The rollers must roll without slipping on both the inner and outer race, otherwise the wear rate would be very high.

Holding the inner race stationery implies the outer race must move at speed

(1)

where subscript o stands for outer, CM stands for the center of mass of the rollers, and r stands for the roller bearing. The w’s are angular rotation rates. r_CM is the radius from the center of the bearing to the center of the roller. r_o is the inside radius of the outer race.

We also have the relations:

(2)

Substituting 2 into 1 we have:

(3)

Solving for w₀ we have:

(4)

Outer race is stationery implies that the inner race (axle) must move at speed:

(5)

Where r_I is the radius of the outside of the inner race.

We also have the relations:

(6)

Substituting 6 into 5 we obtain:

(7)

Solving equation 7 for w_I we have:

(8)

In terms of the diameters, the equations are:

When the inner race rotates and the outer race is stationery:

When the outer race rotates and the inner race is stationery:

is the angle moved by the center of mass of the bearing ball