T-36
4.3 Undercutting
From Figure 4-3, it can be seen that the maximum length of the line-of-contact is
limited to the length of the common tangent. Any tooth addendum that extends beyond the
tangent points (T and T') is not only useless, but interferes with the root fillet area of the
mating tooth. This results in the typical undercut tooth, shown in Figure 4-4. The undercut
not only weakens the tooth with a wasp-like waist, but also removes some of the useful
involute adjacent to the base circle.
Fig. 4-3 Geometry of Contact Ratio
Fig. 4-4 Example of Undercut
Standard Design Gear,
(12 Teeth, 20° Pressure Angle)
From the geometry of the limiting length-of-contact (T-T', Figure 4-3), it is evident that
interference is first encountered by the addenda of the gear teeth digging into the mating-pinion
tooth flanks. Since addenda are standardized by a fixed value (ha = m), the interference
condition becomes more severe as the number of teeth on the mating gear increases. The limit
is reached when the gear becomes a rack. This is a realistic case since the hob is a rack-type
cutter. The result is that standard gears with teeth numbers below a critical value are automatically
undercut in the generating process. The condition for no undercutting in a standard spur gear is
given by the expression:
mz
ü
Max addendum = ha £ –––– sin
2
a
ï
2
ï
ï
and the minimum number of teeth is:
ý
(4-1)
ï
2
ï
zc ³ ––––––
ï
sin2a
þ
This indicates that the minimum number of teeth free of undercutting decreases with
increasing pressure angle. For 14.5° the value of zc is 32, and for 20° it is 18. Thus, 20°
pressure angle gears with low numbers of teeth have the advantage of much less
undercutting and, therefore, are both stronger and smoother acting.
Length-of-Action
AB = Base Pitch
a
Ra
Rb
a
B
A
T'
Z
B'
W
T
rb
a
ra
