## 3 Results
\begin{table}
\begin{tabular}{l c} \hline \hline
**Construction Characteristic** & \(\psi\_{\_}\)**m = b/m,** \\  & **No More than** \\ \hline Highly loaded precision gears, shafts, supports and housings & \\ with increased rigidity at HB: & \\ up to 350 & 45...30 \\ over 350 & 30...20 \\ Conventional gearbox-type transmissions in a separate & \\ housing with sufficiently rigid shafts and supports at the HB: & \\ up to 350 & 30...25 \\ over 350 & 20...15 \\ Rough gears (including those with cantilever shafts) & 15...10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Selection of wheel tooth modulus. Initial bases for the calculation:  
(1) The calculation is made for the moment of force application at the tooth apex (Figure 6). This corresponds to the start of gearing for a wheel tooth and the end of gearing for a gear tooth. Although there is theoretically another pair of teeth in gear at this point, it is assumed that the entire load is transferred by one pair of teeth. Only for precisely manufactured gears (above 6 th degree of accuracy) can it be considered that the load is transmitted by two pairs of teeth  
Let us transfer the force \(F_{n}\) along its line of action on the symmetry axis of the tooth to point C and decompose it into two force components: circumferential \(F_{t}^{\prime}\) and radial \(F_{r}^{\prime}\);  
Then  
\[F_{t}^{\prime}=F_{n}\cos\alpha_{\alpha}=F_{t}\cos\alpha_{\alpha}/\cos\alpha\]  
\[F_{r}^{\prime}=F_{n}\sin\alpha_{\alpha}=F_{t}\sin\alpha_{\alpha}/\cos\alpha\]  
Here \(\alpha_{\alpha}\) is the pressure angle at the tooth apex, which is greater than the pressure (gearing) angle a on the dividing circle. (2) Having plotted the normal stresses (Figure 4), we obtain that the highest stresses occur in the indicated dangerous section. The concentration of stress is also observed here. The calculated stress is the stress on the tensile side of the tooth, i.e.,  
\[\sigma_{F_{n}}=\sigma_{u}-\sigma_{c} \tag{20}\]  
fatigue fracture cracks occur on the tensile side of the tooth. This is not random, because the surface layers of the tooth material, as experiments show, offer less resistance to alternating tensile stresses than to compressive stresses. (3) For the dangerous section \(D\)-\(K\), located near the chord of the basic circle, we write (taking into account the stress concentration)  
\[\sigma_{F}=\sigma_{F_{\text{rem}}}K_{T} \tag{21}\]  
where \(\sigma_{F_{\text{rem}}}\)--calculated nominal stress;  
Figure 6: Calculation scheme for calculation of active surfaces of helical gear wheel teeth on resistance to contact fatigue. \(K_{T}\)--theoretical stress concentration coefficient. \[\sigma_{F}=\left(\frac{M_{x}}{W_{x}}-\frac{N_{z}}{A}\right)\cdot K_{T}=\left(\frac {F_{t}^{\prime}h_{p}}{W_{x}}-\frac{F_{r}^{\prime}}{A}\right)\cdot K_{T} \tag{22}\]  
where \(W_{x}=\frac{b\;S_{p}^{2}}{6}\)--axial moment of resistance of the dangerous section of the tooth stem;  
\(A=S_{p}b\)--tooth stem cross-sectional area;  
\(h_{p}\;\), \(S_{p}\)--design tooth height and thickness, respectively;  
\(b\)--Tooth length (width of the gear wheel ring). Values \(h_{p}\) and \(S_{p}\) can be expressed in fractions of the tooth modulus: \(h_{p}=\gamma_{h}m\);  
\(S_{p}=\gamma_{S}m\);  
Where \(\gamma_{h}\), \(\gamma_{S}\)--coefficients according to design height and design tooth thickness. Figure 4 shows that the force \(F_{t}^{\prime}\) causes transverse bending of the tooth and the force \(F_{r}^{\prime}\) causes compression of the tooth. Thus, in the dangerous cross-section of the tooth (in the stem at the transition point of the involute to the tooth flange) there are three internal force factors: bending moment \(M_{x}\), transverse force \(Q_{y}\) and longitudinal force \(N_{z}\).