1R6. Formulas for Dynamic Analysis. - RL Huston (Dept of Mech, Indust, and Nucl Eng, Univ of Cincinnati, 598 Rhodes Hall, Cincinnati OH 45221-0072) and CQ Liu (DiamlerChrysler, Auburn Hills MI). Marcel Dekker, New York. 2001. 624 pp. ISBN 0-8247-9564-4. $175.00.
Reviewed by J Angeles (Dept of Mech Eng and Center for Intelligent Machines, McGill Univ, 817 Sherbrooke St W, Montreal, PQ, H3A 2K6, Canada).
A collection of formulas for rigid-body dynamics without context would be meaningless, to say the least. Fortunately, the book under review goes beyond compiling such a collection. This book, in fact, provides a comprehensive view of the subject, starting from the very basics. While such a book would normally cater to a specialized readership with an engineering or science background, the authors include definitions as fundamental as those of time, distance, particle, etc, which one would normally assume are well known to this readership. The material in Chapter 1 thus appears somehow out of place for such a book; the untold reader, moreover, may be puzzled by some flawed definitions, like that of particle, which the authors base on the concept of body, not defined in this chapter but not until two chapters later, and only as pertaining to rigid body. The particle the authors intend to define, as understood in engineering circles, is a mechanical entity not necessarily small, for what is at stake here is the inability of this entity to change its attitude, and so, a particle can be the car of a train if the analyst is interested in studying only its vertical vibrations under pure translation. In fact, the authors admit this notion in Chapter 3—the reader has to go two chapters ahead to have an explanation of a definition that is first given in Chapter 1.
Chapter 2 is about elementary vector algebra and vector calculus, while Chapter 3 introduces the kinematics of particles. Here, the authors introduce the concept of rigid body, although the study of this entity is relegated to Chapter 6. It is unclear to this reviewer why the basic concept of angular velocity, proper of rigid bodies, is introduced in Chapter 3, although the authors come back to this concept in Chapter 6, without connecting the two discussions. Chapters 4 and 5 are devoted, respectively, to the kinetics and the dynamics of particles. Kinetics, as is recalled here, refers to the independent study of active and inertia forces, dynamics being the study of the interaction of these two kinds of forces. The downside of devoting one chapter to each of these two items is that the reader has to wait until Chapter 6, which the authors chose to call Kinematics of Bodies, to come across the subject that really matters to engineers, namely, rigid bodies. Furthermore, the authors claim in the opening paragraph of Chapter 6 that the chapter “focuses” on rigid bodies. The fact of the matter is that the authors do not treat other than rigid bodies in this chapter and in the balance of the book.
Rigid bodies, on the other hand, can be regarded either as aggregates of material points (particles) or as continua of matter; in either case, this material entity is constrained to move in such a way that the distance between any pair of its points (the continuum occupies a region of space, and hence, it also involves points) is preserved. The authors chose the traditional first approach, which is not flawless. In this approach, the reader has to believe that a summation of discrete terms can lead to an integral over a continuum. In the second approach, on the other hand, the formulation of dynamics is straightforward, for a whole body of knowledge is available, namely, that pertaining to continuum mechanics. In this context, Newton’s second law and Euler’s equation of balance of moments become natural derivations of Euler’s laws of conservation of momentum and of angular momentum, respectively, which are valid for any continuum, whether fluid or solid and, if the latter, whether deformable or rigid.
A special feature of the book is the extensive discussion on how to derive rotation matrices for different triplets of elementary rotations about the coordinate axes. Here, the authors include what they call configuration graphs, a mnemonic means of producing those matrices by using simple rules. This reviewer is not quite convinced of the usefulness of this extensive discussion. Along these lines, the authors go so far as to include a subsection with the heading “Computation Algorithms.” The problem is that in this half-page subsection, no single algorithm is spotted. Then, the discussion on motion classification has some problems: translation and rectilinear translation are discussed, along with plane motion and general plane motion. However, the authors overlooked a rational analysis of displacements in terms of group theory, as available in the archival literature , and in English in this reviewer’s monograph . In the same chapter, the terminology on screw motion is flawed. Indeed, in the most general case of rigid-body motion, no single point of the body remains fixed with respect to an observer (a coordinate frame essentially), whatever this observer may be. However, a set of points of the body can always be found whose points have a velocity, relative to that observer, of minimum Euclidean norm, all points lying on a line, which is termed the instant screw axis of the motion, all the points on this line having identical velocities. The authors wrongfully speak for this general motion of the existence of a center of rotation.
One plus of the book is the authors’ departure from the usual practice of resorting to what is known as quasi-coordinates to account for putative variables from which non-derivative quantities like the angular velocity is believed to derive upon time-differentiation. Here, as in the whole book, the authors faithfully follow Kane’s approach, who is a pioneer in this regard, and independent of quasi-coordinates and other esoteric quantities like pseudo-derivatives, notorious for plaguing the literature on the subject. Kane’s approach is, in fact, extremely enlightening in treating nonholonomic systems. In Chapter 10, addressing this subject, the authors follow literally an earlier paper (Passerello and Huston) , in which the authors introduced the nonholonomic constraints into the Lagrange equations of the system under analysis, with the purpose of reducing the number of governing equations to a set of independent second-order ordinary differential equations (ODE), free of constraint forces. In doing this, the authors faced the problem of solving for a number of unknown generalized velocities from a smaller number of linear algebraic equations. The way the authors got around this quandary was by adding identities in the same set of generalized velocities, thereby rendering their linear system determined, ie, with as many equations as unknowns, and nonsingular. In the process, the authors rendered their derivations unnecessarily cumbersome. Indeed, as Ostrovskaya and Angeles  demonstrated in 1998, there is actually no need to solve any system of equations in this analysis. It is a pity that the authors did not conduct a literature survey—the number of bibliography items that appeared in the last 10 years that are cited in this book is a mere 11. Eight of those are, in fact, either undergraduate textbooks or monographs; only three are archival publications and, of these, two are the first author’s own work. Chapter 7 includes additional formulas of rigid-body kinematics, while Ch 8 discusses the inertia properties of rigid bodies in great detail. Chapters 9 and 10 are devoted to rigid-body kinetics and dynamics, respectively. Chapter 11, in turn, includes various classroom-type of mechanical systems whose mathematical models are derived using the various formulations discussed in the book: d’Alembert’s Principle; Kane’s equations; and Lagrange’s equations. The last three chapters cover mechanical systems composed of multiple rigid bodies. On a side note, the typesetting could have been much better. Usual practice calls for italics in literals occurring in mathematical relations; the authors used the same roman fonts of the text. The outcome is that the reading becomes rather heavy. It does not help that the authors used a word processor with rather limited typesetting capabilities, and the publisher printed their document without typesetting it.
This book does not contain a collection of end-of-chapter problems, for which reason its use as a textbook is rather limited, but perhaps it was never the intention of the authors to produce a textbook. As a reference document, Formulas for Dynamic Analysis is recommended to the practicing engineer and to the mature graduate student.