Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, And Averaging Methods available in Hardcover
Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, And Averaging Methods
- ISBN-10:
- 981429165X
- ISBN-13:
- 9789814291651
- Pub. Date:
- 01/19/2010
- Publisher:
- World Scientific Publishing Company, Incorporated
- ISBN-10:
- 981429165X
- ISBN-13:
- 9789814291651
- Pub. Date:
- 01/19/2010
- Publisher:
- World Scientific Publishing Company, Incorporated
Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, And Averaging Methods
Hardcover
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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;+axp=0 7
1.4.4 &xuml;+ax+bx1/3=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;+x1/3=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 ak 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;+x1/3=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;+x1/3=0 105
5.2.6 &xuml;+x-1/3 108
5.2.7 &xuml;+x+x1/3=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;+x1/3=-2ε&xdot; 133
6.2.5 &xuml;+x1/3=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 x1/3=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;+x1/3=-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;+x1/3=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