## 10 Identification of Flutter Derivatives from 2 DOF Tests
### Flutter Derivatives from ILS Method  
The ILS method computes flutter derivatives using numerically obtained displacement, velocity, and acceleration responses. However, the properties of these responses may differ from the actual recorded responses due to filtration. The computed mechanical stiffness and damping matrices may also be affected. Table 5 presents the extracted system parameters from the ILS method in wind-off conditions, which show non-zero off-diagonal values for both heave- and pitch-dominant motions, indicating strong mechanical coupling between both motions. The ILS algorithm failed to simulate the response for Setup 2 due to convergence failure in matrix A or failure to identify the dominant motion in the response. This was likely due to the system's flexibility causing an additional mode to be excited. Figure 28 shows all the obtained derivatives, which are compared with the theoretical and experimental reference values. The \(H_{1}^{*}\) values from the heave-dominant motions show very close agreement with the reference experimental values. On the contrary, \(H_{1}^{*}\) values from the pitch-dominant motions follow the same trend, but the values are not close to the theoretical or experimental values. The \(H_{2}^{*}\) coefficients from both motions show the same trend, but the one from the pitch-dominant motions shows good correspondence with the water channel values and wind tunnel values obtained by [PERSON] et al. [2]. Similarly, the \(H_{3}^{*}\) values from both motions show the same decreasing trend as the reference values. In the case of \(H_{4}^{*}\), a completely opposite trend is observed compared with the reference values. The \(A_{1}^{*}\), \(A_{2}^{*}\), and \(A_{4}^{*}\) values from heave-dominant motions are in close agreement with the reference values. For the \(A_{2}^{*}\) function, a close correspondence can only be observed at low reduced wind speeds with the reference values. In the case of pitch-dominant motion, a good agreement can be seen for \(A_{3}^{*}\) and, similarly, for \(A_{2}^{*}\) at low reduced wind speeds. The \(A_{2}^{*}\) values become positive for reduced velocity above six, which indicates an increase in the damping values of the torsional motion. All the other derivatives of \(A_{1}^{*}\) and \(A_{4}^{*}\) show complete disagreement with the reference values. Overall, the derivatives computed for the heave-dominant response from the Setup 1 configuration show better results as compared with the pitch-dominant motions.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \multicolumn{2}{c}{**Motion**} & \multicolumn{2}{c}{**Stiffness Matrix**} & \multicolumn{2}{c}{**Damping Matrix**} \\ \hline Setup 1 & \multicolumn{2}{c}{\(\overline{K}\)} & \multicolumn{2}{c}{\(\overline{C}\)} \\ \hline \multirow{2}{*}{Heave-Dominant} & \(-360.567\) & \(5.885\) & \(-0.8407\) & \(-0.3517\) \\ \cline{2-5}  & \(18.667\) & \(-979.77\) & \(-0.5524\) & \(-0.6884\) \\ \cline{2-5}  & \(-461.608\) & \(-0.7175\) & \(-3.731\) & \(0.0305\) \\ \cline{2-5}  & \(969.21\) & \(-978.34\) & \(-75.761\) & \(-0.0806\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Mechanical system matrices for wind-off conditions from the ILS method.  
Figure 27: Comparison of aerodynamics derivatives from 2 DOF for Setup 2. (o) Polynomial fit from pitch using MITD method, (...) polynomial fit from heave using MITD method, (\(\square\)) wind tunnel test [42], (\(\Delta\)) wind tunnel test [2], (\(-\)) water channel tests, (—) flat plate by [PERSON]. (a) \(H_{1}^{*}\); (b) \(A_{1}^{*}\); (c) \(H_{2}^{*}\); (d) \(A_{2}^{*}\); (e) \(H_{3}^{*}\); (f) \(A_{3}^{*}\); (g) \(H_{4}^{*}\); (h) \(A_{4}^{*}\).