Page 220 - Analysis and Design of Machine Elements

Analysis and Design of Machine Elements
198
Bevel gear teeth are tapped, both in tooth thickness and height, from the large tooth
profile at one end to the small tooth profile at the other end. Customarily, tooth dimen-
sions are specified either at the large end or at the midpoint of face width. Dimensions
at the large end are generally used for geometrical calculation; while dimensions at face
width midpoint are used for strength analysis and design. The following discussion is
limited to straight bevel gears mounted on perpendicular shafts with a pressure angle
∘
of 20 .
1) Pitch diameter, d
When a standard modulus is specified at the large end, the pitch diameter d at the
large end is calculated in the same way as for spur gears, that is,
d = mz (8.64)
2) Gear ratio, u
The gear ratio for bevel gears are determined from the number of teeth, the pitch
diameters or the pitch cone angles, as
n 1 z 2 d 2 sin 2
u = = = = = cot = tan 2 (8.65)
1
n z d sin
2 1 1 1
Therefore, the pitch cone angles for the pinion and gear are determined by the
1
2
ratio of the number of teeth. A pair of bevel gears having a unit ratio is used simply
for the purpose of rotational direction change.
3) Cone distance, R
Figure 8.10, the cone distance is calculated by
√
( ) 2 ( ) 2 √
d 1 d 2 m d 1 √
2
2
2
R = + = z + z = u + 1 (8.66)
2
1
2 2 2 2
4) The relationship between the diameter of virtual spur gear d ,meandiameter d and
v m
pitch diameter d. Similar to obtaining a virtual spur gear on the developed back cone
on the large ends [6, 13], we can have a virtual spur gear at the midpoint of face width
for strength analysis. The relationship between the diameter of virtual spur gear at
the midpoint of face width d (not shown in Figure 8.10) and mean diameter d is
v m
[12]
d m
d = (8.67)
v
cos
and
d m1 d m2 R − 0.5b b
= = = 1 − 0.5 = 1 − 0.5 R (8.68)
d 1 d 2 R R
The face width b is bounded by large and small ends. Let the face width factor of a
bevelgearbe = b/R, which is normally within the range of 0.25∼0.35. The most
R
commonly used value is 1/3. So
d =(1 − 0.5 )d d =(1 − 0.5 )d (8.69)
m1 R 1 m2 R 2
5) The relationship between mean module m and module m
m
Since d m = 1 − 0.5 ,wehave m m = 1 − 0.5 ,so
d R m R
m =(1 − 0.5 )m (8.70)
m R