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The article is devoted to the analysis of the state of the contact surfaces of the higher kinematic pair in the general case of relative motion, that is, in the presence of rolling, sliding, and twisting, which is characteristic of Novikov’s circular-screw gears. The purpose of the work is to assess the impact of friction forces, the state of contact surfaces after tool processing, and the localization of the instantaneous contact spot on the level of contact—fatigue durability of gears. Power contact in the presence of geometric slippage of the mating surfaces leads to a significant change in the initial geometry and the mechanical properties of surface layers. In the existing methods of calculations of contact strength, the effect of running-in is investigated insufficiently, which leads to an incorrect result, especially for gear with high hardness of the teeth. In this work, the conditions of contact interaction close to the real requirements are studied on the basis of experimental material, numerical solution of the contact problem, determination of the terms of the contact areas of slip, and adhesion within the instantaneous spot. The shape of the instant contact spot has asymmetry and can be approximated by an ellipse with the introduction of a correction factor. The running-in period is of a plastic nature with cold deformation and reduction of the roughness of surfaces. As a result of the run-in period, the area of actual contact (tooth height) is increased by 2 or more times. It is not desirable to spread the area of contact at the area of adhesion that initiates the formation of pitting. The presence of defective surface area on the level of contact strength does not have significant influence, because of the running-in period, but increases the risk of spalling and brittle fracture.

Running-in is the process of transition of the properties of the contacting surfaces of the parts from the initial to the operational. There is a change in size and macro - and microgeometry, as well as physical and mechanical properties of the material of the interacting surfaces to optimize their parameters in a relatively short period. For Novikov's transmissions, this process is of particular importance, because due to its initial point contact passes into contact on the surface, which is due to a sharp increase in the contact strength of such transmissions. However, this is a qualitative indicator. The burnishing process is influenced by many factors—geometric, kinematic, power, lubricant properties, and contact surfaces. It is clear that instantaneous contact spots, both calculated and determined experimentally, differ significantly (Figure

The spots of contact in Novikov’s gear: (a) unprocessed single gear line (SGL), model on Plexiglas [

The quasielliptic instantaneous contact spot (IPC) obtained from the initial geometry by the numerical solution of the contact problem [_{0}= 2013 MPa chordal arc length of the active portion of the profile of the head of the tooth L (Figure

Changing the chordal length of the arc_{0}: after running-in at the moment on the output shaft T_{2}=1000 Nm;_{1}: as a result of running-in at the time of failure—a breakdown on the tooth of the tooth fillet at T_{2} = 3550 Nm with the operating time of 7.2⋅10^{6} loading cycles; _{α}: radius of curvature of the tooth; _{к}: pressure angle at the theoretical contact point; _{ρ}: the minimum angle of the profile of the active part of the tooth.

However, if the results of the study are evident, its mechanism is a subject of discussion. Thus, confirmed by many subsequent studies, for example [

Running-in process experiments were conducted on a pin-on-disk tester in [

Running-in attractor was investigated as a stable and time-space ordered structure formed in running-in process in [

The dual-disk model testing concept, shown in [

A model for the simulation of wear particles formation and running-in in mixed lubricated sliding contacts was developed in [

The effect of running-in on surface characteristics of spur gears and on their development during subsequent efficiency testing is studied in [

The conditions of running-in and wear are not exactly identical but interrelated [

Experimental studies [

The general case of relative motion—the combination of rolling, slippage, and rotation—is characteristic of Novikov gearing. The instant contact area is different from the ellipse. The effect of the cantilever application of the load to the tooth is obvious, but little has been studied. Therefore, the characteristics of the stress-strain state (SSS) in the contact area are determined numerically. That is justified in the research plan, but in the solution of applied engineering problems, it is advisable to enlist analytical models, which are corresponding to the practical problem only approximately, significantly simplify the solution, and enhance the universality of the results obtained. Hence, the assessment of the possibilities and conditions of using existing analytical solutions is a necessary stage in the development of engineering methods of calculation. The study of contact interaction provides the determination of the mode of lubricating action in the conditions of actual contact, determination of the characteristics of the contact area, and determination of the components of the stress-strain state (SSS) in the contact area.

Hardening thermal or chemical heat treatment (CHT) makes significant adjustments that affect the changes in the conditions of contact and the properties of the material of the contacting elements. The existing regulations of traditional involute transmissions are not applicable since the stress state in the contact area is different even for Novikov transmissions with different initial contours. This paper analyzes the results of the assessment of SSS in the contact area of gear Novikov-based solutions to problems of solid mechanics— analytical and numerical—and the results of bench testing carbonitriding transmission Novikov with the initial loop of SWG-5 (base for GOST30224-96) “Reducer" (Izhevsk).

Tests were carried out in single-stage gearboxes CU-160 on stands with a closed flow of power, mechanical loading, the number of revolutions of the drive shaft_{1}=1500 min^{−1}, and the circulating oil lubrication system MS-20. The transmissions with parameters were tested: center-to-center distance:_{w} = 160 mm; module:_{1} = 32, z_{2} = 65; angle of inclination of teeth: _{1} =_{2} = 0; width of the rim: b_{1} = 50 mm and b_{2} = 60 mm; material: steel with 0.25% C, 1% Cr, 1% Mn, and 0.2% Mo; CHT: nitrocarburizing with _{a} = 0.45 _{a} = (0.30... 0.35) _{1} =

The study of contact interaction provides the solution of the following tasks: determination of the mode of lubricating action in the conditions of actual contact; determination of the characteristics of the contact site—_{x1} and_{x2} are the curvature radii of the first and second surfaces in the section passing through the direction of rolling or sliding (Figure _{1}/ (1-^{2}) is reduced modulus of elasticity; _{1} +_{2})/2 is average speed of movement of the surfaces of parts;_{1} and_{2} are speed of movement of the first and second surfaces;_{0} is the maximum pressure in contact with the Hertz.

