Mathematical Modeling of Load Capacity and Durability of Metal-Polymeric Bearings with a Composite Bushing Based on Polyamides, Polytetrafluoroethylenes, Polyetheretherketones, or Polyethylene Terephthalates

Abstract

Predictive assessments of the performance characteristics of metal-polymer (MP) plain bearings at the design stage are necessary and important for engineering practice. They include assessments of bearing capacity, linear wear of the bushing, and bearing life. However, no method for their calculation has been developed. The authors created a generalized analytical method for the mathematical modeling of MP sliding bearings as a classical science-based method to study the contact mechanics of conformal cylindrical bodies with consideration of polymer bushing wear. The contact problem of the theory of elasticity takes into account the significant difference in the elasticity characteristics of polymeric materials from those of steel: their Young’s modulus is 60–250 times smaller, and Poisson’s ratio is 1.27–1.37 times larger. The method was used to study MP bearings with bushings made from several common groups of tribopolymers: polyamides (PAs), polytetrafluoroethylenes (PTFEs), polyethylene terephthalates (PETs), and polyetheretherketones (PEEKs). The maximum contact pressures and durability of the dry friction bearings were calculated while taking into account the load; radial clearance; bushing thickness; the tribopolymers’ elasticity characteristics and wear resistance; the sliding friction coefficient; and the type of tribopolymer material. The Young’s modulus of a polymer material had a significant impact on the level of maximum contact pressures. Polymer fillers, depending on their type and the type of polymer, had very different effects on pressure changes (increase, no change, or decrease). An increase in pressure caused an increase in contact pressures. The durability of a mechanical bearing depended not only on the contact pressure but also on the polymer’s wear resistance, Young’s modulus, and friction coefficient. This study determined the quantitative and qualitative effects of these factors on the maximum contact pressures and bearing durability. The presented analytical method provides an effective and reasonable assessment of the specified service characteristics of sliding bearings with consideration of the multifactorial influence of the above factors.

1. Introduction

Metal-polymer (MP) plain bearings are widely used in a wide variety of human activities. They have proven to be a suitable and cost-effective replacement for metal bearings in many cases. Various types of tribological polymeric materials (unreinforced and reinforced) are used for the bushings of this type of sliding bearing for rotary (radial or axial) or reciprocating (gliding) motion. They operate reliably at sufficiently high contact pressures, high sliding speeds, under dynamic loads, in chemically active and excessively humid environments, in water, in a vacuum, with a boundary or dry friction, and do not require additional lubrication or maintenance. The areas of application of plain bearings in various types of equipment are extremely diverse, including the engineering (in machines, machine tools, the automotive industry, instrumentation, aircraft construction, etc.), agricultural, construction, road, rocket and space, military, robotic, technological (in food processing, the pharmaceutical industry, textiles, the pulp industry, woodworking, the chemical industry, etc.), service (in computers, offices, control and measuring instruments, household items, medical equipment, diagnostic equipment, etc.) fields, and much more. It should be noted that among the above-mentioned positive qualities of MP bearings, their ability to operate reliably under dry friction is of particular practical importance. In many cases, it is impossible to use lubrication in sliding friction units for various reasons.

Various crystalline thermoplastic, amorphous, and epoxy polymeric materials are used to manufacture bushings for mechanical bearings. To improve their strength and tribological characteristics, various types of fillers are widely used to form composites. Common ones include fiberglass (GF), carbon fiber (CF), molybdenum disulfide (MoS2), graphite (C), bronze powder, polytetrafluoroethylene (PTFE), aramid (AF), oil, etc. The most commonly used polymers for the sleeves of deep-groove bearings are polyamides, polyacetals (e.g., POM), and polytetrafluoroethylene. The less commonly used polymers are polyethylene terephthalates, polyetheretherketones, polyethylene (PE-HD), etc.

Despite the widespread use of sliding bearings, no relevant standards have been developed for the calculation of their bearing capacity. Effective analytical methods are also absent in scientific publications. The calculations [1,2,3] or numerical methods [4,5,6] used in the literature for studying metal plain bearings are not used in engineering practice to assess the bearing capacity of MP bearings. There has been no attempt to develop analytical methods for calculating the wear and durability of sliding bearings.

In [7], an FEM study was carried out on the mechanical behavior of a large-sized MP sliding bearing with a sleeve made of a phenolic-based composite filled with PTFE and reinforced with polyester fibers. A simplified 3D FEM model was used in [8] to study the effect of clearance on contact stresses and the kinematics of a polymer-composite sliding bearing. Article [9] was a numerical study using the adaptive method of modeling the wear of a bearing made of a laminated polymer composite but it did not provide estimate the durability. Here, local wear was modeled according to the Archard abrasive/adhesive wear law with a linear dependence of wear on contact pressure and sliding speed. However, Ref. [10] indicated that at friction coefficients above 0.3, which is typical for dry friction pairs, the dominant type of polymer wear is the frictional fatigue mechanism under the influence of specific friction forces. Adhesive wear can then be a secondary concomitant process. A sign of adhesive wear as the main process is the transfer of a thin adhesive film of the polymeric material to a metal counterbody.

