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## Overview

Mechanical Vibrations, 5/e is ideal 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.

## ADVERTISEMENT

## Product Details

ISBN-13: | 9780132128193 |
---|---|

Publisher: | Pearson |

Publication date: | 09/21/2010 |

Edition description: | Older Edition |

Pages: | 1104 |

Product dimensions: | 7.10(w) x 9.30(h) x 1.60(d) |

## About the Author

**Dr. Singiresu S. Rao** is a Professor and the Department Chair of the Mechanical and Aerospace Engineering Department at the University of Miami College of Engineering.

## Read an Excerpt

PREFACE:

This text serves as an introduction to the subject of vibration engineering at the undergraduate level. The style of the prior editions has been retained, with the theory, computational aspects, and applications of vibrations presented in as simple a manner as possible. As in the previous editions, computer techniques of analysis are emphasized. Expanded explanations of the fundamentals are given, emphasizing physical significance and interpretation that build upon previous experiences in undergraduate mechanics. Numerous examples and problems are used to illustrate principles and concepts. Favorable reactions and encouragement from professors and students have provided me with the impetus to write the third edition of this book. Several new sections have been added and many topics modified and rewritten. Most of the additions were suggested by those who have used the text and by numerous reviewers. Some important changes should be noted:

- The sections on the history of vibration, harmonic motion and harmonic analysis are expanded in Chapter 1.
- In Chapter 3 the section on self-excitation and stability analysis has been rewritten and expanded.
- A section on earthquake response spectra has been added to Chapter 4.
- Two new sections, Using Newton's Second Law to Drive Equations of Motion and Free Vibration of Undamped Systems, have been added to Chapter 6.
- A section on forced vibration of beams has been added to Chapter 8.
- The sections on isolation and absorbers have been expanded in Chapter 9.
- The section on experimental modal analysis has beenrewritten and a new section on machine condition monitoring and diagnosis has been added to Chapter 10.
- A section on chaos has been added to Chapter 13.
- A section on response of a multidegree of freedom system has been added to Chapter 14.
- Two new appendixes, on mathematical relationships and deflection of beams and plates, are now included.
- Approximately 30 new illustrative examples appear throughout the book.
- More than 220 new problems have been added at the ends of various chapters.
- In several chapters, more project type problems are now included.

### Features

Each topic in Mechanical Vibrations is self-contained, with all concepts explained fully and the derivations presented with complete details. The computational aspects are emphasized throughout the book. Several Fortran computer programs, most of them in the form of general purpose subroutines, are included in the diskette accompanying the book. These programs are given for use by the students. Although the programs have been tested, no warranty is implied as to their accuracy. Problems that are based on the use/development of computer programs are given at the end of each chapter and expose students to many important computational and programming details.

Certain subjects are presented in a somewhat unconventional manner. The topics of Chapters 9, 10, and 11 fall in this category. Most textbooks discuss isolators, absorbers, and balancing in different places. Since one of the main purposes of the study of vibrations is to control vibration response, all topics directly related to vibration control are given in Chapter 9. The vibration-measuring instruments, along with vibration exciters, experimental modal analysis procedures, and machine condition monitoring, are presented in Chapter 10. Similarly, all the numerical integration methods applicable to single- and multi-degree of freedom systems, as well as continuous systems, are unified in Chapter 11.

Specific features include the following:

- Nearly 130 Illustrative examples accompanying most topics.
- More than 50 review questions to help students in reviewing and testing their understanding of the text material.
- Approximately 850 problems, with solutions in the instructor's manual.
- More than 30 design project type problems at the ends of various chapters.
- Twenty-three computer programs to aid students in the numerical implementation of the methods discussed in the text.
- Biographical information about scientists and engineers who contributed to the development of the theory of vibrations given on the opening pages of chapters and appendixes.
- A convenient format for all examples: Following the statement of each example, the known information, the quantities to be determined, and the approach to be used are first identified and then the detailed solution is given.

