Closely Spaced Roots and Defectiveness in Second-Order Systems
Publication: Journal of Engineering Mechanics
Volume 131, Issue 3
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.
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Acknowledgments
The work presented in this paper was partly carried out while the writer was on sabbatical at the Politecnico di Torino in Italy. The financial support provided by the Italian Ministry of Universities and Research (MIUR) during this period is gratefully acknowledged. The writer would also like to acknowledge Professor Alessandro DeStefano from the Politecnico di Torino and Professor Gilead Tadmore from the Electrical Engineering Department at Northeastern University for their valuable comments.
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© 2005 ASCE.
History
Received: Nov 14, 2003
Accepted: Aug 13, 2004
Published online: Mar 1, 2005
Published in print: Mar 2005
Notes
Note. Associate Editor: Joel P. Conte
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