In the literature, more attention is paid to experimental studies of mechanical bearings [11,12,13,14,15,16,17], for example, the works of [7,9]. Paper [11] reports on the tribological behavior of various bearings with polymeric bushings made of polyacetal (POM-Delrin), cast polyamide PA6, etc., under dry friction. Ref. [12] is devoted to the study and evaluation of the wear resistance of bearings based on PE (polyethylene), PA, POM, PTFE, and bakelite polymers. The effect of the sliding speed, pressure, and temperature on the friction and wear of PA66 bearings and PA66+18PTFE and PA66+20GF+25PTFE composites was studied at room temperature in [13]. A tribological analysis of the dry friction behavior of PA6 in a connecting rod plain bearing was performed in [14]. Study [15] evaluated polymeric plain bearings with PET and UHMWPE bushings and an Inconel stainless steel shaft under dry friction and with water lubrication. The effect of radial clearance on the coefficient of friction, mass wear, and temperature of plain bearings with a sleeve made of the NORDEN Marine 605 composite and a shaft made of AISI 316 steel was studied in [16]. Two modes of friction in the bearings were used—dry and with solid lubrication (PTFE). In [17], the wear of a sliding bearing with PEEK bushings and PEEK+30CF, PEEK+30GF, and PEEK+10CF+10C+10PTFE composites was studied.

The authors have solved the problem of calculating the maximum contact pressures, wear, and durability of mechanical bearings by reasonably modifying the developed method of the elasticity theory for the study of metal bearings [18,19,20,21,22]. The developed method has been presented in [23], and the results of studies using it are presented in [24,25,26,27,28]. Below are the results for MP bearings with bushings made of the polyamides PA6, PA6+30GF, PA6+30CF, PA6+MoS2, and PA6+oil; polytetrafluoroethylenes PTFE+10C+10CF and PTFE+15C+5MoS2; polyethylene terephthalates PET and PET+20PTFE; and polyetheretherketones PEEK and PEEK+30GF.

2. Generalized Method for Calculation of Metal-Polymer Bearings

A plain bearing (Figure 1) consisting of a polymer sleeve 1, a steel shaft 2, and a housing 3 was considered. Shaft 2 is loaded with a static load N and rotates with an angular velocity ω = const. The diagram of the bearing (Figure 1b) shows the force reduced to the unit width $l$ of bushing 1: $F=N/l$. When the bearing is loaded, an unknown contact pressure $p_{\alpha }$ occurs, which is distributed along an unknown arc of contact $2R_2{\alpha }_0.$ The maximum contact pressure $p_{max}$ occurs between the shaft and the sleeve along the action line of the force F. A radial clearance $\epsilon =R_1-R_2>0$ must be guaranteed between the polymer sleeve 1 with inner radius $R_1$ and shaft radius $R_2$. The polymer sleeve has a certain thickness S, which is selected structurally.

It is known that the mechanical properties of polymeric materials for sleeves of sliding bearings have significantly lower properties than structural steel for shafts: an 8–10 times lower tensile strength, 60–260 times lower Young’s modulus, and 50–150 times lower wear resistance. Consequently, at the limit friction of such bearings, their load-bearing capacity, wear resistance, and durability will also be significantly lower than those of plain bearings with metal sleeves. Metal bearings, on the other hand, cannot operate under dry friction, in water, or in process liquids. Therefore, MP bearings are indispensable under these conditions, which leads to their widespread use within a reasonable limitation of bearing capacity based on the strength properties of the tribopolymers. However, except for the authors’ method, there are no such methods available for use at the design stage.

The generalized analytical method for studying MP sliding bearings includes the following partial methods for calculation:

• −. Contact pressures $p_{\alpha }$ and the area of their distribution, that is, the initial contact angle $2{\alpha }_0$;

• −. Maximum contact pressure $p_{max}$;

• −. Taking into account bushing wear to reduce contact pressures $p_{\alpha h}$ and increase the contact angle $2{\alpha }_{0h}$;

• −. Taking into account the wear of the bushing to reduce the initial maximum contact pressure;

• −. Durability t of the MP bearing at the accepted permissible linear wear h₁ of the polymer sleeve;

• −. Maximum linear wear h_1max of the bushing at a given bearing life t.

These methods are described in detail in the authors’ previous publications [18,19,20,21,22]. Below are the basic formulas for calculating the initial maximum contact pressure $p_{max}$ in the bearing and its durability at a given wear $h_1=h_{1max}$ of the polymer bushing. Here, a modified method was used to solve the contact problem with bushing wear, which takes into account the effect of the bushing thickness S on the contact parameters.

