AN ARNOLDI-BASED EIGENSOLUTION TECHNIQUE FOR THE NONPROPORTIONALLY DAMPED FREE VIBRATION PROBLEM

摘要

The dynamic response of multi degree-of-freedom linear systems subjected to external loadings is generally determined using the modal superposition method, which employs the modes from an undamped free vibration analysis to decouple the system of equations. The basic assumption in using these undamped free vibration modes is that the system is proportionally damped. However, in several applications, this assumption is not reasonable, and the system must be analyzed as nonproportion-ally damped. Decoupling of such nonproportion-ally damped systems of equations requires use of the (complex) eigenvectors associated with the damped problem. Obtaining these solutions usually involves transforming the original quadratic eigenproblern of order N into an eigenproblern of order 2N, which is unsymmetric and computationally intense to solve. This paper presents an Arnoldi-based eigensolu-tion technique for solving the quadratic eigenvalue problem associated with damped free vibration analysis of skeletal systems. Specifically, an implicitly restarted κ-step Arnoldi method is used which overcomes the loss of orthogonality among the Arnoldi vectors. Examples show that this implicitly restarted Arnoldi technique efficiently obtains the complex eigenvalues and eigenvectors associated with damped free vibration analyses. In addition, a comparison is performed between this method and the more tra- ditionally used dense reduction of the transformed unsymmetric eigenproblern.