T-33
circle is called the base circle of the involutes. Two opposite hand involute curves meeting
at a cusp form a gear tooth curve. We can see, from Figure 3-4, the length of base circle
arc ac equals the length of straight line bc.
bc
rbq
tana = –––– = –––– = q (radian)
(3-5)
Oc
rb
The q in Figure 3-4 can be expressed as inva + a, then Formula (3-5) will become:
inva = tana – a
(3-6)
Function of a, or inva, is known as involute function. Involute function is very important
in gear design. Involute function values can be obtained from appropriate tables. With
the center of the base circle O at the origin of a coordinate system, the involute curve can
be expressed by values of x and y as follows:
rb
ü
x = r cos(inva) = ––––– cos(inva)
ï
cosa
ï
ý
(3-7)
ï
rb
ï
y = r sin(inva) = ––––– sin(inva)
ï
cosa
þ
rb
where, r = ––––– .
cosa
Fig. 3-4 The Involute Curve
y
r
rb
c
b
a
x
aq
inv a
a
O
