## 2 Aero-Hydro-Servo-Elastic Model
### Coupling Model  
The theoretical models for the different subsystems of the FOWT are explained in the above subsections. In order to simulate the coupled dynamics, the coupling between the subsystems must be described. A visualization of the coupling between the different subsystems can be seen in Figure 2.  
The TLP is considered as a moving support for the tower. The effect of the moving support from the TLP to the tower can be described by a force in each node. This force can be can be described by assuming that the acceleration of the TLP is applied to all nodes in their respective DOF [35]. The force vector can then be calculated by using [PERSON]'s second law  
\[\mathbf{F}_{sup}=\mathbf{T}_{tow}\,\mathbf{M}_{tow}\,\mathbf{\hat{y}} \tag{17}\]  
where \(\mathbf{F}_{sup}\in\mathbb{R}^{6N}\) is the force vector arising from the acceleration of the TLP on the tower. \(\mathbf{M}_{t}\in\mathbb{R}^{6N\times 6N}\) is the global mass matrix for the tower and nacelle. \(\mathbf{T}_{tow}\in\mathbb{R}^{6 Nx6}=[\mathbf{I}_{6x6}\,\mathbf{I}_{6x6}\, \ldots\,\mathbf{I}_{6x6}]^{T}\) is a transformation matrix consisting of N-identity matrices. This transforms the TLPs accelerations into each of the nodes FEM model of the tower, in the respective DOFs. The towers relative movement with respect to the TLP will also affect the TLPs movement. The reaction force can be calculated in the bottom node of the FEM model of the tower and applying it to the TLPs external forces. The reaction force can be calculated by  
\[\mathbf{F}_{Rac}=\left(\mathbf{C}_{tow}\dot{\mathbf{Z}}+\mathbf{K}_{tow}6 \mathbf{Z}\right)\mathbf{T}_{Rac} \tag{18}\]  
where \(\mathbf{F}_{Rac}\in\mathbb{R}^{6}\) is the reaction force in the bottom node of the FEM of the tower. \(\mathbf{T}_{Rac}\in\mathbb{R}^{6\times 6N}\) is a transformation matrix, which transforms the global load matrix from the FEM model of the tower to the load matrix for the TLP. The motion of the TLP and tower will affect the rotors aerodynamics by changing the wind speed the wind turbines blades experience. The TLP and towers motion in surge are taken into account for the relative wind speed in Equation (1). The loads from the aerodynamics on the rotor in this paper are the rotor thrust force and moments in pitch and yaw, and a roll moment around the tower is due to the acceleration of the rotor. These loads will be inserted as an external force on the center node of the nacelle in their respective DOF in the FEM code for the tower. The loads arising from the rotor are  
\[\mathbf{F}_{Rot}=\left[F_{Su}\quad 0\quad 0\quad M_{Ya}\quad M_{Pi}\quad M_{Ro} \right]^{T} \tag{19}\]  
where \(\mathbf{F}_{Rot}\in\mathbb{R}^{6}\) is the load vector from the rotor. \(F_{Su}\) are the thrust force on the rotor. \(M_{Pi}\) and \(M_{Ya}\) are the pitch and yaw moment around the tower. These load components from the aerodynamics can be calculated as  
\[F_{Su}=\sum_{j=1}^{NB}\sum_{k=1}^{NS}FT_{j,k} \tag{20}\]  
\[M_{Pi}=\mathbf{Si}\,\mathbf{FT}\,\mathbf{Br} \tag{21}\]  
\[M_{Ya}=\mathbf{Co}\,\mathbf{FT}\,\mathbf{Br} \tag{22}\]  
where \(\mathbf{FT}\in\mathbb{R}^{NB\times NS}\) are the thrust force matrix. \(NS\) is the number of blade sections used in the unsteady BEM model and \(NB\) is the number of blades. In this paper, the blade is split into 19 blade sections on each of the 3 blades. \(\mathbf{Br}\in\mathbb{R}^{NS}\) is a distance vector, which describes the blades spanwise  
Figure 2: Visualization of the coupling terms between the different subsystems. length from center to each of the blade sections. \(\mathbf{Si}=\left[\sin(\phi_{r})\,\sin(\phi_{r}+2\pi/3)\,\sin(\phi_{r}+4\pi/3)\right]\) and \(\mathbf{Co}\)\(=\left[\cos(\phi_{r})\,\cos(\phi_{r}+2\pi/3)\,\cos(\phi_{r}+4\pi/3)\right]\), where \(\phi_{r}\) is the reference blades azimuthal postion. The roll moment induced around the tower top can be calculated as  
\[M_{Ro}=3\,\dot{\omega}_{r}\mathbf{Br}^{T}\,\mathbf{mr} \tag{23}\]  
where \(M_{Ro}\) are the roll moment around the tower.