Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, And Averaging Methods

by Ronald E Mickens

Overview

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.

Product Details

ISBN-13:	9789814291651
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/19/2010
Pages:	260
Product dimensions:	6.20(w) x 9.00(h) x 0.80(d)

Table of Contents

Preface vii

List of Figures xix

List of Tables xxi

1 Background and General Comments 1

1.1 Truly Nonlinear Functions 1

1.2 Truly Nonlinear Oscillators 2

1.3 General Remarks 3

1.4 Scaling and Dimensionless Form of Differential Equations 5

1.4.1 Linear Damped Oscillator 5

1.4.2 Nonlinear Oscillator 6

1.4.3 &xuml;+ax^p=0 7

1.4.4 &xuml;+ax+bx^1/3=0 8

Exactly Solvable TNL Oscillators 9

1.5.1 Antisymmetric, Constant Force Oscillator 10

1.5.2 Particle-in-a-Box 11

1.5.3 Restricted Duffing Equation 12

1.5.4 Quadratic Oscillator 14

1.6 Overview of TNL Oscillator Methods 14

1.6.1 Harmonic Balance 16

1.6.2 Parameter Expansion 16

1.6.3 Averaging Methods 17

1.6.4 Iteration Techniques 18

1.7 Discussion 18

Problems 20

References 21

2 Establishing Periodicity 23

2.1 Phase-Space 23

2.1.1 System Equations 24

2.1.2 Fixed-Points 24

2.1.3 ODE for Phase-Space Trajectories 25

2.1.4 Null-clines 25

2.1.5 Symmetry Transformations 26

2.1.6 Closed Phase-Space Trajectories 26

2.1.7 First-Integrals 26

2.2 Application of Phase-Space Methods 27

2.2.1 Linear Harmonic Oscillator 27

2.2.2 Several TNL Oscillator Equations 31

2.3 Dissipative Systems: Energy Methods 33

2.3.1 Damped Linear Oscillator 35

2.3.2 Damped TNL Oscillator 35

2.3.3 Mixed-Damped TNL Oscillator 36

2.4 Resume 39

Problems 39

References 40

3 Harmonic Balance 43

3.1 Direct Harmonic Balance: Methodology 44

3.2 Worked Examples 46

3.2.1 &xuml;+x³=0 47

3.2.2 &xuml;+x^-1=0 49

3.2.3 &xuml;+x²sgn(x)=0 51

3.2.4 &xuml;+x^1/3=0 54

3.2.5 &xuml;+x^-1/3=0 57

3.3 Rational Approximations 61

3.3.1 Fourier Expansion 62

3.3.2 Properties of a_k 62

3.3.3 Calculation of &xuml; 63

3.4 Worked Examples 63

3.4.1 &xuml;+x³=0 63

3.4.2 &xuml;+x²sgn(x)=0 65

3.4.3 &xuml;+x^-1=0 66

3.5 Third-Order Equations 67

3.5.1 Castor Model 68

3.5.2 TNL Castor Models 69

3.6 Resume 70

3.6.1 Advantages 70

3.6.2 Disadvantages 70

Problems 71

References 72

4 Parameter Expansions 75

4.1 Introduction 75

4.2 Worked Examples 76

4.2.1 &xuml;+x³=0 76

4.2.2 &xuml;+x^-1=0 78

4.2.3 &xuml;+x3/(1+x²)=0 80

4.2.4 &xuml;+x^1/3=0 81

4.2.5 &xuml;+ x³=ε(l-x²)&xdot; 84

4.2.6 &xuml;+sgn(x)=0 85

4.3 Discussion 86

4.3.1 Advantages 87

4.3.2 Difficulties 87

Problems 87

References 88

5 Iteration Methods 89

5.1 General Methodology 89

5.1.1 Direct Iteration 89

5.1.2 Extended Iteration 91

5.2 Worked Examples: Direct Iteration 92

5.2.1 &xuml;+x³=0 92

5.2.2 &xuml;+x³(1+x²)=0 97

5.2.3 &xuml;+x^-1=0 100

5.2.4 &xuml;+sgn(x)=0 103

5.2.5 &xuml;+x^1/3=0 105

5.2.6 &xuml;+x^-1/3 108

5.2.7 &xuml;+x+x^1/3=0 110

5.3 Worked Examples: Extended Iteration 112

5.3.1 &xuml;+x³=0 113

5.3.2 &xuml;+ x^-1=0 115

5.4 Discussion 117

5.4.1 Advantages of Iteration Methods 118

5.4.2 Disadvantages of Iteration Methods 119

Problems 120

References 121

6 Averaging Methods 123

6.1 Elementary TNL Averaging Methods 124

6.1.1 Mickens-Oyedeji Procedure 124

6.1.2 Combined Linearization and Averaging Method 126

6.2 Worked Examples 129

6.2.1 &xuml;+ x³=-2ε&xdot; 129

6.2.2 &xuml;+x³ =-ε&xdot;³ 131

6.2.3 &xuml;+x³=ε(1-x²)&xdot; 132

6.2.4 &xuml;+x^1/3=-2ε&xdot; 133

6.2.5 &xuml;+x^1/3=ε(l - x²)&xdot; 134

6.2.6 &xuml;+x=-2ε(&xdot;)^1/3 135

