© AFM, EDP Sciences 2017

1 Introduction

As an important part of rolling bearing, rolling ball contacts with its friction pairs frequently, the wear in the process of contacting has great effect for the assembly precision of bearing [1,2]. During the running of rolling ball, it not only rotates alone the inner or outer ring but also slides when the friction of contact point between rolling ball and its friction pairs is not enough, which is called the sliding-rolling state [3]. Cheng et al. [4] established a model for calculating surface friction heat in slide state. By analyzing factors which affect the part temperature in contact point, the variation trends of surface temperature was obtained. Sundar et al. [5] measured vibration to estimate coefficient of friction and a mechanical system with combined rolling-sliding contact. An empirical relationship for the coefficient of friction is suggested based on a prior model under a mixed lubrication regime. Šperka et al. [6] put forward amplitude decay theory, established a model of predicting tribology performance according to surface roughness, and analyzed how roughness would affect tribology performance at sliding-rolling state using optical interference film test rig. Ciulli et al. [7] measured film thickness of point contacts under steady state and transient conditions at different sliding-rolling ratios (SRRs). The experiments proved that under transient conditions, high slide-to-roll ratios seem to reduce the squeeze effect on the film thickness.

Although studies about bearing tribology performance in sliding-rolling state exist, researches only involved with wear performance of rolling ball are not sufficient [8,9]. The main contribution of our work lies on the experiments, by collecting real time wear identification with variation of SRRs, load and rotate speed based on self developed test rig. The wear performance in different states has been discussed. The calculation model of wearing rate under the operation condition of sliding-rolling was established based on Archard model.

2 Experiment rig and method

2.1 Experiment rig and material

Figure 1 shows the test rig instrumentation and control system.

Test rig is the self designed and developed three-point contact bearing ball fatigue life testing machine [10], and its structure is shown in Figure 2.

Motor 1 connects support wheels 2, 3, 4 with cog belt. Support wheels as middleware transmit velocity to slave rollers 5, 6. The driven wheel 9 transmit velocity between load wheel 8 and driver 10 which connects with motor 1. In this condition, the rolling ball 7 contacts with slave roller 5, 6 and load wheel 8 in three points.

For implementing sliding-rolling state and collecting real time wear volume, test rig has been improved. By changing transmission ratio between driven wheel 9 and driver 10, the velocity of load wheel 8 can be different from slave rollers 5, 6. At this time, the velocities of three contact points in rolling ball are not same any more, rolling ball is in a certain sliding-rolling state. Adjusting the transmission ratio between driven wheel 9 and driver 10, SRR can be controlled. Displacement sensor 11 is placed above load wheel 8. With proceeding of wear in rolling ball, diameter of rolling ball 7 is smaller and smaller, the change reflects in distance between sensor 11 and wheel, distance data are collected online using DAQ card and calculated to reality wear volume in Labview.

Fig. 1 Test system at the laboratory.

Fig. 2 Structure of three-point contact bearing ball fatigue life testing machine.

2.2 Experiment method

Operation condition is shown in Table 1.

A simple sliding-rolling condition is described by two speeds. (1) (2)

where v_u, v_d are speeds of friction pair and counter-pair in sliding-rolling (m·s⁻¹), v_a is sliding speed (m·s⁻¹), v_r is rolling speed (m;^∙1)

SRR [11] is defined as the ratio of slide velocity to rolling velocity: (3)

The SRRs in three contact points, S₁, S₂, S₃, are defined as: (4) (5) where v₁, v₂, v₃ are the velocities in three contact points, respectively (m·s⁻¹), v_ball is velocity of test ball (m·s⁻¹), shown as Figure 3.

The angles of contact points in v₁ and v₂ or v₁ and v₃ are big (≥160°). For calculating the sliding-rolling simply, each angles are assumed to be 180°, relationship among speeds in test ball is shown in Figure 4.

Due to the existent of sliding, a part of speed transmitted to test ball may lost. (6) (7) where is the real speed what v₁ transmits to test ball (m·s⁻¹), is the real speed what v₂ transmits to test ball (m·s⁻¹), v_s1 is the slide speed in point v₁ (m·s⁻¹), v_s2 is the slide speed in point v₂ (m·s⁻¹).

