Normal modes and localization in nonlinear systems

4 NORMAL MODES AND LOCALIZATION IN NONLINEAR SYSTEMS w WILEY SERIES IN NONLINEAR SCIENCE Series Editors: ALI H. NAY...

Author: Alexander F. Vakakis | Leonid I. Manevitch | Yuri V. Mikhlin | Valery N. Pilipchuk | Alexandr A. Zevin

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4

NORMAL MODES AND LOCALIZATION IN NONLINEAR SYSTEMS

w

WILEY SERIES IN NONLINEAR SCIENCE Series Editors:

ALI H. NAYFEH, Virginia Tech ARUN V. HOLDEN, University of Leeds

Abdullaev Bol oti n Nayfeh Nayfeh and Balachandran Nayfeh and Pai Ott, Saucr. and Yorke

Theory of Solitons in Inhomogeneous Media Stability Problems in Fracture Mechanics Method of Normal Forms Applied Nonlinear Dynamics Linear and Nonlinear Structural Mechanics Coping with Chaos Robust Control of Nonlinear Uncertain Systems Matched Asyinptotics of Lifting Flows Normal Modes and Localization in Nonlinear Systems

QU

Rozhdestvensky Vakakis, ct al.

4

NORMAL MODES AND LOCALIUTION IN NONLINEAR SYSTEMS ALEXANDER F. VAKAKIS Department of Mechanical and Industrial Engineering University of Illinois at Urbana-Champaign Urbana. 1L 61801

LEONID I. MANEVITCH Institute of Chemical Physics Russian Academy of Sciences Moscow. Russia

YURl V. M l K H L l N Department of Applied Mathematics Kharkov Polytechnic University Kharkov. Ukraine

VALERY N. PlLlPCHUK Department of Applied Mathematics Ukrainian State Chemical and Technological University Dnepropetrovsk, Ukraine

ALEXAN DR A. ZEVl N TRANSMAG Research Institute Ukrainian Academy of Sciences Dnepropetrovsk. Ukraine

A Wley- Interscience Publication

JOHNWILEY & SONS, INC. New York

Chichester

0

Brisbane

Toronto

Singapore

This text is printed on acid-frcc paper. Copyright 0 1996 by John Wiley B Sons, Inc

All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department. John Wiley B Sons. lnc., 605 Third Avenue. New York. NY 10158-00 12. Library of Congress Cataloging in Publication Data:

Normal modes and localization in nonlinear systems /Alexander F. Vakakis . . . [et al.1. p. cm. - (Wilcy series in nonlinear science) Includes hibliographical references and index. ISBN 0-47 1- 133 19- I 1. Vibration. 2 . Nonlinear theories. 1. Vakakis, Alexandcr F.. 1961. 11. Series. TA3SS.N668 1996 53 1’.32-dc20 95-26004 Printed i n the United States of America 109876.5432 I

CONTENTS

ix

Preface Acknowledgments 1

introduction

xiii 1

1.1 Concepts of lonlinear Normal Mode (NNM) and Nonlinear Localization, 1 1.2 Example: NNMs of a Two-DOF Dynamical System, 8

2

N NMs in Discrete Oscillators: Qualitative Results 2.1 2.2 2.3 2.4

3

23

Preliminary Formulation, 23 Existence Theorem for NNMs, 35 Applications of the Existence Theorem, 46 NNMs in Systems with Concave and Convex Nonlineari ties. 52

N NMs in Discrete Oscillators: Quantitative Results

69

3.1 Introduction, 69 3.2 Conservative Systems, 72 3.2.1 Trajectories of NNMs in Configuration Space, 72 3.2.2 Similar NNMs, 80 3.2.3 Nonsimilar NNMs and Matched Asymptotic Expansions, 86 3.2.4 Application to a Two-DOF Strongly Nonlinear System, 116 3.3 Invariant Manifold Approaches for NNMs, 124 3.4 Analysis of NNMs Using Group Theory, 130 3.5 Vibro-Impact Systems. 145 4

Stability and Bifurcations of N N M s

157

4.1 General Stability Results, IS8 V

vi

CONTENTS

4.2 Similar NNMs, 169 4.2.1 Analysis of Stability Boundaries, 175 4.2.2 Finite-Zoning Instability Conditions, 186 4.3 Nonsimilar NNMs, 196 4.4 NNM Bifurcations in a System in Internal Resonance, 207 4.5 Stability of Stationary Waves, 219 5

Resonances of Discrete Systems Close to NNMs

229

5.1 Exact Steady State Motions, 230 5.2 Admissible Forcing Functions for Steady State Motions, 238 5.3 Effects of NNM Bifurcations on the Resonances. 253 6

The Method of Nonsmooth Temporal Transformations ( N SlTs)

261

6.1 Preliminaries, 261 6.2 Representations of Functions Using NSTTs, 266 6.3 Analysis of Dynamical Systems, 269 7

Nonlinear Localization in Discrete Systems

285

7.1 Weakly Coupled Oscillators: Qualitative Results, 289 7.1.1 Existence and Stability of Periodic Solutions, 289 7.1.2 Nonlinear Mode Localization, 295 7.2 Mode Localization in Systems with Cyclic Symmetry, 304 7.2.1 Asymptotic Analysis of Modal Curves, 305 7.2.2 Transition from Localization to Nonlocalization, 325 7.3 Mode Localization in a Strongly Nonlinear System, 337 7.4 Localization in Impulsively Forced Systems, 344 8

NNMs in Continuous Systems

8.1 Systems of Finite Spatial Extent, 349 8.1.1 Direct Analysis ofthc Equations of Motion, 352 8.1.2 Analysis by Discretization, 372 8.1.3 Stability Analysis of NNMs. 374 8.2 Systems of Infinite Spatial Extent, 380 8.2.1 Stationary Waves as NNMs, 381 8.2.2 Waves in Attenuation Zones of Monocoupled Nonlinear Periodic Systems, 389

349

CONTENTS

9 Nonlinear Localization in Systems of Coupled Beams

vii 391

9.1 Theoretical Analysis, 391 9.1.1 Nonlinear Mode Localization: Discretization, 391 9.1.2 Passive Motion Confinement of Impulsive Responses, 410 9.1.3 Nonlinear Localization of Forced Steady-State Motions, 424 9.1.4 Nonlinear Mode Localization: Direct Analysis of the Equations of Motion, 444 9.2 Experimental Verification, 462

1 0 Nonlinear Localization in Other Continuous Systems

473

10.1 Multispan Nonlinear Beams, 473 10.1.1 Derivation of the Modulation Equations, 473 10.1.2 Numerical Computations, 480 10.2 Waves with Spatially Localized Envelopes, 496 10.2.1 General Formulation, 499 10.2.2 Application: Localization in an Infinite Chain of Particles, 501

References

51 7

Index

549

4

PREFACE

T h e principal aim of this book is to introduce the reader to the concept and applications of a special class of nonlinear oscillations termed nonlinear normal modes (NNMs). These motions can be regarded as nonlinear analogs of the classical normal modes of linear vibration theory, although NNMs possess some distinctively nonlinear properties; first, the number ofNNMs of a discrete nonlinear oscillator may exceed in number its degrees of freedom; second, in contrast to linear theory, a general transient nonlinear response cannot be expressed as a linear superposition ofNNM responses; third, a subclass of NNMs is spatially localized and leads to nonlinear motion confinement phenomena. Hence, the study ofNNMs and nonlinear mode localization in discrete and continuous oscillators reveals a variety of exclusively nonlinear phenomena that cannot be modeled by linear or even linearized methodologies. As shown in this book, these essentially nonlinear phenomena have direct applicability to the vibration and shock isolation ofgeneral classes of practical engineering structures. O n a more theoretical level, the concept of NNMs will be shown to provide an excellent framework for understanding a variety of distinctively nonlinear phenomena such as mode bifurcations and standing or traveling solitary waves. The material of this book is organized into ten chapters. In the first chapter a general discussion on the concept of NNMs and nonlinear mode localization is given. Lyapunov’s and Rosenberg’s definitions ofNNMs are presented, along with a group-theoretic approach to nonlinear normal oscillations. A motivational example is included to demonstrate the concepts. In Chapter 2 general qualitative results on the existence ofNNMs in a class of discrete conservative oscillators are presented and applications of the general theory are given for systems with convex or convex stiffness nonlinearities. In addition to general existence theorems, theorems regarding the nonlinear mode shapes of NNMs in discrete oscillators are also proved. I n Chapter 3 quantitative analytical methodologies for computing NNMs of conservative and nonconservative discrete oscillators are discussed. NNMs are asymptotically studied by analyzing their trajectories in configuration space or by computing invariant normal mode manifolds in phase space; the later approach due to Shaw and Pierre provides a n analytical framework for extending the concept of NNM in general classes of damped oscillators. In the same chapter, a group-theoretic approach for computing NNMs is presented, along with a discussion of ix

X

PREFACE

NNMs and nonlinear localization in vibro-impact oscillators. The stability and hifurcations of NNMs of discrete oscillators are discussed in Chapter 4. Linearized stability methodologies are considered, and the problem of stability of a NNM is converted to the equivalent problem of determining the stability of the zero solution of a set of variational equations with periodic coefficients. In many cases it is advantageous to transform this variational set to a set of equations with regular singular points. Analytical techniques for computing the instability zones of the transformed variational set are presented. In addition, conditions for the existence of finite numbcrs ofinstability zones in the variational equations are derived (finite-zoning instability). As a demonstrative example, the bifurcations o f NNMs of a discrete oscillator in internal resonance are analyzed in more detail. In Chapter 5 forced resonances occurring in neighborhoods of NNMs are studied. I t is shown that exact steady state motions of nonlinear systems occur close to NNMs of the corresponding unforced systems. Moreover, it is found that NNM bifurcations have profound effects on the topological structure of the nonlinear frequency response curves of the forced system. A new analytical methodology for studying nonlinear oscillations is formulated in Chapter 6, termed the method of nonsmooth temporal transformations (NSTTs). This method is based on nonsmooth (saw-tooth) transformations of the temporal variable and leads to asymptotic solutions that are valid even i n strongly nonlinear regimes where conventional analytical methodologies are less accurate. An application of the NS7T methodology to the problem ofcomputing NNMs in strongly nonlinear discrete systems is presented along with some additional strongly nonlinear (even nonlinearizablc) applications. In Chapter 7 nonlinear mode localization in certain classes of periodic oscillators is discussed. and analytical studies of transitions from mode localization to nonlocalization are given: in addition. NNM bifurcations in a discrete system with cyclic symmetry are analyzed. In the same chapter a numerical example of nonlinear passive motion confinement of responses generated by impulsive loads in a cyclic system is presented. The extension of the concept of N N M in continuous oscillators is performed in Chapter 8. Several quantitative methodologies for studying continuous NNMs are discussed, based on discretization or on direct analysis of the governing partial differential equations of motion. It is shown that the concept of NNM can be employed to study nonlinear stationary waves in partial differential equations, o r waves with decaying envelopes in attenuation zones of continuous periodic systems of infinite spatial extent. In Chapters 9 and 10 nonlinear localization and passive motion confinement in periodic assemblies of continuous oscillators is discussed, and three examples from mechanics are analyzed in detail: a system of coupled nonlinear beams, a multispan nonlinear beam, and a nonlinear periodic spring-mass chain. Experimental studies of nonlinear localization i n systems of coupled nonlinear beams are also presented in Chapter 9, and a new design methodology based o n the nonlinear motion confinement phenomenon is formulated. An interesting conclusion from the applications

PREFACE

xi

presented in Chapter 10 is that the concept of localized NNM can be used to analyze solitary waves or solitons in certain classes of nonlinear partial differential equations. In that context, localized NNMs in discrete oscillators can be regarded as discrete analogs of spatially localized solitary waves and solitons encountered in nonlinear partial differential equations on infinite domains. Many individuals contributed with critical discussions and suggestions in the development of the ideas and methodologies presented in this book. The authors would like to thank Prof. Thomas K. Caughey and Prof. Stephen Wiggins (California Institute ofTechnology), Prof. Richard H. Rand (Cornell University), Prof. Ali H. Nayfeh (Virginia Polytechnic Institute and State University), Prof. R. A. Ibrahim (Wayne StateUniversity), Prof. Stephen Shaw (Michigan State University), Prof. V. Ph. Zuravlev (Russian Academy of Sciences), Prof. A. Bajaj (Purdue University), Prof. I. Adrianov (Prydneprovic State Academy of Civil Engineering and Architecture), and Prof. L. Zhupiev (Mining University ofthe Ukraine) for stimulating discussions, contributions, and suggestions on many topics of this book. The first author would also like to acknowledge the contributions of his current and former graduate students, M. E. King (Boston University), C. Cetinkaya (Wolfram Research Inc.), and T. A. Nayfeh, J. Aubrecht, M. A. F. Azeez, E. Emaci, and J. Brown (Hughes Aircraft Company); their valuable contributions made this book more complete. In addition, the first author would like to acknowledge the past and current support received in the form of research and equipment grants from the National Science Foundation (NSF),the Dow Chemical Company, the Electric Power Research Institute (EPRI), the Hughes Aircraft Company, and IBM. Additional research support was provided by the Center for Advanced Study, the National Center for Supercomputer Applications (NCSA), and the Department of Mechanical and Industrial Engineering and the College of Engineering of the University of Illinois at Urbana-Champaign. This support was instrumental in the development of a major part of the theoretical and experimental results contained in this book. In addition, the authors would like to acknowledge the secretarial support of Mrs. Cel Daniels and Mrs. Tammy Smith of the University of Illinois at Urbana-Champaign. Finally, the authors would like to thank Fotis and Anneta Vakakis, Sotiria Koloutsou-Vakaki, Elpida Vakaki-Emery, and Brian Emery: Elena Vedenova; Olga Lysenko-Mikhlin; Valentina Pilipchuck and Irina Pilipchuck; Aron Zevin and Raisa Phybusovitch; and their extended families in Greece, the United States, and the Commonwealth of Independent States. This book could never have been written or even conceived without their continuous and unconditional love and support. This book is dedicated to them with immense gratitude. A. VAKAKIS Fehruar?, I996

ACKNOWLEDGMENTS

Figures 3.2.4,3.2.5,9.2.1,9.2.2,9.2.3,9.2.4,9.2.5, and 9.2.6 are reprinted with permission of Academic Press Ltd. Figures4.4.1,4.4.2,5.2.1,5.2.2,7.2.1,7.2.7,7.2.8,7.2.9,7.2.10,7.4.1,8.1.1,10.1 10.1.2,10.1.3,and 10.1.4are reprinted with permission ofthe American Society of Mechanical Engineers. Figures 9.1.1, 9.1.2, 9.1.3, 9.1.4, and 9.1.6 are reprinted with permission of Academie Verlag GmbH. Figures 7.2.2,7.2.3,7.2.4,7.2.5, and 7.2.6 are reprinted with permission from the SIAM Journal on Applied Mathematics, pp. 265-282, volume 53, number 1, February 1993. Copyright 1993 by the Socciety for Industrial and Applied Mathematics, Philadelphia, Pennsylvania. All rights reserved. Figures 1.2.1, 1.2.3, 1.2.3, 1.2.4, 1.2.5, 5.3.1, 5.3.2, 5.3.3, 9.1.18, 9.1.19, 9.1.20. 9.1.21, 10.2.1, 10.2.2, and 10.2.3 are reprinted with permission of Elsevier Science. Figures 1.2.6,9.1.10,9.1.11,9.1.12,9.1.13,9.1.14,9.1.15,9.1.16,and9.1.17are reprinted with permission of Kluwer Academic Publishers. Figures 9.1.5,9.1.7,and 9.1.8 are reprinted with permission of the American Institute of Aeronautics and Astronautics.

xiii

NORMAL MODES AND LOCALIZATION IN NONLINEAR SYSTEMS

NORMAL MODES AND LOCALIZATION IN NONLINEAR SYSTEMS ALEXANDER F VAKAKIS, LEONID I MANEVITCH YURl V MlKHLlN VALERY N PlLlPCHUK & ALEXAN DR A ZEVl N Copyright@ 1YYh hy Jolin Wtley & Son$ I ~ L

CHAPTER 1 INTRODUCTION 1.1 CONCEPTS OF IONLINE. R NOR NONLINEAR LOCALIZATION

ODE AND

It is well established that normal modes are of fundamental importance in the theory of linear conservative and nonconservative dynamical systems. This is so because linear normal modes can be used to uncouple the governing equations of motion, and to analytically evaluate the free or forced dynamic response for arbitrary sets of initial conditions. This is performed by employing modal analysis and making use of the principle of linear superposition to express the system response as a superposition of modal responses. In classical vibration theory, the problem of computing the normal vibrations of discrete or continuous oscillators is reduced to the equivalent problem of computing the eigensolutions (natural frequencies and corresponding eigenvectors or eigenfunctions) of linear transformations. Clearly, such an approach as well as the principle of linear superposition are generally inapplicable in nonlinear theory. So, the obvious question arises: I s there a reason to extend the concept of normal modes in the nonlinear case? Naturally, one can define nonlinear normal modes (NNMs) merely as synchronous periodic particular solutions of the nonlinear equations ot motion without seeking any connection of such motions to the (linear) superposition principle. In the framework of such a restricted definition, a nonlinear generalization of the concept of normal mode is possible, and beginning with the works of Lyapunov several attempts were undertaken in this direction. Lyapunov's theorem (1907) proves the existence of n synchronous periodic solutions (NNMs) i n neighborhoods of stable equilibrium points of n degree-of-freedom (DOF) hamiltonian systems whose linearized eigenfrequencies are not integrably related. Weinstein (1973) and Moser (1976) extended Lyapunov's result to systems with integrably related linearized eigenfrequencies (systems in "internal resonance"). Kauderer (1958) studied analytically (and graphically) the free periodic oscillations of a two-DOF system, thus becoming a forerunner in the conceivement of quantitative methods for analyzing NNMs. The 1

2

INTRODUCTION

formulation and development of the theory of NNMs can be attributed to Rosenberg and his co-workers who developed general qualitative (Pak and Rosenberg, 1968), and quantitative (Rosenberg, 1960, 1961, 1962, 1963, 1966; Rosenberg and Hsu, 1961; Rosenberg and KUO, 1964) techniques for analyzing NNMs in discrete conservative oscillators. Rosenherg mnsiderecf n DOF conservative oscillators and defined NNMs as "vibrations in unison," i.e,, synchronous periodic motions during which all coordinates qf the system vibrate equiperiodically, reaching their inuximum and rninirnuin values at the same instant of time. Some additional representative quantitative techniques based on the previous formal definition of NNMs were performed in (Magiros, 1961; Rand, 1971a,b, 1973, 1974; Rand and Vito, 1972; Manevitch and Mikhlin, 1972; Manevitch and Pilipchuk, 198 I ; Mikhlin, 1985; Vakakis, 1990; Caughey and Vakakis, 1991; Shaw and Pierre, 1991, 1992, 1993, 1994; Boivin et al., 1993; Nayfeh and Nayfeh, 1993, 1994; Nayfeh et al., 1992; Pakdemirli and Nayfeh, 1993). Application of the concept of NNM to control theory is studied by Slater (1993). General reviews of analytical and numerical methods for computing NNMs in discrete and continuous oscillators can be found in King (1994) and Slater (1 993, 1994). In linearizable systems with weak nonlinearities it is natural to suppose that NNMs are particular periodic solutions that, as the nonlinearities tend to zero, approach in limit the classical normal modes of the corresponding linearized systems. Evidently the number of these NNMs must be less or equal to the number of DOF of the systems considered. Moreover, when weak periodic forcing is applied, NNMs can be used to study the structure of thc system's nonlinear resonances (Malkin, 1956; HSU,1959, 1960; Kinney, 1965; Kinney and Rosenberg, 1966; Manevitch and Cherevatzky, 1972; Mikhlin, 1974; Vakakis and Caughey, 1992; Vakakis, 1992b). Here rests a ,first practical application of defining NNMs: Although the principle (fl superposition does not hold in the nonlinear case, f o r w d resonunccs of' nonlinear systems occur in neighborhoods of NNMs, in direct analogy to linear theory. Hence, understanding the structure of NNMs of discrete or continuous oscillators enables one to better study the forced responses of these systems to external periodic inputs. In addition, in some of the aforementioned works particular attention was devoted to "homogeneous" systems, i.e., to nonlinearizable (essentially nonlinear) systems whose stiffness nonlinearities are proportional to the

1.1 CONCEPTS OF NNM AND NONLINEAR LOCALIZATION

3

same power of the displacement. It was shown that the NNMs of homogeneous systems can exceed in number their DOF, a feature with no counterpart in linear theory (with the exception of the case of multiple natural frequencies). This is due to NNM bifurcations, which become exceedingly more complicated as the number of DOF of the systems increase. Thus, not all NNMs can be regarded as nonlinear analytic continuations of normal modes of linearized systems; indeed, an accurate computation of NNMs can reveal dynamic behavior that cannot be modeled by conventional linear or linearized approaches. Bifurcations of NNMs in discrete systems were first studied in works by Rand and co-workers (Rand, 1971a; Rand and Vito, 1972; Month and Rand, 1977; Johnson and Rand, 1979; Month, 1979; Rand et al., 1992), and in (Zhupiev and Mikhlin, 1981; Manevitch et al., 1989; Caughey et al., 1990). In these works it was found that bifurcating NNMs are typically localized in a small portion of the dynamical system. It will be shown that such locali~edNNMs lead to nonlinear spatial confinement of motions generated by external inputs, a feature which is one of the most interesting and important applications o j the theory of NNMs. Nonlinear mode localization can be studied in the frumework of NNMs and gives rise to a variety of nonlinear dynamic phenomena that can be used to develop robust shock and vibration isolation designs for certain classes of engineering systems. Some alternative ways of viewing nodinear normal oscillations are formulated in the following exposition. It is known that linear conservative systems possess certain symmetries that reflect on the properties of their normal modes. Every such property can be associated with a specific symmetry of the governing equations of motion, and in classical vibration theory the normal modes of linear conservative systems can be computed by imposing an invariance of the equations of motions with respect to arbitrary temporal shifts (temporal invariance). In particular, for oscillations on a normal mode all position coordinates of a linear system are proportional to the same exponential function, dut, where j=(-l)l'2, co is the frequency of the normal mode oscillation, and t is the temporal variable. Part of the properties of linear normal modes can be extended to the nonlinear case. More specifically, f o r a certain class of nonlinear systems it is possible to define NNMs as special periodic solutions with exponential temporal dependence of all positional variables. The simplest system representative of this class is a system composed of two weakly coupled particles that are

4

INTRODlJCTION

connected to the ground by weakly nonlinear springs. This system is governed by the following equations of motion:

ii + a u v

+av

+ E e (u - v) - E b u3 = 0 + E e (v - u) - E b v3 = O

(1.1 . I )

where a, e, and b are real quantities, and 1 ~ No= Q/a two additional NNMs exist, corresponding to Q = 0 , e = ~4 c o s - l ( ~ / a ~ ) These additional modes bifurcate from the in-phase NNM, which for N > No becomes orbitally unstable. The two bifurcating modes are stable (Kosevich and Kovalev, 1989), and spatially localized, since the energy of each bifurcating mode is found to be predominantly confined to only one of two particles of the system. So one observes two essentially nonlinear features of system (1. 1.1), namely, that its NNMs can exceed in number the DOF of the oscillator, and that some of its NNMs are spatially localized. Spatial nonlinear localization is one of the most important properties encountered in NNMs and provides a link between NNMs and solitary solutions (solitary waves and solitons) in the theory of nonlinear waves. To demonstrate this link one must consider a generalization of system (1.1.1) for arbitrarily large or infinite degrees of freedom (Scott et al., 1985). The analysis then shows that in the limit of weak nonlinearity the n DOF system possesses NNMs in direct analogy to the linear case. Moreover, when the coupling terms become of the same order as the nonlinear terms, there exist numerous mode bifurcations, and the system possesses (3"-1)/2 NNMs, the majority of which are spatially localized; this is in contrast to the corresponding linear n DOF system which only possesses n normal modes. Hence, nonlinear mode localization is a general property of a wide class of weakly coupled oscillators. An additional interesting feature of the n DOF generalization of system (1.1.1), is that as n -+ 00 the system reduces to the discrete approximation of the continuous nonlinear Schr odinger's equation (NSE) with periodic boundary conditions [for an application of Schrodinger's equation to model a linear disordered lattice, see Kuske et al.

6

INTRODUCTION

(1993)l. The important work of Ford (1961) and Waters and Ford (1966) must be mentioned here. They studied recurrence phenomena in the UlarnFermi-Pasta (1955) problem, and showed that lack of equipartition of energy in an infinite nonlinear lattice with periodic boundary conditions is partly due to the existence of stable nonlinear normal modes in this system. The NSE equation is well known (Lamb, 1980; Novikov et al., 1984) to describe a fully integrable dynamical system and to possess soliton solutions of different types in the form of spatially localized waves. Hence, there is a relationship hetween the localized N N M s of certain w e a k l ~coupled ~ mechanical systerns and the soliton solutions of' the NSE. AS shown in (Vedenova et al., 1985; Vedenova and Manevitch, 1981; King and Vakakis, 1994), in the context of NNM theory, stationary periodic solitary waves can be regarded as NNMs of infinite-dimensional systems defined on unbounded domains. A note of caution is appropriate, however, here. If the NSE is regarded as the continuous approximation of an infinite nonlinear lattice of weakly coupled particles, the continuous approxirnation i s only applicable for waves whose wavelengths exceed the distance between adjacent particles. In contrast to such solutions, certain (strongly) localized NNMs of the discrete infinite system are localized predominantly to single particles. Taking this observation into account one notes that rlw concepts of loccdizc~/ N N M s and solitons mutually complement each other. A third distinct formulation of NNMs can be performed by considering symmetries in the configuration space of a nonlinear oscillator. If one expresses the equations of motion of a linear conservative systcm in Jacobi's form (geometric formulation), one finds that these equations are invariant with respect to a continuous group of' extensions or compressions in the configuration space. Linear normal modes (which correspond to straight lines in the configuration space) turn out to be the only possible solutions that are invariant with respect to this group of transformations. Taking this property of normal modes into account, it is possible to construct a systematic analytic methodology for computing normal modes, by reducing the problem to an algebraic eigenvalue one. From this viewpoint, this groupinvariance method is equivalent to the previous approach for computing NNMs based on temporal invariance. However, in contrast to the latter, the former approach provides the eigenvectors or eigenfunctions, but does not compute directly the eigenfrequencies of the normal modes. Invariance with respect to extensions or compressions in the configuration space is not a

1.1 CONCEPTS OF NNM AND NONLINEAR LOCALIZATION

7

distinctive property of linear systems. Considering a nonlinear conservative discrete oscillator with homogeneous potential function of even degree, it can be shown that its equations of motion can be made invariant to extensions or compressions in the configuration space. Hence, for a homogeneous system it is possible to seek NNMs that correspond to straight lines in the configuration space and possess the group-invariance properties of the linear modes. This formulation provides an alternative nonlinear generalization of the concept of linear normal modes. What distinguishes the nonlinear from the linear case is the fact that a nonlinear homogeneous system may possess more straight-line NNMs than its DOF. This feature was also noticed in the previous definitions of NNMs, where it was noted that the majority of the additional NNMs are spatially localized. So, one finds that homogeneous systems (i.e., systems with essential nonlinearities) exhibit nonlinear mode localization. As shown in the following chapters this is not an exclusive feature of homogeneous systems, since localized NNMs will be detected in a wider class of nonlinear oscillators. A last generalization of the concept of normal mode to the nonlinear case can be carried out by noting that the equations of the motion of linear systems possess an additional discrete symmetry group in the configurational space: After transforming to normal coordinates, any Cartesian transformation of coordinates is equivalent to mere inversions of normal coordinates. This reveals that linear normal modes are invariant solutions with respect to the group in Cartesian transformations in the configuration space. This viewpoint turns out to provide a very efficient way of computing normal modes of linear systems with geometric symmetries. In the linear case, there exits a linear vector space that is formed by the linearly independent normal modes; certainly, this is not the case in nonlinear theory. A first attempt was undertaken by Yang (1968) to employ discrete symmetries of certain nonlinear systems for computing NNMs, without resorting to group theoretic techniques. As discussed by Manevitch and Pinsky (1972a), NNMs can be determined in the framework of the theory of invariant-group solutions. In that context, one must classify sets in the configuration space that are invariant with respect to subgroups of admitting groups. This procedure allows one to find the sub-space of the configuration space that contains a certain NNM. If the dimension of this subspace is equal to 1, the subspace coincides wirh a NNM. Since the theory of discrete groupinvariant solutions is applicable to both linear and nonlinear systems, one

8

INTRODUCTION

obtains un additional nonlinear generalization of normal modes. Moreover, considering a general nonlinear conservative system, one can formulate the following "inverse" problem: Is it possible to compute a special set of system parameters that leads to an extension of the admitting group? The answer to this problem allows one (at least in principle) to classify all nonlinear systems possessing specified symmetries in the configuration space and to compute their NNMs (Manevitch et al., 1989; Manevitch and Pinsky, 1972a; Pilipchuk, 1985). The previous exposition shows that there exist several distinct ways for extending the concept of normal mode vibrations to nonlinear system. In that context, NNMs can be regarded, (a) as mere synchronous periodic solutions of the equations of motion (formal approach), (b) as solutions that possess exponential temporal dependence, or (c) as solutions that preserve invariance of the equations of motion with respect to certain continuous or discrete symmetry groups (group-theoretic approach). By extending the notion of normal mode to nonlinear theory one is able to better classify and study the symmetries and the forced resonances of discrete and continuous oscillators. In addition, NNMs provide the necessary framework for studying nonlinear mode localization and motion confinement phenomena in weakly coupled oscillators and can be employed to establish a link between localized periodic responses of discrete or continuous oscillators and solitary waves or solitons in nonlinear wave theory. Additional applications of NNMs on the study of the global dynamics and chaotic responses of nonlinear oscillators are discussed in later chapters. 1.2 EXAMPLE: NNMs OF A TWO-DOP DYNAMICAL SYSTEM

The concept of nonlinear normal modes is now demonstrated by considering the dynamics of a simple nonlinear oscillator. To this end, the two DOF hamiltonian system depicted in Figure 1.2.1 will be studied, with governing equations of motion given by:

( I .2. I )

1.2 EXAMPLE: NNMs OF A TWO-DOF DYNAMICAL SYSTEM

X

9

Y

Figure 1.2.1 The two DOF nonlinear oscillator where the exponent m is assumed to be an odd number. This system possesses similar NNMs, corresponding to the following lineur relation between the depended variables x and y: y=cx

( I .2.2j

The similar modes (1.2.2) are represented by straight modal lines in the configuration plane of the system, and are the only types of normal modes encountered in linear theory. As shown in chapter 3, similar NNMs are not generic in nonlinear discrete oscillators, since they exist only in systems with special symmetries (such as the system depicted in Figure 1.2.1). More typical in nonlinear systems are nonsimilar NNMs, which correspond to nonlinear relations between depended variables of the form y = f(x), and are represented in the configuration space by modal curves. Asymptotic methodologies for computing nonsimilar NNMs are also developed in chapter 3. As shown by Vakakis and Rand (1992), the similar NNMs (1.2.2) are the only type of normal modes that system (1.2.1) can possess. Since the linear relation (1.2.2) is assumed to hold at all times, one can use it to eliminate the y variable from the equations of motion and to obtain the following equivalent set of equations: xi- x

x

+ [l + K (l-c)m]

+ x - (l/c) [K (1-cjm+

xm= 0

cm] xm = 0,

c#0

(1.2.3)

For motion on a NNM both equations (1.2.3) must provide the same response x = x(t), a requirement that is satisfied by matching the respective coefficients of linear and nonlinear terms. Since both equations possess

10

INTRODLJCTION

identical linear parts, one obtains a single equation satisfied by the modal constant c:

K ( l + ~ (c-l)m ) = ~ ( l -~ m - l ) , c # 0

(1.2.4)

As pointed out by Vakakis (1990), the simultaneous matching of d l linear and nonlinear coefficients in a discrete system generally leads to a set of overdetermined algebraic equations governing the modal constants, which can only be solved if the problem under consideration possesses certain symmetries. The algebraic equation (1.2.4) always possesses the solutions c = k l , which correspond to in-phase and antiphase NNMs. These are the o n l y normal modes that the corresponding linear system (with m = 1) can possess. Interestingly enough, the nonlinear system (m = 3 , S , ...) can possess additional NNMs, with modal constants computed by solving the following algebraic equations: (m- 1)/2 dk-1 k= 1

+ K (1-c)m-1

= 0,

c

f

0,

m = 3,5,7 ,... (1.2.5)

It turns out that thc additional normal modes (1.2.5) always occur in reciprocal pairs and bifurcate from the antiphase mode c = -1 at the critical value,

K = Kc = 21-m

(m- 1) / 2 (-1)zk-l k = 1

in hamiltonian pitchfork bifurcations. The stability of the computed NNMs can be studied by performing, a local (linearized) analysis (Rosenberg and Hsu, 1961; Pecelli and Thomas, 1979; Zhupiev and Mikhlin, 1981, 1984; Caughey et al., 1990), an analysis based on Ince-algebraisatioll of the variational equations (Zhupiev and Mikhlin, 198 1,1984) or a global (nonlinear) analysis based on analytical or numerical Poincare' maps (Month, 1979; Hyams and Month, 1984; Vakakis and Rand, 1992). In Figure 1.2.2 the NNMs of systenis with m = 1, 3, S and 7 are depicted. These results are summarized in the following remarks. (1) The additional bifurcating NNMs of the nonlinear systems with m = 3 and 7 exist only at small values of the coupling parameter. The


0.8

K

11

I C

.

0.4 .

2

(b)

Figure 1.2.2 Bifurcations of NNMs for systems with (a) m = 1 (linear case), (b) m = 3 , ( c ) m = 5 , and (d) m = 7. -Stable NNMs, ------ Unstable NNMs.

bifurcating NNMs are essentially nonlinear and cannot be regarded as analytic continuations of any linear modes. This is in contrast to the modes c = k1 which can be regarded as nonlinear continuations of the linear normal modes of the system with m = 1. (2) As K -+ 0, a pair of bifurcating NNMs becomes strongly localized, with modal constants approaching the limits, c + 0 and 00, respectively. It can be shown that these NNMs are orbitally stable and, thus, physically realizable. (3) The bifurcations of NNMs have important implications on the low- and high-energy global dynamics and on the forced nonlinear resonances of system (1.2.1).

12

INTRODUCTION

To demonstrate the effects of the mode bifurcations on the global dynamics, the nonlinear system with m = 3 is considered in more detail. This system is hamiltonian with a four-dimensional phase space (x, x, y, y, and its global dynamics can be studied by constructing numerical or analytical Poincare' maps (Month and Rand, 1977, 1980; Month, 1979). Here only a brief description of the construction of these maps will be given, and for a more detailed discussion, the reader is referred to the aforementioned references. By fixing the total energy of the dynamical system to a constant level, one restricts the flow in the phase space to a threedimensional isoenergetic manifold. This is perfornied by imposing the following condition:

where H(a) is the hamiltonian of the system, and h is the fixed energy level. The hamiltonian H is a first integral of the motion, and for autonomous oscillators represents conservation of energy during free oscillations. 11' an additional independent first integral of motion exists, the two-DOF system is said to be integrable and the isoenergetic manifold H = h is fibered by invariant two-dimensional tori (Guckenheimer and Holnies, 1984). 'This integrability property is not generic in hamiltonian systems, and, in general, one does not expect the existence of an independent second integral of motion. However, for low energies, even nonintegrable oscillators appear to have an approximate second integral of motion. This is because for low energies the isoenergetic manifolds of these systems appear to be fibered by approximate invariant tori which, as the energy increases, "break," giving rise to randomlike chaotic motions (Lichtenberg and Lieberman, 1983). Now suppose that one intersects the three-dimensional isoenergetic manifold defined by (1.2.6) with a two-dimensional cut-plane. If the intersection of the two manifolds is transverse (Guckenheimer and Holmes, 1984; Wiggins, 1990), the resulting cross-section, X, is two-dimensional, and the flow of the dynamical system intersecting the cut-plane defines a 'Poincare' map. Choosing the cut-plane as T:{x=O),the Poincare' section C is defined as Z = {x=O,X>O} n.{H=h}


13

Note that an additional restriction was imposed regarding the sign of the velocities. This condition guarantees that the Poincare' map is orientation preserving. Transverse intersection of the flow on the isoenergetic manifold with the cut-plane occurs when the following condition is satisfied: (X, x,y,

y> (1,0,0,0) # 0

*

x# 0

(1.2.7)

An NNM is a periodic orbit in phase space that pierces the cut-plane only once, and, hence, is represented by a single point in the Poincare' section, C. If the point corresponding to the NNM appears as a center, i.e., surrounded by closed curves resulting from intersections of invariant tori with the cutplane, then the normal mode is orbitally stable. If, on the contrary, the mode appears as a saddle point, then it is orbitally unstable. As the energy level h increases, KAM (Kolmogorov-Arnold-Moser) theory predicts that "rational" tori of the dynamical system break, giving rise to layers of ergodic motion, which fill the phase space between sufficiently "irrational" preserved tori (Guckenheimer and Holmes, 1984). Summarizing, one integrates numerically the differential equations of motion (1.2.1) of the system with cubic nonlinearities (m=3) for fixed total energy, and samples the values (y, y) corresponding to x=o, x>o The resulting Poincare' maps provide a picture of the global dynamics of the oscillator at arbitrary levels of energy. In Figures 1.2.3(a) and (b) the Poincare' maps of systems with K = 0.1 and K = 0.4 are depicted, for a constant low energy h = 0.5. Note the qualitative change of the global flow of the system as the coupling parameter is decreased below the bifurcation value K = 0.25. For K = 0.4 > 0.25 the anti-phase NNM [the lower fixed point in the Poincare' plot of Figure 1.2.3(a)] is orbitally stable. For K=0.1 < 0.25 this mode becomes orbitally unstable, and there exist two closed "loops" connecting the mode to itself (the lower "loop" is difficult to observe in Figure 1.2.3(b) since it lies close to the boundary curve of the Poincare' map). These loops are homoclinic orbits of the Poincare' map of the dynamical system, and correspond to iterates of the map that approach the unstable NNM after an infinite number of positive or negative iterations. Homoclinic orbits are formed when the stable and unstable invariant manifolds of an unstable equilibrium point of the map coalesce, and their

14

INTRODUCTION

0-

-0 2 -0 4

-

-0 8 -

-0.6

-I

0

. . . . . I

I

t

15


I

I

I

I

I

. . . . . . .

Y

1

t

121

......................... I’

.- , .................... ’. . ...................... . . .. . . - .. . ;.: . ... , ,. ...,- .......... ........... ._ -,-, - ’ ..Jf

I

7 .

.

~

- 2 - 0

- c- 2-0-

-8I

I

I

2

0

0

2

I

I

I

I

2.

I

t-

2t-

4

P

I

I

4

f

- OI-12

b

Y

Figure 1.2.4 High-energy Poincare' map (h = 50.0), K = 0.1 < 0.25.

"breakdown" for the nonintegrable case is recognized as a primary mechanism for generation of chaos in hamiltonian systems. It must be noted, that the low-energy plots of Figure 1.2.3 can be deceiving, since they may lead to the impression that the dynamics of the oscillator close to the NNMs are smooth and totally predictable. In fact, since the oscillator under investigation is not integrable, certain invariant tori of the flow "break" according to the KAM theorem, giving rise to randomlike chaotic motions. These complicated trajectories occur in "stochastic layers," which, for low energies, are of small measure, and, thus, not easily observable in numerical simulations. In addition, transverse homoclinic intersections between the stable and unstable manifolds of the unstable NNM occur, which lead to large-scale chaotic motions. Hence, the global dynamics of system (1.2.1 ) are more complicated than what they appear in the low-energy Poincare' plots.

16

INTRODUCTION

In Figure 1.2.4, the high-energy Poincare' map for K = 0.1 < 0.25, and energy h = SO is depicted. One observes essential changes in the global dynamics as the energy increases. First, there exist certain regions of the map where the orbits of the oscillator seem to wander erratically. These socalled seas qf stochasricity (Lichtenberg and Lieberman, 1 983), are regions of chaotic motions of the haniiltonian system, i.e., of motions with extreme sensitivity on initial conditions. One can detect a large chaotic region surrounding the unstable antiphase NNM. In that region, large-scule chaotic m o f i o m occur. Moreover, a careful examination of the plots indicates that there also exist some islurzcls in the stoclzastic seu; these correspond to stubleunstable pairs of subharmonic orbits, surrounded by small-scale chirotic motions. It is now shown that the large-scale chaotic motions occur only qfter the bifurcation of NNMs at K=0.25 (cf. Figure 1.2.2). In Figure 1.2.5, the stable and unstable manifolds of the unstable antiphase NNM are shown. Note the violent windings (Wiggins, 1990) of the manifolds as they approach the unstable mode. It can be proven that there exist an infinite number of these "windings" as the manifolds accumulate on themselves [in accordancc with the "lambda-Ienlma" (Guckenheimer and Holmcs, 1984)l. An infinity of transverse intersections of the two manifolds then occurs forming an infinity of Swzale horsesho~.r. This has interesting implications i n thc dynamics uf the Poincare' map. In fact, using the Snide-Birkhoff homoclinic theorem (Guckenheimer and Holmes, 1984), it can be shown that the Poincare' map contains a countable infinity of periodic orbits, an uncountable infinity of nonperiodic orbits, and dense orbits. Thus, the dynamics of the inap in the vicinity of the unstable NNM possesses sensitive dependence on initial conditions and is virtually unpredictable. It is interesting to note that this type of large-scale, global chaos occurs only when the antiphase N N M is orbitally unstable, since only then one-dimensional global invariant manifolds of this mode exist. For values of the coupling stiffness parameter K greater than 0.25, no such motions can occur, since, then the antiphase N N M is orbitally stable and the corresponding stable equilibrium in the Poincare' map possesses a two-dimensional center manifold. On the other hand, small-scale chaotic motions in the vicinity of the subharmonic orbits result from the destruction of the invariant tori of the haniiltonian system, and are local i n naturc. These motions are independent of the NNM bifurcation. The small-scale chaotic motions of the system of Figure 1.2.1 were analytically studied by Vakakis and Rand (1992) using the subharnionic


17

Figure 1.2.5 Transverse intersections of the stable and unstable invariant manifolds of the unstable antiphase NNM: (a) global Poincare’ map and (b) close to the unstable antiphase NNM.

18

INTRODUCTION

Melnikov techniques developed by Holmes and co-workers (Veerman and Holmes, 1985, 1986; Greenspan and Holmes, 1983). From the previous discussion it can be concluded that the bifurcation of NNMs greatly effects the global dynamics of the two-DOF nonlinear oscillator. In particular, a necessary condition for the existence of large-scale chaos was found to be the orbital instability of the anti-phase NNM, since only then could large-scale transverse intersections of global onedimensional invariant manifolds occur in the Poincare' map. As a result, the bifurcations of NNMs appear to increase the complexity of the high-energy global dynamics of the oscillator and, in fact, one can state that the system after the bifurcation of the NNMs becomes "more chaotic." It is interehting to note that in Child and Lawton (1982) and Child (1993) a similar increase of the complexity of the global dynamics due to normal to local mode bifurcations is detected in a two-DOF model of nonlinearly interacting molecules. It is now shown that NNM bifurcation also affects the forced steady-state response of the system. The dynamical system of Figure 1.2.1 is again considered, with rn = 3, and with weak damping and weak external harmonic forcing. Making the additional assumption of small nonlinearities and weak coupling, the equations of motion of the damped, forced system become:

+ x + E C I X + E ~3 + E K (X - y)3 = 2~ P I coswt y +y +Ec ~ + Y E y3 + E K (y - x)3 = 2~ P2 cos wt X

(1.2.8)

where IEI = a:! cos(t + p2) + O(E)

i1.2.9)


19

where the amplitudes and phases in (1.2.9) are computed by solving the following set of four modulation equations:

+ (3/8) K a2a12 sin$ - (3/8) K ala22 sin241 + (318) K a23 sin$ PI sinpi = 0 c2a2 - (3/8) K ala22 sin$ + (3/8) K am12 sin2$

all= (-112) cia1

-

a2'= (-1/2)

K a13 sin@- P2 sinP2 = 0 a i p i ' = (318) (1+K) a13 + (318) K ala22 cos2$ - (9/8) K a2a12 cos$ - (318) K a23 cos$ + (314) K a12122 - PI cospi - m i = 0 a2B2' = (318) (l+K) a23 + (3/8) K a2a12 cos2@- (9/8) K ala22 cos@ - (318) K a13 cos@+ (314) K a2a12 - P2 cospz - (3a2 = 0 (1.2.10) - (3/8)

where @ = p2-P], and prime denotes differentiation with respect to the "slow time" Et. For Cl=O.05, c2=0.07, P1=0.2, P[=O, and K=O.l, 0.4, the amplitudes and phases of the fundamental resonances are depicted in Figure 1.2.6. In the same plots the backbone curves, i.e., the frequency-amplitude relations for motions on NNMs are also depicted. For the system with K=0.4 [cf. Figure 1.2.6(a)],i.e., before the NNM bifurcation, at most three stable steady-state solutions can exist at any value of the frequency detuning parameter 0,and "jump phenomena" can occur from one resonance branch to another (Szemplinska-Stupnicka, 1980). Note, that for relatively high-frequency detuning values CJ, the ratios of the amplitudes of the forced motions, al/a2, are almost equal to the corresponding ratios of the unforced NNMs. Moreover, all steady-state fundamental solutions are detected in neighborhoods of stable NNMs, and, for sufficiently large frequency detuning values, a stable-unstable pair of fundamental resonances exists close to each NNM. A change in the coupling parameter K introduces qualitative and quantitative changes in the topology of the fundamental resonances. In Figure 1.2.6(b) the steady-state solutions of the system corresponding to K=O.1 0.25) and (b) after (K = 0.1 < 0.25) the NNM bifurcation. -Stable solutions, ----- Unstable solutions, - - - N N M curves.


21

to large values of y. This later result is due to the fact that only one of the masses (corresponding to coordinate x) is excited. These results .show that the topology of the fundamental resonance curves is greatly affected by the number and the stability type of the NNMs of the system. The application considered in this section has demonstrated that NNMs and their bifurcations can greatly influence the local and global, free and forced dynamics of nonlinear oscillators. The fact that nonlinear mode localization was observed even in the simple system considered hints on the general occurrence of mode localization in nonlinear, weakly coupled systems. For the specific oscillator considered, the localized NNMs were found to be the limits of extended branches of NNMs as a coupling parameter tended to zero, a result indicating that an analysis based on NNMs is the appropriate framework for studying localization phenomena in nonlinear oscillators. In the next chapters, analytical and numerical techniques for studying NNMs and nonlinear mode localization in discrete and continuous oscillators will be developed, and applications of NNMs on the free and forced dynamics of these systems will be investigated in detail.


CHAPTER 2 NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS In this chapter general qualitative results regarding the existence and properties of nonlinear normal modes (NNMs) in discrete oscillators are given. Due to their generality, these results complement the qualitative results derived in later sections, where specific types of nonlinear oscillators are considered, and provide valuable insight into the principle and physics of nonlinear normal vibrations. Previous works on the existence of NNMs are reviewed and new existence results are derived. More specifically, autonomous systems are examined, and conditions are obtained that guarantee the analytic continuation up to arbitrary levels of potential energy of Lyapunov periodic motions (NNMs) which exist only locally, in small neighborhoods of stable equilibrium points.

