An iterative method for exact eigenvalues and eigenvectors of general nonviscously damped structural systems

Abstract

A novel and computationally efficient iterative method is proposed for the exact eigenvalues and eigenvectors of nonviscously damped vibration systems. General nonviscous damping model is assumed in which damping forces depend on the past motion history via convolution integrals over exponentially decaying kernel functions. The presence of nonviscous damping leads to complex frequency dependent eigenvalue problem whose solution requires either extensively augmented state-space formulation or iterative full complex eigensolutions on mode by mode basis, both of which are computationally expensive when practical systems with large dimensions are considered. By simply solving the eigenvalue problem of the underlying undamped vibration system, the real eigenvalues and eigenvectors can then be combined with the nonviscous damping matrix to develop an iterative procedure from which required complex eigenvalues and eigenvectors of the damped system can be computed. First order perturbation is employed to further improve the starting estimates of the desired eigenvalues and eigenvectors and the convergence is generally very fast. The proposed method is mathematically developed based on progressive eigensensitivity analysis and convergence is ensured when the norm of the damping matrix is of second order when compared with that of the stiffness matrix. Representative numerical examples of discrete mass spring damping model as well as practical finite element model with nonviscous damping are given to demonstrate and validate the accuracy and efficiency of the proposed iterative method.