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87
Therefore, the efficiency of this train, considering tooth mesh loss only,
is
R
RPP
P
e
tt
t
t
t
t
)1(
)1(
1
00
0
η−+η
η
=
η−
η
+
= (28)
The value of R for this train is given by
)1(
)(
p
K
DK
K
K
mR −−=
ω
ω
+
ω
−
=
ω
ω
′
= (29)
The absolute value of R must be used in Equation 28 for efficiency
calculations.
Sample Calculations: A planetary differential train. Fig. 7a, has
the following design data:
025.1=
KBJA
dddd and η
t
= 0.97. From Equation 20, m
p
= -40;
from Equation 29, R = -41; and from Equation 28, using the absolute
value of R=41, e
t
=0.4409. In a planetary gear train of this type with P
o
=
1 hp, the power developed by gears J and K is more than 41 hp while
that developed by gears A and B is 41/0.97 = 42.26 hp. These conditions
must be taken into account when designing gears for the planetary train.
The relationship between efficiency and speed ratio for, the
differential planetary gear train, Fig. 7a, is shown in Fig. 7b. The left-
hand portion of the curve represents the cases where speed ratio is
negative, and the right-hand portion those with positive speed ratios.
When mp = +1, the entire system is revolving as one unit, without
relative motion of its parts, and et = 1 or 100 per cent. Actually,
however, such a ratio is not possible in practice because it requires that
0=
KBJA
dddd . When, mp = - 1, the absolute value of R = 2, and the
losses in this train are about twice as large as in a conventional train
consisting of the same gears.
The chart indicates that efficiency of this type of planetary train
decreases rapidly with the increasing speed ratio, and that gears in trains
having large speed ratios operate under extremely severe service
conditions. Apparently, trains of this type are most suitable for
applications where speed ratio or power transmitted are relatively low.
Reversed Power Flow: If, in the train depicted in Fig. 7a, the sun
gear K is made the input member and the shaft connected to the planet
cage becomes the output member, a different mechanism will be
obtained, Fig. 8a.
The speed ratio of this modified train is
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