A Relationship Between Defective Systems and Unit-Rank Modification of Classical Damping

A common assumption within the mathematical modeling of vibrating elastomechanical system is that the damping matrix can be diagonalized by the modal matrix of the undamped model. These damping models are sometimes called “classical” or “proportional.” Moreover it is well known that in case of a repeated eigenvalue of multiplicity $m,$ there may not exist a full sub-basis of $m$ linearly independent eigenvectors. These systems are generally termed “defective.” This technical brief addresses a relation between a unit-rank modification of a classical damping matrix and defective systems. It is demonstrated that if a rank-one modification of the damping matrix leads to a repeated eigenvalue, which is not an eigenvalue of the unmodified system, then the modified system is defective. Therefore defective systems are much more common in mechanical systems with general viscous damping than previously thought, and this conclusion should provide strong motivation for more detailed study of defective systems. [S0739-3717(00)00602-4]