# Passive Vibration Control of a Semi-Submersible Floating Offshore Wind Turbine
## 2 Modeling the Dynamics of Semi-Submersible FOWT
### Model Parameters Identification
The exact value of \(I_{\text{t}}\) in the real model is taken as the initial. For the restoring stiffness coefficients, \(K_{\text{P}}^{\text{P}}\), \(K_{\text{P}}^{\text{sg}}\) and \(K_{\text{sg}}^{\text{sg}}\), their approximation are obtained by the quasi-static theory of catenary line. In addition, the hydrostatic stiffness in \(K_{\text{P}}^{\text{P}}\) also needs to be calculated according to Equation (8). The initial values of other coefficients are determined by trial and error. Finally, the initial values and the final estimation values are compiled in the Table 3, in which the metric units of each parameter are omitted for a concise expression. The estimation iteration processes of all the ten parameters are shown in Figure 3. As can be seen, the initial values of \(I_{\text{t}}\), \(K_{\text{P}}^{\text{P}}\), \(K_{\text{sg}}^{\text{sg}}\) are highly close to the estimation respectively, which demonstrates the accuracy of our established model to a certain extent. However, due to a relatively strong non-linearity in the mooring system, the estimation of \(K_{\text{sg}}^{\text{P}}\) remarkably deviates from the initial value. To validate the accuracy of the identified simplified model, we illustrate the free decay responses with inactive TMD obtained by the two approaches in Figure 4. Results of the OC4-DeepCwind surge-pitch motion with a \(5^{\circ}\) initial platform pitch are shown in Figure 4a. Furthermore, the free decay motions with a \(10^{\circ}\) initial pitch are also displayed in Figure 4b. The red solid line denotes the FAST simulation results, while the blue dash-dot line denotes the result of the proposed model. Upon examination of the six pair curves, the proposed simplified model can predict the pitch, surge and TTD in a high overlap ratio with the FAST simulation. To check the model performance with active TMD, a validation case with and without TMD stops is designed. In this test, the TMD mass is \(1\times 104\) kg, the spring and damping coefficients are 1000 N/m and 2000 N/(m/s) respectively. All the other parameters equal to the aforementioned definitions. Figure 5 displays comparisons of the \(5^{\circ}\) pitch free decay motions between the simplified model and the FAST simulation. The displacements of TMD \(x_{\text{TMD}}\) clearly show the effects of stops induced by the limited space in nacelle. For the TMD without stops, since more energy has been dissipated in the first few periods with longer travel, the vibration amplitude of TMD decays slightly faster than the case with stops. In both cases, the simplified model presents a closely agreement with the FAST simulation through the pitch, surge and TTD. In summary, all the four cases manifest the rational representativeness of the established model. The adopted assumptions and derived equations is able to neglect the minor effects but captured the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Initial Value** & **Estimation** & **Parameters** & **Initial Value** & **Estimation** \\ \hline \(I_{\text{t}}\) & \(2.6368\times 10^{9}\) & \(2.9359\times 10^{9}\) & \(D_{\text{t}}\) & \(4.6000\times 10^{7}\) & \(6.9515\times 10^{7}\) \\ \(K_{\text{P}}^{\text{P}}\) & \(-2.9330\times 10^{8}\) & \(-3.0746\times 10^{8}\) & \(D_{\text{P}}^{\text{P}}\) & \(1.5000\times 10^{10}\) & \(3.4079\times 10^{10}\) \\ \(K_{\text{sg}}^{\text{sg}}\) & \(7.0800\times 10^{4}\) & \(6.6026\times 10^{4}\) & \(D_{\text{sg}}^{\text{sg}}\) & \(-4.0000\times 10^{6}\) & \(8.7278\times 10^{6}\) \\ \(K_{\text{sg}}^{\text{P}}\) & \(-1.0800\times 10^{5}\) & \(-2.4605\times 10^{5}\) & \(D_{\text{P}}^{\text{sg}}\) & \(-6.0000\times 10^{7}\) & \(-4.5968\times 10^{7}\) \\ \(K_{\text{t}}\) & \(1.6000\times 10^{10}\) & \(1.5944\times 10^{10}\) & \(D_{\text{sg}}^{\text{sg}}\) & \(3.5000\times 10^{5}\) & \(1.0129\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial values and estimation results of the undetermined parameters. essential dynamics of OC4-DeepCwind system.