Study of the Influence of Temperature on Contact Pressures and Resource of Metal-Polymer Plain Bearings with Filled Polyamide PA6 Bushing

Abstract

It is known that the elastic characteristics of polyamides change with increasing temperature; in particular, the Young’s modulus decreases significantly. This fact is practically not taken into account in design calculations of metal-polymer plain (MP) bearings, operating under conditions of the boundary and dry friction. The purpose of the study was the analysis of the effect of temperature on the change of the Young’s modulus and, accordingly, the contact strength and triboresource according to the developed method of calculating MP bearings. MP bearings with a bushing made of polyamide PA6 reinforced with glass or carbon-dispersed fibers were investigated. Quantitative and qualitative regularities of change of the maximum contact pressures and resource of the bearings at temperature increase under conditions of boundary and dry friction are established. The pressures in the bearing bushing made of PA6 + 30GF will be lower than for the bushing made of PA6 + 30CF. The resource of the bushing made of PA6 + 30CF will be significantly greater than for PA6 + 30GF. For thermoplastic polymers, the increase in temperature will have a useful practical effect due to the decrease in the rigidity of the polymer composites of the bearing bushing.

1. Introduction

Metal-polymer (MP) plain bearings, which have been known since the 1930s, are quite common in modern conditions due to their characteristic positive qualities. These include sufficiently large load capacity, the possibility of use in dry friction and in a variety of special technological and operational conditions and under shock loads, small axial dimensions in comparison with their diameter, low noise level, damping ability, etc. They are able to operate in a wide range of temperatures, both low and high.

Various types of thermoplastic polymers and composites based on them are used as the material of MP bearing bushings, in particular, polyamides PA6, PA46, and PA66. To increase the strength and wear resistance of polymers, fillers of various types and structures (glass and carbon fiber, bronze powder, molybdenum disulfide, graphite, PTFE, polyethylene, and even semiconductor materials) are used [1,2,3,4,5,6,7]. As a result of modification by one or more fillers, significantly higher wear resistance, reduction of the sliding friction coefficient, and increase of strength characteristics in comparison with basic polymers are achieved. It should be noted that these performance characteristics of both unmodified and modified polymers are influenced by temperature. In addition, the temperature affects the elastic characteristics of polymeric materials. It is known that as the temperature decreases, the polymer materials become harder due to an increase in the Young’s modulus and a decrease in the Poisson’s ratio. As it increases, the Young’s modulus will decrease and the Poisson’s ratio will increase, i.e., the polymers become more deformable. However, despite the rather large area of distribution and scope of MP bearing, the question of the impact of changes in these elastic characteristics of polymer composites, in particular polyamide, under the influence of temperature on their life and the bearing capacity remains poorly understood. It is known [1] that the temperature significantly affects the decrease in the Young’s modulus of polyamides.

A priori, it can be argued that with decreasing Young’s modulus and increasing Poisson’s ratio due to increasing temperature, the contact pressures in the MP bearing will be lower because these mechanical characteristics of the polymer bushing are in the expressions for their calculation in contact mechanics. It is logical to assume that the MP bearing life will increase at a temperature higher than normal. However, this type of research for MP bearings is absent in the literature as well as the corresponding effective analytical methods for their implementation. It should be noted that the computational [8,9,10,11,12,13,14,15] or numerical methods for the study of metal plain bearings [1,16,17] have not been used in the calculation of metal-polymer plain bearings. In numerical methods of research of metal-polymer plain bearings [18,19,20], the estimation of contact pressures is carried out; however, there is no possibility of their triboresource determination.

It should be noted that in the experimental study of MP bearings, in addition to the analysis of polymer bushing wear and sliding friction coefficient, some authors also assess the temperature in the bearing and its effect on these tribotechnical characteristics of various polymer composites [2,3,4,5], etc., in particular, PA6-based composites [2,3].

In [2], the tribological behavior of different bearings with polymer bushings was studied in dry friction conditions, where it was shown that friction and wear depend on speed, load, temperature, and duration of operation. In [3], the influence of sliding speeds, pressure, and temperature on friction and wear of metal-polymer bearings made of PA66, PA66 + 18% PTFE, and PA66 + 20% GF + 25% PTFE composites was described. The article [4] was devoted to the study of the influence of the radial clearance of composite plain bearings on the coefficient of friction, temperature, and wear. In [5], the wear under dry friction of bearings with bushings made of modified PEEK + 30% wt. GF, +10% wt. CF, graphite, and carbon fiber was investigated. The tribological properties of three different, highly efficient polymers and their composites were studied at different temperatures in [6].

