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
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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 ~ << 1 1 . It is well known (Nayfeh and Mook, 1984) that the free response of this system is governed by two time scales, t and Et, which exist due to the weak nonlinearity and the weak coupling. Moreover, weak coupling leads to energy redistribution (beating) between the two particles of the system. Hence, a two-scales asymptotic analysis seems to be the natural way to proceed in computing an approximation to the free oscillation (Rand et al., 1992). In this case normal modes correspond to single-frequency motions, which to the leading order of approxiniation possess exponential temporal dependence. However, one can consider an alternative method for solving the problem. Instead of computing an approximate asymptotic solution of system (1.1.1), it is possible to replace it by the following equivalent nonlinear system which adinits exact (closed-form) solutions (Kosevich and Kovalev, 1989):
where j = (-1)1/* and 51 characterizes the coupling term Note that if a=O system (1.1.2) becomes linear. It is easy to show that both linear and nonlinear systems admit single-frequency exponential solutions of the form:
Wi = ai exp(-jot),
i = 1,2
(1.1.3)
where the complex amplitudes aj are expressed as:
(1.1.4)
1.1 CONCEPTS OF NNM AND NONLINEAR LOCALIZATION
5
If 8 = n/4 one obtains at most two single-frequency solutions. When $ = 0 the system vibrates in an inphase NNM and the oscillators undergo in-phase vibrations with frequency equal to o = 00 - a N/2 Similarly, when $ = 71: the system vibrates in an antiphase NNM and the motions of the oscillators are in antiphase, with frequency o = 00 + Q - a N/2 Hence, one obtains a nonlinear generalization of linear normal modes from the viewpoint considered. The previous example illustrates an additional dynamical feature of the nonlinear system. It is easily proven that, if N > 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
1.2 EXAMPLE: NNMs OF A TWO-DOF DYNAMICAL SYSTEM
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}
1.2 EXAMPLE: NNMs OF A TWO-DOF DYNAMICAL SYSTEM
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
1.2 EXAMPLE: NNMs OF A TWO-DOF DYNAMICAL SYSTEM
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
1.2 EXAMPLE: NNMs OF A TWO-DOF DYNAMICAL SYSTEM
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 << 1, is a small parameter, and the excitations are assumed to be harmonic, and of identical phase and frequency. System (1.2.8) was studied by Vakakis ( 1 992b) using the method of multiple-scales. Assuming that the frequency of the external excitations is close to the linearized natural frequency of the oscillator, w = 1 + co, where (T is a frequency detuning parameter, it was shown that the fundamental resonances of the system can be approximated by x(t) = a1 cos(t + p i ) + O(E) y(t> = a:! cos(t + p2) + O(E)
i1.2.9)
1.2 EXAMPLE: NNMs OF A TWO-DOF DYNAMICAL SYSTEM
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 are shown, and it can be seen that an additional branch of steady-state solutions exists, occurring in the neighborhood of one of the (stable) bifurcating NNMs (the one corresponding to large values of x and small values of y). Note that in this case no stable steady solutions exist in the neighborhood of the backbone curve of the unstable antiphase NNM, and that no steady motions occur in the vicinity of the bifurcated mode corresponding
20
INTRODUCTION
a2
I
Figure 1.2.6 Fundamental resonance curves: (a) before (K = 0.4 > 0.25) and (b) after (K = 0.1 < 0.25) the NNM bifurcation. -Stable solutions, ----- Unstable solutions, - - - N N M curves.
1.2 EXAMPLE: NNMs OF A TWO-DOF DYNAMICAL SYSTEM
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.
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 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,
2.1 PRELIMINARY FORMULATION
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
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
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
2.1 PRELIMINARY FORMULATION
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)
2.1 PRELIMINARY FORMULATION
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
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
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)
2.1 PRELIMINARY FORMULATION
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
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
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)]
2.2 EXISTENCE THEOREM FOR NNMs
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).
2.2 EXISTENCE THEOREM FOR NNMs
Y[X(z>l2 0,
39
[Gx](z) > 0 for z E [0,1/4) ,
[-]'(1/4) <0 (symmetric case) (2.2.8)
Hence, the operator [G.] is strictly positive, and, as a result, for z E [0,1/4) it transforms any nonnegative vector ~ ( tinto ) a positive one. Similarly, in view of (2.1.27) and (2.2.7), for T E [T-,T+) and for every y(t) 2 Q the following inequalities hold:
Hence, it is proven that the operator [G,:] is also strictly positive. Positive operators possess special properties (Krasnoselskii, 1968), some of which will be employed in the subsequent analysis. These properties will be applied directly to the integral operators considered herein, along with some simple proofs. Denote by L the linear operator formed by replacing g(x) in (2.1.23) by (kwhere ) R is a nonnegative matrix. The following linear eigenvalue problem is then formed:
x=hh
(2.2.10)
) a positive eigenvector of (2.2.101, and hi the corresponding Let L ~ ( Tbe eigenvalue. Krasnoselskii (1968) has proven the existence of such an eigenvector. Lemma 1. The eigenvalue hi is the smallest positive eigenvalue of problem (2.2.10). Proof. Consider the eigenvalue problem,
xi =hi h i , hi > 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
2.2 EXISTENCE THEOREM FOR NNMs
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)
since by assumption, Cxl(z) 2 Q and L > 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
2.2 EXISTENCE THEOREM FOR NNMs
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
2.2 EXISTENCE THEOREM FOR NNMs
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
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
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
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
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 r22(~,u,T)
Let mi* be the first natural frequency of the system and be the corresponding period. Assuming that inequalities (2.3.5) and (2.3.6) hold for TE [Ti*,Ti+), where T1+=2nlwl, by Theorem 1 there exists a continuum S1 of NNMs satisfying the conditions x(t) = x(-t) = -s(t+T/2) and s(t) > Q for t E (0,T/4) connecting the origin s = Q and an. If then it can be shown that xj(t) vary monotonically between their extreme values Xi(0) = Ai and xi(T/2) = -A, Suppose now that 02/w1 5 2. Then both terms in the second inequality (2.3.6) are positive, and this inequality holds trivially. Clearly, the remaining inequalities (2.3.5) and (2.3.6) hold for TE [Ti*,Ti+), provided that they hold for T = Ti *. Let mi = m2 = m, and n i = n2 = n. Then, setting x l l = x12, x21= -x22 and taking into account (2.3.5) and (2.3.6), leads to the following inequality: 01"
5 (112) (01
+ 02) = (112) [ n1/2 + (n + 2c)l/* ]
rn-ll2
(2.3.7)
52
NNMs IN DISCRETE OSCILLATORS: QLJALITATIVE RESULTS
e.
This condition can be also formulated by means of the elements of matrix In particular, if one sets pi = p2 = p, then m i * = (p/m)l/2, and inequality (2.3.7) becomes (p/n) 5 (1/4) [ l
+ (1 + 2c/n)1/*I2
(2.3.8)
where ( c h ) 5 (3/2) in view of the assumed condition m2/w1 5 2 . The values of p and n depend on the region R . If nu I fl(u) 5 pu for all u > 0 (i = 1,2), then under condition (2.3.8) the continuum S1 of NNMs tends to infinity tor large values of the total energy h. There exists an additional continuum S2 of NNMs corresponding to the Lyapunov family ~ 2 ( t , h ) .These periodic solutions are the nonlinear continuations of the linearized antiphase normal mode x i = - x2, To analyze this family of NNMs it is necessary to introduce a change of sign to coordinate xi or x2 in order to comply with the requirement ~ 2 >0 0. Suppose that nl+c > 0 and n2+c > 0. Then the conditions on the restoring forces (2.3.1b) are satibfied, and the continuum S2 tends to infinity as the total energy of the motion increases. These NNMs correspond to anti-phasc oscillations of masses mi and m2.
2.4 NNMs IN SYSTEMS WITH CONCAVE AND CONVEX NONLINEAKITIES In nonlinear systcms the function T=T(h) relating the period of oscillation and the total energy h is not, in general, monotonous. If function T(h) does increase or decrease monotonically with h, the system will be termed sojrfy ur hardly anisocliroiiic, respectively. The type of anisochronicity is determined by the nature of the nonlinearity. In particular, in accordance with a theorem by Opial (Opial, 1961), the single-DOF system, X+f(x)=O,
X E
R
(2.4.I )
is softly or hardly anisochronic if the function c(x) = f(x)/x decreases or increases with 1x1, respectively, so that a straight line y = kx intersects f(x) not more than in one point for x > 0 and x < 0 (cf. Figure 2.4.1). Consider now the n degree of freedom nonlinear system:
2.4 NNMs IN SYSTEMS WITH CONCAVE AND CONVEX NONLINEARITIES 53
J
Y
0
X
Figure 2.4.1 Nonlinear restoring force for (a) a softly and (b) a hardly anisochronic single-DOF nonlinear oscillator.
MX + f(x) = 0,
f(x) = V,(s),
21 E Rn
(2.4.2)
In this section conditions on the potential function V(x) are obtained, which guarantee soft or hard anisochronicity of nonlinear normal mode oscillations. These conditions provide some additional properties of NNMs, especially for the case of softly anisochronic systems. In compliance with previously introduced notation, A(x) = V,(x) is the Hessian of V(x). Using the integral mean-value theorem, f(x) can be written in the form:
f(x) = C(X>X where C(&> is an (n x n) matrix with elements
(2.4.3) Cik,
defined as
A ( u ) dv
C(x)= 0
Suppose that for inequalities:
x E s1, 2~ f 0, matrices C(x>and A(x) satisfy C(8) > A(x)
(concave nonlinearity)
the
(2.4.4a)
54
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
or
C(x) < A(x)
(convex nonlinearity)
(2.4.4b)
These relations imply that for any vector y # 0 the quadratic forms of these matrices satisfy the inequalities
respectively. Inequalities (2.4.4) characterize the type of nonlinearity as concave or convex. To clarify conditions (2.4.4), consider the nonlinear system depicted in Figure 2.3.l(a). For this system, the quadratic forms of matrices A(x) and C(x) assume the following explicit forms:
Inequalities (2.4.4a) or (2.4.4b) are satisfied if all restoring forcing functions fii(x) and fi(i+l)(z) are of the type depicted in Figure 2.4.1 (a) or 2.4,1(b), respectively. As a second example, consider the transverse vibrations of a string with concentrated masses [cf. Figure 2.3.l(h)]. The potential function of this system is given by
where xi ,...,xI1 are the displacements of the masses, Xn+l = xo = 0, 10 ,...,1“ are the lengths of string segments between masses, E and F are the modulus of elasticity and cross-sectional area, respectively, and To is the initial tension of the string. The corresponding quadratic forms of matrices A(x) and C(x)are given by:
2.4 NNMs IN SYSTEMS WITH CONCAVE AND CONVEX NONLINEARITIES 55
Clearly, for k > 1 (k < 1) the system is softly (hardly) anisochronic. Note that in physical terms k is the relative elongation of the string caused by tension To. Hence, for this system the type of nonlinearity (concave or convex) depends only on the magnitude of the initial tension. Some NNM properties implied by inequalities (2.4.4) are now discussed. First, systems with concave nonlinearity are considered. Suppose that the conditions of Theorem 1 are satisfied, and in addition, that the linearized matrix A(&)satisfies the inequalities:
Inequality (2.4.8) implies that matrix [A(x)-N] is nonnegative. In accordance with Theorem 1, there exists a continuum S of nonlinear periodic solutions (NNMs) satisfying (2.1.12) and connecting the origin x = 0 and the boundary dsZ. The following theorem is proved for this family of NNMs. Theorem 2. Under conditions (2.4.4a) and (2.4.8), the continuum S of NNMs within R consists of the one-parameter family x(t,h); the corresponding period of oscillation T(h) increases monotonically. Proof. Let X(T) E S . It follows that X(T) is a positive solution of the boundary problem (2.1.19) and (2.1.21), with corresponding period of oscillation equal to T. According to perturbation theory, x(T,T) has a unique
56
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
analytic continuation in T if the following boundary problem for the variational equation has only the trivial solution:
&&” + T~A(T,T)Y = 0,
~ ‘ ( 0=) y( 1/4) = Q,
A(t,T) = A[K(T,T)] (2.4.9) Consider now the self-conjugate boundary value problern:
m’+ T~NJ
+ h T 2 R ( ~ ) y= 0,
y’(0)= y(1/4) = 0,
R(t) = RT(T)
(2.4.10) where the components of R(T)are denoted by r,k(T), i,k=l, ..a It can be shown that this problem is equivalent to the following integral equation: y(T) =
[GRY](T) where [GRY](T) = -T2
IN
n
T(~,u,T)E(u)y(u) du (2.4.1 I )
Since x(z,T) satisfies the equation
IV&”+ T2 C(7,T)y = 0
(2.4. I ? )
where C(T,T)= C[x(.r,T)I it follows that problems (2.4.10) and (2.4.11) possess the eigenvalue h= 1 when R(T)= C(7,T)- l3 By (2.4.8) and (2.1.1S), the matrix [C_(T,T)-N] is nonnegativc, and, as ;I result, the corresponding integral operator GR is strictly positive. Hcnce, thc eigenvalue h = 1 corresponding to the positive eigenvector x(7,T) is the least positive eigenvalue of the system. Now, consider system (2.4.10) with R(T,E)= C(T)- + E [A(T)-~(T)] By (2.4.8), R(z,E)is nonnegative for E E [0,1], the first eigenvalue 11= hi(€,)is simple, and the corresponding eigenvector ~ ( T , E )is positive [note that ~ ( T , O )= x(z,T)]. Taking into account that problem (2.4.10) is sclfconjugate, one obtains the following sensitivity function for the eigenvalue h1 = I [ ( & ) :
2.4 NNMs IN SYSTEMS WITH CONCAVE AND CONVEX NONLINEARITIES 57
so hi(&)increases with
and hl(1) > 1. It follows that for R = A(.t,T)-N the least positive eigenvalue of the problem is greater than one. Hence, problem (2.4.9) possesses only the trivial solution, and the uniqueness of the analytic continuation of x(z,T) in T is assured. It can be shown that variable T can be replaced by the total energy h (Zevin, 1985, 1986). This is shown by proving the uniqueness of the analytic continuation of x(t) in the parameter h. Since, as already shown, x(t) is uniquely continuable in T, then T = T(h) is a monotonous function. It is now proven that T(h) > Tj' for small values of h. Denote by h k ( s ) , the positive eigenvalues of the following problem: E,
It is assumed that hk I h k + l . Clearly, T2 is one of the eigenvalues, with x ( ~ ) being the corresponding eigenvector. In view of (2.4.2a), one obtains the inequality:
for x > Q and k > 0. Hence, C(0) = A(0) > C(x>, and Consider now (2.4.14) with
C(z>< A(0)
c = C(2,E)= C(z) + E [A(O)-C(z)]
Then T = T(E), with T(0) = T and T(1) = Tjo. Similarly to (2.4.13) one obtains
58
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
Since C(T)< A(0) one has D < 0. In addition, using (2.4.14) one can express the denominator E of the right-hand side of (2.4.16) as 1/4
j
E=
(C(T)X(.t),X(T))
dT
0
I /4
= T-2
f (M z'(T),~'(T))d~
-
T - 2 ( m f ( 1 / 4 ) , ~114)) ( + T-2(U1(0),x(O))
0
(2.4.17) Since
~ ' ( 0=) X( 1/4) = 0
the boundary terms in (2.4.17) vanish, and it is inferred that E > 0. Hence, T(E) decreases with E, and T(h) > Tj" for small h. Since the function T(h) is monotonous, it follows that T(h) increases for all h in Q. This concludes the proof of the theorem. An application of Theorem 2 is now discussed. Suppose that R = {x : V(x) 5 ho} Then it will be shown that any even symmetric solution x(t) E R, x(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
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
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
2.4 NNMs IN SYSTEMS WITH CONCAVE AND CONVEX NONLINEARITIES 61
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
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
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 <+ 3 ~ 1 1 0
(2.4.25) In particular, for j=n conditions (2.4.25) become
Note that conditions (2.4.25) are analogous to condition (2.1.4) of Lyapunov's theorem. However, unlike conditions (2.1.4), conditions (2.4.25) require the absence of integer ratios not only between the values and Ojo, but also between all points in the intervals [ w p O , ~ p + ] and [Oj', mi+]. The following theorem is proved. Theorem 4. Under conditions (2.4.4b) and (2.4.25) the family of NNMs &j(t,h) is uniquely continuable in h to the boundary of R ; the NNMs satisfy the relations Lj(t,h) = &j(-t,h) = -~j(t+T/2,h) The periods Tl(h) of the NNMs decrease monotonically in h. Proof. Taking into account that for this problem C(&)> 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:
2.4 NNMs IN SYSTEMS WITH CONCAVE AND CONVEX NONLINEARITIES 63
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
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
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)
2.4 NNMs IN SYSTEMS WITH CONCAVE AND CONVEX NONLINEARITIES 65
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
2.4 NNMs IN SYSTEMS WITH CONCAVE AND CONVEX NONLINEARITIES 67
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
NNMs IN DISCRETE OSCILLATORS: QUALITATIVE RESULTS
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.
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 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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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,
3.2 CONSERVATIVE SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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):
3.2 CONSERVATIVE SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
(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
3.2 CONSERVATIVE SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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:
3.2 CONSERVATIVE SYSTEMS
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:
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NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
(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
3.2 CONSERVATIVE SYSTEMS
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)
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NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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)
3.2 CONSERVATIVE SYSTEMS
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
3.2 CONSERVATIVE SYSTEMS
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:
3.2 CONSERVATIVE SYSTEMS
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
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NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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)
3.2 CONSERVATIVE SYSTEMS
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)
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NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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:
3.2 CONSERVATIVE SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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)
3.2 CONSERVATIVE SYSTEMS
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
3.2 CONSERVATIVE SYSTEMS
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:
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NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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 )
3.2 CONSERVATIVE SYSTEMS
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:
3.2 CONSERVATIVE SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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
3.2 CONSERVATIVE SYSTEMS
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)
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NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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
3.2 CONSERVATIVE SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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
3.2 CONSERVATIVE SYSTEMS
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
3.2 CONSERVATIVE SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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
3.2 CONSERVATIVE SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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 ,
3.2 CONSERVATIVE SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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
3.2 CONSERVATIVE SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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
3.2 CONSERVATIVE SYSTEMS
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 << 1, and unless ortherwise stated, all quantities different from E are assumed to be of O( 1). Hence, in this particular problem the small parameter E provides a measure of the weak coupling between the two oscillators of the system. Note, that even under the previously introduced scalings the system under consideration is classified as strongly nonlinear, since the grounding stiffnesses contain O( 1) nonlinear terms. Moreover, assuming that the coefficients 11 in (3.2.96) are O(1) quantities, in the limit E + 0 one obtains a nonhomogeneous generating system consisting of a set of two uncoupled, strongly nonlinear single-DOF oscillators. Following the previously developed methodology, a nonsimilar NNM of oscillator (3.2.96) is sought in the form: x2 = 22(x1)
(3.2.97)
where function 22(*) governs the trajectory of the mode in the configuration plane (xl,x2). Since relation (3.2.97) must hold at every instant of time, the
118
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
derivatives of the positional variables for motion on a nonsimilar NNM can be expressed by the chain rule as follows: A'
x2 = x2xi,
A"
x2 = X2(X1)2
+ X2Xl A'
(3.2.98)
where E d/dxl. Substituting for x2, x2, and x2 into the equations of motion, one obtains the following alternative set of equations of motion that describe the oscillation on the nonsimilar mode: (*)I
An expression for the velocity xi can be obtained by integrating the first of the above equations by quadratures, as follows: (x1)2 = -2
XI
XI
{ 5(1 + EKI)+ p1\3 - ~K1;2(5) + EM^[\ - ?2(5,]')
dk
(3.2.100) where X I is the niaxinium amplitude attained by the parametrizing coordinate xi. Substituting (3.2.100) into the second of equations (3.2.99) and eliminating the variable XI, one obtains the following differential equation governing the function ft2(*):
+
7
{ ~ M 2 [ 5- 22(5)13 - EKiftz(5) 1 dk
]
XI
-
$ [ X I + 1-11 X 3I + E K ~ -x &~ f t ~ K+l E M ~ ( X I22j3 ] -
+ (1 + a 1 ) 2 2 + pi(1 + a3) 2: + ~ K l f t 2- EKixl + E M z ( $ ~- x1)3 = 0 (3.2.101) This differential equation is analogous to the general expression (3.2.10) of section 3.2.1. Note that the coefficient of the second derivative of $2
3.2 CONSERVATIVE SYSTEMS
119
becomes zero at X I = +Xi, which are the instances when the oscillator reaches its maximum potential energy. As a result, a power series approximation to the solution will be valid only in open intervals contained in [-Xi,Xi], and to analytically continue the solution up to the maximum equipotential surface, one imposes the additional boundary condition:
This condition is analogous to the general boundary orthogonality conditions (3.2.11) formulated in section 3.2.1. The nonsimilar NNMs of the problem are sought in the following asymptotic form:
where it is assumed that, A(0)
A(1)
A(1)'
A(
1)"
o ( x 2 ) = 0(1), o(x2 ) = o(x2 ) = o(x2 ) = O(&) The displacement X I is assumed to be of 0(1), and higher approximations
$ik),k 2 2, are assumed to be of O ( E ~or) of higher order. The successive
4) approximations x2 (XI) are computed by substituting the series (3.2.103) into (3.2.100) and (3.2.101) and matching respective powers of E. The zeroth order approximation x2 ( X I ) corresponds to similar NNMs, and assumes the form: A@)
where the values for the modal constant c are the solutions of the following set of algebraic equations: a1c=O Pl[C(C2 - I ) + a&] = 0
(3.2.105)
120
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
0
pi = 0
arbitrary
*
c=o,--M
, +(1+ a3)-1/2
c = 0, --M
c = 0, -a, 1
0
PI #O
#0
c = 0,
--03
c arbitrary
c = 0, -m
If p1=0 (for no nonlinearities in the grounding stiffnesses), only the first of the above equations needs to be satisfied, and the resulting values for c are listed at Table 3.2.1. When pi # 0, depending on the values of the parameters ai there may exist two or four solutions for c (cf. Table 3.2.1). If the linear term a1 is nonzero, then the only possible solutions for c are the degenerate values c = 0 or c = -=. In the following analysis o n l y the nonsimilcir inodes neighboring these degenerate similar inodes will Dr considered. However, a similar analysis can be carried out to compute other classes of nonsimilar NNMs close to nondegenerate values of c (Vakakis, 1992a). A(l)
To compute the O(E) approximation to the nonsimilar trajectory, x2 , the series (3.2.103) is substituted into the differential equation (3.2.101), and only O(E) terms are considered. Taking into account expression (3.2.104) with c=O, the following nonhomogeneous differential equation governing is obtained:
Complementing this equation is the following boundary condition:
3.2 CONSERVATIVE SYSTEMS
12 1 A(1)
Following the general formulation of previous sections, the solution for x2 is expressed in the form:
When this series solution is substituted in the differential equation (3.2.106) and terms proportional to xi are set equal to zero, the following expression of a3(1) in terms of the linear coefficient $1') results:
Similar analytic expressions (albeit of increasing complexity) can be derived (1) (1) for higher-order coefficients, a21, j 2 5. To compute the unknown a21, one must substitute the series (3.2.108) into the boundary condition (3.2.107), (1) taking also into account relation (3.2.109). The resulting expression for a21 is
and the trajectory of the nonsimilar NNM in the configuration plane is approximated as follows:
Note that, since (3.2.111) is a truncated form of the general expression (3.2.103), it only asymptotically approximates the solution for E and X I sufficiently small. However, one can extend the accuracy of the solution by computing additional terms in (3.2.111) which are proportional to higher powers of x i . The following remarks are made regarding the nonsimilar NNM (3.2.1 11).
122
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
0
-20
I
1
I
I
20
40
60
80
. i c
I
100 120-20
I
I
I
I
I
0
20
40
60
80
100
t I
I
I
I
I
I
I
b
'
I
I
I
I
1
I
I
I
I
I
I
I
I
I
-0.8 -0.6-0.4 -0.2 00 0.2 0.4 0.6 0.8 -04-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
(4
(b)
Figure 3.2.5 Localized nonsimilar mode of a weakly coupled system: (a) with mistuning and (b) with no mistuning. The coefficients $2;) in expression (3.2.1 11) depend explicitly on the amplitude X 1, Therefore, the trajectory of the nonsiinilar NNM depends on the amplitude of oscillation (or equivalently on the total energy of the motion). A careful examination of (3.2.11 1) reveals that the computed nonsiinilar NNM is an oscillation where the positional variable x 1 assumes O( 1) values, whereas variable x2 is of O(E). Therefore, the computed nonsiwzilar NNM i.5 localized, since the oscillation is approximately confined to only one of the two masses of the system. In particular, the trajectory (3.2.1 1 1 ) can be viewed as perturbation of the degenerate similar normal mode (c = 0, ki =
3.2 CONSERVATIVE SYSTEMS
123
p2/p1 = 0) of the tuned system, and it is only realized for small values of E, i.e., only for weak coupling stiffness. Setting a1 = a3 = 0 in the above expressions, one obtains the following localized nonsimilar normal mode for the tuned oscillator:
Hence, the tuned nonlinear oscillator possesses a localized nonsimilar NNM. This is a unique feature of the nonlinear system, since mode localization is not realized in the corresponding linear tuned system. Note, however, that the localized mode (3.2.112) can only be realized when pi # 0, i.e., for tuned oscillators with nonlinear end stiffnesses. In the limit p i -+ 0, the coefficient of xi in the above expression becomes a large quantity, thus violating the assumptions of the perturbation theory. Setting K1 = 0 in (3.2.89), the trajectory of the localized mode becomes ? 2 ( ~ 1 )= - (~M2/p1)X I
5 + O ( E 5X ~ , E= ~-)(p2/11) X I + O(Ex1,E2)
(3.2.113) which is an asymptotic approximation for a similar NNM. In fact, the above localized mode is the asymptotic approximation as (p2Ip1) + 0 of one of the two branches of bifurcating NNMs of the system studied in section 1.2. Indeed, in that section it was shown that the symmetric nonlinear oscillator corresponding to a1 = a3 = 0, K1 = 0 possesses a bifurcating pair of localized similar NNMs, with modal constants given by the roots of the equation (p2/p1) (1-c)2 + c = 0 The localized mode (3.2.1 11) is the nonlinear analytic continuation of the degenerate similar mode (c = 0, K1 = p2/p1 = 0) of the uncoupled system. There exists an additional localized mode corresponding to O( 1) oscillations of positional variable x2 and O(E) motions of xi. This mode is the analytic continuation of the degenerate mode (c = -00, K1 = p2/p1 = 0) of the uncoupled system (cf. Table 3.2.1), and is asymptotically evaluated by
124
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
expressing the trajectory in configuration space as x 1 = 2, (x2), and applying the previously described asymptotic methodology (Vakakis, 1992a). A stability analysis of the localized mode (3.2.1 1 1 ) reveals that it is orbitally stable, and thus, physically realizable (Vakakis, 1992a). To check the accuracy of the asymptotic results, numerical computations of nonsimilar NNMs were performed for a system with parameters pi = 1.0, E K =~ EM^ = 0.01 and maximum amplitude X i = 0.6. To verify the existence of the localized modes, the nonlinear equations of motion were numerically integrated using a fourth-order Runge-Kutta algorithm and initial conditions identical to the ones predicted by the analysis. In Figure 3.2.5(a) the nonsimilar NNM of a mistuned system with a1 = 0.05 and a3 = 0.02 is presented, whereas in Figure 3.2.5(b) the nonsimilar localized mode of the tuned system (a1 = a3 = 0) is shown. The numerical integrations verify that the localized modes are orbitally stable.
3.3 INVARIANT MANIFOLD APPROACHES FOR NNMs The methodologies developed in section 3.2 can be used for asymptotically computing NNMs of conservative systems, or of systems that possess a first analytic integral of motion. For systems not possessing such a first integral, a5 systems with dissipative forces, the concept of nonlinear normal mode must be reformulated. This is necessary since, in such cases, the oscillation of the system during a normal mode vibration may not be synchronous, with nontrivial phase differences existing between pairs of positional variables. There exist two early works extending the principle of NNMs to damped nonlinear systems (Morgenthaler, 1966; Chi and Rosenberg, 1985). Morgenthaler investigated a special class of damped systems possessing decaying synchronous NNMs. Chi and Rosenberg formulated necessary and sufficient conditions for the existence of decaying NNMs in a general class of discrete damped oscillators and showed that in the limit of linearity their results degenerated to the known conditions derived by Caughey and O'Kell y (1965) on the existence of linear mode oscillations in damped systems. More recently, Shaw and Pierre (1991, 1992, 1993) reformulated the concept of NNM for a general class of nonlinear discrete oscillators, without assuming the existence of an analytic first integral of motion. Their analysis was carried out in the real domain (real invariant manifold formulation) and was
3.3 INVARIANT MANIFOLD APPROACHES FOR NNMs
125
based on the computation of invariant manifolds of motion on which the NNM oscillations take place. The parametrization of the invariant manifolds of the NNMs was performed by employing two independent reference variables, namely, a reference positional displacement and a reference positional velocity. A drawback of the invariant manifold approach was that the necessary computations became cumbersome, even for relatively simple, lower-dimensional nonlinear systems. A computationally efficient extension of the invariant manifold methodology was proposed by Nayfeh and Nayfeh (1994), who reformulated the invariant manifold method on a complex framework (complex invariant manifold formulation). A further advantage of the complex approach is that it can be easily extended to compute higherdimensional invariant manifolds of NNMs in internal resonance (Nayfeh and Chin, 1993). In the following exposition both invariant manifold approaches are discussed. The existence of invariant integral manifolds of motion in the phase space of discrete nonlinear systems is well known (Hale, 1961; Carr, 1981; Gilsinn, 1975, 1987). For example, the reduction of the dynamics of a system on an invariant center manifold is a well-established technique for studying the stability and bifurcations of dynamical systems close to nonhyperbolic fixed points or periodic orbits (Guckenheimer and Holmes, 1984; Wiggins, 1990; Carr, 1981). An invariant set for a dynamical system is defined as a subset S of the phase space such that, if the initial state of the system is in S, the trajectory of the system will remain in S for all time. In the formulation of Shaw and Pierre (1993), the NNMs are defined as invariant subspaces in phase space of the nonlinear equations of motion governing the dynamical system. Consider an n-DOF nonlinear autonomous oscillator with equations of motion given by: xi
+ fi(s,i) = 0,
i = 1,2,...,n
(3.3.1)
where 1~ denotes the (n x 1) vector of positional variables, and fi(x,i) are assumed to be sufficiently smooth functions of their arguments. No assumption concerning the existence of a first integral of motion of (3.3.1) is imposed. Expressing (3.3.1) in state form, one obtains the following alternative representation of this dynamical system:
126
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
Suppose that there exists a motion for which all displacements and velocities are functionally related to a single reference displacement-velocity pair, say (xi ,yl) = (x,y). In such a motion, the following differential relations hold:
Parametrization (3.3.3) is similar to the parametrizations employed in the conservative case, the main difference being that in (3.3.3) f w o parametrizing reference variables are used, instead of the single positional variable that was employed in the conservative case. Equations (3.3.3) define a constraint two-dimensional surface in the 2n-dimensional phase space of the dynamical system. An NNM cfsystem (3.3.1) is defined as a motion that takes place on the two-dimensional invariant manifold defined by (3.3.3). The NNM invariant manifold is computed by eliminating the explicit time dependence from (3.3.2), thus formulating a set of differential equations governing the functions X,(x,y) and Yi(X,y), i = 2, ...,n. Note that this calculation is similar to that carried out in the conservative case. As noted by Shaw and Pierre (1993) such a parametrization is possible only when no internal resonances exist. This is because in the case of internal resonances a number of NNMs become nonlinearly coupled and the resulting invariant manifolds are of dimensions higher than two. Moreover, as shown in chapter 4, internal resonances lead to bifurcations of NNMs, which further complicate the dynamics and invalidate the parametrization (3.3.3). Using the chain rule of differentiation, the time derivatives in (3.3.2) are expressed as follows:
Substituting (3.3.3) and (3.3.4) into the equations of motion (3.3.2), and rearranging terms, one obtains the following set of differential equations with functions Xi(X,y) and Yi(X,y) as unknowns:
3.3 INVARIANT MANIFOLD APPROACHES FOR NNMs
ax y + dYi
dYi
127
fl(x,x2,. ..,Xn,y,Y2,...,Yn) = fi(x32,...,Xn,y,Y2,. .., Y d ,
i = 2,...,n (3.3.5) Once a solution for Xi(X,y) and Yi(X,y), i = 2, ...A is derived, the time response of the dynamical system is computed by expressing the first pair of equations (3.2.2) in the form:
and solving approximately the resulting differential equation (3.3.6) by the method of averaging or the method of multiple scales. The partial differential equations (3.3.5) are, in principle, as difficult to solve as the original problem (3.3.1). However, an approximate solution to these equations can be developed using power series expansions, in a calculation resembling constructions of Center manifolds (Carr, 1981). To this end, the solutions of (3.3.5) are expressed in terms of power series, as follows :
It is noted that, in contrast to the theory of center manifolds where rigorous theorems regarding their approximation by power series have been formulated (Carr, 198l), no such theorems for approximating invariant manifolds of NNMs by power series (3.3.7) exist yet. In addition, the NNMs computed by the aforementioned methodology are assumed to be nonlinear analytic continuations of the normal modes of the linearized system obtained by setting the nonlinear and nonconservative terms in (3.3.1) equal to zero. Hence, in its present form the invariant manifold approach cannot capture
128
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
essentially nonlinear NNMs, such as the localized ones of the weakly coupled system of section 3.2.4. Substituting (3.3.7) into (3.3.5) and matching the coefficients of the monomials (xmyn) one derives a set of coupled algebraic equations in terms of the coefficients ai(P) and bi('l), which can be solved using standard linear algebra techniques. The necessary computations involve a large amount of algebraic manipulations, which can be conveniently performed using a software manipulation package such as Mathenintica (Shaw and Pierre, 1993). For examples of application of the invariant manifold method in the study of the NNMs of conservative and nonconservative oscillators the reader is referred to the aforementioned works of Shaw and Pierre. A modification of the invariant manifold approach that eliminates certain of the computational limitations of the presented approach was developed by Nayfeh and Nayfeh (1994). Instead of using the reference displacement pair (x,y) to formulate the constraint relations (3.3.5) i n the real domain, they performed an invariant manifold computation in a complex framework. To this end, a transformation of coordinates is introduced, and equations (3.3.1) are placed into the following form:
where functions gi(y,jl) contain nonlinear and nonconservative terms. In writing (3.3.8) it is assumed that all the eigenvalues of the linearized system are purely imaginary conjugate pairs. The following additional coordinate transformation is introduced:
where ci(t) are complex variables, (*)* denotes complex conjugate, and j is Expressing (3.3.8) using (3.3.9) one the imaginary constant, j = (-l)I'*. obtains the following set of first order complex dlfferentiuI equatioiis governing the complex dependent variables l,i(t):
3.3 INVARIANT MANIFOLD APPROACHES FOR NNMs
129
where functions Gi(s,L) are derived from gi(y,y) by imposing the transformation (3.3.9). Employing (3.3.10) the problem is studied in the complex domain, and the parametrization of the NNM invariant manifolds is performed by employing the complex pair, (<,<*) = ( < 1 , < 1 * ) , as parametrizing variables. Expressing the sought-after NNM as
i = 2 ,...,n
(3.3.1 1)
where hi denote complex-valued functions satisfying hi(0, 0) = (0,O) and O(lhil) at least O(ll&l12), i = 2, ...,n, the motion on the invariant manifold is governed by:
where ci) = 01. Equations (3.3.12) can be used to obtain an approximation to the invariant manifold of the NNM and can be viewed as the complex analogs of equations ( 3 . 3 3 , which were formulated in a real framework, Functions hi( <,<*) are approximated by the complex power series expansion:
which is analogous to expansions (3.3.7) of the real formulation, albeit of much more compact form. The unknown coefficients A,(@ of the series are computed by substituting (3.3.13) into (3.3.12) and matching respective powers of complex monomials <m<*n. The use of the complex notation (3.3.9) is natural for systems with linearized parts possessing purely imaginary or complex pairs of eigenvalues, as is the case for the class of systems under consideration. Moreover, the use of complex notation is the natural setting for simplifying the nonlinear terms of equation (3.3.12) using normal form theory [near-identity coordinate transformations (Guckenheimer and Holmes, 1984; Wiggins, 1990)], prior to the calculation
130
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
of the series approximation (3.3.13). This last feature provides a clear computational advantage of performing the invariant manifold calculation in the complex domain. More technical details of the method are given in (Nayfeh and Nayfeh, 1994) were applications of the method are also presented. In (Nayfeh and Chin, 1993) the complex invariant manifold approach is extended to compute higher-dimensional manifolds of NNMs in internal resonance and to study bifurcations of NNMs. More details on these topics will be presented in chapter 4. In recent works, Georgiou and co-workers presented a modification of the previously mentioned invariant manifold methodologies (Georgiou, 1993; Georgiou et al., 1994; Georgiou and Schwartz, 1995: Georgiou and Vakakis, 1996), by showing that when the nonlinear dynamics of discrete oscillators possess fast and slow scales one can develop asymptotic solutions by restricting the motion on slow invariant manifolds (SIVs) where the slow dynamics are dominant and the fast dynamics provide small order perturbations. In the aforementioned works SIVs were employed with computer algebra to study NNMs and other types of nonlinear motions of various discrete and continuous oscillators. An interesting topic of current research is the study of the topological singularities of SIVs, which will lead to a more complete understanding of NNM bifurcations.
3.4 ANALYSIS OF NNMs USING GROUP THEORY The existence of NNMs may be associated with certain symmetries of a dynamical system, that is, with the equations of motion being invariant with respect to a group of transformations. This group-theoretic approach for computing NNMs was followed i n a number of works (Yang, 1968; Manevitch and Pinsky, 1972a,b; Singh and Mishra, 1972, 1974). To a significant extent, this section refers to and extends some previous results reported by Manevitch and Pinsky (1972b). In what follows, the NNMs of certain two-DOF nonlinear systems are examined using mathematical techniques based on the theory of continuous and discrete groups. Aside from conservative systems, these techniques enable the evaluation of the NNMs of a more general class of nonlinear systems, including oscillators with gyroscopic or friction forces. Consider the following two-DOF conservative system:
3.4 ANALYSIS OF NNMs USING GROUP THEORY
13 1
k
Since system (3.4.1) is autonomous, its solutions are invariant with respect to arbitrary time translations. One seeks a complementary continuous group of transformations of these equations. This group is completely defined by its infinitesimal operator (Peter, 1986; Bluman and Kumei, 1989), which is sought in the following form:
where E, q1, q 2 are yet undetermined functions. The condition of invariance of (3.4.1) with respect to a specific group of transformations may then be formulated as follows:
where M is the manifold on @space ( X l , X l ,Xl,X2,X2,X2,t) defined by the equations of motion S = 0, and U" is a twice-continued operator (Peter, 1986; Bluman and Kumei, 1989) computed as:
(3.4.4) where
+ ri
hi ae. (G2 at)
+ r3-i
In its expanded form, condition (3.4.4) becomes
hi aX3-i'
i = 1,2
(3.4.5)
132
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
+[
cc P
k
k=l j=O
(k-j)ckj(2)~1Jx2~-J-l] q21M =
o
(3.4.6)
Decomposing (3.4.6) in pi one obtains
(3.4.7a)
Selecting the function E(t) as
&(t)= qt + p where q and p are constants with q f 0, conditions (3.4.7b) lead to the following expressions for q 1 and q 2 :
Substituting these expressions into (3.4.74 and decomposing in X I and x2, one obtains that the potential energy of the system is a homogeneous function
3.4 ANALYSIS OF NNMs USING GROUP THEORY
133
of degree k of the coordinates with the following values for the various coefficients: q = (1-k)a/2, kl I = k22 = a, k l z = k2l = 0 where cx is an arbitrary constant. The corresponding group is a group uf nonhomogeneous dilatations whose invariant manifolds are straight lines in configuration space. These one-dimensional invariant manifolds define the trajectories of the similar NNMs of the system in configuration space. Suppose now that the time coefficient in the expression of E(t) is q = 0. Then, from expressions (3.4.6) it follows that relations (3.4.7b) hold for q l q 2 # 0. Substituting (3.4.7b) into (3.4.7a) and matching terms of monomials xlmx2n yields r sets of algebraic equations. One can then select the values for coefficients kij which make the derived equations consistent. Consider first sets corresponding to the linear terms of the equations of motion. Their solutions are defined by kll =k22=1. k12=k21=0 yielding an infinitesimal dilatational operator. In the nonlinear case, all other sets of algebraic equations are inconsistent with the linear ones, and, therefore, the governing equations have no solutions if the conditions q 1 # 0, q 2 f 0 hold simultaneously. So, assume that q l = 0, q 2 f 0. Clearly, in that case it must be also satisfied that 61 = 0, and the first of the governing equations (3.4.6) assumes the form: (3.4.921) while the second equation of these equations becomes m262 +
[$
k
k=l j=O
7)21M = 0
(k-j)ckj(2),lJx2k-J-1]
(3.4.9b)
From (3.4.9a) one derives two possible conditions: (3.4.1Oa)
1121M = 0 P
k
C C (k-j)ckj(l)xlJx2k-J-l
k=l j=O
=
o
(3.4. lob)
134
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
Suppose that condition (3.4.10~1)is satisfied. Then, from (3.4.9b) one obtains that m2 62 = 0, and, in view of the fact that 62 is not identically zero, one finds that m2 = 0. The corresponding nontrivial limiting systems are
If condition (3.4.lob) is satisfied, all elastic coefficients in that expression must be zero, i.e., k = 1,...,p, j = 0,...,k Ckj(') = 0, When m2 # 0, the corresponding nontrivial limiting systems are XI
= 0,
1112x2 +
P
C CkO(2)X2k = o
k= 1
(3.4.12)
For the case when q 2 = 0, q l # 0, one obtains the previous results with subscripts 1 and 2 interchanged. Relations (3.4.11) and (3.4.12) represent degenerations of the original two-DOF system and are useful in the study of linear and nonlinear oscillators (3.4.1) with large differences in the values of their masses or grounding stiffnesses. The corresponding NNMs are found to be nonsimilar for (3.4.11) and similar for (3.4.12). By the technique described above one obtains limiting systems that permit nontrivial continuous groups of transformations. These limiting systems are either homogeneous or degenerate: the later may be obtained in the limiting case when either a mass or a stiffness of the system tends to zero. Note that degenerate systems admitting similar or nonsimilar NNMs were for the first time investigated by Manevitch and Cherevatsky (1969). Consider now a more general class of two-DOF dissipative systems with equations of motion given by: (3.4.13) where the potential energy, V, is a polynomial of the positional variables x i and x2. A complementary continuous group of transformations for this
3.4 ANALYSIS OF NNMs USING GROUP THEORY
135
system may be obtained if the potential energy is a homogeneous function of its variables. The infinitesimal operator then assumes the following form:
(3.4.14) where li, i = 1,2,3 are constant coefficients. As previously, the invariant manifolds of the group are straight lines, corresponding to similar NNMs of system (3.4.13). Additional classes of limiting systems possessing NNMs may be obtained by solving the problem of finding a complementary discrete group of transformations. In the following analysis the invariance conditions are formulated considering the Lagrangian function of a conservative system. For equations of the form (3.4.1) and two degrees-of-freedom, the Lagrangian of the system is given by: 2
L(x,i) = ( 1 / 2 ) c mixi2 i= 1
where
P
c (l/k)Bj,(k-j)(k-1)xiJx2k-J (3.4.15) k
k=2 j=O
x = (xi, x2)T. Introduce at this point the matrix representation, (3.4.16)
corresponding to the elements of the discrete group, (3.4.17) The invariance condition for the Lagrangian,
L(x,i) = L[S-l(xAl splits into two conditions, one for the kinetic and one for the potential energy of the system. These conditions are formulated as follows: (3.4.18)
136
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
where T and V denote the kinetic and potential energies of the system, respectively. For the sake of simplicity, assume unit masses for the system, mi = 1, i = 1,2. This condition does not restrict the generality of the analysis since it can be imposed to an arbitrary system of the general form (3.4.1) by a transformation of the temporal variable. Conditions (3.4.18) lead to the following relationships for the elements of matrix S: a I 1 2+ a212 = 1 ,
a I 2 2 + a2*2 = 1,
+ a21a22= 0
(3.4.19)
The above relations are a set of orthogonality conditions for matrix 5. A necessary condition for the existence of nontrivial invariant manifolds is that a12a21= ( a l l -
- 1)
(3.4.20)
From (3.4.19) it follows that
a 1 1 2 = a22 2,a 122 = a2I2
Non-identity (nontrivial) transformations result from the relationships a l l = -a22, a I 2 = a21 Moreover, the invariance condition for the potential function yields a set of nonlinear algebraic equations that relate the parameters of the transformation, aij, and the stiffness coefficients of the system, Bj,(k-j,(k). This set splits into subsets that express the invariance conditions for the orders of the same degree of nonlinearity in the expression of the potential energy. Consider the invariance conditions corresponding to the quadratic components of the potential function. These are given by: a I 2 2 + d2al 12 + D2al a l 12 + d2aI2*- D2al , a l 2 = 1, (1 - d2)aI1al2- (1/2)D2(a122- a,,2) = (1/2)D2
= d2 (3.4.21)
where d2 = B22(1)/B11(1) and D2 = 2B12(1)/Bll(l) Solving (3.4.21) one obtains the following expressions for the parameters of the transformation:
3.4 ANALYSIS OF NNMs USING GROUP THEORY
137
where a = (d2 - 1)D2-1. The relationships expressing the invariance of the nonlinear components of degree of nonlinearity higher than two may now be regarded as sets of linear homogeneous conditions in Bj,(k-j)ck), k > 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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
+ [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)
3.4 ANALYSIS OF NNMs USING GROUP THEORY
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
(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,
3.4 ANALYSIS OF NNMs USING GROUP THEORY
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
(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
3.5 VIBRO-IMPACT SYSTEMS
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
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
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
3.5 VIBRO-IMPACT SYSTEMS
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 << 1 and resorts to an asymptotic approximate analysis. The variable v in the previous expression represents the (constant) velocity of the fictitious mass in the ($1,@2) phase plane during an NNM oscillation. The smallness of parameter ~2 indicates that the coupling parameter y and/or the spacing between the rigid supports, e/v, must be small quantities. The existence of a small parameter in the problem enables the study of the nonlinear dynamics in "slow" and "fast" time scales and permits the performance of an averaging stability analysis. In what follows the stability of each of the previously detected NNMs of the two-DOF vibro-impact system will be examined separately. Considering the in-phase NNM, $ 1 = $ 2 = $, one introduces the perturbations
150
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS $1
= $ + SI(E$),
$2 = $
+ S2(E$),
151.21
<< 101
where $ = (1/2)nve-lt is the fast phase, and E$ the slow phase in the following averaging analysis. Substituting the above expressions in (3.5.3) and carrying out averaging with respect to the fast phase $, one obtains the following averaged equations:
(3.5.4) where function @(u) is 2n-periodic in u, defined as follows: IT
@(u) = (n2/4) < T'($)T($+u) >+= ( ~ 2 / 4 )1( 1 2 ~J) T'($)T($+u) d@ --K
= u (1-u/n),
if 0 I u I n, and u (l+u/n), @(u + 2 ~ =) @(u), 'd u E R
if
-xI u I
0
(3.5.5)
Since the second equation in (3.5.4) possesses orbitally stable (bounded) solutions, the in-phuse NNM is stable in the lineurized sense. A similar averaging analysis for the antiphase NNM, $ 1 = 4 2 = $, leads to the following variational system, which describes the evolution of small perturbations in the vicinity of the antiphase mode:
(3.5.6) where the 2n-periodic function @(u) is defined by (3.5.5). Note that the sign change in the second of the above variational equations leads to unbounded solutions for the perturbations [k 1 (E$)+$~(E$)], and, hence, the untiphuse NNM of the vibro-impact system is unstable. The localized nonsimilar NNMs are now examined. Consider the localized N N M , which degenerates to the straight line $2 = 0 as y -+0. For y # 0, but y sufficiently small, due to continuity it is expected that the curved trajectory
3.5 VJBRO-IMPACT SYSTEMS
151
of the localized mode in the (0 1,$2) plane will lie in a small neighborhood of line $2 = 0. Assume that the NNM trajectory lies in the strip -7d2 I $ 2 I 7~12.The transformed equations of motion (3.5.3) can then be written in the form:
Setting $1 = $ + S I ( E $ )
and averaging (3.5.7) over variational equations:
4 leads to the following averaged set of
(3.5.8) which indicates orbital stability of the "averaged" trajectory $2 = 0, relative to the small perturbations
It follows that the localized nonsimilar NNM in the vicinity of the line 92 = 0 is stable. Similar arguments show that the additional localized NNM close to the line $ 1 = 0 is also stable. As discussed in later chapters, stable localized NNM in vibrating systems lead to passive motion confinement of motions generated by external impulses. For example, when vibrational energy in injected to one of the two particles of the vibro-impact system under consideration, most of this energy is passively confined to the directly excited particle, and only a minimal amount "leaks" to the second particle. Such passive motion confinement phenomena improve the controllability of nonlinear dynamical systems and can be potentially employed in the
152
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
development of refined vibration and shock isolation designs of engineering structures. The previously developed methodology was based on Zhuravlev's transformation and can be easily extended to vibro-impact systems with more that two DOF. For example, for a system with N = 3 particles, application of the described methodology leads to the following smooth equations of motion:
with the understanding that $0 = $4 = 0. Introducing the nondiniensionless phase parameter $ = (1/2).rcve-lt where v is a reference velocity, the equations of motion are written in the following scaled form:
(3.5.10) Equations (33.10) describe the oscillations of a fictitious unit mass in threedimensional space, and in a periodic potential field of force. The equipotential surfaces of this system form a spatial pattern obtained by multiple reflections of the equipotential surfaces lying inside the cube K = {($1,$2,$3) E R3, -7~12I $ j I ~ / 2 i, = 1,2,3) Inside K the dynamical system is linear and the equipotential surfaces possess the three axes of symmetry of a triaxial ellipsoid. Hence, for sufficiently small motions inside K, the system possesses three normal modes, aligned along the aforementioned three axes of 'symmetry. For larger amplitudes of motion, vibro-impacts occur and the configuration space of the system extends beyond K. It turns out that the equipotential surfaces of the extended phase space of the vibro-impact system possesses the 13 axes of symmetry of the cubic lattice; as a result, for sufficiently high energies the three-DOF vibro-impact system possesses 13 NNMs. The spatial distributions of these
3.5 VIBRO-IMPACT SYSTEMS
(PI =-(Pz=
(P3
CPI
=
(P2
=
153
-93
Figure 3.5.3 NNMs of a three-DOF vibro-impact oscillator: (+) signs indicate stability and (-) signs instability. NNMs are depicted in Figure 3.5.3. A linearized stability analysis indicates that only modes (1),(2),(3),and (6) are stable in a linearized sense. Note that NNMs (1) and (2) are strongly localized to a single particle, whereas mode ( 3 ) is weakly localized to two of the three particles of the system. The stable NNM (6) is spatially extended (nonlocalized), since it corresponds to finite, in-phase oscillations of all particles. From these results it becomes clear that vibro-impact chains with N 2 3 DOF possess stable localized NNMs, which are spatially confined either to one (strong localization) or to more than one (weak localization) particles. A detailed analysis of mode localization in discrete nonlinear oscillators is carried out in chapter 7. As a concluding remark, it is noted that the developed analytical vibro-impact solutions can be used as generating solutions for studying the NNMs of strongly nonlinear systems with degrees of stiffness nonlinearities much greater than unity (i.e., 2n - 1 >> 1 according to the previous notation) (Vedenova et a]., 198.5). In chapter 6, a new methodology is described, based on nonsmooth temporal transformations (NSTT). The perturbation schemes associated with this NSTT analysis will be described in detail in that chapter, and in this section only an illustrative example of application of the methodology will be given. Consider the following two-DOF oscillator:
x, + y(2x1- x2) + x12n-1 = 0
154
NNMs IN DISCRETE OSCILLATORS: QUANTITATIVE RESULTS
x,
+ y (2x2 - X I ) + x22n-1
where n >> I. A localized NNM employing a perturbation analysis previously computed localized corresponding to n + 00. To this sought in the form: X l = xo
+ v-2u(xo),
=0
( 3 . 5 11)
of this system will be computed by and using as generating solution the NNM of the vibro-impact system end the localized N N M of (3.5.11) is
x2 = v-2v(xo),
1v-21<< 1
(3.5.12)
where $ = vt/A xo = AT($), T($) is the previously defined sawtooth sine function, v a reference velocity, and A the amplitude of oscillation. Substituting (35.12) into (3.5.11) onc obtains the following equations governing functions U(x0) and V(x0):
U"
+ [ A - l ( ~ 2+ U')(d2~/d$2)]+ 2 ~ x 0+ " y ~ - ~ ( 2 -U V) + ( x o + v - ~ U ) ~=~0- ~ V" + [A-1V'(d%/d@)] - p y x 0 + V - ~ ( ~ -VU) + ( v - ~ V ) ~ ~=-0I
(3.5.13) where primes in (3.4.13) denote differentiations with respect to xo. Setting,
one eliminates the bracketed terms in (3.5.13). This operation is necessary since these terms are proportional to the quantity T", which, in terms, is expressed as a periodic series of discontinuous generalized impulse functions (cf. chapter 6). The first of conditions (3.5.14) is satisfied by appropriate selection of the amplitude A, whereas the second is met by computing appropriate constants of integration when solving the second of equations (3.5.13). The functions U(x0) and V(x0) are now expressed in the following series forms:
u = u1 + v - 2 ~ 2+ 0 ( ~ - 4 ) ,
v = v1 + v - 2 ~ 2+ o ( ~ - ~(3.515) )
3.5 VIBRO-IMPACT SYSTEMS
155
Substituting these expressions into (3.5.13) one obtains the following solutions:
Hence, the localized NNM of system (3.5.1 1) is analytically approximated as, XI
= xo - p 0 3 / ( 3 ~ 2) (k+2)-l(k+l)-1~-~~&+2 + O(vP4) x2 = ~ - ~ y ( x 0 3 /-6 A2x0/2) + O ( V - ~ ) (3.5.17)
with the reference velocity v related to the amplitude of oscillation by the third of expressions (3.5.15). The perturbation analysis can be extended to higher-order approximations by expressing the amplitude in the series form, A = A1 + v-*A2 + O ( V - ~ ) and solving the corresponding equations for Un and Vn, n = 2,3, ... Note that the analytic approximations (3.5.17) are valid for strongly nonlinear oscillations since the generating vibro-impact solutions employed in the analysis are essentially nonlinear.
