# A Rapid and Inexpensive Method for Finding the Basic Parameters of Involute Helical Gears
## 2 Materials and Methods
### Helical Gears
#### 2.3.2 Alternative Method for the Helical Gears  
The methodology outlined above is not applicable to involute helical gears. The main reason for this is the difficulty of employing pins or rollers with this type of gear. As illustrated in Figure 10, the axes of the two pins making contact with the tooth flanks are not coplanar, making this approach impractical. Figure 10: The axes of the pins are not coplanar. Figure 9: The proposed method for finding the characteristic parameters of a spur gear with involute teeth: (**a**) gear with even number of teeth, when the direction of measurement passes through two teeth or two spaces; (**b**) gear with odd number of teeth, when the direction of measurement passes through a tooth and a space. This can be overcome by using ball bearings spheres, keeping in mind that these balls should be in contact with the flanks of the teeth and with another frontal surface. The motivation behind this approach is to ensure that all the sphere centres are situated within the same frontal plane, as demonstrated in Figure 11.  
The geometry of the wheel is characterised by the number of teeth \(z\), the module \(m\) in the normal section, the normal profile shift coefficient \(x\), and the helix angle of the teeth of the base rack \(\beta_{0}\). The gearing is studied in the frontal section, and the profile is also an involute, characterised by the number of teeth \(z\), the frontal module \(m_{t}\), and the frontal profile shift coefficient \(x_{t}\). Considering the relation (25) and the fact that the angle \(\beta_{0}\) may take any value, it results that in the frontal section, the value of the module is not standard and the methodology applied for the spur gears is not effective. Therefore, in this case, the conditions imposed upon the coordinates of the contact points are used. In Figure 12, a frontal section is presented, which contains the centre of a ball of radius \(r_{B}\) which makes contact with the boundary flanks at points \(Y\) and \(Y\). The normal at the contact point is tangent to the base circle in the \(T_{y}\) point. The \(Y\) points are placed on a circle characterised by the pressure angle \(\alpha_{yt}\) of the involute generated using the base circle of radius \(r_{B}\). The pitch circle passes through points \(D\). Applying the involut  
Figure 11: Schematics for the device used for involute helical gears. Figure 12: Finding the coordinates of the contact points. \[ \begin{array}{l}arcY^{Y}Y^{\prime\prime}=r_{y}\phi_{yt}=\frac{1}{2}\frac{mzcosa _{qt}}{cosa_{yt}}\left[\phi_{0t}+2(\theta_{yt}-\theta_{0t})\right]=\frac{1}{2} \frac{mzcosa_{qt}}{cosa_{yt}}\left[\frac{e_{qt}}{d/2}+2(inva_{yt}-inva_{0t}) \right]\\ =\frac{mzcosa_{qt}}{cosa_{qt}}\left[\frac{\frac{\pi}{2}-2x_{t}ttna_{0t}}{z}+( inva_{yt}-inva_{0t})\right]=\frac{mzcosa_{qt}}{cosa_{qt}}\left[\frac{\pi}{2}- \frac{2x_{t}ttna_{0t}}{z}+(inva_{yt}-inva_{0t})\right]\end{array}  \tag{47}\]  
where \(\phi_{y}\) represents the arc of the space width on the circle \(C_{y}\) and \(\phi_{0t}\) is the arc of the space width on the pitch circle. From the above relation, one can obtain  
\[ \begin{array}{l}\phi_{yt}=\frac{arcYY}{r_{y}}=\frac{mzcosa_{0t}}{\frac{cosa _{qt}}{cosa_{yt}}}\left[\frac{\pi}{2z}-\frac{2x_{t}ttna_{0t}}{z}+(inva_{yt}- inva_{0t})\right]=2\left[\frac{\pi}{2z}-\frac{2x_{t}ttna_{0t}}{z}+(inva_{yt}- inva_{0t})\right]\end{array}  \tag{48}\]  
Next, one can obtain  
\[ \begin{array}{l}T_{Y}OT_{Y}=\varphi_{yt}+2\alpha_{yt}=2\left[\frac{\pi}{2z} -\frac{2x_{t}ttna_{0t}}{z}+(inv\alpha_{yt}-inv\alpha_{0t})\right]+2\alpha_{yt} \\ =2\left[\frac{\pi}{2z}-\frac{2x_{t}ttna_{0t}}{z}+(inv\alpha_{yt}+\alpha_{yt}- inv\alpha_{0t})\right]=2\left[\frac{\pi}{2z}-\frac{2x_{t}ttna_{0t}}{z}+tna_{yt}- inv\alpha_{0t}\right]\end{array}  \tag{49}\]  
The angles \(T_{Y}O_{B}T_{Y}\) and \(T_{Y}OT_{Y}\) are supplementary:  
\[ \begin{array}{l}T_{y}O_{B}T_{y}=\pi-2\left[\frac{\pi}{2z}-\frac{2x_{t}ttna_ {0t}}{z}+tna_{yt}-inv\alpha_{0t}\right]\end{array}  \tag{50}\]  
The length of the cord \(YY\) can be expressed in two ways:  
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