Abstract

The mathematical principle for calculating the contacting curve length of the involute Helicon gearing is put forward. The transient contacting curves within the conjugate zone are attained. The approximate analytical formula of the contacting curve length is derived. Based on that, the lengths of the contacting curves are computed by three methods, which are the approximate analytical formula, the numerical integration method, and calculating the distance between the beginning and the end of the contacting curve on the worm gear tooth surface. Besides, to demonstrate the rationality of the third method, two novel formulae for calculating the principal curvatures and directions of the surface are derived from the curvature parameters of two perpendicular directions to each other. These two novel formulae are used to calculate the principal curvatures and directions of the worm gear tooth surface, and evaluate the flatness of the tooth surface quantitatively. The results show that the contacting curve lengths calculated in this paper are generally between 2.7087 mm and 4.4858 mm; most of the contacting curve lengths do not vary much. The contacting curve length calculation principle proposed in this paper has high precision, and the maximum relative error between three methods is not more than −3.8838%. The worm gear tooth surfaces are relatively flat, the minimum of the principal curvature radii is 43.5494 mm, and the maximum is 3.3152 × 104 mm; most of the principal curvature radii are much larger than the contacting curve lengths.

Issue Section:
Research Papers
Keywords:
worm drive, contacting curve length, principal curvature, principal direction, conjugate zone, computer-aided design, computer-aided engineering
Topics:
Transients (Dynamics), Worm gears

References

1.
Johnson
,
K. L.
,
1985
,
Contact Mechanics
,
Cambridge University Press
,
Cambridge
.
2.
Xu
,
Y.
,
Deng
,
X.
,
Wang
,
S.
,
Ren
,
J.
, and
Yang
,
H.
,
2024
, “
Thermal Elastohydrodynamic Lubrication Analysis of the Roller Enveloping Hourglass Worm Drives Considering the Roller Self-Rotation Behavior
,”
Tribol. Int.
,
200
, p.
110142
.
3.
Choe
,
T.
,
Ri
,
C.
,
Jo
,
M.
, and
Ri
,
M.
,
2022
, “
Research on the Engagement Process and Contact Line of Involute Helical Gears
,”
Mech. Mach. Theory
,
171
, p.
104778
.
4.
Zhou
,
C.
, and
Wang
,
H.
,
2018
, “
An Adhesive Wear Prediction Method for Double Helical Gears Based on Enhanced Coordinate Transformation and Generalized Sliding Distance Model
,”
Mech. Mach. Theory
,
128
, pp.
58
–
83
.
5.
Jiang
,
H.
,
Shao
,
Y.
,
Mechefske
,
C. K.
, and
Chen
,
X.
,
2015
, “
The Influence of Mesh Misalignment on the Dynamic Characteristics of Helical Gears Including Sliding Friction
,”
J. Mech. Sci. Technol.
,
29
(
11
), pp.
4563
–
4573
.
6.
Dudas
,
I.
,
2000
,
The Theory & Practice of Worm Gear Drives
,
Penton Press
,
London
.
7.
Wu
,
H.
,
Zhang
,
Y.
, and
Qi
,
L.
,
1986
,
Design of Worm Drives
, Vol.
1
, ed.,
China Machine Press
,
Beijing
.
8.
Dong
,
X.
,
1987
,
Design of Worm Drives
, Vol.
2
, ed.,
China Machine Press
,
Beijing
.
9.
Crosher
,
W. P.
,
2002
,
Design and Application of the Worm Gear
,
ASME Press
,
New York
.
10.
Saari
,
O. E.
,
1960
, Skew Axis Gearing, U.S. Patent No. 2954704.
11.
Sarri
,
O. E.
,
1954
, Speed-Reduction Gearing, U.S. Patent No. 2696125.
12.
Litvin
,
F. L.
, and
De Donno
,
M.
,
1998
, “
Computerized Design and Generation of Modified Spiroid Worm-Gear Drive With Low Transmission Errors and Stabilized Bearing Contact
,”
Comput. Methods Appl. Mech. Eng.
,
162
(
1–4
), pp.
187
–
201
.
13.
Litvin
,
F. L.
,
Nava
,
A.
,
Fan
,
Q.
, and
Fuentes
,
A.
,
2002
, “
New Geometry of Face Worm Gear Drives With Conical and Cylindrical Worms: Generation, Simulation of Meshing, and Stress Analysis
,”
Comput. Methods Appl. Mech. Eng.
,
191
(
27–28
), pp.
3035
–
3054
.
14.
Bohle
,
F.
,
1955
, “
Spiroid Gears
,”
Machinery
,
62
(
2
), pp.
155
–
161
.
15.
Zhang
,
Y.