NORMAL MODES AND LOCALIZATION IN NONLINEAR SYSTEMS ALEXANDER F VAKAKIS, LEONID I MANEVITCH YURl V MlKHLlN VALERY N PlLlPCHUK & ALEXAN DR A ZEVl N Copyright@ 1YYh hy Jolin Wtley & Son$ I ~ L
CHAPTER 4 STABILITY AND BIFURCATIONS OF NNMs Periodic solutions of nonlinear systems are of physical significance only if they are stable, since only then can they be physically realizable. Hence, stability analysis of NNMs is a problem of considerable practical importance. The stability of NNMs of discrete oscillators has been the topic of numerous previous works. Rosenberg (1966) investigated the stability of similar and nonsimilar NNMs by a linearized methodology, reducing the problem to a set of uncoupled Mathieu equations. A similar approach was followed by Month and Rand (1977). In (Pecelli and Thomas, 1979, 1980), the stability of the NNMs of oscillators with combined geometric and kinematic nonlinearities was examined by formulating Lame' variational equations and investigating their stability. Rand (1973) examined the stability of NNMs by adopting a geometric formulation and performing orbital stability analysis in configuration space. His paper was based on previous work by Synge (1926) who established stability criteria "in the kinematostatic sense." Analytical methodologies for studying the stability of NNMs were also developed by Auld (1961) and (Porter and Atkinson, 1962) who related mode stability to certain topological features of the equipotential surfaces of a dynamical system in the configuration space. Zhupiev and Mikhlin (1981, 1984) developed a linearized stability methodology for studying the stability of similar NNMs. Their analysis was based on reducing the stability problem to the computation of degenerate solutions of hypergeometric variational equations. In addition, they formulated conditions for the existence of "finite-zoning instabilities" of similar NNMs in conservative oscillators. The stability results of Zhupiev and Mikhlin will be discussed in section 4.3. Bifurcations of NNMs were studied by Rosenberg (1961) and Anand (1972). In Yen (1974) a physical interpretation of a mode bifurcation was given by considering which of the terms in the expression of the potential function of the system dominate for low or high levels of the total energy of motion. An analytic investigation of the stability "in the kinemato-statical sense" of bifurcated NNMs was carried out by Pak (1989). Studies on bifurcations of NNMs were performed by Johnson and Rand (1979) using 157
158
STABILITY AND BIFURCATIONS OF NNMs
numerical techniques and by Month and Rand (1980) employing analytical or numerical Poincare’ maps. Most of existing studies on NNMs were carried under the assumption of small-amplitude motions. Mikhlin (1985) developed an analytical technique for studying NNM bifurcations in discrete conservative systems with arbitrarily large energies of motion. Caughey et al. (1990) investigated the bifurcations of similar NNMs of oscillators with cubic, quintic, and seventh-order stiffness nonlinearities. Vakakis and Rand (1992) studied the global dynamics of an undamped nonlinear two degree-offreedom system by means of Poincare’ maps. The effects of a mode bifurcation on the global dynamics of this system was investigated by numerical and analytical techniques. At low energies a homoclinic orbit existed in the Poincare’ map generated by an NNM bifurcation. At high energies, large-scale chaotic motions were detected, resulting from the breakdown of the homoclinic orbit, which led to transverse intersections of the stable and unstable manifolds of an unstable antisymmetric NNM (see also the example discussed in chapter 1). In addition, low-scale chaos was found to exist due to the breakdown of invariant KAM-tori. Subharmonic Melnikov analysis was employed to study the creation of subharmonic free motions close to a pair of stable NNMs, and the stability of these orbits was examined by an averaging methodology. Rand et al. (1992) investigated bifurcations of NNMs and elliptic orbits of two-DOF oscillators with degeneracies. Additional works investigated nonlinear mode localization in periodic systems generated by NNM bifurcations; these works will be reviewed in chapter 7.
4.1 GENERAL STABILITY RESULTS In this section general qualitative results are derived concerning the stability of the NNMs of the following n-DOF autonomous nonlinear oscillator:
Mk(t) + V V ( r ) = Q
(4.1.1)
As an introduction to the following analysis, some basic results regarding the stability of periodic solutions of autonomous systems (Urabe, 1967; Nayfeh and Mook, 1984) will be reviewed. Let x(t) be a periodic solution of (4.1.1) with period T, and let Co be the corresponding closed orbit in the 2n-
4.1 GENERAL STABILITY RESULTS
159
dimensional phase space of the system. Let s ( t ) be a perturbed solution of (4.1.1) with initial conditions sufficiently close to the periodic orbit, given by &(O) = [?id0>,&(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
STABILITY AND BIFURCATIONS OF NNMs
(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
STABILITY AND BIFURCATIONS OF NNMs
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
STABILITY AND BIFURCATIONS OF NNMs
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
4.1 GENERAL STABILITY RESULTS
167
As mentioned above, under the conditions of the theorem the multiplicity of the multiplier h = -1 is equal to 2 for all values of h, so that the inequality arg[hk(l)(T/2)] < 7c holds for all values of h. Since A(x) > 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
STABILITY AND BIFURCATIONS OF NNMs
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
STABILITY AND BIFURCATIONS OF NNMs
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
STABILITY AND BIFURCATIONS OF NNMs
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
STABILITY AND BIFURCATIONS OF NNMs
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
STABILITY AND BIFURCATIONS OF NNMs
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 << Iql, luil << Iql, i = 2 ,...,11 and obtains the following system of linearized variational equations:
g + (uV)VV[g(t)] = 0
3
ii + A(t) u = Q
(4.2.17)
where u = (u1, ...,un)T and A(t) is an (n x n) symmetric time-periodic matrix of coefficients. The time-dependent coefficient in the first of relations (4.2.17) is evaluated at y(t) = (q(t),O, ...,O)T, i.e., for motion of the system on the similar NNM whose stability is examined. Expanding the timeperiodic matrix of coefficients as + Alq(t) + &q2(t) +... A(t) = where Ai are constant and symmetric (n x n) matrices, the variational system (4.2.17) is expressed in the following form:
ii + [A0 + &q(t)
+ A2q2(t> +...I
11 = Q
(4.2.1X)
The constant symmetric matrices A, can be diagonalized by a nonsingular transformation of coordinates, 11 = B, provided that they are mutually commuting and none of them possesses multiple eigenvalues (Hsu, 1961). In the following analysis it will be assumed that the variational equations (4.2.18) can be decomposed into the following set of n uncoupled equations:
The stability of (4.2.18) when matrix A(t) is nondiagonalizable was studied by Hsu (1963, 1964) and by Nayfeh and Mook (1984). The first of equations (4.2.19) corresponding to i = 1 governs the variation u l along the straightline trajectory of the NNM in the configuration space, whereas the remaining variational equations determine the evolution of perturbations on directions orthogonal to the mode trajectory. It follows that the first variational equation always indicates neutral stability and that the remaining variational equations are the ones determining the stability of the NNM. Considering the quantity q(t) as the new independent variable, the system of
4.2 SIMILAR NNMs
177
variational equations (4.2.19) is transformed to the following set of singular eauations: i = 2 ,...,n (4.2.20) where Ri(q) is an analytic function with no real roots of modulus smaller than lail or Ibil. As shown previously, the problem of determining T- or 2Tperiodic solutions of the variational equations (4.2.19), where T is the period of the NNM [and also of q(t)], is reduced to finding the degenerate solutions of system (4.2.20). This task is performed by solving Sturm-Liouville problems for functions that either are regular at q = ai and q = bi or possess singularities of the forms (ai - q)"2 or (bi - q)"2 at these points. The significance of computing the degenerate solutions of (4.2.20) stems from the fact that such solutions lie on the stability-instability boundaries of the system of variational equations (4.2.19). If the potential function n(g) in (4.2.16b) is an even analytical function, the periodic coefficients of the variational equations possess half the period of the NNM considered. In that case it can be shown that functions S, i n (4.2.20) are even functions of q and that bi = -ai. Introducing a rescaling of coefficients, and without loss of generality one can set i = 2, ...,n SI = (1 - q')Ri(q), and reduce the problem of computing the stability-instability boundaries of (4.2.20) to the computation of four Sturm-Liouville problems for even or odd functions, which either are regular at q = +1 or possess a singularity of the form (1 - q2)1/2 at these points. In what follows the outlined stability analysis is demonstrated by studying the stability-instability boundaries of variational equations of homogeneous and symmetric systems. It will be shown that for homogeneous systems the degenerate solutions of the singular variational equations can be computed in closed form. Consider a similar NNM of a homogeneous system with equations of motion (4.2.16a) and potential function given by
A rotation of the coordinate axes transforms the equations of motion into the form (4.2.16b). The stability of the similar NNM is performed by analyzing variational equations of the form (4.2.17). A diagonalization of matrix A(t)
178
STABILITY AND BIFURCATIONS OF NNMs
leads to the following uncoupled set of linear variational equations with periodically varying coefficients:
where air is a constant scalar, and q(t) describes the periodic oscillation of the system along the mode trajectory in configuration space. As shown in chapter 3, the derivatives of q(t) can be expressed as follows: q(t) = -S22qr(t),
q2(t) = 2[h - 522(r+l)-lq'+l(t)]
where
Q2=
n k=O
(4.2.22)
blk(1 - Ck)'
is the frequency-squared of the NNM, h is the level of the conserved energy of the motion, and ck are the modal constants defining the mode trajectory in configuration space with co = 0, c l = 1. Replacing the independent coordinate t by q(t), the variational equations (4.2.21) are rewritten into the following form:
Introducing the new independent variable z = R2(r (4.2.23) are expressed as:
+ l)-lqr+I
(4.2.23) equations
i = 1,...,n (4.2.24) where hi = (air/2)(r + l)-lQ-*. Expressions (4.2.24) are hypergeometric equations with regular singular points z = 0 [with corresponding singularity indices 0 and (r + 1)-1] and z = 1 (with indices 0 and 1/2). Hence, the problem of stability of the similar NNM is reduced to finding the values of parameters hi, which lead to degenerate solutions of the variational equations
4.2 SIMILAR NNMs
179
(4.2.24). In particular, regarding z as a complex variable one requires that on following a closed contour containing the singular points z = 0 and z = 1 in the complex plane, the solution of (4.2.24) is multiplied by (+1) or (-1). An analytical treatment of such degenerate solutions of hypergeometric equations can be found in (Whittaker and Watson, 1986). Denoting by T=27c/L2 the period of the similar NNM, the degenerate solutions of (4.2.24) are Gegenbauer polynomials; even TI2-periodic solutions of the variational equations correspond to hi = j[(2j + l)(r + 1) - 21 odd T/2-periodic solutions to hi = (i + 1)[(2j + l)(r + 1) + 21 even T-periodic solutions to hi = (2j + l)[Q + l)(r + 1) - 11 and odd T-periodic solutions to hi = (2j + l)lj(r + 1) + 11 for j = 0,1,2, ... The previous values of coefficients hi establish the boundaries between stability and instability of the similar NNM under consideration. As a specific example, consider the following two-DOF homogeneous conservative oscillator:
To compute the similar NNMs of this system one sets x2 = cxl and substitutes into (4.2.25). A matching coefficients of respective powers of X I in the resulting equations leads to the following equation governing the modal constant c:
where a = a22/a11, K = a12/a11, and p = m2/m1. For system (4.2.25) the hypergeometric variational equation determining the stability of mode x2 = cxi is expressed as follows:
180
STABILITY AND BIFURCATIONS OF NNMs
Figure 4.2.1 Similar mode bifurcations for homogeneous systems (a) of degrees r = 3, 7, and (b) of fractional degrees r.
4.2 SIMILAR NNMs
2)-1 z] r( l+pc)(acr-pc)
h = p( 1-c)(ac'+ 1)
du
t
18 I
hu = 0 (4.2.27)
The solutions of the algebraic equation (4.2.26) are depicted in Figure 4.2.l(a), where
The number of NNMs of the system depend on the system parameters. Curves labeled a correspond to r = 3, a = 1; curves b to r = 7, a = 1; curves c to r = 3, a = 0; and curves d to r = 3, a = 1, 2. All curves correspond to p = 1. Each point of change of stability is marked by symbol (0) for r = 3 and by symbol (x) for r = 7. The number shown near a point of exchange of stability denotes the corresponding value of coefficient h. Employing previous results, one finds that the ranges of h for which one obtains unstable solutions of the variational equation are given by 4j + 1 < h < 4J + 2 and 4j + 3 < h < 4j + 4, j = 0,1,2, ... Figure 4.2.l(b) presents similar mode bifurcations of homogeneous systems with fractional degrees of homogeneity r. Curves labeled a correspond to r = 0, p = 1, a = 1; curves b to r = 1/3, p = 1, a = I ; curves c to 0 < r < 1, p > 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
STABILITY AND BIFURCATIONS OF NNMs
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 << 1
+ q(t), the
variational
4.2 SIMILAR NNMs
183
This is a Lame' equation and is analogous to the variational equations (4.2.20) derived in the previous exposition. A perturbation computation is now performed to analytically approximate the first stability-instability boundary of (4.2.30). Two general classes of oscillators (4.2.28) will be considered in the following analysis. The first class of oscillators analyzed is assumed to possess weak nonlinearities, and to "neighbor" linear homogeneous systems. This condition is imposed by rescaling the coefficients in (4.2.30) associated with nonlinear stiffness terms as follows: p -+ ED and p + &p,where IEI << 1. To compute the stability-instability boundaries of (4.2.30) in parameter space, the various coefficients and the solution u are expanded in power series of increasing powers of E as follows:
c EPup(q), = c co
u = u(q) =
p=o
c
EPYp,
ED
c
m
y=
p=o
m
EP
EPPp, p= 1
o=
c m
p=o
EP o p
m
=
p= 1
EPPp
(4.2.3 1)
Substituting the above expansions in (4.2.30) and retaining only terms of O ( ~ 0 one ) obtains the following zeroth order approximation to the variational equation: (4.2.32)
It can be shown that, correct to O(EO) the first stability exchange of the antiphase NNM takes place at line yo = (30 in parameter space. The corresponding periodic solution for uo is computed as uo(t) = Kqo(t), where qo(t) = cosQt, Q = (2.niyo) + O ( E ~is) the zeroth order approximation to the periodic oscillation of the system for motion on the antiphase NNM. Higherorder approximations to the stability-instability boundary yo = 00 are computed by considering O(E) terms in (4.2.30) and solving the following first-order variational equation:
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STABILITY AND BIFURCATIONS OF NNMs
= (y1q + 2p1q3)
duo
dq - (01
+ P1q2)uo
(4.2.33 )
As mentioned above the stability-instability boundaries correspond to periodic solutions of the variational equation. A periodic solution of (4.2.32) exists only when the right-hand-side of (4.2.33) is orthogonal to the periodic solution uo(t) of the homogeneous equation. The periodicity condition is formulated as follows:
where the integration is carried over one period T = 27c/R of the NNM, and q = so(t)+ O(E) A simple calculation leads to the following O(E) approximation to the first stability-instability boundary:
The stability of the anti-phase NNMs of oscillators (4.2.28) neighboring homogeneous systems of degree r = 3 can be similarly investigated. This class of oscillators is essentially and strongly nonlinear and is obtained by rescaling y -+ EY and o + EO, with IEI << 1, in (4.2.28) and (4.2.30). The corresponding variational equation determining the stability of the antiphase mode assumes the following form:
Introducing the expansions
P=
c
p=O
@Pp,
p=
c
p=o
EPPp
(4.2.37)
and matching zeroth order terms in (4.2.36), one obtains the following variational equation governing uo(q):
4.2 SIMILAR NNMs
I85
(4.2.3 8) In this case the zeroth approximation to the first stability-instability boundary is given by 2p0 = Po, and the corresponding periodic solution of (4.2.38) given by uo(t) = Lqo(t). Variable qo(t) represents the zeroth order response of the system for motion on the antiphase NNM and is computed by solving the following initial-value problem:
;i+ 2 ~ 0 4 3+ O(E) = 0,
q(o) = 1,
q(o) = o
* q(t) = qo(t) + O(E) = cn(st,1/2) + O(E)
(4.2.39)
where cn(*,*) is the elliptic cosine and s = (2p0)"2. The frequency of the antiphase NNM is approximated by Q = ns/[4K(1/2)], where K(-) is the complete elliptic integral of the first kind (Byrd and Friedman, 1954). Considering O(E) terms in (4.2.36), one obtains the following first-order variational equation:
(4.2.40) Imposing the periodicity condition on (4.2.40) [i.e., orthogonality of the right-hand-side term with respect to the homogeneous perodic solution uo(t)], one obtains the following relation:
where, as in the previous case, the integration is carried over one period T = 2n/Q of the NNM, and q = qo(t) + O(E). Carrying out the integration, one obtains the following first-order approximation to the first stabilityinstability boundary:
186
STABILITY AND BIFURCATIONS OF NNMs
(4.2.42) where I-(*) is the Gamma function. The outlined perturbation analysis can be employed to compute higher-order approximations to the first stabilityinstability boundary or to analytically compute additional boundary lines separating regions of stability and instability of the antiphase NNM in parameter space.
4.2.2. Finite-Zoning Instability Conditions In the previous section the stability of NNMs of discrete conservative oscillators was investigated by analyzing linear variational equations possessing periodically dependent coefficients or regular singular points. This type of equations generally possess an in,finite number of instability zones in parameter space. Indeed, the existence of an infinite number of instability zones in the Mathieu equation (the so-called Stutt diagram) or in the Lame' equation is well documented and understood. In this section conditions on the system parameters are derived, which guarantee thr existence of only a finite number of instability zones uf the variational equations derived from perturbations of NNMs (finite-zoning instability). The following analysis can be of considerable practical significance, since it can be used to eliminate potentially troublesome instability regimes and, thus, stabilize NNMs of nonlinear dynamical systems. For the sake of simplicity the analysis will be carried out by considering conservative systems with two-DOF. However, the analytical principles developed herein can be extended to higher-dimensional systems. Consider a conservative oscillator with equations of motion:
It is assumed that the dynamical system (4.2.43) posseses the similar NNM, q i ( t ) = q(t) f 0, q2(t) = 0. Variable q(t) is governed by the following nonlinear differential equation:
4.2 SIMILAR NNMs
187
(4.2.44)
where h is the total energy of motion and Q the maximum amplitude attained by variable q(t) along the mode trajectory. In deriving the last of relations (4.2.44) the initial conditions q(0) = Q, q(0) = 0 were assumed. The potential function of the system, n(ql,q2), is assumed in the form:
where the coefficients ai and ei are real constants. Note that the condition of existence of normal mode oscillations, an (q 1 ,O)/dq2 = 0, is satisfied identically by (4.2.45). As shown in the previous section, the linearized stability of the similar NNM of (4.2.43) is determined by studying the evolution of small variations orthogonal to the mode trajectory. Perturbing the NNM solution by variations, ql(t) + q(t) i ul, ql(t) + 0 + u2, with lul,21 << Iq(t)l, one obtains the following time-variant variational equation governing u2:
ii + [a2n(q(t),O)/aq22]u = 0
(4.2.46)
Stable or unstable solutions of this equation imply stability or instability of the NNM under consideration. Although, equation (4.2.46) generally possesses an infinite number of unstable zones in parameter space (Yakubovich and Starzhinskii, 1975), there still exist certain conditions on the system parameters that guarantee the existence of only a finite number of instability zones. Novikov et al. (1984) formulated such finite-zoning instability conditions by considering the time-variant Schrodinger's equation, and their results are summarized in the following theorem. Theorem 1 (Novikov et al.. 19841. Consider the following Schrodinger's equation with periodic potential: -[-dZ/dt2 + z(t)]yr = eyr, z(t + T) = z(t) V t E R (4.2.47a)
188
STABILITY AND BIFURCATIONS OF NNMs
In order for this equation to possess only n instability zones, the following equation, j i - x 2 = z(t> - e must admit a solution of the form
(4.2.47b)
x
= (D - jp/2)/p where j = (-1)1'2, D is a constant, and p = p(t,e) is a polynomial in e of degree n with time-dependent coefficients,
p(t,e> =
n
C ak(t)ek
k=O
The equation determining the polynomial p(e) is given by:
..,
(4.2.47~) p(t,e) + 4[z(t) - elp(t,e> - 2z(t)p(t) = 0 where overdots denote differentiations with respect to the temporal variable.
To apply Theorem 1 to the variational problem (4.2.46), one must set z(t) - e = -iPn(q(t),O>/aqz* which, taking into account definition (4.2.44), leads to the relation e = -2eo. Substituting these expressions into (4.2.47c), one obtains the following equation governing the polynomial p(t,e):
For motion of the system on the similar NNM (ql,q2) = (q(t>,O>,the time derivatives in the relation above can be expressed by the chain rule as follows:
.. drI(q(t>,O) q=a91 '
... = ----a91 drI(q(t),O)
4
(4.2.49)
4.2 SIMILAR NNMs
189
The expressions above were derived by taking into account relations (4.2.44). Substituting (4.2.49) into (4.2.48) one obtains a time-invariant variational equation with regular singular points of the form:
where primes denote differentiations with respect to q, and variable p is considered to be a function of q and e. Employing Theorem I , one finds that in order for the variational equation (4.2.46) to possess only n zones of instability, the solution of equation (4.2.50) must be in the following polynomial form:
(4.2.51) Substituting (4.2.5 1) into (4.2.50) and matching coefficients of equal powers of e, one obtains a series of problems that provide the necessary values of the system parameters ai and ei [cf. (4.2.45)] for which the variational equation possesses finite-zoning instability. Note that since the potential function and its derivarives in (4.2.50) are polynomials in q, the functions ak(q) in (4.2.5 I ) should also be sought in polynomial form. Certain examples of application of the outlined methodology are now given. Consider a system with potential function given by n(q1,92) = a2 912 + w 914 +q22 (eo + ~ 1 2 ) Substituting this expression into (4.2.50) and taking into account (4.2.5 l), one obtains the following equation governing p(q,e):
190
STABlLlTY AND BIFURCATIONS OF NNMs
(4.2.52) Consider first the case when the variational equation possesses 11 = 0 instability zones. Approximating the polynomial p as p(q,e) = p(q) = ao(q), and matching terms proportional to eo in (4.2.52), one finds the following relation governing ao(q): ao’(q) = 0
=3
ao(q1 = a0
=3
P = a0
(4.2.53)
Applying Theorem 1, and taking into account (4.2.53) and the previous definitions, one concludes that, in order for the variational equation of the NNM not to possess any instability zones (n = 0), the equation jx - x 2 = -d2n(q(t),O)/dq22 must admit the solution = (D/p) = (D/ao) = constant. This leads to the condition -(D/ao)2 = -2e0 - 2e2q2 = constant or to the requirement that e2 = 0 and (D/ao)2 = 2e0 Hence, when the potential .function of system (4.2.43) is o f the form f l ( 4 1 4 2 ) = ~ 2 9 +1 a4q14 ~ +Q2eO theorem I guarantees that the N N M (91,42) = (q(t),O)does not possess an\! instability zones in the linearized sence. Note, however, that this result does not rule out the existence of unbounded instability regions, i.e., of regions in parameter space that are bounded by a single curve and extending to infinity (such is, for example, the first instability region of Mathieu’s equation). Consider now the case of only one instability zone in parameter space, n = 1 . The polynomial p(q,e) is expressed in the form p(q,e) = ao(q) + a l ( q ) e , which upon substitution in (4.2.52) leads to the following relations for the coefficients of p(q,e):
x
4.2 SIMILAR NNMs
191
The expresions above were derived by matching coefficients of eo and e l , respectively, in (4.2.52). Solving for a o ( q ) and a1(q), and applying Theorem 1, one finds that in order for the variational equation to possess a single instability zone (n = l), it must be satisfied that e2 = 2a4, and W q ~ q 2 =) + a4q14 + q22(e0 + 2a4q12) [cf. (Zhupiev and Mikhlin, 1984)l. More generally, it can be shown that, when the parameters e2 and a4 of the potential function n ( q i 4 2 ) = a 2 q 1 ~+ a4qi4 + 922(e0+ e2qi2) of system (4.2.43) are related by the expression e2 = (n + l)na4, n = 0,1,2 the similar NNM under examination possesses precisely n linearized instability zones. When the potential function is in the form n(q142) = a2912 + w q l 4 + a6qi6 + q22(eo + e2q12+ e4q14) it can be similarly proven that the conditions for "finite-zoning" instability of the NNM [only n unstable zones for variational equation (4.2.48) or (4.2.52)], are given by (Manevitch et al., 1989; Zhupiev and Mikhlin, 1984): 4a2aqa6 - q3+ 8ha$ = 0 n = 0,1,2 (4.2.55) In addition, for a system with potential function of the form: n(q1,q2) = a2qi2 + a3qi3 + a4qi4 + q22[eo + eiqi+ e2qi21 the conditions for finite number of instability zones are formulated as follows: e2 = 2n(n+l)a4,
e4 = 4n(n+l)a6,
+ l)q,
el = n(n
+ l)a3/2,
4a2a4 - a32 = 0 n = 0,1,2 (4.2.56) To compute the periodic solutions of the variational equation on the boundaries separating the stability and instability zones computed above, one introduces a transformation of the independent variable in the general variational equation (4.2.46), t + q(t), and derives the following alternative variational problem: e2 = n(n
192
STABILITY AND BIFURCATIONS OF NNMs
where primes denote differentiations with respect to q. For the class of potential functions considered above, (4.2.57) is an equation of the Fuchs type with degrees of singularity equal to 0 or 112 at its regular singular points (as discussed in chapter 3 , the singular points of (4.2.57) are the zeros of the coefficient [h-n(q,O)] and the points q = +o). In particular, for the three potentials considered above, the variational equation (4.2.57) assumes the following form:
2u"[h - a292 - a494 - as$] - u'[-2azq - 4wq3 - 6asqs] + u[eo + 2e2q2 + 2e4q4]= 0 (4.2.58b) when n(q1,q2) = a2qi2 + uqi4+ a6qi6 + q22[e0+ e2qi2+ e4qi41, and 2u"[h - a2q2 - a3q3 - a q 4 ] - u'[-2azq - 4a3q2 - 4qq3] + u [eo + 2elq + 2e2q2] = 0 (4.2.58~) when n(qi,q2) = a2412 + a3qi3 + a q 1 4 + q22[eo + elql+ e2q121. Conditions for the existence of a finite number of instability zones for these potentials were previously derived. It can be shown that by applying suitable transformations of the independent variable q, the above variational equations can be placed into the form of a Lame' equation: 2(z2 - a2)(z2 - b2)u" - z(2z2 - a2 - b2)u' - [n(n + 1)z2 - h]u = 0 (4.2.59) where primes denote differentiations with respect to z, integer n denotes the (finite) number of instability zones, and u is a function of the new independent variable z. Equation (4.2.58a) is brought into the form (4.2.59) by setting z=q, h l a = -a2b2, a2/a4 = -a2 - b2, eolaq = -3L e2/a4 = n(n + 1) (4.2.58b) is transformed by setting z = q2 - a, h/ag = 2a(a2 - b2), a2la6 = Sa2 - b2, aqlag = -4a
4.2 SIMILAR NNMs
193
eo/ag = 4[a2n(n + 1) - h], e2/ag = -8an(n + l), e4/ag = 4n(n + 1) (4.2.58~)is transformed by setting z = qkp, p = IfI 2-1/2(a2 + b2)1/2, h / q = (1/4)(a2 - b2)2 a2/a4 = 2(a2 + b2), a3/a4 = 4p, eo/a4 = p2n(n + 1) - h el/a4 = 2pn(n + l), e2/a4 = n(n + 1) In each case conditions for finite-zoning instability were imposed. Considering the Lame' equation (4.2.59) it is noted that the coefficient of u" has the physical interpretation of kinetic energy [h - n(q,O)] for the periodic motion q = q(t) along the N N M trajectory. The kineric energy of the system vanishes twice per period of oscillation, at points z = *a. It follows that throughout the periodic oscillation the inequality IzI Ia is satisfied, where a > 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
STABILITY AND BIFURCATIONS OF NNMs
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
STABILITY AND BIFURCATIONS OF NNMs
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 << 1 (4.3.1) where p is a perturbation parameter, h a system parameter, and II and F are analytic functions in q1 and 92. It is assumed that for p = 0 system (4.3.1) possesses the similar NNM q1 = q(t) # 0, 42 = 0. This assumption is satisfied provided that XI[q(t),O,h]/aq2 = 0 and does not restrict the generality of the analysis. This is because, as shown in previous sections, such a similar mode representation can be achieved by a suitable rotation of the coordinate axes of the system in the configuration plane. Expanding functions n ( q 1,qa,h) and F(qi,q2,h) in Taylor series about (q1,q2,h) = (qi,O,h), and retaining only the leading nonlinear terms, the equations of motion are expressed as:
The task of the following analysis is to analytically approximate the nonsimilar NNMs of system (4.3.2) which neighbor the generating similar mode q1 = q(t) # 0, 92 = 0 of the system with 1.1 = 0, and to investigate their stability. Nonsimilar NNMs with periods equal to the period, T, of the generating similar mode will be sought. At this point the notation E = pl’2 is introduced, and the sought-after nonsimilar NNMs are expressed in the following series expansions:
(4.3.3)
198
STABILITY AND BIFURCATIONS OF NNMs
where the zeroth order approximations (q io(t),O) represent the “generating” similar mode for E = p1/2 = 0. Note that the system parameter ?L is also expressed in series, and the problem is reduced to computing the values q f A f o r which nonsimilar NNMs or nonsimilar mode bifurcations exist. AS discussed in previous sections, the O(1) term qlo(t) in (4.3.3) is governed by the differential equation q10 + an(q1o,o,Waqi = 0 with initial conditions qlo(0) = Qo, qlo(0) = 0. Note that the amplitude Qo provides the 0(1)approximation to the amplitude of the nonsimilar NNM. The O(E) approximation to the nonsimilar NNM is computed by solving the following equations:
System (4.3.4) is a set of linear equations with periodically varying coefficients. Considering the equation governing qi I(t), it can be shown that it admits the homogeneous solution qii(hi)(t)= A1 qiott) [this can be proved by differentiating the equation governing qIo(t); see also (Vakakis, 1994a)l. An additional linearly independent homogeneous solution ql i(h2)(t) can be found by the method of variation of parameters; however, this second homogeneous solution, generally, is not periodic. It follows that in order for the first of equations (4.3.4) to possess a periodic solution of period T [the period of qlo(t)], the following periodicity condition must be imposed:
J
[a2~(qlo(t),0,~)/aqia?LI qio(t> 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
STABILITY AND BIFURCATIONS OF NNMs
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
STABILITY AND BIFURCATIONS OF NNMs
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).
STABILITY AND BIFURCATIONS OF NNMs
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
STABILITY AND BIFURCATIONS OF NNMs
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) + ...
4.4 NNM BIFURCATIONS IN A SYSTEM IN INTERNAL RESONANCE
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
STABILITY AND BIFURCATIONS OF NNMs
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
STABILITY AND BIFURCATIONS OF NNMs
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):
4.4 NNM BIFURCATIONS IN A SYSTEM IN INTERNAL RESONANCE
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
STABILITY AND BIFURCATIONS OF NNMs
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:
4.4 NNM BIFURCATIONS IN A SYSTEM IN INTERNAL RESONANCE
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
STABILITY AND BIFURCATIONS OF NNMs
(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).
4.4 NNM BIFURCATIONS IN A SYSTEM IN INTERNAL RESONANCE
Y1
(b)
Figure 4.4.2 (Continued)
2 17
218
STABILITY AND BIFURCATIONS OF NNMs
$ 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 $(x) in (4.4.3) was found not to be valid. Moreover, examining the plot of Figure 4.4.1 it is evident that for large or small values of the frequency-detuning parameter (T (i.e., away from conditions of internal resonance), one obtains that a21(0)+0, and solution (4.4.15) converges to the analytical solution for NNM I in (4.4.3). In contrast to perturbation computations of nonsimilar NNMs carried out in previous sections (where mode nonsimilarities were assumed to be small), solutions (4.4.15) represent modes with strong nonsimilarities. These "nonsimilarities" result due to the assumed n o n s e p a r a b l e O( 1 ) approximations ?2(0)(yl) in space and time, which, in terms, are a consequence of the strong nonlinear interaction between the two modes in internal resonance.
4.5 STABILITY OF STATIONARY WAVES In this section an additional application of the previously developed stability methodology is given, addressing the problem of linearized stability of stationary waves in certain nonlinear partial differential equations. The stationary waves considered in this section are defined by second-order
220
STABILITY AND BIFURCATIONS OF NNMs
differential equations with analytic first integrals, in analogy to NNMs of conservative finite-dimensional oscillators. Consider, for example, the nonlinear Klein-Gordon equation: 4 x x - Qtt
(45.1)
= p(@)
where p($) 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<2 = F + P(O), F = const,
Q,
P(@) =
p(z) dz
(4.5.3)
0
where F is the energy of the wave. The linearized stability of the stationary wave is studied by analyzing the following partial differential equation, which governs the evolution of small variations y(5,t):
Introducing separation of variables, the solution of (4.5.4) is represented as y(5,t) = estq(5) where q(5) = eA<w and A = -su/(l - u2) and (4.5.4) is reduced to a linear variational equation with parameterdependent coefficients: ] 0, 2 ~ & ( 1- ~ 2 +) w[B - p ~ ( @ ) =
B = ~ 2 / ( ~ 21)-
(4.5.5)
Similarly to several previous works addressing the linearized stability of nonlinear waves, the stability problem is posed as follows: consider the linearized stability in time of a nonlinear stationary wave over a class of
4.5STABILITY OF STATIONARY WAVES
221
perturbations q(<) that satisfy the same boundary conditions as the stationary waves under consideration. If the perturbations decay at infinity, it is well known that for the problem under consideration the spectrum of (4.5.5) (Schrodinger's equation) is real. In all cases the parameter s2 must be real in the stability problem, otherwise the perturbations fall outside the indicated function class. Note that when the spectrum of eigenvalues s2 is continuous, one should use the Laplace transform with zero boundary conditions instead of the method of separation of variables; in that case variable s will be regarded as the Laplace transform variable. Regarding @ as the independent variable instead of 5 the variational equation (4.5.5) can be alternatively formulated as follows: 2wocp[F
+ P(@>)]+ WQP(@)+ w[B - PQ(@)] = 0
(4.5.6)
Furthermore, introducing the new independent variable z = @2, (4.5.6) can be expressed in the form: 8z[F
+ P(z1/2)]wzz + {4[F + P(z1/2)] + ~ z ~ / ~ P ( z ~ / ~ ) } w , (4.5.7) + [B - P @ ( z ~ / ~ )=] w0
This form of the variational equation is used only for even potential functions P(@). Finally, if one regards the kinetic energy of the stationary wave, k = F + P(@), as the independent variable, the variational equation is reduced to: 2kP2[@(k)lwkk + { 2kPcp[@(k)l + P2[@(k)l} Wk + { B - Pd@(k)l} W = 0 (4.5.8) As in previous sections, it is possible to define certain classes of functions P(@) for which the resulting variational equations are of the Lame' type. For Lame' variational equations it is possible to analytically determine regions in parameter space where the solutions are bounded or unbounded. Potential functions P(@) leading to Lame' variational problems are considered in the following exposition.
(i) P ( @ ) = Dsin@ - G c o s @ (Sine-Gordon equation) For this system, the phase portraits of equation (4.5.2) are depicted in Figure 4.5.1 for various levels of the wave energy
222
STABILITY AND BIFURCATIONS OF NNMs
Q,
=-PFigure 4.5.1 Phase plane portraits of the Sine-Gordon equation: (a) u2 - 1 > 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
4.5 STABILITY OF STATIONARY WAVES
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
STABILITY AND BIFURCATIONS OF NNMs
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,
4.5 STABILITY OF STATIONARY WAVES
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
STABILITY AND BIFURCATIONS OF NNMs
-
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
4.5 STABILITY OF STATIONARY WAVES
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.
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 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
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
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<
5.1 EXACT STEADY-STATE MOTIONS
233
equal to the period of the excitation), and, within half the period of the steady-state motion one can express t as a single-valued function of the displacement x. This idea was first introduced by Rausher (1938). Symbolically the aforementioned inversion can be written as:
x = X(t,E)
3
t = t (X,E)
(5.1.2)
Moreover, at an exact steady-state the equations of motion (5.1.1) become decoupled, and the system resembles an autonomous one. The total (conserved) energy h(E) of the system oscillating at an exact steady-state is expressed by the following symbolic relation: X2/2
+ V(x,O,...,0) + EF(x,E)= h(E)
(5.1.3a)
where the pseudo-potential function F ( x , E ) will be defined later. It is assumed that, within half the period of the steady-state motion, all variables pertaining to the periodic solution are single-valued functions of the parametrizing coordinate x:
The velocity X is defined as a function of x by (5.1.3b), while t is expressed as a function of x by the following quadrature relation [cf. relation (5.1.2)]:
The phase $ is selected so that the initial conditions x(0) = X , X(0) = 0 are satisfied. In this case the energy and amplitude are related by the expression, V(X,0 ,...,0) + EF(X,E)= h(E) Assuming that the series expansion of the potential function begins with even-power terms, at each fixed level of the energy, h(E), there exist two values for the amplitude X = Xj, j=1,2. Employing (5.1.2)-(5.1.3~),one can (at least in principle) eliminate the temporal variable t from the equations of motion (5.1.1) to obtain the
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
234
following pseudo-autonomous system valid at the exact steady-state and at the time interval ttz [O,T/2): xi + aV(x)/axi + Efi[Xl,Xl,...,Xn,xn,t(x)] = 0,
i = 1,2,...,n
(5.1.4)
It follows that an exact steady-state motion of the nunautunumous system (5.1.1)is equivalent to an NNM of the pseudo-autonomous system (5.1.4). This equivalence, however, holds only at an exact steady-state motion. The equations governing the trajectory Xi = Xi(X,E), i = 2, ...A of the NNM of the pseudo-autonomous system may be obtained as in chapter 3, i.e., by introducing the new independent variable x and combining the equations of motion (5.1.1) with the energy integral (5.1.34 and the functional relations (5.1.3b) and (5.1.3~).Then, one obtains the following relations governing functions Xi(X,E): dV xi'' x ~ ( X , E ) + xi' [x,x2(x,E),...,xn(x,E)]
(- ax
}
- E f l rx,x(x,E),x2(x,E),X2(x,E) ,...,t(X,E)] 2V +) x ~ ( x , E ) ]+ E~~[x,X(X.E),X~(X,E>,X~(X,E) ,...,~ ( x , E ) ]= o axi [ x , x ~ ( x , E,..., i = 2 ,...,11 (5.13)
where primes denote differentiation with respect to x. At X=Xj, j=1,2 and X = 0 one obtains the following additional set of orthogonality equations complementing relations (5.1.5) (cf. chapter 3 ) : (Xi'(
-
av [X,X2(X,E),
...,X,(X,E)] - E f l [X,X(X,E),X2(X,&),X2(X,E)
,...,t(X,E)]
]
Solving equations (5.1.5) and (5.1.6) leads to the unique determination of functions xi(x,E). In addition to these relations two more conditions must be considered. First, from ( 5 . 1 . 3 ~ )the following periodicity condition is formulated:
T + 41 = 2-112
8 [h(E)
-
V(t,0 ,...,0) - E F ( ~ , E ) ] - ' /d~ t
(5.1.7)
5.1 EXACT STEADY-STATE MOTIONS
235
where the integral refers to a complete cycle of steady-state oscillation. Once functions xi( x , ~ )are determined, the response of the parametrizing coordinate is computed by solving the following nonlinear ordinary differential equation:
2+
ax [X,XZ(X,E) ,...,Xn(X,E>]
av
+ Ef~[X,X(X,E),X~(X,E),X~(X,E))...)t(X,E)] = 0
(5.1.8)
Comparing (5.1.8) with (5.1.3a), it is found that
from which we derive:
Relation (5.1.10) provides a means for determining the function F(x,E), which represents the total energy of the pseudo-autonomous system (5.1.4). The periodicity condition (5.1.7) must be supplemented by a second relation stating that the work performed by all forces during a complete period of steady-state motion is equal to zero:
+ Efi [S,X(5,E),X2(5,E),Xi2(5,E)
dV ,...,t(5,&)1- dx (6&..$ )
} dk = 0
(5.1.1 1) The two additional conditions (5.1.7) and (5.1.11) enable the evaluation of the phase 4, and energy h(E) of the steady-state motion. These conditions together with the sets of equations (5.1.5) and (5.1.6) determine uniquely the steady-state motion. Since no exact solution of these equations currently exists, an approximate iterative methodology must be employed. To this end,
236
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
the following series approximations for the various unknown variables are introduced: Xi(X,E) = XiO(X) -I-E X i l ( X ) + o(E2), i = 2 ,...,n, = Xo(X) 4- EXl(X) + o(E2) h(E) = ho + &hi + O(E2)
X(X,E)
(S.1.12)
The coefficients of the above approximations are computed by substituting (5.1.12) into the previous relations and matching coefficients of respective powers of E. In the zeroth order approximation one considers the unperturbed system (corresponding to E=O), which, as mentioned previously, possesses the similar NNM, xi0 # 0, X i 0 = 0, i = 2,...,n. For &=O equation (5.1 3)implies that Xo2 = 2[ho - V(X,O,...,0)] whereas the O(1) approximation of the response of the parametrizing variable is given by,
t(O)(x) + Cp(0) = 2-1/2
5 X
[ho - V(k,O, ...,0)]-1/*dc
X(0)
In these expressions, @(o)and ho are the zero-th order approximations for the phase and energy variables of the nonlinear steady-state. At this order the periodicity condition (5.1.7) is written as: T
+
= 2-112
f
[ho - V(k,0,...,0)]-1/2 dk
(5.1.13)
Moreover, since for the unperturbed conservative system the energy condition (5.1. I 1) holds identically, without loss of generality one may assume that @(o)= 0. Then, the periodicity relation (5.1.13) is used to compute the zeroth order approximation to the energy, ho. In addition, at this order of approximation it also holds that V(Xj(O),O,...,0) = ho a relation that can be used for computing the O( 1) amplitude approximations Xj (0). The equations governing approximations Xil, i = 2,...A are obtained by substituting (5.1.12) into (5.1.5) and (5.1.6), and considering O(E) terms:
5.1 EXACT STEADY-STATE MOTIONS
i=2 ,..., n
237
(5.1.14)
The solution of the above sets of equations can be asymptotically computed by employing the perturbation methodology developed in chapter 3. This tedious procedure will not be repeated herein, but an example of its application to a two-DOF oscillator will be given in the next section. The next step in the computation is to estimate the pseudo-potential function F(x,E) from (5.1.10); in the first approximation with respect to E this quantity is computed by the following equation: n dF( 1 a2v x,O,...,0) + f l [x,Xo(x),O,...,O,t(')(x)] ~dx( x , E= ) Xjl(X) -( j=2
C
(5.1.16)
The function F ( ~ ) ( x , Eis) obtained from the above relation by integration. The constant of integration during this operation represents the O(E) correction to the energy ho of the pseudo-autonomous system at the exact steady-state oscillation. Regarding the time evolution of the parametrizing variable x, a new approximation replacing t(O)(x) is computed by the following quadrature:
whereas, the periodicity relation (5.1.7) and the energy integral (5.1.11) lead to the following complementing conditions:
238
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
Condition (5.1.18) is used to compute the O(E) energy correction h i , whereas (5.1.19) yields the new approximation for the phase $ ( I ) of the exact steady-state solution. Improved approximations for the steady-state amplitudes Xj(1)are computed by the following relation: ho + Ehl =
( V [ X , E X ~ ~...,( XE X) ,~ ~ ( X )+]
(5.1.20) Efl[x,xo(x),O ,...> 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)) = &$*(XI,E) is used. The exact nonsimilar steady-states of the original system (5.2.1) correspond to nonsimilar NNMs of the pseudo-autonomous system (5.2.4); hence, the problem of the computing forced exact steady-state motions is converted to the equivalent problem of computing the nonsimilar NNMs of the pseudo-autonomous system. The nonsimilar NNMs of system (5.2.4) are expressed as X i = A A Xi(XI), i = 2,.,.,n, where the nonlinear functions Xi(*) are computed using the methodology of section 5.1. A specific calculation of exact steady-state motions is now carried out considering the following two-DOF system.
