Three laws of gearing are presented in terms of a three link 1-dof spatial direct contact mechanism. The first law of gearing defines the instantaneous relationship between an infinitesimal displacement of an output body to an infinitesimal angular displacement of an input body for a specified tooth contact normal. A system of cylindroidal coordinates are introduced to facilitate a universal methodology to parameterize the kinematic geometry of generalized motion transmission between skew axes. The second law of gearing establishes a relation between the instantaneous gear ratio and the apparent radii of the hyperboloidal pitch surface in contact as parameterized using a system of cylindroidal coordinates. The third law of gearing establishes an instantaneous relationship for the relative curvature of two conjugate surfaces in direct contact and shows that this relation is independent of the tooth profile geometry. These three laws of gearing along with the system of cylindroidal coordinates establish, in part, a generalized geometric theory comparable to the existing theory for planar kinematics.

Issue Section:
Technical Papers
Keywords:
mechanical engineering, kinematics, geometry
Topics:
Displacement, Gear teeth, Gears, Geometry, Kinematics, Rotation, Generators
1.
Grant, G. B., 1899, A Treatise on Gear Wheels, Grant Gear Works, Boston.
2.
Litvin, F. L., 1994, Gear Geometry and Applied Theory, Prentice Hall, Englewood Cliffs, NJ.
3.
Colbourne, J. R., 1987, The Geometry of Involute Gears, Springer-Verlag, New York.
4.
Hunt, K. H., 1978, The Kinematic Geometry of Mechanisms, Clarendon Press, Oxford.
5.
Bottema, O., and Roth, B., 1979, Theoretical Kinematics, North-Holland Publishing Co., Amsterdam.
6.
Sommer
,
H. J.
,
1992
, “
Determination of First and Second Order Instant Screw Parameters from Landmark Trajectories
,”
ASME J. Mech. Des.
,
114
(
2
), pp.
274
–
282
.
7.
Chen
,
N.
,
1998
, “
Curvatures and Sliding Ratios of Conjugate Surfaces
,”
ASME J. Mech. Des.
,
120
, pp.
126
–
132
.
8.
Ko¨se
,
O¨.
,
1999
, “
A Method of the Determination of a Developable Ruled Surface
,”
Mech. Mach. Theory
,
34
, pp.
1187
–
1193
.
9.
Roth, B., 1999, “Second Order Approximations for Ruled-Surface Trajectories,” Tenth World Congress on the Theory of Machines and Mechanisms, Oulu, Finland, June 20–24.
10.
Stachel, H., 2000, “Instantaneous Spatial Kinematics and the Invariants of Axodes,” Ball 2000 Symposium, Cambridge England, July 9–11.
11.
Shtipelman, B. A., 1978, Design and Manufacture of Hypoid Gears, Wiley, New York.
12.
Stadtfeld, H., 1991, Handbook of Bevel and Hypoid Gears, Rochester Institute of Technology, Rochester.
13.
Stadtfeld, H., 1995, Gleason Bevel Gear Technology, The Gleason Works, Rochester.
14.
Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears, Elsevier, Amsterdam.
15.
Lunin, S., 2001, “New Method of Gear Geometry Calculation,” The JSME International Conference on Motion and Power Transmission, Fukuoka Japan, Vol. II, pp. 472–477.
16.
Dudas, I., 2000, The Theory and Practice of Worm Gear Drives, Penton Press, London.
17.
Xiao
,
D. Z.
, and
Yang
,
A. T.
,
1989
, “
Kinematics of Three Dimensional Gearing
,”
Mech. Mach. Theory
,
24
, pp.
245
–
255
.
18.
Figliolini, G., and Angeles, J., 1999, “On the Geometry of Kinematic Synthesis of Gears with Skew Axes,” Proceedings of XIV National Congress of the Italian Association of Theoretical and Applied Mechanics, Meccanica delle Machine-Paper N. 31., Como.
19.
Phillips
,
J.
,
1999
, “
Some Geometrical Aspects of Skew Polyangular Involute Gearing
,”
Mech. Mach. Theory
,
34
, pp.
781
–
790
.
20.
Honda, S., 2001, “A Unified Designing Method Applicable to All Kinds of Gears for Power Transmission,” The JSME International Conference on Motion and Power Transmission, Fukuoka Japan, Vol. II, pp. 506–5512.
21.
Ito
,
N.
, and
Takahashi
,
K.
,
2000
, “
Differential Geometrical Conditions of Hypiod Gears with Conjugate Tooth Surfaces
,”
ASME J. Mech. Des.
,
122
(
3
), pp.
323
–
330
.
22.
Dooner, D. B., and Seireg, A. A., 1995, The Kinematic Geometry of Gearing: A Concurrent Engineering Approach, John Wiley and Sons, Inc., New York.
23.
Ball, R. S., 1900, A Treatise on the Theory of Screws, Cambridge University Press, London.
24.
Beggs, J. S., 1959, “Ein Beitrag zur Analyze Ra¨umlicher Mechanismem,” Doctoral Thesis, Technische Hochschule Hannover, Hanover.
25.
Phillips
,
J. R.
, and
Hunt
,
K. H.
,
1964
, “
On the Theorem of Three Axes in the Spatial Motion of Three Bodies
,”
Australian Journal of Applied Science
,
15
, pp.
267
–
287
.
26.
Wildhaber
,
E.
,
1946
, “
Basic Relationship of Hypoid Gears..II
,”
American Machinist
,
28
, pp.
131
–
134
.
27.
Disteli
,
M.
,
1914
, “
U¨ber des Analogen der Savary schen Formel und Konstruktion in der kinematischen Geometrie des Raumes
,”
Zeitschrift fu¨r Mathematic und Physik
,
62
, pp.
261
–
309
.
28.
Veldkamp
,
G. R.
,
1967
, “
Conical Systems and Instantaneous Invariants in Spatial Kinematics
,”
J. Mec.
,
3
, pp.
329
–
388
.
29.
Struik, D. J., 1961, Lectures on Classical Differential Geometry, 2nd edition, Dover Publications, Inc., New York.
30.
Dooner, D. B., 2001, “Design Formulas for Evaluating Contact Stress in Gear Pairs,” Gear Technology: The Journal of Gear Manufacturing, Randall Publishing Inc., May/June, pp. 31–37.
31.
Grill, J., 1999, “Calculating and Optimizing of Grinding Wheels for Manufacturing Grounded Gear Hobs,” 4th World Congress on Gearing and Power Transmission, Paris France, Mar. 16–18, pp. 1661–1671.
32.
Dyson, A., 1969, A General Theory of the Kinematics and Geometry of Gears in Three Dimensions, Clarendon Press, Oxford.
33.
Wu Da-ren, and Luo Jia-shun, 1992, A Geometric Theory of Conjugate Tooth Surfaces, World Scientific, Singapore.
Copyright © 2002
 by ASME
You do not currently have access to this content.