# A Study on the Effect of Gear Hobbing Process Parameters on the Residual Stress of the Tooth Root
## 2 Kinematic Relation of Gear Hobbing  
Gear hobbing machining is based on the principle of a generating method, and the tooth shape is formed by the meshing movement between the hob and the workpiece. The gear hobbing movement is equivalent to the meshing movement of the gear pair with staggered axes, including the rotation of the hob and the generative motion of the workpiece. The principle diagram of gear hobbing is shown in Figure 1a [18]. Reference [19] clearly explains the spatial relative position and motion relationship between the hob and workpiece during hobbing machining, as shown in Figure 1. \(\text{O}_{\text{h}}X_{\text{h}}Y_{\text{h}}Z_{\text{h}}\) is the fixed coordinate system of the hob, which rotates continuously with the hob around the \(X_{\text{h}}\) axis during the machining process; \(\text{O}_{\text{1}}X_{\text{1}}Y_{\text{1}}Z_{\text{1}}\) is the hob installation coordinate system, which rotates an installation angle \(\delta\) around the \(Y_{2}\) axis relative to the tool frame coordinate system according to the gear helix angle and hob thread lift angle; \(\text{O}_{\text{2}}X_{\text{2}}Y_{\text{2}}Z_{\text{2}}\) is the tool rest coordinate system, parallel to the table coordinate system \(\text{O}_{\text{3}}X_{\text{3}}Y_{\text{3}}Z_{\text{3}}\), which is the stationary reference coordinate system of the hobs and workpieces, respectively; \(\text{O}_{\text{g}}X_{\text{g}}Y_{\text{g}}Z_{\text{g}}\) is the fixed coordinate system of workpieces, which rotates around the \(Z_{\text{g}}\) axis with the hob linkage during the hobbing process [19]. The process parameters of gear hobbing can determine the relative motion relationship between the hob and gear during machining. The parameters are shown in Table 1, and the calculation formulas for each parameter are shown in Equations (1)-(5). Figure 1: (**a**) Principle diagram of hobbing [18]: (**b**) schematic diagram of spatial relative position and motion relationship between hob and gear [19; 20]. \(d_{h}\) is the pitch diameter of the hob, \(d_{g}\) is the pitch diameter of the gear, and the radial installation distance \(\rho\) of the hob is  
\[\rho=\frac{d_{h}+d_{g}}{2} \tag{1}\]  
In order to ensure that the hob helix direction coincides with the gear tooth direction, the hob installation angle \(\delta\) is the algebraic difference between the gear helix angle \(\beta\) and the hob helix lift angle \(\lambda\):  
\[\delta=\beta-\lambda \tag{2}\]  
The hob axial feed rate and feed speed are a pair of related process parameters. Determine the axial feed rate \(f\) (mm/r) (the axial feed distance of the hob in one rotation of the table) during the process design, and then determine the axial feed speed \(v\) (mm/min) according to the rotation speed of the hob and the table. \(N\) is the number of hob heads, and \(Z\) is the number of gear teeth. \[v=\frac{N}{Z}\omega_{h}f \tag{3}\]  
As a function of the hob's axial displacement \(\zeta\) and gear rotation phase angle \(\psi\) with respect to hob rotation phase angle \(\varphi\), the relation is shown in Equations (4) and (5). When the hob is rotating, the \(\zeta\) displacement is fed along the axis of the gear, and the \"+\" is taken when the process is straight-rolling, and the \"\(-\)\" is taken when the process is reverse-rolling. \(F\) is the axial feed speed. The workpiece that is relative to the hob synchronously rotates at \(\psi\) angle, including the development angle \(\psi_{g}\) and differential angle \(\psi_{d}\) of the two components:  
\[\zeta=\pm\frac{F}{2\pi\omega_{h}}\varphi \tag{4}\]  
\[\left\{ \begin{array}{l}\psi=\psi_{g}+\psi_{d}\\ \psi_{g}=\frac{\lambda}{|\lambda|}\frac{f}{Z}\varphi\\ \psi_{d}=\frac{\zeta}{|\zeta|}\frac{f\tan\beta}{\pi\omega_{h}d_{g}}\end{array} \right.