2.1 PRELIMINARY FORMULATION Qualitative analyses of periodic oscillations in autonomous hamiltonian systems have been conducted in several previous works. Desolneux-Moilis (198 l), Rabinowitch (1 982), and Zehnder (1983) have conducted extensive reviews of existing works in this area. The majority of results reported in the literature are pure existence theorems, which, although they guarantee the existence of periodic oscillations with a prescribed total energy or period, do not provide any insight into the corresponding mode shapes of the nonlinear oscillations. The theory of NNMs originated from the works of Rosenberg (1962, 1966). Cooke and Struble (1966) and Pak and Rosenberg (1968) examined the existence and properties of NNMs of two degree-offreedom (DOF) nonlinear conservative systems. Greenberg and Yang (1971 ) used symmetries of the potential function to find subspaces of the configuration space where the trajectories of an NNM are confined and to reduce the dimensions of these "modal subspaces"; a modal subspace of dimension 1 was identified as an NNM. Rosenberg (1966) found that a system with a homogeneous potential function possesses similar normal mode oscillations, i.e., normal modes with straight-line trajectories in the 23

24

NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS

configuration space. In addition, he showed that, in contrast to linear theory, the NNMs of a discrete oscillator may exceed i n number its degrees of freedom. For a general class of discrete oscillators with potential functions that are symmetric with respect to the origin of the configuration space, it was proven that there exist periodic solutions that pass through the origin of the configuration space, and correspond to synchronous motions of all coordinates of the system. Van Groesen (1983) proved the existence of at least n similar normal modes in an n-DOF system with homogeneous potential (though this result, as he put it, is generally accepted for granted). In the same work bifurcations of NNMs for increased energy of oscillation are studied. Although previous works on existence of NNMs give general conditions under which periodic motions passing through the origin of the configuration space of a discrete oscillator exist, they do not provide any information on the relative motion of the coordinates of the system during such an oscillation. This section is concerned with more detailed characterizations of the motions of discrete systems oscillating on N N M s . According to Lyapunov's theorem (1W7), analytic hamiltonians with n DC)F whose linearized eigenfrequencies are not integrably related possess exactly n one-parameter families of periodic solutions close to each stable equilibrium point. Thus, at each level of energy, one finds at least n periodic solutions (NNMs) near each stable equilibrium point. Weinstein (1973) and Moser (1 976) generalized Lyapunov's theorem for systems with integrably related eigenfrequencies, i.e., systems in "internal resonance." They proved that analytic hamiltonians with n DOF whose linearized eigenfrequencies are integrably related, possess at least n one-parameter families of periodic solutions close to each stable equilibrium point. Clearly, the periodic solutions predicted by Lyapunov and Weinstein are qualitatively the same as the normal vibrations of the linearized system, although not all NNMs are analytic continuations of linearized normal oscillations. In particular, if a nonlinear system has a potential function that is symmetric with respect to the origin of the configuration space, its NNMs correspond to motions during which all coordinates keep their signs in the course of half a period of oscillation, or even vary monotonically between their extreme values. Moreover, an NNM preserves the nodal properties (has the same number ol' nodes) of the corresponding linear mode. These are precisely the types o f solutions studied in what follows. Some additional qualitative results on

2.1 PRELIMINARY FORMULATION

25

periodic oscillations of autonomous Hamiltonian systems are contained in (Zevin, 1988, 1993). A first goal of the analysis is to establish nonlocal criteria for the existence of NNMs. Lyapunov's and Weinsten's results hold only close to neighborhoods of stable equilibrium points and do not guarantee the existence of NNMs away from such neighborhoods. Hence, their results are only local in nature and cannot be applied to systems with relatively high energies. A nonlocal existence theorem for NNM can be formulated as follows (the notation used is defined below): Let s2 be a specified region containing the origin in the configuration space of an n DOF hamiltonian oscillator, for example, Q = { x ~ R n : V ( & ) i ) h ~ }or

R = {q~Rn:llxllSA}

where V(x) is the potential energy. Let dQ be the boundary of Q. Obtain criteria that guarantee the existence in s2 of a family of periodic solutions that exist in a neighborhood of q = Q and reach dQ. The last requirement means that for every open region 'P E Q that includes the origin of the configuration space, there exists a solution x(t) such that x(t) E Y for all te R, and q(t*) E 13" for some t*. If such existence criteria can be formulated, one could prove the existence of families of NNMs at arbitrary levels of the total energy, satisfying h 5 ho, where ho is an arbitrary value. In what follows, the derivation of such criteria is carried out. Some preliminary definitions are appropriate at this point. An (n x n) matrix A (underbars denote vectors or matrices) will be called positive (negative), if all the elements aij, i j = 1,...A of this matrix are positive (negative). A similar definition applies to vectors. An (n x n) matrix A will be called positive (negative) definite, if the quadratic form (&,y) is positive (negative) for every nonzero (n x 1) vector y. Using the previous definitions, one can define inequalities between vectors or matrices. For example, consider two (n x 1) vectors X I and g2; the relations ~1 > ~ 2 , or E I 2 zi2 imply that all elements of the difference vector (xI-x~) are positive or nonnegative quantities, respectively. The notation (*,*) is used to denote internal product between two vectors.

26

NNMs LN DISCRETE OSCILLATORS: QUALITATIVE RESULTS

The systems considered in this section are n DOF conservative hamiltonian oscillators. Denote by x = (Xi, ... xn)T the (n x 1) vector of generalized coordinates, by V(x) the potential energy, and by )

K(i) = (1/2)(&,i) the kinetic energy, where M is an (n x n) positive definite symmetric inertia matrix. The potential function V(&) is assumed to be at least twice differentiable and to satisfy the condition V(0) = 0. The corresponding set of differential equations governing the motion of the system are written in matrix form as: Mji + f(x) = 0,

f(x) = V,(X) -

(2.1.1)

where f(x) is the n-vector of nonlinear restoring functions. Equation (2. I . 1 ) admits the first integral of motion:

where x(t) is a solution of (2.1.1), and h is the total (conserved) energy of the motion. Now, denote by O j O and i=l,...,k 5 n x,O = { x l l O , ..., xl,O}T, the k natural frequencies and n mode shapes, respectively, of the following IineariLed system: Mj; + A(0)x = 0

(2.1.3)

where

A(&)= VX&) is the Hessian matrix of V(x). If A(0) is positive-definite then k

=

Consider the j-th linearized normal mode of (2.1.3). Suppose that for all i j the following relation is satisfied: WiO/OjO

z q,

9 integer

11.

f

(2.1.4)

Condition (2.1.4) guarantees that the jth linearized normal mode is not in internal resonance with any other linearized mode of the system. Then,


27

employing Lyapunov's theorem, it is guaranteed that for sufficiently small levels of the energy h system (2.1.1) possesses a unique one-parameter family of nonlinear periodic solutions xj(t;h) with period Tj(h), such that, xj(t;h) + 0 and Tj(h) + 2d0j0 as h + 0 These are NNMs confined locally, i.e., in sufficiently small neighborhoods of the origin of the configuration space of the system. Some general properties of periodic solutions of the hamiltonian system (2.1.1) are now reviewed. Suppose that x(t) is such a T-periodic solution. Since (2.1.1) is autonomous, a translation in time of a solution is also a solution, and, hence, the system also admits the family of T-periodic solutions x(t+c) where c is a constant time translation. If this family of periodic solutions is unique, then, for an appropriate choice of the initial time, the following relation can be satisfied: x(t)

= x(-t)

(2.1.5)

Since x(-t) satisfies equation (2.1. l), it follows that one can find a constant c], such that,

x(-t) = x(t+cl) for some c i [O,T] ~ If one shifts both arguments of this last equation by (ci/2), one finds that the solution x(t) satisfies relation (2.1.5). Therefore, in the following exposition only even periodic solutions will be considered, without restricting the generality of the analysis. Moreover, from (2.1.5) it follows that X(0) = i(T/2) = Q i.e., that system (2.1.1) reaches its maximum potential energy value V(X) = h at the beginning and at the middle of its period. If the potential function V(x) is symmetric with respect to the origin of the configuration space, i.e., if the following relation is satisfied,

then, the nonlinear restoring force in (2.1.1) satisfies the symmetry relation:

28


f(x) = -f(-x)

(2.1.7) If (2.1.7) is satiafied, then it can be easily proven that if L(t) is a solution of (2.1. l), so is -&(t), and that there exists a constant time translation c2 such that, - = ~ ( t + c 2 ) for C ~ [O,T] E -x(t) It follows that x(t+2c2) = x(t) or that 2c2=T Hence, if the restoring function satisfies (2.1.7), the periodic solution x(t) possesses the following additional property: x(t)

= -x(t+T/2)

(2.1.8)

Periodic solutions satisfying (2.1.8) will be termed symmetric. When the system vibrates in a symmetric periodic motion, all coordinates of the system, xi(t), i=l ,...,n, pass simultaneously through the equilibrium state x=Q. Moreover, in view of (2.1.5) and (2.1.8) it is noted that, by shifting time by T/4, the symmetric solution x(t) becomes an odd function of t. Since the ratios (Oi"/Oj"), i = 1,...,k, i # j are assumed by (2.1.4) not to be integer numbers, it can be proven that the NNM xj(t;h) is the only Tj(h)periodic solution of system (2.1.1) in any small neighborhood of the equilibrium state x = 0. Clearly, under an appropriate choice of the time variable, the NNM can be made to satisfy relation (2.1 S),or, if the potential function is symmetric, relation (2.1.8). For sufficiently small values of the total energy h, the NNM can be expanded in Taylor series with respect to the total energy as follows: xj(t;h) = h1'2 xjO(t)

+ O(hl"),

t E [O,T]

(2.1.9)

where x j O ( t ) is the linearized normal mode solution of problem (2.1 3).In view of (2.1.5) this linearized mode is given by,

From (2.1.9) it follows that in small neighborhoods of the equilibrium point the NNM Lj(t;h) is qualitatively the sume to the normal mode zjO(t) of the linearired problem. In particular, assuming that the system is symmetric, for small energies the coordinates xji(t;h) of the NNM are expected to maintain


29

the same signs with the coordinates Xjio(t) of the linearized normal mode in the course of half of a period (the notation Xji denotes the ith element of the n-vector Xj): sgn[xji(t;h)] = sgn[xjio(t)],

t

E (-T/4,T/4)

(2.1.10)

t

(2.1.11)

or sgn[Xji(t;h)] = ~gn[Xjj"(t)],

E (O,T/2)

For small values of the energy h, relation (2.1.1 1) is also satisfied by nonsymmetric systems. For the sake of simplicity, and with no loss of generality, it is additionally assumed that the following relations regarding the signs of xji(t;h) hold: xji(t;h) > 0, t

E

(-T/4,T/4)

or Xji(t;h) < 0, t

E

(O,T/2)

a Aj(t;h) > 0, t E (-T/4,T/4) (2.1.12) a ij(t;h) < 0, t

E

(O,T/2)

(2.1.13)

In the following analysis it is proven that there exists a continuum S of periodic solutions (NNMs) in Q satisfying (2.1.12) or (2.1.13). These solutions coincide with the NNMs Lj(t;h) predicted by Lyapunov in small neighborhoods of x = 0. As the level of energy h increases, bifurcations (branching) of this family of periodic solutions may occur, giving rise to additional (bifurcated) NNMs. The bifurcated modes are not analytic continuations of linearized normal modes, but they satisfy conditions (2.1.12) or (2.1.13) in R. It is now shown that the problem of existence of NNMs in R can be reduced to an equivalent integral equation problem. First, symmetric periodic solutions of system (2.1.1) are considered. Let N and p be (n x n ) symmetric matrices with elements nik and Pik. respectively, such that, for all x > 0, A E R, the following inequalities are satisfied:

Nx 5 f(x) 5 px

(2.1.14)

The matrices N and P in (2.1.14) are defined as follows. Using the integral mean-value theorem, the vector of nonlinear restoring forces f(x) can be expressed in the form:

30

NNMs IN DISCRETE OSCILLATORS: QUALITATIVERESULTS

f(x) = C(x>x where

(2.1.15)

c(x)is an (n x n) matrix with elements Cik, defined as I

C(X) = j A(vx) dv 0

The elements of matrices N and P are then defined as follows:

To illustrate the use of matrices energy given by:

N and P, consider a system with potential

Note that the potential energy of a system of n nonlinear discrete oscillators coupled to each other by means of linear strings is of the form (2.1.17). Suppose that for x > 0 , E ~R,the nonlinear restoring conservative force of the ith oscillator, fi( xi)=dV,(xi)/dxi is bounded by the lines y = njxi and y = pixi (cf. Figure 2.1.1). Then matrices N and P are then defined as:

N = Co + diag(n1, ...,nn), and E = Co + diag(p1 ,...,pn) where diag(') denotes a diagonal matrix. For the special case when I fi(Xi) I 2 I (dfi(O)/dxi) xi I 'd i matrix N simplifies to N = A@) = V,(Q) Alternatively, when I fi(Xi) I I I (dfi(O)/dxi) Xi I 'd i matrix P simplifies to p = A(0) = V,(Q>

(2.1.18)


3I

Figure 2.1.1 Bounds for the nonlinear restoring force.

These special conditions imply that the nonlinear system is more or less stiff, respectively, than the linearized one. Introducing the new time variable z = t/T, the nonlinear system (2.1.1) can be expressed as:

Mx" + T 2 k = -T2y(x),

where y(x) = f(x)- Nx

(2.1.19)

where primes denote differentiation with respect to z.Taking into account (2.1.5), (2.1.8), and (2.1.12), one seeks periodic solutions ~ ( z of ) (2.1.19) of unit period satisfying the relations X(T) = ~ ( - 7 )= - x(z+1/2),

and X(T) > Q for T

E

[0,1/4)

(2.1.20)

From the above conditions, it can be shown that the solution must also satisfy the following relation, which follows from (2.1.20):

-~ ' ( 0=) ~ ( 1 / 4 =) 0

(2.1.21)

Hence, for z E [0,1/4] the desired periodic solution X(T) is a positive solution of the nonlinear boundary problems (2.1.19) and (2.1.2 1). Denote by mi', where i = 1,..., r In, the positive eigenvalues of the matrix M-", and assume that the quantities (Toi/27t ) are not integer numbers. Then, the following linear boundary value problem possesses only the trivial solution:

32


Mx" + T2&

= 0, ~ ' ( 0=) ~ ( 1 / 4 = ) Q

x =Q

(2.1.22)

It follows that there exists an (n x n) Green matrix, E ( T , ~ , Twith ) , elements rik(T,u,T), by use of which the solution X(T) of the boundary value problem (2. I . 19), (2.1.2 1) can be expressed in the following form: X(T)= h[Gx](z), where

5

114

[B](T) = - l-(~,u,T) y [ ~ ( u ) du, ] h = T2 0

[a]

(2.1.23) In (2.1.23), denotes an integral operator. By definition, for t f T matrix E(T,u,T) satisfies (2.1.22), and the limiting relation:

lim [C(T+E,T,T)- C'(Z--E,Z,T)] = M-1

as E+O

(2.1.24)

be the eigenvectors of M-IN, and -h$ the nonpositive eigenvalues, where, v = 1,...,n, k = r+l, ...,n, and the eigenvectors are normalized according to (MxvAv) = 1 It follows, that the ikth component of matrix L(.s,u,T) can be expressed i n the following form:

qv(T,u,T) = -

cos(wvuT)sin[ov( 1/4-z)TJ wvTcos(wiT/4)


33

sinh( hvuT)sinh (hv( 1/~-T)T) for T > u pv(z,u,T) = hvTcosh(hvT/4) (2.1.25b) To prove expression (2.1.25), one takes into account that it satisfies relations (2.1.21) and (2.1.22) and that

Since

MXT& = I

(where, X=[xl,...,xn],(-)T denotes the transpose of a matrix, and 1is the identity matrix), one obtains

21T21--M-l

By a direct calculation one finds that the ik-th element, rik, of matrix XTX is equal to the right-hand side of equality (2.1.26), and, thus, that equality (2.1.24) is also satisfied. In addition, it is easy to show that matrix r ( ~ , u , T ) is symmetric. Nonsymmetric, even periodic, solutions ~ ( z of ) system (2.1.19) are now considered. Employing the previously defined notation, it is assumed that there exist (nxn) matrices N and P, such that the following inequalities hold for all 1~ E R,and all (n x 1) vectors y > 0:

where A(&)= Vxx(x) is the Hessian of the potential function of the system. Denoting the elements of A(&)by aik(&), the elements of matrices N and P are defined as:

It is noted that conditions (2.1.27) are stronger than (2.1.14). In particular, combining conditions (2.1.16) and (2.1.17), one obtains for the symmetric case that

34

NNMs W DISCRETE OSCILLATORS: QUALITATIVE RESULTS

where ni and pi were defined previously (cf. Figure 2.1 .l)). From (2.1.28) and (2.1.17), one obtains

Since an even solution satisfies the conditions, x'(0) =

x'(1/2) = 0

(2.1.29)

it follows that it coincides with a solution of boundary problem (2.1.19) and (2.1.29) for z E [0,1/2]. This last problem can be reduced to an integral equation similar to (2.1.23) by employing the corresponding Green matrix. To this end, denote by H(z,u,T), the Green matrix of the boundary value problem: (2. I .30) + T2& = 0, ~ ( 0=) ~ ( 1 / 2 =) 0 I&'

The components of H(z,u,T) are defined as follows:

where

si n(wvTT)sin[~ov( u- I /2)T] -~~ qv(z'u'T) = w v T s i n ( w v T / 4 )

pv(w,T) =

cosh(hvzT)sinh [hv(u- 1/2)T] . .. hvTsinh(hvT/4)

qv(tAT)=

for 1' : < u

sin(ovuT)sin[wv(z- 1 /2)T] wvTsin(wvT/4)

1/2)T] pv(z,u,T) = - sinh(hvuT)sinh[hv(z~ ~ _ _ _ - - hvTsinh(hvT/4)

for z > u (2.1.3 1 b)

2.2 EXISTENCE THEOREM FOR NNMs

35

The components of Y(T,u,T) satisfy the conditions (2.1.29),and the relations (2.1.26)and (2.1.24)when substituting E+H. For the nonlinear problem (2.1.19),differentiating with respect to 7, and taking into account (2.1.29), one obtains the following problem satisfied by the variable v = -x'(z):

Mv" + T2& = -T2[A(x)-N]v,

~ ( 0=) ~(1/2) =0

(2.1.32)

Employing Green's matrix H(z,u,T) one can reformulate problem (2.1.32) as an equivalent integral equation of the following form:

h = T2 and

[&:I

(2.1.33)

denotes the integral operator for the nonsymmetric case.

2.2 EXISTENCE THEOREM FOR NNMs Before proceeding to the formulation and proof of an existence theorem for NNMs, some preliminary findings concerning the following nonlinear eigenvalue problem will be reviewed (Krasnoselskii, 1956; Rabinowitch, 197 1; Crandall and Rabinowitch, 1971):

u = h b + H(h,u)

(2.2.1)

where L is a compact linear operator on a Banach space E, and H is a nonlinear compact operator defined on (R x 0)where 0 is an open subset in E containing the origin, with llI-i(h,~~)ll = O(llull) for 1 1u11 sufficiently small The solutions of (2.2.1)consist of pairs of nonlinear eigenvalues and eigenvectors ( h , ~with ) IE R, UE E. For any value of h there exits the trivial is called a bifurcation point if every solution (h,Q).The point (A,) neighborhood of (h,Q in (R x 0)contains nontrivial solutions. Such value h is necessarily an eigenvalue of the linearized eigenvalue problem U = p b

36


It can be proven that if a linearized eigenvalue p = p i is of odd multiplicity, then p = p ~is necessarily a bifurcation point. In that case, there exists a continuum of solutions, S, of the nonlinear problem (2.2. I) which emanates from (pi,@, and, either tends to infinity fix., there are solutions ( h , d E S with arbitrarily large values of the norm (IhP + llu112)1/2], or eventually coincides with (p2,Q)where p = p2 is some other distinct eigenvalue of the linearized problem. Consider now the linearized system:

In terms of the previously defined notation, the eigenvalues of (2.2.2) are given by (2xp/wr0)2, r = 1,...,k, p = 1,2,... By condition (2.1.4) the eigenvalue pk = Tj02 = (2n/Oj0)2 is simple. Employing the results of the general theory it follows that there exists a continuum of solutions, S, emanating from (pk,Q), which in small neighborhoods of the equilibrium point coincides with the family of NNMs, xj(t,h), predicted by Lyapunov. Let mi*', and Xi* = { X i ] * , ..., xin*}T be the positive eigenvalues and the corresponding eigenvectors of the matrix M-IP, where matrix P was defined earlier; in accordance with the notation of the previous section, { Oi2,xi} and { WiO',XiO} are the eigenvalues and eigenvectors of matrices M-*N and M--lA(Q),respectively. Suppose that

Condition (2.2.3) definitely holds if niairices N and P are sufficiently close to the matrix of the linearized system, A(0). Note that

f(:(E) A@>x

when the region R is sufficiently small. Since the frequency Oj" is assumed to be simple, the linearized eigenvector x j O changes continuously under a small change of the parameters of the system, so that the elements of the eigenvectors xj, Kj*,and XjO are positive. To prove (2.2.3), consider the matrix R = A(0) + E [P-A(O)]


37

Let hi(E) be the positive eigenvalues of M-lK; clearly, hi(0) = mi"' and hi(1) = Oi*2 It is known that for a simple eigenvalue hi(E) the following relation holds: (2.2.4) where S(E) is the normalized eigenvector (yi(E),Xi(E)) = 1 of the conjugate matrix (M-1R)T. Taking into account that, due to the symmetry of M and R the relation yi(E) = M Xi(&)holds, one has that (2.2.5) By inequalities (2.1.14), the matrix [E-A(Q)] is nonnegative, and the elements of eigenvector &j(E) are positive for p Close to A@) and 0 5 E 5 1. From (2.2.5) it follows that the eigenvalue hj(E> increases with E, and hence, that WjO < mj*. Using a similar reasoning it can be shown that Oj 5 OjO, with the equality holding when = A@). Symmetric periodic solutions are now considered. By previous assumptions the elements of the eigenvector zj are positive. It follows that for T < T+ = 2n/mj, the time history of the jth term, r i k j , of the series expression (2.1.25) is as shown in Figure 2.2.1. Since

Figure 2.2.1 Dependence on variable z of the jth term, r,kJ(T,u,T), of the series for rik(T,u,T).

38

NNMs IN DISCRETE OSCILLATORS: QUALITATIVE: RESULTS

C O S ( W ~ T-+/ ~0) as T + T+ it follows that for small values of IT-T+I the term r j k J is the dominant term in the series of rik(T,u,T). Hence, there exists LI value T- such that, for T E [T-,T+) and z, u E [0,1/4) the following two inequalities hold:

In the nonsymmetric case, the quantity T- is defined in the following manner. The jth item of the series of the ikth component of Green's matrix Hjk(z,u,T) is shown in Figure 2.2.2 for T < T+. Since this term tends to --oo as T + T+, there exists a value T- such that, for T E LT-,T+) and z, LI E (0,1/2) the following inequalities hold: H,k(z,u,T) < 0,

H,k'(O,u,T) < 0,

H,k'(l/2,u,T) > 0,

i, k = I ,....n (2.2.7)

The values T- will be employed in the following analysis. The integral equations (2.1.23) and (2.1.33) for the symmetric and nonsymmetric case, respectively, are now considered. In view of (2. I . 14) and (2.2.6), the following inequalities hold for T E [T-,T+) and for every x(z) 2

0:

Figure 2.2.2 Dependence on variable z of the jth term, H,kJ(z,u,T), of the series for Hik(z,u,T).


Y[X(z>l2 0,

39

[Gx](z) > 0 for z E [0,1/4) ,

[-]'(1/4) 0, Clearly, a scalar C can be found such that:

i+1

40

NNMs W DISCRETE OSCILLATORS: QUALITATIVE RESULTS

for some k E [ l ,...$I, and for some 2* E [0,1/4). In (2.2.1 l ) , yk denotes the kth element of the vector y. The following notation is adopted:

Suppose that hi I hi. Then, taking into account (2.2.1 1) and the inequalities xi(z) > 0, hi > 0 and hi > 0 one obtains that rk(z*) I 0 or rk'(U4) 2 0 where rk denotes the kth element of the vector r. This result, however, contradicts the previous relations:

This contradiction proves that hj > hl. Using the above lemma one can prove the following lemmas. Lemma 2. The eigenvalue problem (2.2.10) has only onc positive eigenvector. Lemma 3. The eigenvalue hi is simple. Consider now the nonhomogeneous equation:

LL =

where

0 < h,

[LXI + e(z>

~ ( 7 2) 0

(2.2. I.?)

for z E [0,1/4]

(2.2.14)

The following lemma is now proved.

) (2.2.13) is positive for z Lemma 4. The solution ~ ( z of only if h < hl.

E

[0,1/4) if and

Proof. Suppose that h < h~ and ~ ( ris)not positive. Since XI(T)> 0 for z [0,1/4), there exists a scalar C 2 0 such that the following relations hold:

E


y ( ~=) Crri(z)

+ x(z) 1 0

for 'I: E [0,1/4),

for some k E [1,...,n], and for some notation is adopted at this point:

yk(z*) = 0

41

or yk'( 1/4) = 0 (2.2.15)

z* E [0,1/4). The following new

Since h < hi, it follows that rk(z*) < 0 or rk'(114) > 0 a result which contradicts the fact that the right-hand side of (2.2.16) is positive and [&1'(1/4) < 0 This contradiction shows that ~ ( z >) Q for z E [0,1/4). Suppose now that h > hi and ~ ( z >) 0; then, relations (2.2.15) hold for some C < 0. One can similarly show that equality (2.2.16) cannot be satisfied in this case. If h = hi, then for any solution ~ ( z one ) obtains rk(z) = 0 or rk'(U4) = 0 a result which also contradicts the equality (2.2.16). Therefore, for h = hi, no solution of (2.2.13) exists. Lemma 5. If L > G, then h*

5

hl(G)> hi.

Proof. Let

x* = A*(&* Then, there exists a scalar C > 0 such that

y(z) = Crri(z) - x*(z) 2 0 for

Then,

z E [0,1/4)

yk(T*) = 0 or yk'(1/4) = 0 for some k and for some z* E [0,1/4)

E

[1,...,n],

( CXl(~)/hI1 - ( x*(wL* 1 = [GY+(L-G)CXll(z:)

Since y(z) 2 0, then

(2.2.17)

42

NNMs IN DISCRETE OSCILLATORS: QCJALITATIVE RESULTS

[@](z)

> Q for z

E

[0,1/4)

and

[&]'(1/4)

G, it follows that (2.2.17) can only hold if h* > hi. The problem of existence of normal mode oscillations of the nonlinear problem is now considered. Employing the previously adopted notation, suppose that T- 5 T" = 27~/0j*

(2.2.18)

In addition, assume that in the frequency interval [ ~ j , ~ j the * ] linearized system does not possess a natural frequency different from Wj" possessing a positive mode. Note, that if &J is a diagonal matrix, this assumption certainly holds. Since distinct modes satisfy the classical orthogonality property (mjo,&ko) = 0 for k # j 0 2 0 is not possible. The following existence theorem is the inequality a now proved. Theorem 1. There exists a continuous branch, S , of even periodic solutions that coincides with the Lyapunov family Lj(t,h) in small neighborhoods of 2~ = 0, and connects the origin x = Q with the boundary an. Every solution x(t) E S satisfies relations (2.1.12) or (2.1.13), for the symmetric or nonsymmetric case, respectively. Moreover, the period T of a solution is bounded by the following upper and lower bounds: T* < T 5 T+ (2.2.19) where the quantities T* and T+ were defined earlier. Proof. Assuming, first, that A(Q)# N,one obtains that TjO < T+. It follows that for small values of the total energy h the family of NNMs x,(t,h) satisfies the conditions of the theorem. It is now shown that under a continuous change of symmetric solutions within the continuum S, inequality (2.1.12) holds for T E (T",T+). This equality must hold, since otherwise, there would exist a solution ~ " ( 7E) S such that


x*(z) 2 0 for T E [0,1/4),

xk*(z*) = 0 or xk*”(1/4) = 0

43

(2.2.20)

where xk* is the kth component of the vector &*, for some k E [ I , ...,n] and for some z* E [O,T/4). Since x*(z) satisfies the integral equation (2.1.23), and the operator G is strictly positive for T E (T*,T+), relations (2.2.20) cannot hold. This contradiction proves that inequality (2.1.12) holds. Using similar arguments one can show that inequality (2.1.13) holds for the nonsymmetric case for T E (T*,T+). It is now proven that inequality (2.2.19) holds. Note that in the limit T -+ T+ one obtains the limiting values i,k = 1,...,n rik(T,u,T) -+ -00, Hik(T,U,T) + -DO, Using these limits one concludes that &(f)+ 0 a result that cannot hold in view of the fact that the quantity Oj = 2n/T+ is not a natural frequency of the linearized system. Hence, T f T+. Suppose now that the symmetric system possesses a solution ~ ( z> ) 0 for T = T* Then, the corresponding eigenvalue of (2.1.23) is denoted by h*= T*2. Since Oj* = 2n/T* is a natural frequency of problem (2.1.1) with f(x) = p x, it follows that (Lj*COST) and hi = T*2 are the positive eigenvector and corresponding eigenvalue, respectively, of the linear problem:

x(z) = h Lx where By relation (2.1.14),

Lx (7) = - I T(T,U,T*)[P-N]K(U) du .,

I/ A.

0

(2.2.21)

[P-N]~2 g(x) for li 2 Q

It follows that L > G. By Lemma 5 , the eigenvalues hi and h* satisfy the inequality hi < h*. The contradiction obtained shows that T > T*. In the nonsymmetric case, zj*sinT and hi = T*2 are the positive eigenvector and corresponding eigenvalue, respectively, of the linear problem: X(T) = h L’x

IR

where

I

L’K(T)= - H(.r,u,T*)[P-N]&(u) du (2.2.22) 0

44

NNMs IN DISCRETE OSCILLATORS: QUALlTATIVE RESULTS

Taking into account that due to (2.1.27)

it follows that

[ p - ~ ] q2 [ ~ ( s ) - ~ ] xfor r 2 o

L' > G, Using an argument similar to that used in the symmetric case, one can prove that the equality T = T* cannot hold. Finally, in the case when = A@), one obtains the relation, T+ = Tj'. By setting N*= A(Q) - &I,I s k l , the corresponding value T+(E) > TI(). Since T+(E)+ T j O as E+O all previous results hold. The previous findings show that in all cases inequality (2.2.19) holds for all solutions of the continuum S within the region Q. Suppose now that the continuum S meets a different periodic solution emanating from point (pi,Q), where pi is a different eigenvalue of the linearized problem (2.2.2). As shown above,

X(Z) > Q on [0,1/4) for X(T)

E

S

so the corresponding eigenvector is positive, x , O > 0. However, by previous ] a assumptions, there exists no natural frequency WiO 011 [ ~ j , ~ j *with positive eigenvector; therefore, the continuum S cannot meet a different periodic solution, and it can only reach the boundary of R . This concludes the proof of Theorem I . The implications of Theorem 1 are now discussed. If the region L2 of the theorem contracts to zero, then the frequencies Oj* and Oj tend to W-j". Hence, ,for .yufficientlysmall regions R the cotzditinn.s qf the theorem ore satisfied, und f o r any tiutural ,freyurricy GO, sutisJ:ving the r ~ ~ ~ r ~ i F i t e ~ ~ r ( i ~ i l i t ? , condition (2.1.4) one can .find u finite value of the total etzergv, h *, such tlrat ,for any h Ih* there exists a solittion x(t) on the energy h that is qunlitatively the same with the solution xj*CosT (for the symmetric case) of the lineurized sysrem. If the conditions of the theorem are met for all & > Q in (2.1.14) or for all x in (2.1.27), then the theorem guarantees the existence of such periodic solutions (NNMs) for any level of the total energy h. Note that in the symmetric case condition (2.1.14) does not guorantcc the monotonicity of the NNM solution x(t) for t E [O,T/Z]. However, this


45

monotonicity takes place if the matrix [A(&)- N 3 is nonnegative for x > 0, X E Q . This holds, since y(z) = -x'(z) satisfies equation (2.1.32) with boundary conditions, ~ ( 0=) ~ ' ( 1 / 4=) 0; the corresponding Green matrix

is negative for T E (T*,T+). So, in analogy to the nonsymmetric case one obtains that x'(z) < Q for z E (0,1/4] Differentiating (2.1.23), one finds that, regardless of the positiveness of matrix [A(&)- N],a symmetric solution ~ ( z )decreases monotonically o n (0,1/4) if C'(z,u,T) is positive, As seen from Figure 2.2.1, the jth term of the series (2.1.254 increases on [0,1/4). It follows that, r ' ( ~ , u , T is ) positive for some T E [T',T+), where T' 2 T-. It is concluded that x(t) changes monotonically between its extreme values. It is necessary to emphasize that the previous analysis was carried out under the assumption that XjO > Q The analysis of a different branch of NNM solutions requires, in general, the change of directions of some coordinates since the corresponding eigenvectors of the linearized problem will not be positive. As a result, the definitions of matrices N and E differ for different families of NNMs. However, in certain systems inequalities (2.1.14) and (2.1.27) (and, therefore, matrices N and P) are invariant with respect to the previously mentioned change of coordinates. This holds, for example, when the potential energy of the system is of the form (2.1.17). In that case, matrices N and P can be defined by (2.1.18) (symmetric case) or by (2.1.28) (nonsymmetric case), for each family of NNMs. As a last comment, it is noted that if the nonlinear system contains nonconservative forces, the vector of restoring forces f ( ~ in ) (2.1.1) can not be represented in terms of partial derivatives of a potential function. It can be shown, that in that case Theorem 1 still remains valid if the Green matrix E(z,u,T) (symmetric case), or H(z,u,T) (nonsymmetric case), is negative for some T E (T-,Tj). This is the case, for example, when the term

46

NNMs IN DISCRETE OSClLLATORS: QUALITATIVE RESULTS

corresponding to frequency Oj is negative, since (as in the conservative case), this term dominates over all other terms for small values of the quantity (Ti - T).

2.3 APPLICATIONS OF THE EXISTENCE THEOREM As an application of Theorem 1, consider a nonlinear system of the form

MX + f(x) = Q

(2.3.1a)

The following conditions concerning the mass matrix and the nonlinear restoring forces are imposed on system (2.3. la):

M = diag(m1 ,...,m,) f(g) 2 0 if

s L 0.

and

,

t(x) = -f(-a) f(g)f Q if x ?t Q

(2.3.1b)

The following results are valid for conservative as well as nonconservative systems. In physical terms, the conditions imposed on the restoring force vector imply that in case of positive displacements all restoring forces Qi = -fi(a) are directed toward the equilibrium point x = 0. Systems governed by differential equations of the form (2.3.1) are depicted in Figure 2.3.1. In Figure 2.3.l(a) a discrete system of oscillators coupled by means of elastic elements with symmetric nonlinearities is shown. For this system, the conditions on the restoring forces, f.. I,(&> = -fij(-x), fij(L) x > 0 for E # Q, i,j = I ,..,,4 are satisfied in a coordinate system where positive displacements of adjacent masses are in opposite directions to each other. For the weightless string or beam supporting lumped masses Lcf. Figure 2.3.l(b)], the conditions on the restoring forces (2.3.1b) are satisfied if positive displacements of adjacent masses are in opposite directions to each other. For the beam with intermediate supports of Figure 2.3.1(c), positive displacements of adjacent masses separated by a support should be of the same direction. In view of the conditions (2.3.1b), the linearized matrix A(0) is nonnegative. Assume at this point that matrix A(0) [and matrix M-lA(Q)]

2.3 APPLICATIONS OF THE EXISTENCE THEOREM 47

3

rLa '"1

Ga

m3

Figure 2.3.1 Nonlinear dynamical systems of various configurations satisfying conditions (2.3.lb). cannot be decomposed, i.e., that there exists no permutation of indices i and k that reduces this matrix into quasi-diagonal form. Then, according to Frobenius's theorem (Frobenius, 1912), the nonnegative nondecomposable matrix M-lA(0) has a unique positive eigenvector XjO, and the corresponding eigenvalue is simple and largest in modulus. It follows that the first eigenvalue of the linearized problem (2.2.2) is simple, and, according to previous findings, gives rise to a manifold S of NNMs satisfying (2.1.12) in the neighborhood of x = 0. It is now shown that every solution ~ ( z E) S is positive for TE [0,1/4). This has to hold since, otherwise,

48


there would exist a solution x*(T) such that x*(T) 2 0 for T E [O, 1/4), satisfying the conditions xk*(T*) = 0 or xk*'( 1/4) = 0 for some k E [ I , ...,n] and for some Z*E [0,1/4). By the assumed conditions on the restoring forces xk*"(zj _I 0 for &*(z) 2 Q

so this would imply that xk*(zj 3 0, and therefore, fk[x*(z)]

e

0. Now, since

x*( 1/4) = 0, for z close to 1/4 one has that Hence, the identity

f(x") = A(0) 21' fk(x*(z)) z 0 for some k

would imply that the (k,p) element of matrix A(,), akp(Q), should be equal to zero, provided that xp*(.t) f 0. Observing that this result would hold fo1 all components Xi*(T) of the solution x*(z) satisfying x,*(z) =_ 0, one concludes that the linearized system would be reduced to a quasi-diagonal form by a permutation of indices. The contradiction obtained shows that every solution &(T) E S is positive for T E [0,1/4). Since x,O is the only positive linearized mode, there are solutions X(T) E S with arbitrarily large values of the norm (A2 + 11~11~)1/2,where h = T2. Moreover, since

x"(z) 211 for TE (0,1/4) all coordinates Xi(T) decrease monotonically on (0, U4). By (2.3.lbj, for any finite region R, there exists a nun-negative matrix f 0, such that, for x 2 0, XE Q, the inequality

f(x) 2 Nx

N

(2.3.1c)

holds. Suppose that (2.3.1~)is valid for all 2~ 2 0. Then, ab seen from the proof of Theorem 1 , the period of a solution X(Z) E S satisfies the inequality T I T+ = 27~/c0j,where W j 2 is the eigenvalue of the matrix M-lN corresponding to the positive eigenvector Zj. Therefore, there exist solutions

2.3 APPLICATIONS OF THE EXISTENCE THEOREM 49

Figure 2.3.2 Two-DOF nonlinear system.

x(z)

E

S with arbitrarily large norm Ilxll, and, provided that system (2.3.la) is conservative, with any large value of the energy h. Employing the assumed conditions on the restoring forces, it can be shown that the linearized matrix A(0) is nonnegative. Now, disregarding the symmetry conditions (for the nonsymmetric case), suppose that A(x) is nonnegative for all x. In particular, for the system depicted in Figure 2.3.l(a), matrix A(x) is nonnegative if all restoring forces fik(X) are monotonic functions of the displacements. Taking into account that ~ ' ( 5=)~ ( 5 satisfies ) the equation: (2.3.1 d) -V " + T ~ A ( T=)0,~ A(T)= A(x(7))

and using similar arguments as before, it can be proved that a11 x(z) E S satisfy the relation x'(z) < Q for T E (0,1/2) Since A(&)is nonnegative, there exist solutions of (2.3.Id) with arbirrary values of the norm llyll, and, therefore, with arbitrary values of the norm Ilx(0) - x( 1/2)ll. In the corresponding oscillations, all coordinates vary monotonically between their extreme values, but, unlike in the symmetric case, they do not pass simultaneously through the origin of the configuration space. To illustrate an application of Theorem 1, consider a system of Figure 2.3.2 consisting of two nonlinear oscillators connected by a means of a linear string. The corresponding governing differential equations of motion are given by mixi

+ fl(x1) + c (xi - x2) = 0

m2x2

+ fz(x2) + c (x2 - x i ) = 0

(2.3.2)

50


Suppose that f,(u) = -fi(-uj, and that the nonlinear restoring forces can be bounded by, X i > 0, i= l, 2 njxi 5 f i ( X i ) I pixi, for Xi E Q, Considering the first mode of the linearized system, one has that xi0 > 0, i.e., that both masses are oscillating in phase. Conditions are now derived that guarantee the continuation of such solutions in R. This will prove the existence of NNMs corresponding to in phase oscillations of all masses of the system. In terms of the notation of the previous sections one defines matrices M,N,and p as follows:

Suppose now, that matrix N is positive-definite. It follows that the components of the Green matrix 1 ( ~ , u , Tare ) given by

where, quantities qi(z,u,T) and q2(z,u,T) are given by (2.1.2Sb), (012 and 0 2 2 are the eigenvalues of the matrix (M-IN), with XI,x2 being the corresponding eigenvectors. The condition

can be considerably simplified. Assuming that 02/w1 < 3, it follows that cos(wiT/4) > 0 and cos(o2T/4) < 0 for TE (27c/o2,2x/w1) In that case it can be shown that, TI 1 (z,u,T) < 0 for TE [0, I /4) provided that TI 1'(1/4,u,T)2 0 and l-1 l(O,u,T) I 0 These last inequalities hold provided that d Q ( o , u , TJ rl11(1/4,0,~) 2 o and { -5 O aU respectively, which imply that

L4

2.3 APPLICATIONS OF THE EXISTENCE THEOREM 5 1

xi 12cos(m2T/4) + x212cos(wlT/4) 5 0

(2.3.5)

Analogous considerations lead to the following conditions which ensure that the inequalities, ) r 2 1 ( ~ , u , T 0, t E [O,T/4) and period T E [T*,T+] belongs to the family Kj(t,h). This can be proven as follows. As shown above, x(t) can be uniquely continued in h to h = 0, with the corresponding period T(h) remaining within [T*,T+]. Since WjO is the only natural frequency within [ ~ j , ~ jcorresponding *] to a positive mode, x(t,h) coincides with xj(t,h) for sniall values of h. Due to the uniqueness of the analytic continuation, it follows that x(t,h) coincides with xj(t,h) for all levels of energy h I ho. If a system satisfies conditions (2.3.lb), then one can apply (2.4.10) with N = 0. Analogously to the proof of inequality E > 0, one can show that the denominator in expression (2.4.13) with R ( w ) = C(T>+ E [A(T)-C(T)l is positive even though the matrix &(T,E) is nonpositive. Therefore, under conditions (2.3. I b), relation (2.4.8) in Theorem 2 [i.e., the positiveness of' matrix A@)] is unnecessary.

