Preface xiii About the Companion Website xxi 1 Introduction 1 1.1 Some Traits of Nonlinear Dynamical Systems 1 1.2 Mathematical Preliminaries 7 1.2.1 Nonlinearity 7 1.2.2 Taylor Series Approximation – Linearization 12 1.2.3 Secular Terms 16 1.2.4 First-Order (State) Form of Differential Equations 17 1.2.5 Hamiltonian Functions 17 1.3 Computer Aided Math Software: Matlab and Maple 20 1.4 Some Machinery Nonlinear Components 21 1.4.1 Flexible Coupling Connecting Rotating Shafts 21 1.4.2 Electric Motor with an Eccentric Shaft and Motor Air Gap 22 1.4.3 Hydrodynamic Journal Bearing 24 1.4.4 Turbocharger Shaft Supported by Floating Ring Bearings 27 1.4.5 Spinning Shaft Supported by a Magnetic Bearing Including Nonlinear B–H Curve Effects 27 1.4.6 Spinning Shaft Supported by a Magnetic Bearing Including Nonlinear B–H Curve Effects 27 Exercises 29 References 39 2 Parametric Vibration 41 2.1 Introduction to Floquet Theory 41 2.2 Usage of Floquet Theory for Evaluating the Stability of Nonlinear System Harmonic Response 49 2.3 Derivation of the Floquet Theorem 51 2.3.1 Nutshell Summary 51 2.3.2 Proof of the Floquet Theorem (FT) 52 2.4 Mathieu Equation 68 2.4.1 Mathieu Stability Boundary Curve Plots 77 2.4.2 Damped Mathieu Equation (DME) 91 2.4.3 Perturbation Solution for Mathieu 2 T min Stability Boundary with Damping 93 2.4.4 Damped Mathieu Equation – Monodromy Matrix Eigenvalues 95 2.4.5 Higher-Order Boundary Curves for the Damped Mathieu Stability Diagram 98 2.4.6 Damped Mathieu Equation Stability Boundary Curve Plotting 100 2.5 Hill’s Equation 103 2.5.1 Hill Equation T min = 2π Periodic Solutions 111 2.6 A Class of Multi-DOF Oscillator Systems with Periodic Stiffness Coefficients 113 2.7 Rotating Asymmetric Shaft Vibrations 119 2.7.1 Pinned (Rigid) Bearing Case 119 2.7.2 Flexible Asymmetric Bearing Case 122 2.8 Autoparametric Vibration – Internal Resonance 123 Exercises 133 References 152 3 Nonlinear Vibration: Concepts 153 3.1 Introduction 153 3.2 Illustrative Nonlinear Mathematical Models 153 3.3 Some Qualitative Aspects of Nonlinear Vibrations 170 Exercises 176 References 182 4 Nonlinear Vibrations: Analytical Solutions for Natural Frequencies 183 4.1 Introduction 183 4.2 Simple Systems with Natural Frequency Formulas 184 Exercises 200 5 Nonlinear Vibrations: Approximate Methods for Autonomous Systems 205 5.1 Introduction 205 5.2 Multiple Time Scales Method (MTSM) 205 5.2.1 Multiple Time Scale Method Using the Complex Variable Approach 215 5.3 Linstedt–Poincare Method (LPM) 221 5.4 Krylov–Bogeliubov (K–B) 236 5.4.1 K–B Method Summary 241 5.5 Harmonic Balance Method (HBM) 250 Exercises 263 References 284 6 Nonlinear Vibrations: Fixed Equilibrium Points and Stability 285 6.1 Introduction 285 6.2 Determination of Equilibrium Points 287 6.3 Equilibrium Point Stability – Lyapunov’s Method 288 6.3.1 EP3: Existence and Stability 293 6.3.2 EP2: Existence and Stability 293 6.4 Types of Fixed Equilibrium Points 296 6.5 Phase (State) Plane Plotting Rules 302 6.6 Equilibrium Point Local Stability vs. Parameter Variation 311 6.7 Heteroclinic and Homoclinic Trajectories, Separatrices and Domains of Attraction 320 6.8 Plotting Heteroclinic Trajectories Utilizing Numerical Integration (NI) 325 6.9 Homoclinic Trajectories – Paths (H o P) 329 6.10 Numerically Integrated Domain of Attraction for Coexisting Limit Cycles (LC) with Different EPS 331 6.10.1 Domain of Attraction Boundaries 332 6.11 Lyapunov’s Second Method (L2M) 334 Exercises 340 References 357 7 Nonlinear Vibrations: Approximate Methods for Non-Autonomous Systems 359 7.1 Introduction 359 7.2 Undamped Duffing Hardening System 360 7.2.1 Harmonic Balance Method Solution 361 7.2.1.1 Slope of f (a) for Various ω R Ranges 366 7.2.1.2 Summary (a)–(g) 367 7.2.2 Additional Findings 370 7.2.2.1 VanderPol Approach to OES Stability Determination 378 7.2.2.2 VanDerPol Phase Plane Trajectories (VPPT) 384 