Conditions revisited for asymptotic stability of pervasive damped linear systems

A linear oscillatory system with a positive definite mass matrix M, a positive definite stiffness matrix K and a positive semi-definite damping matrix D can be asymptotically stable. In this case the damping D of the system is called pervasive. The condition for the damping to be pervasive was given in a paper by Moran in 1970. Moran's result has been reinvented in a recent paper from 2004 motivated by a long discussion on “intuitive” examples of marginal and asymptotic stability of systems with incomplete damping. The aim of the present note is to draw attention of the readers to Moran's result. We reveal the importance of Moran's criterion for suppression of oscillations by incomplete damping in systems with multiple eigenfrequencies. Finally, we provide extensions to the case of indefinite damping.