# Analytical and Graphical Profiling of Thread-Milling Cutters for Forming Internal Threads
## 2 Literature Review
### Methodology for Analytical Determination of the Profile of the Tool Surface  
Figure 5 shows the profile of an internal thread in cross- and axial section. A thread whose profile in axial section is linear is defined as an Archimedean helical surface. Any section of it normal to its geometric axis, for example, section A-A in Figure 5, is an Archimedean spiral with an equation in polar coordinates:  
\[\mathrm{R_{i}}=\mathrm{a.}\varphi_{\mathrm{i}},\mathrm{mm} \tag{1}\]  
where \(\mathrm{R_{i}}\) is the radius vector of the Archimedean spiral in mm; a is parameter of the Archimedean spiral in mm/rad; \(\varphi_{\mathrm{i}}\) is the angle of rotation of the spiral in rad. These cross-sections are identical but are rotated relative to each other by an angle \(\epsilon_{i}\), depending on the helical parameter of the helical surface (Figure 6):  
\[\epsilon_{i}=\frac{\mathrm{z_{i}}}{\mathrm{p}},\mathrm{rad} \tag{2}\]  
where \(\mathrm{p}=\frac{\mathrm{n,P}}{2.\pi}\) is a helical parameter; \(\mathrm{P}\) is the thread pitch in mm; \(\mathrm{n}\) is the number of thread passes; \(\mathrm{z_{i}}\) is the displacement of the secant plane along the \"z\" axis in mm. The secant plane A-A also intersects the rotational tool surface oriented in the \"Oxyz\" coordinate system. Its section in the same secant plane is a circle with diameter \(\mathrm{d_{0}}\), which is tangent to the Archimedean spiral S at point \(\mathrm{i}\), where they have a common normal passing through points \(\mathrm{i}\) and \(1^{\prime}\) (Figure 6). This section is a profiling section of the tool surface. Since the rotary tool surface has a smaller diameter than the inner diameter of the thread, the center distance \(\mathrm{A_{0}}\) is set, which is determined by the following equation:  
\[\mathrm{A_{0}}=\mathrm{R}-\mathrm{r_{0}},\mathrm{mm} \tag{3}\]  
Figure 5: Profile of the formed thread: (**a**) cross-section; (**b**) axial section; (**c**) thread parameters. Figure 6: Scheme for determining the profile of the tool surface when milling internal threads. where R is the radius of the outer diameter of the thread and r\({}_{0}\) is the radius of the circle of the tool tangent to the outer diameter of the thread in mm. The circle drawn with radius A\({}_{0}\) is the trajectory of the circular feed motion 3 (Figures 2 and 3). In order to realize the milling, it is necessary that the normal vector at any point of the spiral describing the cross-section of the thread intersects this circle. Considering that with known pitch P, number of thread passes n, average diameter D\({}_{2}\) and thread profile angle \(\alpha\) each helical surface is determined, then Archimedean spirals are also determined, drawing the different sections that the tool touches. Their parameter \"a\" can be determined by the following equations:  
\[R_{0}=a.\varphi_{0},mm \tag{4}\]  
\[R_{6}=a.\varphi_{6},mm \tag{5}\]  
\[H=R_{0}-R_{6}=\frac{P}{2.\mathrm{tg}\frac{\alpha}{2}},mm \tag{6}\]  
where R\({}_{0}\) is the radius vector of point 0 of the Archimedean spiral drawn at angle \(\varphi_{0}\) in mm; R\({}_{6}\) is the radius vector of point 6 on the same spiral, but at an angle \(\varphi_{6}\) in mm; H is the height of the theoretical thread profile in mm. Then,  
\[H=a.\varphi_{0}-a.\varphi_{6}=a.(\varphi_{0}-\varphi_{6}),mm \tag{7}\]  
The radius vectors of the spirals increase from R\({}_{6}\) to a value of R\({}_{0}\) for an angle \(\varphi_{6}-\varphi_{0}=\frac{\pi}{n}\). Then, from Equation (7), it follows that \(H=\frac{a.\pi}{n}\), and taking into account Equation (6), for the spiral parameter, the following equation is obtained:  
\[a=\frac{n.P}{2\pi.\mathrm{tg}\frac{\alpha}{2}},m/rad \tag{8}\]  
For a certain parameter of the spiral \"a\" and given radius vectors R\({}_{i}\) at different points in the range R\({}_{1}\div R_{6}\), the angle of rotation of the spiral \(\varphi_{i}\) is determined.