y (τ )α (τ ) + εv (τ ) (τ ) = 2 2 2

1 2 2 1 1 1 2 2 4 4 y(τ ) + rα (τ ) + ξy y(τ ) + ξα α(τ ) + 2 2 4 4

464


1 C (y(τ ) − δα(τ ) − v(τ ))4 + 4

(9.74)

The energy dissipated by the viscous damper of the NES is computed by the following expression: τ 2 EdNES (τ ) = ελ (9.75) v (s) + δα (s) − y (s) ds 0

whereas the energy input to the system is the sum of the initial energy provided by the initial conditions and the non-conservative work performed by the flow Input

Ed where y

y

α (τ ) = E Total (0) + Wnc (τ ) + Wnc (τ )

Wnc (τ ) = µCL,α 0 α (τ ) = −γ µCL,α Wnc

τ

y (s) +

τ

(9.76)

α(s) y (s)ds

y (s) +

α(s) α (s)ds

(9.77)

0

As a result, at any time instant the following instantaneous energy balance should hold: E Total (τ ) = E Input (τ ) − EdNES (τ ) (9.78) Figure 9.49 depicts the instantaneous energy exchanges between the two aeroelastic modes (pitch and heave) and the NES (upper parts), and the relation between the energy input fed from the flow to the system in time and the corresponding energy dissipation by the NES (lower part), for each of the three LCO suppression mechanisms. For comparison, the case where the LCO survives the action of the NES (case of no suppression) is also provided. Note that the depicted partition of total energy into each wing mode assumes that the contribution to the potential energy of the essentially nonlinear coupling is assigned entirely to the NES part. The instantaneous energy exchanges depicted in Figure 9.49 demonstrate the transient and broadband nature of the nonlinear modal interaction between the inflow wing and the NES. A study of these plots indicates that the first suppression mechanism exhibits the most vivid transient energy interactions, especially between the pitch mode and the NES (see Figure 9.49a). The form of these modal interactions is quite similar to the corresponding energy exchanges studied in Section 3.4, where the targeted energy transfer (TET) mechanisms were investigated for a two-DOF system with essential stiffness nonlinearity (specifically, TET initiated by nonlinear beat phenomena-resonance captures will be discussed later). In a later section, we will interpret this nonlinear beating behavior in terms of the study of steady state bifurcations, whereby it will be shown that the first LCO suppression mechanism is due to a Neimark–Sacker bifurcation of a stable LCO (which is analogous to a Hopf bifurcation in a codimension-one bifurcation prob-


465

Fig. 9.49 Instantaneous energy exchanges (upper), and comparison of input energy to the system fed from flow in time and corresponding energy dissipation by the NES (lower), for the three LCO suppression mechanisms (a–c) and the case of no suppression (d); the simulations correspond to those depicted in Figures 9.45–9.48.

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lem; see Guckenheimer and Holmes, 1983), yielding quasi-periodic solutions by adding an additional dominant harmonic component in the steady state response. Hence, variation of the total energy shows repeated burst-outs followed by suppressions; eventually, the energy input from the flow to the wing E Input continuously increases nearly at the same rate with the energy dissipated by the NES, EdNES . At the time instant where a complete balance between E Input and EdNES occurs, the total energy balance becomes zero; however, small disturbances from that totally balanced energetic state lead to recurring excitations of aeroelastic instabilities, and, as a result, the alternating series of suppressions and instability burst-outs is continuously repeated. Note that, although the aeroelastic instabilities cannot be completely removed by this suppression mechanism, their corresponding amplitudes are greatly reduced compared to those developed when the NES is absent (see Figure 9.45). The second suppression mechanism involves initial strong modal interactions so that a balance between energy input and dissipation of energy by the NES is quickly reached at the initial early stage of the motion (see Figure 9.49b). This vigorous initial energy exchange behavior resembles the mechanism of fundamental TET mechanism (due to in-phase 1:1 resonance capture) discussed in Section 3.4.2.1. Again, in this case small disturbances can lead to the reappearance of instabilities but with much smaller amplitudes (see Figure 9.46). The (increasing) rates of energy input and energy dissipation by the NES become identical when an energy balance is reached, i.e., when the averaged trends in the energy input and NES energy dissipation follow parallel but non-coinciding paths; the nearly constant steady state difference between these energy rates makes possible the survival of aeroelastic instability in this case, in the form of reduced-amplitude LCOs. In a later section we will establish that this partial LCO suppression mechanism is related to the generation of stable LCOs that bifurcate from a stable trivial equilibrium in a supercritical Hopf bifurcation; however, the Hopf bifurcation point, i.e., the flutter speed, occurs above that of the corresponding system with no NES attached. Moreover, we will show that the robustness of the second suppression mechanism depends on the global bifurcation structure of the steady state dynamics of system (9.71), and that this mechanism can be destroyed under sufficiently large disturbances to yield LCOs with amplitudes greater than those realized in the wing with no NES attached. As for the third LCO suppression mechanism (see Figure 9.49c), most of the total energy of the motion apparently remains confined in the pitch mode so that the erroneous conclusion might be drawn that the NES does not perform efficiently in this case. However, comparing the energy input fed by the flow to the energy dissipated by the NES, we conclude that EdNES increases in a manner that energy balancing can occur only asymptotically for increasing time, thus preventing reappearance of LCOs in the long term. In this case we can obtain complete elimination of LCOs in a robust way, depending on the global bifurcation structure of the steady state dynamics, as discussed later. Figure 9.49d depicts the energy exchanges for a case where the LCO survives the action of the NES. Initially, there occur vigorous modal energy exchanges, but these involve predominantly interactions between the heave and pitch modes, with only

468


secondary involvement of the NES. This means that the action of the NES is not as effective as in the previous three cases; as a result, the NES fails to prevent energy interactions between the heave and pitch modes. Moreover, the energy dissipation by the NES is not sufficiently strong to balance the energy input fed from the flow, which ‘feeds’ directly into the modes of the wing inducing aeroelastic instability. As a result, the LCOs are not only retained in this case, but their amplitudes become even larger than those realized in the wing with no NES attached (see Figure 9.48). Note that at steady state the NES continuously dissipates energy at a constant rate, which is nearly equal to the average rate of increase of the energy input from the flow. This behavior is similar to the second suppression mechanism except for the magnitude of the uniform differences between the two averaged dissipation rates. We now examine the time-frequency behavior of the transient responses by utilizing the numerical wavelet transform (WT). In Figure 9.50 we present the WT spectra corresponding to the transient responses of Figures 9.45–9.48, i.e., the three instability suppression mechanisms and the additional case of no suppression. Comparing the evolutions of the dominant instantaneous frequencies of the heave and pitch mode responses and of the NES response, provides an additional (direct) way to verify the occurrence of resonance captures in the transient dynamics. First, we focus in cases where partial or complete instability suppression is realized, corresponding to Figures 9.45–9.47. The WT spectra depicted in Figures 9.50a–c indicate that primarily there occur 1:1 TRCs between the NES and the heave mode; in addition, there occur 1:1 TRCs followed by transitions to subharmonic resonance captures between the NES and the pitch mode. Moreover, there exists a common strong harmonic component in these plots at a frequency near the natural frequency of the pitch mode ω ≈ 1. Indirectly, the WT spectra suggest that in the study of all three suppression mechanisms one may utilize a reduced-order averaged model possessing two fast frequencies, since at most two dominant frequency components appear in the transient responses governing the resonance interactions between modes; that is, in all cases considered, the responses are dominated by two frequency components with frequencies ωh ≈ = 0.5 and ωα ≈ 1 (i.e., the linearized eigenfrequencies of the heave and pitch modes, respectively). We will make use of this important finding in the next section, in our analytical study of the LCO suppression mechanisms. Focusing now on Figures 9.48 and 9.50d where no LCO suppression takes place, we make the additional remark that when the LCO survives the action of the NES, the interaction between the heave and pitch modes completely resembles the behavior of the LCO triggering mechanism studied in Section 9.2. That is, there occurs a transition from an initial 1:1 TRC to a 3:1 SRC. Moreover, in this case, the NES possesses a higher superharmonic component, so that its dynamic interaction with the pitch mode also involves a transition from a 1:1 TRC to 3:1 SRC. It follows that in this case the resonance interactions between the various modes of the wing-NES assembly are qualitatively different than those taking place in the three cases where LCO suppression is realized. Combining the results of Figures 9.49 and 9.50, we may construct the frequencyenergy plot (FEP) of the transient dynamics of system (9.71), which, as shown in


469

Fig. 9.50 Wavelet transform spectra of transient responses: (a–c) the three suppression mechanisms depicted in Figures 9.45–9.47, respectively; (d) the case of no suppression depicted in Figure 9.48.

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471

previous chapters, can be a useful tool for identifying the essential nonlinear modal interactions and presenting the damped transitions that occur in the dynamics (including 1:1 TRCs, subharmonic and superharmonic TRCs, and escapes from these regimes). In Section 3.3 the FEP for a Hamiltonian system composed of a SDOF linear oscillator with an essentially nonlinear lightweight attachment was computed, and it was shown that such a simple system possesses very complicated dynamics, including a countable infinity of subharmonic tongues. For the corresponding weakly damped system the Hamiltonian FEP was also utilized in conjunction with WT spectra of damped transient responses to study alternative mechanisms for TET (see Section 3.4). Similar use of the FEP was made in other chapters of this monograph for MDOF linear oscillators with SDOF or MDOF essentially nonlinear attachments, as well as for the case of a self-excited system, namely, the van der Pol oscillator with an attached NES (see Section 9.1). However, unlike the above-mentioned cases, the definition of an FEP describing the topological structure of the periodic or quasi-periodic orbits of the underlying Hamiltonian system becomes difficult in the current problem (9.71). This is mainly due to the aeroelastic terms that provide non-conservative work to the system and change the intrinsic eigenfrequencies (and the total energy of the system) of the system when the flow speed varies. Hence, we propose an alternative way of constructing the FEP for aeroelastic systems based on the bifurcation structures of their steady state LCO solutions, obtained either numerically (utilizing a numerical continuation technique) or analytically (employing multi-phase averaging or harmonic balance). We demonstrate the construction of the FEP for system (9.71) by considering the steady state LCOs for parameters δ = 90%, ε = 1%, λ = 0.1, and C = 10.0 (corresponding to transient responses shown in Figure 9.45). In Figure 9.51a we depict the bifurcation diagrams of system (9.71) with respect to the reduced velocity . The stable LCOs generated at = H become unstable at = NS , yielding new periodic components that amount to quasi-periodic motions realized on two-tori in phase space. The resulting branch of quasi-periodic motions is closely associated with the first LCO suppression mechanism (see Figure 9.45). In addition, the periods of the LCOs, TLCO , are computed when performing the numerical continuation of the branches of LCOs (these are depicted as functions of in the upper plot of Figure 9.51b); it follows that the corresponding frequencies of these LCOs can be computed as ωLCO = 2π/TLCO, and are depicted as functions of in the lower plot of Figure 9.51b. It is noted that this numerical method may not provide information regarding the existence of multiple frequencies in the computed responses, if any, since it only provides the dominant frequency of the LCO. On the other hand, we already practiced computing the total energy of the system; in Figure 9.51c we depict the mean value of the total energy of the system as function of reduced speed . We note that the instantaneous total energy of the system is expected to oscillate about this mean value due to nonlinear modal interactions (resonance captures – see Figure 9.49). Finally, combining the lower plot of Figure 9.51b with the plot of Figure 9.51c we compute the FEP shown in Figure 9.51d. By construction, the FEP is parameterized by the reduced speed , and represents the averaged dominant

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Fig. 9.51 Construction of the frequency-energy plot (FEP) of LCOs of system (9.71) for δ = 0.9, ε = 0.01, λ = 0.1, and C = 10: (a) bifurcation diagrams of LCO amplitudes with respect to ; (b) LCO periods and frequencies as functions of ; H, NS, and LPC denote Hopf, Neimark– Sacker, and Limit Point Cycle bifurcations, respectively; solid line indicates stability, dashed line instability, and dash-dotted quasi-periodic solutions.


473

Fig. 9.51 Construction of the frequency-energy plot (FEP) of LCOs of system (9.71) for δ = 0.9, ε = 0.01, λ = 0.1, and C = 10: (c) total energies of the system when LCOs occur as functions of ; (d) FEP for LCO solutions; H, NS, and LPC denote Hopf, Neimark–Sacker, and Limit Point Cycle bifurcations, respectively; solid line indicates stability, dashed line instability, and dashdotted quasi-periodic solutions.

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frequencies of LCOs as functions of the corresponding total energies of the system (9.74). We are now in a position to examine the three LCO suppression mechanisms by means of representations of the dynamics on appropriately defined FEPs, as depicted in the plots of Figure 9.52. To this end, the instantaneous frequencies of the modal responses were computed by applying numerical Hilbert transforms to the transient responses corresponding to the three LCO suppression mechanisms represented in Figures 9.45–9.47. In addition we apply numerical wavelet transform to the transient responses for the case when the LCO survives the action of the NES, as depicted in Figure 9.48. The traces of the instantaneous frequencies are then plotted against the corresponding instantaneous total energies of the system, with time being the parametrizing variable. Finally, these traces are superimposed to the FEP corresponding to the system parameters of the specific case considered. In computing the instantaneous frequencies, we applied the Savitzky–Golay polynomial smoothing filter in order to remove high-frequency noise caused by numerical differentiation of the phase variable. A general conclusion is that nonlinear modal interactions between the heave and pitch modes are realized mainly through 1:1 resonance captures whether or not LCO suppression is realized. The first LCO suppression mechanism (depicted in Figure 9.52a and corresponding to the numerical simulation depicted in Figure 9.45) is characterized by repeated loops of the instantaneous frequencies with time, corresponding to the recurrent burst-outs and suppressions of aeroelastic instability; these loops consist of transitions from 1:1 to subharmonic TRCs (suppression stage), and then reversals back to 1:1 TRCs (burst-out stage). Reflecting the intrinsic LCO solutions depicted in the FEP, we find the following interesting behavior influencing the first suppression mechanism: Once the dynamics exceeds the Neimark–Sacker (NS) bifurcation point NS = 0.89 (note that the instantaneous frequency plot of Figure 9.52a corresponds to the flow speed = 0.9 > NS ) the only choice for the dynamics is to follow the unstable LCO branches which might yield either periodic or quasi-periodic steady state motions, as the heave and pitch modes are in 1:1 TRC (see Figure 9.51d). Hence, the dynamics can only be captured into loops that lead to transitions into subharmonic resonances. These repetitions (bursting outs and consequent suppressions) characterize the first suppression mechanism. In the second LCO suppression mechanism depicted in Figure 9.52b intermediate (partial) suppression of the LCO occurs, and the dynamics on the FEP forms a single loop involving a single transition from 1:1 to subharmonic resonance capture, with a final reversal back to 1:1 SRC when the steady state dynamics is reached. The plots of Figure 9.52b correspond to the transient responses and system parameters of the plots of Figure 9.46, again for flow speed = 0.9 > NS . On the contrary, complete LCO elimination (the third suppression mechanism – Figure 9.47) involves 1:1 TRCs before the dynamics escapes from resonance, at which point the NES has completely exhausted the energy input fed by the flow to the wing (see Figure 9.52c). Finally, we note a transition from 1:1 to superharmonic resonance capture between the NES and the aeroelastic modes when the LCOs survive the action of the


475

Fig. 9.52 Instantaneous frequencies superimposed to FEPs of LCOs of the corresponding systems: (a, b) LCO suppression mechanisms corresponding to Figures 9.45–9.47 (thin solid lines); H, NS, and LPC denote Hopf, Neimark–Sacker, and Limit Point Cycle bifurcations, respectively, of the FEPs of LCOs; bold solid line indicates stable LCOs, dashed line unstable LCOs, and dash-dotted line quasi-periodic solutions.

476


Fig. 9.52 Instantaneous frequencies superimposed to FEPs of LCOs of the corresponding systems: (c) LCO suppression mechanisms corresponding to Figures 9.45–9.47 (thin solid lines); (d) no suppression corresponding to Figure 9.48 (WT spectra); H, NS, and LPC denote Hopf, Neimark– Sacker, and Limit Point Cycle bifurcations, respectively, of the FEPs of LCOs; bold solid line indicates stable LCOs, dashed line unstable LCOs, and dash-dotted line quasi-periodic solutions.


477

NES (Figure 9.52d – the plot corresponds to the time responses and system parameters depicted in Figure 9.47). Comparing the plots of Figures 9.52c and 9.52d (corresponding to the same system parameters but different NES mass ratios), we conclude that whether or not the LCOs are suppressed depends on the state of the dynamics of the system when 1:1 resonance capture occurs. That is, in Figure 9.52c where complete LCO suppression occurs, the dynamics is captured into the domain of attraction of the stable LCO branch at a lower energy regime; whereas, in Figure 9.52d where LCO suppression fails, the dynamics is entrapped in the domain of attraction of the stable LCO branch formed at a higher energy regime and involving superharmonic SRCs between the NES and the aeroelastic modes that yield large-amplitude LCOs. In conclusion, we studied frequency-energy representations of the transient dynamics of system (9.71) superimposed on appropriately defined FEPs of LCOs. The first LCO suppression mechanism exhibits escapes near the NS bifurcation point that drives the dynamics into regimes of quasi-periodic responses; the second suppression mechanism drives the dynamics to a stable steady state (LCO) corresponding to the reduced velocity (and energy) of the system (i.e., = 0.9 ≤ NS = 0.905 in Figure 9.52b). Moreover, the same system parameter values (i.e., the same underlying FEP of LCOs) can yield drastically different LCO suppression results, depending on the specific energy values (or reduced flow speeds) where the resonant interaction phenomena between the flow and the wing modes occur; this last observation raises the issue of robustness of LCO suppression which will be addressed in a later section. These concluding remarks underline the importance of designing the initial entrapment of the wing – NES dynamics into the proper resonance manifolds in order to achieve efficient and robust LCO suppression. To numerically prove that the basic underlying dynamical mechanism for TETbased aeroelastic instability suppression is a series of resonance captures of the wing – NES dynamics, we analyze the dynamics by empirical mode decomposition (EMD). By computing the IMFs of the transient responses we express the heave, pitch, and NES responses as y(t) =

N1 i=1

ci (τ ),

α(t) =

N2

ci (τ ),

v(t) =

i=1

N3

ci (τ )

(9.79)

i=1

where the i-th (complex) IMF, ci (τ ), for each response is Hilbert-transformed and expressed in polar form as ci (τ ) = ai (τ )ej θi (τ ) . The amplitude ai (τ ) and phase θi (τ ) can be computed from the analytic signal, zi (τ ) = ci (τ )+j HT [ci (τ )], where HT [ci (τ )] denotes the Hilbert transform of ci (τ ): (9.80) ai (τ ) = ci (τ )2 + HT [ci (τ )]2 θi (τ ) = tan−1

HT [ci (τ )] ci (τ )

(9.81)

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It follows that the instantaneous frequency of the i-th IMF can be computed by the expression: dθi (τ ) (9.82) ωi (τ ) = dτ To demonstrate the application of EMD in the problem of identification of the nonlinear modal interactions in the in-flow wing-NES dynamics, we will consider only the first LCO suppression mechanism studied in Figures 9.45, 9.49a, 9.50a and 9.52a. We wish to numerically prove that this mechanism is governed by a recurrent series of resonance captures, escapes and transitions to recaptures. In the next section, these numerical results will be compared to, and verified with, analytical models. The other two LCO suppression mechanisms can be analyzed by applying a similar methodology so they will not be considered herein. Figure 9.53 depicts the IMFs of the heave, pitch and NES transient responses shown in Figure 9.45. The value on the upper right part of each plot represents the maximum amplitude of the corresponding IMF, so we conclude that the leadingorder IMF of each of the analyzed responses is the dominant oscillatory component in that specific time series. The plots of Figure 9.54a depict a comparison of the three leading-order (dominant) IMFs of each response with the corresponding exact time series from Figure 9.45. We observe satisfactory match between them, except for an initial period where the end effects of the EMD process pollute the data, and for some lower-frequency intervals where higher-order IMFs may possess enhanced contributions in the responses. This confirms that the leading-order IMFs in Figure 9.53 are dominant and capture the essential dynamics of the transient interactions. Let θi , i = 1, 2, 3 [computed by relation (9.81)] be the phases of the three aforementioned dominant IMFs of the heave and pitch modes, and the NES, respectively. Then, θ12 ≡ θ1 − θ2 denotes the corresponding phase difference between the heave and pitch modes; θ13 ≡ θ1 −θ3, the phase difference between the heave mode and the NES; and θ23 ≡ θ2 − θ3 , the phase difference between the pitch mode and the NES. Figure 9.54b depicts the temporal evolutions of the instantaneous phase differences, θ12 , θ13 , θ23 . If a phase difference exhibits monotonically increasing or decreasing temporal behavior, it is considered to be time-like; otherwise, it is regarded as nontime-like phase difference. For example, a constant or oscillatory phase difference with zero mean is considered to be non-time-like. Following the averaging arguments discussed in Section 2.4, if a phase difference is time-like, it can be considered to be a ‘fast’ phase of the dynamics, and, as a result, it may be removed from the dynamics (as non-essential) by simply averaging it out of the problem; in other words, this phase difference will negligibly influence the slow flow dynamics of the system after averaging and its contribution will not be considered as essential in the corresponding time window of the dynamics. On the other hand, if a phase difference is non-time-like, it may not be averaged out of the dynamics (as it cannot be regarded as ‘fast’ phase), and is expected to influence the slow (essential) dynamics of the system in the specific time interval of the analysis. In the latter case there occurs resonance capture and TET, as the dynamics is captured transiently in the corresponding resonance manifold (Arnold, 1988) defined


479

Fig. 9.53 Intrinsic mode functions (IMFs) of the transient responses depicted in Figure 9.45 (first LCO suppression mechanism) computed by EMD: (a) heave mode; (b) pitch mode; the value shown on the upper right of each IMF plot indicates the maximum amplitude of that IMF.

480


Fig. 9.53 Intrinsic mode functions (IMFs) of the transient responses depicted in Figure 9.45 (first LCO suppression mechanism) computed by EMD: (c) NES; the value shown on the upper right of each IMF plot indicates the maximum amplitude of that IMF.

by an integral relation between the instantaneous frequencies of the corresponding IMFs. From Figures 9.54b, c, we note that there exist three time intervals where nontime-like behavior of certain phase differences occurs, namely, for τ ∈ [20, 60], τ ∈ [110, 400] and τ ∈ [490, 700]. In these time intervals 1:1 TRCs occur (as evidenced by the instantaneous frequency plots, θi , i = 1, 2, 3, of Figure 9.54d), appearing as spirals in the corresponding phase portraits of Figure 9.54c. We note that not only do 1:1 TRCs occur between the heave mode and the NES, and between the pitch mode and the NES, but also between the heave and pitch modes, exactly as in the case of the LCO triggering mechanism (see Section 9.2). In between these recurring 1:1 TRCs the dynamics engage in subharmonic TRCs, which correspond to the loops in the FEPs depicted in Figure 9.52a. These results confirm the conclusions drawn earlier regarding the dynamics governing the first LCO suppression mechanism. Moreover, this analysis can be extended to the other two suppression mechanisms discussed previously.


481

Fig. 9.54 EMD analysis of the transient responses of Figure 9.45 (first LCO) suppression mechanism: (a) dominant IMFs compared to the exact responses; (b) phase differences between the dominant IMFs of the NES and aeroelastic mode responses.

482


Fig. 9.54 EMD analysis of the transient responses of Figure 9.45 (first LCO) suppression mechanism: (c) phase differences between the dominant IMFs of the NES and aeroelastic mode responses; (d) corresponding instantaneous frequencies of the dominant IMFs.


483

9.3.2.2 Analytical Study In the previous section, we found numerically that LCO suppression in the wing under consideration is due to a series of 1:1 or subharmonic resonance captures between the pitch and heave modes and the attached NES. In this section, we analytically prove this result by constructing and analyzing slow flow analytical models using the CX-A technique (see Section 2.4). We will focus only in the first LCO suppression mechanism, as the analysis can be extended similarly to study the other two suppression mechanisms. Based on the WT spectra depicted in Figure 9.50a, we conclude that there exist two dominant (‘fast’) frequencies in the dynamics at normalized frequencies equal to unity and = 0.5, corresponding to the two normalized linearized eigenfrequencies ωh and ωa of the pitch and heave modes, respectively. For notational convenience we designate these components as LF (lower-frequency) and MF (middlefrequency) components, respectively, since this is consistent to the notation introduced in previous sections. There are some additional superharmonic frequency components (realized in between the recurrent 1:1 TRCs), but these will be neglected from the model as non-dominant. Accordingly, following the multi-frequency CX-A methodology we will express the heave, pitch and NES transient responses as follows: y(τ ) = y1 (τ ) + y2 (τ ) a(τ ) = a1 (τ ) + a2 (τ ) v(τ ) = v1 (τ ) + v2 (τ )

(9.83)

where the notation of the previous sections was employed, and the components with subscripts 1 and 2 correspond to slowly varying modulations of the fast frequency components ej τ and ej τ , respectively. In essence, these representations are slowfast, multi-frequency decompositions of the transient responses, with the fast frequencies determined by the dominant harmonic components identified by the WT spectra of Figure 9.50a (in this case two fast frequencies). Introducing the new complex variables ψ1 = y1 + j y1 ≡ ϕ1 ej τ ,

ψ3 = y2 + jy2 ≡ ϕ3 ej τ

ψ2 = α1 + j α1 ≡ ϕ2 ej τ ,

ψ4 = α2 + j α2 ≡ ϕ4 ej τ

ψ5 = v1 + j v1 ≡ ϕ5 ej τ ,

ψ6 = v2 + j v2 ≡ ϕ6 ej τ

(9.84)

expressing the normalized equations of motion (9.71) in terms of the complex variables, and applying two-frequency averaging over the two fast components ej τ and ej τ , we obtain a set of six complex-valued modulation equations governing the slow flow dynamics, (9.85) ϕ = F (ϕ)

484


where ϕ ∈ C 6 and the complex-valued vector F (ϕ) is quite involved and is not explicitly reproduced here. Introducing the polar form decompositions of the slow modulations, ϕi (τ ) = ai (τ ) ej bi (τ ) , ai (τ ), , bi (τ ) ∈ R, i = 1, . . . , 6, we express (9.85) as a set of 12 (real-valued) autonomous modulation equations governing the slow evolutions of the amplitudes and phases, a = f (a, φ),

φ = g(a, φ)

(9.86)

where a ∈ R +6 and φ ∈ S 6 . The slowly-varying amplitudes a1 and a3 (a2 and a4 ; a5 and a6 ) are LF and MF slowly varying amplitudes, respectively, of the heave (pitch; NES) mode. The phase angle vector φ in (9.86) possesses the components φ12 ≡ b1 −b2 (phase difference between LF heave and LF pitch), φ34 ≡ b3 −b4 (MF heave – MF pitch), φ15 ≡ b1 − b5 (LF heave – LF NES), φ25 ≡ b2 − b5 (LF pitch – LF NES), φ36 ≡ b3 − b6 (MF heave – MF NES) and φ46 ≡ b4 − b6 (MF pitch – MF NES). We note that all independent phase interactions occur between same frequency components (LF-LF or MF-MF), and that all other phase interactions can be expressed in terms of the aforementioned six independent phase differences. Comparisons of the transient responses predicted by the slow flow (9.85–9.86) and the exact responses resulting by direct numerical simulations of system (9.71) are depicted in Figure 9.55a, and demonstrate the validity of the slow flow model. Except for very low-frequency variations in the NES response, the slow flow model shows good overall match with the exact transient dynamics. The discrepancies may be improved if we employ additional fast frequency components in the ansatz (9.83– 9.84). That is, recalling that the dominant instantaneous normalized frequency in Figure 9.54d is approximately equal to ω ≈0.85 in the regimes of 1:1 TRCs, we conjecture that single-frequency averaging with respect to that ‘fast’ frequency might be sufficient to capture the important (slow flow) dynamics of the system; nonetheless, special care must be taken when applying single-frequency averaging as the fast frequencies are expected to vary with respect to the flow speed. In fact, some efforts have been made recently to establish sufficiently reasonable slow flow models by utilizing EMD and WTs, showing that multi-frequency averaging is basically equivalent to applying EMD (Kerschen et al., 2006b). From the evolutions of the amplitude components in Figure 9.55b, we verify that the MF components are the dominant ones in the first LCO suppression mechanism; this result is consistent with the WT results depicted in Figure 9.50a. Moreover, similar resonance captures followed by escapes to the ones depicted in Figures 9.55c, d were observed in the numerical EMD results presented in Figures 9.54b–d. An additional interesting remark regarding the first LCO suppression mechanism is that the resonance captures between the heave and pitch modes (characterized by the phase angles φ12 and φ34 ) occur ahead of those between the heave mode and the NES (phase angles φ15 and φ36 ), or those between the pitch mode and the NES (phase angles φ25 and φ46 ). This implies that in the first LCO suppression mechanism there occur nonlinear modal energy exchanges between the heave and pitch modes (i.e., the triggering mechanism for the LCO is activated – see Section 9.2)


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Fig. 9.55 CX-A analysis of the first LCO suppression mechanism shown in Figure 9.45: (a) comparison of analytical and numerical responses; (b–d) instantaneous amplitudes and phase interactions of the slow flow model (9.86).

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Fig. 9.55 Continued


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before TET from the aeroelastic modes to the NES (with the ensuing instability suppression) occurs; this early occurrence of RCs between the heave and pitch modes ‘activating’ the LCO triggering mechanism makes the repetition of suppressions and burst-outs in the first suppression mechanism possible. Moreover, this suggests that for more efficient (and robust) suppression of LCOs, the NES should interact with the aeroelastic modes before energy transfers between these modes are realized, thus preventing the activation of the triggering mechanism for LCO instability. The results of Figure 9.56 depicting the phase interactions of the transient responses of Figure 9.47 corresponding to the third LCO suppression mechanism (resulting in complete LCO elimination) support this argument. Note that in this case the energy transfer from the pitch mode to the NES occurs almost simultaneously with the triggering of the pitch from the heave mode, leading to complete elimination of the aeroelastic instability.

9.3.3 Robustness of LCO Suppression We now investigate the robustness, i.e., the dependence on initial conditions and changes on flow speeds, of the identified aeroelastic instability suppression mechanisms. We will perform this study by means of steady state bifurcation analysis of the dynamics of system (9.71) utilizing the method of numerical continuation of equilibrium and periodic solutions. This bifurcation analysis will examine the possibility of co-existence of stable attractors in the dynamics and will determine the parameter ranges where the dynamics of the system is attracted to steady state solutions that are favorable to the LCO suppression objective. In the process we will explore the effect of offset distance δ on LCO suppression, and confirm the numerical finding that negative offsets generally appear to cause more robust and effective LCO suppression results. From here on, by ‘positive’ or ‘negative’ attachment, we will mean that the NES is connected to the wing ahead or aft of the elastic axis, respectively (that is, closer to the nose or tail of the in-flow wing). The global bifurcation structure of the dynamics will eventually reveal how the previously discussed three LCO suppression mechanisms are related to bifurcations of steady state solutions. This study will provide hints for NES designs that result in efficient and robust aeroelastic suppression. The results depicted in Figure 9.57 indicate that in some cases strong disturbances may eliminate LCO suppression, that is, LCO suppression may be achieved at certain energy levels but not at others. Indeed, as shown in Figure 9.57, by increasing the non-zero initial condition y (0) by a factor of ten compared to the value used for demonstrating the first LCO suppression mechanism in Figure 9.45, we may completely eliminate this LCO suppression mechanism yielding larger-amplitude LCO compared to the system with no NES attached. Hence, the issue of robustness of LCO suppression is raised with respect to the effects of changes in initial conditions or the flow speed with the other system parameters remaining fixed.

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Fig. 9.56 EMD analysis of the third suppression mechanism shown in Figure 9.47: (a) phase interactions θij ≡ θi − θj computed by the Hilbert transform of the dominant IMFs; (b) instantaneous frequencies θi .


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Fig. 9.57 Breakdown of the first suppression mechanism to form a larger amplitude LCO compared to the system with no NES attached; system parameters and initial conditions are identical to those used for the plots of Figure 9.45, except for y (0) = 0.1.

In Figure 9.58 the reduction in the steady state r.m.s. pitch amplitude α [defined by relation (9.76)] with respect to the set of initial conditions {y(0) = 0, α(0) = 0, y (0) ∈ [0, 0.1], α (0) ∈ [0, 0.1]} and various values of the system parameters is computed. For these computations we performed direct numerical simulations of the normalized equations of motion (9.71). Generally, it appears that the reduction of the steady state pitch amplitude due to the action of the NES does not depend in an essential way on the initial pitching velocity. Moreover, for a fixed flow speed increasing the mass ratio and damping broadens the domain of initial conditions where complete elimination or significant reduction of the amplitudes of LCOs occurs. On the contrary, for fixed mass ratio and damping increasing the flow speed can eliminate or significantly reduce the domain where LCO suppression occurs. It will turn out in the later part of this section that robustness of LCO suppression with respect to the initial conditions is highly dependent on the global bifurcation structure of the steady state dynamics of the system. Robustness of LCO suppression with respect to variations of the flow speed also depends on the global features of the steady state dynamics. Figure 9.59 depicts bifurcation diagrams of peak-to-peak steady state amplitudes of system (9.71) against the reduced flow speed , when the flow speed increases slowly with acceleration rate (τ ) = 10−5 . For fixed mass ratio, damping and nonlinear coupling stiffness, the effects of positive or negative offsets δ are compared in Figures 9.59a, b. For a

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Fig. 9.58 The steady state r.m.s. pitch amplitude ratio α , with respect to the set of initial conditions {y(0) = 0, α(0) = 0, y (0) ∈ [0, 0.1], α (0) ∈ [0, 0.1]}, δ = 0.9 and: (a) = 0.9, ε = 0.01, λ = 0.2, and C = 30; (b) = 0.9, ε = 0.02, λ = 0.4, and C = 10.


