# Options on Tooth Profile Modification by Hob Adjustment
## 2 Methodology
### Method for Determination of the Profile Deviations in the Workpiece Tooth  
To evaluate the deviations, which will be projected in the manufacturing of the tooth by the modified toothed rack, it is important to determine the theoretical profile of the tooth (i.e., the involute profile). It can be achieved by rolling the basic rack [15]. If correction is not taken into consideration, the pitch of the straight line in the rack rolls along the pitch circle of the manufactured wheel (Figure 8). Figure 6: The set of axial cuts enter/run through the screw plane of the hob. Figure 7: Influence of the setting of the hobbing mill. Module \(m=10\) mm, \(R_{\rm H}=67.5\) mm, profiled in the reference plane. Mathematical notation is a combination of vectors. Figure 8 shows a definition of the coordinate systems \(O_{k}\) (\(x\), \(y\)), \(O^{\prime}_{k}\)(\(x^{\prime}\), \(y^{\prime}\)), \(A_{\rm H}\) (\(n_{\rm H}\), \(b_{\rm H}\)) and \(A_{\rm K}\) (\(n\), \(b\)). The transformation equations for the dashed-line coordinate system of the forming rack to transform into the fixed coordinate system of the manufactured toothed wheel are as follows [16]:  
\[ \begin{array}{l}\mathbf{i}^{\prime}=\mathbf{i}\cos\lambda+\mathbf{j}\sin\lambda\\ \mathbf{j}^{\prime}=\mathbf{\tilde{j}}\cos\lambda-\mathbf{\tilde{i}}\sin\lambda.\end{array}  \tag{19}\]  
The unit vectors are \(n_{\rm H}=\mathbf{i}^{\prime}\), \(b_{\rm H}=\mathbf{j}^{\prime}\), \(n=-\mathbf{i}^{\prime}\) and \(b=-\mathbf{j}^{\prime}\). After transformation, the final position vector of the basic rack \(\mathbf{r_{s}}\) is given by the total of the partial vectors (Figure 8):  
\[\mathbf{r_{s}}=\mathbf{r_{s_{1}}}+\mathbf{r_{s_{2}}}+\mathbf{r_{s_{3}}}+\mathbf{r_{s_{4}}}, \tag{20}\]  
and the individual partial vectors are as follows:  
\[ \begin{array}{l}\mathbf{r_{s_{1}}}=\mathbf{i}\,\mathbf{R_{1}}\cos\lambda+\mathbf{j}\,\mathbf{R_{1 }}\sin\lambda\\ \mathbf{r_{s_{2}}}=-\mathbf{j}\,\lambda\,\mathbf{R_{1}}\cos\lambda+\mathbf{i}\,\lambda\,\mathbf{R_ {1}}\sin\lambda\\ \mathbf{r_{s_{3}}}=\mathbf{i}\,h_{n}\cos\lambda+\mathbf{j}\,h_{n}\sin\lambda\\ \mathbf{r_{s_{4}}}=-\mathbf{j}\,h_{b}\cos\lambda+\mathbf{i}\,h_{b}\sin\lambda\\ \end{array} , \tag{20a}\]  
and their absolute values are as follows:  
\[ \begin{array}{l}\mathbf{|r_{s_{1}}|}=\mathbf{R_{1}}\,\ |\mathbf{r_{s_{2}}|}=\lambda\ \mathbf{R_{1}}\,\ |\mathbf{r_{s_{3}}}|=h_{n}\,\\ \mathbf{|r_{s_{4}}|}=h_{b}=\frac{\mathbf{p_{3}}}{4}-h_{n}\,\mathbf{f}\mathbf{g}\mathbf{n}.\end{array}  \tag{20b}\]  
Parts of the vector \(\mathbf{r_{s}}\) in the direction of the coordinate axes \(\mathbf{x}\) and \(\mathbf{y}\) are as follows:  
\[ \begin{array}{l}\mathbf{x_{\rm K}}=\mathbf{|r_{s_{x}}|}=\mathbf{R_{1}}\cos\lambda+\lambda \,\mathbf{R_{1}}\sin\lambda+h_{n}\cos\lambda+h_{b}\sin\lambda\\ \mathbf{y_{\rm K}}=\mathbf{|r_{s_{y}}|}=\mathbf{R_{1}}\sin\lambda-\lambda\,\mathbf{R_{1}}\cos \lambda+h_{n}\sin\lambda-h_{b}\cos\lambda\\ \end{array} , \tag{21}\]  
However, the tooth profile must relate to the coordinates \(n\), \(b\) in the coordinate system \(A_{\rm K}\) (\(n\), \(b\)). The coordinates of the theoretical profile of the tooth are consequently given by the following:  
\[ \begin{array}{l}n_{\rm teo}=\mathbf{R_{1}}-x_{\rm K}\,\\ b_{\rm teo}=-y_{\rm K}.\end{array}  \tag{22}\]  
Figure 8: Pitch of the straight line in the rack. These coordinates define the theoretical or actual profile. However, it is still unknown for which of the values \(\lambda\) or \(h_{n}\) is the point described by the position vector \(\mathbf{r_{s}}\), simultaneously by the point in the tooth profile of the manufactured wheel. The toothed wheel profile is formed by the envelope of the profiles in the basic rack. To determine the parameters, i.e., to determine the values \(\lambda\) and \(h_{n}\), the envelope method is used [17] and, in this case, there is two parametric methods. The rack rolling results in the achievement of a set of lines for the diverse value \(h_{n}\) (Figure 9). A single point lies on each of the lines.