## 2 Modeling and Simulation
### Thrust Calculation  
The effective thrust of the controllable pitch propeller \(F_{e}\) and the experimental thrust of the water flow \(F_{s}\) can be calculated according to the following equations:  
\[F_{e}=F_{s}(1-t) \tag{1}\]  
\[F_{s}=K_{T}pn^{2}D^{4} \tag{2}\]  
where \(K_{T}\) stands for the thrust coefficient, \(\rho\) indicates the seawater density (kg/m\({}^{3}\)), \(n\) denotes the diesel engine speed (\(r/s\)), \(D\) marks the pitch paddle diameter (\(m\)), and \(t\) represents the thrust derating factor.  
Aiming to facilitate the calculation, the square coefficient value of the hull \(C_{B}\) is typically adopted in engineering to approximate the thrust reduction factor of a single-propeller ship \(t\), and their relationship is presented in Table 1.  
The thrust coefficient is related to the process ratio of the pitched paddle and the pitch angle. The \(K_{T}\) can be expressed as follows:  
\[K_{T}=f(J,\theta) \tag{3}\]  
\[J=\frac{V_{a}}{nD} \tag{4}\]  
The propeller feed speed \(V_{a}\) can be written as:  
\[V_{a}=V_{s}(1-\omega) \tag{5}\]  
where \(V_{s}\) denotes the ship speed (m/s), and \(\omega\) represents the companion flow coefficient.  
Typically, the effect of the hull's squareness coefficient value \(C_{B}\) is considered, and other factors are ignored. According to the squareness coefficient value \(C_{B}\), Table 2 is checked to obtain the approximate value of \(\omega\).  
The pitch drag torque is affected by the torque coefficient \(K_{Q}\), pitch diameter \(D\), water density \(\rho\), and diesel engine speed \(n\), and it can be expressed as follows:  
\[M_{P}=K_{Q}\rho n^{2}D^{5} \tag{6}\]  
The torque coefficient \(K_{Q}\) is influenced by the pitch paddle process ratio \(J\) and the pitch angle \(\theta\). It can be calculated as follows:  
\[K_{Q}=f(J,\theta)\]  
\begin{table}
\begin{tabular}{c c c c} \hline \(C_{B}\) & \(\omega\) & \(C_{B}\) & \(\omega\) \\ \hline
0.52 & 0.15 & 0.68 & 0.27 \\
0.56 & 0.18 & 0.72 & 0.30 \\
0.60 & 0.21 & 0.76 & 0.33 \\
0.64 & 0.24 & 0.80 & 0.36 \\ \hline \end{tabular}
\end{table}
Table 2: The relationship between \(C_{B}\) and \(\omega\).  
\begin{table}
\begin{tabular}{c c c c} \hline \(C_{B}\) & \(t\) & \(C_{B}\) & \(t\) \\ \hline
0.52 & 0.11 & 0.68 & 0.19 \\
0.56 & 0.13 & 0.72 & 0.21 \\
0.60 & 0.15 & 0.76 & 0.23 \\
0.64 & 0.17 & 0.80 & 0.25 \\ \hline \end{tabular}
\end{table}
Table 1: The relationship between \(C_{B}\) and \(t\).  
Due to the calculation complexity, the flow characteristic curve of the four-blade pitch paddle torque coefficient \(K_{Q}-(J,\theta)\) is introduced to solve the torque coefficient \(K_{Q}\) and plotted in Figure 4.  
During the transmission of the controllable pitch propeller, the friction between the drive shaft and the bearings will generate the friction loss torque \(M_{f}\). This torque is related to the rotational speed of the controllable pitch propeller. Due to the comparison with the diesel output torque \(M_{s}\) and the pitching paddle resistance torque \(M_{P}\), the friction loss torque \(M_{f}\) is small. To simplify the calculations, the friction loss torque can be considered constant, with a magnitude of \(M_{f}=0.02M_{H}\). \(M_{H}\) represents the output torque of the diesel engine under the rated operating conditions.