# A Rapid and Inexpensive Method for Finding the Basic Parameters of Involute Helical Gears
## 2 Materials and Methods
### Aspects Concerning the Geometry of Spur Gears with Involute Teeth
Consequently, it corresponds to half the pitch of the rack (maintaining consistency across any section parallel to the direction of motion):  
\[s_{0}=e_{0}=p_{0}/2 \tag{7}\]  
with  
\[p_{0}=\pi m \tag{8}\]  
where \(m\) is a parameter with dimension of length and taking standard values. The reference straight line divides the tooth into two zones:  
* The addendum, of height \[h_{a0}=h_{a0}^{*}m,h_{a0}^{*}=1\pm^{*}(ISO\pm^{*}value)\] (9)
* The dedendum. As can be noticed from the detail A, the tooth is cut on the inferior zone of a length \[c_{0}=c_{0}^{*}m,c_{0}^{*}=0.25(\text{ISO\,value})\] (10)  
Figure 2: The basic rack with straight teeth. The reference height of the dedendum becomes  
\[h_{f0}=h_{f0}^{*}m=(h_{a0}^{*}+c_{0}^{*})m \tag{11}\]  
For a complete definition of the geometry of the rack, additional specifications must be stipulated:  
* The pressure angle of the straight profile: \[\alpha_{0}=20^{\circ};\] (12)
* The filleting radius \(\rho_{0}\) at the root of the tooth: \[\rho_{0}=\frac{c_{0}^{*}}{1-sina_{0}}m\cong 0.38m\] (13)  
As a conclusion, the profile of the basic rack is defined by the module \(m\), the pressure angle \(\alpha_{0}\), the reference addendum coefficient \(h_{a0}^{*}=1\), and the coefficient of radial backlash \(c_{0}^{*}=0.25\). The gear with involute teeth is defined using the basic rack. The definition parameters require adherence to two specific conditions:  
* The gearing is accomplished without backlash between the flanks;
* The gearing is accomplished with standard radial backlash \(c_{0}\)  
The two conditions are sufficient for the complete definition of the gear. The centroids of the relative motion between the wheel and the base rack are represented (Figure 3) by the pitch circle of the gear \(c_{d}\) of diameter  
Figure 2.0  
and the rolling straight line of the rack \(\Delta_{w}\) parallel to the reference line \(\Delta_{0}\), but at a distance from this, with the quantity \(X\) named profile shift and expressed as  
Figure 2.1  
The profile shift coefficient \(x\) of the gear is considered:  
\(x=0\) when the straight line \(\Delta_{w}\) is tangent to the pitch circle;  
\(x<0\) when \(\Delta_{w}\) intersects the pitch circle;  
\(x>0\) when \(\Delta_{w}\) is external to the pitch circle. In conclusion, with stipulated parameters for the shape and dimensions of the basic rack, the toothed wheel is fully characterized by the next three parameters:  
1. The number of teeth \(z\), which describes the dimension of the wheel;
2. The module \(m\), which describes the dimension of the tooth;  
Figure 2.2 The pitch straight line of the rack rolls over the pitch circle of the gear; the profile shift definition. The profile shift coefficient \(x\), which describes the shape of the tooth. A precise method for manufacturing the involute gears is gear generation, wherein the resultant gear is conceived as the envelope of a distinctive series of positions of a toothed cutting tool. This procedure requires a relative motion between the gear blank and the tool mimicking the motion that would otherwise be present between the tool and the final, machined gear. One manufacturing technique uses a tool which mirrors the profile of the generating rack (conjugate to the basic rack). To highlight the impact of the number of teeth and of the profile shift coefficient on gear machining, a simulation program was developed. This simulation experiment incorporated two different configurations of tooth count, \(z=5\) and \(z=10\), along with three distinct values attributed to the profile shift coefficient. Figure 4 shows the results of the simulation. It is evident that negative values of the profile shift coefficient led to an undercutting phenomenon which involves the removal of tooth material within the dedendum zone. Conversely, positively large values result in overcutting of the tooth. Both phenomena are more pronounced when the tooth count of the gear wheel \(z\) is smaller.