Page 266 - Tribology in Machine Design

Rolling-contact bearings 251

Torque, M gr, increases with an increase in contact angle a and attains a
maximum value when y. = n/2. Thrust bearings, for example, have such an
angle. Radial bearings are characterized by a =0, so that M gr =0. In order
to avoid gyroscopic spin of the balls it is necessary to satisfy the inequality

where P is the axial load. The effect of gyroscopic spin is especially
noticeable in radial thrust bearings.

7.2.3. Friction torque due to elastic hysteresis
The energy losses due to elastic hysteresis in the material of the bearing can
be determined, assuming that during the loading cycle a specific quantity of
kinetic energy is expended. Friction torque caused by elastic hysteresis has
been sufficiently studied for the case of slow-running bearings. In the case of
high-speed running, however, large centrifugal forces arise and they create
an additional load. Hence, in all the expressions given below, friction torque
M must be treated as a function of rotational speed.
When the ball rolls along the raceway, the Hertz contact ellipse is formed
between them as shown in Fig. 7.4. The energy developed during the rolling
of the body about one point of the raceway is proportional to the load and
to the magnitude of deformation. Contact time, f, of the rolling body with
Figure 7.4
the raceway is directly proportional to the extent of the contact surface in
the direction of motion 2b and inversely proportional to the circumferential
velocity, rco (see Fig. 7.5), thus

The energy losses during the contact of the rolling body with one point of
Figure 7.5 the raceway are

where P is the load and d denotes the associated deformation.
According to the Hertz theory, the deformation for point contact is given
by

where Up is the sum of the reciprocal of the principal radii of curvature, i.e.

here PI and p 2 are the reciprocals of the principal radii of curvature of the
bodies at the point of contact and K is a coefficient depending on the
function F(p) defined as