Page 449 - Marks Calculation for Machine Design

P2: Sanjay
P1: Shibu/Rakesh
15:34
January 4, 2005
Brown.cls
Brown˙C10
U.S. Customary MACHINE MOTION SI/Metric 431
Example 4. Determine the output angular Example 4. Determine the output angular
velocity for the planetary gear train as that velocity for the planetary gear train as that
shown in Fig. 10.20, where shown in Fig. 10.20, where
ω B = 1,800 rpm (input) ω B = 1,800 rpm (input)
N A = 64 teeth N A = 64 teeth
N D = 192 teeth N D = 192 teeth
N E = 80 teeth N E = 80 teeth
N F = 48 teeth N F = 48 teeth
solution solution
Step 1. Substitute the given input angular Step 1. Substitute the given input angular
velocity (ω B ) and the number of teeth on gears velocity (ω B ) and the number of teeth on gears
(A), (D), (E), and (F) in Eq. (10.49) to calcu- (A), (D), (E), and (F) in Eq. (10.49) to calcu-
late the output angular velocity (ω F ) as late the output angular velocity (ω F ) as

N A + N E N D N A + N E N D
ω F = − 1 ω B ω F = − 1 ω B
N F N A N F N A
(64 + 80) teeth 192 teeth (64 + 80) teeth 192 teeth
= − 1 = − 1
48 teeth 64 teeth 48 teeth 64 teeth
×(1,800 rpm) ×(1,800 rpm)
= (3)(3 − 1)(1,800 rpm) = (3)(3 − 1)(1,800 rpm)
= (6)(1,800 rpm) = 10,800 rpm = (6)(1,800 rpm) = 10,800 rpm
There is a 6:1 increase in the angular speed, There is a 6:1 increase in the angular speed,
with the direction of gear (F) in the same with the direction of gear (F) in the same
direction as arm (B). direction as arm (B).

Realize that to achieve the same input to output ratio 6:1 using a spur gear train, the
input gear would have to be six times the diameter, or number of teeth, as the output gear.
This would not be a very compact gear train arrangement. Again, this is the advantage of
a planetary gear train; however, its disadvantage is that it is more complex to manufacture
and maintain the close tolerances necessary for its efficient operation.
One last comment on planetary gear trains. To distribute the loading on the ring gear,
many times there are multiple compound gears driving a single sun gear, resulting in a
rotating arm with multiple spokes. However, this would not change the input to output ratio
of the angular velocities; therefore, the principles and formulas presented in this section are
valid for even the most complex configuration of gears and rotating arms.

10.4 WHEELS AND PULLEYS

While the discussion in the previous section involved the motion of gears, which were
treated as both rotating and rolling wheels, here the motion of wheels rolling freely on a
flat rough surface will be discussed. The velocity at a variety of important locations around
the rolling wheel will be presented.