NONLINEAR OSCILLATIONS

Transcription

1 NONLINEAR OSCILLATIONS

2 NONLINEAR OSCILLATIONS ALI HASAN NAYFEH Univmity hlrtinguixhed Rofesaor DEAN T. MOOK Rofaaor Department of Engineering Science and Mechonica Wginia Polytechnic Itmilute and Stote University ~ckabuf& Virginia Wiley Classics Library Edition Published 1995 WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA

3 All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers arc advised to kcep in mind that statements, data, illustrations, procedural details or other items inay inadvertently be inaccurate. Library of Congress Card No.: Applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available froin the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliogratie; detailed bibliographic data is available in the Internet at < by John Wiley & Sons, Inc WILEY-VCH Verlag GinbH & Co. KGaA, Weinheiin All rights reserved (including those of translation into other languages). No part of this book inay be reproduced in any fonn - nor transmitted or translated into machine language without written permission froin the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Rcpublic of Gennany Printed on acid-free paper Printing and Bookbinding buch bucher dd ag, Birkach ISBN-13: ISBN-10:

4 To Our Wives Samirah and Sally

5 PREFACE Recently a large amount of research has been related to nonlinear systems having multidegrees of freedom, but hardly any of this can be found in the many existing books related to this general area. The previously published books emphasized, and some exclusively treated, systems having a single degree of freedom. These include the books of Krylov and Bogoliubov (1947); Minorsky (1947, 1962); Den Hartog (1947); Stoker (1950); McLachlan (1950); Hayashi (1953a, 1964); Timoshenko (1955); Cunningham (1958); Kauderer (1958); Lefschetz (1959); Malkin (1956); Bogoliubov and Mitropolsky (1 961); Davis (1962); Struble (1962); Hale (1963); Butenin (1965); Mitropolsky (1965); Friedrichs (1965); Roseau (1966); Andronov, Vitt, and Khaikin (1966); Blaqui6re (1966); Siljak (1969), and Brauer and Nohel(l969). Exceptions are the books by Evan- Iwanowski (1 976) and Hagedorn (1 978), which treat multidegree-of-freedom systems. However, a number of recent developments have not been included. The primary purpose of this book is to fill this void. Because this book is intended for classroom use as well as for a reference to researchers, it is nearly self-contained. Most of the first four chapters, which treat systems having a single degree of freedom, are concerned with introducing basic concepts and analytic methods, although some of the results in Chapter 4 related to multiharmonic excitations cannot be found elsewhere. In the remaining four chapters the concepts and methods are extended to systems having multidegrees of freedom. This book emphasizes the physical aspects of the systems and consequently serves as a companion to Perturbation Merhods by A. H. Nayfeh. Here many examples are worked out completely, in many cases the results are graphed, and the explanations are couched in physical terms. An extensive bibliography is included, We attempted to reference every paper which appeared in an archive journal and related to the material in the book. However omissions are bound to occur, but none is intentional. Many exercises have been included at the end of each chapter except the first. These exercises progress in complexity, and many of them contain intermediate steps to help the reader. In fact, many of them would expand the state of the art if numerical results were computed. Some of these exercises provide further references. Vii

6 Viii PREFACE We wish to thank Drs. D. T. Blackstock, M.P. Mortell, and B. R. Seymour for their valuable comments on Chapter 8, Fred Pearson for his careful reading of four chapters, and Drs. J. E. Kaiser, Jr., and W. S. Saric for their valuable comments on Chapter 1. A special word of thanks goes to our children Samir (age 7). Tariq (age lo), and Mahir (age 1 I) Nayfeh and Art Mook (age 16) and to Patty Belcher, Hazim Zibdeh, and Tom Dunyak for their efforts in checking the references. Many of the figures were drawn by Chip Gilbert, Joe Mook, and Fredd Thrasher, and we wish to express our appreciation to them. We wish to thank Janet Bryant for her painstaking typing and retyping of the manuscript. Finally a word of appreciation goes to Indrek Wichman, Jerzy Klimkowski, Helen Reed, Albert Yen, and Yen Liu for proofreading portions of the manuscript. Blacksburg. Virginia January 1979 Ali Hasan Nayfeh Dean T. Mook

7 CONTENTS 1. Introduction Preliminary Remarks, Conservative Single-Degree-of-Freedom Systems, Nonconservative Single-Degree-of-Freedom Systems, Forced Oscillations of Systems Having a Single Degree of Freedom, primary Resonances of the Duffing Equation, 7 I Secondmy Resonances of the Du ffing Equation, Systems with Quadratic Nonlinearities, Multifiequency Excitations, Serfsustaining Systems, Parametrically Excited Systems, Systems Having Finite Degrees of Freedom, Continuous Systems, Traveling Waves, Conservative Single-Degreesf-Freedom Systems Examples, A Simple Pendulum, A Particle Restrained by a Nonlinear Spring, A Particle in a Central-Force Field, A Particle on a Rotating circle, Qualitative Analysis, Quantitative Analysis, The Straightforward Expansion, The Lindstedt-Poincart! Method, The Method of Multiple Scales, The Method of Harmonic Bahnce, Methods of Averaging, Applications, The Motion of a Simple Pendulum, 63 ix