Since the problem of estimating the influence of the temperature factor on the bearing capacity and triboresource of MP bearings is of great practical importance, the developed analytical method [21,22,23,24,25] for studying the contact strength and frictional stability of this type of sliding tribosystems was used. It is based on the methodology of research of wear kinetics at sliding friction and the authors’ methods of calculation of sliding bearings with metal elements [22,25,26,27]. The article investigated MP bearings with bushings made of composites PA6 + 30%GF and PA6 + 30%CF, modified with glass and carbon fibers, correspondingly.

2. Materials and Methods

2.1. Influence of Temperature

The influence of temperature on the contact pressures and the resource of MP bearings is based on changes in the elastic characteristics of polymer materials. Such changes for the modulus of elasticity E and Poisson’s ratio μ of polyamide PA6 and PA6-based composites PA6 + 30%GF and PA6 + 30%CF at T = 25–55 °C and relative humidity of 50% are given in

Table 1. Influence of temperature on the modulus of elasticity and Poisson’s ratio.

Materials/Temperature, T °C	25	35	45	55	Decrease E, Times	Poisson’s Ratio 1, μ	Increase μ, Times
PA6	2700	2550	2400	2050	1.32	0.39/0.416	1.067
PA6 + 30%GF	3900	3750	3500	3000	1.3	0.42/0.438	1.043
PA6 + 30%CF	5400	5270	5150	4450	1.21	0.4/0.418	1.045

¹ Poisson’s ratio μ at 25 and 55 °C.

([1,28,29] and our own research results obtained by the Oliver–Farr method).

The operating temperature in the metal-polymer plain bearings should not exceed 60 °C, as this can lead to their thermal aging. The decrease in the modulus of elasticity, i.e., the decrease in rigidity, is more than 30% for PA6 and PA6 + 30%GF, while for PA6 + 30%CF this decrease is smaller. As for the increase in the Poisson’s ratio, i.e., the increase in plasticity, it is insignificant.

2.2. Calculation of Contact Pressures

To conduct a study on the calculation of maximum contact pressures, the authors’ method of contact mechanics [21,22,23,24,25] on the internal contact of close radius cylindrical bodies was used. This contact occurs in the plain bearing. The view and diagram of the MP bearing is shown in Figure 1.

A static load N acts on the shaft 2 of the MP bearing. Since the plane contact problem of the elasticity theory is considered, the total load N is reduced to a single F = N/l, where l is the length of the shaft journal. The bearing has a guaranteed initial radial clearance ε = R₁–R₂ ≥ 0. Accordingly, R₁ is the inner radius of the bushing 1 and R₂ is the radius of the shaft 2. In MP bearings, the shafts are metal and the bushings are non-metallic. It should be noted that the strength characteristics of the shaft and the bushing materials differ significantly (8–10 times) as well as their modulus of elasticity (40–100 times), which has a fundamental effect on the contact parameters. It is also known that in metal-polymer tribocouples, polymer composites have 3–4 orders of magnitude lower wear resistance than steel. The external load leads to the contact interaction of the shaft elements, resulting in the appearance of the contact pressure p(α), unknown by magnitude and distribution acting on the contact arc $2R_2{\alpha }_0$. That is, the task is to determine the specified contact parameters.

In the general equation for determining the contact pressure in the contact area [22,25], we used the following function:

$$p\left(\alpha \right)\approx E_0\epsilon \sqrt{{\mathit{tan}}^2\frac{{\alpha }_0}{2}-{\mathit{tan}}^2\frac{\alpha }{2}},$$

(1)

where $E_0=(e/R_2){\mathit{cos}}^2({\alpha }_0/4)$, $e=4E_1{E}_2/Z,$$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,$ α is the polar angle, $E_1, E_2$ is the modulus of elasticity, ${\nu }_1,{\nu }_2$ are Poisson’s ratios, and $\kappa =3-4\nu$.

