## 4 Results and Discussion  
The dynamic data acquired by the load cell were analysed both in frequency and time domains. In Figure 6, as a general example, the base shear force time histories for the three different bottom shapes with a water depth of 7 cm are illustrated. The experimental data for the base shear force \(F_{\text{base}}\) (light grey line of Figure 6a-c) recall the typical trend of a mass with a single degree of freedom in free vibration in presence of damping. The analytical function which describes the amplitude modulation of a free oscillating damped system (below critical damping) is  
\[F_{\text{base}}(t)=Ae^{-\zeta\omega_{w}t} \tag{16}\]  
where \(A\) is the amplitude, \(\omega_{w}\) is the natural angular frequency and \(\zeta\) is the damping. The power spectra were extracted from the dynamic records of the base shear force in order to estimate the natural angular frequency \(\omega_{w}\), and then the natural sloshing frequency \(f_{w}\), for the different liquid \(h/L\) ratios. Table 1 shows a comparison between the natural frequencies for the flat bottom tanks \(f_{w}^{\text{EXP}}\) obtained as a result of the experimental campaign. As introduced in Section 2, the experimental results were also compared with the analytical formulations related to the linear theory (Equation (3)) and the linear long wave theory (Equation (4)). In the first column of Table 1, aspect ratios are reported. It is clear how their values, apart from the first two, exceed the limit of validity for the linear theory and yet the error with respect to [PERSON]'s expression is under 2% for all the different water depths showing a very good agreement. This condition might be explained by the intrinsic nature of the induced sloshing phenomenon under analysis, which might be perfectly compliant with the standard hypothesis of the linear--shallow waters equations. The impulsive input induced to the tank to make the liquid in free vibrations, together with the geometrical features of the tank, produce a liquid motion that recalls the motion of the _iso-potential_ lines for an irrotational flow, or better put, they generate a set of oscillating waves that closely follow the solution of the system of equations under the potential flow conditions, which is, in fact, the starting point for the analytical solution of [PERSON]'s theory. On the other hand, the comparison with [PERSON]'s formulation [39] points out two peculiar aspects. Firstly, relative errors are larger due to the different approach of [PERSON]'s work who has employed [PERSON]'s equations and [PERSON]'s formula together with the method of virtual work, an approach that shows a higher sensibility to the geometrical parameters. The second notable aspect is related to the evident increase of the error for water depths higher than 90 mm, (ratio \(h/L=0.225\)) showing, in this case, the actual relaxation of the  
Figure 6: Base force time histories with the calculated enveloping exponential interpolating curves for different bottoming bases at h = 7 cm. linear--shallow waters hypothesis, as it is even more evident from Figure 7a. This is a consequence of the nonlinear effect of amplitude dispersion that, indeed, resembles that of a hardening spring [33]. Figure 7 gives a comprehensive picture of the differences between experimental data and the different closed analytical approaches, including the sloped and circular bottom geometries, as reported in Figure 7b,c, respectively. The first thing to be highlighted, similarly to the flat case and as expected, is the change of trend for all the three bottom geometries, at values of \(h/L\) exceeding the limit of validity of the linear theory.