8 X CONTENTS Motion of a Chrrent-Carrying Conductor, Motion of a Particle on a Rotating Parabola, 72 Exercises, Nonconservative Single-Degree-of-Freedom Systems Damping Mechanisms, Coulomb Damping, Linear Damping, Nonlinear Damping, Hysteretic Damping, Material Damping, Radiation Damping, Negative Damping, Qualitative Analysis, A Study of the Singular Points, The Method of Isoclines, Litkrd s Method, Approximate Solutions, The Method of Multiple Scales, The Method of Averaging, Damping Due to Friction, Negative Damping, Examples of Positively Damped Systems Having Nonlinear Restoring Forces, Nonstationary Vibrations, Conservative Systems, Systems with Nonlinear Damping Only, Relaxation Oscillations, 142 Exercises, Forced Oscillations of Systems Having a Single Degree of Freedom Systems with Cubic Nonlinearities, Rimaty Resonances, i2 = a,,, Nonresonant Hard Excitations, Superharmonic Resonances, A2 = 4 a,,, Subharmonic Resonances, 51 = 300, Combination Resonances for Two-Term Excitations, Simultaneous Resonances: The Case in Which w,, = 30, and w,, = 3. a,, An Example of a Combination Resonance for a Three-Tern Excitation, 192

9 4.2. Systems with Quadratic and Cubic Nonlinearities, Rimary Resonances, Superharmonic Resonances, Subharmonic Resonances, Cbmbinatbn Resonances, Systems with Self-Sustained Oscillations, Rimary Resonances, Nonresonant Excitations, Superharmonic Resonances, Subharmonic Resonances, Cbmbination Resonances, Nonstationary Oscillations, Nonideal Systems, 224 Exercises, 234 CONTENTS Xi 5. Parametrically Excited Systems Examples, A Pendulum with a Moving Support, A Mechanical-Electrical System, A Double Pendulum, 262 Stability of EIastic Systems, Stability of Steaay-State Soktwns, The Floquet Theory, Single-Degree-ofFreedom Systems, Multidegree-of-Freedom Systems, Single-Degree-of-Freedom Systems, Hill's Infinite Determinant, The Method of Strained Parameters, l%e Method of Multiple Scales, Hill's Equation, Effects of Viscous Damping, Nonstationaty Excitations, Linear Systems Having Distinct Frequencies, The Straightforward Expansion, First-Order Expansions, Second-Order Expansions, Lateral Vibrations of a Cbkmn Produced by a Follower Force, Effects of Viscous Damping, Linear Systems Having Repeated Frequencies, The Case of w Near 2wl, 323

10 Xii CONTENTS The Case of w Near w1 t w3, The Case of w Near wl, Gyroscopic Systems, The Nonresonant Case, The Case of w Near 2w,, The Case of w Near w1 - wl, Effects of Nonlinearities, The Case of Quadratic Damping, The Rayleigh Oscillator, The Duffng Equation with Small Damping, 343 Exercises, Systems Having Finite Degrees of Freedom Examples, The Spherical Pendulum, The Spring Pendulum, A Restricted Ship Motion, Self-sustaining Oscillators, The Stability of the Triangular Points in the RestrictedProblem of Three Bodies, Free Oscillations of Systems Having Quadratic Nonlinearities, The Nonresonant Case, The Resonant Case (Internal Resonance), Free Oscillations of Systems Having Cubic Nonlinearities, The Nonresonant Case, The Resonant Gse (Internal Resonance), Free Oscillations of Gyroscopic Systems, Forced Oscillations of Systems Having Quadratic Nonlinearities, The Case of S2 Near w2, The Case of S2 Near wl, The Case of Nonresonant Excitations, The Case of 2S2 Near wl, The Case of S2 Near w1 t w2, Forced Oscillations of Systems Having Cubic Nonlinearities, TheGweofS2Nearw1, The Case of S2 Near w2, Parametrically Excited Systems, Zl. The Case A-2 Near 201, 425

11 CONTENTS xiii The Case R Near 2w2, The Gase R Near w1 t w2, 430 Exercises, Continuous Systems Beams, General Introduction, Governing Equations for the Gase u = O(w), 450 Z1.3. Governing Equation for the Case u = O(w2), Coupled Longitudinal and Transverse Oscillations, Hinged-Hinged Beams, Rimaty Resonances, Superharmonic Resonances, Subhumonic Resonances, Combination Resonances, Hinged-Clamped Beams, Rimaty Resonances, The Gase of!2 Near wl, Primary Resonances, The Case of R Near w2, Superharmonic Resonances, Combination Resonances, strings, Governing Equations, Rimaty Resonances, Stability of Possible Solutions, Plates, 500 Z 6.1. Governing Equations, Forced Symmetric Response of Circular Plates, 507 Exercises, Traveling Waves Longitudinal Waves Along a Bar, I. Hyperbolic Equations and Cluracteristics, The Method of Multiple Scales, Shock Fitting by Using Velocity, Shock Fitting by Using Displacement, Effect of Heterogeneity, Finite Uniform Bars, I. 7. Finite Heterogeneous Bars, 576

12 xiv CONTENTS 8.2, Transverse Waves Along a Beam on Elastic Foundation, A Nonlinear Parabolic Equation, Monochromatic Waves and their Stability, Modulational Stability, Modulational Instability, Nonlinear Coupled Parabolic Equations, Interaction of Stationluy Waves, 599 Exercises, 601 References 617 Index 687