## 5 Results and Discussion
### Manoeuvring Tests  
#### 5.2.1 Turning Circle Test  
When the rudder angle was given a constant value, the SFRM would run in a circle with the direction of its course changing \(360^{\circ}\). In the turning circle test, the tactical diameter, advance, transfer, drift angle, and speed loss can be measured. At the beginning, the SFRM  
Figure 11: The trim polygon of the SFRM.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Step** & **Trimming Moment [N\(\cdot\)m]** & **Mass [g]** \\ \hline
1 & 11.76 & 0 \\ \hline
2 & 16.63 & 600 \\ \hline
3 & 13.1 & 1200 \\ \hline
4 & 1.34 & 1200 \\ \hline
5 & \(-4.54\) & 1200 \\ \hline
6 & \(-9.14\) & 600 \\ \hline
7 & \(-5.61\) & 0 \\ \hline
8 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Variations of trimming moment and mass in different steps.  
sailed straight for 100 s. Subsequently, the rudder was then pulled to 20\({}^{\circ}\) immediately for another 100 s for a total simulation time of 200 s. In the turning circle test, the exact turning time of the rudder was set to be 1 s.  
The main purpose of the turning circle test was to evaluate the turning performance of the SFRM. Figure 12 compares the trajectories of the simulated SFRM at three sailing speeds: \(U\) = 0.5, 1, and 1.5 m/s. It is clearly found that the straight parts of these three trajectories are not coincident due to the variation of the critical point [35], which will affect the pitch angle and the straight-line stability. When executing the turning circle tests, the advances of the SFRM are 44.8, 95, and 145.6 m, corresponding to three sailing speeds, i.e., \(U\) = 0.5, 1, and 1.5 m/s. On the other hand, the transfers are identical to 4.61, 4.52 and 4.47 m, whereas the tactical diameters are 9.79, 9.77, and 0.68 m, respectively. In contrast to advance, both transfer and tactical diameter become small with the increase of the sailing speed.  
When the speed was near 1.5 m/s, the SFRM started to exhibit slight vertical vibrations. It is obvious that the distance and trajectory of the SFRM were evidently affected by the sailing speed. With the increase of the sailing speed, the trajectory's vertical vibration becomes evident due to the increase of hydrodynamic trimming moment.  
Figure 13 presents the time series of velocity components at different sailing speeds in each degree of freedom, including surge (\(u\)), sway (\(v\)), heave (\(w\)), roll (\(p\)), pitch (\(q\)), and yaw rates (\(r\)). The variations of the 6-DOF velocity components were found to be dominated by the sailing speeds. This was particularly evident after the rudder was steered, as lateral resistance caused a considerable decrease in surge velocity (Figure 13a). Additionally, different speed settings caused the heave velocity and the pitch velocity in Figure 13c,e, respectively, to undulate slightly when the sailing speed grew high but remained rather stable when the sailing speed was low.  
Figure 12: The SFRM’s trajectories at different sailing speeds in the turning circle test.  
Figure 13: Time series of 6-DOF velocity components during the turning circle test at different sailing speeds. (**a**) Surge rate: \(u\) [m/s], (**b**) sway rate: \(v\) [m/s], (**c**) have rate: \(w\) [m/s], (**d**) roll rate: \(p\) [rad/s], (**e**) pitch rate: \(q\) [rad/s], and (**f**) yaw rate: \(r\) [rad/s].