Repeated eigenvalues and their derivatives of structural vibration systems with general nonproportional viscous damping

Abstract

Repeated vibration modes often occur in practice in structural vibration systems with general nonproportional viscous damping. However, the topic remains perhaps the least understood in vibration analysis. Some researchers have suggested that the inclusion of nonproportional viscous damping renders a system defective in the case of repeated eigenvalues, while others have assumed that a complete set of linearly independent eigenvectors can always be found regardless of the forms and magnitudes of viscous damping. This paper seeks to first establish that for light non-proportional viscous damping, a system with repeated eigenvalues generally does not become defective based on perturbation theory and realistic numerical examples since rigorous theoretical proof is believed to be difficult. Once non-defectiveness is confirmed, a method for computing the eigen derivatives with repeated eigenvalues in the case of general viscous damping is developed. Effect of mode truncation on numerical accuracy has been discussed. When the magnitude of non-proportional viscous damping becomes considerably high however, it is possible for a damped system to become defective when repeated modes occur. This can have profound practical implications where very high damping is desired as in the cases of vibration suspension and absorber designs, since system defectiveness can lead to major difficulties in the applications of conventional vibration analyses.