Machine dynamics is the fundamental basis for optimal machine design. Based on dynamic analysis of the system special control, damping and counterbalance arrangements can be foreseen, which would simplify operations, adjustment and maintenance during the lifetime of the machine. According to the authors’ preface, the aim of writing the book has been to provide practitioners and graduate students with the basic principles of contemporary machine dynamics with emphasis on the qualitative understanding of machine operations. Accordingly, the approach of the book is analytical rather than numerically oriented with the mathematical analysis kept at a minimum.

Dynamic equations of machines with a single driver and a single mechanism (two-degree-of-freedom systems) are thoroughly analyzed in steady state and transient operation modes. The non-linear equations of motion are derived based on classical analytical (Lagrangian) mechanics. The level of generalization of the formulation is relatively high. Illustrative examples representing the derivation of some simple machines with or without compliance would have been useful. The equation of motion is generalized to multi-degree-of-freedom chained systems. The analytical treatment of the equations of motion is based on a $1st$ order perturbation technique, where the $0th$ and $1st$ order problems are solved by classical linear modal analysis.

The book provides an instructive treatment of the principles for control of unintended machine vibrations and performance error reduction. Both passive (tuned mass dampers) and active (feed-back) control strategies are outlined for a number of specific case studies. The approach to control theory is the classical one (cascade of single-input single-output control elements, stability criterion based on Nyquist plots, etc.), although the particular dynamic features of digital control are emphasized. Modern approaches for multi-degree-of-freedom systems (optimal control, adaptive strategies, etc.) based on state vector formulations are not dealt with.

Because a variety of control problems have been worked, the book may be useful for practitioners with basic knowledge in structural dynamics in analyzing machine dynamics and associated control problems. However, it can hardly stand alone as a postgraduate textbook in the field. First, the number of illustrative examples is relatively small, especially in the first half of the book, and exercises are totally lacking. Secondly, the book is based on concepts developed by the authors, previously published in Russian sources. This is reflected by the bibliography, which is dominated by older Russian references. Here an updated literature review of more recent developments in the field would be desirable. Finally, the English translation deviates in several cases from common practice in mechanics. For example, the principle of virtual work is referred to as the *equation of elementary work*, and perturbation analysis is designated as the *method of successive approximations*.

## Theory of Vibro-Impact Systems and Applications

There are many good books on mechanical vibrations, and on impact dynamics. But this seems to be the only monograph in English dealing thoroughly with *vibro-impact* systems and processes. Such processes involve colliding elements, and vibrations with abrupt changes in velocity and force during each cycle. Applications include devices to crush, grind, forge, rivet, drill, punch, tamp, print, tighten, pile, cut, and surface treat a variety of materials and objects, at frequencies ranging from sub-Hz to ultrasonic. Typically, to minimize input force and energy, these systems are driven at resonance, and utilize impacts to maximize output force.

How can such processes be described, quantified, modeled, analyzed, controlled, suppressed, enhanced, and synthesized? Prof. Babitsky’s *Theory of Vibro-Impact Systems and Applications* answers these questions with unquestionable authority, reflecting the author’s position as a leading expert and main contributor in the field. The present edition is a translation of the similar Russian title from 1978. According to its preface no efforts have been made to revise or adapt the original work, other than to append an updated bibliography, two new sections, and a short appendix. Nevertheless, the book appears timely, and probably will appear refreshingly new and inspiring to many non-Russian readers in academia and industry.

There are five chapters, two bibliographies, and two appendices. Chapter 1 gives a summary of basic collision theory, and shows how exact methods based on kinematic impact theory and stitching can be used to analyze simple vibro-impact systems with uni- or bilateral stops. The special character of resonance in vibro-impact systems is discussed, and basic vibro-impact technical devices are described.

