Design method of end worm gear drive, including tooth curve geometry and improved lubrication angle formula

Abstract

End worm gear drives are characterized by their multi-tooth contact, compact contour, and theoretical potential to overcome some inherent flaws of cylindrical worm drives. However, quantitative basic research on end worm gear drives is currently inadequate, which hinders the development of this transmission. This work focuses on the computational design of end worm gear drives and proposes a new Niemann-type design. Meshing models of the proposed drive are established, and its engagement theory is deduced systematically. Based on the derived tooth surface equations, an innovative research methodology for the tooth curve geometry of the end worm gear is created, and the tooth curve in the worm gear reference plane is proved to be a spiral. An improved formula for the lubrication angle is developed, which can provide more rational numerical results for the angle. Theoretically, the modified formula is universally applicable for line contact drives and can be used to quantitatively investigate the lubrication level between the teeth for the proposed drive. Simulation outcomes demonstrate the favorable characteristics of the transmission, including broad conjugate areas, even contact lines, and fine global lubrication state.

Graphical abstract

Keywords

end worm gear / tooth curve geometry / lubrication angle / meshing characteristic / nonlinear equation

References

[1] CrosherW P. Design and Application of the Worm Gear. New York: AMSE Press, 2002, 39‒55

[2] Chen Y H, Zhang G H, Chen B K, Luo W J, Li F J, Chen Y. A novel enveloping worm pair via employing the conjugating planar internal gear as counterpart. Mechanism and Machine Theory, 2013, 67: 17–31

[3] LitvinF L, Fuentes A. Gear Geometry and Applied Theory. 2nd ed. Cambridge: Cambridge University Press, 2004, 112‒118

[4] Meng Q X, Zhao Y P, Cui J, Dou T H. Meshing characteristic of arc-toothed cylindrical worm pair and cutting geometric condition of its worm. Journal of Mechanical Design, 2021, 143(6): 063401

[5] Zhu X Y, Zhao Y P, Chi Y F, Li G F, Chen X Y. Manufacturing principle and meshing limit line of toroidal surface enveloping cylindrical worm drive with arc-toothed worm. International Journal of Precision Engineering and Manufacturing, 2022, 23(3): 319–332

[6] Zhao X, Zhao Y P, Li G F. Novel method to establish equation of ZN helicoid and meshing limit line of related worm drive. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2024, 238(2): 479–490

[7] Cui J, Zhao Y P, Mu S B, Li G F, Chen X Y. Locus of lubricating weak point for the toroidal surface enveloping cylindrical worm drive. Lubrication Science, 2023, 35(1): 56–70

[8] Meng Q X, Zhao Y P, Cui J, Mu S B, Li G F. Kinematic-geometric lubrication theory and its application to arc-toothed cylindrical worm drive. Journal of Tribology, 2022, 144(11): 111602

[9] SaariO E. US Patent, 2 954 704, 1960-10-04

[10] NapauM, Napau I D, NapauI, MatiesV, Napau-Stoica A A, Napau-Stoica R C. Computerized modeling and simulation of idle and loaded multi-tooth contact analysis in worm-face gear drives. In: Proceedings of the ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Las Vegas: ASME, 2007, 1103‒1112

[11] StaniekR. Shaping of the face worm gear by means of the single edge cutting tool. In: Proceedings of the ASME 2008 9th Biennial Conference on Engineering Systems Design and Analysis. Haifa: ASME, 2008, 247‒253

[12] Litvin F L, Argentieri G, De Donno M, Hawkins M. Computerized design, generation and simulation of meshing and contact of face worm-gear drives. Computer Methods in Applied Mechanics and Engineering, 2000, 189(3): 785–801

[13] Meng Q X, Zhao Y P, Cui J, Mu S B, Li G F. Meshing theory of offsetting Archimedes cylindrical worm drive. Mechanics Based Design of Structures and Machines, 2022, 52(1): 152–168

[14] LitvinF L, Nava A, FanQ, FuentesA. New geometry of face worm gear drives with conical and cylindrical worms: generation, simulation of meshing, and stress analysis. Computer Methods in Applied Mechanics and Engineering, 2002, 191(27‒28): 3035‒3054

[15] Napau I D, Maties V, Napau I, Napau-Stoica R C, Napau M. A new orthogonal skew-axis drive with face gear and cylindrical worm: mathematical model, geometric synthesis and generation method. In: Proceedings of the ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Long Beach: ASME, 2005, 5b: 615–624