### Notation and Units

Both the SI and the English system of units have been used in the examples and problems. A list of symbols, along with the associated units in SI and English systems, is given following the Contents. A brief discussion of SI units as they apply to the field of vibrations is given in Appendix E. Arrows are used over symbols to denote column vectors and square brackets are used to indicate matrices.

### Contents

Mechanical Vibrations is organized into 14 chapters and 5 appendixes. The material of the book provides flexible options for different types of vibration courses. For a one-semester senior or duel-level course, Chapters 1 through 5, portions of Chapters 6, 7, 8, and 10, and Chapter 9 may be used. The course can be given a computer orientation by including Chapter 11 in place of Chapter 8. Alternatively, with Chapters 12, 13, and 14, the text has sufficient material for a one-year sequence at the senior level. For shorter courses, the instructor can select the topics, depending on the level and orientation of the course. The relative simplicity with which topics are presented also makes the book useful to practicing engineers for purposes of self-study and as a source of references and computer programs.

Chapter 1 starts with a brief discussion of the history and importance of vibrations. The basic concepts and terminology used in vibration analysis are introduced. The free vibration analysis of single degree of freedom undamped translational and torsional systems is given in Chapter 2. The effects of viscous, Coulomb, and hysteretic damping are also discussed. The harmonic response of single degree of freedom systems is considered in Chapter 3. Chapter 4 is concerned with the response of a single degree of freedom system under general forcing functions. The roles of convolution integral, Laplace transformation, and numerical methods are discussed. The concept of response spectrum is also introduced in this chapter. The free and forced vibration of two degree of freedom systems is considered in Chapter 5. The self-excited vibration and stability of the system are discussed. Chapter 6 presents the vibration analysis of multi-degree of freedom systems. Matrix methods of analysis are used for the presentation of the theory. The modal analysis procedure is described for the solution of forced vibration problems. Several methods of determining the natural frequencies of discrete systems are outlined in Chapter 7. The methods of Dunkerley, Rayleigh, Holzer, and Jacobi and matrix iteration are also discussed. The vibration analysis of continuous systems, including strings, bars, shafts, beams, and membranes is given in Chapter 8. The Rayleigh and Rayleigh-Ritz methods of finding the approximate natural frequencies are also described. Chapter 9 discusses the various aspects of vibration control, including the problems of elimination, isolation, and absorption. The balancing of rotting and reciprocating machines and the whirling of shafts are also considered. The vibration-measuring instruments, vibration exciters, and signal analysis are the topics of Chapter 10. Chapter 11 presents several numerical integration techniques for finding the dynamic response of discrete and continuous systems. The central difference, Runge-Kutta, Houbolt, Wilson, and Newmark methods are summarized and illustrated. Finite element analysis, with applications involving one-dimensional elements, is discussed in Chapter 12. An introductory treatment of nonlinear vibration, including a discussion of subharmonic and superharmonic oscillations, limit cycles, systems with time-dependent coefficients and chaos, is given in Chapter 13. The random vibration of linear vibration systems is considered in Chapter 14. Appendixes A and B focus on mathematical relationships and deflection of beams and plates, respectively. Finally, the basic relations of matrices, Laplace transforms, and SI units are outlined, respectively, in Appendixes C, D, and E.

