0
Research Papers: Design of Direct Contact Systems

The Complete Set of One-Degree-of-Freedom Planetary Gear Trains With Up to Nine Links

[+] Author and Article Information
Wenjian Yang

School of Mechanical Engineering and Electronic
Information,
China University of Geosciences,
No. 388 LuMo Road, Hongshan District,
Wuhan 430074, China
e-mail: ywj19900125@163.com

Huafeng Ding

School of Mechanical Engineering and
Electronic Information,
China University of Geosciences,
No. 388 LuMo Road,
Hongshan District,
Wuhan 430074, China
e-mail: dhf@ysu.edu.cn

1Corresponding author.

Contributed by the Power Transmission and Gearing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 3, 2018; final manuscript received September 11, 2018; published online January 11, 2019. Assoc. Editor: Hai Xu.

J. Mech. Des 141(4), 043301 (Jan 11, 2019) (22 pages) Paper No: MD-18-1356; doi: 10.1115/1.4041482 History: Received May 03, 2018; Revised September 11, 2018

The structural synthesis of planetary gear trains (PGTs) is helpful for innovating transmission systems in machinery. A great deal of research has been devoted to the synthesis of one-degree-of-freedom (1-DOF) PGTs over the past half century. However, most synthesis methods are limited to PGTs with no more than eight links. Moreover, the synthesis results are not consistent with each other. Until now, the inconsistency of synthesis results is still unresolved and exact synthesis results remain elusive. This paper presents a systematic and fully automatic method based on parent graphs to synthesize 1-DOF PGTs. The complete database of rotation graphs (r-graphs) and displacement graphs (d-graphs) of 1-DOF PGTs with up to nine links is established for the first time. All possible reasons for the contradictory synthesis results in the literature are analyzed and the controversy in the existing synthesis results which has lasted for nearly half a century is completely resolved. The exact results of the 6-, 7-, and 8-link r-graphs are confirmed to be 27, 152, and 1070, respectively. The exact results of the 6-, 7-, and 8-link d-graphs are confirmed to be 81, 647, and 6360, respectively. Additionally, the new results of 8654 r-graphs and 71,837 d-graphs of 9-link PGTs are provided for the first time.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Levai, Z. , 1968, “ Structure and Analysis of Planetary Gear Trains,” J. Mech., 3(3), pp. 131–148. [CrossRef]
Buchsbaum, F. , and Freudenstein, F. , 1970, “ Synthesis of Kinematic Structure of Geared Kinematic Chains and Other Mechanisms,” J. Mech., 5(3), pp. 357–392. [CrossRef]
Freudenstein, F. , 1971, “ An Application of Boolean Algebra to the Motion of Epicyclic Drives,” ASME J. Eng. Ind., 93(1), pp. 176–182. [CrossRef]
Ravisankar, R. , and Mruthyunjaya, T. S. , 1985, “ Computerized Synthesis of the Structure of Geared Kinematic Chains,” Mech. Mach. Theory, 20(5), pp. 367–387. [CrossRef]
Tsai, L. W. , 1987, “ An Application of the Linkage Characteristic Polynomial to the Topological Synthesis of Epicyclic Gear Train,” ASME J. Mech., Trans. Autom., 109(3)pp. 329–336. [CrossRef]
Tsai, L. W. , and Lin, C. C. , 1989, “ The Creation of Nonfractionated Two-Degree-of-Freedom Epicyclic Gear Trains,” ASME J. Mech., Trans. Autom., 111(4), pp. 524–529. [CrossRef]
Hsieh, H. I. , and Tsai, L. W. , 1996, “ Kinematic Analysis of Epicyclic-Type Transmission Mechanisms Using the Concept of Fundamental Geared Entities,” ASME J. Mech. Des., 118(2), pp. 294–299. [CrossRef]
Tsai, L. W. , 2000, Mechanism Design: Enumeration of Kinematic Structures According to Function, CRC Press, Boca Raton, FL.
Kim, J. U. , and Kwak, B. M. , 1990, “ Application of Edge Permutation Group to Structural Synthesis of Epicyclic Gear Trains,” Mech. Mach. Theory, 25(5), pp. 563–574. [CrossRef]
Shin, J. K. , and Krishnamurthy, S. , 1993, “ Standard Code Technique in the Enumeration of Epicyclic Gear Trains,” Mech. Mach. Theory, 28(3), pp. 347–355. [CrossRef]
Hsu, C. H. , and Lam, K. T. , 1992, “ A New Graph Representation for the Automatic Kinematic Analysis of Planetary Spur-Gear Trains,” ASME J. Mech. Des., 114(1), pp. 196–200. [CrossRef]
Hsu, C. H. , 1993, “ A Graph Representation for the Structural Synthesis of Geared Kinematic Chains,” J. Franklin Inst., 330(1), pp. 131–143. [CrossRef]
Hsu, C. H. , 1994, “ Displacement Isomorphism of Planetary Gear Trains,” Mech. Mach. Theory, 29(4), pp. 513–523. [CrossRef]
Hsu, C. H. , Lam, K. T. , and Yin, Y. L. , 1994, “ Automatic Synthesis of Displacement Graphs for Planetary Gear Trains,” Math. Comput. Modell., 19(11), pp. 67–81. [CrossRef]
Hsu, C. H. , and Wu, Y. C. , 1997, “ Automatic Detection of Embedded Structure in Planetary Gear Trains,” ASME J. Mech. Des., 119(2), pp. 315–318. [CrossRef]
Hsu, C. H. , and Hsu, J. J. , 1997, “ An Efficient Methodology for the Structural Synthesis of Geared Kinematic Chains,” Mech. Mach. Theory, 32(8), pp. 957–973. [CrossRef]
