An efficient method for determining the critical modification coefficient for cylindrical worm gearing in which the worm is generated by a curve

Abstract

This paper proposes an efficient method for the determination of the critical modification coefficient (CMC) for the cylindrical worm gearing in which the worm is generated by a curve. The term CMC is defined as the modification coefficient that causes the first boundary points to be just on the boundary of the area of meshing. Solved from a system of five non-linear equations, the CMC not only avoids gear undercutting, but also produces the maximum contact ratio. In developing the system equations, the equation of non-undercutting is derived directly from the curvature information of the generating curve and the parameters of relative motion, without the need to derive any partial derivatives to the equation of the generating surface. By using an equivalent generation mechanism rather than the real one, an explicit form of equation of meshing is created. The equation of meshing created can simplify the form of the equation of non-undercutting, reduce the number of non-linear equations and unknowns in the system equations, increase the speed of convergence and decrease numerical instability. Based on the proposed method, a computer program is created and applied to analyse the CMCs of four commonly used worm gearings: ZA, ZN, ZE and ZC₃.

Keywords

cylindrical worm gear undercutting critical modification coefficient

References

Litvin

F. L.

Gear Geometry and Applied Theory, 1994 (Prentice-Hall, New Jersey).

Kin

Limitations of worm and worm gear surfaces in order to avoid undercutting. Gear Technology, November-December 1990, 7, 30–35.

Fong

Z. H.

Tsay

C. B.

The undercutting of circular-cut spiral bevel gears. Trans. ASME, J. Mech. Des., 1992, 114, 317–325.

Colbourne

J. R.

Undercutting in worms and worm-gears. AGMA, Technical Paper, 1993, pp. 1–10.

Chen

C. K.

Tang

Y. Y.

H. Y.

Differential Geometry and Its Application in Mechanical Engineering (in Chinese), 1998 (Harbin Institute of Technology Press, Harbin).

Litvin

F. L.

Kin

Computerized simulation of meshing and bearing contact for single-enveloping worm-gear drives. Trans. ASME, J. Mech. Des., 1992, 114, 313–316.

Seol

I. H.

Litvin

F. L.

Computerized design, generation and simulation of meshing and contact of worm-gear drives with improved geometry. Computer Meth. Appl. Mechanics Engng, 1996, 138, 73–103.

Fang

H. S.

Tsay

C. B.

Effects of the hob cutter regrinding and setting on ZE-type worm gear manufacture. Int. J. Mach. Tools Mf., 1996, 36, 1123–1135.

Fang

H. S.

Tsay

C. B.

Mathematical model and bearing contacts of the ZK-type worm gear set cut by oversize hob cutters. Mech. Mach. Theory, 1996, 31, 271–282.

10.

De Donno

Litvin

F. L.

Computerized design and generation of worm gear drives with stable bearing contact and low transmission errors. Trans. ASME, J. Mech. Des., 1999, 121, 573–578.