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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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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.



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


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


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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