7.3 Undamped Duffing (Cubic) Softening System 390 7.3.1 Softening Duffing Analysis 392 7.3.2 Summary for Duffing Softening Stiffness Case 395 7.3.3 VanderPol Approach to OES Stability Determination for the Softening Duffing 398 7.4 Damped, Duffing Hardening System 399 7.4.1 Phase Lag Angle of Damped System, Steady State Harmonic Responses 411 7.5 Damped Duffing Softening System 414 7.6 Stability of Co-Existing Harmonic Response Using Floquet Theory 416 7.6.1 Stability of Damped Duffing Hardening System OES Obtained via the HBM 416 7.7 Duffing System 1/3 Sub-Harmonic Response 419 7.8 Other Sub-Harmonics of a Damped Duffing System 431 7.9 Superharmonic Response of a Damped Duffing System 433 7.10 Quadratic Nonlinearity and 1 / 2 Sub-Harmonic Response 438 7.10.1 OES of the Quadratic-Cubic Nonlinear Damped System 440 7.11 Multiple Loads with Different Forcing Frequencies 443 7.11.1 Steady State Response with Two Forces 443 7.12 A Comparison of the Multiple Time Scale and Harmonic Balance Methods 449 7.12.1 Near Resonance Condition ω R ≈ 1 451 7.12.1.1 MTSM Steady State Amplitude Equation 452 7.12.1.2 HBM Steady State Amplitude Equation 452 7.13 Natural Frequencies, Mode Shapes and Forced Harmonic Response of a 2 Degree of Freedom, Nonlinear System by the Harmonic Balance Method (HBM) 453 Exercises 456 References 464 8 Numerical Methods for Nonlinear System Steady-State Periodic Response 465 8.1 Introduction 465 8.2 The Time Domain Trigonometric Collocation Method (TCM) 465 8.3 The Shooting Method (SM) 473 8.3.1 Theory 474 8.3.2 Programming the Shooting Method 478 8.3.2.1 Steps 478 8.3.3 Practical Programming Tips for Shooting Algorithm Implementation 479 Conclusions 494 8.4 Poincare/Hayashi Plane Dynamics 494 8.4.1 Poincare Plots 494 8.4.2 Poincare Plot for Response Display and Iterated Map Functions 494 8.4.3 Orbital Equilibrium Types and Domains of Attraction in the Poincare Plane 497 8.4.4 Saddle Eigenvalues in Hayashi Plane 497 8.5 Shooting Method Jacobian Eigenvalues (SMJE) and Bifurcation Type 502 Exercises 507 References 510 9 Advanced Shooting and Arc-Length Continuation Method 511 9.1 Introduction 511 9.2 Shooting Method for Autonomous Systems 511 9.3 Arc-Length Continuation Method 515 9.4 Multiple Shooting Method 518 Exercises 523 References 524 10 Introduction to Chaos 1 525 10.1 Introduction 525 10.2 Iterated Map Function IMF Behavior of Poincare Points for Simple Chaotic Systems 525 10.3 Iterated Map Functions 537 10.3.1 General Properties of Iterated Map Functions 538 10.4 Logistic Iterated Map Function (LM) 542 10.5 Lyapunov Exponents for Iterated Map Functions (IMF) 559 Exercises 565 References 570 11 Homoclinic and Heteroclinic Tangle Path to Chaos 571 11.1 Introduction 571 11.2 Poincare Maps, Tangles, and Chaos 572 11.2.1 Poincare Maps and Invariant Manifolds 572 11.2.2 Poincare Maps 572 11.2.3 Determining the Poincare Map 573 11.2.4 Invariant Manifolds 575 11.2.5 Discussion on Determining the Invariant Manifolds 576 11.2.6 Invariant Manifolds as Boundaries of Behavior 578 11.3 Melnikov’s Method Applied to the Dynamical System 579 11.3.1 The Homoclinic Case 579 11.3.2 Subharmonic Melnikov Function 582 11.3.3 Consideration of Heteroclinic Orbits 582 11.3.4 Resulting Tangle Dynamics Following Intersections 583 11.3.5 Heteroclinic vs. Homoclinic Intersections 583 11.3.6 Simple Examples 584 11.3.7 Further Discussion of More Advanced Topics in This Area 585 Exercises 586 References 587 12 Lyapunov Exponents for Continuous Time Systems 589 12.1 Introduction 589 12.2 Procedure 589 Exercises 596 References 598 Appendix A Some Useful Trigonometric Identities 599 A.1 Trigonometry Identities 599 A.2 Amplitude/Phase – Component Formulae 599 A.3 Law of Sines and Cosines 602 Appendix B The Derivation and Mathematical Details of the Melnikov Function 603 References 605 Index 607

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