During the experiment, centre of test ball is fixed, so the speed of centre of test ball is zero called instantaneous center of velocity. It means the values of , need to be equal, and the directions of , are opposite. According to equations (6) and (7), we can get (8)

According to the equations (1)–(3): (9)

Similarly, S₁ = S₃

Assuming S₁ = S₂ = S₃, v_ball is defined as: (10) where i is transmission ratio, r_s are radius of slave rollers and load wheel (mm), v_s are speeds of slave rollers and load wheel (m·s⁻¹).

SRR S is defined as: (11)

Therefore, the SRR in experiment can be obtained by calculating the transmission ratio.

Due to the existent of vibration and thermal expansion, measured movement of load wheel by displacement sensor is not the real change of rolling ball. By some calibration experiments, real variation of rolling ball in radius is fitted using distance changes of load wheel, and relationship between them is defined as: (12) where Δr_t is the real measured variation of rolling ball in radius (μm), ΔL is the measured movement of load wheel (μm).

Wear volume is calculated by Δr as follows: (13) where ΔV is the real wearing volume (mm³), r_t is the radius of tested rolling ball (mm), Δr_t is the real measured variation of rolling ball in radius (mm).

Wear rate is calculated as follows: (14) where V is the wearing rate (mm³·min⁻¹), ΔV is the real wearing volume (mm³), and t is the wearing time (min).

Fig. 3 Velocities in three points.

Fig. 4 Relationship among speeds in test ball.

3 Results and discussion

3.1 Wear performance in different SRRs

Figure 5 gives results of wear volume of different SRRs in 50 N and 3000 r·min⁻¹.

As shown in Figure 5, when SRR is 0, tested ball is in pure rolling state, wear volume is only 7 × 10⁻⁵ mm³, and it is hard to observe different wear stages at this condition. When SRR is 0.05, compared with pure rolling state, the wear volume increases to 0.0034 mm³, although SRR is still low. At the same time, different wear stages are obvious. From 0 to 1000 min, the wear volume appears very high increase. Within 4000 min, wear volume increases continually but growth rate is lower, and this period is called running-in period. In running-in period, asperity-peak of contact zone is worn or flatted due to local high pressure. Surface topography of friction pairs is improved gradually [12]. After 4000 min, wear level is changed into stabilization period. When SRRs are increased to 0.095 and 0.15, wear volumes are 0.0078 mm³ and 0.01745 mm³, respectively. According to equation (3), the slide velocity is proportional to SRR. So in the same wear time, slide distance is proportional to SRR. According the model described by Archard, wear volume is proportional to slide distance. Therefore, the wear volume is proportional to SRR.

Figure 6 gives results of wear rates of different SRRs in 50 N and 3000 r·min⁻¹.

As shown in Figure 6, when SRR is 0 called pure rolling state, wear rate is only 10⁻¹² mm³·min⁻¹, and variation of wear rate is not obvious during process of wear. When SRR is 0.05, stabilization wear rate is 4.1 × 10⁻¹⁰ mm³·min⁻¹, and much larger than the wear rate in pure rolling period. In the beginning of wear, for forming a favourable surface quality which is suitable for operation condition such as surface pressure and friction coefficient, wear rates are lower and lower and tend to be in stable state. It's called stabilization period. Compared with pure rolling condition, wear rate in SRR is 100 times than one in pure rolling state. In 0.095 and 0.15 SRRs, stabilization wear rates are 5.3 × 10⁻¹⁰ mm³·min⁻¹ and 6.1 × 10⁻¹⁰ mm³·min⁻¹, respectively. As shown in Figure 6, greater the SRR is, the duration reaching stabilization wear stage is shorter, but the stabilization wear rates are almost same. It means that in the same operation condition and tribology performance of surface, wear rate is independent of SRR.

Fig. 5 Effect of sliding-rolling ratio for wear volume.

Fig. 6 Effect of sliding-rolling ratio for wear rate.

