# Research on a Strategy for Prediction Methods of Submarine Self-Propulsion Hydrodynamic Performance
## 5 Propeller Excitation Force Calculation and Analysis under Different Computational Strategies
The fluctuating amplitudes of the propeller bear-ing forces at the BPF were analyzed, with a time-domain selection range of 0.2 s. The analysis results are presented in Table 14.  
As shown in Table 14, the differences in the fluctuating amplitudes of the BPF for the propeller's force in the \(X\)- and \(Z\)-directions, as well as the torque in the \(X\)-, \(Y\)-, and \(Z\)-directions, are minimal across the four computational strategies, with the maximum fluctuation range being 1.81%. However, the fluctuating amplitudes of the force in the \(Y\)-direction show a larger difference between the four strategies, with a maximum fluctuation range of 3.24%. Considering that different time-domain selections during the solution \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \(\mathbf{T_{X}}\) **(BPF)** & \(\mathbf{T_{Y}}\) **(BPF)** & \(\mathbf{T_{Z}}\) **(BPF)** & \(\mathbf{Q_{X}}\) **(BPF)** & \(\mathbf{Q_{Y}}\) **(BPF)** & \(\mathbf{Q_{Z}}\) **(BPF)** \\ \hline strategy 1 & 0.299 & 0.531 & 0.277 & 0.011 & 0.062 & 0.056 \\ strategy 2 & 0.296 & 0.541 & 0.281 & 0.011 & 0.063 & 0.057 \\ strategy 3 & 0.297 & 0.540 & 0.281 & 0.011 & 0.063 & 0.057 \\ strategy 4 & 0.297 & 0.524 & 0.282 & 0.011 & 0.061 & 0.057 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: The fluctuating amplitudes of the BPF in propeller bearing force under different calculation strategies. Figure 19: Time- and frequency-domain curves of blade load (strategy 1). process of the load frequency-domain curve can influence the results, the fluctuations caused by the strategy methods can be considered negligible. Therefore, the effect of different computational strategies on the solution results for the fluctuating amplitudes of the propeller bearing forces at the BPF is not significant. Figure 19 presents the time-domain and frequency-domain curves of the forces and torques in three directions for a single propeller blade under unsteady conditions in strategy 1. As shown in Figure 19, when a single propeller blade completes one full rotation, multiple peaks and troughs appear in the \(X\)-direction force and torque. This phenomenon is primarily due to the formation of multiple high- and low-wake regions by the tail rudder's horseshoe vortex in the propeller disk. When the propeller blade reaches these high- and low-wake regions, the effective angle of attack reaches its maximum and minimum, respectively. In Figure 19, the \(X\)-direction force is the largest, and the torque is the smallest for a single propeller blade, while the differences in the forces and torques in the \(Y\)- and \(Z\)-directions are not significant. Regarding the frequency-domain curves, the primary pulsation frequency of the \(X\)-direction force and torque is four times the first-order shaft frequency (APF), consistent with the main pulsation frequency observed in the literature [34]. In contrast, the main pulsation frequency of the forces and torques in the \(Y\)- and \(Z\)-directions corresponds to the APF. Concerning the magnitude of the fluctuating amplitudes, the \(Z\)-direction force and torque peaks are the largest, with values of 9.310 and 1.460, respectively. The \(Y\)-direction follows, with values of 9.142 and 1.435, while the \(X\) direction has the smallest peaks, with values of 1.073 and 0.039. The differences between the \(Z\)-direction and \(Y\)-direction force and torque peaks are not significant. However, the \(Z\)-direction force and torque peaks are 8.7 times and 37 times larger than those in the \(X\)-direction, respectively, indicating a significant difference. This disparity is closely related to the generation of the tail rudder's horseshoe vortex.