## 3 Method Application
### The Pinion Cutter--Turning Case
#### 3.3.1 Analytical Modeling  
For example, it was considered that the tooth of the pinion cutter (the generating tool) belongs to an ordered vortex of involute profiles--with straight teeth and without correction, respectively. Then, the involute model was represented (Figure 16), and the cases of the two flanks, left and right, were analyzed separately. Figure 15: Variation in the chip section when machining a toothed wheel with the pinion cutter tool (radial feed area); constant feed vs. variable feed (toothing with radial feed motion) [25]. For the involute profile of the tooth's left flank, the reference systems are defined as follows:  
* _xy_--system solidary with the pinion cutter;
* _x:y:i_--system solidary with \"\(i\)-th\" flank tooth of the tool, originally overlapped on _xy_. Starting from the definition of the involute, \[\overline{PT}=\overline{ST}=R_{b}\cdot(\gamma+\varphi),\] (13) and transforming and replacing, the equation of the left involute flank was obtained as the semi-arc of the full tooth on the base circle and of radius \(R_{b}\), respectively. \[\widehat{s}_{b}=\left(\frac{\pi}{2}+z\cdot inv20^{\circ}\right)\cdot m,\] (14) where \(z\) is the teeth number of the plane wheel and \(m\) is the normal module identical to the frontal module (for straight teeth). Similarly, the equation of the right involute flank was obtained. Finally, the circular flank, defined as the upper arch of the tooth on the outer circle of the teeth and delimited by the left flank and the right flank, is described by the equations:  
\[ \begin{array}{l}X_{i}=R_{e}\cdot\sin\theta\\ Y_{i}=R_{e}\cdot\cos\theta^{\prime}\end{array}  \tag{15}\]  
with \(\theta\) as a variable parameter, where \(\theta\in[0,(V_{e})d]\) for the right flank and \(\theta\in[(V_{e})_{s},0]\) for the left flank, respectively. Therefore, the analytical model of teeth that simultaneously detach chips can be written as a set of equations:  
\[ \begin{array}{l}\begin{array}{l}\begin{array}{l}X=X_{i}\cdot\cos(k\delta) -Y_{i}\cdot\sin(k\delta);\\ Y=X_{i}\cdot\sin(k\delta)-Y_{i}\cdot\cos(k\delta),\end{array}&k=\ldots-2,\;-1, \,0,\,1,\,2\,\ldots,\end{array}\end{array}  \tag{16}\]  
where \(X_{i}\) and \(Y_{i}\) are replaced with the tooth apex profile equations to describe the left flank, the right flank, and the tooth apex, respectively. The worm generation process involves a rolling motion between the centrodes of the generated worm and those of the pinion cutter. If \(\Phi_{1}\) and \(\Phi_{2}\) are the angular parameters  
Figure 16: The involute model: (**a**) left flank; (**b**) right flank [28]. of the tool motion, respectively, of the worm motion around its own axes, then the rolling condition between the two centrodes has the expression:  
\[\Phi_{1}=\frac{p_{ax}}{2\pi R_{d}}\cdot\Phi_{2}, \tag{17}\]  
where \(p_{ax}\) is the axial pitch of the generated worm (with a single thread) and \(R_{d}\) is the radius of the tool rolling circle:  
\[p_{ax}=m\cdot\tau\text{and }R_{d}=\frac{m\cdot z}{2} \tag{18}\]  
It was considered that the transverse feed of the pinion cutter was proportional to the rotation angle of the tool, having the ratio \(\lambda\).