Improvement of subspace iteration method and development of technique for checking missed eigenvalues of structures with nonproportional damping

Abstract

In order to obtain the exact dynamic response of a structure or to avoid the resonant response of a structure, it is required to develop a solution method that can evaluate the desired eigenvalues and the corresponding eigenvectors and a technique that can check whether the desired eigenpairs are indeed calculated without any missed ones or not. This dissertation presents a numerically stable eigenproblem solution method by improving the subspace iteration method with shift and a technique for checking the missed eigenvalues of structures with nonproportional damping. The subspace iteration method has hitherto been known to be very efficient for solving large eigenproblems. A major difficulty of the conventional subspace iteration method using shifting technique is that a shift very close to an eigenvalue cannot be used due to the singularity problem, resulting in slower convergence. In this study, the above singularity problem has been solved by introducing side conditions without sacrifice of convergence. The proposed method is always nonsingular even if a shift is on a distinct eigenvalue or multiple ones. This is one of the significant characteristics of the proposed method. The nonsingularity is proved analytically. The convergence of the proposed method is at least equal to that of the conventional subspace iteration method with shift, which is also proved analytically, and the operation counts of the above two methods are almost the same when a large number of eigenpairs are required. To show the effectiveness of the proposed method, four cases are analyzed. They are two structures with distinct eigenvalues such as a three-dimensional frame structure and a simply supported rectangular plate structure and two structures with multiple eigenvalues such as a three-dimensional frame structure with a symmetric cross-section and a simply supported square plate structure. Most of the eigenvalue analysis methods such as the subspace iteration method and the Lancz...