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

U Prells & MI Friswell (University of Wales Swansea)

Journal of Vibration and Acoustics, Vol. 122, No. 2, April 2000, pp. 180-183


A common assumption within the mathematical modelling of vibrating elastomechnical 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 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.

Paper Availability

This material has been published in the Journal of Vibration and Acoustics, Vol. 122, No. 2, April 2000, pp. 180-183, the only definitive repository of the content that has been certified and accepted after peer review. The copyright has been assigned to the American Society of Mechanical Engineers.

Link to paper using doi:10.1115/1.568458

Journal of Vibration and Acoustics