Probabilistic Solutions to Nonlinear Random Ship Roll Motion

The probability density function (PDF) and the mean up-crossing rate of the responses for nonlinear ship roll oscillations excited by random sea waves are examined. The excitation of random sea waves is approximated as white noise. The ship roll motion is described by a nonlinear stochastic differential equation that includes a nonlinear wave drag force and a nonlinear restoring moment. The PDF and mean up-crossing rate solution of the nonlinear oscillator are investigated with a new approximate method that expressed the PDF as an exponential function with an exponent in the form of a polynomial in the response variable and its derivative. A special measure is taken such that the Fokker-Planck-Kolmogorov equation is satisfied in the weak sense of integration with the assumed PDF. Numerical examples and a comparison with Monte Carlo simulation are given to show the effectiveness of the method in the study of randomly excited nonlinear ship roll motion.