+ X I + X 3I + Kl(xl - x2) + K3(xl - x2)3 = Ep(t) 3 x2 + x2 + x2 + Kl(x2 - xi) + K3(x2 - x1)3 = 0
x1
(5.2.5)
where the stiffness coefficients K1 and K3 are assumed to be O(1) positive scalars. When E=O the unforced system possesses the similar NNMs x2 = cxl, with c = +1 (in-phase mode), or c = -1 (antiphase mode). When E # 0, the NNMs are perturbed by the excitation, and the system becomes weakly
240
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
nonconservative. At an exact steady-state the following pseudo-autonomous system is obtained, Xi
+ X I + X 3I + Kl(x1 - ~ 2 +) K3(X1 - ~ 2 = )$(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:
5.2 ADMISSIBLE FORCING FUNCTIONS FOR STEADY-STATE MOTIONS
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 $o(xl) into (5.2.11) and (5.2.12) one computes the firstorder approximation f i 2 ( l ) ( x i ) . It turns out that the resulting expression is
242
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
too complicated to be of any practical use, and a modified approach must be followed. To this end, a transformation of variables is introduced in terms of the amplitude function, am(*,*) (Byrd and Friedman, 1954): cn(u,k) = cos$
+
$ = 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
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
(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
5.2 ADMISSIBLE FORCING FUNCTIONS FOR STEADY-STATE MOTIONS
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
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
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)
5.2 ADMISSIBLE FORCING FUNCTIONS FOR STEADY-STATE MOTIONS
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
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
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
5.2 ADMISSIBLE FORCING FUNCTIONS FOR STEADY-STATE MOTIONS
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
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
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
5.2 ADMISSIBLE FORCING FUNCTIONS FOR STEADY-STATE MOTIONS
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
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
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
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
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):
5.3 EFFECTS OF NNM BIFURCATIONS ON THE RESONANCES
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
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
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
5.3 EFFECTS OF NNM BIFURCATIONS ON THE RESONANCES
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
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
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.
5.3 EFFECTS OF NNM BIFURCATIONS ON THE RESONANCES
259
Figure 5.3.2 Nonlinear frequency response curves of the system with K = 0.4,(two unforced NNMs): -Stable, ------ unstable steady-states.
260
RESONANCES OF DISCRETE SYSTEMS CLOSE TO NNMs
Figure 5.3.3 Nonlinear frequency response curves of the system with K = 0.15 (four unforced NNMs):Stable, ------ unstable steady-states.
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 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 << 1, existing perturbation methods based on small parameters and trigonometric expansions encounter convergence problems and do not lead to accurate results. It will be shown that, by using as generating solutions the nonsmooth functions { T(t),e(t) 1, one can analytically study such essentially nonlinear solutions without encountering convergence problems. The principal aim of the analysis carried in the next sections is to construct a new analytical methodology for studying the dynamics of discrete oscillators in strongly nonlinear regimes. In contrast to the majority of standard methods, which employ the harmonic oscillator as the generating system for computing the dynamics, the new method uses the strongly nonlinear vibro-impact response as generating solution. What is achieved by this, is that, instead of modeling the nonlinear response from the viewpoint of "almost linear motion," the response will be viewed as resulting from a perturbation of a strongly nonlinear one. As a result, the validity of the new methodology for analyzing strongly nonlinear or even nonlinearizable problems will be asserted. The fundamental principles of the NSTT technique and its application in the analysis of the dynamics of discrete and continuous oscillators can be found in (Pilipchuk, 1985,1988; Vedenova et a]., 1985; Manevitch et al., 1989).
266
THE METHOD OF NSTTs
6.2 REPRESENTATIONS OF FUNCTIONS USING NSTTs Transformations of the time variable are often employed in problems in dynamics. In this section the following NSTT is introduced: t
-+z(t) = (2/n)arcsin[sin(nt/2)],
tE R
(6.2.1)
The transformation T(t) is a saw tooth periodic function with discontinuous derivatives, and as mentioned in section 6.1, physically it represents the response of a vibro-impact oscillator. Denote the derivative of the transformation by e(t) = f(t), where derivatives and equalities should be understood in the generalized sense of the theory of distributions. Note that since e2(t) = 1 the metric of time is preserved under the NSTT (6.2.1). The following proposition is made. Proposition 1. Consider the general periodic function x = x(t) with period T = 4. By applying the NSTT this function can be expressed as: x(t) = x(z,e) = X(z)
+ e Y(z),
e(t) = i(t)
(6.2.2)
where
X(z) = (1/2)[x(z)
+ x(2 - z)],
Y(z) = (1/2)[x(z) - x(2 - 211 x(z,e) = x(z(t>,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:
6.3 ANALYSIS OF DYNAMICAL SYSTEMS
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)
6.3 ANALYSIS OF DYNAMICAL SYSTEMS
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) << 1. It can be verified that the computation of such strongly nonlinear motions close to the separatrices is reduced to solving the following nonlinear boundary value problem in terms of the I-component, Y, of the NSTT of x(t):
6.3 ANALYSIS OF DYNAMICAL SYSTEMS
+
x(t) = X(T) + e Y(z) Y" + a2f(Y) = 0, Ylz=+l = 0, Y(-T) = Y(z),
275
X = 0 (6.3.17)
Consider the one-parameter family of closed periodic orbits surrounding the stable equilibrium (x,X) = (0,O) in the phase plane. These orbits are parametrized by the energy E and become strongly nonlinear at the limit E-E,, i.e., as the heteroclinic orbits are approached. Introduce at this point the following representation to the motion, based on the NSTT technique: x(t) = Y(z) Z(t/a) = Y(z) e(t/a) Y(T) = K + Yl(z) + Y ~ ( z +) ...
(6.3.18)
where Yi(T) is the ith approximation, and i = l,Z, ... O(Yi(T)) >> O(Yi+l(T)), As previously, z = z(t/a) and a is the quarter-period of the motion. It is noted that as E-+E,, the summation of higher order terms in (6.3.18) must tend to zero for almost every value o f t , with the exception of time instants in the singular set N, where N = ( t / z(t/a) = k l } : Y ~ ( z+ ) Y2(z) + ... + Ym(z) -+ 0
as E+E, , V m , and V t E R - N
(6.3.19)
Hence, close to the separatrix branches the response of the system tends to the limit, x(t)-+Ke(t), where f = t/a, and the system performs sudden "jumps" between the two unstable equilibrium positions. Expressing the quarter period squared of the strongly nonlinear periodic motion in a series of successive approximations, a2 = ao2(1 - 21 - ~2 -...)-',
O(Zi) >> O(Zi+l),
i = 1,2,... (6.3.20)
and substituting (6.3.20) and (6.3.18) into (6.3.17), one obtains a hierarchy of problems at various orders of approximation. Considering the first-order approximation in (6.3.17), one obtains the following solution for the first approximation Y 1: Y1" - p2hoYl = 0, Y1I,=-j1 = -K =j Y 1 = -Kcosh(paoz)/cosh(pao)
(6.3.21)
276
THE METHOD OF NSTTs
where
ko = a$,
p2 = -f(K),
and
Yi(-z) = Y ~ ( T V ) T
The second-order approximation is governed by the following equation: Y2" - p2lqY2 = -zlp2hgKcosh(paoz)/cosh(pao)
- (1/2)hoK2f"(K)cosh~(pa~z)/cosh~(pao)
(6.3.22)
The first term on the right-hand side of (6.3.22) is secular and is eliminated by setting z1 = 0. This leads to a solution for Y2, which is uniformly valid in time. Solving the resulting ordinary differential equation, and satisfying the boundary conditions (6.3.171, one obtains the following analytic expression for the second-order approximation Y2: Y2 = K2f"(K)(6p2)-1[ 1
+ cosh(2pao)]-l[
3 - cosh(2paoz)
- [3 - cosh(2pao)]cosh(paoz)/cosh(pao)
}
(6.3.23)
Considering terms of O(Y3), one computes the first nonzero correction in the series (6.3.20) as follows: 22
= K2[ 8p2cosh2(pao)]-l[ (5/3)(f"(K)/p)2 + f "( K)]
(6.3.24)
As an example of application of the previous formulas, consider nonlinear motions close to the heteroclinic orbits of the pendulum equation:
x(t)
+ sinx = 0,
XE R
(6.3.25)
Computing the first two steps in expansion (6.3.18) leads to the following analytic approximation for the response: x(t) = K[ 1 - cosh(aoz)/cosh(ao)]e, z = z(t/a), a* = a$/[ 1 - (n2/8)cosh-z(ao)]
e = e(t/a), (6.3.26)
where the physical meaning of the various parameters in the above expression was discussed earlier.
6.3 ANALYSIS OF DYNAMICAL SYSTEMS
277
Example 3: Transient response of a damped nonlinear oscillator To demonstrate the application of the NSTT technique for analyzing the response of nonconservative systems, consider the following strongly nonlinear but weakly damped single-DOF oscillator: X
+ 2pX + f(x) = 0,
0 < p << 1,
XE
R
(6.3.27)
where f(x) is an odd function such that f(x) > 0 for x > 0. To analyze this system a two-scales expansion will be followed (Nayfeh and Mook, 1984). The role of "fast" time scale will be played by the NSTT variable z($), where the derivative of the phase @ (i.e., the angular velocity) will be assumed to depend on the "slow" time scale to = pt: = O(t0)
(6.3.28)
Note that the explicit form of relation (6.3.28) is yet to be determined. Applying the NSTT technique, the solution of equation (6.3.27) is expressed as follows: x(t) = X(z,to) + eY(z,to) (6.3.29) Substituting (6.3.29) into the equation of motion and assuming continuity of the solution at the points of the singular set N (cf. section 6.2), one obtains the following set of partial differential equations governing the variables X and Y:
(6.3.30) These equations are complemented by the following set of boundary conditions (implying continuity of X and Y at points of the singular set N): (6.3.3 1)
278
THE METHOD OF NSTTs
In expressions (6.3.30), terms Rf and If are the R- and I-components, respectively, of function f(x), and their analytic expressions are given by (6.2.6). The differential oDerators H and L are defined as, do a2 H=2o(l+m)+3j and L = -at02 + 2 - - at0
a
a
To facilitate the application of the method of successive approximations, the first two terms on the right-hand sides of (6.3.30) are multiplied by a formal parameter E, and the last terms by ~ 2 The . solution of the problem is then sought in the series expansions: x = XO(T,tO) + EXl(T,tO) + E2X2(T,tO) + ... Y = YO(T,tO) + EY l(T,tO) + &2Y2(T,tO)+ ... 0 = oo(tO) + EO1(tO) + E 2 0 2 ( t 0 ) + ... (6.3.32a) The formal parameter E is introduced only for book-keeping purposes and helps to distinguish the various orders of approximation. At the end of the analysis this parameter is set equal to unity. It turns out that, instead of computing the coefficients of the series expression of the frequency, it is more convenient to determine the coefficients of the series expansion of the frequency squared:
Substituting (6.3.32) into (6.3.30), and considering terms of O( l ) , one obtains the following generating solutions at the zeroth order of approximation:
(6.3.33) In writting (6.3.33), the damped response is solely expressed in terms of the X-component. This assumption does not restrict the generality of the analysis, and alternative solutions in terms of X- and Y-components can be similarly sought. Also note that in this case the constant of integration A0
6.3 ANALYSIS OF DYNAMICAL SYSTEMS
279
depends on the slow time scale to; this constant is determined by eliminating the Y-component of the first-order of approximation. At this point it must be noted that what appear as secular terms in equations (6.3.30) are not secular in terms of the original time variable t ; indeed, since z(t) and e(t) are nonsmooth periodic functions o f t , terms such as zvsin(m/2) are periodic functions of t and do not diverge to infinity as t++--. Hence, the methodology followed herein attempts to improve the smoothness properties of the analytical asymptotic solutions. Retaining O(E) terms in (6.3.30), the following sets of equations result:
(6.3.34) where Ro = Rfle=o, 10 = Ifle=o and the operator Hi is the ith term of the expansion of operator H, H = Ho + E H +~ &2H2+ ... The solutions of equations (6.3.34) can be expressed in closed form as follows: 7
XI = -(l/XO)J
(7 - 5)f(A05)d5 0
Y = (p/23LO)(1 - +)HOAO = o
(6.3.35)
where &jo = wo(0) and A0 = Ao(0). Setting the component Y 1 equal to zero, one obtains the following zeroth order approximation to the amplitude:
ho = 002 = (l/Ao)
1
f(Aoz)dz
(6.3.36b)
0
Taking into account O ( E ~terms, ) one obtains the equations governing the second order of approximation:
280
THE METHOD OF NSTTs
(6.3.37) where At this order of approximation, the elimination of component Y2 leads to the following equation determining the frequency correction o 1 :
The solutions of equations (6.3.37) can then be explicitly expressed by quadratures:
Y2 = -(p/Xo)[Hoj
Xi(6) d t
+ (1/2)H1A0(z2- l)] = 0
(6.3.39)
I
Higher order approximations can be computed in the same manner. As an example of application of the previously derived formulas, consider the following weakly damped oscillator: X+2pX+xm=O,
O
XER
(6.3.40)
where m = 2k + I , k = 1,2,.,. Retaining the first two orders of approximation, the solution of (6.3.40) is analytically approximated as:
x = A*exp[-4pt/(m+3)][z
- zm+2/(m+2)],
z = z(Q,),
m > 1 (6.3.41)
where A* is an arbitrary constant, and the phase Q, is approximately given by:
6.3 ANALYSIS OF DYNAMICAL SYSTEMS
4, = Q ~ 1[- exp(-2yt(m
-
28 1
l)/(m+3))]
4,m = (m + 3)A*(1-m)/2(23/2y)-1(m - l)-l(m
+ l)-”*
(6.3.42) Observe that the solution is derived only in powers of the saw tooth function 7.It is interesting to note that the above analytical expressions predict that the damped oscillator performs afinite number of oscillations as t +m. It is well known that this does not hold in the underdamped linear case (i.e., for m=l). The analogy of the result (6.3.41) and (6.3.42) to the motion of a damped vibro-impact oscillator is clear: if the energy of the vibro-impact oscillator (cf. left Figure 6. I .2) diminishes during each cycle due to viscous dissipation, one anticipates that there will be only afinite number of impacts after which the mass reaches asymptotically one of the two rigid boundaries as t +m. This does not hold for the underdamped harmonic oscillator (cf. right Figure 6.1.2), which undergoes an infinite number of cycles (oscillations) before it reaches its zero equilibrium state. X
1 1 C0
-1 -1
X
0
5
10
15
2U
25
30
35
0
S
10
15
20
2s
30
35
t
1 0
-1
t
Figure 6.3.1 Strongly nonlinear free damped response of an oscillator with (a) cubic and (b) seventh-order stiffness nonlinearity: NSTT technique, ------- numerical simulation.
282
THE METHOD OF NSTTs
In Figures 6.3.l(a)and (b) the damped responses of the strongly nonlinear oscillator (6.3.40)with degrees of nonlinearity, m = 3 and in = 7, respectively, are presented. It is noted that the approximate analytical expressions (6.3.4 1) and (6.3.42) compare relatively well to the numerical simulations, especially in the strongly nonlinear regime.
Example 4: Waves in a one-dimensional nonlinear continuum Finally, it is shown how the NSTT technique can be used for studying stationary or traveling waves in one-dimensional nonlinear continua. Suppose that the wave process is described as: Ll = u($,x,t> (6.3.43) where $ = $(x,t) is a phase variable and x,t are the spatial and temporal variables, respectively. It is assumed that the phase variable varies faster compared to the variation of x and t. This requirement can be mathematically formalized by introducing a small parameter E to the problem, and scaling the various variables as follows:
xo = EX, to = Et,
$ = (l/&)$(xO,tO)
and
u = u($,xO,to)
4
(6.3.44)
Suppose that function u is periodic with respect to the phase and that its period is normalized to T = 4.Applying an NSTT,the response is expressed as follows: u = U(qxO,tO) + eV(z,xO,tO), z = ~($1 (6.3.45) Consider now the one-dimensional nonlinear continuous system governed by the following equation of motion:
(6.3.46) where f(*) is an odd function. Substituting (6.3.46) into the equation of motion, using the chain rule of differentiation to express the derivatives in terms of the NSTT variable z, and setting separately the R- and Icomponents of the transformed equations equal to zero, one obtains the following equations governing U and V:
6.3 ANALYSIS OF DYNAMICAL SYSTEMS
*)2 [(at0
-
aXo
[(%)2
-
($9'1
VIT,+1 = 0
a+
- -E (H = -E (H
where operators H and G are defined as,
H=2
av + Rf) - E ~ G U au + If) - E ~ G V
au xlT=kl =0
and
%&-*%&+
283
a2
G$, G = at02 ---
(6.3.47)
a2 ax02
and Rf, If are the R- and I-components of the restoring function f(x) [cf. expression (6.2.6)]. Problem (6.3.47) can be analyzed as the previous example using a two-scale expansion. Omitting the details of the calculation, it can be shown that the first-order approximation to the solution can be written as:
where o = -a$iat,
k = a$iax,
and
amfax + akiat = o
The last equation in (6.3.48) is recognized as the first-order dispersion relation of the system. The described NSTT formulation will be used in later chapters to study stationary and traveling nonlinear waves in unbounded domains and to extablish a relation between this type of motions and NNMs.
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 7 NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS In a number of studies initiated by Anderson (1958, 1978, 1985), it was shown that weak structural irregularities (which are inherent in all practical engineering structures) may lead to passive confinement of free and forced vibrations of weakly coupled linear periodic systems (Hodges, 1982; Pierre and Dowell, 1987; Bendiksen, 1987; Kramer and MacKinnon, 1993). This is caused by mode localization, i.e., by spatial motion confinement of certain of their normal modes. Mode localization can be beneficial in cases where a confinement of the vibrational energy is required (as, for example, in the problem of vibration and shock isolation of large flexible space structures), or, on the contrary, it may result in failure of critical structural components (as in the case of flow-induced vibrations of mistuned rotating bladed disks). It follows that study of linear and nonlinear mode localization in engineering structures is of great practical interest. Extensive studies on mode localization were carried out under the assumption of linearity. Anderson (1958) studied localized motions in linear lattices with random impurities. Using an analytical perturbation technique, he proved that wave functions of particle chains oscillating in a random potential can become spatially localized; in the context of solid-state physics he interpreted his results as an absence of spin diffusion in certain disordered arrays of "sites" (lattices) [the reader is also referred to the collection of papers on localization by Anderson (1994)l. High-frequency localized eigenfunctions in one-dimensional disordered chains were investigated numerically in (Dean and Bacon, 1963). Payton and Visscher (1967a,b) studied extensively the localized modes of randomly disordered one-, two-, and three-dimensional linear (harmonic) lattices by numerically solving the associated eigenproblems. In addition, localization phenomena in lattices used as models of phonon (or other type of energy) transmission in solid-state physics are reviewed by Wallis (19861, Donovan and Angress (1971), Lee and Ramakrishnan (1985), Erdos and Herndon (1982) and Thomas (1992). Localization and scattering of electromagnetic and acoustic waves in random media is studied in the collection of papers edited by Sheng (1990). Weaver 285
286
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
(1993) and Weaver and Burkhardt (1994) studied Anderson localization in acoustic wave propagation in random media. Pierre and Dowel1 (1987) used a perturbation methodology to study mode localization in a system of weakly coupled, weakly mistuned, linearized pendula. They showed that localized modes exist when the coupling frequency between subsystems was of the order, or smaller than the spread in natural frequencies of the component systems. Wei and Pierre (1988) used the same perturbation methodology to study free and forced IocaliLed motions in linear mistuned assemblies with cyclic symmetry. Hodges (1982) showed that structural irregularities can lead to mode localization in elastic systems. Numerical computations of localized vibrations and wave confinement in models of large periodic space structures were carried out by Bendiksen (1987) and Cornwell and Bendiksen (1989). Keane and Price (1989) analyzed the vibrations of finite, ordered, and disordered systems with axial elements and presented a heuristic explanation for the spatial localization of certain modes possessing frequencies in attenuation zones (AZs) of the infinite ordered system. They showed that a single disorder can have more profound mode localization effects than multiple, random disorders. Luongo (1992) studied mode localization in a beam continuously restrained by imperfect elastic springs by formulating a turning point mathematical problem. Luongo (1 988) studied localized vibrations in disordered structures with high modal densities, and Scott (1985) analyzed localization and scattering of one-dimensional waves propagating in a random medium, using stochastic process theory. Luongo and Pignataro (1988), Pierre (1989), and Luongo (1991), studied mode localization in linear buckling problems. Ishii (1 973) and Kissel (1988) studied localization in one-dimensional lattices with random disorder by employing Furstenberg's theorem ( 1963) on products of random matrices. Additional works on localization phenomena in acoustics and optics are by Baluni and Willemsen (1985), Sheng et al. (1986) and Anderson et al. (1980). Numerous studies on mode localization in nonlinear systems also exist in the physics and engineering literature. Localized or nonlocalized timeperiodic motions in repetitive nonlinear particle chains of infinite spatial extent were studied extensively (Fermi et al., 1955; Jackson, 1963a,b; Ford and Waters, 1963; Jaffe and Brummer, 1980; Rosenau, 1987; Sayadi and Pouget, 1991). In these and other works (Takeno et al., 1988; Takeno, 1990; Dauxois et al., 1992; Dauxois and Peyrard, 1993), one-dimensional chains of
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
287
identical single-DOF nonlinear oscillators were investigated, and it was shown that certain of these systems undergo spatially localized periodic oscillations whose amplitudes decay exponentially in space. In a recent work (MacKay and Aubry, 1994), the existence of such localized modes in weakly coupled nonlinear chains was rigorously proved. Kuske et al. (1993) considered the stationary Schiodinger’s equation in a one-dimensional chain of particles with random potentials (Anderson model). Using singular perturbation expansions they computed the localization length of the localized wave functions. Discreteness effects on envelope solitons propagating in nonlinear lattices were studied in works such as (Peyrard and Pnevmatikos, 1986). Payton et al. (1967) performed numerical simulations of energy transport in one- and two-dimensional randomly disordered nonlinear lattices to study the thermal conductivity of these systems. McKenna et al. (1994) studied the scattering of localized traveling waves in disordered media. Additional works investigated nonlinear localization in lattices with localized impurities. In linear lattices, localized oscillations take place around light impurities; when the concentration of impurities is large, nearly all of the linear modes of the lattice are localized (Toda, 1981) and energy transmission is impaired. Analytical and numerical investigations of scattering of solitons by localized impurities in nonlinear lattices were performed in (Yoshida and Sakuma, 1978; Nakamura, 1978; Yajima, 1978; Nakamura, 1979; Watanabe and Toda, 1981a,b; Klinker and Lauterborn, 1983; Li et al., 1988). It was found that due to soliton-impurity interaction, the incident wave is decomposed to a transmitted and a reflected wave, while in some cases the impurity is excited and a localized impurity mode appears. It was shown that the localized impurity modes are excited more strongly by solitons with widths of the order of the impurity length in the lattice. Moreover (Li et al., 1988), envelope solitons cannot excite significantly a localized mode, whereas, kink solitons interacting with heavy-mass impurities may excite a localized mode with a relaxation time remarkably shorter. Impurity modes in the Toda lattice with an isolated mass disorder were analytically studied in (Yoshida and Sakuma, 1982), by employing a perturbation methodology based on Inverse Scattering Theory (IST). In that work, the localized impurity mode was expressed in terms of a parametrized N-soliton solution, with parameters determined so as to satisfy the equation of motion of the disordered mass.
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NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
In (Wei, 1989), a numerical technique was used to study the response of weakly coupled, nearly cyclic structures with dry friction. It was found that dry friction damped systems were more susceptible to localized modes than viscously damped ones, Thompson and Virgin (1988), Hunt et al. (1989), Hunt and Wadee (1991) studied localized buckling phenomena in forced nonlinear elastic systems. In these works nonlinear localized buckling modes were examined using analytical and numerical methods. In (Vedenova et al., 198.5), localized nonsimilar modes were detected in a two-DOF vibro-impact system. Pierre and Shaw (1991) numerically studied nonlinear mode localization in a disordered two-DOF nonlinear system. In a series of works by Vakakis and co-workers (Vakakis. 1992a; Vakakis and Cetinkaya, 1993; Vakakis et al., 1993a), nonlinear localization i n discrete systems with stiffness nonlinearities was studied. The main finding of these works was that, in contrast to linear mode localization, nonlinear mode localization can occur in perfectly symmetric nonlinear periodic systems, and the onlv prereyuisifefor its existence is weak coupling between subsystems. A similar result was reported by Luongo (1991), where it was found that geometric nonlinearities in systems with high modal density have the same effect, with regard to localization, that structural imperfections have in linear theory. In general, nonlinear mode localization was found to be independent of any mistuning of the subsystems of the nonlinear structure, a novel result, with no counterpart in existing linear theories. Nonlinear mode localization in the context of molecular vibrations was studied in (Child and Lawton, 1982) and (Child, 1993), where normal (extended) to local (localized) mode bifurcations were investigated in a two-DOF model of nonlinearly interacting molecules. In an interesting recent work Achong (1996) investigated theoretically and experimentally the vibrations of a steelpan musical instrument. He showed that nonlinear internal resonances influence the steelpan's dynamic response, leading to numerous nonlinear resonances and to localized (spatially confined) NNMs. In this chapter the concept of NNM is applied to the study of nonlinear mode localization in periodic assemblies of discrete oscillators. In section 7.1 some general qualitative results regarding the existence and stability of NNMs and nonlinear mode localization in weakly coupled oscillators are presented. In section 7.2 a quantitative analysis of localization of NNMs in unforced periodic systems with or without cyclic symmetry is given, whereas in sections 7.3 and 7.4 the forced responses of systems possessing
7.1 WEAKLY COUPLED OSCILLATORS: QUALITATIVE RESULTS
289
localized modes are considered; in these sections it is analytically and numerically proven that nonlinear mode localization gives rise to a variety of steady-state and transient forced localization phenomena, which can be employed in the development of refined vibration and shock isolation designs of mechanical structures.
7.1 WEAKLY COUPLED OSCILLATORS: QUALITATIVE RESULTS In engineering applications one often encounters repetitive systems composed of a number of identical or nearly identical substructures coupled by means of coupling elements. In the following two sections general theorems on the existence and stability of NNMs in weakly coupled repetitive systems will be given. It will be shown that a subset of these modes are strongly or weakly spatially localized to certain regions of the repetitive system. The implications of the nonlinear mode localization phenomenon on the free and forced dynamics of repetitive mechanical structures are discussed in later sections. 7.1.1 Existence and Stability of Periodic Solutions Consider a repetitive system consisting of r weakly coupled subsystems. Considering subsystem k, k = 1,...,I-,denote by nk the number of DOF, by xk the (nk x 1) vector of local generalized coordinates, by Mk the inertia matrix, and by Vk(& the potential function. The equations of motion of the repetitive system are expressed as follows: Mk $
+ fk(xk) + &Qk(x1,...,xr) = Q,
k = 1,...,r
(7.1.1)
where fk and $k are (nkxl) vectors denoting grounding and coupling stiffness forces, respectively, and E is a small parameter indicating weak coupling between substructures. In particular, for nl =...= nr = 1 equations (7.1.1) describe a set of r-coupled single-DOF oscillators. Suppose that for E = 0 each subsystem possesses a periodic solution x k =xk(t) of period T. Without loss of generality it is assumed that xk(t) is an even solution, i.e., that xk(t) = xk(-t) (this can always be achieved by an appropriate time shift,
290
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
cf. chapter 2). The periodic (degenerate) solution of the uncoupled system is expressed as follows:
x(t) = x(t;s) = [xl(t + slT/2),...,xr(t + srT/2)]T
(7.1.2)
where, s = [sl,...,sr]T, integers sp take the values 0 or 1, and, with no loss of generality, it is assumed that s l = 0. Hence, the total number of degenerate periodic solutions x(t;S) of the uncoupled repetitive assembly is 2r-I. Suppose that for E = 0 the set of variational equations corresponding to system (7.1.1),
Mk ji
+ Ak(t)y = 0,
Ak(t) = Ak[xk(t)] = V&'[xk(t)] k = 1,...,r
(7.1.3)
possesses for each k a single T-periodic solution tk(t). Then, the multiplicity of the multiplier h = 1 of (7.1.3) equals to two, and for small energy differences Ih - hkl [where the energy level hk corresponds to the solution xk(t)] each subsystem possesses a one-parameter family of periodic solutions xk(t,h) with period Tk(h) such that xk(t,h)+xk(t) and Tk(h)+T as h+hk. Using arguments similar to those employed in chapter 2 it can be shown that dTk(h)/dh f 0. The sign of dTk(h)/dh defines the type of anisochronicity at a fixed level of energy hk; namely, for dTk(h)/dh greater or less than zero, each subsystem is softly or hardly anisochronic at some energy interval containing hk. The following theorem establishes the existence of a T-periodic solution x(t,E) = [xl(t,&) ,...,xr(t,&)]T for the weakly coupled system, close to each of the generating T-periodic solutions (7.1.2) of the uncoupled system. Theorem 1. For E sufficiently small there exists a unique T-periodic solution &(t,E) of (7.1. I), such that x(t,&)+x(t)as &+O.
Proof. The initial conditions KO(&) corresponding to an even solution
x(t,Xo,&)are determined by the equation ~ ( T / ~ J O ,=E 0. ) Using the implicit function theorem, it can be shown that SO(&) is uniquely continuable in E for small E [otherwise, equations (7.1.3) possess for some k an even periodic solution yk(t) that is linearly independent of &k(t); this, however, contradicts the above supposition].
7.1 WEAKLY COUPLED OSCILLATORS: QUALITATIVE RESULTS
29 1
Note that T is not necessarily the minimal period of each periodic solution of the kth uncoupled substructure, xk(t), k = 1,...,r; generally, the minimal period of this solution is equal to (T/qk) where qk is an integer. If each subsystem possesses a single DOF and q1 =...= qr = 1, the periodic solution x(t,&)of the weakly coupled system describes synchronous oscillations of all positional variables of the system, i.e., an NNM oscillation. In that case, Theorem 1 proves that for sufficiently small coupling the repetitive system possesses 2r-1 NNMs, which for r > 2 exceed in number of DOF, n = r. Weakly and strongly localized NNMs can be studied by selecting a subset of trivial generating solutions &J(t)= 0, j e [1, ...,r] for the subsystems. This is performed in section 7.1.2. In this section only nonlocalized NNMs are considered by requiring nonzero generating solutions, d ( t ) f 0, b' j e [ I , ..., r]. The stability of the periodic solutions &(t,E)is investigated by analyzing the corresponding system of variational equations:
+ A(t,E)Z = 0
(7.1.4)
where
M = diag(M1,...,Mr),
A(t,E) = A[&(t,E)] r
and ED(&) represents the part of the potential energy due to the coupling elements & & k ( x l ,...,XI), k = 1,...,r, in the equations of motion (7.1.1). Introducing a canonical transformation of variables, system (7.1.4) is expressed in the following canonical form:
where, y(t) = [yl(t) ,...,yzn(t)]T, n
=cnk is the total number of DOF, and k
matrices J and I-I(t,E) are defined by: (7.1.6)
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NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
In (7.1.6), In denotes the unit matrix of order n. At this point the following notation is introduced:
Note that the (nk x np) matrices k p ( t ) (k f: p) depend only on the coupling forces between the kth and pth subsystems, and are determined directly by the generating solution x(t,O); matrices &(t) depend additionally on the derivative &(t,E)/&. Define now the following matrix:
D = [ d k ~ I k , ~,.I,, = l,., dkp = (&cp(t>&P,&k), P f k dkk=
(-dkp)
( p = l ,...,r, p#k)
(7.1.9)
Let A k and &L = (ukl,...,ukr)T be the eigenvalues and eigenvectors, respectively, of matrix II (k = 1,...,r; Ak I &+I). Since h p ( t ) = BkpT(t), matrix Q is symmetric and its eigenvalues are real. Moreover, by the last equality (7.1.9) it follows that there exists a zero eigenvalue Ap = 0 with the associated eigenvector up= (1, ...,1)T. Suppose at this point that all subsystems are softly or hardly anisotropic [all dhk/dT = l/(dT/dhk) are positive or negative, respectively] and denote by D, the determinant of the submatrix Dq = [dkp]k,p=l,,,,,9, q E [ I ,...,r-l]. The following stability theorem holds. Theorem 2. The periodic solution x(t,E) of the weakly coupled system is stable if Dq > 0 for dhk/dT < 0 or Dq < 0 for dhk/dT > 0 for q = l , ...,r-1. If D, < 0 for dhk/dT < 0 or Dq > 0 for dhk/dT > 0 for some q E [ l,...,r-11, the solution z(t,E) is unstable. Proof. Note that the T-periodic solution q(t,E) possesses at least two characteristic multipliers equal to unity. Hence, the solution is considered to be stable to a first approximation if the remaining multipliers lie on the unit
7.1 WEAKLY COUPLED OSCILLATORS: QUALITATIVE RESULTS
293
circle in the complex plane; otherwise, the solution is unstable. The proof of the theorem relies on the use of Sylvester's criteria (Bellman, 1960) and is not given here. Note that the proof also follows from the integral stability criteria for weakly connected systems (Blechman, 1971). For a more general case (where some of the values dhk/dT have different signs), the reader is referred to (Nagaev, 1965; Blechman, 197 1) and references therein. From the stability theorem, one notes that there exists the possibility of systems with no stable periodic solutions. To illustrate this result, consider a weakly coupled symmetric system with linear connecting stiffnesses between subsystems. Since in this case the matrices &p are constant and i P ( t ) = -@(t + T/2) [the degenerate solutions of the uncoupled system are given by (7.1.2)], a variation of a parameter sk in (7.1.2) leads to a sign reversal of the elements dqk and dkq (q = 1, ...,r, q # k), and thus, to a variation of the elements dqq. For example, suppose that r = 3 , dhk/dT < 0, and s2 = s3 = 0 in (7.1.2). Then corresponding matrix D is evaluated as follows: D= -
[
5 -10 5 -10 16 -6 5 -6 1 1
(7.1.10)
It is easy to show that D2 = -20 < 0, and the corresponding NNM of the weakly coupled system is unstable. For the remaining NNMs [corresponding to the generating solutions (7.1.2) with ( ~ 2 3 3 )= (l,O),(O,1),( 1,l)], it can be shown that some diagonal elements dkk become negative, a fact which indicates instability by Theorem 2. Hence, for sufficiently small values of the coupling parameter E the coupled system under consideration possesses no stable NNMs. As an additional example of application of the stability theorem, consider a weakly coupled system with nl =...= n r = 1 and q1 =...= q r = 1, i.e., composed of r single-DOF oscillators. Let T be the minimal period of the even periodic solution xk(t) of the kth oscillator of the uncoupled system. Without loss of generality it is assumed that xk(0) > 0 and xk(0) < 0 for t E (O,T/2), k = 1,...,r. The 2r-1 periodic solutions of the coupled system describe NNM oscillations with all positional variables varying monotonically between their extreme values. Suppose, first, that the coupling
294
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
U
Figure 7.1.1 Nonlinear force-displacement relation of a nonmonotonic coupling stiffness. forces between subsystems are monotonic. Suppose that dhk/dT < 0 for all k, i.e., that the coupled system consists of weakly connected hardly anisochronic oscillators; the stability theorem then predicts that the NNMs of the coupled system are stable if all neighboring masses oscillate in phase. If, on the contrary, dhk/dT > 0 for all k (system of softly anisochronic oscillators), only NNMs corresponding to antiphase motions between neighboring masses are stable. For example, oscillations of weakly coupled pendula are stable only when neighboring pendulums rotate in opposite directions. If, in addition, the nonlinearities of the system are also concave, then, as shown in chapters 2 and 4, the antiphase NNMs exist and are stable for any value of the coupling coefficient E. When the coupling forces are not monotonic functions of the displacements, stable periodic solutions may not exist. To illustrate this (perhaps surprising) result, consider a system of two identical oscillators with coupling force characteristic f 12(u) as in Figure 7.1.1. Since dfi2(0)/du < 0, for in phase oscillations (sl = s2 = 0) one finds that d12 > 0 [f12 = f12(xi - x2)l. Clearly, if the interval (cl,c2) is sufficiently small, then for antiphase oscillations (si = 0, s2 = 1) it is also satisfied that d12 > 0; therefore, if the uncoupled oscillators of the system (corresponding to f12 = 0) are hardly anisochronic, the stability condition dl 1 > 0 is not satisfied (dl 1 = -d12) for both the in phase and antiphase NNMs, and both these modes are unstable. For softly anisochronic uncoupled oscillators, the in phase and antiphase NNMs are stable.
7.1 WEAKLY COUPLED OSCILLATORS: QUALITATIVE RESULTS
295
7.1.2 Nonlinear Mode Localization The spatially localized periodic solutions of the weakly coupled system (7.1.1) are now investigated. Assuming that E is sufficiently small, Theorem 1 of section 7.1.1 guarantees the existence of 2r-1 periodic solutions x(t;s,E) neighboring the solutions x(t) = x(t;s) = [xl(t + siT/2),...,xr(t + srT/2)]T of the uncoupled system, with coefficients sk assuming the values of 0 or 1; moreover, it is satisfied that x(t;s,E) + x(t;s) as E+O. In section 7.1.1 the energies corresponding to the generating solutions xk(t) were assumed to be nonzero for k = 1,...,r, so that only nonlocalized periodic solutions of the weakly coupled system were examined. In this section generating solutions x(t) with xk(t) = 0 for some k are considered. For sufficiently small E , Theorem 1 guarantees that the coupled system possesses the localized periodic solution x(t;s,&)= [x'(t + s l T / 2 , ~ )...,xr( , t + srT/2,&)]T,such that the condition xk(t + skT/2,E) + 0 as E+O for some k, is satisfied. Hence, from an asymptotic point of view the existence of localized periodic solutions in the coupled system should present no problems. However, from a practical point of view it needs to be ascertained that these localized oscillations remain localized as the coupling parameter E increases to a given value EO. In what follows both strongly and weakly localized periodic solutions are examined. Strongly localized solutions correspond to localization of vibrational energy to a single subsystem, whereas weakly localized ones to a set of subsystems. As a first step, strongly localized solutions are considered, and the equations of motion (7. I . 1) are expressed in the following form:
z + fo(z) + EQO(ZJ) = 0
(7.1.1 1) where
x = (xl ,...,xn-l)', f(2i) = [ f l ( ~ > , . . . A - l ( 2 i ) I ~ Q(~,L> = [Q1( z , ~ ) .,.,$n. 1(z7x>IT
and the coupled system is assumed to possess n DOF. For E = 0 the system is decomposed into a single-DOF nonlinear oscillator with positional variable Z E R (termed "oscillator 1") and an (n - 1)-DOF system with positional variables ZG Rn-1 ("oscillator 2"). Assuming symmetric nonlinearities fO@) = -fo(-z), f(x) = -f(-x)
296
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
Qo(z,zr>= -QO(-Z,-Zr>, @(z,x, = -@(--G--X) one considers an odd, T-periodic symmetric solution [z(~,E),x(~,E)] of the coupled system, such that z(t,&)+ zo(t) and K(t,E) Q as E+O, where zo(t) is a T-periodic solution of oscillator 1. Suppose that this solution is continuable in parameter E to a certain value EO. The solution will be termed strongly localized for E = EO, if X(t,EO) E R,where Rn ZI R is a given small neighborhood in the space (xi, ...,xn-1). Moreover, it is assumed that if & E R, then also kz E R for k E [0,1). The aim of the following analysis is to derive sufficient conditions that guarantee that the solution x(t,EO) is strongly localized. Bounds for the amplitude A(T) of the solution z(t,&)when x(t,E)E R and E E (O,EO] are now established. Let $-W = min,,~ E$O(Z,X) and @+(4= maXx,E E$O@,X) for 2~ E R and E E (O,EO] and consider the equation: v
dv
& + fo(z) + $(z)
=0
(7.1. 12)
Denote by v-(z,A) and v+(z,A) [where v*(A,A) = 0, z 5 A] the positive solutions of (7.1.12) with $(z) = $-(z) and $(z) = $+(z), respectively. Integrating (7.1.12) one obtains v+(z,A) = 2[Fo(A)
+ @*(A) - Fo(z) - @*(~)]1'2
(7.1.13)
where Z
z
0 0 The phase trajectory v(z,A) of the solution z(t,E) of amplitude A satisfies equation (7,1.12) with $(z) = E$O(Z,X) [here it is assumed that x=~ ( z ) ] By . (7.1.13), it holds that:
A, v-(&A) I v(z,A) I v+(z,A), for z I
v(z,A) 2 0
Suppose that v-(z,A) > 0 for 0 I z < A. Introducing the notation,
(7.1.14)
7.1 WEAKLY COUPLED OSCILLATORS: QUALITATIVE RESULTS
A T(A) = 4
5
297
A and T+(A) = 4
v-'(z,A) dz 0
v+-l(z,A) dz
0
from (7.1.14) one obtains that: T+(A) < T(A) < T-(A)
(7.1.15)
Let A+(T) be the inverse functions of T+(A). If these inverse functions exist when the period is equal to T, the quantities A+(T) and A-(T) may serve as bounds of the amplitude A(T), provided that the solution satisfies X(t,E) E
n for
[A-(T) < A(T) < A+(T) or A-(T) > A(T) > A+(T) when A+(T) increase or decrease, respectively]. Let E E (O,EO]
A*(T) = max(A+(T)], n- 1
and
Ifp(&)lI
k= 1
Cpklxkl
rp = maxx,z I@p(z,x)l
for p=l, ...,n-1,
lzl 5 A*(T),
xE R
In accordance with the integral mean value theorem, the vector of functions
f(x) can be represented in the following form: (7.1.16) where
The coefficients Cpk in the previous expression for the bound of Ifp(x)l can be chosen according to Cpk = maxxE52 Icpk(x)l. Denote by &+(t) the solution of the following boundary value problem:
x+
+ EOL = 0,
~ ( 0 =) t(T/4) = 0
(7.1.18)
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NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
where
r. = (ri,...,rn-i)T , C =
[Cpklk,p=l,,.., n-l. Employing the previous
derivations it can be shown that Cpk 2 0, so that matrix C is nonnegative. It follows that the largest in modulus eigenvalue of C is real and positive; this eigenvalue is denoted by w*2. The following theorem provides sufficient conditions for the existence of the strongly localized solution in the parameter range E E (O,EO].
3. If w * < 27t/T and x + ( T / 4 ) [z(~,Eo),x(~,Eo)] of (7.1.11) is strongly localized.
Theorem
E
R, the solution
Proof. First it is shown that x+(t) > 0 for t E (O,T/4]. Consider (7.1.18) by replacing C with hC, and examine the behavior of the corresponding solution x+(t;h) as h increases from 0 to 1 [such a continuation is possible since the largest eigenvalue of matrix h c is equal to m*2(h) = hm*2 < (2n/T)2, so the homogeneous problem possesses only the trivial solution]. Clearly, x+(t;O) 2 0 for t E (O,T/4]. The inequality x+(t;h) > 0 for t E (O,T/4] is proved by contradiction; suppose that it does not hold for some h
h'; then &+(t;h') 2 0, xk+(t';h') = 0 or i(o)k+(o;h') = 0 for some k and t' E (O,T/4]. This, however, cannot hold, since by (7.1.18) xk(t) < 0 for t E (O,T/4] provided that x(t) 2 0 (note that the elements Cpk of matrix C are nonnegative). In view of the inequality jif(t) < Q for t E (O,T/4], one
=
concludes that X+(t) increases monotonically for t E [O,T/4]. Next, taking into account (7.1.1 l), (7.1.18), the definition for A*(T) and the previously
derived bounds for QP(z,x) and fp(x),one finds that Ig(t;&)I< -h+(t) for Ix(t;&)I< x+(t) for IzI 5 A*(T), E 5 Eo. Taking into account that &(t;O) = x+(O) = i(T/4,&)= &+(T/4) = Q
one finds that the following inequality holds:
Ili(t;&)I < x+(t) I x+(T/4)
(7.1.19)
It follows that X(t,&) E Q for & E (O,&0] and A(&)I A*. However, as shown above the equality A(&)= A* can not be realized, and thus A(&)< A* for x(t,&)E R and the theorem is proved. It is noted that system (7.1.1 1) was not assumed to be conservative, and so, the existence o r not of a potential function bears no relation to the localization phenomena under consideration. If all eigenvalues of matrix C are positive, the solution x+(t) can be expressed in the form:
7.1 WEAKLY COUPLED OSCILLATORS: QUALITATIVE RESULTS
299
where 0 k 2 and a are the eigenvalues and eigenvectors, respectively, of matrix C; the constants ak in (7.1.20) are determined by the equation n- 1
where vector L was defined earlier. The validity of expression (7.1.20) is verified by direct substitution in (7.1.18). From (7.1.20) the vector x+(T/4) is computed as follows: (7.1.21) If the system considered is conservative, matrices A(x) and C(x) are symmetric. In that case, the eigenvectors 2 ~ ksatisfy the orthogonality condition (a,xp)= 0, for k # p. Normalizing according to (a&) = 1, the coefficients ak are computed as ak = EO(I,U). Hence, for conservative systems the condition for strong localization can be easily checked. It is interesting to note that under certain conditions the one-parameter family of localized solutions can be extended up to arbitrarily large values of EO. Sufficient conditions for the existence of such extensions will now be established; since these conditions will not depend on the value EO, i t is assumed at this point that E = 1. Consider system (7.1.1 1) and set
Denote by (A, A l , ..., An-1) the amplitudes of the components of the solution vector [z(t),x(t)l = [z(t), xi(t), ..., ~n-l(t)].The following theorem is proved. Theorem 4. Suppose that W(Z)/Z+ w and z2$k(z,Q)/Y(z) + 0 as z + 00, k = 1,...,n-1. Then, for sufficiently small periods T system (7.1.11) possesses an odd T-periodic solution [z(t,T),x(t,T)] such that
300
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
the corresponding amplitudes satisfy the limiting solutions, A(T) and Ak(T) + 0 as T + 0, k = 1,...,n-1.
