## 2 The Original Dynamic Model of Gear System
### Actual Tooth Surface Contact Analysis
#### 2.1.2 Meshing Performance Analysis of Actual Tooth Surface  
The actual tooth surface was constructed by the superposition of the theoretical tooth surface and normal deviation surface. As shown in Figure 5, mesh was divided on the tooth surface. The position vector \(\mathbf{r_{0}}\ (x,y,z)\) of point \(N\) on the theoretical tooth surface was obtained, based on the space meshing theory [22; 23]. The tooth surface deviation, \(\delta(h,l)\), corresponding to point \(N\) was calculated by the mathematical model of each error. The corresponding relationship between coordinates of the theoretical tooth surface and the deviation surface is shown in Equation (4). Based on the superposition of the deviation \(\delta\) and the position vector \(\mathbf{r_{0}}\) of the theoretical tooth surface, the position vector and normal vector of the actual tooth surface are expressed in Equations (5) and (6).  
\[\left\{\begin{array}{l}x^{2}+y^{2}=h^{2}\\ z=l\end{array}\right. \tag{4}\]  
Figure 4: Different directions of slope errors for the (**a**) tooth profile and (**b**) lead.  
Figure 3: Description of parameter values in the directions of the tooth profile and tooth width.  
Figure 2: Three forms of shape deviations in the direction of the tooth profile.  
\[\mathbf{r}(u,\theta)=\mathbf{r}_{0}(u,\theta)+\delta(u,\theta)\cdot\mathbf{n}_{0}(u,\theta) \tag{5}\]  
\[\mathbf{n}(u,\theta)=\frac{\frac{\partial\mathbf{r}(u,\theta)}{\partial u}\times \frac{\partial\mathbf{r}(u,\theta)}{\partial u}}{\left|\frac{\partial\mathbf{r }(u,\theta)}{\partial u}\times\frac{\partial\mathbf{r}(u,\theta)}{\partial \theta}\right|} \tag{6}\]  
where \(u\) and \(\theta\) are the parameters of the tooth surface, and the relationships between them and tooth surface coordinates \(x,y\), and \(z\) are shown in reference [23]. \(\mathbf{n_{0}}\) is the unit normal vector of the theoretical tooth surface.  
A pair of helix gears was used as an example, whose parameters are shown in Table 1. The value of each error was randomly sampled within the tolerance limit of level 5, as shown in Table 2. In order to better analyze the influence of error, only the tooth surface error of the pinion was considered in this paper. Tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) were carried out based on the actual tooth surface.  
Figure 6 shows the loaded transmission error (LTE) and the unloaded transmission error (TE) of the helical gear pair with errors. TE is mainly caused by the gap between the teeth. LTE includes the gap between the teeth and the bearing deformation of the tooth surface. The influence of errors on the gear meshing performance was analyzed by comparing them with a case with no errors. Figure 7 illustrates the amplitude of LTE and the load distribution of the tooth surface for the two cases. It can be seen from Figure 7 that due to the existence of errors, the amplitude of LTE increased significantly, and the  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & **Pinion** & **Gear** \\ \hline Number of teeth, \(Z_{1}/Z_{2}\) & 19 & 47 \\ Handedness & left & right \\ Normal modulus, \(m_{n}\) (mm) & 6 & \\ Normal pressure angle, \(\alpha\) (\({}^{\circ}\)) & 20 & \\ Helix angle, \(\beta\) (\({}^{\circ}\)) & 9.91 & \\ Tooth width, \(B\) (mm) & 79 & 75 \\ Load torque, \(T_{2}\) (N\(\cdot\)m) & 1000 & \\ Input speed, \(n_{1}\) (rpm) & 2000 & \\ \hline \end{tabular}
\end{table}
Table 1: The main parameters of the gears.  
Figure 5: The schematic diagram of tooth surface meshing.  
phenomenon of uneven load-bearing on the tooth surface occurred. These effects were not conducive to the smooth operation of gears.