## 2 Mesh Relationship Modeling of Internal Spur Gears with Tooth Wear
### Analytical Model
\(\gamma_{2\text{perfect}}\) and \(\gamma_{2}\) represent the angles of the points \(A_{2\text{perfect}}\) and \(A_{2}\), respectively. According to the above two analytical models, the single-tooth meshing analytical model with tooth wear can be established, which is presented in \[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]0 Three coordinates \(XO_{2}Y\), \(X_{1}O_{1}Y_{1}\), \(X_{2}O_{2}Y_{2}\) and two meshing points \(A_{1}\) and \(A_{2}\) are contained in the proposed model, in which \(XO_{2}Y\) is the fixed global coordinate system, which can be established through the axis \(O_{1}O_{2}\) of two gears. For each gear, there is a corresponding local coordinate system rotating with, \(X_{1}O_{1}Y_{1}\) for the driving and \(X_{2}O_{2}Y_{2}\) for the driven gears, respectively. Additionally, \(t_{1}\) and \(t_{2}\) are the tangents of two meshing points. In order to obtain the new mesh relationship of internal spur gear with tooth wear, the following geometric principles need to be satisfied:  
1. The meshing point of the driving gear moves from tooth root to tip as the gear rotates, and the driven gear is the opposite. 2. The gear center distance remains unchanged, which can be written as follows: \[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]1 (9)  
\[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]2 Analytical model of single tooth in the driven gear. \[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]3 Analytical model of singe-tooth meshing with tooth wear. (3) With the meshing points \(A_{1}\) and \(A_{2}\) engaging, the axis \(Y\) and the tangents of two meshing points are the same, which can be expressed as:  
\[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]4  
\[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]5  
Additionally, in order to describe the geometric relationships shown in Figure 5, there are the following formulas:  
\[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]6  
\[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]7  
\[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]8  
\[\gamma_{1\text{perfect}}=\arctan(\alpha_{2}+\alpha_{1})-\alpha_{1} \tag{1}\]9  
\[R_{A1\text{perfect}}=\frac{R_{b1}}{\cos(\alpha_{1}+\gamma_{1\text{perfect}})} \tag{2}\]0  
\[R_{A1\text{perfect}}=\frac{R_{b1}}{\cos(\alpha_{1}+\gamma_{1\text{perfect}})} \tag{2}\]1  
where \(\delta_{1}\) and \(\delta_{2}\) are the angles between the tangent \(t_{1}\) and \(O_{1}A_{1}\), the tangent \(t_{2}\) and \(O_{2}A_{2}\), respectively. \(\psi_{1}\) denotes the angle between \(O_{1}X\) and \(O_{1}A_{1}\), where \(\psi_{1}>0\) when \(A_{1}\) is above \(O_{1}X\) and \(\psi_{1}<0\) when \(A_{1}\) is below \(O_{1}X\). \(\psi_{2}\) denotes the angle between \(O_{2}X\) and \(O_{2}A_{2}\), where \(\psi_{2}>0\) when \(A_{2}\) is above \(O_{2}X\) and \(\psi_{2}<0\) when \(A_{2}\) is below \(O_{2}X\). \(\theta_{1}\) and \(\theta_{2}\) represent the acute angles between the tangent \(t_{1}\) and \(O_{2}X\), the tangent \(t_{2}\) and \(O_{2}X\), respectively. \(\varphi_{1}\) represents the rotational angle of \(X_{1}O_{1}Y_{1}\) relative to \(O_{1}X\), where \(\varphi_{1}>0\) when the axis \(X_{1}\) is above \(O_{1}X\) and \(\varphi_{1}<0\) when the axis \(X_{1}\) is below \(O_{1}X\). \(\varphi_{2}\) presents the rotational angle of \(X_{2}O_{2}Y_{2}\), where \(\varphi_{1}>0\) when the axis \(X_{2}\) is above \(O_{2}X\) and \(\varphi_{1}<0\) when the axis \(X_{2}\) is below \(O_{2}X\). \(\lambda_{1}\) and \(\lambda_{2}\) represent the angles between the tangent \(t_{1}\) and the axis \(X_{1}\) and the tangent \(t_{2}\) and the axis \(X_{2}\), respectively. In addition to the axes \(O_{1}Y_{1}\) and \(O_{2}Y_{2}\) of the meshing points \(A_{1}\) and \(A_{2}\), the slope of the tangents \(t_{1}\) and \(t_{2}\) is also changed by gear wear.