Scheme of the pressure distribution in the contact area of the cylindrical surfaces in elastohydrodynamic regime.

Center distance is_{w}= 160 mm; module is_{1}= 32 and_{2}= 65; angle of inclination of the teeth is _{1}=_{2}= 0; the width of the bases is_{1}= 50 mm_{2}= 60 mm; material is steel with C=0.25%, Si=0.17-0.37%, Mn=0.9-1.2%, Cr=0.9-1.2%, Ni<0.3%, Cu<0.3%, and Mo=0.2-0.3%; chemical-thermal treatment nitrocarburizing is c _{1} = 1500-2000 Nm; mode: 100%; drive shaft speed:_{1} = 1500 min^{−1}; lubrication with MS-20 oil.

Tested gear is as follows: _{1}= _{2}/_{1} is the gear ratio;_{t} is the pressure angle and the distance from the pole to the theoretical contact point in the end plane:^{−1}. In this case (even without taking into account a multiplying factor X) the minimum value of the thickness of the layer

At the same time, similar calculations were carried out in accordance with the works of Hamrok-Dawson [

When the elastic characteristics of the materials of interacting bodies are equal, the jump of normal displacements determining the contact area depends only on the normal load [

In the orthogonal coordinate system OX_{1}X_{2}X_{3}, where X_{3}=0 is the common tangent plane,

The solution obtained in the previous problem—the definition of the contact area and the density of the normal load [_{1}, x_{2})]—allows us to calculate the components of the stress state, in which under the simultaneous action of the normal_{1},x_{2}) and tangent t_{x1}(_{1},_{2}) loads can be, based on the solution of Boussinesq-Cherutti, presented as

Tangential loads are caused by friction forces, which are generally determined by the kinematics of the interacting bodies as absolutely rigid, as well as by elastic deformations that violate the “ideal” kinematics of rolling. Thus, the instantaneous contact spot can represent either the area of total slip_{+} or a combination of slip and clutch zones_{0} with zero relative slip speeds, i.e., _{+}.

The solution of the problem of determining the tangential loads in the contact of a higher kinematic pair (in particular, with respect to Novikov transmissions) was carried out on the basis of the variational approach, which formulated it as the problem of minimizing the Kalker [

The theoretical and experimental studies carried out showed the following:

Contact area (a) and pressure distribution (b) in Novikov’s cylindrical gear: 1: contact of real surfaces; 2: equivalent elliptical contact.

The distribution of tangential stresses over the contact surface in the Novikov transmission: (a) - X_{1} = X_{2} = 0: complete slip; (b) -X_{1} = -0.3; X_{2} = + 0.3: slippage and grip. X_{1} and X_{2}: the coefficients of displacement, respectively, gears, and wheels.

_{0} and E_{+}, zones of local concentration of these stresses appear (Figure

Metallurgical studies checked the quality of the near-surface area of the tested gears. Fractographic analysis, metallographic analysis of the core and hardened layer, measurement of microhardness from the surface to the core, and measurement of the roughness of the contact surface were carried out. The measurements were carried out both on the working side and on the nonworking side of the tooth.

The microstructure is quite stable (Figure

Microstructure of the reinforced layer and core (Sample 3.1). (a) near-surface zone; (b) effective area; (c) the core.

Despite the fact that all samples were made from the same batch steel and HTO passed simultaneously, the spread of hardness values reached 150 HV1 units within one tooth and 250 HV1 for the same points of different samples (Figure

Hardness distribution along the thickness of the hardened layer. 1-3: actual distribution of hardness of samples of one batch (3 samples); 4: probabilistic curve of the actual hardness distribution (for 14 samples with a probability of 90%); 5: distribution according to the approximation.

The destruction of teeth in the majority of cases had the character of fatigue kinks (Figure

Surface of failure (sample 3) and sample microslip.

The qualitative side of the process of running-in is obvious, but quantitative side is practically not studied. In [_{1} is determined by the parameters of the initial contour, the coefficient of displacement by the module, and the number of teeth of the transmission wheels formulas (11-10-11-12), table 17 of work [

Considering the running-in, the dependence for determination of the normal contact stress of the gear, with the original contours type GOST 30224-96 (with the center of curvature of the head of the tooth lying outside the pitch line), takes the form

where

Rx is the reduced longitudinal radius of curvature at the theoretical contact point.

Even the unhandled transmissions worked in a stable elastic hydrodynamic lubrication regime. The evaluation of the stress-strain state at the early stages of operation can be performed on the basis of the initial one, without taking into account the rheology of the lubricant, which greatly simplifies the calculations.

The paper data used to support the findings of this study are included within the article. There are no restrictions.

The authors declare that they have no conflicts of interest.

The work was financially supported by the Russian Foundation for Base Research (Project 18-01-00715-а).