One of the major sources of sliding friction is the sliding that exists on the surface of the elliptical contact area. When a ball rotates relative to the deformed surface, combined rolling and sliding motions occur.^1,12,14 Depending on bearing geometry and operation, one or two PRLs exist in the ball-race contact area.^8,20 In the region other than the PRLs, sliding occurs due to the presence of differential slippage, as shown in Fig. 2. The slip velocities in the rolling directions can be determined as follows¹:

Fig. 2 
Slip velocity distribution on the elliptical contact surface

For inner race,

For outer race,

where ω is the bearing rotational speed, and d_m, D_a are the pitch diameter and ball diameter, respectively. a_i and a_e are the semimajor radii of the contact ellipses on the inner and outer races, respectively. The relative speeds of the inner and outer race to the ball are given as

The defined ω_R and ω_m are the ball pivotal speed and orbital speed, respectively, and R_k is the Hertzian contact radius, or the radius caused by deformation as given by,^1,8

where k = i, e denote the inner and outer race, respectively, and f_k is the conformity factor that is defined as the ratio of raceway radius to ball diameter. β is the pitch angle. Using the equal raceway control theory,^23,24 the pitch angle is represented as

where α_i, α_e are the ball contact angles on the inner and outer races, respectively.

If the rolling motion does not occur on a line exactly parallel to the raceway, a dependent motion, called spinning, occurs. Spinning is a macro-sliding that is responsible for a major portion of the total friction in an ACBB and can contribute to significant heat generation. Any ACBB experiences spinning about the transverse direction to the center of the elliptical contact area. The contact angle inevitably causes ball spinning to take place.^1,6 The spinning motion in the elliptical contact area is depicted in Fig. 3. The pivotal speeds on the races are as follows:

Fig. 3 
Friction force and spinning velocity acting on the elliptical contact area

For inner race,

For outer race,

Another factor that may complicate the ball motion in ACBBs is the existence of gyroscopic moment acting on the balls. In ball bearings with non-zero contact angles between the balls and races, a gyroscopic moment occurs in each ball. This motion creates pure sliding colinear with the major axis of the elliptical contact area.¹ This implies that an additional sliding orthogonal to rolling direction will occur. At low speeds, such gyroscopic moment is negligible but becomes significant at high speeds. Gyroscopic motion may be prevented when the ACBB is sufficiently preloaded.¹⁴

On the other hand, at very high speeds, balls may experience gross sliding over the races.^11-17 Such behavior is called skidding and can significantly degrade bearing endurance and performance. Skidding usually happens in ACBBs operating under a relatively light preload with rapid accelerations and decelerations.¹ Then, the loading is not enough to develop sufficient friction force in ball-race contacts to overcome dragging, churning losses, and to avoid gyroscopic motion. Controlling skidding is an important issue in utilizing bearings.^1,7,14,17

Between the ball and races under loading, there forms an elliptical contact area, in which pure rolling lines (PRLs) may exist. PRLs denote “lines” in the contact area at which pure rolling motion occurs. In the region other than the location of PRLs, sliding dominates due to differential slippage.¹ The elliptical contact area created by ball-race contact is shown in Fig. 4. Sliding within the elliptical contact area is significantly dependent on the PRLs.

Fig. 4 
Elliptical contact area in ball-race contact

For a ball-race contact, one or two PRLs can be found inside the elliptical contact area.^8,20 Fig. 5(a) shows how PRLs are formed in the elliptical contact area. If the generatrix of motion [green-dotted line] is angled with respect to the tangent plane [red-dashed line] at the center of the contact surface, the center of rolling is positioned asymmetrically in the elliptical contact area. Depending on the angle of the generatrix to the contact surface, one or two lines of intersection may occur at which pure rolling can be obtained. In other words, pure rolling may occur in one or two lines contained in the elliptical contact area as illustrated in Fig. 5(b).^1,8,20

Fig. 5 
Description of PRLs in the ball-race contacts

Houpert²⁰ has proposed an analytical calculation method for PRL locations with consideration of the effect of centrifugal force. He considered that the contact angles on the races become unequal during operation. Fig. 6 shows z₁ and z₂ which represent the heights of PRLs of the deformed contact surface caused by the Hertzian contact effect. The spatial shape of the contact ellipse is assumed to be parabolic as shown in Fig. 7.²⁰ Here, the vertex and focus of the parabola are set to be at (0,0) and (0, R/2), respectively.

