Towards incorporating impact losses into random vibration analyses: a model problem

Energy Balance for Random Vibrations of Piecewise-Conservative Systems

“…It may be speculated, however, that their accuracy should be higher than those of the asymptotic stochastic averaging*as the latter requires not only small variations of the response period, but also small variations of the response energy per cycle (and thus, small losses). This general expectation had been con"rmed in reference [1] for the vibroimpact system by direct Monte-Carlo simulation.…”

“…An alternative method for response prediction had been proposed in reference [1] for an s.d.o.f. vibroimpact system, subjected to a white-noise excitation.…”

“…The RHS is the mean energy input per cycle, with ¹ being the expected duration of the cycle. It can be identi"ed as the solution to the relevant "rst-passage problem for the response*namely, as an expected time to arrive at the starting point of the next response cycle after the start of the present cycle with energy H. This (conditionally) expected time satis"es the relevant generalized Pontryagin equation [2], which had been identi"ed and analyzed in reference [1] for the corresponding vibroimpact system. Solution ¹(H) to this PDE, which is to be used in the exact (by itself ) relation (1), with H replaced by its unconditional mean value, is the challenging part of the approach.…”

“…It may be added that, in general, ¹ may also be present in the LHS of relation (1); this will be the case where the magnitude of the energy drop depends not only, say, on the initial energy of the cycle, but on the instantaneous energy as well. A perturbational analysis of the second order PDE for ¹(H) has been made in reference [1], with the excitation intensity D regarded as a small parameter. In a zero order approximation, D"0, the PDE is of reduced ("rst) order, and its exact solution is just the system's natural half-period, or ¹" / , where is the system's natural frequency.…”

“…As long as the solution satis"es both boundary conditions for the original PDE, this is the case of regular rather than singular perturbations. Thus, the solution for the deterministic cycle duration ¹ of the system without excitation may naturally be used in relation (1). Of course, this would imply that the predictions are approximate only for not-very-small D's.…”