Structural dynamics analysis using an unsymmetric block Lanczos algorithm

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Abstract
This paper describes the way in which an unsymmetric block Lanczos algorithm can be employed for approximate eigensolutions or dynamic response solutions for large systems having arbitrary damping and/or repeated (or closely spaced) eigenvalues. To reduce a square, unsymmetric matrix which may have repeated eigenvalues to block‐tridiagonal form, an unsymmetric block Lanczos algorithm is developed. Right and left Lanczos vectors, generated by the reduction process, are employed to define the reduced‐order model based on a discrete analytical model of general linear, time‐invariant dynamic systems. A special type of starting vector is also introduced, which automatically includes the static displacement. This reduced‐order model is employed to compute approximate eigenvalues/vectors and dynamic responses of the original system. To verify the proposed algorithm, examples of an 8‐DOF beam‐rotor system are provided for eigensolutions and dynamic responses due to step and random external forces. These examples showed promising results, indicating that the proposed algorithm is worthy of further study.