Geometric characteristics of involute helicoid

Abstract

The involute helicoid is one of the most commonly-used surfaces in mechanical engineering. Its typical application is to be used as the tooth flank of a cylindrical gear and the helicoid of a kind of cylindrical worm, which is named as the type I worm in both ISO and DIN standards. In history, the earlier research regarding the geometry of an involute helicoid can be dated back to the study of grinding approach for the above type I worm though the related study is qualitative. Nowadays the geometry of an involute helicoid is briefly and superficially discussed in the courses of an undergraduate student. In the current work, the scientific fact is proved that an involute helicoid is the unique developable cylindrical helicoid with constant helix parameter. The geometries of the profiles in different sections are systematically and deeply studied for an involute helicoid. The analytical solution of the geodesic line on an involute helicoid is attained through solving nonlinear ordinary differential equation. The numerical examples are provided to verify and illustrate the theory and method set forward. The whole work aims at supplying more in-depth knowledge in regard to the geometry of the involute helical surface for the readers, mainly including the teacher, trainer and engineers in mechanical engineering. Facing to an involute helicoid, instead of via the screw rack in the classical methodology, the mathematical relationship between the normal and transverse pressure angles is obtained. As to an involute helicoid, the tooth number condition that its root circle is larger than its base circle is given out. The numerical examples are provided to verify and illustrate the theory and method set forward.

Keywords

Involute helicoid developable surface differential geometry helical gear involute worm

Get full access to this article

View all access options for this article.

References

Crosher

. Design and application of the worm gear. New York: The American Society of Mechanical Engineers, 2002.

Poritsky

Dudley

. Conjugate action of involute helical gears with parallel or inclined axes. Q Appl Math 1948; 6: 193–214.

Litvin

. Meshing principle for gear drives. Shanghai: Shanghai Science and Technology Press, 1964.

Litvin

. Meshing principle for gear drives. 2nd ed. Shanghai: Shanghai Science and Technology Press, 1984.

Litvin

Fuentes

. Gear geometry and applied theory. 2nd ed. New York: Cambridge University Press, 2004.

Litvin

Gonzalez-Perez

Fuentes

, et al. Design, generation and stress analysis of face-gear drive with helical pinion. Comput Methods Appl Mech Eng 2005; 194: 3870–3901.

L’ukshin

. Theory of helicoids in cutting tool design (Peng XZ, Trans.). Beijing: China Machine Press, 1984.

Sichuan Provincial Bureau of Machinery Industry. Theoretical foundations of gear cutting tool design. Beijing: China Machine Press, 1982.

. Theory of gear meshing. 2nd ed. Xi'an: Xi'an Jiaotong University Press, 2009.

10.

Yuan

. Metal cutting tools. Shanghai: Shanghai Scientific and Technical Publishers, 1984.

11.

Yuan

. Gear cutting tool design. Beijing: National Defense Industry Press, 2014.

12.

Yuan

. Handbook of metal cutting tool design. 2nd ed. Beijing: China Machine Press, 2019.

13.

Radzevich

. Theory of gearing kinematics, geometry, and synthesis. 2nd ed. Boca Raton: Taylor & Francis Group, CRC Press, 2018.

14.

Zhu

. Involute external meshing cylindrical gear transmission. Beijing: National Defense Industry Press, 1990.

15.

Zhang

. Involute internal meshing cylindrical gear transmission. Beijing: National Defense Industry Press, 1991.

16.

Zhao

. Tooth profile design theory of asymmetrical involute cylindrical worm in face worm gear drive. In: IFToMM world congress on mechanism and machine science. Cham: Springer Nature Switzerland, 2023, pp.56–64.

17.

Han

, et al. Tooth profile design and meshing analysis on a helical gear with asymmetric involute teeth. Chin J Eng 2011; 33: 876–882.

18.

Vullo

. Cylindrical involute helical gears. In: Gears: Volume 1: Geometric and kinematic design. Cham: Springer International Publishing, 2020, pp.301–349.

19.

Qinchuan Machine Tool Plant “721” Workers University, Mechanical Manufacturing Teaching and Research Section of Xi'an Jiaotong University. Principles of gear grinding. Beijing: China Machine Press, 1977.

20.

. Geometric principle and calculation of involute gears. Beijing: China Machine Press, 1985.

21.

Colbourne

. The geometry of involute gears. New York: Springer-Verlag, 1988.

22.

. Principle and application of spatial gearing. Vol. 1. Beijing: China Coal Industry Publishing House, 1987.

23.

Radzevich

. Gear cutting tools (Fundamentals of design and computation). Boca Raton: Taylor & Francis Group, CRC Press, 2010.

24.

Zhao

. Meshing theory of involute worm drive. Mech Mach Theory 2021; 165: 1–23.

25.

Chen

. Gear meshing theory. Beijing: Coal Industry Press, 1985.

26.

. Differential geometry and the theory of gear meshing. Dongying: China University of Petroleum Press, 1999.

27.

Dong

. Theory and application of two-degree-of-freedom gear meshing. Beijing: China Machine Press, 2011.

28.

Zhao

. Engineering differential geometry of curves and surfaces. Beijing: Science Press, 2023.

29.

Litvin

. Development of gear technology and theory of gearing. Cleveland, Ohio: NASA Reference Publication, 1997.

30.

Editorial Group of Mathematics Handbook. Mathematics handbook. Beijing: Higher Education Press, 1979.

31.

Zheng

. Mechanical principles. 7th ed. Beijing: Higher Education Press, 1999.

32.

Sun

. Mechanical principles. 9th ed. Beijing: Higher Education Press, 2021.

33.

Buckingham

. Analytical mechanics of gears. New York: McGraw-Hill Book Company, 1949.