# Analytical and Graphical Profiling of Thread-Milling Cutters for Forming Internal Threads
## 2 Literature Review
### Application of the Methodology for Analytical Determination of the Tool Surface Profile  
According to a developed methodology for analytical determination of the profile of the tool surface, an example of determining the parameters of a thread-milling cutter for cutting internal threads has been solved. The tool surface of a thread-milling cutter M10 \(\times\) 1, with which an internal metric thread M20 \(\times\) 1 is formed, is determined. Table 2 shows the results of solving the problem. \begin{table}
\begin{tabular}{l c} \hline \multicolumn{1}{c}{**Parameter**} & **Equation** \\ \hline Spiral rotation angle & \(\varphi_{\rm i}=\frac{\rm R_{\rm i}}{\rm a},\rm rad\) \\ \hline Abscissa of point i & \(\rm x_{\rm i}=R_{\rm i}.\rm cos\varphi_{\rm i},\rm mm\) \\ \hline Ordinate of point i & \(\rm y_{\rm i}=R_{\rm i}.\rm sin\varphi_{\rm i},\rm mm\) \\ \hline Angular coefficient of the normal in point i & \(\rm k=-\frac{a.cos\varphi_{\rm i}-R_{\rm i}.sin\varphi_{\rm i}}{a.sin\varphi_{ \rm i}+R_{\rm i}.cos\varphi_{\rm i}}\) \\ \hline Parameter a\({}_{1}\) & \(\rm a_{1}=1+k^{2}\) \\ \hline Parameter b\({}_{1}\) & \(\rm b_{1}=2.k.(y_{\rm i}-k.x_{\rm i})\) \\ \hline Parameter c\({}_{1}\) & \(\rm c_{1}=\left(k.x_{\rm i}-y_{\rm i}\right)^{2}-A_{0}^{2}\) \\ \hline Abscissa of point 1\({}^{\prime}\) & \(\rm x_{1^{\prime}}=\frac{-b_{1}+\sqrt{b_{1}^{2}-4a_{1}.c_{1}}}{2.a_{1}},\rm mm\) \\ \hline Abscissa of point 2\({}^{\prime}\) & \(\rm x_{2^{\prime}}=\frac{-b_{1}-\sqrt{b_{1}^{2}-4a_{1}.c_{1}}}{2.a_{1}},\rm mm\) \\ \hline Ordinate of point 1\({}^{\prime}\) & \(\rm y_{1^{\prime}}=k.(x_{1^{\prime}}-x_{\rm i})+y_{\rm i^{\prime}},\rm mm\) \\ \hline Ordinate of point 2\({}^{\prime}\) & \(\rm y_{2^{\prime}}=k.(x_{2^{\prime}}-x_{\rm i})+y_{\rm i^{\prime}},\rm mm\) \\ \hline Distance between point 1\({}^{\prime}\) and point i & \(\rm 1_{1^{\prime}}=\sqrt{(x_{1^{\prime}}-x_{\rm i})^{2}+(y_{1^{\prime}}-y_{ \rm i})^{2}},\rm mm\) \\ \hline Distance between point 2\({}^{\prime}\) and point i & \(\rm 1_{2^{\prime}}=\sqrt{(x_{2^{\prime}}-x_{\rm i})^{2}+(y_{2^{\prime}}-y_{ \rm i})^{2}},\rm mm\) \\ \hline Radius of the profiled circle & \(\rm r_{0i}=min(1_{1^{\prime}},1_{2^{\prime}}),\rm mm\) \\ \hline Applicate of point i & \(\rm z_{i}=\frac{P}{4}-(R_{i}-R_{2}).tg\frac{\alpha}{2},\rm mm\) \\ \hline Distance between point 1\({}^{\prime}\) and point i & \(\rm h_{0i}=r_{0i\rm max}-r_{0i,\rm mm}\) \\ along the “\({}^{\prime}\)” axis of the profiled circle & \(\rm z_{0i}=z_{i}-z_{1,\rm mm}\) \\ \hline Distance between point 1\({}^{\prime}\) to point i along & \(\rm h_{i}=R_{i\rm max}-R_{i,\rm mm}\) \\ the “\({}^{\prime}\)” axis of the thread & \(\rm h_{i}=h_{0i}-h_{i,\rm mm}\) \\ \hline Difference in profile heights & \(\rm\Delta h_{i}=h_{0i}-h_{i,\rm mm}\) \\ \hline Tool profile angle & \(\rm\alpha_{0i}=2.arctg\left(\frac{z_{0i}}{R_{0i}}\right)\), \({}^{\circ}\) \\ \hline Deviation in profile angle & \(\rm\Delta\alpha_{0}=\alpha-\alpha_{0i}\), \({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Profile parameters of the tool surface of a thread-milling cutter when forming an internal thread M20 \(\times\) 1. The calculations were made with the program based on Microsoft Excel 2019. The program contains all the equations from Table 1. The diameter of the thread-milling cutter \(\mathrm{d}_{0}\) and the type of thread that will be formed are entered into the program. After that, the program calculates the parameters of the tool surface. The designation used for the thread-milling cutter M10 \(\times\) 1 means M--metric thread, 10--minimum size of the formed thread and 1--thread pitch, P. The same approach is used for the designations of all thread-milling cutters in the publication. With the values obtained for the points of the tool surface profile, it can be manufactured. Each tooth of the thread-milling cutter must be machined with such a surface. The tool surface obtained by the analytical profiling method is such that when the tool is operated, there is no trimming of the profile of the formed thread.