## 3 Mathematical Model
### Propeller Thrust  
The Quadmaran vessel was equipped with four propellers as shown in Figure 12a: one for each hull. \(X_{P}\) and \(Y_{P}\) as shown in Equation (2) were the resultant propeller thrusts along the \(x\) and \(y\)-axis. From Figure 6, we can see that \(X_{P}\) was the resultant thrust force of propellers No. 1 and 3, while \(Y_{P}\) was the resultant thrust force of propellers No. 2 and 4. For each propeller, the thrust was generally positively correlated with water density, rotational speed, and operating area. Equation (9) is adopted to express the thrust for the propeller in this study [19].  
\[T_{P}=(1-t)K_{T}\rho n_{P}^{2}D_{p}^{4} \tag{9}\]  
Figure 11: (**a**) Variation of the yaw moment due to static rotation motion along the \(z\)-axis with the yaw rate; (**b**) Yaw moment coefficient.  
Here, \(t\) is the thrust deduction coefficient, \(\rho\) is the water density, \(n_{P}\) is the rotational speed, \(D_{P}\) is the diameter of the propeller, \(K_{T}\) is the thrust coefficient, which is represented as a function of the advance ratio \(J\), with \(J\) being defined as follows:  
\[J=\frac{\left(1-w_{p}\right)V_{a}}{n_{P}D_{P}} \tag{10}\]  
\(V_{a}\) is the speed for each hull. \(W_{p}\) is the effective wake fraction factor, which is a coefficient representing the decrease in the flow velocity near the propeller due to the effect of the forward hull. The value measured through the wake fraction measurement test conducted by [PERSON] et al. has been used [7].  
In regards to the thrust deduction coefficient (\(t\) in Equation (9)), since the navigation speed while in DP is slow, the increase in hull resistance due to the operation of the propeller can be ignored. It is appropriate to treat the thrust deduction coefficient as 0 based on the results of the test of the thrust deduction coefficient.  
The thrust coefficients (\(K_{T}\)) were surveyed using a Propeller Open Test (POT). The rotation speed, rotation direction, and traveling direction of four propellers were constantly changing depending on the vessel's traveling state while in DP. Therefore, it was necessary to understand the characteristics of the propeller in all four quadrants. The POT was thus conducted in the circulating water tank as shown in Figure 11(b). The test conditions considered were that the propeller rotation was in a clockwise and a counterclockwise direction, and the hull was traveling in headway and sternway directions. Since the water in the circulating water tank could only flow in one direction, we changed the configuration direction of the propeller as an alternative. Table 4 shows the POT conditions.  
The change of \(K_{T}\) is the independent performance curve of the propeller. As shown in Figure 13, especially in Quad 1 and Quad 2, \(K_{T}\) tended to be smaller as \(J\) (absolute value) increased. The curve of Quad 3 and Quad 4 was much smoother when the propeller reversed. This study used the thrust coefficient at the advance ratio \(J=0\) for Quad 3 and 4, as proposed by [PERSON] et al. [20].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Value** & **Unit** \\ \hline Flow velocity (\(u_{a}\)) & 0.0, 0.1, 0.2, 0.4, 0.5, 0.7, 1.0 (0.7, 1.0 only for Quad 1) & m/s \\ Rotational speed (\(n_{p}\)) & 976 & rpm \\ \hline \hline \end{tabular}
\end{table}
Table 4: Propeller Open Test conditions.  
Figure 12: (a) Propeller; (b) Propeller Open Test.  
\(K_{T}\) is expressed as a function of \(J\), as shown in the following equation:  
\[K_{T}=k_{0}+k_{1}J+k_{2}J^{2} \tag{11}\]  
Here, \(k_{0}\), \(k_{1}\), and \(k_{2}\) are coefficients representing the characteristics of \(K_{T}\) based on the measurement data.