Int. J. Mech. Sci. Vol. 32, No. 3, pp. 169 179, 1990 0020-7403/90 $3.00 + .00 Printed in Great Britain. © 1990 Pergamon Press pie STRUCTURAL MODIFICATION ANALYSIS USING RAYLEIGH QUOTIENT ITERATION W. M. To and D. J. EWlNS Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, London, U.K. Abstract--This paper highlights the application to structural dynamics of the sensitivity analysis methods developed by numerical analysts and presents a historical development of first- and higher- order eigenvalue and eigenvector sensitivities. Different formulae for an eigenvalue sensitivity are presented and it is shown that all of these are implicitly the same. A condition number is presented to give the limited bound of application for the first-order eigenvalue and eigenvector sensitivities. An alternative structural modification method based on Rayleigh Quotient Iteration is presented. A lumped spring-mass system with 7 degrees-of-freedom (7DoF) is used to show the applicability of the Rayleigh quotient iteration method. [A], IdA] [K], [M] [dK], [dM] {y,}, {~,}, ,~, {x'}o, {~'},. .L.,, )~... 0~ I1{ }11~ I1[ ]11~ }T, [ ]T NOTATION system matrix of the eigenvalue problem [A]{x} = 2{x} and the change in [A] system stiffness and mass matrices modifications in system stiffness and mass matrices left-hand and right-hand mass-normalized eigenvectors and eigenvalue of ith mode orthonormalized eigenvectors of the original system, modified system eigenvalues of the original system, modified system design parameter of system matrices 2-norm (Euclidean norm) of a real vector, i.e. It{x} 112 ~- ({x}r{x}) a/2 2-norm of a real matrix i.e. II [A] 112= (max. eigenvalue of [A]T[A]) 1/2 transposes of a vector and a matrix 1. INTRODUCTION Sensitivity analysis has been applied by several workers to the general eigenvalue problem [1-8] and, more specifically, to applications of structural modification in references [9-11]. In this area, both first- and higher-order eigenvalue and eigenvector sensitivities have been investigated with a view to predicting the dynamics of a modified structure from knowledge of its properties in an original, or unmodified, state. As the sensitivity analysis of a mechanical structure is based on a Taylor expansion of the eigenvalues and eigenvectors of the unmodified structure, and the computation of the higher-order terms of this series is difficult and time consuming, the effectiveness of this method is limited to small modifica- tions. However, it is not easy to determine what is "small". In this paper, a condition number is presented to indicate how sensitive the eigenvalues and eigenvectors of a mechanical structure are to small modifications. The value of this condition number is used to determine a limit of applicability for the first-order eigenvalue and eigenvector sens- itivities. The Rayleigh quotient provides a well-known procedure for the approximate evaluation of the eigenvalue of a structure. It is shown that under some conditions the Rayleigh quotient and first-order eigenvalue sensitivity are equivalent. Structural modification problems possess an inherent advantage in that some initial conditions are known and this information is very useful for an iteration method of analysis. Accordingly, an iterative procedure using the Rayleigh quotient would seem to offer advantages for the solution of the modification analysis problem, especially in view of the cubic convergence property of this method. 2. THE ORIGINAL PURPOSE OF DEVELOPING THE FIRST-ORDER PERTURBATION First-order perturbation estimates (i.e. eigenvalue and eigenvector sensitivities) have long been used by numerical analysts and scientists to investigate the stability of the eigenvalue problem [A]{x} = 2{x}. These estimates have the advantage that they provide a quick, 169