Distribution tail structure and extreme value analysis of constrained piecewise linear oscillators

A single-degree-of-freedom random oscillator with a piecewise linear restoring force (experiencing softening after a certain point value of the response, called a "knuckle" point) is studied with the goal of understanding the structure of the distribution tail of its response or (local) maximum. A theoretical analysis is carried out by two approaches: first, by focusing on the maximum and response after crossing the "knuckle" point, where explicit calculations can be performed assuming standard distributions for the derivative at the crossing, and second, by considering the white noise random external excitation, where the stationary distribution of the response is readily available from the literature. Both approaches reveal the structure of the distribution tails where a Gaussian core is followed by a heavier tail, possibly having a power-law form, which ultimately turns into a tail having a finite upper bound (referred to as a light tail). The extent of the light tail region is also investigated, and shown to be the result of the conditioning for the system not to reach the unstable equilibrium. The study is motivated by applications to ship motions, where the considered random oscillator serves as a prototypical model for ship roll motion in beam seas, and estimation of the probabilities of these motions exceeding large critical angles is of interest. Standard statistical methodology for such estimation is based on the peaks-over-threshold approach, for which several lessons are drawn from the analysis of the tail structure of the considered random oscillator.