## 3 Method Application
### The Pinion Cutter--Turning Case
#### 3.3.2 Graphical Modeling  
The graphic solution for determining the shape and cumulative size of the chip areas detached from the pinion cutter teeth is developed for the case of generating a globoid worm with an involute profile of module \(m=5\) mm and outer diameter \(2\cdot r_{e}=34\) mm, with a pinion cutter tool with \(z=20\) teeth. The workpiece from which the worm was obtained had a cylinder shape with a diameter of 34 mm.  
To simulate the motion of the tool and workpiece, rotation steps of the workpiece of 45\({}^{\circ}\) were considered. According to the rolling condition, this corresponds to a rotation angle of the pinion cutter,  
\[\Delta\Phi_{1}=\frac{2\cdot\pi}{8\cdot z}=2.25^{\circ}, \tag{25}\]  
for each axial section analyzed, and a translation of the measurement plane located on the respective section with  
\[\lambda\cdot\Delta\Phi_{1}\ =\ 0.0625\ \text{mm}. \tag{26}\]  
For the processing of a globoid worm (Figure 17) with a pinion cutter (Figure 18), the graph of the variation in detached chips section, measured in the 8 semi-planes, was drawn.  
Finally, the shape and area of the detached chips were obtained by comparing the current surface with the previously measured surface in the same axial section. The sequences of chip formation in a semi-plane are presented in Figure 19 (the successive positions of the tool relative to the workpiece axis are presented in the figure).  
For simplification, without very large deviations, one can choose, for approximating the distribution function of the chip section during the processing of the globoid worm, a much more accessible function of the shape, namely a function of the 1 st degree (see also Figure 20):  
\[f(x)=p_{1}\cdot x+p_{2}, \tag{27}\]  
where \(p_{1}\) and \(p_{2}\) mean the coefficients from the known straight-line equation.  
In Figure 20, the feed of the pinion cutter is represented on the abscissa, starting from the contact with the workpiece. The measured area of the chips corresponds to a feed of 0.0625 mm during the rotation of the workpiece by 45\({}^{\circ}\).  
If the area of the formed chips is required to be approximately 2 mm\({}^{2}\), and the imposed value of the chip section is much lower than the values meas  
Figure 19: Chip formation sequences: hollow formation—globoid worm [28].  
Figure 20: Polynomial function of 1 st degree to approximate the section of detached chips when processing a globoid worm (\(z=20\) teeth, \(m=5\)) [25].  
of a cutter feed of 0.5 mm/workpiece rotation, the number of rotations performed by the workpiece increases from 23 to 744/8 = 93 rotations.  
However, the machine's stress drops significantly compared to the first case when the formed chips had areas between 0 and 17 mm\({}^{2}\) (Figure 21).  
Unlike the situations presented previously, in the case of globoid worm turning, it is no longer possible to act discretely on the feed. Thus, in the globoid worm, which was obtained by turning, the cutter is actuated in correlation with the rotational motion of the workpiece. Therefore, the variation law no longer allows a displacement or angle change/double stroke, but a feed/rotation value or a per-rotation fraction.  
Given the fact that measurement was carried out with a 45\({}^{\circ}\) angular step, the feed on each octave of the workpiece rotation was calculated. Thus, for each relative position of the tool and workpiece, the two models of the tool and workpiece overlap. The chip was detached from the workpiece and the area of the remaining workpiece was measured.  
In conclusion, the chip section area formed when turning a globoid worm using a pinion cutter can be controlled and maintained at imposed values by controlling the feed of the cutting tool. This is possible if the analytical law giving the chip area variation during simulated machining in a graphical environment is known.