Nonlinear oscillations

716 Pages·2007·28.202 MB·English

by  Mook, Dean T.;Nayfeh, Ali Hasan

NONLINEAR
OSCILLATIONS
ALI HASAN NAYFEH
Univmity hlrtinguixhed Rofesaor
DEAN MOOK
T.
Rofaaor
Department of Engineering Science and Mechonica
Wginia Polytechnic Itmilute and Stote University
~ckabuf&Vi rginia
Wiley Classics Library Edition Published 1995
WILEY-
VCH
WILEY-VCH Verlag GmbH & Co. KGaA
This Page Intentionally Left Blank
NONLINEAR
OSCILLATIONS
ALI HASAN NAYFEH
Univmity hlrtinguixhed Rofesaor
DEAN MOOK
T.
Rofaaor
Department of Engineering Science and Mechonica
Wginia Polytechnic Itmilute and Stote University
~ckabuf&Vi rginia
Wiley Classics Library Edition Published 1995
WILEY-
VCH
WILEY-VCH Verlag GmbH & Co. KGaA
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0 1995 by John Wiley & Sons, Inc.
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To
Our Wives
Samirah and Sally
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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 exist-
ing books related to this general area. The previously published books empha-
sized, 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 remain-
ing 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 re-
sults were computed. Some of these exercises provide further references.
Vii
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 (age 1 I) Nayfeh and Art Mook (age 16) and to Patty Belcher,
Mahir
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
to Indrek Wichman, Jerzy Klimkowski, Helen Reed, Albert Yen, and Yen
goes
Liu for proofreading portions of the manuscript.
Ali Hasan Nayfeh
Mook
Dean T.
Blacksburg. Virginia
January 1979
CONTENTS
1. Introduction 1
1.1. Preliminary Remarks, 1
1.2. Conservative Single-Degree-of-FreedomS ystems, 1
1.3. Nonconservative Single-Degree-of-FreedomS ystems, 3
1.4. Forced Oscillations of Systems Having a Single Degree of Freedom, 6
1.4.1. primary Resonances of the Duffing Equation, 7
I. 4.2. SecondmyR esonances of the Duf fing Equation, 1 1
1.4.3. Systems with Quadratic Nonlinearities, 14
1.4.4. MultifiequencyE xcitations, 14
1.4.5. Serfsustaining Systems, 17
1.5. Parametrically Excited Systems, 20
1.6. Systems Having Finite Degrees of Freedom, 26
1.7. Continuous Systems, 30
1.8. Traveling Waves, 35
2. Conservative Single-Degreesf-Freedom Systems 39
2.1. Examples, 39
2.1.1. A Simple Pendulum, 39
2.1.2. A Particle Restrained by a Nonlinear Spring,
40
2.1.3. A Particle in a Central-Force Field, 40
2.1.4. A Particle on a Rotating circle, 41
2.2. Qualitative Analysis, 42
2.3. Quantitative Analysis, 50
2.3.1. The Straightforward Expansion, 5 1
2.3.2. The Lindstedt-Poincart! Method, 54
2.3.3. The Method of Multiple Scales, 56
2.3.4. The Method of Harmonic Bahnce, 59
2.3.5. Methods of Averaging, 62
2.4. Applications, 63
2.4.1. The Motion of a Simple Pendulum, 63
ix
X CONTENTS
2.4.2. Motion of a Chrrent-Carrying Conductor, 67
2.4.3. Motion of a Particle on a Rotating Parabola, 72
Exercises, 77
3. Nonconservative Single-Degree-of-Freedom Systems 95
3.1. Damping Mechanisms, 95
3.1.1. Coulomb Damping, 95
3.1.2. Linear Damping, 96
3.1.3. Nonlinear Damping, 96
3.1.4. Hysteretic Damping, 97
3.1.5. Material Damping, 100
3.1.6. Radiation Damping, 100
3.1.7. Negative Damping, 103
3.2. Qualitative Analysis, 107
3.2.1. A Study of the Singular Points, 110
3.2.2. The Method of Isoclines, 1 17
3.2.3. Litkrd’s Method, 118
3.3. Approximate Solutions, 119
3.3.1. The Method of Multiple Scales, 120
3.3.2. The Method of Averaging, 121
3.3.3. Damping Due to Friction, 122
3.3.4. Negative Damping, 129
3.3.5. Examples of Positively Damped SystemsH aving Nonlinear
Restoring Forces, 13 1
3.4. Nonstationary Vibrations, 136
3.4.1. Conservative Systems, 139
3.4.2. Systems with Nonlinear Damping Only, 141
3.5. Relaxation Oscillations, 142
Exercises, 146
4. Forced Oscillations of Systems a Single of Freedom 161
Having Degree
4.1. Systems with Cubic Nonlinearities, 162
4.1.1. Rimaty Resonances, i2 = 163
a,,,
4.1.2. Nonresonant Hard Excitations, 174
4.1.3. Superharmonic Resonances, = 4 175
A2 a,,,
4.1.4. Subharmonic Resonances, 51 = 300, 179
4.1.5. Combination Resonances for Two-TermE xcitations, 183
4.1.1. SimultaneousR esonances: The Case in Which w,, = 30, and
w,, = 3. a,, 188
4.1.7. An Example of a Combination Resonance for a Three-Tern
Excitation, 192