## 2 Dynamic Model of Vibrating Screen Excitation System
### Verification of Dynamic Model of Excitation System  
Firstly, it is known that the meshing force of helical gears can be calculated as Formula (1) [34]. An individual gear transmission virtual prototype model is used to calculate the meshing force. \[\left\{ \begin{array}{c}F_{t}=\frac{2T_{L}}{d_{1}}=\frac{2.229.200}{\frac{3.28}{3.28}.93}=1389.08N\\ F_{r}=\frac{F_{t}\tan a_{n}}{\cos\beta}=\frac{1389.08-\tan 20}{\cos 32.28}=598.01N\\ F_{\rm a}=F_{t}\tan\beta=1389.08-\tan 32.28=877.46N\\ F_{n}=\frac{F_{r}}{\cos p-\cos a_{n}}=\frac{1389.08}{\cos 32.28-\cos 20}=1748.45N \end{array} \right., \tag{1}\]  
where \(F_{n}\), \(F_{r}\), \(F_{t}\), and \(F_{a}\) are, respectively, normal meshing force, the radial meshing force, the tangential, and the axial component of meshing force, \(T_{L}\) is driven gear torque, \(d_{1}\) is pitch diameter, \(\alpha_{n}\) is the pressure angle of the pitch circle, and \(\beta\) is the helical angle of the pitch circle. ADAMS mainly uses two methods, i.e., the restitution method and impact method, to calculate the contact force, of which the impact function method is more suitable for the helical gear transmission of vibration exciter. As a result, the impact function model is adopted for the normal meshing force \(F_{n}\) calculation, i.e., Formula (2) [35]. It can be seen from the formula that when the distance between two objects at the time of collision minus the distance between objects at the initial time is greater than or equal to 0, the two objects do not contact, and the \(F_{n}\) is 0. In other cases, the specific value of \(F_{n}\) can be calculated by Formula (2). Using the same parameter setting as shown in Figure 1b, the meshing force obtained by simulation calculation and theoretical calculation is shown in Table 1. It can be seen that the meshing force value calculated by simulation is bigger than the theoretical value calculated by Formula (1), and the main reason is that the former considers friction. The maximum deviation between the simulated value and the theoretical value is 3.1%, which is within a reasonable range according to references [36; 37; 38], so the setting of gear contact parameters is reasonable. Figure 1: (**a**) Schematic diagram of mine-used linear vibrating screen and virtual prototype model of excitation system. (**b**) Flowchart of the modeling contents. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Meshing Force** & **Theoretical Value (N)** & **Simulation Value (N)** & **Relative Error** \\ \hline Tangential force \(F_{t}\) & 1389.08 & 1390.54 & 0.11\% \\ Radial force \(F_{r}\) & 598.01 & 616.42 & 3\% \\ Axial force \(F_{a}\) & 877.46 & 905.22 & 3.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between theoretical value and simulation average value of meshing force. Impact method model:  
\[F_{n}=\left\{ \begin{array}{cc}0&q\geq q_{0}\\ K(q_{0}-q)^{\epsilon}-C\cdot\frac{dq}{dt}\cdot step(q,q_{0}-d,\ q_{0}\ )&q\leq q_{0}\end{array} \right., \tag{2}\]  
where \(K\) is the stiffness coefficient of the collision object, \(q_{0}\) is the distance between objects at the initial time, \(q\) is the distance when the object collides, \(e\) is the non-linear index, \(C\) is the damping coefficient, and \(d\) is the penetration depth. Secondly, the correctness of the virtual prototype model of the vibration system in Figure 1a can also be verified from the point of view of the motion trajectory of the mass center (CM) of the vibrating screen. The curve in Figure 2 is obtained by running the model in Figure 1a. It can be seen in Figure 2 that after the system enters a stable working state, the displacement curves in the x and y directions (CM_x and CM_y) change into a simple harmonic curve, which is in line with the mass center motion of the vibration exciter deduced in reference [30]. The vibration amplitudes of CM_x and CM_y are 3.32 mm and 3.34 mm. By merging, the synthesized amplitude (CM_t) is 4.7 mm, which is approximately consistent with that in the actual working condition of 4.8 mm.