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85
For the assumed conditions, the value of ω
K
' is negative. Actually,
in the planetary train, Fig.7a, gear K has a positive direction of rotation
and, being connected to the output shaft, is loaded by an external torque
acting in a direction opposite to its motion. Thus, the external torque
acting on shaft C and gears K and K' is negative. The product of
negative angular velocity and negative torque is positive. Therefore,
gear K' is the driver in the equivalent train obtained by stopping the
planet cage.
This case is an example of an interesting peculiarity in analysis.
The gear that must be considered as the driver for analysis of train
performance is actually connected to the driven shaft of planetary train
and appears, at first, to be a driven member. Consequently, with the
planet cage fixed, gear A', becomes the driven member. Now the
expression for efficiency may be derived. If the power at output shaft C
of the planetary train is P
0
, the power developed in the engagement of
gears J and K is P
0
R. Since gear K' is considered as the driver, the tooth
mesh loss in this train is
)1(
0 tt
RPL
η
−
=
(22)
Efficiency of this train is
)1(1
1
)1(
00
0
tt
t
RRPP
P
e
η−+
=
η−+
= (23)
Planetary velocity ratio R is
(a)
Output shaft, H
Planet gear, J
Planet cage, G
Planet gear, B
Input shaft, D
Sun gear, A
(fixed)
0 40 -80 80 120 160 200-40-120-160-200
40
20
60
80
100
Speed ratio
Efficiency, per cent
(b)
Fig.7–Basic differential planetary gear system showing, a, arrangement of system
elements and, b, relationship between efficiency and speed ratio
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