Critical Shoulder Height of Raceway in Ball Bearing Considering Elastohydrodynamic Lubrication

Abstract

In this study, the effects of Elasto-hydrodynamic lubrication pressure on the critical shoulder height of raceway in an angular contact ball bearing were investigated. Both 3D contact analyses using an influence function and the EHL analysis were conducted for the contact geometry between the ball and raceways. The pressure distributions by 3D contact analysis and EHL analysis for an example bearing were compared. The effect of ellipse truncation on the minimum film thickness also investigated from EHL analysis. The critical shoulder height in the dry contact and the EHL state were compared for various applied loads. It is shown that when the ellipse truncation occurs, the pressure spike for the EHL conjunction is higher than that for the dry contact, and its location moves more inward of the contact center. The steep pressure gradients would increase the flow rate, so in order to maintain flow continuity a significant reduction in film thickness and an abrupt rise in pressure occurs in the edge of shoulder. Significant reduction of the minimum film thickness occurs near the edge of shoulder. The critical shoulder heights in the EHL state are calculated as higher values compared with in the dry contact. This results shows that the determination of critical shoulder height by the EHL analysis is more proper.

1. Introduction

Ball bearings are essential elements of rotating machine systems, and many studies have already been conducted. It is important that ball bearings have high reliability and realize miniaturization and weight reduction at the same time. The design of the appropriate dimensions of the bearings does not cause premature failure of the system while satisfying the required service life is an important issue. In general, ball bearings should be designed so that the ball does not ride the shoulder of the raceway during rotation, and the height of the shoulder that can withstand the axial load should be determined.

If the shoulder height is insufficiently designed, the reliability of the bearing system is degraded due to the formation of pressure spikes due to the edge effect as shown in Fig. 1. If the shoulder height is enough high, the contact shape between the ball and raceway will be an ellipse. Conversely, if the shoulder height is not enough, non-Hertzian contacts may occur at the ball/raceway conjunction which the contact ellipse is truncated. On the other hand, when the shoulder height is designed excessively, there is a problem of an invisible cost increase such as an increase in material cost and a problem of insufficient space for retainer installation [1]. Studies on the height of the shoulder are still insufficient, and most of them have been determined by experience without any special criteria being set. Frantz and Leveille [2] derived a simple and approximate expression relating the mean stress with the fraction truncated at the contact between a ball and raceway of a loaded angular contact ball bearing. Kwak and Kim [3] proposed a method to determine the shoulder height in an angular contact ball bearing using a 3D contact analysis. They calculated the critical shoulder heights for a steel ball bearing subjected to a practical load through a series of simulations such as a static analysis to calculate the internal load induced on the balls and a contact analysis. Choi and Kim[4] presents an optimization method to determine the shoulder height of an angular contact ball bearing by 3D contact analysis using nondimensional-shaped variables. They investigated the relationship between the shoulder height and the radius of curvature of the shoulder under various loading conditions. The study of EHL has been advanced for past several decades. Chang [5] presented a partial elastohydrodynamic lubrication model of line contacts. Jiang et al. [6] investigated a model that solved hydrodynamic and asperity contact pressure simultaneously by utilizing FFT technique and multi-grid algorithm.

Fig. 1. Pressure spike due to the edge effect of shoulder height [3].

Fig. 2. Contact geometry between the ball and the inner/outer raceway [3].

In this study, we investigated the effect of EHL conjunction on the critical shoulder height of an angular contact ball bearing. The surfaces of raceways are numerically generated including the groove, the ridge and the shoulder. Based on the contact geometry, the pressure distributions are obtained by 3D contact analysis using influence functions and EHL analysis for an example bearing. The critical shoulder height calculated by the contact analysis and the EHL analysis are compared for various applied loads.

2. Analysis

2-1. Contact analysis

In this study, three-dimensional contact analysis was basically performed by using the influence function. In order to perform contact analysis, the shape function of the bearing is first required. Analyzing the contact shape between the ball and the raceway, in the case of the inner ring, conformal contact occurs in the axial direction and non-conformal contact occurs in the radial direction, resulting in an elliptical contact shape. For the contact modeling of the ball and the raceway, please refer to the previous work[3]. Through geometric analysis of the ball and the raceway, it is modeled as a contact between one equivalent surface and a rigid plate, and the pressure distribution of the contact surface can be obtained. To solve the contact problem, the Boussnesq equation [7], which expresses the displacement and stress generated in the semi-infinite elastic body as a potential function, was used. The pressure and contact area were calculated through numerical repetitive calculations by applying Love's equation [8], which assumed that the pressure on the rectangular surface has a constant value by discretizing the analysis area into a square piece. It provides useful information on the contact pressure, number of contacts, their sizes and distributions. The numerical analysis method used an algorithm using the Newton-Raphson method, which is robust to a system of nonlinear equations. Please refer to the previous work[3] on more detailed contact analysis methods.