### Acknowledgments

I would like to express my appreciation to the many students and faculty whose comments have helped me improve this edition. I am most grateful to the following people for reviewing the book and/or offering their comments, suggestions, and ideas: Richard Alexander, Texas A&M University; C. W. Bert, University of Oklahoma; Raymond M. Brach, University of Notre Dame; Alfonso Diaz-Jimenez, Universidad Distrital "Francisco Jose de Caldas," Colombia; George Doyle, University of Dayton; Hamid Hamidzadeh, South Dakota State University; H. N. Hashemi, Northeastern University; Zhikun Hou, Worcester Polytechnic Institute; J. Richard Houghton, Tennessee Technological University; Faryar Jabbari, University of California-Irvine; Robert Jeffers, University of Connecticut; Richard Keltie, North Carolina State University; J. S. Lamancusa, Pennsylvania State University; Harry Law, Clemson University; Robert Leonard, Virginia Polytechnic Institute and State University; James Li, Columbia University; Sameer Madanshetty, Boston University; M. G. Prasad, Stevens Institute of Technology; F. P. J. Rimrott, University of Toronto; Subhash Sinha, Auburn University; Daniel Stutts, University of Missouri-Rolla; Massoud Tavakoli, Georgia Institute of Technology; Theordore Terry, Lehigh University; Chung Tsui, University of Maryland-College Park; Alexander Vakakis, University of Illinois-Urbana Champaign; Chuck Van Karsen, Michigan Technological University; Aleksandra Vinogradov, Montana State University; K. W. Wang, Pennsylvania State University; William Webster, GMI Engineering and Management Institute.

It has been gratifying to work with the staff of Addison-Wesley throughout this revision. In particular, the help of Stuart Johnson, Publishing Partner, has been most valuable. Helen Wythe, Senior Production Supervisor, and Marybeth Mooney, Production Coordinator, handled the task of incorporating my corrections and revisions very efficiently. I would like to thank Purdue University for granting me permission to use the Boilermaker Special in Problem 2.82. Finally, I wish to thank my wife, Kamala, and daughters Sridevi and Shobha without whose patience, encouragement, and support this edition might never have been completed.

S. S. Rao

## Table of Contents

**1. Fundamentals of Vibration.**

Origins of vibration. From Galileo to Rayleigh. Recent contributions.

Importance of the Study of Vibration. Basic Concepts of Vibration.

Vibration. Elementary parts of vibrating systems. Degree of freedom. Discrete and continuous systems.

Classification of Vibration.

Free and forced vibration. Undamped and damped vibration. Linear and nonlinear vibration. Deterministic and random vibration.

Vibration Analysis Procedure. Spring Elements.

Combination of springs.

Mass or Inertia Elements.

Combination of masses.

Damping Elements.

Construction of viscous dampers. Complex number representation of harmonic motion. Complex algebra.

Harmonic Motion.

Vectorial representation of harmonic motion. Complex number representation of harmonic motion. Complex algebra. Operations on harmonic functions. Definitions and terminology.

Harmonic Analysis.

Fourier serious expansion. Complex Fourier series. Frequency spectrum. Time and frequency domain representations. Even and odd functions. Half range expansions. Numerical computation of coefficients.

**2. Free Vibration of Single Degree of Freedom Systems.**

Equation of motion using Newton's second law of motion. Equation of motion using other methods. Equation of motion of a spring-mass system in vertical position. Solution. Harmonic motion.

Free Vibration of an Undamped Torsional System.

Equation of motion. Solution.

StabilityConditions. Rayleigh's Energy Method. Free Vibration with Viscous Damping.

Equation of motion. Solution. Logarithmic decrement. Energy dissipated in viscous damping. Torsional systems with viscous damping.

Free Vibration with Coulomb Damping.

Equation of motion. Solution. Torsional systems with Coulomb damping.

Free Vibration with Hysteretic Damping. Computer program. References. Review Questions. Problems. Design Projects.

**3. Harmonically Excited Vibration.**

Total response. Beating phenomenon.

Response of a Damped System Under Harmonic Force.

Total response. Quality factor and bandwidth.

Response of a Damped System Under F(t) = F. Response of a Damped System Under the Harmonic. Motion of the Base.

Force transmitted. Relative motion.

Response of a Damped System Under Rotating. Unbalance. Forced Vibration with Coulomb Damping. Forced Vibration with Hysteresis Damping. Forced Motion with Other Types of Damping. Self-Excitation and Stability Analysis.

Dynamic stability analysis. Dynamic instability caused by fluid flow.

Computer Program. References. Review Questions. Problems. Design Projects.

**4. Vibration Under General Forcing Conditions.**

Response to an impulse. Response to general forcing condition. Response to base excitation.

