Eigenvalue and eigenvector derivatives of second-order systems using structure-preserving equivalences

Abstract

A “structure-preserving equivalence” in the sense intended here is a mapping between the stiffness, damping and mass matrices describing some initial second-order system and the corresponding three matrices of another second-order system having identical spectrum. Most second-order systems can be “diagonalised” through a mapping of this sort. The mapping provides a new approach to the evaluation and the understanding of eigenvalue and eigenvector derivatives. In place of pairs of eigenvalues, we think of real scalar stiffness, damping and mass quantities representing decoupled single-degree-of-freedom systems. In place of pairs of eigenvectors, we think of individual columns of the matrices involved in the mapping. This approach resolves the completely artificial phenomenon that the eigenvalue and eigenvector derivatives become “undefined” at instants when modification of, say, a damping parameter causes a pair of complex eigenvalues to turn into a pair of real eigenvalues or vice-versa. It also has the advantage of being applicable to cases where any one or more of the system matrices are singular.