## 4 Aeroelastic Stability and Response Analyses
### Test Case 1  
The stability scenario of the aircraft model in the frozen liquid configuration (blue diamonds) is compared in Figure 6 with the lateral sloshing considered in the formulation \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode Number** & **Direction** & **Frequency (rad/s)** \\ \hline
1 & \(x\) & 11.80 \\
2 & \(y\) & 18.31 \\
3 & \(x\) & 22.64 \\
4 & \(y\) & 32.05 \\
5 & \(x\) & 29.26 \\
6 & \(y\) & 41.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Linear sloshing natural frequencies—Case study 2. Figure 5: Configuration with two box-shaped tanks placed under the BFF wings. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value (m)** \\ \hline side in direction \(x\) & 0.18 \\ side in direction \(y\) & 0.09 \\ \(h\) (height) & 0.13 \\ \(h_{f}\) (filling level 50\%) & 0.065 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Tank geometry—Case study 2. (red squares). Figure 6b highlights the additional poles representing linear sloshing dynamics, along with rigid-body dynamics. Lateral sloshing slightly worsens the stability of the aircraft with respect to the frozen fuel case. In fact, the flutter speed \(U_{F}\) slightly decreases from \(18.95\,\mathrm{m/s}\) to \(18.8\,\mathrm{m/s}\). This corresponds to a slight change in the critical frequency \(\omega_{F}\), which increases from \(23.53\,\mathrm{rad/s}\) (\(3.74\,\mathrm{Hz}\)) to \(23.82\,\mathrm{rad/s}\) (\(3.79\,\mathrm{Hz}\)). In both configurations, the critical mode occurs due to the coupling between the _short period_ mode and the first vibration mode. The crossing of the imaginary axis occurs with the pole originating from the branch of the first mode (see Figure 6a). Furthermore, sloshing dynamics have a non-negligible effect on _dutch roll_ and _phugoid_, rendering the latter less stable and very close to the imaginary axis. On the other hand, the higher frequency poles are only minimally affected and show no particular changes compared to the frozen case. Two different flight conditions are considered for gust response analysis, that is \(U_{\infty}=$18\,\mathrm{m/s}$\) (i.e., before the flutter speed of the linear sloshing and the frozen fuel models), and \(U_{\infty}=$19.35\,\mathrm{m/s}$\) (after flutter onset). Moreover, dependency of the vertical-sloshing-induced damping by the response magnitude is investigated by taking into account different gust amplitudes. The first analysis is performed at a free-stream velocity of \(U_{\infty}=$18\,\mathrm{m/s}$\), with a gust intensity of \(w_{g_{a}}=$2.5\,\mathrm{m/s}$\). Being slightly under pre-critical conditions (speed equal to \(95\%\) and \(96\%\) of the flutter speeds of the frozen and lateral sloshing models), the response is expected to be slightly damped. Figure 7a compares aircraft responses of the four considered models in terms of local angle of attack (\(\mathrm{AoA}=\Delta w/U_{\infty}\)). Similarly, Figure 7b shows the comparison in terms of the response of the first dry vibration mode \(q_{1}\), whose values are measured in meter units when normalised to unit displacement. A reasonable result is the ability of vertical sloshing dynamics in dampening the aeroelastic response more than that in the frozen case due to the sloshing impacts occuring in the tank. It also leads to a change in the frequency of the response, which is lower than that in the frozen case. Linear sloshing does not introduce dissipative effects into the response, but induces a slight increase in frequency (in the same manner as the critical frequency). Figure 8 shows the time trends of the nonlinear dynamic sloshing force for the vertical and full sloshing models. The forces almost overlap because the neural-network-based reduced-order model receives a similar tank velocity as input for both models. The particularly nonlinear response in the first three cycles is caused by impacts between the fluid and the tank walls.