7R12. Vibration of Strongly Nonlinear Discontinuous Systems. Foundations of Engineering Mechanics. - VI Babitsky (Dept of Mech Eng, Louborough Univ, Louborough, Leicestershire, LE11 3TU, UK) and VL Krupenin (Inst of Machine Stud, Russian Acad of Sci, Moscow, 101830, Russia). Springer-Verlag, Berlin. 2001. 399 pp. ISBN 3-540-41447-9. \$99.00.

Reviewed by J Angeles (Dept of Mech Eng and Center for Intelligent Machines, McGill Univ, 817 Sherbrooke St W, Montreal, PQ, H3A 2K6, Canada).

Strongly nonlinear systems are defined by the authors well into the body of their book, on page 76—while the Index has a pointer to page 34, no mention of the term is found in that page—but this definition is limited to conservative systems. Then a whole section, 4.6, is devoted to this kind of system, which, again, is limited to systems with excitation derived from a potential. Special attention is given in this section to systems of the threshold type. Briefly stated, to understand what a “strongly nonlinear mechanical system” is, the reader must be aware of what a “weakly nonlinear system” is. One thus must conclude that a strongly nonlinear system is one whose nonlinear terms cannot be neglected or approximated to a first order, without losing the essence of the system behavior. The discontinuous nature of the nonlinearity, moreover, prevents the linearization of the nonlinear term.

The book is authoritative, but of a rather limited scope. The main motivation of the authors is the study of what they call “vibro-impact systems,” ie, systems excited by impact loading, as those found in machinery for rock-crunching. However, the authors fail to include examples of a much broader scope and modern interest. Such systems occur in gear transmissions due to the unavoidable backlash of gears. The study of impact in these systems is of current research interest, with books devoted to their study, eg, that by Pfeiffer and Glocker (1996). These systems show, in some instances, a chaotic behavior, which is a recognized source of noise in automotive transmissions. Other instances where impact loading leads to serious performance deterioration is found in mechatronic systems, where frequent velocity reversals make backlash a source of discontinuous perturbations. Moreover, systems with discontinuous characteristics have been studied for some time, within a system-theoretic context, eg, (Flu¨gge-Lotz, 1968), but the authors appear too focused on the Russian literature to cite the rich literature in English on the topic.

The authors focus on methods developed by themselves, apparently, in the seventies. In fact, the book is admittedly the English translation of an original book in Russian, but the authors do not disclose the bibliographical information of that original. Most likely, the original book was published in the early eighties. The methods favored by the authors are of the paper-and-pencil type—some would say “analytic,” but this qualifier is, in this reviewer’s opinion, a misnomer—and hence, limited to models that are crude approximations of actual systems. These methods are of academic interest, but it is dangerous to overstate their relevance in light of contemporary tools for simulation, such as Matlab’s Simulink Toolbox. The book would have a more permanent value if it had taken into account modern software and hardware for scientific computations. Not a word is said of algorithms for simulation, which have been developed first and foremost to cope with nonlinear systems, whether with weakly or with strongly nonlinear features. In this vein, it is worth citing Strang (1988): “Solving a problem no longer means writing down an infinite series or finding a formula like Cramer’s rule, but constructing an effective algorithm.”

In coping with the periodic response of systems, the authors hint to an algorithm for solving the integral Fredholm equation thus resulting. However, they mislead the reader into believing that the solution of the underlying system of N linear equations in N unknowns, “may be solved using the [sic] Cramer’s rule.” In fact, Cramer’s rule is inapplicable in practice, by virtue of the combinatoric nature of the number of floating-point operations it requires. This number, roughly $N+1!,$ becomes prohibitively large for even moderate values of N, like N=25.

Technical issues apart, the layout of the book is unusual, to say the least. For starters, the book is divided into three chapters and 10 sections, with a continuous section numbering across chapters. Moreover, while the equation display is highly acceptable—an unusual feature for books typeset in Word—some pages lack right justification, which makes the layout look weird. The bibliography is admittedly limited to the essential references. However, there is a discontinuity here, for the bibliography, made up of 204 references, is limited and outdated—the former because, with counted exceptions, the bibliography includes only Russian works; the latter because the most recent entry is of 1983. To compensate this, the book includes an “Additional Bibliography,” of 40 additional entries, with some works published outside of Russia. These references appear to be listed without citation, although this feature is difficult to verify, mainly because the reference list follows an awkward ordering: the first reference cited is entry [49], followed by [56], and so on. If a numbering of references is to be followed, then each reference should be numbered in order of citation!

The figures, in turn, appear without a caption, and composed figures, with parts referred to as A), B), etc, sometimes lack their appropriate label.

Otherwise, the English is not standard. For example, what is known as the “impulse response” of a linear system, is referred to as the system “Green function.” This is not technically wrong, but looks unfamiliar, especially because Green functions, in the English literature, usually pertain to the solutions of boundary-value problems in a space domain, more so than to solutions of initial-value problems in the time domain. Then, what is known in English as “rational functions” are labeled in the book as “meromorphic functions.” A search of the term in Maple, for example, under “topic,” led to a bulky number of entries, none of which bears the “meromorphic” qualifier. A few such entries bear the name “rational,” however. It would have been convenient to have the English translation proofread by a specialist with knowledge of the usual terminology in English.

All in all, it is difficult to recommend this monograph as a valuable reference of contemporary relevance.