The maximum contact pressure $p_{max}$ occurs along the force line $F$ and is determined using the following formula:

(1)$p_{max}\approx E_{\lambda }\epsilon \tan \left({\displaystyle \frac{{\alpha }_0}{2}}\right),$

where ${\mathrm{E}}_{\lambda }\approx \left({\mathrm{e}}_{\lambda }/R_2\right){\mathit{cos}}^2\left({\alpha }_0/4\right),$ $e_{\lambda }=eL$, $e_4=4E_1{E}_2/Z$, $L=\left(1+{\lambda }^4\right)$, $\lambda =R_1/\left(R_1+S\right)$, $0<{\alpha }_0<{90}^o$, $Z=\left(1+{\kappa }_1\right)\left(1+{\nu }_1\right)E_2+\left(1+{\kappa }_2\right)\left(1+{\nu }_2\right)E_1$, and $\kappa =3-4\nu$. $E_k\mathrm{and}{\nu }_k$ are the Young’s modulus and Poisson’s ratio of the materials of shaft 2 and sleeve 1 of the bearing, and k is the numbering for the bearing elements (Figure 1).

The unknown angle of initial contact $2{\alpha }_0$ is determined from the equilibrium of the forces applied to shaft 2, i.e., the load F and the reaction to it in the form of contact pressure $p_{\alpha }$:

(2)$F=R_2{{\displaystyle \int }}_{-{\alpha }_0}^{{\alpha }_0}{p}_{\alpha }\mathit{cos}\alpha d\alpha =4\pi R_2{E}_{\lambda }\epsilon {\mathit{sin}}^2({\alpha }_0/4).$

Hence, taking into account the expression for $E_{\lambda }$, the formula for the semi-angle of the angle ${\alpha }_0$ will be

(3)${\alpha }_0=2arc\mathit{sin}\sqrt{F/\pi e_{\lambda }\epsilon }.$

The dependence of (1) on p_max and of (3) on α₀ can be used to determine these parameters in ball joints. For this purpose, instead of the radius R₂ of the shaft, the conventional radius $R^{*}=0.5\sqrt{R_1{R}_2}$ of the joint should be introduced, where R₁ = R₂ + ε is the radius of the ball-shaped skull and R₂ is the radius of the ball-shaped head. The solution of this problem was presented for a ball-shaped hip endoprosthesis in [29,30].

Based on the mathematical model of material wear under sliding friction [18,19] and the method for calculating contact pressures in a bearing [18,19,20,21,22], the durability of a sliding bearing at the maximum linear wear $h_1=h_{1max}$ of the bushing is calculated using the following formula [19,28]:

(4)$t={\displaystyle \frac{-B_1{\tau }_{10}^{m_1}}{v{c}_h{\tau }_h{\Sigma }_1\left(1-m_k\right)K_t^{\left(1\right)}}}\left\{{\left[\tau -{\tau }_{10}\right]}^{1-m_1}-{\left[\left(\tau -{\tau }_{10}\right)+c_h{h}_{1max}{\mathsf{\Sigma }}_1{\tau }_h\right]}^{1-m_1}\right\},$

where $B_k,m_k, {\tau }_{k0}$ are characteristics of the wear resistance of the materials in a hybrid tribopair under the accepted external research conditions, which are determined by the results of model triboexperimental studies [23,28] (Table 1, Table 2 and Table 3); $v=\omega R_2$ is the sliding speed; $c_h$ is the wear rate coefficient; ${\Sigma }_1=\left(-K_t^{\left(1\right)}+h_1^{\prime }\right)$; $K_t^{\left(1\right)}=1$ and $K_t^{\left(2\right)}={\alpha }_0/\pi$ are the coefficients of mutual overlap of the sleeve and shaft, respectively; $h_1^{\prime }=h_2/h_1$ is the relative linear wear of the shaft to the wear of the sleeve.

(5)$h_1^{\prime }={\displaystyle \frac{B_1{\tau }_{10}^{m_1}{\left(\tau -{\tau }_{20}\right)}^{m_2}}{B_2{\tau }_{20}^{m_2}{\left(\tau -{\tau }_{10}\right)}^{m_1}}}{K}_t^{\left(2\right)}$

The relationship between the specific friction force $\tau$ on a tribocontact and the contact pressure $p_{max}$ is taken according to the Amonton–Coulomb law:

(6)$\tau =f{p}_{max},$

where $f$ is the sliding friction coefficient.

Due to the wear of the bearing bushing and shaft, the pressure $p_{max}$ will decrease. In an MP bearing, the polymer bushing wears out 50–100 times faster than the steel shaft, and therefore the shaft’s wear can be neglected. The current pressure $p_{hmax}$ is determined as follows:

(7)$p_{hmax}=p_{max}-\left|p_h\right|,$

where $p_h$ is the change in the initial pressures when the bushing experiences wear in a mechanical bearing.

