## 4 Results and Discussion
### Sloshing Force Decomposition  
According to [8], the sloshing force, either measured in the experiments using Expression (4) or computed using the pressure and velocity fields by Equations (19) and (20), can be divided into three components: the inertial part, which is related to the inertia of the tank motion, and would be present even in the case where there is no relative movement of the liquid in the tank, for example if \(f=1\); a dissipative contribution that is related to the relative movement of the liquid inside the tank and is responsible for the damping increase of the mechanical system; and a high-frequency contribution of the harmonics with higher frequencies than the natural one. This last contribution is filtered out by using a fourth-order Butterworth filter around the natural frequency of the system. \[F_{slosh}=\underbrace{m_{l}a_{t}}_{\text{Inertial}}+\underbrace{\Delta F_{ slosh}}_{\text{Dissipative}}+\underbrace{F_{H\_F\_F\_F\_F}}_{\text{High frequency}} \tag{22}\]  
The part of the force that causes the extra damping in the tank oscillation is clearly the dissipative one; this force \(\Delta F_{slosh}\) can be modeled as an amplitude multiplied by a shifted cosine function with the natural frequency \(\omega_{0}\). Expanding the cosine, two terms are generated which can be related with the typical force decomposition in velocity and acceleration projections [53]. \[ \begin{split}\Delta F_{slosh}=F\cos(\omega_{0}t-\phi)& =\\ F\cos(\omega_{0}t)\cos(\phi)+F\sin(\omega_{0}t)\sin(\phi)& =\\ -\beta m_{l}\omega_{0}^{2}Y\cos(\omega_{0}t)+\gamma\omega_{0}Y \sin(\omega_{0}t)&=\\ \underbrace{\beta m_{l}a_{t}}_{\text{Sloshing effective mass}}+ \underbrace{\gamma\gamma_{l}}_{\text{Sloshing induced dissipation}}\end{split}  \tag{23}\]  
where the \(\beta\) and \(\gamma\) coefficients can be related with the amplitude \(F\) and the shift angle \(\phi\) by the equations:  
\[\beta=\frac{F}{-m_{l}\omega_{0}^{2}Y}\cos(\phi)\quad,\quad\gamma=\frac{F}{ \omega_{0}Y}\sin(\phi)\quad\text{with}\quad Y=y_{0}\text{e}^{-\zeta\omega_{0}t} \tag{24}\]  
Figure 13: Comparison of the evolution of the relative liquid kinetic energy and the analytical periodic approximation and its envelope. The filtered version of the total sloshing force, where the high frequency component is removed, can be expressed using both formulations as:  
\[\widehat{F}_{slosh}=F_{slosh}-F_{\mathcal{H}\mathcal{F}}=(1+\beta)m_{l}a_{t}+ \gamma v_{t}=m_{l}a_{t}+F\cos(\omega_{0}t+\phi) \tag{25}\]  
Another way of understanding the vertical sloshing force is by relating it to the motion of the Center of Mass (CM) of the liquid. Dividing \(\widehat{F}_{slosh}\) by \(-m_{l}\omega_{0}^{2}\), the following equation is obtained:  
\[y_{CM}=\frac{\widehat{F}_{slosh}}{-m_{l}\omega_{0}^{2}}=\frac{m_{l}a_{t}}{-m_{ l}\omega_{0}^{2}}+\frac{\Delta F_{slosh}}{-m_{l}\omega_{0}^{2}}=\underbrace{y_{t}}_{ \text{Tank position}}+\underbrace{y_{rel}}_{\text{Relative motion to the tank}} \tag{26}\]  
Equation (26) identifies the inertial component of the vertical sloshing force with the tank motion and the dissipative component with the relative motion of the liquid with respect to the tank. In Figure 14, the non-dimensional relative position of the liquid center of mass \(y_{rel}\) is compared to the term \(-\Delta F_{slosh}/m_{l}\omega_{0}^{2}\) for both water and glycerine test cases. In this figure, the relative position has been non-dimensionalized with half the liquid height \(h_{l}/2\), which is the initial and final position of the liquid CM relative to the tank. It is worth noting that in Figure 14, the amplitude of the signals has been extracted from the time domain SPH computation in order to perform the comparison. The overall amplitude of the relative motion of the liquid center of mass matches the amplitude of the dissipative vertical sloshing force as Equation (26) indicates. Therefore, in order to understand the dissipation of the vertical sloshing problem, it is only necessary to track the relative motion of the liquid center of mass with respect to the tank.