A Method for Checking Missed Eigenvalues in Eigenvalue Analysis with Damping Matrix

A Method for Checking Missed Eigenvalues in Eigenvalue Analysis with Damping Matrix

In the case of the non-proportionally damped system such as the soil-structure interaction system, the structural control system and composite structures, the eigenproblem with the damping matrix should be necessarily performed to obtain the exact dynamic response. However, most of the eigenvalue analysis methods such as the subspace iteration method and the Lanczos method may miss some eigenvalues in the required ones. Therefore, the eigenvalue analysis method must include a technique to check the missed eigenvalues to become the practical tools. In the case of the undamped or proportionally damped system the missed eigenvalues can easily be checked by using the well-known Sturm sequence property, while in the case of the non-proportionally damped system a checking technique has not been developed yet. In this paper, a technique of checking the missed eigenvalues for the eigenproblem with the damping matrix is proposed by applying the argument principle. To verify the effectiveness of the proposed method, two numerical examples are considered.