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Fig. 9.58 The steady state r.m.s. pitch amplitude ratio α , with respect to the set of initial conditions {y(0) = 0, α(0) = 0, y (0) ∈ [0, 0.1], α (0) ∈ [0, 0.1]}, δ = 0.9 and: (c) = 0.95, ε = 0.01, λ = 0.2, and C = 30; (d) = 0.95, ε = 0.02, λ = 0.4, and C = 10.

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Fig. 9.59 Bifurcation diagrams depicting peak-to-peak steady state amplitudes with respect to the reduced speed , for a slowly accelerating flow, d /dτ = 10−5 and ε = 0.02, λ = 0.4, C = 10: (a) δ = 0.9 and (b) δ = −0.9.


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specific value of reduced flow speed , a single point in a bifurcation plot implies either a stable trivial equilibrium or a stable LCO; and multiple points in a bifurcation plot imply a quasi-periodic orbit or an unstable LCO. Basically, this methodology is analogous to a frequency-sweeping method when performing modal testing of a structure. We note that although system (9.71) is derived under the assumption of fixed reduced flow speed , the bifurcation results for slowly varying are still expected to approximate the true dynamics of the system for | | 1. In general, increasing the flow speed delays the occurrence of the Hopf bifurcation that generates the aeroelastic instability (the stable LCOs), for both ‘positive’ and ‘negative’ attachments. An increase of causes a sudden transition of the dynamics to a stable LCO, which implies that there exists a LPC (limit point cycle) bifurcation at the point of transition; the LPC bifurcation is analogous to the saddlenode bifurcation of equilibrium points. We note that for a ‘negative’ attachment (δ = −90%, Figure 9.59b) quasi-periodicity occurs before the transition. Later, we will verify that this quasi-periodicity is generated through a NS (Neimark–Sacker) bifurcation of a periodic solution which is analogous to the Hopf bifurcation of an equilibrium point. For a ‘positive’ NES attachment (δ = 90%, Figure 9.59a), the LCOs after the transition from the stable equilibrium possess larger amplitudes compared to the system with no NES attached. Decreasing the reduced flow speed reveals the clear difference in the steady state dynamics for systems with ‘positive’ and ‘negative’ attachments. In both cases there exist branches of unstable periodic solutions connecting the upper and lower branches of steady state solutions; in addition, jumps to stable equilibrium positions occur, implying that there exist ‘inverse’ LPC bifurcations. Whereas the ‘negative’ attachment induces a transition of the steady state dynamics close to the Hopf bifurcation point (but above the critical flutter speed F of the system with no NES attached), the ‘positive’ attachment induces a similar transition at a flow speed even less than the flutter speed F . In the latter case the NES clearly introduces additional instability to the system, since small disturbances may generate largeamplitude LCOs at flow speeds in ranges of where only stable trivial equilibrium points exist in the system with no NES attached. To perform a further study of robustness of LCO suppression by means of the bifurcation structure of the steady state dynamics of system (9.71) we employ a numerical continuation method based on MATCONT (Dhooge et al., 2003). Figure 9.60 presents bifurcation diagrams depicting the steady state pitch amplitude |α| against damping λ and offset δ, for fixed mass ratios ε = 0.01, 0.02, and fixed reduced flow speed = 0.9 and nonlinear coefficient C = 10. The contour plots for α extracted from Figure 9.44 are incorporated into the bifurcation diagrams to help the visualization of the steady state amplitudes. We note that by the definition of the normalized offset, only the range −1 ≤ δ ≤ 1 is permissible, although in order to provide a complete picture of the steady state dynamics we allowed values of δ outside this range. Figure 9.60a depicts the bifurcation structure of steady state dynamics for the smaller mass ratio ε = 0.01; the corresponding bifurcation diagrams for two specific damping values are depicted in Figure 9.61a (these may be regarded as two-

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Fig. 9.60 Steady state bifurcation diagrams with respect to δ and λ for = 0.9 and C = 10: (a) ε = 0.01 and (b) ε = 0.02; solid lines indicate stable trivial equilibrium points or LCOs, dotted lines unstable trivial equilibrium points; dashed lines, unstable LCOs; and dash-dotted lines quasiperiodic LCOs; squares indicate Hopf bifurcations, triangles for LPCs, circles for NS (Neimark– Sacker) bifurcations, and diamonds neutral-saddle bifurcations.


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Fig. 9.61 Two-dimensional ‘slices’ of the three-dimensional bifurcation diagrams of Figure 9.60 for fixed damping values λ: (a) ε = 0.01 and (b) ε = 0.02; the branches of stable large-amplitude LCOs comparable to the LCOs of the system with no NES attached are not depicted; thin dashed lines depict the permissible range of the offset, −1 ≤ δ ≤ 1.

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dimensional ‘slices’ for fixed damping of the three-dimensional bifurcation diagram of Figure 9.60a). Note that, due to the required computational complexity of the bifurcation study, the stable large-amplitude LCOs (which are comparable to the corresponding LCOs of the system with no NES attached) are not displayed in these plots. The Hopf bifurcation curves are realized at offsets with absolute values greater than unity, which suggests that we cannot achieve complete elimination of aeroelastic instabilities (the third LCO suppression mechanism) within the ranges of system parameters considered in these diagrams. Moreover, the presence of stable largeamplitude LCOs implies that it is not possible to achieve robust suppression of LCOs in this case. However, the first and second LCO suppression mechanisms can be realized in the ‘permissible’ range −1 ≤ δ ≤ 1, although a disturbance can eliminate LCO suppression and give rise to the stable large-amplitude LCOs and aeroelastic instability. Moreover, weaker damping tends to induce more complicated dynamics (see, for example, the lower plot of Figure 9.61a). The steady state bifurcation structure for the larger mass ratio ε = 0.02 is presented in Figure 9.60b and two ‘slices’ corresponding to fixed damping values are depicted in Figure 9.61b. As in the case of smaller mass ratio, larger damping values tend to eliminate complicated dynamic behavior. Using these bifurcation diagrams we are in the position to explain the three LCO suppression mechanisms; we will discuss this only for the case of ‘positive’ attachment, since similar arguments hold for ‘negative’ attachments. Studying the bifurcation diagrams of Figures 9.60b and 9.61b we conclude that for the larger mass ratio and for a certain range of damping the third LCO suppression mechanism (complete elimination of LCOs) can be realized for offsets greater in magnitude than the offsets corresponding to the Hopf bifurcation points but less than unity; moreover, in this case the complete suppression of the LCO is robust, since the bifurcation diagrams in this case provide information on the global dynamics of system (9.71) (that is, in contrast to the plots of Figures 9.60a and 9.61a they include also the large-amplitude LCOs), so there are no other stable attractors to ‘compete’ with the ones depicted in the bifurcation diagrams. For example, the third LCO mechanism is realized in the intervals 1 and 2 in the upper plot of Figure 9.61b for the system with ε = 0.02. At offset values where the only attractors of the steady state dynamics are quasiperiodic LCOs the second LCO suppression mechanism is realized, which is also robust in this case; this holds, for example, in the offset intervals 1 and 2 in the lower plot of Figure 9.61b. Noting that the NS bifurcation implies the generation of a new periodic solution (LCO), one can draw the conclusion that quasi-periodic behavior is the norm in the first suppression mechanism, interrupted, however, by instances of periodic motions when the frequency of the new periodic solution is in rational relation to the frequency of the pre-existing periodic orbit. Decreasing damping tends to increase the interval where the first LCO suppression mechanism is realized. For sufficiently large values of damping, one cannot observe the occurrence of the first suppression mechanism due to the strong dissipation effects in the steady state dynamics. Clearly, at offset intervals where stable LCOs exist, either the second LCO mechanism is realized (for low-amplitude LCOs), or no suppression is


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possible at all (when the amplitudes of LCOs are comparable to the LCOs of the system with no NES attached). The first LCO suppression mechanism examined in Figure 9.45 is re-examined in Figure 9.62 for the negative offset distance δ = −0.9. One evident result is that more energy transfer from the aeroelastic modes to the NES occurs in this case. Moreover, it appears that a ‘negative’ nonlinear attachment extracts more energy from the heave mode, and the duration of the nonlinear resonant interactions between modes lasts longer than in the case of ‘positive’ attachment. In a previous section we briefly raised the issue of maximizing the energy dissipation by the NES, which is proportional to (y − δα − v )2 . Examining the responses under condition of 1:1 resonance captures, we can approximate each response roughly as y (τ ) ≈ Y sin ωτ , α (τ ) ≈ A sin ωτ and v (τ ) ≈ V cos ωτ , where the amplitudes Y, A and V are assumed to be positive and ω is the frequency where resonance captures occur. Then, we derive the approximation y − δα − v ≈ (Y − δA) sin ωτ − V cos ωτ = X sin(ωτ − θ ), where X2 = (Y − δA)2 + V 2 and θ = tan−1 [V /(Y − δA)]. Clearly, in order to maximize the quantity (y − δα − v )2 , the value of X should be maximum, and negative values of the offset δ provide larger values for X. Although this may not be a mathematically rigorous argument, it helps to get a rough understanding of why negative offsets lead to better and more robust instability suppression compared to positive ones. Finally, we provide an alternative view of the bifurcation structure of the steady state dynamics of system (9.71) by plotting the steady state pitch amplitude as function of the offset δ and the reduced flow speed ; this is performed in Figure 9.63. Since the qualitative features of the steady state dynamics are similar in the two plots corresponding to positive and negative offsets, we focus only in the case of ‘positive’ attachment (δ > 0). We note that for zero offset the bifurcation behavior is that of the system with no NES attached, as the NES is incapable of affecting the dynamics of the wing. For large reduced flow speeds branch point cycle (BPC) bifurcations of large-amplitude periodic solutions (LCOs) occur (denoted by asterisks in the bifurcation diagrams of Figures 9.63 and 9.64); a BPC bifurcation of a periodic solution is similar to a pitchfork or transcritical bifurcation of an equilibrium point, and leads to exchanges of stability and generation of new branches of large-amplitude LCOs. For positive offset values δ > 0 the occurrence of the Hopf bifurcation is delayed for an increase of the flow speed; moreover, the intervals where quasi-periodic responses occur (and the first LCO suppression mechanism is realized) widen, as do the intervals between the two LPC bifurcation points. In this case, the BPC bifurcation point converges to the lower-amplitude LPC bifurcation point, and the Hopf bifurcation curves are almost symmetric with respect to the plane δ = 0. Typical twodimensional ‘slices’ of the three-dimensional bifurcation structure of Figure 9.63 are depicted in Figure 9.64; these results reaffirm the quantitative differences between the dynamics of steady state systems with ‘positive’ and ‘negative’ attachments, discussed in our previous bifurcation study. ‘Positive’ attachments generally lead to LCOs of larger amplitudes compared to those of the system with no NES attached, whereas ‘negative’ attachments yield smaller-amplitude LCOs. In terms of the three

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Fig. 9.62 The first LCO suppression mechanism realized in the system with identical parameters to that corresponding to the responses of Figure 9.45, except for the negative offset distance δ = −0.9: (a) transient responses, (b) instantaneous energy exchanges between the NES and the aeroelastic modes.


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Fig. 9.63 Bifurcation diagrams of steady state dynamics for varying reduced flow speed and offset δ, and ε = 0.02, λ = 0.4, and C = 10: (a) case of positive offset (δ > 0), and (b) case of negative offset (δ < 0); the notation of Figure 9.60 applies, and the asterisk denotes a branch point cycle (BPC) bifurcation.

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Fig. 9.64 Two-dimensional ‘slices’ of the three-dimensional bifurcation diagrams of Figure 9.63 for fixed offsets δ: (a) δ = 0.75 and (b) δ = −0.75.


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LCO suppression mechanisms, we observe the transition from the third to the first LCO suppression mechanism when the flow speed increases. Further increase of the flow speed above the bifurcation point LPC1 produces a sudden transition of the dynamics to LCOs with larger amplitudes, and yields increased aeroelastic instability in the system. Based on our previous discussion we conclude that an in-flow wing-NES system with a bifurcation structure of steady state dynamics similar to that depicted in Figure 9.64b is a configuration suitable for practical applications. Indeed, considering this bifurcation diagram we note that LCOs can be completely and robustly suppressed for reduced flow speeds below the speed H corresponding to the Hopf bifurcation (i.e., a robust third LCO suppression mechanism is realized). For reduced speeds in the range H < < NS1 robust partial LCO suppression is achieved, as a low-amplitude LCO survives the action of the NES (robust second LCO suppression mechanism). For NS1 < < LPC1 there is co-existence of the first and second LCO suppression mechanisms, but, of course, neither of these is robust. Which of the two LCO suppression mechanisms is eventually realized depends on the initial state of the problem, as the dynamics may be attracted either by the quasiperiodic LCO (non-robust first suppression mechanism) or by the low-amplitude stable LCO (non-robust second suppression mechanism), both of which co-exist < LPC1 and LPC2 < < BPC . Of course, for in the intervals NS2 < > BPC the case of no LCO suppression becomes possible, and for > LPC1 LCO suppression is unfeasible as the steady state dynamics is attracted by the largeamplitude LCOs generated after the BPC bifurcation.

9.3.4 Concluding Remarks In this section we investigated passive suppression of aeroelastic instabilities in a two-DOF in-flow rigid wing system by means of passive, broadband, nonlinear targeted energy transfers. The physical mechanism for inducing these transfers was a lightweight, essentially nonlinear SDOF oscillator attachment which acted as nonlinear energy sink. Through numerical parametric studies we found that there exist three suppression mechanisms for suppressing aeroelastic instabilities in this system. We investigated these mechanisms both numerically and analytically, and proved that the underlying dynamics were series of resonance captures, i.e., of transient resonances either between the NES and the heave and/or pitch aeroelastic modes, or between the wing modes themselves. We explored these LCO suppression mechanisms in terms of steady state bifurcation analysis, which also addressed the issue of the robustness of suppression, i.e., of the dependence of LCO suppression on the initial conditions and the parameters of the problem. We found that NESs attached at negative offsets can provide robust aeroelastic instability suppression within relatively wide ranges of system parameters; on the contrary, NESs at positive offsets do not provide robust suppression, as explained by the associated series of bifur-

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cations of steady state dynamics that occur in this case. It follows that lightweight NESs with negative offsets can form the basis of practical, economical, robust and realistic designs for LCO suppression in the in-flow wing under consideration. In the following section we provide experimental verification of these theoretical findings by presenting results of a series of wind tunnel tests of a practical in-flow wing with an attached SDOF NES. In a later section we discuss the issue of extending robustness of TET-based passive LCO suppression, by investigating the dynamics of wings with more complicated NES configurations, namely MDOF NESs. We show that such configurations improve the robustness of LCO suppression compared to the results presented in this section.

9.4 Experimental Validation of TET-Based, Passive LCO Suppression In this section we examine experimental validation of LCO suppression using the SDOF NES design discussed previously. The exposition follows the work by Lee et al. (2007b) where more details of the experimental methodology and results can be found. The potentially unstable aeroelastic structure considered herein is a rigid airfoil in a low-speed wind tunnel located at Texas A&M University. This wing was mounted on separately adjustable springs restraining its motion in heave (plunge) and pitch. The apparatus has been used by Strganac and co-workers in several experiments on passive and active aeroelastic control (Block and Strganac, 1998; Ko et al., 1999; Platanitis and Strganac, 2004), in the course of which it has been very thoroughly studied, modeled and dynamically characterized. In the absence of any corrective measures, this wing has a critical speed of approximately 9.5 m/s. When the flow speed in the wind tunnel exceeds this value, LCOs can readily be induced by a small initial displacement in the heave degree of freedom. A SDOF NES with the configuration examined in the previous section (Configuration II – Section 3.1) was designed to be coupled to the heave mode of the airfoil, with the goal of increasing the critical speed of the combined system above that of the wing without NES attached. The assessment of the LCO suppression capacity of the NES is studied by comparing measured responses of the wing and the integrated wing-NES system configurations to predictions from analysis and simulation in an effort to both quantify the performance improvement due to the action of the NES, and verify our theoretical analysis of the underlying dynamics.

9.4.1 Experimental Apparatus and Procedures The hardware used in the tests reported below is broadly divisible into (i) the wind tunnel, the model wing, and its supporting structure; (ii) the nonlinear energy sink and its support; and (iii) the equipment used to measure the response of both sub-


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structures (Lee et al., 2007b). The nonlinear aeroelastic test apparatus (NATA) at Texas A&M University was developed to experimentally test linear and nonlinear aeroelastic behavior. The device consists of a rigid NACA 0015 wing section capable of movement with two degrees of freedom, pitch and heave, as shown schematically in Figure 9.65a. Stiffness nonlinearity can be introduced to either degree of freedom. The device is mounted in a 0.61 × 0.91 m low-speed wind tunnel capable of speeds up to 45 m/s. Each degree of freedom of the NATA wing is supported by its own set of springs. Heave motion, which mimics out-of-plane bending of the wing, is provided by mounting the wing on a carriage which can slide from side to side on shafts mounted under the wind tunnel. The motion of the carriage is restricted by springs stretching from the rigid frame of the wind tunnel to a rotating cam. The carriage is attached to the same cam such that its movement is resisted by the springs, as shown in Figure 9.65b. The wing section stands vertically in the wind tunnel, spanning the entire tunnel from top to bottom, as shown in Figure 9.65c. The wing is attached to a shaft that exits through the tunnel floor and mounts via rotational bearings to the plunge carriage beneath the tunnel. These bearings allow the wing to pitch (rotate), simulating torsion of the wing. Each of the pitch springs has one end rigidly fixed to the plunge carriage, whereas its other end wraps around a cam on the pitch shaft. Hence, the set of equations of motion of the NATA may be expressed as follows: M x¨ + C x˙ + K x = F c + F a

(9.87)

where x(t) = [h(t), α(t)]T , with h(t) and α(t) being the responses of the heave and pitch modes, respectively; F c the (2 × 1) vector of Coulomb friction forces, and F a = [−L, M]T the 2 × 1 vector of aerodynamic forces and moments (see Figure 9.65a). Moreover, in equation (9.87) M, C and K denote the (2 × 2) mass, viscous damping and stiffness matrices of the system, respectively. These matrices are expressed in terms of the physical parameters of the NATA (Lee et al., 2007b) which are listed in Table 9.1. We only note here that the only nonlinear structural element in NATA is the torsional stiffness in pitch, which is a nonlinear function of the response α(t) as depicted in Figure 9.66 and listed in Table 9.1. Experiments using the NATA are conducted at very low speeds and very low reduced frequencies. The wing section spans the entire wind tunnel, so the flow can be considered as being approximately two-dimensional. For this tame flow environment, lift and drag can be modeled with quasi-steady aerodynamics. This type of aerodynamic model has provided very good agreement with NATA experimental results in the past. However, an element that has not been considered in the theoretical analysis of Section 9.3 is dry friction. Friction has a significant effect on the dynamic behavior of the NATA system, so both viscous damping and Coulomb friction are expected to appear in equations (9.87). For the first proof-of-concept experiments with an NES in an aerodynamic application, the design goals were similar to what would be desired of flight hardware, tempered by the realities of the laboratory environment and the scale of the test program. It was desired to design a lightweight, passive, self-contained, essentially

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Fig. 9.65 Experimental apparatus: (a) schematic of wing with heave and pitch degrees of freedom; (b) schematic of the nonlinear aeroelastic test apparatus (NATA); (c) picture of the NATA, air flow is from left to right, the plunge carriage and pitch cam are visible beneath the tunnel.


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Fig. 9.66 Measured and modeled torsional stiffness in pitch. Table 9.1 NATA parameter values. Parameter

Value

Wing mass, mw Pitch cam mass, mc Total plunging mass, mT Total pitching inertia, Iα Wing mass offset, rcg Wing section semichord, b Non-dimensional elastic axis location, a Pitch cam mass offset, rc Viscous plunge damping coefficient, ch Viscous pitch damping coefficient, cα Plunge spring stiffness, kh Pitch spring stiffness, kα (α)

1.645 kg 0.714 kg 12.1 kg 2 kg m2 0.04561 + mw rcg −b(1 + 0.18) m 0.1064 m −0.4 m 0.127 m 5.0747 kg/s 0.015 kg m2 /s 2537.2 N/m 8.6031 − 27.67α + 867.15α 2 + 376.64α 3 − 7294.6α 4 Nm/rad 0.6 m

Wing section span, s

nonlinear SDOF attachment of Configuration II (see Section 2.6 – Figure 2.27, and Section 3.1 – Figure 3.2) that would significantly improve the dynamic response of the NATA under typical operating conditions. In the present context, the characterization ‘lightweight’ should be construed as a requirement that the NES mass should be small with respect to the total translational mass of the NATA. When the structure supporting the wing section was taken into account, it was found that the NATA mass equaled approximately mT ≈ 12 kg. To make the best use of available hardware, it was convenient to fix the mass of the NES at ms ≈ 1.2 kg, corresponding to a mass ratio, in heave, of ms /mT ≈ 0.1. Because of the manner in which the wing is supported in the NATA, it was possible to regard the NES as interacting

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directly with only the heave degree of freedom of the wing (i.e., it was assumed that the NES was effectively mounted at the elastic axis of the NATA wing). Indeed, the analysis of Section 9.3 suggests that, in general, it would in be preferable to attach the NES away from the wing’s elastic axis, but to achieve this with the NATA would require much more extensive structural modifications so it was not attempted in the experimental work reported herein. With the mass fixed, preliminary design of the NES was reduced to the specification of the linear viscous damping coefficient cs and the coefficient ks of the essentially nonlinear stiffness that couples the NES mass ms to the NATA plunge displacement h(t). The ranges of values of these two parameters that could be readily produced with an existing NES had been established in earlier experiments on other (non-aerodynamic) NES applications (some of which are discussed in other sections of this work), and data were available relating nonlinear stiffness and coupling efficiency at low structural frequencies. On this basis, preliminary values for the NES stiffness and damping were selected, and then refined through a series of numerical simulations carried out in Matlab. With the addition of the SDOF NES the equations of motion of the integrated wing-NES system are given by

M 0 x¨ x˙ x C 0 K 0 Fc Fa Fs + + = + + 0 ms 0 cs 0 ks v¨ v˙ v 0 0 −fs (9.88) where the previously introduced notation holds, v(t) denotes the absolute response of the NES, and F s = [−fs 0]T the force exerted on the NATA by the NES. This force was experimentally identified as follows: ˙ + ks |v − h|γ sgn (v − h) fs = cs (v˙ − h)

(9.89)

where the exponent of the essential stiffness nonlinearity was identified as γ = 2.8 ≈ 3, and is typical of the values experimentally identified for the (theoretically purely cubic) coupling stiffness. The results of simulations indicated that good NES performance could be achieved over a range of damping values. As shown in Chapter 3, the viscous damping coefficient cs is not expected to significantly affect the rate and efficiency of TET in the integrated system, and this was confirmed by the fact that the simulations were relatively insensitive to this parameter; so a value of cs = 0.40 Ns/m was chosen, which was typical of damping levels identified in previous experiments. Additional simulations indicated that values of ks in the range [1 × 106, 2 × 106 ]N/m2.8 led to effective TET, with larger values in this range to be preferred for practical reasons, such as smaller relative displacements during testing. Hence, the value ks = 1.6 × 106 N/m2.8 was selected for the preliminary NES design. The aforementioned parameter values for the NES can be considered as nominal NES parameters, since actual values were identified and adjusted accordingly during the experimental testing.


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Fig. 9.67 NATA-NES configuration: (a) the NES at rest on the air track, with the connecting rod to the NATA visible at the top (wing heave and NES motion are parallel to the track); (b) the NES and air track structure mounted to the wind tunnel frame of the NATA (wing heave and NES motion are orthogonal to the picture).

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The NES used for the experiments described here consisted of a small ‘car’ made of aluminum angle stock. The car was supported on an air track to reduce sliding friction in the NES system, and was connected to the NATA through the essentially nonlinear spring in parallel to a viscous damper. The nonlinear spring was created by securing a pair of thin wires perpendicular to the relative movement of the NATA and the NES (i.e., to the heave motion of the NATA) as shown in Figure 9.67a. The wires were mounted with no initial tension and hence their spring force had no linear component (see Section 2.6). This arrangement of wires ensured that when the NATA moved with respect to the NES, tension was created in the wires, ideally providing a nearly pure cubic restoring force between the two subsystems. The entire NES assembly was attached to the NATA plunge carriage through a rod. Figure 9.67b shows the NES installed on the NATA in line with the plunge carriage. Considering the data acquisition system, the pitch and plunge motions of the aeroelastic system were measured with optical angular encoders attached to the cams of the NATA, whereas the NES response was monitored using an accelerometer, and then integrated numerically to obtain velocity and displacement time series as needed. The force developed between the NES and NATA was sensed by a piezoelectric force transducer, and this signal was used only during some of the system identification procedures [for example, for identifying the nonlinear force (9.89)]. Freestream velocity inside the wind tunnel was determined using a Pitot probe and an electronic pressure transducer. All of these signals were sent to a data acquisition board for recording and post-processing.

9.4.2 Results and Discussion The first step in performing experiments with the integrated NATA-NES system was to set the wind tunnel to the desired freestream velocity. Next, the initial conditions were imposed by manually displacing the NATA plunge carriage (and thus the NES) and waiting until equilibrium was established. The system was then released and the responses measured. In all the experiments conducted with the NATA with NES attached, the dynamic behavior of the system was initiated by such a static heave displacement. As a consequence of the kinematics of the NATA, these initial conditions produced no aerodynamic moment on the wing, and when the NES was attached its initial displacement was equal to that of the plunge carriage, h0 . Considering the wing without NES attached, when the flow speed in the wind tunnel exceeded the NATA’s flutter speed, approximately 9.5 m/s, an LCO could be reliably induced by releasing the wing from an initial heave displacement h0 , with all other initial conditions (pitch angle and pitch and heave rates) being zero. Because of the presence of Coulomb friction in the system, very small values of h0 (up to a few millimeters) could not trigger the wing’s LCO, but for all larger values of h0 the development of the LCO was very consistent. The various combinations of flow speed, initial plunge displacement, and NES characteristics tested are summarized as follows: freestream speed U = 9–13 m/s;


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initial heave displacement h(0) = h0 = 1.27 × 10−2 and 1.91 × 10−2 m; NES mass ms = 1.2 and 1.5 kg; NES nonlinear stiffness coefficient ks = 1.6 × 106 and 2.0 × 106 N/m2.8 ; NES damping cs = 0.015 and 0.040 Ns/m (where the leading values represent baseline values). The mass of the NES was determined by means of a laboratory static balance test, while the stiffness of the essentially nonlinear coupling spring and the associated viscous damping coefficient were identified using the restoring force surface method (Masri and Caughey, 1979). The damping force produced by the motion of the NES car on the air track was by comparison very small, and so has been neglected throughout this work. The ratios of the two NES mass values (1.2 and 1.5 kg) to the translational mass of the NATA were equal to 10 and 12.5%, respectively. Although these may represent unrealistically large values for flight hardware, they were deemed not to be unreasonable for purposes of validating the theoretical results of Section 9.3 (see also Lee et al., 2007a). The results presented there indicate that TET from the wing to the NES will not be degraded as the NES mass is made smaller. Time histories of the heave and pitch responses of the various wing-NES configurations at several flow speeds are shown in insets within Figures 9.68–9.70. In each case, the corresponding response of the NATA with no NES attached is shown for reference and comparison. On these are superimposed responses with the NES attached, which experimentally verify the three LCO suppression mechanisms discussed in Section 9.3, namely: (i) recurring LCO burst out and suppression; (ii) intermediate (partial) LCO suppression; and (iii) complete LCO elimination. The steady state amplitudes of the LCOs, if they survive TET to the NES, are plotted in these Figures as discrete points for each flow speed at which tests were conducted (namely, 9, 10, 11, 12, and 13 m/s). When the amplitude of a ‘surviving’ LCO exhibited amplitude modulation (i.e., for the first LCO suppression mechanism), a pair of points has been plotted to indicate the range of the response envelope. It is clear from these results that, with no NES attached, the NATA does not exhibit aeroelastic instability at a flow speed of 9 m/s. At this speed, the heave and pitch responses following an initial heave displacement of either h(0) = 1.27 × 10−2 m or 1.91×10−2 m (with all other initial conditions zero) decay to the original trivial equilibrium position. When the flow speed is increased to U ≥ 10 m/s or higher, the response of NATA exhibits a highly consistent LCO, with amplitude and frequency depending on the flow speed U . These findings are consistent with the estimate of U = 9.5 m/s for the flutter speed of the NATA reported in previous works. The precise flutter speed is of limited interest here, as it is sufficient to show that robust LCOs exist for U ≥ 10 m/s in the wing with no NES attached. Concerning the LCO triggering mechanism (see Section 9.2) in the NATA, we noted that there occurred a nonlinear interaction between the two aeroelastic modes (heave and pitch) through a 1:1 resonance capture. Although the experimental dynamics did not involve a stage of superharmonic resonance capture for the particular parameters (i.e., frequency ratio) of the NATA as configured for these tests, it did display a phase-locked frequency shift with time (and thus with increased energy fed into the system from the freestream).

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Fig. 9.68 Experimental responses (gray – without NES; black – with NES) obtained for the lighter NES mass (ms = 1.2 kg) and weaker nonlinear coupling stiffness (ks = 1.6 × 106 N/m2.8 ): (a) initial heave h(0) = 1.27 × 10−2 m; (b) h(0) = 1.91 × 10−2 m; steady state amplitudes are indicated by squares () for NATA without NES, triangles ( ) for NATA and NES with lower damping (cs = 0.015 Ns/m), and circles (◦) for NATA and NES with higher damping (cs = 0.04 Ns/m); minimum and maximum values of modulated amplitudes are indicated by bars (‡); M1, M2, and M3 indicate the type suppression mechanism at work.


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Fig. 9.69 Experimental responses (gray – without NES; black – with NES) obtained for the lighter NES mass (ms = 1.2 kg) and stronger nonlinear coupling stiffness (ks = 2.0 × 106 N/m2.8 ): (a) initial heave h(0) = 1.27 × 10−2 m; (b) h(0) = 1.91 × 10−2 m; steady state amplitudes are indicated by squares () for NATA without NES, triangles ( ) for NATA and NES with lower damping (cs = 0.015 Ns/m), and circles (◦) for NATA and NES with higher damping (cs = 0.04 Ns/m); minimum and maximum values of modulated amplitudes are indicated by bars (‡); M1, M2, and M3 indicate the type suppression mechanism at work.

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Fig. 9.70 Experimental responses (gray – without NES; black – with NES) obtained for the heavier NES mass (ms = 1.5 kg) and weaker nonlinear coupling stiffness (ks = 1.6 × 106 N/m2.8 ): (a) initial heave h(0) = 1.27 × 10−2 m; (b) h(0) = 1.91 × 10−2 m; steady state amplitudes are indicated by squares () for NATA without NES, triangles ( ) for NATA and NES with lower damping (cs = 0.015 Ns/m), and circles (◦) for NATA and NES with higher damping (cs = 0.04 Ns/m); minimum and maximum values of modulated amplitudes are indicated by bars (‡); M1, M2, and M3 indicate the type suppression mechanism at work.


