## 4 Aeroelastic Stability and Response Analyses
### Test Case 2  
Figure 11 shows the results of the stability analysis conducted for the second case study, both for the frozen model (blue diamonds) and for the linear sloshing model (red squares). The stability scenario of the frozen model is different from that of Test Case  
Figure 10: \(U_{\infty}=19.35\,\mathrm{m/s}\) and \(w_{g_{\star}}=0.3\,\mathrm{m/s}\)—Test Case 1. (**a**) Local angle of attack. (**b**) First structural mode. Figure 9: Sloshing modes for \(U_{\infty}=18\,\mathrm{m/s}\) and \(w_{g_{\star}}=2.5\,\mathrm{m/s}\)—Test Case 1. (**a**) First mode (dir. x). (**b**) Second mode (dir. x). 1. In fact, to ensure the static stability of the two-tank configuration, it was necessary to place them under the wings with an offset in the longitudinal direction. Such a tank location emphasizes the coupling between the torsional mode (fourth mode) and the anti-symmetrical bending mode (second mode). For the frozen model, flutter instability occurs at \(U_{\infty}=U_{F}=20.86\,\mathrm{m/s}\) with the branch originating from the _short period_ (see Figure 11a), whereas the flutter frequency is \(\omega_{F}=22.7\,\mathrm{rad/s}\) (\(3.61\,\mathrm{Hz}\)). The critical mode occurs due to coupling between the _short period_ and _first vibration_ mode. Note that the third mode becomes unstable at a flight speed of \(U_{\infty}=22.65\,\mathrm{m/s}\), which corresponds to a frequency value equal to \(48.22\,\mathrm{rad/s}\) (\(7.67\,\mathrm{Hz}\)). Lateral sloshing introduces poles which significantly affect the aircraft dynamics. Figure 11b highlights the rigid-body dynamics and the added dynamics of linear sloshing. The latter crosses the imaginary axis at a low velocity equal to \(U_{\infty}=16.2\,\mathrm{m/s}\), rendering the system slightly unstable. The presence of sloshing dynamics increases the stability of the body freedom flutter pole (short-period mode) augmenting the flutter margin to \(U_{\infty}=21.71\,\mathrm{m/s}\) (\(\omega=23.12\,\mathrm{rad/s}\)). However, there is also a deterioration in the stability of the third mode, which crosses the imaginary axis at a speed of \(U_{\infty}=20.94\,\mathrm{m/s}\) (\(\omega=49.13\,\mathrm{rad/s}\)) giving rise to a critical bending-torsional mode. Sloshing dynamics also have an impact at low frequencies, also affecting both the frequency and damping of _dutch roll_ and _phugoid_. Different aeroelastic analyses are considered by varying the speed and gust amplitude. The first analysis is performed at a free-stream velocity of \(U_{\infty}=18\,\mathrm{m/s}\) (speed equal to \(86\%\) and \(83\%\) of the body freedom flutter mode critical speeds of the frozen and lateral sloshing models, respectively), with a gust intensity of \(w_{g_{a}}=3\,\mathrm{m/s}\). In this condition (see Figure 11b), the linear sloshing model is unstable due to the crossing of the imaginary axis by a sloshing pole, whilst the frozen fuel model remains stable. The local angle of attack and the first structural mode responses are, respectively, shown in Figure 12a,b, comparing the four different models considered. Vertical sloshing contributes to dampening of the responses of both quantities shown also providing an increase in response frequency. As can be seen from the structural response, linear sloshing provides a damping contribution, which is, however, less clear than that in the local angle of attack. This highlights the full sloshing model as the one demonstrating the most damped response for the first bending mode. The tail of the response is featured by the trigger of unstable sloshing dynamics that also bears a small effect on the structural and flight dynamics components (see inset plots in Figure 12). Albeit very slowly, vertical sloshing succeeds in containing this instability. Increasing oscillations become unphysical as they are associated with sloshing mode. This means that when accounting for the nonlinear behaviour of lateral sloshing (not considered in this work), the dynamics will be limited in oscillation amplitudes. Figure 11: Root locus with linear sloshing—Test Case 2.