Preface

PREFACE

This book presents material fundamental to a modern treatment of vibrations, placing the emphasis on analytical developments and computational solutions. It is intended as a textbook for a number of courses on vibrations ranging from the junior level to the second-year graduate level; the book can also serve as a reference for practicing engineers. Certain material from pertinent disciplines was included to render the book self-contained, and hence suitable for self-study. Consistent with this, the book beings with very elementary material and raises the level gradually. A large number of examples and homework problems, as well as computer programs written in MATLAB, are provided.

The following review is designed to help the reader decide how best to use the book:

Chapter 1. Concepts from Vibrations - Sections 1.1-1.6 are developed to a review of basic concepts from Newtonian mechanics. Issues concerning the modeling of mechanical systems, from components to assembled systems are derived in Sec.1.10 Section 1.11 and 1.12 are concerned with the nature of the excitations, the system characteristics and the nature of the response; the concept of linearity and the closely related principle of superposition are discussed. Finally, in Sec. 1.13, the concepts of equilibrium points and motions about equilibrium point are introduced.

The whole chapter is suitable for a first course on vibrations at the undergraduate level, but Secs. 1.1-1.6 may be omitted from a first course at the graduate level.

Chapter 2. Response of Single-Degree-of-Freedom Systems to Initial Excitations- This chapter is concerned with the free vibration of undamped, viscously damped and Coulomb damped systems to initial displacements and velocities. It includes a MATLAB program for plotting the response of viscously damped systems.

This chapter is essential to a first course on vibrations at any level.

Chapter 3. Response of Single-Degree-of-Systems to Harmonic and Periodic Excitations- In Secs. 3.1 and 3.2, the response to harmonic excitations is represented in the frequency domain, through magnitude and phase angle frequency response plots. Sections 3.3-3.7 discuss applications such as systems with rotating eccentric masses, systems with harmonically moving support, vibration isolation and vibration measuring instruments. In Sec. 3.8, structural damping is treated by means of analogy with viscous damping. Finally, in Sec. 3.9, the approach to the response of systems to harmonic excitation is extended to periodic excitations through the use of Fourier series. A MATLAB program generating frequency response is provided in Sec.3.10.

The material in Sec. 3.1-3.6 is to be included in a first course on vibrations, but the material in Secs. 3.7-3.9 is optional

Chapter4. Response of Single-Degree-of-Freedom Systems to Nonperiodic Excitations - Sections 4.1-4.3 introduce the unit impulse, unit step function and unit ramp function and the respective response. Then, regarding arbitrary excitations as a superposition of impulses of varying magnitude, the system response is represented in Sec. 4.4 as a corresponding superposition of impulse responses, becoming the convolution integral in the limit. Section 4.5 discusses the concept of shock spectrum. Sections 4.6 and 4.7 are devoted to the system response by the Laplace transformation; the concept of transfer function is introduced. Next in Sec. 4.8, the response is obtained by the state transition matrix. Numerical solutions for the response are carried in discrete time by the convolution sum in Sec. 4.9 and by the discrete-time transition matrix in Sec. 4.10. A MATLAB program for the response using the convolution sum is given in Sec.4.11 and another program using the discrete-time transition matrix is given in Sec. 4.12.

Sections 4.1-4.4 are to be included in a first course on vibrations at all levels. Section 4.5 is optional, but recommended for a design-oriented course. Sections 4.6-4.10 are optional for a junior course, recommended for a senior course and to be included in a first course at the graduate level.

Chapter 5. Two-Degree-of-Freedom Systems - Sections 5.1-5.6 present in a simple fashion such topics as the eigenvalue problem, natural modes, response to initial excitations, coupling, orthogonality of modes and modal analysis. Section 5.7 is concerned with the beat phenomenon, Sec. 5.8 derives the response to harmonic excitations carried out in continuous time in Sec. 5.10 and in discrete time in Sec.5.11. Three MATLAB programs are included, the first in Sec. 5.12 for the response to initial excitations, the second in Sec. 5.13 for producing frequency response plots and the third in Sec. 5.14 for the response to a rectangular pulse by the convolution sum.

The material belongs in an undergraduate course on vibrations, but is not essential to a graduate course, unless a gradual transition to multi-degree-of-freedom systems is deemed desirable.

