Preface......Page 7
Contents......Page 10
Abstract......Page 15
1.2 Brief Review of Nonlinear Oscillations History......Page 16
1.3 Overview of the Book......Page 18
1.4 Nonlinear Dynamical Systems......Page 19
1.4.2 Predator-Prey Dynamics......Page 20
1.4.4 Mathematical Model of Crime in a Society......Page 21
1.4.5 Mathematical Model for Global Warming......Page 22
1.5 Conservative Oscillatory Systems......Page 23
1.5.1 Duffing Equation......Page 24
1.5.2 Oscillator with Fractional Power......Page 25
1.5.4 Oscillator with Discontinuity......Page 26
1.7 Parametrically Excited Vibration......Page 28
1.7.1 Fractional Mathieu Equation......Page 30
1.8 Resonance in Nonlinear Systems......Page 34
References......Page 38
Abstract......Page 42
2.1.1 Singular Points......Page 43
2.1.2 Linearization Around Singular Points......Page 44
2.1.2.1 Example on Damped System......Page 45
2.1.3.2 F(x) with Concave Form......Page 46
2.1.3.3 F(x) with Convex Form......Page 48
2.1.3.4 Turning Point......Page 49
2.2 Perturbation Methods......Page 50
2.2.1 Straightforward Expansion Method (SEM)......Page 51
2.2.2 Lindstedt–Poincaré Perturbation Method (LPPM)......Page 55
2.2.3 Multiple Time-Scales Method (MTSM)......Page 59
2.2.3.1 Vibration of Cantilever Beam Carrying an Intermediate Lumped Mass [9]......Page 62
2.2.3.2 Nonlinear Viscoelastic Plates Subjected to Subsonic Flow and External Loads......Page 64
2.2.3.3 Nonlinear Vibration of Variable Speed Rotating Viscoelastic Beams......Page 66
2.2.3.4 Resonance Analyses of Clamped–Clamped Microbeams......Page 68
2.2.4 Bogoliubov–Krylov Averaging Method (BKAM)......Page 70
2.2.4.1 Linear Differential Equation......Page 71
2.2.4.2 Coulomb Friction......Page 72
2.2.4.3 Van der Pol Equation......Page 73
2.2.4.4 Rayleigh Equation......Page 74
2.3 Parametric Excitation and Hill’s Equation......Page 75
2.3.1 Floquet Theorem......Page 76
2.3.2 Mathieu Equation......Page 77
2.4 Practice Problems......Page 79
References......Page 83
Abstract......Page 85
3.1.2 Solution Procedure......Page 86
3.2.2 Modified Petrov–Galerkin Approach......Page 88
3.4 Hamiltonian Approach......Page 89
3.5.1 Second-Order Hamiltonian Approach......Page 91
3.5.2 Third-Order Hamiltonian Approach......Page 92
3.6.1 Fourier Expansion......Page 93
3.7 Generalized Duffing Equation......Page 94
3.7.1 Nonlinear Oscillations of Single-Walled Carbon Nanotubes......Page 99
3.7.2 Nonlinear Oscillations of Rectangular Plates......Page 101
3.8 Nonlinear Dynamic Buckling of an Elastic Column......Page 103
3.9 Vibrations of Cracked Rectangular Plate......Page 105
3.10 Relativistic Oscillator......Page 108
3.11 Plasma Physics Equation......Page 109
3.12 Nonlinear Oscillator with Discontinuity......Page 111
3.13 Nonlinear Oscillator with Fractional-Power Restoring Force......Page 112
3.14 Generalized Conservative Oscillatory Systems (Type 1)......Page 117
3.15 Generalized Conservative Oscillatory Systems (Type 2)......Page 118
3.16 Duffing Harmonic Oscillator......Page 120
3.17 Helmholtz Duffing Oscillator......Page 121
3.18 Autonomous Conservative Oscillatory System......Page 123
3.19 Nonlinear Oscillation of Rigid Bar on Semi-circular Surface......Page 124
3.20 Nonlinear Oscillations of Centrifugal Governor Systems......Page 126
3.21 Nonlinear Lateral Sloshing in Partially-Filled Elliptical Tankers......Page 127
3.22 Nonlinear Oscillations of Elevator Cable in a Drum Drive Elevator......Page 128
References......Page 132
Abstract......Page 135
4.1.1 The Nine Chapters on the Mathematical Art......Page 136
4.1.2 The Ying Buzu Shu, a Method to Approximate Real Roots......Page 137
4.2 Frequency–Amplitude Formulation......Page 138
4.2.1 The Method of Weighted Residuals......Page 139
4.3 Max-Min Approach......Page 140
4.3.1 He Chengtian Inequality......Page 141
4.3.2 Application to Nonlinear Oscillators......Page 142
4.4 Generalized Duffing Equation......Page 144
