## 1 Introduction  
Aircraft are generally subjected to a wide range of loads during flight and ground operations that cause significant deformation of the wings, especially during the most severe loading phases. This can lead to sloshing of the fuel stowed in the wing tanks, whose weight is comparable to that of the structural components. The interaction between the fuel dynamics and the structure of an aircraft can in turn influence its performance, affecting both stability and response to external loads. This work is framed within the H2020 project SLOshing Wing Dynamics (SLOWD) (Ref. [1]), which studies fuel sloshing as a method to reduce loads in aircraft wings by increasing effective damping. The aim is to investigate the effects of linear and nonlinear sloshing on the stability and response of an aircraft, using reduced-order models (ROMs) capable of reproducing the forces induced by the fuel motion. The linear sloshing detailed in this analysis occurs during rotations or lateral motions of the tank. In aeronautical applications, the anticipated onset of flutter instability can arise due to the coupling between the sloshing waves generated by the structural motion and structural dynamics. Studies on the effects of lateral sloshing on the aeroelastic behaviour of aircraft can be found in Ref. [2], where the lateral sloshing is modelled via the mass-spring-damper mechanical model, in [3] where an hydroelastic added-mass model is used, in [4] where a linearized frequency domain (LFD) is employed, and in [5] where an Equivalent Mechanical Model for sloshing is used by demonstrating the effect on aeroelastic and flight mechanics stability as well. The reduced-order model used to describe linear sloshing in this work relies on the analytical formulation provided in Ref. [6] and its further development in Ref. [7], according to which a realistic representation of linearised liquid dynamics inside containers with simplified geometry can be approximated by an equivalent mechanical model, whose parameters can be suitably related with the physical quantities obtained from the linearised potential flow theory (Ref. [8]). Nonlinear vertical sloshing, on the other hand, represents 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, vertical sloshing can be regarded as a kind of energy sink. For small tank vertical accelerations, the free surface tends to remain flat. As the amplitude of the oscillations increases, symmetrical standing waves appear, while for vertical accelerations greater than the acceleration of gravity, the system shifts into Rayleigh-Taylor instabilities (Ref. [9]), which may determine a turbulent flow regime inside the cavity. The impacts of the liquid with the tank ceiling following free surface fragmentation cause additional dissipation of energy (see Ref. [10] for an energy-dissipated analysis via volume of fluid approach and Refs. [11; 12; 13] via Smoothed-Particle Hydrodynamics (SPH) approach as well), in which the overall balance of elastic potential energy and fluid energy leads to a noticeable increase in the effective damping of structural motion (Refs. [14; 15]). In harmonic motion, this damping behavior depends on the amplitude and frequency of tank oscillations as emphasized in Refs. [16; 17] via numerical simulation and experimental tests. In general, coupling between the tank structural dynamics and fluid sloshing can be solved by exploiting multi-physics interface tools (Refs. [18; 19]) that couple structural finite element solvers with sloshing CFD solvers based on volume-of-fluid (Refs. [20; 21]) or SPH models (Ref. [22]). Reference [23] studied the aeroelastic response of a wing by means of the so-called _bouncing ball_ reduced-order model, in which sloshing performances were characterised as a function of speed and gust intensity. The bouncing ball model provides a quick and effective way of describing this complex phenomenon (Ref.