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84
Assume that input shaft D revolves clockwise. The speed ratio m
p
is
always positive; therefore, the angular velocity of output shaft H,
ApAH
m ω<ω=ω , is also positive.
When angular velocity
pAH
m
ω
−
=
ω
−
is added to the system to
stop the planet cage, the value of ω
A
' which is the new absolute angular
velocity of gear A is given by Equation 10. Because the value of m
p
for
this train is greater than one, ω
A
' is positive if ω
A
is positive. Therefore,
the product of torque and angular velocity for gear A' will be positive,
making this gear the driver in the train with the stopped cage.
Following the same reasoning, it can be shown, that the expression
for efficiency for this planetary train is the same as for the previous two
types (Equation 14). The difference lies in the value of R. For this train,
BCJA
BC
pA
A
dddd
dd
m
R
+
=−=
ω
ω
′
=
1
1
(19)
It can be seen that R for this train is always smaller than one. The
efficiency of this train must therefore be somewhat higher than that of a
comparable conventional gear train consisting of two gear pairs with the
same tooth mesh loss ratios.
Planetary Differential: A typical differential planetary gear train
Fig. 7a, has the planet cage connected to the input shaft, and sun gear K
is driven member. The speed ratio of this train is
JAKB
KB
KB
JA
H
D
p
dddd
dd
dd
dd
m
−
=
−
=
ω
ω
=
1
1
(20)
If the value of m
p
is positive, the output and input shafts rotate in
the same direction. If m
p
is negative, they revolve in opposite directions.
The most interesting peculiarity of this train is that a speed ratio of any
magnitude, even infinity (ω
H
= 0), can theoretically be obtained.
The efficiency of this train may be determined by the methods
discussed previously. Assume that input shaft C revolves in a clockwise
(positive) direction and
1
<
KGJA
dddd so that m
p
>1 and is positive. To
stop planet cage G, let the entire system be revolved at an angular
velocity —ω
D
. With the planet cage stopped, the absolute angular
velocity of gear K becomes
−ω−=ω−
ω
=ω−+ω=ω
′
p
DD
p
D
DKK
mm
1
1)( (21)
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