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Mechanical Vibrations, CourseSmart eTextbook, 5th Edition

By Singiresu S. Rao

Published by Prentice Hall

Published Date: Aug 2, 2011

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Description

For undergraduate courses in Vibration Engineering.

Retaining the style of its previous editions, this text presents the theory, computational aspects, and applications of vibrations in as simple a manner as possible. With an emphasis on computer techniques of analysis, it gives expanded explanations of the fundamentals, focusing on physical significance and interpretation that build upon students' previous experience. Each self-contained topic fully explains all concepts and presents the derivations with complete details. Numerous examples and problems illustrate principles and concepts.

Table of Contents

1.1 Preliminary Remarks

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1.2 Brief History of Vibration

1.2.1 Origins of Vibration

1.2.2 From Galileo to Rayleigh

1.2.3 Recent Contributions

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1.3 Importance of the Study of Vibration

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1.4 Basic Concepts of Vibration

1.4.2 Elementary Parts of Vibrating Systems

1.4.1 Vibration

1.4.3 Degree of Freedom

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1.5 Classification of Vibration

1.4.4 Discrete and Continuous Systems

1.5.4 Deterministic and Random Vibration

1.5.3 Linear and Nonlinear Vibration

1.5.2 Undamped and Damped Vibration

1.5.1 Free and Forced Vibration

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1.6 Vibration Analysis Procedure

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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

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1.8 Mass or Inertia Elements

1.8.1 Combination of Masses

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1.9 Damping Elements 45

1.9.1 Construction of Viscous Dampers 46

1.9.3 Combination of Dampers 52

1.9.2 Linearization of a Nonlinear Damper 52

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1.10 Harmonic Motion 54

1.10.1 Vectorial Representation of Harmonic Motion 56

1.10.2 Complex Number Representation of Harmonic Motion 57

1.10.3 Complex Algebra 58

1.10.4 Operations on Harmonic Functions 59

1.10.5 Definitions and Terminology 62

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1.11 Harmonic Analysis

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

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1.12 Examples Using Matlab7

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1.13 Vibration Literature

References

Chapter Summary

Review Questions

Problems

Design Projects

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CHAPTER 2 Free Vibration of Single Degree of Freedom Systems

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2.1 Introduction 126

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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

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2.3 Free Vibration of an Undamped Torsional System 1¿

2.3.1 Equation of Motion

2.3.2 Solution

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2.4 Response of First Order Systems and Time Constant

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2.5 Rayleigh’s Energy Method

2.6.2 Solution

2.6.1 Equation of Motion

2.6.3 Logarithmic Decrement

2.6.4 Energy Dissipated in Viscous Damping

2.6.5 Torsional Systems with Viscous Damping

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2.7 Graphical Representation of Characteristic Roots and Corresponding Solutions

2.7.1 Roots of the Characteristic Equation

2.7.2 Graphical Representation of Roots and Corresponding Solutions

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2.8 Parameter Variations and Root Locus Representations

2.8.1 Interpretations of and in s-plane

2.8.2 Root Locus and Parameter Variations

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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

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2.10 Free Vibration with Hysteretic Damping

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2.11 Stability of Systems

Chapter Summary

Review Questions

References

Problems

Design Projects

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CHAPTER 3 Harmonically Excited Vibration

3.1 Introduction

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3.2 Equation of Motion

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3.3 Response of an Undamped System Under Harmonic Force

3.3.2 Beating Phenomenon

3.3.1 Total Response

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3.4 Response of a Damped System Under Harmonic Force

3.4.1 Total Response

3.4.2 Quality Factor and Bandwidth

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3.5 Response of a Damped System Under

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3.6 Response of a Damped System Under the Harmonic Motion of the Base

3.6.1 Force Transmitted

3.6.2 Relative Motion

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3.7 Response of a Damped System Under Rotating Unbalance

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3.8 Forced Vibration with Coulomb Damping

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3.9 Forced Vibration with Hysteresis Damping

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3.10 Forced Motion with Other Types of Damping

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3.11 Self-Excitation and Stability Analysis

¿3.11.1 Dynamic Stability Analysis

¿3.11.2 Dynamic Instability Caused by Fluid Flow

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3.12 Transfer Function Approach

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3.13 Solutions Using Laplace Transforms

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3.14 Frequency Transfer Functions

3.14.1 Relation Between the General Transfer function T(s)

and the Frequency Transfer Function

3.14.2 Representation of Frequency Response Characteristics

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3.15 Examples Using Matlab

References

Chapter Summary

Review Questions

Problems

Design Projects

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CHAPTER 4 Vibration Under General Forcing Conditions

4.1 Introduction

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4.2 Response Under a General Periodic Force

4.2.1 First-Order Systems

4.2.2 Second-Order Systems

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4.3 Response Under a Periodic Force of Irregular Form

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4.4 Response Under a Nonperiodic Force

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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

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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

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4.7 Laplace Transform

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

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4.8 Numerical Methods

4.8.1 Runge-Kutta Methods

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4.9 Response to Irregular Forcing Conditions Using Numerical Methods

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4.10 Examples Using Matlab

References

Chapter Summary

Review Questions

Problems

Design Projects

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CHAPTER 5 Two Degree of Freedom Systems

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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

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Chapter Summary

Review Questions

References

Problems

Design Projects

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CHAPTER 6 Multidegree of Freedom Systems

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6.1 Introduction

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6.2 Modeling of Continuous Systems as Multidegree of Freedom Systems

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6.3 Using Newton’s Second Law to Derive Equations of Motion

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6.4 Influence Coefficients

6.4.1 Stiffness Influence Coefficients

6.4.2 Flexibility Influence Coefficients

6.4.3 Inertia Influence Coefficients

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6.5 Potential and Kinetic Energy Expressions in Matrix Form

