Strong Nonlinear Oscillators

Strong Nonlinear Oscillators

Analytical Solutions

Cveticanin, Livija

Springer International Publishing AG

06/2017

317

Dura

Inglês

9783319588254

15 a 20 dias

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.
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-Poincare method . . . . . . . . . . . . . . . . 28 3.4 Modi.ed Lindstedt-Poincare 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 dierential 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 eect . . . . . . . . . . . . . . . 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
Este título pertence ao(s) assunto(s) indicados(s). Para ver outros títulos clique no assunto desejado.