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The dashed lines in Figure 9.68 connect points representing the LCO amplitudes computed for the NATA with of an attached NES with parameters equal to the baseline values mentioned previously. This configuration corresponds to the smaller mass, weaker coupling nonlinear stiffness, and smaller damping coefficient of the values tested for the NES. From the results depicted in that figure, we note significant improvement in flutter speed, from 9 to 11 m/s, irrespective of the initial heave displacement considered. Beyond the flutter speed there is a transition from complete LCO suppression (third suppression mechanism) to partial suppression (i.e., ‘surviving’ LCOs with amplitudes less than those in the NATA with no NES attached – second suppression mechanism). We also note that while the LCO amplitude decays almost exponentially when U = 10 m/s, the decay becomes slower and the dynamics more complicated as the limit U = 11 m/s of the third suppression mechanism is approached, although complete LCO suppression is still achieved in the process. The effects of varying the initial conditions or NES parameters on the response of the integrated NATA-NES system (depicted in the results of Figures 9.68–9.70) are generally in agreement with theoretical predictions, namely, that increasing the NES mass and decreasing its damping or nonlinear stiffness tend to simplify the bifurcation behavior of the steady state dynamics, and thus, enhance the robustness of LCO suppression. However, it is difficult to formulate definitive rules describing these influences because of the high dimensionality of the phase space of the system, and the possible presence of subcritical LCOs, as discussed in Section 9.3 (see also Lee et al., 2007a). Still, based on the results depicted in Figures 9.68–9.70 we arrive at the following conclusions: Stronger nonlinear coupling stiffness ks reduces the dependence of LCO suppression on the amplitude of the initial displacement; weaker nonlinear coupling stiffness can result in higher flutter speeds (i.e., it extends the range of validity of the third suppression mechanism, yielding complete LCO suppression at higher flow speeds); heavier NES damping usually produces better LCO suppression results, although this may come at the expense of increased sensitivity to initial conditions; and finally, larger NES mass improves its TET performance for the smaller initial heave displacement, but yields more complex dynamics (and generally inferior NES performance) for the larger initial condition. This later remark is consistent with our earlier theoretical finding that ungrounded NESs of the type considered in this application (i.e., Configuration II – see Sections 2.6 and 3.1) display better TET performance for small mass ratios, and suggests that the mass of the NES used in this study was rather large compared to the actual optimum value. Of course, because of the complexity of the bifurcation structure of the steady state dynamics introduced by the NES the aforementioned conclusions may not apply under different operating conditions or different NES parameters than those tested in this work. Apart from the experimental affirmation of enhancement of flutter speed in the wing with NES attached, a major result of the experimental work is the verification of the three LCO suppression mechanisms that were theoretically predicted in Section 9.3. Figure 9.71 depicts the experimentally realized first LCO suppression mechanism for U = 11 m/s, the NES with heavier mass, weaker nonlinear cou-

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Fig. 9.71 Experimentally realized first LCO suppression mechanism, for flow speed U = 11 m/s, NES with heavier mass, weaker nonlinear coupling, heavier damping, and h(0) = 1.91 × 10−2 m: (a) transient responses, and (b) wavelet transform spectra; these results correspond to data points in Figure 9.70b.


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Fig. 9.72 Normalized energy exchanges between the aeroelastic modes and the NES during the first LCO suppression mechanism depicted in Figure 9.71.

pling stiffness and heavier damping, and the larger heave initial condition (these results correspond to the data points in Figure 9.70b). In agreement with theory, both aeroelastic modes and the NES response exhibit nonlinear beating, that is, a recurring bursting out of aeroelastic instability followed by suppression. The frequencies of the aeroelastic modes indicate the occurrence of an initial 1:1 transient resonance capture (TRC) around 2 Hz, and later a sustained resonance capture (SRC) at about 3 Hz. The ratio between the NES frequency and that of the heave mode, and the corresponding ratio between the NES and the pitch mode, are nearly 1:2, which suggests that this suppression mechanism consists primarily of recurring transient subharmonic resonance captures between the NES and the aeroelastic modes, while the two aeroelastic modes remain continuously locked in 1:1 SRC. The results depicted in Figure 9.72 indicate that strong energy exchanges between the aeroelastic modes and the NES take place in this case. The instantaneous energies shown in that plot were normalized by the total instantaneous energy of the system. Initially, the triggering mechanism for aeroelastic instability is that discussed in Section 9.2 (see also Lee et al., 2005b), that is, energy from the flow is fed directly into the heave mode, which then excites the pitch mode. After the initial development of instability the NES interacts transiently with the heave mode on the onset of the pitch response (0 < t < 7 s) thus preventing full development of the LCO. Almost half of the total instantaneous energy of the system is transferred to

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the NES within 2 s. Noting that subharmonic resonance capture is generally less efficient than 1:1 resonance capture as far as targeted energy transfer is concerned (see discussion in Sections 3.4.2.1–3.4.2.3), we conjecture that in the responses considered in Figure 9.72 the 1:2 subharmonic resonance captures between the NES and the aeroelastic modes were not sufficient to eliminate the 1:1 internal resonance between these modes (i.e., the triggering of the LCO) and thus prevent the recurrent bursting out of aeroelastic instability. Figure 9.73 depicts the experimentally realized second LCO suppression mechanism, leading to partial LCO suppression; these simulations correspond to flow velocity U = 13 m/s, the NES with lighter mass, weaker nonlinear coupling stiffness and heavier damping, and the smaller initial heave displacement (these results correspond to the data points in Figure 9.68a). After the decay of the initial transients this system exhibits a steady state LCO of reduced amplitude compared to the NATA with no NES attached. Again, there exists strong nonlinear interaction between the heave and pitch modes through 1:1 resonance capture, and the dynamics of the dynamics of the NES undergoes 1:2 subharmonic resonance capture with both aeroelastic modes. The experimentally realized third LCO suppression mechanism, corresponding to complete elimination of the LCO, is shown in Figure 9.74 for a flow speed of U = 10 m/s, the NES with smaller mass, stronger nonlinear coupling and higher damping, and the smaller initial heave displacement. This response is characterized by rapid and complete elimination of the aeroelastic instability. From the WT spectrum depicted in Figure 9.74b, it can be seen that while the aeroelastic modes are again in 1:1 TRC and the NES interacts with these modes in 1:2 subharmonic resonance (as in the previous LCO suppression mechanisms), the NES develops an additional 1:1 resonance interaction with the heave mode. This 1:1 TRC is sufficient to eliminate the aeroelastic instability in this case. Finally, Figure 9.75 depicts another experimental example of complete LCO suppression, i.e., the third LCO suppression mechanism. In this case the flow speed is increased to U = 11 m/s, and the coupling stiffness between the wing and the NES is reduced to its smaller value; as a result, complete LCO suppression is achieved, but only after a modulated initial transient response, which superficially resembles the characteristic LCO bursting and suppression of the first suppression mechanism. However, when the frequency contents of the transient responses are examined, it is found that the NES resonantly interacts with both aeroelastic modes, ultimately overcoming the 1:1 TRC between the heave and pitch modes (i.e., eliminating the LCO triggering mechanism) to stabilize the dynamics. In conclusion, the experiments reported in this section successfully validate the theoretical results of Section 9.3. The experimental results confirm that key to achieving complete LCO suppression by means of passive TET is the design of the NES to resonantly interact with both aeroelastic modes through 1:1 resonance captures. Moreover, increasing the mass of the NES and decreasing its damping and essentially nonlinear stiffness tend, in general, to simplify the bifurcation structure of the steady state dynamics, and thus, to enhance the robustness of LCO suppression. The presented results show that passive TET from the wing to the NES signif-


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Fig. 9.73 Experimentally realized second LCO suppression mechanism, for flow speed U = 13 m/s, NES with lighter mass, weaker nonlinear coupling, heavier damping, and h(0) = 1.27 × 10−2 m: (a) transient responses and (b) wavelet transform spectra; these results correspond to data points in Figure 9.68a.

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Fig. 9.74 Experimentally realized third LCO suppression mechanism, for flow speed U = 10 m/s, NES with lighter mass, stronger nonlinear coupling, weaker damping, and h(0) = 1.27 × 10−2 m: (a) transient responses and (b) wavelet transform spectra; these results correspond to data points in Figure 9.69a.


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Fig. 9.75 Experimentally realized third LCO suppression mechanism, for flow speed U = 11 m/s, NES with lighter mass, weaker nonlinear coupling, weaker damping, and h(0) = 1.27 × 10−2 m: (a) transient responses and (b) wavelet transform spectra; these results correspond to data points in Figure 9.68a.

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icantly improves the stability envelope of the NATA wing, preventing the formation of LCOs for flow speeds up to U = 12 m/s, compared to the flutter speed of 9.5 m/s of the NATA with no NES attached.

9.5 Suppressing Aeroelastic Instability of an In-Flow Wing Using a MDOF NES In this section we consider an alternative design for LCO suppression based on the use of MDOF NESs. We show that simultaneous multi-modal broadband TET from the in-flow wing to the nonlinear normal modes (NNMs) of the MDOF NES improves significantly the robustness of passive aeroelastic instability suppression in this case. In particular, numerical bifurcation analysis of the LCOs of the integrated wing-MDOF NES configuration indicates that the action of the MDOF NES can result in more robust LCO suppression, compared to a SDOF NES of equal total mass. Moreover, compared to the SDOF case, LCO suppression is achieved for lower total mass of the MDOF NES, which from a practical point of view represents an added benefit of the proposed MDOF NES design. In this section we investigate also the nonlinear modal interactions that occur between the aeroelastic modes and the MDOF NES, in an effort to gain a physical understanding of the mechanisms governing aeroelastic instability suppression in this complicated MDOF strongly nonlinear system. Since the issue of robustness of LCO suppression is the main focus of the discussion of this section, we start by revisiting the SDOF NES design of Section 9.3 in order to explore in greater detail its limitations regarding robustness of LCO elimination. This task will be performed by examining the dependence of the bifurcation structure of steady state dynamics on variations of NES parameters, and by relating certain topological features of LCO bifurcations to robustness of instability suppression. Following this preliminary discussion we will be in the position to better assess the benefits in robustness gained by the proposed alternative MDOF NES design.

9.5.1 Revisiting the SDOF NES Design We start by revisiting the SDOF NES aeroelastic suppression design discussed in Section 9.3, and refer to the normalized governing equations of motion (9.71). We recall that the computational study carried out in that section revealed the existence of (at least) three passive LCO suppression mechanisms, resulting in recurring LCO bursting out followed by suppression (mechanism 1), partial LCO suppression (mechanism 2), or complete LCO elimination (mechanism 3). Robustness of LCO suppression can be studied according to whether the three aforementioned TETbased suppression mechanisms can be sustained in the presence of disturbances in parameters and/or initial conditions. In Figure 9.58 we provided a numerical robust-


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ness study by considering the effects on LCO suppression of variations in initial conditions for a system with positive NES offset (δ = 0.9). In Figure 9.76 we depict the results of a similar study for a SDOF NES with negative offset (δ = −0.9), from which we deduce that for the same mass ratio ε, complete elimination of aeroelastic instability can be achieved for a wider set of initial conditions compared to the case of positive offset. Indeed, as Figure 9.76b indicates, complete LCO elimination may be possible over the entire domain of initial conditions considered in the present study. As discussed in Section 9.3.3, the problem of robustness of LCO suppression is closely linked to the bifurcation structure of the steady state nonlinear dynamics of the integrated wing-NES configuration. To remind ourselves of the results reported in Section 9.3.3, in Figure 9.77 we provide an example demonstrating the link between robustness of LCO suppression and bifurcations of steady state dynamics. In particular, in Figure 9.77a we depict the bifurcation diagrams of systems with positive and negative offsets, utilizing the reduced flow speed as bifurcation parameter. These diagrams illustrate clearly the NES configurations that lead to more robust suppression of instability. For example, we note that an NES with either positive or negative offset causes a delay of the occurrence of LCOs by shifting the Hopf bifurcation point H (indicated by squares in Figure 9.77a) to higher reduced speeds. This, however, in itself does not necessarily imply enhancement of robustness of LCO suppression for both NES configurations. Indeed, considering the system with positive offset, the LPC bifurcation point close to pitch amplitude 0.1 (indicated by a triangle in Figure 9.77a) occurs before the Hopf bifurcation, LP C2 < H . It follows that for reduced speeds in the range LP C2 < < BP C < H the stable trivial equilibrium coexists with a stable large-amplitude LCO, thus preventing < H the LCO suprobust LCO suppression; whereas, in the range BP C < pression is robust since the trivial equilibrium is the only stable attractor of the dynamics. On the contrary, for the system with negative offset it is possible to achieve robust and complete LCO suppression in the range < H , as the stable trivial equilibrium is a global attractor of the dynamics in this range; in addition, in the range H < < LP C2 partial LCO suppression is robust as well, as there exists global attraction of the dynamics by a stable, low-amplitude LCO. In Figure 9.77b we demonstrate non-robustness and destruction of the first LCO suppression mechanism for the system with negative offset and reduced speed = 0.95, by applying an impulsive disturbance to the heave mode. This result can be understood by considering the respective ‘slice’ of the bifurcation diagram of Figure 9.77a corresponding to = 0.95. We deduce that the quasi-periodic LCO (which is responsible for the repeated LCO burst-outs and suppressions before the external disturbance is applied) coexists with an unstable low-amplitude LCO and a stable large-amplitude LCO resulting from a branch point cycle bifurcation (i.e., for > BPC ). It follows that by applying disturbances to the initial conditions it is possible to drive the dynamics out of the domain of attraction of the quasi-periodic LCO (i.e., to eliminate the first suppression mechanism), and into the domain of attraction of the stable large-amplitude LCO. This yields the reappearance of strong aeroelastic instability in the system, in spite the action of the NES.

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Fig. 9.76 SDOF NES design, steady-state r.m.s. pitch amplitude ratio α , with respect to the set of initial conditions {y(0) = 0, α(0) = 0, y (0) ∈ [0, 0.1], α ∈ [0, 0.1]}, δ = −0.9 and = 0.9: (a) ε = 0.01, λ = 0.2, and C = 30; (b) ε = 0.02, λ = 0.4, and C = 1.


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Fig. 9.77 SDOF NES design, robustness study for ε = 0.02, λ = 0.4, C = 10.0: (a) (incomplete) bifurcation diagrams for systems with offsets δ = ±0.75, solid (dashed, dash-dotted) lines indicate stable (unstable, quasi-periodic) LCOs, and squares (circles, triangles, asterisks) indicate Hopf (Neimark–Sacker, LPC, BPC) bifurcation points; (b) effect of an impulsive disturbance on the heave mode applied at τ = 300 for the system with = 0.95 and δ = −0.75.

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Fig. 9.77 SDOF NES design, robustness study for ε = 0.02, λ = 0.4, C = 10.0: (c) effect of an impulsive disturbance on the heave mode applied at τ = 250 for the system with = 0.915 and δ = −0.75.

A different scenario is provided in Figure 9.77c, where robustness of the second LCO suppression mechanism is demonstrated for the system with negative offset and reduced speed = 0.915. In this case we note that after the application of the impulsive external disturbance to the heave mode the system returns to its prior state, i.e., to a small-amplitude LCO. Again, this result may be interpreted by examining the ‘slice’ = 0.915 of the corresponding bifurcation diagram of Figure 9.77a. Indeed, we note that for that value of the reduced speed the stable LCO generated after the Hopf bifurcation is the only attractor of the dynamics, as it only co-exists with the unstable trivial equilibrium. It follows that the small-amplitude LCO is the only possible stable steady state solution of the dynamics for that particular reduced flow speed. These results demonstrate that the bifurcation structure of LCOs (and the corresponding nonlinear resonant interactions between the aeroelastic modes and the NES) affects the robustness of instability suppression in the system with the SDOF NES. We recall from Section 9.3.3 that a Hopf bifurcation point signifies the generation of stable LCOs from the trivial equilibrium point. Since in the wing with no NES attached a Hopf bifurcation signifies the generation of aeroelastic instability, it follows that by comparing the relative placement of this point with respect to the corresponding point of the system with NES attached we can obtain a first quantitative measure of the stabilizing effect on the dynamics of the action of the


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NES. On the other hand the LPC bifurcation occurring at = LPC1 represents the critical point above which no LCO suppression is possible by the action of the SDOF NES. As mentioned in the previous example, the location of the other LPC bifurcation point = LPC2 relative to the Hopf bifurcation point = H affects the robustness of instability suppression in the system with SDOF NES attached. Being interested in the location of this LPC bifurcation point relative to the Hopf bifurcation point, in the following exposition we extend the bifurcation analysis of Section 9.3.3 to three-dimensional parameter spaces. On the other hand, a NS bifurcation point generates quasi-periodicity in the dynamics, and is responsible for the recurrent burst-outs and suppressions of aeroelastic instability (i.e., for the first LCO mechanism). In general, a NS bifurcation occurs between a Hopf bifurcation and the higher-speed LPC bifurcation at = LPC1 . Finally, the BPC bifurcation (which is a codimension-2 bifurcation of an LCO) generally occurs at higher fluid velocities that the first LPC bifurcation at = LPC2 and introduces a destabilizing effect in the dynamics, as it gives rise to stable, largeamplitude LCOs. Again, in the following analysis we will be interested in the variations of all these bifurcation points as the parameters of the SDOF NES vary. In Figures 9.78a–c we present bifurcation sets of steady state dynamics of the inflow wing with a SDOF NES attached in the three-dimensional parameter spaces (ε, δ, ), (λ, δ, ) and (C, δ, ), respectively. These bifurcation diagrams can be understood in the following way. We recall that in the bifurcation diagram of Figure 9.77a, the NES parameters were fixed to the values ε = 0.02, λ = 0.4 and C = 10.0. Considering the two-dimensional ‘slice’ ε = 0.02 of the threedimensional bifurcation structure of Figure 9.78a, we obtain the bifurcation diagrams of Figure 9.79, where three curves can be identified resulting from the intersection of the plane of constant mass ratio ε with the three two-dimensional surfaces corresponding to the Hopf and the two LPC bifurcations. If we are interested in studying the bifurcation structure of the dynamics when the SDOF NES is attached at offsets δ = ±0.75, additional two-dimensional ‘slices’ (planes) corresponding to constant offsets should be imposed, as depicted in Figures 9.79a and 9.79b, respectively. Once these two planes (‘slices’) are constructed in the three-dimensional bifurcation diagrams, we can add vertical axes (depicted with solid lines in Figures 9.79a, b) starting at the Hopf bifurcation points = H and being orthogonal to the plane ε = 0.02; on these axes we may represent steady state LCO amplitudes, in which case the bifurcation diagram of Figure 9.77a can be drawn for δ = ±0.75. Moreover, the two dashed vertical lines passing through = LPC1 and = LPC2 in Figures 9.79a, b correspond to points A1,2 and B1,2 , respectively. Points A1,2 (B1,2 ) are obtained as intersections of the planes ε = 0.02, δ = ±0.75, and the LPC1 (LPC2) two-dimensional bifurcation surface. With these constructions, compact information regarding the bifurcation behavior of the dynamics on the three-dimensional parameter spaces of Figures 9.78a–c can be displayed. Note that, in order to clarify the robustness features of the SDOF NES design, the NS and BPC bifurcation surfaces are not included in these diagrams, and neither are additional bifurcations of co-dimension two or of higher co-dimension. Recall

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(a)

(b)

(c) Fig. 9.78 Bifurcation diagrams of steady state dynamics of the in-flow wing with an attached SDOF NES: (a) effect of varying the mass ratio ε for λ = 0.4 and C = 10.0 (diamonds on the LPC2 bifurcation surface imply optimal offset locations for enhancing robustness of LCO suppression); (b) effect of varying damping λ for ε = 0.02 and C = 10.0; (c) effect of varying the coefficient of the essential nonlinearity C for ε = 0.02 and λ = 0.4.


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(a)

(b) Fig. 9.79 In-flow wing with SDOF attached, understanding the three-dimensional bifurcation sets for ε = 0.02 and (a) δ = −0.75, (b) δ = 0.75.

that, in the SDOF NES design, a zero offset (δ = 0) does not affect the aeroelastic responses, since in that case the bifurcation structure coincides with that of the wing with no NES attached. This can be inferred also from the bifurcation results of Figures 9.78a–c, since as δ → 0 the two LPC bifurcation surfaces tend to coalesce with the Hopf bifurcation surface.

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In the following, we provide some remarks regarding the effects of varying the NES mass ratio (ε), damping (λ), and essential nonlinearity (C) on LCO suppression, for fixed offset δ and reduced fluid velocity . Focusing first on the effect of varying the NES mass ratio ε, and referring to Figure 9.78a, we note that this ratio contributes significantly to the shift of the Hopf bifurcation point towards higher reduced velocities (this is contrary to the common view that damping has a greater influence towards that effect). In addition, as the NES mass ratio ε increases along with the offset δ away from the elastic axis, the Hopf and LPC bifurcation points exhibit a monotonic shift towards higher reduced velocities. In terms of robustness of LCO suppression (which depends on the location of the bifurcation point LPC2 relatively to the Hopf bifurcation point), an optimal interval of offset values exists for each mass ratio. That is, if the SDOF NES is attached too far away from the elastic axis, the bifurcation point LPC2 may be realized very close to the Hopf bifurcation point for negative offsets. Even worse from the point of view of robustness, for positive offsets the point LPC2 can be realized at smaller reduced velocities than the Hopf bifurcation point. As was conjectured in Section 9.3, if the NES possesses larger mass, the robustness of LCO suppression may improve within the optimal offset interval. The points denoted by diamonds in Figure 9.78a indicate optimal offsets from the point of view of optimally enhanced robustness of aeroelastic instability suppression. The above-mentioned conclusions are confirmed by the series of bifurcation diagrams presented in Figure 9.80, which depict the steady state amplitude of the pitch mode as function of NES parameters for six values of the mass ratio ε. We note that as the mass ratio increases, the LPC bifurcation curves are realized at increasingly higher pitch amplitudes. This observation indirectly explains why robust suppression of LCO instability is barely observed for very small values of NES mass. Indeed, for small NES masses (see Figure 9.80a) the Hopf bifurcation curve appears to be nearly flat and to lie parallel to the offset axis; this indicates that in this case the offset of the SDOF NES does not significantly affect the critical reduced speed of Hopf bifurcation. Moreover, for small NES masses partially-suppressed LCO branches exist (corresponding to the second LCO suppression mechanism), but they are realized over small intervals of reduced velocities; in addition, small-amplitude LCOs on those branches are vulnerable to disturbances, after which the dynamics may undergo transitions to large-amplitude LCOs, rendering the LCO suppression non-robust. Finally, regarding the overall topology of the LCO branches of Figure 9.80a, we note that, apart from the LCO branches leading to partial instability suppression (second LCO suppression mechanism) or recurring burst-outs and suppressions of instability (first LCO suppression mechanism), it resembles the topology of the bifurcation diagram of the system with no NES attached. Although not as significant as the effect of the mass ratio, increasing the NES damping coefficient λ improves the flutter speed (see Figure 9.78b). The overall robustness behavior of LCO suppression seems to be similar to that obtained when increasing the mass ratio. That is, there exist optimal intervals of the offset for which enhanced robustness of LCO suppression is realized, with negative offsets resulting in improved LCO suppression performance. As depicted in Figure 9.81,


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Fig. 9.80 In-flow wing with SDOF NES attached, bifurcation plots of steady state dynamics for varying mass ratio ε and λ = 0.4, C = 10.0: (a) ε = 0.005, (b) ε = 0.014, (c) ε = 0.023, (d) ε = 0.032, (e) ε = 0.041, (f) ε = 0.05; bold solid (dashed, dash-dotted) line denotes a Hopf (LPC1, LPC2) bifurcation curve.

when damping is increased the LPC bifurcation curves shift towards higher LCO amplitudes. Moreover, low NES damping (see Figure 9.81a) results in inefficient dissipation of the energy transferred from the wing to the NES, which tends to yield non-robust LCO suppression over the reduced fluid velocities of interest.

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Fig. 9.81 In-flow wing with SDOF NES attached, bifurcation plots of steady state dynamics for varying damping λ and ε = 0.02, C = 10.0: (a) λ = 0.1, (b) λ = 0.238, (c) λ = 0.376, (d) λ = 0.514; bold solid (dashed, dash-dotted) line denotes a Hopf (LPC1, LPC2) bifurcation curve.

Considering the effect of varying the coefficient C of the essential stiffness nonlinearity (see Figure 9.78c), we note that the Hopf and LPC bifurcation manifolds seem to be nearly insensitive to variations of this coefficient. It follows that the robustness of LCO instability suppression is not significantly affected by the coefficient of the essential nonlinearity, provided that this coefficient is sufficiently large. In fact, the main role of the essential nonlinearity is to induce broadband nonlinear resonance interaction between the wing and SDOF NES. This conclusion is supported by the bifurcation diagrams of Figure 9.82, where the topologies of all bifurcation diagrams appear to be similar, with the possible exception of the diagram corresponding to small C (C = 1 – see Figure 9.82a). A minor distinction between the bifurcation diagrams is that the LPC bifurcation curves are suppressed to lower pitch amplitudes as C increases. The previous results identify clearly certain limitations of the SDOF NES design from the point of view of robustness of LCO suppression. As a way to enhance robustness we will consider an alternative design based on the use of MDOF NESs, which as shown in Chapters 4 and 5 have the potential to yield enhanced and broadband TET performance. In the next section we will perform a numerical study of the


531

Fig. 9.82 In-flow wing with SDOF NES attached, bifurcation plots of steady state dynamics for varying essential nonlinearity C and ε = 0.02, λ = 0.4: (a) C = 1.0, (b) C = 4.8, (c) C = 8.6, (d) C = 12.4, (e) C = 16.2, (f) C = 20.0; bold solid (dashed, dash-dotted) line denotes a Hopf (LPC1, LPC2) bifurcation curve.

bifurcation structures of the steady state dynamics of an integrated wing – MDOF NES system, in an effort to demonstrate that the MDOF NES can yield considerable improvement of LCO suppression robustness, as a direct result of enhanced passive broadband TET.

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9.5.2 Configuration of a Wing with an Attached MDOF NES The alternative MDOF NES configuration attached to the in-flow wing is depicted in Figure 9.83 (Lee et al., 2008). The MDOF NES consists of three particles coupled in series by essentially nonlinear stiffnesses lying in parallel to viscous dampers; moreover, the NES is coupled to the wing through a linear coupling stiffness. This configuration is identical to the MDOF NES design studied in detail in Chapter 4. Recalling the results of that study we anticipate that the highly degenerate structure of the MDOF NES dynamics will introduce additional features in the dynamics of the integrated wing-NES system, some of which will end up being beneficial to the task of robust passive LCO suppression. Assuming small motions and quasi-steady fluid-structure interaction the equations of motion of this system are expressed in the following form: ˙ ) + k(h − dα − z1 ) = 0 mh¨ + Sα α¨ + Kh (h + c1 h3 ) + qSCL,α (α + h/U ˙ ) + dk(dα + z1 − h) = 0 Iα α¨ + Sα h¨ + Kα (α + c2 α 3 ) − qeSCL,α (α + h/U (ms /3) z¨ 1 + cs (˙z1 − z˙ 2 ) + k(z1 + dα − h) + ks (z1 − z2 )3 = 0 (ms /3) z¨ 2 + cs (˙z2 − z˙ 1 ) + cs (˙z2 − z˙ 3 ) + (ks /50)(z2 − z3 )3 + ks (z2 − z1 )3 = 0 (ms /3) z¨ 3 + cs (˙z3 − z˙ 2 ) + (ks /50)(z3 − z2 )3 = 0

(9.90)

or in non-dimensional form, y + xα α + 2 y + ξy y 3 + µCL,α (y +

α) + C1 (y − δα − v1 ) = 0

rα2 α + xα y + rα2 α + ξα α 3 − γ µCL,α (y +

α) + δC1 (δα + v1 − y) = 0

(ε/3) v1 + ελ(v1 − v2 ) + C1 (v1 + δα − y) + C(v1 − v2 )3 = 0 (ε/3) v2 + ελ(v2 − v1 ) + ελ(v2 − v3 ) + C(v2 − v1 )3 + (C/50)(v2 − v3 )3 = 0 (ε/3) v3 + ελ(v3 − v2 ) + (C/50)(v3 − v2 )3 = 0

(9.91)

where the notation of Sections 9.2 and 9.3 applies for (9.90) and (9.91), and the additional new normalized parameters are defined as follows: C1 = k/mωa2 ,

v1 = z1 /b,

v2 = z2 /b,

v3 = z3 /b

Some interesting features of the MDOF NES design are now discussed. First, we note that the total mass of the MDOF NES is set to be identical to the SDOF NES considered in previous section; that is, in the new design the single normalized mass ε of the SDOF NES is divided into the three masses equal to ε/3 in the MDOF NES. This allows us to make direct comparisons of the LCO suppression capacities of the two designs without any added-mass effects. Second, there exists a linear coupling stiffness between the wing structure and the first mass of the NES; as discussed


533

Fig. 9.83 Two-DOF rigid wing model with an attached MDOF NES.

in Chapter 4, weaker linear coupling facilitates broadband TET from the primary structure (in this case, the wing) to the MDOF NES, and it would be interesting to examine if this enhanced TET yields enhanced and more robust LCO suppression compared to the SDOF NES design. Finally, we make the remark that the upper essentially nonlinear stiffness (which couples the first and second masses) of the NES is chosen to be much stiffer than the lower nonlinear stiffness (which connects the second and third NES masses). This relative scaling is motivated by the results presented in Chapter 4, where it was found that a stiffer upper nonlinear spring enables enhanced nonlinear resonance (modal) interactions between the MDOF NES and the modes of the primary structure, whereas a weaker lower nonlinear spring yields large-amplitude relative motions between the second and the third masses of the NES, and thus facilitates strong dissipation of the energy transferred to the NES from the primary structure through TET. An indication of the improvement in robustness of LCO suppression achieved by the action of the MDOF NES is obtained when considering the preliminary numerical results depicted in Figure 9.84, where the steady state r.m.s. pitch amplitude ratio

α is computed for various combinations of initial conditions and system parameters. These results should be compared to similar results for the SDOF NES design depicted in Figures 9.58 and 9.76. We note that when using the MDOF NES the domain of initial conditions where complete LCO suppression is realized expands,

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Fig. 9.84 MDOF NES design, steady state r.m.s. pitch amplitude ratio α , with respect to the set of initial conditions {y(0) = 0, α(0) = 0, y (0) ∈ [0, 0.1], α (0) ∈ [0, 0.1]}, δ = 0.9, C1 = 0.01: (a) = 0.9, ε = 0.01, λ = 0.02, C = 30.0; (b) = 0.9, ε = 0.02, λ = 0.4, C = 10.0.


535

Fig. 9.84 MDOF NES design, steady state r.m.s. pitch amplitude ratio α , with respect to the set of initial conditions {y(0) = 0, α(0) = 0, y (0) ∈ [0, 0.1], α (0) ∈ [0, 0.1]}, δ = 0.9, C1 = 0.01: (c) = 0.95, ε = 0.01, λ = 0.2, C = 30.0; (d) = 0.95, ε = 0.02, λ = 0.4, C = 10.0.

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even for small total NES mass. For example, the simulations of Figure 9.84a correspond to ε = 0.01, i.e., for NES mass equaling 1% of the total wing mass. Moreover, for a mass ratio ε = 0.02 (see Figure 9.84d) the action of the MDOF NES can completely eliminate aeroelastic instability over the entire domain of initial conditions of interest, even for the reduced velocity = 0.95, which is a relatively high flow speed that produces large-amplitude LCOs (above 0.25 rad in the pitch mode) in the wing without NES. The three LCO suppression mechanisms identified in the SDOF NES design can be detected in the MDOF NES design as well. However, as will become apparent from the following bifurcation analysis, the action of the MDOF NES tends to suppress the occurrence of Neimark–Sacker (NS) bifurcations which lead to quasiperiodic LCOs. As a result, the first LCO suppression mechanism (corresponding to recurrent series of LCO burst-outs and suppressions) is rarely realized in this case. In fact, we conjecture that the MDOF NES renders the NS bifurcation highly degenerate; for example, neutral-saddle singularities (Kuznetsov, 1995) may occur instead, for which the numerical continuation method may fail to accurately compute the resulting bifurcating LCO branches. The rare occurrence of the first LCO suppression mechanism implies that partial or complete suppression of aeroelastic instabilities (i.e., the second and third suppression mechanisms, respectively) will be the main LCO suppression mechanisms observed in the MDOF NES design.

9.5.3 Robustness of LCO Suppression – Bifurcation Analysis The MATCONT numerical continuation technique (Dhooge et al., 2003) was utilized to construct bifurcation diagrams of steady state dynamics (LCOs) of the system (9.91). As discussed in previous sections, these diagrams are important for assessing the robustness of LCO suppression due to the action of the MDOF NES, and for comparing the relative performances of the SDOF and MDOF NES designs. In the following numerical bifurcation study we consider two cases of linear coupling stiffness between the wing and the MDOF NES, namely, C1 = 0.1 (designated as strong coupling) and C1 = 0.01 (designated as weak coupling). In addition, the overall MDOF NES parameters are chosen to be identical to those of the SDOF NES considered in Section 9.5.1, so that direct comparisons between the two NES designs make sense and can be performed.

9.5.3.1 Case of Strong Coupling In Figures 9.85a–c we depict the two-dimensional manifolds of Hopf and LPC bifurcations for the system with strong linear coupling between the wing and the MDOF NES. These diagrams are constructed with respect to the total NES mass ratio ε, the damping coefficient λ, and the coefficient C of the essential nonlinearity, respec-


537

Fig. 9.85 MDOF NES design, sets of Hopf and LPC bifurcations for (a–c) strong linear coupling (C1 = 0.1), and (d–f) weak linear coupling (C1 = 0.01): (a, d) effect of mass ratio ε for λ = 0.4, C = 10.0; (b, e) effect of damping λ for ε = 0.02, C = 10.0; (c, f) effect of essential nonlinearity C for ε = 0.02, λ = 0.4; bifurcation points LPC1 are denoted by triangles (connected by solid line), and bifurcation points LPC2 by squares (connected by dashed line).

tively. The corresponding branches of steady state dynamics (LCOs) for selected parameters are depicted in Figures 9.86–9.88.

538


Fig. 9.86 MDOF NES design, branches of steady state dynamics (LCOs) with respect to the mass ratio ε for strong linear coupling C1 = 0.1 and λ = 0.4, C = 10.0: (a) ε = 0.005; (b) ε = 0.014; (c) ε = 0.023; (d) ε = 0.032; (e) ε = 0.041; (f) ε = 0.05; bold solid line denotes Hopf bifurcations curve, triangles (squares) LPC1 (LPC2) bifurcations, and asterisks degenerate bifurcation points such as ‘neutral-saddles’ or ‘generalized Hopf bifurcations’ (Kuznetsov, 1995).


