Computing eigenvalues, eigenvectors and frequency responses of structures with non-proportional damping

This paper presents algorithms to compute eigenvalues, eigenvectors and frequency response matrix of a linear structure with non-proportional damping. The method is based on derivatives of repeated or unrepeated complex eigenvalues and complex eigenvectors. Since eigenvectors are not unique for a repeated eigenvalue, a special algorithm is developed to select the correct eigenvector for a given non-proportional damping. The advantage of the proposed method is that it only requires the knowledge of natural frequencies and mode shapes of the undamped structure. The validity of this method is established by comparing numerical results from this algorithm to exact answers.