Mechanical vibrations

Front Cover
Equivalent Masses, Springs and Dampers
Title Page
Copyright Page
Contents
Preface
Acknowledgments
List of Symbols
Chapter 1 Fundamentals of Vibration
	1.1 Preliminary Remarks
	1.2 Brief History of the Study of Vibration
		1.2.1 Origins of the Study of Vibration
		1.2.2 From Galileo to Rayleigh
		1.2.3 Recent Contributions
	1.3 Importance of the Study of Vibration
		1.3.1 Conversion of Vibrations to Sound by the Human Ear
	1.4 Basic Concepts of Vibration
		1.4.1 Vibration
		1.4.2 Elementary Parts of Vibrating Systems
		1.4.3 Number of Degrees of Freedom
		1.4.4 Discrete and Continuous Systems
	1.5 Classification of Vibration
		1.5.1 Free and Forced Vibration
		1.5.2 Undamped and Damped Vibration
		1.5.3 Linear and Nonlinear Vibration
		1.5.4 Deterministic and Random Vibration
	1.6 Vibration Analysis Procedure
	1.7 Spring Elements
		1.7.1 Nonlinear Springs
		1.7.2 Linearization of a Nonlinear Spring
		1.7.3 Spring Constants of Elastic Elements
		1.7.4 Combination of Springs
		1.7.5 Spring Constant Associated with the Restoring Force due to Gravity
	1.8 Mass or Inertia Elements
		1.8.1 Combination of Masses
	1.9 Damping Elements
		1.9.1 Construction of Viscous Dampers
		1.9.2 Linearization of a Nonlinear Damper
		1.9.3 Combination of Dampers
	1.10 Harmonic Motion
		1.10.1 Vectorial Representation of Harmonic Motion
		1.10.2 Complex-Number Representation of Harmonic Motion
		1.10.3 Complex Algebra
		1.10.4 Operations on Harmonic Functions
		1.10.5 Definitions and Terminology
	1.11 Harmonic Analysis
		1.11.1 Fourier Series Expansion
		1.11.2 Complex Fourier Series
		1.11.3 Frequency Spectrum
		1.11.4 Time- and Frequency-Domain Representations
		1.11.5 Even and Odd Functions
		1.11.6 Half-Range Expansions
		1.11.7 Numerical Computation of Coefficients
	1.12 Examples Using MATLAB
	1.13 Vibration Literature
	Chapter Summary
	References
	Review Questions
	Problems
	Design Projects
Chapter 2 Free Vibration of Single-Degree-of-Freedom Systems
	2.1 
Introduction
	2.2 Free Vibration of an Undamped Translational System
		2.2.1 Equation of Motion Using Newton’s Second Law of Motion
		2.2.2 Equation of Motion Using Other Methods
		2.2.3 Equation of Motion of a Spring-Mass System in Vertical Position
		2.2.4 Solution
		2.2.5 Harmonic Motion
	2.3 Free Vibration of an Undamped Torsional System
		2.3.1 Equation of Motion
		2.3.2 Solution
	2.4 Response of First-Order Systems and Time Constant
	2.5 Rayleigh’s Energy Method
	2.6 Free Vibration with Viscous Damping
		2.6.1 Equation of Motion
		2.6.2 Solution
		2.6.3 Logarithmic Decrement
		2.6.4 Energy Dissipated in Viscous Damping
		2.6.5 Torsional Systems with Viscous Damping
	2.7 Graphical Representation of Characteristic Roots and Corresponding Solution
		2.7.1 Roots of the Characteristic Equation
		2.7.2 Graphical Representation of Roots and Corresponding Solutions
	2.8 Parameter Variations and Root Locus Representations
		2.8.1  Interpretations of ωn, ωd, ζ, and τ in the s-plane
		2.8.2 Root Locus and Parameter Variations
	2.9 Free Vibration with Coulomb Damping
		2.9.1 Equation of Motion
		2.9.2 Solution
		2.9.3 Torsional Systems with Coulomb Damping
	2.10 Free Vibration with Hysteretic Damping
	2.11 Stability of Systems
	2.12 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Projects
Chapter 3 Harmonically Excited Vibration
	3.1 Introduction
	3.2 Equation of Motion
	3.3 Response of an Undamped System Under Harmonic Force
		3.3.1 Total Response
		3.3.2 Beating Phenomenon
	3.4 Response of a Damped System Under Harmonic Force
		3.4.1 Total Response
		3.4.2 Quality Factor and Bandwidth
	3.5 Response of a Damped System Under F(t) = F0eiwt
	3.6 Response of a Damped System Under the Harmonic Motion of the Base
		3.6.1 Force Transmitted