, and
Xu
,
H.
,
2003
, “
Pitch Cone Design and Avoidance of Contact Envelope and Tooth Undercutting for Conical Worm Gear Drives
,”
ASME J. Mech. Des.
,
125
(
1
), pp.
169
–
177
.
16.
Kirov
,
V.
,
1999
, “
Design of Spiroid Hobs
,”
J. Mater. Process. Technol.
,
88
(
1–3
), pp.
208
–
215
.
17.
Nelson
,
W. D.
,
1961
, “
Spiroid Gearing, Part 1, 2, and 3
,”
Mach. Des.
,
33
(
4
), pp.
136
–
144
; (5) 93–100, (6) 165–171.
18.
De Donno
,
M.
, and
Litvin
,
F. L.
,
1999
, “
Computerized Design, Generation and Simulation of Meshing of a Spiroid Worm-Gear Drive with a Ground Double-Crowned Worm
,”
ASME J. Mech. Des.
,
121
(
2
), pp.
264
–
273
.
19.
Goldfarb
,
V. I.
,
2006
, “
What We Know About Spiroid Gears
,”
Proceedings of the International Conference on Mechanical Transmissions
,
Chongqing, China
,
Sept. 26–30
, pp.
19
–
26
.
20.
Abadjiev
,
V.
, and
Petrova
,
D.
,
1997
, “
Testing of the Kinematic Conjugation of the Flanks Active Surfaces of Gear-Pairs of Type Spiroid
,”
Mech. Mach. Theory
,
32
(
3
), pp.
343
–
348
.
21.
Litvin
,
F. L.
,
1997
,
Development of Gear Technology and Theory of Gearing
,
NASA Reference Publication
,
Cleveland, OH
.
22.
Wang
,
S.
,
Zhou
,
Y.
,
Liu
,
X.
,
Liu
,
S.
, and
Tang
,
J.
,
2021
, “
An Advanced Comprehensive Approach to Accurately Modeling the Face-Milled Generated Spiral Bevel Gears
,”
ASME J. Comput. Inf. Sci. Eng.
,
21
(
4
), p.
041008
.
23.
Litvin
,
F. L.
, and
Fuentes
,
A.
,
2004
,
Gear Geometry and Applied Theory
, 2nd ed.,
Cambridge University Press
,
Cambridge
.
24.
Litvin
,
F. L.
,
Fuentes
,
A.
, and
Howkins
,
M.
,
2001
, “
Design, Generation and TCA of New Type of Asymmetric Face-Gear Drive With Modified Geometry
,”
Comput. Methods Appl. Mech. Eng.
,
190
(
43–44
), pp.
5837
–
5865
.
25.
Chen
,
Z.
,
Zeng
,
M.
, and
Fuentes
,
A.
,
2020
, “
Computerized Design, Simulation of Meshing and Stress Analysis of Pure Rolling Cylindrical Helical Gear Drives With Variable Helix Angle
,”
Mech. Mach. Theory
,
153
, p.
103962
.
26.
Guo
,
H.
,
Gonzalez
,
I.
, and
Fuentes
,
A.
,
2019
, “
Computerized Generation and Meshing Simulation of Face Gear Drives Manufactured by Circular Cutters
,”
Mech. Mach. Theory
,
133
, pp.
44
–
63
.
27.
Tang
,
J.
,
Cui
,
W.
,
Zhou
,
H.
, and
Yin
,
F.
,
2016
, “
Integrity of Grinding Face-Gear With Worm Wheel
,”
J. Cent. South Univ.
,
23
(
1
), pp.
77
–
85
.
28.
Zhou
,
Y.
,
Tang
,
J.
,
Zhou
,
H.
, and
Yin
,
F.
,
2016
, “
Multistep Method for Grinding Face-Gear by Worm
,”
ASME J. Manuf. Sci. Eng.
,
138
(
7
), p.
071013
.
29.
Zhao
,
Y.
, and
Kong
,
X.
,
2018
, “
Meshing Principle of Conical Surface Enveloping Spiroid Drive
,”
Mech. Mach. Theory
,
123
, pp.
1
–
26
.
30.
Huai
,
C.
, and
Zhao
,
Y.
,
2019
, “
Meshing Theory and Tooth Profile Geometry of Toroidal Surface Enveloping Conical Worm Drive
,”
Mech. Mach. Theory
,
134
, pp.
476
–
498
.
31.
Mu
,
S.
,
Zhao
,
Y.
,
Zhang
,
X.
,
Meng
,
Q.
, and
Li
,
G.
,
2021
, “
Meshing Theory of Involute Worm Drive
,”
Mech. Mach. Theory
,
165
, p.
104425
.
32.
Zhu
,
X.
