Response level crossing rate of a linear system excited by a partially specified Gaussian load process

The problem of determining the maximum mean response level crossing rate of a linear system driven by a partially specified Gaussian load process has been considered. The partial specification of the load is given only in terms of its total average energy. The critical input power spectral (PSD) function, which maximizes the mean response level crossing rate, is obtained. The critical input PSD turns out to be highly narrow-banded which fails to capture the erratic nature of the excitation. Consequently, the trade-off curve between the maximum mean response level crossing rate and the maximum disorder in the input process, quantified in terms of its entropy rate, has been generated. The method of Pareto optimization is used to tackle the conflicting objectives of the simultaneous maximization of the mean response level crossing rate and the input entropy rate. The non-linear multi-objective optimization has been carried out using a recently developed multi-criteria genetic algorithm scheme. Illustrative example of determining the critical input of an axially vibrating rod, excited by a partially specified stationary Gaussian load process, has been considered.