+
00
Proof. Consider the following equations:
+ T ~ w ( z J )= 0 X" + pT2[f(x) + Q(z,x)] = 0 Z"
(7.1.22)
where ~ ( Z J ) = fo(z) + E$J(z,x), primes denote differentiation with respect to nondimensional time z = t/T, and p is a parameter. Clearly, equations (7.1.22) reduce to (7.1.1 1) for p = 1. For p = 0 and T < 2x/oo, where 0 0 2 = dy(O,Q)/dz, equations (7.1.22) admit the periodic solution of unit period, (zo(z) = -zo(-T),L(T) = 0). Let A = zo(1/4) be the amplitude of zo(z); then the amplitude-period relation for this system is given by: 1 T(A) = 4 A j
{ 2[Y(A) - Y(sA)]}-1/2 ds
(7.1.23)
0 Since the frequency o = 2 d T > W I for sufficiently small T, it follows that for small p system (7.1.22) possesses the 1-periodic solution [z(T,~),x(T,~)], such that z(z,p) + zo(z) and z(z,p) + 0 as p + 0. In sufficiently small regions R of Rn-1, this solution is continuable in p as long as x(z,p) E Q. Employing the limiting assumptions of the theorem and expression (7.1.23). for x E R it is satisfied that: T2(A)[fk(x)
+ $k(A,&)] + A2$k(A,zr>ck/'Y(A) + 0
as A
+
00
k = 1,...,n-I (7.1.24) where Ck are constants. Hence, for sufficiently large values of A, or equivalently, for sufficiently small periods T, the term
in (7.1.22) becomes arbitrarily small, and there exists an interval T E (O,T*] where the solution X(t,p,E) lies within R for p E [0,1]; thus, within that range of T the localized solution is continuable in p up to p = 1. So, for T E
7.1 WEAKLY COUPLED OSCILLATORS: QUALITATIVERESULTS
301
(O,T*] equations (7.1.11) possess the T-periodic solution [z(t),x(t)], and, as seen from the previous discussion, the corresponding amplitudes satisfy A(T) + 00 and &(T) + 0 as T 0, k = 1,...,n-1. The theorem is proved. The stability of the strongly localized solution [z(~,E),x(~,E)] can be investigated by assuming that for E = 0 the generating solution x(t) = Q is stable, and analyzing the corresponding set of variational equations: y + B(t,E)Y = 0,
B(t,E)
= l3(t
+ T,E)
(7.1.25)
One examines first the degenerate uncoupled system. When E = 0 system (7.1.25) decomposes into the following set of uncoupled variational eauations:
v + A(Q)y= 0
-
(7.1.26)
where u = y 1 and y = (y2,...,yn)T are partitions of the variational vector y, and zo(t) = z(t,O) is the T-periodic solution of the uncoupled oscillator where strong localization takes place. Since the first of equations (7.1.26) possesses the T-periodic solution zo(t), it also possesses a pair of multipliers equal to unity. The multipliers of the second equation are given by h k = exp(jCokoT) and hk+n-l = exp(-jokOT) where o k o 2 , k = 1,...,n-1 are the eigenvahes of matrix A(O), and j = (-1)1'2. The variational equations (7.1.25) of the weakly coupled system are now considered. Since system (7.1.11) is autonomous, the variational equations possess a critical multiplier equal to unity, hi(&)= 1. The strongly localized solution [z(~,E),x(~,E)] is orbitally stable to a first approximation if the remaining multipliers of the variational system (7.1.25) lie on the unit circle in the complex plane. Since the stiffness forces of the system are symmetric [cf. previously imposed symmetry conditions on the restoring forces of system (7.1.1 l)], it is satisfied that B(t,&) = B(-t,&), and, thus, system (7.1.25) is invariant with respect to the temporal transformation t + -t. It is well known (Yakubovich and Starzhinskii, 1975) that such a system possesses ILL). It follows that under an reciprocal pairs of characteristic multipliers (1, increase in E the characteristic multipliers move in pairs, a feature that leads
302
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
to the conclusion that if all multipliers lie originally on the unit circle the only possible way for the occurrence of instability is the coincidence of pairs of multipliers on the unit circle. This result, coupled with the fact that the characteristic multipliers hk(&)depend continuously on E, indicates that if the characteristic multipliers of the uncoupled system (7.1.26) lie on the unit circle and are distinct [with the exception of the critical multiplier hl(0) = 11, then the characteristic multipliers of (7.1.25) also lie on the unit circle and are distinct, a feature which proves orbital stability of the localized solution. Hence, to a first approximation, the conditions,
where q is an integer, are the conditions for stability of the strongly localized solution for sufficiently small &. If system (7.1.11) is conservative, then matrix €3(t,&) is symmetric. In that case, according to Krein's theory (Krein, 1955) the multipliers pk(&) can be classified into those of first and second kind, h k ( l ) ( & )and h k ( 2 ) ( & ) = l / l . k ( i ) ( & ) , respectively (cf. chapter 4). When met on the unit circle, multipliers of the same kind remain on it under a small change in E. Therefore, when the coupled system is conservative the strongly localized solution is stable for small E provided that there are no coinciding multipliers of different kind for E = 0. In this case the stability condition for the strongly localized solution simplifies to: wkoT + opoT f 27cq;
k,p = 1,...,n-1
(conservative system) (7.1.28)
Typically, conditions (7.1.27) and (7.1.28) are satisfied. Hence, as a rule, a strongly localized periodic solution ( N N M ) is stable f o r sufficiently small coupling. The above stability conditions are violated in cases of internal resonances where complicated interactions between NNMs occur. Analysis of NNMs in systems with internal resonance were carried out in section 4.4. It is also noted that when the conditions of Theorem 3 are fulfilled, one can derive bounds for the strongly localized solution, Iz(t,&o)l< A*, &(t,&) E R . Since the previously derived stability criteria are formulated for sufficiently small values of E, the stability of the localized solutions at finite values of E (i.e., in neighborhoods of EO) should be studied via the qualitative and quantitative stability criteria discussed in chapter 4.
7.1 WEAKLY COUPLED OSCILLATORS: QUALITATIVE RESULTS
303
Theorem 4 can be extended to weakly localized solutions, i.e., to oscillations where localization is realized over r coordinates, 1 < r < n. To analyze such motions one rewrites equations (7.1.1) into the following form:
2' + f'(&1,&2) = Q x2 + f2(&1,&2)= 0
(7.1.29)
where ~1 = (xi ,...,xr)T, ~2 = (x,+l ,...,xn)T, fl(&) = (fi ,...,fr)T, f2(x)= (fr+l,...,fn)T, and fk(&',X2) = -fk(-xl,-&2), k=l,...,n. Suppose that the system
xl
+ fl(&l,Q)= 0
(7.1.30)
admits a one-parameter family of odd T-periodic solutions &l(t,T) with corresponding amplitudes satisfying Ak(T) + 00 as T -+ 0, k = 1,...,r. Using similar arguments as before it can be shown that, if the relation T2 f2[xl(t,T),Q] + 0
as T + 0
is satisfied, then, for sufficiently small periods T system (7.1.29) possesses a family of odd periodic solutions [&l(t,T),&2(t,T)] with the component amplitudes of the vectors xl(t,T) and &2(t,T) satisfying the limiting relations Ak(T)
-+
00,
k = 1 ,...,r,
and
Ap(T) + 0,
p=r+l, ...,n
respectively, as T -+0. The stability of the weakly localized solutions can be studied by applying the analytical and numerical techniques discussed in chapter 4. ' Computations of strongly and weakly localized NNMs in systems of coupled nonlinear oscillators are carried out in the following sections. It is shown that the number of localized NNMs may exceed in number the DOF of the systems under consideration. This is because different families of NNMs can be generated or destroyed in NNM bifurcations, which increase in complexity as the DOF of the systems increase. Analytic approximations and bifurcations of different branches of localized NNMs with increasing energy of oscillation can be developed using the techniques presented in
304
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
section 3.1. In addition, NNM bifurcations leading to mode localization can be analyzed using the two-timing asymptotic methodology outlined below. 7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY In the following sections some quantitative techniques for studying localized NNMs in conservative systems with or without cyclic symmetry are presented. The system considered in this section consists of n identical, nonlinear substructures coupled by means of weak coupling stiffnesses (cf. Figure 7.2.1). The governing equations of motion are given by:
where fi(u) is the restoring force of the ith uncoupled substructure, ~ f , ( " )is the weak coupling force acting on the ith substructure, and E is a small parameter of perturbation order (lel << 1). In writing (7.2.1) it is assumed that the ith subsystem interacts only with its two neighboring substructures i-1 and i+l, and that no other interaction between subsystems occurs. The requirement of perfect periodicity is satisfied if
whereas the requirement of cyclic symmetry is met by making the identifications Xn+l = x i , x0 = xn, The restoring functions fi(u) and fi(c) are assumed to possess the following symmetry properties:
With ( l ) , the stiffnesses of the uncoupled substructures are assumed to be symmetric, i.e., to respond to an equal amount to tension and compression. With assumption (2), the total stiffness interaction between three substructures i-1, i, and i+l can be decomposed into mutual individual interactions between neighboring substructures. The individual restoring 304
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
305
1
Figure 7.2.1 The general n-DOF nonlinear system with cyclic symmetry. forces fi(c)(*) are assumed symmetric due to (3). Finally, the functions fi(u) and fi(C) are assumed to be sufficiently smooth and to possess Taylor series expansions about the zero value of their arguments. Under these conditions, the dynamical system (7.2.1) possesses NNMs that are represented in the configuration space of the system by modal curves. According to the classification introduced in earlier chapters, these nonlinear modes can be either similar (with straight modal lines) or nonsimilar (with modal curves). Similar modes can only occur in systems with special symmetries in their potential function and are not generic for the class of systems (7.2.1). Nonsimilar modes are generic for the dynamical systems under consideration, and their modal curves can be computed by the asymptotic methodology developed in section 3.1.2.
7.2.1 Asymptotic Analysis of Modal Curves To study the NNMs of system (7.2.1), the motion of the ith mass, xi, is expressed in terms of the motion of the rth reference mass as follows: Xi
A
= Xi(Xr),
i = 1,..., n,
i#r
(7.2.3)
Hence, coordinate xr is used to paranietrize the nonsiniilar motion. The modal functions f;i(*) (which generally have a nonlinear dependence on xr),
306
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
define the modal curve in the n-dimensional configuration space of the oscillator. As in section 3.1.2, the time derivatives in the equations of motion are expressed in terms of the functions fii(a) and the coordinate Xr, and the following set of (n-1) functional equations governing the modal functions is obtained:
A dCi + fr[;l (xr),.. .,xn(xr)~ = fi[fil (xr)?...A(xr)~
i = 1 ,_.., n, i#r (7.2.4) where i = 1,2,...,n, i f r, h is the total (fixed) energy of the free motion, and V is the potential energy of the system during oscillation. Since the functional equations are singular at the maximum equipotential surface V = h (they possess regular singular points there), it is necessary to develop asymptotic approximations for the modal lines; these approximations will be valid in open neighborhoods of the origin of the configuration space and will be analytically continued to the maximum equipotential surface. Therefore, as in section 3.1.2 it is necessary to impose an additional set of (n - 1) functional boundary conditions that hold at the points of intersection of the modal line with the maximum equipotential surface (Xr is the amplitude of oscillation of the parametrizing coordinate xr):
i = 1,...,n,
i
f
r
(7.2.5)
In what follows, the sets of equations (7.2.4) and (7.2.5) are asymptotically solved using the methodology developed in section 3.1.2. The asymptotic solutions will be developed for the specific set of initial conditions X i ( 0 ) = Xi, ii(0) = 0, i=l,2, ...,n, but the same methodology can be applied for systems with different initial conditions. The modal line (7.2.3) is expressed as:
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
307
where xi (xi) = O(1); for convenience and without loss of generality from now on the parametrizing coordinate is chosen to be Xr = x i . The series expression (7.2.6) is substituted into the functional equations (7.2.4) and (7.2.5), the coefficients of respective powers of E are set equal to zero, and successive approximations to the modal line (7.2.3) are computed.
O ( d ) Approximation The zeroth order approximation is in the form of similar normal modes:
0) xi ( x i ) = a j x l ,
i = 2, ...,n
(7.2.7)
Substituting (7.2.6) into (7.2.4) and (7.2.5) and gathering terms of O( l), one obtains the following relations for the modal constants ai:
At this point the stiffness of the ith uncoupled substructure is expanded in Taylor series about Xi = 0: m
i = 1,2,...,n
(7.2.9)
where the short-hand notation for differentiation, (*),P = dP(*)/dxjP, was used. Expressing fl(u) and fi(u) in (7.2.8) in Taylor series, and taking into account that fl(u) = fi(u) due to symmetry, the following relations for the determination of the modal constants Cri are obtained:
i = 2 ,...,n with solutions:
k =0
* ai = arbitrary,
i = 2, ...,n
(7.2.10)
308
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
k#O*
a i = O , - t 1,
i = 2,...,n
These are the modal constants corresponding to the degenerate structure consisting of n uncoupled identical subsystems, and the corresponding similar modes (7.2.7) are termed degenerate NNMs. When the uncoupled subsystems are linear, the Taylor series (7.2.9) contains only the term corresponding to k = 0 and the modal constants Cri can take any arbitrary real value. In that case, mode IocaliLation can only occur if small imperfections (disorders) are introduced in the periodic system. When nonlinearities exist, expansion (7.2.9) contains nonzero terms with k 2 1, and the values of a i are restricted to the values 0 or + I . For a system consisting of n identical substructures, (3,-1+2,- 1 ) degenerate N N M s exist, corresponding to all possible combinations of coefficients ( a 2 , ..., an).When the weak coupling effects are taken into account [at the O(E) approximation], each degenerate NNM gives rise to an NNM of the system, which depending on the values of coefficients (a2,..., a,) can be strongly, weakly, or nonlocalized. These NNMs are analytically studied by evaluating the firstorder corrections $') in expansions (7.2.6). O ( E ~Approximation )
At this order of approximation, one obtains the following set of functional A(1)
equations for the quantities xi :
(7.2.11)
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
309
where i = 2, ...,n. The quantity V(u) represents the O(1) term of the potential energy V, and the uncoupled restoring force was expressed as fi(u)(aixl + e i ( l ) > = fi(u)(aixl) + E;Zi(l) afi(u)(aixl)/iIxi + 0(&2) Complementing (7.2.1 1) are the following (n - 1) boundary orthogonality relations:
(7.2.12) where i = 2, ...,n. Equations (7.2.11) form a set of (n - 1)-coupled equations in terms of the first-order approximations f;i('); however, a careful examination of the coupling term in (7.2.11) reveals that it is identically equal to zero:
This is because ai fl(u)(xl) - fi(u)(aixl) = 0 for f l ( u ) = fi(u) and ai = 0 or +1. Hence, it is concluded that equations (7.2.I I ) [and apparently (7.2.I 2 ) ] A become uncoupled in the unknowns xi ( I ) , and one can solve each equution separately to obtain the first-order approximations of the corresponding NNMs. To this end, the following Taylor expansions of the coupling forces are performed:
i = 1,2,...,n
(7.2.13a)
310
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
Expressing the quantity V(U) as:
=
p=l
0
c c fp,(2k+l)(u)(0)xp2k+2/(2k+2)! n
m
p=l k=O
(7.2.13b)
and taking into account (7.2.13a), one can express the functional equations (7.2.1 1) and (7.2.12) in the following form:
m
m
(7.2.14a) and
m
(7.2.14b) where i = 2,...,n, and the various coefficients are given by:
The yet undetermined functions ?i(l) are now expressed in the series form:
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
3 1I
$2
where the coefficients are as yet unknown O ( E ) quantities that are determined by substituting (7.2.16) into the functional equations (7.2.14) and balancing coefficients of respective powers of xi. Note that only even powers of xi appear in (7.2.16), a consequence of the symmetries of the problem. Substituting (7.2.16) into the ith functional equation (7.2.14a), the following is derived: relation for the coefficients
2;)
4h(2s + 1)sd;) = 4
c { Ap-laj::-p)[2(s
s- 1
p= 1
- p)
+ l](s - p ) )
(7.2.17) where i = 2 ,...,n, and s = 1,2,... Equation (7.2.17) represents a recursive (1) formula for computing the coefficients ais . Using this expression all unknown coefficients in the series (7.2.16) can be expressed in terms of the (1) first coefficient aiO . This last coefficient can then be evaluated by substituting the series expressions (7.2.16) into the ith boundary functional equation (7.2.14b). Using this methodology, one obtains the following and [which are correct to O(x17)]: analytical expressions for
d:), d:',
4;)
312
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
and quantities Ak, Bk, Cik and Dik, are defined by relations (7.2.15). In section 3.1 it was shown that the asymptotic approximations for the functions k i ( l ) converge in open intervals of the parametrizing coordinate x i E (a,b), which do not contain the limiting values x i = &XI. Combining the previous results, the nonsimilar mode is asymptotically approximated by the truncated series: xi = $i(xl) = a j x l +
(1) 2m+1 + o ( E x ~ ~ M +~ ~2 , ) C Eaimxl m=O
i = 2 ,...,n (7.2.20) Hence, close to each degenerate similar mode ^xi(') = aix 1, a nonsimilar mode exists with modal curve given by (7.2.20). Using the derived asymptotic approximations for the modal curves, one can compute the time response of the Parametrizing coordinate X I = x i ( t ) and the frequency of free oscillation of the nonsimilar mode by substituting (7.2.20) into the equations of motion (7.2.1) and integrating by quadratures. The following remarks can be rriade with regard to the computed nonsimilar NNMs.
( 1 j The derived asymptotic results hold only when the coupling between substructures is weak. Moreover, the coefficients &) in the expressions of the modal curves depend on the amplitude of oscillation X i , and thus, on the total energy of the motion. (2) Suppose that Dik = 0, for some i,k = 0,1,2 ,..., in (7.2.14)-(7.2.17). Then the functional equations of the first approximation (7.2.14a,b) become homogeneous in fii(l) and possess only the trivial solution ki(') = 0, i = 2, ...,n. This result can be obtained by examining the form of the recursive relations (7.2.17). It follows that if Dik = 0 for some i, the ith component of the modal curve is
A necessary and sufficient condition for Dik = 0 is that [cf. expressions (7.2.15)1:
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
3 13
Taking into account that Fi(c)(xl; an,al,a2) = fi(c)[(ai- ai-l)xl] + Pi(c)[(ai - ai+l)xl] (where the definitions of Fi(c) and Pi(') were considered), condition (7.2.21b) is expressed as:
Hence, provided that equation (7.2.22) is satisfied, no O(&)terms exist in the ith component of the modal curve. (3) Suppose now that ai = 0 for i = 2, ...,n. Then equation (7.2.22) is satisfied for i = 3, ..., n-2, and the asymptotic analysis predicts the following motions for the corresponding nonsimilar NNM: X I = O(I),
xn = O(E), xi = O(E') i = 3 ,...,n-2 (7.2.23) Hence, it is proven that the perfectly tuned, weakly nonlinear, cyclic structure possesses a strongly localized nonsimilar NNM corresponding to ,finite motion of only one mass; the remaining masses of the system oscillate with amplitudes of at least O(&). It can be shown that this localized mode exists only in the presence of nonlinearities. When no nonlinearities exist mode localization can occur only when the cyclic periodicity is perturbed by weak structural disorders. From expressions (7.2.23) it is also noted that the coordinates neighboring the 0(1) coordinate oscillate with amplitudes of O(E) and that the remaining coordinates vibrate with amplitudes of at least 0 ( & 2 ) . (4) In addition to the strongly localized nonsimilar mode (7.2.23), there exist additional weakly localized modes corresponding to finite motions of two or more masses, with the remaining masses vibrating with amplitudes of at least O(E). In the following special cases condition (7.2.22) is satisfied, leading to trivial O(E) solutions ?i(l) = 0, and to NNMs of the form Xi = ajxl
+ O(E*):
~2 = O(E),
(a) ai = 0, for some i, and ai-1 = ai+l= 0, or ai-1 = -ai+l (b) ai = +1, for some i, and ai-1 = a n , ai+l= a2, or ai-1 = a', a i + l = an
314
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
(c) ai = -1, for some i, and ai-1 = - a n , ai+l= -a2, or ai-1 = -a2, aI+l = -an
It is noted, however, that only a small subset of these NNMs is expected to be orbitally stable and physically realizable. As far as the stability of the nonsimilar NNMs is concerned, since they are Lyapunov unstable (their frequency of free oscillation depends on the amplitude of the motion), one can expect at most orbital stability. The stability of the NNMs is investigated by perturbing the nonsimilar periodic solutions with small variations:
where xi*(t) is the time response of the ith mass when the system oscillates in the nonsimilar mode (7.2.20). Substituting (7.2.24) into the equations of motion (7.2.1) and retaining only terms of O(cj), the following n variational equations result:
(7.2.25) where i = 1, ...,n. This is a set of n coupled, linear differential equations with periodic coefficients and their stability can be determined by using concepts from Floquet Theory (Nayfeh and Mook, 1984). Note that since no general solutions exist for the set (7.2.25), the stability analysis must be performed on a case by case basis.
Table 7.2.1 Degenerate modes of the weakly coupled system (n = 3)
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
3 15
Figure 7.2.2 The three degree-of-freedom system with cyclic symmetry and cubic nonlinearities. As an application of the previous general analysis, consider the cyclic system with n = 3 degrees of freedom and cubic stiffness nonlinearities (cf. Figure 7.2.2). Note that this system is strongly nonlinear since the nonlinear terms of the substructures are of O(1). When E = 0 (no coupling between masses), there exist 13 degenerate similar NNMs, which due to the cyclic symmetry are reduced to only five. Referring to Table 7.2.1, the degenerate NNM 1 gives rise to strong mode localization [where only one of the masses vibrates with O( 1) amplitude], whereas degenerate mode 2 gives rise to weak mode localization with finite vibrations of only two of the three masses. Mode 3 gives rise to a nonlocalized nonsimilar mode with all masses oscillating with O( 1) amplitudes. The similar modes 4 and 5 will be shown to be exact solutions of the coupled periodic system. Strong Mode Localization (NNM 1 ) The degenerate mode 1 with modal constants a 2 = a 3 = 0 gives rise to a strongly localized nonsimilar mode where x i takes values of O( I ) , whereas x2 and xn are of O ( E ) . The localized nonsimilar modal curve can be computed using the asymptotic formulas derived earlier as follows: A
A
x2(x1) = x3(xl) = E
(R~+E~X:)(~+~EIX?)+~R~ ---
-
2
2
E l X 1(6+3ElX1)
3 16
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
(7.2.26)
For a definition of mnj2 and E, see Figure 7.2.2.
Weak Mode Localization (NNM 2 ) The modal curve of the nonsimilar N N M neighboring degenerate mode 2 (corresponding to a 2 = 1, a3 = 0) is computed as:
(7.2.27) This mode is localized in only two of the masses of the system.
Nonlocalized NNMs 3,s and Weakly Localized NNM 4 For a 2 = 1, and a 3 = -1, a nonlocalized nonsimilar N N M exists with modal curve:
A direct computation shows that the zeroth order degenerate similar N N M s 4 and 5 are exact similar modes of the weakly coupled periodic
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
317
system, i.e., they uncouple the three nonlinear equations of motion. Therefore, it is found that the weakly coupled periodic system possesses two nonlocalized NNMs 3 and 5 , and a weakly localized NNM 4. NNMs 4 and 5 are similar, a feature that is predicted by the asymptotic analysis, which provides identically zero corrections f;i(l) for these modes. The stability of the identified nonsimilar modes is now examined. At this point, it is assumed that the nonlinearities of the individual substructures are of perturbation order, by imposing the rescaling el + E E ~ This . assumption is necessary for the averaging analysis that follows. The variational equations for the system under consideration assume the form:
where i = 1,2,3, xi* is the time response of the parametrizing coordinate x i during the mode under investigation, ai are the modal constants of the zeroth order approximations, and it is assumed that 50 = 53, 5 4 2 51, and CXI = 1. An approximation for the periodic function xl*(t) can be obtained as:
*
x "(t) = p
COSWt
+ O(EP2)
(7.2.30)
and p is the approximate amplitude of motion. Introducing the new time variable T = at, the variational equations (7.2.29) are expressed as:
where i = 1,2,3, ci = {i(T), (*)"= d2(*)/dT2, and
3 18
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
with the understanding that A34 = A31 and A10 = Ai3. Employing the method of multiple scales (Nayfeh and Mook, 1984), it can be shown that the variations ci(7) are approximately given by: E,i(Z) = Bi(T1)eJTo + cc
+ O(E),
i = 1,2,3
(7.2.33)
where T I = &z and To = z are the slow and fast time scales, respectively, (cc) denotes the complex conjugate, and j = (-1)1/2. The complex amplitudes Bi(T1) are determined by eliminating the secular terms in the O(E) approximation (Nayfeh and Mook, 1984). Noting that x1*2= (p2/4)(2 + e*J@+ e-2J@) one obtains the following set of differential equations in terms of the complex amplitudes Bi(q):
where (*)C denotes the complex conjugate, and denotes differentiation with respect to the slow time variable T i . Equations (7.2.34) are a set of linear ordinary differential equations with constant coefficients and complex dependent variables Bi. Writing the complex variables in terms of their real and imaginary parts, Bi = Bir + jBim, and expressing the real and imaginary parts as Bir = bire’YT1, and Bi, = bi,e’YT1 (Nayfeh and Mook, 1977), one obtains the following set of real, simultaneous, homogeneous, algebraic equations in terms of quantities bi, and him: (*)I
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
1 is a (6 x 6)
where [*IT denotes that transpose operation, and matrix of coefficients, given by:
3 I9
constant
(7.2.36)
and Eij are (2 x 2) matrices with analytic expressions: 20-2R2-(3/4)Ai$’ -2Y R2-( 3/4)AijP’
i = 1,2,3
i j = 1,2,3, i
f
j
(7.2.37) For nontrivial solutions of (7.2.35) one requires that:
where the coefficients Ij, j = 1,2,3 were obtained using the symbolic manipulation package Mathematics. The nature of the solutions of (7.2.38) for the exponents y determines the stability or instability of the solutions of the variational equations. In view of the structure of equation (7.2.38), at most one can achieve neutral stability corresponding to purely imaginary exponents y. Employing the outlined methodology, it was found that the strongly localized NNM 1 and the weakly localized NNM 2 are stable, whereas NNMs 3 and 4 are orbitally unstable. As far as nonlocalized NNM 5 is concerned, the linearized stability analysis gives inconclusive results, since all exponents y are zero for this mode. The stability indeterminacy of this mode is due to the degeneracy of the symmetric system under consideration, and by using a nonlinear stability analysis [with numerically computed Poincare’ maps (Month and Rand, 1980; Vakakis, 1990)] it can be shown that NNM 5 is orbitally stable. An interesting remark of the outlined stability analysis is that only the leading terms of the asymptotic expansions for the nonsimilar modal curves were taken into account [i.e., no terms involving
320
NONLINEAR LOCALEATION LN DISCRETE SYSTEMS
the O(r) approximations aij(') are involved in the stability analysis since these corrections give rise to terms of at least O(r2) in the variational equations]. Hence, the stability of the nonsimilur N N M s under consideration is nzainly depended by the structure of the neighboring similar (degenerute) nzodrs, a result in agreement with the conclusions made by Rosenberg (1966), who examined nonlocalized nonsimilar modes using a different perturbation methodology. In Figures 7.2.3-7.2.6, numerical simulations of the N N M s of the system of Figure 7.2.2 are presented. These results were obtained by numerically integrating the equations of motion (7.2.1) with initial conditions corresponding to the theoretically predicted NNMs. For a system with parameters w n j 2 = I , s o n 2 2 = 0.025, € 1 = 0.5, E E =~ 0.01, and X I = 1.0, the strongly localized NNM 1 possesses the theoretical modal curve: ft2(xl) = $3(xl) = -0.083333~1 + 0.003333~13+..., x i = O( 1)
(7.2.39) The numerical simulations of Figure 7.2.3 verify the existence and the stability of this mode. For a system with w n i 2 = 2, ~ 0 n 2 2= 0.05, € 1 = 2.0, E E= ~ 0.04, and X 1 = 2.0, the asymptotic analysis predicts the existence of the weakly localized NNM 2,
; \ X ~ ( X I ) = -0.0622222~1 + 0.000694~13+...,
f t 2 ( ~ 1= ) x i +...,
XI
= O(1)
(7.2.40) which is verified by the numerical results of Figure 7.2.4. Note the orbital stability of this N N M , in accordance with theoretical predictions. The orbitally unstable nonlocalized N N M 3 of a system with an12 = 1, &on$= 0.08, EI = 1.0, E E =~ 0.05, and X i = 1.0 is presented in Figure 7.25. The theoretical modal curve for this mode is: $ 3 ( ~ 1 )= -0.8222222~1
+ 0.008888~13+...,
A
X ~ ( X I )= X I
+..., xi
= 0(1)
(7.2.4 1) and the numerical results indicate mode instability. For the numerical simulations presented in Figure 7.2.5, a 5% perturbation in the theoretically predicted initial condition for x l(0) was given. The resulting response of the system is not close to the theoretically predicted NNM, indicating mode instability. Similarly, the orbital instability of the weakly localized similar
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
321
1.5 x2
1
0.5
0 -0.5 -1 -1.5 -1.5
-1
-0.5
0
0.5
1
1.5 Xl
(a)
1.5 1
0.5 0
-0.5 -1
-1.5
Figure 7.2.3 Strongly localized stable NNM 1: (a) projection of the modal curve on the (x2,xl) plane and (b) time responses xi(t) and x2(t).
322
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
x3
3 2 1
0 -1
-2
-3
-2
-3
-1
0
1
3
2
Xl
(a)
3 2 1
0 -1
-2 - Q
- u
I
0
~
I
I
50
I00 (bl
-
1
15C t
Figure 7.2.4 Weakly localized stable NNM 2: (a) projection of the modal curve on the (x3,xl) plane and (b) time responses xi(t) and x3(t).
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
=3
323
1.5 1
0.5 0
-0.5 -1
-1.5 -1.5
-100
-1
0
-0.5
100
0 (a>
0.5
200
300
0)
1
1.5 Xl
400
500 t
Figure 7.2.5 Nonlocalized unstable NNM 3: (a) projection of the modal curve on the (x3,xl) plane and (b) time response x3(t).
324
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
1 x3
0.5 0
-0.5 -1 -1.5
-1
-0.5
0
0.5
(a)
1
1.5 Xl
1 x3
0.5 0
-0.5 -1
-200
0
200
400 0)
600
800 1000 t
Figure 7.2.6 Weakly localized unstable NNM 4: (a) projection of the modal curve on the (x3,xl) plane and (b) time response x3(t).
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
325
NNM 4, is demonstrated in Figure 7.2.6 for a system with O n 1 2 = 1, E C O E I = 0.5, E E = ~ 0.02, and X I = 1.0. A 5 % perturbation in the theoretically predicted initial condition of xz(0) was considered. Summarizing, it was shown that a periodic discrete system with strong nonlinear grounding stiffnesses and sufficiently weak coupling between subsystems possesses spatially localized NNMs. It was found that nonlinear mode localization exists in the perfectly symmetric structure with no structural imperfections (disorders). Although the asymptotic analysis of this section proves the existence of nonlinear mode localization in the weakly coupled system, it does not show how this localization is generated. In the next section the generation of nonlinear mode localization in the cyclic system will be studied by employing the method of multiple scales. It will be shown that localized NNMs can be generated either through mode bifurcations or as limits of continuous branches of NNMs. Moreover, localization of NNMs will be shown to arise due to the nonlinear dependence of the frequency of oscillation on the amplitude; in that context, nonsymmetric initial conditions lead to effective frequency mistuning between substructures, and, thus, to localization. In linear periodic systems such mistunings can only be generated when the cyclic symmetry is perturbed by parameter irregularities (disorder). = 0.1,
7.2.2 Transition from Localization to Nonlocalization
The cyclic system of Figure 7.2.1 is again considered, and assumed to possesses only nonlinearities of the third order. Nonlinearities of a more general type can be treated similarly. In terms of the previously introduced notation one defines fi(u)(Xi) = -Xi - E F X ~ ~ , Efi(C)(Xi-l,Xi,Xi+l)= -Ek(Xi - X i + l ) - Ek(Xi - xi-]) and fi(C)(Xj-xi.l) = -Ek(Xi - Xi+l) where ~p is the nonlinear coefficient of the grounding stiffness and Ek the linear coupling stiffness between adjacent masses. Note that although the nonlinearities of the grounding stiffnesses ~p and the linear coupling stiffnesses Ek are assumed to be weak, their ratio, R = Ep/&k,can take finite values. For the system under consideration the equations of motion (7.2.1) assume the form:
~ ~ ~
326
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
The method of multiple scales is now applied in order to obtain a uniformly valid, first order approximation to the NNMs of the system (Nayfeh and Mook, 1984; Rand and Armbruster, 1987). Fast and slow variables To = t and Ti = Et, respectively, are introduced, and the time derivatives of X i in expressions (7.2.42) are transformed using the chain rule. Using the new time scales the equations of motion are expressed as follows:
where Di denotes partial differentiation with respect to the time scale Ti, i=O,l. Next, the solution Xi is expressed in the series form:
Substituting (7.2.44) into (7.2.43) and matching coefficients of respective powers of E, one obtains successive approximations to the solution.
O ( d ) Approximation The equations of the O( 1) approximation provide the following solutions:
where j = (-1)1/2 and (cc) denotes the complex conjugate of the preceding term. The unknown (complex) amplitudes Ai(T1) are obtained by considering terms of O(E) in (7.2.43). O ( E ~ Approximation )
Matching terms of O ( E )one obtains the following set of nonhomogeneous linear differential equation governing the corrections Xi 1:
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
327
Eliminating secular terms, i.e., terms proportional to exp(kjT0) on the righthand sides of (7.2.46), one derives the following equations governing the complex amplitudes Ai: -2jDlAi - 3pAi'AiC - k(2Ai - Ai+l - Ai-1) = 0,
i = 1,...,n
(7.2.47)
where (*)C denotes the complex conjugate. Equations (7.2.47) form a set of n complex differential equations of first order, which can be transformed to a set of 2n real differential equations by introducing the polar transformations, i = 1,...,n Aj(T1) = (1/2)ai(Tl)e~pbPi(T1)], where aj and Pi are real amplitudes and phases, respectively. Separating real and imaginary parts in (7.2.47) one obtains the following set of first-order differential equations, which govern the (slow) modulations of the amplitudes and phases: aj' = (k/2)aj+lsin&+l - (W2)ai-lsinQi (7.2.48a) aiPi' = (3p/8)aj3 + kai - (W2)ai+lcosQj+l - (k/2)aj-l~0s@i,i = 1,...,n (7.2.48b) where prime denotes differentiation with respect to T i , and angles Qj are defined as $j = Pj - pj-1. Combining the first n equations (7.2.48a) it can be shown that the following energy-conservation relation holds: alal'
+...+ anan' = O
a12 + ... + an2 = p2
(7.2.49)
Hence, during free oscillations the sum of the squares of the amplitudes is conserved, which at this order of approximation is equivalent to conservation of the total energy of the system (Vakakis and Rand, 1992). By employing relation (7.2.49) the number of independent amplitudes of the problem is reduced by one. Furthermore, using the definition of the angles Qj one combines equations (7.2.48b) and develops the following autonomous set of (2n - 1) differential equations with dependent variables ai, i = 1,...,n, and $i , i = 2, ...,n:
328
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
The remaining phase angle $1 is determined by the compatibility relation n $l=-x$k k=2
Equations (7.2.50) together with the energy-like relation (7.2.49) describe, correct to O(E), the free oscillations on the isoenergetic manifold of the cyclic system of Figure 7.2.1. The localized and nonlocalized NNMs are computed by imposing the stationarity conditions ail = 0, $i' = 0, i = I , ...,n, and computing the resulting periodic, synchronous free oscillations of the system. Before proceeding with the analysis of the stationary values of (7.2.50), it is of interest to examine the NNMs of the uncoupled system corresponding to Ek = 0. In the notation of the previous section these are degenerate NNMs since they correspond to synchronous periodic motions of a set of n uncoupled oscillators. For the uncoupled system equations (7.2.50) are solved exactly as ai = aio, i = l,..*,n (7.2.5121) Cpi = (3p/8)[ai02 - a(i-l)o*]T1 + $io i=2 ,...,n (7.2.51b) and the corresponding oscillations of the masses of the system are given by, ai'= 0
2
(9i' = (3F/8)[aio2 - a(i-1)02]
*
where Pi0 and pi0 are constant phase angles and the quantities Ai are given by: 1
Ai =
k=2
(3p/8)[ako2 - a(k-l)02].
i = 2, ...,n
(7.2.53)
From equations (7.2.52) it is concluded that, .for a specified set of amplitudes aio, and, correct to O(E),the uncoupled nonlinear system is equivulent to ci system of n "disordered" linear oscillators with ,frequency "disorders" equal
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
329
to EAj. It is well known (Pierre and Cha, 1989; Pierre, 1988) that such a system, when coupled by sufficiently weak stiffnesses, possesses localized modes. Therefore, it is expected that when the weak substructure coupling is taken into account in (7.2.52),the nonlinear system will possess a variety c?f strongly and weakly localized NNMs. As in the previous section, these NNMs can be considered as perturbations of the degenerate NNMs of the uncoupled system. From equations (7.2.52), free synchronous motions of the uncoupled system take place when one of the following conditions is satisfied: (1) Ai = 0 3 ai02 = a(i-1)02 for all i E C = [2,...,n] (2) Ai = 0 for all i E S, and ajo = 0 for all j E S, where C 2 S .
( 3 ) aiO = 0 for all i
E
I:
Condition (1) corresponds to nonlocalized in phase and antiphase degenerate NNMs, which after the addition of coupling give rise to nonlocalized NNMs of the cyclic system. Condition (2) corresponds to weakly localized degenerate NNMs, which with the addition of weak coupling give rise to weakly localized similar or nonsimilar NNMs. In such motions only a limited number of masses of the cyclic assembly oscillates with finite amplitudes, with the remaining masses undergoing small motions. Condition ( 3 ) is the most interesting case since it corresponds to a degenerate NNM with motion of only one of the masses of the uncoupled system. With the addition of weak coupling this degenerate mode gives rise to a strongly localized NNM, corresponding to O( 1) oscillation of only one of the masses of the cyclic system. Because of the cyclicity of the system under investigation, there exist n strongly localized NNMs (in the presented results the motion is assumed to localize at mass 1, but this does not restrict the generality of the analysis). As an example, consider a system with only two subsystems (i.e., n = 2). In this case there exists only one equivalent "frequency disorder," A2 = (3W8)(a202 - aio2) and only conditions (1) and ( 3 ) are realizable. Introducing the polar transformation a20 = psiny, a10 = p c o s y [where p is the energy-like quantity defined by relation (7.2.49)], the quantity A2 takes the form: A2 = -(3yp2/8)cos2y,
(n = 2)
(7.2.54)
330
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
Localization
Figure 7.2.7 "Frequency disorders" for a system with (a) n = 2 and (b) n = 3 masses. A graph of 62(3pp2/8)-1 versus polar angle yr is given in Figure 7.2.7(a), and the values corresponding to degenerate in phase (a10 = 4 2 0 , $2 = IT), antiphase (a10 = a20, $2 = n), and strongly localized NNMs (a10 = p and a20 = 0 or a10 = 0 and a20 = p) are indicated. Note that the degenerate tocalized NNMs correspond to extreme values of the 'Ifrequency disorder" A2. Each of these degenerate modes gives rise to (stable or unstable) nonlocalized or localized NNMs of the weakly coupled system. For the system with n = 3 , introducing the polar transformation a10 = pcosyri, a20 = psinyricosyrz, a30 = psinyri sinyr2, the following two "frequency disorders" result: A2 = (3~p2/8)[sin2yricos2W2- cos2W1] A3 = A2
- (3yp2/8)sinzyrlcos2y2,
(n = 3)
(7.2.55)
In this case all conditions (1)-(3) can be realized, as seen from the plots of Figure 7.2.7(b) where the quantities A2(3pp2/8)-1 and A3(3pp2/8)-1 are plotted versus the angles y r l and yr2. In these case one obtains the five
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
33 1
degenerate NNMs listed in Table 7.2.1. Again it is noted that strong mode localization occurs for extremum values of the frequency disorders. The aforementioned analysis shows that correct O(E), the uncoupled cyclic system is equivalent to a set of weakly disordered uncoupled oscillators. The frequency disorders Ai are generated due to the dependence of the frequency of oscillation on the amplitude, a feature that is universal in nonlinear systems. The existence of such "frequency disorders" explains why nonlinear mode localization takes place in perfectly periodic nonlinear systems, i.e., even in the absence of structural disorders. When individual components of such systems oscillate with uneven amplitudes, the nonlinearities give rise to effective frequency disorders which, under weak coupling, lead to spatial localization of the free motions. The stationary values of the coupled system (7.2.50) are now sought. In what follows the strongly localized NNMs of the cyclic assembly are studied, and the transition from mode localization to nonlocalization for varying structural parameters is investigated. Motivated by the asymptotic results of section 7.2.1 (where it was found that a strongly localized NNM corresponds to anti-phase motions of adjacent masses of the system), one imposes the following relations on the amplitudes and phases: For n = ( 2 ~+ 1')
For n=2(t1 - 1)
a2 = an, a3 = an-], ... , ap-1 = ap+l Qi = X, i = 1,...,n, i # p Qp+2 = -2PX Qp = -(2p - 3)n: (7.2.5 6) Imposing the stationarity conditions ail = Qi' = 0 on the system of equations (7.2.50), and taking into account conditions (7.2.56), one obtains the following set of algebraic equations governing the amplitudes of the NNMs of the system: a2 = an, a3 = an-1, ... , ap+l = ap+2
Qi = X,i = 1,...,n, i # p+2
For n = ( 2 +~ 1)
332
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
For n = 2(p - 1)
(7.2.57b) Solving the nonlinear sets of algebraic equations (7.2.57) one determines the localized NNMs of the cyclic system, corresponding to the following physical motions of the masses of the system: For n = (2u + 1)
For n = 2 ( ~- 1)
(7.2.584
where the constant angle p l o is determined by the initial conditions, and = m ( a ] ) is the small amplitude-dependent nonlinear correction to the frequency of oscillation, determined by integrating equations (7.2.48b). Although equations (7.2.57) cannot be solved analytically, the existence of strongly localized NNMs in the cyclic system cun be predicted by meuns o j a simple perturbation analysis. To this end, consider the set of equations (7.2.57) and assume that (k/p) = O ( E ~ )where , &1 is a new perturbation parameter with l ~ i << l 1. Strongly localized solutions are now sought in the
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
333
form, a1 = 0(1), and ai = O(&ii-l), i = 2 ,...,P where P = p+l if n = 2p+l, and P = p if n = 2(pl). Considering the orders of the various terms in equations (7.2.57), taking into account that a12 = p2 + O(&12),and retaining only terms of 0(1), one obtains the following perturbation solutions for the strongly localized NNM: For n = (2p + 1)
For n = 2(u - 1)
a2/al = (4/3p2)(k/p) + O(~12) ai/al = (4/3~2)i-'(k/p)i-' + O(&li) i = 3,...,p+ 1
a2/ai = (4/3p2)(Wp) + O ( ~ i 2 ) ai/a1= (4/3p2)i-'(Wp)i-l + O(&li) ap/a 1 = 2(4/3p 2 ) ~ -1(k/p)~1+O(E1 P) i = 3, ...,p 1 (7.2.59) These solutions indicate that a strongly localized NNM exists corresponding to finite-amplitude motions of only one of the masses of the system. The remaining masses oscillate with amplitudes that rapidly diminish with increasing distance from mass where localization occurs. An interesting observation on the perturbation results (7.2.59) is that the ratio a2/a] during the strongly localized mode is found to be independent of the actual number of degrees of freedom of the system, which shows that it is possible to predict the amplitudes of the strongly localized motions irrespective of the degrees of freedom of the system. These findings are in full agreement with the asymptotic results of section 7.2.1. For n = 2 relations (7.2.57b) degenerate to the following equation, which can be solved exactly: a12=a22 or a1a2 = f 8W3p (7.2.60) Taking into account that a12 + a22 = p2, one obtains the following relations for the NNMs: a1 = a2 = ~12112, $2 = 7c (antiphase mode) a1 = a2 = p/21/*, $2 = n: (in phase mode) a2 = (2-1/2) { p2 f [p4 (16W3p)2]1/2}1/2 a1 = (21/2)(8k/3p){ p2 f Cp4 (16W3p)2]1/2}-1/2 $2 = 7c for ( p k ) 2 (16/3p2) (bifurcating modes) (3p/8k) (a12-a22) - (a12-a22) / ala2 = 0
334
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
The in phase and antiphase NNMs are similar (since they correspond to straight modal lines), nonlocalized, and their modal ratios ai/aZ = -tl do not depend on the energy-like quantity p*. The bifurcating NNMs exist only for weak coupling and/or large substructure nonlinearities and their modal ratios depend on the energy: az/al = (3p/16k){ p2 f [p4 (16M3p)2]1/2},
(3E/8) $ ~ = T c for , (Wp)I
(7.2.61) These NNMs are nonsimilar and bifurcate from the antiphase mode at (k/p) = (3E/8) + O(E). The bifurcating nonsimilar modes give rise to strong nonlinear mode localization since it can be easily shown that lim wp (a2/ai) = 0 or 03 depending on the choice of sign in (7.2.61). In Figure 7.2.8 the NNMs of the system are depicted for pz = 1. Note that nonlinear mode localization takes place only for small values of the ratio kip. As this ratio increases, the localized nonsimilar modes become nonlocalized and eventually coalesce with the antiphase NNM in a hamiltonian pitchfork bifurcation. The stability analysis of the computed NNMs was carried out by introducing small transient perturbations to the NNM periodic solutions and considering the eigenvalues of the corresponding variational matrix. Using this procedure it is found that the localized modes are orbitally stable. --f
az/al = a3/a1 ( 3 DOF)
--
az/al = a3/a1 ( 3 DOF) 0
0.1
0.2
0.3
0.4
0.5
k/P
Figure 7.2.8 Transition from mode localization to nonlocalization for cyclic systems with n = 2,3: -Stable, ( x ) unstable solutions.
