A solution method for some nonlinear eigen-problems in structural dynamics

Frequency-dependent matrices in structural dynamics result in a non-linear eigenvalue 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 eigenvalue can be expanded in terms of its powers. The solution approach consists of first solving the basic linear eigenvalue problem and then using the higher order matrices to converge the eigenvalues one at a time. A perturbation approach is adopted to derive the convergence scheme in which separate iterations are performed for the eigenvalue 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 eigenvalue. Also, a method is introduced to perform dynamic condensation for finding approximate eigenvalues when only a small fraction of the total eigenvalues are wanted. Numerical examples demonstrate the application of the method.