Frequencies of Some Near-Regular Structures: A Combined Graph Product and Bisection Method

Abstract

In this paper, an efficient iterative method is presented to calculate the eigenvalues of the stiffness matrices of some structures having near-regular forms. In the present method, by a suitable nodal ordering of the near-regular structures, partitioning stiffness matrices of special form and applying an effective Schur decomposition, the matrix corresponding matrices become decomposable. Then utilizing the theorem about the relation between the poles of a Schur complement equation and the eigenvalues of the submatrix and using an iteration bisection method, the eigenvalues of the stiffness matrices or the frequencies of the near-regular structure are obtained. Also utilizing the Cauchy interlacing theorem about the relation between the eigenvalues of a matrix A and eigenvalues of the matrix B obtained by striking out a row and the corresponding column of A, the proper bounded interval for eigenvalues of B becomes obtainable. Previously, for solving the eigenvalues of near-regular structures, they were approximately considered as regular ones because graph products were unable to solve the eigenvalues of these near-regular structures in a swift and exact manner. In this paper, using a combination of graph products, Schur decomposition, and the above-mentioned effective theorems together with an iteration bisection method with appropriate initial intervals, more suitable results are obtained.