		3.6.2 Relative Motion
	3.7 Response of a Damped System Under Rotating Unbalance
	3.8 Forced Vibration with Coulomb Damping
	3.9 Forced Vibration with Hysteresis Damping
	3.10 Forced Motion with Other Types of Damping
	3.11 Self-Excitation and Stability Analysis
		3.11.1 Dynamic Stability Analysis
		3.11.2 Dynamic Instability Caused by Fluid Flow
	3.12 Transfer-Function Approach
	3.13 Solutions Using Laplace Transforms
	3.14 Frequency Transfer Functions
		3.14.1 Relation between the General Transfer Function T(s) and the Frequency Transfer Function T (iw)
		3.14.2 Representation of Frequency-Response Characteristics
	3.15 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Projects
Chapter 4 Vibration Under General Forcing Conditions
	4.1 Introduction
	4.2 Response Under a General Periodic Force
		4.2.1 First-Order Systems
		4.2.2 Second-Order Systems
	4.3 Response Under a Periodic Force of Irregular Form
	4.4 Response Under a Nonperiodic Force
	4.5 Convolution Integral
		4.5.1 Response to an Impulse
		4.5.2 Response to a General Forcing Condition
		4.5.3 Response to Base Excitation
	4.6 Response Spectrum
		4.6.1 Response Spectrum for Base Excitation
		4.6.2 Earthquake Response Spectra
		4.6.3 Design Under a Shock Environment
	4.7 Laplace Transforms
		4.7.1 Transient and Steady-State Responses
		4.7.2 Response of First-Order Systems
		4.7.3 Response of Second-Order Systems
		4.7.4 Response to Step Force
		4.7.5 Analysis of the Step Response
		4.7.6 Description of Transient Response
	4.8 Numerical Methods
		4.8.1 Runge-Kutta Methods
	4.9 Response to Irregular Forcing Conditions Using Numerical Methods
	4.10 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Projects
Chapter 5 Two-Degree-of-Freedom Systems
	5.1 Introduction
	5.2 Equations of Motion for Forced Vibration
	5.3 Free-Vibration Analysis of an Undamped System
	5.4 Torsional System
	5.5 Coordinate Coupling and Principal Coordinates
	5.6 Forced-Vibration Analysis
	5.7 Semidefinite Systems
	5.8 Self-Excitation and Stability Analysis
	5.9 Transfer-Function Approach
	5.10 Solutions Using Laplace Transform
	5.11 Solutions Using Frequency Transfer Functions
	5.12 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Projects
Chapter 6 Multidegree-of-Freedom Systems
	6.1 Introduction
	6.2  Modeling of Continuous Systems as Multidegree-of-Freedom Systems
	6.3 Using Newton’s Second Law to Derive Equations of Motion
	6.4 Influence Coefficients
		6.4.1 Stiffness Influence Coefficients
		6.4.2 Flexibility Influence Coefficients
		6.4.3 Inertia Influence Coefficients
	6.5 Potential and Kinetic Energy Expressions in Matrix Form
	6.6 Generalized Coordinates and Generalized Forces
	6.7 Using Lagrange’s Equations to Derive Equations of Motion
	6.8 Equations of Motion of Undamped Systems in Matrix Form
	6.9 Eigenvalue Problem
	6.10 Solution of the Eigenvalue Problem
		6.10.1 Solution of the Characteristic (Polynomial) Equation
		6.10.2 Orthogonality of Normal Modes
		6.10.3 Repeated Eigenvalues
	6.11 Expansion Theorem
	6.12 Unrestrained Systems
	6.13 Free Vibration of Undamped Systems
	6.14 Forced Vibration of Undamped Systems Using Modal Analysis
	6.15 Forced Vibration of Viscously Damped Systems
	6.16 Self-Excitation and Stability Analysis
	6.17 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Projects
Chapter 7 Determination of Natural Frequencies and Mode Shapes
	7.1 Introduction
	7.2 Dunkerley’s Formula
	7.3 Rayleigh’s Method
		7.3.1 Properties of Rayleigh’s Quotient
		7.3.2 Computation of the Fundamental Natural Frequency
		7.3.3 Fundamental Frequency of Beams and Shafts
	7.4 Holzer’s Method
		7.4.1 Torsional Systems
		7.4.2 Spring-Mass Systems
	7.5 Matrix Iteration Method
		7.5.1 Convergence to the Highest Natural Frequency
		7.5.2 Computation of Intermediate Natural Frequencies