7.2 MODE LOCALIZATION IN SYSTEMS WITH CYCLIC SYMMETRY
335
For n = 3, equations (7.2.57a) reduce to a single equation for amplitudes a1 and a2 = a3: (3p/8k)(a12 a22) (1/2)(a12 a22)/ala2 + (1/2)[(a2/al) + 11 = 0
(7.2.62)
with the energy relation a12 + 2a22 = p2. These equations were numerically solved and the NNMs corresponding to p2 = 1 are presented in Figure 7.2.8. Two branches of nonsimilar NNMs can be seen. Branch 1 of NNMs represents orbitally stable, free oscillations and for sufficiently small values of (Wp)is strongly localized. For small values of parameter (Wp) this mode coincides with the strongly localized NNM 1 of the 3-DOF cyclic system examined in section 7.2.1. Branches 2 and 3 represent orbitally unstable NNMs and for small values of (Wp) are identical to the unstable NNMs 3 and 4, respectively, of the cyclic system of section 7.2.1. I t is interesting to note that f o r the 3-DUF cyclic system the branch of NNMs giving rise to strong mode localization does not bqurcate from any other mode; instead, as [he ratio ( W p ) increases the strongly localized nonlinear mode becomes nonlocalized, and its modal ratios a2/al = a3/a1 approach asymptotic limits equal to (U2). As shown in section 7.2.1, the three degree of freedom system possesses two additional stable NNMs, which are not depicted in Figure 7.2.8. The localized NNMs for systems with n = 4 and 5 DOF and p2 = 1 are presented in Figures 7.2.9 and 7.2.10, respectively [for n = 4 3 , the algebraic equations governing the modal amplitudes ai can be found in (Vakakis et al., 1993a)l. For low values of the ratio (Wp) I 0.1, nonlinear mode localization occurs and the numerical solutions are nearly identical to the perturbation results (7.2.59). As the ratio (k/p) increases, the localized branches of the system with n = 4 become nonlocalized and eventually coalesce with the antiphase nonlocalized NNM corresponding to a1 = a2 = a3 = aq and $1 = 92 = $4 = Z, $3 = 37c in a degenerate hamiltonian pitchfork bifurcation, Qualitatively different branches of NNMs exist for the cyclic system with n = 5 , where the branches of NNMs giving rise to mode localization do not bifurcate from any other branch of solutions. For this system (as in the system with n = 3), no mode bifurcations occur and strong nonlinear mode localization is generated at the limit of a continuous branch of NNMs as the ratio (Wp) tends to zero. As the ratio (Up)increases, the strongly localized
336
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
0
0.1
/0.2
0.3
0.4
k/p = 3~92/16
0,5
k/P,
Figure 7.2.9 Transition from mode localization to nonlocalization for a cyclic system with n = 4: -Stable, ( x ) unstable solutions. 1
0.8 0.6 0.4
0.2
0
0
0.4
0.8.
1.2
1.6
2
Figure 7.2.10 Transition from mode localization to nonlocalization for a cyclic system with n = 5 : -Stable, ( x ) unstable solutions.
7.3 MODE LOCALIZATION IN A STRONGLY NONLINEAR SYSTEM
337
modes become nonlocalized and their modal ratios reach the asymptotic limits a2/a1 = a5/al = 0.8 and a3/al = a d a l = 0.3 as (Up)+-. The previous results indicate that the essential control variable regarding the transition from mode localization to nonlocalization in the cyclic system is the ratio (Wp) of linear coupling stiffness to grounding nonlinearity. For sufficiently small values of this ratio nonlinear mode localization always occurs, irrespective of the number of degrees of freedom of the system. As (Wp)increases, the branches of localized modes became nonlocalized in two distinct ways. For systems with odd number of degrees of freedom the branches of NNMs giving rise to nonlinear localization are continuous and their modal ratios reach definite asymptotic limits as (k/p)+-. For systems with an even number of degrees of freedom the branches of localized NNMs exist only for finite ranges of (k/p) and bifurcate from nonlocalized antiphase modes of the cyclic system in degenerate "multiple" pitchfork hamiltonian bifurcations. Hence, two distinct generating mechanisms for nonlinear mode localization are detected in the cyclic system.
7.3 MODE LOCALIZATION IN A STRONGLY NONLINEAR SYSTEM In this section nonlinear mode localization in a strongly nonlinear periodic system is examined, by employing the method of nonsrnooth temporal transformations (NSTT) developed in chapter 6. Since the analytical methodology used in this section is valid for systems with strong stiffness nonlinearities, it can be considered as more general than the techniques employed in section 7.2 (which were applied under the assumption of weak nonlinearities). A two-DOF strongly nonlinear oscillator is considered to illustrate the application of the analytical technique on the mode localization problem. First, consider a system of two strongly nonlinear oscillators coupled by means of a linear stiffness. The particles of the system are of unit mass, and the stiffness nonlinearities are assumed to be of arbitrarily high degree. The equations of motion are given by:
+ y(2x1 x2) + xlm = 0 X2(t) + y(2x2 XI) + x p = 0
Xl(t)
(7.3.1) 337
338
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
where m = 2v 1, v = 2,3,..., is an odd exponent, and y is the strength of the linear coupling stiffness. Applying the NSTT methodology developed in chapter 6, a nonlinear eigenvalue problem is formulated for approximating the amplitudes of coordinates xi (t) and x2(t) during synchronous periodic motion. To this end, the ith coordinate xi(t) is expressed as: xi(t) = Xi(T),
z = 7(ot),
i = 1,2
(7.3.2)
where 7 is the discontinuous transformation of the temporal variable, and o is the frequency of the synchronous oscillation. Substituting (7.3.2) into (7.3.1), the equations of motion are expressed as: d2X 1
o2 __ d72
+ y(2X1 X2) + Xlm = 0 (7.3.3)
Normalizing the period of oscillation to T = 4, the following additional conditions are imposed:
(7.3.4) The solutions of (7.3.3) and (7.3.4) are analytically computed by the method of successive approximations developed in chapter 6. The motions Xi and the frequency o are expressed in the following regular series expansions:
where Ui(i)(Z) and (j are the jth order approximations, which are computed after evaluating all other approximations of lower orders (cf. chapter 6). In the first step of the iteration, only the leading approximations of the series (7.3.5) are taken into account, resulting in the following set of nonlinear
7.3 MODE LOCALIZATION IN A STRONGLY NONLINEAR SYSTEM
339
algebraic equations governing the zeroth order approximations for the amplitudes, AiO:
(y l/ho)A10 (y/2)A20 + AlOm/(m+l) = 0 (y l/ho)A2O (y/2)A10 + A2om/(m+l) = 0
(7.3.6)
Moreover, the following analytical expression for the first-order frequency correction 51 is derived:
+ A202)-1{ (y2/4!)(5A102 - 8A10A20 + 5A202) + (y/3!)m(m+3)-1[ 1 + 3!m-l(m+l)-l(m+2)-1](2Ai0(m+l)+ 2A20(m+l) - AiOA20m - A10mA20) + 2--1m(m+l)-2(m+2)-1(A10(2m) + A20(2m))} 61 = -X$(Alo2
(7.3.7) Consider the set of algebraic equations (7.3.6). When no nonlinearities exist, it can be shown that they admit the solutions (Alo,A20) = (A,A), Lo = 2/y, and (A10,A20) = (A,-A), ho = 2/(3y), where A is an arbitrary scalar. Clearly, these solutions correspond to the in-phase and antiphase linear normal modes of system (7.3.1) with no nonlinearities. Substituting for Lo and Aio, i = 1,2, in (7.3.7), it is found that the first-order approximation to the frequency, L1, is the same for the in-phase and antiphase linear normal modes, and equal to 51 =1/6. Substituting this value into the expression of the frequency squared [the second of equations (7.3.5)], one finds the following approximations for the natural frequencies of the linear modes: 02 = 02 =
(1/h)= (5/12)y ( l / h )= (5/4)y
(In phase linear normal mode) (Antiphase linear normal mode)
(7.3.8)
Note that the exact values of the natural frequencies for the two modes can be computed by multiplying the approximate values (7.3.8) by the numerical factor 48/(5~2). When the nonlinearities of the system are taken into account, it can be shown that the nonlinear amplitude equations (7.3.6) admit as solutions the values (Alo,AzO) = (A,A) and (A1O1A20)= (A,-A). Hence, the nonlinear system possesses in phase and antiphase NNMs, a feature that, as noted in previous sections, is due to its special symmetrical configuration. However, in contrast to the linear case, the frequencies of the in-phase and antiphase
340
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
NNMs are found to depend on the amplitude of oscillation. For example, considering the in-phase N N M one has that (AlO,A20) = (A,A) and the quantities Lo and 61 in the expression of the frequency are computed as follows: ho = 2/(y + 2(m + l)-lAln-l)
+ 3)-1[1 + 3!m-l(m+l)-l(m+2)-l]Am-l (7.3.9) + 12m(m+l)--2(m+2)-1A2m-2}(y + 2(m + l)-lAm-1)-2 = (1/6)( y2 +4ym(m
When A-0, the above expressions reduce to the respective values of the inphase linear normal mode. As A+m or y+O the above expressions asymptotically approach the respective values of an uncoupled oscillator with strong nonlinearity of mth degree. In addition to the in-phase and anti-phase modes, system (7.3.6) and (7.3.7) admits additional localized solutions. These are analyzed by scaling the amplitudes according to (A10,A20) = A(al,a2), where A is a scaling factor and lail = 1, i = 1,2. In terms of the new variables, equations (7.3.6) are written in the following form:
where 6 = (m+l)/(hOAm-l), and p = y(m+l)/Am-l. At this point it is further assumed that either A+m or y+O, so that the coupling parameter p in equations (7.3.10) is small, lpl << 1. At the limit p+O, equations (7.3.10) admit the following four solutions: (i) (ai,a2) = ( l , l ) , 6 = I ; (ii) (al,a2) = (1,-l), 6 = l ; (iii) (al,a2) = (l,O), 6 = l ; and (iv) (al,a2) = (O,I), 6 = l . In accordance to the definitions of section 7.3 these solutions are termed degenerate, since they correspond to a degenerate system of two strongly nonlinear oscillators. Solutions (i) and (ii) correspond to symmetric and antisymmetric NNMs of the degenerate system, respectively. As mentioned earlier, these solutions are due to the symmetry of system (7.3.1) and exist over the entire range of values of parameter p. Degenerate solutions (iii) and (iv) are completely localized to either one of the two particles of the uncoupled system. Clearly, this type of complete (total) mode localization is not possible when p # 0. For nonzero values of p the solutions of (7.3.10)
7.3 MODE LOCALIZATION IN A STRONGLY NONLINEAR SYSTEM
341
neighboring the degenerate focalized solutions (iii) are sought i n the following form,
whereas the frequency parameter 6 is expressed as
6 = 1 + p6(')
+ p26(2) + ...
(7.3.1 Ib)
Substituting (7.3.1 1) into (7.3. lo), and matching respective coefficients of powers of p, one obtains the foilowing analytical approximations for the strongly localized NNMs of the weakly coupled system: (AlO,A20) = A( 1,-p/2 + O(p4)) + p + (~214)+ O(p5)],
ho = [(m + l)Al-m]/[l or,
(A10,A20) = A(l,-p/2
if m = 3
(7.3.12a)
+ (p3/8)+ O(p4))
Lo = [(m + l)Al-m]/[ 1 + p + (~214)- (~4116)+ O(p5)], if m > 3
(7.3.12a) In Figure 7.3.1, the localized NNMs of a system with y = 0.2 are depicted. The results presented in that figure were computed in two distinct ways: by performing two iterations and computing the first two leading terms of the series expressions (7.3.5) (solid lines), and by employing the asymptotic results (7.3.12) (thin lines). In accordance to the findings reported in section 7.2, it is seen that the localized NNMs are bifurcating from the antiphase NNM of the system. After the bifurcation, the antiphase NNM becomes orbitally unstable. As a second example, a strongly nonlinear periodic system with third-, fifth-, and seventh-order stiffness nonlinearities is examined. The equations of motion are written in the form: xl(t) + r(x1) + g(xi - x2) = 0 x2(t)
+ r(x2) + g(x2 - x i ) = 0
(7.3.13)
where the characteristics of the stiffnesses are given by r(u) = riu + r3u3 + rsu5 + r7u7 and g(u) = glu + g3u3 + g5u5 + g7u7
342
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
Figure 7.3.1 Localized NNMs of the system with y = 0.2 and m = 3. Two leading terms of the series (7.3.5) (solid lines), and asymptotic expressions (7.3.12) (thin lines): -Stable modes, ------ unstable modes. In Figure 7.3.2 the localized NNMs of the system are depicted, after performing two iterations and computing the leading two approximations of expansions (7.3.5) for this problem. The parameters of the system were assigned the values rl = 1.0, r3 = 5.0, r5 = 1.0, r7 = 6.0, gi = 0.1296, g3 = 0.3995, gs = 0.68375, and g7 = 0.21875, i.e., the coupling stiffness was assumed to possesses weak third- and seventh-order terms, and strong linear and fifth-order ones. From Figure 7.3.2 it is seen that, for the given values of the system parameters, the antiphase NNM possesses two zones of bifurcation where additional bifurcating NNMs exist and nonlinear mode localization is generated. The first bifurcation zone is associated with the weak cubic coupling terms, and is limited by the strong fifth-order coupling terms. The second zone of NNM bifurcation is associated with the weak seventh-order coupling terms. To analytically study mode localization in this problem, one introduces the following transformation of variables: x = (XI
+ x2)/2,
y=
Xl
- x2)/2
(7.3.14)
Expressing the equations of motion (7.3.13 in terms of the new variables x and y one obtains:
7.3 MODE LOCALIZATION IN A STRONGLY NONLINEAR SYSTEM
343
/ Figure 7.3.2 Bifurcations of NNMs of the nonlinearly coupled system with cubic, quintic and seventh order stiffness nonlinearities. Stable modes, ------- unstable modes.
Working as in the previous example and employing the analytic methodology developed in chapter 4, one finds the following equations governing the amplitudes A,o and AyO, which are the zeroth order approximations to the amplitudes of variables x and y during a synchronous periodic motion of the system: Axo/hj = [R(Axo + Aye) + R(A,o - AyO)]/2 = 0 AyO/ko = [R(Axo+ AyO) - R(A,o- Ayo)]/2 + G(2AyO) = 0 (7.3.16) where, 1
R(u) =
r(u5) d 5 0
and
G(u) =
1.
g(u5) d5
0
Localized NNMs are sought, corresponding to small values of the amplitude AxO, i.e., by imposing the condition lAxOl << 1. Taking this requirement into account, equations (7.3.16) degenerate into the following equation for AyO:
344
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
[R(AyO) + G(2Ayo) - Ay0 R'(Ayo)]Axo = 0
(7.3.17)
In addition to the in-phase and antiphase NNMs Ax0 = 0 and Ay0 = 0, the algebraic equation (7.3.17) provides the approximate NNMs, which bifurcate from the antiphase NNM of the system. For the previous values of the parameters of the system, it can be shown that the solutions of (7.3.17) agree with those depicted in Figure 7.3.2. In this section it was shown that the strongly nonlinear NSTT methodology developed in chapter 6 can be used to study nonlinear mode localization in periodic nonlinear discrete systems. The analytical solutions developed herein can be further refined by carrying out additional iterations and computing high-order successive approximations to the localized NNMs. The advantage of this technique is that it can be applied to systems with strong (essential) nonlinearities, in contrast to the perturbation techniques used in section 7.2, which can only be employed to analyze systems with weak nonlinearities.
7.4 LOCALIZATION IN IMPULSIVELY FORCED SYSTEMS In this section the effects of nonlinear mode localization on the impulsive response of a system are numerically examined. It will be shown that a system with localized NNMs possesses passive motion cunfinernent properties, that is, motions generated by external impulsive excitations remain passively confined close to the point where they are initially generated instead of "spreading" through the entire structure. This passive confinement of impulsive responses will be shown to arise due to the nonlinearities, and to occur even in perfectly symmetric systems. I t is important to note that such a passive confinement phenomenon in perfectly symmetric cyclic systems has no counterpart in linear theory. The results presented herein demonstrate that the novel nonlinear motion confinement phenomenon coupled with appropriate control mechanisms, can constitute a new design tool, capable of enhancing the resiliency of nonlinear periodic assemblies in unpredictable forcing environments. To demonstrate the effect of localized NNMs on the impulsive response of a nonlinear periodic system, consider the cyclic system of Figure 7.2.1 with stiffness elements: 344
7.4 LOCALIZATION IN IMPULSIVELY FORCED SYSTEMS
345
where, as in section 7.2.2, ~p is the nonlinear coefficient of the grounding stiffness, &k the linear coupling stiffness between adjacent masses, and the ratio R = &p/Ek is assumed to take finite values. As shown previously, as (k/p) + 0 a branch of NNMs strongly localizes to a single mass of the system. The responses of a system with 50 DOF excited by an impulsive load on one of its subsystems is now examined. Such computations have been previously performed for weakly disordered linear discrete and distributed cyclic structures using finite-element techniques (Bendiksen 1987; Cornwell and Bendiksen 1989). In these studies it was shown that linear mode localization gives rise to motion confinement of impulsive responses, a fact that can be useful for developing robust control algorithms for the active vibration isolation of such structures. The numerical results reported in this section were obtained by finiteelement computations. The global equations of motion were first assembled, and were subsequently integrated using the Newmark numerical algorithm (Bathe 1982). A force with unit magnitude was applied for a duration of At = 0.2 second at mass 1, and the resulting displacements of the system were evaluated by numerical time integrations. In Figure 7.4.1(a) the impulsive responses of the 50-DOF cyclic system with parameters &p = 0.3, Ek = 0.05, and ratio Mp = 0.166 (weakly coupled system) are depicted, at time instants t = 0.2, 500, and 1000 seconds. In these plots the instantaneous normalized displacements of all masses of the system are presented. For comparison purposes, the responses of the corresponding linear system with E/L = 0 are also shown in these graphs. It is clearly noted that in the nonlinear case the energy of the applied impulse is confined close to the point of its application instead of "leaking" to the entire system; by contrast, in the linear case there is a gradual "spreading" of the injected energy to all masses of the system. This spreading of energy in the linear system becomes more profound as time increases. The motion confinement in the nonlinear system is clearly attributed to the existence of a stable strongly nonlinear localized mode, which, once excited by the impulse, restricts the propagation of energy through the entire system.
346
NONLINEAR LOCALIZATION IN DISCRETE SYSTEMS
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ek4.03 14.2
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1=m
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Ib
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6
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I
I0
36
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COOROINRTE NUMBER
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Figure 7.4.1 Transient passive motion confinement in the system em with 50 DOF, impulse applied at mass 1: (a) weak coupling, k/p = 0.166, and (b) strong coupling, Wp = 1.666: - Nonlinear system, ------- linear system.
7.4 LOCALIZATION IN IMPULSIVELY FORCED SYSTEMS
347
To test the theoretical prediction of section 7.2.2, namely that localized NNMs become nonlocalized as the ratio (k/p) increases, numerical simulations were carried out with a cyclic system with &p = 0.3, Ek = 0.5, and ratio Wp = 1.666 (strongly coupled system), using the same impulsive loading condition with the one used in the weakly coupled case. The impulsive response of the strongly coupled system is presented at Figure 7.4.l(b) at the time instant t = 500 seconds. Note that for the strongly coupled system the energy of the impulse is no longer confined in the directly excited mass, and spreads through the entire system. This result demonstrates that the passive nonlinear motion confinement phenomenon is eliminated as the coupling between adjacent masses increases, a feature that correlates with the fact that nonlinear mode localization is also eliminated for increasing values of (Wp).This is an indication that the passive motion confinement phenomenon of Figure 7.4.l(a) is solely due to nonlinear mode localization and can be realized only for sufficiently small values of the ratio (k/k). An analytical treatment of the passive motion confinement phenomenon due to nonlinear mode localization is postponed until chapter 9, where localization of forced responses in coupled systems of beams is discussed. The results reported in this section indicate that nonlinear mode localization can be a novel design tool for the vibration and shock isolation of mechanical systems. This is because a system whose inherent dynamics leads to motion confinement of external disturbances is expected to be much more amenable to active or passive control than a system with no such d ynami cal properties.
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 8 NNMs IN CONTINUOUS SYSTEMS In previous chapters the concept of nonlinear normal modes (NNMs) was used to study the dynamics of conservative and nonconservative discrete oscillators. In this chapter, the principle of NNMs is extended to continuous (with infinite DOF), one-dimensional systems. In section 8.1 general analytical methodologies for computing NNMs of continuous structural members of finite spatial extent are considered, and a technique for studying their stability is presented. In section 8.2 continuous systems of infinite extent are studied, and an extension of the notion of NNM to this class of systems is considered.
8.1 SYSTEMS OF FINITE SPATIAL EXTENT The NNMs of a class of one-dimensional, conservative or nonconservative, continuous bounded oscillators will be first examined. NNMs in this class of systems are defined as free, periodic motions during which all particles of the system reach their extremum amplitudes at the same instant of time. Various analytical and numerical techniques for analyzing continuous NNMs will be developed, and the stability of the computed NNMs will be investigated by numerical Floquet analysis. There exist numerous techniques for analyzing the normal modes of continuous systems such as beams, shells, and plates, and the most important of them are reviewed in what follows. The flexible systems considered herein are governed by partial differential equations of the following general form:
(8.1.1) B[u(x,~)]= 0,
x = 0,l
(8.1.2)
In the equations above, L[*] is a nonlinear integro-differential operator acting on the displacement variable u(x,t), and B[e] is a boundary condition operator. It is assumed that the nonlinearities of the system are small and proportional to a small parameter E, IE~<< 1. This assumption is necessary 349
350
NNMs IN CONTINUOUS SYSTEMS
for the validity of the perturbation analyses that follow. There are two general classes of methodologies for computing the NNMs of (8.1.1) and (8.1.2). The first class is based on the discretization of the governing partial differential equation and the analysis of the resulting set of weakly nonlinear ordinary differential equations. The second class of methods (direct approaches) relies on the direct analysis of the nonlinear equation of motion, without involving any discretization schemes. Methods based on discretization of the partial differential equation of motion express the displacement u(x,t) in the following series form: (8.1.3) where one of the functions +i(x) or qi(t) is prescribed. If the boundary conditions of the problem are homogeneous, the functions +)(x) are normally taken as the eigenfunctions of the corresponding linearized problem. In other classes of applications (generalized harmonic balance approaches), the functions q,(t) are chosen from a complete set of harmonic functions. Substituting (8.1.3) into (8.1.1) and employing appropriate orthogonality conditions, one reduces the original partial differential equation to a set of n ordinary modal equations. Representative works demonstrating the application of this technique to the vibration analysis of beams, shells, and plates are those of Bennet and Eisley (1970), Chen and Babcock (1975), Datta (1976), Nayfeh and Mook (1984), and Marion and Temam (1989). A basic limitation of the discretization analysis is that, in cases where the @,(x) are prescribed, the mode shape of the nonlinear vibration is defined u priori, and thus, is unaffected by the nonlinearity since it is identical to the normal mode of the associated linear system. Similarly, in cases where the qi(t) are prescribed, the time dependence does not account for nonlinear corrections. In a series of papers, Szemplinska-Stupnicka (1974,1979,1983) developed an approximate perturbation methodology for nonlinear discrete and continuous oscillators (“harmonic balance principle”), which, although based on the discretization of the equations of motion, accounts for nonlinear corrections to the modeshape and the time dependence of the vibration. Considering direct approaches for finding NNMs of continuous systems, Nayfeh, Mook, and co-workers applied the method of multiple scales
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
35 1
directly to the nonlinear partial differential equations of motion, thus accounting explicitly for the boundary conditions of the problem (Nayfeh, 1981; Nayfeh et al., 1992, 1993; Nayfeh and Nayfeh, 1993, 1994; Nayfeh et al., 199.5; Pakdemirli and Nayfeh, 1996). Shaw and Pierre (1994) developed a method for computing NNMs based on invariant manifolds. Their methodology was primarily developed in (Shaw and Pierre, 1991, 1993), where NNMs of damped and undamped discrete oscillators were computed by constructing analytic approximations to invariant manifolds of the free motions. Their formulation resembles the construction of center manifolds in nonhyperbolic discrete nonlinear systems (Wiggins, 1990) and can be effectively used for studying nonlinear normal modes in discrete or continuous systems with no low-order internal resonances (and thus no bifurcating modes). In Shaw and Pierre (1994) and Boivin et al. (1993) onedimensional nonlinear elastic systems are considered, and the displacement of an arbitrary point of the system is expressed as a function of the displacement and velocity of a fixed reference point. The invariant manifolds resulting from this formulation are then analytically approximated using a perturbation analysis. In a hybrid methodology, Shaw and Pierre (1992) used a Galerkin approximation to discretize the partial differential equation of the nonlinear continuum. The NNMs of the resulting set of ordinary differential equations were computed by the invariant manifold approach. King and Vakakis (1993b) developed an alternative approach for computing continuous NNMs by reducing the problem to a set of singular nonlinear partial differential equations and solving them asymptotically by power series expansions. In King and Vakakis (1994) this technique is extended to the study of solitary waves in nonlinear partial differential equations. In the following sections several representative methods for computing NNMs in continuous systems with weak nonlinearities are discussed in detail. These include, asymptotic methods tackling directly the governing nonlinear partial differential equations, methods based on discretizations of the equations of motion using bases of linear eigenfunctions, and a technique employing nonsmooth coordinate transformations. Furthermore, in a recent publication by Nayfeh (199.5), a new direct method for computing NNMs in continuous systems is proposed, based on the method of normal forms. An interesting feature of this method is its applicability to continuous systems with internal resonances. A comparison of the aforementioned
352
NNMs IN CONTINUOUS SYSTEMS
methodologies with the one based on normal forms is presented in the same reference.
8.1.1 Direct Analysis of the Equations of Motion In this section two direct analytical methodologies for computing NNMs of continuous structural components are presented. The common feature of these methods is the elimination of the time variable from the problem by the introduction of a suitably defined transformation of the independent variables. The methodology presented eliminates the need of discretizing the governing partial differential equations of motion, thus improving the accuracy of the results. Energy-Based Formulation The first method for computing NNMs of continuous systems applies to conservative one-dimensional flexible elements and was developed by King and Vakakis (1993b). Consider the vibration of the nonlinear continuous system (8.1.1) occupying the one-dimensional region 0 5 x I 1. Expressing the operator L[.] in terms of separable and nonseparable terms one rewrites the equation of motion as
+
utt= L,to)[~(~,t)]EL,~(')[u(x,~)],
x E (0,l)
(8.1.4)
with the boundary conditions given by (8.1.2). In (8.1.4), L,tO)[*] and &LXt(l)[*] are integro-differential operators acting on the variable u(x,t) and E is a small parameter. The following assumptions are made regarding system (8.1.4): (1) System (8.1.4) is conservative and the boundary conditions (8.1.2) involve no dissipation of energy. (2) For E = 0, the system is separable in space and time and executes free vibrations with bounded amplitudes for any given set of initial conditions. (3) L,(O)[*] consists of differentiations and/or integrations involving only the spatial variable x, whereas L,,(l)[*] involves integro-differential
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
353
operations in terms of the spatial variable x and possibly the time variable t. (4) Only odd-order nonlinearities exist in the system. ( 5 ) The displacement u(x,t) is a sufficiently smooth function of its variables so that all derivatives appearing in the following analysis exist. Assumption (1) is needed in order to impose the condition of preservation of the total energy of oscillation, ( 2 ) , ( 3 ) and ( 5 ) are necessary for carrying out the perturbation analysis that follows, and (4) provides a symmetric structure for the computed NNMs. In addition, the system is assumed to be natural (Nayfeh and Mook, 1984) so that the kinetic energy can be expressed as a quadratic function of the velocity u,(x,t). When this last requirement is not satisfied, a modification of the perturbation method is needed. In the spirit of Rosenberg (1966) and Shaw and Pierre (1994), define continuous nonlinear normal modes as motions during which all material points of the system vary equiperiodically, vanishing or reaching their extremum values at the same instant of time. During a nonlinear normal mode the displacement of the material point at position x = xo is denoted by u,(t) E u(xo,t). Using u,(t) as a reference displacement, we express the displacement of any other point of the system in terms of the reference motion by a functional relation of the form
where U[*;] is a modal function used to describe the mode shape of the nonlinear mode under consideration. Note that in writing (8.1.5) the set oj' independent variables is changed from (x,t) to [x,u,(t)], and thus the explicit time dependence of the motion is eliminated. Such a transformation of coordinates requires x, not to coincide with a node of the nonlinear mode under investigation. Shaw and Pierre (1994) used a similar functional relation to define the nonlinear mode shape, using both the displacement and the velocity of the reference point as new independent variables. Their methodology is discussed later. Since system (8.1.4) is conservative and natural, the total energy, Etot, is conserved during a nonlinear mode, that is,
354
NNMs IN CONTINUOUS SYSTEMS
1
Etot= (1/2)5
ut2 dx
0
+ L@J [u(x,t)] + E h ' l
[u(x,~)]
(8.1.6)
where LW)[el and Lf1)[o] denote the energy terms corresponding to the operators L,(O)[*] and Lxt(l)[.], respectively. Typically, these terms involve integrations with respect to the spatial variable, with the integrands containing partial derivatives of the displacement. As shown later, equation (8.1.6) is used to derive a relation governing the velocity of the reference point xo. Note that since the oscillator under consideration is conservative and natural, the energy relation holds for all times. Using the functional relation (8.1 S ) , we express the partial derivatives of the displacement with respect to time using the chain rule of differentiation as follows: aurat =
azu/at2 = (a2u/au,2)uo,t2
+ (au/au,)u,,,,, ...
(8. I .7) where the subscript t denotes the derivative with respect to the time variable. Using (8.1.7), we express the energy relation as 1
EtOt= ( 1 / 2 ) ~ , , ~(aU/au,)2 2~ dx
+ L(O)[U] + ~L(~)[U,u,(t)l
(8.1.8)
0
where j=O, 1
Due to the previous assumption (3), L(O) does not contain any time derivatives; any time derivatives in EL(') are transformed into partial derivatives with respect to u, using the chain rule of differentiation. Relation (8.1.8) provides a means for computing the velocity of the reference point, that is, ( 8.1.9)
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
355
where O(E) represents corrections to the reference velocity due to the operator ~L(1)[U,u,(t)].Since the general analysis is only carried out up to O(E), expression (8.1.9) is sufficient for the perturbation analysis that follows. However, evaluation of higher-order approximations requires the computation of higher-order corrections in the expression of u,,?. Expressions (8.1.5) and (8.1.9) provide the displacement and velocity of the reference point in terms of the (yet unknown) modal function U[x,uo(t)]. Substituting these expressions into the equation of motion (8.1.4) and expressing the acceleration of the reference point by Uo,tt
=
{ Lx(O)[u(x,t>l+ ELx,(~)[u(x,t)l}x=x0
one obtains the following equation governing the modal function U[x,u,(t)]:
J 0
(au/au,)2dx
=L(O)[U(X,U,)]+ EC(L)[U(X,U,)]+ O(E),
x E [0,1]
(8.1.10)
subject to the boundary conditions B[U(x,u,(t)] = 0,
x = 0, 1
(8.1.1 1)
Operators C(O)[*],L(l)[*],and B[*] are derived from the original operators L,(O)[*], Lxl(l)[*], and B[*], by transforming the set of independent variables (x,t) + [x,uo(t)]. Denoting by uo* the maximum amplitude attained by uo(t), we note that equation (8.1.10) is singular at the maximum potential energy level.
because the coefficient of the highest derivative (d2U/au02) vanishes there. In the following analysis asymptotic approximations to the function U will be developed for values of the potential energy less than E,,, and will be analytically continued up to the maximum potential energy level. A similar asymptotic analysis is carried out in section 3.1, where nonsimilar NNMs of
356
NNMs IN CONTINUOUS SYSTEMS
discrete nonlinear oscillators are computed. The following condition is thus imposed at the maximum potential energy level:
Condition (8.1.12) guarantees the analytic continuation of the asymptotic solution up to the maximum potential energy level. Summarizing, the normal modal shape U[x,u,(t)] is governed by the singular partial differential equation (8.1.10) subject to the boundary condition (8.1.1 1) and the condition for analytic continuation (8.1.12). An asymptotic methodology for solving the system of equations (8.1.10)(8.1.12) is now developed by employing power series expansions. Far from the maximum potential energy level, the solution is represented by the series expansion
The linear.form of lJ(x,u,) is due to the previous assumption that the equation of motion is separable when E = 0. Note that since only nonlinearities of odd power are assumed, only odd terms appear in the series expression (8.1.15). Moreover, the function U(k) and all its partial derivatives are assumed to be of order ~ k that , is, anuU(k) ~~~
ax,,
= O ( E ~ ) , k 2 0,
n = 0,1,2,...
(8.1.16)
This assumption is necessary for ordering the various terms in the perturbation analysis that follows and is verified by the asymptotic results. Finally, note that, due to compatibility, al(o)(x,) = 1 and a2,,1(k)(xo) = 0, m = 0,1,2, ..., k 2 1. The various orders of approximation are computed by
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
357
substituting (8.1.13) into (8.1.10)-(8.1.12) and matching terms of various orders in E. The following subproblems then result. O(EO) Approximation
The equation governing the O( 1) approximation is given by
{ L(o)[al(o)(x)u,(t)l}x=xo al(o)(x)= Uo)[aP)(x)u,(t)l
(8.1.17)
with the boundary condition B[al(o)(x)u,(t)] = 0 and the compatibility relation al(o)(xo)= 1. Since U O ) is linear and does not depend explicitly on t, one can express the operator in (8.1.17) as
L(0)[a 1 (O)( x)uo(t)] = X [a1(O)(x)1u,(t)
(8.1.18)
where X[*]is a spatial operator. Substituting (8.1.18) into (8.1.17) and eliminating the trivial solution uo(t) = 0, one obtains the following equation governing X[*]:
which can be solved for al(o)(x) in compliance with the 0(1) boundary and compatibility conditions. It should be pointed out that since a w - m o 2= 0, the maximum potential energy condition (8.1.12) is identically met by the O( 1 ) solution.
O ( E ~Approximation ) Since u,,~= 0 when the potential energy is maximum, the equation governing the O(E) correction to the modal function is given by
358
NNMs IN CONTINUOUS SYSTEMS
(8.1.20) where u,* is the maximum of u,(t) and satisfies
To obtain (8.1.20) we performed series expansions of the various operators in terms of the small parameter E. Equation (8.1.20) is complemented by the following O(E) condition describing the analytic continuation of the solution at the maximum potential energy level:
The O(E) approximation is computed by truncating the infinite series for U(')(x,u,) to the desired order of approximation and equating like powers of u,(t) in the above equations. The resulting ordinary differential equation in the spatial variable is solved subject to the O(E) boundary conditions. The described analytic methodology can be carried out up to any desired order of approximation, provided that the expression for the velocity of the reference point [i.e., equation.(8.1.9)] is upgraded at each step of the calculation. The modal function is then approximated by
where m = 1, 3, ... Substituting this expression into the equation of motion (8.1.4), evaluating the resulting expression at the reference point x = x,, and taking into account the previous analytic results, one obtains the following nonlinear ordinary differential equation governing the displacement of the reference point:
1
l uo(t) = { ~ [ a l ( o ) ( x >,=,,uo(t)
+E
[cX[~,(')(X>]U,~(~> + L(l)[a~(~)(x)u,(t)]] + O ( E U ~ ~ + ~(8.1.22) ,E~) m
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
359
where X[*] is a spatial operator resulting from the separation of variables of the linear 0(1) operator Lx(0)[*]. Considering only terms up to O ( E U , ~ )one , can solve equation (8.1.22) analytically for a given set of initial conditions [u,(O), uO(0)],using a perturbation methodology (Nayfeh 1973, 1981). This solution will provide the nonlinear dependence of the frequency of the NNM on the amplitude. To demonstrate the application of the method, we analyze two examples involving nonlinear beam vibrations. In both applications, the main algebraic manipulations were performed using symbolic algebra (Mathematics running on HP-Apollo workstations). In the first example we consider the vibration of a linear, simply supported beam resting on an elastic foundation. This structure was first considered by Shaw and Pierre (19941, and its NNMs were obtained by the invariant manifold approach discussed later. The force exerted by the nonlinear foundation is assumed to contain linear and cubic terms, and the equation of motion is given by Utt
= -u,,,,
- ku - ~ y ~ 3 ,x E
[0,1]
(8.1.23)
where u(x,t) is the transverse displacement of the beam, and the boundary conditions are given by u(0,t) = u(1,t) = u,,(O,t) = uxx(l,t)= 0 The NNMs of this system are computed by introducing a modal function defined by u(x,t) = U[x,u,(t)], where uo(t) = u(xo,t) is the displacement of a reference point, which is assumed not to coincide with any node of the nonlinear mode under investigation. The energy of the system during the nonlinear mode oscillation is expressed as 1
Etot= (1/2)1 [ut2 + uXx2+ ku2
+ (1/2)&yu4]dx
0
and the velocity of the reference point is given by
(8.1.24)
360
NNMs IN CONTINUOUS SYSTEMS
I 1
2Et,, -
[Uxx*+ kU* + (1/2)&yU4]dx (8.1.25)
Applying the modal relation to the equation of motion, expressing u,,,2 by (8.1.25), and taking into account the fact that Uo,tt
= { - U X X X X - ku - EyU3 } x=x,,
one obtains the following singular partial differential equation governing the modal function U(x,u,):
This equation is a special case of the functional equation (8.1.10) of the general theory. The condition satisfying analytic continuation of the solution up to the maximum of the potential energy is given by
au + UxxXx+ kU + &yU31uo=uo* = 0 [ { - u x x x x - kU - w31x=xo aU,
(8.1.27) and is similar to condition (8.1.1 1) of the general theory. For potential energy values less than Etot,solutions to (8.1.26) and (8.1.27) are sought in the series form (8.1.13)-(8.1.16) and are computed by solving the following problems at leading orders of approximation. The first approximation to the modal function is computed by solving:
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
361
Imposing separation of variables, U(O)(x,u,) = al(o)(x)u,(t), for nontrivial solutions [u,(t) # 01, one obtains the following ordinary differential equation governing the spatial coefficient al(O)(x):
with the following boundary and compatibility conditions
where the prime denotes differentiation with respect to the spatial variable x. The constant coefficient { a1( o Y x > x=xo = h4
1
depends on the unknown solution and must be positive for bounded free oscillations [assumption (2) of the general formulation]. The O( 1) problem is thus reduced to a fourth-order ordinary differential equation in the spatial variable, which, taking into account the O( 1) boundary conditions, leads to the following solution sin(nnx) al(o)(x) = sin(nnx,)'
n4n4 = h,4,
n = 1,2,...
(8.1.30)
The restriction that x, is not a node of the 0(1) solution leads to the condition sin(nnx0) f 0. Thus, to the first approximation, the nth nonlinear normal mode is given by (8.1.3 I ) which is identical to the separable solution of the linearized problem, corresponding to E = 0. To compute the leading nonlinear corrections to the modal function, one must consider the O(E) terms. The O(E) functional equation governing the term U(1)of the modal function is
362
NNMs IN CONTINUOUS SYSTEMS
which corresponds to expression (8.1.20) of the general theory. This equation is complemented by a condition at the maximum of the potential energy similar to (8.1.21), that is,
(8.1.33) Equations (8.1.32) and (8.1.33) are complemented by the O(E) boundary and compatibility conditions U(l)(O,U,) = U(')(l,u,) = u~~),,(0,u,)= u(l)xx(l,u,)= 0 and
U(')(x,,t) = 0
In this application, only O(E) corrections of third order in the displacement will be considered, and the following truncated series expansion for U(l)(x,u,) is assumed: U(l)(x,u,) = al(l)(x)uo(t) + a$I)(x) uo3(t) +...
(8.1.34)
Substituting (8.1.34) into (8.1.32), applying the appropriate compatibility relations, and matching respective coefficients of uo(t), one obtains the following ordinary differential equation governing the spatial coefficients al(I)(x)and a+I)(x):
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
363
Equation (8.1.35) relates the cubic spatial coefficient a$])(x) to the linear coefficient al(l)(x). Substituting (8.1.34) into condition (8.1.33) and taking into account (8.1.33, one obtains the following nonhomogeneous, linear, ordinary differential equation with constant coefficients, in terms of al(l)(x)
Note that the nonhomogeneous term in (8.1.36) depends on the (already computed) first approximation al (O)(x) and on the (yet undetermined) constant al(l)(xo).In addition, equation (8.1.36) is formulated for the nth nonlinear normal mode (n = 1,2,..,), and is complemented by the boundary conditions: dJal(l)(O)/dxJ= dJalCl)(l)/dxJ= 0
for j = 0,2,4,6
The solution of (8.1.36) is standard, composed by homogeneous and particular solution terms: 3~y(n4n4+k)u,*2 al(l)(x)= 320nW(9n4~4-k) Combining (8.1.37) and (8.1.35), and the cubic spatial coefficient a3(l)(s) is computed as follows a+l)(x) =
[
EY sjn(nnx) sin3(nnx) 8(9n4n4-k) sin(nnx,) - sin'(nnx,)
]
(8.1.38)
Note that the above expressions become singular when k = 9114x4. The same finding was reported by Shaw and Pierre (1994), and it is due to a 3:l internal resonance between modes (n) and (3n). It is therefore concluded, that the aforementioned formulation is only vaEid for computing N N M s that
364
NNMs IN CONTINUOUS SYSTEMS
are not in internal resonance with any other modes. From the aforementioned results it is also noted that for fixed structural parameters k and y, the differences between linear and nonlinear normal modes decrease as the order of the mode, n, increases. Combining the previous solutions, the following asymptotic approximation for the modal function is obtained:
+ o(Eu,~(~),~ 2 )
(8.1.39)
This result is in full agreement with that found by Shaw and Pierre (1994) using the invariant manifold approach, which is discussed later. The response of the reference point, uo(t>,can be analytically approximated by substituting (8.1.39) into the equation of motion (8.1.23) and solving the resulting ordinary differential equation in uo(t):
3
uo(t) = uo*cos(m,t ) + O(E),
n = 1,2,...
(8.1.40)
where the frequency of oscillation of the NNM is given by
(8.1.4I ) with initial conditions uo(0) = uO*and &(O) = 0. It may at first seem counterintuitive that the frequency of oscillation is a function of the point xo. This dependence on xo can be eliminated, however, by considering the following argument presented by Shaw and Pierre (1994). Consider a purely modal motion for an odd-numbered mode that has a maximum midpoint displacement of magnitude A. The maximum displacement at xo is then uo* = Asin(n7cx0),and the frequency of oscillation is given by
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
365
(8.1.42) i.e., is not a function of the reference point x,. A stability analysis of the NNMs (8.1.39) and (8.1.40) is performed in a later section. The second application considers the planar vibrations of a cantilever beam with geometric nonlinearities. Assuming finite deflections and inextensionality, nonlinear terms in the governing partial differential equation arise due to the nonlinear relation between curvature and displacement, and a nonzero longitudinal inertia. For in-plane motions, the equation governing the transverse displacements was developed by Crespo da Silva and Glynn (1 978) (see also section 9.1):
Utt
+ uxxxx= - E { ux[uxuXxIx + (1/2)ux
1
0
(I 5
ux2 dh),, d c
0
}
x E [0,1] (8.1.43) where the previously introduced notation holds and the boundary conditions are u(0,t) = u,(O,t) = uxx(l,t) = uxxx(1,t) = 0 The first term in the right-hand side of (8.1.43) accounts for nonlinear curvature effects, whereas the second represents nonlinear longitudinal inertia effects. Introducing the functional relation u(x,t) = U[x,u,(t)], u,(t) = u(x,,t), the energy for free oscillations is expressed as: (ut2
+ uxx2) dx + ET + EV
(8.1.44)
0
where ET and EV are O(E) contributions to the kinetic and potential energy, respectively, due to the geometric nonlinearities. Substituting the displacement functional relation into the energy equation, the velocity of the reference point x = xo is computed as:
366
NNMs IN CONTINUOUS SYSTEMS
Uxx2dx - ET - EV
2EtO1U",?
0
i
=
(8.1.45)
(au/au,)2 dx
0
Introducing the displacement functional relation into the equation of motion and expressing u",~and uo,ltin terms of U, one obtains the following singular partial differential equation governing the modal function: 1
2E10, -
1 Uxx2dx - ET - EV
= - u x x x x - E[ux[uxux,Ix
0
1
For simplicity, the quantities uo,tand uo,ttin the nonlinear inertial term were not expanded in terms of U. From (8.1.46) one can easily derive the condition that applies at the maximum potential energy level, that is 1
uxx2dx
(1/2) 0
+ EV = El,,
8.1 SYSTEMS OF FINITE SPATIAL EXTENT'
367
Figure 8.1.1 NNMs of a cantilever beam with geometric nonlinearities: (a) n = 1, E = 0.01, x, = 0.5, uo* = 1.6 and (b) n = 2, E = 0.01, xo = 0.5, uo* = -0.6; ___ Nonlinear theory, ------ linear theory. As in the general formulation, one seeks an asymptotic approximation to the modal function in the series form (8.1.13). The first-order approximation is the linearized mode shape al(o)(x)=
[sin(h,,x> - sinh(h,x)] + Pn[cos(hnx) - C O S ~ ( ~ , ~ X ) ] [sin(hnxo)- sinh(3inxO)]+ p,[CoS(l,nXo) - ~ ~ ~ h ( h n ~ o ) ] (8.1.47)
368
NNMs IN CONTINUOUS SYSTEMS
where h, is the n-th root of the characteristic equation cos(h)cosh(h) + 1 = 0
and
PI, = [cos(h,) + cosh(h,)]
[sin&) - sinh(h,)]-1
Again, point x, is restricted from being a node of the 0(1) approximation. The O(E) approximation is expressed in the following truncated series U(l)(x,u,) = al(l)(x) uo(t) + a3(l)(x) u,'(t) +...
(8.1.48)
where the spatial coefficients are computed by a procedure similar to the one used in the previous section for the simply supported beam. The ordinary differential equations evaluating al(l)(x) and a'(l)(x) were derived using Marhematica. Retaining only terms up to O(EU,~) in (8.1.46) one expresses the cubic coefficient a3(1)(x)as follows:
1 0
x
-
[aI(WI
I i
1 0
a~(Q(h)'* dh dc]'}
J! J!
al(o)(x)"*d x
0
with O(E) boundary and compatibility conditions
U0**
al(o)(x)*dx (8.1.49)
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
369
Imposing the condition for analytic continuation up to maximum potential energy and taking into account (8.1.49), one obtains the following nonhomogeneous, linear, ordinary differential equation governing a1(I)(xj:
where
The solution of this equation was explicitly computed using Mathemuticu, but the results are too lengthy to be reproduced here. For E = 0.01 the NNMs corresponding to n = 1 and 2 are presented in Figure 8.1.1. N N M 1 corresponds to u,* = 1.6 and xo = 0.5, whereas NNM 2 to uo* = -0.6 and xo = 0.5. For comparison purposes the linearized modes are also shown in the Figure. It is noted that for the cantilever beam the nonlinear effects are small over the range of amplitudes considered in this study. The stability of the detected modes is investigated in a following section.
