A SOLUTION METHOD FOR SOME NONLINEAR EIGEN-PROBLEMS IN STRUCTURAL DYNAMICS

摘要

Frequency-dependent matrices in structural dynamics result in a non-linear eigenualue problem. The advantage of frequency-dependent matrices is that the order is usually smaller for comparable accuracy of the eigenparameters of a given system. This paper presents a method for solving non-linear eigenvalue problems, in which the matrix function of the eigenualue can be expanded in terms of its powers. The solution approach consists of first solving the basic linear eigenualue problem and then using the higher order matrices to converge the eigenualues one at a time. A perturbation approach is adopted to derive the convergence scheme in which separate iterations are performed for the eigenualue and eigenvector during each step. This method is an alternative to the classical companion matrix method which can only be used for problems that have weak nonlinearity, i.e., which have series expansions up to a small power of the eigenualue. Also, a method is introduced to perform dynamic condensation for finding approximate eigenualues when only a small fraction of the total eigenualues are wanted. Numerical examples demonstrate the application of the method.