3.2 Effect of SRR in different loads

Figure 7 give results of wear rates of different SRRs in 3000 r·min⁻¹ and different loads.

As shown in Figure 7(a), when load is 50 N, due to low load, stabilization wear rates, about 5.9 × 10⁻¹⁰ mm³·min⁻¹, are close for three SRRs. The main wear behavior is slight abrasive wear as shown in Figure 8(a).

In Figure 7(b), when load is elevated to 100 N, stabilization wear rates in 0.05 and 0.095 SRRs are 5.9 × 10⁻¹⁰ mm³·min⁻¹ and 6.1 × 10⁻¹⁰ mm³·min⁻¹, and they are similar with the operation condition in 50 N. When SRR increases to 0.15, wear rate is stable at 8.4 × 10⁻¹⁰ mm³·min⁻¹. It can be seen in Figure 8(b), the wear behavior in this condition is not only abrasive wear but also slight adhesive wear in area A1. Adhesive wear is not a wear performance appearing on oxide layer. Adhesion point breaks the surface of material [13], and metal inside is exposed in air, so wear rate is increased.

As shown in Figure 7(c), when load is changed to 150 N, stabilization wear rate in SRR 0.05 is 5.8 × 10⁻¹⁰ mm³·min⁻¹, and surface is still in slight wear like condition in load 100 N. In 0.095 and 0.15 SRRs, stabilization wear rates are 1.12 × 10⁻⁹ mm³·min⁻¹ and 1.43 × 10⁻⁹ mm³·min⁻¹, respectively. It can be seen in Figure 8(c) which is divided into area B1 and B2. Because of high contact stress, compared with normal contact surface B2, area B1 presents obvious depth differences, called plastic deformation appearance. Plastic deformation causes cold-work hardening in metal [14], and surface of metal is more and more fragile. This characteristic results in that it is easier to form abrasive particles in surface. With the increase of abrasive wear, the wear rate is higher and higher.

The conclusion of these experiments is described. In low load condition, effect of SRR in different load is not obvious. With the load increased, the SRR has a great significance for wear rate. Compared with operation condition in low load, wear performance and shapes in surface are changed from abrasive wear to adhesive wear, and the surface area presents plastic deformation gradually. All changes above reflect in the increase of wear rate.

Fig. 7 Effect of sliding-rolling ratio in different loads.A1

Fig. 8 SEM images from sliding-rolling ratios in different loads.

3.3 Effect of SRR in different rotate speeds

According to effect of SRR in different loads, load is higher and the degree of wear is severer. For avoiding adhesive wear, the experiments about wear rates of different SRRs in various rotate speeds are carried out in load 50 N.

As shown in Figure 9(a), when the rotate speed is 1000 r·min⁻¹, with the SRR increasing, the wear rate achieves the stable state more quickly. All wear still occur on oxide layer with slight abrasive wear, as shown in Figure 10(a). Therefore, three stabilization wear rates, all about 5.7 × 10⁻¹⁰ mm³·min⁻¹ in three different SRRs, are similar.

In Figure 9(b), the variation of SRR affects wear rate in 2000 r·min⁻¹. The stabilization wear rates of SRR 0.05, 0.095, 0.15 are 4.9 × 10⁻¹⁰ mm³·min⁻¹, 5.9 × 10⁻¹⁰ mm³·min⁻¹ and 6.1 × 10⁻¹⁰ mm³·min⁻¹, respectively. In Figure 10(b), compared with low speed, although the speed is twice as much as before, wear volume didn't multiply as speed, and wear performance of this condition are also oxidative wear with slight abrasive wear. With elevated speed, performance of abrasive wear do not be worse. It means that because of limited abrasive particles, in high rotate speed, most of abrasive particles formed by slide wear are threw away by rolling of ball. All of characters result in the phenomenon that wear performance and wear rate are not changed obviously.

