## 1 Introduction
### Equivalent Model of Liquid Sloshing for the Liquid Tank Semi-Trailer  
The schematic diagram of the TP model is shown in Figure 1, and its symbols are explained in Table 1.  
For the TP model, the liquid in the tank is divided into the sloshing part and the static part. The movement of the sloshing part is equivalent to the swing of the pendulum ball. The motion track is circular or elliptical and is similar to the tank section profile, which meets the following requirements:  
\[\frac{l_{a}}{l_{b}}=\frac{l_{acg}}{l_{bcg}}=\frac{l_{ap}}{l_{bp}}=\zeta \tag{1}\]  
Based on Lagrange mechanics, the motion equation of the pendulum ball in the accelerated translational tank can be derived as follows [13]:  
\[\ddot{\gamma}\Big{(}l_{ap}^{2}\sin^{2}\gamma+l_{bp}^{2}\cos^{2}\gamma\Big{)}+ \frac{1}{2}\dot{\gamma}^{2}\Big{(}l_{ap}^{2}-l_{bp}^{2}\Big{)}\Big{(}\sin 2 \gamma-gl_{bp}\cos\gamma-a_{y}l_{ap}\sin\gamma\Big{)}=0 \tag{2}\]  
The lateral sloshing force \(F_{W}\) on the tank caused by liquid sloshing is  
\[F_{w}=m_{p}\Big{(}a_{y}-l_{ap}\ddot{\gamma}\sin\gamma-l_{ap}\dot{\gamma}^{2} \cos\gamma\Big{)}+m_{s}a_{y} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Description** & **Value** & **Unit** \\ \hline \(l_{ap}\) & long semi-axes of the motion track of the pendulum ball & 0.5613 & m \\ \(l_{bp}\) & short semi-axes of the motion track of the pendulum ball & 0.3742 & m \\ \(l_{acg}\) & long semi-axes of the liquid center of mass motion track & \(-\) & m \\ \(l_{bcg}\) & short semi-axes of the liquid center of mass motion track & \(-\) & m \\ \(l_{a}\) & long semi-axes of the tank body section & 1.0925 & m \\ \(l_{b}\) & short semi-axes of the tank body section & 0.7283 & m \\ \(l_{bs}\) & distance from the centroid of the liquid at rest to the bottom of & 0.6939 & m \\ \(m_{s}\) & mass of the liquid at rest & 5631 & kg \\ \(m_{p}\) & mass of the swing ball & 7826 & kg \\ \(m_{w}\) & total mass of liquid in the tank & \(13,457\) & kg \\ \(\zeta\) & ellipticity of the section of the liquid tank & \(1.5\) & \(-\) \\ \(\gamma\) & liquid swing angle & \(-\) & \(\circ\) \\ \(\Delta\) & liquid filling ratio in the tank & 0.6 & \(-\) \\ \(L\) & tank body length & 9 & m \\ \(\rho\) & liquid density in the tank & 997 & kg/m\({}^{3}\) \\ \(a_{y}\) & lateral excitation of the tank body & 1 & m/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Description and value of TP model parameters and basic parameters of the tank.  
Figure 1: Schematic diagram of the TP model.  
The sloshing torque \(M_{w}\) around the lowest point of the tank caused by liquid sloshing is  
\[M_{w}=-m_{s}a_{y}l_{bs}+m_{p}\bigg{|}\begin{matrix}l_{ap}\cos\gamma&l_{b}-l_{bp} \sin\gamma\\ a_{y}-l_{ap}\left(\tilde{\gamma}\sin\gamma+\dot{\gamma}^{2}\cos\gamma\right)&l_ {bp}\left(\dot{\gamma}^{2}\sin\gamma-\tilde{\gamma}\cos\gamma\right)+g\bigg{|} \end{matrix} \tag{4}\]  
The TP model parameters shall be determined using the parameter identification method. According to the equivalence principle of dynamic similarity, kinematic similarity, and geometric similarity [14], the simulation results of an elliptical cylinder tank under step excitation of unit lateral acceleration are simulated by fluid numerical simulation using the Fluent software for parameter identification, and the TP model parameters are obtained as shown in Table 1.