## Mathematical Model and Solution
### Mathematical Model
The external forces on the lumped-mass points are mainly the gravitational forces, the buoyancy forces, and the hydrodynamic forces. The temperature varies with water depth, resulting to the variation of physical and mechanical properties of the sea water and the cable-payload system, such as the density and viscosity. However, the variation of these properties is usually insignificant. For example, the sea water's density variation from 0 to 30 \({}^{\circ}\)C is usually less than 1%, which is neglectable for a cable-payload system. Therefore, temperature variation of the seawater and its effects are neglected in the present study. A constant water density is assumed, and the gravitational (\(F_{g}^{i}\)) and buoyancy (\(F_{b}^{i}\)) forces on the \(i\)th mass point are expressed as  
\[F_{g}^{i}=\left\{ \begin{array}{ll}\rho_{l}\Delta Lg&\quad\quad\quad\quad for \ i=1,\ \ldots,\ N-1\\ \big{(}m_{p}+\frac{1}{2}\rho_{l}\Delta L\big{)}g&\quad\quad\quad for\ i=N \end{array} \right., \tag{6}\]  
\[F_{b}^{i}=\left\{ \begin{array}{ll}\rho_{w}\pi D_{r}\Delta Lg&\quad\quad\quad \quad\quad for\ i=1,\ \ldots,\ N-1\\ \rho_{w}\big{(}V_{p}+\frac{1}{2}\pi D_{r}\Delta L\big{)}g&\quad\quad\quad for\ i=N \end{array} \right., \tag{7}\]  
where \(\rho_{w}\) is the sea water density; \(g\) is the gravitational acceleration; \(D_{r}\) is the outer diameter of the rope; and \(V_{p}\) is the displaced water volume by the payload. As the cross-sectional area of the cable is small, much smaller than the payload, the added mass of the cable is neglected in the model. The hydrodynamic drag force (\(F_{d}^{i}\)) on each segment is also lumped, then, on each mass point:  
\[F_{d}^{i}=\left\{ \begin{array}{ll}\frac{1}{16}C_{d,p}\rho_{w}\pi D_{r} \Delta L\big{[}\big{(}\dot{u}_{i}+\dot{u}_{i+1}\big{)}\big{|}\dot{u}_{i}+\dot{ u}_{i+1}\big{|}+\big{(}\dot{u}_{i}+\dot{u}_{i-1}\big{)}\big{|}\dot{u}_{i}+ \dot{u}_{i-1}\big{|}\big{]}\\ \frac{1}{16}C_{d,p}\rho_{w}\pi D_{r}\Delta L\big{(}\dot{u}_{i}+\dot{u}_{i-1} \big{)}\big{|}\dot{u}_{i}+\dot{u}_{i-1}\big{|}+\frac{1}{2}C_{d,p}\rho_{w}A_{p} \dot{u}_{i}\big{|}\dot{u}_{i}\Big{|}\end{array} \right.,\qquad \begin{array}{ll} for\ i=1,\ \ldots,\ N-1\\ for\ i=N\end{array} \right., \tag{8}\]  
in which, \(C_{d,r}\) and \(C_{d,p}\) are the hydrodynamic drag coefficients of the cable and the payload, respectively, and \(A_{p}\) is the vertical projected area of the payload. The drag coefficient of the payload may vary with water depth. With a lack of drag coefficient data, it is assumed to be constant in the present study, in order to simplify the analysis. However, it should be noted that the present model is capable of considering varying drag coefficients without major modifications. For the payload, there is also an inertia force acting on it, which is also known as the added mass effect. This force (\(F_{a}^{N}\)) is given as  
\[F_{a}^{N}=-m_{a}\ddot{u}_{N}, \tag{9}\]in which, \(m_{a}\) is the added mass due to the inertia effect, and \(\ddot{u}_{N}\) is the acceleration of the payload. Applying [PERSON]'s second law for each mass point, the equilibrium equations are given as  
\[m_{i}\ddot{u}_{i}=F_{g}^{i}-F_{b}^{i}-F_{k}^{i}-F_{c}^{i}+F_{k}^{i+1}+F_{c}^{i+1} -F_{d}^{i},\quad\text{for $i=1,\ldots,\ N-1$,} \tag{10}\]  
\[(m_{N}+m_{a})\ddot{u}_{N}=F_{g}^{i}-F_{b}^{i}-F_{k}^{i}-F_{c}^{i}-F_{d}^{i}\, \quad\text{for $i=N$} \tag{11}\]  
To consider the lowering velocity, the boundary condition at the crane tip is defined as  
\[u_{0}=v_{l}t+S(t), \tag{12}\]  
\[\dot{u}_{0}=v_{l}+\dot{S}(t), \tag{13}\]  
in which, \(v_{l}\) is the lowering velocity, \(t\) is the time, and \(S(t)\) is the crane tip displacement. In the present study, the crane tip motion is not coupled with the cable-payload system to simplify the numerical model, and the neglection of coupling tends to give a conservative result. A sinusoidal movement is assumed for the crane tip, which is given as  
\[S(t)=H_{s}sin(2\pi t/T_{s}), \tag{14}\]  
where \(H_{s}\) the crane tip motion amplitude, and \(T_{s}\) the period.