The matrix analysis of vibration.

*(English)*Zbl 1241.70029
Cambridge: Cambridge University Press. x, 404 p. (1965).

”The introduction of high-speed electronic computers into engineering has opened the way for solution of vibration problems of great complexity. One of the most suitable ways of expressing a problem for computational analysis is to use matrices. This book is concerned with the matrix formulation of the equations of motions of vibrating systems, and with techniques for the solution of matrix equations” (quoted from the preface). The book is divided into two distinct parts. The first (Chapters 1–6) is due to the first two authors and shows the use of matrices in mechanical vibration problems, the second (Chapters 7–9) is due to the third author and is concerned with the numerical solution of matrix equations. The book starts from very elementary premises; practically no prerequisites are required of the reader. In Chapter 1, matrices and determinants are explained in great detail. Each new subject is illustrated with numerical examples, and problems are proposed, with solutions at the end of the book. Chapters 2–4 concern the vibration of conservative systems having a finite number of degrees of freedom. Forced harmonic vibration, free vibration, principal modes, orthogonality, Rayleigh’s principle, semi-definite systems and multiple natural frequencies are carefully discussed and illustrated with numerical examples. Chapter 5 brings in viscous damping and hysteretic damping; similar subjects are treated in The mechanics of vibration by Bishop and Johnson [Cambridge Univ. Press, New York, 1960; Zbl 0090.14202)], but from another point of view. Chapter 6 is concerned with continuous systems without damping. The concept of receptances of continuous systems is used to set up the frequency equation of composite systems. The study of those systems brings in a full discussion of Holzer’s method and its improvements for torsional vibrations. In connection with composite systems, a brief reference is made to the method of transmission matrices developed by the German school of Marguerre, Falk, Fuhrke, etc. A detailed exposition of the approximation of continuous systems by lumped mass systems follows, with evaluation of the errors. Then the direct solution of continuous systems is considered by means of Rayleigh’s theorem and the Rayleigh-Ritz, or assumed modes, method. This chapter, together with the first part of the book, ends with an exposition of the branch mode analysis, which is a mixture of the lumped mass and assumed modes methods.
The authors have developed their subject with the aim of providing the best practical tools for solving vibration problems of complex structures, e.g., airplanes, with the use of digital electronic computers. This is probably the reason why other classical methods are mentioned but not developed, such as, for example, the Stodola method, which is nothing else but an iteration method. This omission is, however, rather strange because Chapter 8 treats in detail the iterative methods for characteristic-value problems from the point of view of matrix calculus. This is perhaps also the reason why interesting theoretical developments are left out, such as the Green’s resolvent associated with the differential equation which throws light on the receptance concept.
The last three chapters of the book form the second part, which is a clear exposition of numerical analysis in matrix calculus. It is completely distinct from the first part, to which it makes no reference. Chapter 7 treats of the techniques of solving linear equations, the difficulties that can occur (e.g., ill-conditioned equations), and the evaluation of bounds on the error of a solution. As already mentioned, Chapter 8 deals with iterative methods and gives bounds for the characteristic values, as well as the iteration techniques for finding a non-dominant characteristic value. Finally, Chapter 9 contains an exposition of direct methods for finding characteristic values, i.e., methods giving the exact value after a finite number of steps. They are based on Householder’s and Lanczos’ reduction to triple diagonal matrices and the study of Sturm’s sequences. Many numerical examples illustrate the different methods discussed. Numerous references are given in these last three chapters, particularly to the contributions of J. H. Wilkinson.

Reviewer: J. Kestens (MR0173384)