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### Theory of Vibration

Theory of Vibration

Kin N. Tong

Professor of Mechanical Engineering

Syracuse University

364 Pages

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Preface:

This book is the outgrowth of lecture notes for a course given to beginning graduate students and qualified seniors. Because of this origin, it is primarily a textbook, although some utility as a reference volume is also intended.

A course in mechanical vibrations can be organized in one of two ways, which may be described as problem-centered and theory-centered. This book is written for a theory-centered course, which develops the basic principles in a logical order, with engineering applications inserted as illustrations. No attempt is thus made to cover all problems of technological importance or to restrict the discussion only to topics having immediate applications. It is felt that a theory-centered course has its place in an engineering mechanics curriculum, since the analytical aspects of the theory have pedagogical values beside their utility in solving vibration problems.

The book is divided into four chapters. Chapter 1 treats systems having a single degree of freedom. All the basic concepts pertaining to mechanical vibrations are presented, with the exception of vibration modes. Chapter 2 introduces the concept of vibration modes in a multi degree-freedom system, using a system with two degrees of freedom as a simple model. The discussion is kept as close as possible to physical aspects of the problem. By means of matrix algebra and generalized coordinates, Chapter 3 extends the results previously obtained. In this way this chapter also lays the foundation for the solution of vibration problems on digital computers and provides a heuristic picture of what is to follow. Chapter 4 discusses the vibration of continuous media. Because only a limited amount of student knowledge in elasticity can be assumed, the systems selected for illustration in this chapter are relatively simple, yet the theory presented is quite general.

The layout of this book is somewhat different from the usual. In the beginning of each chapter fundamental principles are presented in a connected series of articles. Articles dealing with examples, applications, and specialized topics, which are more or less independent of one another, are placed at the ends of the chapters. (In the first three chapters these articles are grouped into two sections, A and B. The same grouping is not indicated in the fourth chapter, since there the demarcation is not so clear.) The purpose of this arrangement is twofold. It emphasizes the structural coherence of the theory, and it affords flexibility in classroom assignments. The instructor can plan his lectures by following the main development of the theory. At intervals appropriate to the level and the interest of a particular class, he may discuss, or assign as home reading, examples, applications, and methods selected from this book or from other sources. A number of exercises is given at the end of each chapter. Many of' these exercises supplement the material in the text.

The students are assumed to have the usual preparation, including a course in differential equations, in undergraduate mechanics and mathematics. Certain fundamental theorems in advanced calculus and in vector analysis are referred to in a few isolated passages; these can be omitted, if necessary, without disrupting the continuity of presentation. An appendix on the basic ideas of matrix algebra is given. The scope of this appendix is limited, but it contains all that is needed for studying Chapter 3. In short, little prior knowledge is required to understand this book, although some degree of maturity is indispensable.

To keep the scope of the book within the limits of a two-semester course and to preserve the unity of the entire presentation, certain topics are omitted. These include nonlinear vibrations and the solution of transient problems by operational calculus. However, seeding ideas pertaining to these topics are planted in Arts. 1.4, 1.10, 1.11, 1.13, 1.14, 2.8 and 3.8, but their complete development is left to other standard courses generally available to advanced students.

Many persons helped to prepare this book. I wish especially to thank Professor Harold Lurie for a thorough reading of the manuscript and for offering valuable suggestions. Thanks are due to Mrs. Patricia Fisch and Mrs. Marilyn Levine for typing the manuscript and to Mr. C. Y. Chia and Mr. K. Ruei for assisting in various other ways.

TOC

CHAPTER 1

Systems with a Single Degree
of Freedom

Theory and Principles

Introduction 3

Simple harmonic motion 3

Complex number and graphical representation of
a sinusoidal function 6

Harmonic oscillation of system with a single
degree of freedom—General discussion 9

Energy relation, Rayleigh's principle, and
phase trajectory 10

Damped vibration with viscous or linear
damping 12

Forced vibration under a harmonic force 17

Complex number representation 24

Steady-state response to periodic forces 28

Work done by external forces and energy
dissipation in vibratory systems 30

Response of linear systems to a general external
force—Superposition theorem 31

Signal-response relation of linear systems in
general 39

Methods and Applications

Examples of linear vibratory systems with a
single degree of freedom 52

1.13 Linearization of systems in small oscillations 55

1.14 Piecewise-linear systems 59

1.15 Theory of galvanometer and moving-coil
instruments 68

1.16 Seismic instruments and transducers 73

1.17 Vehicle suspension 79

1.18 Structural damping and the concept of complex
stiffness 82

Exercises

CHAPTER 2

Systems with Two Degrees of Freedom

Theory and Principles 100

Introduction 100

Free undamped vibration—a model and its
equation of motion 100

Principal or normal modes 101

General Solution 102

Formulation by energy consideration
generalized analysis for the free vibration of
system with two degrees of freedom 105

The use of influence coefficients 108

Rayleigh's quotient 111

Vibration of damped systems 114

Forced vibration 116

Degenerated cases 120

Repeated roots in frequency equations
transverse vibration of rotating shafts 123

Methods and Applications 128

Illustrative examples 128

Application of Rayleigh's method 132

Some principles in vibration control 135

Effects of rotation on critical speeds of shafts 146

Exercises

CHAPTER 3

Systems with Several Degrees
of Freedom

Theory and Principles
Introduction 168

3.1 Generalized coordinates, constraints, and degrees
of freedom 169

3.2 Energy expressions in generalized coordinates for
linear systems 170

3.3 Summation convention and matrix notation 173

3.4 Free vibrations of an undamped system an
eigenvalue problem 175

3.5 Principal coordinates and orthogonal property of
modal vectors 180

3.6 Rayleigh's quotient 183

3.7 Forced vibration of an undamped system 185

3.8 Free and forced vibrations of a damped
system 188

3.9 Semidefinite systems 192

3.10 Repeated roots of the frequency equation 196

Methods and Applications 197

3.11 Solution of eigenvalue problems by matrix
iteration 197

3.12 Additional theorems and methods 208

3.13 Chain systems—Holzer's method 212

3.14 Electrical analog of mechanical systems and
electromechanical systems 221

Exercises 231

CHAPTER 4 Vibration of Elastic Bodies

4.0 Introduction 236

4.1 Coordinates and constraints 237

4.2 Formulation of a problem by differential
equation 239

4.3 Separation of time variable from space variables
reduction to eigenvalue problems 249

4.4 Orthogonal property of eigenfunctions 260

4.5 Formulation by integral equation 271

4.6 Rayleigh's quotient and its stationary values 274

4.7 Rayleigh-Ritz method 279

4.8 Formulation of problem by infinite-series expansions
of energy expressions Rayleigh-Ritz
method re-examined 287

4.9 Forced vibration of elastic bodies 296

4.10 Vibration of an infinite or semi-infinite elastic
body—wave phenomenon 304

4.11 Methods of finite differences 312

Exercises 318

APPENDIX. Outline of Matrix Algebra in Linear Transformation of Vectors

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