## 3 Mesh Stiffness of Internal Spur Gears with Tooth Wear
### Single-Tooth Mesh Stiffness  
Gear mesh stiffness is usually considered to be the main factor affecting gear dynamics with failures [24; 25; 26]. The influence of tooth wear on mesh stiffness will be analyzed with the use of the new mesh relationship gained in Section 2. The gear tooth is considered to be a cantilever beam, and the potential energy method is used to calculate mesh stiffness. In reference [22], the potential energy of a single tooth contains four parts: Hertzian contact energy \(U_{h}\), bending energy \(U_{b}\), shear deformation energy \(U_{s}\), and axial compressive energy \(U_{a}\), and the corresponding stiffness includes Hertzian contact stiffness \(k_{h}\), bending stiffness \(k_{b}\), shear stiffness \(k_{s}\), and axial compressive stiffness \(k_{a}\). The internal gear pair contains external and internal gears. For external spur gears, there are two different situations in calculating mesh stiffness: Situation 1: \(R_{b1}>R_{r1}\) and Situation 2: \(R_{b1}\leq R_{r1}\). In Situation 1, the base circle radius is larger than the root circle radius. Figure 10 shows the cantilever beam model for the gear tooth with wear where wear occurs on both sides of single tooth. It is worth noting that the wear depth on both sides of the gear is consistent, and the tooth wear changes both the height of tooth profile, represented by \(h\) and \(h_{x}\), and the action force direction, represented by \(\lambda_{1}\). These parameters are derived as:  
\[F_{a}=F\sin\lambda_{1},\ F_{b}=F\cos\lambda_{1} \tag{24}\]  
\[h=R_{b1}[(\alpha_{1}+\alpha_{2})\cos\alpha_{1}-\sin\alpha_{1}]-h_{A1\_x1} \tag{25}\]  
\[h_{x}=R_{b1}[(\alpha_{2}-\alpha)\cos\alpha+\sin\alpha]-h_{\text{wear\_x1}} \tag{26}\]  
Figure 10: Beam model for single tooth with worn profile when base circle is larger than root circle. \[d=R_{b1}[(\alpha_{1}+\alpha_{2})\sin\alpha_{1}+\cos\alpha_{1}]-R_{b1}\cos\alpha_{2} \tag{27}\]  
\[I_{x}=\frac{1}{12}(2h_{x})^{3}b,A_{x}=2h_{x}b \tag{28}\]  
\[I_{d1}=\frac{1}{12}(2R_{b1}\sin\alpha_{2})^{3}b,A_{d1}=2R_{b1}\sin\alpha_{2}b \tag{29}\]  
where \(h_{\text{wear\_x1}}\) denotes the wear depth at the axis \(X_{1}\). \(I_{x}\) and \(A_{x}\) depict the moment of inertia and the area in section \(x\), respectively. \(I_{d1}\) and \(A_{d1}\) depict the moment of inertia and the area in section \(d_{1}\), respectively. According to the potential energy theory, the relationship between the potential energy and corresponding stiffness is composed of two parts: from root circle to base circle and from base circle to the meshing point \(A_{1}\), which are expressed as:  
\[U_{h}=\frac{F^{2}}{2k_{h}},\,k_{h}=\frac{\pi Eb}{4(1-v^{2})} \tag{30}\]  
\[U_{b}=\frac{F^{2}}{2k_{b}}=\int\limits_{0}^{d}\frac{\left[F_{b}(d-x)-F_{a}h \right]^{2}}{2 EI_{x}}\text{d}x+\int\limits_{0}^{d_{1}}\frac{\left[F_{b}(d+x)-F _{a}h\right]^{2}}{2 EI_{d1}}\text{d}x \tag{31}\]  
\[U_{s}=\frac{F^{2}}{2k_{s}}=\int\limits_{0}^{d}\frac{1.2F_{b}{}^{2}}{2 GA_{x}} \text{d}x+\frac{1.2F_{b}{}^{2}d_{1}}{2 GA_{d1}} \tag{32}\]  
\[h=R_{b1}[(\alpha_{1}+\alpha_{2})\cos\alpha_{1}-\sin\alpha_{1}]-h_{A1\_x1} \tag{25}\]0  
where \(b\) represents the tooth width. \(v\) represents the [PERSON]'s ratio. \(E\) and \(G\) represent [PERSON]'s modulus and shear modulus, respectively, and \(G=E/(2(1+v))\). \(d_{1}\) is approximately replaced by \(R_{b1}-R_{r1}\). Thus, these stiffness values are derived as follows:  
\[h=R_{b1}[(\alpha_{1}+\alpha_{2})\cos\alpha_{1}-\sin\alpha_{1}]-h_{A1\_x1} \tag{25}\]1  
\[h=R_{b1}[(\alpha_{1}+\alpha_{2})\cos\alpha_{1}-\sin\alpha_{1}]-h_{A1\_x1} \tag{25}\]2  
\[h=R_{b1}[(\alpha_{1}+\alpha_{2})\cos\alpha_{1}-\sin\alpha_{1}]-h_{A1\_x1} \tag{25}\]3  
where \(D=\cos\lambda_{1}[(\alpha_{1}+\alpha_{2})\sin\alpha_{1}+\cos\alpha_{1}-( \alpha-\alpha_{2})\sin\alpha-\cos\alpha]+\sin\lambda_{1}[h_{A1\_x1}/R_{b1}+\sin \alpha_{1}-(\alpha_{1}+\alpha_{2})\cos\alpha_{1}]\), and \(H=\sin\alpha+(\alpha_{2}-\alpha)\cos\alpha-h_{\text{wear\_x1}}/R_{b1}\).