Page 273 - Tribology in Machine Design

258 Tribology in machine design

but not at the outer-race contact C. If the ball has an angular velocity CO B
about the axis OA, then it has a rolling component co r and a spin component
co s>0 relative to the outer race as shown in Fig. 7.13. The frictional heat
generated at the ball-race contact, where slip takes place, is

Figure 7.13 where M s is the twisting moment required to cause slip. Integrating the
frictional force over the contact ellipse gives

whenfe/a = l;a=0°and E = n/2, but when b/a=0; a = 90° and E=\. For
the same P, M s will be greater for the ellipse with the greater eccentricity
because the increase in a is greater than the decrease in E. In a given ball-
bearing that operates under a given speed and load, rolling will always take
place at one race and spinning at the other.
Rolling will take place at the race where M s is greater because of the
greater gripping action. This action is referred to as ball control. If a bearing
is designed with equal race curvatures (race curvature is defined as the ratio
of the race groove radius in a plane normal to the rolling direction to the
ball diameter) and the operating speed is such that centrifugal forces are
negligible, spinning will usually occur at the outer race. This spinning
results from the fact that the inner-race contact ellipse has a greater
eccentricity than the outer-race contact ellipse. The frictional heat gene-
rated at the ball-race contact where spinning takes place accounts for a
significant portion of the total bearing friction losses. The closer the race
curvatures, the greater the frictional heat developed. On the other hand,
open race curvatures, which reduce friction, also increase the maximum
contact stress and, consequently, reduce the bearing fatigue life.

7.4.2. High speeds
At high speeds, the centrifugal force developed on the balls becomes
significant, and the contact angles at the inner and the outer races are no
longer equal. The divergence of contact angles at high speeds tends to
increase the angular velocity of spin between the ball and the slipping race
and to aggravate the problem of heat generation. Figure 7.14 illustrates
Figure 7.14 contact geometry at high speed in a ball-bearing with ball control at the
inner race. The velocity diagram of the ball relative to the outer race
remains the same as in the previous case (normal speed) except that y has
become greater and the magnitude of co s<0 has increased. As the magnitude
of P becomes greater with increasing centrifugal force, ball control
probably will be shifted to the outer race unless the race curvatures are
adjusted to prevent this occurring. Figure 7.15 illustrates ball control at the
outer race. The velocity of the ball relative to the inner race is shown in Fig.
7.16. The inner-race angular velocity co ; must be subtracted from the
angular velocity of the ball CO K to obtain the velocity of the ball relative to
Figure 7.15 the inner race co Bii.