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Analytical geometry method of planetary gear trains

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Abstract

This paper presents an analytical geometry method for kinematics and efficiency of planetary gear trains (PGTs). The novel method which is capable of evolution and contrast analysis of mechanism kinematics, can be applied to any typical one- and two-degree-of-freedom plane PGTs containing any number of simple, compound or complex-compound planetary gear sets. The efficiency analysis of this method features a systematized and programmed process and its independence of the speed ratio. The primary contribution of this work lies in the integration of quantitative calculation, qualitative evolution and comparative analysis of kinematics of PGTs into one diagram, and in the integration of kinematics and efficiency analysis into a single method system. First, the analytical geometry method is defined, its basic properties are given, and the systematization procedure to perform kinematic analysis is demonstrated. As an application, analytical geometry diagrams of common PGTs are exhibited in the form of a list, whose kinematic characteristics and general evolution tendency are discussed. Then, with the mapping of PGTs onto the angular speed plane, the efficiency formula of analytical geometry, which has an extremely concise form, and a simple method for power flow estimation are put forward. Moreover, a general procedure is provided to analyze the efficiency and power flow. Finally, four numerical examples including a complicated eleven-link differential PGTs are given to illustrate the simpleness and intuitiveness of the analytical geometry method.

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References

  1. Tsai L W. An application of the linkage characteristic polynomial to the topological synthesis of epicyclic gear trains. J Mech Transm Autom Des, 1987, 109(3): 329–336

    Article  Google Scholar 

  2. Rao Z G. Design of Planetary Gear Transmission (in Chinese). Beijing: Chemical Industry Press, 2003

    Google Scholar 

  3. Design and Manufacture of involute planetary gear transmission (in Chinese). Beijing: China Machine Press, 2004

  4. Freudenstein F. An application of boolean algebra to the motion of epicyclic drives. ASME J Eng Ind, 1971, 93(Series B): 176–182

    Article  Google Scholar 

  5. Tsai L W. The kinematics of spatial robotic bevel-gear trains. IEEE J Robot Autom, 1988, 4(2): 150–156

    Article  Google Scholar 

  6. Hsieh L C, Yan H S, Wu L I. Algorithm for the automatic kinematic analysis of planetary gear trains by fundamental circuit method. J Chin Soc Mech Eng, T Chin Inst Eng-Ser C, 1989, 10(2): 153–158

    Google Scholar 

  7. Hsu C H, Lam K T. A new graph representation for the automatic kinematic analysis of planetary spur-gear trains. J Mech Des, 1992, 114(1): 196–200

    Article  MathSciNet  Google Scholar 

  8. Hsieh H I, Tsai L W. Kinematic analysis of epicyclic-type transmission mechanisms using the concept of fundamental geared entities. J Mech Des, 1996, 118(2): 294–299

    Article  Google Scholar 

  9. Chatterjee G, Tsai L W. Computer-aided sketching of epicyclic-type automatic transmission gear trains. J Mech Des, 1996, 118(3): 405–411

    Article  Google Scholar 

  10. Liu C P, Chen D Z. On the application of kinematic units to the topological analysis of geared mechanisms. J Mech Des, 2001, 123(2): 240–246

    Article  Google Scholar 

  11. Liu C P, Chen D Z, Chang Y T. Kinematic analysis of geared mechanisms using the concept of kinematic fractionation. Mech Mach Theory, 2004, 39(11): 1207–1221

    Article  MATH  Google Scholar 

  12. Chen D Z, Shieh W B, Yeh Y C. Kinematic characteristics and classification of geared mechanisms using the concept of kinematic fractionation. J Mech Des, 2008, 130(8): 082602

    Article  Google Scholar 

  13. Ma R, Gupta K C. Signal flow graphs for spatial gear trains. J Mech Des, 1994, 116(1): 326–331

    Article  Google Scholar 

  14. Rao A C. A genetic algorithm for epicyclic gear trains. Mech Mach Theory, 2003, 38(2): 135–147

    Article  MATH  Google Scholar 

  15. Kahraman A, Ligata H, Kienzle K, et al. A kinematics and power flow analysis methodology for automatic transmission planetary gear trains. J Mech Des, 2004, 126(6): 1071–1081

    Article  Google Scholar 

  16. Talpasanu I, Yih T C, Simionescu P A. Application of matroid method in kinematic analysis of parallel axes epicyclic gear trains. J Mech Des, 2006, 128(6): 1307–1314

    Article  Google Scholar 

  17. Pennock G R, Alwerdt J J. Duality between the kinematics of gear trains and the statics of beam systems. Mech Mach Theory, 2007, 42(11): 1527–1546

    Article  MATH  Google Scholar 

  18. Tsai M C, Huang C C, Lin B J. Kinematic analysis of planetary gear systems using block diagrams. J Mech Des, 2010, 132(6): 065001

    Article  Google Scholar 

  19. Müller H W. Epicyclic Drive Trains: Analysis, Synthesis and Applications. Detroit: Wayne State University Press, 1982

    Google Scholar 

  20. Hedman A. Transmission analysis-Automatic derivation of relationships. J Mech Des, 1993, 115(4): 1031–1037

    Article  Google Scholar 

  21. Tian L, Lu L Q. Matrix system for the analysis of planetary transmissions. J Mech Des, 1997, 119(3): 333–337

    Article  Google Scholar 

  22. del Castillo J M. Symbolic computation of planetary gear train efficiency. In: European Congress on Computational Methods in Applied Sciences and Engineering CIMNE, Barcelona, 2000

  23. del Castillo J M. The analytical expression of the efficiency of planetary gear trains. Mech Mach Theory, 2002, 37(2): 197–214

    Article  MATH  Google Scholar 

  24. Salgado D R, del Castillo J M. Selection and design of planetary gear trains based on power flow maps. J Mech Des, 2005, 127(1): 120–134

    Article  Google Scholar 

  25. Lin J D, Chen X A. Simplified approach for the determination of the mechanical efficiency in gear trains (in Chinese). Chin J Mech Eng, 2004, 40(9): 33–37

    Article  Google Scholar 

  26. Chen C, Angeles J. Virtual-power flow and mechanical gear-mesh power losses of epicyclic gear trains. J Mech Des, 2007, 129: 107–113

    Article  Google Scholar 

  27. Mantriota G, Pennestrì E. Theoretical and experimental efficiency analysis of multi-degrees-of-freedom epicyclic gear trains. Multibody Syst Dyn, 2003, 9(4): 389–408

    Article  MATH  Google Scholar 

  28. Esmail E L, Hassan S S. An approach to power-flow and static force analysis in multi-input multi-output epicyclic-type transmission trains. J Mech Des, 2010, 132(1): 011009

    Article  Google Scholar 

  29. Pennestrì E, Valentini P P. A review of formulas for the mechanical efficiency analysis of two degrees-of-freedom epicyclic gear trains. J Mech Des, 2003, 125(3): 602–608

    Article  Google Scholar 

  30. Willis J R. On the kinematics of the closed epicyclic differential gears. J Mech Des, 1982, 104: 712–723

    Article  Google Scholar 

  31. Wolfram S. The Mathematica Book. 5th ed. New York: Wolfram Media, 2003

    Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, 400044, China

    XiaoAn Chen & Hong Chen

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  1. XiaoAn Chen
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  2. Hong Chen
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Correspondence to XiaoAn Chen.

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Chen, X., Chen, H. Analytical geometry method of planetary gear trains. Sci. China Technol. Sci. 55, 1007–1021 (2012). https://doi.org/10.1007/s11431-011-4736-y

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  • DOI: https://doi.org/10.1007/s11431-011-4736-y

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