Then, the equation of the parabola is given by

Thus, the heights of the two PRLs can be obtained as

Fig. 6 
Ball-race contact under the deformed contact surface

Fig. 7 
Parabolic representation of ball-race contact

Assume here that c_2ka_k(> 0) and c_1ka_k(< 0) without loss of generality.

From Fig. 6, it can be seen that the two PRLs can be related to the slope of generatrix and the normal plane as follows:

where γ_k is the angle between the red and green lines in Fig. 5(a). By substituting from Eqs. (10) to (12), the following equation can be achieved:

The two PRLs, associated with a geometrical E parameter, are shown in Fig. 8. Here,

Fig. 8 
PRLs as a function of E and e parameters for 2-PRL case

where

When the sliding friction force is very small (e≈0) and E > 1 as illustrated in Fig. 8 with red squares, one PRL is present inside the elliptical contact area while its pair lies outside. On the other hand, the e parameter is defined as the ratio of sliding friction force and maximum friction force, μQ_k, i.e.,

where μ denotes the sliding friction coefficient,²⁸ Q_k being the ball contact load. The sliding friction can be expressed for the differential area of dS as

where b_k is the semi-minor radius of the contact ellipse. Eq. (17) is used to estimate the net sliding frictional force in the rolling direction. We can utilize PRLs to determine such sliding friction force. Note here that the direction of sliding changes across PRLs. In the presence of two PRLs, the sliding direction changes twice, while the sliding direction changes once in the case of one PRL.

It should be noted here that the necessary values such as ball contact loads, contact angles, and ball speeds should be precalculated. To this end, we utilized the quasi-static model of ball bearings based on de Mul.^23,24,26,27 Then we went on an iterative process associated with the equilibrium of frictional moments acting on individual balls.

Any ACBB experiences spinning motion about the normal direction to the center of the elliptical contact area (See Fig. 9).^{8,20,23,24,29} The friction moment with respect to the center of the elliptical contact area is calculated by¹⁰

Fig. 9 
Spinning motion of a ball in ACBBs

which gives the integrated form as

where θ and φ are expressed as

The sliding velocities in the x- and y-directions are represented by v_x,k and v_y,k the sliding velocity in the x-direction on both races are assumed to be zero.

The PRL locations represented by c-values are very useful in estimating sliding friction force. The spinning moment is affected by the locations as well as the number of PRLs in the elliptical contact area. Houpert and Popescu^8,20 suggested the spinning friction torque as a function of c-values as follows:

For 2-PRL case,

For 1-PRL case,

Then the total friction torque due to spinning is written as^24,29

For validation of the PRL determination, we used spinning friction torque as presented here; spinning friction torque by Eqs. (22) and (23) is compared with that by Eq. (19).

The improved spinning friction torque formulation given by Eq. (19) was used to compare the spinning friction formulas as a function of PRL locations. Fig. 11 shows the comparison of spinning friction torques by the two methods with rotational speed from 0 to 15 k rpm. Fig. 11 shows a good match between the current method and the conventional integration method, also showing that the c-value approach as presented in this study is useful in estimating spinning friction torque.

Fig. 11 
Comparison of spinning friction torques calculated by full integration and by c values

The PRL location determination was validated using spinning friction torque estimation. However, the confirmation of the existence of one or two PRLs in the elliptical contact area was not explained. Here, we will address the number of PRLs and differential sliding (Slip) velocity inside the elliptical contact area.

To investigate the number of PRLs and the behavior of c₁ and c₂, the differential sliding (Slip) velocities of the ACBB were plotted with rotational speeds of 5, 10, and 15 k rpm. Fig. 12 shows the differential sliding velocities along with the PRLs calculated using the E parameter, defined in Eqs. (14) and (15). The PRL locations calculated with the E parameter (c₂-Circle, c₁-Square) are well matched with those found at the points of intersection between the differential sliding velocity curves and the x-axis. In Fig. 12, it can be observed that the differential sliding velocities under F_z = 1,000 N intersect the x-axis twice in the dimensionless major diameter of the contact ellipse, showing that the direction of sliding velocity changes twice. In this case, the estimated PRLs show that there are two PRLs in the elliptical contact area as depicted in Figs. 12(a) and 12(b) both for the inner and outer races.