The value of $p_h$ is calculated according to the following dependence [19]:

(8)$p_h\approx E_{\lambda }{\mathrm{c}}_h{\epsilon }_h\tan \left({\displaystyle \frac{{\alpha }_{0h}}{2}}\right),$

where $c_h$ > 0 is the wear rate; ${\epsilon }_h=h_{1max}{\mathsf{\Sigma }}_1$; and ${\alpha }_{0h}>{\alpha }_0$ is an unknown contact angle when the bushing wears.

Therefore, during the wear process, the current contact pressure is determined by the formula

(9)$p_{hmax}\approx E_{\lambda }\left[\epsilon \tan \left({\displaystyle \frac{{\alpha }_0}{2}}\right)-c_h{h}_1{\mathsf{\Sigma }}_1\tan \left({\displaystyle \frac{{\alpha }_{0h}}{2}}\right)\right].$

When the polymer sleeve experiences wear, the current specific friction force is ${\tau }_{hmax}<\tau$. Accordingly, ${\tau }_{hmax}=f{p}_{hmax}$ and ${\tau }_h=f\left|p_h\right|.$

The contact angle ${\alpha }_{0h}$ during wear is calculated according to the condition

(10)$F=4\pi R_2{E}_{\lambda }\left(\epsilon +c_h{\epsilon }_h\right){\mathit{sin}}^2({\alpha }_{0h}/4)=4\pi e_{\lambda }{\mathit{cos}}^2({\alpha }_0/4)\left(\epsilon +c_h{\epsilon }_h\right){\mathit{sin}}^2({\alpha }_{0h}/4).$

3. Results

In the bushings of the MP sliding bearings, shaft 2 was made of normalized C45 steel (E₂ = 210,000 MPa, ν₂ = 0.3) and the tribopolymer materials of bushing 1 are shown in Table 1, Table 2 and Table 3. For the calculations, $N$ = 1500 and 3000 N; $F=N/l$; $l=D_2$; D₂ = 30 mm; $F=$ 50; 100 N/mm; $\epsilon =$ 0.15 mm; S = 2, 4, 6 mm; n₂ = 100 rpm; and h₁ = 1.0 mm.

The wear resistance characteristics $B_1,m_1,{\tau }_{10}$ of the polymeric materials presented in the tables above were determined based on the results of experimental studies under dry friction conditions using the method in [23,28,31]. For this, a pin-on-disk tribometer [28,31] under controlled conditions (temperature T = 23 ± 1 °C, relative humidity HR = 50 ± 5%) was used. The results from calculating the maximum contact pressures p_max are shown in Figure 2 and Figure 3.

The data on the durability t of the bearings are shown in Figure 4 and Figure 5. Here, the diagrams show their durability in black at a constant friction coefficient f = 0.35. However, experimental studies have shown that in each steel–polymer pair, the t value can differ significantly from the specified value (see Figure 6). In addition, the friction coefficient decreases with increasing contact pressure. Therefore, these circumstances were taken into account when calculating the actual durability t, which is shown in the color-coded diagrams below.

4. Discussion

The quantitative and qualitative results of the influence of the load, sleeve thickness, type of tribopolymer material, filler, and Young’s modulus on the value of maximum contact pressures p_max were obtained (Figure 2 and Figure 3).

Reducing the bushing thickness S from 6 to 2 mm resulted in a 1.126-fold increase in the p_max, regardless of the type of polymeric material used for the bushing. The bushing thickness S = 2 mm for a shaft diameter D₂ = 30 mm is indicated for the IGUS deep-groove bearings Iglidur J, Iglidur G, and Iglidur W300. The contact pressures in deep-groove bearings increased in the following order depending on the type of polymeric material used for the bushing: PTFE → PA6+MoS2 → PA6+oil → PA6 → PET+20PTFE → PET → PA6+30GF → PA6+30CF → PEEK+30GF → PEEK. The Young’s modulus of the polymers also increased in this order (Figure 7).

The fillers had a different effect on the change in contact pressures compared to the unfilled (basic) polymeric materials. Fiberglass and carbon fiber in the polyamide group caused the contact pressures to increase because they increased the Young’s modulus of the composite compared to that of PA6 (Table 1). The MoS2 filling reduced the pressures, while the oil filling did not change them. With the introduction of fillers into PA6, the change in the p_max increased up to 1.4 times. The filling of PTFE with polyetheretherketone or PET did not affect the p_max. In the case of filling polyethylene terephthalate and PEEK with glass fiber, the p_max slightly decreased (by a factor of 1.045).

