**Element of Vibration Analysis**

Leonard Meirovitch

McGraw-Hill, 1975

x + 495 Hal.

Rp. 54,000 ,-

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**PREFACE**

The most important single factor affecting recent trends in the field of vibrations **has **been the electronic computer revolution. Indeed, the ability to perform routine computations with incredible speed has been behind the development of relatively new methods of analysis, such as the finite element method, as well as of new computational algorithms for efficient handling of matrices of high dimensions. On the other hand, it has rendered obsolete many methods, including certain graphical procedures.

The ability to solve increasingly sophisticated problems has led to new demands for rigor in analysis. This text addresses itself to this task by adopting an approach that is as mathematically rigorous as possible, while attempting to provide a large degree of physical insight into the behavior of systems. The text covers a broad spectrum of subjects, including matrix methods for discrete systems, various discretization procedures for continuous systems, rigorous qualitative and quantitative treatment of nonlinear oscillations, and statistical analysis of random vibrations. Notwithstanding this breadth, an attempt has been made not to sacrifice depth. Special emphasis has been placed on formulations and methods of solution suitable for automatic computation.

This book is intended as an up-to-date text fora one-year course in vibrations beginning at the junior or senior level. A certain amount of more advanced material has also been included, making the book suitable for a senior elective or a beginning graduate course on dynamics of structures, nonlinear oscillations, and random vibrations. The position of courses in vibrations in engineering curriculums has never been defined clearly. Whereas some curriculums require an elementary course at the junior level and offer an elective course at the senior level, others offer the first course on vibrations as a dual-level course (open to seniors and graduate students), and still others regard vibrations as strictly graduate material. In recognition of this, the text has been designed to cover material from the very elementary to the more advanced ‘in increasing order of difficulty. Moreover, relatively advanced material has been placed at the,end of certain chapters and, can be omitted on a first reading.

To help the instructor in tailoring the material to his needs, the book is reviewed briefly. In the process, the various levels of difficulty are pointed out.

Chapter 1 is devoted to the free vibration of single-degree-of-freedom linear systems. This is standard material fora first course in vibrations.

Chapter 2 discusses the response of single-degree-of-freedom linear systems to external excitation in the form of harmonic, periodic, and nonperiodic forcing functions. The response is obtained by classical and integral transform methods. A large number of applications is presented. If the response by integral transform methods is not to be included in a first course in vibrations, then Sections 2.11 to 2.14 can be omitted.

Chapter 3 is concerned with the vibration of two-degree-of-freedom systems. The material is presented in a way that makes the transition to multi-degree-of-freedom systems relatively easy. The subjects of beat phenomenon and vibration absorbers are discussed. The material is standard fora first course in vibrations.

Chapter 4 presents a matrix approach to the vibration of multi-degree-of-freedom systems, placing heavy emphasis on modal analysis. The methods for obtaining the system response are ideally suited for automatic computation. The material is suitable fora senior level course. Sections 4.8 to 4.13 can be omitted on a first reading.

Chapter 5 is devoted to exact solutions to response problems associated with continuous systems, such as strings, rods, shafts, and bars. Again the emphasis is on modal analysis. The intimate connection between discrete and continuous mathematical models receives special attention. The material is suitable for seniors.

Chapter 6 provides an introduction to analytical dynamics. Its main purpose is to present Lagrange’s equations of motion. The material is a prerequisite for later chapters, where efficient ways of deriving the equations of motion **are **necessary.

Chapter 7 discusses approximate methods for treating the vibration of continua for which exact solutions are not feasible. Discretization methods based

on series solutions, such as the Rayleigh-Ritz method, and lumped methods are presented**. **The material is suitable for seniors.

Chapter 8 is concerned with the finite element method, a relatively new method for structural dynamics. The material is presented in a manner that can be easily understood by seniors. Moreover, the same concepts can be used to apply the method to two- and three-dimensional structures.

Chapter 9 is the first of two chapters on nonlinear systems. It is devoted to **s**uch qualitative questions as stability of equilibrium. The emphasis is on geometric description of the motion by means of phase plane techniques. The material is suitable for seniors or first-year graduate students, but Sections 9.6 and 9.7 can be omitted on a first reading.

Chapter 10 uses perturbation techniques to seek quantitative solutions to response problems of nonlinear systems. Several methods are presented, and phenomena typical of nonlinear systems are discussed. The material can be taught in a senior or a first-year graduate course.,

Chapter 11 ‘is devoted to random vibrations. Various statistical tools are introduced, with no prior knowledge of statistics assumed. The material in Sections 11.1 to 11.11 can. be included in a senior level course. In fact, its only prerequisites are Chapters 1 and 2, as it considers only the response of single-degree-of-freedom linear systems to random excitation. On the other hand, Sections 11.12 to 11.17 consider multi-degree-of-freedom and continuous systems and are recommended only for more advanced students.

Appendix A presents basic concepts involved in Fourier series expansions, Appendix B is devoted to elements of Laplace transformation, and Appendix C presents certain concepts of linear algebra, with emphasis on matrix algebra. The appendixes can be used for acquiring an elementary working knowledge of the subjects, or for review if the material was studied previously.It is expected that the material in Chapters 1 to 3 and some of that in Chapter 4 will be used for a one-quarter, first level course, whether at the junior or senior

level. For a first course lasting one semester, additional material from Chapter 4 and most of Chapter 5 can be included. A second-level course in vibrations has many options. Independent of these options, however, Chapter 6 must be regarded

as a prerequisite for further study. The choice among the remaining chapters depends on the nature of the intended course. In particular, Chapters 7 and 8 are suitable for a course whose main emphasis is deterministic structural dynamics. On the other hand, Chapters 9 and 10 can form the core for a course in nonlinear oscillations. Finally, Chapter 11 can be used for a course on random vibrations.

The author is indebted to many colleagues and students for proofreading the manuscript at various stages and for making extremely valuable suggestions. In this regard, special thanks are due to William J. Anderson, Robert A. Calico, Ching-.Pyng Chang, Earl H. Dowell, D. Lewis Mingori, and Harold D. Nelson, who patiently read the entire manuscript. Thanks are also due to Donald L. Cronin, Robert A. Heller, John L. Junkins, Manobar P. Kamat,

**LEONARD MEIROVITCH**

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