Response Spectrum.

Response spectrum for base excitation. Earthquake response spectra. Design under shock environment.

Laplace Transformation. Response to Irregular Forcing Conditions Using. Numerical Methods. Computer Programs.

Response under an arbitrary periodic forcing function. Response under arbitrary forcing function using the methods of section.

References. Review Questions. Problems. Design Projects.

**5. Two Degree of Freedom Systems.**

**6. Multidegree of Freedom Systems.**

Stiffness influence coefficients. Flexibility influence coefficients. Inertia influence coefficients.

Potential and Kinetic Energy Expressions in Matrix Form. Generalized Coordinates and Generalized Forces. Using Lagrange's Equations to Derive Equations of Motion. Equations of Motion of Undamped Systems in Matrix Form. Eigenvalue Problem. Solution of the Eigenvalue Problem.

Solution of the characteristic (polynomial) equation. Orthogonality of normal modes. Repeated eigenvalues.

Expansion Theorem. Unrestrained Systems. Free Vibration of Undamped Systems. Forced Vibration of Undamped Systems. Forced Vibration of Viscously Damped Systems. Self-Excitation and Stability Analysis. Computer Programs.

Generating the characteristic polynomial from the matrix. Roots of an nth order polynomial equation with complex coefficients. Modal analysis of a multidegree of freedom system. Solution of Simultaneous linear equations.

References. Review Questions. Problems. Design Project.

**7. Determination of Natural Frequencies and Mode Shapes.**

Properties of Rayleigh's quotient. Computation of the fundamental natural frequency. Fundamental frequency of beams and shafts.

Holzer's Method.

Torsional systems. Spring-mass systems.

Matrix Iteration Method.

Convergence to the highest natural frequency. Computation of intermediate natural frequencies.

Jacobi's Method. Standard Eigenvalue Problem.

Choleski decomposition. Other solution methods.

Computer Programs.

Jacobi's method. Matrix iteration method. Choleski decomposition. Eigenvalue solution using Choleski decomposition.

References. Review Questions. Problems. Projects.

**8. Continuous Systems.**

Equation of motion. Initial and boundary conditions. Free vibration of a uniform string. Free vibration of a string with both ends fixed. Traveling-wave solution.

Longitudinal Vibration of a Bar or Rod.

Equation of motion and solution. Orthogonality of normal functions.

Torsional Vibration of a Shaft or Rod. Lateral Vibration of Beams.

Equation of motion. Initial conditions. Free vibration. Boundary conditions. Orthogonality of normal functions. Forced vibration. Effect of axial force. Effects of rotary inertia and shear deformation. Other effects.

Vibration of Membranes.

Equation of motion. Initial and boundary conditions.

Rayleigh's Method. The Rayleigh-Ritz Method. Computer Program. References. Review Questions. Problems. Design Projects.

**9. Vibration Control.**

Single-plane balancing. Two-plane balancing.

Whirling of Rotating Shafts.

Equations of motion. Critical speeds. Response of the system. Stability analysis.

Balancing of Reciprocating Engines.

Unbalanced forces due to fluctuations in gas pressure. Unbalanced forces due to inertia of the moving parts. Balancing of reciprocating engines.

Control of Vibration. Control of Natural Frequencies. Introduction of Damping. Vibration Isolation.

Vibration isolation system with rigid foundation. Vibration isolation system with flexible foundation. Vibration isolation system with partially flexible foundation. Shock isolation. Active vibration control.

Vibration Absorbers.

Undamped dynamic vibration absorber. Damped dynamic vibration absorber.

Computer Program. References. Review Questions. Problems. Design Projects.

**10. Vibration Measurement and Applications.**

Variable resistance transducers. Peizoelectric transducers. Electrodynamic transducers. Linear variable differential transformer (LVDT) transducer.

Vibration pickups.

Vibrometer. Accelerometer. Velometer. Phase distortion.

Frequency Measuring Instruments. Vibration Exciters.

Mechanical exciters. Electrodynamic shaker.

