## 3 Contact Analysis of Topologically Modified Internal Helical Gear Pair  
### Basic Parameters of Internal Helical Gears  
Analysis of the forces acting on the tooth surfaces and the transmission error of internal helical gears pair is carried out in view of the multi-tooth contact during meshing. Finite element analysis is applied to construct the dynamic model to investigate the force mechanism in detail. The parameters of the modified gear are shown in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Gear Parameters** & **Driving Wheel** & **Driven Wheel** \\ \hline Helical direction & left-handed & left-handed \\ Number of teeth \(z\) & 79 & 26 \\ Normal modulus \(m_{n}/\text{(mm)}\) & 3.5 & 3.5 \\ Face module \(m_{t}/\text{(mm)}\) & 3.5782 & 3.5782 \\ Normal pressure angle \(\alpha\) (\({}^{\circ}\)) & 20 & 20 \\ Helix angle \(\beta/\)(\({}^{\circ}\)) & 12 & 12 \\ Tooth width \(b/\text{mm}\) & 60 & 60 \\ Face pressure angle \(\alpha_{t}\) (\({}^{\circ}\)) & 20.410 & \\ Transverse contact ratio & 1.211 & \\ Overlap ratio & 1.135 & \\ Total contact ratio & 2.346 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Basic geometrical parameters of gears.  
Firstly, based on the basic geometric parameters of the internal helical gear pair, combined with the principle of gear meshing kinematics and the theory of spatial surface equation, the corresponding calculation program is developed through a MATLAB numerical calculation platform to realize the accurate coordinate solution of the tooth surface points of the involute main and driven gears. The mathematical tooth surface model is shown in Figure 3a. It is necessary to export the calculated discrete point coordinate values of the tooth surface in DAT format, and then import the data into UG, and model through surface generation, Boolean operation and other operations, and finally generate the gear model. The assembly drawing is shown in Figure 3b.  
Matlab2021 software is applied to solve the tooth profile equations generated based on the topological modification principle, and to calculate the tooth profile data. Tooth profile modification amount: \(\Delta L_{1}=b_{m}r_{b}^{2}(u_{1}-u_{0})^{2}\), Here, \(b_{m}\) is the tooth profile modification coefficient, \(r_{b}\) is the radius of the gear base circle, \(u_{1}\) is the angle of the involute, and \(u_{0}\) is the starting point of the modification area. Tooth modification amount: \(\Delta L_{2}=b_{h}(l_{p}-b_{1}/2)^{2}\), \(b_{h}\) is the tooth modification coefficient, \(lp\) is the wheel longitudinal displacement, and \(b_{1}\) is the tooth width. As shown in Figure 4, it is the mathematical model of the profile of the external helical gear. The relevant modification coefficients are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Shaping**} & **Tooth Profile Modification** & **Toothedness Modification** \\  & **Factor**\(b_{m}\) & **Factor**\(b_{h}\) \\ \hline Case 0 & 0.00003 & 0.0001 \\ Case 1 & 0.00001 & 0.0003 \\ Case 2 & 0.000015 & 0.0006 \\ Case 3 & 0.00007 & 0.000008 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Gear profile coefficients.  
Figure 4: Mathematical model of modified tooth surface.  
Figure 3: Mathematical model and assembly drawing.