2-2. Elasto-hydrodynamic lubrication analysis

In order to know the shoulder height at which pressure spikes occur in the state of elasto-hydrodynamic lubrication for ball bearings, elasto-hydrodynamic lubrication analysis should be performed. A given load Q should be equal to the sum of the developed pressures in the entire analysis area, which can be expressed as following.

Q = ∫∫_ΩP(x, y)(dx)dy       (1)

Where, P is the pressure in the entire analysis area Ω. When the lubricant flows in the x-direction, Reynolds equation to calculate P in the lubrication region in which h(x,y) >0 is given by

\(\begin{aligned}\frac{\partial}{\partial x}\left(\frac{\rho h^{3}}{\eta} \frac{\partial P}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\rho h^{3}}{\eta} \frac{\partial P}{\partial y}\right)=12 \bar{U} \frac{\partial(\rho h)}{\partial x}\\\end{aligned}\)       (2)

Where, h is film thickness, \(\begin{aligned}\bar{U}\\\end{aligned}\) is mean velocity of two surfaces, ρ and η are the density and the viscosity of lubricant, respectively. To solve Eq. (2), a geometrical analysis of the ball and raceway is required. In the case of the raceway, it is divided into three parts: the groove, the ridge, and the shoulder. Please refer to the previous work[3] on the geometry function.

As boundary condition for lubrication region, Reynolds boundary condition is used,

P(x,y) = 0 at x = x_in

\(\begin{aligned}P(x, y)=0\; and \;\frac{\partial P(x, y)}{\partial x}=0 \; at\;x=x_{o u t}\\\end{aligned}\)       (3)

For the viscosity and the density of the lubricant, viscosity-pressure equation developed by Roelands[9] and density- pressure equation by Dowson and Higginson [10] are used. In this study, the solution was obtained by applying the Newton-Rapson method, which has high convergence to nonlinear equations and can reflect the correlation of various variables, to the finite difference method.

3. Results and Discussion

The specification of an angular contact ball bearing used in this study is described in Table 1. The initial shoulder height of the inner raceway is assumed as 4.5 mm. The materials of the ball and the raceway are AISI 52100 with E = 208 GPa and ν = 0.3, and AISI 1045 with E = 200 GPa and ν = 0.3, respectively. The lubricant used in the EHL analysis is a mineral oil with viscosity of 0.022 Pasec and density of 861.6 kg/m³. Non-dimensional viscosity-pressure index is 0.4. It is assumed that the lubricant flows into lubrication region at the velocity of 2.5 m/s, and the ball load applied in the contact angle are from 100 N to 500 N which correspond to the maximum hertz pressure of from 0.909 GPa to 1.802 GPa.

Table 1. Specification of angular contact ball bearing

Fig. 3 shows the distributions of pressures by the contact and the EHL analyses for the raceway with a shoulder height of 4.5mm when the applied load is 300N. The contact pressures have elliptical shapes, not showing any pressure spikes in transversal direction (Y = 0). This means that the present shoulder height is enough to accommodate the applied load. Pressure distribution for the EHL conjunction in Fig. 3(b) show a pressure spike with a value less than the peak Hertzian pressure in the outlet of the film. Maximum pressures for the contact analysis and the EHL analysis are 1.46 GPa and 1.442 GPa, respectively. Fig. 4 shows contact and EHL pressures, and film thickness profiles in the rolling direction(X = 0). Typical pressure and film thickness profiles for an EHL contact are shown. Minimum film thickness is calculated as about 0.18 μm in the outlet.

Fig. 3. Pressure distribution by (a) contact analysis and (b) EHL analysis for a shoulder height of 4.5 mm and an applied load of 300 N.

Fig. 4. Comparison of pressures in the rolling direction (X = 0) by contact analysis and EHL analysis for a shoulder height of 4.5 mm and an applied load of 300 N, and the film thickness profile.

Fig. 5 shows the distributions of pressures by the contact and the EHL analyses for the raceway with a shoulder height of 2.8 mm. For both cases the abrupt pressures are generated near the edge of shoulder. Because the contact area is not sufficient under the applied oad, the pressure spike due to the edge effect occurs.

Fig. 5. Pressure distribution by (a) contact analysis and (b) EHL analysis for a shoulder height of 2.8 mm and an applied load of 300 N.

