Analytical Methods in Nonlinear Oscillations: Approaches and Applications (Solid Mechanics and Its Applications, 252)

This book covers both classical and modern analytical methods in nonlinear systems. A wide range of applications from fundamental research to engineering problems are addressed. The book contains seven chapters, each with miscellaneous problems and their detailed solutions. More than 100 practice problems are illustrated, which might be useful for students and researchers in the areas of nonlinear oscillations and applied mathematics. With providing real world examples, this book shows the multidisciplinary emergence of nonlinear dynamical systems in a wide range of applications including mechanical and electrical oscillators, micro/nano resonators and sensors, and also modelling of global warming, epidemic diseases, sociology, chemical reactions, biology and ecology. Preface 7 Contents 10 1 Introduction 15 Abstract 15 1.1 Why Nonlinear Oscillations? 16 1.2 Brief Review of Nonlinear Oscillations History 16 1.3 Overview of the Book 18 1.4 Nonlinear Dynamical Systems 19 1.4.1 Oscillation of a Pendulum 20 1.4.2 Predator-Prey Dynamics 20 1.4.3 Brusselator System in Chemical Reaction 21 1.4.4 Mathematical Model of Crime in a Society 21 1.4.5 Mathematical Model for Global Warming 22 1.4.6 Mathematical Model of Epidemic Disease 23 1.5 Conservative Oscillatory Systems 23 1.5.1 Duffing Equation 24 1.5.2 Oscillator with Fractional Power 25 1.5.3 Relativistic Oscillator 26 1.5.4 Oscillator with Discontinuity 26 1.6 Non-conservative Oscillatory Systems 28 1.7 Parametrically Excited Vibration 28 1.7.1 Fractional Mathieu Equation 30 1.8 Resonance in Nonlinear Systems 34 References 38 2 Classical Methods 42 Abstract 42 2.1 Nonlinear Differential Equations 43 2.1.1 Singular Points 43 2.1.2 Linearization Around Singular Points 44 2.1.2.1 Example on Damped System 45 2.1.3 Classification of Singular Points 46 2.1.3.1 F(x) with Monotonic Form 46 2.1.3.2 F(x) with Concave Form 46 2.1.3.3 F(x) with Convex Form 48 2.1.3.4 Turning Point 49 2.2 Perturbation Methods 50 2.2.1 Straightforward Expansion Method (SEM) 51 2.2.2 Lindstedt–Poincaré Perturbation Method (LPPM) 55 2.2.3 Multiple Time-Scales Method (MTSM) 59 2.2.3.1 Vibration of Cantilever Beam Carrying an Intermediate Lumped Mass [9] 62 2.2.3.2 Nonlinear Viscoelastic Plates Subjected to Subsonic Flow and External Loads 64 2.2.3.3 Nonlinear Vibration of Variable Speed Rotating Viscoelastic Beams 66 2.2.3.4 Resonance Analyses of Clamped–Clamped Microbeams 68 2.2.4 Bogoliubov–Krylov Averaging Method (BKAM) 70 2.2.4.1 Linear Differential Equation 71 2.2.4.2 Coulomb Friction 72 2.2.4.3 Van der Pol Equation 73 2.2.4.4 Rayleigh Equation 74 2.2.4.5 Generalized van der Pol Type Oscillator 75 2.3 Parametric Excitation and Hill’s Equation 75 2.3.1 Floquet Theorem 76 2.3.2 Mathieu Equation 77 2.4 Practice Problems 79 References 83 3 Energy Balance Methods 85 Abstract 85 3.1 Fundamentals of the Energy Balance Method 86 3.1.1 Collocation Method 86 3.1.2 Solution Procedure 86 3.2 Modified Energy Balance Method: Galerkin Approach 88 3.2.1 Galerkin Approach 88 3.2.2 Modified Petrov–Galerkin