539

Fig. 9.87 MDOF NES design, branches of steady state dynamics (LCOs) with respect to the viscous damping coefficient λ for strong linear coupling C1 = 0.1 and ε = 0.02, C = 10.0: (a) λ = 0.1; (b) λ = 0.169; (c) λ = 0.238; (d) λ = 0.307; (e) λ = 0.376; (f) λ = 0.445; bold solid line denotes Hopf bifurcations curve, triangles (squares) LPC1 (LPC2) bifurcations, and asterisks degenerate bifurcation points such as ‘neutral-saddles’ or ‘generalized Hopf bifurcations’ (Kuznetsov, 1995).

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Fig. 9.88 MDOF NES design, branches of steady state dynamics (LCOs) with respect to the coefficient of the essential nonlinearity C for strong linear coupling C1 = 0.1 and ε = 0.02, λ = 0.4: (a) C = 2.9; (b) C = 6.7; (c) C = 10.5; (d) C = 14.3; (e) C = 18.1; (f) C = 20.0; bold solid line denotes Hopf bifurcations curve, triangles (squares) LPC1 (LPC2) bifurcations, and asterisks degenerate bifurcation points such as ‘neutral-saddles’ or ‘generalized Hopf bifurcations’ (Kuznetsov, 1995).


541

We note that the manifolds of Hopf bifurcations for the system with MDOF NES with strong linear coupling stiffness appear to have similar topological structures for varying NES mass ratio, damping and essential nonlinearity to those of the case of SDOF NES. That is, for fixed offset the critical reduced velocity for Hopf bifurcation (i.e., the flutter speed) increases monotonically with increasing NES mass ratio ε, and seems to be nearly insensitive to variations of damping and essential nonlinearity. This is due to the fact that for strong linear coupling stiffness the upper mass of the MDOF NES is almost rigidly connected to the wing. We note that strong linear coupling may facilitate strong linear resonant interactions between the wing modes and the MDOF NES, and it causes stronger shift of the point of Hopf bifurcation towards higher reduced speeds compared to the SDOF design, even when zero offset is considered. The fact that the Hopf bifurcation points increase monotonically with increasing NES mass ratio ε simply means that a larger NES mass provides better suppression of aeroelastic instability in the strongly coupled MDOF NES configuration (see Figure 9.85a). Moreover, for large offsets the curves of LPC1 and LPC2 bifurcations are closely spaced and are realized at higher reduced velocities than the flutter speeds (i.e., the Hopf bifurcation points). This implies that the unstable LCO branches between the two LPC points exist over short intervals of the reduced velocity, so that the manifolds of LCOs in Figure 9.86 appear as surfaces with no turning points (see Figure 9.86). In addition, as the mass ratio increases the bifurcation point LPC2 is realized at higher fluid speeds than the Hopf bifurcation point (the flutter speed). The interval between these two bifurcation points corresponds to the range of reduced velocities where robust suppression of aeroelastic instability occurs. We note, however, that the nearly linear monotonic increase of the speed for Hopf bifurcation with respect to the mass ratio may not be attractive for practical applications since we generally require light weight for the NES. Hopf bifurcations are mostly of the supercritical type (Guckenheimer and Holmes, 1983), that is, stable LCOs are generated after the Hopf bifurcations, below which there exist only stable trivial equilibrium points (see Figure 9.86). However, Hopf bifurcations near the point of zero offset are of the subcritical type in short intervals of reduced velocities (for example, at δ = −0.2 in Figures 9.85a–c). The LPC1 bifurcation points are aligned almost vertically for all NES parameters near that range of offset values, and they hardly affect the overall robustness of instability suppression. The branches of LCOs depicted in Figure 9.86 support the above arguments. For example, similarly to the SDOF NES, the LPC bifurcations occur at higher amplitudes as the mass ratio increases. Furthermore, the LPC bifurcation points tend to move away from the line of zero offset as the mass ratio increases. Higher damping values shift the LPC bifurcation points to larger offsets (see Figure 9.87). High damping also suppresses the occurrence of bifurcations of LCOs and makes the LCO branches smoother. That is, the two LPC bifurcation points merge at larger offsets as damping increases. Thus, optimal offset intervals for robust instability suppression can be identified for smaller damping coefficients. Figure 9.85b depicts the bifurcation sets in the ( , δ, λ) parameter space. As in the case of the

542


SDOF NES, the manifold of points of Hopf bifurcation exhibits less dependence on the damping value compared to its dependence on the total mass ratio ε. Whereas the manifold of Hopf bifurcations still exhibits insensitivity to changes in the coefficient of the essential nonlinearity C (see Figure 9.85c), the manifolds of LPC bifurcations become separated from each other for large offsets as Cincreases. In fact, larger values of C induce two LPC bifurcation points at lower LCO amplitudes (see Figure 9.88). Similar to the SDOF NES design, negative offsets provide better LCO suppression capacity for the MDOF NES design with strong coupling stiffness; moreover, the LCO suppression is robust in that case.

9.5.3.2 Case of Weak Coupling Weak coupling stiffness between the MDOF NES and the wing removes the monotonic dependence of the position of the Hopf bifurcation point on the mass ratio ε, and shifts the points of Hopf bifurcations (i.e., the points of generation of LCOs) to higher reduced velocities (see Figure 9.85d). An optimal mass ratio can be found near the value ε = 0.02, shifting the point of generation of LCOs to the highest values of reduced speed. Then, the locations of the LPC bifurcation points relative to the Hopf bifurcation points provide an indication of the robustness of instability suppression. A single LPC bifurcation point at lower reduced flow velocity than the Hopf bifurcation point implies that that the Hopf bifurcation is of the subcritical type. It is interesting to note that for negative offsets and in the vicinity of the aforementioned optimal mass ratio, most of LPC bifurcations occur at lower reduced fluid velocities than the Hopf bifurcation points; and at higher reduced velocities, for positive offsets. This behavior appears for varying values of damping and essential nonlinearity, and most LPC bifurcations occur at reduced speeds lower (higher) than the Hopf bifurcation points for negative (positive) offsets. Moreover, the distance between the LPC1 and LPC2 bifurcation points is maximized near the mass ratio where optimal shift to higher fluid velocity of the Hopf bifurcation point is achieved. This means that the robustness of instability suppression may not be optimally enhanced at the optimal mass ratio. Figure 9.89 depicts the branches of steady state dynamics (LCO surfaces) for selective values of the mass ratio of the MDOF NES with weak linear coupling stiffness. Comparing the manifolds of Hopf bifurcations for ε = 0.014 and ε = 0.023 (Figures 9.89b and 9.89c, respectively), we conclude that the value of ε = 0.023 is optimal in terms of shifting the points of Hopf bifurcation at higher reduced velocities. However, these Hopf bifurcations are of the subcritical type, as indicated by the fact that the LPC bifurcation points are realized at lower reduced velocities. On the other hand, the mass ratio ε = 0.014 shifts the Hopf bifurcation points by lesser amounts, but these Hopf bifurcations are supercritical, with the LPC bifurcation points occurring at higher reduced velocities. This is more desirable from a practical point of view, as less NES mass is seen to yield better performance in terms of robustness of instability suppression.


543

Fig. 9.89 MDOF NES design, branches of steady state dynamics (LCOs) with respect to the mass ratio ε for weak linear coupling C1 = 0.01 and λ = 0.4, C = 10.0: (a) ε = 0.005; (b) ε = 0.014; (c) ε = 0.023; (d) ε = 0.032; (e) ε = 0.041; (f) ε = 0.05; bold solid line denotes Hopf bifurcations curve, triangles (squares) LPC1 (LPC2) bifurcations, and asterisks degenerate bifurcation points such as ‘neutral-saddles’ or ‘generalized Hopf bifurcations’ (Kuznetsov, 1995).

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Fig. 9.90 MDOF NES design, branches of steady state dynamics (LCOs) with respect to the viscous damping coefficient ë for weak linear coupling C1 = 0.01 and ε = 0.02, C = 10.0: (a) λ = 0.1; (b) λ = 0.169; (c) λ = 0.238; (d) λ = 0.307; (e) λ = 0.376; (f) λ = 0.445; bold solid line denotes Hopf bifurcations curve, triangles (squares) LPC1 (LPC2) bifurcations, and asterisks degenerate bifurcation points such as ‘neutral-saddles’ or ‘generalized Hopf bifurcations’ (Kuznetsov, 1995).


545

Fig. 9.91 MDOF NES design, branches of steady state dynamics (LCOs) with respect to the coefficient of the essential nonlinearity C for weak linear coupling C1 = 0.01 and ε = 0.02, λ = 0.4: (a) C = 2.9; (b) C = 6.7; (c) C = 10.5; (d) C = 14.3; (e) C = 18.1; (f) C = 20.0; bold solid line denotes Hopf bifurcations curve, triangles (squares) LPC1 (LPC2) bifurcations, and asterisks degenerate bifurcation points such as ‘neutral-saddles’ or ‘generalized Hopf bifurcations’ (Kuznetsov, 1995).

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Whereas the manifold of Hopf bifurcations still exhibits near independence with respect to the coefficient of the essential nonlinearity (see Figure 9.85f), it becomes more dependent on the damping coefficient as noted from the plot of Figure 9.85e. Referring to the variation of the LCO branches with respect to damping and essential nonlinearity (see Figures 9.90 and 9.91), we note that for positive offsets the LPC bifurcations occur either at higher reduced velocities or close to the Hopf bifurcation points; this indicates that the Hopf bifurcations are mainly of the supercritical type, and if they are subcritical their effect on robustness is negligible. For negative offsets the LPC bifurcations occur at much lower reduced velocities. Subcritical LCOs become more prevalent for higher values of damping and essential nonlinearity.

9.5.3.3 Robustness Enhancement From the previous bifurcation analysis we conclude that the MDOF NES with strong linear coupling (whose dynamics is discussed in Section 9.5.3.1) behaves similarly to a SDOF NES, with the exception of improved performance in terms of aeroelastic instability suppression. Regarding the MDOF NES with weak linear coupling, when appropriately optimized (in terms of offset, stiffness and mass parameters), it may yield efficient and robust passive LCO suppression even when it is lightweight. In this section, we demonstrate the enhancement in robustness of instability suppression achieved by the MDOF NES design with weak coupling, by comparing its performance to the SDOF NES design with corresponding parameters. Furthermore, we analyze the underlying dynamic mechanisms governing TET from wing modes to the MDOF NES, as well as the resulting nonlinear modal energy exchanges between these subsystems. In Figure 9.92 we provide a direct comparison of the bifurcation diagrams of systems with MDOF and SDOF NESs for various parameter sets. These diagrams examine the dependence of the steady state pitch amplitude of the wing on the reduced fluid velocity , when (i) no NES is attached; (ii) a SDOF NES with mass ratio ε = 0.02 and offset δ = −1 is attached; and (iii) a MDOF NESs with varying total mass ratio ε and offsets δ = ±1 is attached. In these bifurcation diagrams all other NES parameters such as damping, essential nonlinearity, and weak linear coupling stiffness are kept fixed to the values λ = 0.4, C = 10.0 and C1 = 0.01. The SDOF NES with ε = 0.02 (i.e., 2% ratio of the NES mass with respect to the wing mass) and δ = −1 exhibits good performance in suppressing the aeroelastic instability. Moreover, it yields robust suppression up to the reduced velocity ≈ 0.91. However, the MDOF NES with even smaller total mass ratio ε = 0.005 (0.5% overall mass ratio – see Figure 9.92a) provides similar or slightly better suppression results. If the total mass of the MDOF NES increases to ε = 0.0095 (i.e., slightly below 1% of the wing mass), the robustness enhancement (as denoted by the interval RE in Figure 9.92b) becomes pronounced, extending the regime of robust LCO suppression up to ≈ 0.94 for δ = 1, and ≈ 0.96 for δ = −1. Finally, by increasing the total mass ratio of the MDOF NES to ε = 0.014 (see Figure 9.92c – still less than the SDOF NES mass ratio of 2%), robustness enhancement


547

Fig. 9.92 Bifurcation diagrams of the wing with SDOF and MDOF NESs (with weak linear coupling) for λ = 0.4, C = 10.0 C1 = 0.01, and varying offsets and mass ratios: the SDOF NES is considered for mass ratio ε = 0.02 and offset δ = −1; the MDOF NES is considered for varying total mass ratios and offsets, (a) ε = 0.005, δ = ±1, (b) ε = 0.0095, δ = ±1; dashed lines indicate unstable LCO branches, and squares (circles, triangles, diamond, asterisks) indicate Hopf (Neimark–Sacker, LPC, BPC, neutral-saddle) bifurcation points; the intervals indicated by RE provide measures of enhancement of robustness of LCO suppression.

548


Fig. 9.92 Bifurcation diagrams of the wing with SDOF and MDOF NESs (with weak linear coupling) for λ = 0.4, C = 10.0 C1 = 0.01, and varying offsets and mass ratios: the SDOF NES is considered for mass ratio ε = 0.02 and offset δ = −1; the MDOF NES is considered for varying total mass ratios and offsets, (c) ε = 0.014, δ = ±1, (d) ε = 0.0274, δ = ±1; dashed lines indicate unstable LCO branches, and squares (circles, triangles, diamond, asterisks) indicate Hopf (Neimark–Sacker, LPC, BPC, neutral-saddle) bifurcation points; the intervals indicated by RE provide measures of enhancement of robustness of LCO suppression.


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becomes optimal, with the region of robust suppression extending up to ≈ 1.0 for δ = 1, and up to ≈ 1.01 for δ = −1. We note that for these values of reduced velocities the results are out of the realm of the current aeroelastic model (based on quasi-steady approximation of fluid-structure interaction), since the amplitudes of the uncontrolled aeroelastic responses are in violation of the initial assumption of small motions. However, these results can be interpreted as being indicative (i.e., as providing trends) of improvement of LCO suppression achieved with the lightweight MDOF NES design. It is interesting to note that if the total mass ratio of the MDOF NES becomes greater than that of the SDOF NES (see Figure 9.92d), robustness of LCO suppression by the MDOF NES deteriorates compared to the SDOF NES case, even though the generation of LCOs from the trivial equilibrium position is significantly shifted to higher reduced fluid velocities. In conclusion, optimally designed lightweight MDOF NESs can provide improved and more robust LCO suppression performance, compared to optimally designed SDOF NESs. Figure 9.93 demonstrates by means of numerical simulations of the enhancement in robustness of LCO suppression achieved using the MDOF NES design, compared to the SDOF one. Considering a SDOF NES with 2% mass ratio, we assess its LCO suppression capacity at reduced velocity = 0.98, and compare it to the case of a MDOF NES with 1.4% total mass ratio. Considering the time series of Figure 9.93a, for small initial conditions we note that the SDOF NES can suppress the aeroelastic instability through recurrent burst outs and suppressions, i.e., the first suppression mechanism is activated, corresponding to excitation of a quasi-periodic LCO on a branch between two NS bifurcation points. However, an impulsive disturbance applied to the heave mode at τ = 200 destroys this LCO suppression, as it induces a transition of the dynamics to a branch of stable large-amplitude LCOs, thus reviving the triggering mechanism of LCO formation (see Section 9.2). Hence, lack of robustness of LCO suppression is established in the SDOF NES design. On the contrary, as it can be deduced from the results of Figure 9.93b under the same flow conditions an MDOF NES with even smaller mass ratio maintains robustness of LCO suppression even after an identical impulsive disturbance has been applied to the heave mode of the wing. Moreover, the suppression of the developing instability caused by the impulsive disturbance is achieved due to the capacity of the MDOF NES to efficiently and rapidly absorb the (broadband) energy of the disturbance through TET. Now, we briefly examine the underlying dynamics that make robustness enhancement possible in the MDOF NES design. We will study the dynamics by analyzing the computed times series of the wing and NES responses by means of wavelet transforms (WTs), and by examining the dominant transient resonant interactions (TRCs) between the aeroelastic modes and the masses of the MDOF NES. In addition, instantaneous modal energy exchanges and measures of energy dissipation by the MDOF NES will be computed, in an effort to relate TET to the enhanced capacity of the MDOF NES to robustly suppress aeroelastic instabilities over certain ranges of reduced fluid velocities.

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Fig. 9.93 Demonstration of robustness enhancement of LCO suppression, when an impulsive disturbance is applied to the heave mode at τ = 200, for = 0.98, δ = −1, λ = 0.4, C = 10.0 and C1 = 0.01: (a) when a SDOF NES of mass ratio ε = 0.02 is attached, and (b) when a MDOF NES of total mass ratio ε = 0.014 is attached; thicker (thinner) lines indicate controlled (uncontrolled) responses.

To this end, the instantaneous kinetic energy T¯ (τ ), and potential energy V¯ (τ ), of the integrated wing – MDOF NES system in normalized form (9.91) are computed as follows:


551

T¯ (τ ) = (1/2)y 2 (τ ) + xa y (τ ) α (τ ) + (1/2)α 2 (τ ) + (1/3) v12 (τ ) + v22 (τ ) + v32 (τ ) V¯ (τ ) = (2 /2)y 2(τ ) + (ra2 /2)α 2 (τ ) + (ξy /4)y 4(τ ) + (ξa /4)α 4 (τ ) + (C/4) [v1 (τ ) − v2 (τ )]4 + (C/200) [v2 (τ ) − v3 (τ )]4 2 + (C1 /2) y(τ ) − δα(τ ) − v1 (τ )

(9.92)

Hence, the total instantaneous energy of the system is computed as E Total (τ ) = T¯ (τ ) + V¯ (τ )

(9.93)

The instantaneous energy input in the integrated wing-NES system is computed as the summation of the energy provided by the initial conditions and the nonconservative work performed by the flow. This is expressed as follows: y

α E Input (τ ) = E Total (0) + Wnc (τ ) + Wnc (τ )

where y Wnc (τ )

= µCL,α 0

α (τ ) Wnc

τ

/

y (s) +

= −γ µCL,α

τ

(9.94)

0 α(s) y (s) ds

/ y (s) +

0 α(s) α (s) ds

0

Each of the above expressions represents the instantaneous non-conservative work performed by the heave or pitch mode of the in-flow wing. To assess the efficiency of TET from the aeroelastic modes to the MDOF NES, and relate it to the capacity of the NES for LCO suppression we also compute the following energy dissipation measure (EDM), which represents the total energy dissipated by the two damping elements of the NES (see Figure 9.83) up to time τ : EdNES (τ ) = EdNES1 (τ ) + EdNES1 (τ ) τ 2 = ελ v1 (s) − v2 (s) ds + ελ 0

τ 0

v2 (s) − v3 (s)

2

ds (9.95)

Combining the previous energy measures we may formulate the following instantaneous total energy balance for the integrated wing-NES system: E Total (τ ) = E Input (τ ) − EdNES (τ )

(9.96)

We consider the dynamics of the integrated system for reduced fluid velocity = 0.92, when complete and robust elimination of aeroelastic instability is realized by the action of the MDOF NES with parameters ε = 0.014, λ = 0.4, C = 10.0, C1 = 0.01 (i.e., weak coupling stiffness is considered) and δ = ±1; moreover,

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zero initial conditions are considered, except for the heave velocity. The bifurcation diagrams for δ = ±1 are depicted in Figure 9.92c, together with the comparison to the bifurcation diagram of the system with a heavier SDOF NES. In Figures 9.94a and 9.94b we depict the transient responses of the integrated wing-NES system for initial condition y (0) = 0.02 (i.e., for a relatively small initial energy input to the heave mode) and δ = ±1, respectively; whereas in Figures 9.94c and 9.94d, we depict the corresponding responses for initial condition y (0) = 0.1 (i.e., for a relatively large initial energy input to the heave mode) and δ = ±1, respectively. We note that in both cases of relatively small or large initial input energies the developing aeroelastic instabilities are completely eliminated by the action of the MDOF NES (i.e., the third LCO suppression mechanism is activated), due to TET from the developing instabilities in the aeroelastic modes to the MDOF NES, where the energy is locally dissipated without being ‘fed back’ to the instability. Moreover, it is interesting to note, the developing LCO instability is eliminated on a faster time scale when stronger disturbances are applied to the heave mode (see Figures 9.94c, d); it is also of interest to note that for the particular case of negative offset δ = −1 (see Figure 9.94d) there occur initial nonlinear beat phenomena between the pitch mode and the three masses of the NES, which facilitate energetically-rich modal energy exchanges and transfers from the pitch mode to the NES. We recall from the exposition in Chapter 3 that the excitation of nonlinear beats through excitation of impulsive orbits (IOs) provides the most efficient mechanism for TET; in that context, the specific excitation of the heave mode considered in the simulations of Figure 9.94 amounts to excitation of IOs of the integrated wingNES system. In addition, we recall from the results of Chapter 3 that broadband TET from primary systems to SDOF and MDOF NESs is enhanced when energy exceeds certain energy thresholds. We conjecture that this dynamic phenomenon is also observed in the numerical simulations of Figure 9.94, where the NES is observed to perform better at an increased energy level (i.e., for the case of stronger impulsive disturbance). The WT spectra of the transient responses of Figure 9.94 are depicted in Figure 9.95. Similar to (but in a more efficient way than) the SDOF NES design, subharmonic transient resonant captures (TRCs) between the heave and pitch modes take place, thus replacing (and effectively prohibiting) 1:1 TRCs between the same modes that trigger aeroelastic instabilities (LCOs) in the wing without NES. However, the nonlinear interactions between the pitch mode and the masses of the NES occur through 1:1 TRCs; in addition, the nonlinear beat phenomena occurring between the pitch mode and the NES masses for the case of relatively strong disturbance and offset δ = −1 become apparent by the presence of the two closely spaced harmonics close to the unit normalized frequency in the WT spectra of Figure 9.95d. Finally, in Figure 9.96 we depict the nonlinear modal energy exchanges corresponding to the time responses of Figure 9.94. Since the action of the MDOF NES yields complete elimination of the aeroelastic instability, eventual zero energy balance between the input energy and the energy dissipated by the NES is achieved in all cases considered. Moreover, the state of zero balance is achieved faster in cases


553

Fig. 9.94 Transient responses of the integrated in-flow wing-MDOF NES system for initial impulse applied to the heave mode, and system parameters = 0.92, ε = 0.014, λ = 0.4, C = 10.0 and C1 = 0.01: (a) y (0) = 0.02, δ = 1; (b) y (0) = 0.02, δ = −1; by the notation NES1–3 we denote the three masses in the MDOF NES, respectively, and thicker (thinner) line indicates controlled (uncontrolled) responses.

554


Fig. 9.94 Transient responses of the integrated in-flow wing-MDOF NES system for initial impulse applied to the heave mode, and system parameters = 0.92, ε = 0.014, λ = 0.4, C = 10.0 and C1 = 0.01: (c) y (0) = 0.1, δ = 1; (d) y (0) = 0.1, δ = −1; by the notation NES1–3 we denote the three masses in the MDOF NES, respectively, and thicker (thinner) line indicates controlled (uncontrolled) responses.


555

Fig. 9.95 Wavelet transform spectra of the transient responses of Figure 9.94.

where stronger heave disturbances are applied, which confirms the more efficient LCO suppression performance of the MDOF NES at the higher energy level. All four simulations depicted in Figures 9.94–9.96 display strong initial nonlinear interactions between the aeroelastic modes and the NES. That is, the energy

556



applied to the heave mode by the impulsive disturbance is initially absorbed by the MDOF NES, with its upper mass acting as the main absorber of energy as evidenced by the instantaneous energy plots of Figure 9.96 (with the instantaneous energy levels of the upper mass reaching levels of more than 50% of input energy for strong applied impulses). Also, consistent with the previous bifurcation analysis, weaker


557

Fig. 9.96 Nonlinear modal energy exchanges corresponding to the responses depicted in Figure 9.94; E a and E y denote the percentages of total instantaneous energy stored in the pitch and heave modes, respectively; and E v1 , E v2 and E v3 denote the percentages of total instantaneous energy stored in the first, second and third mass of the NES, respectively.

linear coupling stiffness provides less resistance to this energy transfer from the wing modes to the upper mass of the NES. Considering the energy dissipation ca-

558



pacity of the MDOF NES, it is noted that for strong applied impulses it holds that EdNES2 > EdNES1 , i.e., the weakly coupled pair of masses dissipates a bigger portion of the transferred energy from the wing modes compared to the strongly coupled pair; the opposite holds for the case of weak applied impulse. This implies that, for


559

efficient suppression of aeroelastic instabilities by means of the MDOF NES, the strongly coupled pair of masses of the NES (the one closest to the point of attachment to the wing) should act as effective nonlinear energy absorber, whereas, the weakly coupled pair of masses of the NES (the one farthest from the wing) should act as efficient energy dissipater of this transferred energy. This remark is consistent with the results derived in Chapter 4. In conclusion, the bifurcation analysis of the dynamics of the integrated wing – MDOF NES system indicates that the placement of the lower LPC bifurcation point at reduced fluid speeds above the Hopf bifurcation point is crucial to enhancing robustness of LCO suppression. Moreover, we demonstrated that the proposed MDOF NES design not only enhances the robustness of LCO suppression against strong impulsive disturbances, but also achieves better (or at least comparable) LCO suppression performance than the SDOF NES design, with smaller total mass. These results, when viewed in conjunction with previous theoretical and experimental results concerning the SDOF NES design, indicate that appropriately designed lightweight passive nonlinear absorbers with essential stiffness nonlinearities can passively suppress, effectively and robustly, LCO instabilities by means of broadband TET. The dynamical mechanisms governing passive LCO suppression is a series of transient resonance captures occurring between the wing aeroelastic modes and multiple nonlinear normal modes (NNMs) of the MDOF NES, resulting in passive broadband TET of unwanted vibration energy from the wing to the NES, where this energy is spatially confined and passively dissipated. However, open issues that require further investigation include the study of the complex and highly degenerate structure of the dynamics of the integrated wing – MDOF NES system, as well as the study of the performance of the proposed NES designs under conditions of unsteady fluid-structure interaction. We will provide some preliminary results regarding the later topic in the following section.

9.6 Preliminary Results on LCO Suppression in a Wing in Unsteady Flow Having established effective and robust LCO suppression in a rigid wing by means of passive TET and based on the assumption of quasi-steady aerodynamic theory, one may ask whether a similar passive approach will still be effective when a different aeroelastic model is adopted based on more realistic aerodynamic modeling and yielding unsteady lift force and pitching moment. The purpose of this section is to provide a preliminary answer to this question. A two-DOF rigid wing with nonlinear structural support coupled to a SDOF NES with an offset from the elastic axis will be considered, and, contrary to our previous studies, unsteady aerodynamic theory will be employed to model the fluidstructure interaction. After summarizing the system configuration and certain of its dynamical features a numerical continuation technique will be utilized to examine

560


the dynamical mechanisms governing LCO suppression in this system, and to assess their robustness. Hence, we reconsider the two-DOF rigid wing coupled to a SDOF NES with an offset from the aeroelastic axis (see Figure 9.43); in contrast, however, to the analysis of Section 9.3 (which was based on quasi-steady flow), we now consider unsteady flow-structure interaction. In this case, the non-dimensional equations of motion for the integrated wing-NES system can be expressed as follows: y + xα α + (/ )2 (y + ξy y 3 ) + (ελ/ )(y − δα cos α − v ) + (C/ )(y − δ sin α − v)3 = −(µ/π) CL (τ ) α + (xα /rα2 )y + (1/ )2(α + ξa α 3 ) + (δελ/ rα2 )(δα cos α + v − y ) + (δC/

2

ra )(δ sin α + v − y)3 = (2µ/πrα2 ) CM (τ )

εv + (ελ/ )(v + δα cos α − y ) + (C/

2

)(v + δ sin α − y)3 = 0 (9.97)

where an identical notation to Section 9.3 is employed, with the exception of the normalized time which in this case is defined as τ = U t/b. The unsteady lift force CL (τ ) and pitching moment CM (τ ) for incompressible flow are expressed as follows: CL (τ ) = π[y (τ ) − αh α (τ ) + α (τ )] + 2π[α(0) + y (0) + (0.5 − αh )α (0)]φ(τ ) τ + 2π φ(τ − s)[α (s) + y (s) + (0.5 − αh )α (s)] ds 0

CM (τ ) = π(0.5 + αh )[α(0) + y (0) + (0.5 − αh )α (0)]φ(τ ) τ + π(0.5 + αh ) φ(τ − s)[α (s) + y (s) + (0.5 − αh )α (s)] ds 0

+(π/2)αh [y (τ ) − αh α (τ )] − (π/2)(0.5 − αh )α (τ ) − (π/16)α (τ ) (9.98) The Wagner function φ(τ ) in (9.98) can be expressed by Jones’ approximation (Jones, 1940) as φ(τ ) = 1 − ψ1 e−ε1 τ − ψ2 e−ε2 τ (9.99) where ψ1 = 0.165, ψ2 = 0.335, ε1 = 0.0455 and ε2 = 0.3, and αh denotes the normalized distance over b of the elastic axis ea from the midpoint of the chord length c = 2b (with positive values indicating that the midpoint is on the right of the elastic axis – see Figure 9.43). We note that if we set ψ1 = ψ2 = 0 quasi-steady aerodynamic theory is recovered, but the secondary bifurcation occurring after the Hopf bifurcation is not observed (Liu and Dowell, 2004).


561

In order to avoid dealing with a set of integro-differential equations resulting due to the unsteady aerodynamic force and moment, the following new variables (Lee et al., 1997) are introduced: τ τ e−ε1 (τ −s)α(s) ds, w2 (τ ) = e−ε2 (τ −s) α(s) ds w1 (τ ) = 0

w3 (τ ) =

0

τ

e−ε1 (τ −s)y(s) ds, w4 (τ ) =

0 τ

e−ε2 (τ −s)y(s) ds

(9.100)

0

Then, the equations of motion (9.97) can be expressed by adopting a state-vector formulation as follows, x (τ ) = f (x, τ ) (9.101) where the 10-dimensional state vector of dependent variables is defined as x = [α α y y w1 w2 w3 w4 v v ]T ∈ R 10 . As in Section 9.3, in this model structural nonlinearity exists only in the pitch mode (i.e., the plunge support is linear). Hence, we set ξa = 80 and ξy = 0, and assume that there is no viscous damping in the wing structure. In addition, we consider the following numerical values for the system parameters: µ = 100, ra = 0.5 and ah = −0.5. As for the static unbalance and frequency ratio, the following two different parameter sets are considered: (i) xa = 0.25, = 0.2, with the aeroelastic instability occurring due to a supercritical Hopf bifurcation at F = 6.2851 (Set I); and (ii) xa = 0.10, = 1.2 with the aeroelastic instability occurring due to a subcritical Hopf bifurcation at F = 2.951 (Set II). Finally, the following study will be performed for the following parameters for the SDOF NES: ε = 0.02, 0.05, λ = 0.05, C = 10.0, and δ ∈ [−1, 1]. That is, the NES mass will be assumed to be at most 5% of the wing mass, and to possesses light damping; moreover, the offset of the NES will be allowed to cover almost the entire wing span, that is, from the leading to the trailing edge. The numerical method MATCONT realized through a Matlab package is utilized to investigate the steady state bifurcation structure of LCOs in the unsteady flow model (9.97), and to study the robustness of LCO suppression by means of TET to the attached NES. Figure 9.97 depicts the bifurcation diagram for the wing parameter Set I, and an NES with 5% mass ratio. Unlike the case with the quasisteady results reported in Section 9.3, it is interesting to note that the suppressed LCOs exhibit subcritical behavior, whereas the original aeroelastic system undergoes a supercritical Hopf bifurcation. In addition, the shift of the Hopf bifurcations due to the action of the NES is quite insignificant, which is also not coincident with our earlier observations in Section 9.3. However, we conjecture that better results can be obtained once an optimization of NES parameters is performed following the methodologies described previously. Apart from the subcritical LCO branches near the Hopf bifurcation points, we note that the amplitudes of the suppressed LCOs when the NES is attached at the leading edge of the wing (i.e., δ = 1.0) are smaller than the original LCOs, although this reduction involves a series of very complicated

562


(a)

(b) Fig. 9.97 Bifurcation diagram of the pitch response for the wing parameter Set I with respect to the reduced velocity for NES parameters ε = 0.05, λ = 0.05, C = 10.0: (a) δ = 1.0, (b) δ = 0.3; solid (dashed) line indicates a stable (unstable) LCO branch, dash-dotted line a quasi-periodic LCO, and squares (triangles, circles, asterisks, diamonds) Hopf (Saddle-node, Neimark–Sacker, neutral saddle, branch point cycle) bifurcation points.

codimension-1 (Hopf, Saddle-node, Neimark–Sacker) and codimension-2 (Neutral Saddle, Branch Point Cycle) bifurcations. If, however, our aim is to suppress subcritical LCOs (as for the case of wing parameter Set II), this can be performed by suitably selecting the NES parameters so that supercritical Hopf bifurcations are realized. In particular, focusing in the bifurcation diagrams depicted in Figure 9.98, robust LCO elimination can be achieved


563

(c)

(d) Fig. 9.97 Bifurcation diagram of the pitch response for the wing parameter Set I with respect to the reduced velocity for NES parameters ε = 0.05, λ = 0.05, C = 10.0: (c) δ = 0.0, (d) δ = 0.6; solid (dashed) line indicates a stable (unstable) LCO branch, dash-dotted line a quasi-periodic LCO, and squares (triangles, circles, asterisks, diamonds) Hopf (Saddle-node, Neimark–Sacker, neutral saddle, branch point cycle) bifurcation points.

up to about 5% higher reduced speeds than the linear flutter speed when a 2% NES mass ratio is utilized. If a 5% NES mass ratio is used instead, this robust LCO elimination can be achieved even up to 112% of the linear flutter speed; moreover, robust suppression of aeroelastic instability can be guaranteed up to 130% of the linear flutter speed.