Chapter 2 introduces quite general model systems, characterized by vibro-impact forces and arbitrary excitation. The linear parts of the equations of motion are described using dynamic compliance operators, thus unifying the notion for lumped and distributed parameter systems. Here exact methods are inapplicable or inconvenient. Instead equivalent linearization of the nonlinear impact force characteristic is used, corresponding to employing harmonic balancing with a single harmonic term. This gives reasonably accurate results for systems whose linear part displays strong filtering properties, so that the influence from excluded harmonics is not essential. This approach seems justified for the many weakly damped vibro-impact devices operating at or close to resonance, where the filtering assumption is fulfilled. Ready-to-use formulas are provided for the equivalent (harmonic and statistical) linearisation of basic types of collision links, in the presence of harmonic or random excitation.

Chapter 3 presents specific single-degree-of-freedom models with vibro-impact, considering stability, transients, and parametric, self-sustained, and random vibrations. Emphasis is laid on calculating, presenting, and discussing frequency responses and the different phenomena they reveal. In Chapter 4 these developments are extended to general multi-degree-of-freedom systems, and applied to analyze vibro-impact processes involving axially or transversely vibrating beams.

The final Chapter 5 shows how to use the theory, laid out in previous chapters, for synthesizing controllers to achieve optimal motions of self-sustaining vibro-impact systems in the presence of external disturbances.

The first bibliography contains 227 references from 1978 and earlier, almost exclusively in the Russian language. The second bibliography adds 118 newer references, many in English. Then follows a subject index, rather short, but seemingly well thought out and adequate.

Appendix I considers a simple model for a particle impacting a viscoelastic limiter, deriving exact results and comparing these to experimental results and to numerical simulation. This interesting piece would fit more naturally in the first three chapters; at least the placement in the appendix seems unmotivated.

Finally, Appendix II is a reprint of a journal article, explaining the basic principle of vibro-impact, and how vibro-impact processes can be controlled. It serves as a readable and interesting introduction to the whole field covered by the monograph. Thus it would be natural to merge it into Chapter 1.

The presentation is generally well balanced and structured, progressing from a basic level to complex theory and applications. The book is reasonably self-contained—given a background in basic vibration theory, mechanics, and engineering mathematics—though experience with nonlinear systems and vibrations will certainly facilitate the assimilation process. Many readers will recognize the style as classical Russian: Concise, with a heavy use of mathematics, and a balanced consideration to simple illustrative systems, generalized systems, and practical applications. Though, the lack of helpful redundancy, combined with a tendency to document (rather than communicate) the developments in detail, also makes many sections unnecessarily strenuous to read for newcomers in the field.

To non-Russian readers the book may appear somewhat detached and isolated from the literature they already know, or can acquire and read. They will want to know present state-of-the-art of the field, but are left unsure whether they are rather presented with a picture of Russian state-of-the-art in the 1970s. There might be little difference, but the point is that readers are not explicitly told. Prof. Babitsky, now in England, has a unique position and background to do this, and hopefully it will happen in a future revised edition, integrating and reflecting the last 25 years of developments worldwide.

Many monographs on mathematics, physics, and mechanics translated from Russian have evolved into highly esteemed reference works in the west, and remained so decades after their initial publication. Babitsky’s book has all opportunities to share this destiny. Presently there seems to be no competitors.

This monograph is unavoidable for researchers, advanced students, and professionals working with vibro-impact systems. It provides a theoretical foundation for the understanding and practical utilization of the combined effect of resonant vibrations and impact. Full of interesting examples, it will be relevant reading also for a broader audience interested in vibration theory and applications, nonlinear dynamics, impact processes, and machine dynamics.

## Analytical Mechanics

This monumental treatise on analytical mechanics is a translation of the original Russian edition from 1961. With its some 850 pages it contains a wealth of material on the classic theory of mechanics, and methods of formulating the equations of motion of mechanical systems. This book is from the same period as Goldstein’s *Classical Mechanics* and Pars’ *Analytical Dynamics*, and of approximately the same volume. However, the present book has more focus on the analysis of the equations and the formulation of suitable equations of motion, and pays more attention to mechanical problems in engineering such as e.g. the use and function of gyroscopes and motion related to robotics. The style is analytical with emphasis on deriving special and useful forms of the equations of motion, mainly for systems of rigid bodies. Many examples of practical interest are worked through in detail—illustrating the theory, as well as providing useful results.