[16] Ciotea M, Boloş V. Experimental research concerning the contact patch of the double worm face gear. Procedia Technology, 2016, 22: 55–59

[17] GoldfarbV, Trubachev E, PushkarevaT, SavelyevaT. Comparative investigation of worm and spiroid gears with cylindrical worms. In: Uhl T, ed. Advances in Mechanism and Machine Science. Cham: Springer, 2019, 925‒935

[18] SunY H, Wang S P, YueQ. An analysis on contact line performance of offset cylindrical worm gearing. Materials Science Forum, 2011, 697‒698: 760‒764

[19] Wen J, Zhou Y S, Tang J Y, Dai Y. An effective analytical approach to predicting the surface contact temperature of the face gear drives. Mathematics, 2023, 11(14): 3087

[20] Shen Z X, Yang L H, Qiao B J, Luo W, Chen X F, Yan R Q. Mesh relationship modeling and dynamic characteristic analysis of external spur gears with gear wear. Frontiers of Mechanical Engineering, 2022, 17(1): 9

[21] Chen Y H, Chen Y, Luo W J, Zhang G H. A novel backlash-adjustable and wear-compensable hourglass worm drive: computerized design, simulation of meshing and stress analysis. Journal of Advanced Mechanical Design, Systems and Manufacturing, 2016, 10(2): JAMDSM0029

[22] DIN3975. Definitions and Parameters on Cylindrical Worm Gear Pairs with Rectangular Crossing Shafts — Part 1: Worm and Worm Wheel. Berlin: Beuth Verlag, 2012, 1–36

[23] International Oragnization for Standardization. PD ISO/TR 10828‒2015. Worm Gears — Worm Profiles and Gear Mesh Geometry. Switzerland: BSI Standards Limited, 2015, 25‒28

[24] Boantă C I, Boloş V. The mathematical model of generating kinematic for the worm face gear with modified geometry. Procedia Technology, 2014, 12: 442–447

[25] Deng X Q, Hong L, Li W, Wang S S, Liu Y C. High precision machining of ZC1 worm gear drives with large modulus. Mechanism and Machine Theory, 2021, 165: 104437

[26] Liang D, Chen B K, Hua C L, Liao R J. Generation and analysis of gear drive with tubular tooth surfaces having double contact points. Archives of Civil and Mechanical Engineering, 2017, 17(2): 318–325

[27] HuL R. Principle and Application of Space Meshing. Beijing: China Coal Industry Publishing House, 1988, 193‒195 (in Chinese)

[28] DowsonD, Higginson G R. Elasto-Hydrodynamic Lubrication. New York: Pergamon Press, 1977, 58‒71

[29] Ren W, Li H T, Zhang S L, Li M M, Li J, Wang T Y, Ma M. A method for point-contact conjugate tooth surfaces with a constant instantaneous transmission ratio generated by point-contact conjugate involute helicoids. Mechanism and Machine Theory, 2023, 185: 105324

[30] TappK. Differential Geometry of Curves and Surfaces. Cham: Springer, 2016, 24‒48

[31] Ni G X, Liu Z Q, Song C S, Liu S Y, Dong Y F, Cao Y F. Geometric parameter design and contact characteristics of beveloid gear and involute cylindrical gear transmission with crossed axes. Journal of Mechanical Science and Technology, 2024, 38(2): 815–825

[32] ZhaoY P. Engineering Differential Geometry of Curves and Surfaces. Beijing: Science Press, 2023, 22‒24 (in Chinese)

[33] Cui J, Zhao Y P, Meng Q X, Li G F. Direct method to determine singular point of enveloped surface and its application to worm wheel tooth surface. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2024, 480(2282): 20230369

[34] American Gear Manufacturers Association. ANSI/AGMA 6022: Design Manual for Cylindrical Worm gearing. Alexandria: American National Standards Institute Inc, 2014, 1‒3

[35] Zhao Y P, Kong X W. Meshing principle of conical surface enveloping spiroid drive. Mechanism and Machine Theory, 2018, 123: 1–26

[36] Deng X Q, Wang J G, Horstemeyer M F, Solanki K N, Zhang J F. Parametric study of meshing characteristics with respect to different meshing rollers of the antibacklash double-roller enveloping worm gear. Journal of Mechanical Design, 2012, 134(8): 081004

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