Chapter 6. Elements of Analytical Dynamics- Sections 6.1-6.3 provide the prerequisite material for the development in Sec. 6.4 of the extended Hamilton principle, which permits the derivation of all equations of motion. In Sec. 6.5, the principle is used to produce a generic form of the equations of motion, namely Lagrange's equations.

This chapter is suitable for a senior course on vibrations and is a virtual necessity for a first-year graduate course.

Chapter 7. Multi-Degree-of-Freedom System - Sections 7.1-7.4 are concerned with the formulation of the equations of motion for linear and linearized systems, as well as with some basic properties of such systems. In Sec. 7.5-7.7, some of the concepts discussed in Ch. 5, such as linear transformations, coupling, the eigenvalue problem, natural modes and orthogonality of modes, are presented in a more compact manner by means of matrix algebra. Then, in Sec. 7.8, the question of rigid-body motions is addressed. In Secs. 7.9 and 7.10, modal analysis is first developed in a rigorous manner and then used to obtain the response to initial excitations. Certain issues associated with the eigenvalue problem are discussed in Sec. 7.11 and 7.12. Section 7.13 is devoted to Rayleigh's quotient, a concept of great importance in vibrations. The response to external excitations is obtained in continuous time in Secs. 7.14 and 7.15 and in discrete time in Sec. 7.16. MATLAB problems are provided as follows: the solution of the eigenvalue problem for conservative systems and for nonconservative systems, both in Sec. 7.18, the response to initial excitations in Sec. 7.19 and the response to external excitations by the discrete-time transition matrix in Sec. 7.20.

This chapter, in full or in part, is suitable for a senior course on vibrations, and should be considered as an alternative to Ch. 5. The material rightfully belongs in a first year graduate course.

Chapter 8. Distributed-Parameter Systems: Exact Solutions - In Sec. 8.1, the equations of motion for a set of lumped masses on a string are first derived by the Newtonian approach and then transformed in the limit into a boundary-value problem for a distributed string. The same boundary-value problem is derived in Sec. 8.2 by the extended Hamilton principle. In Sec. 8.3, the boundary-value problem for a beam in bending is derived by both the Newtonian approach and the extended Hamilton principle. Sections 8.4-8.8 are devoted to the differential eigenvalue problem and its solution. Rayleigh's quotient is used in Sec. 8.9 to develop the variational approach to the differential eigenvalue problem. The response to initial excitations and external excitations by modal analysis is considered in Secs. 8.10 and 8.11, respectively. A modal solution to the problem of a rod subjected to a boundary force is obtained in Sec. 8.11. The wave equation and its solution in terms of traveling waves and standing waves are introduced in Sec. 8.12, and in Sec. 8.13 it is shown that a traveling wave solution matches the standing waves solution obtained in Sec. 8.11.

Sections 8.1-8.5, 8.9 and 8.10 are suitable for a senior course or a first-year graduate course on vibrations. The balance of the chapter belongs in a second-year graduate course.

Chapter 9. Distributed-Parameter Systems: Approximate Methods - Sections 9.1- 9.4 discuss four lumped-parameter methods, including Holzer's method and Myklestad's method. The balance of the chapter is concerned with series discretization techniques. Section 9.5 presents Rayleigh's principle, which is the basis for the variational approach to the differential eigenvalue problem identified with the Rayleigh-Ritz method, as expounded in Secs. 9.6-9.8. Sections 9.9 and 9.10 consider two weighted residuals methods, Galerkin's method and the collocation method, respectively. A MATLAB program for the solution of the eigenvalue problem for a nonuniform rod by the Rayleigh-Ritz method is provided in Sec. 9.11.

The material is suitable for a senior or a first-year graduate course on vibrations, with the exception of the second half of Sec. 9.6 and the entire Sec. 9.7, which are more suitable for second-year graduate course.

Chapter 10. The Finite Element Method - Section 10.1 presents the formalism of the finite element method. Sections 10.2 and 10.3 consider strings, rods and shafts in terms of linear, quadratic and cubic interpolation functions. Then, Sec. 10.4 discusses beams in bending. Estimates of errors incurred in using the finite element method are provided in Sec. 10.5. In Sec. 10.6 and 10.7, trusses and frames are treated as assemblages of rods and beams, respectively. Then, systems response by the finite element method is discussed in Sec. 10.8. A MATLAB program for the solution of the eigenvalue problem for a nonuniform pinned-pinned beam is provided in Sec. 10.9.

