# Analytical and Graphical Profiling of Thread-Milling Cutters for Forming Internal Threads
## 2 Literature Review
### Methodology for Graphical Determination of the Tool Surface Profile
The profile of the new tool surface is obtained with a profile angle  
Figure 14: Profile of the surface of the tool obtained after profiling. Figure 13: Determining the points on the tool surface corresponding to the main diameters of the tool profile. \(\alpha_{0}<\alpha\) as well as a width at the tip \(\rm f_{a0}<f_{f}\), where \(\rm f_{f}\) is the width of the groove of the internal thread. The size of the error \(\delta_{0}\) is influenced by the pitch P of the formed thread, the outer diameter of the milling part \(\rm d_{0}\) and the ratio between the diameters of the milling part of the tool and the outer diameter of the thread \(\rm d_{0}/D\), that is, the maximum diameter of the internal thread that can be machined with a given diameter of the milling part. Figure 15 shows the profiling of the tool surface of a thread-milling cutter M10 \(\times\) 1 using the CAD system SolidWorks 2022. The results of the profiling can be seen in Section 5.2. The graphical profiling of the tool surface of milling cutters for forming threads M10 \(\times\) 1, M16 \(\times\) 1.5 and M20 \(\times\) 20 was made. The methodology described in Figure 15 was used for profiling. Figure 15: Cont. Profiling with the SolidWorks 2022 CAD system includes the following stages:  
* A three-dimensional model of a part with an internal thread M10 \(\times\) 1 is drawn. * In a plane built in the cross-section, the profile of the internal thread with diameters D, D\({}_{2}\) and D\({}_{1}\) is drawn. In this way, the position of points 1, 2 and 3 is defined (Figure 15a). * Planes are built in the axial section passing through points 1, 2 and 3 (Figure 15b). * In the plane passing through point 1, a circle with radius r\({}_{0}\) is drawn tangent to a point of the Archimedean spiral and with a center lying on the circle with radius A\({}_{0}\) (Figure 15c). The circle corresponds to the diameter of the instrument d\({}_{0}\). * In the plane passing through point 2, a circle with radius r\({}_{20}\) is drawn tangent to a point of the Archimedean spiral and with a center lying on the circle with radius A\({}_{0}\) (Figure 15d). The circle corresponds to the diameter of the instrument d\({}_{20}\). * In the plane passing through point 3, a circle with radius r\({}_{10}\) is drawn tangent to a point of the Archimedean spiral and with a center lying on the circle with radius A\({}_{0}\) (Figure 15e). The circle corresponds to the diameter of the instrument d\({}_{10}\). * A new sketch is built on which the resulting tangent circles with radii r\({}_{0}\), r\({}_{20}\) and r\({}_{10}\) are drawn, which form the profile of the tool surface in cross-section (Figure 15f). From the sketch, information is obtained about the profile height h\({}_{01}\) and the angle of the tool surface profile \(\alpha_{01}\). Graphical Determination of the Influence of the Rake Angle and the Height of the Backing off on the Tool Surface  
The profile of the tool surface is influenced by the rake angle \(\gamma_{0}\) and the backing off K\({}_{0}\). They lead to a tooth profile that is different from that in the axial section, i.e., a profile with a different height h\({}_{0}\) and a different profile angle \(\alpha_{0}\). The values of the rake angle \(\gamma_{0}\) depend on the mechanical characteristics of the material being processed and are in the range \(0\div 10^{\circ}\), and the height of the backing off K\({}_{0}\) determines the values of the back angle at the top points. Figure 15: Graphical profiling of an M10 \(\times\) 1 thread-milling cutter with SolidWorks 2022: (**a**) Stage 1; (**b**) Stage 2; (**c**) Stage 3; (**d**) Stage 4; (**e**) Stage 5; (**f**) Stage 6. The graphical method for determining the influence \(\gamma_{0}\) and K\({}_{0}\) was developed using the CAD system SolidWorks 2022 [21]. The profile of the tool surface obtained by analytical or graphical profiling methods is taken as a starting point.