# 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
\tag{59}\]  
A system of two transcendental equations (from the first and third equation of system (59) and, respectively, from the second and fourth equation) of unknowns \(\alpha_{y1t}\) and \(\alpha_{y2t}\) results in  
\[\left\{ \begin{array}{l}Y_{B1}cos\big{[}asin\Big{(}\frac{r_{B1}}{R_{B1}}cosa _{y1t}\Big{)}+\alpha_{yt1t}\big{]}-Y_{B2}cos\big{[}asin\Big{(}\frac{r_{B2}}{R_ {B2}}cosa_{y2t}\Big{)}+\alpha_{y2t}\big{]}=0\\ inva_{y1t}-asin\Big{(}\frac{r_{B1}}{R_{B1}}cosa_{y1t}\Big{)}-inva_{y2t}+asin \Big{(}\frac{r_{B2}}{R_{B2}}cosa_{y2t}\Big{)}=0\end{array} \right. \tag{60}\]  
Figure 13: The dimension over balls for the proposed method. The two unknowns \(\alpha_{y1t}\) and \(\alpha_{y2t}\) are found via a numerical method. After the two pressure angles are found, two equations of system (59) are considered (either the first two, or the last two equations). We consider here the first two equations of system (59) and with the relations (25), (33), and (39) between the geometrical parameters from the normal and frontal section; the next system of equations results in  
\[\left\{ \begin{array}{l}\frac{m\kappa z}{2Y_{B1}}cos\alpha_{0t}-cos\Big{[}asin \Big{(}\frac{r_{B1}}{Y_{B1}}cos\alpha_{y1t}\Big{)}+\alpha_{y1t}\Big{]}=0\Big{]} \\ r_{B1}cos\alpha_{y1t}-Y_{B1}\Big{[}\frac{\pi}{2z}-\frac{2\chi_{t}tan\alpha_{0} }{z}+inv\alpha_{y1t}-inv\alpha_{0t}\Big{]}=0\end{array} \right. \tag{61}\]  
Now, we recall Equations (25), (39), and (54) and have  
\[\left\{ \begin{array}{l}\frac{mcos\beta_{0}}{2Y_{B1}}zcos\Big{(}atan\frac{tan \alpha_{0}}{cos\beta_{0}}\Big{)}-cos\Big{[}asin\Big{(}\frac{r_{B1}}{Y_{B1}} cos\alpha_{y1t}\Big{)}+\alpha_{y1t}\Big{]}=0\\ r_{B1}cos\alpha_{y1t}-Y_{B1}\Bigg{[}\frac{\pi}{2z}-\frac{2 xcos\beta_{0}\frac{ tan\alpha_{0}}{cos\beta_{0}}}{z}+inv\alpha_{y1t}-inv\Big{(}atan\frac{tan \alpha_{0}}{cos\beta_{0}}\Big{)}\Bigg{]}=0\end{array} \right. \tag{62}\]  
The resulting system (62) has three unknowns:  
* The normal module \(m\), which takes standard values;
* The helix angle of the flank \(\beta_{0}\) on the pitch cylinder of the wheel;
* The normal profile shift coefficient \(x\) of the wheel. A further equation is needed to solve the system. To this end, we accept the hypothesis (justified in Appendix A) that the helix angle \(\beta_{a}\) of the flank on the addendum cylinder does not differ substantially from the helix angle from the pitch cylinder \(\beta_{0}\). This remark is extremely important because the slant of the tooth on the addendum cylinder can be measured relatively easily, since the addendum surface exists physically while the pitch surface is fictive. Supposing that the inclination angle of the tooth on the addendum cylinder \(\beta_{a}\) is found, this value is replaced into the first equation of system (62):  
\[\frac{mcos\beta_{a}}{2Y_{B1}}zcos\Big{(}atan\frac{tan\alpha_{0}}{cos\beta_{a} }\Big{)}-cos\Big{[}asin\Big{(}\frac{r_{B1}}{Y_{B1}}cos\alpha_{y1t}\Big{)}+ \alpha_{y1t}\Big{]}=0 \tag{63}\]  
and then, the equation is solved with respect to \(m\):  
\[m=2Y_{B1}cos\Big{[}asin\Big{(}\frac{r_{B1}}{Y_{B1}}cos\alpha_{y1t}\Big{)} \Bigg{]}\frac{cos\beta_{a}}{z}\sqrt{1+\frac{tan^{2}\alpha_{0}}{cos^{2}\beta_{ a}}} \tag{64}\]  
The closest standard value is adopted for the normal module:  
\[m_{n}=m_{nsId}\]  
and using this value, it is replaced in Equation (63) and then the equation is solved with respect to the angle \(\beta_{0}\):  
\[\beta_{0}=acos\sqrt{\frac{z^{2}m_{nsId}^{2}}{\Big{\{}2 cos\Big{[}asin\big{(} \frac{r_{B1}}{Y_{B1}}cos\alpha_{y1t}\big{)}\Big{]}Y_{B1}\Big{\}}^{2}-tan^{2} \alpha_{0}}} \tag{65}\]  
Now, the helix angle on the pitch cylinder is known and this value is replaced in the second Equation (62), obtaining an equation of unknown \(x\), the normal profile shift coefficient. The solution is  
\[x=\frac{z}{2 tan\alpha_{0}}\bigg{[}\frac{\pi}{2z}+inv\alpha_{y1t}-inv\bigg{(}atan \frac{tan\alpha_{0}}{cos\beta_{0}}\bigg{)}-asin\big{(}\frac{r_{B1}}{Y_{B1}} cos\alpha_{y1t}\big{)}\bigg{]} \tag{66}\]