The book contains twelve chapters and two appendices, a presentation of basic properties of matrices and a very readable introduction to tensor calculus. The first three chapters are concerned with the kinematics of rigid body motion described in terms of generalized coordinates. The concept of quasi-velocities—i.e. nonintegrable linear combinations—is introduced early on and are followed up throughout most of the book, particularly with a view to formulation of problems with nonholonomic constraints. The description of rigid body motion covers several variants of the Euler angles, the “rotation vector” defined by the tangent to half the rotation angle, and the scalar form of quaternion parameters, here termed the Rodrigues-Hamilton parameters.

The following two chapters introduce the central concepts of kinetic energy, and work and potential energy. The dissipation function is introduced in general form, permitting nonlinear dependence on the speed of the particles.

Chapters 6 to 8 constitute a core part of the book. Chapter 6 deals with the “fundamental equation of dynamics,” which essentially is the statement of virtual work under various constraint conditions. Also here the discussion includes quasi-velocities. Finally we arrive at Lagrange’s equations in Chapter 7. The equations are derived, and the Jacobi, or energy, integral is discussed. This is followed by very readable account of the geometric structure of the equations of motion, with reference to the appendix on tensor calculus, but fairly self-contained. Cyclic coordinates are discussed as well as the Routhian function, in which the explicit dependence on these coordinates is eliminated by a Legendre transformation. In Chapter 8 the Lagrange equations are generalized to the so-called Euler-Lagrange form in terms of quasi-velocities. Additional terms appear in the equations, but the kinetic energy can be formulated directly in terms of nonintegrable quasi-velocities. An alternative form is the so-called Gibbs-Appell equations, where the generalized forces are equal to the derivatives of the so-called acceleration energy. A number of examples demonstrate the general formulation.

After a chapter on relative motion the theory of Legendre transformations, canonical equations and the associated perturbation theory are treated in Chapters 10 and 11. The final chapter of more than 100 pages is devoted to the variational principles of mechanics. General principles of variational calculus are discussed together with the stationary action integrals of Hamilton, Lagrange and Jacobi. This chapter also contains a brief discussion of the equations of motion of distributed systems, notably cables and rods.

This book contains an overwhelming amount of material, and makes good background reading for anyone with an interest in the classical foundations of mechanics. The gradual build up to the general equations and principles, supplemented with many examples, prepares the reader well, but also makes it difficult to read only the latter parts of the book.

## Mechanical Vibrations

This book is a traditional undergraduate mechanical vibration text covering approximately the same range of topics as several other books intended for use in introductory courses. What distinguishes this work, now in its fourth edition, is the depth and uniformity of its exposition. In nearly eleven hundred pages, numerous subjects in vibration theory and practice are introduced, developed, and demonstrated through examples.

Chapter 5 is concerned with 2-DOF systems, and addresses the choice of coordinates and potential coupling of the equations of motion. The matrix formulation favored there is extended to multi-DOF systems in Chapter 6, which is rather more comprehensive than might be expected in a text of this level; for example, equations of motion are developed using both Lagrange’s equations and Newton’s second law. Several techniques for the calculation of natural frequencies and modes are covered in Chapter 7, although not at the depth of the earlier chapters. Chapter 8 examines the vibration of continuous systems, mostly through a sequence of one-dimensional examples (although a section is devoted to the vibration of membranes). The Rayleigh and Rayleigh-Ritz methods are applied to example problems at the end of this chapter.

While the first eight chapters of this book are often exhaustive in their treatment of their respective topics, the remaining six, comprising about a third of the text, tend toward summaries of broader areas. Chapter 9, entitled “Vibration Control,” covers passive techniques of balancing, isolation and absorption in some depth. Vibration measurement is the subject of Chapter 10, which describes equipment for vibration excitation and response measurement and gives a very readable, if brief, summary of some experimental methods. Several historically important techniques for the numerical integration of differential equations of motion are examined in Chapter 11. Finite element analysis is introduced in Chapter 12, which is limited to one-dimensional structures such as beams and shafts. The body of the text is concluded by Chapters 13, an introduction to nonlinear vibration, and 14, an introduction to random vibration, both of which contain numerous definitions and examples at a level consistent with the rest of the book. Six brief appendices provide supplementary information and may be useful for review.

