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80
Velocity
A
ω
′
, which is the pitch-line velocity of gear A', is thus also
the velocity of engagement of gear
A in Fig. 3a. Since the value of m
p
for this planetary train is always greater than one, the value of
A
ω
′
is
always positive if ω
A
is positive. The planetary velocity ratio, R
AB
, for
gears
A and B is
pA
p
A
A
A
AB
m
m
R
1
1
1
1
−=
ω
−ω
=
ω
ω
′
=
(11)
This result means that the velocity of engagement of gear
A in the
planetary train, Fig. 3a, is
R
AB
times the pitch-line velocity of this gear,
and the power developed by gear
A is R
AB
times the power at input shaft
D.
With the planet cage stopped and planet gear
B serving as an idler,
Fig. 3b, all gears in the train have the same pitch-line velocity.
Therefore,
R
BC
, the planetary velocity ratio for gears B and C, is equal to
R
AB
, and R
AB
= R
BC
= R is a constant value for a given planetary train.
To derive an expression for the efficiency of the planetary train, it
is important to know which gear in the train with the stopped planet
cage is the driver. The fact that gear
A is the driver in the planetary
train, Fig. 3a, does not necessarily mean that gear
A' has the same
function in the stopped system, Fig. 3b.
In any gear train, an external torque applied to the input shaft acts
in the direction of rotation of this shaft. Therefore, the product of torque
and angular velocity at the input shaft is always positive. The external
resisting torque acting on the output shaft, however, always acts in a
direction opposite to that of shaft rotation. Therefore, the product of
external torque and angular velocity for the output shaft is always
negative.
For the present case, Fig. 3, let it be assumed that ω
D
, ω
D
', and the
external torque acting on shaft
D' all have positive values. Thus, the
product of torque and angular velocity at shaft
D' is positive, and gear A'
is the driver. In this instance, the position of the driving gear is the same
in both trains, Figs. 3a and b.
The power,
P
AB
', developed between gears A' and B' in the
equivalent train, Fig. 3b, is:
P
AB
' = P
i
R. Equal power is developed by
gears
A and B, Fig. 3a. If the tooth mesh loss ratio for gears A' and B' is
∆
1
, and the for gears B' and C' is ∆
2
, the power delivered at output shaft
H' is
tiiH
RPRPP
η
=
∆
−
∆
−
=
′
)1)(1(
21
(12)
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