2.4 NNMs IN SYSTEMS WITH CONCAVE AND CONVEX NONLINEARITIES 59

Theorem 2 does not exclude the existence of a symmetric, T-periodic solution, x(t), which is sign-variable for t E [O,T/4). It can be proven, however, that the positive solution x(t,T) with period T E [TjO,Tj+] (i.e., the N N M oscillations) possess the following extremal property. Suppose that any line through the origin of the configuration space of the system intersects aQ at one point only. This holds in the case, for example, when the region C2 is convex. The following theorem then holds. Theorem 3. The solution x(t,T) satisfies the inequality

-x(t,T) > I x(t) I,

t

E

[O,T/4)

(2.4.18)

where x(t) is any even symmetric, T-periodic solution with I x(t) I E SZ. The notation I x(t) I implies an (n x 1) vector with elements equal to the absolute values of the corresponding elements of x(t) [this notation should not be confused with the notation for the norm of &(t), llx(t)Il]. Proof. Introduce the following quantities:

(2.4.19) Denote by [G*(T) *] the operator (2.1.23) with Y(X) = T2 [f*(i+-NxI and consider the following sequence: (2.4.20) Since

[G"(T) *] = [G(T) *] for x(t) E R it follows that x(7) =

[G*(T>rr1(7)

60


so that, x~(T) > xl(z). By (2.4.8) and (2.4.19), matrix [A*(zr)-N] is 11011negative, and, thus, operator [G*(T) *] is monotonous, and [G*(T>a](7) > [G*(T)b] (7) for a ( ~>) b(z) It follows that sequence (2.4.20) is increasing. Suppose first that this sequence is also bounded. Then xk(5) + x"(5) as k+-= Since f * ( ~ is ) continuous, the operator [G*(T) *] is uniformly continuous (Krasnoselskii, 1968), and, therefore, x * = [C* (T)x* 1 Clearly, by (2.4.4a) and (2.4.19), one has that c* (x) > A*(x) for all x > 0 Hence, according to the remark of Theorem 2, the positive solution to the problem is unique. Therefore, one obtains that X* = x(t,T) > I x(t) I Suppose now that sequence (2.4.20) is unbounded, so that Xkp(T) +m as k+m where xkp(z) is the pth component of vector &k(2), p = 1,...,n. Assuming that f*(&k>= €.*(a) 2,y = -a, =U + I - a from inequality xk+l (T) > %(T) and relation (2.4.20) one finds that

s

where

5 r(%u,T){c*[Xk(u)l-N}Y(u) du

1/4

[Ly](Tt) = -T2

0

and matrix r ( ~ , u , T )is defined by (2.1.25). By relation (2.4.19), c*(s~) -+ N as I I ~ ( T ) I I+ and by (2.4.8), L < T2G for sufficiently large k By Lemma 5 , it follows that the first eigenvalue hi of operator [L:] satisfies the inequality hi > hl(G)/T2 = 1 Since &(T) > 0, by Lemma 4 the solution of (2.4.21) must be positive. However, this leads to a contradiction with the relation y ( ~=) -xk(T) 4 0. This contradiction shows that sequence (2.4.20) is bounded, and Theorem 3 is proved. 00


If the region R coincides with the entire Rn, then inequality (2.4.18) holds for any even symmetric solution ~(z),and an oscillation corresponding to a NNM has the largest amplitude of any possible oscillation. Now, suppose that instead of condition (2.4.4a) the following inequality holds: ~ ( s x2 ) sf(x)

for s E [0,1], & > 0,

zE

R

(2.4.22)

Then, operator (2.1.23) satisfies the analogous inequality:

[Gsx] 2 s [CX]

for s E [0,1],

x>0 , xE R

(2.4.23)

In the theory of positive operators (Krasnoselskii, 1968), operators satisfying inequality (2.4.23) are called concave. It can be shown that concave operators possess properties similar to those established under the condition (2.4.4a). In particular, the positive solution X(T) of (2.1.23) with a concave operator is unique. If operator [G *] is monotonous, the successive approximations Xk = [GXk-11 converge to x(t) for any xl(z) 2 0; hence, the extremal property of x(t,T)

[inequality (2.4.18)] is valid in this case. A comparison of conditions (2.4.4a) and (2.4.22) is now performed by considering the system depicted in Figure 2.3.l(a). Condition (2.4.22) is satisfied if all functions fpk(x)/x decrease in 1x1. As shown above, in this case inequality (2.4.4a) is also satisfied. However, inequality (2.4.22) does not, in general, hold once a change of coordinates has been introduced; so, when analyzing the family of NNMs zj(t,h) (j#n), the corresponding operator [G -1 will not be concave (the concavity holds if fp(p+l)(X) = Cp(p+l)X i.e., if all springs connecting the masses are linear). By contrast, condition (2.4.4~1)is invariant with respect to changes in the coordinate system, and so, this inequality holds for all families of NNMs zj(t,h). On the other hand, condition (2.4.22) has the advantage over (2.4.4a) that it does not require the existence of a potential function V(x), and, thus, it is applicable for systems with non-conservative coupling forces. The case of convex nonlinearity [condition (2.4.4b)l is now considered. Unlike to the previously discussed case, the uniqueness of the continuation of

62


the NNM zj(t,h) in h to dR requires an additional condition. Suppose that for 5 2 0, x E R,the following inequality holds: 0 < A(&)< A+

(2.4.24)

where A+ is a symmetric constant matrix. Using the previously introduced notation, one finds that matrix C(k&)increases in k, so that, C(0) = A@>< C(&)< A(&) Denoting by wp+2, p = 1,...,n, the eigenvalues of matrix M-lA+, and employing the previously introduced notation, the following conditions are assumed: q

CE

[(o,'/o~+),(o~+/w,'>],

p

# j,

q = 1,3,5 ,...,

~ n A(Q),one can use arguments analogous to the ones used for the proof of Theorem 2 to show that Tj(h) < TjO for sufficiently small energies h. It will now be shown that the NNM solution x(t,T) is uniquely continuable in T. Consider the following eigenvalue problem:


Denote by hpo,A,(&, (2.4.27) for

and

A,+,

p = 1,2,..., the positive eigenvalues of

R =A(Q), R=A(z,T), and 12.=A+ respectively. Clearly, hpo = (2nq/Toko)2, and h,+ = (2nq/Tr~&+)~, where k = 1,...,r, q = 1,3,5,... Since A(0) < A(2,T) < A+ it follows that A,+ < hp(A) < hp0, p = 1,2,... (2.4.28) Let hs+and h,o correspond to k = j and q = 1. Then one obtains that hs+ < 1 and hso > 1, for T E (Tj+,TjO) By (2.4.25), the interval [hs+,hsO]has no common points with the interval [hk+,hko],k f s. Hence, the following inequalities hold:

Let hi(C)denote the eigenvalues of (2.4.27) for 12. = C(z,T). Clearly, hp+ < hp(C)< hpO, p = 1,2,... and, as a result one obtains that

hdC) E @s+,hsO)

Since the solution x(t,T) satisfies (2.4.12), it follows that h&) = 1. By (2.4.4b), one has that C(2,T) < A(z,T) hs(A) < L ( C )= 1 Taking into account (2.4.29), one finds that the variational equation (2.4.9) has only the trivial solution, a result that proves the uniqueness of the continuation of the NNM xj(t,h) in T. One can then use arguments analogous to the ones used for proving Theorem 2 to show that parameter T may be replaced by the energy h, and the theorem is proved. Similarly to the case of concave nonlinearity, one can show that any even symmetric solution x(t) E Q = {x/V(x) 5 h} with period T E [T*,T+]belongs to the family of NNMs Lj(t,h). Note, that if (2.4.25) is not satisfied, then one can find a value o E (UjO,aj+), such that

64


It is clear from the previous proof that the NNM xj(t,h) is uniquely continuable in h as long as xj(t,h) E i2, and the corresponding period satisfies the relation T 2 2 d 0 . Condition (2.4.25), however, does not provide the positiveness of q(t,h) on [O,T/4). This last property is ensured by the inequalities rik(T,u,T) < 0 for t,u E [0,1/4), i.k = 1,...,11 It should also he noted that for positive solutions of systems with convex nonlinearity, Theorem 3 is not true. In a system with concave nonlinearity, the following inequality holds. A(&) < A(@.Suppose that A(x) > A- for 11E R Then one can show that the NNM xj(t,h) is uniquely continuable in T, as long as the frequency w = 27dT satisfies the condition:

where (oP-)2 are the eigenvalues of matrix M-IA-. If condition (2.4.3 1) is satisfied for o = Wj-, then xj(t,h) is continuable up to the boundary of region R.Clearly, the above condition, unlike Theorem 2 , does not provide the positiveness of xj(t,h) for t E [O,T/4). To illustrate the results obtained, consider the transverse vibrations of the string depicted in Figure 2.3.l(b). For this system conditions (2.3.lb) are satisfied, and one can prove that there exists an infinite continuum S, o f even and positive solutions on [O,T/4), corresponding to the largest linearized natural frequency WnO. Under such NNM oscillations adjacent masses of the system always move in opposite directions. Assume, now, that the initial elongation of the string is k > I ; then inequality (2.4.4a) is satisfied. As seen from (2.4.7), in using inequality (2.4.8) one can select N as N = lim 11&. I( + m A(&)

N = [ npk I,

where

np(p+i)= kp,

npp = kp-1 + kp npk = 0 if I p-k I > 1

npk = nkp,

kp = EFAp

(2.4.32)


By Theorem 2, the continuum Sn consists of a one-parameter family of NNMs xn(t,h), whose period Tn(h) increases monotonically with h, i.e., oo Tn(h) = Tn = 27~/0n lim +

where a$, k = 1 ,...,n, are the eigenvalues of matrix M-lN (physically, the quantities o n are the frequencies for longitudinal oscillations of the string). By Theorem 3, the amplitudes An = &n(Oh) exceed those of any even symmetric solution x(t) of the same period T E (TnO,Tn). To consider other families of NNMs, xj(t,h), j # n, one must transform properly the directions of the coordinates of the system, in order to comply with the requirement x;o > 0. Hence, introduce the following new coordinates in (2.4.7); let zp = Xp+l + xp when the signs of coordinates X j , p + l O and Xj,pO coincide (in the above coordinate system), and zp = Xp+l - xp when they are opposite. If for some region R the conditions (2.2.6) are satisfied for T E (TjO,Tj+), then the corresponding continuum Sj of NNMs reaches the boundary dR of the region. If matrix A(&)-Nj is nonnegative, then, by Theorem 2 the family Sj consists of the one-parameter family of NNMs Lj(t,h) whose period T;(h) increases monotonically. The corresponding amplitudes of the NNMs exceed the amplitudes of any other symmetric solution. Suppose that Nj = lim 11 11 + oo A(&) (i.e., np(p+l) = -kp or k, if the positive directions of the axes xp+l and xp coincide or are of opposite sign, respectively). Then, the matrix A(&)- Nj is nonnegative, and, if conditions (2.2.6) are satisfied for T E (2~/0;0, 2 7 ~ / ~ j ) , the corresponding NNMs xj(t,h) exist for any value of h. Conditions (2.4.30) and (2.4.31) will now be used to prove the uniqueness of the continuation of Kj(t,h) in h, and the monotonicity of the period Tj(h) with respect to h [these conditions, however, will not guarantee positiveness of xj(t,h) for t E [O,T/4)]. Suppose that the region R is defined by the condition IXp+l + XpI 2 ClP where C is a parameter. Then, one can define

66

NNMs IN DlSCRETE OSCILLATORS: QUALITATIVE RESULTS

A - = A * f o r k > 1, or A + = A * fork < 1 where the quadratic form (A*y,y) is obtained from the quadratic form (&x)y,y) by replacing of quantities (xp+l + xp)/lp by the constant C. In that case the frequencies wp and wpo are proportional, and one obtains a relation of the form:

wp = dwpO, d =

[ [(k-l)(C2+1)-3/2+1]/k}

1'2

(2.4.33)

Setting wp = dwpO in (2.4.31) for k > 1, or in (2.4.30) for k < 1, for each j (i.e., for each family of NNMs) one can compute the limit dj-(O) < 1 or dj+(w) > 1 at which the condition is violated. Therefore, for d E (dj-,dj+), the family of NNMs zj(t,h) is uniquely extendable in T up to T = ~ K / u .If conditions (2.4.30) or (2.4.31) are satisfied for w = dwjo, the family xj(t,h) is extendable to the boundary of 51, i.e., until the limit IXj,p+l + Xj,pl = Cj lp for some p, where Cj(dj) is defined by (2.4.33). Clearly, the values of dj- or dj+ are equal to the values (qOjO/OpO) or (OpO/qOjO), p = 1,...,n, p f j, q = 1,3,5,..., nearest to the unity on the left or on the right, respectively. Hence, it is concluded that dj- = l/dj+. Numerical calculations were performed for a string with four equally spaced, identical masses, 1, = lo, mp = mo, n = 4. The nondimensional , linearized natural frequencies of the system, 2 ( T o l l o m o ) ~ / 2 w p ~are computed as 0.3090, 0.5878, 0.8090, and 0.95 12. The numerical calculations yielded the following limiting values: dl- = 0.974, d2- = 0.539, d3- = d4- = 0.851 d l + = 1.026, d2+ = 1.855, d3f = d4+ = 1.176 Substituting the resulting values dj into (2.4.33), one obtains the corresponding values Cj, provided that the relative elongation of the string, k, is prescribed. If for a certain value of dj relation (2.4.33) does not provide real values for Cj, this means that the corresponding family of NNMs zj(t,h) is continuable to any value of energy h. As a second example, consider the oscillations of two linearly coupled pendulums. The governing differential equations of motion are given by


m2122x2 + m2g12 sinx2

+ c(x2-xi)

=0

(2.4.34)

where mp and l,, p = 1,2, is the mass and length of each pendulum, respectively, and x i , x2 are angle coordinates. For this example, the previously defined matrices assume the following specific forms:

C(L)=

[ c + mlg!csinxi/xl

c

+ m2gl2sinx2/x2 ] -C

(2.4.35)

Since (sinx/x) > cosx for x < 4.49, one finds that A(&)< C(x) for this problem, and the nonlinearity is concave. The first mode of the linearized system is g o > 0. Suppose that a = {x/lXkl<Xk+}, k = 1,2 To comply with inequality (2.1.14), one can choose

N = A(Xkf),

= A(O)

Let 012 and 0 2 2 be the eigenvalues of M-lN. If 02/01 < 2, and inequalities (2.3.5) and (2.3.6) hold for T = TI" = 2 ~ / ~ 1 0 then the conditions of Theorem 1 are satisfied. Therefore, there exists a continuum of NNMs, S1, corresponding to the family xl(t,h) which reaches the boundary of s1. Under this type of oscillations the pendulums rotate in phase. If inequality (2.4.8) is satisfied, then, by Theorem 2 the family of NNMs xi(t,h) is uniquely continuable in h until the corresponding amplitudes A = xl(0,h) I &+. The period Ti(h) of this family of NNMs increases monotonically. By theorem 3, xi (t,h) possesses the extremal property (2.4.18). To analyze the additional family of NNMs, x2(t,h), one must change the positive direction of xi or x2. As a result one must replace -c + c in (2.4.34) and (2.4.35). Suppose that c > 0.218mkglk, k = 1,2; then conditions (2.3.lb) are satisfied, and the NNMs corresponding to the second normal mode of the linearized system exist for any value of h. Under this type of oscillations the pendulums rotate in antiphase. As mentioned previously, under condition (2.3.18) condition (2.4.8) is unnecessary, so, by Theorems 2

68


and 3, for xk < 4.49, the corresponding continuum S2 consists of the one parameter family x2(t,h), with periods, Tz(h), which increase monotonically. In addition, the extremal property (2.4.18) holds. Considering oscillations of the pendulums about the upper equilibrium points, the quantities sinxk and cosxk in (2.4.34) and (2.4.35) should be replaced by their negatives. Clearly, for such types of motions only the family of NNMs &2(t,h) exists, since the first eigenvalue of the corresponding matrix M-IN is negative. If c > mkglk, k = 1,2, then conditions (2.3.lb) hold, and the NNMs x2(t,h) exist for any value of 11. Under such oscillations the pendulums rotate in antiphase, and meet at the upper equilibrium point. In contrast to the previous case, for this type of rotations one finds that A(x) > C(x)for xh < 4.49, k = 1,2, and so their period T2(h) decreases monotonically.


CHAPTER 3 NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS Quantitative techniques for computing NNMs in discrete, unforced nonlinear oscillators are now developed. It will be shown that, in contrast to linear theory, nonlinear conservative systems can admit two types of NNMs, namely, similar and nonsimilar ones. Similar NNMs correspond to straight lines in the configuration space and are realized only in oscillators possessing special symmetries. Nonsimilar NNMs are represented by curves in configuration space and, as shown, are encountered in general classes of nonlinear systems. Analytical techniques for computing similar and nonsimilar modes will be developed in this chapter. The first class of methods examined relies on the geometrical study of the trajecories of NNMs in configuration space. An alternative class of methods is based on the construction of invariant manifolds of NNMs in real or complex domains. Special methodologies for computing NNMs in vibro-impact systems or in systems with multiple equilibrium configurations will be discussed, and an exposition of a group-theoretic approach for finding general classes of systems admitting NNM oscillations will be given. Before proceeding with a discussion of the aforementioned methodologies, a brief summary of some well-established quantitative methods for studying the nonlinear responses of conservative and nonconservative discrete oscillators is given.

3.1 INTRODUCTION The fundamentals of quantitative methods for analyzing nonlinear oscillations were established in the classical works by Rayleigh, Poincare', Lyapunov, and Lindstedt. Detailed expositions of quantitative nonlinear techniques for analyzing autonomous dynamical systems can be found in the works by Bogoliubov and Mitropolsky (1961), Nayfeh and Mook (1984), Guckenheimer and Holmes (1984), Sanders and Verhulst (1983, and Wiggins (1990). Consider the single-DOF autonomous oscillator:

69

70

NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS

A method to analytically approximate a periodic solution of this system (Lindsted's method) is to introduce a new time variable, f = cot, and to expand x and co in power series with respect to the small parameter E. It can be proved that these power series expansions approximate asymptotically the solution of (3.1.1) as &+O. By employing this technique one not only approximates the nonlinear periodic response x(t), but also studies the nonlinear amplitude-frequency relation. A similar methodology was employed by Lyapunov for a system with an analytical first integral of the form: X =Ax

+ f(x),

x

E

Rn, f : Rn

+ Rn,

f E Cr, r 2 1

(3.1.2)

where A is an (n x n) constant matrix. Assuming that the linearized system, X

= b

possesses periodic solutions, and that it satisfies certain nonresonance conditions, the periodic solutions of the nonlinear system (3.1.2) were obtained by expressing K in series of powers of an amplitude parameter. Lyapunov proved that (3.1.2) possesses a one-parameter family of periodic solutions (NNMs), which he considered as nonlinear analogs of the linear normal modes of classical vibration theory. Although NNMs cannot be used as a basis for analytically computing general solutions of (3.1.2) (in contrast to linear theory where the principle of superposition holds), nevertheless, these modes can be used as generating solutions for studying the responses of much more complicated systems (Malkin, 1949) or for analyzing the forced resonances of (3.1.2) under periodic external excitations (Malkin, 1956). A class of quasi-linear (weakly nonlinear) methodologies extended Lyapunov's analysis of NNMs by computing the local nonlinear dynamics of discrete oscillators in small neighborhoods of fixed points (Moser, 1968,1976; Weinstein, 1973; Siege1 and Moser, 1971). One of the most important quantitative methods for analyzing nonlinear oscillations is the method of averaging. From a mathematical point of view, this method relies on the transformation of the equations of motion to an averaged set which is more easily analyzable. Considering a nonlinear hamiltonian system with hamiltonian function: H(E,Q;E)= Ho@) + E H I ( ~ , Y+) O ( E ~ )

3.1 INTRODUCTION

71

where p and 9 are vectors of generalized coordinates and generalized momenta (Percival and Richards, 1982), the aim is to introduce a nonlinear + (p,Q),such that in the new coordinates the change of coordinates, (p,~) hamiltonian function assumes the simplified form: K(P;E) = KO(€') + E K ~ ( P+) O ( E ~ ) If the hamiltonian H(Q,Q;E)is periodic in 9, then the function Kl(P) is taken as the average over all Q of the function H l ( ~ , q )i.e., , K 1(PI = g where < > denotes the averaging operator. Hence, to order ~ 2 the , transformed hamiltonian function depends only on the new generalized momenta p, a feature that permits the integration of the equations of motion and the analytic approximation of the nonlinear response. A basic limitation in applying the method of averaging is the existence of small divisors in the transformation of coordinates (p,~) + (P,Q),which appear when the hamiltonian system has internal resonances (Moser, 1968; Arnold, 1978). An alternative averaging method exists, developed by Van der Pol (1922), and Bogoliubov and Mitropolsky (1961) to study the response of conservative as well as nonconservative discrete nonlinear systems. Considering system (3.1. l), one imposes the transformation of coordinates,

x = a cos@+ O ( E ) ,

X

= -aoosin@ + O(E)

(3.1.3)

where a and @ are the slowly varying amplitude and phase, respectively. These variables are governed by the following first-order equations: a = -(E/oo)sin@f(acos@,-awosin@)

4

= 00 - (E/oo>cos@f(acos~,-aoosin@)/a

(3.1.4)

Averaging system (3.1.4) with respect to the phase over a period equal to 27c leads to a simplified set of equations with no @-dependenceon the righthand-side terms. The solution of this simplified set of equations provides an approximation to the response of the system. Computations of higher order approximations in (3.1.3) by the method of averaging were carried out in Sanders and Verhulst (1985).

72

NNMs IN DISCRETE OSCILLATORS:QUANTITATIVE RESULTS

The method of multiple scales (Nayfeh and Mook, 1984; Lagerstrom, 1988) is an additional powerful quantitative technique. This method systematically computes the dynamics of the nonlinear system using fast and slow time scales, enabling an increasingly accurate approximation of the dynamic response as additional time scales enter in the analysis. Since this method is applied in subsequent chapters, it will not be described here. Finally, starting with the works of Poincare', an entire class of powerful geometrical methods were developed for studying the geometry of the "flow" of a dynamical system in phase space. In contrast to the previously mentioned qualitative techniques, geometrical methods enable the study of the global dynamics of a system, of local and global bifurcations of various solutions in phase space, and of chaotic responses. In what follows general quantitative methods for analyzing similar and nonsimilar NNMs of conservative or nonconservative discrete oscillators are developed. In section 3.2 the trajectories of NNMs in configuration space are analyzed using a geometric approach. In section 3.3 NNMs are computed by computing invariant manifolds for the motion. In section 3.4 elements of group theory are employed to investigate general classes of two-DOF nonlinear systems possessing similar or nonsimilar NNMs. In the two final sections of this chapter, NNMs of vibro-impact systems and of systems with multiple equilibrium positions are analyzed.

3.2 CONSERVATIVE SYSTEMS 3.2.1 Trajectories of NNMs in Configuration Space As proved by Lyapunov, under certain limitations (absence of internal resonances), a nonlinear finite-dimensional system with an analytic first integral of motion possesses families of periodic solutions (NNMs) close to the origin of the configuration space of the system. To cotnpute such periodic solutions Lyapunov used two distinct approaches. The first approach involved representations of the NNMs in power series in amplitude; the coefficients of the series were assumed to be periodic in time. The second approach was based on the analytic approximation of the phase trajectories of the NNMs. In the following analysis an alternative technique for studying NNMs is formulated, based on the analysis of their trajectories in the configuration space. For the sake of simplicity, throughout the

3.2 CONSERVATIVE SYSTEMS

73

following discussion it is assumed that the first integral of the motion is identical to the energy. Consider an n DOF conservative system with equations of motion given by: xi

+ aV(E)/dxi = 0,

i = 1,2,...,n

(3.2.1)

where V(x) is the potential function of the system, x = (xi, ...,xn)T, and V(z) is assumed be positive definite. Suppose that the potential energy can be expressed as a summation of a quadratic form and of higher order terms: (3.2.2) where function N(*) contains terms of at least O(11~113).The energy integral of the system is given by: n

E (&

= (1/2)

m= 1

n

n

m= 1

m= 1

Xm2

+ V(&) (3.2.3)

where h is the fixed level of the total energy. It is assumed that the region of the configuration space bounded by the closed maximum equipotential surface V(x) = h contains only the trivial equilibrium position x = 0. Clearly, all displacements of the system are bounded, since no trajectory of (3.2.1) can cross the maximum equipotential surface. By applying Hamilton’s principle, t?

6 J (K-V) dt = 0 tl

where K is the kinetic energy of the system and t i < t2 are two time instances, one recovers the equations of motion (3.2.1). However, alternative representations of the equations of motion of the dynamical system are possible, by applying the principle of least action (Jacobi’s form) (Synge, 1926; Rosenberg and Hsu, 1961):

74


6s = 6

P2

(h-V)1/2 ds = 0,

ds2 =

PI

n

C dxm2

m= 1

(3.2.4)

where Pi and P2 are two points on the trajectory of the dynamical system i n configuration space, s is the arc length of the trajectory in Euclidean space, and S is the length of the trajectory in the Riemannian metric defined by dS2 = 2(h-V)

n

C dxm*

m= 1

Since the action integral is homogeneous in the differential terms dxi, the value of the action S is independent from the specific type of parametrization of the trajectories; as a result, one can express (3.2.4) in the following equivalent form:

where the dependent variables were scaled according to xm = x,n(a), m = I , ...,n. Setting a = t, one obtains Euler's equations in the Newtonian form (3.2.1) with the corresponding energy integral given by (3.2.3). If the parametrization a is chosen to be equal to the Euclidean norm of s, i.e., a = IIslI, Euler's equations and the energy integral assume the following forms:

where prime denotes differentiation with respect to a then the corresponding relations become:

E

Ilsll. Choosing a = S,


c (x,')~ n

2[h-V(x)]

m= 1

=1

75

(3.2.7)

Suppose now, that one of the dependent variables, xi = x, plays the role of the parametrization a, i.e., that xi = x = a. For this type of parametrization, Euler's equations for the least action assume the form:

i = 2 ,...,n (3.2.8) where, as previously, prime denotes differentiation with respect to the parametrization variable a = x. In order to write (3.2.8) into the simplest possible form, consider the equations of motion (3.2.1), and express the time derivatives in terms of the new independent variable x using the chain rule of differentiation: d d_ . d2 d2 . d .. and dt2 = dx2 x2 + & x = dx x, The velocity of the parametrizing variable x can then be expressed in closed form by considering the energy integral (3.2.3):

1+

2

m=2

(3.2.9) (XJ*

Expressing the time derivatives in (3.2.1) in terms of x, and taking into account (3.2.9), the equations governing the trajectories of the dynamical system in configuration space are obtained as follows:

76


where the mth dependent variable Xm is considered to be a function of the parametrization variable x, i.e., Xm = Xm(X), m = 2, ...,n. It is easy to show that the set of equations (3.2.10) is equivalent to the set (3.2.8), which wa4 derived by employing Jacobi's principle of least action. The solution of (3.2.10) determines the trajectory of the dynamical system in the ndimensional configuration space. It is noted that although the set (3.2.10) contains one fewer equation than the original set of equations of motion (3.2.1), the analysis of an arbitrary solution is by no means simpler, since equations (3.2.10) are nonlinear, nonautonomous and possess essensial sirigularities if xi'+" and reinnvable (nonessential) ,singularities othrrwrse. This is due to the fact that at the instant when V(x) = h the dynamical system reaches its maximum equipotential surface and the coefficients of the highest derivatives in (3.2.10) vanish simultaneously. Hence, points of niaxinwni potential energy represent removable singularities for the equations of the trajectories, and it is necessary to supplement equations (3.2.10) by the following set of transversality conditions, valid at points of intersection of the NNM trajectory with the maximum equipotential surface:

In (3.2.1 1) X denotes the amplitude attained by the parametrizing coordinate during the motion. Clearly, the equivalence x = x '3 V(x) = 11 holds. By satisfying equations (3.2.10) and conditions (3.2. I I), a trajectory Xnl = Xm(X), m = 2, ...,n, can be analytically continued up to the niaxiinum equipotential surface V(&) = h. Indeed, at the limit x + X one can use Hospital's rule to derive the following expression for the second derivative (Rosenberg and Hsu, 1961):


m=2

as x

+ X,

77

i = 2 ,...,n (3.2.12)

where all derivatives in (3.2.12) are evaluated at x = X. From the above relation it is concluded that the second derivatives Xi", i = 2, ...,n, are defined at the limits x + X provided that { avx=x f 0 i.e., that no equilibrium positions of the system exist at the maximum equipotential surface. Hence, the following analysis is carried under the assumption that no additional equilibrium positions of the system exist at the maximum potential energy level. Equations (3.2.10) and (3.2.11) are well suited for an analysis of an important class of particular solutions of the system, namely, of nonlinear normal modes (NNMs). In classical linear vibration theory it is well known that an n-DOF conservative oscillator possesses precisely n normal modes. If a system oscillates on a normal mode, all positional variables oscillate equiperiodically and reach their extreme values simultaneously. Such motions are represented by straight lines in the n-dimensional configuration space of the system. A nonlinear conservative discrete oscillator also possesses normal modes in the form of synchronous periodic oscillations. However, in contrast to the linear case the trajectories of NNMs i n configuration space can be either straight lines or curves. NNMs represented

1

Xm

(4

(b)

Figure 3.2.1 Trajectories in configuration space of (a) similar and (b) nonsimilar NNMs.

78


(4

(b)

Figure 3.2.2 Trajectories in configuration space of a nonsimilar N N M corresponding to (a) even and (b) odd potential function. by straight lines in configuration space are termed similar NNMs, whereas those represented by curves, nonsimilar (cf. Figure 3.2.1). Nonsimilar NNMs are generic in nonlinear oscillators. Moreover, in contrast to classical linear vibration theory, the number of NNMs may exceed the number of degrees of freedom of the nonlinear system (see, for example, the two-DOF system discussed in section 1.2). The solutions of (3.2.10) and (3.2.1 1) will be denoted as Xm = $m(X), ni = 2,3,. ..,n, where ?m(*) are analytical functions. If the potential energy of the system, V(x), contains only even powers of the variables, all positional variables pass through the equilibrium position simultaneously. In addition, on an NNM all positional variables reach the maximum equipotential surface V(x) = h simultaneously and twice during a period of the normal mode oscillation. At instances of maximum potential energy all velocities of the positional variables vanish simultaneously. A trajectory in configuration space of a (nonsimilar) NNM corresponding to even potential energy is presented in Figure 3.2.2(a). Points P i , P2 where the trajectory intersects the surface V(x) = h are cusps, and the trajectory satisfies the symmetry relations $,(x) = -Gm(-x), and &,-,(0>= 0, m = 2,3 ,...,n If the potential function V(x) contains odd powers of positional variable?, the surface V ( r )= h is no longer symmetric with respect to the origin of the configuration space, and the NNM trajectories do not pass through the origin


79

of the configuration space. An example of such an NNM trajectory, which intersects twice the maximum equipotential surface, is depicted in Figure 3.2.2(b). Note the lack of symmetry of the trajectory in this case. On a NNM, the nonlinear system behaves like a single-DOF conservative system. This can be shown by considering the equations of motion (3.2.1) on A an NNM trajectory, Xm = Xm(X), m = 2,3, ...,n. On an NNM all equations of motion degenerate to the following equation: (3.2.13) where the notation V[x,&?(x),...,?,(x)] 5 P(x) was used. Hence, on an NNM the system is reduced to a single conservative nonlinear oscillator whose response can be computed in closed form by quadratures: X

t

+ $ = 2-l/21

[h-P(5)]-1/2 d5

X

+ x = x(t,X)

(3.2.14)

where the arbitrary phase can be chosen so that x(0) = 0. Expression (3.2.14) is of the form t = t(x,X) and can be inverted in appropriate intervals of time to yield a relation of the form x = x(t,X). As discussed in Vakakis and Caughey (1988), this inversion can be accomplished by defining special functions and their corresponding parameters. Once the time dependence of the reference parametrizing coordinate is obtained, the time dependences of the remaining variables are computed by employing the A solutions of (3.2.10) and (3.2.1 l), X m = Xm(X), m = 2,3 ,...,n. The amplitude of the parametrizing coordinate, X, and the energy of the motion, h, are related by: P(X) = h

( 3 . 2 .I 5 )

For a specified level of total energy h the above equation can be solved in terms of the amplitude X, provided that the equipotential surfaces are closed, and that no equilibrium positions other than the trivial equilibrium, xm = 0, m = I, ...,n, exist. These conditions also assure that the function x = x(t,X) which results from the inversion of (3.2.14), is periodic (Wintner, 1941).

80


Since the leading terms of the potential energy P(x) are even powers of x, one expects that for each value of h there correspond at least two values of the amplitude X, denoted by X = Xi, i = 1,2. When the potential energy contains only even powers of x, due to the symmetry of the solution one obtains that Xi = -X2. From the previous exposition it is noted that NNMs constitute a singleparameter (in the energy h) family of periodic solutions with smooth trajectories in the configurational space of the system. A second arbitrary parameter equal to the phase $ in (3.2.14) may also be considered, but this i q eliminated without loss of generality by imposing the initial condition x(0) = 0. It must be noted that equations (3.2.10) [or equivalently ( 3 . 2 . 8 ) ] do not provide a unique definition of the trajectories of the NNMs. To uniqiiely define these trajectories one must impose the additional boundary conditions (3.2.11). Moreover, since the solution x = x(t,X) resulting from the inversion of (3.2.14) is always periodic (with the trivial exception of equilibrium positions), there is no need to impose additional periodicity conditions to the problem. In the following sections two basic classes o f solutions of equatioris (3.2.10) and (3.2.1 1 ) are studied i n detail, corresponding to similar and nonsimilar NNMs.

3.2.2 Similar NNMs Consider first small vibrations of system (3.2.1) in a m a l l neighborhood of a stable equilibrium position. The corresponding linearized systeni is defined by two positive-definite quadratic forms, one representing the kinetic and onc the potential energy. Introducing the transformation of variables ( X l , ...,X") -+ (y1,...,ytl) where y = (y1 ,...,y,))T denotes the vector of canonical coordinates, the quadratic forms for the kinetic and potential energies arc expressed as: n

I1

Employing canonical coordinates the equations of motion (3.2. I ) are decoupled and written as:


81

The linearized system possesses n normal modes of vibration, each mode corresponding to the vanishing of all but one canonical coordinates: yi(t) = Yi cosoit,

Yp(t) = 0, p = 1,...,n, i # p (ith linearized normal mode) (3.2.18)

with initial conditions yi(0) = Yi, yi(0) = 0. In terms of the original (noncanonical) coordinates, x = (XI,...,xn)*, the ith linearized normal mode is represented as:

xm = cm(i)x, XI

m = 2 ,...,n, i = 1,..., n, x = Xcosojt (ith linearized normal mode) (3.2.19)

From (3.2.19) it is concluded that the linearized normal modes are ,simi/cir, since they are represented by straight lines in the configuration space. If the linearized natural frequencies Wi, i = 1,...A are incommensurable, the set of normal modes coincides with the set of periodic solutions. Otherwise any superposition of normal oscillations with commensurable frequencies will also be a periodic solution (generalized normal mode). However, only normal modes are represented by straight line trajectories in the configuration space. The nonlinear system (3.2.1) is now considered. Rosenberg (1966) first investigated similar NNMs in classes of essentially nonlinear discrete systems. On a similar NNM the positional variables of the system are related by linear expressions of the form: Xm

= c ~ x , m = 2 ,...,n,

XI = X

(3.2.20)

where it is assumed that the parametrizing coordinate x is not identically equal to zero, and that cm # 0, m = 2 ,...,n. Substituting (3.2.20) into (3.2.1) one obtains the following set of n differential equations:

82


(3.2.21) All equations (3.2.21) are in terms of the parametrizing variable x, and are, thus, equivalent. It follows that for compatibility reasons the following algebraic relations must hold: m = 2, ...,n (3.2.22) These equations can be used to determine the coefficients of the similar NNM, Cm. An examination of set (3.2.22) reveals that it constitutes an overdetermined set of algebraic equations. To show this, denote by V(r)(x) the terms of the potential function or order r in x, i.e., V(')(Z) = O(llXII') Employing this notation, and noting that the compatibility conditions should hold for any value of x, one obtains the following modified set of equations representing compatibility of the various powers of x in (3.2.21):

avm ax (I,c2,...,cn) = (l/cm) avw -r (l,c2 ,...,cn), Xm

m = 2 ,...,n,

r = 2,3,4,...

(3.2.23) Denoting the highest order of nonlinearity in V(x) by R, equations (3.2.22) form a set of (R - 1) x (n - 1) simultaneous algebraic equations with (n - I ) unknowns. Therefore, the problem of finding similar NNMs in system (3.2.1) is overdetermined, a feature that limits the class of nonlinear discrete oscillators that possess similar NNMs. As discussed in (Rosenberg, 1966) and (Vakakis, 1990), similar NNMs are not generic in nonlineur discrete oscillators, and exist only in systems with special configurutionul syvnmetries. It is concluded that similar NNMs form a very restricted class of NNMs and reflect certain symmetries of the system. Generically, nonlinear discrete oscillators possess nonsimilar NNMs. Certain properties


83

of similar NNMs are now mentioned; for a more general exposition the reader is referred to (Rosenberg, 1962, 1966). The straight-line trajectories of similar NNMs in configuration space intersect all equipotential surfaces orthogonally. This can be proven by noting that the trajectories defined by (3.2.22), can be rewritten as:

av av ax ( x , x ,..., ~ xn) =

( x , x,..., ~ xn), m = 2,...,n, where xP = cPx axm These equations, however, constitute the conditions for orthogonality of lines xp = xp(x) with the family of surfaces V ( X , X..., ~ ,xn) = d I h There exist classes of nonlinear systems for which the general free response (with arbitrary initial conditions) can be expressed as linear combination of the responses of similar NNMs. As an example of such a system, consider a symmetric nonlinear system of two-DOF with equations of motion given by: xln'

~

x2 + bx2

+

afi) (x2 - x1)J = 0

fi=1,3,5,...)

(3.2.24)

where b > 0 and a(J) > 0, j = 1,3,5,... This is a system of two linear oscillators coupled by a nonlinear stiffness. Denote the initial conditions of the problem by: xm(O) = xm(O), Xm(O> = .im(O), and introduce the transformation of variables

m = 1,2

(3.2.25) Equations (3.2.24) can then be expressed as:

v

+ bv +

C

2aO)vJ = 0

Cj=1,3,5,...)

(3.2.26)

84


The solution of system (3.2.26) can be written in the form:

The solution for v is periodic in t and can be expressed in closed form by quadratures. Evidently, through relations (3.2.25), the general solution to the initial value problem (3.2.24) can be obtained by a superposition of the two periodic solutions (3.2.27), which correspond to the two similar NNMs of the system: [u = u(t), v = 01 and [u = 0, v = v(t)]. A remarkable property of NNMs (not just of similar ones) is that the>l might exceed it1 number the DUF of the nonlinear oscillutor. As shown in the example of section 1.2, in repetitive structures with spatial symmetry and weak coupling, similar NNMs undergo bifurcations, leading to mode localization. Hence, not all similar NNMs of a dynamical system may be considered as analytical continuations of corresponding linearized modes, since certain NNMs are essentially nonlinear, with no counterparts in linear theory. In addition, NNM bifurcations may generate orbitally unstable modes, a feature that is in contrast to linear theory where all normal modes are neutrally stable. As shown in section 1.2, orbitally unstable NNMs affect the global dynamics of a nonlinear oscillator. ln later sections it will be shown that orbitally unstable NNMs influence the stability of steady-state periodic solutions (resonances) of a discrete system under the application of external forcing. Rosenberg ( 1966) identified certain classes of nonlinear oscillators that possess similar NNMs, which are discussed in the following exposition. The first class of nonlinear oscillators that possess (exclusively) similar NNMs are the so-called homogeneous ,systrm,s, i.e., systems whose potential function is a homogeneous function of the displacement. Consider a homogeneous system (3.2.1) with degree of homogeneity equal to (r + 1) and potential function given by:

Vine 10,...,r+ll,

m = 1,...,11

(3.2.28)


85

Introducing the following set of generalized polar coordinates,

the homogeneous potential function is expressed in the simplified form: (3.2.30) Evidently, the homogeneous system admits similar normal vibration modes if the following conditions are satisfied:

av

-=O aem

*

aa, 3%

-=O,

m = 1 , 2,...,n-1

(3.2.31)

From the last condition, the values of em necessary for similar mode oscillations can be determined. The solution p = p(t) then governs the motion of the homogeneous system along a similar NNM, with variables 61 being the angles between the coordinate axes and the straight-line trajectory of the mode in configuration space. In terms of Cartesian coordinates the trajectory of the similar mode is computed by the following linear relations: Xm = CmX, X l E X

1

c m = [sinen+i-m nj,l,...,,-,cos~~] [nj,, ,.,,,n-l cosejl- ,

m = 2,...,n

(3.2.32) The problem of estimating the number of similar NNMs in a homogeneous system was addressed in various works (Rosenberg, 1966; Zhupiev and Mikhlin, 1981; Van Groesen, 1983; Manevitch et al., 1989; Caughey et al., 1990). It was shown that in homogeneous systems bifurcations of similar NNMs occur giving rise to NNMs that exceed in number the degrees of freedom of the system. An existence study of similar NNMs in general classes of homogeneous systems was performed by Van Groesen (1983) and is briefly outlined in what follows. Consider the equations of motion of a homogeneous nonlinear dynamical system (3.2.1) and seek a solution of the form x = ca(t), where x = (xi,x2,...,xn)T, c = (cl,c2,...,cnIT, and a(t) is a scalar time-dependant function. Substituting the expression for x in (3.2. l ) , one obtains:

86

NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS a(t)

c + VV(c)ar(t) = 0

(3.2.33)

where the homogeneity condition ~ v [ c a ( t ) ]= VV(c)ar(t) was used. Assuming that VV(c) = -0c equation (3.1.33) reduces to a single-DOF conservative nonlinear oscillator, a(t) + o ar(t) = 0 which possesses periodic solutions denoted by a = a(t). Hence, the problem of existence of similar normal modes in a homogeneous system (3.2.1) reduces to the problem of existence of real solutions for o of the problem VV(c) = - og Van Groesen (1983) proved that, for homogeneous and even functions V(x), this problem has at least n solutions for o (n being the dimension of the response vector x). Therefore, he proved that an n-DOF homogeneous system possesses at least n similar NNMs. An additional class of nonlinear oscillators admitting similar N N M s as solutions are symnzetrical systems (Rosenberg, 1964) with potential energy: m

m

These systems admit always solutions of the form Xm = It: x i , 111 = 2,...,n, which correspond to in-phase or antiphase similar NNMs. Additional results regarding bifurcations of similar N N M s will be presented in chapter 4. The condition of straight-line trajectories in configuration space confines substantially the class of nonlinear systems possessing similar NNMs. Of more interest is to study nonsiinilar NNMs. since they represent the class of normal modes that is typically encountered in nonlinear discrete oscillators. This is addressed next.

3.2.3 Nonsimilar NNMs and Matched Asymptotic Expansions In this section general asymptotic methodologies for studying the nonsiinilar NNMs of linearizable and nonlinearizable systems will be developed. For sufficiently small oscillations, lineari7ahle systems can be regarded as


87

perturbations of linear ones. The nonsimilar NNMs of this type of systems can be viewed as nonlinear perturbations of the normal modes of the corresponding linearized ones. Nonlinearizable systems are essentially nonlinear systems, which in the limit of small oscillations do not degenerate into linearized ones. I n the following analysis, linearizable and nonlinearizable systems are considered separately. In (Happawana et al., 1995) the nonsimilar NNMs of a two-DOF strongly nonlinear system are asymptotically analyzed by constructing expansions of the modal curves about their regular singular points at the maximum equipotential energy surface. This elegant analysis has the drawback of requiring a considerable amount of algebraic manipulations, which render it impractical for analyzing multi-DOF oscillators. In this section an alternative asymptotic analysis for computing the modal curves is formulated based on Taylor expansions of the modal curves in neighborhoods of the origin of the configuration space, and analytic continuations of these expansions up to the maximum potential energy surface. It will be shown that such an analysis is of practical use, since it can be applied to the study of the nonsimilar NNMs of multi-DOF systems.

Nonsimilar NNMs of Linearizable Systems Consider the n-DOF conservative nonlinear system (3.2. l), and assume that the system is linearizable in the limit of small oscillations. If one disregards nonlinear terms, the resulting linearized system possesses n natural frequencies, mi, i = 1, 2, ..., n, and to each of these frequencies corresponds a linearized normal mode. A basic assumption made by Lyapunov in his construction of NNMs was that there are no linearized normal modes with natural frequencies that are integrably related, i.e., that satisfy relations of the form zpmp = zkok, where zp and zk are nonzero integers. The same assumption will be imposed in the following analysis; for the sake of convenience canonical coordinates are introduced. Expressing the nonlinear equations of motion (3.2.1) in terms of canonical coordinates, one obtains a set of equations that are uncoupled in the linear approximation:

88

NNMs IN DISCRETE OSCILLATORS: QUAN'I'ITATIVE RESULTS

where function N(y) represents the nonlinear terms of the potential energy. If the nonlinear terms are set equal to zero, dN(x)/dyi = 0, equations (3.2.35) possess the linearized (similar) normal mode:

where the notation

~ i3 )

01

was introduced, and initial conditions, y(0) = Y,

y(0) = 0, were assumed. By Lyapunov's results, in the absence of internal resonances and sufficiently close to the origin of the configuration space, system (3.2.35) possesses exactly n (nonsimilar) NNMs. The aim of the following analysis is to compute the nonsimilar NNM that "neighbors" the linearized similar mode (3.2.36). The computed nonsimilar NNM can then be regarded as the nonlinear analytic continuation of mode (3.2.36). However, the same analysis can be performed to compute additional nonsimilar NNMs "neighboring" linearized modes different than (3.2.36). Following Lyapunov's analysis, the amplitude of the parametrizing coordinate Y is rescaled according to Y + EY, where E is a small parameter [ ~ = y ( 0 )was used in Lyapunov's original work], and the canonical coordinates are similarly rescaled a9 y + EY, Ym + E y m , m = 2 ,...,n (in Lyapunov's work the small parameter was defined as, E = y(0) = Y). These rescalings lead to the following form for the potential energy:

where function N(k) represents the O(llyllk) terms of the potential function. It is seen that every homogeneous term of V(y) is proportional to the same power of the small parameter E. Note that, as E + 0, the nonlinear system (3.2.35) degenerates to a linearized system; this system provides the generating solutions for the asymptotic methodology developed in the following exposition. The nonsimilar NNMs of the canonical equations (3.2.35) are now investigated. Their trajectories in configuration space are parametrized as: yi = yi(y), i = 2 , . ~ yl(t) = y(t) where the (nonlinear) functions Ym(*) are computed by solving the following set of singular differential equations:


89

Complementing (3.2.38) is the following set of (n- 1 ) boundary orthogonality conditions at the maximum equipotential surface:

It will be assumed that in the limit E -+ 0 the nonsimilar NNMs under consideration degenerate to the (similar) linearized modes (3.2.36). The solutions of (3.2.38a,b) are denoted by $i(y), i = 2, ...,n, and are sought in the series form:

Before proceeding to construct the solution it is worth noting that the singular points of equations (3.2.38a) are roots of the equation: h - V(y) = 0

(3.2.40a)

As the order of approximation varies so does the estimate of the energy h, since the solution yi = $i(y), i = 2, ...,n, becomes more accurate through the inclusion of higher order terms. For example, if one considers terms of 0 ( & 2 ) in (3.1.38a), it is found that at this order of approximation the singular points are the roots of the equation:

The roots of this equation clearly differ from those of (3.2.40a). In order to avoid this contradiction, one allows the total energy level to change at different orders of approximation and computes the corresponding energies at the various orders of approximation. In particular, having estimated the functions ?i(y), i = 2, ...,n, up to O(&P),relation (3.2.40a) leads to

90


For a given amplitude of vibration y = Y, the above relationship provides an O(EP) estimate for the total energy, as follows:

where ho and hl(P) are energy approximations used in the following derivation. In what follows an asymptotic analysis for approximating the modal curves $(y) is carried out. O(E) Approximation

Substituting the series (3.2.39) into (3.2.38a), and considering terms of O ( E ~ ) one , obtains the following expression governing the first-order approximation:

where N(3) represents O(&3)terms in the potential energy (3.2.37), and aN(3)/ayi = aiY2, i = 2,...,n In (3.2.41), the energy is related to the amplitude of the pararnctrizirig coordinate by, ho = (1/2)02Y2. Employing the boundary conditions (3.2.38b), and retaining the lowest order terms in E one derives the ) conditions which complement the set following set of (n-1) O ( E ~boundary (3.2.4 1): aN(3,

- 02Y$i(I)'(Y) + wi2;j(l)(Y> + ~ ( Y ,,...,o0 ) = 0, ayi

i = 2 ,...,n (3.2.42)


91

The homogeneous equations (3.2.41) are hypergeometric equations with two regular singular points, and their solutions have been thoroughly studied in the literature. One way of solving these equations is by expressing their solutions in series expansions about the regular singular points. However, the corresponding analytical expressions are too mathematically involved to be of practical importance. Alternatively, the analytical solution of (3.2.41) can be represented in terms of Taylor series about the origin of the configuration space (i.e., away from the regular singular points), which can then be analytically continued up to the maximum equipotential surface by satisfying boundary conditions (3.2.42). This methodology enables the derivation of more compact asymptotic approximations for the solution. To this end the first order approximations $i(')(y) are expressed as: (3.2.43) where coefficients aip(l) are computed by substituting (3.2.43) into (3.2.41) and matching respective powers of y. The following binomial recurrent relationships for the coefficients then result:

(3.2.45)

92


Hence, it is proved that the Taylor series (3.2.43) converges in open intervals y E (+Y*,-Y*), which do not contain the regular singular points y = +Y, i.e., IYI > IY*l. The recursive formulas (3.2.44) provide nonunique solutions for the coefficients of the series (3.2.43). Uniqueness of the solution is obtained by imposing the (n-1) boundary conditions (3.2.42), i.e., by continuing the Taylor series expansions up to the maximum equipotential surface. Using (3.2.44), one expresses the arbitrary coefficient aip(l) in terms of coefficients aiO(l) and a , l ( l ) . These last coefficients are computed by employing relations (3.2.42). On substituting (3.2.43j into (3.2.42), one obtains the additional algebraic relations:

Introducing at this point the quantities (p2-Ai2) wi Ai = Kip = (p+2)(p+l) ' the solutions to the recursive relations (3.1.44) are expressed as:

w2

ai(2k+l)(l)= Ki(zk-l)K1(2k-3)...K,3KIla,l(')( --)k, 2hO

k = 1,2,...,

i = 2 ,..., n

(3.2.47) Expressions (3.2.47) relate an arbitrary coefficient a l p ( ] )to the leading coefficients aiO(l) and a i l ( l ) . These last coefficients are determined by substituting (3.2.47) into the boundary conditions (3.2.46), resulting in two sets of n nonhomogeneous algebraic equations of the following form:

where the computation of coefficients RiO and Ril requires some algebraic manipulations:


Ril = 1 + 3Kil

93

+ 5Kj1Ki3 +...- Ai’(l+Kil+KilKi3 +...) + C (2m+l-Ai’)KilKi3 ...Ki(2m-1) m

= (1-Ai’)

m= 1

m

= 6Kil

+ C [-(2m+1)2m + (2m+1)2 - Ai’]KilKi3...Ki(2m-]) m= 1

m

= 6Ki1

+ C [-(2m+1)2m + (2m+3)(2m+2)Ki(2m+1)]KjlKi3 ...Ki(2m-1) m= 1

= nm=om Ki(2rn+l)

(3.2.49) In order to obtain unique and nontrivial solutions for the coefficients aiO(1) and ail(l), it is necessary that the coefficients of the homogeneous parts of (3.2.48) satisfy the conditions Rjo f 0 and Rjl # 0, i = 2, ...,n. Examining the analytical expressions (3.2.49) it is concluded that in the critical case when RjO = Ril = 0, a subset of coefficients p2-Ai2 Kip = (p+2)(p+l) vanishes, or equivalently, that the linearized natural frequencies of the system satisfy resonance relations of the form Wi = pw for some positive integers p E [1,2, ...), and i E [ 1 , ...,n]. It is concluded that if the lineurized system has natural frequencies that are integrably related, conditions of internal resonance occur and the previous asymptotic analysis is not valid. These were precisely the cases eliminated from consideration in Lyapunov’s analysis (Lyapunov, 1907,1947); an extension of the present analysis for cases of internal resonance will be given in chapter 4. If no internal resonances occur, the algebraic set of 2(n-1) equations (3.2.48) can be solved in terms of the 2(n-1) coefficients aio(1) and ail(l), i =

94


2, ...,n, leading to an analytical approximation for the O(E) terms ;i(')(y), i = 2 ,...,n of the nonsimilar NNM (3.2.39). O ( E ~ Approximation )

The computation of higher order approximations is now considered. Suppose that the leading (k-l), k = 2,3,..., approximations have been computed in expression (3.2.39), and one is interested in approximating the ) in (3.2.38~1) k-th order terms ?i(k)(y), i = 2, ...,n. Considering O ( E ~terms one obtains the following set of (n-1) singular equations:

s = 1,2,3

(3.2.5 la)

and the summation corresponding to the second summation sign in the above expression is carried out over all positive integer solutions of the equation:

c (bw1+2bw2+...+rbwr) = r, n

w=2

and 6(Y)=

with

c bwm n

m= 1

= arw,

r! n m = l r (arw>!(m!Iarw

c arw n

w=2

=

Y

(3.2.52)

Functions Pir(s), s = 1,2,3, are computed by:

(3.2.53)


95

Equations (3.2.50) are complemented by the following set of O ( E ~ ) boundary orthogonality conditions:

i = 2 ,...,n

(3.2.54)

where Y is the amplitude of the parametrizing coordinate y. Equations (3.2.50) are now represented in the following simplified form:

where functions Fi(k)(y) consist of terms that are already computed functions of y. Similarly to the case of the O(E) approximation, the solution of set (3.2.55) is sought in the following series form, (3.2.56) which, upon substitution into (3.2.50) leads to a nonhomogeneous recurrent set of linear equations governing the coefficients aip(k). This set is similar in structure to the recurrent set (3.2.44), and by solving it, one expresses the general coefficient aip(k) in terms of the leading coefficients aiO(k) and ai 1(k) of (3.2.56). These last coefficients are then approximated by substituting (3.2.56) into the (n-I) boundary conditions (3.2.54), and solving a set of 2( n- 1) nonhomogeneous algebraic equations, similar i n structure to (3.2.48). It can be shown that when no internal resonances occur one obtains a set of unique values for the unknown coefficients. Moreover, the radius of convergence of the series (3.2.56) can be determined as in the case of the O(E) approximation. After approximating the modal functions $i(y), i = 2, ...,n, the problem of computing the nonsimilar NNM reduces to the integration by quadratures of a conservative system with one degree of freedom [cf. relations (3.2.13) and (3.2.14)]. Employing the computed approximations (3.2.56), and imposing the condition for maximum potential energy,