Hsu, C. H. , 2002, “ An Analytic Methodology for the Kinematic Synthesis of Epicyclic Gear Mechanisms,” ASME J. Mech. Des., 124(3), pp. 574–576. [CrossRef]
Chen, D. Z. , and Yao, K. L. , 2000, “ Topological Synthesis of Fractionated Geared Differential Mechanisms,” ASME J. Mech. Des., 122(4), pp. 472–478. [CrossRef]
Chen, D. Z. , Liu, C. P. , and Duh, D. W. , 2003, “ A Modular Approach for the Topological Synthesis of Geared Robot Manipulators,” Mech. Mach. Theory, 38(1), pp. 53–69. [CrossRef]
Liu, C. P. , Chen, D. Z. , and Chang, Y. T. , 2004, “ Kinematic Analysis of Geared Mechanisms Using the Concept of Kinematic Fractionation,” Mech. Mach. Theory, 39(11), pp. 1207–1221. [CrossRef]
Del Castillo, J. M. , 2002, “ Enumeration of 1-DOF Planetary Gear Train Graphs Based on Functional Constraints,” ASME J. Mech. Des., 124(4), pp. 723–732. [CrossRef]
Salgado, D. R. , and Del Castillo, J. M. , 2005, “ A Method for Detecting Degenerate Structures in Planetary Gear Trains,” Mech. Mach. Theory, 40(8), pp. 948–962. [CrossRef]
Prasad Raju Pathapati, V. V. N. R. , and Rao, A. C. , 2002, “ A New Technique Based on Loops to Investigate Displacement Isomorphism in Planetary Gear Trains,” ASME J. Mech. Des., 124(4), pp. 662–675. [CrossRef]
Rao, A. C. , 2003, “ A Genetic Algorithm for Epicyclic Gear Trains,” Mech. Mach. Theory, 38(2), pp. 135–147. [CrossRef]
Rao, Y. V. D. , and Rao, A. C. , 2008, “ Generation of Epicyclic Gear Trains of One Degree of Freedom,” ASME J. Mech. Des., 130(5), p. 052604. [CrossRef]
Hsieh, W. H. , 2009, “ Kinematic Synthesis of Cam-Controlled Planetary Gear Trains,” Mech. Mach. Theory, 44(5), pp. 873–895. [CrossRef]
Xie, T. L. , Hu, J. B. , Peng, Z. X. , and Liu, C. W , 2015, “ Synthesis of Seven-Speed Planetary Gear Trains for Heavy-Duty Commercial Vehicle,” Mech. Mach. Theory, 90, pp. 230–239. [CrossRef]
Ngo, H. T. , and Yan, H. S. , 2016, “ Configuration Synthesis of Parallel Hybrid Transmissions,” Mech. Mach. Theory, 97, pp. 51–71. [CrossRef]
Kamesh, V. V. , Rao, K. M. , and Rao, A. B. S. , 2017, “ Topological Synthesis of Epicyclic Gear Trains Using Vertex Incidence Polynomial,” ASME J. Mech. Des., 139(6), p. 062304. [CrossRef]
Kamesh, V. V. , Rao, K. M. , and Rao, A. B. S. , 2017, “ An Innovative Approach to Detect Isomorphism in Planar and Geared Kinematic Chains Using Graph Theory,” ASME J. Mech. Des., 139(12), p. 122301. [CrossRef]
Kamesh, V. V. , Rao, K. M. , and Rao, A. B. S. , 2017, “ Detection of Degenerate Structure in Single Degree-of-Freedom Planetary Gear Trains,” ASME J. Mech. Des., 139(8), p. 083302. [CrossRef]
Li, M. , Xie, L. Y. , and Ding, L. J , 2017, “ Load Sharing Analysis and Reliability Prediction for Planetary Gear Train of Helicopter,” Mech. Mach. Theory, 115, pp. 97–113. [CrossRef]
Barhoumi, T. , and Kum, D. , 2017, “ Automatic Enumeration of Feasible Kinematic Diagrams for Split Hybrid Configurations With a Single Planetary Gear,” ASME J. Mech. Des., 139(8), p. 083301. [CrossRef]
Wei, J. , Zhang, A. Q. , Qin, D. T. , Lim, T. C. , Shu, R. Z. , Lin, X. Y. , and Meng F. M. , 2017, “ A Coupling Dynamics Analysis Method for a Multistage Planetary Gear System,” Mech. Mach. Theory, 110, pp. 27–49. [CrossRef]
Gao, M. F. , and Hu, J. B. , 2018, “ Kinematic Analysis of Planetary Gear Trains Based on Topology,” ASME J. Mech. Des., 140(1), p. 012302. [CrossRef]
Yang, W. J. , Ding, H. F. , Zi, B. , and Zhang, D. , 2018, “ New Graph Representation for Planetary Gear Trains,” ASME J. Mech. Des., 140(1), p. 012303. [CrossRef]
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]
Esmail, E. L. , 2018, “ Influence of the Operating Conditions of Two-Degree-of-Freedom Planetary Gear Trains on Tooth Friction Losses,” ASME J. Mech. Des., 140(5), p. 054501. [CrossRef]
Ding, H. F. , Huang, Z. , and Mu, D. J , 2008, “ Computer-Aided Structure Decomposition Theory of Kinematic Chains and Its Applications,” Mech. Mach. Theory, 43(12), pp. 1596–1609. [CrossRef]
Ding, H. F. , and Huang, Z. , 2007, “ The Establishment of the Canonical Perimeter Topological Graph of Kinematic Chains and Isomorphism Identification,” ASME J. Mech. Des., 129(9), pp. 915–923. [CrossRef]
Ding, H. F. , Yang, W. J. , Zi, B. , and Kecskemethy, A. , 2016, “ The Family of Planar Kinematic Chains With Two Multiple Joints,” Mech. Mach. Theory, 99, pp. 103–116. [CrossRef]
Ding, H. F. , Huang, P. , Yang, W. J. , and Kecskemethy, A. , 2016, “ Automatic Generation of the Complete Set of Planar Kinematic Chains With Up to Six Independent Loops and Up to 19 Links,” Mech. Mach. Theory, 96, pp. 75–93. [CrossRef]
Yang, W. J. , and Ding, H. F. , 2018, “ Automatic Detection of Degenerate Planetary Gear Trains With Different Degree of Freedoms,” Appl. Math. Modell., 64, pp. 320–332. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