	7.6 Jacobi’s Method
	7.7 Standard Eigenvalue Problem
		7.7.1 Choleski Decomposition
		7.7.2 Other Solution Methods
	7.8 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Projects
Chapter 8 Continuous Systems
	8.1 Introduction
	8.2 Transverse Vibration of a String or Cable
		8.2.1 Equation of Motion
		8.2.2 Initial and Boundary Conditions
		8.2.3 Free Vibration of a Uniform String
		8.2.4 Free Vibration of a String with Both Ends Fixed
		8.2.5 Traveling-Wave Solution
	8.3 Longitudinal Vibration of a Bar or Rod
		8.3.1 Equation of Motion and Solution
		8.3.2 Orthogonality of Normal Functions
	8.4 Torsional Vibration of a Shaft or Rod
	8.5 Lateral Vibration of Beams
		8.5.1 Equation of Motion
		8.5.2 Initial Conditions
		8.5.3 Free Vibration
		8.5.4 Boundary Conditions
		8.5.5 Orthogonality of Normal Functions
		8.5.6 Forced Vibration
		8.5.7 Effect of Axial Force
		8.5.8 Effects of Rotary Inertia and Shear Deformation
		8.5.9 Beams on Elastic Foundation
		8.5.10 Other Effects
	8.6 Vibration of Membranes
		8.6.1 Equation of Motion
		8.6.2 Initial and Boundary Conditions
	8.7 Rayleigh’s Method
	8.8 The Rayleigh-Ritz Method
	8.9 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Project
Chapter 9 Vibration Control
	9.1 Introduction
	9.2 Vibration Nomograph and Vibration Criteria
	9.3 Reduction of Vibration at the Source
	9.4 Balancing of Rotating Machines
		9.4.1 Single-Plane Balancing
		9.4.2 Two-Plane Balancing
	9.5 Whirling of Rotating Shafts
		9.5.1 Equations of Motion
		9.5.2 Critical Speeds
		9.5.3 Response of the System
		9.5.4 Stability Analysis
	9.6 Balancing of Reciprocating Engines
		9.6.1 Unbalanced Forces Due to Fluctuations in Gas Pressure
		9.6.2 Unbalanced Forces Due to Inertia of the Moving Parts
		9.6.3 Balancing of Reciprocating Engines
	9.7 Control of Vibration
	9.8 Control of Natural Frequencies
	9.9 Introduction of Damping
	9.10 Vibration Isolation
		9.10.1 Vibration Isolation System with Rigid Foundation
		9.10.2 Vibration Isolation System with Base Motion
		9.10.3 Vibration Isolation System with Flexible Foundation
		9.10.4 Vibration Isolation System with Partially Flexible Foundation
		9.10.5 Shock Isolation
		9.10.6 Active Vibration Control
	9.11 Vibration Absorbers
		9.11.1 Undamped Dynamic Vibration Absorber
		9.11.2 Damped Dynamic Vibration Absorber
	9.12 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Project
Chapter 10 Vibration Measurement and Applications
	10.1 Introduction
	10.2 Transducers
		10.2.1 Variable-Resistance Transducers
		10.2.2 Piezoelectric Transducers
		10.2.3 Electrodynamic Transducers
		10.2.4 Linear Variable Differential Transformer Transducer
	10.3 Vibration Pickups
		10.3.1 Vibrometer
		10.3.2 Accelerometer
		10.3.3 Velometer
		10.3.4 Phase Distortion
	10.4 Frequency-Measuring Instruments
	10.5 Vibration Exciters
		10.5.1 Mechanical Exciters
		10.5.2 Electrodynamic Shaker
	10.6 Signal Analysis
		10.6.1 Spectrum Analyzers
		10.6.2 Bandpass Filter
		10.6.3 Constant-Percent Bandwidth and Constant-Bandwidth Analyzers
	10.7 Dynamic Testing of Machines and Structures
		10.7.1 Using Operational Deflection-Shape Measurements
		10.7.2 Using Modal Testing
	10.8 Experimental Modal Analysis
		10.8.1 The Basic Idea
		10.8.2 The Necessary Equipment
		10.8.3 Digital Signal Processing
		10.8.4 Analysis of Random Signals
		10.8.5 Determination of Modal Data from Observed Peaks
		10.8.6 Determination of Modal Data from Nyquist Plot
		10.8.7 Measurement of Mode Shapes
	10.9 Machine-Condition Monitoring and Diagnosis
		10.9.1 Vibration Severity Criteria
		10.9.2 Machine Maintenance Techniques
		10.9.3 Machine-Condition Monitoring Techniques
		10.9.4 Vibration Monitoring Techniques
		10.9.5 Instrumentation Systems
		10.9.6 Choice of Monitoring Parameter
