A reformulated Arnoldi algorithm for non-classically damped eigenvalue problems

In applying Arnoldi method to non-symmetric eigenvalue problems for damped structures, a structure of the projected upper Hessenberg matrix is obtained in this paper. By exploiting the structure of the upper Hessenberg matrix and taking advantages of the block properties of system matrices, the Arnoldi reduction algorithm is reformulated for less computation and higher accuracy. In conjunction with the reformulated Arnoldi algorithm, real Schur decomposition instead of Jordan decomposition is adopted aiming at non-complex arithmetic, non-discriminative processing of defective and non-defective systems and numeric stability. A concise reduction algorithm for eigenproblems for undamped gyroscopic systems is obtained by directly degenerating from the reformulated Arnoldi algorithm. For safely solving engineering problems without omitting eigenvalues, a restart reduction procedure is proposed in terms of the reformulated reduction algorithm with deßation developed in this paper. Numerical examples once solved with algorithms originated from Lanczos methods were re-solved. In addition, the non-symmetric eigenvalue problem for a shear wall by BEM modeling and a damped gyroscopic system with eigenvalues of high multiplicity were also used to demonstrate the e¦cacy of the presented methods