## Mathematical Model and Solution
### Numerical Solution  
The discrete system is comprised of a series of elements (e.g., segments) and nodes (e.g., mass points). The above equations can be reformulated into a set of nonlinear ordinary differential equations. They can then be reformatted into the matrix form as Equation (15), which can be efficiently solved with the fourth-order Runge-Cutter algorithm implemented in MATLAB:  
\[[M]\big{[}\ddot{u}\big{]}=[D][u]+[B_{c}]\big{[}\ddot{u}\big{]}-[B]\big{[} \widetilde{\dot{u}}\big{]}\star\big{[}\widetilde{\dot{u}}\big{]}+[f], \tag{15}\]  
in which, \([M]\) is the mass matrix; \([D]\) is the rigidity matrix; \([B_{c}]\) is the damping matrix; \([B]\) is the viscous matrix; \([f]\) is the load vector; \([u]\), \(\dot{u}\big{]}\), \(\big{[}\widetilde{\dot{u}}\big{]}\) are the displacement, velocity, and acceleration vectors of the nodes, respectively; \(\widetilde{\dot{u}}\big{]}\) is the vector of the average velocity of each element; and \(\big{[}\widetilde{\dot{u}}\big{]}\) is the vector of the absolute average velocity of each element. Detailed expressions of these matrixes and vectors are listed in Appendix A.  
Once the problem is numerically solved, the traction in the cable can be obtained. As half of the mass and the drag force of the first segment is lumped into \(m_{0}\), the traction on the top of the cable (\(F_{top}\)) should be calculated by  
\[F_{top}=k_{i}(u_{1}-u_{0})+\frac{1}{2}\rho_{l}\Delta L+\frac{1}{16}C_{d,r}\rho _{w}\pi D_{r}\Delta L\big{(}\dot{u}_{1}+\dot{u}_{0}\big{)}\big{|}\dot{u}_{1}+ \dot{u}_{0}\big{|}, \tag{16}\]  
Similarly, the traction on the top of the payload should be calculated as  
\[F_{top}=k_{i}(u_{N}-u_{N-1})-\frac{1}{2}\rho_{l}\Delta L-\frac{1}{16}C_{d,r} \rho_{w}\pi D_{r}\Delta L\big{(}\dot{u}_{N}+\dot{u}_{N-1}\big{)}\big{|}\dot{u}_ {N}+\dot{u}_{N-1}\big{|}, \tag{17}\]  
As the wire rope is released through the crane tip, the suspended cable length increases. Therefore, the cable discretization should be adjusted during the solution process. In the present study, the variable-domain method is adopted for the numerical solution. That is to say, the length of each element is increased as the cable length increases.  
Each element has an initial length that is one \(N^{th}\) of the whole cable length at the beginning of the solution. Once the wire rope is released, the cable length increases, resulting to an increase of the element length. At each increment of the solution, the element length should be adjusted according to the whole cable length. The solution technique involved is sketched in Figure 3, and the detailed solution process is as follows.  
1. At \(t_{0}\), the beginning of the solution, Equation (15) is solved with the fourth-order Runge-Cutter method to get the displacement and velocity of each mass point, e.g., \(u_{i}^{t0}\) and \(u_{i}^{\cdot t0}\).
2. At \(t0+\Delta t\), the whole length of the cable is increased by \(v_{l}\Delta t\), and then the element length is increased by \(v_{l}\Delta t/N\). Equation (15) is updated according to Appendix A, and solved to get \(u_{i}^{t0+\Delta t}\) and \(\dot{u}_{i}^{\cdot t0+\Delta t}\). Then the traction in the cable can be calculated through Equations (16) and (17).
3. Step (2) is repeated until the dedicated lowering depth is achieved.  
As the wire rope cannot be compressed, the snap load will be induced if the cable is slack. To calculate the possible snap load, the relative displacement between two connected nodes is carefully monitored during the solution. If the relative displacement between the two nodes of the \(i\)th element is negative, then the spring force \(F_{k,i}\) is set to zero.