## Mathematical Model and Solution
### Model Validation  
To validate the proposed model, it is compared with the widely recognized commercial software Orcaflex. For the following comparison, as well as for later analysis, a typical subsea equipment installation configuration is used, of which the physical properties are listed in Table 1.  
Figure 3: Schematic diagram of the variant-domain method. (**a**) Initial configuration of the discrete system; (**b**) adjusted configuration of the discrete system with segment length increased.  
In the validation case, the system's response, considering a lowering velocity and a sinusoidal displacement imposed on the crane tip, is evaluated. The cable-payload system is discretized in 30 elements, and the hydrodynamic force on the payload is considered. The response obtained with the proposed model is compared with that from Orcaflex.  
The imposed sinusoidal displacement oscillates, with a period (\(T_{s}\)) of 9 s and an amplitude (\(H_{s}\)) of 0.3 m. Figure 4a,b shows the equipment's motion response at 1500 m and 3000 m water depth, respectively. Comparison of both the amplitude and the phase of payload's response between the two models show that the present model agrees well with the widely recognized software Orcaflex. Figure 5 presents the traction force on the top of the cable, and on the payload when the equipment is lowered to 1500 m depth. It also shows good agreement between the two models. The traction in the cable oscillates as the cable-payload system is imposed with a sinusoidal excitation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Parameter** & **Value** \\ \hline \multirow{6}{*}{Cable} & Linear mass, \(\rho_{l}\) & 24.6 kg/m \\  & Cross-section area, \(A\) & 4098 mm\({}^{2}\) \\  & Axial stiffness product, \(EA\) & 315 MN \\  & Equivalent diameter, \(Dr\) & 72.23 mm \\  & Drag coefficient, \(C_{d,r}\) & 0.7 \\  & Damping ratio, \(\zeta\) & 0.1 \\  & Mass, \(m_{p}\) & 60 t \\  & Volume, \(V_{p}\) & 7.63 m\({}^{3}\) \\ \multirow{2}{*}{Equipment} & Vertical projected Area, \(A_{p}\) & 56.95 m\({}^{2}\) \\  & Added mass, \(m_{a}\) & 300 t \\  & Drag coefficient, \(C_{d,p}\) & 7 \\ \hline \hline \end{tabular}
\end{table}
Table 1: System parameters for the model validation and analysis.  
Figure 4: Comparison of the payload motion under the sinusoidal crane tip excitation (\(H_{s}\) = 0.3 m, \(T_{s}\) = 9 s) with a payout speed of 0.2 m/s at different water depths: (**a**) 1500 m depth, (**b**) 3000 m depth.  
Figure 6 represents the envelopes of the oscillating forces on the top of the cable and on the payload. As for the traction development in the cable, along with the payload lowering, the present model agrees well with the Orcaflex. It is found in Figure 6a that the traction on the top of the cable increases with the lowered depth. This is because the cable length increases when lowering the payload, increasing cable-payload system's weight hung at the top of the cable. It can be seen in Figure 6b that the traction on the equipment significantly increases at around 1500 m depth. At this depth, the first-order natural frequency of the cable-payload system is about 0.12 Hz (period of 8.37 s), which is very close to the excitation frequency, so the resonance occurs for the system.