Short-term dynamic instability of a system with randomly varying damping

A single-degree-of-freedom system with temporal random variations of its total apparent damping is considered. It is demonstrated that the dynamic response of the system exhibits spontaneous transient outbreaks, which are induced by brief periods when the damping coefficient becomes negative. The analysis is based on a parabolic approximation for the random temporal variations of the damping coefficient during these excursions into the domain of dynamic instability, together with the asymptotic Krylov-Bogoliubov method of averaging over the response period. It results in an explicit relation for the response amplitude in terms of the peak value of the damping coefficient during the corresponding downcrossing of the zero level. In this way the reliability analysis for the system as based on a solution to the first-passage problem for the response or on evaluating the probability density (pdf) of the response peaks is reduced to the corresponding problems for the damping coefficient; the accuracy of the derived expression for the above pdf is verified by direct Monte-Carlo simulation of the basic equation. Furthermore, the above analytical solution is used also to derive a simple identification procedure for the system from its measured (on-line) response. Specifically, the mean value of the damping coefficient can be estimated, as can its standard deviation and the mean frequency of its temporal variations.