4.5 Generalized Conservative Oscillatory Systems (Type 1)......Page 145
4.5.3 Plasma Physics Equation......Page 146
4.6 Generalized Conservative Oscillator Systems (Type 2)......Page 147
4.7 Nonlinear Oscillator with Fractional Power......Page 148
4.8 Helmholtz Duffing Oscillator......Page 149
4.9 Relativistic Oscillator......Page 150
4.10 Autonomous Conservative Oscillatory System......Page 151
4.11 Nonlinear Oscillation of a Mass Attached to a Stretched Elastic Wire......Page 153
4.12 Nonlinear Schrödinger Equation......Page 155
4.13 Rigid Frame Rotates at a Fixed Rate \varOmega......Page 157
4.14 Conservative Lienard Type Equation......Page 158
References......Page 160
Abstract......Page 163
5.1 Variational Principle......Page 164
5.2 Semi-inverse Method......Page 166
5.3 Variational Approach......Page 169
5.4 Hamiltonian Approach—Comparison with Variational Approach......Page 171
5.5 Relationship Between Hamiltonian and Variational Approaches......Page 174
5.6 Generalized Duffing Equation......Page 177
5.7 Elastic Force with Rational Characteristic Equation......Page 178
5.8 Elastic Force with Non-integer Fractional Characteristic Equation......Page 179
5.9 Higher Order Hamiltonian Approach to Duffing Equations......Page 181
5.10 Hamiltonian Approach to Rational and Irrational Oscillator......Page 185
5.11 Hamiltonian Approach to Nonlinear Oscillator with Discontinuity......Page 189
5.12 Nonlinear Oscillator with Quintic Nonlinearity......Page 192
5.13 Nonlinear Schrodinger’s Equation......Page 193
5.14 Thomas–Fermi Equation......Page 195
5.15 Heat Conduction Equation......Page 196
5.16 Lane–Emden-Type Equation......Page 198
5.17 Dynamic Analysis of Centrifugal Governor System......Page 199
5.18 Duffing Harmonic Equation......Page 201
References......Page 206
Abstract......Page 208
6.1.1 Background......Page 209
6.1.2 Description of Adomian Decomposition Method......Page 210
6.1.2.2 Laplace Adomian Decomposition Method......Page 212
6.2.1 Background......Page 213
6.2.2 Description of Method......Page 214
6.2.2.3 Algorithm III......Page 215
6.2.2.4 Application to the Fractional Differential Equation......Page 216
Optimal Variational Iteration Methods......Page 217
Fractional Variational Iteration Method with Adomian Series......Page 218
Laplace Variational Iteration Method......Page 219
Variational Homotopy Perturbation Method......Page 220
6.2.4 Description of Method......Page 221
6.3 Optimal Homotopy Asymptotic Method (OHAM)......Page 223
6.4 Volterra-Integro Differential Equations—(History, Development and Applications)......Page 225
6.4.1 Example 1......Page 226
6.4.2 Example 2......Page 228
6.5 Nonlinear Schrödinger Equations—(History, Development and Applications)......Page 230
6.5.1 Example 1......Page 231
6.5.2 Example 2......Page 232
6.5.3 Example 3......Page 234
6.6 Van der Pol Equation (History, Development and Applications)......Page 235
6.6.1 Example 1......Page 236
6.6.2 Example 2......Page 238
6.7 Korteweg-de Vries Equation (History, Development and Applications)......Page 239
6.7.1 Example 1......Page 242
6.7.2 Example 2......Page 243
6.7.3 Example 3......Page 244
References......Page 252
Abstract......Page 259
7.1.1 Mass Sensors......Page 260
7.1.2 Vibration of Carbon Nanotubes......Page 265
7.1.3 Vibration of Microtubules......Page 267
7.2.1 Electrically Actuated Microbeams......Page 269
7.2.2 Micro-Gyroscope......Page 271
7.3.1 Doubled-Walled Carbon Nanotubes (DWCNTs)......Page 273
7.3.2 Higher Mode Vibration of Single-Walled Carbon Nanotubes......Page 276
7.3.3 Nonlinear Oscillations of Nanowire Resonators......Page 280
7.3.4 Applications of Timoshenko Beam Theory in Nanoscale Systems......Page 282
7.4.1 Quadratic–Cubic Nonlinearity in Curved Nano–Micro Structures......Page 284
7.4.2 Rydberg–Varshni Potentials and Casimir Force......Page 287
7.4.3 Non-natural Oscillations......Page 288
References......Page 292