370
NNMs IN CONTINUOUS SYSTEMS
Invariant-Manifold Formulation f o r NNMs The invariant manifold formulation was developed by Shaw and Pierre (1992) for computing NNMs of damped or undamped one-dimensional continuous systems. Its main advantage is that it can be applied even for systems with viscous damping, contrary to the previously outlined energybased technique. A disadvantage of the invariant manifold approach is that it requires a greater amount of analytical computations, and thus is more cumbersome. The method is developed under the assumption that the weakly nonlinear equations of motion (8.1.1) can be expressed in the following form: u,(x,t> = 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:
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
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
NNMs IN CONTINUOUS SYSTEMS
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:
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
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
NNMs IN CONTINUOUS SYSTEMS
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
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
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
NNMs IN CONTINUOUS SYSTEMS
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)
8.1 SYSTEMS OF FINITE SPATIAL EXTENT
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
NNMs IN CONTINUOUS SYSTEMS
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
NNMs IN CONTINUOUS SYSTEMS
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
8.2 SYSTEMS OF INFINITE SPATIAL EXTENT
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
NNMs IN CONTINUOUS SYSTEMS
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 <,, i > 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 <,) 1=
1
Clearly such stationary waves can be regarded as NNMs of the continuou,s system (8.2.2); hence, the previous analytic formulation can be used to extend the notion of nonlinear normal mode to systems of infinite spatial extent. Note that traveling waves with nonzero wavenumbers (k f 0) do not lead to synchronous motions of the particles of the chain, and, thus, cannot be studied in the framework of NNMs. In what follows, the stationary waves of (8.2.2) are studied in more detail.
Stationary Waves Considering the discrete chain of oscillators governed by (8.2. I ) , one seeks stationary periodic waves corresponding to synchronous periodic motions of all masses of the system. Applying the definitions introduced in previous chapters such motions can be regarded as NNMs defined over an infinite spatial extent. For a stationary wave the nth displacement un(t) is expressed as
8.2 SYSTEMS OF INFINITE SPATIAL EXTENT
385
where T is the nonsmooth temporal transformation introduced in chapter 6 and o is the frequency of oscillation of the stationary wave. By defining T according to (8.2.9) the assumption of synchronicity of the motions of all masses of the system is imposed. Substituting (8.2.9) into (8.2.1) the equations of motion are expressed as
d2U,
a2
+ ~ U , ( T )- U,-l(T)
-
Un+,(T) + f(U,) = 0,
n = O,*1,*2, ... (8.2.10)
under the following condition (cf. chapter 6) (8.2.11) One analytically computes stationary waves of the system (8.2.10) and (8.2.11) by successive approximations of the following form (cf. chapter 4):
where E is a scaling parameter introduced for book-keeping purposes. In the first step of the iteration only the leading approximations in representations (8.2.12) are taken into account and the following equations are obtained:
(8.2.13) where n = 0,&1,&2,.., Equations (8.2.13) represent the O(EO) terms in equations (8.2.10) and (8.2.1 1). Integrating the first of equations (8.2.13) one obtains the following solution for the first-order approximations:
386
NNMs IN CONTINUOUS SYSTEMS
Imposing the conditions on the derivatives of U,(l) at z = I one obtains the following set of algebraic equations in terms of the variables A,(O): -2A,(o)
+ A,-l(o) + An+l(0)+ 2[An(o)/b- Il(A,(O))/A,(O)] = 0
(8.2.15) where Il'(u) = f(u). Using (8.2.15) the solution for U,(1) assumes the form Un(1) = -b( [A,(O)/ho - n(An(O))/A,(O)](~3/3)
+
j
(z
- t)f(A,(O)C)
dc),
n = O,+l,f2,...
(8.2.16)
0
At this point one applies the continuum limit approximation to study the vibrations of the chain. For this approximation to hold the wavelength of the envelope of the motion must be larger that the distance between neighboring particles of the chain, and the variable A,(') must depend smoothly on the index n. Substituting the difference term in (8.2.15) by an ordinary derivative,
in the continuum limit one reduces the set of equations governing the variables A,@)to d2A,(o) dn2 + 2[A,(o)/ho - ll(An(o))/An(o)] =0 (8.2.17) In the special case when A,@) does not depend on the index n, one finds that d2An(0)/dn2= 0 and from (8.2.17) computes ho as ho = A(O)WI(A(O)) where A,(') = A@), n = O,fl,f2, ... In this case all particles of the chain execute uniform in phase vibrations with identical amplitudes and the system oscillates in an in phase NNM. In general, the quantities A,(") are expected to depend on n, in which case for a given value of ho equation (8.2.17) governes the envelope of the stationary wave. It can be shown that provided that (I/ho)> f(0)/2, in a sufficiently small neighborhood of A,W = 0 there exists a family of periodic solutions of (8.2.17) parametrized by ho. How large this neighborhood is, is dictated by the nearest distance of the roots of the equation
8.2 SYSTEMS OF INFINITE SPATIAL EXTENT
387
Equation (8.2.18) possesses nontrivial roots only when the function f(x) = II'(x) increases faster than linearly, or equivalently, when the nonlinear chain possesses stiffening nonlinear characteristics. In that case there exist homoclinic or heteroclinic orbits in the phase plane of system (8.2.17) separating periodic from nonperiodic solutions of A,(@. The family of periodic solutions of (8.2.17) corresponds to a family of stationary timeperiodic waves whose envelopes vary periodically in space. These motions can be regarded as NNMs of the infinite chain. Homoclinic or heteroclinic orbits can be regarded as limiting cases of periodic orbits with infinite wavenumbers. As shown in (Vedenova and Manevitch, 1981) and (Vakakis et al., 1993b) these special orbits correspond to solitary waves with spatially localized envelopes or envelope slopes. An analysis of solitary waves by NNM techniques is postponed until chapter 9. As an application of the previous methodology consider the case of a nonlinear restoring force f(*) proportional to the mth power of the displacement, f(U,) = y U p , where y > 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
NNMs IN CONTINUOUS SYSTEMS
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
8.2 SYSTEMS OF INFINITE SPATIAL EXTENT
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
NNMs IN CONTINUOUS SYSTEMS
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 u<,(z) is introduced defined as the displacement of the kth layer at a reference position x = x,, U"(T)
= U(k)(X,,T)
f
0
(8.2.26)
To study standing waves in AZs of the nonlinear periodic assembly one introduces the change of variables, ( x , ~ + ) [x,u,,(z)] and seeks solutions in the form i = 0,+1,+2, ...,
U(k)[u,(z),x,] = u,(T) (8.2.27) The functions U(ij(.,*) determine the spatial distribution of the synchronous motion using the reference displacement as parametrizing coordinate. Following the energy-based methodology described in the previous sections the displacement field in the layers of the periodic system during a nonlinear standing wave can be analytically approximated as u(l)(x,z) = U(i)[u,(z),x],
where the coefficients a,(l)n(x) are determined by solving infinite systems of homogeneous and nonhomogeneous difference equations (Vakakis and King, 1995; Mickens, 1987). It can be shown that, in contrast to linear theory the nonlinear standing waves (8.2.28) possess multiple wavenumbers and propagation constants. Also, a prerequisite of the analysis is the nonexistence of resonance conditions between modes that lead to complicated nonlinear modal interactions that render the analytical representation (8.2.28) invalid.
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 9 NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS The analytical methods developed in chapter 8 for computing NNMs in continuous systems are now extended to the study of nonlinear mode localization and passive motion confinement in systems of coupled nonlinear beams. In section 9.1 analytical and numerical investigations of transient and steady state localized motions are presented. In section 9.1.1 discretization of the governing partial differential equations of motion is carried out and nonlinear mode localization is studied. In sections 9.1.2 and 9.1.3 it is analytically and numerically shown that nonlinear mode localization leads to passive motion confinement of disturbances generated by external harmonic or impulsive loads. In section 9.1.4 an alternative method for studying nonlinear mode localization in continuous oscillators is presented, by asymptotically analyzing the equations of motion with no discretization involved. An experimental verification of nonlinear motion confinement (localization) in an experimental fixture of coupled beams with actively induced stiffness nonlinearities is carried out in section 9.2; to the authors' knowledge these experimental results are the first to confirm the existence of the nonlinear mode localization phenomenon in a practical flexible structure.
9.1 THEORETICAL ANALYSIS 9.1.1 Nonlinear Mode Localization: Discretization The first system considered in this section consists of two weakly coupled flexible cantilever beams and is depicted in Figure 9.1.1. Geometric nonlinearities arising due to the nonlinear relation between curvature and transverse displacement and longitudinal inertia are considered, and a variety of localized NNMs is studied. A variety of works on mode localization in linear periodic structures exists, but there are only a few works dealing with nonlinear mode localization in continuous oscillators. Localized modes in linear cyclic assemblies of cantilever beams modeling large space reflectors were investigated in (Bendiksen, 1987; Cornwell and Bendiksen, 1989; 391
392
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
Figure 9.1.1 The system of weakly coupled nonlinear beams.
Pierre and Cha, 1989). Linear mode localization in such flexible structures occurred only in the presence of structural disorders and for sufficiently weak coupling between substructures. Linear mode localization was investigated both analytically and numerically. It was shown that higher flexible modes were more susceptible to localization than lower ones. Moreover, the strength of the localization phenomenon was found to depend on the position of the coupling stiffness. A recent experiment (Levine and Salama, 1991, 1992) proved the existence of certain linear localized modes in a circular antenna with 12 flexible ribs and a gimballed central hub. To the authors' knowledge, this study was the first experimental verification of the linear mode localization phenomenon in a practical engineering structure. In agreement with existing theories, the "second bending group" of the antenna (ribs oscillating in their second flexible mode) was more effectively localized than the first group. Moreover, stronger localization was observed with increasing modal band. In an additional recent work (Brown, 1993), the vibrations of rotating turbine-bladed disk assemblies were experimentally investigated. Blades of relatively large length (15 inches) were considered, and it was observed that under certain operational speeds a single blade (blade 11 of the tested bladed assembly) vibrated with an amplitude much greater than that of the other blades. It is the authors' belief that this vibration localization was due to the geometric nonlinearities of the (flexible) blades, and thus that (Brown, 1993)
9.1 THEORETICAL ANALYSIS
393
is the first reported experimental confirmation of nonlinear vibrational localization in rotating cyclic assemblies. The structure of Figure 9.1.1 consists of two homogeneous, linearly elastic beams of identical material properties and dimensions, which are coupled by means of a linear stiffness. The following analysis follows closely (King and Vakakis, 1995b). Assuming no out-of-plane components of motion, and finite-amplitude oscillations, the nonlinear relation between curvature and transverse displacement, and the longitudinal inertia of the beams give rise to geometric nonlinearities that can greatly influence the dynamic response. Considering initially a single beam, the governing equation of motion can be expressed as: PVTT
PUTT
+ uX)* + EIuxxxvx(1 + UX) - Tvx]x = [-EIvxxx (1 + uX)vx+ EIuxxx(vx)2+ T( 1 + uX)]x
= [-EIvxxx(l
(9.1.1)
where x is the arclength of the beam per unit length, z is the time, v and u are the transverse and longitudinal displacements of the beam, respectively, E is the modulus of elasticity of the material, I is moment of inertia of the cross section of the beam about an axis orthogonal to the plane of the motion, p is the density per unit length, T is the internal axial force in the beam and the short-hand notation for partial differentiation is used, for example, (*)XX = @(*)/dx2. Equations (9.1.1) are derived using the exact nonlinear relation between the curvature K and the displacement v: K = vxx[1 + (Vx)2]-3/2 and by assuming that no extension takes place in the midplane axis of the beam, i.e., that (1 + ux)2 + (vx)2 = 1 (condition of inextensionality). Moreover, it is assumed that the beams under consideration are slender and that shear deformations of the cross section or rotary inertia effects are negligible. Equations (9.1.1) are not amenable to a perturbation analysis, so one expands the various powers of the displacements on the right-hand side terms and retains only terms up to O(v3). A similar procedure was followed in previous works by Crespo Da Silva and Glynn (1978), Crespo Da Silva and Zaretzky (1990>,and Crespo Da Silva (1991), where the nonlinear equations of motion were derived by Hamilton's extended principle, and nonlinear curvature and inertia terms were taken into account; in these works the interaction between torsional and flexural motions was also investigated. A
394
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
more general formulation for studying the nonlinear flexural and torsional vibrations of homogeneous and composite beams can be found in the work by Pai (1992). In the present formulation it is assumed that the beams have large torsional rigidities and that no internal resonance occurs between torsional and flexural modes. Therefore, only transverse and axial displacements are considered and the following approximate equations describing the motion of each beam result:
Vtt
+ vxxxx +
x
s
I
0
{ v x ~ v x v x x+~(1/2)vXj x [I X
u(x,t) = (-1/2)j v,*(s,t) ds
vx2 dultt ds},
= o + o(V5)
+ O(v4) = O(v2)
(9.1.2)
0
where the time t is nondimensionalized according to the formula t = z(EI/pL4)1/2. A similar expression can be derived for the internal axial force, T, indicating that T = O(v2). The first nonlinear term in (9.1.2) is due to the nonlinear relation between the curvature and the transverse displacement of the beam, whereas the second represents the dynamic effect of the nonlinear longitudinal inertia of the beam. Expressions (9.1.2) indicate that the longitudinal displacement u(x,t) is of an order higher than the transverse displacement v(x,t), and therefore of much smaller magnitude. Before formulating the equations governing the motion of the system it is necessary to evaluate the vertical and horizontal components of the coupling forces that are exerted from the linear stiffness to the beams. Denoting the displacements at the ends of the coupling stiffness by up(c,t) and vp(c,t), p = 1,2, letting c = 1/L, and referring to Figure 9.1.2, the vertical and horizontal components of the coupling force are expressed as:
F, = K[vz(c,t)
- vl(c,t)]
+ O(kvi2)
C
Fu = -(W2D)[v2(c7t) - vl(c,t)lJ [v2x2(Lt) - v1x2(5,t)I dk+ 0(vi4) 0
(9.1.3) where K is the stiffness constant and D is the distance between the beams. At this point it is additionally assumed that the transverse displacements ure
9.1 THEORETICAL ANALYSIS
395
D Figure 9.1.2 Deformation of the coupling element during the oscillation of the system. I small and that the stiffness coupling the beams is weak. These assumptions are met by introducing the following scalings for the transverse displacement and coupling stiffness, v -+ E * ’ ~ vand K -+ Ek, where E is a parameter of perturbation order. Taking these scalings into account, the components (9.1.3) of the coupling force are expressed as:
F, = ~3/2k[v2(c,t)- vl(c,t)]
+ O(E~),
F U = O(~5’2)
(9.1.4)
Taking into account the previously introduced scalings and expressions (9.1.2) and (9.1.4), the equations describing the oscillation of the coupled beams take the form: x Vptt
+ vpxxxx= -E{ vpx[vpxvpxxlx+ ( W p x J up(x,t) = O(E),
s
[ J vpx2 dultt ds}x p,m=1,2,
pzm (9.1.5)
where vp and up denote the transverse and longitudinal displacements of beam i, a common notation, x, for the arclengths per unit length of the two beams is used, and all quantities appearing in (9.1.5) other that E are assumed to be of O(1). It must be noted that expressions (9.1.5) are valid only for weak coupling stiffnesses (K = Ek), since only then the vertical component of the coupling force is of higher perturbation order and thus can be neglected
396
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
from the first-order averaging analysis that follows. Hence, the remaining analysis of this section is carried out under the assumption of weak coupling. For small E equations (9.1.5) form a set of weakly nonlinear and weakly coupled partial differential equations; in what follows these equations are discretized and then analyzed by the method of multiple scales. Assuming identical beams, one approximates the transverse displacements Vj(X,t) by the following series expression: vp(x,t>=
n
C $i(x)qpi(t)
(9.1.6a)
i= 1
where the functions $i(X) are the normalized eigenfunctions of the linear part of (9.1.5) (corresponding to E = 0) and are given by:
where
and Wi is determined by solving the characteristic equation,
The ith linearized natural frequency is then given by
Oi =
2 vi. Substituting
(9.1.64 into (9.1S ) , premultiplying by the normalized eigenfunctions Qi(x) and using the orthogonality relations for the linearized eigenfunctions by integrating from x = 0 to x = 1, one obtains the following set of ordinary differential modal equations with qpi(t) as dependent variables:
(9.1.7) where first subscript of qpi represents the beam number, p = 1,2, p = 3 = 1, and the second subscript denotes the order of the linearized mode-shape, i = 1,2,...,n. The various coefficients in (9.1.7) are defined as follows:
9.1 THEORETICAL ANALYSIS
397
where prime denotes differentiation with respect to the argument. The numerical values for some of coefficients (9.1.8) can be found in (King and Vakakis, 1995b). Equations (9.1.7) form a set of weakly nonlinear ordinary differential equations. The nonlinear localized free motions are periodic solutions of these equations and are now investigated using the method of multiple scales (Nayfeh and Mook, 1984). To this end, "fast" and "slow" time scales, To = t and Ti = Et, respectively, are introduced and the responses qpi are expressed in the series form: m
qpi(T03T1)=
C Ekqpik(To,Tl>
k=O
(9.1.9)
Substituting (9.1.9) into (9.1.7), and matching coefficients of respective powers of E, one obtains the following problems at various orders of approximation:
O ( E ~ Approximation ) 2
2
Do qpil + oi qpil = -2DODlqpio-
3
3
3
C [aibk1qpboqpkoqp10 1=1
b=l k=l 3 2 + ( b i b k l ~ ~ ) ~ p b o ~ O [ ~ p-k o ~Yiblqpbo pl~l~ b= 1
c
F)
p = 1,2,
- q(p+l)boI i = 1,..., n
(9.1.11)
is the complex conjugate, and j = (-1)1/2. where Dj(*) = a(*)/aTj, j = 0,1, The complex amplitudes Api(T 1) are evaluated by substituting solutions (9.1.10) into (9.1.11) and eliminating the secular terms, i.e., terms on the right-hand sides of (9.1.1 1) of frequency Oi. At this point, it is noted that a low-order internal resonance exists between the second and third flexural
398
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
cantilever modes @2(x)and @3(x), since their corresponding linearized natural frequencies are nearly integrably related, 0 3 302. Such a loworder internal resonance is well known (Bogoliubov and Mitropolsky, 1961; Nayfeh and Mook, 1984) to lead to nonlinear transfer of energy between the corresponding linearized modes. This internal resonance is expected to influence the structure of the nonlinear mode localization in the flexible system under consideration. To study the effects of the low-order "internal resonance" one introduces a detuning parameter, 0,defined as:
-
03=
302 + E(T
(9.1.12)
Parameter 0 quantifies the closeness of the multiples of the natural frequencies of the linearized modes in internal resonance. At this point the series expression (9.1.6a) is truncated to n = 3, i.e., it is assumed that only the first three linearized modes participate in the beam responses. Taking into account (9.1.12) and eliminating secular terms in equations (9.1. I l), one obtains a set of first order differential equations in terms of the complex amplitudes Api(Tl), p = 1,2, i = 1,2,3. Introducing the polar transformation, Api(T1) = ( l/2)api(T1)J0Pi(T1)
(9.1.13)
where api and 0pi are real amplitudes and phases, substituting into the equations of the complex amplitudes Api(Tl), and separating real and imaginary parts, one obtains a set of first-order modulation equations for the modal amplitudes and phases of the two beams:
n
9.1 THEORETICAL ANALYSIS
399
where primes denote differentiation with respect to Ti, and the coefficients appearing in (9.1.14)-(9.1.17) are defined as: qpp
2
2
2
2bipip(oi + 0 1 - 2(aiipp + aippi + aipip) P 6 = b2232(03 - 0212 + 20ib2322 - a2223 - a2232 - a2322
hpbpppp - 3apppp,
Yip
2
6 202b3222 - a3222
(9.1.1 8)
Equations (9.1.14c), (9.1.15c), (9.1.16c), and (9.1.17) are derived by combining the modulation equations for the phases 0pi and using the following definitions for the variables aPand Y 1:
400
NONLINEAR LOCALEATION IN SYSTEMS OF COUPLED BEAMS
Variable Q i , i = 1,2,3 denotes the phase difference between the ith linearized modes of beams 1 and 2, whereas variable "1 is a measure of the phase difference between the internally resonant modes 2 and 3 of beam 1 (a similar variable for beam 2 can be expressed in terms of the previously defined angles, "2 = "1 + a3 - 3@2, but it is not used in the analysis). In writing (9.1.14)-(9.1.17), all amplitudes api are assumed to be nonzero quantities. The periodic solutions of the system are obtained by imposing the stationarity conditions "pi' = 0, @ i f = 0, and " 1 ' = 0, p = 1,2, i = 1,2,3. The resulting free motions of the beams associated with the stationary solutions api*, @pi*(Tl)are given by:
p = 1,2 (9.1.20) where the previously introduced scaling of the transverse displacements, v + &1'2v, was taken into account. The stability of the stationary solutions of (9.1.14)-(9.1.17) can be studied by forming the appropriate system of linear variational equations and computing the eigenvalues of the associated matrix of coefficients (Nayfeh and Mook, 1984). In what follows the localized stationary solutions of system (9.1.14)(9.1.17) are numerically computed. Two classes of these solutions are investigated. In the first class of localized motions only the first linearized cantilever modes of the two beams participate. The second class involves participation of the second and third linearized cantilever modes of the beams; it is shown that the internal resonance between these two modes affects in an essential way the structure of nonlinear mode localization and gives rise to a very complicated series of NNM bifurcations. Case I : Nonlinear Mode Localization in the Absence of Internal Resonance Assuming that both beams oscillate in their first cantilever modes, one sets ap2 = ap3 = 0, api f 0, p = 1, 2. Such an assumption is valid since the first linearized natural frequency is not commensurable with the second and third linearized natural frequencies (i.e., no low-order internal resonance exists involving the first mode), and, thus, no nonlinear coupling between the first
9.1 THEORETICAL ANALYSIS
401
and higher modes exists. Under these assumptions, one obtains the following simplified set of modulation equations: o l a l l ' = (1/2)ylla2lsin@i, wi@i' =
oia21'=-(1/2)yllaiisin@l
( 1 / 2 ) ~ i i [ ( a-~a:l)l(a~ia2i)]cos@i ~ + (1/8F111[a?1 (9.1.21)
where y11 and q 11 are given by,
y11 = (kL4/EI)@12(1/L) 2 q i i 5 2 0 1 P i i i i - 3ai111
1
= 2o:j
x i
01[01*5 0
1
5 01'01' d h d i l ' d x - 35 01[~1'(01'01">'1'dx 0
1 0
(9.1.22) Parameter y11 is the ratio of the coupling stiffness versus the linearized beam stiffness, whereas ~ 1 is1 a measure of the strength of the geometric nonlinearities. A relation describing the conservation of energy is obtained by multiplying the first two equations in (9.1.21) by a1 1 and a21, respectively, adding them, and integrating the resulting expression: 2
2
a l l +a21 = P 2
(9.1.23)
where p > 0 is the energy-like constant of integration. The periodic motions of the system are studied by computing the stationary conditions of (9.1.2 1). The following solutions for the amplitudes and phase result: @ I = 0, a l l = a21 = 2-1/33 (9.1.24a) a1 1 = a21 = 2-1/2 p (9.1.24b) @ I = n,
The solutions (9.1.24a,b) correspond to in-phase and antiphase NNMs, respectively. Solutions (9.1.244 represent two branches of NNMs that bifurcate from the antiphase mode through a pitchfork bifurcation (cf. Figure 9.1.3). A stability analysis of the variational equations associated with
402
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
system (9.1.2 1) reveals that the in-phase and bifurcating modes are orbitally stable; the antiphase mode is stable up to the mode bifurcation, and looses stability afterward. For constant energy (p constant), as the ratio (yl i/Iq 1 1I ) tends to zero the bifurcating modes become localized and the energy of the corresponding free motion is mainly confined to only one of the two beams [cf. Figure 9.1.3(a)]. lim (r11/11111 I)+O
(a11/a21] = 0
or
60
(nonlinear localization)
(9.1.25) In physical terms, the parameter (yli/Iq 1 1 I) represents the ratio of the coupling versus nonlinear forces. Result (9.1.25) then indicates that when both beams oscillate in their primary bending mode, nonlinear mode localization occurs when the linear coupling stiffness is weak and/or the nonlinear stiffness forces are relatively strong. Moreover, the point where the NNM bifurcation occurs depends upon the relative stiffnesses of the beams and coupling element, the position of the coupling element, c = 1/L, and the energy-like term, p. From Figure 9.1.3(b) it is seen that as c increases, the bifurcation occurs at increasingly lower values of the ratio (kL4IEI)Ilq 111. Note that since the beams in the system under consideration are assumed to be identical, nonlinear mode localization takes place even irz the absence of any mistuning (structural asymmetries) of the system. This is in contrast to linear mode localization which occurs only in weakly asymmetric and weakly coupled systems (Cornwell and Bendiksen, 1989; Pierre and Cha, 1989). The results presented in this section agree qualitatively with those derived in section 8 for discrete systems. This is due to the fact that, since the beams were assumed to oscillate only in their first flexural mode, the resulting modulation equations are similar to those of a two degree-of-freedom nonlinear oscillator. In what follows it is shown that when the internal resonance between the second and third modes is taken into account, the localization picture changes completely, and a series of very complicated NNM bifurcation phenomena occur. Case 2: Nonlinear Mode Localization in the Presence of Internal Resonance Localized solutions are now sought satisfying the conditions ap2, ap3 f 0, apl = 0, p = 1,2, i.e., with the two beams oscillating only in their second and
9.1 THEORETICAL ANALYSIS
403
third flexural modes. This assumption is made in order to examine the effect of the internal resonance on nonlinear mode localization. The modulation equations in this case are given by (9.1.15)-(9.1.17). Combining equations (9.1.15a,b) and (9.1.16a,b), one can show that the following energy-like relation is satisfied by the modal amplitudes: (9.1.26) This relation is similar to equation (9.1.23), which holds when the beams oscillate in their first flexural mode. The free periodic motions of the system are obtained by solving the set of stationary algebraic equations, which is obtained by setting the time derivatives of the amplitudes and phases in the modulation equations equal to zero. Since no analytic solutions of the stationary equations exist, these equations were solved numerically and the localized oscillations of the system were studied. Although a variety of nonlocalized NNMs exist in the system under consideration, in this work only localized NNMs are investigated, i.e., free oscillations during which only one of the beams oscillates with finite amplitude, whereas the other remains nearly motionless. For the numerical calculations the parameters p and E were assigned the values p = 0.5, E = 0.001, whereas the coupling parameters rij, were varied by changing the position of the coupling stiffness c = 1/L and the strength of the stiffness parameter kL4/EI. An interesting feature of mode localization in the presence of internal resonance is its essential dependence on the position of the coupling stijjizess c = l/L. Indeed, when c is close to 0.783 [the value corresponding to the node of the second (lower)flexural mode], a complicated sequence of bifurcations of certain solution branches takes place. This result can be associated with the fact that, when the connecting stiffness is positioned close to the node of the second mode, the coupling forces between the two beams become extremely weak and transfer of energy is restricted. On the contrary, when the position of the coupling stiffness is away from the node, the magnitudes of the coupling forces are relatively larger and nonlinear transfer of energy between modes due to internal resonance is facilitated. An interesting observation is that no similar bifurcation phenomena are observed when the position of the coupling is close to the node of the third (higher)flexible mode. For c = 1/L = 0.7650, the bifurcation diagrams for the localized
404
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
om
Figure 9.1.3 Bifurcation of NNMs and localization for the first beam cantilever mode: (a) modal ratio versus (yl lilq 1 1 I) and (b) modal ratio versus (kL4/EI)/lq 11I: _ _ Stable, ------- unstable motions. modes appear in Figure 9.1.4. In each of the two diagrams the ratio of the modal amplitudes (alp/azp), p = 2,3 is plotted versus the parameter h = (kL4/EI)/a3223. Such bifurcation diagrams are typical for systems with the coupling stiffness positioned away from the node of the second flexural mode; the following observations are made concerning the nonlinear mode localization. The branches of localized modes bifurcate from the in-phase modes of the system in Hamiltonian pitchfork bifurcations (Guckenheimer and Holmes,
9.1 THEORETICAL ANALYSIS
405
1984). This is in contrast to the bifurcation diagrams of Figure 9.1.3 (absence of internal resonance effects), where the localized modes bifurcated from the antiphase mode. There exists a nonzero value of h = for which (a13/a23) = 0, or -. For this value of h the amplitude of the third mode in one of the two beams is identically zero, and perfect localization of the third flexural mode occurs. This result is in contrast to that found in the nonresonant case (Figure 9. I .3), where perfect localization of the first mode was achieved only asymptotically in the limit h+0. Such perfect localization for a nonzero value of the coupling stiffness is solely due to internal resonance and occurs only for the higher (third) mode, but not for the lower (second) one. From Figure 9.1.4 it is observed that mode localization occurs over a higher range of values of the coupling stiffness, k, than when the beams oscillate in their first mode (Figure 9.1.3). This indicates that high-mode beam oscillations are more susceptible to nonlinear mode localization than lower-mode ones. In Figure 9.1.5, the values of the coupling stiffness at the points of generation of the localized mode branches are plotted as functions of the position of the coupling stiffness for the same value of the energy-like quantity p. Both categories of localized solutions (cases 1 and 2) are depicted, and it can be seen that, for the same value of p nonlinear localization of modes 2 and 3 is generated at much higher values of the coupling stiffness than for mode 1. Similar results hold for higher modes. It is concluded that, if the coupling stiffness is low enough to localize the first cantilever mode, then it is sufficient to localize all higher modes; this result is in full agreement with existing theories on mode localization of disordered, linear beam assemblies (Wei and Pierre, 1988). Numerical integrations of the discretized differential equations of motion (9.1.7) using the analytically predicted initial conditions confirm the existence and the stability types of the theoretically predicted localized nonlinear modes. When c = cn = 0.783, the position of the coupling stiffness is close to the node of the second cantilever mode, complicated bifurcation phenomena occur caused by nonlinear interactions of different branches of NNMs. The series of bifurcation phenomena is depicted in Figures 9.1.6(a)-(g), for values of c above and below the node value cn; the "primary" branches of NNMs identified in Figure 9.1.4, interact nonlinearly with "secondary" branches when the position of the coupling stiffness is sufficiently close to
a*
406
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
4n
¶.a M -20
-*O -ao
-1.0
-M
h
a ............................................
+
~
1
I
i I
M
00
1.0
r
h
13
Figure 9.1.4 Bifurcation of NNMs and localization in the presence of internal resonance: (i) a12/a22 versus h, and (ii) a13/a23 versus h: Stable, and ------- unstable motions.
10
-'0.00
0.21
0.50
0.75
1.no
Figure 9.1.5 Coupling strength at the point of generation of nonlinear mode localization, plotted against the nondimensional position of the coupling stiffness.
9.1 THEORETICAL ANALYSIS
407
the node of the second mode. Referring to Figure 9.1.6, the interaction between the various solutions can be described as follows. For c = 0.7950 > Cn [Figure 9.1.6(a)], the secondary branch of NNMs bifurcates from the in-phase mode at point B and is everywhere unstable. The two unstable secondary branches meet at point A, which, however, is not a bifurcation point. The primary branch has the form of Figure 9.1.4, and bifurcates in a pitchfork bifurcation (point C) from the in-phase mode. The bifurcating primary NNMs localize as h+O. When c is slightly decreased to c = 0.7945 > Cn [Figure 9.1.6(b)], the primary and secondary branches of NNMs converge to each other, and branch B1B2 of the secondary solution becomes orbitally stable. The change of stability at points B 1 and B2 occurs through saddle-node bifurcations. Further decrease of c to 0.7900 > cn [Figure 9.1.6(c)] destabilizes a portion of the bifurcating "primary" solutions, The exchange of stability is of particular interest since it occurs through Humiltonian Hopf bfurcations (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 two pairs on the Imaginary axis, and then split producing four complex eigenvalues with nonzero real parts. The Hamiltonian Hopf bifurcation leads to amplitude-modulated instabilities of the dynamic responses of the system. In addition to the bifurcation described above, the unstable branch of the secondary solution approaches the unstable branch of the primary one, and the bifurcation at point B occurs at a high value of h. As c + cn [Figure 9.1.6(d)], the stable branches of the (a12/a22) plot become strongly localized, indicating that the second mode is nearly confined to only one beam. This is in agreement with physical intuition, since as c 3 cn the second cantilever modes of the beams cease to be directly coupled. However, even in this case, there is an indirect transfer of energy between the second cantilever modes, due to their internal resonance with the third cantilever modes, which at c = cn are still directly coupled. Therefore, a nonlinear interaction between two nondirectly coupled lower modes is observed, caused by their internal resonance with a pair of directly coupled higher modes. For c = 0.7725 [Figure 9.1.6(e)], c = 0.7700 [Figure 9.1.6(f)J and c = 0.7675 [Figure 9.1.6(g)], the position of rhe coupling stiffness is below the node of the second mode, and the secondary branch of NNMs diverges away from the primary one. A reverse series of bifurcation phenomena takes place, during which the primary branches
408
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS 1
\
h
..............................
.......
,o
Hoof, 00
-1.0
-2.0
.
\
1 0 ,
2 , 2
cd
..............c:..................................
00
yi
*
I
I)o
I 0
I0
30
I D
50
I D
7.0
11 I D
h
10
(.I
.... Cl
.............
8.Y.
f.C..d.,,
/.
h
Figure 9.1.6 Mode bifurcation and localization in the presence of internal resonance, coupling position close to the node of the second.cantilever mode: (a) c = 0.7950, (b) = 0.7945, (c) c = 0.7900, (d) c = 0.7850, (e) c = 0.7725, (f) c = 0.7700, (g) c = 0.7675; (i) a12/a22 versus h, (ii) a13/a23 versus h, (iii) Stable, -------- unstable motions. Detail of plot (ii): ~
9.1 THEORETICAL ANALYSIS
h
h
Figure 9.1.6 (Continued)
409
410
NONLTNEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
become totally stable and the secondary ones totally unstable. For values of c sufficiently lower than cn [Figure 9.1.6(g)], the primary solution regains the form of Figure 9.1.4, and the branches of NNMs are similar to those depicted in Figure 9.1.6(a). Summarizing, when only the primary modes of the two beams of Figure 9.1.1 are taken into account, no internal resonance affects the dynamics, an$ the structure of the branches of localized NNMs is similar to that of a system of two weakly coupled, discrete oscillators: branches of modes bifurcating from the antiphase mode of the system localized when the coupling stiffness decreased, and/or the nonlinear stiffness forces increased. However, when the beams oscillate in their second and third linearized cantilever modes, the picture of nonlinear mode localization is radically different. In that case, the beams are more susceptible to nonlinear mode localization, since localized motions exist over a wider range of values of the coupling stiffness. Moreover, a very complicated series of bifurcations occurs when the coupling stiffness is positioned sufficiently close to the node of the second cantilever mode. It must be pointed out that, in addition to the localized NNMs found herein, a variety of additional stable and unstable nonlocalized solutions exist in the system under consideration. Examples of such modes are reported in later sections. In the following section the transient responses of the coupled system to impulsive external forces will be investigated. It will be analytically and numerically shown that, for weak coupling and/or relatively strong geometric nonlinearities, vibrational energy impulsively injected into the system localizes close to the point of its application, instead of spreading into the entire structure. This passive transient motion confinement of disturbances generated by impulsive loads is solely due to the localized NNMs in the continuous assembly and becomes more profound as the nonlinear effects increase and/or the coupling stiffness connecting the two beams decreases.
9.1.2 Passive Motion Confinement of Impulsive Responses A number of previous works investigated the spatial confinement of propagating disturbances in linear structures with localized modes. The confinement of transient responses (Anderson localization) due to linear mode localization was first reported in the classical work of Anderson
9.1 THEORETICAL ANALYSIS
4 11
(1958), who investigated the responses of disordered chains of discrete oscillators. In (Hodges and Woodhouse, 1989), localization of propagating disturbances in one-dimensional disordered coupled oscillators and in beams with irregularly spaced constraints is studied using ensemble averaging procedures. A work investigating scattering of propagating disturbances due to Anderson localization was carried out by (He and Maynard, 1986). Analytic and numerical logarithmic averages for the transmission of disturbances along such systems were given. In (Kissel, 1988), a wave propagation formulation for studying transmission in disordered periodic systems is adopted. Multiplication of random transmission matrices is carried out in order to compute the “localization factors” inside the passbands of the unperturbed system. An extension of these statistical analyzes was given by Pierre (1993), where theoretical results on the localization factors were confirmed by Monte Carlo simulations. The effect of disorder on the dynamics of infinite and finite discrete and continuous monocoupled linear systems was treated by Mead and Bansal (1978) and Mead and Lee (1984), using the “characteristic receptance“ method. It was found that a single disorder resulted in reduced transmission of flexural waves in a beam when the frequency of the wave was in propagation zones of the corresponding periodic system. In some cases increased transmission was detected in attenuation zones, and conditions producing resonance of the disorder against the periodic system were determined. Localization of flexural propagating waves along a fluid-loaded plate with an irregular array of line attachments was investigated by Photiadis (1992,1993). A structural acoustics formulation was adopted in his works, and localization is studied analytically and by means of numerical simulations. Lu and Achenbach (1991) investigated ultrasound propagation in disordered layered media using a transfer matrix approach. Vakakis et al. ( 1 9 9 3 ~ )studied stress propagation in an impulsively excited linear disordered layered medium and, under certain conditions, observed transient localization of stress close to the disorder. Additional numerical computations of motion confinement of impulsive responses due to mode localization were carried out with models of linear continuous structures (Bendiksen, 1987; Cornwell and Bendiksen, 1989; Lust et al., 1991). In section 8.4.1 the impulsive responses of a discrete nonlinear periodic system with cyclic symmetry were examined. The numerical simulations performed in that section indicated that nonlinear mode localization can lead
412
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
to passive motion confinement of disturbances generated by external impulsive loads. In this section the assembly of coupled beams of Figure 9. I. 1 is considered again, and analytical/numerical techniques are employed to prove that when the structure possesses strongly localized NNMs its impulsive responses remain passively confined close to the point were they are initially generated. The following exposition follows closely (Vakakis, 1994b). Assuming that spatially distributed impulsive excitations are acting on each of the two beams of the system, the governing equations of motion take the form: x
vptt + vpxxxx = -E{
vpx[vpxvpxxlx + (U2)Vpxj 1
s
[j
vpx2 dultt ds},
0
vm(l/L,t)]G(x-l/L) + Fp(x,t) up(x,t) = O(E), p,m = 1,2, p # m (9.1.27)
- ~(kL'/EI)[vp(l/L,t) -
where as in the previous section, vp and up denote the transverse and longitudinal displacements of beam p, and x the arclength per unit length of the two beams. In (9.1.27), t is the scaled time defined by t = z(EI/pL4)1/2, where T represents physical time, and Fp(x,t) is the distributed impulsive excitation acting on beam p. For small values of E , equations (9.1.27) form a set of weakly nonlinear, weakly coupled nonhomogeneous partial differentia1 equations. As in section 9.1.1, this set of equations is discretized by expressing the transverse displacements vp(x,t> in the following series form:
where +m(x) are the linearized normalized cantilever eigenfunctions, and q p m (t) the new generalized coordinates, which are governed by the following discretized set of nonhomogeneous ordinary differential equations:
-
r n
n
n
(9.1.29) where, as previously, the first subscript of qpi represents the beam number, p = 1,2, p = 3 3 1, and the second subscript denotes the order of the
9.1 THEORETICAL ANALYSIS
41 3
linearized mode shape, i = 1,2..A. The coefficients in (9.1.29) are defined by expressions (9.1.8a), and the forcing functions Fpi(t) by: (9.1.30) The objective of the analysis is to study the response of systems (9.1.27) and (9.1.29) for a general class of impulsive excitations Fpi(t). As proved in section 9.1.1, for Fpi(t) = 0 (no external forcing) the system possesses stable, strongly localized NNMs (cf. Figures 9.1.3-9.1.6). Moreover, the topological picture of the localized branches of NNMs depends on the spatial waveform of the beams during oscillation. Motivated by the findings of the previous section, one examines two cases of external forcing. In the first case, excitation of only the first linearized cantilever modes of the two beams is considered. In the second case, the external forcing functions are assumed to excite the second and third cantilever modes of the beams, and a 1:3 internal resonance influences the impulsive response of the system.
Case I : Impulsive Response in the Absence of Internal Resonance First, only the first cantilever modes of the beams are assumed to participate in the forced motion, and for the sake of simplicity only forcing of beam 1 is considered: Fl(x,t) = (l/&)fl(t)$l(x) for 0 I t < ED, and Fl(x,t) = 0 for t 2 ED (9.1.3 1) F2(x,t) = 0, 0 I t < 00 Hence, a general impulsive excitation of duration ED is assumed to act on beam 1, with a spatial distribution identical to that of the first linearized cantilever mode. Forcing functions with more general spatial distributions will be considered later. Taking into account definitions (9.1.30), the nonhomogeneous terms in equations (9.1.29) assume the form, 1
(l/E)Fpi(t) = (l/&)J fl(t)$i*(x) dx 6pl6il 0
Fpi(t) = 0
for 0 I t < ED
for t 2 ED (9.1.32)
414
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
where Sij is Kroenecker's symbol. It is of interest to study the transfer of the impulsively injected energy from the directly excited mode to the remaining modes of the system; clearly, spatial motion confinement of the external impulse is achieved only if minimal amounts of vibrational energy eventually "leak" to the not-directly forced beam 2. In the previous section, it was found that a low-order internal resonance exists between the second and third cantilever modes, and that the first cantilever mode does not possess any nonlinear coupling with other higher modes. Hence, no energy transfer can occur between the directly excited first mode of beam 1 and any other higher modes of the system, and the only possibility for energy exchange between the two beams is between their first cantilever modes. This theoretical prediction will be verified in the following analysis. The dynamics of the forced system (9.1.29) will now be analyzed, by examining the modal responses in two distinct phases. Only n = 3 cantilever modes per beam are considered in the following analysis.
First Phase of the Motion: 0 5 t c ED During this phase the external impulse is applied to beam 1, and the response is asymptotically approximated by introducing the new time T, defined by t = ET. The range of values of the new time variable during this phase of the motion is 0 5 T D. Expressing the time derivatives in (9.1.29) in terms of the new variable T, the equations governing the amplitudes of the first modes of the beams are written as:
-=
p = 1,2 (9.1.33) with similar expressions holding for the modal displacements qp2, and qp3, p = 1,2, of the higher modes. Since at t = T = 0 the system is assumed to be at rest, equations (9.1.33) are complemented by the set of initial conditions qpi(0) = 0, d(qpi(O))/dT = 0, p = 1,2, i = 1,2,3. In equations (9.1.33), all depended variables are functions of T, and the new forcing function is defined as,
9.1 THEORETICAL ANALYSIS
415
The response of system (9.1.33) is approximated using regular perturbation expansions, i.e., by expressing the responses in the form, m
qpi(T) =
C
m= 1
~m qpi(m)(T)
By matching the coefficients of respective powers of E, one determines the solution at various orders of approximation. Omitting the calculations, and transforming in terms of the original time variable, the response of the system during this phase is computed as:
The analysis shows that for 0 It < ED (the duration of the impulse) the response of the system is mainly determined by the impulse itself and not by any structural parameters (the system “does not have time to oscillate“). Note that the velocity of the directly excited mode is of 0(1), whereas the response of all unforced modes are orders of magnitudes smaller than that of the directly forced mode. The physical transverse displacements are: vi(x,t) = E1’2($l(X)qll(t) + O(~7/2), v2(x,t) = O(&9/2),
for 0 I t < ED (9.1.36) where the scaling v + E ~ / Z V , introduced in section 9.1.1, was taken into account.
Second Phase of the Motion: t 2 ED During this phase of the motion, the impulse ceases to apply, and the system performs a free oscillation, with initial conditions determined from (9.1.35). Introducing the time translation = t - ED, the governing equations (9.1.29) are expressed as: 2
qpi + ai qpi
416
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
(9.1.37) where differentiation is carried out with respect to i, and qpi = qpi(i). The set of initial conditions complementing (9.1.37) is
0
0 0
qpi(0) = O(E4),
and Gpi(0) = O ( E ~ )otherwise
Since equations (9.1.37) represent free oscillations of the system, their solutions can be analytically approximated by a multiple-scales perturbation analysis. To this end, the modal responses are expressed as,
with time scales To =
and Ti =
E{.
Introducing the transformations
Api(T1) = (l/2)api(Tl)ejeP'(T'), p = 1,2, i = 1,2,3, the modulation equations (9.1.14)-(9.1.17) governing the real amplitudes api(T1) and angles Opi(T1) are obtained. In order to compute the amplitude and phase modulations of the response, one should solve the modulation equations with initial conditions, D
a11(0) = -(l/wi>
81 i(0) = +7t/2,
0
PI(s) ds,
Ozl(0) = 0,
a21(0) = 0
ap,(0) = 0 otherwise
The resulting physical motions of the beams are then approximated by: 3
r
t 2 & D , p = 1, 2 (9.1.38)
9.1 THEORETICAL ANALYSIS
417
Considering the structure of the modulation equations (9.1.14)-(9.1.17), it can be proven that, if a12(0) = a13(0) = a22(0) = a23(0) = 0, then a12(T1) = a13(Tl) = a22(T1) = a23(T1) = 0, V T i . Therefore, in this case, the modulation equations are reduced as follows: a1 1' = (y/2w)a2lsin(821- 81 1) a21' = -(y/2o)ai1sin(821 - 811) a i l e l l ' = [(3a/8oi) - (Pwi/4)]aii3 + (y/2o)ai1 - (y/2w)a2icos(02i- 811) a 2 1 =~[ ( 3 a / 8 ~-) (Pw/4)1a213 + (y/2w)a21- (~/20)aiicos(e21-01 1) (9.1.39) where primes denote differentiation with respect to the "slow time" Ti, and the simplified notation, a E a1 1 11, p = b 1 1 1 1, and y = y11, is adopted from now on. Hence, the only possible energy transfer is between the first modes of the two beams, and no other energy exchange between higher modes of the system occurs. In order to investigate the energy transfer between the first modes of beams 1 and 2, one must integrate equations (9.1.39). One defines at this point the quantity:
Parameter J can be recognized as the ratio of the coupling over the nonlinear forces. It will be now shown that, for J << 1, the response of the system can be analytically approximated. This is achieved by expressing the amplitudes and angles as, 00
apl(T1) =
W
Jm a p ~ ( ~ ) ( T i ) epi(T1) , = Jm epl(*)(Ti) m=O m=O
p = 1,2, and substituting into (9.1.39). Matching terms proportional to the same power of J leads to the following analytical expressions for the amplitude modulations: all(T1) = i ( 2 p - (J2/p)sin2([(3a/4~01)- (fioi/2)]p2Ti)
+ O(J3))
(J <
(9.1.40)
418
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
Similar analytical expressions hold for the angles OP1(T1). Solutions (9.1.40) indicate that, when J<<1, the transient impulsive response of the system is mainly confined to the directly excited beam, since in that case a l l ( T 1 ) >> a21(T1). Note, that based on the results of section 9.1.1, the condition J<
0
pi(s> ds = -2p
Hence, for J << 1, passive motion confinement of the impulse to the directly excited beam is observed, in accordance with the analytic result
9.1 THEORETICAL ANALYSIS
4 19
(9.1.40). As J increases, the amplitude a21 of the unforced beam also increases, indicating an enhanced transfer of energy from the directly excited beam to the unforced one, or equivalently, a diminishing of the motion confinement capacity of the system. At a critical value, J = Jcr = p/2, all vibrational energy of the directly excited beam is eventually transferred to the unforced one. For values of J above the critical value, energy is continuously transferred between the two beams in a beat phenomenon, and the system does not possess any passive motion confinement properties anymore. To verify the results of the multiple-scales analysis, an impulsive distributed excitation of magnitude Fl(x,t) = 5$1(x) for a duration of ED = 0.1 second was applied to beam 1, and the structural parameters were assigned the values E = 0.5, a1 11 1 = 40.44, bi 111 = 4.59, y1i = 0.02, 01 = 3.5 1 (rad/sec), J = 0.0367 << 1; the response was numerically computed by direct integration of the differential equations of motion (9.1.29) assuming zero initial conditions at t = 0. In Figures 9.1.8(a) and (b) the responses of the two beams are shown, both as functions of time and as projections of the phase space of the system. It is clear that the energy of the injected impulse is passively confined to the directly forced beam, in full agreement with theoretical predictions. The amplitude of the unforced beam can be further diminished by decreasing the coupling stiffness and/or increasing rhe nonlinear coefficients. For comparison purposes, the theoretically predicted amplitude modulations (9.1.40) are presented in Figure 9.1.8(c). Clearly, the asymptotic theory agrees well with the numerical computations for this low value of J. Note that in the absence of nonlinearities, all injected energy is continuously transferred between the forced and unforced beams, in the well-known beat phenomenon. Hence, the detected passive motion confinement phenomenon is solely attributed to the geometric nonlinearities of the system.