96

NNMs IN DISCRETE OSCILLATORS: QUAN‘IITATIVE RESULTS

(3.2.57) one obtains an O ( E ~estimate ) for the total energy, for a specified value for the amplitude Y. Expressing this energy estimate in a series of increasing orders of E , h(k) = E2hO(k) + ~ 3 ~h( k ) one computes the energy terms h,,(k), which appear in the calculations of the next order of approximation, ?i(k+’)(y), i = 2, ...,n. Needless to say, an alternative series of calculations would be also acceptable, namely, determining the amplitude Y given a fixed level of total energy h. At this point the convergence of the series approximation (3.2.39) is addressed. It was previously shown that a series of the form (3.2.39) represents a single-valued formal solution of the boundary problem (3.2.38a,b), provided that no internal resonances exist in the system. The initial conditions of the problem can be made arbitrarily small by selecting a sufficiently small value of the small scaling parameter E. Over a domain V < h all functions involved in (3.2.38a,b) are analytical in y. Tliereforc, it follows from Poincare’ ‘s theorem on small-parameter series expansions (Poincare’, 1899) that there exists a value EO > 0 such that, for all IEI c: E O , the series (3.2.39) converges in the domain V < h, and represents a unique solution of (3.2.38a,b). This solution is analytical in E and y and satisfies the initial conditions of the problem. Since the series (3.2.39) also satisfies the boundary conditions (3.2.38b), the solution can be analytically continued up to the boundary of the domain under consideration, V = h. Concluding this treatment of similar and nonsimilar NNMs of linearizable systems with no internal resonances, one notes the following. The trajectories yi = $i(y), i = 2,..,n can be derived not only in terms of power series of y, but also by the method of successive approximations when the generating systems possess similar NNMs (Manevitch et al., 1989). For similar NNMs the trajectories in the configuration space are independent of the level of total energy of the system (note that this is also a feature of linear normal modes). On the contrary, the trajectories qf nonsimilar NNMs depend on the specific value ofthe total energy h, i.e., one finds that $i = ?i(y;h); as a result, the nonsimilar trajectories change when the energy of oscillation changes [cf. also (Kauderer, 1958) and (Rosenberg


97

and Kuo, 1964)l. This property of nonsimilar NNMs introduces additional complications in their calculation. The requirement that there exists the energy integral (3.2.3) is not essential. Consider an autonomous system in noncanonical coordinates: xi

+ fi(z)= 0,

i = 1,2,.._, n

(3.2.58)

Considering a NNM of this system in configuration space, the trajectory is parametrized as: ?l(x) = x xp = cp(x), p = 2 ,...,n, and equations (3.2.58) assume the form:

If the system has an energy integral, by eliminating X2 from this integral, one obtains equation (3.2.10). If an analytical first integral H(x7X,x2,X2,...,xn,xn) exists, all that is needed is, using this integral, to express x2 as a singlevalued analytical function of X $ ~ ( X ) , ...,in(x). Substituting the derived expression for X2 into (3.2.59) leads to a set of (n-1) equations for the unknown functions kP(x), which can be solved using the methodologies described above. Nonsimilar NNMs of Nonlinearizable Systems Consider now the n-DOF conservative nonlinear system (3.2. I), and assume that the system is nonlinearizable in the limit of small oscillations. The equations of motion are expressed in the form:

where the potential energy is expressed as:

98


V(zr> = Vo(x) + EV I (XI and E is a small parameter. In contrast to the linearizable case, it is assumed that for E = 0 system (3.2.60) is nonlinear. However, in analogy to the linearizable case, the assumption is made that the generating nonlinear system possesses similar NNMs of the form x,,, = cmx, m = 2 ,...,n, xi = x The modal constants cm are obtained by the methodology presented in section 3.2.2. Consider one of these similar NNMs as the generating solution for determining the nonsimilar modes of (3.2.60). Rotating the coordinate axes, so that the new q-axis is directed along the trajectory of the generating similar mode and the remaining coordinate axes are orthogonal to it, the generating solution in the new coordinates is represented as: qm = 0, m = 2,...,n,

q1 = q = q(t) (generating similar NNM) (3.2.61)

Expressing the equations of motion in terms of the new coordinates, one obtains

where the quantities no(y) and n 1(y) are derived from the potential energy terms Vo(x) and V1 (x),respectively, by imposing the coordinate transformation x = (XI )...,X")T + (q1,... q,)T = CJ Since, by assumption system (3.2.62) possesses the generating similar NNM (3.2.61) at E = 0, it must be additionally satisfied that, dno(q,O,...,0)/dqi = 0, i = 2 ,...,n It is assumed that the unperturbed system corresponding to E = 0 is homogeneous, and that I7o(q)is an evenfunction cfO(lIqll'+1). Note that, in similarity to the linearizable case, the small parameter of the problem could also be chosen to scale the amplitudes of the positional variables during n nonsimilar NNM oscillation. That is, an alternative way to scale the equations of motion would be to introduce the rescalings: qi -+ Eqi, i = 2, ...,n, q + Eq In that case the (nonlinear) generating homogeneous system in (3.2.60) would be chosen as the one containing the smallest powers of the positional )


99

variables, and the nonlinear perturbing terms would contain higher powers of E. The trajectories of the nonsimilar NNMs of system (3.2.62) are determined by solving the following sets of singular functional equations and boundary orthogonality conditions:

where the mth dependent variable qm is considered to be a function of the parametrization variable q, i.e., qm = qm(q), m = 2, ...,n, Q is the amplitude attained by the parametrizing coordinate q, and

n(g)= no(g>+ En l(9) A solution qi = Gi(S), i = 2, ...,n of (3.2.63a,b) is sought in the small parameter series:

Although in this case the generating system is essentially nonlinear, all computations are similar to those performed for the linearizable case. As in the linearizable case, for a fixed amplitude of motion, Q, the total energy, h, of the system can be expressed as h = ho + &hi,where ho corresponds to the energy of the generating system. O ( E ~Approximation )

Suppose that one has computed the terms of the series (3.2.64) correct to O(&k-l), and is interested to compute the next order of approximation. Substituting (3.2.64) into (3.2.63a), and matching the coefficients of O(&, one obtains the following set of kth order singular equations:

s=

where the

1,2,3 (3.2.66a)

sign in the above expression is carried over all positive integer

solutions of the equation n- 1

(bw1+2bw2+...+rbwr) = r, with

w= 1

r

bwm = arw,

m= 1 ..I

n- I

a,

w= 1

= y,

(3.2.66b)

Functions Pir(')), s = 1,2,3, are computed by:

Equations (3.2.65) are complemented by the following set of boundary orthogonality conditions:


101

k- 1

where Q is the amplitude of the parametrizing coordinate q. This amplitude is related to the total energy by: h = n(q742,...,&d where the functions $i should be expressed by the series (3.2.64) [whose terms are assumed to be known up to O(&k-l)]. Since the unperturbed system is homogeneous, the matrix of second partial derivatives, B = [bikli,k,l,,,,n = [a2nO(4,0,...,0)/3qi3qk]i,k=l,.,n may be written as

= [Pikli,k=l,.,,n qr-l

Note that, owning to the conservative nature of systems (3.2.60) and (3.2.62), it is satisfied that Pik = Pki, and, hence, the symmetric matrix B can be reduced to a diagonal form by an invertible linear transformation of coordinates. Therefore, without loss of generality, one can assume that a2no(q,o,...,O)/aqidqk = 0, for i # k, i,k = 1,...,n It follows that the summations in (3.2.65) and (3.2.67) involving second partial derivatives of the potential function can be simplified as follows:

i = 2 ,...,n

(3.2.68)

A consequence of relation (3.2.68) is that the sets of equations (3.2.65) and A (3.2.67) become uncoupled in the unknowns qi(k),and as a result, each equation with its accompanying boundary condition can be solved independently from the others. This feature greatly simplifies the asymptotic analysis. Assuming the solution in the form:

(3.2.69)

102


substituting into (3.2.65), and matching coefficients of different powers of q, one obtains the following recursive relations for the coefficients aip(k):

where iJ(k) denote terms depending on the solutions of preceding approximations. The recurrent relationships (3.2.70) can be used to express all coefficients aiJ(k) in terms of the leading coefficients of the series (3.2.69), a&) and a,l(k). It can be shown that series (3.2.69) convergences in open domains n ( q , q 2 , ...,qn) < h. An analytical continuation of the solution up to the maximum equipotential surface n ( q , y 2 , ...,q,) = h is achieved by imposing the boundary conditions (3.2.67). Substituting the series (3.2.69) into these boundary conditions, and in view of the recurrent relationships (3.2.701, one obtains the equations governing aiO(k) and a, 1 (k). These equations have a form similar to (3.2.38). It can be shown that unique and nontrivial solutions for a,@) and all@)exist, provided that the following determinants are nonzero:

I denotes the determinant, and 6ij is Kronecher's delta. where detl Conditions (3.2.7 1) are formulated for a nondiagonal symmetric matrix: B = [b.Ik ]i,k=l,..,n = [a2nO(q,o,...,o)/a4iaqkli,k=,,,,,,

If a transformation of variables is imposed that renders B diagonal, the last term in the expression of element Zij in (3.2.71) should be replaced by 6ij [a2no(l ,0,...,0)/aqi2] When the generating system is linear (r = l), the solvability conditions (3.2.71) can be shown to degenerate to the conditions of absence of internal resonances. Hence, conditions (3.2.71) can be viewed as generalizations qf the conditions on absence of internal resonances derived in the linearizuble


103

A

case, and ensure that the analytical, asymptotic solutions qi = qi(q), i = 2,...,n are unique and single-valued. Finally, it is noted that a generalization of the asymptotic analysis presented in this section can be found in (Manevitch et al., 1989), where an iteration method was employed to compute NNMs of systems neighboring "generating" systems with nonhomogeneous potential functions no(@.In addition, in (Mikhlin et al, 1984) quasi-normal oscillations of nonlinearizable viscoelastic systems are analyzed.

Matched Asymptotic Expansions The previously derived analytical expressions can be used to compute the nonsimilar NNMs of linearizable or nonlinearizable discrete conservative oscillators. These results hold, generaly, for small levels of the total (conserved) energy of motion, h. To construct analytic approximations of NNMs valid over the entire energy range 0 < h < m, one needs to resort to matched asymptotic expansions. The resulting analytical expressions allow the computation of nonsimilar NNMs of systems oscillating at arbitrarily high levels of energy, and, thus, extend the previous "local" analyses, which are only valid in small neighborhoods of stable equilibrium points. The construction of matched asymptotic expansions for computing nonsimilar NNMs is developed by considering the following n-DOF conservative oscillator: Xi

+ aV(L)/aXi = 0,

i = 1,2,...,n

(3.2.72)

where the potential function V(x) is assumed be a positive definite polynomial of the n-vector of positional coordinates x, having terms of minimum order O(Ix12) and maximum order 0 ( 1 ~ 1 2 m ) ,m > 1. Introducing the rescaling of coordinates xi -+ Exi [note that Xi = O(l)], system (3.2.72) is rewritten as: Xi

where

+ aV(L,E)/axj

= 0,

i = 1,2,...,n

(3.2.73)

104


2m-2

and V ( k + 2 ) ( ~represents ) terms of the potential energy of O(lxlk+2). The amplitude parameter E in (3.2.73) can assume values in the entire range EE [O,+m). Note that other possible admissible definitions of the small parameter could be E = xl(0) or E = h (total energy of the motion). Indeed, as the energy of oscillation increases, so does parameter E. Since system (3.2.73) is conservative, the following energy relation is satisfied: n

2m-2

i= I

k=O

(1/2)&*c Xi2 +

&kV(k+*)(%) =h

*

h=

2m k=2

Ekhk

(3.2.74)

where hk is the O(&k)term of the total energy h. The trajectories of the nonsimilar NNMs of system (3.2.73) are computed by solving the set of singular differential equations (3.2.10) subject to the boundary orthogonality conditions (3.2.1 1). At the limit E+O (small amplitudes of motion) the system is close to a "generating" linear oscillator with potential function equal to Vgen(X>= V ( 2 ) ( ~ ) For large amplitudes of motion, E + W , system (3.2.73) is close to a generating homogeneous nonlinearizable system of degree 2m [the maximum order of the potential function V(X,E)]with potential function equal to vgen(x) = V ( 2 m ) ( ~ ) In both limiting cases, the "generating" systems possess similar normal modes of the form Xm = CmX,

m = 2 ,...,n,

XI

=x

(3.2.75)

where the modal constants cm # 0 are computed by solving the set of algebraic equations:

In writting (3.2.76) the homogeneity of the potential function of the generating systems, Vgen, was taken into account. Note that, as discussed in


105

earlier sections, the number of similar NNMs of the nonlinear generating system obtained in the limit E+- may exceed the number of the normal modes of the linear generating system corresponding to the limit E+O (in fact, the number of modes of the linear generating system is equal to n, the number of DOF of the system). It follows that certain brunches of N N M s qf’ (3.2.73) are eliminated as the amplitude of motion increases. At the limit E+O, the trajectories of the nonsimilar NNMs of (3.2.73) are analyticaly approximated by power series expansions in X I = x and E (cf. earlier analysis of this section):

(3.2.77) whereas, at the limit m

E+W,

by the following series expansions in x and &-I: m

m

(3.2.78) The computation of the coefficients of the above series was consider earlier. Note that the series (3.2.77) correspond to nonsimilar NNMs of a linearizable system, whereas (3.2.78) to NNMs of a nonlinearizable one. Also note that coefficients amk(X) and Pmk(x) can be computed in explicit form by quadratures, since the differential equations governing the trajectories of the nonsimilar modes under consideration can be converted to standard hypergeometric problems by suitable changes of coordinates (cf. chapter 4). For values of E between the two limiting values, 0 and +w, the trajectories of the corresponding nonsimilar NNMs are computed by contructing matched asymptotic expansions and joining the local expressions (3.2.77) and (3.2.78). Without loss of generality, the initial conditions for the parametrizing coordinates of the sought-after nonsimilar modes are chosen as: x(0) = 1 [from here on it is supposed that x(O)=E]

106


This initial set defines the NNM trajectory completely. The following matched asymptotic expansions will be derived at the time instant of maximum potential energy of the system, i.e., when x = 1 and X = 0. This, however, does not restrict the generality of the asymptotic analysis since a similar asymptotic analysis can be followed for different values of the parametrizing coordinate x. For simplicity the following notation is introduced at this point:

M

(3.2.79) The matched asymptotic expansions joining the local solutions (3.2.77) and (3.2.78) are performed using Pade' approximations (Baker and GravesMorris, 1981; Mikhlin, 1985, 1995). To this end, one considers the following fractional rational Pade' approximants:

c Ekamk 5

P p )=

k=O

c

s = 1,2,3,...;

,

m = 2 , 3,...,n (3.2.XOa)

s

p&n)

&k-'a,x, k

k=O = ____ k=O

~

,

s = 1,2,3,...;

m = 2,3,...,n (3.2.80b)

Ek+brnk

Employing the above expansions, quantities pm(0)(&)and (3.2.79) are analytically approximated as follows:

P ~ ~ ( ~ ) defined (E)

in


W

S

* [k=O C E k a m k ] [k=O C ckbmk]

c S

k=O

Ekarnk

107

(3.2.8 la)

Considering only terms of O(E~)in the above expressions, where -s I r I s, and matching coefficients of respective powers of E, one obtains (n - 1) sets of 2(s + 1) linear algebraic equations in terms of coefficients aInk and bmk, m = 2, ...,n, k = 0,1,2,... The solution of these sets of equations determines the coefficients of the Pade' approximations (3.2.80) in terms of the (known) coefficients of the local nonsimilar NNMs a m k and Pmk. Omitting cumbersome calculations, it can be shown that the determinants associated with the solution of the simultaneous sets of algebraic equations for a,k and bmk, are of the following form: -s+lI Dm(a)s+l

where

(3.2.82)

I-s+l, Dm(a)s+ll and Dm(pls+1are (s + 1) x (s + 1) matrices given by:

(3.2.83)

108

NNMs nV DISCRETE OSCILLATORS: QUANTITATIVE RESULTS

Since the determinants Ams are generally not identical to zero, the systems of algebraic equations possess the single trivial solution amk = bmk = 0, m = 2 ,...,n, k = 0,1,2,... Consider now a Pade' approximant satisfying the relations (3.2.8 I), with nonzero coefficients alk and b l k . Assume that bmo # 0, ni = 2, ...A since, otherwise, it would be satisfied that X,nl(l) --i, m as & -3 Without loss of generality it is also assumed that bmo = 1, m = 2, ...,n Then, the aforementioned systems of algebraic equations for computing q l l k and bmk become overdetermined; all unknown Pade' coefficients anlo, a m i , ..., ams, b m i , bm2 ,..., bms, m = 2 ,...,n, are determined from the first (2s + 1) equations, while the "error" of this approximation is estimated by substituting all coefficients in the remaining equation. Clearly, this crror is determined by the value of the determinant Ams, since for Ams = 0 nonzeio solutions (and, thus, exact Pade' approximations) are obtained for (3.2.8 I ) at the given order of approximation of E . It follows that a necessary condition for convergence of the successive Pade' approximants (3.2.80a) to fractional rational functions as se-, m

lims+- Ps(m) = P(m) =

k=O

Ekhk

, blnO = 1;

m = 2,3,...,n (3.2.84)

is that the following limiting conditions are satisfied:

lims+- Ams = 0,

m = 2,3 ,...,n

(3.2.85)

Indeed, if the limiting conditions (3.2.85) do not hold, one cannot find nonzero values for the coefficients amk and bmk in relation (3.2.84). Note that the limiting Pade' approximations P(m) are suitable for describing the NNM at any value of E, only if they do not contain any poles. In addition, note that although relations (3.2.85) are necessary conditions, they are not sufficient for the convergence of the Pade' approxiniants (3.2.80) to the


109

limiting functions (3.2.84). This will become clear in the following numerical application. At the limit as SJM, provided that (3.2.85) holds, the limiting functions P(m) provide analytic approximations to the nonsimilar NNMs at maximum potential energy, for arbitrary values of E E ( 0 , ~ ) .From the previous exposition it should be clear that at the limits as E+O and E+M, functions P(m) tend to the generating local solutions: Pm(O)(E) xm(O)(1,E) and Pm(-)(E) xm(-)( 1 , ~ ) It will be shown that the limiting relations (3.2.85) are essential for matching such pairs of local solutions [ ~ ~ ( O ) ( E ) , P ~ ( " ) ( E ) ] . Since the number of normal modes of the limiting generating systems corresponding to &-+O and &+is not equal, conditions (3.2.85) are necessary ,for determining belong to the same branch which pairs of local solutions [pm(o)(~),p,,CW)(&)] of N N M s . As shown in the following application certain local NNMs (3.2.81) do not possess analytic continuations for arbitrary values of E, since they are terminated at mode bifurcation points. The procedure for matching asymptotic solutions is more clearly demonstrated in the following example. Consider a two-DOF conservative oscillator with potential energy

The equation governing the trajectories, x2 = ?2(x), x i z x, of the nonsimilar NNMs of this system takes the form

where prime denotes differentiation with respect to x. Equation (3.2.86) is complemented by the following boundary orthogonality condition holding at points of maximum potential energy:

110


In writting the second of above relations, it was assumed that at maximum potential energy the parametrizing coordinate assumes the value x = f l . Introduce at this point the small parameter v , defined as v = ~2 for the small-amplitude NNM asymptotic expansion, and v = E - ~for the largeamplitude NNM asymptotic solution. Employing parameter v one expresses the solution of (3.2.86) in the following form: m

For small amplitudes, v = ~ 2E,-+ 0, the potential and total energies in (3.2.86) and (3.2.87) are expressed as: v = vo + VV] = V(2)(Xl,X2)+ VV(4)(Xl,X2) h = ho + vhl = V(2)[ 1$2( l)] + vV(4)[ l&( 1 )] Nonsimilar modes based on the low-amplitude linearized generating solution are denoted by fi2(o)(x), and the corresponding terms in series (3.2.88a) by 22k(o)(x). For large amplitudes, v = E - ~ E, + +M, the energy expressions are given by: v = vo + vv1 = V(4)(XI,X2)+ V V ( 2 ) ( X l , X 2 ) h = ho + vhl = V(4)[ l,;2(1)] + ~V(2)[1,;2(1)] Nonsimilar modes based on the large-amplitude essentially nonlinear generating solution are denoted by ft2(m)(x),and the corresponding terms in series (3.2.88b) by ;2k(")(x). Substituting relations (3.2.88) into (3.2.86) and (3.2.X7), and taking into account the aforementioned expressions for the energies, one obtains analytic approximations for the nonsimilar NNMs at various orders of v . I n the O(v0) approximation, one obtains two limiting generating systems: a lowamplitude linear one with potential energy V ( ~ ) (I X ,x2), and a largeamplitude homogeneous nonlinear system with potential energy V(4)(xI ,x2). Both limiting systems admit similar normal modes of the form: &(OP)(X)

= C(0P)X


11I

with modal constants determined by solving the following algebraic equations (cf. (3.2.86)): c ( O ) ~ V ( ~,c(O)]/axi )[~ + aV(2)[l,c(o)]/axz = 0 (limit of small amplitudes) (3.2.89a) ~(~)~V(~)[l,c("+ ) ]aV(4)[ / ~ x i l,c(")]/dx2 = 0 (limit of large amplitudes) (3.2.89b) The similar modes of the two limiting systems differ in number. To demonstrate this feature of the limiting systems, the coefficients of the potential energy are assigned the values, d l = d2 = 1 + y, d3 = -y y1 = 1, y2 = 0 , y 3 = 3, y4 = 0.2091, 75 = y The equations of motion of the system then assume the form:

+ X I + y(xyx2) + &2(x13+3x~x22+0.2091x2~) =0 X2 + x2 + y(x2-xl) + ~ 2 ( 2 ~ 2 3 + 3 ~ 1 2 ~ 2 + 0 . 6 2 7 3 ~ =10~ 2 (3.2.90) 2) XI

Note that y i s a linear coupling parameter. In the linearized limiting case (i.e., as E 0) the system possess two similar normal modes: 220(0)(x) = C(O)X, c(0) = &l In the nonlinear limiting case (i.e., as & + m ) one obtains a homogeneous system with cubic stiffness nonlinearities, possessing four similar NNMs: ?~o(")(x) = c(")x, c(") = 1.496, 0, -1.279, -5 Note that these limiting similar modes do not depend on the linear coupling parameter y. Proceeding to the next order of approximation, O(v) corrections to the trajectories of the nonsimilar NNMs are computed by solving the following O(v) singular differential equation:

112


The above equation is complemented by the following O(v j boundary orthogonality condition:

Similar equations governing higher order corrections to the nonsimilar trajectories are obtained by considering O(VP) terms, p 2 2, in equations (3.2.86) and (3.2.87). Asymptotic approximations are now derived for nonsimilar N N M s close to the limiting similar mode solutions. Using the notation p2(O)(v) = x@)( 1,v) and p2(")(v) x2(")( 1 ,v) setting y = 0.5, and employing the asymptotic analysis outlined in previous sections, one obtains analytical solutions for the O(v) solutions $2 1 ( O , M ) i n (3.2.91) and (3.2.92). Combining these solutions with the limiting similar modes 220(o~")(x) derived earlier, leads to the following branches of nonsimilar N N M s valid for small or large values of amplitude parameter V . NNMs close to low-amplitude linear limiting solution Mode I: p2(o)'(v) = 1 - 1 . 0 1 3 ~ 1 . 1 4 0 ~ 2-k O(v3) Mode 11: p2(0)[1(v) = -1 - 0 . 4 4 5 ~- 0 . 6 8 6 ~ 2+ O ( V ~ ) ,(v = ~ 2E ,+ 0) NNMs close to laree-amplitude nonlinear limiting solution Mode 111: pz(")"'(v) = 1.496 + 0.1X ~ + V 0 . 0 2 9 ~ 2+ O ( V ~ ) Mode IV: p2(")1v(v) = 0 + 0 . 3 3 3 ~+ 0 . 0 9 8 ~ 2+ O(v3) ( v-1.279 ) + 0 . 1 9 2 ~- 0 . 1 5 8 ~ 2+ O(v3) Mode V: ~ 2 ( ~ ) v = Mode VI: p2(-)vI(v) = -5 - 0 . 7 9 1 + ~ 0.696~2+ O(v3), (v = E -+ + m ) The above limiting branches of NNMs are now matched using Pade' approximations. The analysis indicates that only the pairs of nonsimilar modes (1,IV) and (I1,V) satisfy condition (3.2.86), i.e., only for these mode pairs an increase of the order of the Pade' approximarits, s, is aconipanied by a decrease of the determinant A ~ s Therefore, . each of rhe mode puirs (1,IV)and (I1,V)belong to the same brunch of izonsinzilar NNMs. Analytical expressions for NNMs valid over the entire range of amplitudes, 0 < E < 00, are constructed by computing the corresponding & C 2 ,


113

Pade' approximants. By matching the local NNM expansions (1,IV) and (ILV), one obtains the following Pade' approximations: Matching: the local expansions 1 and IV 1+1.06~2+0(~~) p2'-1V(v) E 1+2.06~2+3.20~4+0(~6) s p2I-'V(v) p2(O)I(v)as E +,0, and p21-'V(v)

-

- p2(")Iv(v)

Matching the local expansions I1 and V -1-2.76~2- 1 . 3 6 ~ 4 + 0 ( ~ 6 ) p2"-V(v) z 1+2.31~2+1.04~4+0(~6) + p2II-V(v) p2(O)II(v) as v + 0, and p2I'-v(v)

-

- p2(")V(v)

as E -+ +m (3.2.93)

as v + (3.2.94) 00

The two remaining local NNM solutions, 111 and VI, exist only for relatively large values of E and do not possess analytical continuations as E decreases. It will be shown that, at a critical value of E, NNMs 111 and VI coalesce in a Saddle-node bifurcation, after which no analytic continuations of these modes exist. To analyze this NNM bifurcation, one introduces the new variable: O(V)= [p(v)-1.496I/[p(~)+51 Setting p(v) = ~ ~ ( " ) I I I ( v )= 1.496 + 0 . 1 8 3 ~+ 0 . 0 2 9 ~ 2+ O(v3) or p(v) = p2(")vI(v) = -5 - 0 . 7 9 1 ~ + 0 . 6 9 6 ~ 2+ O(v3) one obtains the following expressions of v = v(0) (NNM HI), or v = v(0-I) (NNM VI). The resulting asymptotic expansion for mode 111 is valid for CT 0, whereas that of mode VI for (T + +m. Mode 111: v(")III(o) = 35.4970 - 164.18402 + 1882.64803 + O(04), o+O Mode Vl: v(")vI(o) = 8.2120 - 67.5560-'+ 982.913C2 + O(O-~>, 0 -+ +a Hence, by employing the new variable 0 , one obtains asymptotic approximations for the two bifurcating NNMs, each valid for sufficiently small or sufficiently large values of o. Fractional Pade' approximations can now be introduced to match the aforementioned asymptotic solutions, in order to obtain analytic expressions that are uniformly valid in the entire parameter range 0 < 0 < The Pade' approximations matching the NNMs 111 and IV are computed as follows: 00.

I14


Matching the local expansions 111and VI 35.4970+5.10802 vIII-VI(0) 1+3.0210-0.79402+0.62203 =$ vIII-VI(O) v ( - W ( 0 ) as o + 0, and vIII-VI(o)

-

- V ( ~ ) V I ( O ) as CT +

M

(3.2.95) To obtain an analytic approximation to the point of the Saddle-node bifurcation of NNMs I11 and VI, one imposes the condition on the Pade’ approximation (3.2.95), from which the bifurcation value Vbif 11.10 3 &bif = Vbif-1‘2 0.30 is obtained. As mentioned earlier, this bifurcation point corresponds to a value of the coupling parameter y = 0.5. It can be shown (Manevitch et al., 1989), that as the coupling parameter tends to zero, the bifurcation value

also tends to zero, indicating that in the weakly coupled system the t w o NNMs III and VI exist even at small amplitudes of vibrution. Note that thew NNMs exist only in the nonlinear system and cannot be detected b y a linearized analysis. In the limit y = 0 the system possesses four similar NNMs given by: 22(x) = cx, where c = 1.496, 0, -1.279, -5 These are the NNMs of the homogeneous nonlineur system (clearly, for y = 0 the limiting linear system possesses an infinite number of (degenerate) normal modes). In Figure 3.2.3(a) the fractional Pade’ approximatons (3.2.93)-(3.2.95) of the NNMs of system (3.2.90) are presented. In these plots, the N N M amplitude parameter 4 = tan-l(p2) [cf. previous definition of p2(v)] is depicted as a function of the scaled energy parameter 6 = ln(1 + &2h), for varying values of the linear coupling parameter y. The depicted graphs are periodic in 9 with period equal to 2x. In Figure 3.2.3(b) a comparison between analytical (solid lines) and numerical results (dashed lines) is presented for the case when y = 2. Good agreement is observed between the asymptotic and numerical solutions, even at large energies of oscillation. The results depicted in Figure 3.2.3 demonstrate that the construction of matched asymptotic expansions by means of Pade’ approximations is a powerful analytical technique for computing nonsimilar NNMs of systems oscillating at arbitrarily large energies. The described technique is of wider


1It

Figure 3.2.3 Nonsimilar NNMs at arbitrarily energies: (a) matched asymptotic expansions using Pade’ approximations, y = 0, 0.2, 0.5, 2.0 and (b) analytical (--) and numerical (-----) results for y = 2.0.

116


applicability compared to alternative local techniques which computc NNMs in small neighborhoods of a stable equilibrium of a conservative system. Additional analytical methods for analyzing nonsimilar NNMs of conservatives systems can be found in (Kauderer, 1958; Rosenberg and Kuo, 1964; Rosenberg, 1966; Rand, 197 lb; Manevitch and Mikhlin, 1972; Manevitch et al., 1989; Rand et al., 1992; Vakakis, 1992a; Vakakis and Cetinkaya, 1993; Vakakis et.al, 1993a). Slater (1993, 1994) studied applications of NNMs to control, and developed an efficient numerical methodology for computing NNMs. In the following section an additional application of the asymptotic methodologies described in this section is given, by studying the nonsimilar NNMs of a strongly nonlinear two-DOF oscillator with cubic stiffness nonlinearities.

3.2.4 Application to a Two-DOF Strongly Nonlinear System Consider the conservative oscillator depicted in Figure 3.2.4, consisting of two unit masses connected by means of three strongly nonlinear stiffnesses with cubic nonlinearity. The equations of motion are given by:

+ X I i-~ 1 x3 1+ kl(xl - ~ 2 +) ~ 2 ( x -l ~ 2 ) 3= 0 3 x 2 + (1 + a l ) x 2 + pl(1 + a3)x2 + kl(x2 - X I ) + p2(x2 - x1)3 = 0 Xl

(3.2.96) and the initial conditions are chosen as,

Xl(0) = XI, Xl(0) = 0, x2(0) = x2, i2(0) = 0 The scalars pi and p2 are the nonlinear stiffness terms of the grounding and coupling stiffnesses, respectively, and a1, a 3 are mistuning parameters. adjusting the symmetry of the system. When a1 and a2 are equal to zero, the oscillator is said to be t l l l l p c l (symmetric), and, as can be easily proved by direct computation, possesses similar NNMs given by x2 = cxl. Using the methodology described in section 3.2.2, the modal constant c is found to depend on the ratios of the linear and nonlinear coefficients of the coupling and grounding stiffnesses, k] and p2/p1. For nonzero coupling, kiF2 f 0, the only possible values for c are c = f l , and the tuned system possesses only two similar NNMs, namely, a symmetric and an antisymmetric one. It is interesting to note that in the limits of zero coupling


X I

117

x2

Figure 3.2.4 The two-DOF strongly nonlinear system. between the two oscillators, i.e., as kl --+ 0, ~2 0, there exist two additional degenerate similar NNMs corresponding to c = 0, and c = -00. It will be shown that, for weak coupling, localized nonsimilar N N M s exist in the neighborhoods of these degenerate similar modes. When ala2 # 0, the symmetry of the system is perturbed, and the symmetric and antisymmetric similar NNMs can no longer be realized. In the following computations it will be assumed that the coupling stiffnesses between the two oscillators are weak; however, a similar analysis can be carried out for the case of strong coupling. For weak coupling the following scalings are introduced, kl = E K1, p2 = EM^, where E is a small parameter, IEI 2, since the parameters aij, which contribute nonlinear terms in these equation,s are already computed by (3.4.22). As an example, considering the fourth-order terms of the potential function, one obtains the following relationships (Manevitch and Pinsky, 1972b):

4B1 1(3)a11a123- 4B22(3)a1 + 2B12(3)allaI2(a112 - a122) + B3i(3)a122(a,22- 3a1]2) + B32(3)[al12(3a122- al ] 2 ) - I ] = 0 (3.4.23) Viewing the variables Bij(3) as unknowns, for nontrivial solutions one imposes the condition that the determinant of the matrix of the coefficients of Bij(3) in (3.4.23) must vanish. This condition provides an additional relationship that the elements aij must satisfy in order for the system to possess a discrete group of transformations. As an example, consider a system of two nonlinear oscillators coupled by means of a linear stiffness. For this system the Lagrangian assumes the form:

138

NNMs IN DISCRETE OSCLLLATORS: QUANTITATIVE RESULTS

In this case one obtains the following relationships between the elements of matrix 5: mla122 + m2a222 = m2 mlaI12 + m2a2,2 = mi, m1alla12+ m2a21a22= 0

(3.4.25)

from which the following expressions result: a l l = -a22,

pal2 = aZ1,

where p = ml/m2

(3.4.26)

Taking into account (3.4.26), the invariance conditions for the quadratic terms of the potential energy assume the form: c11al12+c22p*a122+c12(a11- p a I 2 P = c i i +c12 c l ia122 + c229 12 + c12(a12 - a l l P = c22 + c12 clialla12+c22pa11a12 +c12(a11- pal21 ("11 +a12)=-c12

(3.4.27) For a specific transformation, these equations yield the following relationships:

(CII

+ c12Y(c22 + c12) = p,

where a l = 0

(3.4.28~~)

From the invariance conditions of the nonlinear terms in (3.4.24) one obtains the additional relations:

Clearly, conditions (3.4.28a,b) do not exhaust all cases of complementary discrete groups of transformations. However, when the above relations between the parameters are satisfied, the system under consideration permits a group of transformations, and the NNMs of the system coincide with the invariant manifolds of this group, given by x l = ~ 1 1 2 x 2 and xi = -p112x2

3.4 ANALYSIS OF NNMs USING GROUP THEORY

139

In addition, the previously described technique for searching for complementary discrete groups of transformations can be conveniently extended to dissipative systems and to systems with gyroscopic forces. In such cases, however, the analysis results in cumbersome sets of nonlinear algebraic equations, and the task of finding limiting systems with similar NNMs becomes demanding even for oscillators with two-DOF. To find classes of nonlinear oscillators that are more general than conservative systems and possess similar NNMs, an approach involving the solution of an inverse problem can also be employed. To this end, the straight-line trajectory corresponding to a similar NNM in configuration space can be regarded an being the invariant manifold of a continuous group, while the system possessing this mode will be regarded as the differential invariant of that group. For a two-DOF system, consider the one-dimensional manifold x2 = cxl as the invariant of a continuous group defined by the infinitesimal operator: (3.4.29) where it is assumed that E(t) = at + p, r l l = y(t)x1, 7 2 = y(t)x2 and y(t) is an arbitrary function of time. The group under investigation is more general than the group of dilatations. The corresponding twicedifferentiable operator, U", is in the form (3.4.4), where the various coefficients are given by:

where, pi = Xi, ri following form:

=

Xi,

i = 1,2. Hence, operator U" is expressed in the

140


+ [y"xi + 2y'pl + ri(y-2a)I

a

a

+ [y"x2 + W p 2 + r 2 ( ~ - 2 a ) l ~2

(3.4.3 I ) One now seeks a set of equations of motion with positional velocities and accelerations remaining invariant with respect to the group U". Denote the sought-after set of equations of motion by:

Fi ( X I , X ~ , X I , X ~ , X ~=, 0, X ~ , ~ )i = 1,2

(3.4.32)

The invariance conditions to be satisfied by these equations are then given by (Peter, 1986; Bluman and Kumei, 1989): i = 1,2

(3.4.33)

In order to compute the invariants of zeroth, first, and second orders, the following set of differential equations must be solved:

The first two differential equations yield the following invariants of the zeroth order: C I = xiexp[-j

( y k ) dt],

C2 = x2exp[-j

(Y/E)

dt]

(3.4.35)

where C1 and C2 are arbitrary constants, and E(t) = at + p. Solving the next two differential equations one derives invariants of the first order of the form: C3 = (pie: - xly) exp(-Il),

C4 = (p2c: - x2y)exp(-li)

(3.4.36)


141

where C3 and C4 are arbitrary constants, and the exponent is given by:

Finally, solving the two remaining equations in (3.4.35), one finds the invariants of the second order as follows:

and the constants of integration in the evaluations of the integrals were set equal to zero. In general, the set of equations of motion (3.4.32) which remains invariant with respect to the group specified by operator (3.4.29), can be expressed as:

where the constants in the above expression are evaluated by (3.4.35) (3.4.37). Introducing the new set of coordinates (group coordinates), dz = d t k , zl = xiexp(-Ii), and z2 = x2exp(-I2) the equations of motion can be expressed in the following simplified form: Fi ( ~ l , z 2 , ~ 1 ' , ~ 2 ' , ~ 1 '=' ,0, ~ 2 ' ' ) i = 1,2

(3.4.39)

where prime denotes differentiation with respect to the new variable z.To proceed with more concrete results, one must consider specific classes of nonlinear oscillators,

Conservative Systems The equations of motion of a two-DOF conservative system are expressed in the form:

142


(3.4.40) where the potential function depends only on the positional coordinates. It turns out that the general equations (3.4.32) can be put into the form (3.4.39) only when variable y of the previous analysis is a constant quantity. In this case the equations of motion (3.4.41) assume the form:

(3.4.41)

In the above expressions, h(6) is an arbitrary function of 6 # 1. Hence, as before one obtains limiting homogeneous systems admitting similar NNMs, with the degree of homogeneity, however, not restricted necessarily to integer values (i.e., exponent 6 may assume real, noninteger values).

Dissipative Systems Consider a two-DOF nonlinear oscillator of the form: (3.4.42) The inverse analysis indicates that this system possesses NNMs provided that the potential energy is a homogeneous function of the positional variables, and the dissipative terms may be chosen in the form, R(xi,t) = Gixi(at + b)-' Note that time-dependent dissipative terms are encountered in may physical applications (Nashif et al., 1985; Zhang, 1992).

Gyroscopic Systems Two-DOF systems of the following general Lagrangian form,


143

(3.4.43) with L = al(xllx2)x12 + p(x17x2)xlx2+ a2(xIJ2)x22

+ a1(x 1,x2)x1 + a2(x 1 ~ 2 1 x 2+ V(x 1 ~ 2 ) can also be considered for computing limiting nonlinear systems possessing similar NNMs. Restricting the analysis to the case when V(xl,x2) is a homogeneous function of its variables, and setting a1 = a 2 = constant, p = 0 a1 = a2 = C (1+6)-1~,6+1~2-6-aly, 6 z-1 6 the corresponding class of limiting systems possessing similar NNMs is of the following form:

Note, that the described inverse technique can be readily extended to systems of more than two DOF. Moreover, the method can be applied for finding classes of limiting systems possessing nonsimilar NNMs. To show this, consider a two-DOF system and seek nonsimilar NNM solutions of the form, x2 = C ( x l + px13) where C is an arbitrary constant. It follows that during such motions, the differentials of the positional coordinates are related by the expression: (3.4.45)

The infinitesimal operator of the group for which a cubic parabola is invariant, is written in the form,

144


(3.4.46) and the equation defining the invariants is expressed as, dxl dx2 ~~~

Considering (3.4.43, and setting

5 = v ~~

5 = X l + pX13,

'Il=

x2(1

+ 3pX12)

one writes the twice-continued operator in the form:

a axl

U" = 5 -+

a a a ax2 + 6 ap + ag

q~

-

(3 -

(3.4.47)

where

p = dx2Idx 1, q = d2xzIdx 12, 6 = 6px 1 x2 (r = 6Dx2 + 6 0 x 1 ~ q(l + 3pxi2) The differential invariants of the first and second order, Ci and C2, are found by solving the set of equations, (3.4.48) and are computed as

(3.4.49) Hence, the general form of the second-order equation, which is invariant with respect to the group specified by operator (3.4.461, and permits nonsimilar NNMs with trajectories in configuration space i n the form of the cubic parabola, is given by:

3.5 VIBRO-IMPACT SYSTEMS

145

(3.4.50) where F(*) is an arbitrary function. Equation (3.4.50) governs the trajectory of the NNM in configuration space and is derived by eliminating the time variable from the equations of motion.

3.5 VIBRO-IMPACT SYSTEMS In previous sections quantitative techniques for constructing the similar and nonsimilar NNMs of discrete nonlinear oscillators were presented. In this section a special class of nonlinear systems with nonsmooth nonlinearities is considered, namely vibro-impact oscillators. The study of nonlinearities due to vibro-impacts is of significant practical importance since they are often encountered in engineering practice. For example, elastic structures with loosely connected components possess clearances in their joints, which under vibration lead to vibro-impacts between structural members and, thus, to strongly nonlinear dynamical response. The analysis of the dynamics of such strongly nonlinear (and nonlinearizable) problems requires the development of special analytical techniques suitable for handling strong nonlinearities. Analytical and numerical studies of vibro-impact oscillations were carried out by Masri and Caughey (1966), by matching linear solutions computed before and after the time instants of impacts. The same authors investigated the implementation of vibro-impact dampers as vibration isolators. Studies of piecewise linear and vibro-impact oscillations with analytical/numerical Poincare' maps and geometrical techniques were performed by Shaw and Holmes (1982), Moon and Shaw (1983), Shaw (1985), Shaw and Rand (1 989), and Shaw and Shaw (1989). The Poincare' maps constructed i n these works were discontinuous and sampled the dynamics at the time instants of impact; moreover, the applications considered were basically single-DOF oscillators. Ivanov (1 993) studied vibro-impact oscillations by introducing auxiliary phase planes. A strongly nonlinear analytical method for analyzing vibro-impact oscillators was developed by Zhuravlev (1 976,1977), who introduced nonsmooth spatial transformations of variables to eliminate the discontinuities in the equations of motion of the vibro-impact system. In this section the nonsmooth spatial transformations developed by Zhuravlev

146

NNMs LN DISCRETE OSCILLATORS:QUANTITATIVE RESULTS

(1976,1977) are used to study localized and nonlocalized NNMs in multiDOF vibro-impact oscillators. The corresponding analytical solutions will be expressed in closed form by employing an essentially nonlinear new technique, termed the method of nonsmooth temporal transformutions (NSTT). In chapter 6 a detailed formulation of the NSTT method will be presented, and additional applications of the method will be given. The following exposition follows closely the work of Vedenova et al. (1985). Consider the transverse vibrations of a symmetric chain of N particles coupled to each other by a massless elastic string, and connected to the ground by strongly nonlinear elastic supports [cf. Figure 3.5.l(a)]. The restoring forces exerted by the supports are assumed in the form: fi = c(xi/e)2n-l where xi denotes the transverse displacement of the string at the position of particle i, e is a reference displacement, c a stiffness constant, and n a positive integer [Figure 3.5.l(b)]. Note that, at the limit n + the elastic supports become rigid boundaries with gaps equal to 2e, and, for sufficiently large amplitudes, the particles of the chain undergo vibro-impact oscillations. Depending on the strength of the nonlinear elastic supports one distinguishes between two cases. When 1 < n < 00 and the coefficient of the nonlinearity c is small, the system under consideration is weakly nonlinear and its NNMs can be analytically computed employing the perturbation techniques developed in previous sections. The weakly nonlinear system will not be further analyzed. In the second case, one assumes that the supporting stifnessess exert strongly nonlinear forces on the particles, of greater magnitude than the linear coupling forces generated by the connecting string. The dynamics of the strongly nonlinear system cannot be analyzed by standard perturbation techniques, especially in cases when the exponent of the nonlinearity is much greater than unity, 211-1 >> 1; hence, a new technique must be followed, capable of accounting for the vibro-impacts of the problem under consideration. Consider a vibro-impact system (in the limit n -+ m), possessing N = 2 particles. The equations of motion governing the transverse oscillations of this system are expressed as: M


147

Figure 3.5.1 The strongly nonlinear chain of particles: (a) configuration of the system, and (b) nonlinear restoring forces exerted by the elastic supports. where y = S/1, S is the tension of the connecting string, 1 is the distance between particles (cf. Figure 3.5. l), and q(x1,2,x1,2)denote the vibro-impact forces exerted by the rigid supports on the particles of the system. Clearly, the temporal derivatives of function q(*,*) possess singularities (discontinuities) at time instants of impact, i.e., when x1 = +e, or x2 = ke. When y = 0, and for perfectly elastic vibro-impacts, the two particles undergo independent oscillations, given by:

where z(Qi) is the sawtooth sine of period 2n and amplitude 1 , $i is the phase and Vi the (constant) velocity of the ith particle, i = 1,2. Clearly the velocities vi are prescribed by the initial conditions of the problem. More details concerning the sawtooth sine and its derivatives can be found in chapter 6. When the coupling between particles is nonzero, y # 0, one introduces nonsmooth changes of the dependent coordinates by employing Zhuravlev's

148


transformation (Zhuravlev, 1976, 1977, 1978), x i = e ~ [ $ l ( t ) ]and x2 = ez[$2(t)]. Substituting these expressions into (3.5. l), and taking into account properties of the derivatives of the sawtooth sines, one obtains the following transformed set of equations of motion:

It is interesting to note that the transformed set of equations (3.5.3) does not contain functions with singularities in contrast to the original set (3.5.1). In addition, equations (3.5.3) may be viewed as describing the motion in ($1,$2) configuration plane of a fictitious unit mass lying on a periodic restoring force potential. When y = 0 (no coupling) one obtains inertial motion of this mass. A set of equipotential lines and four trajectories of the fictitious mass in the configuration plane ($1 4 2 ) for y f 0 appear in Figure 3.5.2. The physical motions of the particles of the system in the (xl,x2) plane can be deduced froin Figure 3.5.2, by employing the previously introduced Zhuravlev's transformation: X I,2(t) = eT($1,2) = (2e/n)arcsin[sin($1,2)] For small energies of oscillation, no impacts between the particles and the rigid supports exist, and the trajectory of the system lies inside the square: K = { ( $ 1 4 2 ) E R2, -n/2 5 $i I n/2, i = 1,2} Note that the pattern of the equipotential lines inside K (which correspond to equipotential lines of a linear system) possesses an axis of rotational symmetry of second order. Hence, for low-energy motions restricted inside K, the system is linear (no vibro-impacts exist), and possesses precisely two normal modes lying along the two diagonal symmetry directions of the square: an in-phase mode, $1 = $2, and an antiphase mode, $ 1 = -92. As the energy of vibration increases, the system undergoes vibro-impacts and the configuration plane of the system extends beyond the boundaries of square K. The new equipotential lines of the system are obtained by multiple reflections of the equipotential lines inside K, and generate the pattern depicted in Figure 3.5.2. Each vibro-impact generates an additional reflection of the equipotential lines. Interestingly enough, the extended configuration plane of the vibro-impuct oscillator possesses u fourth-order axis o j rotutionul symmetry, and, as a result, the vibro-impact systern


149

Figure 3.5.2 Equipotential lines and NNMs in the configuration plane of the transformed equations of motion (3.5.3). possesses four NNA4.s. Two of these modes are similar and are the nonlinear extensions of the in-phase and antiphase modes of the low-energy linear system with trajectories inside K. The additional two NNMs are nonsimilar, and spatially localized to one of the two particles of the system (cf. Figure 3.5.2). At the limit of no coupling, y -+0, the localized lionsimilar modes degenerate to the straight lines $1 = 0 and $2 = 0, and correspond to one of the particles being at rest. To study if the previously detected NNMs are physically realizable, a stability analysis must be undertaken. Due to the complicated nature of the dynamics one introduces the perturbation parameter ~2 = 4e2y/(zv)2, ~2 ,&(0>1= + El

[m

where ~ ( 0=) [&(O),i(O)] and g = ( E ~ , . . . , E ~ ~The ) T . solution x(t) is said to be l distance between orbitally stable if, for sufficiently small values of l l ~ lthe Co and the trajectory of zc(t)remains small for t > 0. In particular, denoting by a -( t ) and z(t) the (2n x 1) vectors of phase variables evaluated on the perturbed and periodic solutions, respectively, the periodic solution is orbitally srable if, for any 6 > 0, there exists an E * = ~ ~ ( such 8 ) that l E*. If in addition it is satisfied that dist[%(t),Co] < 6 whenever l l ~ l < limt+, dist[zg(t),Co] = 0 then the periodic orbit is said to be asymptotically stable. Necessary conditions for orbital stability can be derived by applying a linearized stability analysis, and solving the following set of linear variational equations with periodically varying coefficients:

Mu(t)

+ A(t)g(t) = 0

(4.1.2)

where 11 is an n-vector, A(t) = [aik], aik = (d2v/dXidXk),,x(t), _ i,k = 1,..., n and A(t) = A(t + T) 'd te R. An analytical methodology for computing the stable, unstable, and periodic solutions of (4.1.2) will be developed in a following section. Here, only some qualitative results concerning the solutions of (4.1.2) are reviewed. Introducing the (2n x 1) state vector defined by y = (u.MU)T, the variational equation is reduced into the following state form: h ( t ) = H(t)y(t),

J=

[r0_ -I-61,

$11

H(t) = ['$ I - -

(4.1.3)

where 1 and 0 denote the (nxn) unit and zero matrices, respectively. Assuming initial conditions p(0) = yo, the solution of (4.1.3) is expressed as follows:

160

STABILITY AND BIFURCATIONS OF NNMs

The time-dependent matrix K ( t ) is called the matrizant, and the constant matrix W(T) is called the monodromy matrix of (4.1.3). A well-established result from Floquet theory is that the eigenvalues AP, p = 1,...,2n, of the monodromy matrix (Floquet multipliers) determine the stability of the zero solution of (4.1.3) or, equivalently, of the periodic orbit x(t) of (4.1.1). If the modulus of any multiplier exceeds unity, the periodic solution x(t) is unstable. Considering the structure of system (4.1.3) it can be easily shown that all Floquet multipliers exist in reciprocal pairs. Hence, if there exists a multiplier satisfying Ih,l < 1, then there also exists an additional multiplier with modulus I1 /hPl> 1. The necessary condition for the stability of solution x(t) is that all Floquet multipliers lie on the unit circle. Note that this condition may provide merely neutral stability but not asymptotic stability of the solution. For more rigorous stability results one must resort to KAM theory (Guckenheimer and Holrnes, 1984; Wiggins, 1988,1990) [cf. (Month, 1979) and (Vakakis and Rand, 1992)l. In the following analysis, a NNM will be denoted as stable if all the corresponding Floquet multipliers lie on the unit circle; such a stable mode will be said to be stable to L[ first approximation. Note that a nonlinear periodic solution x(t) usually belongs to a one-parameter family x(t;h) with period T(h); as a rule dT(h)/dh f 0. The last inequality prevents closeness of periodic solutions initiated at neighboring values of h, and, hence, the orbitally stable solution is not stable in Lyapunov's sense. In the following analysis some properties of the Floquet multipliers established by Krein (1955) are utilized [cf. also (Yakubovich and Starzhinskii, 1975)l. The Floquet multipliers of (4.1.3) will be separated into multipliers cffirst und second kind, denoted by h k ( l ) and Ak(2) = l/kk(I), k = 1 ,...,n, respectively. Suppose that the periodic matrix H in (4.1.3) depends on the small parameter E , i.e., H = H(t,&),and that the quadratic form (H(t,E)y,y) increases with E for any vector y, i.e., [ d H ( t , ~ ) / d > ~ ]0. Then, it can be proven that if multipliers h k ( l ) ( & )and hk(Z)(&)lie on the unit circle, they transverse along it in counterclockwise and clockwise directions, respectively. A multiplier of the first kind, h k ( l ) , may leave the unit circle at E = E" Only if hk(l)(E*) = A,@)(&*), where is some multiplier of the second kind. Hence, if for E = 0 all multipliers lie on the unit circle and for increasing E in the range 0 I E 5 1 no multipliers of different kinds

4.I GENERAL STABILITY RESULTS

161

meet, then it must be satisfied that Ihk(l)(&)I= lhk(2)(E)l = 1, for E E [0,1], and k = 1,...,n. It is also noted that since system (4.1.1) is autonomous,

vector i ( t ) is a T-periodic solution of (4.1.3). It follows that there exist at least two multipliers satisfying h p ( l )= hp(2)= 1, and hence rhe periodic solution of (4.1.1) can be at most orbitally stable (but not asymptotically stable). The aim of the following analysis is to obtain sufficient conditions satisfied by the potential function V(x) of system (4.1. l), which guarantee stability of the corresponding NNMs. These conditions will not require any actual computations of the NNMs of the system. Denote at this point an NNM of (4.1.1) by x(t) E R, where R is a region of the configuration space (cf. chapter 2) and x(t) = x(t+T), V t E R. In chapter 2 existence theorems regarding NNMs of conservative systems were established, and certain symmetry properties of the mode shapes of the NNMs were proven. In the first part of the following exposition no symmetry restrictions on the potential function V(x) and on the NNM solution x(t) are imposed and general stability results are derived. In the second part of the analysis it is shown how any additional symmetry properties of NNMs enable a relaxation of the derived general stability conditions. Suppose that for all x E R the Hessian matrix of V(x), A(x) = Vxx - (x), satisfies the following inequalities:

< A+

A- I A(&)

(4. .5)

where A- and A+ are symmetric positive-definite constant matrices In applications the potential function is often of the following form: n

n

where yo is a constant symmetric matrix, and (*,*) denotes the internal product. Assuming that the system possesses nonlinear elastic constraints, Vp(wp) denotes the potential energy of the pth nonlinear constraint and wp its deformation. For such a system it is satisfied that

162


(4.1.7)

where y E Rn is an arbitrary vector. Denote by ap- and ap+ the minimum and maximum values of ap(wp) for E R. Making the substitutions ap(wp) -+ ap- and ap(wp) + ap+, p = 1,...,n, in (4.1.7) one obtains expressions for (A-y,y) and (A+y,y), respectively, from which the matrices A- and A+ are evaluated. Denote by wk-2 and wk+2, k = 1,...,n, wkf I wk+li, the eigenvalues of matrices M-1A- and M-lA+, respectively. This notation will be employed throughout the following analysis. Note, that by Rayleigh's theorem one has that 6&+ 2 wk-, k = 1,...,n. Suppose that the frequency of the NNM, o = 2 d T , is contained in the interval w E [ ~ j - , ~ j +for ] some j E [l,n] (it will be shown that w belongs necessarily to at least one such frequency interval). Then the following theorem is proven. Theorem 1 . If qw @ [Op-+Ok-,Op++-Ok+], p, k = 1,...,n, k f j , q = 1,2,..., then the NNM 21 = x(t) of (4.1.1) is orbitally stable to a first approximation. Proof. Consider the linear system:

Clearly, the corresponding natural frequencies wp(p), p = 1 ,...,n, increase from wp- to wp+ for p increasing in the interval [0,1]. Assuming that &(p) is a T-periodic matrix, one finds that the multipliers of system (4.1.8) are given by r p ( l ) ( p ) = expljwp(p)T] and rp(*)(p)= exp[-jwp(p)T]. As p increases, multipliers rp(l)(p)and rp(2)(p) move counterclockwise and clockwise, respectively, along the unit circle. For p E [0,1] multipliers = [rp(l)(0),rp(l)(l)]and rp(2) = rp(l)(p) and rp(2)(k) lie on the arcs rp(l) lrp(2)(0),rp(2)(I)], respectively. Since 9 0 6Z [Wp- + Ok-,Op+ + Ok+], p, k = 1,...,n, k # j, q = 1,2,... the arcs rP(l)and Ti&*),p,k = 1,...,n, have no common points, with the and rj(2). Consider now the variational system exception of r&l)

4.1 GENERAL STABILITY RESULTS

Mu(t)

+ A(t;p)u(t) = 0

163

(4.1.9)

with A(t,p) = A- + p [A(t) - A-1. If the variational equation corresponding to p = 1 is unstable, then for some p = p* E (0,l) at least one pail- of multipliers hk(l)(p*) and hp(2)(p*)meet at a point A* of the unit circle, which does not belong to any arcs r,(l)and T@), p,k = 1,...,n, p,k # j. The following self-conjugate boundary value problem is now examined:

where R(t) is a symmetric matrix with R(t) > A-. Since hk(l)(p*) = h*, then for R(t) = A(t) problem (4.1.10) possesses the eigenvalue ps(A) = p*. Because the positive eigenvalues decrease as R(t) increases, for R(t) = A+ there exists an eigenvalue satisfying ps(A+) < F* < 1. Since the quantities eXp('jC0k-T) and exp(-jok-T) belong to the arcs r k ( 1 ) and Tk(2), it follows that p = 0 is not an eigenvalue of (4.1.10). Hence, ps(A+) E (0,l) and, as a result, one of the corresponding multipliers rp(l)(ps)or rp(2)(ps)is equal to h*. However, this result contradicts with the fact that rp(l)(p) E Tp(l) and rp(2)(p)E rp(2). This contradiction shows that all multipliers h, lie on the unit circle, and the theorem is proved. As shown by the proof of the theorem, all multipliers of system (4.1.2) are definite (i.e., only multipliers of the same kind may coincide), with the exception of the indefinite multiplier h = 1, which is of multiplicity two. It is, thus, proven that under the conditions of the theorem the zero solution of (4.1.2) is neutrally stable or that the NNM x(t) is orbitally stable. Suppose now that the potential function V(x) and the corresponding NNM x(t) are symmetric, i.e., that V(x) = V(-x) and x(t) = -x(t + T/2). It follows that V,,(x) = Vxx(-&) and A(t) = A(t + T/2), and the minimal period of A(t) is equal to (T/2). In that case the condition of Theorem 1 can be relaxed and replaced by the condition 2qw P [wp-+ok-,op++ok+l, p, k = I , ...,n, k # j, q = 1,2,... This condition guarantees that all multipliers h k ( T / 2 ) corresponding to the matrix W(T/2) lie on the unit circle. Since in this case the monodromy matrix is given by K ( T ) = [W(T/2)]2, the Floquet multipliers are computed by hk = hkz(T/2) and all lie on the unit circle.

164


Note that the larger the number j in the condition o E [ ~ j - , ~ j + ] where , w is the frequency of the NNM, the larger is the number of conditions of Theorem 1 that are deliberately satisfied. So, in general, the likelihood uf stubility of small-period NNMs is higher than that of large-period ones. Next suppose that the condition of Theorem 1 is satisfied for all o E [ ~ j - , ~ j + ] , i.e., that

1

, p , k = 1,...,n,

Oj-

k#j,

q = 1,2,... (4.1.11)

Since A- I A(Q)I A+, the natural frequencies of the linearized system &(t)

+ A(Q)u(t) = Q

are in the interval COko E [ O k - , w k + ] , k = 1, ...,n. Taking into account (4.1.1 1) one finds that (wkO/Ojo) it q for k # j. Hence, in a neighborhood of the equilibrium point there exists the local Lyapunov family of NNMs yi(t,h) satisfying the limiting conditions 0, Tj(h) -+ 27~/0jO as h + 0 zj(t,h) As shown in chapter 2 modes xj(t,h) are uniquely continuable in h provided that the multiplicity of the multiplier h = I of the corresponding variational problem is equal to 2. As seen from the proof of Theorem 1, this is exactly the case when condition (4.1.11) is satisfied and Oj- 5 o(h) 5 Wj+. This last inequality is not violated for increasing h [this is because, assuming that w = wj- - E or o = Oj+ + E, one finds that for small E the arcs Ti(')and T,(2) do not contain the point h=1, i.e., that equation (4.1.2) has no multiplier equal to unity, and, thus, no periodic solutions; this result, however, contradicts the fact that vector i(t,h) is a periodic solution of (4.1.2)]. Hence, under condition (4.1.1 1) the family of NNMs xj(t,h) is uniquely continuable up to the boundary of the region R,and orbitally stable to a first approximation. Employing similar arguments one can show that the condition

Oj-

1

,

p , k = l , ...,n,

k#j,

q = 1 , 2,... (4.1.12)

4.1 GENERAL STABILITY RESULTS

165

guarantees the unique continuation and stability of the symmetric solution xj(t,h) in Q. Suppose now that the system has concave or convex nonlinearities (cf. section 2.4). Since in that case A(x) < A@) or A(&)> A(@, one can use the inequality condition of Theorem 1 with o k + = o k 0 or wk- = o k o , respectively. As shown in section 2.4, under condition (2.4.30) (convex nonlinearity) or (2.4.31) (concave nonlinearity) the NNMs xj(t,T) = xj(-t,T) = -&j(t + T/2,T) are uniquely continuable in T from TjO = 2l~i0jOto T = 2x/o. The corresponding variational equation has the unique solution y(t) =

x(t,T) satisfying the condition y(t) = -y(t+T/2). Hence, the multiplicity of the characteristic multiplier h(T/2) = -1 of the matrix W(T/2) is equal to 2, and the remaining (2n - 2) multipliers are not equal to (-1). It follows that the parts of the inequalities involved in the above stability condition which exclude the possibility that the arcs r j ( S ) and rk(S) (s = 1,2; k=l, ...,n, k#j) have the common point h = -1, are no longer necessary. So, for a system with concave or convex nonlinearity the stability condition becomes 2q0 P [Op- + Ok-, Op+ + Wk+], p,k = 1,...,n, p,k # j , q = 1,2,... (i.e., unlike the general case, both indices k and p need not take the value j). Hence, convexity and concavity of the nonlinearity diminishes the number of inequalities involved in the stability condition. Suppose now that j = n. Taking into account that On(h) 2 On-, from (4.1.12) it follows that in the general case the family of NNMs xn(t,h>E52 is stable if

This inequality excludes the coincidence of the multipliers hn(T/2) = -1 and hn-l(T/2). If the nonlinearity is convex or concave, such coincidence is excluded by the prolongability conditions On-1' < O n o , 30nO > O n f or O n - > ~ ~ - 1 respectively. 0 , Hence, in a system with concave or convex nonlinearity the stability conditions for the fumily of NNMs x,(t,h) coincide with its existence conditions. Until this point only the symmetry properties of the periodic solutions x(t) were utilized. Such solutions can be determined by employing the integral equation (2.1.23) of section 2.1. Suppose that the Green matrix r(z,u,T) of that equation satisfies the conditions

166


rik(Z,u,T) < 0, rik'(l/4,u,T) > 0

for

Z,

u

E

[0,1/4), T E (Tj*,Tj+] i,k = 1 ,...,n (4.1.14)

where Tj" = 27c/wJ*, Tj+ = 2n/wJ, Oj*2 and Oj2 are the eigenvalues of matrices M-lp and n/l-*N, respectively, where matrices P and kl are chosen to satisfy relations (2.1.14). Then, by Theorem 1 of section 2 the continuum Sj of periodic solutions corresponding to the family of NNMs xj(t,h) reaches the boundary of Q. Any solution x(t) E Sj is even, symmetric, and satisfies the inequality x(t) > 0 on [O,T/4) (i.e., it represents NNM oscillations as defined in chapter 2). If, in addition, matrix [A(x) - Ill is nonnegative, then

one obtains that k(t) < 0 on (O,T/4). In a system with concave nonlinearity the continuum Sj of NNMs consists of the one-parameter family q ( t , h ) , whose period T,(h) increases monotonically with h, as proven by Theorem 2 of chapter 2. In the proof of' that theorem it was shown that the only solution of the variational equation satisfying the condition y(t) = -y(t + T/2) is y(t) = i(t,h). It follows that the inultiplicity of the eigenvalue = -1 of matrix W(T/2) is equal to 2, and the stability conditions for NNM oscillations of a system with concave nonlinearity regardless of condition (2.4.31) takes the form: , p,k = 1,...,n;

p,k # j ,

q = 1,2 ,... (4.1.15)

The stability of the NNMs xn(t,h) corresponding to the largest linearized natural frequency WnO is established by the next theorem.

Theorem. Suppose that matrix A(&)is positive definite, A(x) - N is nonnegative, A(&)< C(x), and condition (4.1,14) is satisfied for j = n; then the NNMs xn(t,h) with period T to a first approximation.

E

(TnO,Tn+] are orbitally stable

Proof. For small values of the total energy h, the characteristic multipliers of the variational problem associated with the family of NNMs xn(t,h) assume the values hn(l)(T/2) = hn(2)(T/2) = -1, lhk(l)(T/2)1= 1 arg [hk(l)(T/2)] E 0k0T/2 < 7c for k Q, there exists a positive-definite matrix A- satisfying (4.1S ) . As seen from the previous proof of Theorem 1, arg[&(l)(T/2)] > ol-T/2 > 0 where is the smallest eigenvalue of M-lA-. Therefore, for all h, multipliers of the first and second kind, hk(l)(T/2) and hk(2)(T/2)= l/hk(l)(T/2) lie on the upper and lower unit semicircle, respectively. Hence, all multipliers h k ( l ) = [hk(l)(T/2)]2 and &(2) = 1/&(1) lie on the unit circle and the theorem is proved.

The previously proven results are now illustrated by means of simple examples. Consider first the transverse oscillations of a string [cf. Figure 2.3.l(b)]. As shown in section 2.4, for this system, A(x) < C(x) or A(&)> C ( x ) if k > 1 or k < 1, respectively, where k is the relative elongation of the string due to initial tension. In the first case one can set Ok+ = Oko, whereas in the second, wk- = oko. Suppose that k > 1. Then, as shown in section 2.4 the family of NNMs xn(t,h) is uniquely continuable in h up to infinite values of h. Under such oscillations all neighboring masses move always in opposite directions and the period Tn(h) increases monotonically, with Tn(h) -+ 27~/0n as h + 0. In this limiting relation o k , k = I , ...,n, denotes the kth linearized natural frequency for longitudinal oscillations of the string. The conditions of Theorem 2 are satisfied, and the NNMs xn(t,h) are orbitally stable to a first approximation. For k < 1 the stability of modes xk(t,h) is determined by conditions (4.1.12) where p,k f j. In particular, if Wn-1 < O n 0 and 3wn0 > O n , then h ( t , h ) is orbitally stable for all h. The corresponding period Tn(h) decreases monotonically to 27~/wnas h -+ 00. As a second example, consider the system of two coupled oscillators with equations of motion given by:

+ fll(x1) + f12(x1 + x2) = 0 m2x2 + f22(x2) + f12(x1 + x2) = 0

mixi

(4.1.16)

where fpk(u) = -fpk(-u), fpk(u) > 0 for u > 0, u E SZ. One defines the following matrices:

168


where Supposing that A- > 0, consider the existence and stability of the families of NNMs xl(t,h) and x2(t,h). For j = 1, condition (4.1.12) becomes 611++02+]

[,I-+,,-

2qe

wl+

'

01-

'

qE

[q+'ml~

02-

0?+] __ ,

q = 1,2,... (4.1.18)

Under the above conditions, the family of NNMs xl(t,h) is uniquely continuable in h up to the boundary of the region Q, and the modes are orbitally stable to a first approximation. The uniqueness and existence of the family x2(t,h) is guaranteed by the condition (4. I . 13), which in this case takes the form: 202- > ( 0 2 +

+ Wl+)

(4.1.19)

Note, that since conditions (2.3.lb) are satisfied, as shown in section 2.3 there exists a continuum S2 of even symmetric periodic NNMs, which are positive on [O,T/4). As shown in later sections, under certain conditions bifurcations (branching) of this family may occur giving rise to additional NNMs in S2. Condition (4.1.19) prevents the occurrence of such bifurcations in the example considered. If all functions fpk(u)/u decrease or increase with u (cf. Fig. 2.4.1), the nonlinearity of the system is concave or convex and one can set wk+ = wko or wk- = wko, respectively. In the case of convex nonlinearity the existence and stability of the NNM xl(t,h) are guaranteed by the condition: (4.1.20) For the family of NNMs x2(t,h) the corresponding conditions are:

4.2 SIMILAR NNMs

169

In the case of concave nonlinearity the family &l(t,h) is uniquely continuable in h and stable provided that (4.1.22) Combining the previous results and those of chapter 2, it can be shown that for concave nonlinearity the family x2(t,h) is uniquely continuable and stable regardless of the values of the quantities Ok- and o k + . As a numerical example, let mi = m2, apk- = a- > 0, apk+ = ra-, p,k = 1,2, r > 1. Then 0 2 - = 31/2W1-, o k + = r1/2ok-. The stability of &l(t,h) is ensured in all cases [conditions (4.1.18), (4.1.20), and (4.1.22)] if r < 4/3. The stability of x2(t,h) in the general case [condition (4.1.19)] is guaranteed by the inequality r < 12/(1 + 31/2)2 = 1.602. In the case of convex nonlinearity [condition (4.1.2 I)] stability of the highest NNMs is ensured by the inequality r < 3. Since in this application A- > 0, in the case of concave nonlinearity the family x2(t,h) is stable for any value of r. As a second example, consider the system (4.1.16) with f l l(u) = cisinx, f22(u) = c2sinx, f12(u) = cu. These equations model the oscillations of a system of two linearly coupled pendulums. As shown in section 2.4, for Ixkl < 4.49, k =1,2, the nonlinearity is concave. Clearly, for c > ck, k=1,2, the corresponding matrix A(x) is positive definite. Hence the family of NNMs x2(t,h) (i.e., antiphase rotations of the two pendulums) are stable until their amplitudes reach the value Ak = maxt Ixk(t)l < 4.49, k = 1,2. The previous analysis enables the study of existence and stability of NNMs of systems with strong nonlinearities; moreover,the stability results a r e solely based on the analysis of the restoring nonlinear forces, without any computation of the NNMs. In the following sections studies of similar and nonsimilar mode bifurcations of specific discrete oscillators are performed. 4.2 SIMILAR NNMs

Before proceeding with the stability analysis some additional results from the theory of linear equations with periodic coefficients are reviewed. Consider

170


the following second-order differential equation with periodic coefficients (Hill's equation): x(t)

+ p(t)x(t) = 0,

p(t) = p(t+T) V t E R

(4.2.1)

where for simplicity it is assumed that x and p(t) are scalars in R (note that the following results can be easily extended to multidimensional systems with x E Rn and p(t) an (n x n) matrix of periodic coefficients). Let xl(t) and x2(t) be a set of (linearly independent) fundamental solutions of (4.2.1) corresponding to initial conditions: xl(0) = 1, xi(0) = 0, and x2(0) = 0, X2(0) = 1 Since the coefficient p(t) is periodic, it can be easily proved that xl(t + T) and x2(t + T) are also solutions of the equation under consideration. Moreover, since the equation is linear, one can express any solution as a linear combination of the fundamental ones. Hence, the following relations hold:

The normal solutions wl(t) and w2(t) of (4.3.1) satisfy the relations wl(t + T) = pwl(t) and w2(t + T) = pw2(t) where p = exp(pT) is the characteristic multiplier and p. the characteristic exponent. Clearly, the characteristic multipliers are identical to the eigenvalues of the matrix of coefficients of (4.2.2) and are computed by solving the following characteristic equation: (4.2.3) From (4.2.3) one computes two characteristic multipliers, p i and p2, and, provided that the multipliers are distinct, expresses the corresponding sets of normal solutions in the form:

4.2 SIMILAR NNMs

I7 I

where p 1 and p2 are the corresponding characteristic exponents. Similar expressions can be derived when the multipliers are repeated. Evidently, if Re(p1) > 0 and Re(p2) > 0, as time increases the normal solutions increase with no bounds, and every solution of (4.2.1) is unstable. Hence, for stability one requires that Re(p1) and Re(p2) are nonpositive quantities. Returning to the characteristic equation (4.2.3) it can be shown that the characteristic multipliers occur in reciprocal pairs, i.e., p i p 2 = 1. From the previous discussion it follows that (4.2.1) has bounded solutions only if the characteristic multipliers are purely imaginary and lie on the unit circle: lpil = Ip21 = 1

and

Re(p1) = Re(p2) = 0

Moreover, setting wl(T) + w2(T) = A, one can show that the characteristic multipliers are computed by p1,2 = (A/2) [(A/2)2 - 111’2 When IAl = 2 (4.2.1) possesses only periodic solutions, and the system is on a boundary between stability and instability. It can be easily shown that, when IAl = 2 the characteristic multipliers assume the values p i = p 2 = 1 [corresponding to T-periodic solutions of (4.2.1)] or p1 = p 2 = -1 [corresponding to 2T-periodic solutions of (4.2. l)]. A transformation of coordinates is now imposed on (4.2.1) by means of which this set of equations (with periodic coefficients) is transformed to a set of equations with singular points. This procedure is referred to as Ince algebraization (Ince, 1926) of the variational equations governing the linearized stability of the dynamical system under consideration. A similar nonlinear transformation was introduced in chapter 3 for studying NNMs, whereby the equations of motion were transformed to a set of singular functional equations governing the relative displacements of positional variables during the normal mode. That previous transformation was performed by replacing the temporal variable with a reference displacement x(t), and eliminating the time dependence from the problem. Consider Hill’s equation (4.2.1) with the periodic coefficient expressed as p(t) = so + 2SlCOS2t + s2cos4t + ...:

+

X(t) + [so

+ 2s1cos2t + s2cos4t + ...I x(t) = 0

(4.2.5)

172


Ince algebraization of (4.2.5) is performed by introducing the new independent variable z = cost, leading to the following alternative equation: (1 - 22)

d2x

dz2 - z

dx

+ (n=O bn z2n)x

=0

(4.2.6)

The transformed equation is linear and possesses regular singular points at z = +_1 [with singularity indices 0 and 1/2 (Whittaker and Watson, 1986)] and an essential singularity at infinity. In the vicinity of z = +1 the following set of fundamental solutions is derived: xl(1 - z) = fl(1 - z)

and

x2(1 - z) = (1 - z)”2f2(1 - z)

(4.2.7)

where functions fl(*) and f2(*) are analytical in the region I1 - zI I 2. Since equation (4.2.6) is invariant under the transformation z + -z, one obtains the following additional set of fundamental solutions valid in the vicinity of z = -1: xl(l

+ z) = f l ( l + z)

and

x2(1 + z) = (1

+ z)”2fi(l + z)

(4.2.8)

2. Within where functions fl(*) and f2(*) are analytical in the region I 1 + zI I the common domain of convergence of solutions (4.2.7) and (4.2.8), the following relations hold:

+ z) + P X 2 ( 1 + z) x2(1 - z) = yXl(1 + z) + 6X2(l + z)

(4.2.9a)

+ z) = ax1(l - z) + P X 2 ( 1 - z) x2(1 + z) = yXl(1 L ) + 8x2(1 - z)

(4.2.9b)

X l ( 1 - z) = ax1(l

and,

Xl(l

-

where the coefficients a, p, y, and 6 are constants. Manipulating relations (4.2.9a) one obtains the following alternative expressions for xl(1 + z) and x2(1 + z): Xl(1 + z) = ( a 2 + Py)x1(1 + z) + P(a + 6)x2(1 + z) x2(1 + z) = ?(a + 6)x1(1 + z) + (Py+ @)x2(1 + z)

(4.2.10)

4.2 SIMILAR NNMs

173

which combined with (4.2.9b) lead to the algebraic equations a 2 + p y = 6 2 + p y = 1 and p ( a + 6 ) = y ( a + 6 ) = O From these relations the following two sets of solutions for the coefficients are obtained: (i) a = 6 = f l , p = y = 0 and (ii) a = -6 = +1, py = 1 - a 2 . A closer examination of relations (i) reveals that they lead to fundamental solutions of (4.2.6) satisfying xl(1 - z) = xl(1 + z), x2(l - z) = x2(1 + z). Point z = 0, however, is a regular point for equation (4.2.6), and no two independent even solutions can exist in any small neighborhood of this point. It follows that the set of coefficients (i) cannot be realized, and that the only possible values for the coefficients satisfy relations (ii). Two possible sets of values for the coefficients then exist. Consider first the case when a = -6 = k l , p = 0, corresponding to solutions of (4.2.6) satisfying xl(1 - z) = +xl(l + z). It is seen that this type of solution is even for a = + I and odd for a = -1. Taking into account that z = cost, for a = + I the even solutions can be represented as cosine-series expansions of even multiples of t, whereas for a = -1 the odd solutions can be expressed as cosine-series expansions of odd multiples of t. An alternative type of solutions exists for a = -6 = k l , y = 0, satisfying the relations x2(1 - z) = fx2(1 + z) for I1 f ZI < 2 These solutions are the products of term (1 - z2)1'2 with analytic functions that change their signs for complete revolutions along closed contours in the complex plane surrounding the singular points z = + 1 or z = - 1. The analytic functions in question are even if 6 = +1 and odd if 6 = - 1 . Substituting z = cost, it follows that these functions can be expressed as sineseries expansions of odd or even multiples of t, for 6 = +1 or 6 = - 1 , respectively. The aforementioned special solutions of the singular variational equation (4.2.6) will be termed degenerate solutions and correspond to periodic solutions of Hill's equation (4.2.5) with periods .n and 2.n. Summarizing, a change of independent coordinates can be imposed that transforms Hill's equation to a linear equation with regular singular points. Degenerate solutions of the singular equation can then be analytically computed. These solutions correspond to periodic solutions of Hill's equation and lie on boundaries in parameter space separating bounded (stable) from unbounded (unstable) solutions. As an example of application of the previously described methodology, consider the following Mathieu's equation:

174


X(t) + [6 + 2E cos2t]x(t) = 0

(4.2.1 1 )

Introducing the transformation z = sin%, one expresses this equation in the following form: d2x dx + (6 + 2~ - ~ E Z x) = 0 4~(1 Z) dz2 + 2(1 - 22)

(4.2.12)

The transformed equation (4.2.12) possesses two regular singular points at z = 0 and z = 1 with singularity indices equal to 0 and 112, respectively, and an essential singularity at infinity. Employing the previous analysis, the periodic solutions of Mathieu's equation separating stability and instability regions are expressed as follows: m

m

m

m

where each series converges in domains of the complex plane, which contain both regular singular points. As a second application, consider the following Lame' equation: x(t) + [h - n(n+l)k2sn2(t,k)]x(t) = 0

(4.2.14)

where sn(a,*) is the Elliptic sine function. This equation can be transformed to an equation with regular singular points employing various coordinate transformations. For example, introducing the transformation z = snz(t,k), one reduces (4.2.14) to the following equation with three regular singular points:

dz2

x =0

(4.2.15)

4.2 SIMILAR NNMs

175

The solutions of this singular equation will be studied in the following exposition. The aforementioned analysis was carried out considering linear differential equations with periodic coefficients and no dissipative terms. The effects of dissipation or nonlinearity on the solutions and on the stabilityinstability boundaries were investigated in various works of the literature (Bogoliubov and Mitropolsky, 1961; Nayfeh and Mook, 1984) by applying perturbation techniques. As indicated above, solutions in stability regions of Hill's equation correspond to pairs of purely imaginary characteristic exponents [p1,2 = exp(fpt), Re(p) = 01. Within such stability regions, the main effect of dissipation is to introduce nonzero real parts in the characteristic exponents. If the real parts of the characteristic exponents are negative, the stability regions contain asymptotically stable solutions (in contrast to the nondissipative case where the solutions can be at most neutrally stable). The effects of nonlinearities and of external forcing on the solutions were discussed at length by Nayfeh and Mook (1984).

4.2.1 Analysis of Stability Boundaries The previously outlined stability methodologies will now be applied to study the stability of the similar NNMs of conservative systems. Consider a conservative n-DOF nonlinear oscillator with equations of motion given by:

where the notation of chapter 3 is employed. Suppose that this system possesses the similar NNM xm(t) = cmx(t), m = 2,,..,n, xi(t) 5 x(t), where x(t> is a periodic motion. Rotating the coordinate axes so that the trajectory of the similar mode coincides with a coordinate axis and the remaining axes are orthogonal to it, the NNM under consideration is represented as qm = 0, m = 2,...,n, q1 = q = q(t). Expressing the equations of motion in terms of the new coordinates, one rewrites (4.2.16a) as: (4.2.16b)

176


where the quantity n(q) is derived froin the potential energy V(x) by imposing the coordinate transformation li z ( x i ,...,xn)T 4 (91, ...,qdT= y. To study the stability of the similar mode. one introduces the variations 41 + q + ~ 1 , qi + 0 + ui, lull a , pr> a ; curves d to 0 < r < 1, p < a , pr < a. From Figures 4.2.l(a) and 4.2.l(b) one notes that for positive values of the coupling parameter K in-phase and antiphase similar NNMs exist corresponding to c = +1. Depending on the values of the coupling parameter K and the degree of homogeneity r, additional branches of bifurcating antiphase NNMs exist, generated from the anti-phase mode c = -1 through a mode bifurcation. Note that as K + 0 (i.e., as the coupling stiffness tends to zero), there exist only two localized bifurcating NNMs with c + 0- and c + --oo. For negative values of the coupling parameter K < 0, one obtains a single antiphase mode and multiple in-phase NNMs generated through additional mode bifurcations of the in-phase mode c = + l . The outlined stability methodology can be employed to study the stabiIity of similar NNMs of nonhomogeneous oscillators. In that case one obtains hypergeometric variational equations of the form (4.2.20) whose stability regions can be studied by employing a perturbation methodology. In the zeroth order approximation the variational equations resemble those of a

182


homogeneous generating system, and their solutions can be determined exactly. Higher order approximations to the stability boundaries are computed by perturbing the zeroth order solutions and imposing periodicity conditions on the solutions. This procedure is demonstrated by means of the following example. Consider a symmetric conservative nonlinear system containing linear and cubic stiffness terms and equations of motion given by:

This system possesses in phase and antiphase similar NNMs x2 = cxi, with c = +1, and the following analysis investigates the stability of the antiphase mode c = -1. The variational equation governing the evolution of perturbations orthogonal to the antiphase is given by:

u + [o + (3q2(t)]u = 0

(4.2.29)

where o = kl l/m and p = 3k13/m. Variable q(t) describes the periodic oscillation of the system along the trajectory of the antiphase mode in the configuration plane and is governed by the differential equation

q + ( y + 2pq2)q = 0 where

y = (kii + 2k21)/m

and

p = (k13 + 8k23)/m

Note, that variables q and u are related to the positional variables x i and x2 by

q = (xi - x2)/2

and

u = 6(x1 + x2)/2,

Introducing the change of independent variables t equation (4.2.29) is expressed as:

161 0. The degenerate solutions of (4.2.59), which correspond to T- or T/2-periodic solutions of the original variational equation (4.2.46) (where T is the period of the N N M whose stability is examined), are then computed as follows: N o instability zones (n = 0) uo = c, ho = 0 One instability zone (n = 1) uo = C(z2 - b2)1/2, ho = a2 if b k 0 and uo = C(b2 - z2)1/2, ho = a2 if @>a2 u l = Cz, hi = a2+ b2 if b k 0 and u i = C(a2 - z2)1/2, h i = b2 if b b a 2 u2 = C(a2 - z2)1/2, h2 = b2 if b k 0 and u2 = Cz, h2 = a2 + b2 if b b a 2

194


and u3 = Cz(a2 - z2)1/2, A3 = a2 + 4b2 if b b a 2 u4 = C[z2- (2a2b2/h4)1112 if b k 0 and u4 = C[z2 - (2a2b2/hq)]l/2 if b b a 2 h4 = 2[a2 + b2 - [(a2 + b2)2 - 3a2b2]1/2 if b k 0 and h4 = 2[a2 + b2 + [(a2 + b2)2 - 3a2b2]1/2 if b b a 2 ) where C is an arbitrary real constant, and the instability zones are defined in the following intervals of the h-axis: (--,Lo] (first instability region), [hi,A21 (first instability zone), and [h3,h4] (second instability zone). In Figures 4.2.2(a) and (b) the instability zones of the Lame’ equation are graphically depicted for n = 3 and n = 3, respectively. In these Figures IQI denotes the maximum amplitude of the periodically varying parametrizing coordinate q = q(t) during the NNM oscillation. Note that when the conditions for finite-zoning instability are exactly met, all higher instability zones are eliminated by degenerating into lines [cf. Figure 4.2.2(a)]. When the finite-zoning conditions are only approximately satisfied, there exists an infinite number of higher instability zones of diminished thickness [cf. Figure 4.2.2 (b)]. As a final example of application of the previous analysis, consider the plane vibrations of two masses coupled by a linear spring of stiffness k2 and undeformed length L, and connected to the ground by linear springs of stiffnesses kl and undeformed lengths 1. It is assumed that the springs are prestrained by a constant (tensile or compresive) force of magnitude T. The hamiltonian of this system is given by (Zhupiev and Mikhlin, 1984): 2

2

+ k2[[(xl - x 2 ) 2 + L2]1/2-

Lo]1/2 (4.2.60) where 10 = 1 - T/kl, Lo = L - Tk2, and X i is the transverse displacement of the i-th mass. Expanding the hamiltonian in Taylor series about X I = x2 = 0 and retaining only the leading nonlinear terms, one obtains equations of motion of the following form: Xi2

+ kl[(l

-

Io/l)Xi + (l0/2)(~i/l)3- (310/8)(xi/1)5]

+ k2[( 1 - Lo/L)(Xi - Xj) + ( L o / ~ ) L - ~ -( xxj)3 ~ - (3L0/8)L-’(xi

xj)5] = 0 i = 1,2, j = 3-i (4.2.61) -

4.2 SIMILAR NNMs

195

Figure 4.2.2 Instability zones of the Lame' equation for (a) n = 3 and (b) n = 3. Introducing the new dependent variables 92 = ( X I + x2)/(21) 41 = ( X I - x2)/(21), one transforms the equations of motion into a form similar to (4.2.43), admitting the antiphase similar NNM solution ql(t) = q(t) f 0, q2(t) = 0. The linearized variational equation governing the stability of the antiphase NNM is analogous to (4.2.46) and given by:

u + 2[e0 where q(t) is computed by,

+ e2q2(t) + e4q4(t)]u = 0

(4.2.62)

q + 2a2q + 4a4q3 + 6a6q5 = 0 and the various coefficients are defined as: eo = T/21, e2 = (310/41)kl, e4 = --(151o/161)k~, a2 = (T/21) + (T/L) a4 = (lo/81)kl + (lLo/L3)k2, and a6 = -(lo/I6l)k1 - 2(13Lo/L5)k2 Since the potential function of system (4.2.61) is of the form:

196


one can make use of conditions (4.2.55) in order to obtain finite-zoning instability of equation (4.2.62). Applying (4.2.55), one derives the following finite-zoning instability conditions: n(n + 1)[1 + 8(13Lo/L310)(k2/ki)l = 3 4n(n + 1)[1 + 32(ljLo/Lslo)(k2/ki)] = 15 S[(T/l) + (2T/L)] [l + S(13Lo/L310)(k2/ki)l[ 1 + 32(1~Lo/L510)(k2/ki)l + [ 1 + 8(13L0/L310)(k2/ki)]3= 4h [ 1 + 32(lsLo/Lj10)(k2/k~)l2 n = 0,1,2 ,... (4.2.63) where n is the number of instability zones. When conditions (4.2.63) are exactly satisfied, the antiphase NNM possesses precisely n instability zones. When these conditions are only approximately satisfied, the antiphase mode possesses an infinity of instability zones, with the higher ones being narrow. It is concluded that by applying conditions (4.2.63) one achieves stabilization of the antiphase NNM of the system.

4.3 NONSIMILAR NNMs In previous sections the linearized stability and bifurcations of similar NNMs were considered. However, as shown in chapter 3, similar modes represent special types of free oscillations since they can only occur in restricted classes of nonlinear oscillators. Nonsimilar NNMs, on the other hand, are generic in nonlinear discrete conservative systems, and, therefore a study of their stability and bifurcations is of great practical importance. Preliminary stability analyzes of the stability of nonsimilar NNMs were performed by Rosenberg and Kuo (1964), who considered weakly nonsimilar NNMs, i.e., nonsimilar modes resulting from small perturbations of generating similar NNMs. They provided a general stability result, namely, that the stability of a weakly nonsimilar N N M is identical to the stability of the generatirig similar N N M to which if neighbors. In this section an analytical methodology is developed for studying bifurcations of nonsimilar NNMs occurring close to bifurcation points of similar modes. The stability of the nonsimilar modes prior and after the bifurcation can then be deduced by invoking the aforementioned stability result. The following bifurcation analysis is carried

4.3 NONSIMILAR NNMs

197

out for a two-DOF oscillator, but it can be conveniently extended to multiDOF systems. Consider the following conservative system:

41 + aH(q1,q2,h)/aql = paF(ql9q2,3L)/~ql q2 + a n < q 1 J l 2 J w q 2= paF(ql392&/aq2 0 c p dt = 0

T

* I [a2n(qio,0,ho)/aqia3Ll dqio = 0

(4.33 )

T

where the integration is carried out over the time interval 0 C. t 5 T. Since the integrand in the second of expressions (4.3.5) is an analytic function of q 10, the contour integruls (4.3.5) vanish identically, and the periodicity conditions (4.3.5) are identically met. Therefore, the solution q1 i(t) of the

4.3 NONSIMILAR NNMs

199

first of equations (4.3.4) is always unique and periodic with period T. Consider now the second of equations (4.3.4), governing the approximation q21(t). This is identical in form to the variational equations of Hill-type encountered in section 4.2, where the linearized stability of similar NNMs was examined. As shown in that section, depending on the specific value of parameter ho,the second of equations (4.3.4) can possess stable, unstable, or periodic solutions, the later solutions occurring on stability-instability boundaries in parameter space. It follows that, by assigning appropriate values on the structural parameter ho, the equation governing q21(t) can also be made to possess periodic solutions of period T. Such solutions will be denoted by q21N = KlO(t) where K1 is a constant and cJ(t) = o(t + T) V t c R. It must be pointed out, however, that such periodic solutions are special and "critical" solutions, since they only occur on stability-instability boundaries in the parameter space of the second of equations (4.3.4). The values of ;lo for which periodic solutions 9 2 1 ( t ) exist, are parameter values for which system (4.3. I ) possesses nonsimilar NNMs neighboring the generating similar normal mode ( q / , q 2 )= (qro(t),O). As shown below, the perturbation analysis provides a means for analytically studying nonsimilar NNM bifurcations occurring close to bifurcations of the generating similar mode of the system with p = 0. Considering O ( E ~terms ) in (4.3.2), one obtains the following system of equations governing the terms of the second-order approximation to the nonsimilar mode: q12 + 912 a2n[qlo(t)>o,hol/aq12 = - q112 ~ 3 ~ I [ q l o ( t ) ~ 0 , ~ ol /h~l q l l3 a3n[q10(~)t0,hol/a~12ah - A12 a2n[q10(t)j0,hol/ah2- A2 a2~[qlo(t),0,3iollaqlah - (921~12) a3n[q1~(t>,o,h~l/aq1aq22 + aF[q10(t),O,h01/aql

As in the previous order of approximation, it can be shown that the periodicity condition for the first of equations (4.3.6) is identically met;

200


therefore, there always exists a unique and periodic solution q21(t) of period T. Considering the second of equations (4.3.6), a periodic solution q22(t) exists, provided that the following periodicity condition is satisfied:

T

[ -91 iKio(t) a3n(q~o(t),0,ho)laqiaq22 - hiKio(t) a3n(q10(t),o,ho)laq22ah

[Ki2o2(t)/21a3n(q~o(t),0,h~)/aq23 + Wqio(t),O,ho)/aq2 } o(t) dt = 0 (4.3.7) where the integration is carried over a period of the O(E) periodic solution q21(t) = Kio(t) [which is also a homogeneous solution of the second of equations (4.3.6)]. As discussed previously, the quantities q21(t) and a(t)can be rendered periodic, of period T (the period of the generating similar NNM), by an appropriate choise of parameter ho in the O(E) approximation. Relation (4.3.7) evaluates the O(E) approximation hi of the parameter in terms of the amplitude K1 of solution 42 1(t). Taking into account the previous discussion, one concludes that at parameter values d = A.0 + + O ( E ~nnnsimilar ) NNMs of s y s t m (4.3.1) exist. Points of nonsimilar NNM bifurcations are now sought close to bifurcation points of the generating similar NNM of the system with 1-1 = 0. Analytical approximations to the bifurcated nonsimilar NNMs in the neighborhoods of bifurcation points can be computed by analyzing equations (4.3.4) and (4.3.6). As mentioned earlier, and without loss of generality, the periods of the sought-after nonsimilar NNMs will be assumed to be identical to the period T of the corresponding generating similar NNM. Special consideration will be given to the O(E) periodic solution q21(t) = Kio(t) of the second of equations (4.3.4), since it plays an important role to the construction of the bifurcated nonsimilar NNMs. It can be shown that two types of solutions for q21(t) exist. Type-I solurions are of the form q21W = Klc[qlo(t)l where qio(t) is the generating similar mode response and 5[*]an analytic function. Type-II solutions are expressed as -

q21(t) = KIillO(t) rl[qio(t)l

4.3 NONSIMILAR NNMs

201

where q[qlo(t)] is an analytic, single-valued function of qlo(t). The amplitudes K1 of these solutions are computed by imposing the (nontrivial) periodicity condition (4.3.7), which can be expressed implicitly as F(Ki,ho,hi) = 0 Considering type-I solutions, nonzero multiple roots K1 = K l ( b ' f , # 0 of this equation, correspond to nonsimilar mode bifurcation points where multiple branches of nonsimilar NNMs coalesce. In order for multiple nonzero amplitude solutions to exist, the following condition must be imposed:

Condition (4.3.8a) provides the values of the amplitude K1 at which nonsimilar mode bifurcations occur. For type-I1 solutions, it can be shown that points of multiple zero roots Kl(bif)= 0 of the relation F(Ki,ho,hi) = 0 correspond to intersection (bifurcation) points of multiple nonsimilar NNMs. Hence, points of bifurcation of type-I1 nonsimilar NNMs are points at which the following condition is satisfied:

Depending on the type of nonsimilar mode bifurcations encountered, the zero amplitude solution ~ ~ ~ ~ = ( ob i f ) can be of multiplicity two or three. The previous general results are now applied to the study of the nonsimilar NNM bifurcations of a nearly homogeneous, two-DOF conservative oscillator with equations of motion of the following form:

202


These equations are of the general form (4.3.1) with n ( q 1 ,q2) being a polynomial of least degree r. Introducing the rescaling p = ~ 2 expressing , the nonsimilar NNM solution of (4.3.9) and the corresponding values of the system parameter h in the series form (4.3.3), and substituting into (4.3.9) one obtains a series of problems at various orders of E. The zeroth order approximation is the similar NNM of the generating homogeneous system of degree r:

The solution of the first of relations (4.3.10) can be expressed in explicit form in terms of a cam-function, as defined by Rosenberg (1963). The O(E) approximations are computed by solving the following two equations:

Note that the period of the coefficient hoqlor-1 of the second of the above equations is half the period T of the generating similar N N M (4.3.10). As shown in section 4.2, periodic solutions q21(t) having periods equal or twice the period of coefficient hoqlor-1, are critical solutions since they lie on stability-instability boundaries in the parameter space of the system and correspond to points of exchange of stability of the similar NNMs of the generating problem. It will be assumed that coefficient ho is chosen so that the corresponding solution q21 (t) = K1(3(t) is of period T, i.e., of the period of the generating similar NNM. As discussed previously, there exist two permissible types of periodic solutions for approximation q2 1(t): type-1 solutions: 921(t) = K i W ) , where o(t>= Qqio(t>l and type-I1 solutions, q2i(t) = K i W ) , where W) = qio(t)q[qio(t)l The O ( E ~approximation ) to the nonsimilar mode is computed by analyzing the following system of equations:

4.3 NONSIMILAR NNMs

203

The T-periodic solution of the first of the above equations is written in the form q12(t) = A241o(t) + U2[910(t)l where the first term represents the periodic homogeneous solution and the second term a particular integral, with A2 a constant and U2(*) an analytic function. The second of equations (4.3.12) admits two types of solutions. Type-I solutions are of the form q22(t) = K20(t) + V2[qlo(t)l whereas type-I1 solutions are expressed as q22(t) = K20(t) + V2[q10(t)l + qlo(t) W2[q10(t)l In these expressions K2 is a constant, and V2(*), W2(*) are analytic functions. T-periodic solutions for q22(t) exist, provided that the following conditions are satisfied:

where, as in the general formulation, the integrations are carried over a period of the T-periodic solutions qlo(t) and cT(t). Relations (4.3.13a,b) represent the periodicity conditions for the first and the second of equations (4.3.12), respectively. Condition (4.3.13a) holds identically (since the integrand is an analytic function), and, hence, amplitude A1 in the first of expressions (4.3.11) is undetermined at this order of approximation. Consider now the nontrivial periodicity condition (4.3.13b). Here one must distinguish between type-I and type-I1 solutions regarding q2 1(t).


