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

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World Scientific, 2010 - Mathematics - 238 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.
 

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Contents

1 Background and General Comments
1
2 Establishing Periodicity
23
3 Harmonic Balance
43
4 Parameter Expansions
75
5 Iteration Methods
89
6 Averaging Methods
123
7 Comparative Analysis
155
Appendix A Mathematical Relations
183
Appendix C Fourier Series
193
Appendix D Basic Theorems of the Theory of SecondOrder Differential Equations
199
Appendix E Linear SecondOrder Differential Equations
203
Appendix F LindstedtPoincare Perturbation Method
213
Appendix G A Standard Averaging Method
217
Appendix H Discrete Models of Two TNL Oscillators
221
Bibliography
227
Index
237

Appendix B Gamma and Beta Functions
191

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