Marginal Instability and Intermittency in Stochastic Systems—Part I: Systems With Slow Random Variations of Parameters

Abstract

Dynamic systems with lumped parameters, which experience random temporal variations, are considered. The variations may “smear” boundary between the system’s states, which are dynamically stable and unstable in the classical sense. The system’s response within such a “twilight zone” of marginal instability is found to be of an intermittent nature, with alternating periods of zero (or almost-zero) response and rare short outbreaks. As long as it may be impractical to preclude completely such outbreaks for a designed system, subject to highly uncertain dynamic loads, the corresponding system’s response should be analyzed. Results of such analyses are presented for cases of slow and rapid (broadband) parameter variations in Parts I and II, respectively. In the former case, the “nominal” system—one without variations of parameter(s)—is stable in the classical sense. Its transient response during the “slow” short-term excursions of the parameter(s) into the instability domain is described by a linear model. The analysis is based on Krylov–Bogoliubov averaging over “rapid” time within the response period together with parabolic approximation for the parameter variations in the vicinity of their peaks (so-called Slepian model). Solution to the resulting deterministic transient response problem with random initial condition(s) at the instant of upcrossing the stability boundary yields a relation between peak value(s) of the response(s) and that of the parameter(s); in this way, reliability study for the system is reduced to a probabilistic analysis of the parameter variations. The solutions are obtained for the cases of negative-damping-type instability in a SDOF system and for TDOF systems with potential dynamic instability due to coalescing or merging of natural frequencies; the illustrating examples of applications are rotating shafts with internal damping, two-dimensional galloping of a rigid body in a fluid flow and a row of tubes in a cross flow of fluid. The response is of the intermittent nature due to the way it is generated, with alternating relatively long periods of zero (or almost-zero) response and short outbreaks due to temporary excursions into the instability domain.