564


(a)

(b) Fig. 9.98 Bifurcation diagram of the pitch response for the wing parameter Set II with respect to the reduced velocity for NES parameters ε = 0.02, λ = 0.05, C = 10.0: (a) δ = 1.0 (the LCO branch for ε = 0.05 is superimposed for the purpose of comparison), (b) δ = 0.1; solid (dashed) line indicates a stable (unstable) LCO branch, dash-dotted line a quasi-periodic LCO, and squares (triangles, circles, asterisks, diamonds) Hopf (Saddle-node, Neimark–Sacker, neutral saddle, branch point cycle) bifurcation points.

Figure 9.99 demonstrates the suppressed aeroelastic responses due to the action of the SDOF NES for the following cases: (a) complete elimination at 112% of UL∗ , and (b) recurring burst-out and suppression of aeroelastic instability at 130% of UL∗ . The wavelet transforms of the aeroelastic responses depicted in Figure 9.100 clearly demonstrate that the underlying dynamic mechanisms for passive LCO suppressions


565

Fig. 9.99 Suppressed aeroelastic responses for parameter Set II and NES parameters ε = 0.05, λ = 0.05, C = 10.0 and δ = 1.0: (a) / F = 1.12 (third suppression mechanism – complete elimination of aeroelastic instability); (b) / F = 1.30 (first suppression mechanism – recurring burst-out and suppression of aeroelastic instability).

is a series of 1:1 to subharmonic transient resonance captures (TRCs), which was also the case in our previous results reported of this chapter. The three LCO suppression mechanisms identified in our previous studies of the quasi-steady aerodynamic model are also realized in the unsteady aeroelastic model

566


Fig. 9.100 Wavelet transform spectra of the transient responses depicted in Figure 9.99.

(9.97). Although feasibility of TET-based LCO suppression can be demonstrated by the preliminary results reported in this section, further (analytical and numerical) investigation should be performed in order to gain better understanding of the nonlinear modal interactions that lead to LCO suppression in the unsteady model, and to perform optimization of the considered NES design.


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Chapter 10

Seismic Mitigation by Targeted Energy Transfer

In this chapter we examine the application of NESs to the problem of seismic protection of frame structures, though other infrastructural systems such as towers, bridges, and so forth will likely benefit from this technology. As has often been noted, infrastructure in the United States alone constitutes a societal investment counted in the trillions of dollars. While the fraction of infrastructure vulnerable to large-scale earthquakes is relatively limited by geography (i.e., the west coast, midwest and southeast, in addition to Alaska and Hawaii, are historically the most vulnerable areas), seismic activity capable of causing property damage has been recorded throughout the country and around the world. This provides the impetus to develop effective strategies to protect not only new construction but also existing structures likely to be subjected to seismic effects. Various passive methods for mitigating the effects of earthquakes have been applied to large scale structures, including auxiliary dampers, base isolation systems, tuned mass dampers, as well as active, semi-active and hybrid systems. Detailed descriptions and a comparision of the performance and limitations of each of these are beyond the scope of this chapter. Rather, the interested reader should refer to several excellent monographs, including Soong and Constantinou (1994), Soong (1990), Skinner et al. (1993), Chu et al. (2005), and recent review articles (Housner et al., 1997; Spencer and Sain, 1997; Soong and Spencer, 2002; Spencer and Nagarajaiah, 2003), as well as references therein for details. It seems clear, given the recent extensive body of literature and burgeoning number of isolated structures,1 that the need exists for a fully passive isolation strategy, lightweight and inexpensive but capable of high performance over an extensive range of earthquakes of different properties. The aim of this chapter, then, is to demonstrate through several case studies that it is possible to design and implement one or more NESs in a primary linear system with multiple degrees of freedom (DOF) that will passively absorb and dissipate seismic energy drawn from the primary system as well as advantageously redistrib1 According to Spencer and Nagarajaiah (2003), by 2003 more than 40 buildings and 10 bridges were constructed with integral active or hybrid seismic isolation systems. This does not include statistics for passive base isolation systems employing laminated rubber bearings, which for lowrise buildings have become ubiquitous in seismically-active regions.

571

572


ute seismic energy within the modes of the primary system, thus enhancing its reliability and enabling performance under widely varying seismic conditions. For an example of an earlier implementation of configurations of linear springs exhibiting geometric nonlinearities for passive seismic attenuation we refer to DeSalvo (2007) and references therein. Building upon the theoretical underpinnings presented in earlier chapters, we now demonstrate the application of TET to the seismic protection of flexible structures. In the first section, we continue the discussion of the two-DOF linear primary system examined in Chapter 7, assessing through simulation and optimization the capability of the nonlinear energy sink concept for seismic protection. Evaluation procedures, including the choice of historic earthquakes and system performance criteria to be employed throughout this chapter, are defined. Finally, the ability of a single VI NES to mitigate seismic effects is assessed. In the next section, we examine a more relevant problem: seismic protection of a model three-story, single-bay, two-dimensional steel frame structure using, first, a single VI NES at the top floor and, second, a VI NES at the first floor and a smooth NES at the top floor. The design and optimization of each protective system configuration are discussed, and its performance is assessed both computationally and experimentally. The final section provides a brief evaluation of protective system performance and offers some insights concerning possible full-scale implementation of the NES concept for seismic protection of civil infrastructural systems.

10.1 The Two-DOF Linear Primary System with VI NES 10.1.1 System Description The efficacy of TET for reduction of the seismic response of a primary structure depends on the ability of one or more attached NESs to passively absorb and dissipate a significant portion of the seismic energy at a sufficiently fast time scale. This ensures that the response of the primary structure is significantly reduced during the crucial initial few cycles of the strong motion. Thus, we will first perform a study of the proposed concept by considering a simple linear primary system and developing an optimization procedure for choosing the parameters of the protective system so that its action is compatible with our design objectives. For this, we employ a two degree of freedom linear primary system attached to an ungrounded VI NES. This type of NES, using clearances and impact to achieve the essential nonlinearity, promotes strongly nonlinear behavior of the full system (Georgiades, 2006; Karayannis et al., 2008; Lee et al., 2008). Following Nucera (2005) and Nucera et al. (2007), the integrated system consisting of the linear primary structure connected to the VI NES is shown in Figure 10.1. The NES mass m3 is connected to the primary structure through a weak


573

Fig. 10.1 Two-DOF linear system with VI NES.

linear spring2 k3 . Two rigid stops constrain the relative displacement between m2 and m3 to be less than or equal to the specified clearance e. For simplicity, let the primary two degree of freedom linear system be proportionally damped, and let ζ1 and ζ2 be the assumed (small) viscous damping ratios. In modal coordinates, the diagonal damping matrix will have the form

0 2ζ1 ω1 (10.1) Cˆ = 0 2ζ2 ω2 and, in physical coordinates, ˆ −1 = MC ˆ TM C = MC

(10.2)

where M is the mass matrix and is the normalized modal matrix. Within the NES, the relation for the absolute velocities, with respect to a fixed reference frame, of the colliding masses before and after each impact is given by the expression v2 − v3 = rc(v3 − v2 ) (10.3) where v1 = u˙ ai is the absolute velocity of mass mi (i = 2, 3) before impact, the prime denotes ‘after impact’, and rc is the coefficient of restitution. While energy is not conserved through each impact due to the required condition rc < 1, momentum is conserved. Thus, (10.4) m2 v2 + m3 v3 = m2 v2 + m3 v3 The equations of motion of the system depicted in Figure 10.1 are given by M u¨ + C u˙ + Ku = −MI0 u¨ g

(10.5)

where M, C, and K are, respectively, the mass, damping and stiffness matrices given by ⎤ ⎡ ⎤ ⎡ m1 0 0 λ1 + λ2 −λ2 0 λ2 0 ⎦ , M = ⎣ 0 m2 0 ⎦ , C = ⎣ −λ2 0 0 m3 0 0 0

2

The weak linear spring functions strictly as a centering device for the VI NES. The interpretation of essential nonlinearity in this case should be modified to include ‘nearly’ essential nonlinearity.

574


⎤ −k2 0 k1 + k2 K = ⎣ −k2 k2 + k3 −k3 ⎦ 0 −k3 k3 ⎡

The displacement vector, relative to ground, is u = [u1 u2 u3 ]T , u¨ g is the specified ground acceleration, and I0 = [1 1 1]T is a distribution vector. In all of the simulations to follow in this section, we will assume that m1 = m2 = 2.9 × 106 kg, k1 = k2 = 2.5 × 107 N/m, ζ1 = ζ2 = 0.01, λ1 = 7.62 × 104 N-sec/m, and λ2 = 1.52 × 105 N-sec/m.

10.1.2 Simulation and Optimization An extensive series of simulations employing Matlab as the computational engine was completed. The simulation code determined precisely when impacts occurred in the VI NES and solved a series of linear problems between them, accounting for dissipation due to inelastic impacts between masses m2 and m3 . The response of the full system, including the VI NES, was determined for four historic earthquakes as the excitation source. These included: • • • •

El Centro, N-S component – May 18, 1940; Hachinohe, N-S component – May 16, 1968; Kobe, N-S component – January 17, 1995; and Northridge, N-S component – January 16, 1994.

As discussed in significant detail in Nucera (2005) and Nucera et al. (2007), these four were chosen as representative of two distinct classes of earthquakes. The first, containing the El Centro and Hachinohe records, are characterized by longer effective ground motion duration and smaller peak ground acceleration and velocity, while the second, containing Kobe and Northridge, exhibit shorter duration but larger peak ground acceleration and velocity. As noted in the references, Kobe has the highest energy content and destructive capacity of the four records. The design space for the system optimization encompassed three VI NES parameters: mass m3 , weak linear stiffness coefficient k3 , and clearance e. The eight evaluation criteria, Ji , i = 1, . . . , 8, employed to assign quanitative measures of performance to the computed seismic response of the system, were those introduced by Spencer et al. (1998a, b) in the context of a moderated benchmark control problem for a seismically excited structure. ⎧ ⎫ max |ui (t)| ⎪ ⎪ ⎪ ⎪ t ⎨ i∈η ⎬ J1 = max (10.6) earthquakes ⎪ umax ⎪ ⎪ ⎪ ⎩ ⎭ The first criterion (10.6) is a non-dimensional measure of the displacement relative to ground motion. Here, η represents the set of computed relative displacements,


575

and umax is maximum relative displacement for the uncontrolled (no NES) linear structure. ⎧ ⎫ |di (t)| ⎪ ⎪ ⎪ ⎪ max ⎨ ⎬ t,i hi (10.7) J2 = max ⎪ earthquakes ⎪ dnmax ⎪ ⎪ ⎩ ⎭ The second criterion (10.7) is a normalized interstory drift ratio, where di (t) = ui (t) − ui−1 (t) is the time history of the i-th interstory drift, hi is the i-th interstory height, and dnmax is the maximum interstory drift for the uncontrolled structure. ⎧ ⎫ max |u¨ ai (t)| ⎪ ⎪ ⎪ ⎪ t ⎨ i∈η ⎬ (10.8) J3 = max ⎪ earthquakes ⎪ u¨ max ⎪ ⎪ a ⎩ ⎭ The third criterion (10.8) is a normalized maximum absolute acceleration. Here, u¨ ai (t) is the time history of absolute acceleration for the i-th degree of freedom, and u¨ max is the maximum acceleration over all degrees of freedom for the uncontrolled a structure. ⎧ ⎫ ⎪ ⎪ max m u ¨ (t) i ai ⎪ ⎪ ⎪ t ⎪ ⎨ ⎬ i∈η (10.9) J4 = max ⎪ earthquakes ⎪ Fbmax ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ The fourth criterion (10.9) is a normalized inertial force ratio, where Fbmax is the maximum base shear force for the uncontrolled structure. ⎧ ⎫

ui (t) ⎬ ⎨ max i∈η J5 = max (10.10) earthquakes ⎩ umax ⎭ The fifth criterion (10.10) is the L2 -normed measure of structural response. Here,

tf

ui (t)} = 0

1/2 u2i (t)dt

tf is a sufficiently large time permitting the response of the structure to attenuate to less than 0.1% of its maximum value, and umax = maxi∈η ui (t) is the maximum normed uncontrolled displacement. ⎫ ⎧

di (t) ⎪ ⎪ ⎪ ⎪ max ⎨ i,j hi ⎬ (10.11) J6 = max earthquakes ⎪

dnmax ⎪ ⎪ ⎪ ⎭ ⎩

576


The sixth criterion (10.11) is a normed interstory drift ratio, where dnmax is the maximum normed interstory drift for the uncontrolled structure. ⎧ ⎫

u¨ ai (t) ⎬ ⎨ max i∈η (10.12) J7 = max ⎭ earthquakes ⎩

u¨ max a The seventh criterion (10.12) is the normed absolute acceleration ratio, where

u¨ max a is the maximum normed absolute acceleration of the uncontrolled structure. ⎫ ⎧ ⎪ ⎪ m u ¨ (t) i aηi ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ i∈η (10.13) J8 = max earthquakes ⎪

Fbmax ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ The final criterion (10.13) is a normed inertial force ratio, where Fbmax is the maximum normed base shear force for the uncontrolled structure. We note here that all of the criteria are applied over the array of four historic earthquakes. The method employed to optimize the parameters of the VI NES is Differential Evolution (Storn and Price, 1997), a global evolutionary procedure. In all of the analyses performed, the weak linear spring associated with the VI NES was fixed at k3 = 0.02k1. While each of the afore-mentioned criteria was evaluated over the array of four historic earthquakes, sufficient additional computations were completed to facilitate the determination of an optimal solution corresponding to each historic earthquake. Following Nucera (2005), a series of Pareto optimations were performed utilizing the objective function OF = J1 + J2 + J5 + J6

(10.14)

10.1.3 Computational Results In results that follow, a subset of cases considered most significant are given. Table 10.1 summarizes the optimal clearance values over the earthquake array, mass ratio, and coefficient of restitution, and Tables 10.2 through 10.5 provide numerical values of the evaluation criteria corresponding to the computed optimal clearances. Evaluation of the ability of the system to effectively dissipate energy was accomplished through an energy balance. Components considered included the seismic energy input to the system, the kinetic and strain energies, the energy dissipated in the linear primary structure, and the energy dissipated by the NES through vibroimpacts. The last of these is a consequence of two effects: (i) transient resonance capture, during which significant dissipation occurs due to repeated strong inelastic impacts (as discussed in Chapter 7); and (ii) energy spreading from high amplitude, low frequency modes of the system to low amplitude, high frequency modes, where


577

Table 10.1 Optimal VI NES clearances (m), two-DOF system.

Table 10.2 El Centro earthquake evaluation criteria, two-DOF system.

it is more readily dissipated by viscous effects. Both of these are beneficial effects in the seismic environment, particularly for strong motion earthquakes. Of interest here is the energy ratio ESS /EI , where ESS is the energy dissipated by the NES through both vibro-impacts and viscous dissipation and EI is the total input seismic energy. This ratio must be computed at each vibro-impact. We first compute the total input energy less the energy dissipated by the NES relative to the fixed frame, given by (10.15) EI −S = EK + ED + EP Letting ua , u˙ a , u¨ a be the absolute displacement, velocity and acceleration vectors and u, u, ˙ u¨ the displacement, velocity and acceleration vectors relative to ground, t [(M u¨ a )T + (C u˙ + Ku)T ]u˙ a dt (10.16) EI −S = 0

The energy dissipated by the VI NES for each impact is given by

578

10 Seismic Mitigation by Targeted Energy Transfer Table 10.3 Hachinohe earthquake evaluation criteria, two-DOF system.

Table 10.4 Kobe earthquake evaluation criteria, two-DOF system.

ES =

1 1 (m2 v22 + m3 v32 ) − (m2 v22 + m3 v32 ) 2 2

(10.17)

Thus, ESS = ES + ED , with EI = EI −S + ES evaluated immediately after each impact. The responses of the system with the VI NES compared to those without the VI NES, the relative motion between mass m2 and the VI NES m3 , and the energy ratio ESS /EI are shown in Figures 10.2 through 10.4, for three specific cases sampled from Table 10.1. In general, over the range of cases examined, we noted that the optimal clearance decreases with increasing mass ratio. Furthermore, at each mass ratio, the optimal clearance appeared to be not sensitive to the coefficient of restitution but, rather, to the inherent characteristics of the individual historic earthquakes. As noted in Chapter 7, there are two fundamental dynamic mechanisms that govern the interaction between the seismically excited primary structure and VI NES. The first of these is targeted energy transfer (TET), which leads to dissipation of a significant portion of input energy early in the regime of strong motion. This is due to a 1:1 transient resonance capture, and as shown earlier there is a time window dur-


579

Table 10.5 Northridge earthquake evaluation criteria, two-DOF system.

Fig. 10.2 Responses of the two-DOF system with VI NES to the El Centro earthquake: controlled and uncontrolled relative displacements (top); relative displacement of the NES and energy dissipation due to vibro-impacts (bottom).

ing which the VI NES oscillates with frequency approximately equal to one of the modes of the primary structure until a sufficient fraction of energy is dissipated, and escape from capture occurs. In fact, the fast scale of the VI NES dynamics enables the absorption and dissipation through inelastic impacts during the initial phase of

580


Fig. 10.3 Responses of the two-DOF system with VI NES to the Northridge earthquake: controlled and uncontrolled relative displacements (top); relative displacement of the NES and energy dissipation due to vibro-impacts (bottom).

the high-energy strong ground motion, the critical first few cycles of seismic response. This results in a significant reduction of the maximum levels of seismic response of the primary structure, which is a necessary condition for prevention of catastrophic failure in, for example, connections between floor beams and columns of the structure. These fast scales cannot be realized with smooth (i.e., continuous) nonlinearities; thus, the discontinuous nature of the VI NES is key to the successful application of TET to seismic mitigation in large structures. The second mechanism acting between the primary system and VI NES is energy redistribution from the lower modes of the integrated system to the higher modes due to vibro-impacts in the VI NES. The consequence of this energy transfer from high-amplitude, low-frequency to low-amplitude, high-frequency modes is more effective dissipation due to viscous damping. While this effect is likely to be more pronounced in large scale systems with many degrees of freedom, it is also visible in the energy balance previously discussed. An examination of the performance of the system, as shown in Figures 10.2 through 10.4, reveals that a significant amount of energy is dissipated within the first five vibro-impacts. These vibro-impacts involve sudden changes in velocity and, hence, often result in large accelerations. Thus, the passively controlled accelerations, reflected in evaluation criteria J3 , J4 , J7 and J8 , were not included in


581

Fig. 10.4 Responses of the two-DOF system with VI NES to the Kobe earthquake: controlled and uncontrolled relative displacements (top); relative displacement of the NES and energy dissipation due to vibro-impacts (bottom).

the objective function. However, the remarkably short reaction time of the VI NES, affecting the amplitude of even the first cycle of response of the primary system, makes it a valuable protective component. As demonstrated in Tables 10.2 through 10.5, use of the VI NES provided reductions in maximum displacement (J1 ) and maximum interstory drift (J2 ) of 40% and 50%, respectively, for both the El Centro and Hachinohe earthquakes. For the most severe historic earthquake studied, the Kobe, the figures were 25% and 37%; and for the Northridge, 34% and 36%.

10.2 Scaled Three-Story Steel Frame Structure with NESs In this section we examine the application of TET to a scaled three-story frame structure, subjected to the same four historic earthquakes introduced in Section 10.1. The structure was designed and built in the Linear and Nonlinear Dynamics and Vibrations Laboratory (LNDVL) at the University of Illinois at Urbana-Champaign. The problem was motivated by a recent series of benchmarks designed to challenge the structural control community (Spencer et al., 1998a, b, c; Ohtori et al., 2004),

582


Fig. 10.5 Sketch of the three-story, one-bay steel and polypropylene frame structure shown mounted to an electromechanical shake table.

though we were unable to duplicate and test any of those structures in our laboratory due to physical limitations in our shake table capability. The design and characterization of the structure was accomplished through a combination of finite element computations and experimental modal analysis. Once the model was complete, several protective system configurations employing NESs were designed, optimized, and verified experimentally. One configuration in particular was shown to provide a significant level of protection for all four earthquakes examined, remarkable in view of the fully passive design. The mechanics of the system will be discussed in what follows.

10.2.1 Characterization of the Three-Story Linear Frame Structure Following Nucera (2005) and Nucera et al. (2008a, b), the structure (i.e., the linear primary system), shown in Figure 10.5, is a three-story, one-bay structure with spring steel columns and polypropylene floor slabs, approximately 60 cm tall, 20 cm wide and 10 cm deep. The floors slabs are bolted directly to the columns, through small aluminum plates to increase rigidity, using 4 cap screws per connection. The foundation, also polypropylene, was bolted directly to a mechanical shake table (Figure 10.6) through which the historic earthquake time histories were applied. The floor slabs were sufficiently thick in dimension to ensure adequate rigidity against bending; thus, the frame was designed to respond as a shear beam, with each column modeled as a clamped-clamped Euler–Bernoulli beam with lateral stiffness k = 12EI / h3 . Here, E is Young’s modulus for spring steel, I is the area moment


583

Fig. 10.6 Detail showing mounting of the steel frame to the locked-down shake table.

of inertia of the column about its bending axis, and h is the effective column length of a story. The resulting floor masses are assumed to be equal to 1.127 kg, and the column stiffnesses, also assumed equal, are rounded off to 5000 N/m. The frame is governed by the equation of motion M u¨ + C u˙ + Ku = −MI0 u¨ g

(10.18)

where the mass and stiffness matrices, displacement vector relative to ground motion, and distribution vector are given, respectively, by ⎤ ⎡ ⎡ ⎤ 0 m1 0 2k −k 0 M = ⎣ 0 m2 0 ⎦ , K = ⎣ −k 2k −k ⎦ , 0 0 m3 0 −k k ⎡

⎤ 1 I0 = ⎣ 1 ⎦ , 2

⎤ u1 u(t) = ⎣ u2 ⎦ u3 ⎡

and u¨ g (t) is the ground acceleration. We considered the case where the system is proportionally damped and determined the modal damping factors from an experimental modal analysis of the structure. Here, the mechanical shaker was locked down, and an impact hammer with an integral piezoelectric force transducer was employed to provide the necessary impulsive excitation at each story; responses were measured using a piezoelectric accelerometer at each story, as shown in Figure 10.7. All transducers used were manufactured by PCB Piezotronics, Inc. Data

584


Fig. 10.7 The three-story uncontrolled frame undergoing modal testing using an impact hammer to determine its eigenspectrum and modal damping factors: (a) excitation using a modal hammer; (b) typical accelerometer installation.

was subsequently acquired through a SigLab system, and both impulsive responses and complex frequency responses were saved for further analysis. The experimental modal analysis was performed in the time domain using the Ibrahim Time Domain (ITD) method (Ibrahim and Pappa, 1982; Kerschen, 2002)


585

and frequency domain using the Rational Fraction Polynomial (RFP) method, the latter implemented in the Diamond software package made available by the Los Alamos National Laboratory (Doebling et al., 1997). The average of the two methods gave natural frequencies of 4.6, 12.8, and 18.3 Hz, compared with computational values of 4.7, 13.2, and 19.1 Hz, resulting in a consistently high error ranging from 2 to 4% , which we attribute to arithmetic round-off and, perhaps, to compliance in the ball-screw mechanism of the shaker. However, the close agreement in natural frequency makes the damping estimates, again the average of ITD and RFP results, credible; the modal damping factors were found to be 0.00275, 0.00313, and 0.00236 for modes one through three, respectively. The modal damping matrix Cˆ was then determined, from which the viscous damping matrix was computed from (10.2). This completed the determination of the linear portion of the simulation model.

10.2.2 Simulation and Optimization of the Frame-Single VI NES System A VI NES was attached to the top (third) floor of the primary system in order to utilize the large building displacements at that height to maximize its authority, as shown in the schematic of Figure 10.8. Development of the equations of motion followed that of Section 10.1, where the NES degree of freedom was added to the equations of the primary system. Thus, (10.18) applies with ⎡ ⎡ ⎤ ⎤ m1 0 λ1 + λ2 0 0 −λ2 0 0 ⎢ 0 m2 0 ⎢ 0 ⎥ λ2 + λ3 −λ3 0 ⎥ ⎥ ⎥ , C = ⎢ −λ2 M=⎢ ⎣ 0 ⎣ 0 m3 0 ⎦ 0 −λ3 λ3 0 ⎦ 0 0 0 m4 0 0 0 0 and

⎡

⎤ 2k −k 0 0 ⎢ −k 2k ⎥ −k 0 ⎥ K=⎢ ⎣ 0 −k k + kNES −kNES ⎦ , 0 0 −kNES kNES

⎡

⎤ 1 ⎢ 1 ⎥ ⎥ I0 = ⎢ ⎣ 1 ⎦, 1

⎡

⎤ u1 ⎢ u2 ⎥ ⎥ u(t) = ⎢ ⎣ u3 ⎦ uNES

As before, kNES is small compared with k. As noted earlier, adding the VI NES makes the combined system piecewise linear; i.e., between any two consecutive impacts of mNES the system is linear. Hence, the numerical integration of the equations of motion requires the solution of a sequence of linear initial value problems, each of which is bounded by successive vibro-impacts of the NES. The precise computation of the times at which vibroimpacts occur is necessary for accurate simulation of the transient dynamics of the system, as they determine the temporal boundaries of the linear computations. When a vibro-impact occurs, the computation is halted, the initial conditions are modified

586


Fig. 10.8 Sketch (a) and schematic (b) of the three-story, one-bay steel and polypropylene frame structure with a single VI NES attached to the third floor.

to account for the state of the system post-impact, and the computation then resumes. The relation for the velocities of the affected masses before and after impact is v3 − vNES = rc(vNES − v3 ) (10.19) and, as before, while energy is not conserved through the inelastic impact, momentum is conserved leading to m3 v3 + mNES vNES = m3 v3 + mNES vNES

(10.20)


587

Fig. 10.9 Comparison of uncontrolled and controlled floor displacements relative to the ground and the VI NES displacement relative to the ground for the system optimized for and subjected to the Northridge earthquake.

The system of equations was reduced to state-space form and integrated using function ODE45 in Matlab, taking advantage of its adaptive time-stepping to achieve the required accuracy in the vicinity of each impact. The optimization parameters were those of the VI NES; i.e., mNES , kNES , e, and rc, and the evaluation criteria, objective function, and procedure were identical to those employed in Section 10.1. The optimization was again performed over the array of four historic earthquakes named earlier, with an optimal solution determined for each particular earthquake. Both the Northridge and Kobe results will be discussed further as they represent the more severe case within each class.

10.2.2.1 Optimization Design for the Northridge Earthquake Prior to initiating the study, examination of the Fourier spectrum of the Northridge acceleration record revealed that a mismatch existed between the frequency bandwidth of maximum energy content and the eigenspectrum of the three story structure. In order to place the maximum energy of the earthquake in a frequency band consistent with the eigenspectrum of the structure, the duration of the earthquake was scaled by a factor of one half to 25 seconds from its original 50, thus doubling

588


Fig. 10.10 Performance of the VI NES optimized for and subjected to the Northridge earthquake; (a) comparison between absolute displaclements of the third flor and NES; (b) phase plot of relative velocity versus relative displacement between the third floor and NES; and (c) energy dissipation by the VI NES showing the portion of instantaneous seismic energy dissipated during each vibroimpact.

its effective bandwidth and ensuring that the computed response represents the most severe condition. The acceleration amplitude, however, was left unscaled. It was observed throughout the ensuing simulations that during the strong motion portion of the earthquake, the VI NES was able to respond quickly resulting in vibro-impacts that dissipate a significant portion of the input energy. This is a signal advantage of the VI NES over the smooth NES, which acts more slowly and is unable to affect structural response during the first few critical cycles of strong motion. The optimal VI NES parameters for the Northridge record were determined to be: mNES equal to 4% of the total mass of the primary system; kNES = 0.005k; rc = 0.40; and e = 0.02 m. The natural frequencies of the system including the VI NES were computed to be 2.2, 4.6, 12.8, and 18.3 Hz, reflecting the addition of the new, low-frequency mode due to the weakly coupled NES mass at the top floor. Figure 10.9 provides a comparison between the controlled and uncontrolled (no NES) relative displacements with respect to ground of each floor of the frame, while Figure 10.10 compares the absolute displacement of the third floor with that of the NES and gives the phase plot of the response of the NES relative to the third floor as well as the instantaneous total energy dissipated at each vibro-impact and the


589

Fig. 10.11 Comparison between the wavelet spectra of controlled and uncontrolled relative displacements for a primary system with Northridge-optimized VI NES attached to the third floor and Northridge seismic excitation: (a) u1 (t) − ug (t), (b) u2 (t) − u1 (t), (c) u3 (t) − u2 (t).

vibro-impact time history. The last of these provides a measure of the severity of vibro-impacts, indicating whether there is sufficient interaction between the primary system and NES. Wavelet spectra of the relative displacements between the first floor and ground, the second and first floors, and the third and second floors are shown in Figure 10.11, clearly depicting the scattering of energy to all structural modes due to vibroimpacts of the NES. We note that the uncontrolled structure responds primarily at its first mode of 4.6 Hz, which leads to large relative displacements. In the presence of the NES, however, seismic energy is spread to as many as four linear modes, with the spreading becoming more pronounced at the higher floors. This is a two-fold advantage from the mitigation standpoint and explains the reduced levels of structural response observed with the NES in place. First, due to the vibro-impacts, seismic energy is transferred from the low frequency, high amplitude first structural mode

590


Fig. 10.12 Comparison of uncontrolled and controlled floor displacements relative to the ground and the VI NES displacement relative to the ground for the system optimized for and subjected to the Kobe earthquake.

to the lower amplitude, higher frequency structural modes where it is more readily dissipated by internal damping effects; second, energy ‘leaking’ to the lowest mode at 2.2 Hz results in even greater mitigation as that mode is localized to the NES. The performance of the optimized system can be visualized through the evaluation criteria (J1 , J2 , . . . , J8 ) = (0.61, 0.59, 0.80, 0.58, 0.37, 0.39, 0.57, 0.39). Thus, a 39% reduction in maximum displacement and a 41% reduction in maximum interstory drift are realized compared with the uncontrolled system, with the normed criteria reduced even more. Examination of the controlled responses of Figures 10.9 and 10.10 reveals three distinct phases. During the first three seconds, the relative motion between the NES and third floor is less than the clearance so that no vibro-impacts occur and the system is linear. In the second phase, from three to eight and a half seconds, strong vibro-impacts occur due to a 1:1 transient resonance capture between the first structural mode and the VI NES, resulting in dissipation of approximately 87% of the input seismic energy. The final phase occurs after escape from resonance capture; however, a series of vibro-impacts occurs between 10 and 12 seconds, though efficient energy dissipation is not achieved since the earthquake has released nearly all of its energy by eight seconds. This confirms that the VI NES is effective from the first cycle of response and is able to dissipate seismic energy at a sufficiently fast


591

Fig. 10.13 Performance of the VI NES optimized for and subjected to the Kobe earthquake: (a) comparison between absolute displacements of the third floor and the NES; (b) phase plot of relative velocity versus relative displacement between the third floor and NES; and (c) energy dissipation by the VI NES showing the portion of instantaneous seismic energy dissipated during each vibro-impact.

time scale to mitigate large responses at early time typical of near-field events. This fundamental mechanism, in conjunction with the spreading phenomenon already discussed, provides an effective mitigation strategy for large scale structures.

10.2.2.2 Optimization Design for the Kobe Earthquake Optimization of the VI NES system for the Kobe earthquake followed that of the Northridge earthquake, just discussed. Again, in order to tune the earthquake record to the eigenspectrum of the structure, the record was scaled to 25 seconds from the original duration of 50 seconds, doubling its frequency bandwidth; acceleration amplitude was not scaled. The optimal VI NES parameters for the Kobe record were determined to be: mNES equal to 4.5% of the total mass of the primary system; kNES = 0.004k; rc = 0.40; and e = 0.018 m. The natural frequencies of the system including the VI NES were computed to be 1.7, 4.6, 12.8, and 18.3 Hz, reflecting the addition of the new, low-frequency mode due to the weakly coupled NES mass at the top floor.

592


Fig. 10.14 Comparison between the wavelet spectra of controlled and uncontrolled relative displacements for a primary system with Kobe-optimized VI NES attached to the third floor, and Kobe seismic excitation: (a) u1 (t) − ug (t), (b) u2 (t) − u1 (t), (c) u3 (t) − u2 (t).

