Truly Nonlinear Oscillations

Harmonic Balance, Parameter Expansions, Iteration, And Averaging Methods

By: Ronald E Mickens

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Hardcover | 19 January 2010

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262 Pages

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This unique book provides a concise presentation of many of the fundamental strategies for calculating approximations to the oscillatory solutions of "truly nonlinear" (TNL) oscillator equations. The volume gives a general overview of the author's work on harmonic balance, iteration and combined linearization-averaging methods. However, full discussions are also presented on parameter expansion procedures and a first-order averaging technique for TNL oscillators. The calculational basis of each method is clarified by applying them to a set of standard TNL oscillator equations. This allows a direct comparison to be made among the various methods.The book is self-contained and therefore suitable for both classroom use and self-study by students and professionals who desire to learn, understand, and apply these technique to the field of nonlinear oscillations.

Industry Reviews

The book is self-contained and suitable for both classroom use and self-study The author has definitely succeeded in presenting a readable book useful to students and professionals who are concerned with the field of nonlinear oscillations. -- Mathematical Reviews "Mathematical Reviews"

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Applied Mathematics Nonlinear Science Differential Calculus & Equations Engineering & Technology Mechanical Engineering & Materials Mechanical Engineering Discrete Mathematics Booktopia Publisher Services

World Scientific Publishing

Preface	p. vii
List of Figures	p. xix
List of Tables	p. xxi
Background and General Comments	p. 1
Truly Nonlinear Functions	p. 1
Truly Nonlinear Oscillators	p. 2
General Remarks	p. 3
Scaling and Dimensionless Form of Differential Equations	p. 5
Linear Damped Oscillator	p. 5
Nonlinear Oscillator	p. 6
&xuml;+ax^p=0	p. 7
&xuml;+ax+bx^1/3=0	p. 8
Exactly Solvable TNL Oscillators	p. 9
Antisymmetric, Constant Force Oscillator	p. 10
Particle-in-a-Box	p. 11
Restricted Duffing Equation	p. 12
Quadratic Oscillator	p. 14
Overview of TNL Oscillator Methods	p. 14
Harmonic Balance	p. 16
Parameter Expansion	p. 16
Averaging Methods	p. 17
Iteration Techniques	p. 18
Discussion	p. 18
Problems	p. 20
References	p. 21
Establishing Periodicity	p. 23
Phase-Space	p. 23
System Equations	p. 24
Fixed-Points	p. 24
ODE for Phase-Space Trajectories	p. 25
Null-clines	p. 25
Symmetry Transformations	p. 26
Closed Phase-Space Trajectories	p. 26
First-Integrals	p. 26
Application of Phase-Space Methods	p. 27
Linear Harmonic Oscillator	p. 27
Several TNL Oscillator Equations	p. 31
Dissipative Systems: Energy Methods	p. 33
Damped Linear Oscillator	p. 35
Damped TNL Oscillator	p. 35
Mixed-Damped TNL Oscillator	p. 36
Resume	p. 39
Problems	p. 39
References	p. 40
Harmonic Balance	p. 43
Direct Harmonic Balance: Methodology	p. 44
Worked Examples	p. 46
&xuml;+x³=0	p. 47
&xuml;+x^-1=0	p. 49
&xuml;+x²sgn(x)=0	p. 51
&xuml;+x^1/3=0	p. 54
&xuml;+x^-1/3=0	p. 57
Rational Approximations	p. 61
Fourier Expansion	p. 62
Properties of a_k	p. 62
Calculation of &xuml;	p. 63
Worked Examples	p. 63
&xuml;+x³=0	p. 63
&xuml;+x²sgn(x)=0	p. 65
&xuml;+x^-1=0	p. 66
Third-Order Equations	p. 67
Castor Model	p. 68
TNL Castor Models	p. 69
Resume	p. 70
Advantages	p. 70
Disadvantages	p. 70
Problems	p. 71
References	p. 72
Parameter Expansions	p. 75
Introduction	p. 75
Worked Examples	p. 76
&xuml;+x³=0	p. 76
&xuml;+x^-1=0	p. 78
&xuml;+x3/(1+x²)=0	p. 80
&xuml;+x^1/3=0	p. 81
&xuml;+ x³=¿(l-x²)&xdot;	p. 84
&xuml;+sgn(x)=0	p. 85
Discussion	p. 86
Advantages	p. 87
Difficulties	p. 87
Problems	p. 87
References	p. 88
Iteration Methods	p. 89
General Methodology	p. 89
Direct Iteration	p. 89
Extended Iteration	p. 91
Worked Examples: Direct Iteration	p. 92
&xuml;+x³=0	p. 92
&xuml;+x³(1+x²)=0	p. 97
&xuml;+x^-1=0	p. 100
&xuml;+sgn(x)=0	p. 103
&xuml;+x^1/3=0	p. 105
&xuml;+x^-1/3	p. 108
&xuml;+x+x^1/3=0	p. 110
