# Mathematics of random phenomena: Random vibrations of mechanical structures

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<ul><li><p>BOOK REVIEWS 195 </p><p>of the gravitational physics are the best parts of the book. The principle of equivalence is used in order to define a metric in the presence of a gravitational field via the local geodesic frames. It is interesting to point out that by using the same approach one can actually define metric connection on an arbitrary (pseudo-) Riemannian manifold (cf. R. D. Richtmyer, Principles of Advanced Mathematical Physics, vol. 2, Springer-Verlag, 1981). Further, the book contains neat reports on Schwartzehild metric, gravitational waves, cosmological models, and other related topics. The opening chapters are devoted to an accurate treatment of the special theory including Maxwell's equations and the energy- momentum tensor. </p><p>Goodbody's book is an interesting mixture of three-dimensional linear algebra and theoretical mechanics. It fills a psychological gap between the 'abstract' linear algebra and dynamics via the systematic use of tensor products (of vectors). How many (under)graduates know that the tensor products of vector spaces have something to do with, say, inertia tensor? The author shows convincingly that the simple formula (a(~b) .c= a(b, c) (a definition of the decomposable second-order Cartesian tensor!) can be successfully used for concrete calculations in mechanics, fluid mechanics, and elasticity theory. The book starts with a definition of three-dimensional vectors and proceeds to the calculations of inertia tensors for various solid bodies, analysis of fluid motion (including a derivation of the Navier-Stokes equation), and to the discussion of linear elasticity theory. Interestingly enough, the introductory 'algebraic' chap- ters naturally comprise such topics as vector fields and 'rate of change of vector in rotating frame of reference' (p. 63). </p><p>And finally, one general remark. Many scientists now share a common nostal- gia for the good old (Newtonian?) times when mathematics was a part of science. This good feeling begins to influence the university curricula as well. This means in particular, that (Einsteinian?) 'physical geometry' will sooner or later become a part of the primary education for any student of mathematics. Both books under review may be helpful in the development of a relevant standard course for undergraduates. </p><p>Riga, U.S.S.R. A.H. KUSHKULEI </p><p>Paul Kr~e and Christian Soize: Mathematics of Random Phenomena: Random Vibrations of Mechanical Structures, D. Reidel, Dordrecht, 1986, $98, 69.75 Dfl. 210,-. </p><p>This is a curious book. It is a well-organized, advanced presentation of most of the basics needed for the analysis of mechanical systems subject to forced random oscillations. </p><p>The first three chapters contain a crash-course in stochastic processes, in the fourth the tools for writing down the solutions of the equations of motion of a </p><p>Acta Applicandae Mathematicae 11 (1988) </p></li><li><p>196 BOOK REVIEWS </p><p>system of coupled damped oscillators in terms of the random forcing term are spelled out, and in the fifth chapter the techniques for analysing the random response are presented. The basic issue to which the material in this chapter applies is related not only to the testing of the models but also to problems like the determination of safety margins. </p><p>The four chapters of applications are there basically to show the need for the study of the topics in probability that comprise the rest of the book. They describe the relevant problems arising in modeling randomness coming from the three natural sources: aeolic, marine, and telluric. Besides that, there is a chapter on mechanical fatigue under random loading and a chapter containing the analysis of nonlinear systems in terms of multivalued (stochastic) differential equations. </p><p>Why did I say the book is curious? Well, because relative to the presentation of stochastic processes in the third part of the book, there is something missing in the list of examples: a truly infinite-dimensional problem. For example, a con- tinuous system for which the analysis of a few relevant normal modes would not be adequate. </p><p>It would be good if the authors, or somebody else, would write another volume, equally well organized, addressed to the problems of stability, resonances, etc. </p><p>Do not get annoyed at the few misprints nor at the few hard-to-read left-most columns of letters on several pages. </p><p>The book is quite a nice addition to the series of Mathematics and its Applications. Do not let your library miss it. </p><p>Universidad Central de Venezuela, Caracas, Venezuela </p><p>HENRYK GZYL </p></li></ul>

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