# Passive Vibration Control of a Semi-Submersible Floating Offshore Wind Turbine
## 2 Modeling the Dynamics of Semi-Submersible FOWT
### Simplified Dynamic Model for Semi-Submersible FOWT
#### 2.1.1 Inertia Force Terms  
The first step is to determine all motion acceleration components of platform, tower and TMD device in the established coordinate system. Apart from the surge acceleration \(\ddot{x}_{\text{sg}}\) and pitch acceleration \(\ddot{\theta}_{\text{p}}\) of platform, first fore-aft bending's acceleration \(\ddot{\theta}_{\text{t}}\) of tower and acceleration \(\ddot{x}_{\text{TMD}}\) of TMD device, the tangential acceleration \(a_{\text{t}}^{\text{t}}\), \(a_{\text{p}}^{\text{t}}\), \(a_{\text{TMD}}^{\text{t}}\), and the normal acceleration \(a_{\text{t}}^{\text{n}}\)\(a_{\text{p}}^{\text{n}}\) of each body as illustrated in the Figure 2, and we can use the following formula to calculate these acceleration. \[ \begin{split} a_{\text{t}}^{\text{T}}&=\ddot{ \theta}_{\text{t}}\cdot R_{\text{t}}+\ddot{\theta}_{\text{p}}\cdot H_{\text{c}} \\ a_{\text{p}}^{\text{T}}&=\ddot{\theta}_{\text{p}} \cdot R_{\text{p}}\\ a_{\text{TMD}}^{\text{T}}&=\ddot{\theta}_{\text{t}} \cdot R_{\text{TMD}}+\ddot{\theta}_{\text{p}}\cdot H_{\text{c}}\\ a_{\text{t}}^{\text{N}}&=R_{\text{t}}\cdot\dot{ \theta}_{\text{t}}^{2}+H_{\text{c}}\cdot\dot{\theta}_{\text{p}}^{2}\\ a_{\text{p}}^{\text{N}}&=R_{\text{p}}\cdot\dot{ \theta}_{\text{p}}^{2}\\ \end{split}  \tag{2}\]  
where \(H_{\text{c}}\) denote the distance between the bottom of tower and SL. Due to the presence of radiation effect, we should consider the added mass in the inertia force term. The added mass coefficient \(\mathbf{C}_{t}^{j}(\infty)\) can be calculated by potential theory at infinite frequency, and added mass \(\mathbf{A}_{t}^{j}(\infty)\) can be obtained using the follow formula:  
\[\mathbf{A}_{t}^{j}(\infty)=\rho\mathbf{C}_{t}^{j}(\infty) \tag{3}\]  
For OC4-DeepCwind, we can get the added mass value in [17]. Therefore, we can integrate and list the inertial force term in the follow:  
\[ \begin{split} M_{\text{sg}}^{\text{sg}}&=A_{\text {sg}}^{\text{sg}}+m_{\text{t}}+m_{\text{p}}+m_{\text{TMD}}\\ I_{\text{p}}^{\text{sg}}&=A_{\text{p}}^{\text{sg}}+ \left(m_{\text{twr}}+m_{\text{TMD}}\right)\cdot H_{\text{c}}-m_{\text{p}} \cdot R_{\text{p}}\\ I_{\text{t}}^{\text{sg}}&=m_{\text{t}}\cdot R_{\text {t}}+m_{\text{TMD}}\cdot R_{\text{TMD}}\\ M_{\text{sg}}^{\text{sg}}&=H_{\text{TMD}}\\ M_{\text{sg}}^{\text{P}}&=A_{\text{sg}}^{\text{P}}-R_ {\text{p}}\cdot m_{\text{p}}+\left(R_{\text{t}}+H_{\text{c}}\right)\cdot m_{ \text{t}}+\left(R_{\text{TMD}}+H_{\text{c}}\right)\cdot m_{\text{TMD}}\\ I_{\text{p}}^{\text{P}}&=A_{\text{p}}^{\text{P}}+I_{ \text{p}}+m_{\text{t}}\cdot\left(R_{\text{t}}+H_{\text{c}}\right)\cdot H_{ \text{c}}+m_{\text{TMD}}\cdot\left(R_{\text{TMD}}+H_{\text{c}}\right)\cdot H_{ \text{c}}\\ I_{\text{t}}^{\text{P}}&=R_{\text{t}}\cdot m_{\text {t}}\cdot H_{\text{c}}+I_{\text{t}}+\left(R_{\text{TMD}}+H_{\text{c}}\right) \cdot m_{\text{TMD}}\cdot R_{\text{TMD}}\\ M_{\text{tMD}}^{\text{T}}&=\left(R_{\text{TMD}}+H_{ \text{c}}\right)\cdot m_{\text{TMD}}\\ M_{\text{sg}}^{\text{TMD}}&=R_{\text{t}}\cdot m_{ \text{t}}+R_{\text{TMD}}\cdot m_{\text{TMD}}\\ I_{\text{p}}^{\text{t}}&=R_{\text{t}}\times m_{\text {t}}\cdot H_{\text{c}}+R_{\text{TMD}}\cdot m_{\text{TMD}}\cdot H_{\text{c}}\\ I_{\text{t}}^{\text{t}}&=I_{\text{t}}+R_{\text{TMD}} \cdot m_{\text{TMD}}\cdot R_{\text{TMD}}\\ M_{\text{tMD}}^{\text{t}}&=m_{\text{TMD}}\cdot R_{ \text{TMD}}\\ M_{\text{t}}^{\text{TMD}}&=m_{\text{TMD}}\end{split}  \tag{4}\]  
The centripetal forces and moments caused by the rotation of platform and tower about the reference point \(P\) are calculated by the following expressions  
\[