In Figure 9(c), the variation of SRR affects wear rate in 3000 r·min⁻¹ before 6000 min. The stabilization wear rates in three SRRs are 5.6 × 10⁻¹⁰ mm³·min⁻¹, 6.3 × 10⁻¹⁰ mm³·min⁻¹ and 6.5 × 10⁻¹⁰ mm³·min⁻¹, respectively. Compared with the experiment results in 1000 r·min⁻¹ and 2000 r·min⁻¹, wear rates are changed a little with increase of rotate speeds. Within the range of the temperature which wouldn't influence surface tribology performance [15,16], the increase of rotate speed can only cause the decline of ability of heat dissipation on surface [17] and then affect the wear rate. But in this experiment, the influence is limited.

In the process of rolling-wear, the main failure mode is fatigue failure in high stress cycles. So in this favorable operation condition, time of experiment is lengthened to research the relationship between sliding-rolling and fatigue failure.

As shown in Figure 9(d), experiments data are all after 60 000 min. In 0.15 SRR, wear rate keeps stable before 66 000 min. After 66 000 min, wear rate has a sudden increase from 6.2 × 10⁻¹⁰ mm³·min⁻¹ together with the increase of temperature. At this time, the surface of tested ball presents obvious spalling appearance, called fatigue wear phenomenon. In 0.095 SRR condition, wear rate is changed in 87 720 min suddenly. Compared with condition in 0.15 SRR, time of appearing fatigue wear phenomenon is 21 720 min longer. In 0.05 SRR, required duration time of appearing phenomenon of fatigue wear is too long, so this experiment is terminated. As seen in Figure 10(c), fatigue spalling [18] can be observed in surface area C1 in high stress cycles. Around the spalling area, surface of ball is hard to be observed the obvious wear performance in area C2. Unless fatigue wear occurs, otherwise fatigue spalling data is not easy to obtain from variation of wear rates.

According to experiment results, in normal wear condition, SRR affects the wear rate a little, but has a significant influence for the time of fatigue wear. With the increase of SRR, sliding friction accelerates crack formation and crack propagation on ball surface, and shortens the time of occurring fatigue wear.

Fig. 9 Effect of sliding-rolling ratio in different rotate speeds.C2

Fig. 10 SEM images from sliding-rolling ratios in different rotate speeds.

4 Wear performance model

By analyzing the experiment data, a model of variation of wear rate with time in sliding-rolling condition can be established. According to the conclusion of experiments above, variation of wear rate is divided into two periods, running-in period and steady-state period. Firstly, the influence of SRR to steady wear rate needs to be figured out.

4.1 Steady wear rate model based on Archard

According to Archard theory [19]: (15) where L = ∫ v_adt, ΔV = ∫ V_Adt. v and V_A is constant at steady state, so L = v_a ∫ dt, ΔV = V_A ∫ dt. Equation (15) is changed to: (16)where ΔV is wear volume (mm³), L is wear distance (m), W is load (N), σ_s is yield strength of softer material (Pa), V_A is theoretical wear rate (mm³·min⁻¹), t is wear time (min), v_a is slide speed (m·s⁻¹), and K_s is coefficient of adhesive wear. Assuming equation (16) is changed to: (17)where V_S is steady wear rate (mm³·min⁻¹). Wear coefficient in lubrication condition is K_s = 10⁻⁷, yield strength σ_s = 518.42 MPa, so K = 6.43 × 10⁻¹⁷.

4.2 Corrected steady wear rate model

Archard theory is adhesive wear theory [20], due to the existence of SRR in these experiments, wear performance is not adhesive wear any more, and it is changed with the variations of load, speed and SRR. Even so, Archard model is able to expound the relationships among factors affecting wear rate [21]. Considering this, the equation (17) is corrected in sliding-rolling operation condition.

According to experiment result above, with the increase of load, wear performance is changed from abrasive wear to adhesive wear, and the wear rate is enhanced gradually, but the rate of increase is lower and lower. The approximate trend is shown in Figure 11.

The load W is corrected with adding SRR S: (18) where W′ is the corrected load (N), W is the real load (N), S is SRR, a, b, c are related correct coefficient.

At S = 0, equation (18) is defined as: (19) where is the ratio of corrected load to real load at time zero.

At S = ∞, equation (18) is defined as: (20) where is the ratio of corrected load to real load at steady state.

