T-109
Table 11-1 Equations of Radial Contact Ratio on Parallel Axes Gear, ea
An example of helical gear:
mn = 3
an = 20°
b = 30°
z1 = 12
z2 = 60
x1 = +0.09809
x2 = 0
ax = 125
at = 22.79588°
awt = 23.1126°
mt = 3.46410 da1 = 48.153
da2 = 213.842
db1 = 38.322
db2 = 191.611
ea = 1.2939
Note that in Table 11-1 only the radial or
circular (plane of rotation) contact ratio is
considered. This is true of both the spur and
helical gear equations. However, for helical
gears this is only one component of two. For
the helical gear's total contact ratio, eg , the
overlap (axial) contact ratio, eb , must be
added. See Paragraph 11.4.
11.2 Contact Ratio Of Bevel Gears, ea
The contact ratio of a bevel gear pair
can be derived from consideration of the
eqivalent spur gears, when viewed from the
back cone. See Figure 8-8.
With this approach, the mesh can be treated as spur gears. Table 11-2 presents
equations calculating the contact ratio.
An example of spiral bevel gear (see Table 11-2 on the following page):
m = 3
an = 20°
b
= 35°
z1
= 20
z2 = 40
at = 23.95680°
d1
= 60
d2
= 120
Rv1 = 33.54102
Rv2 = 134.16408
Rvb1 = 30.65152
Rvb2 = 122.60610
ha1 = 3.4275
ha2 = 1.6725
Rva1 = 36.9685
Rva2 = 135.83658
ea = 1.2825
Type of Gear Mesh
Formula of Radial Contact Ratio, ea
Spur Pair
Spur Gear
and Rack
External and
Internal Spur
Helical Pair
––––––––––––– –––––––––––––
da1
2
db1
2
da2
2
db2
2
Ö
(–––) – (–––) +
Ö
(–––) – (–––) – ax sinaw
2 2 2 2
–––––––––––––––––––––––––––––––––––––––––
p m cosa
––––––––––––
da1
2
db1
2
ha2 – x1m d1
Ö
(–––) – (–––) + ––––––––– – ––– sin
a
2 2 sina 2
–––––––––––––––––––––––––––––––––––
p m cosa
––––––––––––– ––––––––––––
da1
2
db1
2
da2
2
db2
2
Ö
(–––) – (–––) –
Ö
(–––) – (–––) + ax sinaw
2 2 2 2
––––––––––––––––––––––––––––––––––––––––
pm cosa
–––––––––––– ––––––––––––
da1
2
db1
2
da2
2
db2
2
Ö
(–––) – (–––) +
Ö
(–––) – (–––) – ax sinawt
2 2 2 2
––––––––––––––––––––––––––––––––––––––––
pmt cosat
Gear
a
Gear
b
Gear
a
Rack
b
External Gear
a
Internal Gear
b
Gear
a
Gear
b
Fig. 11-2 Radial Contact Ratio
of Parallel Axes Gear ea
da1
dw1
db1
Contact
Length
aw
da2
dw2
db2
aw
