Strong Nonlinear Oscillators: Analytical Solutions

ISBN-13: 9783319588254 / Angielski / Twarda / 2017 / 317 str.

Livija Cveticanin

Opis

This book outlines an analytical solution procedure of the pure nonlinear oscillator system, offering a solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameter. Includes exercises.

Spis treści

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 

O autorze

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.

Blurb

This textbook presents the motion of pure nonlinear oscillatory systems and various solution procedures which give the approximate solutions of the strong nonlinear oscillator equations. It presents the author’s original method for the analytical solution procedure of the pure nonlinear oscillator system. After an introduction, the physical explanation of the pure nonlinearity and of the pure nonlinear oscillator is given. The analytical solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameters is considered. In this second edition of the book, the number of approximate solving procedures for strong nonlinear oscillators is enlarged and a variety of procedures for solving free strong nonlinear oscillators is suggested. A method for error estimation is also given which is suitable to compare the exact and approximate solutions.

Besides the oscillators with one degree-of-freedom, the one and two mass oscillatory systems with two-degrees-of-freedom and continuous oscillators are considered. The chaos and chaos suppression in ideal and non-ideal mechanical systems is explained.

In this second edition more attention is given to the application of the suggested methodologies and obtained results to some practical problems in physics, mechanics, electronics and biomechanics. Thus, for the oscillator with two degrees-of-freedom, a generalization of the solving procedure is performed. Based on the obtained results, vibrations of the vocal cord are analyzed. In the book the vibration of the axially purely nonlinear rod as a continuous system is investigated. The developed solving procedure and the solutions are applied to discuss the muscle vibration. Vibrations of an optomechanical system are analyzed using the oscillations of an oscillator with odd or even quadratic nonlinearities. The extension of the forced vibrations of the system is realized by introducing the Ateb periodic excitation force which is the series of a trigonometric function.

The book is self-consistent and suitable for researchers and as a textbook for students and also professionals and engineers who apply these techniques to the field of nonlinear oscillations.