This chapter is suitable for a senior or a first-year graduate course on vibrations, with the exception of Sec. 10.3, which is optional, and Secs. 10.6 and 10.7, which are more suitable for a second-year graduate course.

Chapter 11. Nonlinear Oscillations - Sections 11.1-11.3 are concerned with qualitative aspects of nonlinear systems, such as equilibrium points, stability of motion in the large. Section 11.4 discusses the van der Pol oscillator and the concept of limit cycle. Sections 11.5-11.7 introduce the perturbation approach and how to obtain periodic perturbation solutions by Lindstedt's method. Using the perturbation approach, the jump phenomenon is discussed in Sec. 11.8, subharmonic solutions Sec. 11.9 and linear systems with time-dependent coefficients in Sec. 11.10. Section 11.11 is devoted to numerical integration of differential equations of motion by the Runge-Kutta methods. A MATLAB program for plotting trajectories for the van der Pol oscillator is provided in Sec. 11.12.

The material is suitable for a senior or a graduate course on nonlinear vibrations.

Chapter 12. Random Vibrations - Sections 12.1-12.3 introduce such concepts as random process, stationarity, ergodicity, mean value, autocorrelation function, mean square density functions. Properties of the autocorrelation function are discussed in Sec. 12.6. Sections 12.7-12.11 are devoted to the response to random excitations using frequency domain techniques. Sections 12.12-12.15 are concerned with joint properties of two random processes. The response of multi-degree-of-freedom systems and distributed systems to random excitations is discussed in Secs. 12.6 and 12.17, respectively.

The material is suitable for a graduate course on random vibrations.

Appendix A. Fourier Series - The material is concerned with the representation of periodic functions by Fourier series. Both the real form and the complex form of Fourier series are discussed.

Appendix B. Laplace Transformation - The appendix contains an introduction to the Laplace transformation and its use to solve ordinary differential equations with constant coefficients, such as those encountered in vibrations.

Appendix C. Linear Algebra - The appendix represents an introduction to matrices, vector spaces and linear transformations. The material is indispensable to an efficient and rigorous treatment of multi-degree-of-freedom systems.

In recent years, computational algorithms of interest in vibrations have matured to the extent that they are now standard. Examples of these are the QR method for solving algebraic eigenvalue problems and the method based on the discrete-time transition matrix for computing the response of linear systems. At the same time, computers capable of handling such algorithms have become ubiquitous. Moreover, the software for the implementation of these algorithms has become easier to use. In this regard, MATLAB must be considered the software of choice. It is quite intuitive, it can be used interactively and it possesses an inventory of routines, referred to as functions, which simplify the task of programming even more. This book contains fourteen MATLAB programs solving typical vibrations problems; they have been written using Version 5.3 of MATLAB. The program can be used as they are, or they can be modified as needed, particularly the data. In addition, a number of MATLAB problems are included. Further information concerning MATLAB can be obtained from:

The Math Works, Inc.

3 Apple Hill Drive

Natick, MA 01760

It should be stressed that the book is independent of the MATLAB material, and can be used with or without it. Of course, the MATLAB material is designed to enhance the study of vibrations, and its use is highly recommended.

The author wishes to express his appreciation to William Atherton, Cleveland State University; Amr Baz, University of Maryland; Robert Lipp, University of New Orleans; Hayrani Oz, Ohio State University; and Alan Palazzolo, Texas A&M University; for their extensive review of the manuscript and their many useful suggestions. He also wishes to thank Timothy J. Stemple, Virginia Polytechnic Institute and State University, for producing the computer-generated figures and for reviewing an early version of the manuscript. Special thanks are due to Ilhan Tuzcu, Virginia Polytechnic Institute and State University, for his major role in developing the MATLAB programs, as well as for his thorough review of the manuscript. Last but not least, the author would like to thank Norma B. Guynn for typing the book essentially as it appears in its final form; the book places in evidence the excellent quality of her work.

Leonard Meirovitch

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