## 2 Materials and Methods
What has been described is valid exclusively for symmetrical containers concerning a single axis. Otherwise, for each main axis of the section of the container, there is a fundamental mode [25; 26]. Based on the hypotheses made, a one-dimensional model was developed consisting of a trolley subject to motion that translated along the horizontal direction, visible in Figure 2b, in which a linear damper was connected to the pendulum to add to the model the dissipative characteristics given by the viscosity and by the friction created between the fluid and the container [27]. Figure 2: First three modes of the free surface of the fluid (**a**) and lumped-parameter model used in the numerical activity (**b**). For the definition of the equations of motion that governed the chosen 2-DOF model, the Lagrange energy method, reported in (3), was used. \[\frac{d}{dt}\Big{(}\frac{\partial E_{c}}{\partial\dot{q}}\Big{)}-\frac{\partial E _{c}}{\partial q}+\frac{\partial D}{\partial\dot{q}}+\frac{\partial V}{\partial q }=Q \tag{3}\]  
where \(q\) indicates the following vector of Lagrangian coordinates \( \begin{bmatrix}x\\ \theta\end{bmatrix} \), where \(x\) the one-dimensional motion of the container is indicated, and where \(\theta\), the oscillatory motion of the equivalent pendulum which represents the motion of the fluid, is indicated. Furthermore, \(E_{c}\) is the total kinetic energy of the system, \(D\) is the dissipative function equal to half of the power dissipated by the damper during its operation, \(V\) instead describes the potential energy of the system. Finally, \(Q\) represents the active component of forces and torques applied to the system. \[(M+m)\dot{x}+mL\ddot{\theta}\cos(\theta)-mL\dot{\theta}^{2}\sin(\theta)=F(t) \tag{4}\]  
\[mL\ddot{x}\cos(\theta)+mL^{2}\ddot{\theta}+\sigma L^{2}\dot{\theta}\cos^{2}( \theta)+mgL\sin(\theta)=0 \tag{5}\]  
The two equations of motion are presented in a non-linear form, but for small oscillations, it is possible to linearize them [28]. In the following expression the linearized equations are reported in matrix form:  
\[\left[ \begin{array}{cc}(M+m)&mL\\ mL&mL^{2}\end{array} \right]\left[ \begin{array}{cc}\dot{x}\\ \ddot{\theta}\end{array} \right]+\left[ \begin{array}{cc}0&0\\ 0&\sigma L^{2}\end{array} \right]\left[\dot{x}\right]+\left[ \begin{array}{cc} 0&0\\ 0&mgL\end{array} \right]\left[ \begin{array}{c}x\\ \theta\end{array} \right]=\left[ \begin{array}{c}F(t)\\ 0\end{array} \right] \tag{6}\]  
The container's acceleration becomes the system's control variable and is consequently the variable we focused on in this study. Specifically, by rearranging the terms known to the second member, and dividing the expression by the inertia of the pendulum, the following relation was obtained:  
\[\ddot{\theta}+\frac{\sigma}{m}\dot{\theta}+\frac{\xi}{L}\theta=-\frac{1}{L} \ddot{x} \tag{7}\]  
Therefore, it is clear from (7) that to calibrate the model, it is necessary to define the equivalent mass of the system \(m\), the equivalent length of the pendulum \(L\), and the equivalent damping \(\sigma\) of the liquid. \[m_{n}=m_{liq}\Big{[}\frac{d\tanh(2\xi_{n}h/d)}{\xi_{n}(\xi_{n}^{2}-1)h}\Big{]} \tag{8}\]  
The mass of the equivalent pendulum corresponding to the sloshing mass of the fluid \(m_{n}\) can be estimated by using Equation (8), while the equivalent length of the pendulum can be estimated by using relation (9)  
\[L_{n}=\frac{d}{2\xi_{n}\tanh(2\xi_{n}h/d)} \tag{9}\]  
For both relationships, the integer \(n\) indicates the mass and equivalent length of the pendulum representative of the \(n\)th generic mode of vibration. Experimental testing or a CFD analysis may be used for estimating equivalent damping [29]. In this work, it was decided to test the numerical approach. A CFD analysis makes it possible to find a solution for discretized Navier-Stokes partial differential equations because they do not have an analytical solution.