So equation (18) is changed to: (21)

According to the experiment result about effect of SRRs in different rotate speeds, rotate speed has a little influence to wear rate in low load condition. Some of wear particles formed in sliding wear the surface twice, called three-body abrasive wear, so wear rate is increased. But with the increase of rotate speed, most of wear particles are threw out, and variation trend is not obvious any more. With continuous increase of speed, high temperature caused by high speed may make the state of contact area [22] worse, and the wear rate increases substantially. The approximate trend is shown in Figure 12.

This experiment is not in very high speed. The relationship between SRR S and is assumed to be linear simply. The rotate speed v is corrected with adding SRR S: (22) where v ′ _r is the corrected rotate speed (r·min⁻¹), v_r is the real rotate speed (r·min⁻¹), S is SRR, d and f are related correct coefficient.

The speed v_r is not the slide speed directly in this experiment. The slide speed needs to be calculated indirectly, according to equation (4). (23) (24) (25) where v_ball is speed of test ball (m·s⁻¹), S is SRR, v₁ is speed of slave pairs (m·s⁻¹), v_a is sliding speed (m·s⁻¹), v_r is real rotate speed (r·min⁻¹), r_s is radius of slave roller (mm), v_a′ is corrected sliding speed (m·s⁻¹), v_r′ is the corrected rotate speed (r·min⁻¹).

Wear rate equation, equation (17) after being corrected is: (26)

Equation (26) is fitted using experiment data by least square method. Known from equation (17), K = 6.43 × 10⁻¹⁷. The wear rate equation is corrected as: (27) or the equation can be described with rolling speed v_r in this experiment as: (28)

Fig. 11 Relationship between corrected load and SRR.

Fig. 12 Relationship between corrected speed and SRR.

4.3 Verification of steady wear rate model

The accuracy of the corrected model of steady wear rate is verified using relative error: (29) where δ is relative error, V_f is predicted wear rate, V_t is real measured wear rate.

Test operation condition is shown in Table 2.

Comparison of relative errors between Archard model and corrected model are shown in Figure 13.

Using root-mean-square error as standard of model-error, Archard model's error is 3.775, and corrected model's error is 1.197. As shown in Table 2 and Figure 13, in low load, predictions of two models are both accurate. With the load increasing, considering the influence of SRR to contact area, the corrected model is more precise than Archard model. Therefore, the corrected model is more effective in sliding-rolling condition, and wear rate of corrected model is applied to be the steady wear rate in whole model next.

Fig. 13 Comparison of relative errors between Archard mode and corrected model.

4.4 Relationship between wear rate and time

The variation of wear rate with time is shown schematically in Figure 14. Initially the wear rate is high which reduces exponentially with time and reaches steady-state after some time. This trend is expressed mathematically as follows [23]: (30) where V is wear rate (mm³·min⁻¹), V₀ is wear rate at time zero (mm³·min⁻¹), V_S is wear rate at steady state (mm³·min⁻¹), t is time of wear (min), and h is related coefficient.

The equation (30) is fitted using experiments data by least square method, and related correct coefficient h = 0.0009. The wear rate changed with time is defined as follows: (31)

Fig. 14 The variation of wear rate with time.

4.5 Model verification

Model verification operation condition is shown in Table 3.

As shown in Table 3, average relative errors of four groups are not larger, all errors are under 0.2, so it can be identified that fitting results are credible. The specific fitting results are shown in Figure 15.

In Figure 15, overall, fitting curves are similar with the real wear rate. Especially in steady wear period, precision is higher. Due to variation of machine or unstable state of ball's rolling in the beginning of experiment, the wear rate at time zero is hard to be obtained accurately, and the fitting results at the beginning of experiments or running-in period are barely satisfactory.

The corrected model may be unable to be applied directly in complex practical production, but it describes the change trend of wear rate and the relationship among load, rotate speed and SRR. By amending the relative coefficient of equation (31), the corrected model can be applied in other operation condition with sliding-rolling state.

Fig. 15 Specific fitting results.

5 Conclusion

• A test rig is designed and improved. SRR can be changed and wear data is collected online.