Fig. 12 
Differential sliding (Slip) velocity distributions with varying rotational speed

Fig. 13 shows the effects of centrifugal force and gyroscopic moments on the locations of PRLs. As rotational speed increases, the effects of centrifugal force and the gyroscopic moment become more significant. Without these effects, c₁ and c₂ on both races lie around -0.7 and 0.1, respectively, for the entire speed range.

Fig. 13 
PRL locations under varying pure axial loads and rotational speeds

The effects of centrifugal force and gyroscopic moment tend to move the two PRLs from the original position to a lower position on the inner race and a higher position on the outer race. The E parameter is a good index to correlate c₁ and c₂. Due to the centrifugal force and gyroscopic moment, the E parameter increases on the inner race but decreases on the outer race. Note here that the E parameter is closely tied with the contact angle on both races; the contact angle on the inner race increases while the contact angle on the outer race decreases, as the rotational speed increases.

Thus, the direction of PRL movement should be dependent on the contact angles.

Changing axial load also affects the location of PRLs. Fig. 14 depicts the locations of PRLs as a function of axial load and rotational speed. Fig. 14(a) shows the PRL locations on the inner race. The E parameter increases and forces c₁ to approach -1. c₁ approaches -1 quicker with increasing the rotational speed, when a relatively low axial load is applied.

Fig 14 
PRL locations under varying axial load and rotational speed

Fig. 14(b) depicts the PRL locations on the outer race which behave oppositely to the inner race. Because the E parameter decreases, c₁ tends to increase towards positive side.

Unloaded contact angles, which inevitably affect the contact angles, change PRLs during operation. Fig. 15 shows that the unloaded contact angle gives a significant effect on the number, as well as the locations of PRLs. Figs. 15(a) and 15(b) show the PRLs on the inner and outer races, respectively. As the unloaded contact angle increases, the E parameter also increases. For α₀ = 15^o, two PRLs are present in both races. However, when the unloaded contact angle increases to α₀ = 25^o, the inner race has one PRL, while the outer race has only one PRL for low speed but two PRLs for high speed. Lastly, for the biggest unloaded contact angle, α₀ = 40^o, only one PRL exists for both races throughout the entire rotational speed range.

Fig. 15 
PRL locations under varying unloaded contact angles

This section presents simulation results for radial load and unloaded contact angle effects subject to combined loading of axial and radial loads. In this case, axial load of 500 N is assumed.

Fig. 16 shows PRL locations on individual balls under the effect of combined load with various radial loads of 100, 500, and 1,000 N at 15 k rpm. Fig. 16(a) shows that the radial load can alter the number of PRLs on individual balls. In this case, increasing radial load can cause some balls, unlike the other balls, to operate with only one PRL on the inner race. Balls placed near 0^o angle have more consistent behavior owing to sufficient contact load because the downward radial load is applied. However, balls around 180^o angle are likely to experience a different condition that may lose one of the PRLs. Here, loss of c₁ in a ball implies the possible occurrence of skidding at the ball.

Fig. 16 
PRL locations on each ball under varying radial load; axial load F_z = 500 N, rotational speed of 15 k rpm, and unloaded contact angle of 15^o

On the other hand, the PRLs on the outer race, shown in Fig. 16(b), manifest to be less sensitive to radial load. Such insensitive nature of the outer race against radial load comes from the decreased contact angle due to centrifugal force. Indeed, the balls around 180^o are quite different from those on the inner race.

Fig. 17 shows the effect of unloaded contact angle on the PRL locations under the combined loading. Fig. 17 shows that a significant change occurs in the behavior of PRLs with varying unloaded contact angle. With a high unloaded contact angle, only one PRL is likely to exist both on the inner and outer races even under a small amount of radial load.

Fig. 17 
PRL locations on each ball under varying unloaded contact angle; axial load F_z = 500 N, radial load F_r = 1,000 N, and rotational speed of 15 k rpm