204

Type-I solutions 92 1(t) lead to the following periodicity condition:

I [ K i 2 ~ q ~ ~ r - 2 ( t ) S 2 [ q ~-o (dF(qio,O)/dqz} t)l C[qio(t>l dt =3

h1=

T

T

(-Kiqi~~-~(t)S[qio(t)l } S[sio(t)l dt

(4.3.14) where amplitude the A 1 of approximation ql I ( t ) was chosen equal to zero. The expression above relates the amplitude of the type-I solution 92 1 (t) to the parameter hi for motion on a nonsimilar NNM of period T. TO compute bifurcation points of nonsimilar modes one imposes condition (4.3.8a), and obtains the following additional relation between K1 and hi:

Combining equations (4.3.14) and (4.3.15). one obtains the bifurcation values for hi and K1, for type-I solutions q21(t):

(4.3.16) T

Type-I1 solutions q2l(t) are now considered. In this case the periodicity condition (4.3.13b) becomes

J' T

[-hiK~qior-l(t)o(t)lo(t) dt = 0

=

hi = 0 and K I undetermined (4.3.17)

4.3 NONSIMILAR NNMs

205

Hence, it is necessary to resort to higher-order approximations in order to compute the relation that must be satisfied between the amplitude K1 and parameter h, in order for nonsimilar NNMs of type-I1 to exist. In addition, higher-order calculations will provide analytic approximations for points of bifurcations of nonsimilar NNMs. Considering O(&3) terms in (4.3.9) one obtains the following set of equations:

q23 + hoq10'-~qn = 911 d2F(qlo,0)/dqldq2 + 421 a2F(q10,0)/aq22 (r - l)hoq10'-2q11q21- [h2910'-1 + X O ( ~ - l)q10'-2q12 + hoqll26r31q11 - 2~qior-2q2iq22- 2 ~ ( -r 2)ql0'-3q11q212 - yqior-3q2i3 (4.3.18) The periodic solution of the first of the above equations can be expressed in the form -

q13(t) = A3qIO(t) U3[qlO(t)] where A3 a constant, and U3(*) an analytic function. Type-I1 periodic solutions of the second of equations (4.3.18) are given by q23(t) = K3o(t) + V3[910(t)] + q10(t) W3[910(t)l where K3 is a constant and V3(*), W3(a) are analytic functions. Imposing the periodicity conditions on the above equations one obtains the relations:

206

STABILITY AND BIFURCATIONSOF NNMs

E2 = - j ho(r - ~ ~ ~ l o ' - ~ q l o ( ~ dt ~ ~ 2 ~ q l o ~ ~ ~ l ~ T

E3 = T

{ a2F(qio,0)/dq22 o(t) - (I. - l)h0qio~-~U2[qio(t)l- hoqi i2tjr3

E4 =

-I

T

- 2K.910'-20(t)v2[q10(t)l}o(t) dt E5 = yqior-3o4(t) dt K(r - 2)qlor-391103(t) dt,

-5

T

E6 =

I a2F(qio90)/aqiaq2 q 1 (t)O(t) dt

(4.3.19c)

1

T

From relation (4.3.19a) it is found that, A1 = (ZlKl+Z2K13) / (Z3+ZqK12) where the quantities Zi, i = 1,2,3,4, are constants independent of K1. The amplitude A1 is defined only when the denominator (Z3 + ZqK12) of the previous expression is nonzero; this condition is explicitly written as:

-

(r - 1)/2 ho(r - 2)q10'-3(t)qio(t)K1262(t)]qio(t)

dt

f

0

(4.3.20)

From the previous expressions it is evident that when K1 = 0, it is also satisfied that A1 = 0. Expression (4.3.19b) relates the parameter h2 to the amplitude K1 for nonsimilar NNM solutions. Bifurcations of type-I1 nonsimilar NNMs occur at points where multiple zero roots K I = 0 exist. To compute such bifurcation points one must impose condition (4.3.8b), which, taking into account (4.3.19b), leads to the following bifurcation values for h2:

T

K l = Kl(bif) = 0

(4.3.21) This calculation provides an analytic approximation to the bifurcation points of nonsimilar NNMs for type-I1 periodic solutions q2p(t), p = 1,2,3,... It must be noted that the previously presented analysis is only valid for systems (4.3.9) with parameters K # 0 (Manevitch et al., 1989). When K = 0, parameter p in (4.3.9) must be related to the perturbation parameter E by p = ~ 3 and , the nonsimilar mode solutions by the series representations (4.3.3).

4.4 NNM BIFURCATIONS IN A SYSTEM IN INTERNAL RESONANCE

207

Calculations of nonsimilar mode bifurcations of systems with K = 0 have been performed by Manevitch et al. (1989). Summarizing, when K # 0 the previous analysis indicates that, for sufficiently small values of p, bifurcations of nonsimilar NNMs of system (4.3.9) occur at the critical parameter values h = + p'/2h&bif) + O(p) for type-I solutions, and A = + p h 2 w + o(p3/2) for type-I1 solutions. Analytic expressions for terms ([qio(t)] and q[qlo(t)] of the O(E )approximation q21(t), and for terms U2[qio(t)], V2[qio(t)] and W2[qio(t)] of the O ( E ~approximations ) q12(t) and q22(t) can be obtained by substituting the independent time variable t in the differential equations (4.3.11) and (4.3.12) by q l o , and analyzing the resulting sets of linear singular equations by the methodology described in section 3.1.3. Analytic computations of these terms have been performed by Manevitch et al. (1989). Finally, it should be clear that the presented analysis still holds if the "nonsimilar" perturbation in (4.3.1) is set equal to zero, i.e., if F(qi,q2,h) = 0. In that case the analytically approximated bifurcating NNMs are either nonsimilar or similar, the later case occurring, for example, when the dynamical system corresponding to F(q1,q2,h) = 0 is homogeneous. The problem of the nonlinear stability of NNMs in conservative systems may be solved only after the corresponding linearized analysis is conducted. The nonlinear stability of NNMs was discussed by Pecelli and Thomas (1979,1980) using an approach first introduced by Arnold and Mozer. Employing essentially nonlinear techniques, it was found that additional instability regions for the NNMs exist, albeit of much smaller dimensions in parameter space than those predicted by the linearized stability analysis.

4.4 NNM BIFURCATIONS IN A SYSTEM IN INTERNAL RESONANCE An application of the previous analytical findings will be given by considering a two-DOF oscillator in internal resonance. It will be shown that due to internal resonance the methodology for computing nonsimilar NNMs in configuration space fails, and an alternative formulation must be adopted. Rand (1995) has also studied the NNMs of a two-DOF discrete system with

208


cubic nonlinearities and a 1:3 internal resonance and observed N N M bifurcations; in his work the analysis was performed by employing the method of multiple scales. The system under consideration in the present work possesses cubic stiffness nonlinearities and equations of motion given by:

The above system was first considered by (Shaw and Pierre, 1991,1993,1994) who investigated the effects of internal resonance of their invariant NNM manifold constructions. They found that the invariant manifold approach (as other standard approaches) fails when applied to systems with interacting NNMs in internal resonance. This feature is illustrated in the following calculation. To compute the NNMs of system (4.4.1), one designates variable xi = x as the parametrizing variable, and expresses x2 as a function of x as follows, x2 = 22(x). The following singular equation governing 22(x) then results:

which is complemented by the following boundary orthogonality condition at points of maximum potential energy x = X: ;;(X)

[-X

-k

(X - ?z(X))

-

&gX3]

+ ft2(X) - k(X - 22(X)) = 0

(4.4.2b)

In the expressions above, X denotes the maximum amplitude attained by the parameterizing coordinate x during the N N M oscillation. As in previous sections, the NNM solution of system (4.4.2) is sought in the series form A(0)

22(x) = x2 (x)

+ A(1) x2 (x) + ...


209

where

(1) (1) (1) x2(0) (x) = cx and ^xr’(x) = a21 x + a23x3 + a25x5 + ... Substituting these expressions in (4.4.2) and solving the resulting sets of algebraic equations, one obtains the following asymptotic approximations for the two nonsimilar NNMs of the system: A

(4.4.3) Nonsimilar mode I1 is in the neighborhood of the antiphase similar mode x2 = - xi, and its asymptotic expression is valid for arbitrary 0(1) values of the stiffness parameter k. On the other hand, mode J is close to the in-phase similar mode x2 = xi, and its asymptotic approximation is valid only if k is not in the neighborhood of the critical value kcr = 4. When k = 4 the two denominators in the expression of 22I(x) nearly vanish, and the O( 1) term of the asymptotic expression ceases to dominate over the O(E)terms. T h r failure of the asymptotic analysis regarding mode I f o r k = 4 is due to an internal resonance of the system, since for such values of k the two linearized natural frequencies of system (4.4.1) are nearly integrably related, and equal to the ratio 1:3. This fact, combined with the existence of cubic stiffness nonlinearities in the system, gives rise to a NNM bifurcation at k = 4, which renders invalid the previous NNM analysis. To investigate the bifurcation associated with NNM I for k 4, one introduces the change of coordinates, x2 = y1 - y2 X I = y1 + y2, and expresses (4.4.1) in the following canonical form: L-

Confining the analysis in the neighborhood of the critical value kcr = 4, one introduces the notation (1 + 2k) = 9 + EB, where o is an O(1) frequencydetuning parameter indicating the closeness of the paremeter (1 + 2k) to the critical value (1 + 2kcr) = 9. The bifurcation associated with NNM ],for k =

210


4 will be studied by studying nonsimilar NNMs qf the canonicul system (4.4.4), i.e., by seeking solutions where the canonical coordinates are related byfunctionul expressions of the form y 2 = ;2(y1). Note, that NNMs of the canonical system (4.4.4) do not necessarily correspond to NNMs of the original system (4.4.1). This will become apparent in the following exposition. Before proceeding with the perturbation analysis, it is pointed out that NNM I in (4.4.3) corresponds to (yi,y2) = ( 0 ( 1 ) , 0 ( ~ ) ) ,whereas NNM I1 to (yl,y2) = (O(&),O(l)).Since the following analysis is carried out under the assumption of y~ being O( 1) and k = 4, the derived solutions can be used to describe the bifurcations associated with NNM I for conditions of internal resonance where solution (4.4.3) fails. Substituting the functional relation y2 = $2(y,) into (4.4.4), and following the methodology developed in section 3.1.3, one obtains the following singular functional equation governing $2:

-

$2'{ Y1

+ (&g/2)[Yl + $2(Y1)I3}

+ @+&0)$2(YI)

+ W 2 ) [ Y 1+$2(Y 1)13 = 0

(4.4.5) This equation is complemented by the following boundary orthogonality condition:

where Y1 is the maximum amplitude of the parametrizing canonical coordinate y 1 during the NNM oscillation. The yet undetermined function $2(y I ) is asymptotically approximated by the series expression A(0)

A(l)

$2(Y1) = Y2 ( Y l ) + Y2 (Yl) + ... where o($)) = 0(1),

O($:")

A(

= 0(y2

1)'

A(1)"

= 0(y2

)=

o(&),y1 is of 0(1)

4.4NNM BIFURCATIONS IN A SYSTEM IN INTERNAL RESONANCE

21 1

$ik),

k 2 2, are of O ( E ~or ) of higher order. The 0) successive approximations y2 (yl), k = O , l , ... are computed by substituting and higher approximations

the above series into (4.4.5)-(4.4.6) and matching respective powers of

E.

O(EO) Approximation

The 0(1) approximation to the modal function, $2(0), is governed by the following set of equations:

In contrast to NNM computations performed in previous sections, in the case AfO)

of internal resonance the approximation y2 is not assumed to be separable

in space and time. Hence, the solution of system (4.4.7) is sought in the form: m

Expression (4.4.8) accounts for the nonlinear interaction between the modes of system (4.4.1) due to internal resonance. Substituting (4.4.8) into (4.4.7) and matching coefficients of equal powers of yl, one obtains the following expressions for coefficients a2(zp+1)(0):

The linear coefficient a21(0) remains undetermined at this order of approximation and is computed by considering O(E) terms in (4.4.5). O ( E ~ Approximation )

The set of equations governing the O(E) approximation, $2(1), is as follows:

21 2


The solution of (4.4.10) is expressed in the series form:

Substituting (4.4.11) into (4.4.1 O), taking into account the O( 1 ) solutions (4.4.8) and (4.4.9), and matching coefficients of respective powers of y l , one obtains the following relations evaluating the O(E) coefficients a2(2p+1)(l),p = 1,2,..., and the undetermined O(1) linear coefficient a21 (0):


213

The O(E) linear coefficient, a21(1), remains undetermined at this order of approximation, since its evaluation requires the consideration of O ( E ~terms ) in (4.4.5). The algebraic equation (4.4.12a) governs the O( 1) coefficient a21(0), In Figure 4.4.1 the dependence of a21(0) on the frequency-detuning parameter o is depicted, for g = 1.0 and Y1 = 0.5. From this plot it is inferred that depending on the value of o the system possesses one, two, or three NNMs. A stability analysis indicates that for (3 2 Ocr = 3.14147gY12 3 0.78536 a NNM branch becomes orbitally unstable and the corresponding solutions are not physically realizable (cf. Figure 4.4.1). All other branches of NNMs depicted in Figure 4.4.1 are stable. At 0 = obif 5 0.648 1 two solution branches are generated in a bifurcation, which are both stable for 0 sufficiently close to zero (i.e., close to the internal resonance). Hence, the present analysis reveals that f o r k = 4 (where NNM I in (4.4.3) ,ftrils], LI bifurcation occurs leading to nzultiple solutions which are take the form of' nonsimilar NNMs when expressed in canonical coordinates; his blfurcation can not be captured by a NNM analysis in physical coordinates. In addition, it is noted that in contrast to the 0(1) analysis where only two terms were required to compute ;2(0), the computation of the O(E) approximation $2(1)(y1) requires the consideration of six nonzero terms. All coefficients in (4.4.11) can be computed by (4.4.12), with the exception of the leading linear coefficient, which is computed on the next order of approximation. Moreover, the above O( 1) and O(E) computations provide exact solutions for the NNM functions 92(0) and $*(I). O(E2)

Approximation

The O ( E ~approximation ) ;2(2) is computed in a similar fashion. However, the required algebraic manipulations become cumbersome and resort to symbolic algebra (package Mathernatica) is required. The resulting

214


Figure 4.4.1 Coefficient a2 I(*) versus frequency-detuning parameter g = 1.0 and Y 1 = 0.5: ( x ) unstable branch.

0 , for

expressions are too lengthy to be reproduced herein, and only a summary of results will be given. It turns out that ten terms are required to compute function 3 2 ) : ) ( 2 ) = 0,

p = 10,11,...(4.4.13)

Moreover, the linear coefficient a21 ( 1 ) of the O(E) approximation is computed by the following relation:


2 I5

(4.4.14) Hence, coefficient a21(1) is expressed in terms of the previously determined O(I) linear coefficient a21(0). Coefficients aqq,+1)(2), p = I , ...,9, are computed explicitly at this order of approximation, but their analytic expressions are not reproduced here. As previously, the linear coefficient a21(2) is not determined at this stage, and O ( E ~terms ) should be taken into account for its evaluation. Combining all previous results, it is concluded that for conditions of 1:3 internal resonance, the system possesses the following nonsimilar NNMs in canonical coordinates:

The first bracket in the above expression contains 0(1)terms, and the second O(E) ones. Note that expression (4.4.15), combined with the previous relations (4.4.8)-(4.4.14), solves completely the problem up to O(E). In Figure 4.4.2 the analytical approximations of the three modes of a system with parameters g = 1.0, Y1 = 0.5, E = 0.01, and 0 = 1.0, are presented (cf. Figure 4.4.1). The depicted modes correspond to a21(0) = 0.2344, 2.32517, and -0.97 134, respectively. Direct numerical simulations of the equations of motion (4.4.4) confirm the existence of the analytical solutions. The following remarks are made regarding solutions (4.4.15). Although solutions (4.4.15) are in the form of NNMs in canonical coordinates, these solutions are nor NNMs when expressed in terms of physical coordinates. Indeed, expressing (4.4.15) by means of transformations xi = y1 + y2, x2 = y1 - y2, one obtains the relations: (4.4.16) X l = y1 + ?2(Yl)% x2 = Y l - $2(y1) To obtain a NNM in physical coordinates, a preliminary inversion of the first of relations (4.4.16) is required, y1 = Y l ( X l > = [ I + ?2(91+1

2 16


(a)

Figure 4.4.2 Analytical approximations $2(o)(yl), ?2(l)(yl) and $2(y1) for the three modes of a system with g = 1.0, Y1 = 0.5, E = 0.01, o = 1.0, and a21(())= 0.2344 (a), 2.32517 (b), -0.97134 (c).


Y1

(b)

Figure 4.4.2 (Continued)

2 17

218


$ m y 1)

Y1

0.015

cf

Y1

-0.005

-0.01 -0.015

Yl

Figure 4.4.2 (Continued)

4.5 STABILITY OF STATIONARY WAVES

219

and then a substitution of the resulting expression into the second of the above equations must be performed, to obtain a functional relation of the form, x2 = [ 1 + $2(*)]-lx1 - $2[(1 + $2(*))-lx1] = ft2(x1) Considering the analytical expression (4.4.15) for function $2(*), it can be shown that inversion [ 1+:2(*)]-1 is, generally, a multivalued quantity, and, hence, does not have the representation of a function. It follows that solution (4.4.15) does not correspond to an NNM x2 = $ z ( x ~ )of the system. [according to Rosenberg's (1966) definition]. Thus, taking into account (4.4.3), it is concluded that the system possesses a single, predominantly antiphase NNM x2 = ;2II(x) in the neighborhood of the 1:3 internal resonance. The derived solutions compute functional relations y2 = y2(y 1) between canonical coordinates, where y1 is assumed to be an O( 1) quantity. It follows that the previous analysis in terms of canonical coordinates examines exclusively motions and bifurcations associated with NNM I, close to the point of internal resonance k = 4. Close to internal resonnace, analytical expression $(x) in (4.4.3) was found not to be valid. Moreover, examining the plot of Figure 4.4.1 it is evident that for large or small values of the frequency-detuning parameter (T (i.e., away from conditions of internal resonance), one obtains that a21(0)+0, and solution (4.4.15) converges to the analytical solution for NNM I in (4.4.3). In contrast to perturbation computations of nonsimilar NNMs carried out in previous sections (where mode nonsimilarities were assumed to be small), solutions (4.4.15) represent modes with strong nonsimilarities. These "nonsimilarities" result due to the assumed n o n s e p a r a b l e O( 1 ) approximations ?2(0)(yl) in space and time, which, in terms, are a consequence of the strong nonlinear interaction between the two modes in internal resonance.

4.5 STABILITY OF STATIONARY WAVES In this section an additional application of the previously developed stability methodology is given, addressing the problem of linearized stability of stationary waves in certain nonlinear partial differential equations. The stationary waves considered in this section are defined by second-order

220


differential equations with analytic first integrals, in analogy to NNMs of conservative finite-dimensional oscillators. Consider, for example, the nonlinear Klein-Gordon equation: 4 x x - Qtt

(45.1)

= p(@)

where p($) is an analytic function. Stationary waves are particular solutions of (4.5.1) of the form (I = @(5),where 5 = x - ut, @(5)is bounded at infinity, and u is the wave velocity. Substituting this expression for @ i n (4.5.1) one obtains the following equation governing @: (1 - u 2 > @ g = p(@)

(4.5.2)

(Ufl)

which possesses the analytical first integral, (1/2)(1

-

uz)O 0, -1 < F < 1 (periodic waves), (b) u2 - 1 > 0, F = -1 (solitary wave), (c) u2 - 1 > 0, F < -1 [spiral waves (Whitam, 1974)], (d) u2 - 1 < 0, -1 < F < 1 (periodic waves), ( e ) u2 - 1 < 0, F = 1 (solitary wave), (f) u* - 1 < 0, F > 1 (spiral waves).

F = (1/2)(1 - u2)@~,2+ 1 - 2sin2(@/2) Depending on the values of F and u, this system possess periodic waves (u2 1 > 0, -1 < F < 1 and u2 - 1 < 0, -1 < F < l), solitary waves (u* - 1 > 0, F


223

= -1 and u2 - 1 < 0, F = l), or spiral waves (Whitam, 1974) (u2 - 1 > 0, F < -1, u2 - 1 < 0, F > 1). It turns out that the most convenient variational form for studying the Sine-Gordon equation is (4.5.8) (i.e., using the kinetic energy as independent variable), which can be directly reduced to the standard form of the Lame' equation (Bateman and Erdelyi, 1952):

(4.5.9) where y1 = 0, y2,3 = F If: 1, H = 2(F - B), n = 1. From section 4.2.2, these parameter values imply that there exist a finite unstable zone and an unbounded unstable region in the parameter plane (F,B) (Bateman and Erdelyi, 1952).

(ii) P(@) = Dsinh@

+ Gcosh@

Considering the variational equation (4.5.8), one obtains a variational problem of the form (4.5.9) with parameters yi = 0, y2,3 = F k 1, H = 2(F + B), n = 1 Again, the parameter plane (F,B) possesses a finite zone and an unbounded region where unstable solutions of the variational equation exist.

(iii) P(@) = a@2/2 + b@4/4 In this case it is appropriate to consider the variational equation in the form (4.5.7). The problem then can be reduced to the standard Lame' equation (4.5.9), with z _= k, y1 = 0, y2,3 = -(ah)k [(a/b)2 - 4F/b]1/2 H = 2(B - a)/b, n=2 Recalling the results of section 4.2.2, the parameter plane (F,B) possesses two instability zones and one unbounded unstable region.

For this type of potential function the variational equation (4.5.6) is considered, which can be reduced to the standard Lame' form (4.5.9) by setting

224


Figure 4.5.2 Stability-instability boundaries of the standard Lame' equation (4.5.9) with, (a) n = 1, and (b) n = 2 (parameters are defined in text); regions of instability are indicated by shadowing. n=2 H = B + 2(y1 + y2 + y3), As in case (iii), the parameter plane (F,B) possesses two instability Lones and an unbounded unstable region. Additional classes of potentials P(CD), which under appropriate coordinate transformations lead to Lame' variational equations are discussed i n (Manevitch et al., 1989). Knowledge of the regions where the solutions of the Lame' equation (4.5.9) with n = 1 or n = 2 are bounded or unbounded enable one to study the stability in time of the corresponding nonlinear stationary waves. In Figures 4.5.2(a) and (bj the stability-instability boundaries of the Lame' equation with n = 1 and n = 2 are depicted. The parameters 3, = (e2 - e3)/(ei - e3) and p = [H - n(n + l)e3]/(el- e3) are employed in these graphs, with e l > e2 > e3 representing the ordered parameters y1, y2, y3. The stability of the corresponding nonlinear stationary wave is determined as follows. For each stationary wave solution one

CD

I k,


225

Figure 4.5.3 Stability-instability boundaries of the standard Lame' equation (4.5.9) with n = 1 in the (B,F) parameter plane; regions of instability are indicated by shadowing. establishes if bounded solutions of the associated variational equation exist with s2 > 0; the existence of such solutions is an indication of wave instability. When no bounded variational solutions exist with s* > 0, one establishes linearized stability of the stationary wave under consideration. As an example, consider the Sine-Gordon equation [case (i)] with D=O and G=l. The domains in which the solutions of the variational equation (4.5.9) are bounded or unbounded are presented in Figures 4.5.2(a) [in terms of (h,p) coordinates] and 4.5.3 [in the (B,F) plane]. It was previously pointed out that, depending on the value of F and the sign of (1 - u2), the solutions of the Sine-Gordon equation can be in the form of periodic waves, spiral waves or solitons. From Figure 4.5.3 it is seen that when F > -1 bounded solutions exist for B < 0 and B > 0 alike (leading to positive and negative corresponding values of s2), which indicates that this class of periodic and spiral waves [cases (a),(d) and (f) in Figure 4.5.11 are unstable with respect to perturbations that decay at infinity. However, for spiral waves with F < -1 and (1 - u2) < 0 [case (c) in Figure 4.5.11, bounded solutions of the variational equation exist only for B < 0 [corresponding to s* I 0 since (1 u2) < 01 indicating linearized stability of these waves when perturbed by small variations that decay at infinity.

226


-

I

Figure 4.5.4 Solitary wave of the Sine-Gordon equation corresponding to F = -1. The linearized stability of the solitary waves of the Sine-Gordon equation corresponding to F = +1 [cases (b) and (e) in Figure 4.5.11 is now examined. The solitary wave corresponding to F = -1 appears in Figure 4.5.4. The corresponding variational equation is considered in the form (4.5.8) and is expressed as follows:

2k2(2 f k)W&

+ k(4 f 3k)Wk -+_ (-B F I

- k)w

=0

(4.5.10)

The hypergeometric equation (4.5.10) possesses the regular singular points k = 0, +2, 00. In the neighborhood of the regular singular points the solution of the variational problem is expressed as follows: Point k = 0 3 w - ko, 01,2 = fp/2 where p = (1 - B)1/2 for F = 1, and p = (1

+ B)1'2

for F = -1

* -

Point k = 5 2 w (k 5 2)0,(31 = 0, (32 = 112 Point k = 00 3 w k-o,01 = -1, 02 = -1/2 In the above expressions the quantity B is defined as B = s2/(u2 - 1). Since the solution of the variational equation is required to be bounded at infinity, the singularity index (3 = -p/2 of solutions in the neighborhood of k = 0


227

must be dropped. Moreover, it can be shown that the index 0 = 0 of solutions close to k = +2 (which represent the maximum values attained by the kinetic energy) corresponds to arbitrary shifts along the c-axis; since the original Sine-Gordon equation is invariant with respect to such shifts, the index (3 = 0 may also be neglected with no loss of generality. Hence, the general solution of (4.5.10) is expressed in the form: w(z) = zP/2 (z+2)1/2 f(z)

(4.5.1 1)

where f(z) is an analytic function. This represents a degenerate solution of the hypergeometric equation (4.5.10) (Bateman and Erdelyi, 1952). Since the behavior of w(z) at infinity is prescribed (w z-0 with 01 = -1 or 0 2 = -1/2), function f(z) may, at most, be an mth order polynomial. Comparing the order of the solution as Izl+w with the indices of the singular points at infinity, one obtains the following relations satisfied by m:

-

p/2

+ 1/2 + m = 1

or

pi2

+ 1/2 + m = -1/2

(4.5.12)

Since p > 0, it is evident that only the first of the above equalities can be realized at m = 0, whence p = 1. This, however, is possible only when s = 0, and, thus, the solitary waves under consideration are linearly stable.


CHAPTER 5 RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs In many engineering applications, such as modal analysis and vibration isolation, there is a need for determining the steady-state responses of periodically forced mechanical structures. Information on the steady dynamical motion of a forced structural component is important for designing against large-amplitude resonant motions, which, if uncontrolled, may result in early failure. Numerous approximate analytic techniques for computing the steady dynamic response of nonlinear discrete oscillators exist in the literature. In contrast to linear systems where the dynamic response can be expressed as a simple superposition of individual normal modal responses, when nonlinearities are present transfer of energy may occur between NNMs resulting in a variety of nonlinear resonance phenomena having no counterparts in linear theory. Extensive studies of the dynamic responses of harmonically excited, nonlinear, discrete or continuous systems can be found in (Nayfeh and Mook, 1984) and (Bogoliubov and Mitropolsky, 1961), where different types of nonlinear forced resonances have been analyzed. Denoting by W j the jth linearized natural frequency of an n-DOF nonlinear system and by 03 the frequency of the external excitation, a variety of resonances can occur: fundamental (co = q,for some p), subharmonic (o = kwp for some k), superharmonic (03 = cop/k), combination (mco = k l w l +...+ knWn for a set of integers m,kl,...,kn), and internal ( k i o l + ... + knOn = 0). The majority of perturbation techniques for analyzing nonlinear resonances are based on the assumption of weak nonlinearity and regard the nonlinear response as a perturbation of a (linear) harmonic one. Under this assumption, the nonlinear forced motion is, in general, approximated in a series whose leading term corresponds to the unperturbed linear solution. It should be clear that such approximate solutions fail to accurately model the dynamics in cases of strong nonlinearities, since then, the nonlinear solution cannot be treated as a mere perturbation of a linear response. To analyze such strongly nonlinear cases alternative analytical techniques should be developed. 229

230

RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs

In the aforementioned references special forcing functions were considered for analyzing exact steady-states. A basic feature of nonlinear undamped discrete systems is that, depending on the form of the excitation, they may possess multiple steady-state solutions. A basic general question therefore arises: Suppose that a nonlinear discrete oscillator is forced b y a periodic excitation. Under what conditions will this force produce an exact steady-state motion? Evidently, the required conditions should depend on the degree of the nonlinearity and on the structural parameters of the system. In addition, steady-state motions of undamped nonlinear systems are materialized only for specific sets of initial conditions, since in systems with no damping initial transients do not decay with time. As a result, the exact steady-state motion can be realized only with appropriate selection of the initial conditions. Hence, the following two additional sub-problems arise: first, derive the necessary and sufficient conditions that n periodic force must satisfy to produce an exact steady-state; second, given such an admissible periodic excitation, compute the specific set of initial conditi0n.s that eliminate the initial transients in the response and give rise to a periodic steady-stare motion. In section 5.1 a general methodology for addressing these problems is outlined. Some applications with forced two-DOF systems are presented in section 5.2. Finally, in section 5.3 the effects of NNM bifurcations on the topological structure of the forced resonances of a discrete oscillator are examined.

5.1 EXACT STEADY-STATE MOTIONS In this section the concept of exact steady-state motion is employed to study the forced response of nonlinear discrete oscillators. This concept was first introduced by Rosenberg ( 1966); some preliminary definitions are appropriate. A function s(t) is said to be cosine-like if (Rosenberg, 1966): s(t) is analytic in --oo < t < +m s(0) # 0, S(0) = 0 s(t) is periodic in t with least period T, and s(T/4) = 0 s(t) = -s(T/2 - t) for every t, and s(t + 6) < s(t) for 0 < t < t Consider now the following n-DOF nonautonomous system:

ji + f(x) = g(t),

f, g E Cr,

r21

+ 6 < T/4

5.1 EXACT STEADY-STATEMOTIONS

where 21, f, g

E

23 1

Rn, and g(t) is a cosine-like vector. The solution x ( t )

corresponding to initial conditions ~ ( 0=) X,$0) = Q, is said to be an exact steady-state motion (Rosenberg, 1966), if and only if, all elements of x(t) are cosine-like of the same period as g(t). Then the system vibrates in unison, and all positional variables reach their extreme values or pass through zero at the same instant of time. Hence, on an exact steady-state the system vibrates, in essence, as in a NNM, and the forced problem is transformed to a pseudo-autonomous one. An interesting question concerns the effect that the NNMs have on the exact steady-state motions. In classical linear theory, any forced response can be expressed as a superposition of modal responses. Moreover, linear steadystate motions (resonances) always occur in neighborhoods of normal modes. In the nonlinear case the principle of superposition generally fails. However, as shown by Rosenberg (1966), and Yang and Rosenberg (1968), resonant motions in multi-DOF systems also occur in neighborhoods of NNMs. Hence, although forced nonlinear responses cannot be expressed as superpositions of modal responses, certain forced nonlinear resonances occur close to NNMs, in direct analogy to linear theory. This feature indicates that the computation of NNMs is of significant practical importance in the theory of nonlinear oscillators. Rosenberg also showed that, for a system vibrating in an exact steady-state, the oscillation is represented by a single line or curve in the configuration space. Depending on the form of the corresponding modal line, the steady-state is termed similar (corresponding to a straight modal line), or nonsimilar (possessing a curved modal line). The general problem of the existence of similar exact steady-states was addressed in (Kinney, 1965) and (Kinney and Rosenberg, 1965, 1966), where special cam-functions were used as exciting forces. Subsequently, geometrical methods were used in the configuration space to detect and compute the modal lines of the forced motion. In the same references, a homogeneous two-DOF system with cubic nonlinearity was examined. Elliptic forcing functions were applied to the system, and it was shown that as many as five steady-states may exist at certain ranges of the frequency of the external excitation. The extension of these results to nonhomogeneous systems was presented in (Caughey and Vakakis, 1991), where it was found that the topological portrait of the resonance curves representing similar steady-state motions changes when a bifurcation of the normal modes of the

232


unforced system occurs. In such systems, a variation of a structural parameter may lead to an increase of the number of resonance branches. The only works that the authors were able to find on the problem of norzsinzilur steady-states were those by (Kinney, 1965) and (Mikhlin, 1974); in these works a set of functional equations for the derivation of the curved modal line describing an exact steady-state was given. These equations become singular at the end points of the modal lines, and an asymptotic methodology for approximating the modal curve at low amplitudes is needed. In the same references, specific applications of the asymptotic technique were given for two-DOF systems with cubic nonlinearity, excited by elliptic-cosine functions. As mentioned previously, forced systems undergoing exact steady-state motions behave similarly to single-DOF conservative systems. Rauscher's (1938) ideas are used in the construction of the nonlinear resonances, and the corresponding trajectories of the forced responses in configuration space are analytically approximated. Consider the following nonautonomous n-DOF system:

xi + dV(x)/dxj + Efi(x1,il,...,xn,xn,t) = 0,

i = 1,2....,n (S.1.I)

The functions f j are assumed to be analytical in xk, xk, k = l , ...,n, and periodic in t with period T; the potential function V(x) is subject to the limitations listed in chapter 3, and lel o J ( o ) ( x ) l I+(I) This can be regarded as the O(E) correction of the O(1) equation V(Xj(O),O,...,0) = ho which was used previously for computing Xj(0). Note that the periodicity condition (5.1.19) is not solvable unless the energy estimate ho and the amplitudes Xj(0) are simple roots of the corresponding O( 1) equations. This requirement, however, implies that the unperturbed system cannot be linear (otherwise, due to the isochronicity of linear vibrations the energy-amplitude relations possess multiple roots). Moreover, the case of additional equilibrium points occurring at the maximum potential surface of the steady-state motions should also be excluded. Higher order approximations are computed by extending the previous iterative procedure and considering terms of higher order in E. In the following sections applications of the methodology are given by analyzing the resonances of two-DOF forced nonlinear oscillators. The interesting problem concerning the class of "admissible" periodic excitations leading to exact steady-state motions is also addressed in these sections. 5.2 ADMISSIBLE FORCING FUNCTIONS FOR STEADY-STATE MOTIONS As an example of application of the analytic methodology developed i n

section 5.1, consider the following n-DOF undamped nonlinear system, forced by n external excitations Epi(t):

5.2 ADMISSIBLE FORCING FUNCTIONS FOR STEADY-STATE MOTIONS

xi = fi(x1, ...,xn) + EPi(t), subject to the set of initial conditions: xi(0) = Xi,

ii(0) = 0,

239

i = 1,...,n

(5.2.1)

i = 1,...,n

(5.2.2)

This system is of the general form (5.1.1). Forces EPi(t) are assumed to be weak and periodic with least common period T. Suppose that system (5.2.1) oscillates at an exact steady-state. Then, one can eliminate the time variable by symbolically writing: x i = xi(t,&)

*

t = t (XI,&), t E [ O,T/2 )

(5.2.3)

As a result, the following pseudo-autonomous system (cf. section 5. I ) replacing (5.2.1) is obtained:

x.1 - f.,(XI,...,xn) + E P ~ [ ~ ( X ~=, Efi(xl,...,xn> )] + EGi(X1,E) i=l, ...,n

(5.2.4) where the notation Epi(t(x1,E)) = &$*(XI,E) is used. The exact nonsimilar steady-states of the original system (5.2.1) correspond to nonsimilar NNMs of the pseudo-autonomous system (5.2.4); hence, the problem of the computing forced exact steady-state motions is converted to the equivalent problem of computing the nonsimilar NNMs of the pseudo-autonomous system. The nonsimilar NNMs of system (5.2.4) are expressed as X i = A A Xi(XI), i = 2,.,.,n, where the nonlinear functions Xi(*) are computed using the methodology of section 5.1. A specific calculation of exact steady-state motions is now carried out considering the following two-DOF system.

+ X I + X 3I + Kl(xl - x2) + K3(xl - x2)3 = Ep(t) 3 x2 + x2 + x2 + Kl(x2 - xi) + K3(x2 - x1)3 = 0

x1

(5.2.5)

where the stiffness coefficients K1 and K3 are assumed to be O(1) positive scalars. When E=O the unforced system possesses the similar NNMs x2 = cxl, with c = +1 (in-phase mode), or c = -1 (antiphase mode). When E # 0, the NNMs are perturbed by the excitation, and the system becomes weakly

240


nonconservative. At an exact steady-state the following pseudo-autonomous system is obtained, Xi

+ X I + X 3I + Kl(x1 - ~ 2 +) K3(X1 - ~ 2 = )$(XI)~ 3 x2 + x2 + x2 + Kl(x2 - X I ) + Kg(x2 - x1)3 = 0

tE [O,T/2) (5.2.6) and the forced motion is represented in the configuration plane by the modal curve x2 = 22(x1), governed by the following equations:

This equation is complemented by the orthogonality condition:

$m){ X I + x: + ~ 1 x 1f-t 2 ( ~ 1 ) ~ 1~+ 3 ~ x:2(x1)13 1-

- EE;(xI)

+ 22(X1) + ?2(X1)3 + K122(X1) - KlXl + K3 [ft2(X1) - X1]3

=0

1

(5.2.8) In what follows, the solution of (5.2.7) and (5.2.8) is asymptotically approximated by a series expansion. The zeroth order approximation 22(o)(x 1 ) corresponds to the similar NNMs of the unforced oscillator with E = 0:

Moreover, the zeroth order approximation for the time response, xl = xl(t), is computed in closed form in terms of an elliptic cosine:


24 1

The quantity Xi0 denotes the zeroth order approximation to the amplitude of oscillation X i , and is a yet undetermined quantity. To compute Xi0 one requires that oscillation (5.2.10) is of period T: 0 = xq/2K(k) =

2dT

(5.2.1 1)

where o is the frequency of oscillation in (rad/sec) and K(*) is the complete elliptic integral of the first kind. Relation (5.2.11) is a transcendental equation determining the amplitude X 10. Considering O(E) terms in (5.2.7) and (5.2.8), one obtains the following functional equations governing the first-order approximation, $2(l)(x I):

Term Fo in the equations above denotes the first-order approximation to function f; and is explicitly computed by performing the following inversion of the zeroth order solution (Byrd and Friedman, 1954): xl/Xlo = cn(qt,k) =+ t = t(x1) = F ( sin-I[1 - (xl/Xl0)~]1/2,k)/q (5.2.14) and fio(x1) = p[t(xl)] = p[F{ sin-l[l - (~1/Xlo)2]1/2,k}/q]

(5.2.15)

In the expressions above, F(*,*)is the incomplete elliptic integral of the first kind. Substituting $o(xl) into (5.2.11) and (5.2.12) one computes the firstorder approximation f i 2 ( l ) ( x i ) . It turns out that the resulting expression is

242


too complicated to be of any practical use, and a modified approach must be followed. To this end, a transformation of variables is introduced in terms of the amplitude function, am(*,*) (Byrd and Friedman, 1954): cn(u,k) = cos$

+

$ = am(u,k)

(5.2.16)

The variable $ (not to be confused with the phase variable of section 5.1) will be regarded as the new independent variable of the problem, and will replace the parametrizing variable xi. Making use of (5.2.16) one obtains the following relation between $ and t: cn(qt,k) = cos$

3

$ = am( qt,k )

(5.2.17)

which, when inverted leads to the following analytic expression for t:

Employing the above relation one can eliminate the time variable from the expression of the forcing function as follows: (5.2.19) Expression (5.2.19) represents the first-order approximation to the periodic forcing function. Variable xi can also be expressed in terms of the new variable $: * X l =x1($) = XlOCOS$ (5.2.20) Hence, variable $ eliminates entirely x i from the problem and can be regarded as a new time-like independent variable. An alternative analytical representation for the forcing function is obtained by expanding &)($) in generalized Fourier series with respect to variable $ (Bejarano and Sanchez, 1988,1989; Margallo et al., 1988). Referring to equation (5.2.18) and taking into account the properties of the incomplete elliptic integral of the first kind, it can be shown that the following correspondence between the variables t and $ exists:

5.2 ADMISSIBLE FORCING FUNCTIONS FOR STEADY-STATE MOTIONS t E [O,+T/2)

j

$ E [O,+n), and

t

E

[-T/2,0)

243

$ E [-n,O) (5.2.21)

j

Clearly, in each of the above time intervals, representation t = t(x1) has meaning, i.e., is a single-valued function. Moreover, the above relations coupled with the assumption that the forcing function Ep(t) is periodic with period T, indicate that function &)($) is periodic in $ with period 2 x . Therefore, ;o($) can be expanded in generalized Fourier series as follows:

where coefficients An and Bn are computed by the well-known Fourier series formulas: x n An = (I/x) ;o($)cosn$ d$ AO = (112x1 SO($> d$,

J

J

-x

-x

n

Bm = (l/n)

$o($)sinm$ d@

(5.2.23)

-x

The function k2(l)(xl) is now expressed in the following series form:

which, in view of (5.2.20) can be rewritten as:

Substituting expressions * X l = Xl($>, bO(X1) = $o($) and $2(1)(x1) = Z2(1)($) into the functional equations (5.2.12) and (5.2.13), one obtains the following set of equations containing only trigonometric terms in terms of @:

244


(5.2.27) In the expressions above, terms of O(x17) = O(cos7$) or of higher order are omitted, and coefficients TiCi) are defined as:

For an exact steady-state motion to occur, the above expressions should lead to real solutions for coefficients a2j(l). These coefficients are computed by matching coefficients of respective powers of cos@ and sin$. Before proceeding to this matching, however, there is a need to expand the trigonometric terms cosn@and sinn@ in powers of cos@and sin@. The following are noted regarding the transformed set (5.2.26) and (5.2.27). (1) For an exact steady-state to occur, the coefficients of the sine ternis of the generalized series (5.2.22) must be equal to zero:

Bm = 0,

m = 1,2,3,...

(5.2.29a)

This is due to the fact that terms in (5.2.26) proportional to powers of sin$ cannot be balanced by any real values of the coefficients ,2j(l). In fact, condition (5.2.29a) is equivalent to the statement that the steady-state response of the oscillator is either in-phase or completely out-of-phase with the excitation in the absence of damping (Vakakis, 1990). (2 A second restriction on the coefficients of the Fourier series (5.2.22) results from the fact that there exist only odd powers of cos$ in the


245

functional equations (5.2.26) and (5.2.27). Hence, it is necessary that the Fourier series of &)($) not contain any even cosine terms: A2, = 0,

J=0,1,2 ,...

(5.2.29b)

This condition is due to the existence of only odd stiffness nonlinearities in (5.2.5) and (5.2.6) and leads to the elimination of even powers of cos$ in (5.2.26) and (5.2.27). In particular, for j = 0 condition (5.2.29b) leads to the expression: n n 60($) d$ = j p(F($,k)/q) d$ = 0 (5.2.30)

J

-n

-ll

This equation is the nonlinear equivalent of the following (trivial) condition that is satisfied by periodic forces in linear steady-state motions: TI2

p(t) dt = 0

(linear theory)

(5.2.3 1 )

-TI2

In fact, one can show that when no nonlinearities in the equations of motion (5.2.6) exist, condition (5.2.30) degenerates to relation (5.2.3 1). Note, however, that condition (5.2.30) does not imply (5.2.31). Summarizing, for an exact steady-state motion to exist, certain restrictions on the form of the periodic excitations should be imposed. These are necessary conditions for the existence of a steady-state motion and are given by (5.2.29a,b). It can be also shown that once these conditions are met, one can always compute real values for the coefficients a2j(l). TO this end, suppose that the system is forced by a periodic excitation satisfying conditions (5.2.29a,b). Then it will be shown that, sufficiently close to a similar NNM of the unforced (unperturbed) system, an exact steady-state motion occurs. Hence, sufficiently close to an NNM of the unforced system, relations (5.2.29a,b) can also be proven to be sllfficient for the realization of an exact steady-state motion. It must be noted, however, that in systems with stiffness nonlinearities a variety of additional dynamic responses is possible, such as aperiodic, subharmonic or ultra-harmonic motions (Vakakis, 1992b; Vakakis and Rand, 1992). However, only the special class of exact steady-state motions are considered herein.

246


For weak excitations and sufficiently close to the NNM, x2 = cxl, an exact steady-state motion is described in configuration space by:

The coefficients a2j(l) are evaluated by matching coefficients of the various powers of cos$ in expressions (5.2.26) and (5.2.27). Details of this computation can be found in (Vakakis, 1990), and a synopsis of the analytic results is given below:

(5.2.33)

S1(') = A1 - 3A3 + SA5 -...,

S3(1)= 4A3 - 20A5 + ... S 5 ( l ) = 16A5 +... (5.2.34)


247

In the expressions above, A1, A3, A5, ... are the generalized Fourier coefficients defined in (5.2.23), whereas TiGI are defined in (5.2.28). Since constant c in the above expressions can take the values ( k l ) , two possible exact nonsimilar steady-state motions exist, each occurring in a neighborhood of a NNM of the unforced system. The responses of the system on an exact steady-state can be evaluated by substituting the modal relation (5.2.32) into the first of the equations of motion (5.2.6), and integrating the resulting expression by quadratures. To perform this computation one must eliminate all trigonometric terms in the expression of &)(@) by expanding them in powers of cos@ and making use of formula (5.2.20). The following asymptotic approximation for the forcing function is then obtained:

Finally, an improved approximation to the amplitude of the steady-state oscillation, X 1, can be derived by imposing the requirement that the period of the steady motion be equal to T. Details for this computation can be found in (Vakakis, 1990). The linearized stability of the computed steady-state motions can be examined by numerically computing its Floquet multipliers (Vakakis, 1990). The results of this section can be summarized in the form of a theorem. Theorem 1. Consider the dynamical system (5.2.5). Provided that the excitation is sufficiently small, and the initial conditions are given by (5.2.2), a necessary and sufficient condition for the existence of exact steady-state motions in neighborhoods of NNMs of the unforced system is that the generalized Fourier series of the excitation is of the form: m

(5.2.36a) where 7c

A2j+l = ( l h )

-n

$o($)cos(2j

+

l)$ d$

(5.2.36b)

248


and function &J($) is evaluated by the expression:

In the above equations, F(*;) is the incomplete elliptic integral of the first kind, and quantities q and k depend on the structural parameters of the oscillator and on the period T of the external force. Moreover, at the exact steady-state the system generally oscillates as in a nonsimilar NNM. The following remarks are made regarding Theorem 1 . Implicit in the proof of the theorem is the requirement that the NNMs of the unperturbed system are not in a low-order internal resonance. If conditions for internal resonance exist, the unperturbed NNMs become nonlinearly coupled and cannot be expressed by simple functional relations of the form x2 = x2(xl) (cf. section 4.4). Although the theorem is stated for a specific set of initial conditions, this does not restrict its validity. Indeed, an identical analysis can be carried for different sets of initial conditions, with different restrictions, however, on the "admissible" forcing functions &)($). The theorem can be generalized easily to systems with more than two DOF. In that case more than one functional equation and boundary orthogonality conditions must be considered, but the basic steps for the analysis remain unaltered. In addition, the theorem can be extended to systems with a more general class of odd stiffness nonlinearities. In that case, the incomplete elliptic integral of the first kind in the argument of the forcing function &p(t)should be replaced by a (possibly untabulated) incomplete integral. Numerical applications of the theorem were carried out by considering the following two specific forms for the forcing function: Eplqt) = EPlcoswt and ~p2II(t)= &(P2/2)tan-l[2a(l - a2)-kosot]

(5.2.37a)

Both forcing functions satisfy the conditions of Theorem 1 and lead to exact steady-state motions. To compute the asymptotic approximations to the steady-states, one must first evaluate the generalized Fourier coefficients o f


249

3

t (sec)

Figure 5.2.1 Nonsimilar steady-state oscillation with forcing function Epl(t) = EplI(t): response in (a) configuration plane and (b) in time domain. the excitations. The first-order representations of the forcing functions are

(5.2.37b) The generalized Fourier coefficients of (5.2.37b) were computed by numerically integrating expressions (5.2.37b) for E P =~ 0.1, c = +1 and 61 =

250


A

Excitation Response

vvvvvvvvvvvvvv 10

i0

30

10

i (sec)

Figure 5.2.2 Nonsimilar steady-state oscillation with forcing function &pI(t)= &plII(t):response in (a) configuration plane and (b) in time domain. were 1.25. The leading nonzero generalized Fourier coefficients of &$(I$) computed as A1 = 0.101498, A3 = -0.001543, A5 = 0.000046, etc. The amplitudes of the positional variables for the steady-state motion are then computed as (XlI,X21) = (0.930049,0.901222). To check the accuracy of the asymptotic solution, the nonautonomous system (5.2.5) was numerically integrated using the theoretically predicted initial conditions, (xi (O),x i ( 0 ) ) = (XlI,O), (x2(0),x2(0)) = (X21,0), the forcing function ~ p ~ (=t &piI(t), ) and

parameters K1=1.3, K3=0.7. The results of the numerical simulations are depicted in Figure 5.2.1, from which the existence of an exact nonlinear steady-state motion is confirmed. Similar calculations were performed for


25 1

forcing function EplII(t). The results of the numerical integration of the equations of motion using the theoretically predicted initial conditions are depicted in Figure 5.2.2, for K1 = 1.3, K3 = 0.7, E P =~ 0.15, c = +1, o! = 0.5, and o = 1.25. As an additional application of the theorem, consider the system (5.2.5) with a periodic forcing function proportional to displacement x 1 , in the form &p(t)= &(P/Xl)xl(t).This type of excitation can be shown to satisfy the conditions of the theorem. The corresponding pseudo-autonomous system assumes the following form: XI

+ X I + X 3I + Kl(x1 - ~ 2 +) K3(XI - ~ 2 ) =-&XI 3 3 x2 + x2 + x2 + Kl(x2 - XI) + K3(x2 - x1)3 = 0

tE [O,T/2) (5.2.38) where d = -(P/Xl). The nonsimilar exact steady-state motions of (5.2.38) are computed as: 22(x1) = (c + &a21(l))xi+ &a23(1)x13+ ~a25(1)x15+ o ( E x ~ ~ , E ~ ) (5.2.39) where the coefficients a2j(l) are evaluated by (5.2.33) and (5.2.34) with generalized Fourier coefficients given by A1 = -d, A2j+l = 0, j = 1,2,... Setting c = +1, one obtains exact steady-state motions in neighborhoods of the in-phase and antiphase NNMs of the unforced system. The oscillation of the parametrizing variable xi = xl(t) on an exact steady-state is obtained by substituting (5.2.39) into (5.2.38) and integrating by quadratures: XI

t = t(x1) = k J [I1(Xl2 -

c2)+ (13/2)(X14- c4)+ (15/3)(xl6

XI

+ O(&x17,~2)]-”~d~ =k

I G(Xl,d,c) d5

- 56)

XI

(5.2.40)

XI

where I1 = 1 + d - a21(1) + K1(1 - c), 13 = 1 - K1a23(1)+ K3(l - c - a21(1))3 Is = - Kla25(l) - 3K3[1 - c - a21(’)l2a23(1) In the above expression the response in given in the “inverted” form t = t(x1). The frequency response curves associated with the nonsimilar steadystate motions are computed by expressing the frequency of the forced oscillation as a function of the displacement ampiitude XI:

252


w (radlsec)

w (rad/sec)

Figure 5.2.3 Frequency response curves for the nonsimilar steady-state motions: _ _ asymptotic solutions, (*) numerical (exact) solutions.

n:

o = o(Xl,d) = -

5 XI

(5.2.41)

G(X 1A C ) dk

where o is the frequency in (radhec). Details of this calculation can be found in (Vakakis, 1990). In Figure 5.2.3 the frequency response curves of a system with P = 0.1, K1 = 1.3, and K3 = 0.6 are depicted. To check the

5.3 EFFECTS OF NNM BIFURCATIONS ON THE RESONANCES

253

accuracy of the asymptotic solutions the equations of motion (5.2.5) were numerically integrated using a fourth-order Runge-Kutta algorithm and the exact initial conditions for the nonsimilar steady-state motions were obtained. Satisfactory agreement between the asymptotic and numerical results was observed. A numerical stability analysis based on computations of Floquet multipliers indicates that branches (BA) and (ED) represent unstable steady-state motions. A general conclusion of the previous analysis is that the concept of NNM can be extended to study the forced response of nonlinear discrete oscillators. The exact steady-state motions of these systems can be regarded as perturbations of NNMs, provided that the system is excited by suitable admissible periodic forcing functions and is initiated at appropriate initial conditions. Although harmonic functions are included in the general class of such "admissible" excitations, these are not the only possible forcing functions giving rise to exact steady-state motions. This fact reveals a limitation of the majority of standard analytical methods, which consider mainly harmonic excitations and assume predominantly harmonic responses. No such assumptions were made in the previous analysis. A further advantage of the presented method over other standard techniques lies on the use of elliptic functions (instead of harmonic ones) as zeroth order approximations of the steady motions. Hence, the nonlinearities of the system are taken into account in the zeroth order approximation, a feature that is expected to lead to more accurate results compared to alternative averaging or asymptotic methods that assume harmonic zeroth order approximations.

