# 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
#### 2.1.5 TMD Force Terms  
The external force of TMD device \(F_{\text{TMD}}^{\text{ex}}\) includes three parts, the linear spring force, the damping force, and a stop force. We have  
\[F_{\text{TMD}}^{\text{ex}}=-K_{\text{TMD}}x_{\text{TMD}}-D_{\text{TMD}}\dot{x} _{\text{TMD}}-F_{\text{stop}} \tag{12}\]  
where \(K_{\text{TMD}}\) and \(D_{\text{TMD}}\) are the stiffness and damping coefficients respectively. Due to the space restriction of nacelle cabin, the TMD device is subjected to a stop force \(F_{\text{stop}}\) as the mass hit on the wall. According to the TMD module [20] integrated in FAST, \(F_{\text{stop}}\) is given by  
\[F_{\text{stop}}=\left\{ \begin{array}{ll}0,&\big{|}x_{\text{TMD}}\big{|}<L\\ -K_{\text{stop}}\cdot\Delta x,&\big{(}x_{\text{TMD}}\geq L\,\wedge\,\dot{x}_{ \text{TMD}}\leq 0\big{)}\vee\big{(}x_{\text{TMD}}\leq-L\,\wedge\,\dot{x}_{\text{ TMD}}\geq 0\big{)}\\ -K_{\text{stop}}\cdot\Delta x-D_{\text{stop}}\cdot\dot{x}_{\text{TMD}},&\big{(} x_{\text{TMD}}\geq L\,\wedge\,\dot{x}_{\text{TMD}}>0\big{)}\vee\big{(}x_{\text{TMD}} \leq-L\,\wedge\,\dot{x}_{\text{TMD}}<0\big{)}\end{array} \right. \tag{13}\]  
where \(L\) denotes the maximum TMD displacement from the origin of the nacelle coordinate system. According to the size properties of NREL-5 MW [16], \(L=5\) m is adopted in this work. \(K_{\text{stop}}\) and \(D_{\text{stop}}\) denote a huge constant stiffness and damping coefficients respectively. To be consistent with the work of [PERSON] [6], we take \(K_{\text{stop}}=5\times 10^{5}\) N/m, \(D_{\text{stop}}=5\times 10^{5}\) N/(m/s), and \(\Delta x\) denotes the part of the TMD displacement that exceeds the limit value. Following [12; 21], the connection between the tower and the foundation is simplified to a hinge joint. The first fore-aft bending mode of tower could be equivalently modeled by a linear spring and damper. Therefore, the additional bending moments at the tower bottom is given by  
\[T_{\text{t}}=K_{\text{t}}\big{(}\theta_{\text{t}}-\theta_{\text{p}}\big{)}+D_{ \text{t}}\Big{(}\dot{\theta}_{\text{t}}-\dot{\theta}_{\text{p}}\Big{)} \tag{14}\]  
To ensure the precision of this linear modeling, the tower top displacement (TTD) is validated with FAST output in the following section. Consequently, we can establish the simplified dynamic model by uniting the above equations. \[\left[ \begin{array}{cccc}\text{M}_{\text{sg}}^{\text{sg}}&\text{I}_{\text{ p}}^{\text{sg}}&\text{I}_{\text{t}}^{\text{sg}}&\text{M}_{\text{TMD}}^{\text{sg}}\\ \text{M}_{\text{sg}}^{\text{sg}}&\text{I}_{\text{p}}^{\text{p}}&\text{I}_{\text{ t}}^{\text{p}}&\text{M}_{\text{TMD}}^{\text{sg}}\\ \text{M}_{\text{sg}}^{\text{st}}&\text{I}_{\text{t}}^{\text{t}}&\text{I}_{\text{ t}}^{\text{t}}&\text{M}_{\text{TMD}}^{\text{tMD}}\\ \text{M}_{\text{sg}}^{\text{TMD}}&\text{I}_{\text{p}}^{\text{TMD}}&\text{I}_{ \text{t}}^{\text{TMD}}&\text{M}_{\text{TMD}}^{\text{TMD}}\\ \end{array} \right].\left[ \begin{array}{c}\ddot{x}_{\text{sg}}\\ \dot{\theta}_{\text{p}}\\ \dot{\theta}_{\text{t}}\\ \dot{x}_{\text{TMD}}\\ \end{array} \right]=\left[ \begin{array}{c}F_{\text{sg}}^{\text{restore}}+F_{ \text{sg}}^{\text{damp}}+F_{\text{sg}}^{\text{ctr}}\\ \text{I}_{\text{p}}^{\text{gr}}+T_{\text{p}}^{\text{restore}}+T_{\text{p}}^{ \text{damp}}+T_{\text{p}}^{\text{ctr}}\\ T_{\text{p}}^{\text{gr}}+T_{\text{p}}^{\text{restore}}+T_{\text{p}}^{\text{damp}}+T _{\text{p}}^{\text{ctr}}\\ T_{\text{f}}^{\text{gr}}+T_{\text{t}}^{\text{retr}}+T_{\text{t}}\\ F_{\text{f}}^{\text{gr}}+T_{\text{f}}^{\text{ex}}\\ \end{array} \right] \tag{15}\]In addition, according to [16], the center of gravity of the rotor nacelle assembly (RNA) is deviated about 0.4 m from the centerline of the tower. This deviation induces a minor initial inclination of the entire system, and its magnitude can be estimated by simulating the undisturbed model in the FAST program.