Distributions and extreme value analysis of critical response rate and split-time metric in nonlinear oscillators with stochastic excitation

Abstract

A single-degree-of-freedom oscillator with a nonlinear restoring force and stochastic external forcing is studied with the goal of understanding the distributions of certain response rates. Motivated by applications to ship motions, the response rates concern the minimum response rate leading to capsizing (the critical response rate) and a measurement comparing an observed response rate to the critical response rate (the split-time metric), whose distributions are investigated both analytically and numerically. Three nonlinear restoring forces are considered: piecewise linear (experiencing linearly softening stiffness above a “knuckle” point), doubly piecewise linear (experiencing piecewise linearly softening stiffness above a “knuckle” point), and the cubic softening restoring force of the Duffing oscillator. In the first two cases, an efficient numerical simulation of the critical response rate and split-time metric is proposed from a derived distribution; in the latter case, the density of the critical response rate is approximated assuming white noise excitation. A key interest is in the nature of the right tail of the split-time metric, specifically as it relates to its extrapolation through extreme value analysis in estimating the probability of capsizing. The distribution is found to have a “light” tail, which motivates the use of exponential rather than the generalized Pareto distribution for exceedances above threshold in extreme value analysis. Finally, threshold selection through a prediction error criterion for the exponential distribution is examined, and the Weibull distribution tail is suggested as a useful means for a more refined examination of the distribution tail.