Page 223 - Analysis and Design of Machine Elements

Therefore, the contact stress on a bevel gear surface is calculated by Gear Drives 201
√
4KT 1
= Z Z H (8.77)
E
H
2 3
(1 − 0.5 ) d u
R
R
1
8.5.3.2 Contact Strength Analysis
For a pair of bevel gears to work safely, the contact stress should not exceed the allowable
stress, that is,
√
4KT 1
= Z Z H ≤ [ ] (8.78)
H
H
E
2 3
(1 − 0.5 ) d u
R R 1
The derived equation from Eq. (8.78) can be used for the design of bevel gear teeth for
surface durability, thus the design equation is
√
( ) 2
Z Z 4KT
3 E H 1
d ≥ (8.79)
1
2
[ ] (1 − 0.5 ) u
H
R
R
8.5.4 Tooth Bending Strength Analysis
Likewise, bending stress analysis for bevel gear teeth is also similar to that already pre-
sented for spur and helical gear teeth, with minor alterations according to bevel gear
geometry. The maximum bending stress is calculated at the tooth root fillet of the mid-

point virtual spur gear.
8.5.4.1 Bending Stress Calculation
From the Lewis formula Eq. (8.27), the maximum bending stress occurs at the root of
the tooth is computed from
KF t
= Y Y
Fa Sa
F
bm m
The tangential force is computed at the midpoint of the tooth, thus,
2T 1 2T 1 2T 1
F = = =
t1
d m z m(1 − 0.5 )z
m1 m 1 R 1
Face width, b, by definition is
d 1 √ mz 1 √
2
b = R = 1 + u = 1 + u 2
R R R
2 2
The mean module m is
m
m =(1 − 0.5 )m
m R
Integrating these equations into Lewis formula, therefore, the bending stress can be
calculated by
4KT 1
= √ Y Y (8.80)
F
Fa Sa
3 2
2
m z (1 − 0.5 ) 1 + u 2
1 R R