The Young’s modulus E₂ had a significant effect on the maximum contact pressure. In polyamides, it increased by up to 2-fold (PA6+MoS2 → PA6+30CF), which led to a 1.4-fold change in the p_max. In the studied polymers, the p_max increased by 4.29 times (PTFE → PEEK+30GF) (Table 2 and Table 3). From the analysis of Figure 2 and Figure 3, it can be seen that in this case, the p_max increased by about 1.86 times (1.86–1.88 times at N = 1500 N, and 1.85–1.86 times at N = 3000 N).

According to the criterion of a minimal contact pressure (p_max), the bearings with a bushing made of PTFE+10C+10GF or PTFE+15C+5MoS2 are optimal compared to the other groups of polymers with a bushing made of PA6+MoS2, PET, or PEEK+30GF.

As mentioned above, there are no effective calculation methods in the literature for estimating contact pressures in MP plain bearings. On the other hand, in designing metal plain bearings, the bearing load-carrying capacity is calculated based on the average pressure p that is uniformly distributed over the contact surface 2R₂l, i.e.,

(11)$p={\displaystyle \frac{F}{2R_{2}l}}={\displaystyle \frac{N}{D_2}}\le \left[p\right]$

where [p] is the permissible contact pressure of the less durable material of the bearing elements. The significant disadvantage of this conventional method is that it does not take into account the radial clearance in the joint, which must be greater than zero for normal bearing operation. It also does not take into account the elastic characteristics E and ν of the shaft and sleeve materials. However, these parameters of the tribosystem and its materials have a decisive influence on the value of the contact parameters (maximum contact pressure and the contact zone). In the indicated conventional calculation, the contact arc has a constant length of 2α₀R₂ = D₂ and a correspondingly constant contact angle of 114.6°.

A modified formula for calculation p_max is also known [32].

(12)$p_{max}={\displaystyle \frac{4}{\pi }}{\displaystyle \frac{F}{D_{2}l}}={\displaystyle \frac{4}{\pi }}{\displaystyle \frac{N}{D_2}}$

It has the same disadvantages as the above method. In addition, a cosine law for the contact pressure distribution is assumed. Here, a cosine function of contact pressure distribution is adopted, rather than a uniform distribution as previously. Therefore, the contact angle $2{\alpha }_0$ must be 180° to ensure that the pressure is zero at the extreme contact points ${\alpha }_0=\pm 90°$. However, according to (12) $2{\alpha }_0$ = 145.9°, this condition is not fulfilled either.

From the above, it is clear that there is no objective assessment of the load-carrying capacity of metal plain bearings using these conventional (conventional) methods. In MP plain bearings, the Young’s modulus of steel (shaft) and polymer composites (bushing) differ very significantly (up to 250 times). Therefore, the use of these conventional methods to determine the resistance is unreasonable in this case. There are no examples of their use in the literature.

In [28], the authors made a comparative evaluation of the maximum contact pressure p_max according to the developed method using pressures p acc. (11) and p_max acc. (12). The following data were used for the calculations: N = 2000 N; F = N/l = 20 N/mm, l = 100 mm; ε = 0.1 mm; and D₂ = 40 mm. The results of this evaluation are presented in Table 4.

The analysis of the results presented in Table 4 indicated a significant difference between the contact pressure p_max using the authors’ method of calculating the contact mechanics and the results for p and p_max using the conventional simplified methods for MP sliding bearings (4.23–6.28 times) and metal (18.05, 23.0 times) sliding bearings. The load capacity, according to the permissible pressure [p] in MP bearings using conventional calculation methods, is exhausted much later than that when using the authors’ method.

A comparative evaluation of the contact pressures p_max in the MP bearing with a PA6 sleeve that was calculated in [6] (for data: N = 100N, D₂ = 19 mm, l = 10 mm, S = 6.25 mm, e = 0.25 mm) using the FEM method and those calculated with the method described above was carried out. Accordingly, the p_max values were 2.8 MPa and 2.95 MPa. This confirms the correctness of the solution using the authors’ method (the discrepancy in the results is 5.9%). Thus, p = 0.53 MPa according to Formula (11), and p_max/p = 5.28.

Using the data [8] (N = 56,000 N; D₂ = 300 mm; l = 60 mm; S = 25 mm; and ε = 0.45, 0.85, 1.25, 1.65, 1.85, and 2.05 mm) from a phenolic composite-filled bushing, calculations of the maximum contact pressures p_max were carried out for a large-scale MP plain bearing. The pressures (MPa) at the indicated radial clearance values were ε = 0.45 (6.6/7.9—< 19.7%), 0.85 (9.0/9.6—< 6.7%), 1.25 (10.8/10.9—< 0.9%), 1.65 (12.4/12.1—> 2.4%), 1.85 (13.15/12.5—> 4.9%), and 2.05 (13.8/12.8—> 7.8%). The first number is the pressure according to the presented method, the second is the pressure according to the FEM in the indicated work, and the third is the convergence of the pressures. The convergence of the results depends on the amount of radial clearance. On the other hand, the mean pressures p, calculated using (11), reached a value of 3.1 MPa. This indicates a lack of reliability of the assessment of pressures using the conventional method.

