The improvement of stochastic averaging method to solve the ship rolling response excited by narrow-band waves

The stochastic averaging method of energy envelope can accurately solve the random motion response under the excitation of white noise. The noise had to fulfill the condition of strong mixing which roughly means that the process has to be broad-band. However, the sea conditions in practical engineering conditions are narrow-band spectra. Research on applicability of the stochastic averaging method of energy envelope shows that for narrow-band excitation, the damping of the system and excitation intensity should be small, and the excitation bandwidth should be large. Based on the above background, the improvement of stochastic averaging method of energy envelope under narrow-band waves is studied. The phase angle θ is introduced to characterize the relationship between kinetic energy and potential energy of the system. The drift coefficients and diffusion coefficients are averaged with respect to the period of θ. When the excitation frequency is in the lock-in field, large parametric rolling appears. The stationary probability density function (PDF) of the roll angle obtained by solving the Fokker-Planck-Kolmogorov (FPK) equation compares well with Monte Carlo simulation in the lock-in field. The improved method can relax the limitation of broad-band excitation required by stochastic averaging method. It can also consider both the nonlinear damping and nonlinear restoring force of roll motion. Therefore, the improved method is an efficient way to calculate the nonlinear roll response of ships in real sea conditions.