Introduction to Engineering Nonlinear and Parametric Vibrations with MATLAB and Maple

1. Auflage 2025
ISBN: 978-1-119-31954-2
Verlag: Wiley

Buch, Englisch, 640 Seiten, Format (B × H): 185 mm x 257 mm, Gewicht: 1247 g

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.

Autoren/Hrsg.

Palazzolo, Alan B

Shin, Dongil

Falzarano, Jeffrey

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen
• Technische Wissenschaften Maschinenbau | Werkstoffkunde Maschinenbau Konstruktionslehre, Bauelemente, CAD
• Technische Wissenschaften Maschinenbau | Werkstoffkunde Technische Mechanik | Werkstoffkunde Statik, Dynamik, Kinetik, Kinematik

Inhaltsverzeichnis

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 = 2p 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

Über Autor(innen)

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.