Textbook on nonlinear and parametric vibrations discussing relevant terminology and analytical and computational tools for analysis, design, and troubleshooting
Introduction to Engineering Nonlinear and Parametric Vibrations with MATLAB and MAPLE is a comprehensive textbook that provides theoretical breadth and depth and analytical and computational tools needed to analyze, design, and troubleshoot related engineering problems.
The text begins by introducing and providing the required math and computer skills for understanding and simulating nonlinear vibration problems. This section also includes a thorough treatment of parametric vibrations. Many illustrative examples, including software examples, are included throughout the text. A companion website includes the MATLAB and MAPLE codes for examples in the textbook, and a theoretical development for a homoclinic path to chaos.
Introduction to Engineering Nonlinear and Parametric Vibrations with MATLAB and MAPLE provides information on:
Natural frequencies and limit cycles of nonlinear autonomous systems, covering the multiple time scale, Krylov-Bogellubov, harmonic balance, and Lindstedt-Poincare methods Co-existing fixed point equilibrium states of nonlinear systems, covering location, type, and stability, domains of attraction, and phase plane plotting Parametric and autoparametric vibration including Floquet, Mathieu and Hill theory Numerical methods including shooting, time domain collocation, arc length continuation, and Poincare plotting Chaotic motion of nonlinear systems, covering iterated maps, period doubling and homoclinic paths to chaos, and discrete and continuous time Lyapunov exponents Extensive MATLAB and MAPLE coding for the examples presented
Introduction to Engineering Nonlinear and Parametric Vibrations with MATLAB and MAPLE is an essential up-to-date textbook on the subject for upper level undergraduate and graduate engineering students as well as practicing vibration engineers. Knowledge of differential equations and basic programming skills are requisites for reading.
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 BH
Curve Effects 27
1.4.6 Spinning Shaft Supported by a Magnetic Bearing Including Nonlinear BH
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 Hills Equation 103
2.5.1 Hill Equation T min = 2 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 LinstedtPoincare Method (LPM) 221
5.4 KrylovBogeliubov (KB) 236
5.4.1 KB 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 Lyapunovs 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 Lyapunovs 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 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 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 Melnikovs 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
Alan B. Palazzolo, James J. Cain Professor of Mechanical Engineering, Texas A&M University, USA. Professor Palazzolo has extensive industrial, research, and teaching experience in vibrations. He has taught graduate level courses in Nonlinear and Parametric Vibrations (MEEN 649) and Rotordynamics (MEEN 639). In addition, he has also held industrial positions at Bently Nevada, Southwest Research Institute, and Allis Chalmers Corporation in these areas, and has performed approximately $21M in funded research.
Dongil Shin, Lead Research Engineer at GE Vernova Advanced Research in Niskayuna, New York. Dongil has extensive experience in nonlinear vibration analysis of turbomachinery systems and has published multiple journal papers in this field. At GE Vernova, he specializes in tackling practical nonlinear vibration challenges in turbomachinery components, including blades, dampers, and bearings, with a focus on gas and steam turbine systems.
Jeffrey Falzarano, Professor of Ocean Engineering, Texas A&M University, USA. Professor Falzarano has extensive research, teaching, and industry/government experience. He has taught undergraduate and graduate courses in vibrations and ship dynamics (seakeeping and ship maneuvering). He has held engineering and research positions in both government and industry. He has performed research funded by the Office of Naval Research, National Science Foundation, and other government and industry entities. He is also the 2022 recipient of the Society of Naval Architects and Marine Engineers Davidson Medal for excellence in ship research.
