Abstract: The International Maritime Organization (IMO) developed the second generation of intact stability criteria, which includes the parametric rolling. In the direct stability assessment, numerical simulation is the most direct method to study the rolling response and stability. However, the analytical analysis method is still necessary because ship capsize is a rare event. The stochastic averaging method of energy envelope can accurately solve the random motion response under the excitation of white noise. However, real waves in the practical engineering correspond to a narrow-band spectrum. The research on the applicability of the stochastic averaging method shows that the system damping and the excitation intensity should be small enough and the excitation bandwidth should be large enough for the non-white noise excitation. An improved method of the stochastic averaging method of energy envelope under the narrow-band spectral excitation is studied based on the above situation. The phase angle is introduced into the original stochastic averaging method of energy envelope by referring to the standard stochastic averaging method. The nonlinear restoring force can be considered, and the limitation of the ideal white noise excitation required by the original method is relaxed to make it possible for the narrow-band excitation. The stationary probability density function (PDF) of the roll angle obtained by solving the Fokker-Planck-Kolmogorov (FPK) equation agrees well with that of the Monte Carlo method. The improved stochastic averaging method of energy envelope can accurately calculate the rolling response of the ship under the excitation of real waves and evaluate the ship roll stability. It can provide guidance for the safe navigation of the ship.

Key words: parametric rolling, narrow-band spectrum, stochastic averaging method of energy envelope, Fokker-Planck-Kolmogorov (FPK) equation

摘要： 国际海事组织（IMO）制定了第二代完整稳性衡准,其中包括参数激励横摇。在第3层直接稳性评估中,研究横摇运动响应和稳性最直接的方法是数值模拟。但由于船舶倾覆是小概率事件,因此解析分析手段依然十分必要。能量包线随机平均法能够准确求解白噪声激励下的随机运动响应,但实际工程中真实的海浪为窄带谱。针对随机平均法求解的适用性研究表明,对于非白噪声激励,要满足系统的阻尼和激励强度足够小、激励带宽足够大的条件。基于以上背景,研究了窄带谱激励下能量包线随机平均法的改进方法。在原有的能量包线随机平均法基础上借鉴标准随机平均法引入相角,保留了原有的能量包线随机平均法可以考虑非线性恢复力的优点,同时放宽了原有方法所要求的理想白噪声激励限制,使其可以为窄带随机激励。求解Fokker-Planck-Kolmogorov（FPK）方程得到的横摇角稳态概率分布与蒙特卡洛法对比良好。改进后的能量包线随机平均法能够准确计算真实海浪激励下船舶随机参数激励横摇响应,对船舶横摇稳性进行评估,为船舶安全航行提供指导。

关键词: 参数激励横摇, 窄带谱, 能量包线随机平均法, FPK方程