Noteworthy features of this text include a large number of example problems with detailed solutions, references listed by chapter, and numerous exercises, some identified as design problems. Many examples of numerical solutions using Matlab, $C++$ or Fortran are given (although source code for most of these evidently will be available only via the book’s web site, which was incomplete at the time of this review). Each chapter begins with a brief biography of a figure from the history of mechanics or mathematics. At the end of each chapter are well-chosen review questions with which a student may check his or her understanding of fundamental ideas.

This book is nearly comprehensive at the advanced undergraduate level. The instructor who wishes to cover a particular set of topics in an introductory vibration course will likely find plenty of material here. The mathematical level of the text does require a certain degree of preparation, and the later chapters are not self-contained, but as a course text or as a reference for someone beginning advanced study in the field this book would be an excellent choice.

## Passive Vibration Isolation

The basic principles of vibration isolation have been known for centuries and this topic is covered in every vibration text, although rarely thoroughly and not always from a realistic viewpoint. Not since the publication of C. E. Crede’s 1951 classic *Vibration and Shock Isolation* has there appeared a book that treats all of the important aspects of this subject. The present book does this and more; it summarizes the current state of the technology and also discusses the salient characteristics of the available hardware and materials.

*Passive Vibration Isolation* presents a rather comprehensive treatment of its subject matter, including analysis of basic systems and their dynamic processes, criteria for isolation of many sensitive and vibration-producing items, materials employed in isolators, and the design of isolators and isolation systems. As its title implies, the book does not address active vibration isolation.

The first of the book’s four chapters deals with the basics of isolation of rigid bodies, including consideration of all six degrees of freedom. It presents the relevant equations, but dwells more extensively on simpler situations and practical real-life applications. It also addresses two-stage isolation, nonlinearity effects, wave effects in isolators, and isolation in the presence of random excitation and of shock excitation.

The second chapter begins with a discussion of isolation of vibration-sensitive items. It puts forth vibration limits for a number of instruments and machines and for some of these items also lists the inertial properties one would need to design an isolation system. It indicates the selection criteria for suitable isolation systems and discusses the practical selection of isolators. It includes consideration of the effects of motions induced by the isolated items and of nonrigidity of the isolated item. It also touches briefly on vibration protection of buildings. Having addressed isolation of sensitive items, the chapter continues similarly with isolation of vibration-producing items, also taking into account non-rigidity of supporting structures. Finally, it provides guidelines for the practical isolation of commonly encountered machines that are neither particularly sensitive nor the sources of considerable vibrations, and it includes a section on mounting of engines and machinery in vehicles.

The third chapter deals with isolator elements. It describes metal springs of various kinds and presents their design equations, and it deals similarly with a variety of damping elements. It discusses the dynamic behaviors of elastomeric materials and components in considerable detail, and also addresses fibrous and wire-mesh mesh materials, high-damping metals, and pneumatic isolators.

The final chapter focuses on practical isolation means, beginning with isolating mats and pads, then proceeding to isolators with rubber elements, including a discussion of the generally under-appreciated constant-natural-frequency isolators. It continues with coil spring isolators and with isolators incorporating wire mesh and cables. It describes low-stiffness isolators that include buckled elements and a variety of pneumatic isolation systems. It concludes with sections that deal with means for avoiding short-circuiting of isolation by piping and cables and with power transmission couplings.

Each chapter includes an extensive list of references, many of which were written by the author. The reference lists include a considerable number of older publications in Russian, as well as some of the newest technical and trade literature, reflecting the author’s vast background and experience. The little extra patience and careful study one needs to exert in order to cope with the book’s dense packing of theoretical information and quantitative practical data, coupled with some slightly awkward English, generally is well worthwhile.

This book represents an invaluable resource for engineers concerned with the design and selection of isolation systems and isolators. It deserves thorough study and belongs in the library of every vibration engineer.

Aalborg University, Denmark

Technical University of Denmark jjt@mek.dtu.dk

Technical University of Denmark

University of Illinois at Urbana-Champaign

Acentech, Incorporated, Cambridge, MA