## 2 Theoretical Aspects  
The theoretical background of the sloshing motion dynamics has been largely explored in the literature. For the scope of this work, the theoretical aspects related to the analysis of the natural sloshing frequencies for three bottom geometries have been considered, namely the flat, the sloped and the circular shape. The physics of the sloshing problem is represented for the general case in Figure 1, with \(Oxy\) being the considered coordinate system. The total mass of the liquid is denoted by \(m_{w}\). A motion of the internal liquid generates a variation the free surface, whose kinematics is described by \(\eta(x,t)\) denoting the vertical movement over the horizontal line. Consequently, fluid horizontal pressures \(p_{x}(t)\) and shear actions \(\mu_{x}(t)\) over the sides of the tank produce an overall horizontal force, \(F_{\text{base}}(t)\), which represents the base shear force developing between the tank and the substructure, obtained by their integral over the dynamic wet perimeter \(\mathcal{S}\)  
\[F_{\text{base}}(t)=\int_{\mathcal{S}}\Big{[}p_{x}(t)+\mu_{x}(t)\Big{]}\text{d}s \tag{1}\]  
For the flat bottom case, a tank with base length \(L\) partially filled with a liquid having depth \(h\) has been considered. The value of the ratio \(h/L\) approaching zero determines the well-known range of validity of the small amplitude wave theory [44]. Within this range, linear long wave theory can be applied. The classical reference studies, based on the linear wave theory, date back to the end of the nineteenth century, and in particular to the seminal work of [PERSON] on the mathematical theory of fluid motion [45]. The starting principle underneath the sloshing dynamics theoretical background is the wave motion dynamics described by the classical Laplace equation  
\[\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2 }}=0 \tag{2}\]  
Following a standard solving procedure of linear theory, [PERSON] reports a development through the Bessel harmonics, so that the natural frequency of sloshing for a prismatic tank can be retrieved as a function of tank dimensions and water height [16]  
\[f_{w}^{\text{LT}}=\frac{1}{2\pi}\sqrt{\frac{\pi g}{L}\tanh\left(\frac{\pi h}{L }\right)} \tag{3}\]  
[PERSON] and [PERSON] [39] derived the expressions of the natural frequency for tanks with different bottom shapes also starting from the linear long wave theory and with the assumption of small free surface fluid response amplitude. In this case, the analogous formulation for the flat bottom geometry results as follows  
\[f_{w}^{\text{LW}}=\frac{1}{2}\sqrt{\frac{gh}{L^{2}}} \tag{4}\]  
It is evident how the dependence from the aspect ratio of volume of fluid inside the tank, \(h/L\), represents a crucial parameter. It should be noted that, as stated in [39], the linear long wave theory is valid when \(h/L\) approaches zero. In technical applications,  
Figure 1: Schematic illustration of the sloshing phenomenon and the horizontal dynamic pressures generating on the sides of a tank having generic bottom shape. i.e., related to analysis of sloshing frequencies of water tanks and or TSD, Equation (4) gives accurate values only for \(h/L<0.15\). Otherwise, the possible risk is that the values of natural frequency, effective mass, and damping ratio may be overestimated, as this theory assumes that the horizontal component of the velocity is uniform through the liquid depth. In this work, both the Equations (3) and (4) were used for comparison to verify this limitation. When dealing with different bottom shapes, the contact areas between the liquid and the tank change and the natural sloshing frequency must be differently evaluated. Concerning the sloped bottom case, the estimation can be obtained through a transformation of the potential in the Laplace equation exploited to extract the free surface height (Figure 2), as proposed by [PERSON] and coauthors [46].