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Research Papers: Design of Direct Contact Systems

A Gear Load Distribution Model for a Planetary Gear Set With a Flexible Ring Gear Having External Splines

[+] Author and Article Information
Yong Hu, Ahmet Kahraman

Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210

David Talbot

Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210
e-mail: talbot.11@osu.edu

1Corresponding author.

Contributed by the Power Transmission and Gearing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 30, 2018; final manuscript received September 17, 2018; published online January 11, 2019. Assoc. Editor: Hai Xu.

J. Mech. Des 141(5), 053301 (Jan 11, 2019) (12 pages) Paper No: MD-18-1604; doi: 10.1115/1.4041583 History: Received July 30, 2018; Revised September 17, 2018

In order to accurately predict ring gear deformations and to investigate the effects of ring gear flexibility on quasi-static behaviors of planetary gear sets, a complete load distribution model of planetary gear sets having flexible ring gears will be formulated here based on the baseline model proposed by the same authors (Hu, Y., Talbot, D., and Kahraman, A., 2018, “A Load Distribution Model for Planetary Gear Sets,” ASME J. Mech. Des., 140(5), p. 053302). Direct comparisons to published experiments are provided to assess the accuracy of the proposed load distribution methodology. Example analyses with flexible ring gear rims are performed indicating that ring gear flexibility could influence gear mesh-level and planetary gear set system-level behaviors. Influence of spline supporting a ring gear is also investigated revealing that positions of planet branches with respect to external splines could influence ring deflections and resultant gear mesh load distributions.