The following quantitative and qualitative effects of the load; sleeve thickness; types of tribopolymer and filler materials; Young’s modulus; and wear resistance indices B₁, m₁, and t₁₀ of the sleeve tribopolymers on the durability t of the bearings were determined (Figure 4 and Figure 5).

When the bushing thickness S decreased from 6 to 2 mm, the bearing durability t decreased by 1.14 times for all the types of polymeric materials. The bearing with a PET+20PTFE bushing had the longest durability, and the bearing with a PA6+30GF bushing had the shortest durability. The ratio of their durability values was 5.33 at N = 1500 N and 5.7 at N = 3000 N. Bearings with a bushing made of PA6+MoS2, PA6+oil, and PET had approximately the same durability at N = 1500 N. At a load of N = 3000 N, the durability was the same for bushings made of PA6+MoS2 and PA6+oil and slightly higher than that of PET. The lowest durability of the bearings made of PA6, PA6+30GF, and PEEK at N = 1500 N were similar, and at N = 3000 N, the bearings made of PA6+30GF and PEEK had the lowest durability. The durability of the bearings with PA6+30CF and PEEK+30GF bushings were somewhat higher.

Depending on the type of polymeric material used for the bushing, the durability t of the MP bearings increased in the following order: PA6+30GF → PEEK → PA6 → PEEK+30GF → PA6+30CF → PA6+oil → PA6+MoS2 → PET → PTFE+10C+10GF → PTFE+15C+5MoS2 → PET+20PTFE. The wear resistance of the tribopolymers and the level of specific friction force $\tau =f{p}_{max}$ at the tribocontact (see Equation (4)) [19], rather than the contact pressures p_max, had a decisive influence on the durability. In the authors’ methodology for testing the kinetics of material wear under sliding friction [19], it is assumed that the material wear rate is functionally related to the level of the friction force.

The increase in the linear wear resistance characteristic B₁ of the polymeric materials was in the following order: PA6 → PEEK → PA6+30GF → PA6+MoS2 → PEEK+30GF → PA6+30CF → PET → PA6+oil → PET+20PTFE → PTFE (Table 1, Table 2 and Table 3). On the other hand, the trend in the change in the unit value of the frictional force in the tested MP bearings with different polymers was different: PA6+MoS2 → PTFE+15C+5MoS2 → PA6+oil → PTFE+10C+10GF → PET → PA6 → PET+20PTFE → PA6+30CF → PEEK → PEEK+30GF → PA6+30GF. At the same time, the trend in the changes in the value of the specific friction force $\tau$ in the studied MP bearings with different polymers was also different: PA6+MoS2 → PTFE+15C+5MoS2 → PA6+oil → PTFE+10C+10GF → PET → PA6 → PET+20PTFE → PA6+30CF → PEEK → PEEK+30GF → PA6+30GF. Therefore, this order of increasing service life t of the MP bearings differs significantly from the above sequence of increased contact pressures p_max: PTFE → RA6+MoS2 → PA6+oil → PA6 → PET+20PTFE → PET → PA6+30GF → PA6+30CF → PEEK+30GF → PEEK (Figure 2 and Figure 3).

According to the calculations carried out using the criterion of the calculated maximum life t_max, the best MP bearings in each of the tested tribopolymer groups were PA6+MoS2 (polyamides), PTFE+15C+5MoS2 (polytetrafluoroethylenes), PET+20PTFE (polyethylene terephthalates), and PEEK+30GF (polyetheretherketones). There are no computational methods similar to the one presented in the literature.

A comparison of the pattern of change in p_max in the bearings with different polymer bushings and the durability t of the bearings is shown in Figure 6. Here, it is easy to track the patterns of change in the contact pressure and service life of the bearings with different bushings.

The Young’s modulus E₁ (Figure 7) and friction coefficients f in these hybrid tribo-conjugations of the polymers (Figure 8) have different sequences of change from the minimum to the maximum value. Therefore, the calculated durability t of a bearing depends on the complex effect of the maximum contact pressure p_max, the wear resistance characteristics of polymeric materials ($B_1,m_1,\mathrm{and} {\tau }_{10}$), Young’s modulus E₁, and friction coefficient f. All of these factors are included in Formula (4) for calculating the bearing’s mechanical life.

According to the criterion of maximum durability t, bearings with a PET+20PTFE bushing are optimal compared to the other groups of polymers with a bushing made of PA6+MoS2 or PA6+oil, PTFE+15C+5MoS2, PET+20PTFE, or PEEK+30GF.