(a) A parent graph and its adjacency matrix and (b) a geared graph and its adjacency matrix

Grahic Jump Location
Fig. 2

The main steps of the synthesis method

Grahic Jump Location
Fig. 3

The graph representation and adjacency matrix of Simpson gear train

Grahic Jump Location
Fig. 4

The r-graph of a 7-link 1-DOF PGT

Grahic Jump Location
Fig. 5

(a) An 8-link 1-DOF PGT and (b) the tree

Grahic Jump Location
Fig. 6

The subgraphs derived in the detection process

Grahic Jump Location
Fig. 7

(a) A 7-link parent graph, (b) and (c) canonical graphs

Grahic Jump Location
Fig. 8

(a) The perimeter loop graph and (b) the characteristic graph

Grahic Jump Location
Fig. 9

Fractionated parent graphs

Grahic Jump Location
Fig. 10

Automatic synthesis of 6-link 1-DOF parent graphs

Grahic Jump Location
Fig. 11

An invalid geared graph

Grahic Jump Location
Fig. 12

Automatic synthesis of r-graphs

Grahic Jump Location
Fig. 13

Edge level solutions for the r-graph in Fig. 3(b)

Grahic Jump Location
Fig. 14

The d-graphs corresponding to Fig. 13

Grahic Jump Location
Fig. 15

Automatic synthesis of d-graphs

Grahic Jump Location
Fig. 16

(a) The 6-link d-graph, (b) the 7-link d-graphs, (c) a part of 8-link d-graphs, and (d) a part of 9-link d-graphs that cannot be generated via the recursive method

Grahic Jump Location
Fig. 17

The process of acquiring the adjacency matrix

Grahic Jump Location
Fig. 18

The graph corresponding to the adjacency matrix

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In