# A Rapid and Inexpensive Method for Finding the Basic Parameters of Involute Helical Gears
## 3 Exemplification of the Methodology  
The methodology we use with the measuring device is as follows: For the teethed wheel to be measured as 1, the axis of measurement is found; it is defined by two spaces between teeth or by two teeth diametrically opposite. For an odd number of teeth, the axis is defined by a space and a tooth placed diametrically opposite. Then, two pairs of balls 2 are introduced in the spaces placed symmetrically with respect to the measuring axis. One of the pairs of balls is brought into contact with the surface of the fixed prism 3, which is assembled by screws on the base plate 4. Next, the mobile prism 5 is carefully moved until its surface meets the second pair of balls. In order to diminish the risk of locking the mobile prism, it has ball bushings which ensure a smooth and noiseless motion along the guiding rods 7, fixed on the base plate by the support parts 8. When all the balls have made firm contact with the prisms, the assembly is fixed using the screw 9. Then, the distance D is measured and the angles \(\psi_{up}\) and \(\psi_{down}\) are calculated. The methodology is repeated for another configuration of the balls, symmetrically placed with respect to the measuring axis. Next, Table 1 is completed and the relations presented in the paper are applied. The method is exemplified by applying it to an actual teethed wheel with the number of teeth \(z=14\). The device presented in [25] is used, but the pins were replaced by balls of radii \(r_{B1}\) and \(r_{B2}\); for the two sets of balls (Figure 14), the dimension over balls \(D_{1}\) and \(D_{2}\) were measured. The values of the involved parameters for two values of the radii at which the balls' centres are positioned-- the radii of balls \(r_{B1}\) and \(r_{B2}\), distances \(D_{1}\) and \(D_{2}\), positioning angles \(\psi_{up}\) and \(\psi_{down}\)--and the values resulting from the calculation with relations (55) and (56) are presented in Table 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(r_{B}\) & \(\psi_{up}\) & \(\psi_{down}\) & \(D\) & \(\mathbf{Y_{B}}\) \\  & **[mm]** & **[deg]** & **[deg]** & **[mm]** & **[mm]** \\ \hline entry 1 & 5.760 & \(2\pi/z\) & 0 & 91.27 & 41.952 \\ entry 2 & 4.75 & \(\pi/z\) & \(\pi/z\) & 87.51 & 40.008 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measurements results. Figure 14: _Cont._With these values, a system of transcendental Equation (60) is obtained which is solved using the Newton-Raphson method. The convergence of the solutions of the system as function of the number of iterations is presented in Figure 15. It can be observed that the method is rapidly convergent and conducts to the values of the pressure angles of the involute profile on the circles where the ball-tooth flank contact points are placed. In order to find the helix angle \(\beta_{a}\) of the teeth on the addendum cylinder, two methods were used: first, the wheel was scanned using a laser profilometer (Nanofocus uscan), shown in Figure 16, and then, a photo of the wheel was taken from the side, shown in Figure 17. The helices from the addendum cylinders and the helix angles with respect to the axis of the gear are identified in both images. Figure 15: The fast convergence of the Newton–Raphson method applied for finding the pressure angles \(\alpha_{y1}\) and \(\alpha_{y2}\) on the circles of the contact points. By comparing the measured angles from the two figures, equal values \(\beta_{a}=23^{\rm o}\) were found. Taking a photo is a much simpler method compared to scanning. Now, with found values \(\beta_{a}\) and \(a_{y1t}\) one can apply Equation (64) and the normal module of the gear results in \(m=4.949\). Next, the standard normal module is adopted \(m_{std}=5\); angle \(\beta_{0}\) is obtained by applying relation (65), \(\beta_{0}=21.344^{\rm o}\), and next, by applying relation (66), the profile shift coefficient is found \(x=0.105\). Subsequently, it is proven that the values \(a_{y1t}\), \(a_{y2t}\), \(m_{std}\), \(\beta_{0}\), and \(x\) verify the system of Equation (59).