## 3 Results  
During the dynamic calculation of the bending and contact stiffness of the teeth of the pinion gear and the wheel of the double-sided helical traction gear of the electric locomotive VL-80s, the first stage is the calculation of contact shear stresses of helical gear, taking into account diagnostic parameters according to operation data. The gear is forged from chromium-nickel alloy steel 20 XNZA with subsequent carburizing and hardening of tooth surfaces along the entire contour. Inside there is a tapered hole with a slope of 1:10 for pressing onto the shaft of the TEE armature. There is a 15 mm wide notch on the side of the hole for the nut. The gear is cut into 21 helical teeth at an angle of 24\({}^{\circ}\). The gear wheel is forged from 55X carbon steel and consists of a hub and a middle part in the form of a disk with holes and a crown with teeth. The forged gear is ground on all sides, 16 relief holes are drilled in the center section and 88 oblique teeth are cut on the crown at an angle of 24\({}^{\circ}\). The teeth are cemented and hardened. I. Let us calculate the contact fatigue resistance of the active surfaces of wheel teeth. The purpose of calculation is to establish the dependence of serviceability index (contact stress \(\sigma_{\rm H}\)) on the external load and geometric dimensions of helical gear wheels. The calculation scheme for the calculation of the wheel tooth surface for contact strength is shown in Figure 4.  
Figure 4: Calculation scheme for calculation of wheel teeth for bending fatigue resistance (**a**) for a gear wheel in the wheel-motor block (WMB) of electric locomotive VL-80s and stress diagrams in the tooth stem of the wheel (**b**). Based on the Hertz formula [2; 3] for the case of compression along the formations of two cylinders, we take the calculation formula as  
\[\sigma_{H}=\sqrt{\frac{q_{n}E_{pr}}{2\pi\rho_{red}(1-\mu^{2})}} \tag{11}\]  
where \(q_{n}=\frac{F_{n}}{\ell}\)-specific normal stress (intensity of interaction between cylinders of length \(\ell\), compressed by force \(F_{n}\));  
\(\rho_{red}=\frac{\rho_{1}\,\rho_{2}}{(\rho_{2}+\rho_{1})}\)--reduced radius of curvature of cylinders;  
\(E_{red}=\frac{2\,E_{1}\,E_{2}}{(E_{1}+E_{2})}\)--the reduced modulus of longitudinal elasticity of the gear and gear cylinder material;  
\(E_{1}\) ii\(E_{2}\)--longitudinal elastic modulus of the material of the first and second cylinder, respectively; \(\mu\)--Poisson's coefficient. It has been experimentally established that the lowest contact strength has the near-pole zone of the working surfaces of the teeth, in which the highest loads act on the teeth (the full load is transmitted by one pair of teeth) and the sliding speed of the teeth V_ck is not equal to zero [7]. The teeth are regarded as two cylinders of length b (gear ring width) with a radius equal to the radius of curvature of the involutes of the gear and wheel tooth profiles at the meshing pole, i.e., \(\rho_{1}\), \(\rho_{2}\). From the triangles O\({}_{1}\) N\({}_{1}\) P and O\({}_{2}\) N\({}_{2}\) P (see Figure 5) the radiuses of curvature can be written down \(\rho_{1}=N_{1}\)P = \(\frac{d_{1}}{2}\) sin \(\alpha\) ; \(\rho_{2}=N_{2}\)P = \(\frac{d_{2}}{2}\) sin \(\alpha\)  and through them the reduced radius of curvature  
\[\rho_{red}=\frac{2\,d_{1}d_{2}(\sin\alpha)^{2}}{4\,\left(d_{2}\pm d_{1}\right) \sin\alpha}=\frac{d_{2}\,\sin\alpha}{2\,\left(u\,\pm 1\right)}=\frac{d_{1}\,u \sin\alpha}{2\,\left(u\,\pm 1\right)} \tag{12}\]  
here the sign \"plus\" is for external, \"minus\" for internal gearing. Figure 5: Calculation scheme for calculation of wheel teeth surface on contact strength of the gear wheel in the wheel-motor block (WMB) of electric locomotive VL-80s. The specific load on the contacting teeth can be calculated as  
\[q_{n}=\frac{F_{n}}{\ell_{\Sigma}}=\frac{F_{n}}{be_{a}}=\frac{F_{t}}{be_{a}\text{ cos}\,\alpha} \tag{13}\]  
where \(\ell_{\Sigma}=be_{a}\)--total length of contact lines at the boundary of one- and two-pair gearing.