It is shown that the equivalent non-linear equation method (ENLE) can be used to obtain the approximate solutions of the non-linear rolling equation of ship motion using the results of the average method and that the proposed scheme is to be preferred over the average method when the damping effects become dominant.
The present study investigates a method of predicting nonlinear rolling motion of a ship in irregular waves. Since the nonlinear nature of the rolling motion of a ship in waves is very complicated, a complete analytic solution of the problem has not been proposed so far. In the past, an equivalent linearization approach has been widely adopted in random vibration problems for nonlinear systems. But when the nonlinearity is strong, the higher order moment statistics may not be reliable. A better approximate method known as the equivalent nonlinear equation method has been introduced. Most of the proposed equivalent non-linear equation methods have treated non-linear damping, but only in the special case of linear restoring. In this study, the equivalent nonlinear equation method largely developed in the field of random vibration was adopted to solve the problem of nonlinear rolling motion of a ship with non-linear restoring in irregular waves.
When a dynamic system is subjected to white noise excitation, the exact solution can be obtained through the Fokker -Plank- Kolmogorov (FPK) equation. A few exact analytical solutions to FPK equations exist for random vibration problems.
The difference between the ENLE method and the average method becomes large. The differences between the ENLE and the average method, in the case of large damping, are thought to be due to the fact that the average method assumes light damping. Similar features are also shown in the figures 8 and 9.