Case 2: Impulsive Response in the Presence of Internal Resonance In order to investigate the effects of internal resonance on the passive motion confinement properties of the system, the following applied impulsive forces are now considered: Fi(x,t) = (l/&)fi(x,t) for 0 I t < ED, and Fl(x,t) = 0 for t 2 ED F2(x,t) = 0, 0 5 t < 00 (9.1.41)
420
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
0 -0.1
-0.2 -0.3
................
L
-0.4 0
50 100 150 200 250 300 350 400 Slow Time
Figure 9.1.7 Amplitude modulations by numerical integration of the set (9.1.39): J = 0.0142 ( I ) , 0.0426 (2), 0.0924 (3), Jcr = 0.071 1 (------).
where Fl(x,t) is assumed to excite the first three cantilever modes of beam 1. Equations (9.1.4 1) lead to the following modal excitations for the discretized equations (9.1.29): 1
(l/E)Fli(t) = (I/&)! fl(x,t)oi(x) dx for 0 for t 2 ED Fli(t) = 0 F ~ i ( t= ) 0, i = 1,2,3
0 I t <ED
(9.1.42)
The modal responses can be computed by solving the set of equations (9.1.29), with zero initial conditions at t = 0. Once the modal amplitudes qij are determined, the forced responses of the beams are approximated as:
9.1 THEORETICAL ANALYSIS
421
In (Vakakis, 1994) an analytical methodology similar to the one followed in the previous case was used to analytically study the forced transient response of the system. For sufficiently weak coupling and/or relatively strong nonlinear effects, passive motion confinement was detected. Moreover, it was found that when three modes of the directly forced beam are simultaneously excited, spatial localization of the impulsive response occurs over a wider range of values of the coupling parameter, and, thus, the system possesses more profound motion confinement characteristics. Hence, higher modes provide much more effective passive confinement of the induced motion, a feature that is to be expected in view of the fact that higher modes localize much easier than lower ones (cf. Figure 9.1.5). In Figure 9.1.9 numerical integrations of the discretized equations of motion (9.1.29) are depicted for a system with parameters (kL41EI) = 0.01, 1/L = 0.65 and E = 0.1. It was assumed that all three modes of beam 1 are excited by identical modal forces of the form (l/E)Fli(t) = (I/&) for 0 I t < 0.1 I and Fli(t) = 0 for t 2 0.1 1, for i = 1,2,3. Projections of the phase space of the system are presented, and passive motion confinement of the energies of all three modes is detected. Although energy is continuously exchanged between the second and third modes within each beam due to internal resonance, this energy cannot be transferred between beams since nonlinear mode localization restricts such energy transfers. Summarizing, it was analytically and numerically shown that nonlinear mode localization in a geometrically nonlinear, weakly coupled system of beams can lead to passive motion confinement of disturbances generated by impulsive loads. For motions of the beams in their first cantilever mode, it was found that spatial motion confinement of the impulsive response occurs only when the coupling is sufficiently small and/or the nonlinear effects large. When higher modes are taken into account, the motion confinement becomes more evident. Passive motion confinement due to nonlinear mode localization is expected to be the property of a general class of nonlinear, weakly coupled flexible systems with or without symmetry. By designing such systems so that coupling between components is weak or/and their nonlinearities relatively strong, it is possible to ensure that disturbances generated by external impulsive loads do not "spread" through the entire structure but remain confined close to the component where they were initially generated. Note that such a Confinement of vibrational energy is
422
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
q1(t)
0.4
I
I
I
0.3
I I
i i
i i
I
i
1
i
i i
i
~
0.2 0.1 0
-0.1
-0.2
-0.3 -0.4
I
0
I
I
50
I I
1 1
1 1
j
1
1
1 1
1 1
100 150 200 250 3 0 0 3 5 0 400
t
0
5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 350 4 0 0
(4
t
Figure 9.1.8 Amplitude modulations of single-mode impulsive responses: (a,b) direct integrations of the equations of motion and (c) theoretical results, equation (9.1.40).
9.1 THEORETICAL ANALYSIS
1.5 42
I
I
I
I
0.1
0.3
1 I
/
0.5
0 -0.5 - 1
-1.5
-0.3
-0.5
-0.1
0.5 41 .q2
(b)
0.4
I
I
I
I
I I
I I
I I
I
I
I
0.3 0.2 0.1
__-
0
-
-0.1
-
-0.2
-0.3
-0.4
I
0
I
1
I
I I
I I
I I
I I
5 0 100 150 200 2 5 0 300 350 400 A
(c)
Figure 9.1.8 (Continued)
t
423
424
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
purely passive, and is solely due to strong localization of the NNMs of the system. The implications of such a dynamical feature are profound, since a system whose inherent dynamics lead to motion confinement of external disturbances is expected to be much more amenable to active or passive isolation than a structure possessing "extended" dynamic modal responses. In what follows, the forced resonances of a periodically forced system consisting of n-coupled beams is examined in order to investigate the effect of nonlinear mode localization on its forced nonlinear resonances. Previous sections established the existence of nonlinear mode localization and nonlinear passive motion confinement in coupled continuous oscillators. The results reported in the next section prove the existence of numerous forced localized steady-state motions in such systems in neighborhoods of localized NNMs. 9.1.3 Nonlinear Localization of Forced Steady-State Motions In this section it is shown that localized NNMs can lead to passive motion confinement of steady-state periodic oscillations of coupled flexible systems. The assembly examined in this section is depicted in Figure 9.1.10. It consists of N identical cantilever beams coupled by means of weak linear stiffnesses and can be considered as a cyclic periodic extension of the system depicted in Figure 9.1.1. It is assumed that each beam is excited by a distributed harmonic excitation Pi(X,t), i = 1,...,N. Denoting by R the common frequency of the external excitations, and by Oj the jth linearized natural frequency of the system, it is well known (Nayfeh and Mook, 1984) that a variety of nonlinear resonances is possible: fundamental (when R = Wj for some j ) , subharmonic (when 51 = k o j , for some integer k), superharmonic (when Cl = O j k , for some integer k), and internal (when k i w i +...+ knOn = 0). The aim of the analysis is to study the forced fundamental and subharmonic resonances of the cyclic system in order to establish the existence of steady-state nonlinear localized motions. It will be shown that the system exhibits a very complicated structure of fundamental and subharmonic resonance curves. The resonance branches of the nonlinear system will be shown to exceed in number the resonance branches of the respective linear system, a feature that is due to mode bifurcations that occur in the unforced system.
9.1 THEORETICAL ANALYSIS
425
1.50 I q11.421
0.90 -
0.30 -0.30
-
-0.90
-
-1.50 -0.50
911.921
I
I
i
i
-0.30
-0.10
0.10
0.30
0.50
1.50
0.90 0.30
-0.30 -0.90
-1 -50 -0.50
1.50
0.90
.-
0.30
-
-0.30
-
-0.90
-
-1.50
I
I
1
1
-0.30
-0.10
0.10
0.30
9 12.922
0.50
413. 423
0 I
I
1
1
9 13423
Figure 9.1.9 Amplitude modulations of multimode impulsive responses by numerical integration of the discretized equations of motion (9.1.29): projections of the phase space.
426
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
Figure 9.1.10 The cyclic assembly of N geometrically nonlinear beams. The analysis follows closely (King and Vakakis, 1995a). Using the notation and the assumptions of the previous two sections regarding the motions of the beams and assuming viscous damping, the equation governing the ith substructure of the cyclic assembly can be expressed as (cf. Figure 9.1.10):
where Ci is the coefficient of distributed viscous damping, 6(*) is Dirac’s delta function, and vo = V N , VN+I = v i due to cyclicity. As in previous sections, it is assumed that the coupling linear stiffnesses and the viscous damping coefficients are small (of O ( E ) ,IEI << I), and that the beam deflections are of 0(&ll2).Imposing the rescalings Vi + E1”Vi, Ci + E C i , i = 1,...,N and K + E k, equations (9.1.44) are transformed into the following form: x
vitt
+ vixxxx + ECiVit + E{ vix[vixvixx]x + (112) J’ I
[
s
vit2(<,t) dk]tt ds}x 0
9.1 THEORETICAL ANALYSIS
427
+ Pi(X,t)/E1/2 i = 1,...,N (9.1.45) Since the forcing term has yet to be scaled, the presence of (1/&1’2) terms in (9.1.45) is not problematic. Indeed, investigations of the fundamental and subharmonic resonances of the assembly require different scalings of the excitation (Nayfeh and Mook, 1984; Nayfeh et al., 1974). The distributed harmonic excitations are of the form: = -E(KL~/EI)[~v~(I/L) - Vi-l(l/L) - Vi+l(l/L)]6(X-l/L)
Pi(x,t) = Fi(x)CosSlt,
0 < x < 1,
i = 1,...,N
(9.1.46)
where Fi(x) denotes the spatial distribution of the force acting on the ith beam and C2 is the (common) excitation frequency in radsec. Discretizing the system of equations (9.1.45) as in sections 9.1.1 and 9.1.2, one obtains the following set of ordinary differential equations governing the evolution of the modal coefficients qpi(t):
where p = 1,...,M, i = 1,...,N, the coefficients aivmq, bivmq and Yip are computed by (9.1.8a) and the modal forces Qpi by (9.1.48) where qP(x) is the pth linearized cantilever mode. To simplify the analysis it is now assumed that M = 1, i.e., that each beam oscillates in its principal cantilever mode. This is a justifiable assumption since no low-order internal resonance exists involving the primary cantilever mode and other low-order modes. In order to prevent the direct excitation of higher modes, one requires that Fi(X) = $i(x) in (6), and expresses equations (9.1.47) in the following simplified form:
428
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
where the notation qli = qi, a1 11 1 = a,bl111 = p, YI1 3 Y,Qli = Qi, and WI= w is introduced. The unforced system (9.1.49) possesses a variety of stable and unstable localized NNMs [the unforced system with N = 2 beam was studied in section 9.1.1 (case l)]. It is anticipated that, when harmonic excitations act on the cyclic assembly, spatially localized forced resonances exist, which result as perturbations of the localized NNMs of the unforced system. Moreover, since the NNMs of the cyclic assembly may exceed in number those of the respective linear system, it is anticipated that the topological structure of the nonlinear frequency response curves will be qualitatively different and much more complicated than that predicted by linear theory.
Fundamental Resonances To study fundamental resonances, it is necessary to assume weak excitations (Nayfeh and Mook, 1984). The excitations are scaled as Fi(x) + ~”’fi(x), and the frequency detuning parameter, 0,is introduced defined by: n = w + E O
(9.1.50)
The method of multiple scales is now applied to analyze system (9.1.49). Rescaling time variable by Q t = (w + m ) t + t the equations of motion are written as:
where qi = qi(T), i = I , ...,N and differentiation is carried out with respect to the rescaled time variable. The modal responses are expressed as,
A Cartesian transformation of coordinates is now introduced, whereby the complex amplitudes are expressed in terms of their real and imaginary parts as, Ai(T1) = (1/2)[Xi(Tl) - jYi(Tl)], i = I , ...,N
9.1 THEORETICAL ANALYSIS
429
with Xi and Yi being real, slowly varying quantities. Eliminating secular terms in the O(E) order of approximation (King and Vakakis, 1995a), one obtains the following set of first-order ordinary differential equations, which govern the slow evolution of the real and imaginary parts of the complex amplitudes Ail
where i = I , ...,N. Note that instead of using the Cartesian transformation, one could (as in previous sections) use the alternative polar transformation: Ai(T1) = (1/2)ai(Tl)e-Jpi(Tl) A drawback of the polar transformation is that it is singular (noninvertible) when ai(T1) = 0, which renders it unsuitable for studying periodic motions where one of the beams possesses a zero modal amplitude. When ai(T1) # 0, the Cartesian and polar transformations are equivalent, and are related by the expressions a
i
=
d
m
and
Yi Xi
pi = tan-l(-)
(9.1.53)
In (9.1.53), ai(T1) and Pi(T1) represent the real modal amplitude and phase of the first mode of ith beam. In what follows, all numerical computations are performed employing the Cartesian transformation, but all results are presented in terms of the real amplitudes and phases (9.1.53). Fundamental resonances of the cyclic assembly are stationary solutions of the set of equations (9.1.52). These are computed by setting all the time derivatives equal to zero, Xi' = Yi' = 0, i = 1,...,N, and solving numerically the resulting set of simultaneous nonlinear algebraic equations. Note that in the absence of forcing and damping, equations (9.1.52a,b) reduce to:
Xi-1) = 0
(9.1.54) A manipulation of equations (9.1.54) indicates that the following "energytype" relation is satisfied by the modal amplitudes:
(9.1.%a) where p2 is related to the total energy of the system, which is conserved during free undamped oscillations (when no damping and forcing exists). The stationary solutions of equations (9.1.54a,b) provide the backbone curves for the unforced, undamped cyclic system. The backbone curves provide the relations between the amplitudes of the beams and the corresponding frequencies of free oscillation during a nonlinear oscillation of the unforced, undamped cyclic system. These curves are computed by numerically solving the following set of nonlinear simultaneous algebraic equations:
(9. I .55c) For small damping and forcing, the backbone curves computed by (9.1.55b,c) provide good initial estimates for the numerical computation of the stationary values of equations (9.1.52). The fundamental resonances of cyclic assemblies with two, three, and four beams were numerically studied. Due to the complicated nature of the corresponding sets of equations no analytical solutions are feasible, and thus one must resort to a Newton-Raphson numerical algorithm. The stability of the detected nonlinear resonances are investigated by introducing small perturbations in the computed periodic solutions, and computing the eigenvalues of the resulting variational matrix. In all applications considered
9.1 THEORETICAL ANALYSIS
429
with Xi and Yi being real, slowly varying quantities. Eliminating secular terms in the O(E) order of approximation (King and Vakakis, 1995a), one obtains the following set of first-order ordinary differentia1 equations, which govern the slow evolution of the real and imaginary parts of the complex amplitudes Ai:
where i = 1,...,N. Note that instead of using the Cartesian transformation, one could (as in previous sections) use the alternative polar transformation: Ai(T1) = (1/2)ai(Tl)e-Jpi(Tl) A drawback of the polar transformation is that it is singular (noninvertible) when ai(T1) = 0, which renders it unsuitable for studying periodic motions where one of the beams possesses a zero modal amplitude. When ai(T1) # 0, the Cartesian and polar transformations are equivalent, and are related by the expressions
In (9.1.53), ai(T1) and Pi(T1) represent the real modal amplitude and phase of the first mode of ith beam. In what follows, all numerical computations are performed employing the Cartesian transformation, but all results are presented in terms of the real amplitudes and phases (9.1S3). Fundamental resonances of the cyclic assembly are stationary solutions of the set of equations (9.1.52). These are computed by setting all the time derivatives equal to zero, Xi’ = Yi’ = 0, i = 1, ...,N, and solving numerically the resulting set of simultaneous nonlinear algebraic equations. Note that in the absence of forcing and damping, equations (9.1.52a,b) reduce to: Yi+l - Yi-1) = 0
430
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
O2Yi' - O O x i
+ (3a -82pw2> (Xi3 + XiYi2) + (~/2)(2Xi- Xi+] - Xi-1)
=0
(9.1.54)
A manipulation of equations (9.1 S 4 ) indicates that the following "energytype" relation is satisfied by the modal amplitudes:
where p2 is related to the total energy of the system, which is conserved during free undamped oscillations (when no damping and forcing exists). The stationary solutions of equations (9.1.54a,b) provide the backbone curves for the unforced, undamped cyclic system. The backbone curves provide the relations between the amplitudes of the beams and the corresponding frequencies of free oscillation during a nonlinear oscillation of the unforced, undamped cyclic system. These curves are computed by numerically solving the following set of nonlinear simultaneous algebraic equations:
(9.1.55b) (9.1 S5C) For small damping and forcing, the backbone curves computed by (9.1 .S5b,c) provide good initial estimates for the numerical computation of the stationary values of equations (9.1 S2). The fundamental resonances of cyclic assemblies with two, three, and four beams were numerically studied. Due to the complicated nature of the corresponding sets of equations no analytical solutions are feasible, and thus one must resort to a Newton-Raphson numerical algorithm. The stability of the detected nonlinear resonances are investigated by introducing small perturbations in the computed periodic solutions, and computing the eigenvalues of the resulting variational matrix. In all applications considered
9.1 THEORETICAL ANALYSIS
43 1
it is assumed that only beam 1 of the periodic assembly is directly excited by an external harmonic force, and that no other force is applied to the system, i.e., that Q1 # 0, Qj = 0, j = 2 ,...,N. For all cases considered, the system parameters were assigned the following values: Qi = 1.0, Qj = 0, ci = 0.02, cj = 0.03, j = 2 ,...,N, 1/L = 0.35, kL4/EI = 4.0. The coupling parameter for all applications was y = 0.5235, a value that guaranteed the existence of localized modes in the undamped, unforced assemblies. The fundamental resonance curves for assemblies consisting of two, three, and four beams are presented in Figures 9.1.1 1, 9.1.12, and 9.1.13, respectively. In each case, the amplitudes and phases of the fundamental steady-state motions are shown and the stable branches are numbered for the sake of clarity. It can be seen that as the number of beams increases the topological picture of the resonance curves becomes more complicated. This complicated structure of the resonance curves is caused by nonlinear normal mode bifurcations taking place in the unforced, undamped system, which increase in complexity as the number of beams increases (Vakakis et al., 1993a) . For a system consisting of two, three and four beams, one obtains at most four, eight, and twelve stable co-existing fundamental resonances, respectively. Of particular interest are strongly localized fundamental resonances during which the response is mainly confined to the directly forced beam 1 (labeled as branches 1 in Figures 9.1.11-9.1.13). These strongly localized, orbitally stable solutions occur in the neighborhoods of localized nonlinear normal modes of the undamped, unforced systems, and have no counterparts in linear theory. There also exist weakly localized fundamental resonances during which the forced beam and adjacent beams oscillate with comparable amplitudes, whereas all other beams undergo much smaller vibrations (branches 3 and 4 in Figure 9.1.12, and branches 7 and 8 in Figure 9.1.13). Also note, that for small ranges of the detuning parameter (T there exist fundamental solutions during which the motion is mainly confined to an unforced beam (branch 3 in Figure 9.1.11, branches 5 and 6 in Figure 9.1.12, and branches 5 and 6 in Figure 9.1.13). An interesting result is that, for all assemblies considered, the strongly localized branch 1 always loses stability via a Hopf bifurcation, which occurs at relatively low values of the frequency detuning 0.Hence, it was determined that strong forced nonlinear localization in the cyclic assembly is always eliminated by modulation instability (Hopf bifurcation) as the frequency detuning decreases. Additional
432
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
0.5
0.0
1.0
1.0
1.1 0
a 2.0
Id 1.o
*,;
Od 0.0
z
// 3,,'
-'.- - - _ _ _ 4
0.0
0.1
4
1.0
1.1
0
2.0
I - - - - - - - - - - . - . . . _ _ _ _
Figure 9.1.11 Fundamental resonances of the assembly with N = 2 beams. ____- Stable solutions, ------ unstable solutions, Hopf bifurcations.
~
9.1 THEORETICAL ANALYSIS
433
a1 w
1s
i,,
a75
Mo
am
oso
1.00
lso
0
zm
,,
am am
m
im
lso
am
050
too
lso
,d d
m
w -
o.m
050
tm
iso
zm
m
Figure 9.1.12 Fundamental resonances of the assembly with N = 3 beams. Stable solutions, ------- unstable solutions, Hopf bifurcations.
434
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
,
111,
u82
-
-
*I.
I
/
l a -
-2 Y d
u
a
a
a
Figure 9.1.13 Fundamental resonances of the assembly with N = 4 beams. Stable solutions, ------ unstable solutions, Hopf bifurcations.
9.1 THEORETICAL ANALYSIS
435
points of Hopf bifurcations were detected and are indicated in the graphs of Figures 9.1.11-9.1.13.
Subharmonic Resonances A similar analysis can be performed to study the subharmonic resonances of the cyclic assembly. In order to achieve 0 ( ~ 1 / 2subharmonic ) beam response, the amplitudes of the external harmonic excitations in equations (9.1.49) are scaled as Fi(x) + E’/’fi(x), and the discretized equations of motion are written as:
To analyze subharmonic responses, the frequency of the excitation is related to the linearized beam natural frequency by: Q = 3w
+ E(3
(9.1.57)
where (3 is the frequency detuning parameter. Hence, synchronous steadystate periodic motions of the cyclic assembly with frequencies equal to onethird that of the periodic excitations will be considered. The time variable is rescaled as Qt = (3w + m ) t + t, the ith modal amplitude is expressed as qi = qio + E qil +..., and the method of multiple scales is again applied. The following hierarchy of problems at various orders of approximation is then obtained:
+ e-JTo) qio = Ai(Tl)dTo/3 + RjdTo + cc + O’qil = -18~02D0Dlqio- 6~0oDo’qio
9dDo’qio 9dDo’qi1
+ O’qio
= (Qi/2)(dT0
(9.1.58)
-3ci~Doqio- aqi03 - (9/2)Pw’qioDo2(qio2) - Y(2qio - q(i-110 - q(i+1)0) (9.1.59) b
b
b
436
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
where, as before, i = 1,...,N, To = t, Ti = &t, the variables Ai(T1) are complex amplitudes, Ai = -(Qi/1602), cc denotes complex conjugate, and j = (-1)1’2. Eliminating the secular terms in equations (9.1.S9), and expressing the complex amplitudes in Cartesian form, Ai(T1) = (1/2)[Xi(Tl) - jYi(T1)], one obtains the following set of first-order modulation equations governing the (slow) evolution of Xi and Yi:
+ Y2 (2Xi - Xi+l - Xi-1)
=0
(9.1.60)
where i = 1,...,N. Equations (9.1.60) are analogous to equations (9.1.54), which were developed earlier for studying fundamental resonances. The stationary values of Xi and Yi (computed by setting Xi’ = Yi’ = 0, i = 1, ...,N) provide the amplitudes and phases of the periodic responses of the beams during subharmonic resonance. For zero forcing and damping and imposing stationarity conditions, equations (9.1.60) lead to the following frequency-amplitude relations (which, indeed, are the backbone curves for subharmonic vibrations of the system) :
(9.1.6 1b) The subharmonic resonance solutions for systems consisting of two, three, and four beams are provided in Figures 9.1.14, 9.1.15, and 9.1.16, respectively. As in the case of fundamental response, branches 1 in these figures represent strongly localized subharmonic motions, during which the
9.1 THEORETICAL ANALYSIS
0.5
0.0
1.0
1.1
2.0
437
2.1
0
2.1
0.0
ao
0.5
1.0
1.1
IS
2.0
3.0
0.0
1:: 3.1
2.1
1.0
I
1'
0.0
'
'
'
' ".
0.5
' 1
1.0
"
' ' 1
1.5
'
' '
'
'
2.0
'
' '
u
'
'
2.5
1
Figure 9.1.14 Subharmonic resonances of the assembly with N = 2 beams. Stable solutions, ------ unstable solutions.
438
a'
-
m
o
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
1
nm
lm,
I
im
M
M
rm
a
I
UI,
---I
_ _ - - - --2-- - -
1
..........................
L
W In am
tm
LOO
ara
a
rm
Figure 9.1.15 Subharmonic resonances of the assembly with N = 3 beams. Stable solutions, ------ unstable solutions.
9.1 THEORETICAL ANALYSIS
439
"i----?
Figure 9.1.16 Subharmonic resonances of the assembly with N = 4 beams. Stable solutions, ------- unstable solutions.
440
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
forced responses are mainly confined to the directly excited beam 1. A variety of additional stable subharmonic solutions exist. For a system consisting of two, three, and four beams, one obtains at most two, four, and eight stable co-existing subharmonic resonances, respectively. In contrast to the fundamental resonance case, no stable subharmonic resonances corresponding to motions mainly confined to unforced beams were detected. Moreover, no Hopf bifurcations were detected in this case. To verify the results of the asymptotic analysis, direct numerical integrations of the equations of motion (9.1.51) and (9.1.56) were performed using a fourth-order Runge-Kutta numerical algorithm. The single-mode responses of an assembly consisting of N = 2 beams is considered, with governing equations of motion given by:
+ c(q13+ Pqld2(ql2)/dt2 + y(2qi - 2q2) ) + EmQicOSQd t j 2 + 02q2 = -E { c242 + aq23 + Pq2d2(q22)/dt2 ql
+ w2ql = -E{
clql
+ '/(2q2 - 2q1))
(9.1.62)
To carry out the numerical integrations, the following values for the various parameters were assumed: cl = 0.02, c2 = 0.03, Q I = 1.O. For fundamental excitation, m = 1, L21 = w + ED, and for subharmonic excitation, m = 0, Ro = 3 0 + ED. Trajectories of the dynamical system (9.1.62) exist in the fivedimensional phase space (q 1,q2,ql ,q2,t). Four-dimensional Poincare' maps of the responses were computed by "sampling" the responses qi and velocities q, at multiples of the minimum period T = 2x/RIn of the external excitation. Two-dimensional projections of the Poincare' maps in the (qi,qi), i = 1,2, planes were then constructed, which were amenable to graphical presentation. Fundamental and subharmonic resonances appear as fixed points in the projections of the Poincare' maps. Trajectories of the dynamical system initiated close to stable resonances remain close to those resonances; on the contrary, trajectories initiated in neighborhoods of iinstable resonances are observed to follow paths away from the unstable solutions. Therefore, the two-dimensional projections of the Poincare' maps can be employed to verify the existence and the stability of the analytically predicted resonance points. In all cases considered, the existence of the
9.1 THEORETICAL ANALYSIS
44 I
theoretically predicted stable and unstable fundamental and subharmonic solutions was verified by the numerical simulations. As an example of the numerical simulations performed, in Figure 9.1.17(a) the Poincare’ map for fundamental excitation of the two beam system with frequency detuning CT = 0.9 is shown. In agreement with the analytical predictions, three stable and five unstable fixed points are detected. Two stable resonances are in the neighborhoods of the in-phase normal mode and one localized normal mode (the one which localizes in the unforced beam 2). The third stable resonance is strongly localized in the forced beam and occurs in the neighborhood of the normal mode which localizes in beam 1. In Figure 9.1.17(b) the Poincare’ map for subharmonic excitation of the two beam assembly with frequency detuning (3 = 2.0 is presented. In agreement with the analytical predictions, two stable, and four unstable fixed points are detected. One stable subharmonic resonance corresponds to a perturbation of an in phase nonlinear normal mode. The other stable solution is strongly localized in the forced beam 1 and results as perturbation of a strongly localized nonlinear normal mode of the unforced system. In contrast to the fundamental resonance case discussed above, no subharmonic solutions were obtained corresponding to motion localization in the unforced beam 2. The results of this section prove the existence of weakly and strongly localized fundamental and subharmonic resonances in a weakly coupled nonlinear continuous system. The assembly studied in this work can be considered as a model of a space antenna with coupled flexible ribs and cyclic symmetry. It is anticipated that the results reported in this section will have direct applicability on the design and control of this class of structures. The unforced and undamped cyclic assembly possesses localized and nonlocalized nonlinear normal modes, which, for sufficiently weak coupling and/or strong nonlinearities, exceed in number the modes predicted by linear theory. When harmonic forcing is applied, fundamental and subharmonic resonances occur in the neighborhoods of nonlinear normal modes. The topological structure of these resonances was found to become increasingly complicated as the number of beams increased. For an assembly consisting of four beams, as many as twelve fundamental and eight subharmonic coexisting stable resonances were computed. The results reported in this section complement the findings of section 9.1.2 where the transient response of a weakly coupled system was considered. It was shown that nonlinear mode localization in a perfectly periodic assembly of continuous oscillators
442
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
9.00
,
L
7
I
a
-2.00
9.00
-1.20
-0.40
0.40
1.20
-1.20
-0.40
0.40
1.20
91
2.00
L
42
anF
5.00
1 .oo
-3.00 0.0
0.3
1.0
I1
B
LO
-7.00 ..
-2.00
q2
2.00
(4 Figure 9.1.17 Projections of Poincare' maps, N = 2: (a) fundamental resonances for (3 = 0.9, and (b) subharmonic resonances for CJ = 2.0.
9.1 THEORETICAL ANALYSIS
-6.00
-2.00
7.00
-1.20
-0.40
c
0.4C
...
443
41
1.20
.. .
41
3.75 LO
L
I
0.50
-2.75
-
Figure 9.1.17 (Continued)
.. .. .
.
.
2.00
444
NONLWEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
can lead to passive motion confinement of disturbances generated by impulsive and periodic external excitations. The previous Gallerkin approximations enabled the reduction of the continuous nonlinear problem to a discretized one. This approach is common in studies of free and forced oscillations of structural components (Nayfeh et al., 1974; Nayfeh and Mook, 1984). Although the discretization technique provides nonlinear corrections to the modal amplitudes of the superimposed linearized eigenfunctions, it does not provide any nonlinear corrections to the mode shapes of the ensuing nonlinear free oscillations. The analytical technique employed in the next section directly deals with the partial nonlinear equations of motion instead of discretizing them. Modal functions are introduced to express the motion of an arbitrary particle of a structural assembly in terms of the motion of a reference point, and spatially localized solutions are computed.
9.1.4. Nonlinear Mode Localization: Direct Analysis of the Equations of Motion An alternative method for studying nonlinear mode localization i n continuous systems will be developed by considering the free vibration of a general periodic structural assembly composed of N identical onedimensional continuous substructures coupled by means of elastic elements. All substructures are assumed to possess identical geometrical characteristics, and the equilibrium position of each substructure is parametrized by the same normalized spatial variable x, 0 Ix I 1. Displacements from equilibrium of material points of the ith substructure are denoted by variables ui(x,t), i = 1,...,N. It is assumed that the free motions of the periodic assembly are governed by a set of N nonlinear partial differential equations of the following form:
These equations are complemented by the set of homogeneous boundary conditions: Bi[uIfx,t),...,UN(X,t); &] = 0,
x = 0, 1,
i = 1,...,N
(9.1.64)
9.1 THEORETICAL ANALYSIS
445
Li[*] is an integro-differential operator acting on variables up(x,t), p =
1,...,N, whereas Bi[*] is a boundary condition operator. The nonlinear terms in operators Li[*] and Bip] are assumed to be small and proportional to a small parameter E, E I <<1. The following assumptions are now imposed on the boundary value problem (9.1.63)-(9.1.64).
I
( I ) The total energy of system (9.1.63) is conserved, and boundary conditions (9.1.64) involve no dissipation of energy. (2) For E = 0, the system is separable in space and time and admits bounded periodic solutions for any set of initial conditions. (3) Operators Li[*] involve integro-differential operations in terms of the spatial variable. (4) Only odd-order nonlinearities are present in the system. ( 5 ) The displacements uj(x,t) are sufficiently smooth functions of their variables, so that all derivatives appearing in the following analysis exist. Assuming that the assembly is oscillating in a nonlinear normal mode, the displacement of the kth substructure at position x = xo is denoted by uo(t) = uk(xo,t) (cf. section 8.1.1). The kth substructure will be termed reference substructure, and the quantity uo(t) reference displacement. Since the free oscillation is assumed to be synchronous, the displacement of an arbitrary material point of an arbitrary substructure can be parametrized in terms of the reference displacement uo(t) as follows:
The function Ui(.,*) is termed the ith modal function and describes the mode shape of the ith substructure during a nonlinear normal mode of the structural assembly. In writing (9.1.65) the set of independent variables is changed from (x,t) to [x,u,(t)], and the explicit time dependence of the motion is eliminated from the expression of ui. This transformation of coordinates requires xo not to coincide with a node of the nonlinear mode under investigation. One formulates a condition for strong nonlinear mode localization in the periodic assembly by requiring that the displacement of point x* of the kth (reference) substructure is much larger in magnitude than the corresponding point of any other substructure of the system:
446
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
IUk[x*,uo(t)]l >> IUj[X*,Uo(t)]I, i = 1,...,N, i # k, 0 5 x * l 1 Uk[X*,uo(t)] # 0, ‘d t 2 O (Strong Nonlinear Mode Localization) (9.1.66) Conditions (9.1.66) are imposed away from nodes of the nonlinear normal mode. These conditions guarantee spatial confinement of the free oscillation to the kth substructure. Conditions for weak nonlinear mode localization can be formulated in a similar way. To this end, one requires the nonlinear oscillation to be mainly confined to a subset of substructures (instead of only the reference substructure): IU,[x*,uo(t)]l >> lUi[~*,~o(t)]l,mES1, iES1, [ I , ...,N] 3S1,O I x * l 1 vt20 U,[X*,uo(t)]#O, (Weak Nonlinear Mode Localization) (9.1.67) Since the total energy, Etot, during free oscillation is conserved, one can formulate an energy conservation relation of the following form: N
Etot =
1
C { (1/2)5
i= 1
Uit2
dx
+ Li [Ul(X,t),...,UN(X,t); E])
(9.1.68)
0
where Li [*I is a spatial operator acting on the displacements; this operator represents potential energy terms arising from operator Li[*] in equation (9.1.63). Typically, Li [*I involves integrations with respect to the spatial variable, with integrands containing partial derivatives with respect to the displacements. Since the periodic assembly under consideration is conservative, relation (9.1.68) holds for all times. The partial derivatives of the displacements with respect to time are now expressed as follows:
where subscripts t denote derivatives with respect to the time variable. Using (9.1.69), the energy conservation relation is written as:
9.1 THEORETICAL ANALYSIS
447
where the integro-differential operator Li is derived from Li by substituting up(x,t) = Up[x,uo(t)], p = 1,...,N. From (9.1.70), the velocity of the reference point, uot, is expressed in terms of the modal functions Ui and the total energy Etot:
Expressions (9.1.65) and (9.1.7 1) evaluate the displacement and velocity of the reference point in terms of the (yet unknown) modal functions Up[x,uo(t)]. Substituting these expressions into the equations of motion (9.1.63), and noting that the acceleration of the reference point is computed by Uott = Lk[U1(xo,t),...,UN(xo,t); E l , one obtains the following equations governing the modal functions Ui[s,uo(t)]:
+ Ck[U 1(Xo,Uo>,...,uN(Xo,Uo);E l
aUi
aU, = ci[u 1(x,uo),...,UN(x,uo); E l
x E [O,l] (9.1.72) where i = 1,...,N. The following boundary conditions complement (9.1.72): Bi[Ui[x,uo(t)],...,U ~ [ x , u ~ ( t )=] ]0,
x = 0,1,
i = 1 ,...N
(9.1.73)
In (9.1.72) and (9.1.73), operators b[*] and &[*I are derived from the original operators Li[*] and Bi[*] by transforming the set of independent variables from (x,t) to [x,uo(t)]. Denoting by uo* the maximum amplitude attained by uo(t) during free oscillation, and taking into account that the motion of the system is synchronous, it is noted that equations (9.1.72) become singular when the periodic assembly reaches its maximum potential energy level, i.e., when
This is due to the fact that, at maximum potential energy, the coefficients of the highest derivatives (d’Ui/dUo’) in (9.1.72) vanish. Therefore, for values
448
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
t
I
,
I
I
. .
Nonlinear
I
/////////‘/////////I/////////////’
Figure 9.1.18 The flexible system of coupled simply supported beams on nonlinear foundations. of the potential energy less than Etot, asymptotic approximations to the functions Ui are constructed and then analytically continued up to the maximum potential energy level Etot. A similar asymptotic analysis was carried out in section 8.1.1 to study the NNMs of continuous oscillators. Analytic continuations of the asymptotic solutions up to the maximum potential energy level are achieved by complementing equations (9.1.72) with the following set of relations which hold at the maximum potential energy level: ( ~ ~ [ u l ( x o , u ,...,u o > N(X0,uo); E l
au, aU,
- k[ul(x,uo>,...,uN(x,uo>; E] } uo=uo* -0 i = l ,...,N (9.1.74) Asymptotic approximations to the solutions of equations (9.1.72)-(9.1.74) are sought in the form of a power series (as in section 8.1.1). Once analytic: approximations for the modal functions Ui are obtained, the time response uo(t) of the reference point is analytically approximated by considering the kth equation of motion at the reference point x = xo:
For a prescribed set of initial conditions uo(0) and uot(0), this equation can be solved either by quadratures or by an approximate nonlinear perturbation technique (Nayfeh and Mook, 1984). Once uo(t) is determined, the
9.1 THEORETICAL ANALYSIS
449
oscillation of an arbitrary point of the periodic assembly can be computed using relations (9.1.65). If a subset of the computed modal functions Ui[x,uo(t)] satisfies criteria (9.1.66) or (9.1.67), mode localization occurs in the structure. These criteria guarantee the existence of strong or weak nonlinear mode localization in a mathematical sense. However, not all mathematically computed nonlinear localized modes are expected to be physically realizable, since some may be orbitally unstable. To determine if the computed localized NNMs are physically attainable, one performs a numerical stability analysis based on Floquet theory, similar to the one performed in section 8.1.3. As an example of application of the asymptotic methodology, the localized NNMs of the structure depicted in Figure 9.1.18 will be computed. The structure consists of two coupled simply supported beams resting on nonlinear elastic foundations. Assuming that the distributed linear coupling stiffness and the nonlinearities of the elastic foundations are weak [of O(E), IEI << 11, the governing equations of motion are given by:
Assuming simply supported beams, the boundary conditions of the problem are: Ul(0,t) = ulxx(0,t) = Ul(l,t>= ulxx(1,t) = 0 (9.1.77) uz(O,t) = u2xx(O,t) = u2(l,t) = u2xx(l,t) = 0 The localized NNMs of this system are now sought. Following the previously outlined general methodology, it is assumed that ui(x,t) = Ui[x,uo(t)], u2(x,t) = U2[x,uo(t)], where U1 [xo,uo(t)] E uo(t) is the displacement of the reference point x = xo of beam 1. Omitting the details of the calculation [which can be found in (King and Vakakis, 1995c)], it can be shown that the modal functions are governed by the following equations:
450
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
= -U2xxxx - kU2 - ~yU23- E K ( U ~ Ul)
(9.1.78) These equations are complemented by the following relations holding at maximum potential energy level, uo(t) = uo* (which represents a regular singular point of the above equations):
{ [-U~XXXX - kUi
-
au I
~yU13- W U I - U 2 ) ] x = x o ~+ U I X X X + ~ kU I
+ Eyu13 + ~
u- ~ i2 ) } ~ ~ ( ~- 0) = ~ ~ *
{ [-U~XXXXkUi - q U i 3 - W -
+E
m 3
+~
au2
U 1 - U 2 ) ] x = x o ~+ U Z ~ X + X kU2 X
u - ~2l ) ) . ~ ( ~ ) = , ~ *=
o
(9.1.79)
The boundary conditions (9.1.77) of the problem are expressed in terms of the modal functions as follows:
Moreover, the additional compatibility condition is imposed:
Equations (9.1.78) and (9.1.79) correspond to equations (9.1.72) and (9.1.74), respectively, of the previous general formulation. Following the
9.1 THEORETICAL ANALYSIS
45 1
asymptotic methodology outlined in section 8.1.1, the modal functions are expressed in the series forms, ui[x,~o(t)l= Ui(O)(x,~o) + EUi(l)(X,Uo)+ &2Ui(')(X,uo) + O(E') i = 1,2 and are substituted into (9.1.78)-(9.1.80). In accordance with assumption (2) of the general formulation, the O( 1) approximations are separable in space and time, Ui(O)(X,Uo)= ai I(0)(x)uo(t). By matching coefficients of respective powers of E, a series of subproblems at various orders of E is formulated. The symbolic manipulations required for computing the asymptotic approximations to the modal functions were performed using the software package Mathernatica.
O(&O) Approximation Retaining only O( 1) terms in (9.1.78)-(9.1 .SO), one obtains the following equation governing the first-order corrections Ui(O)(X,Uo):
[-U l(O)xxxx - kU 1(O)] x=xo
au2(0) ~
aUo
= -U2(O)xxxx - kU2(o)
(9.1.8 1) Setting Ui(O)(X,uo) = ail(O)(x)uo(t), i = 1,2, the following ordinary differential equations governing ai 1(O)(x) are derived:
-
=$
a1 1(O)""(x) - a1 I(O)""(X~) a1 1(O)(x) = 0 a1 1(o)(x) = Asinhx + Bcoshx + Csinhhx + Dcoshhx a2 1 (O)""(x) - a1 1(0)""(xo) a2 1(O)(x) = 0 a21(O)(x) = Esinhx + Fcoshx +,Gsinhhx + Hcoshhx
(9.1.82) where h4 = a1 ~(O)""(XO) is a solution-dependant parameter, and primes denote differentiation with respect to the spatial variable x. Imposing the boundary conditions,
452
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
and the compatibility relation all(0)(xo) = 1, solutions (9.1.82) are expressed as: a1 i(O)(x) = sin(nxx)/sin(nxxo) a2i(0)(x) = a~~~~~(x,)sin(nxx)/sin(nxxo), h = nx
(9.1.83)
where n = 1,2,... denotes the order of the mode. The amplitude of beam 1 (the reference beam) is completely determined at this order of approximation, whereas the amplitude of the nonreference beam 2 is determined within a multiplicative constant, a21(0)(xo), which is evaluated at the next order of approximation. O ( E ~ Approximation )
Considering O(E) terms one obtains the following equations governing the first-order corrections Ul(l)(x,uo)and U2(1)(x,uo): 2 1
9.1 THEORETICAL ANALYSIS
453
where j = 1,2. The maximum potential energy value, Etot, in (9.1.84) can be expressed in terms of the maximum displacement of the reference point uo*:
(9.1.86) The corrections to the modal amplitudes Ui(')[x,uo(t)] are now expressed in the series form, Ui(l)[x,~o(t)I= ail(l)(x)uo(t)+ ai3(1)(~)~03(t) + 0(uO5(t)), i = 1,2 (9.1.87) Although the analysis can be carried to any order of approximation, only terms up to O(uo3(t)) are retained in the expressions for Ui(1). Substituting (9.1.87) into (9.1.84) and (9.1.85), taking into account (9.1.86), and matching coefficients of respective powers of uo(t), one obtains the following ordinary differential equations governing the spatial distributions ail(')(x) and ai3(1)(~): sin(nnx) 6 ( n W + k)uo*2ai3(')(x) - a1 i(l)""(x~)sin(nnxo)
- n4,4a11(1)(x) = -all(l)""(x) sin(nnx) 6(n4n4 + k)uo*2a23(l)(x) - a1 1(l)""(xo)a21(0)(xo)sin(nnxo) - n4x4a21W) = ~
sin( nxx) [ l a21(0)(xo>l(l + a21(0)(x0>)sin(nxxo)
(9.1 .88) Equation (9.1.88) can be used to evaluate the cubic spatial coefficients ai3(1)(~)in terms of the (yet unknown) linear spatial coefficients ail(l)(x). These coefficients are governed by the following eigth-order differential equations:
- 6y(n4d
+ k)uo*2] + 6y(n4n4 + k)uo*2
a} sin3 nnx
(9.1.89)
and
454
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
(9.1.90) The nonhomogeneous terms in the above expressions are solution d e p e n d a n t , since they contain the (yet unknown) first approximation a21(0)(xo) and various derivatives of a1 lCl)(x) evaluated at xo. Moreover, these equations depend on the order n of the mode under examination. Equations (9.1.89) and (9.1.90) are complemented by the boundary conditions. and i = 1, 2, and by the compatibility relations, all(l)(xo)= ai3(l)(xo) = 0. The solutions of (9.1.89) and (9.1.90) are composed of homogeneous and particular solution terms, and their derivation is standard. Equation (9.1.89) coupled with the boundary conditions and compatibility relation yields the following solution for a1 i(l)(x):
which, in view of the first of equations (9.1.88) leads to the following solution for the cubic spatial coefficient of the reference beam:
Considering the last two expressions, it is noted that they become singular when k = 9114x4. This is due to a 3:l internal resonance between modes (n) and (3n) (Shaw and Pierre, 1994). Hence the outlined formulation [as well as
9.1 THEORETICAL ANALYSIS
455
the invariant manifold formulation of Shaw and Pierre (1994)l is only valid for computing nonlinear normal modes that are not in low-order internal resonance with other modes. Results (9.1.91) and (9.1.92) compute the mode shape of the reference beam 1, during the nonlinear mode oscillation. The spatial coefficients a21(l)(x) and a23(1)(x) for beam 2 are computed by considering equations (9.1.90) and (9.1.88). Satisfying the boundary conditions, one obtains the following three possible solutions for the undetermined coefficient a2 ~ ( o ) ( xof ~ )the O( 1) approximation: a21(0)(x0) = +1,
a21(0)(xo) = -16Ksin2(nxxo)/(9yuo*2) (9.1.93) in terms of which one obtains the following analytic expressions for coefficients a2i(l)(x) and a23(1)(x): a21(l)(x) = Csin(nnx) +
or
-6po*2(n47c4+k)a2 1(0)3(x0) sin(3nnx) (9.1.94) 2560n4x4(9n4d--k)sin3(n~~x~)
ai i~l~""(xo)a~i~o~(xo)+K(l-a~~~~~~(x,)) . sin(nxx) 6uo*2(n4x4+k)sin(nxxo) 480n4n4yuo*2(n4n4+k)a21 (0)3(xo) sin(3nxx) (9.1.95)
a23(1)(x) =
+
15360uo*~n~x~(n~x~+k)n~x~(9n4x~-k)sin~(nxx~)
The coefficient a23(1)(x) is completely determined at this order of approximation, whereas the expression for a2 1(l)(x) contains the yet undetermined coefficient C . This coefficient can only be determined by considering O ( E ~terms ) in equations (9.1.78)-(9.1.80). This is due to the fact that no compatibility relations for a21(1)(x) and a23(1)(x) exist; by contrast, a1 l(l)(x) and a13(1)(x) satisfy the compatibility conditions a1 l(l)(xo) = a13(l)(x0) = 0, which were employed to compute expressions (9.1.91) and (9.1.92). The analytic expression for C was computed using Mathematica as follows:
456
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
(9.1.96)
(9.1.97)
and a21(0)(x0)is computed by (9.1.93). Combining all previous results, it is concluded that, depending on the value of a21(0)(xo), there exist three possible NNMs .for the system of Figure 9 .1 .18. On an NNM, the modal functions of the two beams are asymptotically approximated as follows:
(9.1.98a)
and
9.1 THEORETICAL ANALYSIS
+ &Csin(nnx)+ E
457
-6&yuo*2(n4n4+k)a21(o)3(x0) . sin(3nnx)}uo(t) 2560n4n4(9n4n4-k)sin3(nnxo)
[a1 1( l)""(xo)a2l(0)(xo)+K[1-a2 1(0)2(xo)]] 6uo*2(n4n4+k)sin(nnxo)
+ O(EU,~(~>,E~)
sin(nnx)
(9.1.98b)
From the above expressions it can be easily shown that, when a21(0)(xo) = +1 the modal functions are either equal or opposite in sign, i.e., Ui[x,uo(t>] = IfrU2[x,uo(t)]. In that case the structure vibrates in a nonlocalized in phase or antiphase NNM. When a2 ~ ( ~ ) ( =x -16Ksin2(nnxo)/(9yu0*2) d
and
(K/po*2) << 1
the modal function of beam 1 is much greater in magnitude than that of beam 2; in the limit (K/yuo*2)+0, it can be shown that Ul[x,uo(t)]+O( 1) and U ~ [ x , u o ( t ) ] - - + O ( ~ u o ~ ( t ) It , ~ is 2 ) concluded . that, as the ratio ( W p ,* 2 ) tends to zero, criterion (9.1.66) of the general analysis is satisfied and the nonlinear normal mode becomes strongly localized to the reference beam 1. Parameter K is the coefficient of the coupling stiffness, whereas product (yuo*2) is indicative of the strength of the nonlinearities of the elastic foundations. Hence, the previous analysis shows that when the ratio of the coupling over nonlinear terms is small, the system possesses a strongly localized nonlinear normal mode. This conclusion is in agreement with the findings of the previous sections. Note that due to the symmetry of the structure under consideration an additional strongly localized mode exists, involving localization of the free vibration in beam 2. This mode can be studied by considering beam 2 to be the reference substructure. The corresponding localized modal functions for the second localized mode are computed by setting a21(0)(xo)= -16Ksin2(nnxo)/(9yuo*2) and interchanging subscripts 1 and 2 in expressions (9.1.98).
