## 2 Methodology
### Numerical Methodology
#### 2.2.2 Quasi-Steady Model of the Transversal Sloshing Force  
To capture dissipative effects arising from the fluid during interaction with the structure, a fluid model must be chosen that can apply representative forcing onto the structure based on the kinematic excitation within each tank cell. Classically, reduced-order models (ROM) of fluid behaviour are employed based on linearised fluid equations or equivalent mechanical models (EMM) such as swinging pendula or mass-spring-damper systems that represent free-surface displacement and fluid impacting [21, 22]. However, these models are often limited to lateral fluid motion or linear and weakly-nonlinear free-surface motions. In response to this, EMMs have recently been developed to capture fluid response under large amplitude vertical excitation, such as the bouncing ball model [5, 23] and ballistic-harmonic model [7]. These provide a good representation of dissipative and interactional behaviour of a fluid during the violent impacting regime but struggle to capture low-amplitude parametric sloshing. More recently, high-fidelity computational fluid dynamics approaches have been used, which are not limited by the magnitude of excitation nor tank geometry effects but, in turn, suffer from complex software coupling requirements and long execution times. Therefore, this section describes a recently developed vertical sloshing \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \(\mathbf{\kappa_{1}}\) & \(\mathbf{\kappa_{2}}\) & \(\mathbf{\kappa_{3}}\) & \(\mathbf{\beta_{2}}\) & \(\mathbf{\beta_{3}}\) \\ \hline Value & 4.71 & 0.70 & 1.29 & 2.64 & 3.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimised damping parameters of the numerical model. Figure 6: Tank tip acceleration comparison between experiment and model. Deflection at the tip of the wing is 2.7% of the semispan: (**a**) acceleration time series for the dry case; (**b**) amplitude of the first bending component versus time. ROM, which can capture accurate hydrodynamic forces over a range of kinematic excitation magnitudes and sloshing response regimes with minimal computational overhead. The reduced-order sloshing model subsequently discussed is based upon the direct application of experimentally identified sloshing forces in a manner comparable to surrogate modelling; therefore, it is prudent to introduce the experimental studies that are the basis of this model. Previous work has focused on identifying sloshing motions and damping characteristics from prescribed, vertical, harmonic excitation of a rectangular fluid-filled tank [7], where the vertical harmonic motion was chosen to replicate the out-of-plane bending dynamics that are typically seen in wing-like structures. In doing so, the total force acting on the fluid and tank structure was recorded to analyse the hysteretic force-displacement loops that define energy dissipation within the system. By removing the hydrostatic and inertial component from the total force, a resultant \"sloshing\" force remains due to the presence of a dynamic medium inside the tank. Specifically,  
\[F_{s}=F_{l}-\dot{y}(m_{l}+m_{s})-g(m_{l}+m_{s}) \tag{6}\]  
where \(m_{l}\) and \(m_{s}\) are the liquid and structural masses, respectively. For a fixed amplitude and frequency of excitation, this sloshing force was measured and averaged across multiple cycles (following the initial transient), resulting in the \"quasi-steady\" sloshing force, which acts at each kinematic condition within the prescribed cycle,  
\[y=A\sin(\omega t),\quad v=A\omega\cos(\omega t)\quad\Rightarrow\quad\overline{ F}_{s}=g(y,v,A,\omega)\sin(\omega t) \tag{7}\]  
Each prescribed cycle can be visualised as a single circle in the \(x\)-\(y\) plane of Figure 7. When the experimental analysis is performed over multiple amplitudes, it forms a series of discrete circles, and therefore, a 3D landscape of the sloshing force.