Worked Examples: Extended Iteration	p. 112
&xuml;+x³=0	p. 113
&xuml;+ x^-1=0	p. 115
Discussion	p. 117
Advantages of Iteration Methods	p. 118
Disadvantages of Iteration Methods	p. 119
Problems	p. 120
References	p. 121
Averaging Methods	p. 123
Elementary TNL Averaging Methods	p. 124
Mickens-Oyedeji Procedure	p. 124
Combined Linearization and Averaging Method	p. 126
Worked Examples	p. 129
&xuml;+ x³=-2¿&xdot;	p. 129
&xuml;+x³ =-¿&xdot;³	p. 131
&xuml;+x³=¿(1-x²)&xdot;	p. 132
&xuml;+x^1/3=-2¿&xdot;	p. 133
&xuml;+x^1/3=¿(l - x²)&xdot;	p. 134
&xuml;+x=-2¿(&xdot;)^1/3	p. 135
General Comments	p. 137
Cveticanin's Averaging Method	p. 138
Exact Period	p. 139
Averaging Method	p. 140
Summary	p. 142
Worked Examples	p. 142
&xuml;+ x x ^¿-1=-2¿&xdot;	p. 142
&xuml;+x x ^¿-1=-2¿(&xdot;)x003C;sup>3	p. 144
&xuml;+x x ^¿-1=¿(1-x x003C;sup>2)&xdot;	p. 145
Chronology of Averaging Methods	p. 147
Comments	p. 149
Problems	p. 151
References	p. 152
Comparative Analysis	p. 155
Purpose	p. 155
&xuml;+x³=0	p. 156
Harmonic Balance	p. 156
Parameter Expansion	p. 158
Iteration	p. 158
Comments	p. 159
&xuml;+x x^1/3=0	p. 160
Harmonic Balance	p. 160
Parameter Expansion	p. 161
Iteration	p. 162
Comment	p. 162
&xuml;+xx003C;sup>3=-2¿&xdot;	p. 163
Mickens-Oyedeji	p. 163
Combined-Linearization-Averaging	p. 165
Cveticanin's Method	p. 166
Discussion	p. 167
&xuml;+x^1/3=-2¿&xdot;	p. 167
Combined-Linearization-Averaging	p. 167
Cveticanin's Method	p. 168
Discussion	p. 170
&xuml;+x³=¿(l-x ²)&xdot;	p. 170
Mickens-Oyedeji	p. 170
Cveticanin's Method	p. 171
Discussion	p. 172
&xuml;+x^1/3=¿(l-x²)&xdot;	p. 175
General Comments and Calculation Strategies	p. 175
General Comments	p. 176
Calculation Strategies	p. 177
Research Problems	p. 179
References	p. 181
Mathematical Relations	p. 183
Trigonometric Relations	p. 183
Exponential Definitions of Trigonometric Functions	p. 183
Functions of Sums of Angles	p. 183
Powers of Trigonometric Functions	p. 183
Other Trigonometric Relations	p. 184
Derivatives and Integrals of Trigonometric Functions	p. 185
Factors and Expansions	p. 186
Quadratic Equations	p. 187
Cubic Equations	p. 187
Differentiation of a Definite Integral with Respect to a Parameter	p. 188
Eigenvalues of a 2 x 2 Matrix	p. 188
References	p. 189
Gamma and Beta Functions	p. 191
Gamma Function	p. 191
The Beta Function	p. 191
Two Useful Integrals	p. 192
Fourier Series	p. 193
Definition of Fourier Series	p. 193
Convergence of Fourier Series	p. 194
Examples	p. 194
Convergence Theorem	p. 194
Bounds on Fourier Coefficients	p. 195
Expansion of F(a cos x, −a sin x) in a Fourier Series	p. 195
Fourier Series for (cos ¿)^¿ and (sin ¿)^¿	p. 196
References	p. 198
Basic Theorems of the Theory of Second-Order Differential Equations	p. 199
Introduction	p. 199
Existence and Uniqueness of the Solution	p. 200
Dependence of the Solution on Initial Conditions	p. 200
Dependence of the Solution on a Parameter	p. 201
References	p. 202
Linear Second-Order Differential Equations	p. 203
Basic Existence Theorem	p. 203
Homogeneous Linear Differential Equations	p. 203
Linear Combination	p. 204
Linear Dependent and Linear Independent Functions	p. 204
Theorems on Linear Second-Order Homogeneous Differential Equations	p. 204
Inhomogeneous Linear Differential Equations	p. 205
Principle of Superposition	p. 206
Solutions of Linear Inhomogeneous Differential Equations	p. 207
Linear Second-Order Homogeneous Differential Equations with Constant Coefficients	p. 207
Linear Second-Order Inhomogeneous Differential Equations with Constant Coefficients	p. 208
Secular Terms	p. 210
References	p. 211
Lindstedt-Poincaré Perturbation Method	p. 213
References	p. 216
A Standard Averaging Method	p. 217
References	p. 220
Discrete Models of Two TNL Oscillators	p. 221
NSFD Rules	p. 221
Discrete Energy Function	p. 222
Cube-Root Equation	p. 223
Cube-Root/van der Pol Equation	p. 225
References	p. 226
Bibliography	p. 227
Index	p. 237
Table of Contents provided by Ingram. All Rights Reserved.

Truly Nonlinear Oscillations

Harmonic Balance, Parameter Expansions, Iteration, And Averaging Methods

At a Glance

Hardcover

Industry Reviews

Shipping

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