Approximate eigensolutions for arbitrarily damped nearly proportional systems

Two efficient methods for determining the approximate eigenvalues and eigenvectors for arbitrarily damped nearly proportional systems are developed. Both approaches are formulated by means of a first-order perturbation technique, whereby the real modes of vibration of the undamped system are used to derive approximate expressions for the complex eigenvalues and eigenvectors of a nearly proportionally damped system. Using either approach, the unperturbed configuration corresponds to a damped one whose damping matrix can be diagonalized by the same transformation that uncouples the undamped system, and the perturbation consists of the deviation of this diagonalizable damping matrix from the actual damping matrix. The proposed approaches are easy to code, implement and solve, and do not require forming state equations. Numerical examples are presented to validate the effectiveness of the current methods.