We characterize the complex, heavy-tailed probability distribution functions(pdf) describing the response and its local extrema for structural systemssubjected to random forcing that includes extreme events. Our approach is basedon the recent probabilistic decomposition-synthesis technique, where wedecouple rare events regimes from the background fluctuations. The result ofthe analysis has the form of a semi-analytical approximation formula for thepdf of the response (displacement, velocity, and acceleration) and the pdf ofthe local extrema. For special limiting cases (lightly damped or heavily dampedsystems) our analysis provides fully analytical approximations. We alsodemonstrate how the method can be applied to high dimensional structuralsystems through a two-degrees-of-freedom structural system undergoing rareevents due to intermittent forcing. The derived formulas can be evaluated withvery small computational cost and are shown to accurately capture thecomplicated heavy-tailed and asymmetrical features in the probabilitydistribution many standard deviations away from the mean, through comparisonswith expensive Monte-Carlo simulations.
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