## 10 Identification of Flutter Derivatives from 2 DOF Tests
### Flutter Derivatives from MITD Method  
The flutter derivatives obtained from the MITD method used filtered acceleration data from both Setups 1 and 2. Initially, raw acceleration data were used, but high-frequency noise and short data length at high wind speeds caused the MITD to produce a singular matrix, making it impossible to simulate time histories. The shifted response matrices were calculated using a fixed time shift, and the mechanical stiffness and damping matrices were extracted for both setups but showed non-zero off-diagonal values, which indicate a mechanical coupling between modes, which is unexpected at zero wind speed. This could be caused by the poor quality of the recorded data, which was shorter in the heave mode, or by small, unmeasured vertical displacements during the pitch-dominant motion. The extracted modal parameters, such as frequency and damping, also showed discrepancies, indicating the recorded data are of poor quality, as shown in Figure 25. These discrepancies can cause large errors in the flutter derivatives, making it important to accurately identify the mechanical parameters of the system. The MITD method itself is not responsible for the discrepancies. The mechanical stiffness and damping matrices extracted from the MITD method for both Setup 1 and Setup 2 are presented in Table 4.  
The flutter derivatives obtained using the MITD method from the 2 DOF test in Setup 1 are shown in Figure 26 and are compared with the wind tunnel tests of reference studies. The results indicate that the \(H_{3}^{*}\), \(A_{2}^{*}\), and \(A_{3}^{*}\) fitted derivatives from both dominant motions show good correspondence with each other. On the other hand, \(A_{1}^{*}\), \(A_{4}^{*}\), and \(H_{1}^{*}\) show the same pattern but with large differences in values. \(H_{4}^{*}\) coefficient values from both dominant motions show disagreement with each other. Overall, the fitted derivatives from both heave- and pitch-dominant motions show well-defined patterns, and their values are fairly close to the reference wind tunnel and water channel values. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Motion** & **Stiffness Matrix** & \multicolumn{2}{c}{**Damping Matrix**} \\ \hline Setup 1 & \multicolumn{2}{c}{\(\overline{K}\)} & \multicolumn{2}{c}{\(\overline{C}\)} \\ \hline \multirow{3}{*}{Heave-Dominant} & 361.09 & \(-\)0.8605 & 0.698 & \(-\)0.0421 \\ \cline{2-5}  & \(-\)11.517 & 984.73 & 0.1694 & 0.1783 \\ \cline{2-5}  & 491.62 & 0.4308 & 0.2169 & \(-\)0.0099 \\ \cline{2-5}  & 1036.9 & 9928.83 & \(-\)31.79 & 0.1751 \\ \hline Setup 2 & \multicolumn{2}{c}{\(\overline{K}\)} & \multicolumn{2}{c}{\(\overline{C}\)} \\ \hline \multirow{3}{*}{Heave-Dominant} & 454.06 & \(-\)3.61 & 0.2117 & \(-\)0.0896 \\ \cline{2-5}  & \(-\)13.63 & 1011.6 & 0.1871 & 0.174 \\ \cline{2-5}  & 698.76 & 1.078 & 0.9287 & \(-\)0.0614 \\ \cline{2-5}  & 4.947 & 1003.8 & 0.1224 & 0.0734 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mechanical system matrices for wind-off conditions from the MITD method. Figure 26: _Cont._  
The flutter derivatives obtained using the MTD method from the 2 DOF test for Setup 2 are shown in Figure 27. The derivative values obtained from the heave-dominant motions show a large scatter as compared with the pitch-dominant motions. The large discrepancies can be linked to the bad quality of the recorded data that were used to calculate the mechanical parameters. The \(A_{1}^{*}\) values from the pitch-dominant motions show good agreement with the experimental reference values. Overall, for Setup 2, the obtained derivatives from the pitch-dominant motions show a similar trend to those depicted by theoretical and experimental values. Figure 27: _Cont._  
Figure 26: Comparison of aerodynamics derivatives from 2 DOF test for Setup 1. (o) Polynomial fit from pitch using MITD method, (... ) polynomial fit from heave using MITD method, (... ) wind tunnel test ([PERSON] et al.) ([PERSON] et al.) ([PERSON] et al.) ([PERSON] et al.) ([PERSON] et al.) wind tunnel test ([PERSON] et al.) ([PERSON] et al.)