## 2 The Analytical Model of Mesh Stiffness of Helical Gear
### The Stiffness of Each Sliced Gear
#### 2.2.2 Fillet-Foundation Stiffness  
As in the tooth, the fillet-foundation undergoes deformation with the axial force. The fillet-foundation can also be considered as a cantilever beam, as shown in Figure 5. The bending moment of the fillet-foundation with the axial force can be calculated as  
\[M_{\text{af}}=F_{\text{a}}\cdot(r\cdot\cos\theta_{\text{c}}-x) \tag{18}\]  
Figure 4: The sketch of the tooth with the transverse mesh force.  
The bending energy stored in the fillet-foundation with the axial mesh force can be written as  
\[U_{\text{af}}=\int_{0}^{r_{\text{f}}}\frac{M_{\text{af}}^{2}}{2 EI_{\text{af}}}dx \tag{19}\]  
where \(I_{\text{afx}}\) denotes the cross-sectional moment of inertia to the neutral axis.  
\[L=\left\{\begin{array}{ll}2(\sqrt{r_{\text{f}}^{2}-x^{2}}-\sqrt{r_{\text{ int}}^{2}-x^{2}})&0<x<r_{\text{int}}\\ 2\sqrt{r_{\text{f}}^{2}-x^{2}}&r_{\text{int}}<x<r_{\text{f}}\end{array}\right. \tag{20}\]  
Thus, the fillet-foundation stiffness with the axial mesh force can be obtained by  
\[\frac{1}{k_{\text{af}}}=\int_{0}^{r_{\text{f}}}\frac{\sin^{2}\beta_{\text{b}} \cdot(r\cos\theta_{\text{c}}-x)^{2}}{EI_{\text{afx}}}dx \tag{21}\]  
There is deformation of the fillet-foundation due to the transverse force. For every sliced gear in the engagement region, one or two teeth participate in engagement, as shown in Figure 6.  
In a single tooth engagement region, only one tooth bears the load, and the corresponding displacement of the fillet-foundation could be expressed as \(\delta_{\text{tf11}}\). Therefore, the corresponding stiffness can be expressed as  
\[k_{\text{tf1}}=F_{\text{t1}}/\delta_{\text{tf11}} \tag{22}\]  
For the double-tooth engagement region in Figure 6b, two teeth bear the load simultaneously. The displacement of the fillet-foundation is not equal to the sum of the displacement in single-tooth engagement because there exists structure coupling. That is, when \(F_{\text{t1}}\
eq 0\) and \(F_{\text{t2}}=0\), there is a deformation \(\delta_{\text{f11}}\) of point \(P_{1}\) and displacement \(\delta_{\text{t21}}\) of point \(P_{2}\). Similarly, when \(F_{\text{t1}}=0\) and \(F_{\text{t2}}\
eq 0\), there is a deformation \(\delta_{\text{t22}}\) of point \(P_{2}\) and displacement \(\delta_{\text{f12}}\) of point \(P_{1}\). The displacement of \(\delta_{\text{t21}}\) and \(\delta_{\text{f12}}\) is detailed in Ref. [22].  
Figure 5: The sketch of the fillet-foundation with the axial mesh force.  
Figure 6: The sketch of the fillet-foundation with the transverse mesh force (**a**) single tooth in engagement (**b**) double teeth in engagement.  
Thus, the fillet-foundation stiffness with transverse force can be derived:  
\[k_{\text{tf1}}=F_{\text{t1}}/(\delta_{\text{f11}}+\delta_{\text{f12}}),\ \ \ \ \ k_{\text{tf2}}=F_{\text{t2}}/(\delta_{\text{f22}}+\delta_{\text{f21}}) \tag{23}\]  
Similar to the calculation process of total tooth stiffness, the total fillet-foundation stiffness when one tooth bears the load can be derived as  
\[\frac{1}{k_{\text{f1}}}=\frac{1}{k_{\text{af}}}+\frac{1}{k_{\text{tf1}}} \tag{24}\]  
The total fillet-foundation stiffness when two teeth bear the load can be derived as  
\[\frac{1}{k_{\text{f1}}}=\frac{1}{k_{\text{af}}}+\frac{1}{k_{\text{tf1}}},\ \ \ \ \frac{1}{k_{\text{f2}}}=\frac{1}{k_{\text{af}}}+\frac{1}{k_{\text{tf2}}} \tag{25}\]