# Research on Influence of Tank Sloshing on Ship Motion Response under Different Wavelengths
## 5 Simulation of Rolling Motion of Liquid Carrier in Transverse Waves Condition
### Moment Analysis of Liquid-Carrying Ship during Rolling Motion  
When the liquid carrier rolls, it is simultaneously affected by the external wave moment and the sloshing moment of the internal tank. The moment experienced by the ship under different working conditions is similar: the sloshing moment of the tank is always in the opposite direction to the external wave moment, and the periods of the two are consistent and equal to the rolling period of the ship. Figure 12 shows the moments experienced by the ship under some conditions. It can be seen from Figure 12 that when the liquid loading rate is non-zero, the sloshing moment of the tank is always opposite in phase to the external wave moment, and the amplitude of the tank sloshing moment is smaller than that of the external wave moment. The superimposed resultant moment is also smaller than the external wave moment. When the liquid loading rate is 0, the sloshing moment of the tank is close to 0, and the external wave moment and the resultant moment curve basically coincide. During the ship's rolling process, the liquid flow direction in the tank is always the same as the ship's rolling direction, the fluid in the tank is still dominated by static pressure, and the generated moment is consistent with the ship's rolling direction. Due to the large roll angle of the ship, the external moment mainly provides the restoring moment of the ship's roll, and its direction is opposite to the direction of the ship's rolling motion, so the sloshing moment of the tank is always opposite to the direction of the external wave moment. Figures 13 and 14 respectively show the changes of liquid level in the tank and the pressure in the tank when the ship rolls under some conditions. Figure 12: Moment of a liquid-carrying ship when rolling. (**a**) Liquid loading rate 0.8 and wavelength 3 m; (**b**) liquid loading rate 0.8 and wavelength 6 m; (**c**) liquid loading rate 0.8 and wavelength 11 m; (**d**) liquid loading rate 0 and wavelength 11 m.  
Figure 15: Rolling free decay curve of ships with different liquid loading rates. (**a**) Free decay curves for carrier ratios of 0 and 0.2, respectively; (**b**) Free decay curves for carrier ratios of 0.2 and 0.4, respectively; (**c**) Free decay curves for carrier ratios of 0.4 and 0.6, respectively; (**d**) Free decay curves for carrier ratios of 0.6 and 0.8, respectively; (**e**) Free decay curves for carrier ratios of 0.8 and 0.98, respectively. Figure 16 shows the variation of the natural frequency of the ship's roll with the liquid loading rate. It can be seen that the sloshing of the tank reduces the natural frequency of the ship's roll, and the natural frequency of the ship's roll gradually decreases during the change of the liquid loading rate from 0.2 to 0.8. This can be explained from the moment curve in Figure 12: when the roll angle of the ship is not very small, the change of the sloshing moment of the tank is always opposite to the external wave moment, and the superposition of the two reduces the restoring moment of the ship. This results in reduced roll angular acceleration, thus longer roll period and lower natural frequency. The higher the liquid loading rate (0.2-0.8), the greater the sloshing moment of the tank, the smaller the restoring moment after superposition, and the smaller the roll angular acceleration, so the lower the natural frequency of roll. However, when the liquid loading rate is 0.98, the tank is basically in a fully loaded state. Figure 17 shows the change of liquid level in the tank under this liquid loading rate. Since the free liquid surface is close to the top of the tank, the sloshing moment of the tank will be at a lower level, which has little effect on the restoring moment. Therefore, the natural frequency of ship roll will increase. Figure 16: Variation curve of ship rolling natural frequency with liquid loading rate. Figure 17: Rolling free decay motion of ship with a liquid loading rate of 0.98. (**a**) Vertical view; (**b**) midship section.