Figure 10.12 provides a comparison between the controlled and uncontrolled (no NES) relative displacements with respect to ground of each floor of the frame, while Figure 10.13 compares the absolute displacement of the third floor with that of the NES and gives the phase plot of the response of the NES relative to the third floor as well as the instantaneous total energy dissipated at each vibro-impact and the vibro-impact time history. The last of these provides a measure of the severity of vibro-impacts, indicating whether there is sufficient interaction between the primary system and NES. Wavelet spectra of the relative displacements between the first floor and ground, the second and first floors, and the third and second floors are shown in Figure 10.14, again depicting the scattering of energy to all structural modes due to vibro-impacts


593

Fig. 10.15 Uncontrolled and controlled displacements relative to the ground when a Kobeoptimized VI NES is attached to the third floor of the frame, subjected to the Northridge excitation.

of the NES. In the presence of the VI NES, seismic energy is spread to the linear modes as before, resulting in dissipation and reduction in response. It would be desirable if a protective system designed and optimized specifically for a particular, severe historic earthquake acting on a known structure functioned in near-optimal fashion for a range of historic earthquakes. Thus, we examined the system resulting from the Kobe analysis in terms of its performance under the El Centro, Hachinohe and Northridge records. Both the El Centro and Hachinohe acceleration records were scaled in time to 50% of their original length, with the amplitude left unscaled. Comparisons of controlled and uncontrolled displacements relative to ground, and the NES displacement relative to ground, for the El Centro, Hachinohe and Northridge earthquake records are given in Figures 10.15, 10.16 and 10.17, respectively. That there is a dramatic reduction in response for all three systems is clear from the plots. The eight evaluation criteria for the Kobe-designed system subjected to all four historic earthquake records are given in Table 10.6. Reductions in maximum displacement and maximum interstory drift were 36% and 38%, respectively, for Kobe, 36% and 36% for Northridge, 50% and 51% for El Centro, and 39% and 42% for Hachinohe. In all cases, reductions in the normed criteria were equally impressive. This cursory study demonstrates that the VI NES can be an efficacious element of a control strategy for seismic protection of shear beam structures. We noted in the course of this study, however, that the VI NES functions most effectively during the strong motion segment of the earthquake, losing its ability to undergo vibro-

594


Fig. 10.16 Uncontrolled and controlled displacements relative to the ground when a Kobeoptimized VI NES is attached to the third floor of the frame, subjected to the El Centro excitation.

Fig. 10.17 Uncontrolled and controlled displacements relative to the ground when a Kobeoptimized VI NES is attached to the third floor of the frame, subjected to the Hachinohe excitation.


595

Table 10.6 Evaluation criteria for four historic earthquakes; linear frame with Kobe-designed single-VI NES at floor 3.

impacts as the input energy decreases and, in the limit, reaching a no-impact condition where it becomes ineffective. This condition will be exacerbated for the case of earthquakes of low-to-moderate intensity, with seismic energy well below that of the design earthquake. Thus, we now consider a second, alternate design based upon the combined utilization of a VI NES and a smooth NES, in order to extend the useful range of the TET strategy for seismic protection of structures.

10.2.3 Simulation and Optimization of the Frame-VI NES-Smooth NES System We now attach a smooth NES incorporating an essentially nonlinear cubic spring to the top (third) floor of the primary structure and a VI NES to the first floor, as shown in Figure 10.18. The intent of this design is to combine a lightweight smooth NES design which will perform adequately for moderate-to-severe earthquakes and a relatively heavy VI NES for protection from severe, possibly near field, events. As the more massive VI NES is positioned lower in the structure, its effect upon the structural design will be minimized; the smooth NES at the top floor will continue to function even as the available seismic energy input becomes relatively small. The equations of motion are now given by M u¨ + C u˙ + Ku = −MI0 u¨ g + f

(10.21)

where, following the previous section, the mass, damping and stiffness matrices and displacement vector relative to ground are given by ⎡ ⎤ m1 0 0 0 0 ⎢ 0 m2 0 0 0 ⎥ ⎢ ⎥ ⎥, 0 0 0 0 m M=⎢ 3 ⎢ ⎥ ⎣ 0 0 0 mNES1 0 ⎦ 0 0 0 0 mNES2

596


Fig. 10.18 Sketch (a) and schematic (b) of the three-story, one-bay steel and polypropylene frame structure with a single-VI NES attached to the first floor and a smooth NES attached to the third floor.

⎡ ⎢ ⎢ C=⎢ ⎢ ⎣ and

λ1 + λ2 −λ2 0 λ2 + λ3 −λ3 −λ2 0 −λ3 λ3 + λ4 0 0 0 0 0 −λ4

⎤ 0 0 0 0 ⎥ ⎥ 0 −λ4 ⎥ ⎥ 0 0 ⎦ 0 λ4


597

Fig. 10.19 Uncontrolled and controlled displacement relative to the ground when a Kobeoptimized VI NES is attached to the first floor and a smooth NES is attached to the third floor of the frame, for the Kobe excitation.

⎡ ⎢ ⎢ K=⎢ ⎢ ⎣

2k + kNES1 −k 0 −kNES1 −k 2k −k 0 0 −k k 0 −kNES1 0 0 kNES1 0 0 0 0

0 0 0 0 7

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

⎧ ⎫ u1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u2 ⎪ ⎬ u= u3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uNES1 ⎪ ⎪ ⎩ ⎭ uNES2

and f represents the restoring force vector associated with the essentially nonlinear, cubically-varying stiffness of the smooth NES, ⎧ ⎫ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎨ ⎬ 3 f = −kNES2 (u3 − u5 ) ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ −kNES2 (u5 − u3 ) Clearly, NES1 and NES2 refer to the VI NES and smooth NES, respectively; as before, u¨ g is the ground acceleration and I0 is the distribution vector. As in the previous section, simulation was performed within Matlab where now, however, the system is strongly nonlinear between impacts of NES1 . Precise calculation of the times at which vibro-impacts occur remains a critical part of the

598


Fig. 10.20 Performance of the VI NES optimized for the Kobe earthquake: (a) comparison between absolute displacements of the first floor and the NES; (b) phase plot of relative velocity versus relative displacement between the first floor and NES; and (c) energy dissipation by the VI NES showing the portion of instantaneous seismic energy dissipated during each vibro-impact.

algorithm, though the problem is no longer piecewise linear. Optimization of this system proceeded in precisely the same way as the previous one, although the number of optimization parameters expanded due to the addition of the smooth NES; i.e., mNES , kNES , e, and rc for NES1 and mNES2 , kNES2 , and λ4 for NES2 . For this system, the optimization study was completed for the Kobe record alone, with the additional constraint that neither NES mass could exceed 2.5% of the total mass of the linear primary structure. The optimal parameters were determined to be: mNES1 equal to 2.4% of the total mass of the linear primary structure, e = 0.011 m, rc = 0.426, kNES1 = 0.001k; mNES2 equal to 2% of the total mass of the linear primary structure, kNES2 /k = 19 m−2 , and λ4 = 3.277λ3. Figure 10.19 provides a comparison between the controlled and uncontrolled relative displacements with respect to ground of each floor of the frame along with the absolute displacements of the third floor and of NES2 . Figure 10.20 depicts the absolute displacement of the first floor and that of NES1 and gives the phase plot of the response of NES1 relative to the first floor as well as the instantaneous total energy dissipated at each vibro-impact and the vibro-impact time history. The frame-dual NES system, optimized for the Kobe record, was also subjected to the Northridge earthquake in order to examine the robustness of the protective system, with the results shown in Figures 10.21 and 10.22. From these, we observe


599

Fig. 10.21 Uncontrolled and controlled displacement relative to the ground when a Kobeoptimized VI NES is atatched to the first floor and a smooth NES is attached to the third floor of the frame, for the Northridge excitation.

that the portion of seismic energy dissipated by the VI NES (NES2 ) is on the order of 25–30%. While smaller than the dissipation realized in the single-VI NES design, it should be noted that here the mass ratio is smaller, and the device is located at the first floor where frame velocity, and consequently momentum exchange, is also smaller. The eight evaluation criteria for the Kobe-designed dual-NES system subjected to all four historic earthquakes are given in Table 10.7. For the Kobe record, the dual-NES system clearly outperforms the single-NES system in terms of all eight criteria, while for the Northridge record there is improvement in the third and seventh criteria but deterioration in the first, though still satisfactory. The situation for the El Centro and Hachinohe records appears somewhat less impressive, with improvements noted only in the normed criteria. Note, however, that these earthquakes are less energetic than the Kobe and Northridge and, consequently, the opportunity for transient resonance capture, which requires exceedance of an energy threshold in order to occur, is reduced with the VI NES located at the first floor and at a reduced mass ratio. These conjectures are confirmed by Figures 10.23 and 10.24, the wavelet spectra of the floor displacements relative to ground, showing that scattering toward higher modes of the system is less pronounced in the dual-NES design. However, reductions in the normed criteria demonstrate that the smooth NES continues to operate in low energy regimes, long after the VI NES becomes ineffective. In Figures 10.25 and 10.26, the wavelet spectra of the relative displacements between NES1 and the

600


Fig. 10.22 Performance of the VI NES optimized for the Kobe earthquake, subjected to the Northridge earthquake: (a) comparison between absolute displacements of the first floor and the NES; (b) phase plot of relative velocity versus relative displacement between the first floor and NES; and (c) energy dissipation by the VI NES showing the portion of instantaneous seismic energy dissipated during each vibro-impact. Table 10.7 Evaluation criteria for four historic earthquakes; linear frame with Kobe-designed single-VI NES at floor 1 and smooth NES at floor 3.

first floor and between NES2 and the third floor, it is clear that seismic energy is pumped more vigorously from the first mode, particularly for the Northridge record. The high amplitude of the wavelet spectum in this case indicates the occurrence of a transient resonance capture accompanied by TET at the first linearized natural frequency of the system. The capture and resulting TET persists for several seconds, after which escape occurs. The result of this effort, then, is a functional, fully passive, lightweight and inexpensive protective system that can be designed for peak performance when exposed to highly energetic earthquakes in the near field but that will also perform well for less energetic earthquakes or those that occur at some distance from the structure.


601

Fig. 10.23 Wavelet spectra of controlled relative displacements for the frame with Kobe-optimized VI NES attached to the first floor, smooth NES at the top floor, and Kobe excitation: (a) u1 (t) − ug (t), (b) u2 (t) − u1 (t), (c) u3 (t) − u2 (t).

602


Fig. 10.24 Wavelet spectra of controlled relative displacements for the frame with Kobe-optimized VI NES attached to the first floor, smooth NES at the top floor, and Northridge excitation: (a) u1 (t) − ug (t), (b) u2 (t) − u1 (t), (c) u3 (t) − u2 (t).


603

Fig. 10.25 Wavelet spectra of controlled relative displacements for the frame with optimized VI NES attached to the first floor, smooth NES at the top floor, and Kobe seismic excitation: (a) uNES1 (t) − u1 (t), (b) uNES2 (t) − u3 (t).

The results of several experiments, to follow, will confirm the capability of both single-NES and dual-NES systems.

10.3 Experimental Verification The single-VI NES and dual-NES systems were built and tested in the Linear and Nonlinear Dynamics and Vibrations Laboratory (LNDVL) at the University of Illinois at Urbana–Champaign. The laboratory is jointly managed by the departments of aerospace engineering and mechanical science and engineering.

604


Fig. 10.26 Wavelet spectra of controlled relative displacements for the frame with optimized VI NES attached to the first floor, smooth NES at the top floor, and Northridge seismic excitation: (a) uNES1 (t) − u1 (t), (b) uNES2 (t) − u3 (t).

The three-story frame was mounted to an electromechanical shake table, basically a servomotor-driven ball-screw device3 that provided sufficient authority for open-loop application of the Kobe and Northridge earthquake records at full amplitude (and scaled by 50% in time as discussed previously) to the frame and protective system. Figures 10.27 and 10.28 illustrate the displacements with respect to ground, at floors one, two and three, for the Kobe and Northridge earthquake records obtained from double integrating the measured accelerations, from which the severity of the response to the Kobe compared with the Northridge is apparent. Henceforth, these will be denoted the uncontrolled responses.

3

The authors are indebted to Professor Bill Spencer of the civil and environmental engineering department at UIUC for the long-term loan of his shake table and associated electronics.


605

Fig. 10.27 Experimentally obtained uncontrolled response to the Kobe earthquake, displacements with respect to the ground of the shear frame: (a) bottom floor 1, (b) middle floor 2, (c) top floor 3.

Fig. 10.28 Experimentally obtained uncontrolled response to the Northridge earthquake, displacements with respect to ground of the shear frame: (a) bottom floor 1, (b) middle floor 2, (c) top floor 3.

606


Fig. 10.29 Frame-single VI NES system: (a) the full system mounted on the shake table; and (b) a detail of the VI NES.

10.3.1 System Incorporating the Single-VI NES The single-VI NES was attached to the third story of the frame, as shown in Figure 10.29. As can be seen in the figure, the NES consists of a small mass clamped to a shaft that is free to move horizontally in linear bearings until either of the two restrictors inelastically impacts its respective bearing housing. Note that mNES consists of the washer, collars, shaft and restrictors, while the bearings and bearing housings are included in m3 . Two discrete values, 2.5% and 3.5%, were placed on the ratio of NES mass to the total mass of the frame in an effort to make the design realistic, and the design optimization was repeated for the Kobe earthquake employing the identical evaluation criteria and algorithm used previously. The optimal solutions, corresponding to the two mass ratios, are: System 1: mNES /mTF = 0.025,

kNES /k = 0.004,

e = 0.024 m,

rc = 0.43

System 2: mNES /mTF = 0.035,

kNES /k = 0.005,

e = 0.016 m,

rc = 0.42

Figures 10.30 and 10.31 show the relative displacements obtained experimentally for Systems 1 and 2, respectively, when subjected to the Kobe record. In System 1, reductions of 31% in maximum displacement relative to ground and 30% in maximum interstory drift are realized, while in System 2, the reductions are 46% and 37%, respectively.


607

Fig. 10.30 Experimentally obtained response of the Kobe-optimized, frame-single VI NES system with mass ratio of 2.5% to the Kobe earthquake: displacements with respect to ground of the shear frame, (a) bottom floor 1, (b) middle floor 2, (c) top floor 3.

Fig. 10.31 Experimentally obtained response of the Kobe-optimized, frame-single VI NES system with mass ratio of 3.5% to the Kobe earthquake: displacements with respect to ground of the shear frame, (a) bottom floor 1, (b) middle floor 2, (c) top floor 3.

608


Fig. 10.32 Experimentally obtained response of the Kobe-optimized, frame-single VI NES system with mass ratio of 2.5% to the Northridge earthquake: displacements with respect to ground of the shear frame, (a) bottom floor 1, (b) middle floor 2, (c) top floor 3.

Fig. 10.33 Experimentally obtained response of the Kobe-optimized, frame-single VI NES system with mass ratio of 3.5% to the Northridge earthquake: displacements with respect to ground of the shear frame, (a) bottom floor 1, (b) middle floor 2, (c) top floor 3.

The Kobe-designed system was also subjected to the consistently scaled Northridge earthquake. Figures 10.32 and 10.33 show the relative displacements obtained


609

Table 10.8 Experimental evaluation criteria; linear frame with Kobe-designed single-VI NES at floor 3.

experimentally for Systems 1 and 2, respectively, for this input record. As expected, the results were not as impressive as for the Kobe, for reasons explained earlier. The computed evaluation criteria corresponding to the two systems for each of the two earthquake records are summarized in Table 10.8. The spreading of seismic energy due to vibro-impacts, predicted in our earlier analyses, was verified in the experimental results. Figures 10.34 and 10.35 show the wavelet spectra of the interstory displacements RD1 = u1 , RD2 = u2 − u1 , RD3 = u3 − u2 of the uncontrolled frame structure subjected to the Kobe and Northridge records, respectively. Examination of these plots reveals that the response is limited nearly completely to the first linear mode of the structure. Significantly different results are obtained when the larger of the two VI NESs (System 2) is installed at the third floor, as indicated in Figures 10.36 and 10.37. Here we see that, for the Kobe earthquake, significant levels of seismic energy are spread from the first to the second and third linearized modes of the system while, for the Northridge record, spreading occurs mainly from the first to the second linearized mode. From these plots we see clearly that the VI NES acts at a sufficiently fast time scale to affect early time response, thus limiting peak values that are major sources of damage. We also note that the effect of the NES is negligible once the seismic energy is reduced to the point that vibro-impacts occur infrequently or not at all.

10.3.2 System Incorporating Both VI and Smooth NESs A VI NES (NES1 ) was attached to the first floor and a smooth NES (NES2 ) was bolted to the third floor of the frame as shown in Figure 10.38. NES1 was described previously; however, now the mass of the bearing housings is added to that of the first floor, m1 . From the figure, we see that NES2 consists of a mass fixed to a shaft supported on bearings, connected to an extension of the third floor of the primary structure through a pair of thin wires without pretension, providing a geometrically nonlinear force-displacement characteristic. Further details of this construction may be found in McFarland et al. (2005a, b) and Kerschen et al. (2007). The force exerted by the mass-wire assembly on the frame is essentially nonlinear due to the absence of pretension and has been shown to be approximately cubic.

610


Fig. 10.34 Experimentally derived wavelet spectra of interstory displacements of the uncontrolled shear frame subjected to the Kobe earthquake: (a) RD1 , (b) RD2 , (c) RD3 .

Fig. 10.35 Experimentally derived wavelet spectra of interstory displacements of the uncontrolled shear frame subjected to the Northridge earthquake: (a) RD1 , (b) RD2 , (c) RD3 .


611

Fig. 10.36 Experimentally derived wavelet spectra of interstory displacements of the Kobeoptimized, frame-single VI NES system with 3.5% mass ratio subjected to the Kobe earthquake: (a) RD1 , (b) RD2 , (c) RD3 .

Fig. 10.37 Experimentally derived wavelet spectra of interstory displacements of the Kobeoptimized, frame-single VI NES system with 3.5% mass ratio subjected to the Northridge earthquake: (a) RD1 , (b) RD2 , (c) RD3 .

612


Table 10.9 Experimental evaluation criteria; linear frame with Kobe-designed VI-NES at floor 1 and smooth NES at floor 3, both with 2.5% mass ratios.

Fig. 10.38 Frame-dual NES system with the VI NES attached to floor 1 and the smooth NES attached to floor 3: (a) the full system mounted on the shake table; and (b) a detail of the smooth NES.

Fixing each of the two NES mass ratios at 2.5% of the total mass of the frame, the system optimization was repeated employing the Kobe earthquake as input. The optimal parameters for NES1 and NES2 were found to be: mNES1 /MTF = mNES2 /mTF = 0.025, kNES1 /k = 0.003, e − 0.012 m, rc = 0.43, kNES2 /k = 16 m−2 , and λ4 /λ3 = 2.8. Displacements with respect to ground of the frame are given by Figures 10.39 and 10.40 for the Kobe and Northridge earthquakes, respectively, and a summary of the eight mitigation criteria for both records is given in Table 10.9. Examination of the Kobe responses reveals dramatic reductions at early time during the strong motion portion of the earthquake and later, around 15 seconds, after which nearly all of the seismic energy has been released by the ground motion. Reductions of 41% and 38% are realized for maximum displacement (J1 ) and maximum interstory drift (J2 ), repectively. Comparison of the dual-NES and single-VI NES systems during the latter stage of response show that, while the maximum displacement is slightly higher, the normed criteria are made uniformly and markedly


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Fig. 10.39 Experimentally obtained responses of the Kobe-optimized, frame-dual NES system with a mass ratio of 2.5% subjected to the Kobe earthquake: displacements with respect to ground of the shear frame, (a) bottom floor 1, (b) middle floor 2, (c) top floor 3.

Fig. 10.40 Experimentally obtained responses of the Kobe-optimized, frame-dual NES system with a mass ratio of 2.5% subjected to the Northridge earthquake: displacements with respect to ground of the shear frame, (a) bottom floor 1, (b) middle floor 2, (c) top floor 3.

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Fig. 10.41 Experimentally derived wavelet spectra of the interstory displacements of the Kobeoptimized, frame-dual NES system with mass ratios of 2.5% subjected to the Kobe earthquake: (a) RD1 , (b) RD2 , (c) RD3 .

Fig. 10.42 Experimentally derived wavelet spectra of the interstory displacements of the Kobeoptimized, frame-dual NES system with mass ratios of 2.5% subjected to the Northridge earthquake: (a) RD1 , (b) RD2 , (c) RD3 .

lower by the presence of NES2 . Furthermore, the wavelet spectra of Figures 10.41 and 10.42 clearly show that spreading of seismic energy to the higher linearized modes of the system is significantly enhanced in the dual-NES design.


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10.4 Observations, Summary and Conclusions That seismic protection of buildings and other civil infrastructure will become an imperative in this century seems obvious. Construction costs associated with repair and replacement, insurance premiums, and the potential for loss of life will jointly drive the development of technologies needed to achieve cost- and performanceeffective mitigation. The necessary technologies are available; however, certain of them, particularly those involving active and hybrid systems, are not viable for reasons including reliability, maintainability, cost, and energy requirements, all of which are well-documented in the literature. Passive systems, then, are most likely to gain broad acceptance, especially if performance can made competitive with that realized by the active and hybrid systems. Structural designers have long been wary of nonlinearity because of its association with damage. Currently, however, there is general understanding that damping in structural systems, much of it derived from friction and hysteresis occurring at many scales, is desirable, though difficult to model and accurately predict. Because of this, the constructive use of stongly nonlinear passive systems to achieve necessary levels of performance and reliability needed for effective mitigation is more likely to be accepted by the community at this time. The case studies presented in this chapter fully demonstrate the ability of targeted energy transfer (TET), implemented through the use of nonlinear energy sinks (NESs), to mitigate seismic effects in frame structures. Efficacious design ensures that the mitigation system performs effectively over a wide range of earthquakes and under conditions of peak input acceleration ranging from extreme to moderate. The protective systems are relatively lightweight, inexpensive to fabricate and maintain, and fully passive; and unlike classical tuned mass dampers they are broadband devices that do not require frequency tuning. We have demonstrated, in the case of the three-story frame, that multiple NESs can be deployed throughout the structure and that, with judicious design of the system, mitigation levels consistent with those reported in the literature using active and hybrid systems can be achieved passively. Furthermore, there are no scalability issues associated with passive TET; thus, application to full-scale structures should offer no significant challenges beyond those already addressed.

References Chu, S.Y., Soong, T.T., Reinhorn, A.M., Active, Hybrid, and Semi-active Structural Control: A Design and Implementation Handbook, Wiley, Hoboken, NJ, USA, 2005. DeSalvo, R., Passive, nonlinear, mechanical structures for seismic attenuation, J. Comp. Nonlinear Dyn. 2, 290–298, 2007. Doebling, S.W., Farrar, C.R., Cornwell, P.J., A computer toolbox for damage identification based on changes in vibration characteristics, in Structural Health Monitoring: Current Status and Perspectives, F-K. Chang (Ed.), Technomic Publishing Co., Lancaster, PA, USA, pp. 241–254, 1997.

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Georgiades, F., Nonlinear Localization and Targeted Energy Transfer Phenomena in Vibrating Systems with Smooth and Non-smooth Stiffness Nonlinearities, Ph.D. Thesis, National Technical University of Athens, Athens, Greece, 2006. Housner, G.W., Bergman, L.A., Caughey, T.K., Chassiakos, A.G., Claus, R.O., Masri, S.F., Skelton, R.E., Soong, T.T., Spencer, B.F., Yao, J.T.P., Structural control: Past, present, and future, ASCE J. Engrg. Mech. 123(9), 897–971, 1997. Ibrahim, S.R., Pappa, R.S., Large modal survey testing using the Ibrahim Time Domain identification technique, AIAA J. Spacecraft Rockets 19(5), 459–465, 1982. Karayannis, I., Vakakis, A.F., Georgiades, F., Vibro-impact attachments as shock absorbers, Proc. IME C J. Mech. Eng. Sci. (Special Issue on ‘Vibro-impact Systems’, V. Babitsky, editor), 2008 (in press). Kerschen G., On the Model Validation in Non-linear Structural Dynamics. Ph.D. Dissertation, University of Liege, Department of Aerospace, Mechanics and Materials Engineering, 2002. Kerschen, G., McFarland, D.M., Kowtko, J., Sup Lee, Y., Bergman, L.A., and Vakakis, A.F., Experimental demonstration of transient resonance capture in a system of two coupled oscillators with essential stiffness nonlinearity, J. Sound Vib. 299(4–5), 822–838, 2007. Lee, Y.S., Nucera, F., Vakakis, A.F., L.A. Bergman, D.M. McFarland, Periodic orbits and damped transitions of vibro-impact dynamics, Physica D, 2008 (submitted). McFarland, D.M., Bergman, L.A., Vakakis, A.F., Experimental study of nonlinear energy pumping occurring at a single fast frequency, Int. J. Non-lin. Mech. 40(6), 891–899, 2005a. McFarland, D.M., Kerschen, G., Kowtko, J., Bergman, L.A., and Vakakis, A.F., Experimental investigation of targeted energy transfers in strongly and nonlinearly coupled oscillators, J. Acoust. Soc. Amer. 118(2), 791–799, 2005b. Nucera F., Nonlinear Energy Pumping as a Strategy for Seismic Protection, Ph.D. Thesis, University of Calabria at Arcavacata of Rende, Cosenza, Italy, 2005. Nucera, F., Vakakis, A.F., McFarland, D.M., Bergman, L.A., Kerschen G., Targeted energy transfers in vibro-impact oscillators for seismic mitigation, Nonlin. Dyn. (Special Issue on ‘Discontinuous Dynamical Systems’) 50, 651–677, 2007. Nucera, F., McFarland, D.M., Bergman, L.A., Vakakis, A.F., Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Part I. Computational results, J. Sound Vib., 2008a (in press). Nucera, F., Lo Iacono, F., McFarland, D.M., Bergman, L.A., Vakakis, A.F., Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Part II. Experimental results, J. Sound Vib. 313(1–2), 57–76, 2008b. Ohtori, Y., Christenson, R.E., Spencer, Jr., B.F., Dyke, S.J., Benchmark control problems for seismically excited nonlinear buildings, J. Eng. Mech. (Special Issue on ‘Benchmark Control Problems’) 130(4), 366–385, 2004. Skinner, R.I., Robinson, W.H., McVerry, G.H., An Introduction to Seismic Isolation, Chichester/Wiley, New York, 1993. Soong, T.T., Active Structural Control: Theory and Practice, Longman Scientific/Wiley, New York, 1990. Soong, T.T., Constantino, M.C., Passive and Active Structural Vibration Control in Civil Engineering, Springer-Verlag, New York, 1994. Soong, T.T., Spencer, Jr., B.F., Supplemental energy dissipation: State-of-the-art and state-of-thepractice, Eng. Struct. 24, 243–259, 2002. Spencer, Jr., B.F., Dyke, S.J., Deoskar, H.S., Benchmark problems in structural control: Part I – Active mass driver system, Earthquake Eng. Struct. Dynam. (Special Issue on ‘Benchmark Structural Control Comparison’) 27, 1127–1139, 1998a. Spencer, Jr., B.F., Dyke, S.J., Deoskar, H.S., Benchmark problems in structural control: Part II – Active Tendon system, Earthquake Engineering and Structural Dynamics (Special Issue on the Benchmark Structural Control Comparison) 27, 1141-1147, 1998b. Spencer, Jr., B.F., Christenson, R.E., Dyke, S.J., Next Ggeneration benchmark control problem for seismically excited buildings, in Proceedings of the Second World Conference on Structural Control, Kyoto, Japan, pp. 1351–1360, 1998c.


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Spencer, Jr., B.F., Nagarajaiah, S., State of the art of structural control, ASCE J. Struct. Eng. 129(7), 845–856, 2003. Spencer, Jr., B.F., Sain, M.K., Controlling buildings: A new frontier in feedback, IEEE Contr. Systems Mag. (Special Issue on ‘Emerging Technology’) 17(6), 19–35, 1997. Storn, R., Price K., Differential evolution – A simple and efficient adaptive scheme for global optimization over continuous spaces, J. Global Optimiz. 11, 341–359, 1997.

Chapter 11

Suppression of Instabilities in Drilling Operations through Targeted Energy Transfer

Our final demonstration of application of TET to engineering problems concerns instability suppression in models of deep drilling of oil and gas wells. In particular, in deep-drilling operations it can be difficult to maintain a smooth cutting process at bit-rock interfaces. Among other disturbances and uncertainties in such operations, we mention the non-homogeneity of rock, the unavoidable friction effects (which are further complicated by fluid flow), and the spatially asymmetric, time-varying character of forces introduced during ‘steering’ of the drill into the well bore. All these effects interact with the drill string, which itself can be several thousand meters long (and, hence, very flexible), to produce a dynamic environment which makes the operation susceptible to instabilities. In this chapter we address the problem of stabilizing the dynamics of a model of the drill-string system by means of nonlinear passive targeted energy transfer (TET), i.e., by adding a passive, local nonlinear energy sink (NES) to an existing configuration. In the particular application considered herein the NES takes the form of a discrete torsional oscillator consisting of a disk coupled to the drill string system through an essentially nonlinear spring and a viscous damper. Following a precise statement of the problem to be studied, we will review some of the distinct numerical challenges associated with the study of the drill-string model. Most of these arise from the need to efficiently compute the dynamics of this model in the presence of friction and related non-smooth effects. A bifurcation diagram depicting the different qualitative dynamics of the system over a realistic range of inputs will be produced, and will be used to assess the influence of the NES parameters on the dynamics of the integrated system. A numerical study and the resulting NES design will be described next. Finally, a detailed analysis of the integrated system composed of the drill string system and the attached NES will be presented, including some remarks on the robustness of the passive control of instabilities achieved with this device. The results presented in this chapter follow closely the work by Viguié et al. (2008).

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11 Suppression of Instabilities in Drilling Operations through TET

Fig. 11.1 Schematic view of the drill-string structure (Mihajlovic, 2005).

11.1 Problem Description Deep wells for the exploration and production of oil and gas are drilled with a rotary drilling system (see Figure 11.1). A rotary drilling system creates a borehole by means of a rock-cutting tool, called the bit. The torque driving the bit is generated at the surface by a motor with a mechanical transmission box. Via the transmission, the motor drives the rotary table that consists of a large disk acting as a kinetic energy storage unit. The medium used to transport mechanical energy from the surface to the bit is a drill-string, mainly consisting of drill pipes. The drill-string is a long elastic medium, reaching in some cases up to 8 km long. The lowest part of the drill-string is the bottom-hole-assembly (BHA) consisting of drill collars and the bit. This structure undergoes different types of vibrations during the drilling operation, including torsional (rotational) vibrations caused by the interaction between the bit and the rock, or the drill-string and the borehole wall; bending (lateral) vibrations often caused by pipe eccentricity and yielding centripetal forces during rotation; axial (longitudinal) vibrations during the rock cutting process (an extreme form of such axial vibrations is called bit bouncing); and hydraulic vibrations taking place in the circulation system, coming from pump pulsations. In addition, there exist coupling effects between torsional, lateral and longitudinal vibrations, as highlighted in Mihajlovic (2005). In this study, our efforts are only devoted towards vibration mitigation of torsional vibrations. Numerous studies were undertaken to gain knowledge of the origins of those vibrations (Brett, 1992; Cunningham, 1968; Jansen and van den Steen, 1995; Kreuzer and Kust, 1996; Leine et al., 2002; van den Steen, 1997). It was established that a possible mechanism for torsional vibration is the stick-slip phenomenon generated by the friction force between the bit and the well (Jansen and van den Steen, 1995;


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Fig. 11.2 Experimental fixture of a drill-string set-up (Mihajlovic, 2005).

