Vibration Theory and Applications with Finite Elements and Active Vibration Control

Based on many years of research and teaching, this book brings together all the important topics in linear vibration theory, including failure models, kinematics and modeling, unstable vibrating systems, rotordynamics, model reduction methods, and finite element methods utilizing truss, beam, membrane and solid elements. It also explores in detail active vibration control, instability and modal analysis. The book provides the modeling skills and knowledge required for modern engineering practice, plus the tools needed to identify, formulate and solve engineering problems effectively.

Preface Acknowledgements About the companion website Chapter 1 Background, Motivation and Overview 1.1 Introduction 1.2 Background 1.3 Our Vibrating World 1.4 Harmful Effects of Vibration 1.5 Stiffness, Inertia and Damping Forces 1.6 Paradigms for Obtaining the Differential Equations of Motion 1.7 Finite Element Method 1.8 Active Vibration Control 1.9 Chapter 1 Exercises 1.10 References Chapter 2 Preparatory Skills: Mathematics, Modeling and Kinematics 2.1 Introduction 2.2 Getting Started with MATLAB and MAPLE 2.3 Vibration and Differential Equations 2.4 Taylor Series Expansions and Linearization 2.5 Complex Variables (CV) and Phasors 2.6 Degrees of Freedom (dof), Matrices, Vectors and Subspaces 2.7 Coordinate Transformations 2.8 Eigenvalues and Eigenvectors 2.9 Fourier Series 2.10 Laplace Transforms, Transfer Functions and Characteristic Equation 2.11 Kinematics and Kinematic Constraints 2.12 Dirac Delta and Heaviside Functions 2.13 Chapter 2 Exercises 2.14 References Chapter 3 Equations of Motion by Newton's Laws 3.1 Introduction 3.2 Particle Motion Approximation 3.3 Planar (2D) Rigid Body Motion Approximation 3.4 Impulse and Momentum 3.5 Variable Mass Systems 3.6 Chapter 3 Exercises Chapter 4 Equations of Motion by Energy Methods 4.1 Introduction 4.2 Kinetic Energy 4.3 External and Internal Work and Potential Energy 4.4 Power and Work-Energy Laws 4.5 Lagrange s Equations for Particles and Rigid Bodies 4.6 Lagrange's Equations for Flexible, Distributed Mass Bodies - Assumed Mode Approach 4.7 Lagrange's Equations for Flexible, Distributed Mass Bodies - Finite Element Approach - General Formulation 4.8 Lagrange's Equations for Flexible, Distributed Mass Bodies - Finite Element Approach -Bar/Truss ModesElement Approach - Bar/Truss Modes 4.9 Chapter 4 Exercises 4.10 References Chapter 5 Free Vibration Response 5.1 Introduction 5.2 Single Degree of Freedom (SDOF) Systems 5.3 Two Degree of Freedom (2DOF) Systems 5.4 N Degree of Freedom (NDOF) Systems 5.5 Infinite Degree of Freedom dof, Continuous Member Systems 5.6 Unstable Free Vibrations 5.7 Summary 5.8 Chapter 5 Exercises 5.9 References Chapter 6 Transient Vibrations due to General Loading 6.1 Introduction 6.2 Single Degree of Freedom (SDOF) Transient Response 6.3 Modal Condensation of NDOF, Transient Forced Vibrating Systems 6.4 Numerical Integration of NDOF Transient Vibration Response 6.5 Summary 6.6 Chapter 6 Exercises 6.7 References Chapter 7 Steady State Vibration Response to Periodic Loading 7.1 Introduction 7.2 Complex Phasor Approach 7.3 Single Degree of Freedom Models 7.4 Two Degree of Freedom (2DOF) Response 7.5 NDOF Steady State Harmonic Response 7.6 Other Phasor Ratio Measure of Steady State Harmonic Response (SSHR) 7.7 Summary 7.8 Chapter 7 Exercises 7.9 References Chapter 8 Approximate Methods for Large Order Systems 8.1 Introduction 8.2 Guyan Reduction-Static Condensation 8.3 Substructures - Super Elements 8.4 Modal Synthesis 8.5 Eigenvalue/Natural Frequency Changes for Perturbed Systems 8.6 Summary 8.7 Chapter 8 Exercises 8.8 References Chapter 9 Beam Finite Elements for Vibration Analysis 9.1 Introduction 9.2 Modeling 2D Frame Structures with Euler Bernoulli Beam Elements 9.3 Three Dimensional 3D Timoshenko Beam Elements Introduction 9.4 3D Timoshenko Beam Elements - Nodal Coordinates 9.5 3D Timoshenko Beam Elements - Shape Functions , Element Stiffness and Mass Matrices 9.6 3D Timoshenko Beam Element Force Vectors 9.7 3D Frame - Beam Element Assembly Algorithm 9.8 Two Dimensional 2D Frame Modeling with Timoshenko Beam Elements 9.9 Summary 9.10 Chapter 9 Exercises 9.11 References Chapter 10 2D Planar Finite Elements for Vibration Analysis 10.1 Introduction 10.2 Plane Strain ( ) 10.3 Plane Stress ( ) 10.4 Plane Stress and Plane Strain: Element Stiffness and Mass Matrices and Force Vector 10.5 Assembly Procedure for 2D, 4 Node, Quadrilateral Elements 10.6 Computation of Stresses in 2D Solid Elements 10.7 Extra Shape Functions to Improve Accuracy 10.8 Illustrative Example 10.9 2D Axisymmetric Model 10.10 Automated Mesh Generation Constant Strain Triangle Elements 10.11 Membranes 10.12 Banded Storage 10.13 Chapter 10 Exercises 10.14 References Chapter 11 3D Solid Elements for Vibration Analysis 11.1 Introduction 11.2 Element Stiffness Matrix 11.3 The Element Mass Matrix and Force Vector 11.4 Assembly Procedure for the 3D, 8 Node, Hexahedral Element Model 11.5 Computation of Stresses for a 3D Hexahedral Solid Element 11.6 3D Solid Element Model Example 11.7 3D Solid Element Summary 11.8 Chapter 11 Exercises 11.9 References Chapter 12 Active Vibration Controls 12.1 Introduction 12.2 AVC System Modeling 12.3 AVC Actuator Modeling 12.4 System Model with an Infinite Bandwidth Feedback Approximation 12.5 System Model with Finite Bandwidth Feedback 12.6 System Model with Finite Bandwidth Feedback and Lead Compensation 12.7 Sensor / Actuator Non-Collocation Effect on Vibration Stability 12.8 Piezoelectric Actuators 12.9 Summary 12.10 Chapter 12 Exercises 12.11 References Appendix A Fundamental Equations of Elasticity Appendix B 3D Timoshenko Beam MAPLE Code Appendix C 3D Piping Model MATLAB Code Appendix D 2D Auto Paint Frame MATLAB Code Appendix E 2D Solid Element Beam Model MATLAB Code Appendix F 3D Solid Element Beam Model MATLAB Code Index