Computation of Extreme Clustered Natural Frequencies of Damped Second-Order Linear Systems

Abstract

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 its mass, 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.Palavras-chave: second-order, power-method, subspaceiteration

Refining natural frequencies of linear second-order vibrating systems

“…This strategy is closely related to the one in [1]. It is shown in [2] that, in order to construct a sequence {α k } converging to a real single isolated eigenvalue λ i that is the closest to a given real number σ, one can shift and invert (12) in order to derive the recurrence formulas 4…”

“…Only eigenvalues with positive imaginary parts were refined. The experiments were done in an Intel Dual Core Pentium G630 2.7GHz Desktop under Ubuntu Linux and software Matlab 2012b, which is able to carry out double complex arithmetic operations with almost no adjust at all to the real data algorithms from [2]. Figures 1 and 2 show that the eigenvalues could be successfully improved through iterative refinement, yielding to nullity ratios varying from 1.1 to 2.8, regardless of the reduction formula that was used.…”