Introduction to Random Vibrations

N. C. Nigam The MIT Press, Cambridge, Massachusetts, 1983. 341pp. Price $35.00.

Introduction to Random Vibrations by N. C. Nigam is the most recent unified text on the subject of random vibration. It includes traditional mate- hal found in earlier introductions to the subject •'2 and unified texts 3-• as well as more recent developments. Among these recent developments is the treatment of nonstationary processes using the evolutionary spectral den- sity introduced by Phestley, 7-s and some matehal on first-passage times.

The first six chapters of the book, which constitute approximately one- half of its length, deal with the basic concepts of probability and random processes, including such topics as the mean rate of process crossings through a specified level and the mean number of process peaks per unit time. The remaining four chapters deal with the response of systems to random excitation.

Among the previous books dealing with random vibration, Nigam's book most closely resembles Lin's excellent text? This resemblance is par- ticularly close in the case of the matehal covered in the first five chapters of Nigam's book which closely parallel the first four chapters and Appendices I-IV of Lin's book. The titles of the first five chapters of Nigam's book are: "Introduction," "Probability Theory," "Theory of Random Vahables," "Theory of Random Processes," and "Some Important Random Pro- cesses." These five chapters are 106 pp. in length whereas the corresponding matehal in Lin's book occupies approximately 116 pp.

Included in these five chapters of Nigam's book are a set theoretic introduction to probability, definitions of probability density and disthbu- tion functions in one and several vahables, treatments of moments, charac- teristic functions, the Chebyshev inequality, and functions of random vaha- bles. Specific probability densities discussed include the binomial disthbution and Bernoulli thals, the single and multidimensional normal disthbutions, and tho log-normal and extreme value disthbutions. The chapter on random processes includes definitions of the stationary and er- godic properties, a discussion of processes with independent increments and the Wiener process, as well as discussions of continuity, differentiation, and integration of random processes. Normal or Gaussian processes are dis- cussed along with simple and compound Poisson processes and Markov processes. The Fokker-Planck equation is dehved and discussed.

Treatments of Bernoulli thals, the log-normal disthbution, extreme value disthbutions, processes with independent increments, and the Wiener process are not found in Lin's book; on the other hand, Lin's book includes a proof of the central limit theorem and a more complete discussion of the ergodic property.

The power spectral density ofstatioaary processes is dealt with in more detail in Lin's book than in Nigam's. However, Lin's treatment of nonsta- tionary processes is less detailed than Nigam's. When charactehzing non- stationary processes in the frequency domain, Lin uses the generalized spec- tral density 9 introduced by Bendat et al., which is a function of two frequency vahables, whereas Nigam for the most part uses the evolutionary spectral density TM which, thus far, has been used mainly for deschbing uni- formly modulated processes.

Chapter 6 of Nigam's book is entitled "Further Properties of Random Processes." It includes discussions of the number of crossings of a specified level, the number of peaks above a specified level, the fraction of time spent above a specified level, envelope statistics, first-passage times, and the maxi- mum value achieved by a process in a specified interval. This chapter covers approximately the same matehal as Chap. 9 of Lin's book which is entitled "Structural Failures Resulting from Dynamic Response and Related To- pics." Nigam's Chap. 6 contains a number of results for nonstationary pro- cesses, whereas Lin's Chap. 9 contains a section on cumulative fatigue dam- age.

Chapters 7 and 8 of Nigam's book treat the response of single and multiple degree-of-freedom discrete parameter systems to random excita- tions, whereas these same topics are treated in Chaps. 5 and 6 of Lin's book. Lin's treatment places somewhat more emphasis on structural models made

up of mass and stiffness elements and includes a section on the Holzer- Myklestad method for torsional and bending analyses. Nigam's emphasis is somewhat less tailored to structural systems.

Nigam's and Lin's books both contain chapters on continuou. s struc- tures. Both books deal with influence function and frequency domain char- acterizations of structures and both contain sections on the normal mode

method. The book by Nigam contains brief treatments of the Rayleigh-Ritz method and Galerkin's method. Both books contain treatments of the trans-

fer matrix method in the chapters on discrete parameter structures and continous parameter structures.

Both books also contain chapters on nonlinear structures. Both include sections on the Markov-vector approach to single and multiple degree-of- freedom structures, perturbation methods, and "equivalent" lineahzation methods. Nigam's book contains a section on iterative solutions of the inte- gral equation formulation of the Fokker-Planck equation and vahous other approximation methods.

The book by Nigam is 341 pp. in length and the book by Lin is some- what longer at 386 pp. although printed on thinner paper. Lin's book con- tain numerous footnotes citing original and additional sources of informa- tion on the vahous topics treated. Additional sources are cited in the text in Nigam's book with somewhat less consistency. Both books contain good indices.

Both books are probabilistic treatments of random vibration theory. Thus, neither book deals with fundamental statistical aspects of the subject such as the vahance of power spectrum estimates of ergodic Gaussian re- cords of finite length--a problem which cannot be avoided by the random vibration practitioner. The book by Newland 6 treats such topics together with digital spectrum analysis and the fast Fourier transform, but New- land's treatment of the remaining topics of random vibration covered by Lin and Nigam is far less complete. Statistical aspects of power spectrum esti- mation also are treated in the books by Bendat and Piersol. •ø'•

The books by Crandall and Mark 3 and Robson 4 are less comprehensive and less advanced than those by Lin and Nigam, but are useful introduc- tions to the subject of random vibration.

Dr. Nigam should be complimented for writing a useful addition to the subject of random vibration which includes a number of developments that have occurred since the appearance of Y. K. Lin's book.

WILLIAM D. MARK

Physical Sciences Bolt Beranek and Newman, Inc. Cambridge, Massachusetts 02238

•Random Vibration, edited by S. H. Crandall (MIT, Cambhdge, MA, 1958).

2Random Vibration, edited by S. H. Crandall (MIT, Cambridge, MA, 1963), Vol. 2.

3S. H. Crandall and W. D. Mark, Random Vibration in MechanicalSystems (Academic, New York, 1963).•

4j. D. Robson, An Introduction to Random Vibration (Edinburgh U.P., Ed- inburgh, and Elsevier, Amsterdam, 1964).

5Y. K. Lin, Probabilistic Theory of Structural Dynamics (McGraw-Hill, New York, 1967). Republished with corrections by Robert E. Krieger, Malabar, Florida, 1976.

6D. E. Newland, An Introduction to Random Vibrations and Spectral Anal- ysis (Longman, London, 1975). 7M. B. Priestley, "Evolutionary Spectra and Non-Stationary Processes," J. R. Statist. Soc., B 27, 204-237 (1965).

SM. B. Priestley, "Power Spectral Analysis of Non-Stationary Random Processes," J. Sound Vib. 6, 86-97 (1967).

9j. S. Bendat, L. D. Enochson, G. H. Klein, and A. G. Piersol, "Advanced Concepts of Stochastic Processes and Statistics for Flight Vehicle Vibra- tion Estimation and Measurement," ASD-TDR-62-973, Aeronautical Systems Division, Wright Patterson AFB, OH, 4-7 (1962).

•oj. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971).

•J. S. Bendat and A. G. Piersol, Engineering Applications of Correlation and Spectral Analysis (Wiley, New York, 1980).

1871 d. Acaust. Sac. Am. 76(6), Dec. 1984; 0001-4966/84/121871-01500.80; ¸1984 Acaust. Sac. Am.; Book Reviews 1871

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