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Research Papers: Design of Direct Contact Systems

Investigation on the Backlash of Roller Enveloping Hourglass Worm Gear: Theoretical Analysis and Experiment

[+] Author and Article Information
Xingqiao Deng

School of Mechanical Engineering,
Xihua University,
999 Jinzhou Road,
Chengdu 610039, Sichuan, China
e-mail: dxq_zyc@hotmail.com

Jie Wang

School of Mechanical Engineering,
Xihua University,
999 Jinzhou Road,
Chengdu 610039, Sichuan, China
e-mail: 1291039637@qq.com

Shike Wang

School of Mechanical Engineering,
Xihua University,
999 Jinzhou Road,
Chengdu 610039, Sichuan, China
e-mail: 1558796839@qq.com

Shisong Wang

School of Mechanical Engineering,
Xihua University,
999 Jinzhou Road,
Chengdu 610039, Sichuan, China
e-mail: 935589973@qq.com

Jinge Wang

School of Mechanical Engineering,
Xihua University,
999 Jinzhou Road,
Chengdu 610039, Sichuan, China
e-mail: 13708066974@163.com

Shuangcen Li

School of Mechanical Engineering,
Xihua University,
999 Jinzhou Road,
Chengdu 610039, Sichuan, China
e-mail: 576801973@qq.com

Yucheng Liu

Fellow ASME
Center of Advanced Vehicular Systems,
Department of Mechanical Engineering,
Mississippi State University,
P. O. Box 9552,
Starkville, MS 39762
e-mail: liu@me.msstate.edu

Ge He

Mem. ASME
Center of Advanced Vehicular Systems,
Department of Mechanical Engineering,
Mississippi State University,
P. O. Box 9552,
Starkville, MS 39762
e-mail: gh663@msstate.edu

1Corresponding author.

Contributed by the Power Transmission and Gearing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 15, 2018; final manuscript received November 16, 2018; published online January 11, 2019. Assoc. Editor: Mohsen Kolivand.

J. Mech. Des 141(5), 053302 (Jan 11, 2019) (11 pages) Paper No: MD-18-1693; doi: 10.1115/1.4042155 History: Received September 15, 2018; Revised November 16, 2018

This paper proposes a single-roller enveloping hourglass worm gear design and verifies its advantages compared to the existing double-roller worm gear system and the conventional worm gear set. Our hypothesis is that the single-roller worm gear with appropriate configurations and parametric values can eliminate the backlash in mating gear transmission while maintaining advantages of the double-roller worm gears. Also, the self-rotation of the rollers when they are in the worm tooth space (TS) will help the gear system to avoid jamming and gear tooth scuffing/seizing problems caused by zero backlash and thermal expansion. In order to test that hypothesis, a mathematical model for the single-roller enveloping hourglass worm gear is developed, which includes a gear engagement equation and a tooth profile equation. Using that model, a parametric study is conducted to inspect the influences of center distance, roller radius, transmission ratio, and the radius of base circle on the worm gear meshing characteristics. It is found that the most effective way in eliminating the backlash is to adjust the roller radius and the radius of base circle. Finally, a single-roller enveloping hourglass worm gear set is manufactured and scanned to generate a 3D computer model. That model is compared with a theoretical model calculated from the developed mathematical model. Comparison results show that both models match very well, which verifies the accuracy of the developed mathematical model and our initial hypothesis that it is possible to achieve transmissions with zero backlash by adjusting the design parameters.

