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78
The essential difference between Equations 1 and 5 is in the
magnitude of the velocity of engagement. In a conventional gear train
with fixed axes of gear rotation,
v
e
is equal to pitch-line velocity v of the
gear, and the maximum limit of the product
Fv is equal to the input
power of the train. In a planetary gear train, however, the velocity of
engagement is affected by the relative motion of the planet cage. Thus,
velocity of engagement
v
e
is not equal to pitch line velocity v; and
product
Fv
e
is not equal to the input power at the driving gear of the
pair.
For comparison, two gear pairs will be considered, Fig. 2: one in a
planetary gear train, the other in a conventional train. Assume that the
geometry, size, and other characteristics of the corresponding gears are
the same. Tangential forces
F acting on both gear pairs are equal, and
the pitch-line, or absolute, velocity of the sun gear in the planetary train
is the same as that for the corresponding gear in the conventional train.
The ratio of tooth mesh losses in these "equivalent" gear pairs is
R
v
v
Fv
Fv
l
L
ee
==
∆
∆
= (6)
where
R is defined here as the planetary velocity ratio. Therefore, the
tooth mesh loss in a pair of gears in a planetary train is
(
)
η
−
=
∆
=
∆== 1RPRPFvRlRL
ii
(7)
where
l is the loss in an equivalent gear pair with fixed centers and the
same input power as the planetary pair. Fig. 2.
Sun gear
(input)
Planet pinion
Planet carrier
Ring gear
(fixed)
Input
gear
a) b)
Fig. 2—Comparison of equivalent gear pairs in, a, simple planetary system and b
conventional gear train
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