Page 447 - Marks Calculation for Machine Design

P2: Sanjay
P1: Shibu/Rakesh
15:34
January 4, 2005
Brown.cls
Brown˙C10
429
MACHINE MOTION
As stated earlier, the number of teeth (N) on a spur gear is directly related to its radius,
or diameter; therefore, the ratio of the radius (r D ) to (r A ) in Eq. (10.43) must be the same
as the ratio of the number of teeth (N D ) on the fixed ring gear (D) to the number of teeth
(N A ) on the planet gear (A). Therefore, Eq. (10.43) can be rewritten as

N D
ω A = − 1 ω B (10.44)
N A
Basedontherelativesizesoftheplanetgear (A) and the ringgear (D) shown in Fig. 10.19,
the number of teeth on gear (A) is less than the number of teeth on gear (D). Therefore,
the output angular velocity (ω A ) could be greater than the input angular velocity (ω B ) of
the arm; however, it is always in the opposite direction.
U.S. Customary SI/Metric
Example 3. Determine the output angular Example 3: Determine the output angular
velocity for the planetary gear train as that velocity for the planetary gear train as that
shown in Fig. 10.19, where shown in Fig. 10.19, 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
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) and (D) in Eq. (10.44) to calculate the out- (A) and (D) in Eq. (10.44) to calculate the out-
put angular velocity (ω A ) as put angular velocity (ω A ) as

N D N D
ω A = − 1 ω B ω A = − 1 ω B
N A N A

192 teeth 192 teeth
= − 1 (1,800 rpm) = − 1 (1,800 rpm)
64 teeth 64 teeth
= (3 − 1)(1,800 rpm) = (3 − 1)(1,800 rpm)
= (2)(1,800 rpm) = 3,600 rpm = (2)(1,800 rpm) = 3,600 rpm
There is a 2:1 increase in the angular speed, There is a 2:1 increase in the angular speed,
with the direction of gear (A) opposite to the with the direction of gear (A) opposite to the
direction of arm (B). direction of arm (B).

Suppose another gear is added to the axis of the planet gear (A) and moves with the
same angular velocity (ω A ) forming what is called a compound gear set. Also, suppose this
additional gear is in contact with another gear, called a sun gear, mounted on the fixed axis of
the rotating arm (B); however, it is free to rotate at its own angular velocity. This new more
complex arrangement is shown in Fig. 10.20, where (E) is the additional gear on the axis
of the planet gear (A) forming the compound gear set, and (F) is the sun gear. Point G is
the point of contact between gears (E) and (F), and must have a velocity that has the same
magnitude and direction whether related to gear (E) or gear (F). This is a similar condition
already placed on point C, the axis of the compound gear set on the rotating arm (B).
For this configuration, the angular velocity (ω B ) of arm (B) is still the input; however,
now the output is the angular velocity (ω F ) of the sun gear (F). Notice that the direction