The bearing capacity of the bearing is characterized by the level of maximum contact pressures p(0), and they occur at $\alpha =0$. That is,

$$p\left(0\right)\approx E_0\epsilon \mathit{tan}{\left({\alpha }_0/2\right)}_{},$$

(2)

The contact semiangle ${\alpha }_0$ is determined from the equilibrium condition of external load F and contact pressures p(α). Taking into account expression (2), it is:

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

(3)

When the shaft rotates at a constant angular velocity, ${\omega }_2$, the parameters of the static contact, i.e., the contact pressures p(α) and the contact angle, $2R_2{\alpha }_0$, will change due to wear of the composite bushing. Accordingly, there will be a decrease in p(α) and an increase in the contact area, $2R_2{\alpha }_0$. According to [22,23,24,25,26], wear contact pressures is

$$p\left(\alpha ,t,h\right)=p\left(\alpha \right)+p\left(h\right),$$

(4)

where p(h) is the change in initial pressures due to wear.

The law of change p(h) is chosen as follows

$$p(h)=E_h{\epsilon }_h\sqrt{{\mathit{tan}}^2\frac{{\alpha }_{0h}}{2}-{\mathit{tan}}^2\frac{\alpha }{2}},$$

(5)

where $E_h=c_h(e/R_2){\mathit{cos}}^2({\alpha }_{0h}/4)$ and $c_h$ > 0 is the wear rate indicator.

The tribocontact semiangle ${\alpha }_{0h}$ characterizing the wear zone during wear is determined by condition (6)

$$F=4\pi R_2{E}_0(\epsilon +c_{\alpha h}{\epsilon }_h){\mathit{sin}}^2({\alpha }_{0h}/4),$$

(6)

where ${\epsilon }_h=h_{k\mathit{max}}\left(-K_t^{\left(\mathrm{k}\right)}+{h^{\prime }}_k\right),$ ${h^{\prime }}_1=h_2/h_1$, ${h^{\prime }}_2=h_1/h_2$ is the relative wear in the tribosystem, $K_t^{\left(1\right)}=1,K_t^{\left(2\right)}={\alpha }_0/\pi$–coefficients of mutual overlap of bearing elements, $c_{\alpha h}$ is the indicator of the tribocontact angle growth rate, and $h_{k\mathit{max}}$ is the acceptable wear of elements 1 and 2 of the bearing:

$${h^{\prime }}_1=\frac{B_1{\tau }_{10}^{m_1}{\left(\tau -{\tau }_{20}\right)}^{m_2}}{B_2{\tau }_{20}^{m_2}{\left(\tau -{\tau }_{10}\right)}^{m_1}}{K}_t^{\left(2\right)},{h^{\prime }}_2=\frac{B_2{\tau }_{20}^{m_2}{\left(\tau -{\tau }_{10}\right)}^{m_1}}{B_1{\tau }_{10}^{m_1}{\left(\tau -{\tau }_{20}\right)}^{m_2}}{K}_t^{\left(1\right)},$$

(7)

where $\tau =fp\left(0\right)$ is the Coulomb specific friction force, $f$ is the sliding friction coefficient, $B_k, m_k, {\tau }_{k0}$ is the wear resistance characteristics of the tribocouple materials under the accepted external conditions of triboexperimental research, and k is the numbering of the bearing elements (Figure 1b).

2.3. Calculation of Resource

The calculation of the MP bearing service life when the bushing wears to $h_1=h_{k\mathit{max}}$ is carried out according to the formula:

$$t=\frac{-B_k{\tau }_{ko}^{m_k}}{v{c}_h\tau \left(h\right){\Sigma }_k\left(1-m_k\right)K_t^{\left(k\right)}}\left\{{\left[\tau -{\tau }_{k0}\right]}^{1-m_k}-{\left[\left(\tau -{\tau }_{k0}\right)+c_h{h}_{k\mathit{max}}{\Sigma }_k{\tau }_h\left(h\right)\right]}^{1-m_k}\right\},$$

(8)

where $v={\omega }_{2}R$ is the sliding speed, c_h is the coefficient of wear rate, ${\Sigma }_1=(-K_t^{\left(1\right)}+{h^{\prime }}_1),{\Sigma }_2=\left(K_t^{\left(2\right)}-{h^{\prime }}_2\right),$ and $\tau \left(h\right)=fp(0,t,h)=f{E}_h\mathit{tan}({\alpha }_{0h}/2)$ is the maximum specific friction force in tribocontact.

2.4. Conditional Calculation of Bearing Capacity

In modern mechanical engineering, in the design of metal plain bearings, there are two main criteria. The average pressure p (as its bearing capacity) and Zeiner’s pv are used. The average contact pressure p is considered to be evenly distributed over the contact area 2R₂l″

$$p=\frac{N}{2R_{2}l}=\frac{F}{D_2}\le \left[p\right],$$

(9)

where [p] is the allowable value of the contact pressure of less durable material (reference parameter).

