## 3 Mesh Stiffness of Internal Spur Gears with Tooth Wear
### Multi-Tooth Mesh Stiffness
#### 3.2.2 Results of Multi-Tooth Mesh Stiffness with Tooth Wear  
Once the single-tooth mesh relationship and the multi-tooth mesh rule are gained, we can obtain the multi-tooth mesh relationship with non-uniform tooth wear. The parameters of the internal spur gears used in the simulation for calculation of multi-tooth mesh stiffness with tooth wear are given in Table 1. Taking Wear03 as an example, the results are presented in Figure 15. Figure 15a shows the mesh relationship between \(\varphi_{1}\) and \(\varphi_{2}\) of the first, second, and third tooth pairs. Figure 15b is the zoomed picture of Figure 15a, which depicts the meshing ranges of the first and second tooth pairs, and there is a difference \(\Delta\varphi_{2}\) between two pairs. In this paper, the _Threshold_ in the proposed multi-tooth mesh rule is set to \(10^{-4}\) rad and then the influence of tooth wear on meshing range of internal spur gear can be acquired. Figure 15c shows the new meshing range of double teeth and the meshing range of double/single/double meshing process is changed in Figure 15d. Compared to the results shown in Figure 14, the meshing range of the second tooth-pair reduces obviously, and the main reason is that when the gear suffers tooth wear, the double-tooth meshing range decreases, while that of the single-tooth increases.  
The results above show that the multi-tooth mesh relationship affects the meshing range of single tooth in Section 3.1. With the new obtained single-tooth meshing range, the new single-tooth mesh stiffness is calculated, which is shown in Figure 16. The results indicate that with the degree of wear increasing, the meshing range of the new mesh stiffness decreases gradually, and correspondingly, the amplitude of the meshing stiffness increases slightly.  
Figure 14: Tooth mesh relationship with perfect tooth profile: (**a**) single-tooth; (**b**) multi-tooth.  
Figure 16: Single-tooth mesh stiffness with the new single-tooth mesh relationship.  
Figure 15: Multi-tooth mesh relationship with non-uniform wear (Wear03): (**a**) multi-tooth; (**b**) zoom of (**a**); (**c**) double-tooth meshing range of (**b**); (**d**) double/single/double meshing range.  
Therefore, the multi-tooth mesh stiffness can be evaluated by superimposing the new single-tooth mesh stiffness as the multi-tooth mesh relationship, which is written as follows:  
\[k=\sum_{i=1}^{2}\frac{1}{\frac{1}{k_{h,i}}+\frac{1}{k_{h,i}}+\frac{1}{k_{s1,i}}+ \frac{1}{k_{s1,i}}+\frac{1}{k_{s1,i}}+\frac{1}{k_{f1,i}}+\frac{1}{k_{h2,i}}+ \frac{1}{k_{s2,i}}+\frac{1}{k_{s2,i}}+\frac{1}{k_{s2,i}}} \tag{60}\]  
where the subscript \(i=1,\ 2\) is the number of the meshing tooth-pair.  
Figure 17 shows the multi-tooth meshing stiffness. It is clearly visible that the appearance of tooth wear mainly affects the meshing range of the mesh stiffness, in which the single-tooth meshing range decreased first and then increased with the increasing wear depth. However, the effect of tooth wear on the mesh stiffness amplitude is not significant, and only the amplitude of double-teeth meshing range is slightly increased with the increasing wear depth.