Page 200 - Analysis and Design of Machine Elements

Analysis and Design of Machine Elements
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18) Tooth face, tooth flank and tooth surface. The tooth face is the surface between the
pitch line and the top of tooth, while tooth flank is the surface between the pitch
line and the bottom land. Therefore, tooth surface is tooth face and tooth flank
combined.
19) Root fillet. This is the portion of tooth flank joined to the bottom land.
8.2 Working Condition Analysis

8.2.1 Kinematic Analysis
8.2.1.1 Speed Ratio and Pitch Line Velocity
Gears are employed to produce rotational speed change in the driven gear relative to the
driving gear. This change is measured by speed ratio, defined as the ratio of the rotational
speed of driving gear to the driven gear. For proper rotational speed transmission, speed
ratio should be maintained at a constant value continuously through meshing.
To satisfy this kinematic requirement, it is required that, from Figure 8.1b, as gears
rotate, the common normal N N to the tooth surfaces at contact point must intersect
1
2
thelineofcentres O O at pitch point P, where the pitch line velocities of the pinion
1
2
and gear are identical and each may be calculated by
d n d n
1 1
2 2
v = = (8.8)
60 × 1000 60 × 1000
The speed ratio, therefore, is
n 1 d 2 z 2
i = = = = u (8.9)
n d z
2 1 1
Equation (8.9) is equally valid for spur, helical and bevel gear drives.
8.2.1.2 Contact Ratio
Figure 8.2 shows a pinion centred at O , rotating clockwise at an angular velocity of ,
1
1
drives a gear centred at O , rotating counterclockwise at an angular velocity of .As
2
2
gears rotate, smooth and continuous motion transfer from one pair of meshing teeth to
the succeeding pair is especially important. This is achieved by keeping the first pair in
contact until the following pair establishes initial contact. Or in other words, a degree
of contact overlapping must be maintained.
Contact ratio is an effective means for measuring this overlapping tooth contact. It is
defined as contact length divided by base pitch p . During tooth meshing, contact points
b
move along the line of action N N or the pressure line. The contact length is the dis-
1 2
tance on the line of action located between the two addendum circles. From Figure 8.2,
we have [2]
√ √
2
2
2
2
r − r + r − r − a sin
B B a1 b1 a2 b2
1 2
= = (8.10)
p p cos
b c
To ensure a smooth and continuous meshing, the contact ratio must be greater than
one. Contact ratio can be regarded as the average number of teeth in contact or, on the
time basis, the number of pairs of teeth simultaneously engaged. For example, a contact