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Exponential-polynomial closure method; filtered white noise; Fokker–Planck–Kolmogorov equation; nonlinear ship rolling.

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.

WEN-AN JIANG¹ XIU-JING HAN¹ LI-QUN CHEN^{2 3} QIN-SHENG BI¹

1. Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China
2. School of Science, Harbin Institute of Technology, Shenzhen 518055, China
3. Department of Mechanics, Shanghai University, Shanghai 200072, China

Published

8 July 2020

Manuscript received

17 November 2019

Manuscript revised

9 February 2020

Accepted

18 February 2020