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6.6 Generalized Coordinates and Generalized Forces

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6.7 Using Lagrange’s Equations to Derive Equations of Motion

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6.8 Equations of Motion of Undamped Systems in Matrix Form

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6.9 Eigenvalue Problem

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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

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6.11 Expansion Theorem

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6.12 Unrestrained Systems

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6.13 Free Vibration of Undamped Systems

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6.14 Forced Vibration of Undamped Systems Using Modal Analysis

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6.15 Forced Vibration of Viscously Damped Systems

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6.16 Self-Excitation and Stability Analysis

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6.17 Examples Using Matlab

References

Chapter Summary

Review Questions

Problems

Design Project

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CHAPTER 7 Determination of Natural Frequencies and Mode Shapes

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7.1 Introduction

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7.2 Dunkerley’s Formula

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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

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7.4 Holzer’s Method

7.4.1 Torsional Systems

7.4.2 Spring-Mass Systems

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7.5 Matrix Iteration Method

7.5.1 Convergence to the Highest Natural Frequency

7.5.2 Computation of Intermediate Natural Frequencies

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7.6 Jacobi’s Method

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7.7 Standard Eigenvalue Problem

7.7.1 Choleski Decomposition

7.7.2 Other Solution Methods

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7.8 Examples Using MATLAB

References

Chapter Summary

Review Questions

Problems

Design Projects

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CHAPTER 8 Continuous Systems

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8.1 Introduction

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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

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8.3 Longitudinal Vibration of a Bar or Rod

8.3.1 Equation of Motion and Solution

8.3.2 Orthogonality of Normal Functions

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8.4 Torsional Vibration of a Shaft or Rod

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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 Other Effects

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8.6 Vibration of Membranes

8.6.1 Equation of Motion

8.6.2 Initial and Boundary Conditions

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8.7 Rayleigh’s Method 742

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8.8 The Rayleigh-Ritz Method 745

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8.9 Examples Using Matlab 748

References

Chapter Summary

Review Questions

Problems

Design Project

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CHAPTER 9 Vibration Control

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9.1 Introduction

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9.2 Vibration Nomograph and Vibration Criteria

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9.3 Reduction of Vibration at the Source

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9.4 Balancing of Rotating Machines

9.4.1 Single-Plane Balancing

9.4.2 Two-Plane Balancing

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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

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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

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9.7 Control of Vibration

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9.8 Control of Natural Frequencies

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9.9 Introduction of Damping

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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

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9.11 Vibration Absorbers

9.11.1 Undamped Dynamic Vibration Absorber

9.11.2 Damped Dynamic Vibration Absorber

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9.12 Examples Using Matlab

References

Chapter Summary

Review Questions

Problems

Design Project

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CHAPTER 10 Vibration Measurement and Applications

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10.1 Introduction

10.2.1 Variable Resistance Transducers

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10.2 Transducers

10.2.2 Piezoelectric Transducers

10.2.3 Electrodynamic Transducers

10.2.4 Linear Variable Differential Transformer Transducer

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10.3 Vibration Pickups

10.3.1 Vibrometer

10.3.2 Accelerometer

10.3.3 Velometer

10.3.4 Phase Distortion

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10.4 Frequency-Measuring Instruments

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10.5 Vibration Exciters

10.5.1 Mechanical Exciters

10.5.2 Electrodynamic Shaker

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10.6 Signal Analysis

10.6.1 Spectrum Analyzers

10.6.2 Bandpass Filter

10.6.3 Constant Percent Bandwidthand Constant Bandwidth Analyzers

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10.7 Dynamic Testing of Machines and Structures

10.8.2 The Necessary Equipment

10.8.1 The Basic Idea

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10.8 Experimental Modal Analysis

10.7.2 Using Modal Testing

10.7.1 Using Operational Deflection Shape Measurements

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

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10.9 Machine Condition Monitoring and Diagnosis

10.9.2 Machine Maintenance Techniques

10.9.1 Vibration Severity Criteria

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

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10.10 Examples Using Matlab

References

Chapter Summary

Review Questions

Problems

Design Projects

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CHAPTER 11 Numerical Integration Methods in Vibration Analysis

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11.1 Introduction

11.2 Finite Difference Method

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11.3 Central Difference Method for Single Degree of Freedom Systems

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11.4 Runge-Kutta Method for Single Degree of Freedom Systems

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11.5 Central Difference Method for Multidegree of Freedom Systems

11.6.1 Longitudinal Vibration of Bars

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11.6 Finite Difference Method for Continuous Systems

11.6.2 Transverse Vibration of Beams

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11.7 Runge-Kutta Method for Multidegree of Freedom Systems

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11.8 Houbolt Method

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11.9 Wilson Method

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11.10 Newmark Method

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11.11 Examples Using Matlab

References

Chapter Summary

Review Questions

Problems

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CHAPTER 12 Finite Element Method

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12.1 Introduction

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12.2 Equations of Motion of an Element

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12.3 Mass Matrix, Stiffness Matrix, and Force Vector

12.3.1 Bar Element

12.3.2 Torsion Element

12.3.3 Beam Element

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12.4 Transformation of Element Matrices and Vectors 998

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12.5 Equations of Motion of the Complete System of Finite Elements

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12.6 Incorporation of Boundary Conditions

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12.7 Consistent and Lumped Mass Matrices

12.7.2 Lumped Mass Matrix for a Beam Element

12.7.1 Lumped Mass Matrix for a Bar Element

12.7.3 Lumped Mass Versus Consistent Mass Matrices

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12.8 Examples Using Matlab

References

Chapter Summary

Review Questions

Problems

Design Projects

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Mathematical Relationships

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

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Mechanical Vibrations, CourseSmart eTextbook, 5th Edition
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