Page 443 - Marks Calculation for Machine Design

P2: Sanjay
P1: Shibu/Rakesh
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January 4, 2005
Brown.cls
Brown˙C10
425
MACHINE MOTION
There are two main categories of gear trains: spur gear trains and planetary gear trains.
Here, the term spur refers to the fact that all the shafts in the assembly are assumed to be
fixed, whereas planetary refers to the fact that some of the gears rotate about their own axis
while rotating about another axis in a planetary motion.
One of the primary principles of gear train analysis is that the radius, or diameter, of a
gear is directly related to the number of teeth. Therefore, the formulas that will be presented
that relate an input angular velocity to an output angular velocity will depend only on the
number of teeth of the gears in the gear train assembly.
10.3.1 Spur Gears
The most basic of spur gear trains is shown in Fig. 10.17 where a single spur gear (A) on
one fixed shaft drives a single spur gear (B) on another fixed shaft.
w B
w A
A r A B r B

C (contact point)
FIGURE 10.17 Basic spur gear train.

If the angular velocity (ω A ) is considered the input, then the output is the angular velocity
(ω B ). Note that if the angular velocity (ω A ) is clockwise, then the angular velocity (ω B )
will be counterclockwise. This is due to the fundamental principle that the velocity of
point C, the point of contact between the two gears, must have the same magnitude and
direction whether determined from gear (A) or gear (B). This means that the relationship
in Eq. (10.33) must govern the motion of the two gears.
v C = r A ω A = r B ω B (10.33)

Solving for the output angular velocity (ω B ) gives
r A
ω B = ω A (10.34)
r B
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 A ) to (r B ) in Eq. (10.34) must be the same
as the ratio of the number of teeth (N A ) on gear (A) to the number of teeth (N B ) on gear
(B). Therefore, Eq. (10.34) can be rewritten as

N A
ω B = ω A (10.35)
N B
Based on the relative sizes of gears (A) and (B) shown in Fig. 10.17, the number of teeth
on gear (A) is less than the number of teeth on gear (B). Therefore, the output angular
velocity (ω B ) will be less than the input angular velocity (ω A ).