Several solution methods for the generalized complex eigenvalue problem with bounded uncertainties

Abstract

The aim of this paper is to evaluate the effects of uncertain-but-bounded parameters on the complex eigenvalues of the non-proportional damping structures. By combining the interval mathematics and the finite element analysis, the mass matrix, the damping matrix and the stiffness matrix were represented as the interval matrices. Firstly, with the help of the optimization theory, we presented an exact solution—the vertex solution theorem, for determining the exact upper bounds or maximum values and exact lower bounds or minimum values of complex eigenvalues of structures, where the extreme values are reached on the boundary of the interval mass, damping and stiffness matrices. Then, an interval perturbation method was proposed, which needs less computational efforts. A numerical example of a seven degree-of-freedom spring-damping-mass system was used to illustrate the computational aspects of the presented vertex solution theorem and the interval perturbation method in comparison with Deif’s method.