REFINING COMPLEX FREQUENCIES OF VIBRATING STRUCTURES - Carvalho, J. B.; Claeyssen, J. C. (Revista Interdisciplinar de Pesquisa em Engenharia, 2017)
In structural analysis, and related, the computation of natural frequencies of vibrating systems given by its mass, damping and stiffness matrices, when at least one of this matricesÂ is dense and the matrix size is not prohibitively large, is usually done through reduction to a generalizedÂ first order system. The QZ iteration method is then applied to achieve the GeneralizedÂ Schur form, from where all the natural frequencies, and associated mode shape vectors, can be taken from, as eigenvalues and eigenvectors of some mathematical problem. As a drawback, this reduction not only increases by a factor of 2 the sizes of the working matrices, but also allowsÂ important properties like positive definiteness (or semi-definiteness), and even symmetry, to beÂ lost. Even further, it must be considered the fact that the condition number of the new problemÂ (as an upper bound for the ratio between changes in the solution and changes in the data, by means of matrix and vector norms) usually increases, sometimes by a large amount. This workÂ investigates how the final computation of complex-valued system eigenvalues (and then complexÂ system frequencies) can take advantage of an iterative refinement technique, based in a shiftedÂ and inverted Krylov-type strategy that uses the original linear second-order matrices, in theÂ same computational environment. The proposed strategy takes advantage that, in its originalÂ second-order form, the eigenvalue problem seams to be far better conditioned and well suitedÂ for high performance computations than in the generalized first-order form. The proposed strategyÂ is applied to some test matrices from the Harwell-Boeing Collection of the Matrix Market website. Numerical examples using professional software are also provided. Keywords: second-order, complex frequency, vibration, structure