From Formula (9), it follows that the contact arc of length 2R₂ = D₂ corresponds to the contact angle $2{\alpha }_0=360{}^{°}/\pi \approx$114.6° = const, which does not depend on the properties of the bearing materials. It should be noted that such a significant contact angle is achieved at significant bearing loads and minimal radial clearances. Accordingly, this is confirmed in the works available in the literature [1,8,9,10,11,12,13,14,15,16,17,18,19,20] and the authors’ works [22,23,24,25,26,27], based on the methods of contact elasticity theory. It is also known that in plain bearings the magnitude and distribution of contact pressures are significantly influenced by the radial clearance and the elastic characteristics of the shaft and bushing materials. Actually, these significant factors of influence are absent in Formula (9).

Additionally, also known is a modified formula [30] for determining the maximum pressure p_max in the bearing:

$$p_{{}_{\mathit{max}}}=\frac{4}{\pi }\frac{N}{D_{2}l}=\frac{4}{\pi }\frac{F}{D_2},$$

(10)

Here it is assumed that the pressure varies according to the law of cosine and the contact arc $2{\alpha }_0$= 180°. The structure of Formula (10) is the same as Formula (9). The maximum pressure p_max according to Formula (10) will be 1273 times higher than the average pressure p. It was also established [11] that at zero clearance in the cylindrical conjugation at static contact, $2{\alpha }_0$≈ 170°. However, there must be a certain radial clearance in the plain bearing because without it, its reliable operation is impossible.

That is, the simplified conditional methods for calculating plain bearings according to the criterion of contact pressure p do not allow objectively assessing the actual pressure level. Their use in the case of MP bearings seems unreasonable. For this purpose, it is reasonable to use the above, presented in Section 2.2, the classical analytical method of the contact mechanics of close radii cylindrical bodies with internal contact.

3. Solution, Results, and Discussion

Data for the calculation of MP bearings were N = 500, 1000, and 1500 N; F = N/l; D₂ = 30 mm; $l=D_2$; $\epsilon =$ 0.2 mm; n₂ = 60 rpm is the shaft rotational speed; f_GF = f_CF = 0.3 under dry friction; and h_1max = 1.0 mm is the acceptable bushing wear.

Materials of the metal-polymeric bearings were shaft-steel 45 (0.45% C) normalized; grinding; E₂ = 210 GPa, μ₂ = 0.3; $B_2$ = 10¹³; $m_2$ = 2; and ${\tau }_{20}$ = 0.1 MPa. The bushings were (1) carbon-filled polyamide PA6 + 30%CF, E_CF = 5.40 GPa, μ_CF = 0.4, $B_{1CF}^{}$ = 24 × 10¹⁰, $m_{1CF}$ = 1.9, and ${\tau }_{10}$ = 0.05 MPa and (2) glass-filled polyamide PA6 + 30%GF, E_GF = 3.90 GPa, μ_GF = 0.42, $B_{1GF}^{}$ = 6.67 × 10¹⁰, $m_{1GF}$ = 1.9, and ${\tau }_{10}$ = 0.05 MPa [29,30]. Volume filler content was 30%.

The calculation was performed according to the given flow diagram (Figure 2).

The example of the calculation examines friction both with greasing (boundary) and without greasing (dry). The mode of dry friction is widespread and dominant in MP bearings due to self-lubrication of polymeric materials [1,2,3,4,5,6,7,18,19]. It is possible, if necessary, to use in this type of bearings greasing with the implementation of boundary friction. In this case, for these hybrid combinations, a steel-polymer composite in the calculated ratios of the presented method, you only need to use the sliding friction coefficients corresponding to this type of lubrication. Instead, the established wear resistance characteristics, $B,m,{\tau }_0$, of these polymer composites remained the same because they were set for the range of change of 0.2–4 MPa.

Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 present the results of calculations. Accordingly, Figure 3 shows the effect of temperature on the change of the maximum contact pressures at different levels of external load.

Quantitative growth of p(0) is almost the same at different temperature levels. It is established that, in the case of MP bearing with a bushing made of carbon composite PA6 + 30%CF, the pressure at temperature T = 25 °C will be 1.1 times lower than at T = 55 °C. For fiberglass bushings, this difference is almost the same, as it is 1085 times. In the case of a bushing made of carbon composite, the maximum contact pressures in the bearing will be 1.17–1.18 times higher than for a glass composite bushing at all investigated operating loads. The maximum contact pressures p(0) in the MP bearings depend nonlinearly on the load N. An increase in load N by three times causes an increase in p(0)by 1736 times, i.e., ≈$\sqrt{3}$.

