Page 445 - Marks Calculation for Machine Design

P2: Sanjay
P1: Shibu/Rakesh
January 4, 2005
15:34
Brown.cls
Brown˙C10
427
MACHINE MOTION
This means that the relationship in Eq. (10.36) will govern the motion of these two gears.
v E = r B ω B = r D ω D (10.36)
Solving for the angular velocity (ω D ) gives
r B
ω D = ω B (10.37)
r D
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 B ) to (r D ) in Eq. (10.37) must be the same as
the ratio of the number of teeth (N B ) on gear (B) to the number of teeth (N D ) on gear (D).
Therefore, Eq. (10.37) can be rewritten as
N B
ω D = ω B (10.38)
N D
Substituting the angular velocity (ω B ) from Eq. (10.35) in Eq. (10.38) gives
N B N B N A N A
ω D = ω B = ω A = ω A (10.39)
N D N D N B N D
where the number of teeth (N B ) on gear (B) cancels. This is why gear (B) is called an idler
gear, as its only purpose is to keep the output angular velocity in the same direction as the
input angular velocity.
Based on the relative sizes of gears (A), (B), and (D) shown in Fig. 10.18, the number of
teeth on gear (A) is less than the number of teeth on gear (D). Therefore, the output angular
velocity (ω D ) will be less than the input angular velocity (ω A ) and in the same direction.

U.S. Customary SI/Metric
Example 2. Determine the output angular Example 2. Determine the output angular
velocity for the triple spur gear train as that velocity for the triple spur gear train as that
shown in Fig. 10.18, where shown in Fig. 10.18, where
ω A = 600 rpm (input) ω A = 600 rpm
N A = 15 teeth N A = 15 teeth
N B = 45 teeth N B = 45 teeth
N D = 30 teeth N D = 30 teeth
solution solution
Step 1. Substitute the given input angular Step 1. Substitute the given input angular
velocity (ω A ) and the number of teeth on gears velocity (ω A ) and the number of teeth on gears
(A) and (D) in Eq. (10.39) to calculate the out- (A) and (D) in Eq. (10.39) to calculate the out-
put angular velocity (ω D ) as put angular velocity (ω D ) as
N A (15 teeth) N A (15 teeth)
ω D = ω A = (600 rpm) ω D = ω A = (600 rpm)
N D (30 teeth) N D (30 teeth)
1 1
= (600 rpm) = 300 rpm = (600 rpm) = 300 rpm
2 2
Note that the direction of gear (D) will be the Note that the direction of gear (D) will be the
same as the direction of gear (A). same as the direction of gear (A).