## 4 Hydrodynamic Characteristics of FOWT-SFFC
### Natural Periods and RAOs of Motions  
For a linear system, the characteristics of the response can be represented by normalizing the response amplitude with the input amplitude. This normalized response function is called the response amplitude operator (RAO). The RAO is unique for a specific structure and can be used to show seakeeping performance of a floating offshore structure. In this section, the motion RAOs of FOWT-SFFC are calculated by WAMIT and then compared with counterparts of OC3 Hwind [32] and OC4 DeepCwind [14] offshore wind turbines. In WAMIT, the substructure of FOWT-SFFC is considered as a rigid body with 6 degrees-of-freedom (DOF). The amplitudes of the body's motions \(\xi_{j}\) are obtained from the solution of the following linear system by applying [PERSON]'s law. \[\left[-\omega^{2}\Big{(}M+M^{E}+A\Big{)}+i\omega\Big{(}B+B^{E}\Big{)}+\Big{(} C+C^{E}\Big{)}\right]\xi_{j}=X \tag{1}\]  
where \(M\) is \(6\times 6\) inertia matrix of FOWT-SFFC, \(M^{E}\) is \(6\times 6\) external mass matrix, \(A\) and \(B\) are \(6\times 6\) added mass matrix and added damping matrix respectively, \(C\) is \(6\times 6\) hydrostatic and gravitational restoring matrix, \(C^{E}\) is \(6\times 6\) external stiffness matrix and \(X\) is \(6\times 1\) load vector induced by waves. Note that the total inertia matrix specified in WAMIT calculation should include the body inertia matrix. It is known that damping has a significant effect on the RAOs. As the WAMIT uses the linear potential-flow method to calculate the motion of the structure, only the radiational damping induced by oscillation of the cage is considered. For the FOWT-SFFC system, there are many small cylindrical members and copper alloy nets. The total damping from linear potential-flow theory must be augmented with viscous drag damping induced by flow separation when it comes to computing its RAOs. Hence, a linear damping matrix determined by free decay tests is added to WAMIT input file to consider the viscous damping. The free decay tests are conducted by FAST [44]. The drag coefficients \(C_{d}=0.7\) for cylindrical members are determined by [PERSON]'s method [45]. Because rectangular members so far cannot be modeled by FAST, the pontoons are modeled by equivalent cylinders with a diameter of 4 m. The corresponding drag coefficient is 2.42 [45] which somehow might slightly amplify the damping in surge and sway motions. The results of free decay tests of FOWT-SFFC with mooring lines but without fishing nets are shown in Table 5.  
The viscous damping of the copper alloy nets is difficult to quantify as it is associated with complicated flow discharge and separation at small holes. To simplify the hydrodynamic analysis, the damping ratios in Table 5 are multiplied by 1.2 so as to calculate the overall damping constant:  
\[c=2\xi m\omega \tag{2}\]  
where \(c\) is damping constant, \(\xi\) is damping ratio, \(m\) is mass, \(\omega\) is natural frequency. Then the linear damping matrix needed in WAMIT input file is obtained by subtracting the added damping calculated by WAMIT from damping constants calculated by Equation (2). The heave and pitch RAOs of FOWT-SFFC and other two offshore wind turbine concepts (OC3 Hywind and OC4 DeepCwind) in their freely floating states (without mooring forces) are all analyzed by WAMIT. The comparisons are given in Figures 5 and 6. Note that in WAMIT, the RAO \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Properties** & **Surge** & **Sway** & **Heave** & **Roll** & **Pitch** & **Yaw** \\ \hline Damping Ratio & 4.89\% & 4.84\% & 3.79\% & 2.45\% & 2.34\% & 2.97\% \\ Natural Period (s) & 81.33 & 80.70 & 26.40 & 27.65 & 27.75 & 78.23 \\ Natural Frequency (Hz) & 0.0123 & 0.0124 & 0.0379 & 0.0362 & 0.0360 & 0.0128 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Damping ratios and natural periods derived from FAST decay tests for the moored FOWT-SFFC without fishing nets. is defined as response amplitude per unit wave amplitude.