# Analytical and Graphical Profiling of Thread-Milling Cutters for Forming Internal Threads
## 7 Discussion  
The shaping of internal threads by cutting is performed by various methods using tools such as thread-turning tools, taps, thread-milling cutters and thread-cutting heads. Among them, the highest productivity and efficiency is achieved when milling with thread-milling cutters. The design and manufacture of such tools requires an accurate definition of the tool surface. In this regard, two approaches are presented--analytical and geometric profiling. A comparison of the results shows that both methods give very close values when determining the points of the tool surface. An analytical method has been developed to determine the profile of the working part of the thread-milling cutters. The developed methodology calculates the points of the profile of each tooth that build the tool surface. For this purpose, a Microsoft Excel 2019-based program has been developed that automatically calculates these points. The obtained parameters allow the design and manufacture of a thread-milling cutter for a specific internal thread or for a group of threads. In parallel, a graphical method for profiling the tool surface using a CAD system is proposed. This method provides a profile in which the helical surfaces of the formed thread are not trimmed. The implementation using SolidWorks 2022 allows for the assessment of the resulting profile deviations when milling internal threads. The graphical method based on enveloping curves forms the profile by multiple circles that are tangent to an Archimedean spiral in axial section and intersect the thread profile in cross-section. Compared with classical analytical approaches, the proposed graphical methodology allows for a more intuitive verification of the profile and easier application when working with non-standard thread and/or helical profiles. One of the main advantages of the proposed graphical method is that it is easy to use and implement in CAD software. This makes it accessible even to engineers who may not have much experience in mathematical modeling. Although analytical methods remain more precise from a theoretical perspective, the graphical method is accurate enough for practical applications and helps speed up the design process. The influence of the design parameters rake angle \(\gamma_{0}\) and the backing off \(\mathrm{K}_{0}\) on the profile of the cutting edge has been assessed. The analysis shows that at constant values of the outer diameter \(\mathrm{d}_{0}\) and the backing off \(\mathrm{K}_{0}\), increasing the rake angle \(\gamma_{0}\) leads to an increase in the height of the profile \(\mathrm{h}_{0}\) and an increase in the deviation of the profile angle \(\Delta\alpha_{0}\). When the thread diameter \(\mathrm{D}\) increases, at a constant pitch \(\mathrm{P}\) and tool diameter \(\mathrm{d}_{0}\), an opposite trend is observed--the height of the profile \(\mathrm{h}_{0}\) and the deviation of the profile angle \(\Delta\alpha_{0}\) decrease. It has also been established that to achieve an accurate profile, it is necessary to correct the profile of the tool surface by \(\Delta\alpha_{0}\) and \(\Delta\mathrm{K}_{0}\). The verification of the results obtained by the analytical method was carried out with the graphical method. Both approaches achieved similar values for the profile parameters. The accuracy of the obtained results is sufficient for the design and subsequent production of a specific thread-milling cutter. A program has been created in Microsoft Excel 2019 to determine the design parameters of the thread-milling cutter--including the milling part, transition part and the shank. On this basis, designs of thread-milling cutters for internal threads have been developed. Geometric and design parameters have been determined using the analytical and graphical methods. Parametric three-dimensional models have been created in the CAD system and the process of creating design documentation automated using template files. User and system attributes are entered into these files, which are also used in the drawings.