General treatment of the evaluation of tri-diagonal secular determinants

Secular determinants associated with eigenvalue problems are tri-diagonal (i) for discrete linear systems if the point-masses are subject to only the nearest-neighbor interaction forces and (ii) for certain distributive systems in one and two dimensions if the partial differential equations governing such systems are transformed into a set of simultaneous linear equations by means of difference techniques. If a vibrating system consists of N sub-units each containing p different masses, the secular determinant has the rank pN and its principal diagonal elements repeat themselves with a periodicity p Such determinants are encountered in a wide range of problems dealing with vibrating systems. A method of evaluation is presented here for such secular determinants for any value of p. Thus, the present paper is an extension and generalization of earlier work published in this journal. © 1973, Academic Press Inc. (London) Limited. All rights reserved