On the Symmetrization of Asymmetric Finite Dimensional Linear Dynamic Systems

Abstract

A linear dynamic system with constant mass, and asymmetric damping and stiffness coefficient matrices, may be transformed to a symmetric system where all coefficient matrices are symmetric. This transformation makes it possible to take advantage of the well-developed theories that use the properties of the symmetric coefficient matrices. Some previous studies have suggested a decomposition method associated with rank checking of a rectangular matrix to determine if such transformations exist. However, these methods were only applicable to coefficient matrices with distinct eigenvalues, and they are computationally intensive. In this paper, a general discussion of the symmetrization problem is presented. A new method for assessing the symmetrizability and a way to find one of such transformations (if they exist) is also proposed. The method needs a fraction of the computations needed for published methods. The proposed method is tailored to treat coefficient matrices with repeated eigenvalues as well as distinct eigenvalues. Four examples, of which three are from previous studies, are used to demonstrate the proposed method. The results show that the proposed method is more computationally efficient than previously published methods, and accommodates the repeated eigenvalue problem.