## 2 Numerical Modeling of Ropes and Large Floating Structures Based on the ANCF
### Numerical Solution of the Equation of Motion  
According to linear structural dynamics, internal forces vary with structural deformation, while external forces remain constant. The variation in internal forces is linearly related to structural coordinates and is calculated using the tangent stiffness matrix. In each iteration, internal forces and tangent stiffness matrices for subsequent time steps must be computed separately. Therefore, during the analysis process, the residual of each iteration step is crucial for evaluating convergence, numerical computation accuracy, and robustness. The expression for the residual is provided by the following equation:  
\[\left\{ \begin{array}{c}rs_{t}=\left(\mathbf{F}_{W}+\mathbf{F}_{el}\right)_{t }-\left(\mathbf{F}_{E}+\mathbf{F}_{Q}\right)_{t}-\left(\mathbf{M}\mathbf{ \tilde{q}}\right)_{t}\\ rs_{t+At}=\left(\mathbf{F}_{W}+\mathbf{F}_{el}\right)_{t+At}-\left(\mathbf{F} _{E}+\mathbf{F}_{Q}\right)_{t+At}-\left(\mathbf{M}\mathbf{\tilde{q}}\right)_{ t+At}\\ ||rs||\leq tol\end{array} \right. \tag{49}\]  
where \(rs\) represents the residual and \(tol\) is the residual tolerance. The iteration convergence criterion of the program is established by the displacement-based method, which is predicated on the condition that the variation in node positions falls below the permissible limit, implying that the convergence of the residual at time \(t+\Delta t\) must be satisfied by the residual at time \(t\). Due to the nonlinear variations in structural internal forces, the dynamic time-domain analysis of the equilibrium equations of ANCF flexible cables is necessitated. This paper utilizes the generalized-\(\alpha\) method [43] for time-domain analysis, which exhibits second-order accuracy, unconditional stability, and optimal numerical dissipation performance, rendering it highly suitable for time-domain analysis of nonlinear systems. The generalized-\(\alpha\) integration scheme is a highly effective and robust integration method in structural dynamics finite element analysis, which utilizes the kinematic response \(u^{t},\dot{u}^{t},\dot{u}^{t}\) at time \(t\) and the displacement increment \(\Delta u^{t+\Delta t}_{t}\) at time \(\Delta t\) to derive the dynamic response at the subsequent time step \(t+\Delta t\). The depiction of its solution process is outlined as follows. \[\left\{ \begin{array}{c}u^{t+\Delta t}=\Delta u^{t+\Delta t}_{t}+u^{t}\\ \dot{u}^{t+\Delta t}=\frac{\gamma}{\beta\Delta t}\Delta u^{t+\Delta t}_{t}- \left(\frac{\gamma}{\beta}-1\right)\dot{u}^{t}-\left(\frac{\gamma}{2\beta}-1 \right)\Delta t\ddot{u}^{t}\\ \ddot{u}^{t+\Delta t}=\frac{1}{\beta\Delta t^{2}}\Delta u^{t+\Delta t}_{t}- \frac{1}{\beta\Delta t}\ddot{u}^{t}-\left(\frac{1}{2\beta}-1\right)\ddot{u}^{ t}\end{array} \right. \tag{50}\]  
In the time-domain analysis process utilizing the generalized-\(\alpha\) method, the dynamic calculations for time steps \(t\) to \(t+\Delta t\) must satisfy the convergence of the residual at time \(\tau\) (Equation (49)). In each time step's Newton-Raphson iteration process, the calculation of the residual during the \(i-th\) iteration is as follows:  
\[\left\{ \begin{array}{c}\left(rs^{t+(1-\alpha)\Delta t}\right)_{i}=\alpha \cdot rs^{t}+(1-\alpha)\left(rs^{t+\Delta t}\right)_{i}\\ \alpha\cdot rs^{t}=\alpha_{f}\left(\mathbf{F}^{t}_{ext}-\mathbf{F}^{t}_{\text{ int}}\right)-\alpha_{m}\mathbf{M}\ddot{u}^{t}\\ \left(1-\alpha\right)\left(rs^{t+\Delta t}\right)_{i}=\left(1-\alpha_{f}\right) \left[\left(\mathbf{F}^{t+\Delta t}_{ext}\right)_{i}-\left(\mathbf{F}^{t+ \Delta t}_{\text{int}}\right)_{i}\right]-(1-\alpha_{m})\mathbf{M}\left(\ddot{u }^{t+\Delta t}\right)_{i}\end{array} \right. \tag{51}\]  
where \(\alpha\) is the introduced parameter. Once the displacement increment \(\left(\Delta u^{t+\Delta t}_{t}\right)_{i}\) at time \(\Delta t\) is obtained, the displacement, velocity, and acceleration at the next time step can be derived through Equation (50).