Gyroscopic stabilization and indefimite damped systems

An important issue is how to modify a given unstable matrix in such a way that the resulting matrix is stable. We investigate in general under which condition a matrix M+A is stable,where M is an arbitrary matrix and A is skew-Hermitian. We show that if trace(M) > 0 it is always possible to find a class of feasibel skew-Hermitian matrices A depending on the choise of M.

The theory can be applied to dynamical systems of the form x''(t) + ( dD + g G) x'(t) + K x(t) = 0 where G is a skew symmetric gyrocopic matrix, D is a symmetric indefinite damping matrix and K > 0 is a positive definite stiffness matrix. d and g are scaling factors used to control the stability of the system.

It is quite astonnishing that when the damping matrix D is indefinite the system can under certain conditions be stable even if there are no gyroscopic forces G present The Lyapunov matrix equation is used to predict the stabilty limit for pure dissipative systems as well as for dissipative systems with gyroscopic stabilization.