## 3 Method Application
### Pintion Cutter--Slotting Case
#### 3.2.2 Graphical Modeling  
The modeling was carried out in a CATIA graphical design environment due to the development of a numerical application, which was designed to assess both the shape and area of the chip detached from the teeth of the pinion cutter in the chipping process, which is particularly complex and laborious at the same time. The case of toothing a pinion with \(z_{p}=30\) teeth or \(m=10\) mm with a pinion cutter having \(z_{s}=20\) teeth was addressed for sampling the method application. The rotational increment of the workpiece was considered \(\Delta\varphi=1^{\circ}\) and, therefore, the rotational increment of the tool was \(\Delta\varphi=1.5^{\circ}\). Using CATIA facilities, the variation in the size of the detached chip was determined, depending on the relative position of the tool to the workpiece (Figure 9). Figure 10 shows the modeling of the toothing process with the pinion cutter, the variant with constant circular feed and the toothing stage when the pinion cutter works with both radial and circular feeds. According to the specifics of the toothing by pinion cutting method, two phases of the process are distinguished, namely, toothing with radial feed motion (penetration) and toothing without radial feed motion. The evolution of the chip size, depending on the relative position of the tool and workpiece, can be seen in Figure 11.  
When analyzing Figure 11, two regions can be distinguished. The first region shows a variation in the size of the detached chip due to the interaction of the tool with the workpiece, the tool having both a circular feed, to achieve the rolling motion, and a radial penetration motion (up to the tangency of the rolling circles, which happened in the example given previously, at the 90 th double stroke). Figure 10: Toothing with a pinion cutter (step 92—toothing with constant circular feed) [25]. Figure 9: The relative position of the tool and workpiece models in the toothing process [27]. The second region starts after the 90 th double stroke and shows an oscillatory variation in the chip size from a median size until the end of the entire tooth generation. For the first region, Function (12) was identified in the Matlab environment, as shown in Figure 12, to approximate the graphic distribution of the determined values of the detached chip area:  
\[f(x)=a_{1}\cdot\sin(b_{1}\cdot x+c_{1})+a_{2}\cdot\sin(b_{2}\cdot x+c_{2})+ \ldots+a_{7}\cdot\sin(b_{7}\cdot x+c_{7}), \tag{12}\]  
where \(a_{1}\), \(b_{1}\), \(c_{1}\), \(a_{2}\), \(b_{2}\), \(c_{2}\), \(\ldots\)\(a_{7}\), \(b_{7}\), \(c_{7}\) are the calculated coefficients. The oscillation of the cumulative area of the chip size, corresponding to the second area of the graph, from the angular interval \(90^{\circ}\ldots 102^{\circ}\), is, for the given example, between 18 and 37.5 mm\({}^{2}\), the variation being from single to more than double. Figure 11: Variation in detached chip area when toothing with the pinion cutter [25]. Figure 12: Approximation function of the detached chips section in the machining of a toothed wheel with the pinion cutter tool (radial feed zone) [25]. By processing the data regarding the variation in the chip area measured in the range of 90... 102\({}^{\circ}\), a polynomial approximation function is obtained (see Figure 13), which will be used to calculate the feed required to be applied to the tool, to obtain chips cumulative area of about 10 mm\({}^{2}\). The comparison between the size of the chips obtained in the two alternatives, with constant feed and with variable feed, is presented in Figure 14.  
Then, the application was used to determine the values of the feed per double stroke in order to obtain a uniformity of the detached chip area.