## 1 Introduction
[24]), but fails in accurately representing the fluid dissipative capacity at different amplitudes and frequencies [25]. Regarding the study of vertical sloshing, a neural-network-based nonlinear ROM was generated for the wing model prototype based on the approach described in Ref. [25]. An artificial neural network (ANN) is an effective tool for reproducing the dynamic behaviour of unknown nonlinear systems (Ref. [26]). Based on time series generated by a scaled experiment, the vertical sloshing ROM is built using a nonlinear finite impulse response (NFIR) network model (see Ref. [16]). The model scaling procedure introduced in Ref. [25] is employed to extend the use of the identified ROM to applications with different tank sizes. The final perspective offered in the present work is the integration of both linear and nonlinear sloshing ROMs into a flexible aircraft model, where flight dynamics and aeroelasticity are involved. It is assumed that sloshing forces can be decomposed into a lateral part with linearised behavior and a vertical part with extremely nonlinear behavior by neglecting their mutual interactions. The present work mainly relies on the approach described in [27], where a fully coupled equation of the aircraft motion was derived by assuming practical mean axes (PMA) constraints. The structure is modelled using the finite element method (FEM), whereas the doublet lattice method (DLM) is used to describe the unsteady aerodynamics (see Ref. [28]). Using the rational polynomial function approximation, the unsteady aerodynamics are recast into a pure differential expression that requires a new set of aerodynamic state-space variables (see Ref. [29]). The integration of linear sloshing, described by means of an EMM, introduces additional state variables (analogous to aerodynamics), thus representing latero-rotational sloshing dynamics. Vertical sloshing interacts with the aeroelastic model by providing a nonlinear force contribution that depends on the vertical excitation of the aircraft tanks. The combined effect of linear and nonlinear sloshing dynamics is modelled by adding the two contributions that they provide to the system. The reference aircraft is the Body Freedom Flutter (BFF) (whose model is proposed in Ref. [30]) and a flutter suppression strategy also proposed in Ref. [31]) in two different cases or configurations. The first contains only one tank that is partially filled with a low-density liquid and is placed beneath the aircraft's center of mass whereas the second case has two rectangular tanks symmetrically placed beneath the wings. Sloshing and frozen mass (in which liquid is simply a ballast) models are compared to quantify the sloshing effects on aircraft aeroelastic behavior. Stability analyses show how the coupling between linear sloshing, aircraft aeroelasticity, and flight dynamics contribute to modification of the overall aircraft stability scenario. Nonlinear response aeroelastic analyses demonstrate that dynamic load alleviation is achieved when vertical (nonlinear) sloshing is also included in the overall aeroelastic model. Moreover, limit cycle oscillations arise after the onset of flutter due to the dissipative characteristics of vertical sloshing. The paper is organised as follows. The flexible aircraft and sloshing modeling are introduced in Section 2. The case studies are introduced in Section 3 whilst the stability analysis and aeroelastic responses are illustrated in Section 4, comparing the performance of sloshing models with the frozen configuration. Finally, concluding remarks are provided at the end of the paper.