Investigation of the E ect of Young’s Modulus on the Contact Strength of Metal Polymer Plain Bearings

The load carrying capacity (contact pressures) of hybrid (metal-polymer) plain bearings is calculated in the article according to the author’s research method, considering the diff erences of the Young’s modules of the bushing made of polymeric materials. Studies have been conducted for metal-polymer bearings with a bushing of two types of polymer composites: epoxy based antifriction composite materials DIAMANT Moglice (DIAMANT Metallplastic GmbH) and DK6 (PT) (fi llers – graphite, MoS2, Zr). The elastic constants of composite materials, in particular the Young’s modulus, have a noticeable eff ect on the contact pressures in metal-polymer bearings. The Young’s modulus is signifi cantly diff erent in these composite materials. In accordance with this factor, the infl uence of load, bushing diameter and radial clearance on the maximum contact pressures was studied. Quantitative and qualitative regularities of maximum contact pressures from the accepted factors of infl uence: Young’s modulus, loading, bushing diameter, radial clearance, type of bushing material are established. The obtained results are compared with the results obtained by known conventional calculating methods of contact pressures


INTRODUCTION
Plain bearings of various types are used in mechanical engineering and in other various fi elds due to a number of advantages over rolling bearings. In particular, these are: high load capacity, a wide range of shaft diameters, a very large range of rotational speeds, good susceptibility to dynamic and shock loads, small lateral dimensions compared to their diameter, low noise level; possibility of operation with dry friction, in water or aggressive media, etc.
Along with bearings made of metallic materials, various non-metallic materials (porous and cermet, carbographite, layered composites, polymers, fi lled composites on a polymer or other basis) are widely used for the bushings. The number of new types of composite materials that can be used in metal-polymer bearings is increasing.
A feature of metal-polymer (MP) bearings is a signifi cant diff erence in the strength characteristics of the shaft and bushing materials (8-10 times) and Young's modulus (40-100 times), which fundamentally aff ects the contact pressure and bearing capacity. The calculation methods [1][2][3][4][5][6]  bearings. Only in [8][9][10] the contact pressures in metal-polymer plain bearings were investigated by numerical methods. Also in [7] the analysis of tribological efficiency of polymeric composite materials -steel hybrid joint at their application in tribosystems of dry sliding friction is carried out. Although they do not take into account the aspect of the effect of the Young's modulus of polymer materials on contact pressures when designing MP plain bearings.
This article is devoted to the study of this issue. Known methods [11][12][13] for calculating metal plain bearings are modified for the study of MP bearings [14,15]. With their use, the calculation of the arising contact pressures was carried out taking into account the peculiarities of the contact interaction of a hybrid pair of materials with significantly different mechanical properties and elastic characteristics. It also effectively provides a qualitative and quantitative analysis of the arising maximum contact pressures and the establishment of regularities of their change from the influence of various factors. The influence of Young's modulus of polymeric materials on contact pressures is investigated in the article.

METHODS
The plain problem of the theory of elasticity about the internal contact of cylindrical bodies of close radii is to be studied. Below is a method for solving it. The bearing analytical model is shown in Figure 1b.
The bearing shaft is subjected to a static radial load F. In order to bring the 3D bearing assembly (Fig. 1a) to the 2D assembly (Fig. 1b), the total load is assigned to the length l of the bushing. That is, in the plane task reduced concentrated load N = F / l will act on disk 2 (unit width of the shaft journal). A small radial clearance ε = R 1 − R 2 ≥ 0 << R is provided between the shaft journal 2 and the bushing l to ensure bearing lubrication. Shaft 2 with a radius R 2 is made of metal (steel), and the bushing l with a radius R 1 is made of polymer composites. The composite bushing is located in housing 3. When the bearing is loaded with force N, unknown contact pressures occur in the contact area 2R 2 α 0 . They are supposed to be found as a result of solving the problem.
Substituting in (3) the expression for E 0 it is obtained that: The contact pressure depends on the radius of the shaft, the radial clearance, the contact angle and the elastic characteristics of the materials.
To determine the unknown semiangle of contact α 0 , the equilibrium condition of the forces applied to the shaft is used Given the expression for the collocation coefficient E 0 , condition (5) to determine the contact semiangle will take the form Accordingly, after the transformations Analysis of this dependence shows that the contact angle in the joint does not depend on the shaft radius, but on the load, radial clearance and elastic characteristics of the materials.
The analytical solution is simple and can be implemented by various software tools, including Microsoft Excel. That is, the given calculation method is easy to implement in engineering design calculations. The sequence of the solution is as follows: 1. According to Eg. (6) at the given values of parameters the contact semiangle α 0 is determined. Accordingly, the contact arc (area) will be 2R 2 α 0 . 2. According to Eg. (3) or (4), the maximum contact pressure is calculated.
In engineering practice in the design calculation of metal plain bearings as a criterion for their load carrying capacity the average pressure p is widely used. It is taken to be evenly distributed over the contact area 2R 2 l, ie From the analysis of Eg. (8) it follows that the formula lacks such parameters as the radial clearance ε = R 1 − R 2 > 0 and the elastic characteristics E, ν of the shaft and bushing materials. As mentioned above, these parameters have a decisive influence on both the contact pressure and the contact arc. In this conventional calculation method, the contact arc has a length of 2R 2 = D 2 corresponding to a constant contact angle of 2α 0 = 360° / π ≈ 114.6° = const. However, such a significant contact angle is achieved with significant loads and minimal radial clearances. This is known from the literature [1-2, 11-13], etc.] and the author's works [11][12][13], obtained by the methods of contact theory of elasticity.
A slightly different modified formula for determining the maximum contact pressure р max in a bearing has been proposed in [16] It is assumed that the contact angle 2α 0 = 180°, and the pressure is distributed according to the cosinusoidal law. The maximum pressure р max according to (9) will be 1.273 times higher than the average pressure p according to (8). It also does not take into account the elastic characteristics of materials. And at a contact angle 2α 0 = 180° the radial clearance is zero. Such initial assumptions in this technique do not correspond to reality, because in the plain bearing must be a certain radial clearance, because without it its reliable operation is impossible.
Therefore, the simplified (conventional) methods of calculating metal plain bearings according to the criterion of contact pressure p do not allow objectively assessing the actual pressure level. With regard to metal-polymer plain bearings, the use of these techniques is virtually unfounded given that they do not take into account the significant differences in the elastic characteristics of steel (shaft) and composite bushing, as indicated in the introduction. For this purpose, it is reasonable to use the above classical analytical method of the mechanics of contact of close radii cylindrical bodies with internal contact, where these simplifications are absent.  Figure 2 shows the dependence of the maximum contact pressures p(0) on the load N when changing the radial clearance e in the joint.

