## 2 Aero-Hydro-Servo-Elastic Model
### Hydrodynamic Model  
The hydrodynamic model used in this project is the same as previously used and described in [24], see this for more information. To model the movement of the TLP in six DOF in the time domain, a modified version of [PERSON]'s equation [25] is utilized. The movement of a rigid floating structure was described by [PERSON] by a second order differential equation with a convolution integral. The model is modified by incorporating a linear and quadratic viscous damping term into [PERSON]'s equation yielding a linear vector integro differential equation  
\[\left(\mathbf{M}_{TLP}+\mathbf{A}(\infty)\right)\mathbf{\hat{y}} (t)+\int_{0}^{z}\mathbf{B}_{quad,\varphi}\left(\left(\mathbf{u}(t,z)-\mathbf{ \hat{y}}(t,z)\right)\left|\mathbf{u}(t,z)-\mathbf{\hat{y}}(t,z)\right|\right)dz \tag{8}\] \[+\int_{0}^{z}\mathbf{B}_{lin,\varphi}(\mathbf{u}(t,z)-\mathbf{ \hat{y}}(t,z))dz+\int_{0}^{t}\mathbf{R}(t-\tau)\mathbf{\hat{y}}(\tau)d\tau+ \mathbf{K}_{\mathbf{TLP}}\mathbf{y}(t)=\mathbf{F}_{wave}(t)\]  
where \(\mathbf{\hat{y}}\), \(\mathbf{\hat{y}}\) and \(\mathbf{y}\), denote the acceleration, velocity and position of the TLP, respectively in \(\mathbb{R}^{6}\). \(\mathbf{u}\in\mathbb{R}^{6}\) denotes the velocity of the water particles, which decays exponentially towards the bottom. The viscous forces arising from the relative velocity therefore needs to be integrated along the height of the TLP. \(\mathbf{M}_{TLP}\in\mathbb{R}^{6\times 6}\) is the inertia matrix of the TLP. \(\mathbf{A}(\infty)\in\mathbb{R}^{6\times 6}\) is a constant positive-definite added mass matrix at an infinite frequency. \(\mathbf{B}_{lin,\varphi}\) and \(\mathbf{B}_{quad,\varphi}\) denote the linear and quadratic viscous damping matrices respectively both in \(\mathbb{R}^{6\times 6}\). The linear and quadratic viscous damping matrices are found by analyzing experimental data from the free decay motion of the TLP, the data for this can be seen in [24]. \(\mathbf{K}_{\mathbf{TLP}}\in\mathbb{R}^{6\times 6}\) is the combined stiffness matrix from the hydrostatic stiffness and mooring stiffness. The hydrostatic stiffness arises due to changes in the water-plane area and the interaction between the forces from gravity and buoyancy, this is described in [26]. The mooring stiffness is found by analytical force/moments equilibrium as described in [27]. \(\mathbf{F}_{wave}\in\mathbb{R}^{6}\) describes the wave induced forces and moments on the TLP, in this paper only the first order wave forces are taken into account. \(\mathbf{R}\in\mathbb{R}^{6\times 6}\) is a matrix consisting of retardations functions, which describe the radiation impulse responses in the different DOFs. The radiation impulse response can be described in the time domain as  
\[\mathbf{R}(t)=\frac{2}{\pi}\int_{0}^{\infty}\mathbf{B}_{pot}(\omega)\cos( \omega t)d\omega \tag{9}\]  
where the retardation function is a conversion of the frequency dependent potential damping matrix \(\mathbf{B}_{pot}(\omega)\in\mathbb{R}^{6\times 6}\) from the frequency domain to the time domain through the Fourier transform. This effect captures the energy transfer from the motion of the structure to the surrounding fluid. The hydrodynamic coefficients \(\mathbf{A}(\infty)\), \(\mathbf{B}_{pot}(\omega)\) and the wave height normalized frequency dependent first order wave loads \(\mathbf{F}_{wave}(\omega)\) are calculated by solving the velocity potential of the diffraction problem when appropriate boundary conditions are applied. This is done by loading the geometry of the TLP into the software NEMOH [28], which uses the boundary element method to solve the diffraction problem. The convolution integral formulation in Equation (8), describes the change in the fluid momentum due to the motion of the structure at a particular time instant, which will affect the motion of the structure at all subsequent times. The convolution integral formulation requiresa significant amount of computer memory if directly implemented.