Probability density function solution to nonlinear ship roll motion excited by external Poisson white noise

Abstract

The stationary probability density function (PDF) solution to nonlinear ship roll motion excited by Poisson white noise is analyzed. Subjected to such random excitation, the joint PDF solution to the roll angle and angular velocity is governed by the generalized Fokker-Planck-Kolmogorov (FPK) equation. To solve this equation, the exponential-polynomial closure (EPC) method is adopted. With the EPC method, the PDF solution is assumed to be an exponential-polynomial function of state variables. Special measure is taken such that the generalized FPK equation is satisfied in the average sense of integration with the assumed PDF. The problem of determining the unknown parameters in the approximate PDF finally results in solving simultaneous nonlinear algebraic equations. Both slight and high nonlinearities are considered in the illustrative examples. The analysis shows that when a second-order polynomial is taken, the result of the EPC method is the same as the one given by the equivalent linearization (EQL) method. The EQL results differ significantly from the simulated results in the case of high nonlinearity. When a fourth-order or sixth-order polynomial is taken, the results of the EPC method agree well with the simulated ones, especially in the tail regions of the PDF. This agreement is observed in the cases of both slight and high nonlinearities.