## 3 Results
### Numerical Results  
Similarly to the experimental study, the structural model was loaded downward and released. Following this event, the structure vibrates freely and decays according to its dry damping and interaction with the fluid force model. As described in Section 2, the sloshing force acts on each tank cell depending on the local instantaneous displacement and velocity. Figure 14 shows the 7.3 Hz interpolated sloshing force map (surrogate model) and the trajectory of the recovered sloshing force in the displacement-velocity domain during the transient decay of tank cell number 8. The starting value is at the maximum displacement and 0 velocity, after which the system starts decaying in the clockwise direction. In various parts of the tank cycle, the liquid acts against the tank with different intensities. The steady-state hysteresis cycles that result from this interaction were studied in-depth in [7], and the collection of such cycles for multiple amplitudes of excitation determine the sloshing force magnitude in the different parts of the map shown in Figure 14. The colour map in Figure 14 also indicates that the sloshing liquid maximises the sloshing force in the vicinity of extreme \(y/h\) values, with phasing between it and the tank velocity depending on the amplitude and frequency of excitation, consistent with previous observations [7]. Note that the \(F_{s}\) values shown here are adjusted with the help of the force smoothing function during the first part of the transient response in order to take into account the region where the sloshing is initially formed (see Section 2, Figure 8). Let us consider the comparison between the properties of the first bending component identified experimentally and those found using the numerical model. Figure 15 shows the evolution of the three response characteristics of the first signal component: amplitude versus time (upper side), damping ratio versus amplitude (left) and frequency versus amplitude (right), where these are given for the numerical and experimental realisation of the dry and wet cases. In order to facilitate the comparison of all data in one graph, a reduction in the data was made by averaging the acceleration-dependent quantities (e.g., the experimental data shown in Figure 13) inside bins of length 0.3 g. The mean values inside each bin are shown in Figure 15.  
Figure 14: Interpolated sloshing force map and trajectory of sloshing force for tank cell 8. The numerical prediction of the variation of tank tip acceleration \(A1(t/T1)\) agrees very well with the experimental data. There is a difference in the initial acceleration, even though the same initial displacement was used and the first bending frequency is the same. An initial assessment of the structural model under the transient conditions with geometrical nonlinearities included (Nastran SOL400) has shown that a reduction in the peak accelerations similar to what is seen in Figure 15 is obtained, suggesting that this difference has its source in the use of the linear modal model. While this difference does not impede other comparisons or validity of the numerical simulations, the adoption of a nonlinear model in the future might provide an improved acceleration amplitude match for cases similar to the one analysed here. The dry and wet damping ratio is shown as well in comparison in Figure 15. The numerical nonlinear dry damping was tuned based on the experimental data using the model described in Section 2, with the resulting damping model parameters shown in Table 2. There is a very good agreement in wet case damping ratios throughout the full acceleration amplitude range. At high amplitudes, the damping of the system is dominated by the structural effects and the sloshing force smoothing function (Equation (10)) improves the agreement in the 4-6 g range.