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79
The efficiency of this planetary gear pair is given by
()
η−−=∆−=
−
== 111
0
RR
P
LP
P
P
e
i
i
i
(8)
This same method may be applied for determining losses in any
planetary gear pair and, consequently, for determining the efficiency of
the entire planetary train.
3. Simple Planetary Train
This method may be best illustrated by an example. Consider the
simple planetary gear train, Fig. 2a, which is represented schematically
in Fig. 3a. This system is one of the most compact planetary gear
arrangements having a wide range of application possibilities in design.
Design of an actual planetary train of this type is shown in Fig. 4. In this
train, Fig. 3a, planet gear
В engages simultaneously with sun gear A,
which is keyed to input shaft
D, and with fixed internal (ring) gear C.
Planet gear
A rotates freely around its own axis on a shaft mounted to
the planet cage,
G, which is keyed to and drives output shaft H.
The speed ratio of this train is
A
C
G
A
H
D
p
d
d
m +=
ω
ω
=
ω
ω
= 1
(9)
where subscripts identify the corresponding system elements. The value
of
m
p
is always positive in this arrangement and input and output shafts
always rotate in the same direction.
Assume now that the planet cage is "stopped" by adding to the
entire system an additional angular velocity, — ω
H
, equal in absolute
value to the angular velocity of the output shaft, but opposite in
direction. The relative motions of all members of the planetary train are
unchanged. However, planet gear
A now becomes an idler, and the
entire train may be considered as a conventional gear train with fixed
axes of rotation. Since the external torque applied to input shaft does not
change, the tangential forces acting on the gears are the same as those in
the actual planetary train. The only change is in the pitch-line velocities
of the gears; they are now equal to the velocities of engagement of the
gears in the planetary train. This train with "stopped" planet cage is
depicted in Fig. 3b.
In Fig. 3b, the absolute angular velocity,
A
ω
′
, of gear A' is given by
−ω=
ω
−ω=ω−+ω=ω
′
p
A
p
A
AHAA
mm
1
1)(
(10)
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