## 3 Ruled Surface for Face Gears
### Analysis of Tooth Surface Deviations  
The conventional tooth surface is regarded as a theoretical reference for comparison and analysis with the ruled surface processed by a conical cutter. The definition for the deviation is shown in Figure 7a. Any point \(R_{\mathrm{f}}\) on the conventional tooth surface intersects with the ruled surface along its normal vector direction \(n_{\mathrm{f}}\) at a point \(R_{2}\), and the distance between the two points can be expressed as follows:  
\[\Delta=(R_{2}-R_{\mathrm{f}})\cdot n_{\mathrm{f}}. \tag{27}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameters** & **Symbol** & **Unit** & **Case 1** & **Case 2** \\ \hline Tooth number of face gear & \(N_{\mathrm{f}}\) & - & 90 & 180 \\ Tooth number of pinion & \(N_{\mathrm{c}}\) & - & 30 & 30 \\ Normal module & \(m_{\mathrm{n}}\) & mm & 4 & 4 \\ Pressure angle & \(\alpha\) & deg & 25 & 25 \\ Shaft angle & \(\gamma\) & deg & 90 & 90 \\ Face gear inner radius & \(L_{1}\) & mm & 170 & 347 \\ Face gear outer radius & \(L_{2}\) & mm & 200 & 402 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters for the face gears. Figure 6: The deviation between the tooth surface machined by conical cutters and the theoretically developed tooth surface: (**a**) case 1, and (**b**) case 2. Figure 7b shows the morphology of the tooth surface deviation, and the parameters are displayed in case 1 of Table 1. In the figure, the black mesh represents the standard surface and the red mesh represents the ruled surface. It can be seen that the deviation of the tooth surface at the contact line position is almost zero, the deviation increases as the tooth surface expands from the contact line to two sides, the maximum deviation \(\delta_{1}\) is mainly at the root of the inner diameter tooth, followed deviation \(\delta_{2}\) by the top of the outer diameter tooth, and the other two parts are almost zero. In order to study the influence of transmission ratio on tooth surface deviation, the number of teeth of cylindrical gears is a constant value, and the number of teeth of face gears varies with the transmission ratio from 1 to 12. Deviation calculations are conducted on 12 sets of calculation examples. The variation of deviation between the inner and outer diameters of the tooth surface at the top and root of the tooth is shown in Figure 8a, when the transmission ratio is small, the inner diameter tooth root and outer diameter tooth top are still the main deviation positions. However, as the transmission ratio increases, their deviation gradually tends to zero. While the deviation at the inner diameter tooth top and outer diameter tooth root is almost always zero, independent of the transmission ratio. In order to study the influence of modules on tooth surface deviation, the number of teeth of the pinion and the face gear remains unchanged, and the tooth surface deviation is calculated with the module from 1 to 12, the results are shown in the Figure 8b, the ratio of tooth surface deviation to modules is a certain value, that is, under fixed transmission ratio conditions, and the tooth surface deviation is independent of modules. Figure 8: The influence of basic parameters on the deviation of face gears: (**a**) transmission ratio and tooth surface deviation, and (**b**) module and tooth surface deviation. Figure 7: Tooth surface deviations of the ruled surface: (**a**) definition of the tooth surface deviation, and (**b**) tooth surface deviation morphology. This section presents the equations of ruled surfaces and conical cutters, and proposes a machining method of ruled surfaces with conical cutters, and the tooth surface deviation between ruled surfaces and conventional surfaces is analyzed.