## 2 Materials and Methods  
The analysis of the dynamic behavior of a fluid inside a container necessarily passes through the resolution of the Navier-Stokes equations [17]. The accurate estimation of the vibration characteristics of the sloshing motion is crucial for analyzing the behavior of tanks or containers experiencing base motion. The virtual mass force, drag and stagnation pressure force, lift force, and others should all be evaluated to analyze the behavior of a fluid in a container [18]. Such modeling can be simplified when the oscillations are confined and non-linear sloshing phenomena are avoided [19]. Under these conditions, it is possible to use the equivalent mechanical model shown in Figure 1a, which takes into account the motion of the fluid directly proportional to the acceleration of the rigid container, described by a mass \(m_{0}\) integral to the container, and the superposition of \(j\) different modes of oscillation described using a series of simple pendulums in which the remaining mass of fluid is concentrated, linearized to the vertical position [20]. In (1), from the linear wave theory obtained by the linearization of the Navier-Stokes equations, the equation of the natural frequency \(\omega_{n}\) of the \(j\)-th oscillation mode has been derived:  
\[\omega_{n,j}=\sqrt{\left(\frac{g\epsilon_{j}}{R}\right)tanh\left(\frac{ \epsilon_{j}h_{s}}{R}\right)} \tag{1}\]  
with:  
* \(g=\) gravitational acceleration;
* R = container radius;
* \(h_{s}=\) static height of the liquid;
* \(\epsilon_{j}=j\)-th root of the derivative of the Bessel function of the first kind (for the first fundamental mode, it is equal to 1.841). Figure 1: Free-surface motion equivalent mechanical models: (**a**) multi-DOF pendulum-based model; (**b**) 1-DOF pendulum-based model. The dependence of the damping coefficient \(\sigma_{j}\) of the _j-th_ mode of oscillation on the internal forces of the system means that \(\sigma_{j}\) can be calculated by observing how the wave motion decreases according to a logarithmic decay due to the dissipation of energy. In addition, only the first asymmetric mode was considered because it is dominant over higher modes. The damping factor from [PERSON]'s theory was obtained by an empirical law reported in (2) that considered the kinematic viscosity \(\
u\), the depth of the liquid \(h\), and the radius of the container \(R\). \[\xi_{1}=\frac{2.89}{\pi}\sqrt{\frac{\
u}{\sqrt{R^{3}g}}}\Bigg{[}1+\frac{0.318 }{sinh\left(\frac{1.34h}{R}\right)}\frac{1-\frac{h}{R}}{cosh\left(\frac{1.34h} {R}\right)}\Bigg{]} \tag{2}\]  
From [PERSON]'s theory, it was possible to derive the relationships between the different physical parameters, where it was found that the values of \(L_{j}\) were reduced by approaching the points where they are bound to make the _j-th_ in pendulum oscillate around the free surface. Furthermore, there is an inverse proportionality between the indices \(j\) for the lengths \(L_{j}\) and masses \(m_{j}\). Such a result allowed us to simplify the problem further, as shown in Figure 1b, and model the system as a single pendulum of mass \(m_{1}\) and length \(l_{1}\), bounded to oscillate around a point at the center of the free and undisturbed surface of the liquid [21; 22]. The plane of the free surface was always orthogonal to the pendulum and planar during the motion. Further simplifying hypotheses were as follows:  
* The incompressibility of the liquid;
* A non-viscous flow rate;
* No environmental speed;
* A two-dimensional system;
* Small amplitudes of the wave. With this model, only the fundamental oscillation mode (mode 1) is adequately described, which, among the various _j_-th modes, is the one with the lowest frequency and characterized by the most significant amplitude (see Figure 2a). In the literature, it has been demonstrated experimentally that the influence of the other higher frequency modes is not particularly significant, and therefore, their inclusion in the model would lead to excessive complications in the control system [23; 24].