Approach 88 3.3 Modified Energy Balance Method: Least Square Method 89 3.4 Hamiltonian Approach 89 3.5 Modified Hamiltonian Approach 91 3.5.1 Second-Order Hamiltonian Approach 91 3.5.2 Third-Order Hamiltonian Approach 92 3.6 Rational Energy Balance Method 93 3.6.1 Fourier Expansion 93 3.7 Generalized Duffing Equation 94 3.7.1 Nonlinear Oscillations of Single-Walled Carbon Nanotubes 99 3.7.2 Nonlinear Oscillations of Rectangular Plates 101 3.8 Nonlinear Dynamic Buckling of an Elastic Column 103 3.9 Vibrations of Cracked Rectangular Plate 105 3.10 Relativistic Oscillator 108 3.11 Plasma Physics Equation 109 3.12 Nonlinear Oscillator with Discontinuity 111 3.13 Nonlinear Oscillator with Fractional-Power Restoring Force 112 3.14 Generalized Conservative Oscillatory Systems (Type 1) 117 3.15 Generalized Conservative Oscillatory Systems (Type 2) 118 3.16 Duffing Harmonic Oscillator 120 3.17 Helmholtz Duffing Oscillator 121 3.18 Autonomous Conservative Oscillatory System 123 3.19 Nonlinear Oscillation of Rigid Bar on Semi-circular Surface 124 3.20 Nonlinear Oscillations of Centrifugal Governor Systems 126 3.21 Nonlinear Lateral Sloshing in Partially-Filled Elliptical Tankers 127 3.22 Nonlinear Oscillations of Elevator Cable in a Drum Drive Elevator 128 References 132 4 Residual Methods 135 Abstract 135 4.1 Basic Ideas 136 4.1.1 The Nine Chapters on the Mathematical Art 136 4.1.2 The Ying Buzu Shu, a Method to Approximate Real Roots 137 4.2 Frequency–Amplitude Formulation 138 4.2.1 The Method of Weighted Residuals 139 4.2.2 Geng and Cai Modification 140 4.2.3 Ren and Gui Modification 140 4.3 Max-Min Approach 140 4.3.1 He Chengtian Inequality 141 4.3.2 Application to Nonlinear Oscillators 142 4.4 Generalized Duffing Equation 144 4.5 Generalized Conservative Oscillatory Systems (Type 1) 145 4.5.1 Linear Harmonic Oscillator 146 4.5.2 Duffing Harmonic Oscillator 146 4.5.3 Plasma Physics Equation 146 4.6 Generalized Conservative Oscillator Systems (Type 2) 147 4.7 Nonlinear Oscillator with Fractional Power 148 4.8 Helmholtz Duffing Oscillator 149 4.9 Relativistic Oscillator 150 4.10 Autonomous Conservative Oscillatory System 151 4.11 Nonlinear Oscillation of a Mass Attached to a Stretched Elastic Wire 153 4.12 Nonlinear Schrödinger Equation 155 4.13 Rigid Frame Rotates at a Fixed Rate \varOmega 157 4.14 Conservative Lienard Type Equation 158 References 160 5 Semi-inverse and Variational Methods 163 Abstract 163 5.1 Variational Principle 164 5.2 Semi-inverse Method 166 5.3 Variational Approach 169 5.4 Hamiltonian Approach—Comparison with Variational Approach 171 5.5 Relationship Between Hamiltonian and Variational Approaches 174 5.6 Generalized Duffing Equation 177 5.7 Elastic Force with Rational Characteristic Equation 178 5.8 Elastic Force with Non-integer Fractional Characteristic Equation 179 5.9 Higher Order Hamiltonian Approach to Duffing Equations 181 5.10 Hamiltonian Approach to Rational and Irrational Oscillator 185 5.11 Hamiltonian Approach to Nonlinear Oscillator with Discontinuity 189 5.12 Nonlinear Oscillator with Quintic Nonlinearity 192 5.13 Nonlinear Schrodinger’s Equation 193 5.14 Thomas–Fermi Equation 195 5.15 Heat