0.1 Preface to Second Edition . . . . . . . . . . . . . . . . . . . . . . viii
1 Introduction 1
2 Nonlinear Oscillators 5
2.1 Physical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Pure Nonlinear Oscillator 19 3.1 Qualitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.1 Exact period of vibration . . . . . . . . . . . . . . . . . . 22
3.2 Exact periodical solution . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1 Linear case . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.2 Odd quadratic nonlinearity . . . . . . . . . . . . . . . . . 26
3.2.3 Cubic nonlinearity . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Adopted Lindstedt-Poincaré method . . . . . . . . . . . . . . . . 28
3.4 Modi.ed Lindstedt-Poincaré method . . . . . . . . . . . . . . . . 31
3.4.1 Comparison of the LP and MLP methods . . . . . . . . . 32
3.4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Exact amplitude, period and velocity method . . . . . . . . . . . 34
3.6 Solution in the form of Jacobi elliptic function . . . . . . . . . . 35
3.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7 Solution in the form of a trigonometric function . . . . . . . . . . 39
3.7.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8 Pure nonlinear oscillator with linear damping . . . . . . . . . . . 42
3.8.1 Parameter analysis . . . . . . . . . . . . . . . . . . . 44
3.8.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.9 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Free Vibrations 49
4.1 Homotopy-perturbation technique . . . . . . . . . . . . . . . . . 51
4.1.1 Duffing oscillator with a quadratic term . . . . . . . . . . 54 4.1.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Averaging solution procedure . . . . . . . . . . . . . . . . . . . . 57
4.2.1 Solution in the form of an Ateb function . . . . . . . . . . 57 4.2.2 Solution in the form of the Jacobi elliptic function . . . . 64
4.2.3 Solution in the form of a trigonometric function . . . . . . 70
4.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Hamiltonian Approach solution procedure . . . . . . . . . . . . . 75
4.3.1 Approximate frequency of vibration . . . . . . . . . . . . 75
4.3.2 Error estimation . . . . . . . . . . . . . . . . . . . . . . . 78 4.3.3 Comparison between approximate and exact solutions . . 79
4.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Oscillator with linear damping . . . . . . . . . . . . . . . . . . . 86
4.4.1 Van der Pol oscillator . . . . . . . . . . . . . . . . . . . . 88
4.4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 Oscillators with odd and even quadratic nonlinearity . . . . . . . 93
4.5.1 Qualitative analysis . . . . . . . . . . . . . . . . . . . . . 95
4.5.2 Exact solution for the asymmetric oscillator . . . . . . . . 97
4.5.3 Solution for the symmetric oscillator . . . . . . . . . . . . 99
4.5.4 Oscillations in an optomechanical system . . . . . . . . . 104
4.5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Oscillators with the time variable parameters 115
5.1 Oscillators with sl
About the Author:
Livija Cveticanin is Professor of Mechanics and Theory of Machines and Mechanisms. She got her PhD at the University of Novi Sad in Novi Sad, Serbia, and the degree of the Doctor of Hungarian Academy of Sciences in Budapest, Hungary. She published more than 300 papers: more than 120 in the journals which have impact factors and are cited by Scopus and Web of Science. Livija Cveticanin was the lecturer at the CISM International Centre for Mechanical Sciences. The number of citations according to Google Scholar is more than 1800. She is one of the Editors of the journal Mechanism and Machine Theory.
