Mechanical Vibration: Theory and Application

Cover
Title
Copyright
Preface
Acknowledgments
Dedication
Contents
Chapter 1: Introduction and Background
1.1 Challenges and Examples
1.2 Systems and Structures
1.3 Basic Concepts of Vibration
1.3.1 Modeling for Vibration
1.3.2 Idealization and Formulation
1.3.3 Inertia, Stiffness, and Damping
1.3.4 Properties of Keyboard Keys
1.3.5 Computational Aspects
1.3.6 Is Vibration Good or Bad?
1.3.7 Vibration Control
1.4 Types of Vibration
1.4.1 Motion Classification
1.4.2 Deterministic Approximations
1.4.3 Probability
1.4.4 System Model Uncertainty
1.4.5 Random Vibration
1.5 Types of System Models
1.5.1 Linear Approximation
1.5.2 Dimensionality
1.5.3 Discrete Models
1.5.4 Continuous Models
1.5.5 Nonlinear Models
1.6 Basic Dynamics
1.6.1 Statics and Equilibrium
1.6.2 The Equations of Motion
1.6.3 Linear Momentum and Impulse
1.6.4 Principles of Work and Energy
1.7 Units
1.7.1 Mars Orbiter Loss
1.7.2 U.S. Customary and SI Systems
1.7.3 The Second
1.7.4 Dimensional Analysis
1.8 Concepts Summary
1.9 Some Sound & Vibration Facts
1.10 Quotes
Chapter 2: Single Degree-of-Freedom Undamped Vibration
2.1 Motivating Examples
2.1.1 Transport of a Satellite
2.1.2 Rocket Propulsion
2.2 Deterministic Modeling
2.2.1 Problem Idealization
2.2.2 Mass, Damping, and Stiffness
2.2.3 Deterministic Approximation
2.2.4 Equations of Motion
2.2.5 Energy Formulation
2.2.6 Representing Harmonic Motion
2.2.7 Solving the Equations of Motion
2.3 Undamped Free Vibration
2.3.1 Alternate Formulation
2.3.2 Phase Plane
2.4 Harmonically Forced Vibration
2.4.1 A Note on Terminology
2.4.2 Resonance
2.4.3 Vibration of a Structure in Water
2.5 Concepts Summary
2.6 Quotes
2.7 Problems
Chapter 3: Single Degree-of-Freedom Damped Vibration
3.1 Overview
3.2 Introduction to Damping
3.3 Damping Models
3.3.1 Viscous Damping
3.3.2 Coulomb Damping
3.4 Free Vibration with Viscous Damping
3.4.1 Critically Damped and Overdamped Systems
3.4.2 Underdamped Systems
3.4.3 Logarithmic Decrement
3.4.4 Time Constants
3.4.5 Phase Plane
3.5 Free Vibration with Coulomb Damping
3.6 Forced Vibration with Viscous Damping
3.7 Forced Harmonic Vibration
3.7.1 Response to Harmonic Excitation
3.7.2 Harmonic Excitation in Complex Notation
3.7.3 Harmonic Base Excitation
3.7.4 Rotating Unbalance
3.8 Forced Periodic Vibration
3.8.1 Harmonic/Spectral Analysis
3.8.2 Fourier Series
3.9 Damping Loss Factor
3.10 Concepts Summary
3.11 Quotes
3.12 Problems
Chapter 4: Single DOF Vibration: General Loading and Advanced Topics
4.1 Arbitrary Loading: Laplace Transform
4.2 Step Loading
4.3 Impulsive Loading
4.4 Arbitrary Loading: Convolution Integral
4.5 Introduction to Lagrange’s Equation
4.6 Notions of Randomness
4.7 Notions of Control
4.8 The Inverse Problem
4.9 A Self-Excited System and Its Stability
4.10 Solution Analysis and Design Techniques
4.11 Model of a Bouncing Ball
4.11.1 Time of Contact
4.11.2 Stiffness and Damping
4.11.3 Natural Frequency & Damping Ratio
4.11.4 Approximations
4.12 Concepts Summary
4.13 Quotes
4.14 Problems
Chapter 5: Variational Principles and Analytical Dynamics
5.1 Introduction
5.2 Constraints
5.3 Virtual Work
5.3.1 Work and Energy
5.3.2 Principle of Virtual Work
5.3.3 D’Alembert’s Principle
5.4 Lagrange’s Equation
5.4.1 Lagrange’s Equation for Small Oscillations
5.5 Hamilton’s Principle
5.6 Lagrange’s Equation with Damping
5.7 Jourdain’s Principle
5.7.1 Jourdain’s Principle from d’Alembert’s Principle
5.8 Concepts Summary
5.9 Quotes
5.10 Problems
Chapter 6: Multi Degree-of-Freedom Vibration
6.1 Motivating Examples
6.1.1 Periodic Structures
6.1.2 Inverse Problems