Signal Analysis.

Spectrum analyzers. Bandpass filter. Constant percent bandwidth and constant bandwidth analyzers.

Dynamic Testing of Machines and Structures.

Using operational deflection shape measurements. Using modal testing.

Experimental Modal Analysis.

Representation of the frequency response of a system. Testing and analysis. Test preparation and setup. Measurement of frequency response functions. Identification of modal parameters. Computer-aided modal testing.

Machine condition monitoring and diagnosis.

Vibration severity criteria. Machine maintenance techniques. Machine condition monitoring techniques. Vibration monitoring techniques. Instrumentation systems. Choice of monitoring parameter.

References. Review Questions. Problems. Design Projects.

**11. Numerical Integration Methods in Vibration Analysis.**

Longitudinal vibration of bars. Transverse vibration of beams.

Runge-Kutta Method for Multidegree of Freedom Systems. Houbold Method. Wilson Method. Newmark Method. Computer Programs.

Fourth order Runge-Kutta method. Central difference method. Houbold method.

References. Review Questions. Problems.

**12. Finite Element Method.**

Bar element. Torsion element. Beam element.

Transformation of Element Matrices and Vectors. Equations of Motion of the Complete System of Finite Elements. Incorporation of Boundary Conditions. Consistent and Lumped Mass Matrices.

Lumped mass matrix for a bar element. Lumped mass matrix for a beam element. Lumped mass versus consistent mass matrices.

Computer Program. References. Review Questions. Problems. Design Projects.

**13. Nonlinear Vibration.**

Simple pendulum. Mechanical chatter, belt friction system. Variable mass system.

Exact Methods. Approximate Analytical Methods. Basic philosophy.

Lindstedt's peturbation method. Iterative method. Ritz-Galerkin method.

Subharmonic and Superharmonic Oscillations.

Subharmonic oscillations. Superharmonic solution.

Systems with Time-Dependent Coefficients (Mathieu Equation). Graphical Methods.

Phase Plane Representation. Phase velocity. Method of constructing trajectories. Obtaining time solution from phase plane trajectories.

Stability of Equilibrium States.

Stability analysis. Classification of singular points.

Limit Cycles. Chaos.

Functions with stable orbits. Functions with unstable orbits. Chaotic behavior of Duffing's equation without the forcing term. Chaotic behavior of Duffing's equation with the forcing term.

Numerical Methods. Computer Program. References. Review Questions. Problems. Design Projects.

**14. Random Vibration.**

Fourier series. Fourier integral.

Power Spectral Density. Wide-Band and Narrow-Band Processes. Response of a Single Degree of Freedom System.

Impulse response approach. Frequency response approach. Characteristics of the response function.

Response Due to Stationary Random Excitations.

Impulse response approach. Frequency response approach.

Response of a Multidegree of Freedom System. References. Review Questions. Problems. Design Project.

**Appendix A: Mathematical Relationships.**

**Appendix B: Deflection of Beams and Plates.**

**Appendix C: Matrices.**

**Appendix D: Laplace Transform Pairs.**

**Appendix E: Units.**

**References.**

**Answers to Selected Problems.**

**Index.**

## Preface

This text serves as an introduction to the subject of vibration engineering at the undergraduate level. The style of the prior editions has been retained, with the theory, computational aspects, and applications of vibrations presented in as simple a manner as possible. As in the previous editions, computer techniques of analysis are emphasized. Expanded explanations of the fundamentals are given, emphasizing physical significance and interpretation that build upon previous experiences in undergraduate mechanics. Numerous examples and problems are used to illustrate principles and concepts. Favorable reactions and encouragement from professors and students have provided me with the impetus to write the third edition of this book. Several new sections have been added and many topics modified and rewritten. Most of the additions were suggested by those who have used the text and by numerous reviewers. Some important changes should be noted:

- The sections on the history of vibration, harmonic motion and harmonic analysis are expanded in Chapter 1.
- In Chapter 3 the section on self-excitation and stability analysis has been rewritten and expanded.
- A section on earthquake response spectra has been added to Chapter 4.
- Two new sections, Using Newton's Second Law to Drive Equations of Motion and Free Vibration of Undamped Systems, have been added to Chapter 6.
- A section on forced vibration of beams has been added to Chapter 8.
- The sections on isolation and absorbers have been expanded in Chapter 9.
- The section on experimental modal analysis hasbeenrewritten and a new section on machine condition monitoring and diagnosis has been added to Chapter 10.
- A section on chaos has been added to Chapter 13.
- A section on response of a multidegree of freedom system has been added to Chapter 14.
- Two new appendixes, on mathematical relationships and deflection of beams and plates, are now included.
- Approximately 30 new illustrative examples appear throughout the book.
- More than 220 new problems have been added at the ends of various chapters.
- In several chapters, more project type problems are now included.

### Features

Each topic in Mechanical Vibrations is self-contained, with all concepts explained fully and the derivations presented with complete details. The computational aspects are emphasized throughout the book. Several Fortran computer programs, most of them in the form of general purpose subroutines, are included in the diskette accompanying the book. These programs are given for use by the students. Although the programs have been tested, no warranty is implied as to their accuracy. Problems that are based on the use/development of computer programs are given at the end of each chapter and expose students to many important computational and programming details.

Certain subjects are presented in a somewhat unconventional manner. The topics of Chapters 9, 10, and 11 fall in this category. Most textbooks discuss isolators, absorbers, and balancing in different places. Since one of the main purposes of the study of vibrations is to control vibration response, all topics directly related to vibration control are given in Chapter 9. The vibration-measuring instruments, along with vibration exciters, experimental modal analysis procedures, and machine condition monitoring, are presented in Chapter 10. Similarly, all the numerical integration methods applicable to single- and multi-degree of freedom systems, as well as continuous systems, are unified in Chapter 11.

Specific features include the following:

- Nearly 130 Illustrative examples accompanying most topics.
- More than 50 review questions to help students in reviewing and testing their understanding of the text material.
- Approximately 850 problems, with solutions in the instructor's manual.
- More than 30 design project type problems at the ends of various chapters.
- Twenty-three computer programs to aid students in the numerical implementation of the methods discussed in the text.
- Biographical information about scientists and engineers who contributed to the development of the theory of vibrations given on the opening pages of chapters and appendixes.
- A convenient format for all examples: Following the statement of each example, the known information, the quantities to be determined, and the approach to be used are first identified and then the detailed solution is given.

### Notation and Units

Both the SI and the English system of units have been used in the examples and problems. A list of symbols, along with the associated units in SI and English systems, is given following the Contents. A brief discussion of SI units as they apply to the field of vibrations is given in Appendix E. Arrows are used over symbols to denote column vectors and square brackets are used to indicate matrices.

### Contents

Mechanical Vibrations is organized into 14 chapters and 5 appendixes. The material of the book provides flexible options for different types of vibration courses. For a one-semester senior or duel-level course, Chapters 1 through 5, portions of Chapters 6, 7, 8, and 10, and Chapter 9 may be used. The course can be given a computer orientation by including Chapter 11 in place of Chapter 8. Alternatively, with Chapters 12, 13, and 14, the text has sufficient material for a one-year sequence at the senior level. For shorter courses, the instructor can select the topics, depending on the level and orientation of the course. The relative simplicity with which topics are presented also makes the book useful to practicing engineers for purposes of self-study and as a source of references and computer programs.

