## 3 Method Application
### The Rack Tool Case
#### 3.1.2 Graphical Modeling  
The tool and part models were generated in CATIA by successively following the stages presented in Section 2 in the toothing process with the rack tool (see Figures 2 and 3).  
The motion increments \(\Delta\lambda\) and \(\Delta\varphi\) for the two moving centrodes are as follows:  
\[\Delta\lambda=R_{r}\cdot\Delta\varphi;\ \Delta\varphi=\varphi_{i+1}-\varphi_{i}. \tag{7}\]  
The detached chip area variation, which resulted in the addressed case of simulating the toothing process, is presented in Figure 4 [25].  
Both the variation in the detached chip cumulative area in the toothing process and the variation in the area of the chip detached by each tooth of the tool can be observed.  
After each repositioning of the rack tool, there will be the same variation law for the chip cumulative area, which has a periodicity of \(360^{\circ}/z=12^{\circ}\), in the presented case.  
Figure 3: Four-tooth rack—step \(1\), after indexing [26].  
Figure 2: Simulation of toothing with the rack tool [25].  
Using the data regarding the evolution of the total section of the chips, it is possible to identify a function that best approximates the function shape. Thus, the most suitable of the approximation variants, which are available in Matlab, was as follows:  
\[f(x)=a_{1}sin(b_{1}x+c_{1})+a_{2}sin(b_{2}x+c_{2})+...+a_{8}sin(b_{8}x+c_{8}), \tag{8}\]  
where the values of the coefficients \(a_{1}\), \(a_{2}\), \(\ldots\), \(a_{8}\), \(b_{1}\), \(b_{2}\), \(\ldots\), \(b_{8}\) and \(c_{1}\), \(c_{2}\), \(\ldots\), \(c_{8}\) are calculated by the program for approximating the graph of the given function. The values determined by knowing the form of the function f and the coefficients are presented in Figure 5, as detailed in previous work [25].  
Using a dedicated application in Matlab [24,25], the feed value \(\Delta\varphi_{i}\) was determined to obtain a quasi-constant area of the chip section.  
By running the application for \(f\) function smoothing, the values of the angular step of the workpiece rotation were obtained, for which, in compliance with the condition of rolling without slipping, the values of the chip section area are close to the ones imposed by the operator.  
Figure 4: The detached chip area variation when toothing with the rack tool [25].  
Figure 5: Approximation function of the chip section variation [25,26].  
The value of the total chip section area detached at a double stroke of the generating tool was limited to 6 mm\({}^{2}\), the machine stress being at the level of teething with a single-tooth rack tool that has a fixed angular pitch of 0.4\({}^{\circ}\)/double stroke, as shown in Figure 6. By double stroke, here, we mean the back-and-forth displacement of the tool between two given points.  
The number of double strokes the machine must complete to reach this result increases from 110 to 165, without being relevant, because by setting a higher value for Ac, the required number of double strokes will diminish.  
The variation in the angular step in the range 0... 44\({}^{\circ}\) is shown in Figure 7. The value of 44\({}^{\circ}\) means a rotation corresponding to a complete passage of the rack tool.  
Figure 6: The variation in the chip section area when toothing with the rack tool: constant step vs. variable step cases [25].  
Figure 7: The variation in the angular pitch of the workpiece to obtain a constant cumulative chip area of 6 mm\({}^{2}\)[25].  
Finally, it can be concluded that the results presented above support the viability of the imagined solution, which, implemented on a specialized numerical control machine that works with a circular feed of the variable workpiece, can lead to the uniformity of the size of the detached chip section area at a double stroke of the tool and, therefore, of the main cutting force magnitude.