• Compared with pure rolling state, even though in low speed, wear rate of sliding-rolling state is considerably higher than pure rolling state. Sliding-rolling state presents obvious running-in wear and stable wear performance.

• Within the certain range of speed, in low load, SRR affects the wear rate in a low extent. With load increase, SRR has a significant influence for wear rates. High SRR and heavy load change the form of contact area from only abrasive wear to adhesive wear, and the wear rate increases obviously.

• In non-heavy load condition, variation of SRRs and velocities have a limited influence for wear rate, but has a great effect for time of fatigue wear appearance. The increment of SRR accelerates crack formation and crack propagation on ball surface, and shorten the time of fatigue wear appearance.

• According to the relationship among load, velocity, SRR, a wear rate prediction model based on Archard model is corrected in sliding-rolling state. The veracity of corrected model shows great improvement over Archard model.

Nomenclature

a, b, c, d, f, h: Related correct coefficient

i: Transmission ratio

r_s: Radius of slave rollers and load wheel (mm)

r_t: Radius of tested rolling ball (mm)

t: Wear time (min)

v_a: Sliding speed (m·s⁻¹)

: Corrected sliding speed (m·s⁻¹)

v_ball: Velocity of test ball (m·s⁻¹)

v_r: Rolling speed (r·min⁻¹)

: Corrected rotate speed (r·min⁻¹)

v_s: Speeds of slave rollers and load wheel (m·s⁻¹)

v_s1: Slide speed in point v₁ (m·s⁻¹)

v_s2: Slide speed in point v₂ (m·s⁻¹)

v_u, v_d: Speeds of friction pair and counter-pair in sliding-rolling (m·s⁻¹)

v₁, v₂, v₃: Velocities in three contact points, respectively (m·s⁻¹)

: Real speed what v₁ transmits to test ball (m·s⁻¹)

: Real speed what v₂ transmits to test ball (m·s⁻¹)

K_s: Coefficient of adhesive wear

L: Wear distance (m)

S: Sliding-rolling ratio

S₁, S₂, S₃: Sliding-rolling ratios in three contact points, respectively

V: Wear rate (mm³·min⁻¹)

V_f: Prediction wear rate (mm³·min⁻¹)

V_t: Real measured wear rate (mm³·min⁻¹)

V_A: Theoretical wear rate (mm³·min⁻¹)

V_S: Wear rate at steady state (mm³·min⁻¹)

V₀: Wear rate at time zero (mm³·min⁻¹)

W: Load (N)

W′: Corrected load (N)

Δr_t: Real measured variation of rolling ball in radius (mm)

ΔL: Measured displacement of load wheel (mm)

ΔV: Real wearing volume (mm³)

σ_s: Yield strength of softer material (Pa)

δ: Relative error

: Ratio of corrected load to real load at time zero

: Ratio of corrected load to real load at steady state

Acknowledgments

This work was carried out with financial support from the National twelfth five-year project of China for science and technology under Contract D.50-0109-15-001.

References

All Tables

All Figures

	Fig. 1 Test system at the laboratory.
In the text

	Fig. 2 Structure of three-point contact bearing ball fatigue life testing machine.
In the text

	Fig. 3 Velocities in three points.
In the text

	Fig. 4 Relationship among speeds in test ball.
In the text

	Fig. 5 Effect of sliding-rolling ratio for wear volume.
In the text

	Fig. 6 Effect of sliding-rolling ratio for wear rate.
In the text

	Fig. 7 Effect of sliding-rolling ratio in different loads.A1
In the text

	Fig. 8 SEM images from sliding-rolling ratios in different loads.
In the text

	Fig. 9 Effect of sliding-rolling ratio in different rotate speeds.C2
In the text

	Fig. 10 SEM images from sliding-rolling ratios in different rotate speeds.
In the text

	Fig. 11 Relationship between corrected load and SRR.
In the text

	Fig. 12 Relationship between corrected speed and SRR.
In the text

	Fig. 13 Comparison of relative errors between Archard mode and corrected model.
In the text

	Fig. 14 The variation of wear rate with time.
In the text

	Fig. 15 Specific fitting results.
In the text