## 2 Mesh Relationship Modeling of Internal Spur Gears with Tooth Wear
### Analytical Model  
Figure 2 shows the gear transmission of perfect involute internal spur gear pair, where the ideal meshing line \(t\) is tangent to the base circle of two gears at points \(N_{1}\) and \(N_{2}\), respectively. Moreover, \(O_{1}\) and \(O_{2}\) are the rotation centers of the driving and driven gears, \(R_{b1}\) and \(R_{b2}\) are the base circle radii of two gears, respectively. According to the relative kinematic relation of gears, the meshing point appears on the meshing line \(t\) and moves as the gear rotates when the perfect internal spur gear pair are engaged. Generally, the meshing condition is not changed by local gear faults such as tooth crack, pitting, and spalling, which means that the meshing point on the tooth profile keeps moving along the ideal meshing line. However, tooth wear changes the tooth involute profile, resulting in the formation of a new mesh relationship when the gear pair engages. In such condition, the meshing point is not on the ideal meshing line \(N_{1}N_{2}\). Therefore, it is vital to deduce a new mesh relationship describing the tooth wear of internal spur gear. Figure 2: Gear kinematic relation for perfect involute internal spur gears. Figure 3 presents the single-tooth analytical model in the driving gear, where \(X_{1}O_{1}Y_{1}\) denotes the local coordinate system and it rotates with the driving gear. Moreover, \(A_{1\text{perfect}}\) represents the meshing point on the prefect tooth profile. When tooth wear occurs, the position of the meshing point changes from \(A_{1\text{perfect}}\) to \(A_{1}\), and the wear depth between the two meshing points \(A_{1\text{perfect}}\) and \(A_{1}\) is \(h_{A1,x1}\). In addition, \(R_{A1\text{perfect}}\) and \(R_{A1}\) are the corresponding radii of the two meshing points. In the local coordinate system \(X_{1}O_{1}Y_{1}\), the single-tooth geometric relationships of the driving gear can be written as follows:  
\[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]  
\[R_{A1\text{perfect}}=\frac{R_{b1}}{\cos(\alpha_{1}+\gamma_{1\text{perfect}})} \tag{2}\]  
\[\gamma_{1}=\arctan\left(\frac{R_{A1\text{perfect}}\sin\gamma_{1\text{perfect} }-h_{A1,x1}}{R_{A1\text{perfect}}\cos\gamma_{1\text{perfect}}}\right) \tag{3}\]  
\[R_{A1}=\frac{R_{A1\text{perfect}}\cos\gamma_{1\text{perfect}}}{\cos\gamma_{1}} \tag{4}\]  
where \(\alpha_{1}\) represents the angle of the perfect meshing point \(A_{1\text{perfect}}\) on the prefect tooth profile, and \(\alpha_{2}\) represents the half tooth angle on the base circle. \(\gamma_{1\text{perfect}}\) and \(\gamma_{1}\) represent the angles that correspond to points \(A_{1\text{perfect}}\) and \(A_{1}\), respectively. As shown in Figure 4, the local coordinate system of the driven gear \(X_{2}O_{2}Y_{2}\) rotates with the driven gear. Moreover, the meshing point on the perfect profile of the driven gear is represented by \(A_{2\text{perfect}}\), and represented by \(A_{2}\) on the worn tooth profile. \(h_{A2,x2}\) is the wear depth between the two meshing points. \(R_{A2\text{perfect}}\) and \(R_{A2}\) represent the corresponding radii of the two meshing points \(A_{2\text{perfect}}\) and \(A_{2}\). Given the above definitions, the single-tooth geometric relationships of the driven gear can be expressed as follows:  
\[\gamma_{2\text{perfect}}=\arctan(\alpha_{2}\prime+\alpha_{1}\prime)-\alpha_{1}\prime \tag{5}\]  
\[R_{A2\text{perfect}}=\frac{R_{b2}}{\cos(\alpha_{1}\prime+\gamma_{2\text{perfect }})} \tag{6}\]  
\[\gamma_{2}=\arctan\left(\frac{R_{A2\text{perfect}}\sin\gamma_{2\text{perfect} }+h_{A2,x2}}{R_{A2\text{perfect}}\cos\gamma_{2\text{perfect}}}\right) \tag{7}\]  
Figure 3: Analytical model of single tooth in the driving gear. \[R_{A2}=\frac{R_{A2\text{perfect}}\cos\gamma_{2\text{perfect}}}{\cos\gamma_{2}} \tag{8}\]  
where \(\alpha_{1}t\) represents the angle of the perfect meshing point \(A_{2\text{perfect}}\) on the single-tooth profile, and \(\alpha_{2}t\) represents the half tooth angle on the base circle.