## Mathematical Model and Solution  
### Mathematical Model  
With the lifting installation method, the subsea equipment is usually lowered through a crane mounted on the dedicated vessel. The whole system is comprised of the vessel, the wire rope, and the equipment, forming a vessel-cable-payload system (see Figure 2a). Usually, the equipment's mass  
Figure 1: Schematic diagram of installation methods for subsea equipment: (**a**) installation with drilling strings; (**b**) lifting installation method. is less than 1% of the vessel, so the vessel motion is considered to be independent from the wire rope and subsea equipment [12]. Generally, the cable-payload system, including the wire rope and equipment, is chosen as the research object, while the vessel motion is set as its boundary condition. Due to the relatively small equipment mass and the hydrodynamic damping on the cable-payload system in deep water, the coupling of the payload and the vessel in the horizontal direction is not likely to be significant. Previous work has confirmed that in the absence of time-varying currents or large horizontal excursions of the vessel, the ship and the equipment are only coupled vertically [9; 25]. Therefore, only the vertical movement is considered in the present study. As shown in Figure 2b, the cable-payload system is divided into \(N\) segments. In the discretized model, each segment is massless, as its mass is lumped equally to the two mass points at both ends. The system is comprised of a spring and a damper, taking into account the elastic and structural damping behavior of the wire rope. Each mass point is also externally connected to a damper modelling the hydrodynamic damping, e.g., the nonlinear drag force acting on the cable-payload system. The first mass point \(m_{0}\) is located at the releasing point at the sea surface, e.g., the crane tip. Based on the lumped-mass assumption, the mass of each point (\(m_{i}\)) is expressed as  
\[m_{i}=\left\{ \begin{array}{l}\rho_{l}\Delta L\\ m_{p}+\frac{1}{2}\rho_{l}\Delta L\end{array} \right.\quad,\quad \begin{array}[] {l}for\ i=1,\ \ldots,\ N-1\\ for\ i=N\end{array} \right., \tag{1}\]  
in which, \(\rho_{l}\) is the linear density of the wire rope, \(\Delta L\) is the segment length, and \(m_{p}\) is the payload. The displacement and velocity at the crane tip (the \(m_{0}\) mass point) are denoted as \(u_{0}\) and \(i_{0}\), respectively, expressing the boundary conditions of the discrete system. In addition, the displacement and velocity of the \(i\)th mass point are denoted as \(u_{i}\) and \(i_{i}\), respectively. As shown in Figure 2c, the \(i\)th mass point bears internal forces from the springs and dampers connected at both sides, e.g., \(F_{k}^{i}\) and \(F_{c}^{i}\) from the upper side and \(F_{k}^{i+1}\) and \(F_{c}^{i+1}\) from the lower side. It also endures external forces due to the gravity, buoyancy, and hydrodynamic drag, e.g., the gravitational force \(F_{g}^{i}\), the buoyance force \(F_{b}^{i}\), and the drag force \(F_{d}^{i}\). The exception is that on the \(N\)th mass point, only the internal forces from the spring and damper connected at the upper side are considered. Figure 2: Mechanical model of the subsea equipment installation. (**a**) Schematic diagram of the operation; (**b**) lumped-mass model; (**c**) force balance analysis. For each segment, the spring and damper impose spring force and damping force on the mass points. The spring force in the \(i\)th spring is  
\[F_{k}^{i}=k_{i}(u_{i}-u_{i-1}), \tag{2}\]  
where \(k_{i}\), the axial stiffness of the \(i\)th segment, is defined as  
\[k_{i}=EA/\Delta L, \tag{3}\]  
in which \(E\) is the Young's Modulus of the cable and \(A\) is the cross-section area of the cable. The damping force (\(F_{c}^{i}\)) in the \(i\)th segment is  
\[F_{c}^{i}=c_{i}\big{(}u_{i}-\dot{u}_{i-1}\big{)}, \tag{4}\]  
where \(c_{i}\), the damping coefficient of \(i\)th segment, is defined as  
\[c_{i}=2\zeta\sqrt{k_{i}\rho_{l}\Delta L}\, \tag{5}\]  
in which, \(\zeta\) is damping ratio of the cable [26].