# Options on Tooth Profile Modification by Hob Adjustment
## 2 Methodology
### Position Vector of the Hob Profile  
A description of the screw plane and of the planar areas of the teeth sides in the basic rack is presented as follows: Figure 1 shows vectors \(\mathbf{r}_{1}\), \(\mathbf{r}_{2}\) and\(\mathbf{r}_{3}\). Three coordinate systems are defined in Figure 1, as follows: the fixed coordinate system \(O\) (\(x\), \(y\), \(z\)) with the coordinate system \(O^{\prime}\)(\(x^{\prime}\), \(y^{\prime}\), \(z^{\prime}\)) turned at an angle of \(\gamma_{f}\) or \(\gamma_{f}\pm\Delta\gamma\) and the coordinate system of the hob \(O^{\prime\prime}\)(\(x^{\prime\prime}\), \(y^{\prime\prime}\), \(z^{\prime\prime}\)). Figure 1: Insertion of the hobbing mill in the basic rack. When Equations (3), (3a) and (4) are substituted into Equation (5), and when Equation (2) is modified and when Equation (2) is used, \(t\) is expressed as follows:  
\[\mathbf{t}=\mathbf{i}^{\prime}\,\sin\gamma_{f}-\mathbf{j}^{\prime}\,\cos\kappa\,\cos\gamma_ {f}+\,\,\mathbf{k}^{\prime}\,\sin\kappa\,\cos\gamma_{f}. \tag{6}\]  
The unit vector \(\mathbf{n}\) is given as follows:  
\[\mathbf{n}=-\frac{\mathbf{r}_{t_{1}}}{|\mathbf{r}_{t_{1}}|}=-\frac{\mathbf{r}_{t_{1}}}{\sqrt{ \mathbf{r}_{t_{1}}^{2}}}, \tag{7}\]  
After substitution and modification of [10] the following is applicable:  
\[\mathbf{n}=\,\mathbf{j}\,\prime\,\sin\kappa+\,\,\mathbf{k}^{\prime}\cos\kappa. \tag{8}\]  
The unit vector \(\mathbf{b}\) is the product of vector multiplication between \(\mathbf{n}\) and \(\mathbf{t}\):  
\[\mathbf{b}=\mathbf{n}\times\mathbf{t}, \tag{9}\]  
and after the substitution of Equations (6) and (8) with Equation (9), and after modification the following is applicable:  
\[\mathbf{b}=\,\mathbf{i}^{\prime}\,\cos\gamma_{f}+\,\mathbf{j}^{\prime}\,\sin\gamma_{f}\, \cos\kappa-\,\mathbf{k}^{\prime}\,\sin\gamma_{f}\,\sin\kappa. \tag{10}\]  
The individual partial vectors of the final position vector (Figures 1 and 3) \(\mathbf{r}\) are as follows:  
\[ \begin{array}{l}\mathbf{r}_{1}=\mathbf{k}\,\,R,\\ \mathbf{r}_{2}=\,\,\mathbf{i}^{\prime}R(2\pi i+\kappa)\,\,\mathbf{t}g\gamma_{f},\\ \mathbf{r}_{3}=-\,\mathbf{n}\,\,R,\\ \mathbf{r}_{4}=\mathbf{n}\,\,\mathbf{l}_{\mathbf{n}},\\ \mathbf{r}_{5}=\mathbf{b}\,\,\mathbf{h}_{\mathbf{m}}=\mathbf{b}\,\left(h_{\mathbf{n}}\,\,\mathbf{t}g\alpha _{\mathbf{n}}+\frac{\mathbf{r}_{5}}{4}\right).\end{array}  \tag{11}\]  
The final position vector \(\mathbf{r}\) is given by the total, as follows:  
\[\mathbf{r}=\mathbf{r}_{1}+\mathbf{r}_{2}+\mathbf{r}_{3}+\mathbf{r}_{4}+\mathbf{r}_{5}. \tag{12}\]  
Figure 3: Partial vectors of the position vector. After transformation and substitution, the following is applicable:  
\[ \begin{array}{l}\mathbf{r}=kR+R(2\pi i+\kappa)tg\gamma_{f}\Big{(}i\,\cos\gamma_{f} -j\,\sin\gamma_{f}\Big{)}+\\ +R\Big{(}-i\,\sin\gamma_{f}\,\sin\kappa-j\,\cos\gamma_{f}\,\sin\kappa-k\cos \chi\Big{)}+\\ +h_{n}\Big{(}i\,\sin\gamma_{f}\sin\kappa+j\cos\gamma_{f}\sin\kappa+k\cos\kappa \Big{)}+\\ +h_{m}\Bigg{[}\begin{array}{c}\mathbf{i}\cos^{2}\gamma_{f}-j\cos\gamma_{f}\sin \gamma_{f}+\\ +\sin\gamma_{f}\cos\kappa\Big{(}i\sin\gamma_{f}+j\cos\gamma_{f}\Big{)}-\mathbf{k} \sin\gamma_{f}\sin\kappa\end{array}\Bigg{]}.\end{array}  \tag{13}\]  
Position vector Equation (13) is described by the point in the screw plane of the hob in the case its turning corresponds with the angle of gradient of the screw \(\gamma_{f}\)[11]. In the case of the angular rotation of the hob in the basic rack by the angle of:  
\[\gamma_{h}=\gamma_{f}\pm\Delta\gamma, \tag{14}\]  
the final position vector will be given by the following:  
\[\mathbf{r}=i\,x+j\,y+\mathbf{k}\,z, \tag{15}\]  
with  
\[ \begin{array}{l}x=|\mathbf{r}_{x}|=R(2\pi i+\kappa)\,tg\gamma_{f}\cos\gamma_{h} -R\sin\kappa\sin\gamma_{h}+\\ +h_{n}\sin\kappa\sin\gamma_{h}+h_{m}\left(\cos\gamma_{f}\cos\gamma_{h}+\cos \kappa\sin\gamma_{f}\sin\gamma_{h}\right)\,.\end{array}  \tag{15a}\]  
\[