## 2 Numerical Modeling of Ropes and Large Floating Structures Based on the ANCF
### Mass Matrix and External Load Matrix
#### 2.3.3 Morison Equation  
Hydrodynamic forces are evaluated at each Gauss quadrature point and then mapped onto the nodes using shape functions; Equation (36) derives the expression for hydrodynamic forces as Expression \(\mathbf{F}_{el}\):  
\[\mathbf{F}_{el}=\int_{0}^{l_{0}}\mathbf{N}^{T}fdx=\int_{0}^{l_{0}}\mathbf{N}^{ T}\mathbf{F}_{p}dx \tag{38}\]  
where \(\mathbf{F}_{p}\) represents the hydrodynamic forces at point \(p\) on the cable structure. Gauss-Legendre integration is utilized for the numerical computation of the nonlinear integration of fluid resistance:  
\[ \begin{array}{ll}\mathbf{F}_{el}&=\int_{0}^{1}l_{0}\mathbf{N}^{T}(\xi) \mathbf{F}_{p}(\xi)d\xi\\ &=\int_{-1}^{l_{0}}\frac{1}{2}\mathbf{N}^{T}\bigg{(}\frac{1+\xi}{2}\bigg{)} \mathbf{F}_{p}\bigg{(}\frac{1+\xi}{2}\bigg{)}d\bigg{(}\frac{1+\xi}{2}\bigg{)}\\ &=\int_{-1}^{1}\frac{l_{0}}{2}\mathbf{N}^{T}(\zeta_{i})\mathbf{F}_{p}(\zeta_{i} )d\zeta_{i}\end{array}  \tag{39}\]  
where \(\zeta_{i}\) is the orthogonal node and also a Gauss integration point. The Gauss integration function is \(f(\zeta)=\frac{l_{0}}{2}\mathbf{N}^{T}\mathbf{F}_{p}(\zeta)\). To calculate the fluid dynamics acting on the inclined cable segment, the orientation of the cable axis relative to the fluid must be considered. \(\hat{\mathbf{r}}\) and \(\hat{\mathbf{r}}\), respectively, represent the absolute velocity vector and absolute acceleration vector of the point in the global coordinate system, being the first and second differentials with respect to time. The acceleration \(\mathbf{a}_{w}\) and velocity \(\mathbf{v}_{w}\) of water are decomposed into tangential and normal components relative to the cable axis, as illustrated in Figure 4. According to the Morison equation [40], the hydrodynamics \(\mathbf{F}_{p}\) at point \(p\) on a rope structure can be decomposed into drag \(\mathbf{F}_{d}\), inertial force \(\mathbf{F}_{i}\), and added mass force \(\mathbf{F}_{a}\). Therefore, the hydrodynamics experienced at point \(p\) on the rope structure can be written as follows. \[\mathbf{F}_{p}=\mathbf{F}_{d}+\mathbf{F}_{i}+\mathbf{F}_{a} \tag{40}\]The terms of the Morison equation are decomposed and aligned along the tangential \(\tau\) and normal \(n\) directions. Given that the cable structure is characterized as a slender member, the influence of tangential fluid forces on the structure is negligible and typically ignored. There is \(\left\{ \begin{array}{ll}C_{\alpha\tau}=0,&C_{a}=C_{\alpha m}\\ C_{d\tau}=0,&C_{d}=C_{dn}\end{array} \right.\). Consequently, the total hydrodynamics can be re-expressed using the Morison equation as follows:  
\[ \begin{array}{ll}\mathbf{F}_{p}&=\mathbf{F}_{d}+\mathbf{F}_{i}+\mathbf{F}_{a }=\mathbf{F}_{dn}+\mathbf{F}_{in}+\mathbf{F}_{an}\\ &=\frac{1}{2}\rho_{w}C_{dn}A_{w}(\mathbf{v}_{wn}-\tilde{\mathbf{r}}_{n}) \big{|}V_{wn}-\tilde{\mathbf{r}}_{n}\big{|}+\rho_{w}V_{w}\dot{\mathbf{v}}_{wn}+ \rho_{w}C_{an}V_{w}(\dot{\mathbf{v}}_{wn}-\tilde{\mathbf{r}}_{n})\end{array}  \tag{41}\]  
where \(C_{a}\) and \(C_{d}\) represent the added mass coefficient and drag coefficient, respectively; \(\rho_{w}\) represents the fluid density; and \(V_{w}\) and \(A_{w}\) represent the main volume and area perpendicular to the direction of water flow. Additionally, it is essential to clarify the direction of point \(p\) on the cable, which is modeled as a cylinder, represented by the unit tangent vector \(\mathbf{\tau}_{p}=\frac{\dot{\mathbf{r}}_{p}}{\big{|}\dot{\mathbf{r}}_{p}\big{|} }=\frac{\frac{\dot{\mathbf{w}}(x-p)}{2n}}{\big{|}\frac{\dot{\mathbf{w}}(x-p) }{2n}}\), which is the derivative of the finite element interpolation function with respect to the local coordinate \(x\). The tangential projection matrix \(\mathbf{T}_{p}\) and normal projection matrix \(\mathbf{L}_{p}\) at point \(p\) can be defined by \(\mathbf{\tau}_{p}\).