6.1.3 Vehicle Vibration Testing
6.1.4 Scope
6.2 The Concepts of Stiffness and Flexibility
6.2.1 Influence Coefficients
6.3 Equations of Motion
6.3.1 Mass and Stiffness Matrices
6.4 Undamped Vibration
6.4.1 Two Degree-of-Freedom Vibration: Direct Method
6.4.2 Harmonically Forced Vibration: Direct Method
6.4.3 Undamped Vibration Absorber
6.4.4 Beating Oscillations
6.5 Free Vibration with Damping: Direct Method
6.6 Modal Analysis
6.6.1 Modal Orthogonality
6.6.2 Modal Analysis with Forcing
6.6.3 Modal Analysis with Proportional Damping
6.7 Nonproportional Damping
6.7.1 Phase Synchronization
6.8 Real and Complex Modes
6.8.1 Modal Analysis vs. Direct Method
6.9 Special Cases
6.9.1 Unrestrained Systems
6.9.2 Rigid-Body Mode
6.9.3 Repeated Frequencies
6.10 Eigenvalue Geometry
6.11 Periodic Structures
6.11.1 Perfect Lattice Models
6.11.2 Effects of Imperfection
6.12 Inverse Vibration Problem
6.13 Fluid Sloshing in Container
6.14 Stability of Motion
6.15 Rayleigh’s Quotient
6.16 Concepts Summary
6.17 Quotes
6.18 Problems
Chapter 7: Continuous Models for Vibration
7.1 Discrete to Continuous
7.2 Vibration of Strings
7.2.1 Wave Propagation Solution
7.2.2 Wave Equation via Hamilton’s Principle
7.2.3 Boundary Value Problem
7.2.4 Modal Solution for Fixed-Fixed Boundary Conditions
7.3 Axial Vibration of Beams
7.3.1 Axial Vibration: Newton’s Approach
7.3.2 Axial Vibration: Hamilton’s Approach
7.3.3 Simplified Eigenvalue Problem
7.3.4 Eigenfunction Expansion Method
7.4 Torsional Vibration of Shafts
7.4.1 Torsion of Shaft with Rigid Disk at Free End
7.5 Transverse Vibration of Beams
7.5.1 Timoshenko Beam
7.5.2 Boundary Conditions
7.5.3 Bernoulli-Euler Beam
7.5.4 Orthogonality of the Modes
7.5.5 Nodes and Antinodes
7.6 Other Transverse Beam Vibration Cases
7.6.1 Beam with Axial Forces
7.6.2 Beam with Elastic Restraints
7.6.3 Beam on an Elastic Foundation
7.6.4 Beam with a Moving Support
7.6.5 Different Boundary Conditions
7.6.6 Beam with Traveling Force
7.7 Concepts Summary
7.8 Quotes
7.9 Problems
Chapter 8: Continuous Models for Vibration: Advanced Models
8.1 Vibration of Membranes
8.1.1 Rectangular Membranes
8.1.2 Circular Membranes
8.2 Vibration of Plates
8.2.1 Rectangular Plates
8.2.2 Eigenvalue Problem
8.3 Approximate Methods
8.3.1 Rayleigh’s Quotient
8.3.2 Rayleigh-Ritz Method
Untitled
8.4 Variables Not Separating
8.4.1 Nonharmonic, Time-Dependent Boundary Conditions
8.4.2 Pipe Flow with Constant Tension
8.5 Concepts Summary
8.6 Quotes
8.7 Problems
Chapter 9: Random Vibration: Probabilistic Forces
9.1 Introduction
9.2 Motivation
9.2.1 Random Vibration
9.2.2 Fatigue Life
9.2.3 Ocean Wave Forces
9.2.4 Wind Forces
9.2.5 Material Properties
9.2.6 Statistics and Probability
9.3 Random Variables
9.3.1 Probability Distribution
9.3.2 Probability Density Function
9.4 Mathematical Expectation
9.4.1 Variance
9.5 Useful Probability Densities
9.5.1 Uniform Density
9.5.2 Exponential Density
9.5.3 Normal (Gaussian) Density
9.5.4 Lognormal Density
9.5.5 Rayleigh Density
9.6 Two Random Variables
9.6.1 Covariance and Correlation
9.7 Random Processes
9.7.1 Random Process Descriptors
9.7.2 Ensemble Averaging
9.7.3 Stationarity
9.7.4 Power Spectrum
9.7.5 Units
9.7.6 Narrow-Band and Broad-Band Processes
9.7.7 White-Noise Process
9.8 Random Vibration
9.8.1 Formulation and Preliminaries
9.8.2 Mean-Value Response
9.8.3 Response Correlations
9.8.4 Response Spectral Density
9.9 Stochastic Response of a Linear MDOF System
9.10 Lunar Seismic Structural Analysis
9.11 Random Vibration of Continuous Structures
9.12 Monte Carlo Simulation
9.12.1 Random Number Generation
9.12.2 Generating Random Variates