When the bushing wears during the operation, the initial pressures p(0) will decrease. Accordingly, Figure 4 shows the tribocontact pressures when reaching the accepted allowable bushing wear h_1max.

Regarding the patterns of maximum wear contact pressures, p(0, t, h) changes due to polymer bushing wear or with increasing load. They will be similar both qualitatively and quantitatively, as in the case of the initial maximum contact pressures p(0) (Figure 3).

The initial semiangles ${\alpha }_0$ were also calculated (Figure 5). It was established that, in a bearing with a glass-composite bushing, they will be, accordingly, up to 1.18 times smaller than in the presence of a carbon-composite bushing. With increasing load, ${\alpha }_0$ increases as p(0), i.e., by 1736 times.

Wear of the composite bushing leads to an increase in the initial contact semiangles ${\alpha }_0$ (Figure 6). Qualitative and quantitative patterns of their increase are preserved and inverted to those that occur when the wear contact pressures p(0, t, h) decrease.

As a result of solving the problem by Equation (8), a predictive estimation of the resource of the MP bearing with bushings PA6 + 30%CF and PA6 + 30%GF for both types of friction was performed.

Figure 7 and Figure 8 show the dependence of the bearing resource on the temperature at the acceptable wear h_1max = 1.0 mm for the boundary friction at different load levels.

Quantitative growth of the resource with increasing ambient temperature will be 22% in the bearing with a carbon composite bushing and 20% with a glass composite bushing. At all loads, such dependences are practically identical.

Figure 8 shows the resource of both MP bearings. The resource of a bearing with a carbon composite bushing is much longer.

The ratio of resources of both MP bearings under boundary friction will be approximately the same for all loads at T = 25–45 °C: At N = 500 N it is 240–256%; at N = 1000 N it is 245–261%; and at N = 1500 N it is 247–264%. At T = 55 °C for all loads N, this ratio is higher, up to 270%.

Accordingly, Figure 9 and Figure 10 show the bearing resource under dry friction.

When the acceptable wear h_1max = 1.0 mm is achieved, there is an increase in the resource with increasing ambient temperature: by 20% in the bearing with a carbon-composite bushing and by 17% in the bearing with a glass-composite bushing. At all loads, such dependence is practically identical.

The resource of a bearing with a carbon composite bushing is significantly greater than with a glass composite (Figure 10).

The ratio of resources of both MP bearings under dry friction will be approximately the same for all loads at T = 25–45 °C: At N = 500 N it is 248–264%; at N = 1000 N it is 250–264%; and at N = 1500 N it is 250–260%. At T = 55 °C for all loads N, this ratio is higher, up to 271%.

Figure 11 shows the calculated resources of MP bearings for both types of friction.

A decrease in the coefficient of friction five times under boundary friction compared to dry friction causes an increase in the resource of the MP bearings by several times. In particular, at N = 1500 N it is 25–25.6 times with increasing temperature, at N = 1000 N it is 25.9–26.8 times, and at N = 500 N it is 28.1–29.8 times.

4. Conclusions

1. An increase in the ambient temperature causes a decrease in the value of the Young’s modulus of polyamide composites PA6 + 30%CF and PA6 + 30%GF, resulting in a decrease in the level of the initial maximum contact pressures p(0). That is, the increase in temperature causes a decrease in the rigidity of polymer composites, which has a positive effect on the bearing capacity of MP bearings (up to 10%). When the bushing wears, the patterns of further reduction of the initial maximum contact pressures p(0) remain similar to the initial pressures.

2. With increasing ambient temperature, there is an increase in the resource of MP bearings. That is, the higher temperature has a positive effect on the resource of MP bearings (up to 22%). The calculated resource under boundary friction of the bearing with the carbon composite bushing will be higher, by 240–270%, than with the glass composite bushing. Under dry friction, the similar correlation of resources takes place.

3. It is established that the resource of MP bearings increases approximately on a square dependence on decrease in coefficient of sliding friction

4. The developed analytical method of calculation of MP sliding bearings provides at a design stage an effective predictive estimation of their contact parameters and the resource, taking into account influence of ambient temperature. The presented method will be used for research of MP bearings made of other types of polymeric materials.