,
Zhao
,
Y.
,
Chi
,
Y.
,
Li
,
G.
, and
Chen
,
X.
,
2022
, “
Meshing Theory of Toroidal Surface Enveloping Cylindrical Worm Drive With Arc-Toothed Worm
,”
Mech. Mach. Theory
,
171
, p.
104780
.
33.
Meng
,
Q.
,
Zhao
,
Y.
,
Cui
,
J.
,
Mu
,
S.
, and
Li
,
G.
,
2024
, “
Meshing Theory of Offsetting Archimedes Cylindrical Worm Drive
,”
Mech. Based Des. Struct. Mach.
,
52
(
1
), pp.
152
–
168
.
34.
Zhu
,
X.
,
Zhao
,
Y.
,
Chi
,
Y.
,
Li
,
G.
, and
Chen
,
X.
,
2023
, “
Meshing Principle and Geometry of Tooth Profile for Offset Enveloping Cylindrical Worm Drive
,”
Mech. Mach. Theory
,
180
, p.
105171
.
35.
Mu
,
S.
,
Zhao
,
Y.
,
Cui
,
J.
,
Meng
,
Q.
, and
Li
,
G.
,
2023
, “
Meshing Theory of Face Worm Gear Drive With Hardened Cylindrical Worm
,”
Mech. Mach. Theory
,
185
, p.
105323
.
36.
Wu
,
D.
, and
Luo
,
J.
,
1992
,
A Geometric Theory of Conjugate Tooth Surfaces
,
World Scientific Publishing
,
Singapore
.
37.
Zhao
,
Y.
,
2023
,
Engineering Differential Geometry of Curves and Surfaces
,
Science Press
,
Beijing
.
38.
O'Neill
,
B.
,
2006
,
Elementary Differential Geometry
, 2nd ed.,
Academic Press
,
Burlington
.
39.
Zheng
,
C.
,
1988
,
Spiral Bevel Gear and Hypoid Gear—Meshing Principle, Wheel Blank Design, Machining Adjustment and Calculation Principle of Tooth Surface Analysis
,
China Machine Press
,
Beijing
.
40.
Yu
,
Y.
,
Zhao
,
Y.
,
Ma
,
J.
,
Meng
,
Q.
, and
Li
,
G.
,
2023
, “
Curvature Interference Characteristic Analysis of Offset Involute Cylindrical Worm Drive
,”
Math. Meth. Appl. Sci.
,
47
(
6
) pp.
4058
–
4075
.
41.
Dong
,
X.
,
1989
,
Foundation of Meshing Theory for Gear Grives
,
China Machine Press
,
Beijing
.
42.
Do Carmo
,
M. P.
,
2016
,
Differential Geometry of Curves & Surfaces
, 2nd ed.,
Dover Publications
,
New York
.
43.
Dong
,
X.
,
2003
,
Design and Manufacture of Cycloid Bevel and Hypoid Gears
,
China Machine Press
,
Beijing
.
44.
Zeng
,
T.
,
1989
,
Design and Processing of Spiral Bevel Gear
,
Harbin Institute of Technology Press
,
Harbin
.
45.
Liu
,
S.
,
Zhu
,
C.
,
Fuentes
,
A.
, and
Song
,
C.
,
2021
, “
Computerized Approach for Design and Generation of Face-Milled Non-Generated Hypoid Gears With Low Shaft Angle
,”
Mech. Mach. Theory
,
155
, p.
104084
.
46.
Zhang
,
B.
,
Wei
,
B.
, and
Zhang
,
R.
,
2022
, “
Research on Evolution Laws of Pitch Cones and Characteristic Values on Whole Tooth Surface of Hypoid Gears
,”
Mech. Mach. Theory
,
174
, p.
104915
.
47.
Cui
,
J.
,
Zhao
,
Y.
,
Mu
,
S.
,
Meng
,
Q.
, and
Li
,
G.
,
2023
, “
Meshing Limit Line of Niemann Face Wormgear Drive
,”
Forsch Ingenieurwes
,
87
(
3
), pp.
901
–
912
.
48.
Yu
,
Y.
, and
Zhao
,
Y.
,
2023
, “
Tooth Profile Design Theory of Asymmetrical Involute Cylindrical Worm in Face Worm Gear Drive
,”
Proceedings of the 16th IFToMM World Congress 2023
,
Tokyo, Japan
,
Nov. 5–10
, pp.
56
–
64
.
49.
Meng
,
Q.
,
Zhao
,
Y.
, and
Yang
,
Z.
,
2020
, “
Meshing Limit Line of the Conical Surface Enveloping Conical Worm Pair
,”
Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
,
234
(
2
), pp.
693
–
703
.
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