Closely Spaced Roots and Defectiveness in Second-Order Systems

Abstract

When two closely spaced eigenvalues merge the associated eigenvectors can either (1) form a subspace where every vector in the span is an eigenvector or (2) coalesce into a single eigenvector. In the second alternative the repeated eigenvalue is associated with a bifurcation point in the eigenvector space and the system is said to be defective. In defective systems a set of coordinates that uncouple the dynamics does not exist and the closest thing possible is the basis of eigenvectors and generalized eigenvectors (sometimes called power vectors) that lead to the Jordan form. Although true defectiveness does not occur in practice, because eigenvalues are never exactly repeated, one anticipates that the features associated with defective conditions will have a bearing on the behavior of systems that are perturbed versions of defective ones. In viscously damped second order systems with symmetric matrices the potential for defectiveness is determined by the structure of the damping. This paper focuses on identification of conditions connecting the damping matrix with defectiveness. A numerical example of a two degree-of-freedom system that varies from being classically damped, to nonclassical, to defective, depending on the position of a dashpot, is used to illustrate the features of the eigensolution as defectiveness is approached.