Vibrations

Contents
List of Examples
List of Interactive Graphics
List of Symbols
Preface to the Third Edition
1 Introduction
	1.1 Introduction
	1.2 A Brief History of Vibrations
	1.3 About This Book
2 Modeling of Vibratory Systems
	2.1 Introduction
	2.2 Inertia Elements
	2.3 Stiffness Elements
		2.3.1 Introduction
		2.3.2 Linear Springs
		2.3.3 Nonlinear Springs
		2.3.4 Other Forms of Potential Energy Elements
		2.3.5 Summary of Equivalent Spring Constants
	2.4 Dissipation Elements
		2.4.1 Viscous Damping
		2.4.2 Combinations of Viscous Dampers and Linear Springs
		2.4.3 Other Forms of Dissipation
	2.5 Model Construction
		2.5.1 Introduction
		2.5.2 A Few Simple Models
		2.5.3 A Microelectromechanical System
		2.5.4 The Human Body
		2.5.5 A Ski
		2.5.6 Cutting Process
	2.6 Design for Vibration
	2.7 Summary
	Exercises
3 Single Degree-of-Freedom Systems: Governing Equations
	3.1 Introduction
	3.2 Force-Balance and Moment-Balance Methods
		3.2.1 Force-Balance Methods
		3.2.2 Moment-Balance Methods
	3.3 Natural Frequency and Damping Factor
		3.3.1 Natural Frequency
		3.3.2 Damping Factor
	3.4 Governing Equations for Different Types of Damping
	3.5 Governing Equations for Different Types of Applied Forces
		3.5.1 System with Base Excitation
		3.5.2 System with Unbalanced Rotating Mass
		3.5.3 System with Added Mass Due to a Fluid
	3.6 Lagrange’s Equations
	3.7 Summary of Natural Frequency Equations for Single Degree-of-Freedom Systems
	3.8 Summary
	Exercises
4 Single Degree-of-Freedom Systems: Free-Response Characteristics
	4.1 Introduction
	4.2 Free Responses of Undamped and Damped Systems
		4.2.1 Introduction: Damping Cases
		4.2.2 Free Response of Underdamped Systems: Kelvin–Voigt Model
		4.2.3 Free Response of Underdamped Systems: Maxwell Model
	4.3 Stability of a Single Degree-of-Freedom System
	4.4 Single Degree-of-Freedom Systems with Nonlinear Elements
		4.4.1 Nonlinear Stiffness
		4.4.2 Nonlinear Damping
	4.5 Summary
	Exercises
5 Single Degree-of-Freedom Systems Subjected to Periodic Excitations
	5.1 Introduction
	5.2 Response to Harmonic Excitation
		5.2.1 Excitation Applied from t = 0
		5.2.2 Excitation Present for All Time
		5.2.3 Response of Undamped System and Resonance
		5.2.4 Magnitude and Phase Information: Mass Excitation
		5.2.5 Magnitude and Phase Information: Rotating Unbalanced Mass
		5.2.6 Magnitude and Phase Information: Base Excitation
		5.2.7 Summary of Results of Sections 5.2.4, 5.2.5, and 5.2.6
		5.2.8 Harmonic Excitation of a System with a Maxwell Model
	5.3 Response to Excitation with Harmonic Components
	5.4 Frequency-Response Function
		5.4.1 Introduction
		5.4.2 Curve Fitting and Parameter Estimation
		5.4.3 Amplitude Response Function and Filter Characteristics
		5.4.4 Relationship of the Frequency-Response Function to the Transfer Function
		5.4.5 Alternative Forms of the Frequency-Response Function
	5.5 Acceleration Measurement: Accelerometer
	5.6 Vibration Isolation
	5.7 Energy Dissipation and Equivalent Damping
	5.8 Influence of Nonlinear Stiffness on Forced Response
	5.9 Summary
	Exercises
6 Single Degree-of-Freedom Systems: Subjected to Transient Excitations
	6.1 Introduction
	6.2 Response to Impulse Excitation
	6.3 Response to Step Input Excitation
	6.4 Response to Rectangular Pulse Excitation
	6.5 Response to Other Excitation Waveforms
		6.5.1 Significance of the Spectral Content of the Applied Force: An Example
	6.6 Impact Testing
	6.7 Summary
	Exercises
7 Multiple Degree-of-Freedom Systems: Governing Equations, Natural Frequencies, and Mode Shapes
	7.1 Introduction
	7.2 Governing Equations
		7.2.1 Force-Balance and Moment-Balance Methods
		7.2.2 General Form of Equations for a Linear Multi-Degree-of-Freedom System
		7.2.3 Lagrange’s Equations of Motion
	7.3 Free Response Characteristics
		7.3.1 Undamped Systems: Natural Frequencies and Mode Shapes
		7.3.2 Natural Frequencies and Mode Shapes: A Summary
		7.3.3 Undamped Systems: Properties of Mode Shapes
		7.3.4 Characteristics of Damped Systems
		7.3.5 Conservation of Energy
	7.4 Rotating Shafts on Flexible Supports
	7.5 Stability
	7.6 Summary
	Exercises
8 Multiple Degree-of-Freedom Systems: General Solution for Response and Forced Oscillations
	8.1 Introduction
	8.2 Normal-Mode Approach
		8.2.1 General Solution
		8.2.2 Response to Initial Conditions
	8.3 Response to Arbitrary Forcing and Initial Conditions: Direct Numerical Approach
	8.4 Response to Harmonic Forcing and the Frequency-Response Function
		8.4.1 Frequency-Response Function
	8.5 Vibration Absorbers
		8.5.1 Undamped Vibration Absorber
		8.5.2 Damped Linear Vibration Absorber
		8.5.3 Centrifugal Pendulum Vibration Absorber
		8.5.4 Bar Slider System
		8.5.5 Pendulum Absorber
		8.5.6 Particle Impact Damper
		8.5.7 Vibration Absorbers: A Summary
	8.6 Vibration Isolation: Transmissibility Ratio
	8.7 Systems with Moving Base
	8.8 Summary
	Exercises
9 Vibrations of Beams
	9.1 Introduction
	9.2 Governing Equations of Motion
		9.2.1 Preliminaries from Solid Mechanics
		9.2.2 Potential Energy, Kinetic Energy, and Work
		9.2.3 Derivation of the Equations of Motion
		9.2.4 Beam Equations for a General Case
	9.3 Free Oscillations: Natural Frequencies and Mode Shapes
		9.3.1 Introduction
		9.3.2 General Solution for Natural Frequencies and Mode Shapes for Beams with Constant Cross-Section
		9.3.3 Orthogonality of the Mode Shapes
		9.3.4 Natural Frequencies and Mode Shapes of Constant Cross-Section Beams Without In-Span Attachments: Effects of Boundary Conditions
		9.3.5 Effects of Stiffness and Inertial Elements Attached at an Interior Location
		9.3.6 Effects of an Axial Force and an Elastic Foundation on the Natural Frequency
		9.3.7 Tapered Beams
	9.4 Forced Oscillations
	9.5 Summary
	Exercises
Appendices
	A Preliminaries from Dynamics
	B Laplace Transform Pairs
	C Solutions to Ordinary Differential Equations
	D Matrices
	E Complex Numbers and Variables
	F State-Space Formulation
	G Natural Frequencies and Mode Shapes of Bars, Shafts, and Strings
	H Evaluation of Eq. (9.120)
Answers to Selected Exercises
Glossary
Index