## 5 Concluding Remarks  
In this work, the effects of linear and nonlinear sloshing dynamics of fuel present inside the tanks were studied. Specifically, the investigation focused on the stability and response of a flexible flying wing model involving flight dynamics and aeroelasticity. The aircraft equations of motion were linearised around a stationary level-flight trim condition and recast in time-domain state-space form, thus obtaining an aeroelastic model that includes rigid-body, elastic, and aerodynamic states. Furthermore, the combined effects of linear and nonlinear sloshing dynamics were modelled by directly adding the two lateral and vertical contributions, respectively, to the aeroelastic system. Linear sloshing was described using an equivalent mechanical model (EMM) capable of restoring the actions exerted by the liquid following latero-rotational perturbations of the tank. Similar to the modeling of unsteady aerodynamics, the EMM introduced additional states related to liquid dynamics. Nonlinear vertical sloshing is a phenomenon caused by accelerations perpendicular to the liquid free surface. Rather than providing a coupling between the dynamics of the free surface and the rest of the system, it can instead be considered a kind of mechanical energy sink. Thus, vertical sloshing dynamics was modelled with a neural-network-based ROM driven by experimental data. Two case studies were considered; one with a single tank placed beneath the aircraft's center of mass, and the second with two tanks placed symmetrically underneath the wings. For the two case studies, the effects of sloshing dynamics on the stability and response were investigated by comparing cases where the liquid is assumed to be a frozen mass, and the other where the liquid is allowed to slosh freely. The results showed how linear sloshing dynamics can significantly influence aircraft stability as a result of their coupling with aeroelastic dynamics. Gust response analyses revealed that the performance of vertical sloshing provides additional damping in both pre-critical and post-critical conditions. Indeed, the presence of limit-cycle oscillations after the flutter guarantees an enhanced margin of stability for the system. The modeling presented herein will allow for the verification of the effects of sloshing on the stability and response of commercial aircraft, including next-generation configurations that may include large tanks within the fuselage. In addition, the high scalability of the presented sloshing reduced-order models will enable the usage of such models in the design phase of preliminary aircraft.  
Data curation, F.S.; Formal analysis, M.P.; Funding acquisition, F.M.; Investigation, F.S.; Methodology, M.P., F.S. and F.M.; Supervision, F.M.; Validation, F.M.; Writing - original draft, M.P. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by European Union's Horizon 2020 research and innovation programme grant number 815044.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Acknowledgments:** This paper was supported by the SLOWD project. The SLOWD project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 815044.  
**Conflicts of Interest:** The authors declare that they have no conflict of interest.