## 3 Results and Analysis
### Motion Response Characteristics of SPM-GDW Cage  
It is noted that relatively small lateral sway, roll, and yaw motions are exhibited in the single-point mooring cage under wave action. However, there is a strong correlation between the pitch and heave motions of the cage. Towards this end, this section focuses on studying the motion response characteristics of the cage under irregular waves from the perspectives of heave and surge. In Figures 8 and 9, the time history curves of the heave and surge motions of the cage under the combined action of irregular waves and currents are presented, respectively. It should be noted that for ease of analysis and comparison, all time-domain data in this study adopt one-third significant value as the statistical peak value. Table 3 summarizes the peak pitch and heave values under different wave and current parameters. As observed from Figures 8 and 9 and Table 3, with the increase of current velocity, the heave motion of the cage remains relatively unchanged. The statistical results also indicate that as the current velocity increases from 0 m/s to 1.24 m/s, the peak have value changes by only 1%, which suggests that the current velocity has a minimal impact on the heave motion of the cage, which is primarily determined by wave conditions. On the other hand, for the surge motion of the cage, a significant increase is observed with increasing current velocity. There is a strong positive correlation between surge and current velocity. As the current velocity increases from 0 m/s to 1.24 m/s, the peak surge value increases by 189%. The above results proved that during the design of such a novel SPM cage, special attention should be paid to the mooring design at high velocity because the increase of the cage surge motion usually would lead to a significant increase in the initial mooring tension. The response characteristics of cage heave and surge motions under irregular wave and current conditions were analyzed. The time history data in the range from 1000 s to 1250 s were extracted, as shown in Figures 10 and 11. The cage motion curve and wave elevation curve were compared. By observing the time history curves of heave motion and wave elevation in Figure 10, it can be seen that the cage's heave motion is closely related to the changes in wave height, and the cage's vertical movement aligns with the wave elevation. With increasing current velocity, there is a slight delay in the heave motion response relative to the wave elevation, but the impact is minimal. For the cage's surge motion, the lag effect relative to the wave elevation is significant, and it becomes more pronounced as the current velocity increases. The motion response of the cage is shown in Figure 12. The spectral density curves are obtained from the time history curves of the cage's motion. It can be observed that the heave and surge motions of the cage are influenced by wave frequency loads ranging from 0.05 Hz to 0.1 Hz. High-frequency loads (>0.1 Hz) have almost no impact on both the heave and surge motions of the cage. Low-frequency loads (<0.05 Hz) have no influence on the cage's heave motion, but as the current velocity increases, they exert a significant and increasing influence on the cage's surge motion. Moreover, Figure 12 shows that the peak values of the spectral density for heave motion remain virtually unchanged under different current velocities. In contrast, the peak values of the spectral density for surge motion increase significantly with increasing current velocity. This observation agrees with \begin{table}
\begin{tabular}{c c c} \hline
**Case** & **Peak Value of Heave/m** & **Peak Value of Surge/m** \\ \hline Case 1 & 3.91 & 6.09 \\ Case 2 & 3.87 & 11.48 \\ Case 3 & 3.86 & 17.60 \\ \hline \end{tabular}
\end{table} 
Table 3: Statistics for heave and surge motion of cage under irregular wave and current. Figure 9: Time domain curves of cage surge motion response under irregular wave and current. (a) U = 0 m/s; (b) U = 0.62 m/s; (c) U = 1.24 m/s.