Strong Nonlinear Oscillators
-15%
portes grátis
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.
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.