Leine et al., 2002; van den Steen, 1997). Other studies showed that the cause of torsional vibrations is velocity weakening in the friction force (i.e., the Stribeck effect) due to the contact between the bit and the borehole (Brett, 1992; Kreuzer and Kust, 1996). In Richard et al. (2004) and Germay et al. (2007) it was shown that this velocity weakening effect originates from the coupling between the torsional and axial dynamics through the bit/rock interaction. Ultimately, the velocity weakening effect plays a very important role in the occurrence of (unstable) limit cycle oscillations (LCOs) in drill-string systems (Mihajlovic et al., 2006). To examine this instability, a prototypical drill-string system was built at the Technische Universiteit Eindhoven (Mihajlovic, 2005; Mihajlovic et al., 2006). Figure 11.2 depicts the experimental set-up, whereas Figure 11.3 presents its schematic representation. The system consists of two discs that model the inertia effects created by the rotating components in the upper (i.e., the rotary table) and lower (i.e., the BHA) parts of the system. For further details about the experimental fixture, the interested reader may refer to Mihajlovic (2005) and Mihajlovic et al. (2006). Because the focus of this study is in the steady state torsional vibration of the drill-string system, it will be assumed that lateral movements of this system are restrained so they can be neglected from further consideration. This results in a reduced-order two-DOF analytical model, the equations of motion of which are given by Ju ω˙ u − kθ α + Tf u (ωu ) = km uc Jl (α¨ + ω˙ u ) + kθ α + Tf l (ωu + α) ˙ =0

(11.1)

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Fig. 11.3 Schematic view of the drill-string set-up (Mihajlovic, 2005). Table 11.1 Definitions ωu ωl Ju , Jl km Tf u , Tf l kθ θu , θl α = θl − θu uc

Velocity of the upper disk Velocity of the lower disk Moment of inertia of the disks Motor constant Friction torque at the upper and lower disks Torsional spring stiffness Angular displacements Relative angular displacement Input voltage at the DC-motor

where the definition of the parameters in these equations is given in Table 11.1. The model consists of two linearly coupled disks, with the upper disk driven by a DCmotor, and both disks being affected by friction effects. Modeling of friction is an important issue in this problem, so we will discuss it in detail. In order to model friction in the set-up, either static or dynamic friction models can be considered. Because the main objective of the present study is the analysis of the steady state dynamical behavior, a detailed modeling of friction for small (initial) angular velocities is not necessary. Therefore, a static friction model will be adopted for our study. Both disks of the system are subject to friction torques that originate from different sources. On the upper disc, the applied friction torque Tf u is due to friction in the bearings, as well as electro-magnetic effects in the driving DC-motor. At the lower disk, the friction torque Tf l is due to friction in the bearings, as well as due to a brake mechanism that aims to reproduce the Stribeck effect. It follows that both disks are subject to a torque which results from the combination of static friction (denoted by Tsu for the upper disk and Tsl for the lower) and viscous friction (applied through the viscous damping coefficients bu and bl for the upper and lower disks, respectively). Moreover, at the lower disk the presence of the Stribeck effect imposes the


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Fig. 11.4 Different regimes in the plot of the friction force Ff as a function of the relative velocity vr .

combination of the previous contributions in a Stribeck model. This model introduces new parameters, such as the Stribeck velocity ωsl , the Stribeck shape parameter δsl , and the Coulomb friction coefficient Tc . The resulting combined friction effect is depicted in Figure 11.4, which presents the plot of the dependence of friction force on relative velocity. This plot can be divided into four different regimes: sticking, boundary lubrication, partial fluid lubrication and full fluid lubrication. All these regimes, as well as further details about this friction model, are discussed in Armstrong-Helouvry (1991), Canudas de Wit et al. (1995), Olsson (1996), Hensen (2002), Mihajlovic (2005) and Viguié (2006). Summarizing, the friction torques acting on the upper and lower disks can be expressed by the following set-valued force laws: Tcu (ωu ) sgn(ωu ), for ωu = 0 Tf u (ωu ) = (11.2) [−Tsu, Tsu ], for ωu = 0 Tf l (ωl ) =

for ωl = 0 Tcl (ωl ) sgn(ωl ), − + [−Tcl (0 ), Tcl (0 )], for ωl = 0

(11.3)

where subscripts u and l refer to upper and lower parts of the set-up depicted in Figure 11.3, respectively. In these relations the terms Tcu (ωu ) and Tcl (ωl ) express the velocity dependencies of the friction torques acting on the upper and lower disks, respectively, and are defined by the following expressions: Tcu (ωu ) = Tsu + bu |ωu |

(11.4)

Tcl (ωl ) = Tcbl + (Tsl − Tcbl ) exp(−|ωl /ωsl |δsl ) + bl |ωl |

(11.5)

where Tsl and Tcbl refer, respectively, to the static friction and Coulomb friction torque acting on the lower disk, and similar notation is used for the upper disk. Figures 11.5 and 11.6 depict the combined friction torques acting on the upper and lower disks as functions of the corresponding angular velocities.

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Fig. 11.5 Friction torque acting on the upper disk.

Fig. 11.6 Friction torque acting on the lower disk.

The present study focuses on numerical simulations of the two-DOF reducedorder model (11.1). The numerical values of the parameters of this model, however, are taken to be identical to the corresponding system parameters identified in the experimental fixture developed by Mihajlovic (2005); the numerical values of the parameters of the model are listed in Table 11.2. Hence, the computational results


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Table 11.2 Parameters of the two-DOF drill-string system for the bifurcation diagram presented in Figure 11.7. Ju km Tsu bu kθ Jl Tsl csl ωsl δsl bl Jadd Ca kθ nl

0.4765 kg·m2 /rad 4.3228 N·m/V 0.37975 N·m 2.4245 N·m·s/rad 0.0775 N·m/rad 0.0414 kg·m2 /rad 0.2781 N·m 0.0473 N·m 1.4302 rad/s 2.0575 0.0105 N·m·s/rad 0.025895 kg·m2 /rad 0.0105 N·m·s/rad 0.0025 N·m/rad3

presented in this chapter should be directly comparable to the experimental results reported in the above-mentioned work by Mihajlovic.

11.2 Instability in the Drill-String Model The objective of this section is to study the steady state behavior of the reduced twoDOF drill-string model for constant input voltage uc . The steady state behavior is of particular interest, since this system is typically driven by a constant torque while aiming at a constant velocity at the lower part of the set-up. The presence of friction and related discontinuities in the equations of motion dictate the use of appropriate numerical algorithms to efficiently compute the dynamical responses. In this study, the so-called switch model is used. It aims in approximating a differential inclusion by sets of ordinary differential equations. This concept is explained in great detail in Leine and Nijmeijer (2004), and its application to the drill-string system is described in Viguié et al. (2008). The numerical integration process provides the possible steady state solutions of the dynamics, such as equilibrium points, and periodic or quasi-periodic orbits. Typically, these solutions are presented in bifurcation plots using selected system parameters as bifurcation parameters. We now proceed to construct the bifurcation plot of steady state responses of the two-DOF drill-string model considering the input voltage uc as bifurcation parameter; the numerical values of the remaining parameters are listed in Table 11.2. In order to obtain accurate approximations of these solutions and their related stability, specific numerical methods and stability analysis techniques must be considered. Stable equilibrium points can be easily computed using numerical simulations, but a more general method consists of the resolution of the algebraic inclusion of the discontinuous system. Moreover, the related stability (local and global) can be deter-

626


mined using Lyapunov’s indirect and direct methods. For the computation of stable and unstable periodic solutions, a shooting method is used in this study (Keller, 1976). This method relies on an iterative process and requires an initial guess. The local stability of the periodic solutions is determined using Floquet theory. The association of the previous methods with the arc length continuation method enables us to compute the bifurcation diagrams of the system for varying values of the constant input voltage uc . Throughout this study, we adopt the same terminology for the bifurcation diagrams. The periodic solutions leading to limit cycle oscillations (LCOs) are denoted by ‘p’, whereas the branches of equilibrium positions by ‘e’. Solid and dotted lines refer to stable and unstable branches, respectively. The bifurcation diagram of the steady state dynamics of system (11.1–11.5) is depicted in Figure 11.7. The following remarks concern the structure of the branches of equilibrium points of the system. For input voltages below a critical value uc ≤ uA (bifurcation point A in Figure 11.7b) there is a single branch of asymptotically stable (trivial) equilibrium points e1 , and therefore the operation of the drill-string is stable. For input voltages in the range uA ≤ uc ≤ uh1 the trivial branch becomes unstable, and gives rise to the branch of asymptotically stable (non-trivial) equilibrium points e2 . This branch loses stability in the interval uh1 ≤ uc ≤ uh2 and appears as a branch of unstable equilibrium points denoted by e3 in the bifurcation diagram. Finally, for relatively high voltages, uc > uh2 this branch of equilibrium points regains asymptotic stability and appears as branch e4 in the bifurcation diagram. Hence, there occur two main bifurcations corresponding to points B and C in the bifurcation plots of Figure 11.7, associated with exchanges of stability of the branch of stable equilibrium positions e2 − e4 . As shown below, these bifurcations give rise to stable LCOs which adversely affect the stability of the operation of the drill-string; however, for sufficiently high voltages (uc > uE in Figure 11.7a) branch e4 is the only stable steady state solution of the system. In this system LCOs are generated due to a steady state balance between the ‘stabilizing’ effect of viscous friction (at higher velocities) and the ‘destabilizing’ effect of the Stribeck effect (at lower velocities). Considering the bifurcation plots of Figure 11.7 we note that for input voltages higher than uh1 (i.e., above point B in Figure 11.7b), in addition to the branch e3 of unstable equilibrium points, there exists the branch p1 of periodic solutions consisting of unstable LCOs without stick-slip. It follows that point B is a subcritical Hopf bifurcation point. The branch of unstable LCOs p1 is connected to branch p2 of stable LCOs at point D (see Figure 11.7b). The branch p2 consists of LCOs during which the drill-string undergoes torsional vibration with stick-slip, so that D is a point of (discontinuous) saddle-node (SN) bifurcation. Moreover, at point C a subcritical Hopf bifurcation takes place that generates the branch p3 of unstable LCOs. This unstable LCO branch merges with the stable LCO branch p2 at point E, in a SN bifurcation of LCOs (see Figure 11.7a). For voltages uc > uE no LCOs are possible in the drill-string model. To illustrate some of the above-mentioned steady state solutions, in Figure 11.8 we present direct numerical integrations of the system on a branch of stable LCOs, as well as, a case of attraction of the dynamics by a stable equilibrium position.


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(a)

(b) Fig. 11.7 Steady state dynamics of the two-DOF drill-string model for varying voltage uc : (a) bifurcation diagram, (b) zoom in the low voltage range; solid lines correspond to stable solutions and dotted lines to unstable ones.

11.3 Suppression of Friction-Induced Limit Cycles by TET We now study passive suppression of destabilizing LCOs in the drill-string model through TET. To this end, we attach an essentially nonlinear, passive torsional non-

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Fig. 11.8 Direct numerical simulation of the drill-string model: (a) stable LCO with stick-slip on branch p2 (uc = 2 V); (b) transition of the dynamics to a stable equilibrium pointon branch e2 (uc = 0.16 V).

linear energy sink (NES) to the lower disk and study its effect on the dynamics. From previous applications it has been shown that appropriately designed NESs are capable of absorbing and locally dissipating significant portions of vibration energy from the systems to which they are attached; moreover, this type of passive device is capable of absorbing energy at extended frequency ranges through transient or sustained resonance captures with multiple structural modes. Additional advantages


629

of the proposed design is that it brings only relatively minor structural modification to the drill-string system, and that it does not require any external energy supply for its operation.

11.3.1 Addition of a NES to the Drill-String System As discussed in previous chapters, the tuned mass damper (TMD) is a simple and efficient device, but it is only effective when it is precisely tuned to the frequency of a vibration mode of the structure to which it is attached. In the drill-string problem under consideration, the nonlinear primary system possesses a single vibration mode, but the LCOs during operation have frequencies that vary with the input voltage uc . It follows, that in this case, the TMD would have to be tuned to a particular frequency (i.e., on a specific LCO of the system), which would clearly limit its efficiency and robustness. To overcome the limitations of the TMD, an essentially nonlinear attachment (characterized by the absence of a linear term in the force- displacement relation) is added to the system, and designed to act as a nonlinear energy sink (NES) and local dissipater of undesired vibrations. As shown in previous chapters an NES possesses certain important properties. Indeed, since the NES does not possesses any preferential resonant frequency, it is capable of engaging in resonance capture with any structural mode of the system to which it is attached, irrespective of the frequency range where that mode lies (as long, of course, as the mode has no node close to the point of attachment of the NES). For instance, an NES may resonate with and extract energy through TET from any mode of a primary structure through either isolated transient resonance captures (TRCs) or resonance capture cascades (RCCs). This versatility of the NES makes it particularly suitable for passive elimination of the stick-slip LCOs described in the previous section, whose frequencies vary with varying input voltage (the passive elimination of a different type of aeroelastic LCO through passive TET was already demonstrated in Chapter 9). The use of an NES therefore seems promising for vibration mitigation of nonlinear primary systems such as a drill-string system, since passive TET from the drill-string to the NES may reduce the amplitude of vibration, or even completely eliminate the LCO instabilities. However, the increased complexity of the dynamical behavior of the integrated system with the attached NES dictates that the dynamics be carefully studied, and the robustness of the proposed design to changes of initial conditions be addressed. The addition of the NES yields an additional degree-of-freedom in the system; the NES consists of a disk with moment of inertia Jadd , coupled to the lower disc of the drill-string by a cubic stiffness, kθnl , which lies in parallel to a dashpot with damping coefficient ca (see Figure 11.9). The only degree-of-freedom of this disk is its rotation θa about its geometric center, whereas any lateral motion is assumed to be negligible. Hence, the equations of motions of the three-DOF integrated system with NES attached are given by

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Fig. 11.9 Integrated drill-string system with torsional NES attached.

Ju (ω˙ u − α) ¨ − kθ α + Tf u (ωl + α) ˙ = km u Jl (ω˙ l + kθ α − kθnl (αa )3 − ca α˙ a + Tf l (ωl ) = 0 Ju (ω˙ l + α¨ a ) + kθnl (αa )3 + ca α˙ a = 0

(11.6)

where αa = θa − θl . A parametric study of the dynamics of the integrated system was performed to study the effect of the NES on the steady state responses. In particular, of significant practical interest is to investigate the efficacy of reducing or even completely eliminating the domain of existence of unwanted stable LCOs through passive TET from the drill-string to the NES, as this would stabilize the operation of the integrated system and enhance its performance.

11.3.2 Parametric Study for Determining the NES Parameters In the parametric design we performed a study on the effect of the NES parameters, namely the nonlinear stiffness kθnl , the moment of inertia Jadd , and the damping coefficient ca , on TET and drill-string stabilization. This study provided a set of values for these parameters; we mention at this point that the design objective was not to


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Table 11.3 Initial set of NES parameters. Jadd ca kθ nl

0.025895 kg·m2 /rad 0.0105 N·m·s/rad 0.0025 N·m/rad3

determine the optimal set of values for the NES parameters, but rather to establish the efficacy of using an NES in order to extend the range of working input voltages for which stick-slip LCOs in the integrated system could be avoided. Therefore, an initial set of parameter values was utilized, and then adapted to obtain the largest range of input voltages leading to stable equilibrium solutions and avoidance of stick-slip limit cycling (i.e., of periodic solutions). For obvious practical reasons, we selected the mass ratio between the NES and the drill-string system to be as small as possible, and followed the same design rules as those considered in our previous applications. To this end, the NES inertia was initially set equal to 5% of the total inertia of the system. In practice, this value may seem to be large, but the focus of our study was on the new dynamics introduced by the addition of the NES to the nonlinear (discontinuous) primary structure, instead of providing an NES design that could be directly applied to practical applications. As the inertia of the additional NES disk was close to that of the lower disk, the NES dashpot was chosen so that its viscous damping coefficient be of value close to that of the lower disk b1 . Finally, the determination of the nonlinear (cubic) stiffness was based on the linear stiffness of the string of the primary system. The aim was to find a value of the nonlinear stiffness that creates an elastic torque of the same order of magnitude to that in the primary system. Based on this rationale, the selected initial set of NES parameters is listed in Table 11.3. A series of numerical simulations was carried out by varying a single parameter of the initial set in order to assess the impact of the nonlinear structural modification on the dynamical behavior of the integrated system; this impact was studied through bifurcation diagrams. The results related to these numerical experiments are available in Viguié (2006), and are summarized in Tables 11.4–11.6 for modifications of nonlinear stiffness kθnl , moment of inertia Jadd and viscous coefficient ca , respectively. Depending on the selected parameter values, the existence of periodic solutions is restricted to wider or narrower ranges of input voltage. For instance, the selection of the nonlinear stiffness coefficient is crucial, as judged from the results listed in Table 11.4; indeed, for some values of the nonlinear stiffness coefficient time-periodic solutions (LCOs) are realized up to 3.5 V, whereas for other values of this parameter LCOs exist only up to 1.7 V. The parametric study in Viguié (2006) showed that the sets of parameters in Table 11.7 provide interesting results regarding LCO suppression in the drill-string system. An illustrative example of the effect of the NES on the dynamics is presented in Figure 11.10; the results clearly show that the effect of the NES is to stabilize the operation of the drill-string system for an input voltage equal to 2 V. It is worth mentioning that the equilibrium of the system without NES, ωl = ωu = ωeq and

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Table 11.4 Changes in the bifurcation diagram for variation of the nonlinear stiffness kθ nl . kθ nl (Nm/rad3 )

kθ nl (n × kθ nl init )

Stable equilibria (range)

Other solutions (range)

Other solutions (type)

0.0025 1 1 × 10−2 1 × 10−3 1 × 10−7

1 4 × 102 4 4 × 10−1 4 × 10−5

[1.7 →] V [3.5 →] V [2.9 →] V [2.1 →] V [2.6 →] V

[0.3 → 1.7] V [0.3 → 3.5] V [0.3 → 2.9] V [0.3 → 2.1] V [0.3 → 2.6] V

Periodic solutions Periodic solutions Periodic solutions Periodic solutions Periodic solutions

Table 11.5 Changes in the bifurcation diagram for variation of the inertia Jadd . Jadd (kg·m2 )

Jadd (% of Jtot )




0.25895 0.05179 0.03107 0.02072 0.01294 0.00259

50 10 6 4 2.5 0.5

[2.5 →] V [2.1 →] V [1.9 →] V [1.6 →] V [2.8 →] V [3.6 →] V

[0.3 → 2.5] V [0.3 → 2.1] V [0.3 → 1.9] V [0.3 → 1.6] V [0.3 → 2.8] V [0.3 → 3.6] V

Periodic solutions Periodic solutions Periodic solutions Periodic solutions Periodic solutions Periodic solutions

Table 11.6 Changes in the bifurcation diagram for variation of the damping coefficient ca . ca (Nms/rad)

ca (n × ca,init )




0.1050 0.0210 0.00525 0.00105

10 2 0.5 0.1

[2.8 →] V [1.7 →] V [2.1 →] V [3.5 →] V

[0.3 → 2.8] V [0.3 → 1.7] V [0.3 → 2.1] V [0.3 → 3.5] V

Periodic solutions Periodic solutions Periodic solutions Periodic solutions

Table 11.7 Selected NES parameters. Set

kθ nl (Nm/rad3 )

Jadd (kg·m2 )

ca (Nms/rad)

#1 #2 #3

0.002515 0.002515 0.002515

0.025895 0.020716 0.025895

0.0105 0.0105 0.0210

α = αeq , is preserved also in the system with NES attached as the new equilibrium ωl = ωu = ωa = ωeq and α = αeq . We note that the new equilibrium solution provided by the NES (and depicted in Figure 11.10b) might not be the only possible steady state solution at this particular value of the voltage. This issue is addressed by a detailed study of the topology of the steady state solutions of the integrated system performed in the next section.


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(a)

(b) Fig. 11.10 Direct numerical simulation of the drill-string system for input voltage uc = 2 V: (a) without NES; (b) with NES (parameter set 1).

11.4 Detailed Analysis of the Drill-String with NES Attached We now carry out a detailed analysis of the nonlinear dynamics of the drill-string system coupled to the NES. The main result of this analysis consists of bifurcation diagrams of the integrated system for the three sets of NES parameters listed in Table 11.7. Moreover, the basins of attraction of the stable solutions (for given input voltages) are presented. The purpose of the study is to investigate the NES efficacy and robustness together with the complexity of the resulting dynamical behavior. Finally, the wavelet transform is applied to the resulting time series in order to iden-

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tify TRCs between the drill-string system and the NES. As discussed in previous chapters, this type of transient resonance interactions is responsible for passive TET in the integrated system.

11.4.1 NES Efficacy To examine the efficacy of using an NES to stabilize the drill-string system, in Figure 11.11 we depict the comparison of bifurcation diagrams of the steady state dynamics of the system with and without NES attached. In these diagrams we adopt the notation of Figure 11.7 to denote the branches of equilibrium points and periodic orbits. These results clearly demonstrate the uniform improvement of the steady state dynamical behavior brought by the addition of the NES, for all three parameter sets considered (see Table 11.7). In particular, we note the following: (i) the ranges of input voltages corresponding to stable equilibrium points (i.e., the nominal behavior of the drill-string system) increase; (ii) the LCOs of the drill-string system can be completely eliminated in certain voltage ranges; and (iii) the addition of the NES can result in partial LCO suppression, i.e., it can reduce the amplitude of the surviving LCOs in the regions where complete elimination is not possible (mainly for voltages below 1.8 V). Moreover, a comparison of the bifurcation diagrams presented in Figure 11.11 reveals that depending on the NES parameters the dynamics of the system in regions of partial LCO suppression can be either relatively simple (e.g., parameter set 3 – Figure 11.11d), or more complicated (e.g., parameter set 1 – Figure 11.11b). For the quantitative assessment of NES efficacy, three different criteria are considered: (i) the percentile reduction of the range of voltages leading to unstable equilibrium points and their stabilization to asymptotically stable equilibrium points; (ii) the percentage of the voltage range leading to stable equilibrium points in the range [0, 3.83 V]; and (iii) the percentage of the input voltage range [1.69 V, 3.83 V] for which locally stable equilibrium solutions are transformed to globally stable equilibrium solutions (or equivalently, complete LCO suppression by the NES). The numerical values related to these criteria are listed in Table 11.8, and show that the NES parameter set 3 (see Table 11.7) provides a wider range of voltages leading to globally stable equilibrium solutions; hence, for this parameter set better stabilization of the drill-string operation results. Only NES parameter sets 1 and 3 are considered in detail in this study, as their respective bifurcation diagrams present clear differences due to the differences of the corresponding damping coefficients. Another advantage of using the NES parameter set 3 is that it leads to a less complex bifurcation diagram, as seen from the plots depicted in Figure 11.11. A complete characterization of the different bifurcation diagrams is beyond the scope of this work. However, we note that transitions between different branches may be realized through discontinuous SN bifurcations and subcritical Hopf bifurcations.


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Fig. 11.11 Bifurcation diagrams of steady state solutions for the system with (a) no NES attached; (b) NES with parameter set 1 (see Table 11.7); solid lines correspond to stable solutions and dotted lines to unstable ones.

An additional aim of the passive vibration mitigation design is to structurally perturb the primary (drill-string) system to the lowest possible extent. Accordingly, the NES was kept as light as possible. The second parameter set in Table 11.7 corresponds to rotating inertia for the NES equal to 4% of the total rotating inertia, instead of 5% used for parameter sets 1 and 3. Comparing the bifurcation diagrams

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Fig. 11.11 Bifurcation diagrams of steady state solutions for the system with (c) NES with parameter set 2; (d) NES with parameter set 3; solid lines correspond to stable solutions and dotted lines to unstable ones.

corresponding to NES parameter sets 1 and 2 (which differ only slightly by the NES mass moment of inertia – depicted in Figures 11.11b and 11.11c, respectively), we note that, although the corresponding global topologies of the branches of steady state solutions appear to be similar, the slightly larger rotating inertia introduces added complexity to the steady state dynamics in the local range [0.7 V, 1.7 V].


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Table 11.8 Stabilization of the drill-string system (quantitative assessment). Criterion

Set 1 (%)

Set 3 (%)

Difference (%)

(i) (ii) (iii)

11.04 65.04 94.95

15.89 66.95 97.66

4.85 1.91 2.71

This confirms the added complexity of the dynamics of the integrated system, due to the strongly nonlinear stiffness characteristic of the attached NES.

11.4.2 Robustness of LCO Suppression The previous parametric study indicated that there are ranges of input voltages at which multiple stable and unstable steady state solutions co-exist. Indeed, a key feature of nonlinear dynamical responses is that multiple periodic, quasi-periodic, equilibrium or even chaotic solutions may coexist in certain parameter ranges. For instance, in the drill-string system a stable LCO may coexist with a stable equilibrium point in certain ranges of input voltage. What determines the attraction of the dynamics to either one of these co-existing steady state solutions is the specific set of initial conditions of the system, which places the dynamics in the domain of attraction of either one of the stable attractors. Therefore, to study the robustness of the passive vibration control realized by the addition of the NES, it is meaningful to study the basins of attraction of each of the computed co-existing steady state solutions. The integrated system (11.6) consisting of the drill-string and the attached NES possesses a five-dimensional phase space, meaning that the corresponding basins of attraction of its stable solutions are five-dimensional. It follows that for graphical representation of the basins of attraction, only two phase variables will be varied, namely the velocity of the lower disc ω1 and the deformation of the string of the primary system α. All other phase variables will be assumed to possess zero initial values. The basins of attraction of the attractors of the integrated system for input voltages uc = 2 V and uc = 1.65 V are depicted in Figures 11.12 and 11.13, respectively; these plots enable us to get insight into the variations of the basins of attraction when the NES is added to the drill-string system. Circles and stars refer to initial conditions leading to stable LCOs and stable equilibrium points, respectively. It is noted that for an input voltage uc = 2 V (see Figure 11.12), the transformation of locally stable equilibrium points into globally stable equilibrium points due to the action of the NES, is evident for the three parameter sets considered in this study. For an input voltage of uc = 1.65 V (see Figure 11.13) we note the complicated dynamics introduced by the NES for parameter set 1; this does not hold, however, for parameter set 3. These diagrams confirm that the LCOs originally present in

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Fig. 11.12 Domains of attraction of the dynamics of the integrated system for input voltage uc = 2 V, system with (a) no NES attached; (b) NES parameter set 1; (c) NES parameter set 2; (d) NES parameter set 3; circles denote attraction by stable periodic solutions and stars denote attraction by stable equilibrium points.

the drill-string system without NES attached, have been transformed into locally asymptotically stable equilibrium points in the integrated system. A final remark is that these numerical results confirm that the branch e4 of stable equilibrium points of the bifurcation diagrams in Figures 11.11b and 11.11d consists indeed of locally (for uc = 1.65 V) or globally (for uc = 2 V) asymptotically stable equilibrium points.


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Fig. 11.13 Domains of attraction of the dynamics of the integrated system for input voltage uc = 1.65 V, system with (a) no NES attached; (b) NES parameter set 1; (c) NES parameter set 3; circles denote attraction by stable periodic solutions and stars denote attraction by stable equilibrium points.

11.4.3 Transient Resonance Captures The purpose of this section is to study the dynamical mechanisms that govern LCO suppression in the integrated drill-string-NES system. In particular, motivated by similar studies carried out in previous chapters we are interested to identify possible TRCs in the dynamics leading to TET from the drill-string to the NES. To address this issue we apply the wavelet transform (WT) to the time series of the responses of

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the integrated system, as this will help us reveal the possible TRCs that occur in the transient dynamics. As shown in previous chapters, the WT is a suitable technique for analyzing the temporal evolution of the dominant frequency components of nonlinear signals. The comparison of the instantaneous frequencies of the velocities of the lower disk of the drill-string and the NES provides a robust means for verifying the occurrence of TRCs (or frequency locking) in the transient dynamics. Figures 11.14 and 11.15 depict the instantaneous frequencies of the velocities of the lower disk (ω1 ) and the NES (ωa ) for input voltages leading to attraction of the dynamics to either a stable equilibrium point (for uc = 2 V – Figure 11.14), or a stable periodic solution (uc = 1.5 V – Figure 11.15), respectively. These Figures confirm that 1:1 transient or sustained resonance captures leading to fundamental TET from the drill-string to the NES are responsible for the beneficial action of the NES, leading to enhanced instability mitigation. In the results depicted in Figure 11.14 there occurs a 1:1 TRC at frequency ≈ 0.13 Hz in the initial stage of the dynamics, after which the dynamics settles to a stable equilibrium (complete LCO suppression); during the initial 1:1 TRC the NES resonantly interacts with the developing LCO instability of the drill-string and suppresses it by means of TET. In Figure 11.15 there occurs a 1:1 SRC at a frequency f ≈ 0.175 Hz as the dynamics settles into a stable LCO, albeit of smaller amplitude compared to the LCO of the drill-string with no NES attached; in this case the NES is capable of only partially suppressing the developing LCO instability. These numerical results confirm once more that, since the essentially nonlinear NES possesses no preferential resonant frequency, it is capable of passively tuning itself to (and transiently resonating at) the frequency of the developing drill-string instability irrespective of the characteristic frequency of this instability; hence, the NES passively ‘tracks’ the varying frequency of the developing instability. This is demonstrated by the fact that the described 1:1 resonance captures occur at different frequencies in the WT plots of Figures 11.14 and 11.15. In accordance with the applications discussed in previous chapters, these resonance interactions lead to TET from the drill-string system to the NES, where the energy is confined and locally dissipated.

11.5 Concluding Remarks Self-sustained vibrations may appear in mechanical systems for various reasons and often limit the performance of such systems or even cause damage and system failure. In this chapter, the focus was on friction-induced vibrations in drill-string systems. As a benchmark we considered a rotor-dynamic system with (set-valued) friction and flexibilities. We investigated the possibility of passively mitigating these friction-induced vibrations using a nonlinear absorber or NES, characterized by essential stiffness nonlinearity. The motivation for using an NES is its absence of preferential resonance frequency, which enables it to resonate with and extract energy from the drill-string system at arbitrary frequency ranges. Indeed it is this passive


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(a)

(b) Fig. 11.14 Instantaneous frequencies of the responses of the integrated system via WT, for uc = 2 V and NES parameter set 1: (a) WT spectra; (b) plot of local frequency maxima.

adaptive (self-tuning) capacity of the NES that makes it suitable for suppressing time-varying instabilities in a wide range of applications. This was demonstrated not only in the application considered in this chapter, but also in the applications involving vibration and shock isolation, aeroelastic LCO suppression and seismic mitigation considered in previous chapters. The parametric study demonstrated that the NES can completely eliminate LCO instabilities over a relatively wide range of parameters. In other parameter ranges, the action of the NES results in only partial LCO suppression; i.e., in a mere reduc-

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(a)

(b) Fig. 11.15 Instantaneous frequencies of the responses of the integrated system via WT, for uc = 1.5 V and NES parameter set 2: (a) WT spectra; (b) plot of local frequency maxima.

tion of the amplitudes of the surviving LCOs in parameter ranges where complete LCO elimination is not possible. As a general conclusion, however, the addition of an NES to a drill-string system has the potential to improve the global dynamical behavior of the system, and to substantially extend its domain of stable operation. Further optimization studies have to be carried out in order to not only improve the vibration mitigation action of the NES, but also to ensure the robustness and effectiveness of the NES-based passive vibration mitigation design to changes in initial conditions and system parameters.


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References Armstrong-Helouvry, B., Control of Machines with Friction, Kluwer Academic Publisher, Boston, 1991. Brett, J.F., Genesis of torsional drill-string vibrations, SPE Drilling Eng. 7(3), 168–174, 1992. Canudas de Wit, C., Olsson, H., Aström, K.J., Lischinsky, P., A new model for control of systems with friction, IEEE Trans. Aut. Control 40(3), 419–425, 1995. Cunningham, R.A., Analysis of downhole measurements of drill string forces and motions, J. Eng. Ind. 90, 208–216, 1968. Germay, C., van de Wouw, N., Nijmeijer, H., Sepulchre, R., Nonlinear drill string dynamics analysis, SIAM J. Appl. Dyn. Syst., 2007 (in press). Hensen, R.H.A., Controlled Mechanical Systems with Friction, PhD Thesis, Eindhoven University of Technology, the Netherlands, 2002. Jansen, J.D., van den Steen, L., Active damping of self-excited torsional vibrations in oil well drill-string, J. Sound Vib. 179, 647–668, 1995. Keller, H.B., Numerical Solution of Two-Point Boundary Value Problems, Society of Industrial and Applied Mathematics, Philadelphia, 1976. Kreuzer, E., Kust, O., Analyse selbsterregter Drehschwingugnen in Torsionsstäben, ZAMM – J. Appl. Math. Mech. 76(10), 547–557, 1996. Leine, R.I., Nijmeijer, H., Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Springer Verlag, Berlin/New York, 2004. Leine, R.I., van Campen, D.H., Keultjes, W.J.G., Stick-slip whirl interaction in drill-string dynamics, J. Vib. Acoust. 124, 209–220, 2002. Mihajlovic, N., Torsional and Lateral Vibrations in Flexible Rotors Systems with Friction, PhD Thesis, Technische Universiteit Eindhoven, the Netherlands, 2005. Mihajlovic, N., van de Wouw, N., Hendriks, M.P.M., Nijmeijer, H., Friction-induced limit cycling in flexible rotor systems and experimental drill-string set-up, Nonl. Dyn. 46, 273–291, 2006. Olsson, H., Control Systems with Friction, PhD Thesis, Lund Institute of Technology, Sweden, 1996. Richard, T., Germay, C., Detournay, E., Self-excited stick-slip oscillations of drill bits, Com. Rendus Méc. 332, 619–626, 2004. Van den Steen, L., Suppressing Stick-Slip-Induced Drill-String Oscillations: A Hyper Stability Approach, PhD Thesis, University of Twente, the Netherlands, 1997. Viguié, R., Passive Vibration Mitigation in Rotor Systems Using Nonlinear Energy Sinks, M.Sc. Thesis, University of Liège, Belgium, 2006. Viguié, R., Kerschen, G., Golinval, J.C., McFarland, D.M., Bergman, L.A., Vakakis, A.F., van de Wouw, N., Using passive nonlinear targeted energy transfer to stabilize drill-string systems, Mech. Syst. Signal Proc., 2008 (in press).

