## 3 Research Method
### Tooth Profile Precise Functions Solving
The X\({}_{1}\)-O\({}_{1}\)-Y\({}_{1}\) coordinates also move to the left with the hob, and pure rolling is maintained between the gear pitch circle and the hob hobbing straight line. The tooth profile can be divided into the involute part and the root fillet curve. Among them, the involute part is formed by the straight part LM of the blade. Point M is the lowest point of the straight part of the blade, and point T is the intersection point between the meshing line and the straight blade. When the X\({}_{1}\)-O\({}_{1}\)-Y\({}_{1}\) coordinate system is at the position of X\({}^{\prime}\)\({}_{1}\)-O\({}^{\prime}\)\({}_{1}\)-Y\({}^{\prime}\)\({}_{1}\), the meshing line is just past the M point, and the lowest point of the tooth-shaped involute part is cut out at this time. As the hob moves to the left, the cutting point also moves upwards. When the X\({}_{1}\)-O\({}_{1}\)-Y\({}_{1}\) coordinate system is at the position of X\({}^{\prime\prime}\)\({}_{1}\)-O\({}^{\prime\prime}\)\({}_{1}\)-Y\({}^{\prime\prime}\)\({}_{1}\), the T point is exactly on the addendum circle, and the involute part of the tooth profile is completed at this time. While cutting the involute part, the blade filled MN forms a fillet curve, and its instantaneous cutting point is the intersection of the extension line of PQ and the arc MN. When the X\({}_{1}\)-O\({}_{1}\)-Y\({}_{1}\) coordinate system is at the position of X\({}^{\prime}\)\({}_{1}\)-O\({}^{\prime}\)\({}_{1}\)-Y\({}^{\prime}\)\({}_{1}\), the extension line of PQ intersects the arc MN at point M, and the tooth fillet curve starts. As the rack moves to the left, the angle \(\psi\) between PQ and PX gradually increases; when \(\psi=90^{\circ}\), the PQ and PY axes coincide to complete the cutting of the involute part. The analytical formula of the meshing line TM is  
\[x_{1}+y_{1}\cot\alpha_{n}=-S \tag{5}\]  
Figure 5:0 Geometric relationship of the hobbing process. The tooth profile of a helical gear is a set of spatial curves which are parallel to the end face with the helix angle of \(\beta\). Combine the vertical formulas above to obtain the coordinates of point B (\(x_{B}\),\(y_{B}\)):  
\[\left\{ \begin{array}{c}x_{B}=-S-\frac{\cos\alpha_{n}}{\tan\alpha_{n}+\cot \alpha_{n}}\left(\frac{2\pi n}{4\cos\beta}-S+xm\tan\alpha_{n}\right)\\ y_{B}=\left(\frac{\frac{7m}{\cos\beta}-S+xm\tan\alpha_{n}}{(\tan\alpha_{n}+ \cot\alpha_{n})}\right)\end{array} \right. \tag{6}\]  
As the tool gradually moves to the left, the \(S\) value continuously changes, which corresponds to a series of continuously changing B point coordinates. The radius of the hob blade edge becomes an ellipse on the conjugate plane, the radius of the minor axis is still \(r\), and the radius of the major axis becomes \(r/\cos\beta\). From this, the expression of the blade ellipse can be obtained as Equations (7) and (8):  
\[\frac{\left(x_{1}+\frac{\pi m}{2\cos\beta}-\frac{\delta}{\cos\beta}\right)^{2} }{\left(\frac{r}{\cos\beta}\right)^{2}}+\frac{\left(y_{1}+1.25m-r\right)^{2}} {r^{2}}=1 \tag{7}\]  
\[l=\frac{r\cos\alpha_{n}}{\cos\beta} \tag{8}\]  
Then, the coordinates of M point can be obtained as (\(x_{M}\), \(y_{M}\)):  
\[\left\{ \begin{array}{c}x_{M}=\frac{-\left(\frac{\pi m}{2}-\delta-r\cos \alpha_{n}\right)}{\cos\beta}\\ y_{M}=-[1.25m-r(1-\sin\alpha_{n})]\end{array} \right. \tag{9}\]  
According to the literature [28], the normal of the cutting point at every moment in the cutting process must pass through the meshing node, thereby determining the position of the K point.