Pressures and film thickness profiles in the rolling direction in Fig. 6(a) are similar to the results with a shoulder height of 4.5 mm in Fig. 4(a). However, higher pressure spikes than the central pressures in contact area are shown in the transversal direction in Fig. 6(b). The maximum pressure for the EHL analysis is calculated as 1.68 GPa, which is higher than that of 1.56 GPa for the contact analysis. Compared with the location of pressure spike by contact analysis, the location of the spike by EHL analysis moved inward the center of contact as shown in Fig. 6(b). In the EHL conjunction which is more realistic, the higher pressure spike is generated more inward of contact, compared with in the dry contact. Therefore, we can say that the determination of optimal shoulder height should be performed by the EHL analysis.

Fig. 6. Comparison of pressures in the rolling (X = 0) and the transversal (Y = 0) directions by contact analysis and EHL analysis at a shoulder height of 2.8 mm under an applied load of 300 N and the film thickness profiles.

Fig. 7 shows contours of the film for the raceways with shoulder heights of 4.5 mm and 2.8 mm under an applied load of 300 N. For the shoulder height of 4.5 mm typical symmetric ‘horse-shoe’ shaped film is generated, whereas a non-symmetric film shape due to the short shoulder height is shown for the shoulder height of 2.8 mm.

Fig. 7. Contours of the film for shoulder heights of 4.5 mm and 2.8 mm under an applied load of 300 N.

Note that the minimum film thickness occurs near the edge of shoulder, not in the outlet region of rolling direction, as shown in Fig. 6. It is shown that the minimum film thickness in the outlet region of rolling direction in Fig. 6(a) is about 0.19 μm, whereas that in the edge of shoulder in Fig. 6(b) is about 0.03 μm. This result can be explained from the difference of contact shape between two regions. The fluid film boundary near the edge of shoulder diverges more rapidly than in the outlet region of rolling direction. In addition, the pressure spike in the edge of shoulder occurs due to the ellipse truncation. Consequently, a larger pressure gradient to reduce the pressure to the ambient value exists in the edge of shoulder. The steep pressure gradients would increase the flow rate, so in order to maintain flow continuity a significant reduction in film thickness and an abrupt rise in pressure occurs in the edge of shoulder. The minimum film thickness of 0.03 μm is the order of the equivalent surface roughness of the general ball and raceway. The significant gap closing in the edge of the contact ellipse truncation can lead to solid to solid contact. Even if the solid to solid contact does not occur, the fatigue life would be remarkably reduced by the rise of internal stress.

Fig. 8 shows maximum pressures by the contact and the EHL analyses as a function of the shoulder height for various applied loads. If there is no effect of the ellipse truncation, the maximum pressures are the pressures in the center of contact. In this figure, maximum pressures increase abruptly at certain values of shoulder height with decrease of the shoulder height. It means the pressure spikes near the edge of shoulder are higher than the central pressure. The results show the initial shoulder height of 4.5 mm in this bearing was designed to be excessively high.

Fig. 8. Maximum pressures by the contact and the EHL analyses as a function of the shoulder height for various applied loads.

In the contact analysis the maximum pressures for the applied loads of 100, 300, and 500 N start to increase in the shoulder heights of 2.7, 2.9, and 3.0 mm, respectively. In the EHL analysis, however, the changes occur when the shoulder heights are 2.7, 3.0, and 3.2 mm, respectively. For the applied load of 100 N, the critical shoulder heights which is defined as the lowest shoulder height on the limit that the maximum edge pressure does not exceed the central pressure at a given load are the same between the dry contact and the EHL state. For the higher applied loads, however, the critical shoulder heights in the EHL state are calculated as higher values because the pressure spike is generated more inward of contact, compared with in the dry contact. Therefore, it is conclude that the critical shoulder height should be designed considering EHL effects. It should be noted that the spinning effect of ball bearing was not considered in this paper. Spinning effect can not be ignored in some elastohydrodynamically lubricated contacts. It has been generally known that the film thickness decreases with increasing spinning, and the symmetry of the film shape about the central entrainment gets lost [11].

4. Conclusions

In this paper, we compared the results of the critical shoulder height calculated by the contact analysis and the EHL analysis. As an example problem, a numerical simulation was performed for an angular contact ball bearing with an inner raceway of an initial shoulder height of 4.5 mm. The results showed that the initial shoulder heights of the example bearing were designed to be excessively high under the applied loads simulated in this paper. Compared with the results by contact analysis, the results in the EHL conjunction which is more realistic showed that the higher pressure spike is generated more inward of contact center when the ellipse truncation occurs. From the EHL analysis it is shown that the minimum film thickness occurs near the edge of shoulder, not in the outlet region of rolling direction, and the value is the order of the equivalent surface roughness of the general ball and raceway, which can lead to adhesional failure. In addition, as the applied load increases the critical shoulder heights in the EHL state are calculated as higher values compared with in the dry contact. Therefore, it is concluded that the determination of optimal shoulder height should be performed by the EHL analysis.