Efficient Extreme Value Prediction for Nonlinear Beam Vibrations using Measured Random Response Histories

Predicted extreme exceedance probabilities associated withexperimental measurements of highly non-linear clamped-clamped beamvibrations driven by band-limited white-noise, are compared using twodifferent approaches for application to short data sets. The firstapproach uses response history measurements to calibrate a discretedynamic model using a Markov moment method appropriately matched toextreme value prediction via finite element solution of theFokker¿Planck (FPK) equation. The dynamic model is obtained via theWoinowsky¿Krieger equation with added empirical damping. Stationary FPKsolutions are used to obtain mean crossing rates, and for the purpose ofextreme value prediction, crossings are assumed to be independent. Thesecond approach uses a Weissman type I asymptotic estimator, justifiedby use of the Hasofer¿Wang hypothesis test. Both methods are comparedwith exceedance probabilities obtained using data from long experiments in which dependence between extreme values is excluded. Thepaper shows that by exploiting the Weissman estimator in a forwardpredictive mode, very accurate exceedance probabilities can be obtainedfrom relatively small amounts of measured data. The calibrated modelbased predictions are consistently in error as a result of non-linearcoupling effects not included in the model ¿ this coupling isimplicitly accounted for in the Weissman predictions.