Chapter 1 starts with a brief discussion of the history and importance of vibrations. The basic concepts and terminology used in vibration analysis are introduced. The free vibration analysis of single degree of freedom undamped translational and torsional systems is given in Chapter 2. The effects of viscous, Coulomb, and hysteretic damping are also discussed. The harmonic response of single degree of freedom systems is considered in Chapter 3. Chapter 4 is concerned with the response of a single degree of freedom system under general forcing functions. The roles of convolution integral, Laplace transformation, and numerical methods are discussed. The concept of response spectrum is also introduced in this chapter. The free and forced vibration of two degree of freedom systems is considered in Chapter 5. The self-excited vibration and stability of the system are discussed. Chapter 6 presents the vibration analysis of multi-degree of freedom systems. Matrix methods of analysis are used for the presentation of the theory. The modal analysis procedure is described for the solution of forced vibration problems. Several methods of determining the natural frequencies of discrete systems are outlined in Chapter 7. The methods of Dunkerley, Rayleigh, Holzer, and Jacobi and matrix iteration are also discussed. The vibration analysis of continuous systems, including strings, bars, shafts, beams, and membranes is given in Chapter 8. The Rayleigh and Rayleigh-Ritz methods of finding the approximate natural frequencies are also described. Chapter 9 discusses the various aspects of vibration control, including the problems of elimination, isolation, and absorption. The balancing of rotting and reciprocating machines and the whirling of shafts are also considered. The vibration-measuring instruments, vibration exciters, and signal analysis are the topics of Chapter 10. Chapter 11 presents several numerical integration techniques for finding the dynamic response of discrete and continuous systems. The central difference, Runge-Kutta, Houbolt, Wilson, and Newmark methods are summarized and illustrated. Finite element analysis, with applications involving one-dimensional elements, is discussed in Chapter 12. An introductory treatment of nonlinear vibration, including a discussion of subharmonic and superharmonic oscillations, limit cycles, systems with time-dependent coefficients and chaos, is given in Chapter 13. The random vibration of linear vibration systems is considered in Chapter 14. Appendixes A and B focus on mathematical relationships and deflection of beams and plates, respectively. Finally, the basic relations of matrices, Laplace transforms, and SI units are outlined, respectively, in Appendixes C, D, and E.

### Acknowledgments

I would like to express my appreciation to the many students and faculty whose comments have helped me improve this edition. I am most grateful to the following people for reviewing the book and/or offering their comments, suggestions, and ideas: Richard Alexander, Texas A&M University; C. W. Bert, University of Oklahoma; Raymond M. Brach, University of Notre Dame; Alfonso Diaz-Jimenez, Universidad Distrital "Francisco Jose de Caldas," Colombia; George Doyle, University of Dayton; Hamid Hamidzadeh, South Dakota State University; H. N. Hashemi, Northeastern University; Zhikun Hou, Worcester Polytechnic Institute; J. Richard Houghton, Tennessee Technological University; Faryar Jabbari, University of California-Irvine; Robert Jeffers, University of Connecticut; Richard Keltie, North Carolina State University; J. S. Lamancusa, Pennsylvania State University; Harry Law, Clemson University; Robert Leonard, Virginia Polytechnic Institute and State University; James Li, Columbia University; Sameer Madanshetty, Boston University; M. G. Prasad, Stevens Institute of Technology; F. P. J. Rimrott, University of Toronto; Subhash Sinha, Auburn University; Daniel Stutts, University of Missouri-Rolla; Massoud Tavakoli, Georgia Institute of Technology; Theordore Terry, Lehigh University; Chung Tsui, University of Maryland-College Park; Alexander Vakakis, University of Illinois-Urbana Champaign; Chuck Van Karsen, Michigan Technological University; Aleksandra Vinogradov, Montana State University; K. W. Wang, Pennsylvania State University; William Webster, GMI Engineering and Management Institute.

It has been gratifying to work with the staff of Addison-Wesley throughout this revision. In particular, the help of Stuart Johnson, Publishing Partner, has been most valuable. Helen Wythe, Senior Production Supervisor, and Marybeth Mooney, Production Coordinator, handled the task of incorporating my corrections and revisions very efficiently. I would like to thank Purdue University for granting me permission to use the Boilermaker Special in Problem 2.82. Finally, I wish to thank my wife, Kamala, and daughters Sridevi and Shobha without whose patience, encouragement, and support this edition might never have been completed.

S. S. Rao