REFINING COMPLEX FREQUENCIES OF VIBRATING STRUCTURES

Resumo

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 matrices is dense and the matrix size is not prohibitively large, is usually done through reduction to a generalized first order system. The QZ iteration method is then applied to achieve the Generalized 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 allows important properties like positive definiteness (or semi-definiteness), and even symmetry, to be lost. Even further, it must be considered the fact that the condition number of the new problem (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 work investigates how the final computation of complex-valued system eigenvalues (and then complex system frequencies) can take advantage of an iterative refinement technique, based in a shifted and inverted Krylov-type strategy that uses the original linear second-order matrices, in the same computational environment. The proposed strategy takes advantage that, in its original second-order form, the eigenvalue problem seams to be far better conditioned and well suited for high performance computations than in the generalized first-order form. The proposed strategy 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