Page 199 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 185
From the basic planetary gear relationship derived above, we can write the following for
planetary stage 4,
N 4,1 N 4,4
w = ⋅ w + ⋅ w (3.291)
4,2 4,1 4,4
N + N N + N
4,1 4,4 4,1 4,4
w out = w 5,1 (3.292)
= w (3.293)
4,2
where equations are simply modified by their subscripts to indicate the planetary stages.
Given the three relationships above, the output speed w out = w 5,1 = w 4,2 can be calculated
as function of input speed w . In other words, we can eliminate w 4,4 from the above
4,1
equations, resulting in a gear ratio relationship between w 4,1 (the input shaft speed) and
w 4,2 which is the same as the output shaft speed of the transmission. It can be shown that
the resuting gear ratio relationship is
N 4,1 N 4,4
w 4,2 = ⋅ w 4,1 + ⋅ w 4,4 (3.294)
N 4,1 + N 4,4 N 4,1 + N 4,4
N 4,1 N 4,4 N 5,1
= ⋅ w 4,1 − ⋅ w 4,2 (3.295)
N 4,1 + N 4,4 N 4,1 + N 4,4 N 4,4
[ ] −1
N 4,4 N 5,1 N 4,1
= 1 + ⋅ w 4,1 (3.296)
N 4,1 + N 4,4 N 4,4 N 4,1 + N 4,4
where the gear ratio at this gear is
[ ] −1
N 4,4 N 5,1 N 4,1
GR = 1 + (3.297)
1
N 4,1 + N 4,4 N 4,4 N 4,1 + N 4,4
N 4,1
= (3.298)
N 4,1 + N 4,4 + N 5,1
Example: Planetary Gears Consider Figure 3.25a with the following gear param-
eters: N = 25, N = 50, N = 100 gears. Inputs are specified as: sun gear speed w =
4
1
1
3
100 rev∕min and the ring gear is stationary, w = 0.0rev∕min. Find the speed of the carrier
4
and the planetary gear and absolute gear ratios between the sun gear and the carrier, and
the sun gear and the planetary gear.
For the direct application of the two equations we have above, first we will find the
absolute velocity of the two components, then calculate the absolute gear ratios.
w 42 w − w 2 N 1 N 3 N 1
4
N r41 = = = sgn( ⋅ ) ⋅ = sgn( ⋅ ) (3.299)
w 12 w − w 2 N 3 N 4 N 4
1
0 − w 2 25
= =− (3.300)
100 − w 2 100
w = 20 rev∕min same direction as w 1 (3.301)
2
w 32 w − w 2 N 1
3
N r31 = = = sgn( ⋅ ) (3.302)
w 12 w − w 2 N 3
1
w − 20 25
3
= =− (3.303)
100 − 20 50
w =−20 rev∕min opposite direction to w 1 (3.304)
3
Notice that w is the absolute angular velocity of the planetary gear. It has two components
3
contributing to its absolute angular velocity: angular velocity of the carrier (w , component
2
2) and angular velocity of the planetary gear with respect to the carrier about it own rotation