# 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
### Influence of Liquid Loading Rate on Rolling Motion of Ship  
In 5.4, when the liquid loading rate is constant (the natural frequency of rolling is constant), the influence of different incident wave wavelengths on the rolling motion of the ship is analyzed. Next, when the wavelength of the incident wave is constant (the frequency of the incident wave is constant), the difference in the rolling motion of the ship with different liquid loading rates is analyzed. Cases 1, 5, and 9 in Table 4 explore the law of ship rolling motion when the incident wave wavelength is 3 m and the liquid carrier rates are 0.8, 0.4, and 0, respectively. At this wavelength, the stability stage of the time history curve of the ship's rolling motion with different liquid loading rates is compared in Figure 20.  
From Figure 20, it can be seen that the amplitude of the roll angle changes with different liquid loading rates is different: the roll amplitude at 0 liquid loading rate is the largest, and the roll amplitude values when the liquid loading rate is 0.4 and 0.8 are close to and smaller than the roll amplitude value when the liquid loading rate is 0. It can be seen that under this wavelength of the incident wave, the roll angle of the ship becomes smaller after loading the tank with the liquid loading rate of 0.4 and 0.8. At this time, the tank sloshing in the two liquid-carrying states restrains the ship's rolling, and the tank sloshing is beneficial to the ship's rolling motion. This is mainly because the natural frequency of the rolling motion of the ship has changed after loading the tank. The natural frequency of ship roll with a liquid loading rate of 0 is the closest to the incident wave frequency, and the natural frequency of ship roll with a liquid loading rate of 0.4 and 0.8 is far from the incident wave frequency; see Table 9. The following conclusions can be drawn: the effect of tank sloshing on the ship's rolling motion is achieved by changing the natural frequency of rolling to make it different from the frequency of the incident wave. In addition, cases 2, 6, and 10 in Table 4 explore the law of ship rolling motion when the incident wave wavelength is 6 m and the liquid carrier rates are 0.8, 0.4, and 0, respectively. Cases 3, 7, and 11 in Table 4 explore the law of ship rolling motion when the incident wave wavelength is 11 m and the liquid carrier rates are 0.8, 0.4, and 0, respectively. Cases 4, 8, and 12 in Table 4 explore the law of ship rolling motion when the incident wave wavelength is 15 m and the liquid carrier rates are 0.8, 0.4, and 0, respectively. Figure 21 compares the roll time history curves of ships with different liquid loading rates. \begin{table}
\begin{tabular}{c c c} \hline \hline \begin{tabular}{c} **Wavelength (m)** \\ \end{tabular} & \
\begin{tabular}{c} **Incident Wave Frequency** \\ **(rad/s)** \\ \end{tabular} &
\begin{tabular}{c} **Ratio of Incident Wave Frequency** \\ **to Natural Frequency** \\ \end{tabular} \\ \hline
3 & 4.532 & 1.247 \\
6 & 3.205 & 0.882 \\
11 & 2.367 & 0.651 \\
15 & 2.027 & 0.558 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The ratio of the incident wave frequency to the roll natural frequency when the loading rate is 0. Figure 20: Rolling time history curves of ships with different liquid loading rates when the incident wave wavelength is 3 m.  
It can be seen from Figure 21a that when the wavelength of the incident wave is 6 m, the order of the roll angle of the ship with different liquid loading rates is: \(\theta_{0}>\theta_{0.4}>\theta_{0.8}\). Under this condition, the sloshing of the tank restrains the rolling motion of the ship. In addition, it can be seen from Figure 21b that when the wavelength of the incident wave is 11 m, the order of the ship roll angle with different liquid loading rates is: \(\theta_{0.8}>\theta_{0.4}>\theta_{0}\). Under this condition, the sloshing of the tank promotes the rolling motion of the ship.