The elastic Young’s moduli of the studied polymeric materials, determined using the Oliver–Farr method, are shown in Figure 7.

The Young’s modulus of the investigated polymeric materials differed by a maximum of 4.29 times (PEEK+30GF/PTFE+10C+10GF). In the group of polyamides, the filling with CF and GF significantly increased the Young’s modulus compared to the base PA6 (Table 1). Filling PA6 with oil reduced it, and MoS2 did not change it. In the polyetheretherketone, a PTFE filling had a slight reducing effect on the modulus, and in polyethylene terephthalate, a GF filling slightly increased it (Table 2 and Table 3). In all the groups of polymers, fillers changed the Poisson’s ratio slightly compared to the base materials (Table 1 and Table 3).

Figure 8 shows the experimentally determined sliding friction coefficients necessary to evaluate the life of MP bearings at the calculated contact pressures p_max (Figure 2 and Figure 3).

The dry friction coefficients depend not only on the type of base polymer but also on the type of filler. The introduction of fiberglass into the base polymer increases the coefficient of friction of composites under dry friction on steel 45. Instead, the fillers with MoS2, oil, and PTFE as lubricants have little effect on it. The increase in load and contact pressures causes a decrease in the coefficient of friction. This effect is reported in the literature. According to the well-known and thoroughly experimentally verified molecular-mechanical theory of sliding friction of rough bodies developed by Kragielski I., the total coefficient of friction is the sum of the adhesive (molecular) and deformational (mechanical) components. At low loads to the tribological system, the mechanical component of friction caused by the deformation of micro-irregularities of the contacting surfaces dominates. It decreases with increasing load, while the adhesion of the surface layers plays a greater role. Correspondingly, changing the role of both of these friction mechanisms causes a decrease in the coefficient of friction.

As stated above, the characteristics of the studied tribopolymer materials had different values and the hierarchy of their changes was different for each characteristic. The hierarchy of changes from the minimum to the maximum value for each characteristic (contact pressure p_max, friction coefficient f, unit friction force $\tau =f{p}_{max}$, Young’s modulus E₁, wear resistance characteristic B₁, and bearing life t) is presented below (Table 5).

5. Conclusions

Using a generalized analytical method, dry friction sliding bearings with a bushing made of one of four groups of polymeric materials for tribological purposes were studied. The influence of the following factors on the maximum contact pressure and durability of MP bearings was analyzed: load; radial clearance; bushing thickness; the tribopolymer’s elasticity characteristics and wear resistance; sliding friction coefficient; type of tribopolymer material; and type of filler. It was found that the Young’s modulus of the polymeric material of the bushing was, in addition to the load and geometric parameters of the bearing, the main factor that had a significant effect on the maximum contact pressure. Fillers made of base polymers, depending on the type, had different effects on the change in the p_max: in the PA composites, they increased the p_max (except for PA6 + oil); in the PET composite, they had no effect; and in the PEEK composite, they slightly decreased the p_max. Instead, the bearing’s durability was influenced by a combination of several factors: the contact pressure, the polymer’s wear resistance and Young’s modulus, and the coefficient of dry friction on steel. The fillers made of all types of bushing polymers (except for the PA6+30GF composite) always increased the bearing life. The quantitative and qualitative effects of these factors on the maximum contact pressures and bearing life were determined.

Since the contact problem in the theory of elasticity for mechanical bearings in the study of their bearing capacity and durability is presented in a closed form, it is easy to solve it using the appropriate computer programs. This method is therefore suitable for engineering applications in order to predict the indicated performance characteristics of MP plain bearings. The results of these tests can be used in the calculations for this type of bearing operating under dry friction conditions. Also, the developed method can be used to optimize a bearing at the design stage according to the criteria of minimal contact pressure or maximum durability. According to the above results, this task has been solved for the tested design data.

Future research will aim to investigate the maximum contact pressures and MP life of bearings with an extended range of shaft diameters, angular velocities, radial clearances in the bearing, radial loads, and tribopolymer elastic moduli. For example, the diameter of Iglidur polymer bushings for bearings manufactured by the well-known German company IGUS^® GmbH is in the range of 1 to 195 mm. Polymer materials such as IglidurX and IglidurTX1 are characterized by a high modulus of elasticity E and a permissible contact pressure [p]: E = 8100 and 12,000 MPa (DIN 53457), and [p] = 150 and 200 MPa, respectively.

Author Contributions

Conceptualization, M.C. and J.Z.; methodology, M.C.; validation, J.C. and J.Z.; formal analysis, M.C.; investigation, J.C.; writing—original draft preparation, M.C. and J.Z.; writing—review and editing, J.C. and J.Z.; funding acquisition, J.Z. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Metal-polymer plain bearing: (a) general view and (b) diagram.