5.3 EFFECTS OF NNM BIFURCATIONS ON THE RESONANCES In previous sections it was established that exact steady-state motions of forced oscillators occur in neighborhoods of its NNMs. As a result, NNM bifurcations affect significantly the topological structure of the nonlinear frequency response curves associated with the steady-state motions. In what follows, this feature will be demonstrated by analyzing the similar steadystate motions of a periodically excited two-DOF with bifurcating NNMs. The equations of motion of the system under consideration are assumed in the form:

254


where p(t) is a periodic force of period T, the stiffness coefficients fij are assumed to be nonnegative quantities, and m is an odd positive integer. At a critical value of the stiffness ratio K = f2m/fl m the unforced system undergoes a NNM bifurcation, which affects the forced response. To study the exact steady-states, the following functional relation between force and displacement is assumed:

Substituting (5.3.2) into (5.3.1), and imposing the condition for similar NNMs x2 = cxl on the resulting pseudo-autonomous system, one obtains the following expressions:

The equations above are solved with initial conditions X I ( 0 ) = XI, x2(0) = cx1, XI(0) = 0, X2(0) = 0 and lead to identical solutions for xi provided that the following algebraic condition is satisfied:

The roots of the above relation are the required values of c for similar exact steady-state motions. Once these values are determined, the oscillation of the parametrizing variable xl is determined by integrating any one of equations (5.3.3). The frequency-response curves are determined by the following relation (Caughey and Vakakis, 1991):


255

Figure 5.3.1 Roots of equation (5.2.6) for (a) P = 0, (b) P/X1 > 0, and (c) PIX1 < 0.

Expression (5.3.5a) relates the amplitude Xi of the steady-state motion to the frequency of oscillation o for a given value of the forcing amplitude P, and it represents the nonlinear resonances of the system (which are analogous to the linear resonance curves of classical vibration theory). For P = 0 (no forcing), (5.3.5a) provides the so-called backbone curves. These relate the amplitude to the frequency of NNM motions and are equal in number to the NNMs of the unforced system. As discussed in section 1.2, for P = 0, equation (5.3.4) always possesses the in-phase and anti-phase NNMs c = f l . In addition, when K = f2m/f1m < Kc = 21-m

(m- 1)/2

C (-1)’k-l

k= 1

an additional pair of reciprocal NNMs bifurcate from the antiphase NNM, after which the bifurcation becomes unstable. Hence, for sufficiently small coupling forces, system (5.3.1) possesses more NNMs than DOF, with at

256


least one of the NNMs (the antiphase one) being unstable; these features have no counterparts in linear theory. For cubic nonlinearity, 111 = 3, the NNMs of the unforced system are depicted in Figure 1.2.2(b). Analytical and numerical computations of the nonlinear resonance curves were performed for a system with cubic nonlinearities. For m = 3 relation (5.3.4) assumes the following form: f13

+ f23(1 - c)3 - (p/x1)3 = f13c2 - f23(1 - c)3/c = p3

(5.3.6)

In Figure 5.3.1 the roots of this equation are schematically presented as functions of the coupling parameter K = f23/f13, for fixed values of P and X 1. For zero forcing a Pitchfork bifurcation of NNMs at K = 1/4 exists, and, depending on the value of K , the oscillator possesses two or four NNMs. For nonzero forcing the Pitchfork bifurcation is perturbed, and for sufficiently small values of the ratio (P/Xl) there exist two (for K large) or four (for K small) roots for c. This result indicates that the ropologicnl portrait of the resonance curves of the forced oscillator change3 03 the coupling parameter K varies. This feature is verified by numerical computations. The solutions of equations ( 5 . 3 3 ) for m = 3 and initial condition? x l ( 0 ) = X I , Xi(0) = 0, are analytically computed as follows:

where cn(*,*) and sn(*,*) are elliptic functions (Byrd and Friedman, l956), K(=) is the complete elliptic integral of the first kind, and kl,k2 are elliptic

It will be shown that there are no values of p3 in the range p3X12/fl 1 < -1, and, hence, the possibility of unbounded responses is eliminated. This is in


257

accordance to physical intuition. The frequency-response curves of the system are determined by employing (5.3.5a) with m = 3: (5.3.9a)

In Figures 5.3.2 and 5.3.3 the nonlinear frequency response curves are presented for P = 0.5, and f l l = f13 = 1. The plots of Figure 5.3.2 correspond to K = 0.4 (where only two NNMs exist), whereas those of Figure 5.3.3 to K = 0.15 (where the unforced system has four NNMs). Note the difference in the topology of the two sets of response curves. In the plots of Figure 5.3.2 there exist, at most, five steady-state motions for any given value of the frequency w, of which at most three are orbitally stable (i.e., physically realizable) motions. By contrast, in the response curves of Figure 5.3.3 as many as nine steady-state motions can occur at any given frequency, of which at most four can be stable. The stability of the steady-state motions was detected by a linearized analysis (Vakakis, 1990). The plots of Figures 5.3.2 and 5.3.3 show that the NNM Pitchfork bifurcation of the unforced oscillator greatly affects the topological structure of the nonlinear resonances of the forced system. At higher frequencies, two branches of nonlinear steady-state motions are detected close to each backbone curve, of which only one is stable. Exceptions to this rule are the two resonance branches occurring in the neighborhood of the unstable antiphase NNM of Figure 5.3.3, which are both unstable. Hence, no stable forced resonances are found to exist close to the unstable NNM. As far as the steady-state responses of the system are concerned, one has to consider three distinct cases, depending on the value of the ratio (olfl11’2). If w/fl I 1/2 > 1, then p3 > 0 and the response is hardening [cf. equations (5.3.6) and (5.3.7)]. If O)/f111/2 < 1, then -f11/X12 < p3 < 0, and the response is softening. One obtains the limiting relations

from which one finds that

258


It follows that no unbounded motions can exist, since this would require that p3 < -fli/X12, which, in view of the previous limiting relations, would imply negative (and, thus, physically unrealizable) frequencies of oscillation. Finally, it is interesting to note that when w/fl I 1'2 = 1, one obtains p3 = 0, and the strongly nonlinear oscillator possesses a harmonic exact steady-state response:

The necessary forcing giving rise to this steady-state is also harmonic:

It must be pointed out, however, that this harmonic steady-state, although an exact nonlinear solution, is valid only at the specific frequency o = fl1112. Small perturbations of the frequency away from this value lead to hardening or softening nonharmonic responses. The effects of viscous damping on the nonlinear resonance plots of Figures 5.3.2 and 5.3.3 were investigated in (Vakakis, 1992b) (a summary of results of that work was given in section 1.2). In the same work subharmonic resonances of the forced oscillator were also examined and found to be affected by the mode bifurcation. Concluding, the results presented in this section demonstrate that a bifurcation of NNMs can significantly affect the exact steady-state motions of a forced system, a result that to the authors' knowledge has received little attention in the existing literature. In section 1.2 it was also shown that at large energies NNM bifurcations may give rise to large-scale chaotic motions of undamped, unforced systems, that are not possible before the mode bifurcation takes place. Hence, bifurcations of NNMs can have global effects on the dynamics of forced and unforced systems, since they may lead to new types of dynamic behavior.


259

Figure 5.3.2 Nonlinear frequency response curves of the system with K = 0.4,(two unforced NNMs): -Stable, ------ unstable steady-states.

260


Figure 5.3.3 Nonlinear frequency response curves of the system with K = 0.15 (four unforced NNMs):Stable, ------ unstable steady-states.


CHAPTER 6 THE METHOD OF NONSMOOTH TEMPORAL TRANSFORMATIONS (NSTTs) In this chapter, an analytical methodology for analyzing the free and forced dynamics of strongly nonlinear discrete systems is formulated. The method is based on nonsmooth temporal transformations (NSTTs), and, in contrast to existing perturbation nonlinear techniques, its applicability is not restricted to only weakly nonlinear systems. In the following sections a detailed development of the NSTT technique is carried out, and some examples of its applicability to strongly nonlinear problems are given. In chapters 7 and 8, the NSTT technique is employed to analyze localized and nonlocalized NNMs of certain strongly nonlinear discrete and continuous oscillators. 6.1 PRELIMINARIES There exist numerous quantitative techniques for computing nonlinear dynamic responses. The majority of these techniques are carried out under the assumption of weak nonlinearity. Assuming that the nonlinear system "neighbors" a linear one, a perturbation parameter, E, is introduced to denote the small magnitudes of the nonlinear terms, and the nonlinear response is constructed "close" to a linear generaring solution. Since the generating functions are harmonic, the weakly nonlinear responses are constructed using complete bases of trigonometric functions. An obvious disadvantage of such techniques is that they cannot be used for studying strongly nonlinear or nonlinearizable oscillators. To circumvent this deficiency of weakly nonlinear techniques, an alternative class of strongly nonlinear ones was developed. These techniques relax the assumption of weak nonlinearity by utilizing nonlinear generating systems, thereby assuming that the strongly nonlinear systems under consideration neighbor simplified, but otherwise, nonlinear systems. These strongly nonlinear techniques are highly specialized and cannot be employed for analyzing general classes of nonlinear problems. The main reason is that multi-dimensional nonlinear systems are generically nonintegrable, and, 26 1

262

THE METHOD OF NSTTs

hence, nonlinear generating solutions are seldom available in closed form (Arnold, 1978). From the above remarks it is concluded that a strongly nonlinear analytical technique with wide range of applicability must employ nonlinear generating systems that:

(1) Are sufficiently general so that they can be used in a broad range of nonlinear applications. (2) Possess a sufficiently simple structure that enables the construction of efficient iterative perturbation schemes for computing the nonlinear response. (3) Capture a wider range of nonlinear complicated dynamic phenomena compared to the generating functions employed in quasi-linear techniques. It must be especially noted that requirement (1) seems to contradict the well-known "individuality" of nonlinear systems, which generally prohibits the concievement of analytical methodologies applicable to general classes of strongly nonlinear systems. In what follows a Flew nonsmooth terizporal transjormation (NSTT)technique is formulated, which, although sufficiently simple to allow analytic computations of stronglv nonlinear free and forced dynamic responses, applies to a wide range of nonlinear problems. The harmonic oscillator is probably the most fundamental model in vibration analysis. The linear ordinary differential equation governing its motion generates the pairs of trigonometric functions { sint, cost], which is the basis of most fundamental linear theories dealing with oscillatory or wave phenomena. The reason for the wide applicability of the harmonic oscillator is that the generated trigonometric functions possesses a number of convenient mathematical properties associated with the group of motions in Euclidean space, such as, invariance with respect to the rotation subgroup. In the same spirit, one could introduce an additional pair of (nonsmooth) functions, { T(t),e(t)), which have relatively simple forms and possess invariance properties with respect to translation- and refection-subgrouiI,s in the group of Euclidean motions. These nonsmooth functions will be termed the saw tooth sine T(t), and the right-angled cosine e(t), respectively, and are defined as

6.1 PRELIMINARIES

263

Figure 6.1.1 The variables z(t) and e(t) employed in the NSTT technique.

{sint,cost}

-

{t(t),e(t)}

Figure 6.1.2 Generating solutions resulting from harmonic and impact oscillators. z(t) = (2/~c)arcsin[sin(~ct2)] and

e(t) = k(t)

(cf. Figure 6.1.1). The derivative of the nonsmooth function z(t) is defined in the generalized sense, using the theory of distributions (Richtmyer, 1985). The mechanical model that generates these functions is the vibro-impact oscillator moving with constant velocity between two rigid barriers, and is depicted in Figure 6.1.2. Interestingly enough, there is a remarkable relation

264

THE METHOD OF NSTTs

between the harmonic oscillator and the vibro-impact system, since both can be viewed as limiting cases of the savne nonlinear oscillator. To show this consider the single-DOF dynamical system: x(t)

+ xm = 0,

~ ( 0=) 0,

X(0) = 1,

x

E

R

(6.1.1)

where m is an arbitrary positive odd integer. The solution of (6.1.1) can be expressed in closed form using special cam-functions (Rosenberg, 1963), but these expressions are too mathematically complicated to provide useful insight into the dynamics. Considering the range 1 5 m 500 for the exponent of (6.1.1), one obtains the following limiting cases for the solutions: {x, XI = (sint, cost},

if m = I

(6.1.2a)

(x, X) = {z(t), e(t)},

if m -+ 00

(6.1.2b)

Hence, the harmonic oscillator can be considered as the limiting case of (6.1. I ) as the nonlinearity tends to zero, whereas the vibro-impact oscillator is the limiting case of (6.1.1) as the degree of the nonlinearity tends to infinity. Note that the limiting case (6.1.2b) cuiz be considered as N generalized solution of (6.1.1) as m+-. Imposing the condition that the solution of (6.1.1) satisfies the first integral of motion,

E = X2/2 + xIn+l/(m + 1) = 1/2 it is evident that the generalized solution (6.1.2b) satisfies the first integral almost everywhere as m -+ 00. The singularities of (6.1.2b) occur at time instants corresponding to z(t) = +I (i.e., at the instances of contact of the vibro-impact oscillator of Figure 6.1.2 with its rigid boundaries); such discontinuities give rise to convergence problems when conventional analyzes based on trigonometric expansions are applied to strongly nonlinear problems. As a second example demonstrating the physical significance of the pair of functions { z(t),e(t) ) consider the Duffing oscillator: x(t) + x - ~3 = 0, x E R (6.1.3) Denote by T = T(E) the period of oscillation of this system, which depends on the total energy (i.e., the first integral of motion) E. When the energy is in the interval, 0 < E < 1/4, the system performs periodic oscillations with

6.1 PRELIMINARIES

265

amplitude A in the neighborhood of the stable fixed point (x,x) = (0,O). For this type of motions, the exact solution of (6.1.3) can be expressed in terms of Jacobian elliptic functions, and it can be proven to satisfy the following asymptotic relations:

T

+ 2n,

x/A -+ cos[n(i' + a)/2)], if E -+ O+ T -+ 00, x -+ e(f + a), if E -+ 1/4

(6.1.4a) (6.1.4b)

where i' = 4t/T is a nondimensional time, and a is an arbitrary phase. Solution (6.1.4b) is written in terms of the previously defined right-angled cosine e(t), and corresponds to motion of the system on a heteroclinic orbit in phase space (Guckenheimer and Holmes, 1984). In terms of the nondimensional time i' the system performs jumps between the two unstable equilibrium points (x,x) = (+1 ,O). Increasing the energy above the critical value E = 1/4, leads to strongly nonlinear nonperiodic motions outside the heteroclinic loop of (6.1.3). For values of the energy E in the range 0 < 1 4E ,e(t)> (6.2.3)

Proof. In an interval equal to a period, functions z(t) and e(t) assume the values, for 1 < t < 3 (z,e} = ( t , l } for -1 < t < 1, and (z,e} = ( 2 - t,-l] Hence, taking into account definitions (6.2.3) it can be easily proven that X(T) + e Y(z) = x(z) in the open time intervals -1 < t < 1, and 1 < t < 3. The points of singularities N = { t / z(t) = k l } need special examination. If function x(t) is continuous in the neighborhoods of the singular points of set N, then it follows that, Ylte N = Ylz=i-l = 0, and equality (6.2.2) holds for all t E N. If function x(t) has discontinuities for t E N, then these discontinuities are accounted for by function e(t) on the right-hand-side of (6.2.2).

6.2 REPRESENTATIONS OF FUNCTIONS USING NSTTs

267

Note that for values of t in small neighborhoods of t = 0, x(t) can be expressed as a summation of even and odd terms in the well-known formula:

x = x(t) = (1/2)[x(ltl)

+ x(-ltl)] + (1/2)[x(ltl) - ~(-ltl)] Itl'

(6.2.4)

Hence, representation (6.2.2) can be viewed as the periodic analog of the well-known decomposition of a general function into even and odd components. From that point of view, term X(z) is the odd component of the periodic function x(t) over a quarter of the period, and term eY(z) as the even component. For example, the cosine and sine functions are expressed in terms of even and odd periodic components, respectively: sin(nt/2) = X(z) = sin(nz/2),

cos(nt/2) = eY(z) = e cos(nz/2) (6.2.5)

The following proposition regarding the representation of an arbitrary function f(x) in terms of the components of the NSTT of its argument x(t) (X(z) and Y(z)) is now proved. Proposition 2. Any function f(x) can be represented as follows: f(x) = f(X where, Rf = (1/2)[f(X

+ Y) + f(X - Y)]

+ eY) = Rf + Ife and

If = (1/2)[f(X

(6.2.6)

+ Y) - f(X

-

Y)]

Proof. The representation (6.2.6) can be easily verified by expressing x(t) in terms of X(z) and Y(z), and evaluating X(z) and Y(z) in a course of a period of 2 , i.e., by setting {.t,e} = ( t , l } for -1 < t < 1, and {z,e} = ( 2 - t,-l} for 1 < t < 3. Points of singularity are treated as in the proof of Proposition 1. The terms Rf and If in (6.2.6) are termed the R- and I-parts of function f(x) in the plane (X,Y). Certain analogies with complex algebra can be noted: The R- and I-parts of f ( x ) can be regarded as analogs to the real and imaginary parts of a complex-valued~inction,with e being the analog of the imaginary constant j = (-1)1/2 of complex algebra. Clearly, a function f(x) is zero if and only if both its R- and I-parts are zero. A simple example of representation (6.2.6) is given by considering the exponential function:

268

THE METHOD OF NSTTs

exp[x(t)] = exp[X(z)

+ eY(t)] = exp[X(z)] [cosh(Y(z)) + esinh(Y(z))]

(6.2.7) The clear analogy to the complex-valued exponential function can be noted. The following proposition discusses the evaluation of the time derivatives of f(x) in terms of the NSTT variables.

Proposition 3 . The derivative of x(t) with respect to time t can be expressed in terms of the generalized derivatives of X(T) and Y(z) in terms of the new variables T and e, as follows: x(t) = Y'(T)

+ eX'(z) + e Y ( t )

(6.2.8)

where prime denotes differentiation with respect to T. The last term in (6.2.8) can be discarded if function x(t) is continuous at the points of singularity, tfz N = ( t / z ( t ) = f l ) , i . e . , i f Y l t € N =Ylz=fl = O If the function x(t) has discontinuities at time instants t

E

N, the last term in

(6.2.8) must be included, and the generalized derivative eft) is computed in terms of a periodic set of impulsive functions that are localized at time instants t E N: Do

e(t) = 2

C [6(t + 1 - 4k) - S(t

k=-m

-

1 - 4k)]

(6.2.9)

A generalization of expression (6.2.9) for higher derivatives can be easily carried out. For example, the second derivative of x(t) is expressed as:

x(t) = X"(z) + eY"(x)

(6.2.10)

provided that the continuity conditions X'lT=+l = 0 are satisfied. In case of discontinuities, expression (6.2.10) must be modified accordingly. Proof. The differentiation formulas are verified as previously by direct computation of the right-hand sides in the duration of a period of 5 . Points of discontinuity are treated as in the proof of Proposition 1.

6.3 ANALYSIS OF DYNAMICAL SYSTEMS

269

Proposition 4. An integration of x(t) with respect to time t can be expressed in terms of variables z and e by employing the following expressions:

5

5(

x(t) dt =

X(z(t))

+ e(t)Y(z(t))) dt = Q + eP

(6.2.1 1)

where T

Q=

Y(u) du

+ C,

P=

0

T

J

X(U) du

-I

C is a constant of integration and the following condition is assumed:

X(U) du = 0

(6.2.12)

-1

Proof. Expression (6.2.11) can be verified by direct differentiation with respect to t.

6.3 ANALYSIS OF DYNAMICAL SYSTEMS The NSTT technique is now applied to the study of the nonlinear dynamics of discrete oscillators in strongly nonlinear regimes. Consider the ndimensional dynamical system: x(t)

+ f(x,i,t) = 0,

ZE Rn

(6.3.1)

where function f(:) is assumed to be sufficiently smooth, and to either depend periodically on time with period equal to T = 4a or to have no time dependence. Expressing the vector of responses x(t) in terms of the NSTT variables T = T(t/a) and e = e(t/a), one obtains a representation of the Tperiodic solution in the form 1z = zL(T,e) = X(T)+ eY(z), where the vector functions X ( T )and Y(z) are the (yet unknown) R- and I-components of the solution vector. Substituting this expression into (6.3.1) and employing the properties of NSTT developed in section 6.2, one obtains the following alternative representation of the equations of motion of the dynamical system:

270

THE METHOD OF NSTTs

As discussed in the previous section, representation (6.3.2) holds provided that the continuity conditions YlZ=+i = 0 and X'I.t=ki = Q and If in (6.3.2) are n-dimensional vectors are satisfied. The quantities computed by:

Setting separately the R- and I-components of equation (6.3.2) equal to zero, and taking into account the continuity conditions for X and Y, one obtains the following set of boundary value problems for the vectors and y:

Although the transformed equations (6.3.4) appear to be of a more complicated form than the original set of equations, they possess certain significant advantages. Indeed, it will be shown that one can solve the transformed equations (6.3.4a) and (6.3.4b) by employing the solutions of the simplified equations X"= Q and Y" = 0, as generating solutions, and applying the perturbation method of successive approximations. This leads to a very simplified perturbation solution for the strongly nonlinear response. Application of the aforementioned procedure is illustrated by the following examples. Example 1: N N M s of an n-DOF conservative system

To demonstrate the use of the NSTTs for solving strongly nonlinear problems, suppose that equation (6.3.1) does not depend explicitly on the velocity vector and on time, and that the vector function f(x) is analytic and odd in x, i.e., that f(-x) = -f(x), KE Rn. Then, the differential equation (6.3.1) assumes the form:


x(t)

+ f(x)= 0,

XE R"

271

(6.3.5)

The NNMs of equation (6.3.5) are now sought by the method of NSTTs. Due to the symmetries of the system, the normal mode solutions are expected to be symmetric with respect to the origin of the configuration space, and, hence, can be expressed as: z = T(t/a) x(t) = ~ ( z = ) X(z), where a is equal to one quarter of the period of the NNM under consideration. Considering system (6.3.5) it can be shown that the R- and Icomponents of the transformed equation (6.3.2) assume the following simplified expressions:

where the vanishing of vector Y is due to the symmetric structure of the restoring forces and the corresponding NNMs. The solution of the nonlinear boundary value problem (6.3.6) is sought by the method of successive approximations. The generating solution vector, Xo, for problem (6.3.6) is chosen as the solution of the following simplified problem:

The vector of amplitudes, ,40,in the generating solution can assume arbitrary values and is computed by the initial conditions of the problem. Note that the above generating solution is the response of an n-DOF vibroimpact oscillator with two rigid barriers. The solution of the boundary value problem (6.3.6) and the quarter-period, a, of the NNM are expressed in series of successive approximations:

X(z) = Xo(z) + &(T) + X2(z) +...,

a2 = ho [ I

+ yi + y2 +...I

(6.3.8)

where it is assumed that O(il&(~)ll) >> O(llX~+~(T)ll), O(yi+l) >> O(yi+2), i = 0,1,2,... and O(ho) = O(1) The next successive approximation to the solution, Xi, is governed by the following set of ordinary differential equations:

272

THE METHOD OF NSTTs

Note that the first term in the expression of Xi is identical to the generating solution (6.3.7), and thus, can be taken equal to zero, 41 = 0. Combining solutions (6.3.8) and (6.3.9a), and imposing the boundary conditions in (6.3.6), one obtains the following expression relating the to the first correction to the quarter period squared Lo: amplitude vector

where (*)T denotes the transpose of a vector. In the next step of the perturbation analysis, terms of the next order of approximation are taken into account, resulting in the following problem:

where A2 is an arbitrary constant vector, and Dxf(*) denotes the (n x n ) matrix of first partial derivatives off(*) with respect to the n components of vector x. Taking into account the boundary condition in (6.3.6), one computes vector A2 as follows:

The correction to the period squared, yi, is yet undetermined. However, by imposing the orthogonality between vector A2 and the corresponding vector of the O( 1) approximation Ao, one obtains the following expression for y1: AoT& = 0 1

3

yi = -[A)'rj

0

1

Dxf(AoT)Xl(z) d.51 [AoT! f(Ao~)dr]-' 0

(6.3.12)


273

Hence, the second approximation to the solution is completely determined, and the solution of the boundary value problem (6.3.6) is completed up to O(X3). Similar calculations can be performed to compute higher order approximations. Combining the previous results, the NNM of the strongly nonlinear autonomous system (6.3.5) is approximated as follows:

f

-

ho j 0

- hoi (z - {)[yif(A05) 0

-

(7 - 5)f(Ao5)d5

5

hoD~f(Ao5)j(5 - u>f(Aou>du] d5 0

(6.1.14a) where quantities ho and y~ are computed by (6.3.9b) and (6.3.12), respectively. The corresponding period of the NNM is approximated by:

I

d A 0T

j 0

-f(A0z) dz1-l

+ ...}

1'2

(6.1.14b)

Note, that if system (6.3.5) has a single degree of freedom (n = l), it can be shown that the boundary conditions in (6.3.6) are satisfied by appropriately computing the scalars ho, yi, y2, ..., and setting the higher order approximations to the amplitude equal to zeio, Ai = 0, i=1,2, ... In this case the zeroth order amplitude vector A0 degenerates to a scalar, A0 + A0 = A, and all the terms in the series expansion (6.3.8) can be explicitly computed by quadratures: XO=AT, AER

i

i- 1

(6.3.15a)

274

THE METHOD OF NSTTs

where A is an arbitrary constant scalar depending on the initial conditions, and quantities Cti and Ri are computed as follows: I

ai =

R,dk 0

Ri = (l/i!)[dif(Xo(z)

[i

Rodk1-l

0

+ EX1(T) + E2x2(T) + O(E3))/dEi]E,0

i=0,1,2 ,... (6.3.15b) Expressions (6.3.15) compute the free periodic response of the single-DOF strongly nonlinear system (6.3.5). Example 2: Strongly nonlinear motion close to a separatrix An additional class of problems arising from the general equations (6.3.4a,b) is concerned with nonlinear free motions close to separatrices of dynamical systems. For example, consider a single-DOF, conservative nonlinear oscillator with an odd restoring force f(x): x(t)

+ f(x) = 0,

XE

R

(6.3.16)

Assume that the phase plane of the dynamical system possesses a stable equilibrium (x,X) = (O,O), two unstable saddle points (x,X) = (fK,O). In addition, suppose that there exist two heteroclinic orbits (separatrices) connecting the two unstable equilibria. The heteroclinic orbits form boundaries to the regime of periodic motions that surround the stable equilibrium position (x,X) = (0,O). Motions on the heteroclinic orbits occur when the energy of the system is equal to the critical value: E = E, =

K

J

f(x)dx

0

It is now shown that the NSTT technique can be used to compute strongly nonlinear periodic vibrations in small neighborhoods of the heteroclinic orbits, i.e., in the energy range, 0 < 1 - (E/Ec) = v(x,t> v,(X,t) = F[u(x,~),v(x,~)], x

E

s1 - an

(8.1 S 1 )

with boundary conditions identical to (8.1.52). F[*] denotes an integrodifferential operator, s1 the one-dimensional region occupied by the system, and an the boundary of this region. One assumes that system (8.1.51) possesses the stable equilibrium position (u,v) = (0,O). An NNM of system (8.1.51) is defined as a nonlinear motion taking place on a two-dimensional invariant manifold in phase space. This invariant manifold passes through the stable equilibrium (u,v) = (0,O) of the system and at (u,v) = (0,O) is tangent to the eigenspace of the associated linearized system (Shaw and Pierre, 1994). To compute this manifold a parametrization similar to that followed in the energy-based approach is performed. Defining again a reference point XO, one expresses the dependent variables in (8.1.51) in terms of the displacement u,(t) and the velocity v,(t) of the reference point as follows:

Using the chain rule of differentiation the equations of motion (8.1.51) are expressed in terms of the modal functions U and V

The boundary conditions are written as

8.1 SYSTEMS OF FINITE SPATIAL EXTENT

371

B[U,V] = 0 , x and the following compatibility conditions are imposed:

Equations (8.1.53) govern the nonlinear functions U and V and can be asymptotically approximated (at least locally) by series expansions. To this end, one assumes the following series representations for the modal functions U and V: U(uo,vo,x,xo)= al (x,xo)uo+ a2(x,xo)vo+ a3(x,xo)uo2

+ a~(x,xo)uovo + a5(x,xo)v,2 + ag(x,x,)uo3 +...

v(uo,vo,x,xo) = bt(X,Xo)Uo + b ~ ( x ~ o ) +v ob3(X,X0)U02

+ ~~(x,x,)u,v,+ bg(X,Xo)Vo2+ bg(X,Xo)Uo3+...

(8.1.54) Substituting (8.1.54) into (8.1.53) and its complementing boundary and compatibility conditions and matching respective coefficients of monomials uoPvo9, one obtains a sequence of boundary value problems in terms of the spatial coefficients a,(x,x,) and b,(x,x,). As pointed out by Shaw and Pierre (1994), these problems are uncoupled in sequential order allowing a direct solution strategy. This series approximation to the two-dimensional invariant manifold resembles the construction of center manifolds in dynamical systems theory (Carr, 198 1; Wiggins, 1990; Guckenheimer and Holmes, 1984). In (Fenichel, 1971) and (Carr, 1981) general theorems on the existence of invariant manifolds and constructive theorems for their analytic approximation are discussed. These theorems can be extended to prove the existence of two- or higher-dimensional NNM invariant manifolds for a general class of conservative or nonconservative vibrating systems. Once an asymptotic approximation to the invariant manifold is constructed, the motion of the reference point is computed by evaluating (8.1.51) at the reference point and combining the two equations to derive the following nonlinear ordinary differential equation governing uo(t): Uo(t) -

( F[u(x,t),au(x,t)/atl] x=xo = 0,

v,(t) = uo(t)

(8.1.55)

372

NNMs IN CONTINUOUS SYSTEMS

Solving this equation by approximate perturbation techniques completes the calculation of the motion on the NNM invariant manifold. Although analytically cumbersome, the outlined methodology enables the calculation of damped NNMs by accounting for nontrivial phase differences between the motions of different points of the structure. In (Shaw and Pierre, 1994) the invariant manifold method is employed to compute NNMs of linear and nonlinear problems involving simply supported beams. In that work they consider a simply supported beam on a nonlinear elastic foundation and provide results that are identical to those obtained by the energy-based formulation. Moreover, they found that the invariant manifold approach is not valid in cases of internal resonance where the dimensionality of the NNM invariant manifold increases. A third direct approach for computing NNMs is based on the method of multiple scales and is developed by Nayfeh and Nayfeh (1994). Applying the method of multiple scales to the nonlinear partial differential equation one constructs a series of boundary value problems at successive orders of approximation. These problems are solved by imposing appropriate solvability conditions, i.e., by eliminating secular terms and rendering the derived approximations uniformly valid in time. An advantage of the method of multiple scales over the previous methods is its applicability in cases of internal resonance (Nayfeh and Mook, 1984). For a general exposition of this method the reader is referred to (Nayfeh and Nayfeh, 1994).

8.1.2 Analysis by Discretization An alternative class of methodologies for computing NNMs of weakly nonlinear continuous oscillators relies on discretizations of the partial differential equations of motion. In the following exposition the formulation adopted by (Nayfeh and Nayfeh, 1994) is followed. Consider a weakly nonlinear continuous system of the following general form Utt

= L[u(x,t)l

+ ~“u(x,t)l,

x

(0,l)

(8.1.56)

where L[*] is a linear self-adjoint spatial operator, N[*] is a nonlinear spatialtemporal operator odd on u(x,t), and the notation introduced in section 8.1.1 applies. Suppose that the problem possesses homogeneous boundary conditions of the form:


B[u(x,t)] = 0,

x = 0,l

373

(8.1.57)

In addition, assume that the linearized problem corresponding to E = 0 has distinct eigenfrequencies Wn and real eigenfunctions @ n ( ~ )where n = 1,2,... Define at this point the inner product between two functions f(x) and g(x) by

Because the linear operator L[*] is assumed to be self-adjoint, the eigenfuctions @n(x)of the linearized problem form an orthogonal set. To normalize the linearized eigenfunctions one imposes the following additional conditions

< @n(X>t@m(X) > = 6nm

and

< L[$n(x>I7$m(x>> = a n 2 & ,

(8.1.59) where 6, is Kronecker's symbol. To discretize equation (8.1.56) the continuous displacement variable u(x,t) is expressed as a nonlinear superposition of the modal responses of the linearized problem (8.1.60) Substituting (8.1.60) into (8.1.56), taking the inner product with respect to on(x) of both sides of the resulting expression, and taking into account the orthonormal conditions (8.1.59), one obtains the following set of discretized ordinary differential equations in terms of the modal amplitudes qn(t):

where the nonlinear functions Gn(q) are derived in terms of the nonlinear operator N[*] as follows:

(8.1.62)

374


To compute the NNMs of the continuous system, one truncates the infinite set of equations (8.1.61) to only N terms and applies the analytical methods discussed in chapter 3 to investigate the NNMs of the truncated system. Hence, the original problem of computing the NNMs of the continuous system is reduced to the problem of computing the modal curves or the modal invariant manifolds of the dicretized set of equations (8.1.61). For an application of the discretization approach to the NNM computation the reader is referred to (Nayfeh and Nayfeh, 1993,1994; Boivin et al., 1993; Pakdemirli and Nayfeh, 1996; Shaw, 1994; Shaw and Pierre, 1992) where various continuous oscillators with geometric nonlinearities or nonlinear boundary conditions are analyzed. An advantage of the discretization approach is its simplicity, whereas its main drawback is the truncation of the modal equations which is necessary to perform the numerical computations. In addition, contrary to the direct approaches of section 8.1.1 the discretization technique does not provide any nonlinear corrections to the mode shape of the NNM vibration; throughout the discretization analysis one uses the linearized eigenfunctions Qn(x) and takes into account only the nonlinear effects on the modal amplitudes qn(t).

8.1.3 Stability Analysis of NNMs As shown in chapter 5 , contrary to linear theory nonlinear normal modes are not always orbitally stable. Indeed, mode bifurcations can occur that give rise to additional normal mode branches or to orbitally unstable free normal oscillations. In this section an approximate analysis is carried out to study the stability of the continuous NNMs which were computed in the previous sections. If the continuous system under consideration is conservative then its NNMs can be at most neutrally stable (Nayfeh and Mook, 1984). In the case of nonconservative systems the NNMs can be asymptotically stable. Consider a general system defined on the one-dimensional region 0 I x I 1 with governing equation (8.1.4) and boundary conditions (8.1.2). Analytical procedures for computing the NNMs of this system were presented in section 8.1.1, and the stability type of the computed NNMs is now examined. Adopting the energy-based direct methodology, one expresses the motion of the system during an NNM as follows:


ii(x,t) = u[x,u,(t>] = al(o)(x)uo(t)+ E

C

a,(I)(x)u,m(t)

(m=1,3,..)

375

+ O(EU,~+~,E~) (8.1.63)

To investigate the stability of the NNM (8.1.63), one introduces arbitrary small perturbations EG(x,t) to the solution as follows: u(x,t) = u"(x,t) + ES(X,t)

(8.1.64)

Substituting (8.1.64) into the equation of motion (8.1.4) and retaining only terms up to O(E),one obtains the following variational equation:

ktt= L,(O)[c] + E DLXt(1)[U",5] +O(E~)

(8.1.65)

is a linear operator in the variation 6, which results from where DLXt(l)[u",5] expanding L,,Cl)[C + ~ 5 in 1 power series in terms of the small parameter E . The solution of (8.1.65) is now approximated in the series expansion (8.1.66) where { Pm(x)}m=lmis a complete family of orthogonal polynomials satisfying orthogonality relations of the form

where w(x) is an appropriately defined weighting function. Polynomials Pm(x) need not satisfy the boundary conditions at x = 0,1, but the numerical convergence of the stability analysis is greatly improved if they do so. Substituting (8.166) into (8.1.65), premultiplying both sides by w(x)Pn(x),n = 1,2,..., integrating from x = 0 to x = 1, and neglecting terms of O ( E ~or) higher, one obtains the following infinite set of linear, second-order ordinary differential equations governing the coefficients an(t):

376


where n = 1, 2, ... and X[*] is a spatial operator acting on Pm(x). For the carrying out the numerical calculations the infinite set (8.1.68) is truncated to only N terms. Taking into account that DL,,(I)[G,c] is linear in 6,the truncated set of equations (8.1.68) can be written in the following matrix form [I] { bi(t)} = [A(o)] { a(t)} + &[A([)($] { a(t)}

+ E[h(2)(U")] { &(t)} + E[A(~)(U")] { bi(t)} (8. I .69) where { a ( t ) }= {a1 a2 . . . ~ N } T ,[A(o)] is a matrix of constant coefficients due to the linear operator, and [A(l)(U")],[A@)($], and [AW(G)] are timeperiodic matrices having as minimum period the period of the NNM under investigation. The time-dependent matrices [A(*)(G)]and [AC?)(u")]are due to the spatial-temporal operator LXt(I)of the equation of motion. Floquet analysis is now applied to the truncated system. A justification for this truncation analysis can be found in (Yakubovich and Starzhinskii, 1975). To this end, the new scalar functions P n = &,, n = 1, 2, ..., N are introduced and equations (8.1.69) are rearranged and written as a set of 2N first-order ordinary differential equations

where {y} { al 01 a2 p 2 ... a N P N ) T. Equations (8.1.70) are integrated for the set of 2N initial conditions Y k = 6,,k, k,n = 1,...,N where Y k denotes the kth element of the vector ( y } and the Floquet matrix is constructed by recording the solutions of (8.1.70) at t = T, where T is the period of the NNM [and the mininum period of the time-varying matrix [A]]. One then determines the stability of equation (8.1.70) by computing the eigenvalues of the Floquet matrix: eigenvalues with modulus greater than unity indicate


377

orbital instability, whereas eigenvalues with modulus equal or less than unity correspond to neutral or asymptotic stability, respectively. Neutral stability does not imply stability, but combined with the fact that the system under consideration is conservative neutral stability implies small bounded oscillations of the perturbation &$fx,t) close to the steady-state solution il(x,t), and thus, a stable NNM. As mentioned earlier, no asymptotic stability is expected for the systems under consideration. Since (8.1.70) represents a truncation of the series expansion (8.1.66), a convergence study must be performed on the eigenvalues, i.e., one must detect the minimum number of terms of the truncated series for which convergence of the Floquet eigenvalues occurs. Stability or instability of equations (8.1.70) implies orbital stability or instability of the NNM under consideration. We now use the previous analysis to study the stability of the NNMs of the two flexible systems considered in section 8.1.1 (energy-based formulation). In both cases we choose orthogonal functions Pn(x) in the expansion (8.1.66) and identical to the eigenfunctions of the linearized problem; for the sake of simplicity we only examine the stability of the first two NNMs of the systems. For the simply supported beam lying on the nonlinear elastic foundation the nth NNM is asymptotically approximated by equation (8.1.39) and the reference motion by (8.1.40). For the nth NNM the variational equation (8.1.65) takes the form

where UfO) is the 0 ( 1 ) approximation to the modal function (8.1.31). Expressing c(x) by the series expansion (8.1.66) with P,(x) = sin(mnx), m = 1,2, ... one obtains the following set of equations governing the amplitudes an(t): N

bin(t) + (k + n2n2)an(t) + 6 ~ y C X,m(t)am(t) = 0 m= 1

where Xnm(t)= COS’(Wnt)Uo*’

i 0

al(o)(x)2sin(nnx)sin(mnx)dx

(8.1.72)

378


and al(o)(x)is given by equation (8.1.30). For n = 1, 2 (the first two NNMs) we expressed equations (8.1.72) as a set of 2N first-order equations and we applied numerical Floquet analysis. In all cases considered the Floquet matrix possessed pairs of complex conjugate eigenvalues of unit modulus indicating orbital (neutral) stability for the normal modes under consideration. A parametric study indicated that at least five terms were needed in expansion (8.1.72) in order to achieve convergence of the eigenvalues of the Floquet matrix. A similar stability analysis was performed to study the stability of the first two NNMs of the nonlinear cantilever beam discussed in section 8.1.1. In this case the spatial coefficients of the modal functions are determined by (8.1.47), (8.1.49), and (8. I S O ) and the reference displacement ug(t) is computed by solving the following equation

1 0

Using the method of multiple scales the modal frequency of the nth mode is approximated as follows: wn=w+

8

-

w2c3u,'z) 2

+...

and uo(t) is computed by (8.1.73) using the set of initial conditions

For this problem the variational equation assumes the form: r

(8. I .75)


1

1

0

379

0

(8.1.76) where U@)(x,t)= al@)(x)uo(t) is the linearized mode shape computed by (8.1.47), and the truncated set of variational equations is given by ([I1 +E[V(t)I) {

I + [Ll{N t ) 1 + E[X(t)l

1 - E[T(t)l{

1=0

(8.1.77) where [I] is the (N x N) unit matrix. The elements at the pth line and mth column of the (N x N) matrices of the variational equation are defined as 1

0

1 0

(8.1.78) An additional complication of system (8.1.78) is that the time-dependent inertial matrix (which is due to the nonnegligible longitudinal inertia of the cantilever beam) must be inverted prior to setting (8.1.78) into a form amenable to Floquet analysis. This complication can be partially removed by

380


noting that for approximation:

E

sufficiently small one can impose the following

([I]

+ E [V(t)])-'

= [I] - E [V(t)l

+ O(E2)

(8.1.79)

Using relation (8.1.79) one can easily perform the required matrix multiplications and reduce the variational system to the required form (8.1.70). The numerical Floquet analysis was performed for the first two nonlinear normal modes of the cantilever beam (n = 1,2), which were found to be orbitally stable. In all cases examined the Floquet multipliers occurred in complex conjugate pairs of unit modulus. The outlined procedure can be extended to the study of the stability of motions on the NNM invariant manifolds de.fined by Shaw and Pierre (1994). Also, more involved problems can be analyzed with the described methodology, such as the problem of stability of localized NNMs in assemblies of coupled continuous nonlinear oscillators, the investigation of NNM bifurcations and the stability of stationary solitary waves in nonlinear partial differential equations defined on infinite domains. Problems of this type are discussed in chapter 9.

8.2 SYSTEMS OF INFINITE SPATIAL EXTENT The system considered in this section is an infinite one-dimensional chain of elastically coupled nonlinear oscillators. Although this system is discrete, its dynamics can be studied by imposing a continuum limit approximation, thereby reducing the infinite set of ordinary differential equations of motion to a single nonlinear partial differential equation. Stationary and traveling waves of the chain will be analytically investigated. It will be shown that stationary waves can be regarded as NNMs of the infinite chain and that the computation of stationary waves in the discrete system is equivalent to computing the NNMs of a continuous nonlinear oscillator of infinite spatial extent. The analysis is performed by employing the sawtooth transformations of the temporal variable introduced in chapter 6. Periodic systems are often encountered in physical and engineering applications, as in studies of the dynamics of crystal lattices in solid-state physics, in modeling power transmission in electric lines, or in investigations of sound propagation in air. Ferguson et al. (1982) studied nonlinear

8.2 SYSTEMS OF INFINITE SPATIAL EXTENT

381

periodic motions (they termed them nonlinear normal modes) in ordered or disordered Toda chains, whereas Ermentrout ( 1992) investigated the existence and stability of nonlinear periodic solutions in chains of weakly coupled neural oscillators. Although a nonlinear spring-mass chain is a relatively simple system, nevertheless, it can be used for studying dynamic phenomena encountered in systems of much more complicated configuration [for example, the reader is referred to the works by Van Gils and Valkering (1986) and Van Opheusden and Valkering (1989)l. Solitons are such essentially nonlinear phenomena that can be studied by considering the relatively simple systems analyzed herein. Solitons are essentially nonlinear waves that can be used for the construction of more general wave solutions, in some analogy to the classical normal modes of linear vibration theory, which can be employed to compute free and forced response of linear oscillators using the principle of superposition. Generally, the nonlinear chain of masses and elastic springs being free of complicating geometrical details, provides a good framework for studying some important dynamical properties of essentially nonlinear periodic systems.

