# 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
\begin{array}{l}r_{B}sin\left[\frac{1}{2}\left(\pi-2\left[\frac{\pi}{2z}- \frac{2x_{t}ttna_{0t}}{z}+tna_{yt}-inv\alpha_{0t}\right]\right)\right]\\ =r_{y}sin\left[\frac{1}{2}\left(2\left[\frac{\pi}{2z}-\frac{2x_{t}ttna_{0t}}{z }+(inv\alpha_{yt}-inv\alpha_{0t})\right]\right)\right]\end{array}  \tag{51}\]  
And from here, the following equation is obtained:  
\[ \begin{array}{l}r_{B}cos\left(\frac{\pi}{2z}-\frac{2x_{t}ttna_{0t}}{z}+tna_ {yt}-inv\alpha_{0t}\right)=\frac{1}{2}mz\frac{cosa_{0t}}{cosa_{yt}}sin\left[ \frac{\pi}{2z}-\frac{2x_{t}ttna_{0t}}{z}+(inv\alpha_{yt}-inv\alpha_{0t})\right] \end{array}  \tag{52}\]  
Additionally, the radius of the centre of the ball can be found:  
\[ \begin{array}{l}Y_{B}=r_{B}cos\left[\frac{1}{2}\left(\pi-2\left[\frac{\pi}{2z }-\frac{2x_{t}ttna_{0t}}{z}+tna_{yt}-inv\alpha_{0t}\right]\right)\right]\\ \hskip 113.811024 pt+r_{y}cos\left[\frac{1}{2}\left(2\left[\frac{\pi}{2z}-\frac{2 x_{t}tna_{0t}}{z}+(inv\alpha_{yt}-inv\alpha_{0t})\right]\right)\right]\end{array}  \tag{53}\]  
From here, the following is obtained:  
\[ \begin{array}{l}Y_{B}=r_{B}sin\left(\frac{\pi}{2z}-\frac{2x_{t}ttna_{0t}}{z}+ tna_{yt}-inv\alpha_{0t}\right)+\frac{1}{2}mz\frac{cosa_{0t}}{cosa_{yt}}cos\left[ \frac{\pi}{2z}-\frac{2x_{t}ttna_{0t}}{z}+(inv\alpha_{yt}-inv\alpha_{0t})\right] \end{array}  \tag{54}\]  
Figure 13 shows that the measured dimension over balls \(D\) using the proposed device can be expressed as  
\[D=Y_{B}cos\psi_{up}+Y_{B}cos\psi_{down}+2r_{B} \tag{55}\]  
where \(Y_{B}\) is the radius of the circle where the centres of the balls (of radii \(r_{B}\)) are positioned in the space between teeth, contacting both flanks of the teeth. As previously stated, the balls are positioned symmetrically with respect to the diameter parallel to the direction of measurement. From relation (55), the following is obtained:  
\[ \begin{array}{l}Y_{B}=\frac{D-2r_{B}}{cos\psi_{up}+cos\psi_{down}}\end{array}  \tag{56}\]The relations (54) and (56) result in the equation:  
\[r_{B}sin\bigg{(}\frac{\pi}{2z}-\frac{2x_{t}tana_{0t}}{z}+tanx_{yt}-inva_{0t} \bigg{)}+\frac{1}{2}mz\frac{cosa_{0t}}{cosa_{yt}}cos\bigg{[}\frac{\pi}{2z}- \frac{2x_{t}tana_{0t}}{z}+(inva_{yt}-inva_{0t})\bigg{]}=\frac{D-2r_{B}}{cosa_{ yt}+cosa_{yt}} \tag{57}\]  
Now, considering the Equations (52) and (57), the following system is obtained:  
\[ \begin{cases}r_{B}cos\big{(}\frac{\pi}{2z}-\frac{2x_{t}tana_{0t}}{z}+tanx_{yt} -inva_{0t}\big{)}=\frac{1}{2}mz\frac{cosa_{0t}}{cosa_{yt}}sin\Big{[}\frac{\pi} {2z}-\frac{2x_{t}tana_{0t}}{z}+(inva_{yt}-inva_{0t})\Big{]}\\ r_{B}sin\Big{(}\frac{\pi}{2z}-\frac{2x_{t}tana_{0t}}{z}+tanx_{yt}-inva_{0t} \big{)}+\frac{1}{2}mz\frac{cosa_{0t}}{cosa_{yt}}cos\Big{[}\frac{\pi}{2z}- \frac{2x_{t}tana_{0t}}{z}+(inva_{yt}-inva_{0t})\Big{]}=Y_{B}\end{cases}  \tag{58}\]  
The unknowns of the system are  
* the frontal module \(m_{t}\);
* the frontal pressure angle \(\alpha_{yt}\) on the circle passing through the contact points;
* the frontal profile shift coefficient of the gear \(x_{t}\);
* The frontal pressure angle \(\alpha_{0t}\) on the pitch cylinder of the gear. Considering that the method is applied using two sets of balls of radii \(r_{B1}\) and \(r_{B2}\) positioned at the angles \(\psi_{down1}\), \(\psi_{up1}\), \(\psi_{down2}\), and \(\psi_{up2}\), respectively, the system (58) written for the two cases leads to a system of four equations which, after some calculations, takes the form  
\[\left\{ \begin{array}{l}\frac{m_{2}^{2}}{2Y_{B1}}cosa_{0t}-cos\big{[}asin \Big{(}\frac{r_{B1}}{R_{B1}}cosa_{y1t}\Big{)}+\alpha_{yt1t}\big{]}=0\\ r_{B1}cosa_{y1t}-Y_{B1}\Big{[}\frac{\pi}{2z}-\frac{2x_{t}tana_{0t}}{z}+inva_{y 1t}-inva_{0t}\Big{]}=0\\ \frac{m_{2}^{2}}{2Y_{B2}}cosa_{0t}-cos\big{[}asin\Big{(}\frac{r_{B2}}{R_{B2}} cosa_{y1t}\Big{)}+\alpha_{y2t}\big{]}=0\\ r_{B2}cosa_{y2t}-Y_{B2}\Big{[}\frac{\pi}{2z}-\frac{2x_{t}tana_{0t}}{z}+inva_{y 2t}-inva_{0t}\Big{]}=0\end{array} \right.