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89
Efficiency of the train may be expressed by
t
tt
i
i
t
R
P
LP
e
η
η
−
−
η
=
−
=
)1(
(33)
where
−−−=
ω
ω
′
= 1
1
pK
K
m
R (34)
Introducing the absolute value of R into Equation 33 gives
t
p
tt
t
m
e
η
−η−−η
=
1
1
)1(
(35)
It is interesting to note that e
t
=0 when
−η−=η 1
1
1
p
tt
m
and this mechanism becomes self-locking.
Consider now a train of this type which has ratio
1>
KBJA
dddd .
Such a mechanism will increase speed and reverse the direction of
rotation when
21
<
<
KBJA
dddd
. It will reduce speed and reverse the
direction of rotation if
2>
KBJA
dddd . This latter case will be assumed
for illustration. In this reduction drive, mp is negative. Angular velocity
of the planet cage is ω
G
= ω
K
/m
p
and the angular velocity of gear K with
the planet cage stopped becomes
−ω=
ω
−+ω=ω
′
p
K
p
K
KK
mm
1
1 (36)
Because m
p
in this case has a negative value, ω'
K
is positive. The
product of torque and velocity at gear K' is positive, making this gear
the driver in the equivalent train. Therefore, the tooth mesh loss in this
train is given by Equation 13, and the efficiency of planetary trains of
this type, in which the speed ratio is negative (
1>
KBJA
dddd ), can be
determined from Equation 14.
When a
1)1( =η
−
t
R
, e
t
= 0. The value of R for this train is given by
Equation 11, where m
p
is a negative value. The relationship between
efficiency and speed ratio for this planetary train arrangement, Fig. 8a,
is shown in Fig. 8b. This chart is based on the assumption that η
t
= 0.97.
When m
p
= +1, the entire system revolves as a unit and η
t
=1, or 100 per
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