## 2 Mesh Relationship Modeling of Internal Spur Gears with Tooth Wear
### Analytical Model
In general, the relationship between slope \(\mathrm{d}y_{1}/\mathrm{d}x_{1}\) and angle \(\lambda_{1}\) can be written as follows:  
\[R_{A1\text{perfect}}=\frac{R_{b1}}{\cos(\alpha_{1}+\gamma_{1\text{perfect}})} \tag{2}\]2  
where the slope of the axis \(O_{1}Y_{1}\) in \(X_{1}O_{1}Y_{1}\) can be expressed by the wear depth \(h_{A1\_x1}\) and the perfect profile function \(g_{1}(  
Similarly, substituting Equation (22) into Equation (21), the new geometric relationship between the slope \(\mathrm{d}h_{A2\_x2}/\mathrm{d}x_{2}\) and angle \(\lambda_{2}\) can be obtained as:  
\[R_{A1\text{perfect}}=\frac{R_{b1}}{\cos(\alpha_{1}+\gamma_{1\text{perfect}})} \tag{2}\]3  
Therefore, with the established analytical model of single-tooth meshing, the new mesh relationship for internal spur gears with tooth wear can be derived. It should be noted that wear depths \(h_{A1\_x1}\) and \(h_{A2\_x2}\) are denoted in Equations (3) and (7), and the slopes of wear depth \(\mathrm{d}h_{A1\_x1}/\mathrm{d}x_{1}\) and \(\mathrm{d}h_{A2\_x2}/\mathrm{d}x_{2}\) are denoted in Equations (20) and (23). Next, these geometric equations are solved to acquire the new mesh relationship of internal spur gears with tooth wear. To derive the new mesh relationship, the algorithm flow for solving the analytical model is written in detail as follow (Algorithm 1):  
```
1. Initialize geometric parameters: \(R_{b1\_}R_{b1\_}R_{b2}\), \(R_{b2}\), \(\alpha_{2}\), \(\alpha_{2}\), \(\mu_{a\_}\), \(h_{A1\_x1}\), \(h_{A2\_x2}\), \(\mathrm{d}h_{A1\_x1}/\mathrm{d}x_{1}\), \(\mathrm{d}h_{A2\_x2}/\mathrm{d}x_{2}\)
2. Initialize tolerable error of center distance and number of meshing points: \(\varepsilon_{a}\), NUM
3. Get meshing range of the driving gear: root angle \(c_{1}\) and tip angle \(a_{1}\)
4. Get meshing range of the driven gear: tip angle \(a_{2}\) and root angle \(c_{2}\)
5. For\(i=1,2,\cdots,\mathrm{NUM}\)
6. From tooth root to tip \(\alpha_{1}=c_{1}+(a_{1}-c_{1})(i-1)/(\mathrm{NUM}-1)\)
7. Solve directly \(\alpha_{1}\rightarrow\gamma_{1}\mathrm{perfect}\), \(R_{A1\_perfect}\), \(\gamma_{1}\), \(R_{A1}\), \(\lambda_{1}\), \(\delta_{1}\)
8. For\(j=1,2,\cdots,\mathrm{NUM}\)
9. From tooth tip to root \(\alpha_{1}t=a_{2}+(c_{2}-a_{2})(j-1)/(\mathrm{NUM}-1)\)
10. Solve directly \(\alpha_{1}\rightarrow\gamma_{2}\mathrm{perfect}\), \(R_{A2\_perfect}\), \(\gamma_{2}\), \(R_{A2}\), \(\lambda_{2}\), \(\delta_{2}\)
11. Solve \(\varphi_{1}=\arctan\bigl{(}\frac{R_{A1}\sin\gamma_{1}+R_{A2}\sin(\gamma_{2}+ \lambda_{2}-\lambda_{1})}{R_{A1}\cos\gamma_{1}-R_{A2}\cos(\gamma_{2}+\lambda_{ 2}-\lambda_{1})}\bigr{)}\), \(\varphi_{2}=\lambda_{2}-\lambda_{1}+\varphi_{1}\)

\(\psi_{1}=\gamma_{1}+\varphi_{1}\), \(\psi_{2}=\gamma_{2}+\varphi_{2}\), \(\theta_{1}=\delta_{1}-\psi_{1}\), \(\theta_{2}=\delta_{2}-\psi_{2}\)
12. End For
13. Find corresponding meshing point: \(\alpha_{1}t=\underset{\alpha_{1}t}{\mathrm{argmin}}\left\|R_{A2}\cos(\varphi_ {2}+\gamma_{2})-R_{A1}\cos(\gamma_{1}+\varphi_{1})-a_{\omega}\right\|_{2}\)
14. If\(\left\|R_{A2}\cos(\varphi_{2}+\gamma_{2})-R_{A1}\cos(\gamma_{1}+\varphi_{1})-a_{ \omega}\right\|_{2}<\varepsilon_{a}\)
15. Get corresponding meshing point \(\alpha_{1}t/\gamma_{2}\mathrm{perfect}\), \(R_{A2\_perfect}\), \(\gamma_{2}\), \(R_{A2}\), \(\lambda_{2}\), \(\delta_{2}\)
16. Else
17. Fail to get corresponding meshing point
18. End If
19. End For
```  
**Algorithm 1**Algorithm of the analytical model solution