Strong Nonlinear Oscillators: Analytical Solutions

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 slow time variable parameters . . . . . . . . . . . 116

5.2 Solution in the form of the Ateb function . . . . . . . . . . . . . 116

5.2.1 Oscillator with linear time variable parameter . . . . . . . 119

5.3 Solution in the form of a trigonometric function . . . . . . . . . . 121

5.3.1 Linear oscillator with time variable parameters . . . . . . 122

5.3.2 Non-integer order nonlinear oscillator . . . . . . . . . . . 123

5.3.3 Levi-Civita oscillator with a small damping . . . . . . . . 124

5.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.4 Solution in the form of a Jacobi elliptic function . . . . . . . . . 128

5.4.1 Van der Pol oscillator with time variable mass . . . . . . 130

5.4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.5 Parametrically excited strong nonlinear oscillator . . . . . . . . . 137

5.5.1 Solution procedure . . . . . . . . . . . . . . . . . . . . . . 139

5.5.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . 146

5.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.6 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6 Forced Vibrations 151

6.1 Oscillator with constant excitation force . . . . . . . . . . . . . . 152<

6.1.1 Solution of the odd-integer order oscillator . . . . . . . . . 154

6.1.2 The oscillator with additional small nonlinearity . . . . . 158

6.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.2 Harmonically excited pure nonlinear oscillator . . . . . . . . . . . 163

6.2.1 Pure odd-order nonlinear oscillator . . . . . . . . . . . . . 163

6.2.2 Bifurcation in the oscillator . . . . . . . . . . . . . . . . . 166

6.2.3 Harmonically forced pure cubic oscillator . . . . . . . . . 169

6.2.4 Numerical simulation and discussion . . . . . . . . . . . . 173

6.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.3 Forced vibrations of the pure nonlinear oscillator . . . . . . . . . 179

6.3.1 Design of excitation and derivation of amplitude-frequency

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.3.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.4 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7 Two-Degree-of-Freedom Oscillator 185

7.1 System with nonlinear viscoelastic connection . . . . . . . . . . . 186

7.1.1 Model with strong nonlinear viscoelastic connection . . . 187

7.1.2 Solution procedure . . . . . . . . . . . . . . . . . . . . . . 188

7.1.3 Pure nonlinear viscoelastic connection . . . . . . . . . . . 191

7.1.4 Special case . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.1.5 .Steady-state.solution . . . . . . . . . . . . . . . . . . . . 195

7.1.6 Mechanical vibration of the vocal cord . . . . . . . . . . . 198

7.1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.2 System with nonlinear elastic connection . . . . . . . . . . . . . . 203

7.2.1 Two-degree-of-freedom Van der Pol oscillator . . . . . . . 205

7.2.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 212

7.3 Complex-valued di¤erential equation . . . . . . . . . . . . . . . . 213

7.3.1 Adopted Krylov-Bogolubov method . . . . . . . . . . 214

7.3.2 Method based on the first integrals . . . . . . . . . . . . . 216

7.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 226

7.4 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

8 Chaos in Oscillators 231

8.1 Chaos in ideal oscillator . . . . . . . . . . . . . . . . . . . . . . . 232

8.1.1 Homoclinic orbits in the unperturbed system . . . . . . . 233

8.1.2 Melnikov.s criteria for chaos . . . . . . . . . . . . . . . . . 235

8.1.3 Numerical simulation . . . . . . . . . . . . . . . . . . . . 238

8.1.4 Lyapunov exponents and bifurcation diagrams . . . . . . 241

8.1.5 Control of chaos . . . . . . . . . . . . . . . . . . . . . . . 242

8.1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 244

8.2 Chaos in non-ideal oscillator . . . . . . . . . . . . . . . . . . . . . 245

8.2.1 Modeling of the system . . . . . . . . . . . . . . . . . . . 246

8.2.2 Asymptotic solving method . . . . . . . . . . . . . . . . . 247

8.2.3 Stability and Sommerfeld e¤ect . . . . . . . . . . . . . . . 248

8.2.4 Numerical simulation and chaotic behavior . . . . . . . . 253

8.2.5 Control of chaos . . . . . . . . . . . . . . . . . . . . . . . 257

8.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 258

8.3 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

9 Vibration of the Axially Purely Nonlinear Rod 263

9.1 Model of the axially vibrating rod . . . . . . . . . . . . . . . . . 263

9.2 Solving procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

9.2.1 Solving of the equation with displacement function . . . . 266

9.2.2 Solving of the equation with time function . . . . . . . . . 269

9.3 Frequency of axial vibration . . . . . . . . . . . . . . . . . . . . . 270

9.4 Solution illustration and simulation . . . . . . . . . . . . . . . . . 272

9.5 Period and frequency of vibration of a muscle . . . . . . . . . . . 274

9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

9.7 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

A Periodical Ateb functions 279

B Fourier series of the ca Ateb function 283

C Averaging of Ateb functions 287

D Jacobi elliptic functions 291

E Euler's integrals of the first and second kind 293

F Inverse incomplete Beta function 295