The Role of Clifford Algebra in Structure-Preserving Transformations for Second-Order Systems

Abstract

Second-order dynamic systems described by the equation Kx+Cx'+Mx''=F (where the dots indicate differentiation with respect to time) are of immense importance in engineering. The system matrices, K, C, M, are real (N × N) matrices and the vectors of displacement and force x, F each contain N functions of time. For many of the analyses performed on these systems, a generalised eigenvalue problem involving two (2N × 2N) matrices is set up and solved. It is common that most or all of the resulting eigenvalue-eigenvector pairs are complex. The numerical methods currently used for solving this generalised eigenvalue problem (GEP) do not take full advantage of its very particular structure. In particular, they do not provide any way to capitalise on the symmetry very often present in K, C, M [1]. Moreover, the structure in this problem results in constraints on the eigensolutions which make it possible to store those solutions more compactly but these constraints are generally ignored. There is compelling evidence that a more natural approach is possible. The role of Clifford Algebra in this more natural approach is examined.