## 2 Analytical Model of Mesh Stiffness of the Internal Gear Pair
### Calculation of Slice Mesh Stiffness
#### 2.2.1 Tooth Stiffness  
The internal and external tooth stiffness includes the bending stiffness, shear stiffness and compressive stiffness. When calculating, the section from the root to the top of the  
Figure 1: Diagram of the internal mesh gear pair.  
Figure 2: Slice diagram: (**a**) external helical gear slice; (**b**) internal helical gear slice.  
tooth is regarded as a variable cross-section cantilever beam. Due to the differences in the tooth profiles of the internal and external gears, the cantilever beam model needs to be constructed separately. The cantilever beam model of the internal and external gear is shown in Figure 3.  
Based on the potential energy method in elasticity, the relationship between the energy stored in the tooth and its stiffness can be expressed as follows:  
\[\left\{\begin{array}{l}U_{b}=\frac{F^{2}}{2k_{b}}=\int_{0}^{d}\frac{M^{2}}{2 EI _{x}}dx\\ U_{s}=\frac{F^{2}}{2k_{s}}=\int_{0}^{d}\frac{1.2F_{b}^{2}}{2 GA_{x}}dx\\ U_{a}=\frac{F^{2}}{2k_{a}}=\int_{0}^{d}\frac{F_{a}^{2}}{2 EA_{x}}dx\\ \end{array}\right. \tag{2}\]  
where \(E\) denotes the elastic modulus, \(G\) represents the shear modulus (\(G=\frac{E}{2(1+v)}\)), and \(v\) is the Poisson's ratio. The torque \(M\) and the mesh force components \(F_{a}\), \(F_{b}\) can be expressed as:  
\[\left\{\begin{array}{l}F_{b}=F\cos\alpha_{1}\\ F_{a}=F\sin\alpha_{1}\\ M=F_{b}(d-x)-F_{a}h\\ \end{array}\right. \tag{3}\]  
where \(I_{x}\) is the moment of inertia at any point and \(A_{x}\) is the cross-sectional area of the microelement. The calculation formula is as follows:  
\[I_{x}=\frac{2}{3}h_{x}^{3}dB,A_{x}=2h_{x}dB \tag{4}\]  
where \(h_{x}\) is the half-width of the microelement.  
According to the involute tooth profile equation, the parameters \(d\), \(h\), \(lx\), \(x\), \(dx\) are expressed as an expression with the mesh angle \(\alpha\) as the variable. Substituting into  
Figure 3: Internal and external gear cantilever beam models: (**a**) external gear cantilever beam model; (**b**) internal gear cantilever beam model.  
Equation (2), the calculation formulae of the bending stiffness, shear stiffness, and compression stiffness of the internal and external gear slice elements can be obtained as follows:  
\[\begin{split}\frac{1}{k_{b}^{p(g)}}=\int_{-\alpha_{1}(\gamma)}^{ \theta_{b}(\alpha_{1})}\frac{3\Big{[}\pm 1+\cos\alpha_{1}(\pm(\theta_{b(a)}- \alpha)\sin\alpha\mp\cos\alpha)\Big{]}^{2}(\theta_{b(a)}-\alpha)\cos\alpha}{2 EdB[(\theta_{b(a)}-\alpha)\cos\alpha+\sin\alpha]^{3}}d\alpha\\ \frac{1}{k_{s}^{p(g)}}=\int_{-\alpha_{1}(\gamma)}^{\theta_{b}( \alpha_{1})}\frac{1.2(1+v)(\theta_{b(a)}-\alpha)\cos\alpha\cos^{2}\alpha_{1}} {EdB[(\theta_{b(a)}-\alpha)\cos\alpha+\sin\alpha]}d\alpha\\ \frac{1}{k_{a}^{p(g)}}=\int_{-\alpha_{1}(\gamma)}^{\theta_{b}( \alpha_{1})}\frac{(\theta_{b(a)}-\alpha)\cos\alpha\sin^{2}\alpha_{1}}{2 EdB[(\theta_{b(a)}-\alpha)\cos\alpha+\sin\alpha]}d\alpha\end{split} \tag{5}\]  
where \(p\) and \(g\) are the external and internal helical gears, respectively. \"()\" outside and inside are the calculated parameters for the external and internal helical gears, respectively, and \"\(\pm,\mp\)\", where the top refers to external helical gears and the bottom refers to internal helical gears. The meanings of the other parameters are shown in Figure 3.  
The tooth stiffness expression of the internal and external gear slice elements can be obtained by summing Formula (5). Its independent variable is the mesh angle \(\alpha\). The calculation formula of gear tooth stiffness at any point on the tooth profile is determined through polynomial fitting:  
\[\left\{\begin{array}{l}\frac{1}{k_{t}^{p}(\alpha)}=polyfit\bigg{(}\frac{1}{k _{b}^{p}}+\frac{1}{k_{s}^{p}}+\frac{1}{k_{s}^{p}}\bigg{)}\\ \frac{1}{k_{t}^{p}(\alpha)}=polyfit\bigg{(}\frac{1}{k_{b}^{p}}+\frac{1}{k_{s}^{ p}}+\frac{1}{k_{s}^{p}}\bigg{)}\end{array}\right. \tag{6}\]