Page 265 - Analysis and Design of Machine Elements

Wormgear Drives
Based on the required capability of wormgears to operate without significant damage 243
from pitting, it gives a design formula from Eq. (9.15) as
( Z ) 2
2
m d ≥ 9.4KT 2 E (9.16)
1
z [ ]
H
2
where [ ] is allowable contact stress of a wormgear, which will be introduced in
H
Section 9.4.3.1.
9.3.2 Tooth Bending Strength Analysis
According to the Lewis formula, which is applied to a helical gear in Eq. (8.57), the bend-
ing stress at the root fillet of wormgear teeth can be similarly expressed as
KF t2 2KT 2
= Y Y Y Y = Y Y Y Y (9.17)
F ⁀ Fa2 Sa2 ⁀ Fa2 Sa2
b m n b d m n
2
2 2
where
⁀
b – curved length of wormgear tooth.
2
m – normal module, m = m cos ,mm;
n
n
Y – contact ratio factor, select Y = 0.667;

Y – helix angle factor, calculated by Y = 1 − ;

120 ∘
Incorporating these into Eq. (9.17), the following equation can be obtained and used
for rough estimates of bending strength as [6]
0.666KT 2
Y
Y ≤ [ ]
= Y Fa2 Sa2 F (9.18)
F
d d m
1 2
Where wormgear tooth form factor Y Fa and stress correction factor Y Sa canbefound
3
from Table 8.4 according to the virtual number of teeth z = z /cos and profile shift
v2
2
coefficient x (limit to x =±0.5 in the table). More data can be found in design hand-
2 2
books [6] for wormgears with other profile shift coefficients.[ ] is the allowable bend-
F
ing stress of wormgear, which will be introduced in Section 9.4.3.2.
The computed values of tooth bending stresses from the left side of Eq. (9.18) are
compared with the fatigue strengths of wormgear material for strength evaluation. The
design formula can be derived from Eq. (9.18) as
0.666KT
2 2
Y
Y
m d ≥ Y Fa2 Sa2 (9.19)
1
z [ ]
F
2
Combined with the value estimated by Eq. (9.16), the design variables for the worm
and wormgear can be decided from Table 9.1.
9.3.3 Rigidity Analysis
The shaft carrying a worm must be sufficiently rigid to limit deflection at pitch point.
Otherwise, the worm and wormgear could not mesh properly. Since a worm usually
integrates with a shaft, the dedendum diameter of worm d is used as shaft diameter for
f 1
rigidity analysis. Therefore, the deflection and rigidity of worm can be calculated by [10]
√
2
F + F 2
t1 r1
3
y = l ≤ [y] (9.20)
48EI