Low-rank damping modifications and defective systems

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 diagonalizes 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 diagonalize 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.