Joint PDF parametric models for the rate of upcrossings of the acceleration for a non-linear stochastic oscillator

The problem considered in this paper is the approximation, with two parametric models, of the tail of the joint probability density function of the acceleration and its derivative for a non-linear stochastic oscillator. The system includes a cubic damping and a cubic restoring force and is excited by wave forces of first and second order. The joint density enables, with the Rice formula, estimation of the rate of upcrossings of high levels for the acceleration. This rate is used to get information about the comfort of a high speed ship.