## 2 Mathematical Model
### Mooring System Modelling
#### 2.4.2 Dynamic Mooring Model  
In the dynamic mooring model, the lumped-mass method [9] is used to calculate the dynamic simulation of the interconnected cables, the weight and the buoyancy. In this method, as shown in Figure 4a, the mooring cable is separated into \(N\) equally-sized line segments connecting \(N+1\) node points. The index \(i\) starts from the anchoring node at the bottom, in which the value of the anchoring node is given as zero. The index value of the line segment between nodes 0 and 1 is 1/2, and the node at the top of the farlead node is \(N\).  
In the right-handed inertial reference coordinate system, the \(z\)-axis is measured in a positive upward direction from the water plane. The position of each node \(i\) is defined by the position vector \(r_{i}\), containing the x, y and z coordinates of the node position. Each mooring line segment \(i+\frac{1}{2}\) has the same properties of un-stretched length \(L\), volume-equivalent diameter \(d\), Young's modulus \(E\), and the inertial damping coefficient \(C_{int}\). The dynamic mooring model combines internal stiffness, damping force, weight, buoyancy, hydrodynamic forces from Morison equation, and contact force from the seabed, as illustrated in Figure 4b. The internal stiffness and damping force of the line segment \(i+1/2\)are expressed as \(T_{i+(1/2)}\) and \(C_{i+(1/2)}\), respectively. Meanwhile, the weight of the line segment lumped at each node \(i\) is denoted as \(W_{i}\), whereas \(B_{i}\) is the net buoyancy at node \(i\).  
In the lumped-mass (LM) method, the mass of the mooring cable is discretized into point masses at each node by assigning each node half the combined mass of the two adjacent line segments. The \(3\times 3\) mass matrix for node \(i\) can be represented by  
\[m_{i}=\frac{\pi}{4}d^{2}L\rho I \tag{31}\]  
where \(\rho\) is the density of the mooring cable; \(I\) is the identity matrix.  
The added-mass matrix, which is included in the mass matrix, for node \(i\) is expressed as  
\[a_{i}=a_{p_{i}}+a_{q_{i}}=\rho\frac{\pi}{4}d^{2}L\left[C_{an}\left(I-\hat{q}_ {i}\hat{q_{i}}^{T}\right)+C_{at}\left(\hat{q_{i}}\hat{q_{i}}^{T}\right)\right] \tag{32}\]  
where \(a_{p_{i}}\) and \(a_{q_{i}}\) are the corresponding transverse and tangential added-mass matrices, respectively. \(C_{an}\) (=0.865) and \(C_{at}\) (=0.269) are added-mass coefficients in the transverse and tangential directions [9; 14]. \(\hat{q_{i}}\) is defined as the tangent direction at each node and can be approximated as the direction of a line passing between the two adjacent node points by  
\[\hat{q_{i}}=\frac{r_{i+1}-r_{i-1}}{\|r_{i+1}-r_{i-1}\|} \tag{33}\]  
where \(r_{i}\) is the position vector at node \(i\).  
Finally, the motion equation of the mooring cable for each node \(i\) is  
\[\begin{array}{l}F_{M}\\ =\sum\limits_{i=1}^{N}(m_{i}+a_{i})\tilde{r}_{i}\\ =\sum\limits_{i=1}^{N}\left[T_{i+(1/2)}-T_{i-(1/2)}+C_{i+(1/2)}-C_{i-(1/2)}+W_ {i}+B_{i}+D_{pi}+D_{qi}\right]\end{array} \tag{34}\]  
where \(D_{pi}\) and \(D_{qi}\) are drag forces in the transverse and tangential directions, respectively.  
Figure 4: Description of dynamic mooring model: (**a**) the discretization of a mooring cable; (**b**) internal and external forces of a dynamic mooring cable.