Page 195 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 181
If the carrier is fixed (not moving), the planetary gear reduces to a standard gear,
with the gear ratio being the ratios of each gear (Figure 3.26c),

( )
w 3 w 3 N 1 N 1
N = = sgn =− (3.271)
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w w N N
1 1 3 3
( ) ( )
w 4 w 4 N 1 N 3 w 4 N 1 N 1
N 41 = = sgn ⋅ = sgn =− (3.272)
w 1 w 1 N 3 N 4 w 1 N 4 N 4
where the gear that is connected to both input shaft and output shaft (in this case the planetary
gear) acts as an idler gear. This is called the “idler” or “idling” mode of operation for a
planetary gear. The idler gear does not change the gear ratio, but changes the direction of
rotation only. The contribution of the idle gear to the gear ratio is multiplication by −1.
The sign of rotation in the gear ratio can also be incorporated into the definition of the gear
ratio. It is customary to consider the direction of rotation in counter-clockwise as positive,
and the rotation in clockwise direction as negative. The sgn( ⋅ ) is negative for a planetary
gear with internal ring gear (Figure 3.25a) for both equations. The sgn( ⋅ ) is negative for
the first, and positive for the second, equation for the planetary gear type with external ring
gear shown in (Figure 3.25b).
If the planetary gear is fixed about its local axis of rotation on the carrier, the relative
rotational speed of the planetary gear with respect to the carrier is zero, (w = 0.0), then
32
the gear mechanism is in locked condition, where the angular speed of the sun, carrier and
the ring gear are all same, in other words, have all gear ratio of 1:1 with respect to each
other (Figure 3.26d).
w = w = w (3.273)
1 2 4
N = N = 1 (3.274)
2,1 4,1
w = w + w (3.275)
3 2 32
w = w + 0 (3.276)
3 2
w = w (3.277)
3 2
In the planetary gear mode, the carrier is not fixed, but moving. This mode is also
referred to as the “walking” mode. In walking mode, either the sun gear or the ring gear
is held stationary. In the case of the stationary sun gear, the planetary gear always turns
in the same direction on its pin as the planet carrier rotation (Figure 3.26a). In the case of
the stationary ring gear, the planetary gear turns in the opposite direction on its pin as the
planetary carrier (Figure 3.26b).
In planetary mode, the gear ratio relationships are applicable in terms of the relative
velocities of the components with respect to the carrier. The relative gear ratio definition with
respect to the planetary carrier is the key principle in defining the gear ratio relationship
of planetary gears, analogous to the non-planetary case where the carrier is fixed. Below
are the two fundamental kinematic equations for planetary gear ratio calculations
( )
w 32 w − w 2 w 32 N 1 N 1 w − w 2 N 1
3
3
N = = = sgn =− → =− (3.278)
r31
w w − w w N N w − w N
12 1 2 12 3 3 1 2 3
w 42 w − w 2
4
N r41 = =
w 12 w − w 2
1
( ) ( )
w 42 N 1 N 3 w 42 N 1 N 1 w − w 2 N 1
4
= sgn ⋅ = sgn =− → =− (3.279)
w 12 N 3 N 4 w 12 N 4 N 4 w − w 2 N 4
1
where we define the above gear ratios as relative, as indicated by the subscript “r.”