Low Rank Modifications of Classical Damping and Defective Systems

MI Friswell (University of Bristol), U Prells & SD Garvey (University of Nottingham)

Journal of Sound and Vibration, Vol. 279, No. 3-5, January 2005, pp. 757-774


The structural modification of dynamical systems is an important issue in a wide range of applications, for example in vibration suppression or in active and passive control. It is well known that for a proportionally (or classically) damped system there always exists a real matrix of eigenvectors which simultaneously diagonalises the three system matrices of inertia, damping and stiffness, even if the system possesses repeated eigenvalues. For general viscously damped systems the eigenvalue analysis must be performed in state space, and for systems with distinct eigenvalues the corresponding eigenvectors diagonalise the state space matrices. However, with general viscous damping, systems with repeated complex eigenvalues may have insufficient linearly independent complex eigenvectors. These systems are termed defective. In contrast to non-defective (or simple) systems, for defective systems only a Jordan decomposition exists. In this paper conditions on rank 1, rank 2 and higher rank modifications are derived which ensure that the modified system is simple. If none of the eigenvalues of the unmodified system is an eigenvalue of the modified system then every rank 1 modification that produces a pair of repeated complex eigenvalues leads to a defective system. Under the same assumptions there exist higher rank modifications which lead to simple systems. Either these modifications produce a proportionally damped system, or the restrictions on these modifications are rather strict which suggests that in practical cases every rank 2 or higher modification that produces repeated pairs of complex eigenvalues will lead to a defective system. The findings are demonstrated by simulated examples.

Paper Availability

This material has been published in the Journal of Sound and Vibration, Vol. 279, No. 3-5, January 2005, pp. 757-774, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by the Elsevier.

Link to paper using doi:10.1016/j.jsv.2003.11.042

Journal of Sound and Vibration on ScienceDirect