Chapter 12

Postscript

For more than a century, beginning with the pioneering work of Lord Rayleigh, a fundamental issue addressed in the field of vibration engineering has been the protection of critical systems subjected to destructive input forces and/or motions. Much of this effort has been directed toward narrow-band excitations, harmonic or at least periodic in nature, thus facilitating the development of classical passive vibration isolation and vibration absorption strategies. In the former case, a compliant suspension was integrated in the system to be protected thus reducing its natural frequencies well below the lowest frequency component of the excitation. The greater this difference, the greater the attenuation of the response of the system achieved by the vibration isolation design. In the latter case, an additional degree of freedom, a vibration absorber, was attached to the primary system and tuned to one of its natural frequencies. Input forces and motions at or near that frequency were attenuated to a degree defined by the inherent damping of the absorber, resulting in localization of vibration to the absorber. In general, neither of these approaches provides a solution for the case of broadband excitations. The difficulty arises when the bandwidth of the excitation encompasses one or more natural frequencies of the primary system, resulting in a resonant condition. This is typical for, say, stationary white noise input, which renders both vibration isolation and vibration absorption strategies ineffective; the former due to magnification of the response of the integrated system over a frequency band defined by the damping of the isolation system; and the latter due to magnification at side-bands formed at the new added natural frequencies of the integrated system. The situation is made even more challenging when the wide-band excitation is transient in nature, such as the case of impulsive excitation, since initial conditions become the predominant factor in determining the magnitude of the peak response which is nearly insensitive to damping. While tremendous strides have been made in the field of vibration engineering over the past one hundred years, the analyst and designer of protective systems are still largely constrained to reducing steady-state responses to narrow-band excitations. Within the past 30 years, there has been a synthesis of control engineering, materials engineering, and vibration engineering, resulting in active vibration control

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12 Postscript

systems often employing smart materials. These systems can provide significant advantages in performance due to their ability to provide optimal and adaptive solutions to a wide range of problems, including those involving wide-band and transient excitations. However, the price paid for flexibility and performance is cost, weight, maintenance and reliability. For these reasons, most vibration engineers prefer a passive to an active solution, if necessary performance can be achieved. In this monograph, the authors have attempted to document the genesis and subsequent development of a different, yet still fully passive, paradigm for protecting critical systems from destructive force and motion inputs: targeted energy transfer (TET). The novelty of this approach, though, is its applicability to wide-band and transient inputs, as well as the usual narrow-band and steady state excitations. The method represents a new and unique application of strong nonlinearity, the nonlinear energy sink (NES), a local, simple, lightweight subsystem capable of completely altering the global behavior of the primary system to which it is attached. The underlying mechanism, a series of transient resonance captures and escapes, combined with nonlinear beating phenomena associated with excitation of special impulsive orbits which comprise the bridges to TET, provides an entirely different passive approach to quickly, efficiently, and nearly irreversibly moving vibrational energy through a preferred, a priori selected direction from the primary system to an NES where it can be localized and harmlessly dissipated. The application of TET through the use of both smooth and discontinuous NESs has been demonstrated herein for shock and vibration isolation, passive aeroelastic instability control, seismic hazard mitigation, and stabilization of long, slender drill strings. Additional applications of the technology currently under investigation by the authors, their colleagues, and others include vibration control of bluff bodies in flow, protection of complex structures from large magnitude shocks, vibration reduction in gear sets, broadband acoustic attenuation, and broadband vibration amplification for energy harvesting. Moreover, we envision that the generality of the methods discussed herein may also lead to fruitful applications of TET in diverse fields of science and engineering. In the field of acoustics, TET can be employed in designing acoustic NESs for passively reducing engine noise or improving the acoustic performance of closed champers and halls. In biomedical applications, electromagnetic TET could be applied to non-intrusive detection of abnormalities, such as cancers. In electronics, TET could find application in wireless energy/power transfer in portable devices. Moreover, TET-based designs can find application in micro/nano devices for enhanced passive vibration mitigation of sensitive components or for achieving directed energy transfers between components, favorable to the design objectives. Additional potential applications of nonlinear TET could be in the fields of solid state physics (e.g., studying the dynamics of lattices or superlattices with local nonlinear defects), sensing technology (e.g., developing sensors with enhanced sensitivity towards ambient energy variations), turbulence modeling and chaotic dynamical systems (that is, studying targeted energy transfers between temporal or spatial scales and relating these to bifurcations and complexity), bioengineering (understanding, for example,


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the role of TET in stiff/soft dynamical interactions in cell dynamics leading to complex dynamic phenomena such as relaxation oscillations) and in other areas. It should also be stated that despite the amount of work already devoted to the development of a theory for nonlinear TET, some of it reported in this monograph and some included in references herein, certain challenging issues still remain unresolved or at least not completely resolved. Among important open issues one should mention the need for designing practical, essentially nonlinear NESs suitable for engineering applications; current designs, some of which are reported in this work, are sufficient under laboratory conditions but are hardly applicable yet to engineering practice. Another challenge is the need to extend low-dimensional analytical models of TET-related phenomena so that they are applicable to realistic high-dimensional problems. The authors hope that this monograph will assist those interested in applying TET in their own work and, in the mean time, raise awareness of the potential benefits to be gained through intentional, constructive and careful application of nonlinearity in applications in science and engineering.

Index

Absorber linear – I 262; II 9, 117, 153, 155, 161, 202, 206–209, 223, 224, 226, 241–249, 251, 253, 255–258, 343 nonlinear – I 104; II 155, 161, 202, 203, 207, 210, 213, 226, 559, 640 shock – II 116, 241, 245, 259 tuned – II 67, 161, 223 vibration – I 1, 5, 134, 257; II 9, 67, 117, 153, 161, 201, 202, 206–210, 213, 226, 241–245, 248, 249, 253, 255–258, 343, 645 Action-angle variable I 38, 40, 41, 48 Acoustics I 95; II 646 Adiabatic I 172, 272, 289; II 241 Aeroelastic excitation II 462 instability I vii, 70; II 311, 353, 397, 398, 402, 428, 438, 452, 456, 459–463, 464, 474, 477, 487, 493, 496, 501, 509, 515, 516, 520, 521, 524, 525, 528, 536, 541, 546, 549, 551, 552, 559, 561, 563, 564, 565, 646 mode II 399, 452, 453, 462–464, 474, 477, 481, 482, 487, 497, 498, 501, 509, 515, 516, 520, 524, 549, 551, 552, 555, 559 response II 398, 404, 527, 549, 564, 565 suppression II 311, 353, 397, 487, 520 Asymptotic expansion I 48, 64, 333 limit I 26, 99, 310, 332; II 4, 16, 140–142, 153, 233, 244, 327, 330, 336 method, methodology I 54; II 263 stability I 20; II 183, 626 Attenuation zone (AZ) I 269, 271, 275, 277– 289, 294; II 42, 66, 67, 70, 80, 82, 95, 98, 103 Attraction

domain of – I 47, 53, 54, 169, 184, 185, 197, 202, 203, 286; II 226, 315, 320, 340, 390, 477, 521, 637 Attractor I 47, 52, 53, 193; II 61, 187–191, 194, 196, 207, 210-214, 225, 226, 382, 394, 453, 459, 462, 487, 496, 521, 524, 637 Averaging method of – I 341 multi-phase – II 408, 410, 414, 452, 471 simple – I 341 Backbone curve I 97, 123, 166, 171, 179, 183, 306, 324, 327, 328, 333, 334, 352, 354, 356; II 39, 120, 127, 259, 319, 342 Beam I 10; II 1-6, 8-12, 67, 236, 242, 349, 580, 582, 593 Beat I 4, 40, 72, 94, 97, 102-105, 114, 134, 137, 141, 142, 146-149, 156, 157, 163, 164, 167– 171, 184–186, 203, 208–212, 216–218, 222, 233, 245, 246, 250, 251, 285, 349; II 11, 16, 17, 21–26, 30, 37, 39, 52, 209, 215, 220, 269, 280, 286, 288, 315, 316, 318–320, 339, 340, 342, 438, 461, 464, 515, 552, 646 Bifurcation branch point cycle (BPC) – II 497, 499, 501, 521, 523, 525, 547, 548, 562–564 co-dimension of – I 303; II 174, 525 global – II 192, 203, 467, 487, 489 Hopf – I 4; II 164, 168–177, 203, 218, 219, 221, 222, 382, 387, 388, 390, 394, 396– 398, 418, 419, 464, 467, 472, 473, 475, 476, 493, 494, 496, 497, 501, 521, 523– 525, 527–531, 536–548, 559–564, 626, 634 limit point cycle (LPC) – II 382, 390, 396, 472, 473, 475, 476, 493, 494, 497, 501, 521, 523, 525–531, 536–548, 559 manifold II 530

369

370 Neimark–Sacker (NS) – II 382, 474, 477, 496, 525, 536, 549 neutral-saddle – II 494, 539, 540, 543-545, 547, 548, 562-564 pitchfork – I 19, 22; II 382, 384, 497 saddle-node (SN) – I 120, 123, 161, 163, 189, 203, 213, 218, 224, 233, 246, 249; II 164, 181, 266, 382, 493, 562–564, 626 Boundary value problem linear (BVP) – II 102 nonlinear (NLBVP) – I 109-114; II 41, 101, 262, 266 Broadband I vii, 3, 5, 37, 72, 74, 86, 156, 170, 233, 237, 257, 258, 262, 264, 304, 313, 349, 363, 365; II 6, 9, 11, 12, 21, 22, 37, 109, 117, 120, 123, 126, 127, 131–133, 137, 158, 161, 202, 206, 226, 230, 231, 239, 241, 245, 254, 257, 312, 322, 323, 331, 353, 354, 452, 459, 461, 464, 501, 520, 530, 531, 533, 549, 552, 559, 615, 645, 646 Chaotic dynamics I 2, 8, 319; II 241 Clearance I 97; II 229, 230, 231, 233–236, 239– 243, 245, 247, 249–251, 253, 256–260, 264, 271, 301, 572–574, 576–578, 590 Configuration plane I 24–26, 112–114, 118, 119, 332, 121, 123, 130, 132, 137, 138; II 264, 269– 271, 321 space I 16, 17, 239, 240, 244; II 67, 320 Complexification-averaging method (CX-A) I 10, 15, 54, 56, 58, 67, 69, 70, 93, 124, 125, 128, 130, 157, 173, 176, 212, 213, 238, 269, 276, 291, 293–295, 342, 343, 346; II 88, 91, 162, 175, 176, 216, 218, 344, 345, 359, 361, 398, 402, 409, 453, 483, 485 Confinement energy – I 37 motion – I 16, 31, 37; II 351 passive – I 31, 32, 37 Continuation analytic – I 17, 74, 239, 352 numerical – II 354, 357, 381, 382, 386, 394, 418, 447, 452, 471, 487, 493, 536, 559 Continuum approximation I 272, 284 limit I 32, 272, 297, 298; II 98 Coupling strong – II 100, 101, 105, 111, 113, 116, 127, 342, 536, 542 weak – I 20, 22, 28, 31, 41, 42, 270, 306, 311; II 67, 71, 74, 101, 113, 116, 230, 245, 259, 322, 331, 348, 536, 542, 546, 551

Index Cutting process, tool II 619, 620 Damping inherent – I 81; II 645 viscous – I 5, 17; II 2, 13, 17, 18, 67, 88, 99, 100, 110, 134, 140, 155, 230, 256, 258, 260, 261, 274, 280, 285, 291, 297, 298, 300, 313, 314, 322, 331, 356, 503, 506, 509, 539, 544, 561, 573, 580, 585, 622, 631 Differential evolution II 576 Dissipative system I 4, 5, 7, 17, 18, 25, 45, 46, 48, 49, 51, 52, 129, 163, 165, 315 Duffing oscillator I 55, 151; II 60 Earthquake II 571, 572, 574, 576–582, 587– 593, 595, 598-600, 604–615 Empirical mode decomposition (EMD) I 15, 70, 77–80; II 12, 15–19, 30–33, 35–37, 40, 56– 66, 99, 107, 109, 110, 117, 118, 127, 132, 133, 141, 142, 147, 151, 153, 198, 282, 292, 293, 295, 301, 302, 331, 336, 338, 453, 463, 477, 478–484, 488, 520 Energy absorber I 308, 311, 365; II 462, 559 dissipater I 289; II 21, 559 dissipation measure (EDM) I 99, 170, 201, 203, 205, 262, 263, 264, 310, 311, 312, 314, 318, 347, 356, 359, 361; II 4–7, 11, 16, 19-22, 24, 33, 109, 111, 112, 115, 117–119, 140, 142, 144, 147, 151, 153, 156, 230, 232, 233, 239, 244–251, 318, 325–330, 340, 342, 354, 453, 551 flow of – I 5, 272, 273; II 17, 24, 26, 141 harvesting I vii, 1, 3; II 646 threshold I 157, 163, 169, 170, 184, 208, 242, 246, 250, 256, 267, 282, 284, 313, 315; II 265, 271, 552, 599 transaction measure (ETM) II 16, 17, 25–27, 52, 53, 55, 141, 153–155 transfer broadband – I 3, 37 multi-frequency – I 304 Equilibrium stable – I 16, 44, 47, 52, 53, 161, 190, 191, 201; II 493, 625, 626, 628, 631, 632, 634, 637–640 unstable – I 44, 47, 52, 53, 161, 190; II 626, 634 Experimental TET I 254 TRC II 333 Fast dynamics I 151, 174, 278; II 93 Fluid-structure instability II 418

Index Fluid-structure interaction II 369, 399, 400, 411, 532, 549, 559 Fluorescence I 3 Floquet multiplier I 109, 120 Floquet theory I 109, 334; II 626 Flutter analysis II 397, 398, 401 instability II 402 merging frequency II 402 speed II 397, 401, 402, 404, 409, 418, 455, 467, 493, 508, 509, 513, 520, 528, 541, 563 suppression I vii Frequency instantaneous – I 79, 80, 237, 256, 258, 259, 261, 262, 273, 274, 285–290, 340; II 56, 60, 61, 78–81, 83, 85, 97, 98, 147, 198, 200, 444, 474, 478, 480 response I 27, 105, 261; II 39, 173, 202, 203–210, 218, 221, 223, 224, 349, 350, 584 Frequency-energy plot (FEP) I 24–26, 70, 95, 97, 119, 120–124, 127, 128, 130, 131, 135, 147, 149, 152, 155–161, 166–179, 183, 186, 213, 218, 224, 237–260, 266, 269, 318, 320– 336, 343, 346, 347, 352–362; II 39, 40, 43– 56, 99, 101, 104–109, 117–130, 134, 259, 260, 263–271, 273–275, 277, 279, 283–286, 288, 290–292, 294, 297–301, 319, 320, 342, 365, 367–371, 373, 374, 376–378, 380, 468, 471–477, 480 Friction I 264, 266, 267; II 241, 331, 503, 508, 615, 619–627, 640 Fundamental resonance capture I 172 TET I 105, 171–181, 185–187, 205, 208, 209, 213, 240, 241, 246–250, 254, 266– 268, 290; II 80, 280, 285, 287, 290, 292, 294, 319, 335, 336, 438, 467, 640 Hamiltonian dynamics I 4, 20, 70, 106, 108, 157, 163, 165, 166, 179, 208, 209, 237, 246, 247, 316, 318, 321, 328, 330, 346; II 39, 40, 278, 292, 356, 374 perturbation I 8, 39 system I 4, 7, 8, 16, 17, 19, 22, 25, 38–40, 43, 44, 47, 48, 51, 58, 94, 95, 97, 106, 108, 109, 119–123, 134, 137, 147–149, 156, 157, 163–166, 171–179, 184–187, 191, 208, 212, 213, 219, 233, 237–241, 260, 315–321, 326, 341, 346, 347, 352– 365; II 39, 40, 42-46, 50, 51, 54, 101, 109, 118, 120, 259–261, 263, 265, 269,

371 274, 285, 320, 342, 365, 369, 371, 374, 394, 471 Harmonic balance I 9, 23, 154; II 398, 471 Heat transport I 1 Heave mode II 398, 404–408, 422, 424, 427, 438, 453, 455, 460–462, 468, 478–480, 483, 484, 487, 497, 502, 515, 516, 521, 523, 524, 549, 550, 552–557 Heteroclinic orbit I 4; II 271 Homoclinic loop, orbit I 20, 22, 45, 47, 48, 51– 54, 157, 163, 213, 218, 219, 224, 226–233, 246; II 192, 271, 273, 288 Impulsive disturbance II 521, 523, 524, 549, 550, 552, 556, 559 excitation, force I 31, 72, 135, 156, 163, 184, 242, 251, 254, 257, 261, 265–267, 272, 289, 290, 291, 309, 349, 356, 359; II 13, 15, 69, 71, 72, 231, 244, 256, 312– 314, 322, 324, 326–329, 339, 342, 583, 645 orbit (IO) I 97, 108, 114, 120, 123, 133–137, 141, 142, 148, 154, 161, 164, 170, 203, 207, 208, 212, 237, 242–245, 311; II 39, 259, 269, 552, 646 Instability aeroelastic – I vii, 70; II 311, 397, 398, 402, 438, 452, 453, 456, 459–463, 467, 468, 474, 477, 487, 493, 496, 501, 509, 515, 516, 520, 521, 524, 525, 528, 536, 541, 546, 549, 551, 552, 561, 563–565, 646 drill-string – II 640 global – I 22 mode – I 20 orbital – I 22 Integrability I 8, 39, 278 Integral of motion I 33, 58, 67, 143, 159, 188, 279 Interaction modal – I 4, 15, 17, 18, 25, 41, 45, 53, 70, 77, 80, 81, 165, 171, 173; II 12, 17, 27, 37, 101, 109, 117, 126, 127, 130–133, 147, 153, 282, 293, 418, 452, 453, 461– 464, 467, 471, 474, 478, 520, 533, 566 nonlinear – I 2, 3, 17, 27, 43, 49, 242, 258, 270, 307, 336, 349, 365; II 15, 60, 86, 94, 111, 127, 128, 417, 463, 509, 516, 552, 555 phase – II 417, 427, 431-433, 436, 437, 444, 445, 448, 450, 484, 485, 487, 488 Internal resonance I 4, 8, 15–18, 20, 25–27, 37– 52, 94, 97, 102, 112, 113, 121, 123, 131, 134, 135, 137, 141, 143, 145–147, 152, 155–161,

372 166, 170, 172, 176, 184, 185, 213, 237, 238, 242, 245, 306, 307; II 61, 369, 398, 404, 432, 433, 438, 439, 516 Intrinsic mode function (IMF) I 71, 77–80; II 15, 18, 31–38, 40, 56–66, 99, 117, 127–132, 134, 146–148, 198, 200, 201, 293, 294, 301, 302, 336–339, 477–482, 488 Invariant manifold I 17, 20, 22, 53, 97, 114, 190, 191, 203, 204, 207, 224, 365; II 177, 179, 182, 183, 200, 280 damped NNM – I 17, 18, 27, 52, 53, 171, 172, 190, 192, 193, 209, 213, 216, 247–249; II 50 NNM – I 17, 18, 97, 171, 179; II 50 torus I 8, 20–22, 39, 40 Isolation vibration – I vii, 10, 36, 81; II 161, 173, 202, 203, 206, 207–210, 212, 213, 218, 221– 224, 226, 227, 242, 641, 645, 646 seismic, base – II 571 shock – I vii, 10, 70, 366; II 37, 109, 229, 230, 232, 233, 237, 239, 243, 251–256, 641, 646 KAM theorem, theory I 8, 21, 39 KAM tori I 122 Limit cycle oscillation (LCO) generation of – II 542, 549 suppression of – II 462, 487, 496, 528 Localization I vii, 1–4, 10, 15, 16, 20, 25, 28– 31, 33, 37, 68, 74, 105, 124, 130, 134, 137, 160, 161, 170–172, 179, 304, 349; II 285, 343, 344, 645 Map horseshoe – I 22 one–dimensional – II 187, 188, 190, 193– 196, 198, 220 Poincaré – I 20, 21, 22, 40; II 187, 189, 197, 198, 199, 220, 271, 272 Mitigation LCO – II 428, 463 seismic – I vii, 10, 70, 81, 85; II 157, 229, 259, 302, 305, 311, 571, 580 shock – II 158 vibration – II 223, 226, 620, 629, 635, 641, 642, 646 Modal analysis I 27, 28, 251, 264, 265; II 313, 322, 333, 348, 582–584 curve I 16, 24, 25 function I 33–35

Index line I 16, 17 oscillator I 35; II 6, 65 relation I 33, 332, 333 response I 16, 250, 307, 308; II 432, 474 series I 260 TET I 267 Mode localized – I 45, 289; II 66, 67, 82, 84, 85 nonlinear normal – (NNM) I 4, 9, 10, 15, 16, 94, 113, 166, 289, 306, 310, 332; II 50, 86, 109, 118, 123, 261, 265, 273, 280, 288, 320, 520, 559 structural – I 3–5, 365; II 9, 11, 12, 24, 37, 158, 397, 400, 589, 590, 592, 628, 629 Modulation I 4, 55, 56, 63, 64, 67, 78, 80, 94, 102, 125, 126, 129, 143–145, 147, 148, 157, 158, 173, 175, 177, 180, 182, 187, 188, 192, 193, 195, 197, 198, 200, 208, 213–215, 219, 221, 222, 238, 276, 278–280, 283, 295, 297, 340–344; II 54, 93–97, 161, 163, 168, 176, 177, 179, 209, 216, 217, 315, 316, 336, 361, 364, 404, 410, 415, 417, 422, 439, 447, 461, 483, 484, 509 Multiple scales I 9, 21, 29, 44, 50, 54–59, 62, 64–66, 187, 188, 190, 193, 196, 341; II 180, 182, 415 Narrowband I vii, 5, 169, 237, 257, 363; II 9, 117, 161, 202, 209, 241, 245, 253 NATA II 503–514, 520 Non-integrability I 8, 122, 178; II 271 Nonlinear energy sink (NES) Configuration I (grounded) – I 96; II 311, 320, 330, 342, 344, 356 Configuration II (ungrounded) – I 96; II 311, 356, 502, 505, 513 MDOF – I 98, 303, 305, 307–311, 313–320, 324, 325, 330–332, 336, 338–340, 347, 349, 352–356, 359, 363, 365, 366; II 1, 99–101, 103–105, 109–117, 120, 123, 126–128, 132, 133, 140–142, 150–156, 226, 353, 397, 502, 520, 530–559 piecewise–linear – II 230 SDOF – I 97, 98, 103–105, 108, 165, 233, 237, 265, 269, 303, 305, 313, 316, 325, 359, 365; II 1, 2, 12, 14, 66, 99, 112, 132, 133, 140, 142, 145–157, 311, 320, 353, 394, 453, 455, 502, 520–533, 536, 541, 542, 546–550, 552, 559–561, 564 vibro–impact (VI) – I 10, 85, 86; II 241– 260, 264–268, 270, 274, 275, 277, 279– 282, 284–305, 572–574, 576–581, 584– 612

Index Nonlinear modal interactions I 4, 15, 17, 18, 25, 41, 45, 53, 70, 165, 171, 173; II 12, 17, 27, 37, 101, 109, 117, 127, 130, 131, 133, 147, 153, 282, 418, 453, 462–464, 471, 474, 478, 520, 566 Nonlinearity essential – I 94, 96, 121, 134, 173, 185, 239, 265; II 4, 9, 20, 46, 78, 87, 105, 111, 213, 312, 313, 320, 344, 349, 526, 528, 530, 531, 536, 537, 540, 541, 542, 545, 546, 572, 573 geometric – I 3, 81, 82; II 349 non–smooth – I 10, 29, 311 smooth – II 262, 269, 273, 285, 287, 288, 302 vibro–impact (VI) – I 10, 86; II 242, 271 Nonlinear Schrödinger equation (NLS) I 2 Oil-drilling I 10 Optimal TET I 10, 93, 97, 163, 165, 170, 185, 207, 212, 213, 216–218, 220, 224, 232, 233, 264; II 112, 151, 273 Oscillation periodic – I 16, 17, 20, 131, 171, 280, 330; II 70, 80, 85, 88, 91, 104, 123, 167, 173, 220, 353, 382 relaxation – II 161, 182, 184–191, 196, 197, 220, 647 Perturbation adiabatic – II 241 regular – I 45, 111, 224, 333 singular – I 8, 45, 335; II 179, 200 Pitch mode II 398, 399, 404–408, 418, 424, 427, 428, 432, 433, 438, 439, 452–456, 458, 460– 463, 467, 468, 474, 478–480, 484, 487, 503, 515, 516, 528, 536, 551, 552, 561 Plate I 10, 264; II 132–158, 582 Propagation zone (PZ) I 269, 271–273, 275, 280, 289, 290, 294, 297; II 42, 66, 70, 80, 86–88, 90, 98, 103 Proper orthogonal decomposition (POD) I 304 Proper orthogonal mode (POM) I 264 Quasi-periodic beat I 147 function I 340, 342 LCO II 521, 523, 549, 562–564 motion, response I 8, 39, 70, 148, 173, 342; II 210, 226, 260, 301, 471, 474, 477, 497 orbit I 4, 7, 40, 44, 45, 97, 108, 135–137, 146, 148, 155, 156, 162, 241, 242; II 39, 269, 382, 471, 493, 625 oscillation I 72, 349; II 174, 176

373 TET I 10; II 201, 203 Quasi-periodicity II 493, 525 Reduced order I 27, 264, 304; II 353, 439, 444, 447, 463, 468, 621 Resonance band I 22, 123, 172 fundamental (1:1) – I 56, 62–65, 172; II 173, 198, 202, 203, 206, 207, 216, 218–221, 224, 225 interaction I 4, 5, 45, 81, 254, 255, 267, 271, 275, 276, 278, 279, 303, 308, 336, 340; II 18, 30, 31, 33, 37, 58, 61, 86, 89, 90, 98, 114, 120, 123, 126, 127, 147, 152, 167, 213, 216, 292, 294, 336, 374, 432, 447, 468, 516, 530, 634, 640 manifold I 40, 41, 43, 45–50, 52, 64, 65, 69, 94, 167, 172, 179, 184, 186, 187, 192, 197, 198, 208, 212–15, 218, 246, 247, 297, 304, 341, 365; II 39, 95, 280, 288, 336, 340, 369, 371, 408, 477, 478 nonlinear – I 2, 5, 25, 45, 81, 172, 246, 254, 275; II 13, 18, 31, 37, 58, 147, 152, 162, 177, 264, 292, 293, 530, 533 passage through – I 9, 48, 52 scattering by – I 8 subharmonic – I 64, 97, 103, 105, 179; II 366, 374, 468, 474, 483, 515, 516 superharmonic – I 192; II 463, 474, 509 transient – I 15, 37, 47, 48, 52, 55, 64, 65, 72, 80, 93, 94, 102, 166, 171, 173, 175, 197, 237, 255, 258, 303, 304, 307, 336, 342, 356, 365; II 9, 13, 18, 55, 115, 120, 128, 156, 274, 319, 365, 371, 394, 452, 501, 515, 559, 565, 576, 578, 590, 599, 600, 629, 634, 639, 646 Resonance capture cascade (RCC) I 5, 233, 237, 246, 247, 255– 262, 267–269, 298, 303–305, 347, 365; II 24, 30, 55, 98, 311, 453, 629 escape, escape from – I 48, 173; II 9, 65, 316, 433, 438, 444, 452 permanent – I 47; II 365 sustained (SRC) – I 15, 37, 38, 46, 47, 52, 53, 64, 74, 172, 349, 352, 356; II 365, 399, 408, 428, 452, 515, 628, 640 transient (TRC) – I 15, 37, 47, 48, 52, 55, 72, 80, 93, 94, 102, 166, 171, 173, 175, 197, 237, 303, 304, 307, 336, 342, 356, 365; II 9, 13, 18, 55, 115, 128, 274, 319, 341, 365, 371, 394, 399, 428, 452, 515, 559, 565, 576, 578, 590, 599, 600, 628, 629, 639, 640, 646 Robustness

374 – of LCO suppression II 353, 354, 357, 359, 391, 453, 467, 477, 487, 489, 493, 501, 502, 513, 516, 520, 521, 524–526, 528, 530, 533, 536, 541, 542, 546–549, 559, 561, 637 – of LCO elimination II 381, 382, 390, 391, 520 – of TET I 7, 9, 205, 303; II 11, 86, 157, 502 Rod I 10, 81; II 12–63, 66, 69–76, 78–82, 84– 88, 90, 91, 93–120, 123, 126–128, 130, 132, 134, 155 Shock spectrum II 232, 238–241, 251, 253, 255–258 Sifting process I 77, 78 Signal processing I 15, 70, 106 Slow dynamics I 151, 198, 216–218; II 90, 183, 187, 362, 363, 366, 444, 447 Slow-fast partition I 56, 57, 67, 124, 141, 146, 155, 197, 276, 291; II 90–93, 402, 410, 177, 216 Slow flow I 44, 51–53, 57, 58, 66–68, 80, 97, 125, 126, 132, 143, 151, 154, 157, 159, 162, 163, 173, 175, 176, 177, 180–184, 187–189, 191, 197–198, 205–209, 212–232, 279, 282, 297, 342–345; II 15, 93, 94, 161, 163, 164, 168–170, 173–175, 177–180, 182, 184–189, 191–197, 203, 216, 218, 220, 295, 336, 359, 361, 365, 371, 383–385, 389, 398, 399, 402, 404, 408, 410–419, 422, 428, 432, 434–437, 439, 440, 444, 447, 452, 463, 478, 483–485 Slow invariant manifold (SIM) II 182–190, 192, 196, 201, 224 Soliton I 270; II 66, 67 Stability asymptotic – I 20; II 183, 626 orbital – I 20 Steady state motion, response I 9, 55, 285; II 87, 161, 162, 173, 177, 178, 191, 201, 203, 205, 206, 208, 209, 212, 216, 343, 344, 351, 354, 357, 359, 386, 390, 391, 394, 396, 408, 422, 447, 455, 467, 474, 625, 630, 645 TET I 54; II 167, 203, 212, 218, 342, 347, 351 Strongly modulated response (SMR) II 161, 177–180, 182, 183, 185–189, 191–194, 196– 201, 203, 207, 209–214, 219–226, 342 Subharmonic motion I 122, 130, 323, 324, 326, 328; II 43, 48, 105, 120, 268, 269

Index TET I 97, 105, 171, 176, 178, 179, 181–186, 208, 246, 249, 250; II 276, 280, 301, 319, 412 tongue I 25, 121, 124, 125, 128, 131–133, 157, 171, 177, 179, 241, 243, 245, 250, 254, 322, 323, 326, 327, 342, 352; II 39, 46, 48, 49, 51, 52, 54, 55, 58, 104, 107, 120, 123, 126, 259, 268–270, 275, 276, 280, 285, 288–291, 320, 471 transient resonance capture (TRC) I 176, 249, 285; II 55, 201, 275, 276, 474, 480 Switch model II 625 System identification I 28, 72, 264; II 65, 293, 295, 312, 313, 331, 398, 508 Transversality I 47; II 185 Transform Fast Fourier (FFT) – I 71, 307, 324 Hilbert – I 15, 38, 71, 77, 79, 80, 247, 258, 259; II 15, 78, 198, 200, 201, 294, 331, 336, 337, 463, 474, 477, 488 wavelet (WT) – I 15, 70, 71, 95, 166, 247, 304, 305, 307, 309, 315–317, 347; II 12, 15–17, 19, 24, 27–31, 33, 37, 39, 40, 50, 51, 54–57, 59, 61–66, 105, 107, 109, 117–120, 122, 123, 125–127, 129, 130, 133, 141, 147, 148, 259, 260, 273–280, 282–285, 287, 288, 290, 291, 293, 294, 296, 297, 299, 302, 319, 335, 336, 342, 363, 365–373, 376, 378, 380, 398, 404, 406, 408, 432, 433, 453, 463, 468, 471, 474, 476, 483, 484, 514–519, 549, 552, 564, 633, 639–642 Transient TET II 177, 201, 462 Transition multi-frequency – II 39, 40, 126, 260 multi-modal – II 259 Triggering II 319, 397, 398, 408, 417, 418, 422, 428, 433, 439, 444, 447, 452–454, 462, 468, 480, 484, 487, 509, 515, 516, 549 Tuned mass damper (TMD) I 5, 98, 103–107, 262–264; II 140, 153–158, 161, 202, 207, 226, 571, 615, 629 Twist II 399, 400 Van der Pol (VDP) oscillator II 353–357, 359 361, 364–366, 369, 371, 374, 390, 391, 394 Vibro-impact (VI) NES I 85; II 157, 241–260, 264–270, 274, 275, 277, 279–282, 284–305, 311, 572– 574, 576–581, 585–604, 606–609, 611, 612 oscillation II 241 seismic mitigation I 86

Index Vortex shedding I vii; II 400 Wave localized – I 33 solitary – I 2, 35; II 66 standing – I 270, 272, 275, 277, 278, 280, 284–287, 289, 290; II 42, 66, 70, 87, 98, 103 traveling – I 36, 271, 273, 274, 280, 298; II 42, 66, 70, 72, 73, 86–88, 95, 97, 98, 103 Wavelet (WT) spectrum II 15, 17, 24, 27–31, 33, 37, 39, 40, 51, 54–57, 59, 61–63, 65, 66,

375 117–120, 122, 123, 125–127, 129, 130, 134, 147, 148, 259, 260, 273–280, 282–285, 287– 291, 293, 294, 296, 297, 299–302, 319, 335, 365, 367–370, 373, 376, 378, 380, 404, 406, 407, 468, 471, 476, 483, 516, 552, 641, 642 Wing II 353, 354, 397–402, 404, 408, 409, 418, 428, 438, 447, 452–455, 458–464, 467, 468, 474, 477, 478, 483, 487, 497, 501, 502–509, 513, 516, 520, 521, 524–527, 529–533, 536, 541, 542, 546–554, 557–564

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