Figure 2. Maximum contact pressures in MP bearing with different types of polymers (N = 1500 N).

Figure 3. Maximum contact pressures in MP bearing with different types of polymers (N = 3000 N).

Figure 4. Bearing durability at N = 1500 N: black colour—S = 6, 4, 2 mm, f = 0.35 = const; light green—S = 6 mm, blue—S = 4 mm, red—S = 2 mm, f ≠ 0.35.

Figure 5. Bearing durability at N = 3000 N: black colour—S = 6, 4, 2 mm, f = 0.35 = const; light green—S = 6 mm, blue—S = 4 mm, red—S = 2 mm, f ≠ 0.35.

Figure 6. Contact pressures (a) and durability (b) of deep-groove bearings: black colour—S = 6, 4, 2 mm, f = 0.35 = const; light green—S = 6 mm, blue—S = 4 mm, red—S = 2 mm, f ≠ 0.35.

Figure 6. Contact pressures (a) and durability (b) of deep-groove bearings: black colour—S = 6, 4, 2 mm, f = 0.35 = const; light green—S = 6 mm, blue—S = 4 mm, red—S = 2 mm, f ≠ 0.35.

Figure 7. Modulus of elasticity of tribopolymers.

Figure 8. Sliding friction coefficients.

Table 1

Characteristics of polyamides.

Characteristic	Polyamides
	PA6	PA6+30GF	PA6+MoS2	PA6+30CF	PA6+oil
B1 × 1010	2.26	4.12	5.58	6.53	7.03
m 1	1.09	1.09	1.1	1.1	1.1
τ10, MPa	0.05	0.05	0.05	0.05	0.05
Young’s modulus E1, MPa	2000	2700	1660	3300	1960
Poisson’s ratio ν1	0.4	0.41	0.4	0.41	0.4

Table 2

Characteristics of polytetrafluoroethylenes.

Characteristic	Polytetrafluoroethylenes
	PTFE+10C+10CF	PTFE+15C+5MoS2
B1 × 1010	8.2	8.2
m 1	1.1	1.1
τ10, MPa	0.05	0.05
E1, MPa	850	850
ν 1	0.46	0.46

Table 3

Characteristics of polyethylene terephthalates and polyetheretherketones.

Characteristic	Polyethylene Terephthalates and Polyetheretherketones
	PET	PET+20PTFE	PEEK	PEEK+30GF
B1 × 1010	6.81	11.45	3.76	5.95
m 1	1.09	1.09	1.09	1.09
τ10, MPa	0.05	0.05	0.05	0.05
E1, MPa	2300	2260	3400	3650
ν 1	0.43	0.44	0.38	0.4

Table 4

Contact parameters comparison.

Shaft–Bushing	Calculated by
	Authors’ Method	Method from (11)	Method from (12)
Steel − PA6+30GF[p] = 11 MPa	2.73/26.9	0.5/114.6 (5.46 times)	0.636/145.9(4.29 times)
Steel − PA6+30CF[p] = 11 MPa	3.14/23.4	0.5/114.6(6.28 times)	0.636/145.9(4.94 times)
Steel−Bronze[p] = 50 MPa	11.66/6.3	0.5/114.6(23.3 times)	0.636/145.9(18.05 times)

Note: PA6+30GF − E_GF = 3900 MPa, $\nu$_GF = 0.42; PA6+30CF − E_CF = 5200 MPa, $\nu$_CF = 0.42; E_Br = 114,000 MPa, ν_Br = 0.34. PA6+30GF and PA6+30CF composites filled with dispersed fibers were dried. Numerator is contact pressure (MPa) and denominator is contact angle (degrees).

Table 5

Hierarchy of tribopolymer materials.

Factor	Tribopolymers
	PA6	PA6+30GF	PA6+MoS2	PA6+30CF	PA6+oil	PTFE+10C+10CF	PTFE+15C+5MoS2	PET	PET+20PTFE	PEEK	PEEK+30GF
pmax, MPa	5	8	3	9	4	1; 2	1; 2	7	6	11	10
f	3; 4	11	2	6	5	9	7	1	3; 4	8	10
τ, MPa	6	11	1	8	3	4	2	5	7	9	10
E1, MPa	5	8	3	9	4	1; 2	1; 2	7	6	10	11
B1·1010	1	3	4	6	8	10,11	10; 11	7	9	2	5
t, hours	3	1	6; 7	5	6; 7	9	10	8	11	2	4

Note: The tribopolymers (Table 1, Table 2 and Table 3) were assigned the following numbers: 1—PA6; 2—PA6+30GF; 3—PA6+MoS2; 4—PA6+30CF; 5—PA6+oil; 6—PTFE+10C+10CF; 7—PTFE+15C+5MoS2; 8—PET; 9—PET+20PTFE; 10—PEEK; 11—PEEK+30GF.

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