## 3 Mathematical Model
### Water Loads Generated from Parallel Motion  
We surveyed the water loads acting on the four hulls in the \(x\) and \(y\)-axis, and the moment around the \(z\)-axis due to parallel motion by conducting oblique towing tests using the \(1/3\) scaled model. Equation (5) expresses \(X_{H}\), \(Y_{H}\), and \(N_{H}\).  
\[\left\{\begin{aligned} & X_{H(u,\,v,\,0)}=\frac{1}{2}\rho LdU^{2}X^{ \prime}_{H}(u^{\prime},\,v^{\prime},\,0)\\ & Y_{H(u,\,v,\,0)}=\frac{1}{2}\rho LdU^{2}Y^{\prime}_{H}(u^{ \prime},\,v^{\prime},\,0)\\ N_{H(u,\,v,\,0)}=\frac{1}{2}\rho L^{2}dU^{2}N^{\prime}_{H}(u^{ \prime},\,v^{\prime},\,0)\end{aligned}\right. \tag{5}\]  
Here \(\rho\) is the density of water, and \(L\) is the representative length of the vessel (the diagonal length while in the DP configuration, Figure 5). \(d\) is the planned draft, and \(U\) is the advanced speed defined as \(U\equiv\sqrt{u^{2}+v^{2}}\). \(X^{\prime}_{H}\), \(Y^{\prime}_{H}\), and \(N^{\prime}_{H}\), which correspond to the hydrodynamic derivatives, \"\({}^{\prime}^{\prime}\)\" means a dimensionless value. In this study, these hydrodynamic derivatives were considered functions of the dimensionless surge (\(u^{\prime}\)) and sway (\(v^{\prime}\)) velocities.  
Since these forces vary based on the drift angle, the fluid force was measured in the oblique towing test at two towing speeds for a range of drift angles. Figure 7 shows the tank test setup, and Table 3 outlines the test conditions.  
Figure 7: Oblique towing test.  
Furthermore, when the dimensionless forces and velocities are considered:  
\[X_{H}^{\prime},\ Y_{H}^{\prime}=X_{H},\ X_{H}/\frac{1}{2}\rho LdU^{2},\ u^{ \prime},\ v^{\prime},=u,\ v/U \tag{6}\]  
Figure 9 shows the dimensionless forces as the vertical axis and \(u^{\prime},v^{\prime}\) as the horizontal axis. The dotted line approximates the linear relationship, which sufficiently outlines the experimental results. We treat the line slope as the hydrodynamic derivatives by the least-square method.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Value** & **Unit** \\ \hline Towing speed (\(U\)) & 0.2, 0.4 & m/s \\ Drift angle (\(\beta\)) & 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 60, 75 & deg. \\ \hline \hline \end{tabular}
\end{table}
Table 3: Oblique towing test conditions.  
Figure 8: Variation of the fluid force with the drift angle: (**a**) \(x\)-axis; (**b**) \(y\)-axis.  
The moment due to parallel motion was thought to be generated. Figure 10a shows the measurement results of the moment in the oblique towing test. It can be clearly seen that the moment fluctuated wildly at a drift angle between 20 and 45 degrees, and peaked at a maximum of around 35 degrees. Also, when the moment was set as dimensionless, we used the following equation:  
\[N_{H}^{\prime}=\frac{N_{H}}{\frac{1}{2}\rho L^{2}dU^{2}} \tag{7}\]  
Here we use the dimensionless velocity component \(v^{\prime}\) as the horizontal axis, and Figure 10b shows their relationship. The dimensionless moments have the same trends and roughly the same value at a \(v^{\prime}\) independent of advancing speed. The dotted line is a cubic curve approximation based on the value of practical velocity (\(U\) = 0.2 m/s). It has been used to calculate the part of the moment generated from parallel motion.  
Figure 10: (**a**) Variation of the yaw moment along the \(z\)-axis with the drift angle; (**b**) Variation of the dimensionless yaw moment along the \(z\)-axis with \(v^{\prime}\).  
Figure 9: (**a**) Variation of the dimensionless fluid force along the \(x\)-axis with \(u^{\prime}\); (**b**) Variation of the dimensionless fluid force along the \(y\)-axis with \(v^{\prime}\).