9.12.3 Generating Time History for Random Process
9.13 Inverse Vibration with Uncertain Data
9.14 Concepts Summary
9.15 Quotes
9.16 Problems
Chapter 10: Vibration Control
10.1 Motivation
10.2 Approaches to Controlling Vibration
10.2.1 Why Active Control
10.3 Feedback Control
10.3.1 Disadvantages of Feedback
10.4 Performance of Feedback Control Systems
10.4.1 Poles and Zeros of a Second-Order System
10.4.2 System Gain
10.4.3 Stability of Response
10.5 Control of Response
10.5.1 Control Actions
10.5.2 Control of Transient Response
10.6 Parameter Sensitivity
10.7 State Variable Models
10.7.1 Transfer Function from State Equation
10.7.2 Controllability and Observability
10.7.3 State Variable Feedback
10.8 Multivariable Control
10.8.1 State and Output Equations
10.8.2 Controllability and Observability
10.8.3 Closed-loop Feedback of MIMO Systems
10.9 Stochastic Control
10.10 Concepts Summary
10.11 Quotes
10.12 Problems
Chapter 11: Nonlinear Vibration
11.1 Introduction
11.2 Physical Examples
11.2.1 Simple Pendulum: Approximate Solution
11.2.2 Simple Pendulum: Exact Solution
11.2.3 Duffing and van der Pol Equations
11.3 The Phase Plane
11.3.1 Stability of Equilibria
11.4 Perturbation or Expansion Methods
11.4.1 Lindstedt-Poincaré Method
11.4.2 Forced Oscillations of Quasi-Harmonic Systems
11.4.3 Jump Phenomenon
11.4.4 Periodic Solutions of Nonautonomous Systems
11.4.5 Subharmonic and Superharmonic Oscillations
11.5 Mathieu Equation
11.6 Van der Pol Equation
11.6.1 Unforced van der Pol Equation
11.6.2 Limit Cycles
11.6.3 Forced van der Pol Equation
11.7 Motion in the Large
11.8 Nonlinear Control
11.9 Random Duffing Oscillator
11.10 Nonlinear Pendulum: Galerkin Method
11.11 Concept Summary
11.12 Quotes
11.13 Problems
Appendix A: Mathematical Concepts for Vibration
A.1 Complex Numbers
A.1.1 Complex Number Operations
A.1.2 Absolute Value
A.1.3 Equivalent Representation
A.2 Matrices
A.2.1 Matrix Operations
A.2.2 Determinant and Matrix Inverse
A.2.3 Eigenvalues and Eigenvectors
A.2.4 Matrix Derivatives and Integrals
A.3 Taylor Series & Linearization
A.4 Ordinary Differential Equations
A.4.1 Solution of Linear Equations
A.4.2 Homogeneous Solution
A.4.3 Particular Solution
A.5 Laplace Transforms
A.5.1 Borel’s Theorem
A.5.2 Partial Fraction Expansion
A.5.3 Laplace Transform Table
A.5.4 Initial-Value, Final-Value Theorem
A.6 Fourier Series & Transforms
A.6.1 Fourier Series
A.6.2 Fourier Transforms
A.7 Partial Differential Equations
Appendix B: Viscoelastic Damping
B.1 Viscoelastic Materials
B.1.1 Work Done Per Cycle
B.2 Viscoelastic Material Models
B.2.1 Maxwell Model
B.2.2 Voigt Model
B.2.3 Maxwell Standard Linear Model
B.2.4 Stress-Strain Equivalent Model
B.2.5 Boltzmann Superposition Model
B.2.6 General Nonviscous Damping
B.3 Causality Issues in Damping Models
B.4 Concepts Summary
B.5 Quotes
Appendix C: Solving Vibration Problems with MATLAB
C.1 Introduction
C.2 SDOF Undamped System
C.3 SDOF Damped System
C.4 SDOF Overdamped System
C.5 SDOF Undamped System with Harmonic Excitation
C.6 SDOF Damped System with Harmonic Excitation
C.7 SDOF Damped System with Base Excitation
C.8 SDOF Damped System with Rotating Unbalance
C.9 SDOF Damped System with Impulse Input
C.10 SDOF Damped System with Step Input
C.11 SDOF Damped System with Square Pulse Input
C.12 SDOF Damped System with Ramp Input
C.13 SDOF System with Arbitrary Periodic Input
C.14 MDOF Undamped System
C.15 MDOF Damped System
C.16 General Vibration Solver
C.17 Van der Pol Oscillator
C.18 Random Vibration
C.19 Duffing Oscillator with Random Excitation
C.20 Monte Carlo Simulation of a Random System
Index