	10.10 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Projects
Chapter 11 Numerical Integration Methods in Vibration Analysis
	11.1 Introduction
	11.2 Finite Difference Method
	11.3 Central Difference Method for Single-Degree-of-Freedom Systems
	11.4 Runge-Kutta Method for Single-Degree-of-Freedom Systems
	11.5 Central Difference Method for Multidegree-of-Freedom Systems
	11.6 Finite Difference Method for Continuous Systems
		11.6.1 Longitudinal Vibration of Bars
		11.6.2 Transverse Vibration of Beams
	11.7 Runge-Kutta Method for Multidegree-of-Freedom Systems
	11.8 Houbolt Method
	11.9 Wilson Method
	11.10 Newmark Method
	11.11 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
Chapter 12 Finite Element Method
	12.1 Introduction
	12.2 Equations of Motion of an Element
	12.3 Mass Matrix, Stiffness Matrix, and Force Vector
		12.3.1 Bar Element
		12.3.2 Torsion Element
		12.3.3 Beam Element
	12.4 Transformation of Element Matrices and Vectors
	12.5 Equations of Motion of the Complete System of Finite Elements
	12.6 Incorporation of Boundary Conditions
	12.7 Consistent- and Lumped-Mass Matrices
		12.7.1 Lumped-Mass Matrix for a Bar Element
		12.7.2 Lumped-Mass Matrix for a Beam Element
		12.7.3 Lumped-Mass Versus Consistent-Mass Matrices
	12.8 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Projects
Chapter 13 Nonlinear Vibration
	13.1 Introduction
	13.2 Examples of Nonlinear Vibration Problems
		13.2.1 Simple Pendulum
		13.2.2 Mechanical Chatter, Belt Friction System
		13.2.3 Variable Mass System
	13.3 Exact Methods
	13.4 Approximate Analytical Methods
		13.4.1 Basic Philosophy
		13.4.2 Lindstedt’s Perturbation Method
		13.4.3 Iterative Method
		13.4.4 Ritz-Galerkin Method
	13.5 Subharmonic and Superharmonic Oscillations
		13.5.1 Subharmonic Oscillations
		13.5.2 Superharmonic Oscillations
	13.6 Systems with Time-Dependent Coefficients (Mathieu Equation)
	13.7 Graphical Methods
		13.7.1 Phase-Plane Representation
		13.7.2 Phase Velocity
		13.7.3 Method of Constructing Trajectories
		13.7.4 Obtaining Time Solution from Phase-Plane Trajectories
	13.8 Stability of Equilibrium States
		13.8.1 Stability Analysis
		13.8.2 Classification of Singular Points
	13.9 Limit Cycles
	13.10 Chaos
		13.10.1 Functions with Stable Orbits
		13.10.2 Functions with Unstable Orbits
		13.10.3 Chaotic Behavior of Duffing’s Equation Without the Forcing Term
		13.10.6 Chaotic Behavior of Duffing’s Equation with the Forcing Term
	13.11 Numerical Methods
	13.12 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Projects
Chapter 14 Random Vibration
	14.1 Introduction
	14.2 Random Variables and Random Processes
	14.3 Probability Distribution
	14.4 Mean Value and Standard Deviation
	14.5 Joint Probability Distribution of Several Random Variables
	14.6 Correlation Functions of a Random Process
	14.7 Stationary Random Process
	14.8 Gaussian Random Process
	14.9 Fourier Analysis
		14.9.1 Fourier Series
		14.9.2 Fourier Integral
	14.10 Power Spectral Density
	14.11 Wide-Band and Narrow-Band Processes
	14.12 Response of a Sngle-Degree-of-Freedom System
		14.12.1 Impulse-Response Approach
		14.12.2 Frequency-Response Approach
		14.12.3 Characteristics of the Response Function
	14.13 Response Due to Stationary Random Excitations
		14.13.1 Impulse-Response Approach
		14.13.2 Frequency-Response Approach
	14.14 Response of a Multidegree-of-Freedom System
	14.15 Examples Using MATLAB
	Chapter Summary
	References
	Review Questions
	Problems
	Design Project
Appendix A Mathematical Relations and Material Properties
Appendix B Deflection of Beams and Plates
Appendix C Matrices
Appendix D Laplace Transform
Appendix E Units
Appendix F Introduction to MATLAB
Answers to Selected Problems
Index
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