In this paper, the probability density function (PDF) and the mean up-crossing rate of nonlinear ship rolling in random beam seas are investigated. The excitation of stationary random sea waves is approximated as a second-order linear filtered white noise. The Fokker–Planck–Kolmogorov (FPK) equation governing the probability density function of ship rolling is a four-dimensional linear partial differential equation with varying coefficients, and obtaining its exact solution is much more sophisticated. The exponential-polynomial closure (EPC) method is applied to solve the corresponding FPK equation of the system. In numerical examples, linear-plus-cubic damping model and linear-plus-quadratic damping model with three different sea states are further examined. Comparison with the equivalent linearisation (EQL) method and Monte Carlo simulated results show that the proposed procedure is effective to obtain a satisfactory probability density function solution, especially in the tail region.

中文翻译：

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随机波束海中非线性船舶横摇的概率解

本文研究了随机波束海中非线性船舶横摇的概率密度函数(PDF)和平均横渡率。静止随机海浪的激励近似为二阶线性滤波白噪声。控制船舶横摇概率密度函数的 Fokker-Planck-Kolmogorov (FPK) 方程是一个具有变系数的四维线性偏微分方程，获得其精确解要复杂得多。应用指数多项式闭包(EPC) 方法求解系统的相应FPK 方程。在数值例子中，进一步研究了具有三种不同海况的线性加三次阻尼模型和线性加二次阻尼模型。