SOLUTIONS
There is an almost linear dependence of pressures p(0) on the load N = 5… 20 N/mm (F = 500, 750, 1000, 2000 N) at different radial clearances ε and the studied shaft diameters D 2 . When the load N in the specified range is increased 4 times, the maximum contact pressures p(0) increase 2 times regardless of the change in the values ε of the radial clearance ε and the shaft diameter D 2 . The diameter D 2 increase by 1.25 times causes a proportional decrease in pressure p(0). In a bearing with a Moglice composite bushing, the pressures will be about 1.3 times higher than in a bearing with a DK6 (PT) composite bushing. This is due to the greater rigidity of Moglice due to the higher value of Young's modulus.
The dependence of the maximum contact pressures p(0) on the radial clearance ε when the load N changes is shown in Figure 3.
Increasing the radial clearance ε leads to an almost linear increase in the maximum contact pressures p(0) at all loads N and shaft diameters D 2 . The doubling of the radial clearance ε leads to a √2 -fold increase in the pressure p(0) regardless of the change in the magnitude of the load N and the shaft diameter D 2 . Table 2 summarizes the maximum contact pressures р(0) М and р(0) DК . The relative increase p̃ of pressures р(0) М in relation to р(0) DК is indicated.
These results show that the ratio is almost unchanged (p̃ = const) with changes in load, radial clearance in the joint and shaft diameters. We also compared the maximum contact pressures p(0) with the pressures p and p max calculated by Eg. (8) and Eg. (9) correspondingly [14,15] in MP bearings with bushings made of Moglice, DK6, PA6+30%GF (polyamide filled with 30% dispersed fiberglass), PA6+30%CF (polyamide filled with 30% dispersed carbon fiber) composites. The following data are accepted for calculation: F = 2000 N; N = F/l = 20 N/mm, l = 100 mm; ε = 0.1 mm; D 2 = 40 mm. In addition to these metal-polymer bearings, a metal bearing with a pair of steel 45 -bronze (Sn -6%, Zn -6%, Pb -6%, Cu -82%) was also investigated. The results of the calculations are given in Table 3.
Accordingly, Fig. 4 shows the dependence of the maximum contact pressures in the investigated bearings with increasing Young's modulus of the bushing materials.
The data obtained by the given author's method show that the Young's modulus has a significant effect on the contact parameters. The Poisson's ratio may also have a slight effect. Instead,  according to simplifi ed (conventional) methods, the pressures do not depend on the combination of shaft and bushing materials, which, of course, does not correspond to reality, and, moreover, the regularities of linear contact mechanics of elastic deformed bodies. The load carrying capacity at the allowable contact pressure of the bearings with polymer bushings (Table 3) according to the conditional calculation is exhausted much later than it will be according to the above method according to the results of research.

CONCLUSIONS
Based on the results of the numerical solution, the maximum contact pressures in MP bearings with an epoxy composite bushing are determined. The quantitative and qualitative regularities of their change have been established, taking into account the following factors: Young's modulus, load, shaft diameter, radial clearance in bearings, type of composite materials.
An increase in the load N on the shaft by a factor of 4 leads to a decrease in p(0) by a factor of √4 = 2, and an increase in the shaft journal diameter D 2 by a factor of 1.5 leads to a decrease in p(0) by a factor of 1.5 √1.5. The established ratios are independent of radial clearance and bushing material. The contact pressure p(0) will increase in proportion to the increase in the radial clearance ε. Since the Moglice composite has a higher Young's modulus than the DK6 (PT) composite, the ratio of maximum pressures p(0) in bearings with such bushing materials is approxi-  (Table  3): with a bushing made of Moglice (Е М = 11200 MPa) in 8.7 times, DK6 (PT) (Е DK = 6500 MPa) in 6.9 times, PA6+30%СF (Е CF = 5200 MPa) in 6.28 times, PA6+30%GF (Е GF = 3900 MPa) in 5.46 times. In the case of a metal bearing with a bronze bushing (Е Br = 114000 MPa), this ratio is very large (23.3 times). is the allowable value of contact pressure of less durable material (reference parameter).