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81
where η
t
= the overall efficiency of a conventional gear train consisting
of the same gears as the planetary train. Therefore, in the planetary train
and its equivalent, which have equal losses,
)1(
titiit
RPRPRPL
η
−
=
η
−
= (13)
The efficiency of a planetary train of this type is
)1(1
)1(
t
i
tii
i
ti
t
R
P
RPP
P
LP
e η−−=
η
−
−
=
−
= (14)
From Equations 11 and 14
−η−−=
p
tt
m
e
1
1)1(1 (15)
Ring gear C is always larger than sun gear A so that the value of R
for this planetary train is always smaller than one but larger than 0.5.
The tooth mesh losses in this train, then, are smaller than, but not less
than half, the tooth losses in a conventional train consisting of the same
gears. Consequently, the efficiency of this type of planetary train is
always higher than the efficiency of a corresponding conventional train
but lower than the efficiency of one pair of gears with fixed centers and
the same unit tooth mesh loss ratio, ∆.
Generally speaking, this efficiency may be considered as highly
satisfactory for planetary gear systems. However, this type of train can
be de-signed only for relatively small speed ratios.
Sample Calculations: A planetary gear train of the type shown in
Fig. 3 has the following design data: d
C
== 12 in., d
A
= 2 in., d
B
== 5
in., and m
p
= 7. For gears A and B, efficiency η
1
= 0.98, and for gears B
and C, η
2
= 0.99.
The efficiency of a conventional train with the same gears is η
t
=
η
1
η
2
= 0.9702. For the planetary train, Fig. 3a, using Equation 11, R
AB
=
R
BC
= R = 1 - (1/7) = 6/7. From Equation 14, e
t
= 1 - (6/7)(1 - 0.9702) =
Fig. 3—Simple
planetary gear system
showing, a, schematic
arrangement of system
elements and, b,
equivalent system
obtained by "stopping"
the planet cage. Prime
notations identify
corresponding
members
Sun gear, A
Planet gear, B
Ring gear, C
(fixed)
Input shaft, D
Output
shaft, H
Planet
ca
g
e, G
A’
B’
C’
D’
H’
G’
(a) (b)
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