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References

Hu, Y. , Talbot, D. , and Kahraman, A. , 2018, “ A Load Distribution Model for Planetary Gear Sets,” ASME J. Mech. Des., 140(5), p. 053302. [CrossRef]
Kahraman, A. , and Vijayakar, S. , 2001, “ Effect of Internal Gear Flexibility on the Quasi-Static Behavior of a Planetary Gear Set,” ASME J. Mech. Des., 123(3), pp. 408–415. [CrossRef]
Kahraman, A. , Kharazi, A. A. , and Umrani, M. , 2003, “ A Deformable Body Dynamic Analysis of Planetary Gears With Thin Rims,” J. Sound Vib., 262(3), pp. 752–768. [CrossRef]
Ligata, H. , 2007, “ Impact of System-Level Factors on Planetary Gear Set Behavior,” Ph.D. dissertation, The Ohio State University, Columbus, OH. http://rave.ohiolink.edu/etdc/view?acc_num=osu1172599656
Kahraman, A. , Ligata, H. , and Singh, A. , 2009, “ Closed-Form Planet Load Sharing Formula for Planetary Gear Set Using a Translational Analogy,” ASME J. Mech. Des., 131(2), p. 021007. [CrossRef]
Kahraman, A. , Ligata, H. , and Singh, A. , 2010, “ Influence of Ring Gear Rim Thickness on Planetary Gear Set Behavior,” ASME J. Mech. Des., 132(2), p. 021002. [CrossRef]
Hidaka, T. , Terauchi, Y. , and Nagamura, K. , 1979, “ Dynamic Behavior of Planetary Gear—7th Report, Influence of the Thickness of the Ring Gear,” Bull. JSME, 22(170), pp. 1142–1149. [CrossRef]
Hidaka, T. , Ishida, T. , and Uchida, F. , 1984, “ Effects of Rim Thickness and Number of Teeth on Bending Stress of Internal Gears,” Bull. JSME, 27(223), pp. 110–116. [CrossRef]
Lewicki, D. G. , and Ballarini, R. , 1997, “ Effect of Rim Thickness on Gear Crack Propagation Path,” ASME J. Mech. Des., 119(1), pp. 88–95. [CrossRef]
Oda, S. , Nagamura, K. , and Aoki, K. , 1981, “ Stress Analysis of Thin Rim Spur Gears by Finite Element Method,” Bull. JSME, 24(193), pp. 1273–1280. [CrossRef]
Chang, S. H. , Huston, R. L. , and Coy, J. J. , 1983, “ A Finite Element Stress Analysis of Spur Gears Including Fillet Radii and Rim Thickness Effects,” ASME J. Mech., Transmissions, Autom. Des., 105(3), pp. 327–330. [CrossRef]
Oda, S. , Miyachika, K. , and Araki, K. , 1984, “ Effects of Rim Thickness on Root Stress and Bending Fatigue Strength of Internal Gear Tooth,” Bull. JSME, 27(230), pp. 1759–1763. [CrossRef]
Oda, S. , and Miyachika, K. , 1987, “ Root Stress of Thin-Rimmed Internal Spur Gear Supported With Pins,” JSME Int. J., 30(262), pp. 646–652. [CrossRef]
Bibel, G. D. , Reddy, S. K. , Savage, M. , and Handschuh, R. F. , 1994, “ Effects of Rim Thickness on Spur Gear Bending Stress,” ASME J. Mech. Des., 116(4), pp. 1157–1162. [CrossRef]
Chen, Z. , and Shao, Y. , 2013, “ Mesh Stiffness of an Internal Spur Gear Pair With Ring Gear Rim Deformation,” Mech. Mach. Theory, 69, pp. 1–12. [CrossRef]
Chen, Z. , Zhu, Z. , and Shao, Y. , 2015, “ Fault Feature Analysis of Planetary Gear System With Tooth Root Crack and Flexible Ring Gear Rim,” Eng. Fail. Anal., 49, pp. 92–103. [CrossRef]
Chen, Z. , and Su, D. , 2015, “ Dynamic Simulation of Planetary Gear Set With Flexible Spur Ring Gear,” J. Sound Vib., 332(26), pp. 7191–7203. [CrossRef]
Gasmi, A. , Joseph, P. F. , Rhyne, T. B. , and Cron, S. M. , 2010, “ Closed-Form Solution of a Shear Deformable, Extensional Ring in Contact Between Two Rigid Surfaces,” Int. J. Solids Struct., 48(5), pp. 843–853. [CrossRef]
Vijayakar, S. , 1991, “ A Combined Surface Integral and Finite Element Solution for a Three-Dimensional Contact Problem,” Int. J. Numer. Methods Eng., 31(3), pp. 525–545. [CrossRef]
Abousleiman, V. , and Velex, P. , 2006, “ A Hybrid 3D Finite Element Lumped Parameter Model for Quasi-Static and Dynamic Analyses of Planetary Epicyclic Gear Sets,” Mech. Mach. Theory, 41(6), pp. 725–748. [CrossRef]
Abousleiman, V. , Velex, P. , and Becquerelle, S. , 2007, “ Modeling of Spur and Helical Gear Planetary Drives With Flexible Ring Gears and Planet Carriers,” ASME J. Mech. Des., 29(1), pp. 95–106. [CrossRef]
Chapron, M. , Velex, P. , Bruyere, J. , and Becquerelle, S. , 2016, “ Optimization of profile modifications With Regard to Dynamic Tooth Loads in Single and Double-Helical Planetary Gears With Flexible Ring-Gears,” ASME J. Mech. Des., 138(2), p. 023301. [CrossRef]
Wu, X. , and Parker, R. G. , 2006, “ Vibration of Rings on a General Elastic Foundations,” J. Sound Vib., 295(1–2), pp. 194–213. [CrossRef]
Bodas, A. , and Kahraman, A. , 2004, “ Influence of Carrier and Gear Manufacturing Errors on the Static Load Sharing Behavior of Planetary Gear Sets,” JSME Int. J., Ser. C, 47(3), pp. 908–915. [CrossRef]
Bodas, A. , 2001, “ Influence of Manufacturing Errors and Assembly Variations on the Load Sharing Behavior of Planetary Gear Sets Under Quasi-Static Conditions,” Master's thesis, University of Toledo, Toledo, OH.
Singh, A. , 2005, “ Application of a System Level Model to Study the Planetary Load Sharing Behavior,” ASME J. Mech. Des., 127(3), pp. 469–476. [CrossRef]
Bogner, F. K. , Fox, R. L. , and Schmit, L. A. , 1967, “ A Cylindrical Shell Discrete Element,” Am. Inst. Aeronaut. Astronaut. J., 5(4), pp. 745–750. [CrossRef]
Meck, H. R. , 1980, “ An Accurate Polynomial Displacement Function for Unite Ring Elements,” Comput. Struct., 11(4), pp. 265–269. [CrossRef]
Pandian, N. , Rao, T. A. , and Chandra, S. , 1989, “ Studies on Performance of Curved Beam Finite Elements for Analysis of Thin Arches,” Comput. Struct., 31(6), pp. 997–1002. [CrossRef]
Yamada, Y. , and Ezawa, Y. , 1997, “ On Curved Finite Elements for the Analysis of Circular Arches,” Int. J. Numer. Methods Eng., 11(11), pp. 1635–1651. [CrossRef]
Palaninathan, R. , and Chandrasekharan, P. S. , 1985, “ Curved Beam Element Stiffness Matrix Formulation,” Comput. Struct., 21(4), pp. 663–669. [CrossRef]
Friedman, Z. , and Kosmatka, J. B. , 1998, “ An Accurate Two-Node Finite Element for Shear Deformable Curved Beams,” Int. J. Numer. Methods Eng., 41(3), pp. 473–498. [CrossRef]
Ashwell, D. G. , Sabir, A. B. , and Roberts, T. M. , 1971, “ Further Studies in the Application of Curved Finite Elements to Circular Arches,” Int. J. Mech. Sci., 13(6), pp. 507–517. [CrossRef]
Ashwell, D. G. , and Sabir, A. B. , 1971, “ Limitations of Certain Curved Finite Elements When Applied to Arches,” Int. J. Mech. Sci., 13(2), pp. 133–139. [CrossRef]
Sabir, A. B. , and Ashwell, D. G. , 1971, “ A Comparison of Curved Beam Finite Elements When Used in Vibration Problems,” J. Sound Vib., 18(4), pp. 555–563. [CrossRef]
Sabir, A. B. , Djoudi, M. S. , and Sfendji, A. , 1994, “ The Effect of Shear Deformation on Vibration of Circular Arches by Finite Elements Method,” Thin-Walled Struct., 18(1), pp. 47–63. [CrossRef]
Zienkiewicz, O. C. , 2013, The Finite Element Method: Its Basis and Fundamentals–7th Edition, Butterworth-Heinemann, Waltham, MA.

