## 4 Dynamic Analysis of the PIM System
### Motion Response of the PIM  
The deployment process is depicted in Figure 10, with the parameters of the cable and the floating structure detailed in Tables 5 and 6. To analyze the dynamic characteristics of the suspension system, the cable is segmented into 30 elements using the ANCF method. Using the catenary method, the cable is segmented into 100 elements. Assuming the initial ocean current velocity is 0.1 m/s, with the direction along the \(x\)-axis oriented horizontally to the right, the variations of the coordinates x and y of the floating structure's deployment position, and the axial force at the connection point between the cable and the structure are simulated and depicted in Figure 11.  
Figure 10: Model coordinate system. The analysis of the free-fall stage concentrates on the overall response of the suspension system. Consequently, the detailed local responses of the payload during free fall are omitted. This limitation manifests in the simplification of coupled modeling, specifically, as follows. (1) Initially, the cable assumes an ideal fully stretched state before the suspension swings. (2) The influence of the workboat's motion response on the multibody system is excluded. (3) Water flow is modeled as an ideal gradient from large to small. (4) The impacts of wind and waves are disregarded. From Figure 11, the displacement of the structure as calculated by the ANCF method closely resembles that of the catenary method. The error in the displacement of the structure's center of mass between the two methods amounts to only 8%. In the first 75 s, the vertical displacement rate of the structure in the ANCF model lags slightly behind that of the catenary model, and exceeds that of the catenary model between 75 and 95 s. Additionally, the structure reaches its lowest point for the first time at approximately 112 s, followed by a small amplitude asymmetric harmonic oscillation (the \"oscillation zone\"), and achieves stability at the lowest point at 350 s. In summary, the modeling method proposed in this paper proves to be feasible and stable, suitable for simulating the deployment process of structures. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Definition** & **Numerical** & **Unit** \\ \hline Top Diameter & 4 & m \\ Bottom Diameter & 5.5 & m \\ Height & 3.5 & m \\ Mass & \(5.5\times 10^{5}\) & t \\ Initial Coordinates \([\mathrm{X_{a}},\mathrm{Y_{a}},\mathrm{Z_{a}}]\) & [1700, 0, 0] & m \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of the large floating structure. Figure 11: (**a**) Time history diagram of axial force at the structural connection and horizontal displacement at the center of mass. (**b**) Vertical displacement at the structure’s center of mass. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Definition** & **Numerical** & **Unit** \\ \hline Axial Stiffness & \(3.93\times 10^{4}\) & kN \\ Bending Stiffness (EI) & 0.982 & kN-m\({}^{2}\) \\ Initial Length & 1700 & m \\ Outer Diameter & 0.05 & m \\ Inertia Coefficient (Ca) & 1.2 & / \\ Drag Coefficient (Cd) & 1.2 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of the cable. The simulation of the deployment process illuminates the variation in tension at the connection point between the cable and the floating structure and compares this variation with the displacement of the floating structure (see Figure 12). The analysis indicates that, in the initial stage of deployment, the cable rapidly tightens, resulting in significant tension at the connection point. The tension predicted by the ANCF sharply increases to a peak within 10 s, followed by fluctuations within 100 s, culminating in a \"resonance zone\". During this period, the system may encounter challenges from the dynamic amplification effect, leading to significant displacement and tension fluctuations in the equipment.