## 2 Mesh Relationship Modeling of Internal Spur Gears with Tooth Wear
### Results of Mesh Relationship with Tooth Wear  
This case will introduce a new mesh relationship with non-uniform wear. Figure 6 shows the non-uniform wear on tooth profile, where non-uniform wear indicates that the wear depth of tooth profile is different, while the wear distribution for each tooth in gears is the same. In addition, Wear01, Wear02, and Wear03 are used to represent different degrees of non-uniform wear, where Wear01 is slight, Wear02 is moderate, and Wear03 is severe. It should be noted that slight, moderate, and severe wear refer only to the relative severity level among three wear conditions instead of the actual definition of wear degrees. Meanwhile, the basic parameters of internal spur gear pair employed in both cases are listed in Table 1.  
Figure 7 shows different non-uniform wear depths of the driving and driven gears tooth profiles. It can be seen that the wear depth does not vary linearly from root to tip, but decreases first and then increases, which is consistent with the trend of gear wear evolution. Figure 8 depicts different slopes of non-uniform wear depth of the internal spur gear pair tooth profiles, represented by \(\mathrm{d}h_{A1\_x1}/\mathrm{d}x_{1}\) and \(\mathrm{d}h_{A2\_x2}/\mathrm{d}x_{2}\).  
Given the wear depths and the slopes obtained above, we can obtain the mesh relationship through Algorithm 1, which includes two parts: one is the position of two meshing points on the driving and driven gears tooth profiles, represented by \(\alpha_{1}\) and \(\alpha_{1}\prime\), and the other one is the rotational angles of driving and driven gears, represented by \(\varphi_{1}\) and \(\varphi_{2}\). The results are shown in Figure 9. The influence of non-uniform wear on the single-tooth mesh relationship between \(\alpha_{1}\) and \(\alpha_{1}\prime\) is presented in Figure 9a. Compared with the perfect condition, the tooth wear changes the meshing points position obviously and the single-tooth meshing range of \(\alpha_{1}\) and \(\alpha_{1}\prime\) also decreases. Meanwhile, the relationship between \(\alpha_{1}\prime\) and \(\alpha_{1}\) is no longer linear. The single-tooth mesh relationship between \(\varphi_{1}\) and \(\varphi_{2}\) with non-uniform wear is shown in Figure 9b. It can be observed that the rotational meshing range of \(\varphi_{1}\) and \(\varphi_{2}\) is increased with respect to the perfect condition, and \(\varphi_{2}\) changes approximately linearly with \(\varphi_{1}\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & **Driving/Driven Gear** & **Items** & **Driving/Driven Gear** \\ \hline Tooth number & 40/100 & Base circle radius (mm) & 112.8/281.9 \\ Tooth width (mm) & 50 & Root circle radius (mm) & 112.5/307.5 \\ Pressure angle (\({}^{\circ}\)) & 20 & Center distance (mm) & 180 \\ Module (mm) & 6 & Tooth root angle (rad) & 0.1540/0.3872 \\ Poisson’s ratio & 0.3 & Tooth tip angle (rad) & 0.4444/0.0542 \\ Young’s modulus (Pa) & 2.05 \(\times\) 10\({}^{11}\) & Number of meshing points & 1849 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of internal spur gear pair.  
Figure 6: Diagram of non-uniform wear.  
Figure 7: Non-uniform wear depths on tooth profile: (**a**) driving gear; (**b**) driven gear.  
Figure 8: Non-uniform wear depth slopes on tooth profile: (**a**) driving gear; (**b**) driven gear.  
Figure 9: Single-tooth mesh relationship with non-uniform wear: (**a**) relationship between \(\alpha_{1}\) and \(\alpha_{1}\); (**b**) relationship between \(\varphi_{1}\) and \(\varphi_{2}\).