458
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
Figure 9.1.19 Asymptotic approximations of U I and U2 for the in phase mode: ----- nonlinear theory (E # 0), ------- linear theory (E = 0).
Figure 9.1.20 Asymptotic approximations of U1 and U2 for the antiphase mode:----nonlinear theory (E f 0), ------- linear theory (E = 0).
9.1 THEORETICAL ANALYSIS
459
Figure 9.1.21 Asymptotic solutions for the strongly localized mode: (a) spatial coefficients aii(o)(s), aii(l)(S), and ai3(1)(~), i = 1,2, and (b) modal functions U1 and U2.
460
NONLINEAR LOCALIZATlON IN SYSTEMS OF COUPLED BEAMS
In Figures 9.1.19 and 9.1.20 the asymptotic approximations to the modal functions of the nonlocalized in phase and antiphase normal modes of the structure are depicted for n = 1, xo = 0.5, uo* = 1.0, K = 25, k = 850, y = 100, and E = 0.5. For coniparison purposes the respective linearized modes corresponding to E = 0 are also shown. In Figure 9.1.21(a) the spatial coefficients ail(O)(x), ail(')(x), and ai3(1)(~),i = 1,2, for the strongly localized mode are depicted for n = 1, xo = 0.5, uo* = 1.0, K = 10, k = 850, y = 100, and E = 0.5, whereas in Figure 9.1.21(b) the complete asymptotic approximations U1 and U2 to the localized modal functions are presented. Note that during the strongly localized mode the free periodic oscillation of reference beam 1 is of much larger amplitude than that of the nonreference beam 2. Hence, when the system oscillates in a strongly localized mode, the energy of the motion is mainly confined to only one of the two beams. Applying formula (9.1.75) of the general analysis and assuming that the system oscillates in a strongly localized mode, one obtains the following differential equation governing the motion of the reference point during nonlinear mode localization:
..
+
&{
uo(t) + (n47c4 + k)u,(t) 3yuo*2(n47c4+k)(4sin2n7cxo-3) 16Ksin2(nnx0) 16(9n4n4-k)sin2nnxo + ~ [ 1 + (9ru0*2) +E
y( 15 n4x4-2( n47c4+ k)sin2n7cxo) uo3(t) + O ( E U ~ ~ ( ~ (9. ),E I .99) ~) 2(9n47c4-k)s in2n7cxo
For a specified set of initial conditions, the solution of (9.1.99) can be expressed in terms of elliptic functions. Assuming that uo(0) = uo*, uot(0) = 0, i.e., that the system is initiated from a position of maximum potential energy, the solution is given by: uo(t) = uo*cn(plt,ki), uo(t) = uo*sn[p2t
+ K(kz),kz],
when
p >0
when -1 < puo*2/a < 0
(9.1.100)
where cn(*;) is the elliptic cosine, with argument pi and modulus kl given by:
9.1 THEORETICAL ANALYSIS
461
and sn(*,*) is the elliptic sine, with argument pi and modulus kl given by:
The quantities a and P are the linear and cubic coefficients of the modal oscillator (9.1.99), and are expressed as:
a = n4n4 + k +
3&yuo*2(n4n4+k)(4sin2nnx0-3) 16(9n47c4-k) sin2nnxo
P=
+
E K [ ~+
&yuo*2[ 15n4n4--2(n4n4+k)sin2n7cxo]
2(9n4x4-k)sin2nnxo
16Ksin2(nnxo) (gwO*2)
1
(9.1.10 lc)
The frequency of oscillation o of the strongly localized mode is computed as: o = o(u,*) = np1/2K(kl), when p > 0 0 = Nuo*) = np2/2K(k2), when -1 < Puo*2/a < 0 (9.1.102) where K(*) is the complete elliptic integral of the first kind. Employing (9.1.100) and (9.1.101) the responses of the two beams on the strongly localized NNM can be computed using the previously derived expressions (9.1.98). The stability of the computed localized mode can be examined by performing a numerical study similar to the one of section 8.1.3. The numerical stability analysis indicates orbital (neutral) stability for the nonlinear localized mode (King and Vakakis, 199%). Hence, the detected strongly localized mode is stable and, thus, physically realizable. The outlined analytical methodology provides an alternative way to study nonlinear mode localization in continuous systems. It is based on the direct asymptotic analysis of the governing nonlinear partial differential equations of motion and does not involve Gallerkin expansions or any other discretization scheme. As a result, the analysis provides nonlinear corrections to the modal amplitudes and to the localized mode shapes of the structure. The asymptotic analysis can be carried out up to any desired order of approximation, although for higher approximations the associated analytical computations become cumbersome and computer algebra is required.
462
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
9.2 EXPERIMENTAL VERIFICATION
An experimental study of nonlinear confinement of transient motions in a flexible structure consisting of two weakly coupled cantilever beams with active grounding nonlinearities is carried out in this section, in order to experimentaly verify the previously derived theoretical predictions. The experimental work described in this section was carried out with the help of Mr. J . Aubrecht and of Professors T.-C. Tsao and J . Bentsman of the University of Illinois (Aubrecht, 1994; Aubrecht et al., 1996). Moog, Inc., kindly provided on loan to the University of Illinois the actuators used for the experimental fixture. In section 9.1.2 passive confinement of transient responses in a system of coupled beams with geometric nonlinearities was theoretically and numerically demonstrated. The type of nonlinearities considered in that section can only be realized for relatively large beam deflections, a condition that is not met in the experimental fixture considered herein. As a result, an alternative source of beam nonlinearities was considered, in the form of actively induced nonlinear grounding "stiffness" forces generated by control actuators, and it is shown that, using active control, the system cun be designed to possess (active) motion confinement properties. The experimental apparatus consists of two coupled cantilever beams and is depicted in Figure 9.2.1. Aluminum beams of identical material properties and dimensions were used; each beam was 1 m in length, 0.0445 m in width and 0.00635 m in thickness. Vises were employed to simulate rigidly clamped boundary conditions, and each cantilever beam was rigidly attached to a steel plate that was firmly fixed to a rigid concrete foundation. A series of preliminary tests was conducted to verify that indirect coupling between beams due to the dynamics of the concrete foundation was negligible. Modal testing was employed to establish that the beams were linear and possessed nearly identical dynamic properties (Aubrecht, 1994). Nonlinearities in the system were actively induced by means of two control actuators simulating cubic grounding stiffnesses, and direct coupling between beams was achieved using an elastic cord. The equipment used for data acquisition and the control of the actuators consisted of a computer (data processor and collector), an RTI input/output board, amplifiers, actuators, displacement transducers, and force transducers. The analog input/output board was used both to collect data as
9.2 EXPERIMENTAL VERIFICATION
463
well as to provide control signals to the actuator amplifiers. Two actuators, one for each beam, were used in conjunction with these amplifiers. These were electro-mechanical linear force motors supplied by Moog, Inc., and produced a force output proportional to a current command. Two displacement transducers (DCDTs) were attached to the armature of each actuator, through the back ends of the actuators. Collocated with the DCDTs and the actuators were force transducers, which were placed between each beam and its corresponding control actuator. This permitted direct measurements of the forces applied by the actuators to the beams, enabling these forces to be used for control purposes. A computer code in C language was used to interface with the RTI board (the input/output board), enabling both data acquisition and control of the actuators. The basic structure of the program is an infinite loop. At the beginning of the loop the program collects data from the DCDTs and force transducers via the RTI board. Then, the appropriate calculations for implementing the active nonlinear stiffness law are performed, and the RTI board is commanded to send the resulting control voltage to the actuator amplifiers. It should be noted that at the beginning of each experimental test the DCDTs inevitably possess a DC offset voltage, corresponding to the static displacement of the actuators. For the DCDT data to accurately reflect the displacement from equilibrium of the beam at the position of the actuator, this offset must be removed. Hence, at the start of the program and at the beginning of each experimental run, the DC offset voltages for each DCDT are calculated and subtracted from the measured DCDT data. A complete listing of the C code can be found in (Aubrecht, 1994). In the experimental fixture the nonlinearities required to produce the transient motion confinement phenomenon were actively induced using the control and data acquisition apparatus described earlier. This was performed by using the actuators to simulate cubic nonlinear grounding stiffnesses. Hence, the overall objective of the employed control methodologies was to achieve a cubic relationship between the displacements of the actuators at their points of attachments to the beams and the forces exerted by the actuators on the beams. In the control methodology employed, the displacement of the actuator was sensed, cubed, multiplied by a gain, and fed as a control voltage to the actuator amplifiers (cf. Figure 9.2.2). The position loop, therefore, was closed; however, in this approach the force loop was not closed, and the technique was termed the open-loop control.
464
NONLINEAR LOCALIZATION W SYSTEMS OF COUPLED BEAMS
Figure 9.2.1 The experimental fixture. Finally, notch filters for eliminating unwanted unstable dynamics in the integrated controller-beam system had to be designed and implemented. These filters suppressed unstable dynamic response due to the excitation by the controllers of the second and third cantilever modes of the two beams. The existence of nonlinear motion confinement in the experimental fixture was experimentally verified by performing a series of transient vibration tests. In each test an initial deflection was imparted on beam 2, termed "directly excited beam," by imposing a fixed initial tip displacement equal to 4 inches; no initial deflection was imposed on beam 1, termed "nondirectly excited beam." The ensuing free transient vibrations of beams 1 and 2 were analyzed by recording the force and displacement signals from the force transducers and DCDTs, respectively, located at the positions of the actuators. This enabled the recording of the transient responses of the beams and, in addition, the experimental determination of the (linear or nonlinear) relationship between the force applied by the actuator and the corresponding beam deflection at the position of the actuator. These experimentally determined force-deflection relationships were used to quantify the active grounding stiffness nonlinearities induced by the actuators to the beams.
9.2 EXPERIMENTAL VERIFICATION
- - Fl
Open LOOP
Gain
b P
Fa
465
GaW
Figure 9.2.2 Open-loop control methodology. A typical set of linear and nonlinear results is presented in Figures 9.2.3 and 9.2.4, where the transient responses of the two beams with coupling stiffness k = 1.01 lb/in., and actuators off and on, respectively, are depicted. Considering the envelopes of the beam responses of the linear assembly [system with actuators off, Figure 9.2.3(d)], it is observed that at t 6.5 seconds near equipartition of instantaneous vibrational energy occurs between beams 1 and 2, indicating that a significant amount of vibrational energy "leaks" from the directly excited to the unexcited beam in the classical (linear) beat phenomenon. In addition, a careful examination of the time traces of Figure 9.2.3(c) reveals that a continuously varying phase difference between the transient motions of the two beams exists. This nonconstant phase difference is due to the participation of both in-phase and out-of-phase (linear) normal modes of the assembly in the transient responses. The transient responses of the system with actuators on (and active grounding stiffness nonlinearities) are depicted in Figure 9.2.4. It is noted that no equipartition on of instantaneous energy occurs in this case, since smaller amounts of vibrational energy leak to beam 1. Hence, a much greater portion of the induced vibrational energy remains confined to beam 2 and nonlinear transient motion confinement occurs. The energy leakage from the directly excited to the unexcited beam can be further reduced by increasing the gain (strengthening) of the active nonlinear stiffnesses. Comparing the envelopes depicted in Figures 9.2.3(d) and 9.2.4(d) one observes a diminishing of the period of the beat of the nonlinear response of
-
466
NONLLNEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
F vs. X for Unexcited Beam
F vs. X for Excited Beam
1
- -50 0
-lo802 a.
-0.01 0 0.01 displacement (in)
0
C.
d.
b.
-0.01 0 0.01 displacement (in)
I
I
2
I
4
I
6
I
0 time (sec)
I
10
0.02
,
I
12
I
I
14
Time Trace Envelopes of Beams
0.021
-0.02L 0
-Yo2
Time Trace of Beams
0.021
-0.021
0.02
I
,
2
4
6
I
8 time (sec)
10
12
14
Figure 9.2.3 Transient measurements at actuator positions, coupling stiffness k = 0.752 lb/in., actuators off (a), (b) force vs. deflection relations, (c) beam deflections, and (d) envelopes of beam vibrations.
9.2 EXPERIMENTAL VERIFICATION
100
-
50
-
O
-e VI
--i__l F vs, X for Unexcited Beam
F vs. X for Excited Beam
I?
0
m
v
m
O
.$
-50 -%02
-1P602 -50
-0.01 0 0.01 displacement (in)
0.02
-0.01
0
0.01
0.02
displacement (in)
a.
b. Time Trace of Beams
0.021
4.02;
467
I
I
2
4
1
6
8 time (sac)
10
12
14
I
C.
0.02-
I
= - -___- __--_---_-_--__--__ -_- _- -_- -_- -_- -_- _- -_- -_- -_- -- --:- - - - - - - 1
-0.02
I
Figure 9.2.4 Transient measurements at actuator positions, coupling stiffness k = 0.752 lbhn., actuators on: (a), (b) force vs. deflection relations, (c) beam deflections, and (d) envelopes of beam vibrations.
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
468
FFT of Displacement Signal of Beam1
2 c U
Y
0
-20
-40 -60 I
-80
‘I
I
10’ .-
a.
1.0‘ -
frequency (Hr)
FFT of DisplacementSignal of Beam2 ..
.
.. .. .. .. .. . . . . . . . . .. ..
.. . . .
-80 b.
.
.
. . . . . .. . .. . .. . .. . . . . . . . .. .. . . . . . . . .. ... ... . . . . :
...
...
... ...
.
.
:
...
:
:
:
:
. . . . . . . . . . . . . . . .,... .. . ..
I
I
10’
frequency (Hz)
.
.;
...........
.. ..
.
.. .
.. .
. . .. .. . 10’
Figure 9.2.5 FFT of time traces of Figure 9.2.3(c): (a) response of beam 1, and (b) response of beam 2.
9.2 EXPERIMENTAL VERIFICATION
469
FFT of Displacement Signal of Beam1
.. . ..
-80 a.
.. . ..
...
... .. .
. .. . ...
. . . .. . . . . . .. . . ..
. .. ...
.. . ..
.. .. . . ..
. ......I.! . .... ..
. . . .. . . . . . . . . . . .. . . . . . .. . . . . . .
-6O-..... : . . ...... .: ..;..
..
.. . .. ..
.
..
.. in' sv
.. .. .. .. . 10' frequency (Hz)
Figure 9.2.6 FFT of time traces of Figure 9.2.4(c): (a) response of beam 1, and (b) response of beam 2.
470
NONLINEAR LOCALIZATION IN SYSTEMS OF COUPLED BEAMS
beam 1, compared to that ofthe linear one. This feature is due to hardeningstiffness effects in the transient nonlinear response and is in full qualitative agreement with the theoretical predictions. Moreover, consideration of the relative phase differences between the time traces of beams 1 and 2 [cf. Figure 9.2.4(c)] indicates that, in contrast to the linear case, the nonlinear beams vibrate in a nearly antiphase transient oscillation, a strong indication that a strongly localized out-of-phase NNM participates in the nonlinear transient responses. In Figures 9.2.5 and 9.2.6 the Fast Fourier Transforms (FFTs) of the linear and nonlinear transient responses are depicted. The frequency spectra of the linear responses (Figure 9.2.5) possess dominant peaks close to the frequency of the first cantilever mode. Each of these peaks is composed of two nearly overlapping peaks, corresponding to the two closely spaced inphase and antiphase extended linear modes of the linear (actuators off) assembly. Considering the spectra of the nonlinear transient responses (Figure9.2.6), it is observed that close to each dominant peak a distinct strong secondary peak exists. It was previously mentioned that the beam responses of the nonlinear system are nearly out-of-phase; therefore, it is conjectured that the double dominant peaks in the spectra of Figure 9.2.6 correspond to the antiphase and strongly localized NNMs of the nonlinear assembly, which dominate the response in that .frequency range. Note that, since the strongly localized NNM exists only in the (nonlinear) system with actuators on, no secondary peaks are observed close to the dominant peak of the frequency spectra of Figure 9.2.5. A conjecture, thvefore, is made that the frequency spectra of the nonlinear transient responses possess distinct peaks corresponding to strongly localized and antiphase NNMs. Similar double peaks in frequency spectra of nonlinear transient responses were observed in experimental tests with weaker and stronger coupling stiffnesses. Two extensions of the previous experimental work were recently performed. King et al. (1995) investigated steady-state nonlinear motion confinement in the experimental fixture of Figure 9.2.1 with one of the beams undergoing periodic excitation by means of a modal shaker. Localized periodic responses in neighborhoods of localized NNMs of the system were detected, along with jumps between localized and nonlocalized branches of co-existing solutions. The experimental results reported in that work were in agreement with theoretical predictions. In a second extension of the experimental work, Emaci et al. (1996) replaced the active actuation in the
9.2 EXPERIMENTAL VERIFICATION
471
experimental fixture with rigid body constraints placed symmetrically with respect to each of the two beams. This arrangement gave rise to strong vibro-impact (clearance) nonlinearities during the vibrations of the beams, which, in turn, led to strong nonlinear mode localization in the system. Both transient and steady-state tests were conducted and strong nonlinear motion confinement in the directly excited beam was detected. For a detailed description of the vibro-impact localization experiments the reader is referred to the aforementioned work by Emaci et al. As mentioned previously, the nonlinear motion confinement phenomenon is of considerable practical importance since the controllability of a structure possessing localized NNMs is greatly enhanced. The results reported in this section indicate that the nonlinear motion confinement phenomenon can be induced in practical flexible structures through active means. In such structures vibration and shock isolation can be achieved by employing the following new design concept: Initially, disturbances generated by external excitations become spatially localized by means of actively induced "stiffness" nonlinearities. Once spatial motion confinement is achieved, the confined unwanted disturbances are eliminated through active control. Current research focuses on the implementation of the above design concept with flexible structures of more complicated geometry. Future applications are anticipated in the areas of vibration and shock isolation of mechanical components, rotordynamics (cyclic bladed assemblies), large space structures, and layered composite media.
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 10 NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS In this chapter we extend the localization results of chapter 9 to noncyclic continuous oscillators governed by nonlinear partial differential equations. In section 10.1 the localized and nonlocalized NNMs of a geometrically nonlinear multispan beam are considered. It is shown that as the number of beam segments increases, the structure of the NNMs of the structure becomes increasingly more complicated, and numerous NNM bifurcations take place. These mode bifurcations give rise to a variety of stable or unstable, weakly or strongly localized modes, having no counterparts in linear theory. In section 10.2, a nonlinear partial differential equation on an infinite domain is studied. It is shown that stationary and traveling solitary waves of this system can be studied by employing techniques previously developed for studying NNMs in bounded domains. Hence, stationary solitary waves can be regarded as localized NNMs of infinite spatial extent. The analytical methodology presented in that section can be used to study standing solitary waves of a general class of nonlinear partial differential equations. 10.1 MULTISPAN NONLINEAR BEAMS
In this section an additional paradigm of nonlinear mode localization in a periodic continuous system will be performed by analyzing the NNMs of a geometrically nonlinear multi-span beam, consisting of n segments that are coupled by means of torsional stiffeners. This problem is studied by the discretization technique used in sections 9.1.1-9.1.3, although the direct asymptotic approach of section 9.1.4 could also be applied. 10.1.1 Derivation of the Modulation Equations Linear systems consisting of n simply supported beam segments coupled by means of torsional stiffeners were studied both analytically (Yang and Lin, 1975; Mead, 1986; Pierre, 1988; Bouzit and Pierre, 1993; Lust et al., 1991), 473
474
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
and experimentally (Pierre et al., 1987). It was found that when the coupling between adjacent beam segments is weak, small perturbations of the periodicity of the system induce linear mode localization. A numerical study of the transmission of energy through the linear multi-span beam was carried out in (Zhang and Zhang, 1991). No analytic study of localization in nonlinear multispan beams appears to exist in the literature. In the following analysis it will be shown that a nonlinear multispan beam can possess a very complicated structure of stable and unstable localized NNMs. The mode bifurcations that give rise to nonlinear localization will be numerically investigated, and the ranges of values of the structural parameters for which localization occurs will be identified. The overall goal of the work presented in this section is to demonstrate that the presence of even weak nonlinearities in periodic structural components can give rise to very rich dynamic phenomena, which have no counterparts in linear theory. The flexible periodic assembly of Figure lO.l.l(a) is considered. This stiffened structure is representative of rib-skin structures encountered in airplane applications, consisting of flexible panels (skins) supported by arrays of periodically spaced stiffeners (ribs). Tail planes and fin structures usually have stiffened skins to carry the overall bending loads (Clarkson and Mead, 1973). Under broadband jet noise excitation the panels vibrate as interconnected stiffened plates between the stiffeners. The stiffeners also undergo vibrations caused by rotations of their connections to the panel, and under excess vibration there is a danger of fatigue failure in the rib in the attachment point to the skin (Clarkson, 1972; Clarkson and Mead, 1973). Previous works dealt with the linear dynamics of stiffened panels (Mead, 1971,1986; SenGupta, 1980). In the application to be considered herein, the stiffeners possess large in-plane stiffnesses, so that no t r a n s v e r s e displacements at the connecting points to the panel occur. Then, the panelstiffener intersections only possess elastic rotational freedoms (Mead, 197 1; SenGupta, 1974). Assuming uniform motion of the panel in the y-direction [cf. Figure lO.l.l(a)], the y-dependence is eliminated from the problem and the simplified system of Figure 10.1.1(b) is considered, consisting of simply supported beams coupled by torsional stiffnesses. The system depicted in Figure lO.l.l(b) is monocoupled, since there is a single coordinate (the local slope) connecting adjacent substructures. Coupling between adjacent beam segments is provided only through the elastic rotational properties of the stiffeners. In addition, it is assumed that
10.1 MULTISPAN NONLINEAR BEAMS
475
each beam segment possesses immovable (pinned) ends and that geometric nonlinearities exist due to nonlinear stretching of its midplane section. The equation of motion governing the transverse vibrations of the ith beam segment, is (Nayfeh and Mook, 1984):
where vj(s,t) is the transverse displacement, p the density per unit length, E and I the modulus of elasticity and moment of inertia, A the cross section, L the length, n the number of beams, and s the local position coordinate [cf. Figure 10.1.1(b)]. In deriving (10.1. l), it is assumed that the slope of the deformation is small, that no shear deformation of the cross section occurs, and that the rotary and longitudinal inertia effects are negligible. The term on the right-hand side of equation (10.1.1) quantifies the nonlinear effect of the stretching of the midplane section of the beam during transverse vibration. Complementing the equations of motion there is a set of boundary conditions of the form: vi(0,t) = vi(L,t) = 0 EI[viss(O,t) - V(i-l)ss(LJ)I = KTvis(O,t) EI[v(i+l)ss(O,t) - viss(L,t)l = KTvis(L,t) i = 1,...,n (10.1.2) Flexible panel (skin)
Stiffener (rib)
Figure 10.1.1 The nonlinear flexible system under consideration: (a) the rib-skin assembly, and (b) the monocoupled nonlinear multispan beam.
476
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
where KT denotes the coupling torsional stiffness at a panel-stiffener connection. Equations (10.1.2) are written with the understanding that v(,+l)(s,t) = vo(s,t) = 0. Equations (10.1.2) ensure zero displacements and balance of moments at all intersections between elastic elements. Introducing the new set of variables, x = s/L, Wi = Vi/L, z = L-Z(EI/p)”2t, and the rescaling Wi+El/2Wi, where E is a small (perturbation) parameter, Id << 1, equations (10.1.1) and (10.1.2) can be expressed into the following nondimensional form:
where wi = w i ( x , ~ and ) r2 = ILZ/A. Equations (10.1.3) and (10.1.4) form a set of weakly nonlinear partial differential equations with linear homogeneous boundary conditions. The nonlinear normal modes of this system will now be sought. In the following analysis, it will be assumed that the stiffeners possess large torsional stiffnesses. This assumption is satisfied by rescaling KT as follows: KT = kT/E (10.1.5) where kT is an O(1) quantity and E is the previously introduced small parameter. It will be shown that large torsional stiffeness of the stiffeners leads to weak coupling between adjacent beam segments, which, as shown in the previous sections, is a prerequisite for mode localization in the flexible assembly of nonlinear oscillators. The system (10.1.3)-(10.1.4) is not amenable to an exact analysis. Following (Timoshenko et al., 1974), the solution Wi = Wi(X,Z) is expressed as:
where wi(S)(X,T) and wi(f)(x,T) are static and flexible components of the transverse displacement, respectively. The static displacement components are computed by solving the equations
10.1 MULTISPAN NONLINEAR BEAMS
477
1
0 = -wixxXx(s)+ ~ ( 2 r 2 ) - l w i ~ ~ ( sWix(S)’ )J dx 0
i = 1,...,n
(10.1.7a)
subject to the boundary conditions: wi(S)(O,z)= wi(S)(l,z) = O Wix(’)(O,Z) = (EIL/KT)[wixx(O,z)- W(i-l)xx(l,z)I wix(’)(1 ,z> = (EIL/KT)[w(i+l)xx(O,z>- Wixx(1 i = 1 ,...,n (10.1.7b) Hence, the static displacement component Wi(S)(X,.t) represents the instantaneous static displacement of the ith segment, when the local beam slopes at its boundaries satisfy relations (10.1.7b). Successive integrations of (10.1.7a) and evaluations of the constants of integration by imposing (lO.1.7b), lead to the following analytic expression for the ith static displacement: wi(s)(x,z) = E ( E I L ~ T() [w(i+l)xx(07z)
- wixx(1,2>1+[wixx(o,z) - w(i-1)xx(1>2>1)x3
]
- E ( E I L ~ T )( [(w(i+l)xx(o,z) - wixx(l9z)l + 2[wixx(o,z) - w(i-1)xx(19z)~ x2 + E(EIWkT)[wixx(O,z)- W(i-~)xx(192)I + WE’) (10.1.8) i = 1,...,n with the understanding that ~ ( ~ + l ) ( S ) ( x= , zwo(S)(x,z) ) = 0. Note that the static displacements wi(s)(X,Z) depend on the instantaneous values of the second partial derivatives of the overall beam displacements Wi(X,T) with respect to x. Moreover, from expressions (10.1.8) one concludes that when the torsional stiffnesses of the stiffeners are large [of 0(1/&)],the static displacements of the beams, W i ( S ) ( X ,z), are small [of O ( E ) ] . Substituting (10.1.6) into the equations of motion (10.1.3) and the boundary conditions (10.1.4), and taking into account (10.1A), one obtains the following set of partial differential equations, which govern the flexible displacement components wi(f)(x,z):
478
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
Thus, the problem of determination of the nonlinear normal modes of system (10.1.3)-(10.1.4) is converted to an equivalent one, namely, of determining the nonlinear normal modes of system (10.1.9)-(10.1.10). The fundamental difference between systems (10.1.3)-( 10.1.4) and (10.1.9)(1 0.1.10) is that in the original formulation coupling between adjacent beam segments entered the problem through the bounduly conditions, whereas in the new formulation the coupling terms exist only in the governing equations of motion. In addition, the new set of boundary conditions (10.1.10) is in much simpler form than the original set (10.1.4), since the new boundary conditions correspond to clumped-clamped beam segments. Once the synchronous free periodic solutions (normal modes) for wi(f)(X,T) are obtained, the corresponding nonlinear normal modes of the assembly are determined by computing the overall beam displacements (10.1.6), taking into account (10.1.8) and the previously introduced scaling transformation wijE1’2wi. The nonlinear normal modes of system (10.1.9)-( 10.1.lo) can be studied, either by discretizing the partial differential equations of motion and analyzing the resulting set of modal ordinary differential equations or by using the direct asymptotic approach described in section 9.1.4. In this section the discretization methodology is employed, and the flexible components of the displacements Wi(f)(X,T)are expressed in the following series form:
10.1 MULTISPAN NONLINEAR BEAMS
479
where $j(x) is the jth normalized eigenfunction of the linearized clampedclamped beam segment:
where N(vj) = (coshvj - cosvj)/(sinhvj - sinvj), and vj is determined by solving the characteristic equation, CosvjcoshVj = 1. The jth linearized natural frequency is given by Oj = v j 2 . It is noted that only the first Mlinearized modes are taken into account in the Gallerkin analysis. Substituting (10.1.1 la) into (10.1.9), premultiplying both sides by the ith eigenfunction $i(x), integrating from x = 0 to x = 1, and employing eigenfunction orthogonality relations, one obtains the following set of ordinary differential equations governing the modal amplitudes qij(7):
480
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
In the equations above, the first subscript of the modal displacement qij specifies the beam segment under consideration, i = 1,...,n, whereas the second denotes the order of the mode in the Gallerkin approximation, j = l , ...,M. Overdots denote differentiation with respect time, and primes with respect to the spatial variable s. Equations (10.1.12) form a set of weakly coupled, weakly nonlinear ordinary differential equations; the free periodic solutions of these equations provide the NNMs of the periodic flexible assembly of Figure lO.l.l(b). As in the previous section, the free periodic solutions of (10.1.12) will be investigated using the method of multiple scales. In this work periodic systems consisting of n = 2, 3 , and 4 periodic segments are considered. Since the perturbation analysis becomes increasingly cumbersome as the number of periodic segments increases, details of the application of multiple scales will only be given for the case n = 2 (two periodic segments).
10.1.2 Numerical Computations First, a two-span nonlinear beam is considered. Setting n = 2 in the previous findings, and introducing "fast" and "slow" time-scales, To = T and TI = ET, respectively, the responses qij are expressed in the series form:
Substituting (10.1.14) into (10.1.12), using the chain rule to replace differentiations with respect to time with differentiations with respect to To and Ti, and matching coefficients of respective powers of E, the following subproblems at various orders of approximation are obtained:
10.1 MULTISPAN NONLINEAR BEAMS
O(EO) Approximation
2
48 1
2
DOqijo + 0 j qijo = 0 (10.1.15)
where i = 1,2, j = 1,...,M, Aij are complex amplitudes, Dn(') 0,1,
= a('>/aT,, n =
(5> denotes the complex conjugate, and j = (-1)1'2.
i = 1,2, j = 1,...,M (10.1.16) with the understanding that q3p0 = qip0. The complex amplitudes Aij(T1) of the O( 1) approximation are evaluated by substituting (10.1.15) into (10.1.16), and eliminating "secular terms," i.e., terms on the right-hand side of ( 10.1.16) of frequency Oj. At this point, it is noted that a low-order internal resonance exists between the first and second linearized modes, $ I ( s ) and $2(s), since their corresponding linearized natural frequencies are nearly integrably related: 0 2 301. As shown in sections 9.1.1 and 9.1.2, such a low-order "internal resonance'' can lead to nonlinear transfer of energy between resonating modes and introduce an additional complication in the asymptotic analysis (Crespo Da Silva and Zaretzky, 1990; King and Vakakis, 1995b). To study the effects of this low-order internal resonance, the Gallerkin approximation is truncated to the first two modes (M = 2), and a frequency detuning parameter, G, is introduced, defined by the relation:
-
0 2 = 301
+ EG
(10.1.17)
Parameter G quantifies the closeness of the multiples of the natural frequencies of the resonating modes. Substituting (10.1.15) into (10.1.16), taking into account (10.1.17), and eliminating secular terms in (10.1.16), one
482
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
obtains the set of first order differential equations that govern the complex amplitudes Aij(T1), i = 1,2, j = 1,2. For the first beam segment (i = 1), these equations take the form:
where prime denotes differentiation with respect to the "slow" time T i . Similar equations hold for modes 1 and 2 of the second beam segment but these are not reproduced here. Numerical computations of the coefficients ajpk] show that a1212 = a1221 = a1112 = a1121 = a1211 = a2112 = a2121 = a2111 = 0 ( 10.1.19)
A careful examination of equations (10.1.18) reveals that, due to (10.1.19), all coupling terms due to internal resonance between modes I and 2 are eliminated from the modulation equations (10.1.18). Hence, although a near-integral relation exists between the natural frequencies of modes 1 and 2, all coupling terms in the modulation equations accounting for the internal resonance vanish identically. It is concluded that no nonlinear transfer of energy due to internal resonance can occur between modes 1 and 2 of the first beam segment, and, as a result, each of these modes can be examined in isolation from the other. A similar analysis can be performed to show that no nonlinear transfer of energy occurs between the first three linearized modes of any beam segment of the periodic assembly. Therefore, each of these modes can be considered in isolation from the others. The elimination
10.1 MULTISPAN NONLINEAR BEAMS
483
of low-order internal resonances in the beam segments greatly simplifies the multiple-scales analysis. Expressing the complex amplitudes in polar form: Aij(T1) = (1/2)aij(T I >expliPij(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
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
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; ~
10.1 MULTISPAN NONLINEAR BEAMS
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
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
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)
10.1 MULTISPAN NONLINEAR BEAMS
’
-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
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
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.
10.1 MULTISPAN NONLINEAR BEAMS
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
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
-
,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.....\
.'
...
---
,/
,,.5
I
I
I
I
0.00001
0.00002
0.00003
0.00004
-
........... ......................... /9 ............
-
0.00005
s........
-2.50
-3.75 -5.00
(b)
I
0.00000
K1 la1111
I
I
I
1
0.00001
0.00002
0.00003
0.00004
Figure 10.1.4 (Continued)
0.00005
10.1 MULTISPAN NONLINEAR BEAMS
5.00
,
1
3.75
e I
2
.x
A
0.00 -1.25
-
13
..................
-2.50 -3.75
-5.00 0.00000 5.00 1.75
0.00
0.00005
1
I
I
0.00015
0.00020
r
I
19 T20 0 21
-
!?... ......
.....................................................
..........a: .-
- =-:.-
-3.75
-
i
16
, . a
I
-5.00
.
,
I
I
I
-
.1.7...................................... n
-1.25 -2.50
-3.75
-5.00
0.00026
v
-1.25
2.50
0.00010
-
13..........
16
K1 I a i i i i I
I
I
Figure 10.1.4 (Continued)
I
493
494
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
5.00 .
1
-
3.75 2'50
9#.
1 ........1 "i",E
-
1.25
'
p
0.00 -1.25
-=
-2.50
-
-3.75
-
P
h
__..... ...
8f'
. -
2.50
-
1.25
o,oo
-e! -e!
I
0.00
I
m
-1.25
.:o
'*._
"* '*
-
-3.75
-
-5.00
I
-,. ... -.._._ .................... ........
..... _.- .................... - .-:zL.-.::2.. \ 0 -
-2.50
I
.
- g. ....... s/ c,n - -\ /
t
5
I
I
3.75
A
n.&&"A
-5.00 5.00
"
.......-
I
v 22 V 23 0 24 25 §2e
-
-- + I
:;:
~ ~
27
c 30
€
n 31
u 32 E 33
e34
c
I
I
I
I
I
I
I
I
0.00005
0.00010
0.00015
0.00020
5.00 3.75
. 2 ; ; d
2.50 1.25
0.00
- 1.25 -2.50
K i la1 1 1 1
-3.75
(d)
-5.00
0.00000
35
Z 36
sl 37
Figure 10.1.4 (Continued)
0.00025
10.1 MULTISPAN NONLINEAR BEAMS
495
Modes 3, 4, 5 , 8, 9, 12, 15, 20, 21, 31, 35, 36, and 37 are weakly localized for small values of (Kllal111)corresponding to finite oscillations of three beam segments of the periodic assembly. Of these modes, only 3 and 4 are orbitally stable at certain ranges of the coupling parameter; again, exchanges of stability occur through Hamiltonian Hopf bifurcations. The remaining nonlinear modes of Figure 10.1.4 are extended, involving finite oscillations of all four beam segments of the assembly. Of the extended modes, modes 1, 2, and 22 are orbitally stable only at certain ranges of the coupling parameter. Summarizing, the NNMs of a geometrically nonlinear multispan beam consisting of n = 2,3, and 4 segments, coupled by means of torsional stiffeners were examined. Numerous stable and unstable, localized and nonlocalized nonlinear normal modes were detected. As the number of periodic segments increases, the topological structure of the branches of NNMs becomes increasingly more complicated. All systems examined possessed orbitally stable, strongly localized nonlinear normal modes. Nonlinear localization in the nonlinear multispan beam occurred only when the torsional stiffnesses of the stiffeners and/or the effects of the geometric nonlinearities were relatively large. Since large torsional coupling stiffness implies weak coupling between adjacent beams, it can be concluded that nonlinear localization occurs in multispan beams with weak coupling between adjacent beam segments and/or relatively strong geometric nonlinearities. This result is in agreement with the results reported in section 9.1. Two distinct mechanisms for generation of strongly nonlinear mode localization were detected. In assemblies consisting of n = 2 and 4 beam segments, strongly localized modes were generated through pitchfork or saddle-node bifurcations; in the system with n = 3 segments, strong localization was generated through saddle-node bifurcations, or was the limit of a continuous NNM branch as the coupling parameter tended to zero. Hence, assemblies consisting of even or odd number of segments appear to possess different mechanisms for generating strongly nonlinear localized modes. Finally, it is noted that nonlinear mode localization was found to take place in the perfectly periodic system, i.e., even in the absence of any structural disorder. This is in contrast to linear theory. The effect of structural disorder on nonlinear mode localization is an interesting topic of research that has not yet been pursued. A nonlinear multispan beam with sufficiently
496
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
weak structural disorder is expected to possess identical localization properties with the ordered system. This due to the fact that, although weak disorder will destroy nongeneric mode bifurcations such as the pitchfork bifurcation of Figure 10.1.2, it will not affect the generic saddle-node mode bifurcations of Figures 10.1.2 and 10.1.4, or the localized mode branches existing at small values of the coupling parameter ( K i l a 1 1 I 1 ) (Guckenheimer and Holmes, 1984). Hence, weakly disordered multispan nonlinear beams are expected to exhibit nonlinear localization phenomena similar to those reported herein. However, additional work is needed to confirm this conjecture and to investigate the effect of moderate or strong disorder on nonlinear mode localization.
10.2 WAVES WITH SPATIALLY LOCALIZED ENVELOPES In this final section, it will be shown how the asymptotic methodology previously developed for studying NNMs (section 8.1.1) and nonlinear mode localization (section 9.1.4) can be further extended to analyze standing and traveling waves with spatially localized envelopes in a class of nonlinear partial differential equations. As an example of application of this methodology, a specific nonlinear partial differential equation is considered and the amplitude distributions of stationary (nonpropagating) localized waves is asymptotically computed. Propagating, weakly modulated waves are computed by imposing Lorentz coordinate transformations. The technique presented in this section relates to methods developed previously for analyzing NNMs in discrete and continuous nonlinear oscillators. In the context of these previous works, the stationary wave solutions computed in this section can be regarded as localized nonlinear normal modes of unbounded, continuous, one-dimensional systems. Waves in nonlinear partial differential equations have been extensively studied. A solitary wave is a spatially localized traveling wave that decays or reaches a constant limit for large positive or negative values of the spatial independent variable. As first observed by Zabusky and Kruskal (1965), a reinarkable property of some solitary waves is their property to interact "elastically" with each other: after a collision of two or more of such waves, each wave retains its original form, and the only effect of the collision is an alteration of the relative phasing between participating waves. Such elastically interacting solitary solutions are termed .yo/itons. Certain
10.2 WAVES WITH SPATIALLY LOCALIZED ENVELOPES
497
nonlinear partial differential equations possess only solitary waves (but not solitons), whereas other (as the KdV and sine-Gordon equations) possess soliton solutions. The theory of solitons and solitary waves has applications in various engineering fields, including plasma physics (Hasegava, 1975), superconducting Josephson junctions (Lomdahl, 1 9 8 3 , and solid-state physics (Thomas, 1992). Whitam (1974) and Novikov et al. (1984) applied the Inverse Scattering Method (ISM) to the study of single- and multiplesoliton solutions in nonlinear partial differential equations. McKenna et al. (1992) experimentally studied nonlinear effects on Anderson localization in a one-dimensional system, and showed that localized states persist even in the presence of nonlinearity. This is one of the few works in the literature considering Anderson localization in nonlinear systems. In addition to solitons and solitary waves nonlinear partial differential equations also admit envelope solitons. These are stationary or travelingwave motions whose envelopes decay [breather solitons (Lamb, 1980)] or reach nonzero limits [dark or hole solitons (Pnevmatikos, 198.5>]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
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
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 <<1). The following assumptions are made. (1) Equation (10.2.1) possesses a time-independent first integral of motion. (2) For E = 0 the system is separable in space and time. ( 3 ) Function u(x,t) is sufficiently smooth, so that all derivatives appearing in the following analysis exist. Note that operator L(o)[u(x,t)] in (10.2.1) may be linear or nonlinear; the only requirement posed is that is separable in space and time. Stationary (nontraveling), time-periodic solutions of equation (10.2.1) are sought that satisfy conditions of the form: lim x-++
u(x,t) = u-,+
u(x,t+T) = u(x,t)
(10.2.2)
where T is the period of the oscillation. It is noted that only a very restricted class of equations (10.2.1) admit solutions that satisjj conditions (10.2.2) (Weinstein, 1985; Kruskal and Segur, 1987). The stationary periodic solutions (10.2.2) are effectively NNMs of (10.2.1); hence, u(x,t) can be related to the motion uo(t) = u(xo,t) of a prescribed reference point x = xo, by a modal functional relation of the form: u(x,t) = Ulx,uo(t)l
(10.2.3)
In writing (10.2.3) it is assumed that uo(t) is not identically equal to zero. The function U[x,uo(t)] is the modal function of this problem. As in section 9.1.4, the transformation of independent variables, (x,t)+ [x,uo(t)], is
500
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
introduced, and a functional equation is constructed satisfied by the modal function U[x,uo(t)]. The first integral of (10.2.1) is symbolically expressed as,
5 Ut2 ds + t(o)[u(x,t)j +
+m
E = (1/2)
E e(l)[u(~,t)]
(10.2.4)
-ca
where L@)[u(x,t)jand &l)[u(x,t)j denote the terms of the first integral due to operators L(O)[*] and L(1)[.], respectively. Typically, these terms involve integrations with respect to the spatial variable, with integrands containing partial derivatives of the displacement. Taking into account (10.2.3), the first integral can be rewritten as:
where This relation can be used to express the time derivative of the motion of the reference point in terms of the modal function U[x,uo(t)]. This expression can be symbolically written as: (10.2.6) uo,t(t) = f[U[x,uo(t>l; El Note that when system (10.2.1) reaches its maximum potential energy, V = Vmax = E, the time derivative uo,t(t) vanishes. Therefore, assuming that the motion of the reference point attains the value uo(t) = uo * at maximum potential energy, it must be satisfied that: f[U[x,uo*]; E l = 0 (10.2.7) E = Vmax = C{O)[U(X,UO*),U~*] + E C(~)[U(X,UO*),UO*] (10.2.8) Substituting (10.2.3) into the equation of motion (10.2. l), using the chain rule of differentiation, and noting that the second time derivative of u(x,t) at the reference point x = xo can be expressed as Uo,tt = { L(O)[u(x,t)] + EL(~)[U(X,t)l}X=XO one obtains the following equation governing the modal function U[x.uo(t)]:
10.2 WAVES WITH SPATIALLY LOCALIZED ENVELOPES
501
with the additional conditions: Iim x++_m ~[x,u,(t)l = u+
(10.2.10)
In (10.2.9), operators L(O)[*] and !=(I)[*] are derived from the original operators L(O)[.] and L(l)[*] by transforming the set of independent variables (x,t)+[x,uo(t)]. Zf equation (10.2.1) admits a stationary time-periodic wave as a solution, the amplitude modulation of this solution is computed by solving (10.2.9) with (10.2.10). Therefore, the problem of existence of stationary time-periodic waves in (10.2.1) is reduced to the equivalent problem of determining the solutions of equations (10.2.9) and (10.2.10). Since f2[U[x,uo*]; E ] = 0, equation (10.2.9) is singular when uo(t) = uo*, i.e., when the system reaches its maximum potential energy value, and the asymptotic methodologies of previous sections is employed to compute the solution. Once U[x,uo(t)] is asymptotically approximated, the reference displacement uo(t) is computed by substituting (10.2.3) into (10.2. l), and evaluating the resulting expression at x = xo. This leads to the complete description of the stationary time-periodic wave by means of (10.2.3). Nonstationary (traveling) envelope solitons can then be computed from the derived expressions of stationary solitons if appropriate Lorentz coordinate transformations can be applied to the system.
10.2.2 Application: Localization in an Infinite Chain of Particles An application of the methodology will now be given by analyzing the spatially localized vibrations of the periodic system depicted in Figure 10.2.1. This system consists of an infinite number of particles (labeled n = O,+1 ,+2, ...), connected to each other by weak linear springs and attached to a rigid foundation by weak nonlinear stiffnesses of the third degree. Assuming that no external excitations exist, the equations of motion of the chain are given by:
502
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
Figure 10.2.1 Infinite nonlinear chain of particles. yn + Yn + W 2 y n - yn-1- yn+l) + &pyn3= 0 n = O,+l,f2,... (10.2.11) where overdot denotes differentiation with respect to the time variable z, b > 0, p 2 0, and I d << 1. Equations (10.2.11) form an infinite set of ordinary differential equations. Under certain conditions (Sayadi and Pouget, 1991: Rosenau, 1987), a continuoum approximation can be imposed, and the infinite set (10.2.1 1) is replaced by a single nonlinear partial differential equation of motion. The continuum approximation is only valid if the wavelengths of the envelopes of the motions of the chain are much greater than the average distance between adjacent particles, i.e., if the envelopes of the examined motions are sufficiently smooth. Motivated by previous results on nonlinear mode localization, localized solutions of equations (10.2.11) corresponding to antiphase motions of adjacent particles are sought. Following Kosevich and Kovalev (1975), the following antiphase condition is imposed: yn(z> = (- 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
-
10.2 WAVES WITH SPATIALLY LOCALIZED ENVELOPES
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
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
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
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
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)]
10.2 WAVES WITH SPATIALLY LOCALIZED ENVELOPES
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
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
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 $(z) are satisfied. System (1 0.2.37) is a linear, parameter-dependant, nonhomogeneous ordinary differential equation. Due to limiting conditions (10.2.20), the solution of
10.2 WAVES WITH SPATIALLY LOCALIZED ENVELOPES
509
(10.2.37a) must reach constant limits as z+ltm. It can be shown (Vakakis, 1994a) that this solution is of the following form:
where @h(i)(q),i = 1,2, are two linearly independent homogeneous solutions of (10.2.37), defined as: @h(l)(z) = sinhz/cosh2z @h(2)(z)= @h(1)(z)[(3/2)z+ (1/4)sinh2z - cothz] (10.2.39) Note that lim z+km$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)
NONLINEAR LOCALIZATION IN OTHER CONTINUOUS SYSTEMS
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):
10.2 WAVES WITH SPATIALLY LOCALIZED ENVELOPES
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)
10.2 WAVES WITH SPATIALLY LOCALIZED ENVELOPES
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.
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
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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
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