Conduction Equation 196 5.16 Lane–Emden-Type Equation 198 5.17 Dynamic Analysis of Centrifugal Governor System 199 5.18 Duffing Harmonic Equation 201 References 206 6 Integral Based Methods 208 Abstract 208 6.1 Adomian Decomposition Method 209 6.1.1 Background 209 6.1.2 Description of Adomian Decomposition Method 210 6.1.2.1 Wazwaz’s Modification 212 6.1.2.2 Laplace Adomian Decomposition Method 212 6.2 Variational Iteration Method 213 6.2.1 Background 213 6.2.2 Description of Method 214 6.2.2.1 Algorithm I 215 6.2.2.2 Algorithm II 215 6.2.2.3 Algorithm III 215 6.2.2.4 Application to the Fractional Differential Equation 216 6.2.2.5 Application to the Fractional Differential Difference Equation 217 6.2.2.6 Modified Variational Iteration Method 217 Optimal Variational Iteration Methods 217 Fractional Variational Iteration Method with Adomian Series 218 Wu–Lee’s Fractional Variational Iteration Algorithm 219 Laplace Variational Iteration Method 219 Variational Homotopy Perturbation Method 220 6.2.3 Homotopy Analysis Method 221 6.2.4 Description of Method 221 6.3 Optimal Homotopy Asymptotic Method (OHAM) 223 6.4 Volterra-Integro Differential Equations—(History, Development and Applications) 225 6.4.1 Example 1 226 6.4.2 Example 2 228 6.4.3 Example 3 230 6.5 Nonlinear Schrödinger Equations—(History, Development and Applications) 230 6.5.1 Example 1 231 6.5.2 Example 2 232 6.5.3 Example 3 234 6.6 Van der Pol Equation (History, Development and Applications) 235 6.6.1 Example 1 236 6.6.2 Example 2 238 6.7 Korteweg-de Vries Equation (History, Development and Applications) 239 6.7.1 Example 1 242 6.7.2 Example 2 243 6.7.3 Example 3 244 References 252 7 Nonlinearities in Nano- and Microsystems 259 Abstract 259 7.1 Duffing Equation in NEMS and MEMS 260 7.1.1 Mass Sensors 260 7.1.2 Vibration of Carbon Nanotubes 265 7.1.3 Vibration of Microtubules 267 7.2 Parametric and Self-Excited Oscillations 269 7.2.1 Electrically Actuated Microbeams 269 7.2.2 Micro-Gyroscope 271 7.3 Nonlinear Coupled Oscillators 273 7.3.1 Doubled-Walled Carbon Nanotubes (DWCNTs) 273 7.3.2 Higher Mode Vibration of Single-Walled Carbon Nanotubes 276 7.3.3 Nonlinear Oscillations of Nanowire Resonators 280 7.3.4 Applications of Timoshenko Beam Theory in Nanoscale Systems 282 7.4 Other Types of Nonlinearities 284 7.4.1 Quadratic–Cubic Nonlinearity in Curved Nano–Micro Structures 284 7.4.2 Rydberg–Varshni Potentials and Casimir Force 287 7.4.3 Non-natural Oscillations 288 References 292 Front Matter ....Pages i-xv Introduction (Ebrahim Esmailzadeh, Davood Younesian, Hassan Askari)....Pages 1-27 Classical Methods (Ebrahim Esmailzadeh, Davood Younesian, Hassan Askari)....Pages 29-71 Energy Balance Methods (Ebrahim Esmailzadeh, Davood Younesian, Hassan Askari)....Pages 73-122 Residual Methods (Ebrahim Esmailzadeh, Davood Younesian, Hassan Askari)....Pages 123-150 Semi-inverse and Variational Methods (Ebrahim Esmailzadeh, Davood Younesian, Hassan Askari)....Pages 151-195 Integral Based Methods (Ebrahim Esmailzadeh, Davood Younesian, Hassan Askari)....Pages 197-247 Nonlinearities in Nano- and Microsystems (Ebrahim Esmailzadeh, Davood Younesian, Hassan Askari)....Pages 249-286