6.2.7 General Comments 137

6.3 Cveticanin's Averaging Method 138

6.3.1 Exact Period 139

6.3.2 Averaging Method 140

6.3.3 Summary 142

6.4 Worked Examples 142

6.4.1 &xuml;+ x|x|^α-1=-2ε&xdot; 142

6.4.2 &xuml;+x|x|^α-1=-2ε(&xdot;)³ 144

6.4.3 &xuml;+x|x|^α-1=ε(1-x ²)&xdot; 145

6.5 Chronology of Averaging Methods 147

6.6 Comments 149

Problems 151

References 152

7 Comparative Analysis 155

7.1 Purpose 155

7.2 &xuml;+x³=0 156

7.2.1 Harmonic Balance 156

7.2.2 Parameter Expansion 158

7.2.3 Iteration 158

7.2.4 Comments 159

7.3 &xuml;+x x^1/3=0 160

7.3.1 Harmonic Balance 160

7.3.2 Parameter Expansion 161

7.3.3 Iteration 162

7.3.4 Comment 162

7.4 &xuml;+x³=-2ε&xdot; 163

7.4.1 Mickens-Oyedeji 163

7.4.2 Combined-Linearization-Averaging 165

7.4.3 Cveticanin's Method 166

7.4.4 Discussion 167

7.5 &xuml;+x^1/3=-2ε&xdot; 167

7.5.1 Combined-Linearization-Averaging 167

7.5.2 Cveticanin's Method 168

7.5.3 Discussion 170

7.6 &xuml;+x³=ε(l-x ²)&xdot; 170

7.6.1 Mickens-Oyedeji 170

7.6.2 Cveticanin's Method 171

7.6.3 Discussion 172

7.7 &xuml;+x^1/3=ε(l-x²)&xdot; 175

7.8 General Comments and Calculation Strategies 175

7.8.1 General Comments 176

7.8.2 Calculation Strategies 177

7.9 Research Problems 179

References 181

Appendix A Mathematical Relations 183

A.1 Trigonometric Relations 183

A.1.1 Exponential Definitions of Trigonometric Functions 183

A.l.2 Functions of Sums of Angles 183

A.l.3 Powers of Trigonometric Functions 183

A.1.4 Other Trigonometric Relations 184

A.1.5 Derivatives and Integrals of Trigonometric Functions 185

A.2 Factors and Expansions 186

A.3 Quadratic Equations 187

A.4 Cubic Equations 187

A.5 Differentiation of a Definite Integral with Respect to a Parameter 188

A.6 Eigenvalues of a 2 x 2 Matrix 188

References 189

Appendix B Gamma and Beta Functions 191

B.1 Gamma Function 191

B.2 The Beta Function 191

B.3 Two Useful Integrals 192

Appendix C Fourier Series 193

C.l Definition of Fourier Series 193

C.2 Convergence of Fourier Series 194

C.2.1 Examples 194

C.2.2 Convergence Theorem 194

C.3 Bounds on Fourier Coefficients 195

C.4 Expansion of F(a cos x, −a sin x) in a Fourier Series 195

C.5 Fourier Series for (cos θ)^α and (sin θ)^α 196

References 198

Appendix D Basic Theorems of the Theory of Second-Order Differential Equations 199

D.1 Introduction 199

D.2 Existence and Uniqueness of the Solution 200

D.3 Dependence of the Solution on Initial Conditions 200

D.4 Dependence of the Solution on a Parameter 201

References 202

Appendix E Linear Second-Order Differential Equations 203

E.1 Basic Existence Theorem 203

E.2 Homogeneous Linear Differential Equations 203

E.2.1 Linear Combination 204

E.2.2 Linear Dependent and Linear Independent Functions 204

E.2.3 Theorems on Linear Second-Order Homogeneous Differential Equations 204

E.3 Inhomogeneous Linear Differential Equations 205

E.3.1 Principle of Superposition 206

E.3.2 Solutions of Linear Inhomogeneous Differential Equations 207

E.4 Linear Second-Order Homogeneous Differential Equations with Constant Coefficients 207

E.5 Linear Second-Order Inhomogeneous Differential Equations with Constant Coefficients 208

E.6 Secular Terms 210

References 211

Appendix F Lindstedt-Poincaré Perturbation Method 213

References 216

Appendix G A Standard Averaging Method 217

References 220

Appendix H Discrete Models of Two TNL Oscillators 221

H.l NSFD Rules 221

H.2 Discrete Energy Function 222

H.3 Cube-Root Equation 223

H.4 Cube-Root/van der Pol Equation 225

References 226

Bibliography 227

Index 237