Roller Bearing equations
The rollers must roll without slipping on both the inner and
outer race, otherwise the wear rate would be very high.
Inner race stationary
Holding the inner race stationery implies the outer race
must move at speed
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(1.1)
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where subscript o stands for outer, CM stands for the center
of mass of the rollers, and r stands for the roller bearing. The ω’s
are angular rotation rates. rCM
is the radius from the center of the bearing to the center of the roller. ro is the inside radius of the
outer race.
We also have the relations:
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(1.2)
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We can solve equations(1.1)
and (1.2)
for the spin rate of the rollers:
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(1.3)
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Substituting(1.3) into (1.1)
we have:
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(1.4)
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Solving for equation (1.4)
for ωo we have:
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(1.5)
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Outer race stationary
Outer race is stationery implies that the inner race (axle)
must move at speed:
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(1.6)
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Where rI is the radius of the outside of the
inner race (or axle).
We also have the relations:
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(1.7)
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and we can solve (1.6)
for
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(1.8)
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Substituting (1.8)
into (1.6)
we obtain:
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(1.9)
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Solving equation (1.9)
for ωI we have:
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(1.10)
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Conclusions:
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