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References

Simon, V. , 2008, “ Influence of Tooth Errors and Misalignments on Tooth Contact in Spiral Bevel Gears,” Mech. Mach. Theory, 43(10), pp. 1253–1267. [CrossRef]
Deng, X.-Q. , Wang, J.-L. , and Wang, J.-G. , 2015, “ Parametric Analysis of the End Face Engagement Worm Gear,” Chin. J. Mech. Eng., 28(6), pp. 1177–1185. [CrossRef]
Fujisawa, Y. , and Komori, M. , 2015, “ Surface Finishing Method for Tooth Flank of Heat-Treated Surface-Hardened Small Gears Using a Gear-Shaped Tool Composited of Alumina-Fiber-Reinforced Plastic,” Precis. Eng., 39(3), pp. 234–242. [CrossRef]
Kacalak, W. , Majewski, M. , and Budniak, Z. , 2015, “ Worm Gear Drives With Adjustable Backlash,” ASME J. Mech. Rob., 8(1), p. 014504. [CrossRef]
Dudas, L. , 2012, “ Modeling and Simulation of a New Worm Gear Drive Having Point-Like Contact,” Eng. Comput., 29(3), pp. 251–272. [CrossRef]
Medvedev, V. I. , Volkov, A. E. , and Volosova, M. A. , 2015, “ Mathematical Model and Algorithm for Contact Stress Analysis of Gears With Multi Pair Contact,” Mech. Mach. Theory, 86(2), pp. 1561–1571.
Chen, Y.-Z. , Luo, L. , and Hu, Q. , 2009, “ The Contact Ratio of a Space-Curve Meshing-Wheel Transmission Mechanism,” ASME J. Mech. Des., 131(7), p. 074501. [CrossRef]
Lumpkin, T. , and Wolf, T. , 2006, “ Methods and Apparatus for Minimizing Backlash in a Planetary Mechanism,” Woodward HRT, Santa Clarita, CA, U.S. Patent No. 7121973B2. https://patents.google.com/patent/US7121973B2/en
Simon, V. , 1988, “ Double Enveloping Worm Gear Drive With Smooth Gear Tooth Surface,” International Conference on Gearing, Zhengzhou, China, Jan. 1, pp. 191–194.
Simon, V. , 1993, “ Stress Analysis in Double Enveloping Worm Gears by Finite Element Method,” ASME J. Mech. Des., 115(1), pp. 179–185. [CrossRef]
Simon, V. , 1993, “ Load Distribution in Double Enveloping Worm Gears,” ASME J. Mech. Des., 115(3), pp. 496–501. [CrossRef]
Deng, X.-Q. , Wang, J.-G. , and Horstemeyer, M. F. , 2013, “ Modification Design Method for an Enveloping Hourglass Worm With Consideration of Machining and Misalignment Errors,” Chin. J. Mech. Eng., 26(5), pp. 948–956. [CrossRef]
Deng, X.-Q. , Wang, J.-G. , and Horstemeyer, M. F. , 2013, “ Parametric Study of Meshing Characteristics With Respect to Different Meshing Rollers of the Anti-Backlash Double-Roller Enveloping Worm Gear,” ASME J. Mech. Des., 134(8), p. 081004. [CrossRef]
Litvin, F. L. , and Fuentes, A. , 2005, Gear Geometry and Applied Theory, Cambridge University Press, Cambridge, UK.
Oberg, E. , Jones, F. , Horton, H. , Ryffel, H. , and McCauley, C. , 2016, Machinery's Handbook, 30th ed., Industrial Press, South Norwalk, CT.

Figures

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Fig. 1

Double-roller enveloping hourglass worm drive developed by Deng et al. [12]

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Fig. 2

Installation of single rollers onto the worm gear

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Fig. 3

Coordinate systems associated with the worm and the worm wheel

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Fig. 4

Dynamic coordinate system on the roller of the worm wheel

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Fig. 5

Relationship between the vectors involved in this parametric study

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Fig. 6

Transverse section of the worm with coordinate systems ST and S1

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Fig. 7

Tooth profile on the axial symmetric cross section of worm

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Fig. 8

Influence of roller radius on contact curves: (a) R = 3 mm, (b) R = 5 mm, (c) R = 7 mm, and (d) R = 9 mm

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Fig. 9

Influence of transmission ratio on contact curves: (a) G = 10, (b) G = 20, (c) G = 30, and (d) G = 40

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Fig. 10

Influence of center distance on contact curves: (a) A = 80 mm, (b) A = 125 mm, (c) A = 160 mm, and (d) A = 220 mm

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Fig. 11

Influence of base circle radius on contact curves: (a) rb = 64.05 mm, (b) rb = 64.15 mm, (c) rb = 64.25 mm, and (d) rb = 64.35 mm

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Fig. 12

Three-dimensional model of the single-roller enveloping hourglass worm gear system: (a) worm gear system and (b) reducer

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Fig. 13

Prototype of the final design of single-roller enveloping hourglass worm gear set: (a) worm, (b) worm gear set, and (c) reducer

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Fig. 14

Scan worm tooth profile using the ROMER Absolute Arm CMM

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Fig. 15

Scan worm tooth profile using the ROMER Absolute Arm CMM

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Fig. 16

Three-dimensional model for the single-roller enveloping hourglass worm: (a) generated by scanning the prototype and (b) predicted using Eqs. (11)(16)

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Fig. 17

Process of comparing the two 3D models for the roller enveloping hourglass worm using geomagic

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Fig. 18

Comparison between the two 3D models for the roller enveloping hourglass worm

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Fig. 19

Deviation analysis results of the two 3D worm models

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Fig. 20

Contact between the worm and the single-roller worm gear: (a) before parameters adjustment and (b) before parameters adjustment

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