T-32
3.2.1 Contact Ratio
To assure smooth continuous tooth action, as one pair of teeth ceases contact a
succeeding pair of teeth must already have come into engagement. It is desirable to have
as much overlap as possible. The measure of this overlapping is the contact ratio. This is
a ratio of the length of the line-of-action to the base pitch. Figure 3-3 shows the geometry.
The length-of-action is determined from the intersection of the line-of-action and the outside
radii. For the simple case of a pair of spur gears, the ratio of the length-of-action to the
base pitch is determined from:
–––––––––– ––––––––
Ö
(Ra
2
– Rb2) +
Ö
(ra
2
– rb2) – a sina
eg = –––––––––––––––––––––––––––––––––
(3-4)
p cosa
It is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances
should the ratio drop below 1.1, calculated for all tolerances at their worst-case values.
A contact ratio between 1 and 2 means that part of the time two pairs of teeth are in
contact and during the remaining time one pair is in contact. A ratio between 2 and 3
means 2 or 3 pairs of teeth are always in contact. Such a high contact ratio generally is
not obtained with external spur gears, but can be developed in the meshing of an internal
and external spur gear pair or specially designed nonstandard external spur gears.
More detail is presented about contact ratio, including calculation equations for specific
gear types, in SECTION 11.
3.3 The Involute Function
Figure 3-4 shows an element of involute curve. The definition of involute curve is the
curve traced by a point on a straight line which rolls without slipping on the circle. The
a
Ra
Rb
a
B
A
T'
Z
B'
W
T
rb
a
ra
Fig. 3-3 Geometry of Contact Ratio
WZ = Length-of-Action
B'Z = AB = Base Pitch