Figures

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Fig. 1

(a) A three-node circular beam element, and (b) finite element model of a ring gear segment

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Fig. 2

External spline tooth l¯ splits the element l between ring gear tooth l and (l + 1) into two elements

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Fig. 3

Definition of contact point m in the local circular coordinate system of tooth l of the ring gear and the global coordinate system of the planetary gear set

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Fig. 4

Measured and predicted radial ring gear deflection ΔR of the three-planet gear set for ring gears having (a) Γ = 0.058, (b) Γ = 0.084, and (c) Γ = 0.112 at Ts=1000 N·m

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Fig. 5

Ring gear deflections of the gear set of Table 1 with Γ=0.058 under Ts=1000 N·m: (a) N = 3, (b) N = 4, (c) N = 5, and (d) N = 6 (radial rim deflections are magnified 200×)

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Fig. 6

(a) Peak contact stress of the sun-planet meshes versus Γ, and (b) maximum contact stress distribution over the line of action for a rigid ring as well as ring gears having Γ = 0.0543, 0.0373 and 0.0237

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Fig. 7

(a) Peak contact stress of the ring-planet meshes versus Γ, and (b) maximum contact stress distribution over the line of action for a rigid ring as well as ring gears having Γ = 0.0543, 0.0373, and 0.0237

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Fig. 8

(1) Deformed ring gear and (2) contact stress [MPa] distribution within plane of action of all gear meshes at multiple positions (a,b,c,d)

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Fig. 9

Maximum contact pressures of sun-planet 1 and ring-planet 1 meshes with respect to carrier rotation spanning three consecutive splines

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Fig. 10

Load sharing of (a) a four-planet gear set and (b) a five-planet gear set with rigid (dashed lines) and flexible (solid lines) ring gear with respect to Ts under E1=70μm and other Ei=0

Tables

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