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Strong Nonlinear Oscillators

Strong Nonlinear Oscillators

Analytical Solutions

by Livija Cveticanin
Hardback
Publication Date: 10/07/2017

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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

ISBN:
9783319588254
9783319588254
Category:
Applied mathematics
Format:
Hardback
Publication Date:
10-07-2017
Language:
English
Publisher:
Springer
Country of origin:
United States
Edition:
2nd Edition
Dimensions (mm):
235x155mm
Weight:
0.8kg

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