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Resumo: In recent contribution, we have proposed methods for computing the largest and the smallest isolated natural frequencies, either real or a complex pair, of a second-order mechanical linear system described by itsmass, damping and stiffness matrices. Those methods were presented as a generalization of the well-known direct and inverse power methods for computing dominant and sub-dominant eigenvalues of a matrix; they applied results on fundamental solutions for second-order systems, as well as the widely used technique of subspace iteration. The present work addresses the case where those extreme frequencies, instead of being isolated, come in clusters. This situation is very common when system matrices come from the Finite Element Method. The strategy we propose can be applied to both sparse and dense problems, and strategies using specialized software libraries as LAPACK or SPARSEKIT can be deployed for achieving outstanding performance. Numerical examples with test matrices from real structural vibration problems are provided.

COMPUTATION OF EXTREME CLUSTERED NATURAL FREQUENCIES OF DAMPED SECOND-ORDER LINEAR SYSTEMS