Non-proportional damping and defective systems

A common assumption within the mathematical modelling of vibrating elastomechanical system is that the damping matrix can be diagonalised 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 a full sub-basis of linearly independent eigenvectors may not exist. These systems are generally termed defective. This paper addresses a relation between a non-proportional damping matrix and defective systems. 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. Further examples are given of systems with general damping that are also defective. The conclusion is that defective systems are much more common in mechanical systems with general viscous damping than previously thought. This conclusion should provide strong motivation for more detailed study of defective systems.