8.2.1 Stationary Waves as NNMs In this section an infinite chain of nonlinear oscillators coupled by means of linear springs is considered. Assuming that only coupling between neighboring oscillators exists, the equations governing the motion of this system are given by:

The function f(un) denotes the nonlinear restoring grounding force acting on the nth oscillator. This function is assumed to be analytic and odd and to possesses a single zero at the equilibrium position un = 0. Traveling and stationary wave solutions of (8.2.1) will be computed by imposing continuum approximations and reducing the infinite set of equations to a single approximate nonlinear partial differential equation. It will be shown that stationary waves of the reduced continuous system can be regarded as NNMs defined over an infinite spatial extent.

382


Travelling Waves Equations (8.2.1) form an infinite set of homogeneous, nonlinear ordinary differential equations. Under certain conditions (Wadati, 1975; Peyrard and Pnevmatikos, 1986; Rosenau, 1987; Sayadi and Pouget, 1991; Aceves and Wabnitz, 1993), a continuum approximation can be imposed, whereby, the infinite set of equations (8.2.1) is replaced by a single nonlinear partial differential equation. The continuum approximation is only valid for longwuve motions, i.e., when the wavelengths of the envelopes of the motions of the chain are much larger than the average distance between adjacent particles. In the continuum limit the displacements un become continuous functions of the spatial and temporal variables, i.e., un(t) + u(s,t) and the finite difference term appearing in equation (8.2.1) is approximately replaced by a second partial derivative, i.e., ( 2 ~ n Un-1 - u,,+I) + -u,,h2 + O(h4), where h is the distance between adjacent particles. Taking into account these approximations, and introducing the new spatial variable x = s/h the infinite set of equations (8.2.1) is replaced by the following KleinGordon nonlinear partial differential equation

utt - u,,

+ f(u) = 0 + O(h4),

t > 0,

--w

< x < +m

(8.2.2)

In writing (8.2.2) we take into account only the leading discrete effects of the chain. Equation (8.2.2) represents an continuous dynamical system of infinite spatial extent. We now seek traveling wave solutions of (8.2.2) in the form u(x,t) = u($),

$ = kx - Ot

(8.2.3)

where k and o are the wavenumber and frequency of the traveling wave, respectively. Considering $ as the new independent variable, (8.2.2) is expressed as an ordinary differential equation as follows (8.2.4) At this point (and without loss of generality) the period T of the traveling wave is normalized to T = 4. Employing the nonsmooth temporal


383

transformation (NSTT) z = z(@) first introduced in chapter 6, the displacement is expressed in terms of the new variable z z = z(@) u(@)= u(@(z))= U(z) (8.2.5)

*

Recall that z(@) is the sawtooth sine function. Taking into account the normalization of the period of the wave and the assumed symmetries of function f(*), one can express the problem in the following final form

where h = (02 - k2)-1. As discussed in chapter 6, the solution to problem (8.2.6) can be analytically constructed using the method of successive approximations. In that chapter it was shown that variables U and h can be approximated by the following regular power series expansions:

where the approximations Ui(2) and si are evaluated by employing already computed approximations of previous orders. All approximations depend on the arbitrary parameter A(@,which is linked to the maximum amplitude of the wave by the relation m

Once h is analytically estimated one obtains the following dispersion relation for the traveling wave:

(8.2.8) Note that the dispersion relation depends on the maximum amplitude of the wave [parameter A@)], a feature that is encountered in the theory of nonlinear waves (Whitam, 1974). As an example of application of the previous methodology consider the case when f(u) = p m , y > 0, m = 2v - 1, v = 2,3 ,..., i.e., a restoring force proportional to the mth power of the displacement. Considering the leading

384


terms in expansions (8.2.7) and employing the results derived in chapter 6, the dispersion relation of the traveling wave is 02

= k2 + (1/2)yA(O)(m-l)(m+ 4)(m + l)-](m

+ 2)-1

where A(())is related to the maximum amplitude of the wave by the relation,

These analytical results can be further improved by computing additional high-order approximations U,(T) and 1. It was mentioned that solution (8.2.7) represents a traveling wave whose wavenumber k and frequency o are related by the dispersion relation (8.2.8). Of special interest is the wave corresponding to k = 0, i.e., the wave with in-phase spatial variation of its envelope. This represents a stationary wave solution of equation (8.2.2) and corresponds to synchronous periodic oscillations of all particles of the chain, with frequency 0 2

= ho-'(l -

2 0,m = 2v - 1, v = 2,3, ... For this system equation (8.2.17) assumes the form (8.2.19) There exist the trivial stable equilibrium A,@) = 0 and the unstable equilibria A,(O) = +[(m + I)/yho]l/(m-l) = f K . For m = 3 the phase plane of (8.2.19) contains a heteroclinic loop consisting of two heteroclinic orbits connecting the unstable equilibria. Motions corresponding to the heteroclinic orbits are stationary waves whose amplitudes reach the asymptotic limits, +K as n + f m and 0 as n+O; these waves possess spatially localized envelope slopes. Periodic solutions inside the heteroclinic loop are stationary waves with spatially periodic envelopes and can be regarded as NNMs of the infinite periodic chain. Aperiodic solutions outside the heteroclinic loop correspond to stationary waves with expanding envelopes as n+,+m. Hence, for f(U,) = y U m the nonlinear chain possesses an infinite family of NNMs parametrized by the maximum amplitude of motion. For increasing maximum amplitude the wavenumbers of the stationary waves increase until the wave with maximum amplitude +K is reached. The transition from

388


equation (8.2.15) to (8.2.17) was accomplished by assuming that the variables A,(o) have a smooth dependence on the index n and imposing the continuum approximation. When neighboring particles of the system oscillate in antiphase motions, one introduces the transformation:

A,,@)= (-1)" a,,

n = O , i l , i 2,...

(8.2.20)

Substituting (8.2.20) into the infinite discrete set (8.2.15) one obtains the following set of equations governing the amplitudes a,:

where n'(u)= f(u). By employing the transformation (8.2.20) and assuming that an depends smoothly on the index n, one can use the continuum approximation to study the antiphase vibrations of the chain. The continuum limit of the set of equations (8.2.21) is given by d*a, __ dn2 - (2110 - 4)an

+ 2FI(an)/an= 0

(8.2.22)

The analysis of (8.2.22) is performed as previously. The trivial equilibrium position a, = 0 is stable if llho < 2 + f(0)/2 and unstable if 1 / 1 0 > 2 + f(0)/2. In the later case, equation (8.2.22) possesses an additional pair of stable equilibrium positions that are the roots of the following equation: (1 /Lo - 2)a, - n(a,)/a, = 0

(8.2.23)

When f(U,) = y U p , y> 0, m = 2v - 1, v = 2,3,..., (8.2.22) takes the form: d*an - (21x0 - 4)a, dn2

+ 2ya,m/(m

-I1) = 0

(8.2.24)

The periodic solutions of equation (8.2.24) correspond to stationary waves (NNMs) of the chain with adjacent particles vibrating in antiphase motions. For m = 3 and (l/ho- 2) > 0 the phase plane of (8.2.24) possesses a pair of homoclinic orbits corresponding to stationary waves that are spatially localized close to particle n = 0 and decay to zero as n + kce. It can be shown that there exist two families of stationary waves with spatially periodic envelopes corresponding to periodic orbits of an lying inside or


I---

389

d i ) ( s.t)

Figure 8.2.1 The layered nonlinear periodic assembly of infinite extent.

outside the homoclinic loops. A more detailed analysis of such localized stationary waves is carried out in chapter 9. When ( l / h o - 2) < 0, no homoclinic loops exist, and the chain possesses a single family of stationary waves with spatially localized envelopes.

8.2.2 Waves in Attenuation Zones of Monocoupled Nonlinear Periodic Systems We finally note that, in addition to computing stationary waves in nonlinear partial differential equations, the concept of NNMs can be used to study attenuating waves in continuous nonlinear periodic systems of infinite spatial extent. For example, considering the system depicted in Figure 8.2.1 composed of an infinite number of nonlinear layers coupled by means of linear stiffnesses, it can be shown (Vakakis and King, 1995) that in analogy to linear theory this system possesses nonlinear attenuation and propagation zones (AZs and PZs) in the frequency domain. Responses in AZs correspond to standing waves with spatially attenuating envelopes and correspond to synchronous m o t i o n s of all points of the periodic system (Mead, 1975). In the terminology introduced previously, these standing waves can be regarded as NNMs of the infinite periodic system and hence, they can be analytically examined by employing previously formulated asymptotic techniques. By contrast, waves in PZs are nonsynchronous motions of the system and are investigated by alternative analytical methodologies (Vakakis and King, 1995). A detailed computation of nonlinear standing waves in the system of Figure 8.2.1 is carried out in (Vakakis and King, 1995), and only a brief description

390


of the analytical results will be presented here. The equation governing the motion of the ith layer can be expressed in the following form:

where normalized coordinates are used and 6(*) is Dirac's generalized impulse function. Following the NNM methodology outlined in previous sections, a reference displacemerit uexpliPij(T1>I substituting into (10.1.18) and the respective modulation equations for beam segment 2, and setting separately real and imaginary parts equal to zero, one obtains the following modulation equations governing the (slow) evolution of the modal amplitudes aij(T1) and phases Pij(T1):

(10.1.20c)

Mode 2, Beam Segments 1 and 2 a12' = -202(EIL/k~)a22sin@2 a22' = 202(EIL/k~)al2sin@2 @2' = -202(EIL/k~){ [a22 2 - a12]/(a22 2 a12)}cos@2

( 10.1.2 1a) (10.1.21b)

2 2 2 2 - (3/802)a2222[a12 - a221 - (1/402)a2211[all - a211

(10.1.2 lc) Equations (10.1.20) and (10.1.2 1) are derived by combining the modulation equations for the phases Pij. Variables CDi, i = 1,2, denote the phase differences between the ith linearized modes of beam segments 1 and 2, respectively: @1=P11-P2l3

@2=P12-P22

(10.1.22)

Combining equations (10.1.20a) and (10.1.20b), it can be shown that, a1 12 + a212 = p12, where p1 is a constant of integration depending on the initial conditions. This relation indicates that the summation of the energies of first modes of beam segments I and 2 is a conserved quantity. Such a conservation relation only holds in the absence of internal resonances, i.e., when no nonlinear transfer of energy exists between the first and higher

484

NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS

order modes. A similar conservation relation can be shown to hold for the second modes of beam segments 1 and 2, a122 + a222 = hi2. The free oscillations of the periodic assembly can be studied by considering individually the sets of autonomous equations (10.1.20) and (10.1.2 1). In what follows the stationary solutions of these equations are studied. For each set, the stationary solutions for aij and @i are sought by imposing the conditions aij' = 0 and @i' = 0. The stability of the periodic solutions (10.1.23) and (10.1.24) can be examined by considering the stability of the stationary solutions of equations (10.1.20) and (fO.l.Zl), i.e., by forming the appropriate systems of linear variational equations and computing the eigenvalues of the associated matrices of coefficients (Nayfeh and Mook, 1984). Once the stationary solutions of (10.1.20) and (10.1.2 1) are determined, the resulting flexible displacement components of the beam segments of the two-span beam can be computed then by the following relations:

wi(x,z) = E1/2alj$j(x)cos[mjt + P11(~t)]+ o(~312) = ~112w i(f)(x,z) + 0 ( ~ 3 / 2 ) , i j = 1,2 (1 0.1.24)

From (1 0.1.24) it is concluded that, correcf fo 0(&1/2), the overall fransverse displacements of the system are identical to their flexible components. Moreover, the static displacement components introduce at most O(~312) corrections, and, therefore, can be neglected at the first order of approximation. Periodic oscillations of the flexible assembly where both beam segments oscillate in their first linearized modes are computed by considering equations (10.1.20) and requiring that a l l ' = a21' = @ I ' = 0 and a12 = a21 = 0. Taking into account the previously derived relation a1 12 + a212 = p12, the

10.1 MULTISPAN NONLINEAR BEAMS

485

solutions of the resulting algebraic equations can be written in the following explicit form: (in phase mode) (10.1.25a) 0 1 = 0, a l l = a21 = 2-112~1 = I T ,a l l = a21 = 2-112~1 (antiphase mode) (10.1.25b) =IT

for Ki

< (3al111/32)p12

(bifurcating modes)

(10.1.25~)

where K1 is a nondimensional torsional stiffness parameter, defined as, K1 = 0 ] 2 ( E I L / k ~ )From . (10.1.25), it is concluded that when both beam segments oscillate in their first linearized modes, the flexible assembly can possess as many as four normal modes. The nonlinear modes (10.1.25a) and (10.1.25b) are similar to those predicted by linear theory. However, when K l = ( 3 a i 1 1 1/32)pi2, the antiphase normal mode (10.1.25b) becomes unstable, and the two bifurcating nonlinear normal modes (10.1.25~)are generated. These modes have no analogy in linear theory. As (Kilai 11 1)+0, the ratios of the amplitudes (all/a21) of the bifurcating modes tend to 0 or 00, indicating that only one of the two beam segments oscillates with finite amplitude. Hence, as ( K l l a 11 11)+0, i.e., when the torsional stiffnesses of the stiffeners and/or the nonlinear effects due to geometric nonlinearity become large, nonlinear mode localization occurs and the energy of the free periodic oscillation becomes spatially localized to only one of the two beam segments. Note that, as shown previously, large torsional stiffness of the stiffeners is equivalent to weak coupling between adjacent beam segments. The previously described nonlinear mode bifurcation and localization is depicted in Figure 10.1.2(a), where the modal ratio (all/a21) is plotted versus the coupling parameter ( K l l a l I 11). Note, that an analogous bifurcation picture was detected in section 9.1.1, where the flexible system of two coupled nonlinear cantilever beams was examined. Analogous results can be obtained when one considers oscillations of both beam segments in their second linearized modes. Figure 10.1.2(b) depicts the nonlinear normal modes of the assembly for a212 + a222 = L l 2 = 1. These modes were computed by examining the stationary values of equations

486


4.00

3.00

\

2.00

2 . -;;

M

1.00 0.00

-1 .00

4

-2.00 -3.00 -4.00 0.0000

I

I

I

I

0.0002

0.0004

0.0006

0.0008

(a)

0.0010

Ki l a i i i i

4.00 3.00 2.00

2 . -2 rJ

1.00 0.00

-1.00 -2.00 -3.00 -4.00 0.0000

0.0002

0.0004

0.0006

(3)

0.0008

0.0010

Ki l a i i i i

Figure 10.1.2 NNMs for oscillations in: (a) first and (b) second cantilever Stable modes, ------- unstable modes. modes; ~


487

(10.1.21) with a l l = a21 = 0. For motions of the beam segments in their second linearized modes, the nonlinear mode bifurcation occurs at K1 = ( 0 1 2/w22)(3a2222/32)~~2, and the bifurcating modes branch-off the in-phase mode. A similar analysis can be performed to investigate the nonlinear normal modes of periodic assemblies consisting of more than two beam segments. In what follows, all beam segments are assumed to oscillate in their first linearized mode, an assumption that is justified due to lack of internal resonance; higher-mode oscillations can be treated in a similar way. For a system consisting of n = 3 beam segments, the modulations of modal amplitudes and phases during free nonlinear oscillations are governed by the following set of autonomous differential equations:

where, @ I 2 = P I 1 - p21,

@23 = p 2 l - p31

~ 1 = Y 1 1 ~ 0 ~ - ~ 1 1 ~~20= ~- Y71 1 ( ~ ) + 2 ~ 1 1 ( ~ ) - W 1 1 ( 1 ) The various coefficients appearing in (10.1.26) are defined by relations (10.1.13), and the notation introduced in the previous section holds (for example, aj 1 and pj 1 denote the amplitude and phase of the first mode of the jth beam segment, j = 1,2,3). As previously, equations (10.1.26a,b,c) can be combined to give the energy conservation relation, a112 + a212 + a312 = p22. The stationary values of the variables in equations (10.1.26) correspond to nonlinear normal modes of the periodic assembly, i.e., to synchronous nonlinear free oscillations.

488


In Figure 10.1.3 the nonlocalized and localized branches of nonlinear normal modes are depicted, for p2 = 1, and varying values of the parameter ( K 1 / a 1 1 1 1 ), In this case twelve nonlinear mode branches exist, corresponding to stable and unstable synchronous oscillations of the multispan beam. The corresponding nonlinear mode shapes are also depicted in that figure. As (Kllal11 l)+0, the three-beam assembly possesses strongly localized (branches 1, 10, 12) and weakly localized (branches 3, 5 , 9, and 11) nonlinear normal modes. When the assembly oscillates in a strongly localized mode, only one beam segment oscillates with finite motions, whereas during a weakly localized mode two beam segments oscillate with finite amplitudes. Note that in this case the strongly localized modes are either generated through saddle-node mode bifurcations (modes 10 and 12) or are just the limits for small ( K l / a l l l l ) of continuous branches of stable nonlinear normal modes (mode 1). In addition to the localized modes, extended modes exist involving finite oscillations of all three beam segments (branches 1, 2, 4, 6, and 7). A linearized stability analysis indicates that all strongly localized nonlinear normal modes are orbitally stable and, hence, physically realizable. From the weakly localized and extended modes, only 2, 3 , and 5 are orbitally stable. An interesting feature of the solutions depicted in Figure 10.1.3(a), is the existence of destabilizing and stabilizing Harniltonian Hopf btfurcations (Van der Meer, 1985). These bifurcations can be studied in the complex plane by following the paths of the eigenvalues of the variational equations of the system: at the points of the bifurcations, four eigenvalues coalesce in pairs on the Imaginary axis and then split producing four complex eigenvalues with nonzero real parts. The Hamiltonian Hopf bifurcations are nongeneric bifurcations and lead to amplitude-modulated instabilities in the free dynamic responses of the assembly. Note that similar Hamiltonian Hopf bifurcations were observed in the dynamics of the system of coupled cantilever beams of section 9. I , 1. The final system to be considered in this section consists of n = 4 beam segments. Assuming that all segments oscillate in their first linearized mode, the differential equations governing the modal amplitudes and phases are given by:

Mode 1, Beam Segments 1, 2, 3, and 4 a1 1’ = -(wl/2)q1a2lsinOl2 2121’ = -(oi/2)qla31sin@23 + (ol/2)q2allsin@l2

(10.1.27a) (10.1.27b)


’

-5.00 0.00000

5.00

I

I

I

I

0.00005

0.00010

0.0001 5

0.00020

489

0.00025

;

3.75-

:. 8 *b*

2.50 -

-

c

N

I

CCI

1.25

_____-. /’. -.~:~:-7i1:i----.~-~.---..6 9 ______ L..-.-.___-.--

0.00 . -1.25

--

-2.50

-

-3.75

-

-5.00

A-iP

1

‘10

‘

Y--

2

3

Ki laiiii I

I

I

I

Figure 10.1.3 NNMs for oscillations in the first cantilever mode: -Stable modes, ----- Unstable modes, Hamiltonian Hopf bifurcations.

490


where, @12 = p11 - P21, 0 2 3 = P2l - p31, 0 3 4 = p31 - p41 q 2 = -y1 l(1) + 281 I( 1) - yf1 I ( 1) ( I 0.1.28) rl 1 = Y1 I(0) - 81 I(O),

and the various coefficients in the equations above are defined by relations (10. I . 13). Combining the first four equations (10.1.27a-d), one obtains the energy conservation relation, a1 12 + ,3212 + a312 + a412 = ~ 3 The ~ various . branches of nonlocalized and localized nonlinear normal modes for singlemode oscillations of this system are presented in Figures 10.1.4(a)-(d), for p3 = 1 and for varying values of coupling parameter (Kl/ai 1 1 I ) . In this case as many as 37 stable and unstable branches of nonlinear normal modes were detected, most of them existing at low values of the coupling parameter (K1/a1111). Modes 23, 24, 26, and 28 are orbitally stable and become strongly localized as (Kl/a1111)+0. Modes 23 and 24 branch off mode 22 through a Hamiltonian pitchfork bifurcation, whereas modes 26 and 28 are generated through saddle-node bifurcations. Modes 13, 16, 18, 25, 27, 29, 30, 33, and 34 are weakly localized for low values of ( K i l a i I 1 I ) , and correspond to finite oscillations of only two segments of the four-span beam. Modes 13, 16, and 18 are orbitally stable at certain ranges of parameter (Kilal111) and gain or loose stability through Hamiltonian Hopf bifurcations (cf. Figure 10.1.4); all other modes are orbitally unstable.


2.50

1.88

-0.63 -1.25

-1.88 -2.50

c

tt '

0,00000

'2.50

49 1

1

\ I

I

I

I

J

0.00005

0.00010

0.00015

0.00020

0.00025

1

.'

1

I .25

- I.88 -2.50 0.00000

0.00005

0.00010

0.00015

0.00020

0.00025

0.63

-1.25

-1.88 -2.50 (a) 0.00000

Ki laiiii 0.00005

0.00010

0.00015

0.00020

0.00025

Figure 10.1.4 NNMs for oscillations in the first cantilever mode: -Stable modes, ------ unstable modes, Hamiltonian Hopf bifurcations.

492


-

,m

;a

3.75

-

0.00

-

-1.25 -2.50

-3.75

y

.....5

8

................ ......... ,9,10 Y;,

-

-

-5.00 0.00000

I

I

I

I

0.00001

0.00002

0.00003

0.00004

0.00005

5.00

. -2

2.50

-

1.25

-

0.00

-

-1.25

-2.50 -3.75

0.00000

5.00 -

-

3.75

-

2.50

-

1.25

-

0.00

N

-1.25

..........-...!?. ................" ............. ?!.-.

-

-5.00

-c!

9,lO

-=++LL.....\]as x - + ~ M , where x is the spatial coordinate. An envelope soliton is a stationary or traveling periodically oscillating wave packet, which possesses a spatially localized envelope or a spatially localized slope envelope. Envelope solitons can interact elastically with other envelope solitons or with other types of solitons. Eleonsky (1991) investigated the problem of existence of breathers in the Klein-Gordon nonlinear wave equation, utt - uxx - g(u) = 0, g(0) = 0, gu(0) < 0, where x and t are the spatial and time variables respectively (with x in the range --oo < x < +m). The problem of existence of nontopological solitons in the Klein-Gordon equation was reduced to the equivalent problem of determining conditions for existence of homoclinic orbits in an appropriately defined Hamiltonian system. It was found that only a very restricted class of functions g(u) give rise to nontopological solitons. The same equation was considered by Weinstein (1983, where it was shown that if gu(0) > 1 and g(a) = 0, there exist solutions u(x,t), which are 2n-periodic in t, and decay as x + +w (or as x -+ --); however, the existence of breather solutions that decay as x + _+= remained an open question. Kruskal and Segur (1987) examined the ‘‘$4 model,” utt - uxx - u + u3 = 0. A technique based on Fourier series expansion was implemented in order to prove that no envelope solitons which decay as x + h (breathers) exist for this equation. Extended envelope solitons (dark or hole solitons) in a KleinGordon equation with quartic potential were examined by (Pnevmatikos,

498


1 9 8 3 , whereas more complicated asymmetric envelope solitons were investigated by Flytzanis et al. (1985). Whitam (1974) analytically investigated weak modulations of propagating waves by a variational approach. The Lagrangian density of the partial differential equation was first averaged over a wavelength, and the equations describing the wave modulations were subsequently obtained by imposing an "average variational principle." A "reductive" perturbation methodology for analyzing weakly modulated propagating solutions of systems of nonlinear wave equations was presented by Taniuti and Yajima (1969) and Taniuti and Wei (1968). Stretching transformations of the independent variables were introduced using a small parameter E, and the slowly varying wave modulations were computed by expanding the dependent variables in powers of E and imposing appropriate solvability conditions (i.e., eliminating self-resonances of the solutions at the various orders of approximation). Application of this technique to the study of weakly modulated breathers in the Klein-Gordon equation was given by Asano and Taniuti (1969). A similar analysis was performed in (Nayfeh and Mook, 1984), where the method of multiple scales was employed to obtain the amplitude and phase modulations of weakly modulated propagating wave packets. In an additional series of works, waves with spatially localized envelopes in infinite chains of coupled nonlinear oscillators were examined. Kosevich and Kovalev (1975) analyzed the localized normal oscillations in an infinite chain of unforced coupled nonlinear oscillators, by employing a "continuumlimit approximation," thereby reducing the problem to an unforced KleinGordon partial differential equation. The envelope solitons of this equation were then computed by a harmonic balance method. Using the same technique, Vakakis et al. (1993b) investigated forced localized motions occurring in an infinite nonlinear periodic chain. Traveling waves with spatially localized envelopes in infinite chains of vibro-impact systems were investigated in (Vedenova and Manevitch, 1981) and (Vedenova et al., 1985). Pnevmatikos et al. (1986) studied the dynamics of solitons in a nonlinear one-dimensional diatomic lattice system. Employing the continuum-limit approximation, they reduced the problem to the modified-KdV equation, utt = c*uxx + a(d),, + buxxxx, which admits breather solitons. Discreteness effects on envelope solitons propagating in a nonlinear lattice were studied in (Peyrard and Pnevmatikos, 1986), whereas Li et al. (1988) investigated the scattering by small disorders of breathers propagating in a discrete chain of

10.2 WAVES WITH SPATIALLY LOCALIZED ENVELOPES

499

oscillators and computed the secondary breathers, which were generated due to the disorder.

10.2.1 General Formulation Consider a nonlinear partial differential equation of the general form: Utt

= L(O)[U(X,t)]+ &L(I)[U(X,t)],

--oo

< x < +-oo

(10.2.1)

where L(o)[*] and &L(I)[*]are integro-differential operators acting on the dependent variable u(x,t), x and t are the spatial and temporal variables, respectively, and E is a perturbation parameter ( I E I 0, p 2 0, and I d = (- l>"un(Z) (10.2.12) Substituting this relation into (10.2.1 1), one obtains:

+ (1 + 4 ~ b ) u n- ~ b ( 2 u n- un-1

un+l) + &pun3= 0 n = O,+l,f2,... (10.2.13) In the continuum limit, the displacements and forces become continuous functions of z and x, un(Z)+U(X,T), and the finite difference term appearing in equation (10.2.13) is replaced by a second partial derivative: un

-


503

(2un - Un-1- un+l) + -uxxh' where h is the distance between adjacent particles (cf. Figure 10.2.1). Taking into account these transformations, the infinite set of equations (10.2.13) is replaced by the following nonlinear partial differential equation: uTT

+ Ebh2Uxx + (1 + 4 ~ b ) u+ E F U ~= 0,

-M

< x < +-

(10.2.14)

Introducing the new time variable t = (1 + 4&b)1%, this equation can be written into the following final form, which is of the general form (10.2.1):

where h = bh2/(l + 4 ~ b >) 0 and a = ~ / ( +l 4 ~ b >) 0. Equation (10.2.15) is the starting point of the analysis. A first integral of the motion can be easily computed as:

E = (1/2)

+-

I

[ut2

+ ~2 - & ~ L U X ~+ ( & a / 2 ) ~ 4dx ]

(10.2.16)

It is assumed that E < -OO for the types of motions considered in this problem. Stationary, time-periodic solutions are now sought in the form (10.2.3), satisfying conditions (10.2.2), with the limiting values uf to be determined. Introducing the change of coordinates (x,t)+ [x,uo(t)], and taking into account (10.2.16), the time derivatives uo,t and uo,tt are evaluated as follows:

Uo,tt =

{ - u - EhUxx - &aU3} x=x0

(10.2.18)

Equation (10.2.17) is analogous to expression (10.2.6) of the general formulation of the previous section. Introducing the change of coordinates (x,t)+ [x,uo(t)], transforming the time derivatives using the chain rule, and taking into account expressions (10.2.17) and (10.2.18), one obtains the

504


equation governing the amplitude modulation function U[x,uo(t)] (which corresponds to the functional relation (10.2.9) of the general formulation):

{

+r

[[U2(X,U0*)- U*] - Eh[UX2(X,U0*)- U,2]

-W

+ [- u - EhUxx - EaU3]x,xo aau = - u - EhUxx - EaU3 U0 ~-

( 1 0.2.19) Equation (10.2.19) must be solved with the following two additional conditions: ( 1 0.2.20)

lim x+.+m ~ [ x , u ~ ( t=) u+ l

[(-u - EhU,,

au - EaU3),--XO aUo+ u + EAU,, + EaU3]Uo=uo*= 0 ~

(10.2.21) Equation (10.2.21) guarantees analytic continuation of the solution up to maximum potential energy value. Note that the coefficient of (d2U/auo*) in equation (10.2.19) vanishes when uo = uo*, i.e., when system (10.2.15) reaches its maximum potential energy. Therefore, the value uo = uo* represents a regular singular point for equation (10.2.19). As in previous sections, the solution is sought in the series form: U[x,uo(t)] =

c EkU(k)(X,UO)

(10.2.22)

k=O

Due to the compatibility relation, U[x,uo(t)] = uo(t), the following additional conditions are imposed on the terms of the series: U(0)(xo,uo) = uo,

and

U(k)(xo,uo)= 0,

k2 1

(10.2.23)

Upon substitution of the series expression into (10.2.19)-(10.2.21), a matching of coefficients of equal powers of E leads to a series of subproblems at successive orders of approximation.

10.2WAVES WITH SPATIALLY LOCALIZED ENVELOPES

505

O(EO) Approxima tion Since (10.2.15) is separable in space and time for E = 0, the O(1) approximation U(0)(x,uo) is expressed in the linear form, U(o)(x,uo) = al(0)(x)uo(t). Coefficient al(O)(x) satisfies the following relation:

Assuming that the product [a1(0)(x)uo(t)] is not identically zero, it follows that al(0)(xo) = 1. This result is equivalent to the first compatibility relation in equations (10.2.25). Hence, the solution of the O(EO) problem does not provide any additional information regarding the form of coefficient al(O)(x). This coefficient is computed by considering the O(E) terms of the problem. O ( E ~ Approximation ) Expressing the modal function as U[x,uo(t)] = al(0)(x)uo(t)+ EU(l)[x,uo(t)], and retaining only terms of O(E), one obtains the following equation

506


In the above equations, a prime denotes differentiation of a function with respect to its argument. The solution for U(l)[x,uo(t)]is sought in the form, m

u(l)(x,u0) =

C am(l)(x)uo*(t)

m=O

(10.2.27)

which upon substitution into (10.2.25) and (10.2.26), and matching of respective powers of uo(t), leads to the following equations governing the coefficients am(l)(x):

For simplicity, in the equations above only terms up to O(uo3(t)) were included. Complementing the above set of equations is the following set of compatibility and limiting conditions:

Conditions (10.2.29) guarantee that, to this order of approximation, U[xo,uo(t)] = uo(t), whereas relations (10.2.30) guarantee constant limits for U[x,uo(t)] as x+b=. Note that the linear coefficient a ~ ( I ) ( xis) not computed at this order of approximation, since it cancels from the O(E) equations (10.2.25). To compute al(l)(x) one must resort to the 0 ( & 2 ) problem. From (10.2.28), it is noted that all odd powers of uo(t) vanish in the expression of U(l)(x,u,), since the equation of motion contains only even (symmetric) nonlinearities. Equations (10.2.28) admit the following solutions: ai(O)(x) = sech[(3a/8h)1/2uo*(x - xo)] a3(l)(x) = -(a/8)sech[(3a/8h)1/2uo*(x- xo)]


x

{ 1 - sech2[(3a/8h)1/2uo*(x - xo)] )

507

(10.2.3 1)

which satisfy the limiting conditions: lim x++m al(O)(x) = lim x++m a3(l)(x) = 0 Similar expressions can be derived for higher approximations am(l)(x),m = 5,... if terms of O(uo5(t)) or higher are included in the O(E)analysis, but this is not performed in this work. To compute the coefficient ai(l)(x), one considers O ( E ~terms ) in (10.2.19)-(10.2.21). O ( E ~ Approximation ) The amplitude modulation is expressed in the form,

and only terms of O ( E ~are ) considered. Expanding the O(c2) term of the modal function in the form,

the following governing equation for al(l)(x) is obtained:

where the scalars K1 and K2 are independent of x and are defined in terms of the previously computed approximations a1(O)(x) and a3(1)(x) as follows:

508


5

+m

K2 =

-ca

[2al(o)(~)a3(l)(x)+(a/2)ai(~)~(x)] dx

+=J

I a1(0)'2(x) dx

(10.2.34)

--Do

Equation (1 0.2.33) is the equation determining the yet unknown coefficient al(I)(x) and is solved subject to the additional conditions:

At this point the following coordinate transformations are introduced:

z = (3a/8h)1/2 uo*(x - xo), $(z) = al(l)[z(3a/8h)-1/2uo*-1

+ x,]

0(z) = a1(0)[z(3a/8h)-1/2uo*-1 + xo] = sechz v(z) = a3(1)[z(3a/8h)-1/2uO*-l + xo] = -(a/8)sechz(l - sechzz) (10.2.36) and equations (10.2.33) and (10.2.35) assume the form: $"(z) - (1 - 6sech2z)@(z) = Fiv"(z) + F2v(z) + F3v(z)@(z) + F40(z) = G(z) Q(0) = 0, and lim z++ca $(z) = c(ll)*

(10.2.37a)

In the expression above, the quantities Fi, i = 1,2,3,4 are defined according to the formulas F1= -3~0*2/4

~*2 + (9u0*2/4)e"(0) + 6uo*2] F2 = -( l / a ~ ~ * 2 ) [ 4 ( K ~+u K2uo*4> F3 = -6~0*2, F4 = $"(0) + ( 3 / 4 ) ~ " ( 0 ) ~ 0 * 2 (10.2.37b) The coefficients F1, F2, and F3 depend on the parameters uo*2, a,and on the previously computed approximations v(z> and @(z),whereas coefficient F4 depends on the solution itseK und thus is yet undetermined. As shown in the following analysis, the term $"(0) in the expression of F4 is computed so that the limiting requirements (10.2.37) on $(z) are satisfied. System (1 0.2.37) is a linear, parameter-dependant, nonhomogeneous ordinary differential equation. Due to limiting conditions (10.2.20), the solution of


509

(10.2.37a) must reach constant limits as z+ltm. It can be shown (Vakakis, 1994a) that this solution is of the following form:

where @h(i)(q),i = 1,2, are two linearly independent homogeneous solutions of (10.2.37), defined as: @h(l)(z) = sinhz/cosh2z @h(2)(z)= @h(1)(z)[(3/2)z+ (1/4)sinh2z - cothz] (10.2.39) Note that lim z+km$h(l)(q) = 0, and, more importantly that $h(2)(z) diverges as z++m. The constants y and 6 in expression (10.2.38) are computed by imposing the limiting conditions (10.2.37). The two integrals in expression (10.2.38) were analytically evaluated using Mathematics, as follows:

I G(q)@h(')(q)dq = [(aF1 + aF2 - 8F4)/16]sech2z 2

0

-

[a(llFi

+ F2 - F3)/32]sech4z

[a(-12F1+ F3)/48]sech6z

+ [(3aF1-

Z

G(q)@h(2)(q)dq = -[(aFl 0 -

3aF2 - a F 3 + 48F4)/96] (10.2.40)

+ C X F-~8F4)/16]~

+ [(3aF1 + 3aF2 - 24F4)/32]z

[3a(llFi

sech*z

+ F2 - F3)/64] z sech4z - [a(-12F1 + F3)/32]z

sech6z

- [(51aF1 + 9aF2 - 5aF3)/192]sinhz sech3z - [a(-12F1+ F3)/32]sinhz sechsz -

[(-24aFi - 6aF2 + aF3 - 72F4)/96]tanhz (10.2.41)


5 10

From the expressions above it can be shown that

and that I2

+ -[(-24aF,

- 6aF2 + aF3 -

72F4)/96] - [(aFl + aF2 - 8F4)/16]z for large IzI (10.2.43)

Taking into account the limiting behaviors for large IzI of the various terms in (10.2.38), it can be shown that, 0

i.e., that the first summation term in expression (10.2.38) decays for large values of IzI. Considering the second summation term, for large values of IzI the integral Z

G(qf@h(l)(T)dq 0

reaches a constant limit, whereas the homogeneous solution @ h ( 2 ) (z ) diverges. Therefore, in order for (10.2.38) to be bounded as z + h , it must be satisfied that

6 = -1im

z

z+fmJ 0

G(q)@h(I)(q)dq = (-3aF1

+ 3aF2 + aF3 - 48F4)/96

(10.2.44) Imposing the boundary condition @(O) = 0 on equation (10.2.38) leads to the relation 6 = 0. This result, combined with relation (10.2.44) leads to an expression relating F I , F2, F3, and F4; this expression can be used to evaluate the yet undetermined quantity F4 as follows: -3aF1

+ 3aF2 + aF3 - 48F4 = 0 =+ F4 = (1/48)[-3aF1 + 3aF2 + aF3]

(10.2.45) Once F4 is determined, its definition from (10.2.37b) is employed to evaluate the unknown quantity @"(O):


5 11

The remaining unknown constant in (10.2.38) is y, which cannot be determined only by considering the limiting conditions (10.2.37). It will be shown that the computation of y can be accomplished only when the complete solution for the amplitude modulation is examined. Combining all previous findings, the expression for $(z) becomes: $(z) = y sinhz/cosh%

+ { [(auo*2 + K1 + uo*2K2)/24]z coshz

- (auo*2/48)sinhz

]tanhz sech2z

(10.2.47) Taking into account the previously defined transformations (10.2.36), solution (10.2.47) enables the computation of the O(E) linear approximation al(l)(x) as z = (3a/81L)l/2u0*(x - xo) al(l)(x) = $(z), By employing all previous analytical findings, one can now derive an asymptotic approximation for the amplitude modulation (modal function) U[x,uo(t)] of the stationary, localized wave. By construction, the asymptotic solution is valid only for sufficiently small values of the perturbation parameter E and for sufficiently small amplitudes of the reference point, , O ( E ~calculations, ) the uo(t). Combining the results of the O(EO), O ( E ~ )and amplitude modulation U[x,uo(t)] is expressed as: ~[x,u,(t)l= [al(O)(x) + ~ a l ( l ) ( x ) ] u ~ (+t )~ a j ( l ) ( x ) u ~ 3 (+t )o ( E u ~ ~ ( ~ ) , E ~ ) =

{ sechz +

Ey sinhzkosh2z

+ E[ [(auo*2 + K1+ uo*2K2)/24]z coshz

- (auo*2/48)sinhz] tanhz sech2z}uo(t) - ~(a/8)sechz( 1 - sech2z)uo3(t) + O ( E U ~ ~ ( ~ > , & ~ ) z = (3~./8A)1/2~0*(~ - xO)

(10.2.48)

Note that the constant y still needs evaluation. To accomplish this, one employs the identity, sinhz/cosh*z = -d(sechz)/dz, in terms of which the two leading summation terms on the right hand side of (10.2.48) can be expressed as sechz + &y sinhz/cosh2z = sech[z - ~y+ 0(&2)]

5 12

NONLLNEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS

Substituting this result into (10.2.48), evaluating the resulting expression at the reference point x = xo (or equivalently, at z = 0), and employing the compatibility identity, U[xo,uo(t)] = uo(t), one obtains the following relation which evaluates the scalar sech[-ey

+ O ( E ~ )=] 1

y= 0

(10.2.49)

In view of (10.2.49), the asymptotic expression of the modal function can be expressed in its final form: U(x,u,(t)) =

{ sechz +

E[

[(auo*2 + K1 + uo*2K2)/24]z coshz

- (ctuo*2/48)sinhz] tanhz

sech2z }uo(t)

- ~(a/8)sechz(1 - sech2z)uo3(t)

+O(EU~~(~),E~)

z = (3~t/8h)~"uo*(x - xO) (10.2.50) The analytical result (10.2.50) is now employed to compute the reference response uo(t). To perform this calculation, the relation u(t) = U[x,uo(t)] is substituted into the governing partial differential equation ( 10.2.15), and the resulting expression is evaluated at the reference point x = xo. The following ordinary differential equation governing uo(t) is then obtained: uo"(t) + [ I + &hal(0)"(xo)+ ~ 2 ~ a ~ ( 1 ) " ( x ~ ) ] u ~ ( t ) + [&a+ &2ha3(1)"(~o)]~03(t) + O ( E ~ U ~ ~ (=~ 0) , & ~ ) (10.2.51) where the quantities al(0)"(xo), al(l)"(xo) and a3(l)"(x0) can be computed using previously derived analytic expressions. Assuming the set of initial conditions uo(0) = uo*, uo'(0) = 0, i.e., initiation of the system from a position of maximum potential energy, the solution of equation (10.2.5 1) is

10.2 WAVES WITH SPATIALLY LOCALIZED ENVELOPES 1.20 L

-30 -25

0.10

0.0s

-0.70

I

-20

-IS -10

-I

0

. m o l VOriiOM..

S

.

I0

I5

20

25

YI

1

1

-30 -25

5 13

-20

-I5

-10

-5

*pow

0

5

.

*ow..

10

IS

20

25

YI

Figure 10.2.2 Leading spatial coefficients of U[x,uo(t)]. Results (10.2.50) and (10.2.52) complete the solution, which represents a stationary, spatially localized periodic motion of the system (stationary breather), with frequency of oscillation given by w = o(u0*) = np/2K(k), where I((.) is the complete elliptic integral of the first kind. In Figure 10.2.2, the spatial coefficients al(O)(x), al(l)(x), and a3(l)(x) for a system with a = 1.2, = 0.9, uo* = 0.25, xo = 0, and E = 0.01 are depicted. Note that in contrast to al(O)(x), the higher order approximations ai(I)(x) and a3(1)(x) possess three local extremes. Moreover all three spatial coefficients decay as x++. In Figure 10.2.3, the expression (10.2.50) for the envelope of the stationary wave is presented. The solutions (10.2.50) and (10.2.52) indicate that on a stationary localized wave, the system vibrates “in unison, ” i.e., all points vibrate equiperiodically, reaching their extreme values at the same instant of time. Hence, in the terminology of Rosenberg (1966) and Vakakis (1990), the stationary wave corresponds to a localized nonlinear normal mode defined in the infinite domain -00 < x < +m. Hence, standing waves

-x)

-25

-20

-15

-10

-5

0

5

spoliol wriobk. 1-2

-50

-25

-20

-15

-10

-5

0

10

15

20

25

30

10

15

20

25

M

I

S

~poijotial variable. I

'

i

E

=P

ax) 0.15

0.00 -0.15 -0.30

1-J

I

-30 -25

___---_-----.-. -20

-15

-10

-5 0 S *pOtial variable. I

I 10

15

20

25

30

Figure 10.2.3 Envelope of the stationary wave.

with spatially localized envelopes can be regarded as infinite domain extensions of localized NNMs defined for discrete and bounded continuous oscillators. In addition to stationary waves, equation (10.2.15) admits solutions in the form of traveling waves with localized envelopes. However, these waves correspond to nonsynchronous motions of the system and cannot be considered as being extensions of NNMs. Traveling waves with localized envelopes can be asymptotically approximated by imposing a Lorentz coordinate transformation on the stationary solution (10.2.50) and (1 0.2.52) (Kosevich and Kovalev, 1975). A direct calculation shows that if (10.2.15)


515

admits a nontraveling solution of the form u(x,t) = U[x,u,(t)], then it also admits a class of traveling solutions of the general form:

(10.2.53) where v is the traveling wave velocity (group velocity) and is related to the frequency o and wavelength k of the propagating wave packet by the wellknown relation v = (do/dk), where o and k are related by a dispersion relation of the form co = o ( k ) . Equation (10.2.53) combined with expressions (10.2.50) and (10.2.52) provides a means for the asymptotic approximation of the solitary wave solutions h(x,t). The frequency of oscillation o of a solitary wave is related to its group velocity v by the relation: 0 = O ( V , U ~= * )7~p(1

+ v2)1/2/2K(k)

(10.2.54)

where p and k were defined earlier. Note that the stationary wave can be regarded as a special case of the solitary wave solution with group velocity v = 0. The stationary and traveling solitary waves of the partial differential equation (10.2.15) correspond to stationary or traveling localized oscillations of the chain of particles of Figure 10.2.1. It is interesting to note that, in view of the previously imposed anti-phase transformation (10.2.12), these localized vibrations correspond to antiphase motions between adjacent particles of the system. Forced localization in the periodic chain of Figure 10.2.1 was examined by Vakakis et al. (1993b) using a harmonic balance technique. In that work, harmonic forces with various spatial distributions were applied to the chain, and a variety of stable and unstable, forced localized oscillations were asymptotically computed.


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Admissible forcing functions, 238253 Anderson localization, 4 10

Continuous systems: of finite extent, 349-380, 391, 473 of infinite extent, 380-390, 389, 496-5 15 NNMs Of, 349-390, 352, 381

Asymptotic stability, 159, 175 Attenuation zones, 389, 390

Continuum approximation, 380, 38 1, 386,388,498 Convex nonlinearities 54, 165

Backbone curves, 19, 255 Cosine-like functions, 230 Bifurcations of NNMs, 11, 84-85, 181, 201, 207, 253, 334-336, 343, 404-41 0, 486-496

Degenerate modes, 308, 320, 328330

Breathers, 497-499 Canonical coordinates, 2 15 Chaotic motions, 16-18

Existence of NNMs: applications, 46-52, 64-68 criteria for, 25, 35-46, 52-68, 290-292 localization, 289-304

Concave nonlinearities 53, 165 Conservative systems: group theory analysis, 141, 142 NNMs Of, 72-76, 201, 270-274 stability of NNMs of, 175, 179, 197

Fast Fourier transforms, 470, 471 Floquet matrix, 376 Floquet multipliers, 160, 167, 380

549

550

INDEX

Fourier series (generalized), 242-253

KAM theorem, 15 KdV equation, 497

Group theory, 7, 130-144 Gyroscopic systems, 142-144

Heteroclinic orbits, 265, 274-276, 387

Klein-Gordon equation, 382, 497, 498

Lame’ equation, 192, 193, 221-224 Linearizable systems, 87-97, 105

Homoclinic orbits, 13, 387 Homogeneous systems, 84,98, 117, 104, 114, 177, 182, 183

Impurity modes, 287 Ince algebraization, 171, 172 Internal resonances, 93, 207-219, 351, 398, 400, 402, 413, 419, 48 1-483 Invariant manifold approach, 124130. 370-372

Localization: examples, 122-124, 315-325, 341344, 449-461 experiments, 462-47 1 impulsive, 344-347, 4 10-424 in discrete systems, 5, 123, 285347, 295, 304, 325, 337, 501 in continuous systems, 39 1-471, 473-5 15 in vibro-impact systems, 149-153 theorems, 298, 300 transition to nonlocalization, 325337 using NSTTs, 337-347 Lorentz transformations, 501, 5 14

Invariant manifolds, 16-18, 125, 127, 37 1 Matched asymptotics, 103-116 Integrable systems, 12 Modal curves, 9, 305-325 Jump phenomena, 19,275

Monte Carlo simulations, 41 1

INDEX

Multi-span beams, 473-496

Nonlinearizable systems, 97- 103 Nonsmooth transformations: applications, 337 localization, 337-344 method of, 146, 153, 261-283, 337, 383,385 vibro-impact systems, 147 Nonsimilar NNMs: definition, 9, 78 examples, 116-124, 207 group theory, 134, 143 localization, 305-325 matched asymptotics, 86-116 stability, 196-207 trajectories, 77, 78, 86 vibro-impact systems, 149

551

260 fundamental, 18-21, 229, 424, 428-435 subharmonic, 229, 424, 435-444

Schrodinger’s equation, 187, 221 Similar NNMs: definition, 9, 78 group theory, 134 stability, 169-196 trajectories, 77, 78, 80-86 vibro-impact systems, 149 Sine-Gordon equation, 225-227, 497 Smale horseshoes, 16 Solitary waves, 219-227,496-5 15 Spiral waves, 222, 225

Normal forms. 129

Orbital stability, 161, 162, 461 Pade ’ approximations: NNMs, 106-116 Poincare’ maps, 12-18, 440-443

Resonances: effects of NNM bifurcations, 253-

Stability of NNMs: asymptotic, 159 boundaries, 175, 186, 224, 225 continuous systems, 374-380 discrete systems, 289-294 examples, 167-169 finite-zoning, 186-196 general results, 158-169, 196 orbital, 161, 162 theorems, 162, 166, 187, 292 vibro-impact systems, 150-153 Stationary waves:

552

INDEX

as NNMs, 381-390 stability of, 219 Steady-state motions: admissible forces, 238-253 examples, 18-2 1, 253-260 exact, 230-253 localization of, 424-444 theorem, 247, 248

Vibro-impact systems: NNMs, 145-155 sawtooth variables, 147, 262-264 Zhuravlev’s transformation, 147- 155

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