Automatic Structural Synthesis of Planetary Geared Mechanisms ‎using Graph Theory

Document Type : Research Paper


Department of Mechanical Engineering, University of Al-Qadisiyah, Diwaniyah, 58001, Iraq‎


Graphs are an effective tool for planetary gear trains (PGTs) synthesis and for the enumeration of all possible PGTs for transmission systems. In the past fifty years, considerable effort has been devoted to the synthesis of PGTs. To date, however, synthesis results are inconsistent, and accurate synthesis results are difficult to achieve. This paper proposes a systematic approach for synthesizing PGTs depending on spanning trees and parent graphs. Trees suitable for constructing rooted graphs are first identified. The parent graphs are then listed. Finally, geared graphs are discovered by inspecting their parent graphs and spanning trees. To precisely detect spanning trees, a novel method based on two link assortment equations is presented. Transfer vertices and edge levels are detected without the use of any computations. This work develops the vertex matrix of the rooted graph, and its distinctive equation is used to arrange the vertex degree arrays according to the vertex levels and eliminate the arrays that violate the distinctive equations. The precise results of the 5-link geared graphs are confirmed to be 24. The disparity between the recent and previous synthesis results can be attributed to the fact that the findings of the current method, which employs rooted graphs, are more comprehensive than those obtained with graphs lacking multiple joints. A novel algorithm for detecting structural isomorphism is proposed. By comparing the vertex degree listings and gear strings, non-isomorphic geared graphs are obtained. The algorithm is simple and computationally efficient. The graph representation is one-to-one with the vertex degree listing and gear string representation. This allows for the storage of a large number of graphs on a computer for later use.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Tsai, L.W., Mechanism Design Enumeration of Kinematic Structures According to Function, CRC Press LLC, Boca Raton, 2001.
[2] Döring, U., Brix, T., Reeßing, M., Application of computational kinematics in the digital mechanism and gear library DMG-Lib, Mech. Mach. Theory, 41(8), 2006, 1003-1015.
[3] Pennestrì, E., Belfiore, N.P., On Crossley's contribution to the development of graph based algorithms for the analysis of mechanisms and gear trains, Mech. Mach. Theory, 89, 2015, 1003-1015.
[4] Ding, H., Yang, W., Kecskeméthy, A., Automatic Structural Synthesis and Creative Design of Mechanisms, Springer Nature Singapore, 2022.
[5] Robin, J.W., Introduction to Graph Theory, Addison Wesley Longman, 1996.
[6] Esmail, E.L., Juber, A.H., An Application of Graph Theory for the Detection of Degenerate Structures in Planetary Gear Trains, Proceedings of the ASME 2021 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 8A: 45th Mechanisms and Robotics Conference (MR), Virtual, Online, August 17–19, V08AT08A025, ASME, 2021.
[7] Shanmukhasundaram, V.R., Number synthesis and structure based rating of multilinkepicyclic gear trains satisfying gruebler’s degree of freedom equation, Ph.D. Thesis, BITS Pilani – Hyderabad Campus, India, 2020.
[8] Marciniec, A., Sobolak, M., Połowniak, P., Graphical method for the analysis of planetary gear trains, Alex. Eng. J., 6, 2022, 4067-4079
[9] Drewniak, J., Kopec, J., Zawilak, S., Kinematical Analysis of Variants of Wind Turbine Drive by Means of Graphs, Graph-Based Modelling in Engineering, Mechanism and Machine Science Series, 42, 2016, 81-95.
[10] Drewniak, J., Zawilak, S., Linear-Graph and Contour-Graph based models of planetary gears, J. Theor. Appl. Mech., 48, 2010, 415-433.
[11] Tan, W. , Wu, J., Ni, D., Yan, H., Xiang, E., Liu, S., Dynamic Modeling and Simulation of Double- Planetary Gearbox Based on Bond Graph, Math. Probl. Eng., 2021, 2021, 1-14.
[12] Buchsbaum, F., Freudenstein, F., Synthesis of kinematic structure of geared kinematic chains and other mechanisms, J. Mech., 5(3), 1970, 357–392.
[13] Freudenstein, F., An application of Boolean algebra to the motion of epicyclic drives, ASME J. Eng. Ind., 93(1), 1971, 176–182.
[14] Chatterjee, G., Tsai, L.W., Enumeration of Epicyclic-Type Automatic Transmission Gear Trains, SAE International Congress and Exposition, Paper No. 941012, Transmission and Driveline Developments, SP-1032, 1994.
[15] Chatterjee, G., Tsai, L.W., Computer-aided sketching of epicyclic-type automatic transmission of gear trains, ASME J. Mech. Design, 118, 1996, 405–411.
[16] Hsu, C.H., Lam, K.T., Automatic analysis of the kinematic structure of planetary gear trains, ASME J. Mech. Design, 115(3), 1993, 631-638.
[17] Hsu, C.H., Lam, K.T., A New Graph Representation for the Automatic Kinematic Analysis of Planetary Gear Trains, ASME J. Mech. Design, 114, 1992, 196–200.
[18] Yang, W.J., Ding, H.F., Zi, B., Zhang, D., New graph representation for planetary gear trains, ASME J. Mech. Design, 140(1), 2018, 012303.
[19] Yang, W.J., Ding, H.F., Kecskeméthy, A., Automatic Structural Synthesis of Non-Fractionated 2-DOF Planetary Gear Trains, Mech. Mach. Theory, 155, 2021, 104125.
[20] Shanmukhasundaram, V.R., Rao, Y.V.D., Regalla, S.P., Enumeration of displacement graphs of epicyclic gear train from a given rotation graph using concept of building of kinematic units, Mech. Mach. Theory, 134, 2019, 393–424.
[21] Shanmukhasundaram, V.R., Rao, Y.V.D., Regalla, S.P., Varadaraju, D., Pennestrì, E., Structural Synthesis and Classification of Epicyclic Gear Trains: An Acyclic Graph-Based Approach, In: Rao Y.V.D., Amarnath C., Regalla S.P., Javed A., Singh K.K. (eds) Advances in Industrial Machines and Mechanisms, Lecture Notes in Mechanical Engineering, Springer, Singapore, 2021.
[22] Hussen, H.A., Esmail E.L., Al-Mayali, M.F., Structure synthesis of planetary gear trains using graph partitioning, AIP Conference Proceedings, 2386, 2022, 040030.
[23] Tsai, L.W., An Application of the Linkage Characteristic Polynomial to the Topological Synthesis of Epicyclic Gear Train, ASME J. Mech. Transm. Autom. Des., 109(3),1987, 329-336.
[24] Tsai, L. W., Lin, C.C., The Creation of Non-fractionated Two-Degree-of-Freedom Epicyclic Gear Trains, ASME J. Mech. Transm. Autom. Des., 111(4), 1989, 524-529.
[25] Kim, J.U., Kwak, B.M., Application of Edge Permutation Group to Structural Synthesis of Epicyclic Gear Trains, Mech. Mach. Theory, 25(5), 1990, 563-574.
[26] Hsu, C.H., A Graph Representation for the Structural Synthesis of Geared Kinematic Chains, J. Frankl. Inst., 330(1), 1993, 131-143.
[27] Hsu, C.H., Lam, K.T., Yin, Y.L., Automatic Synthesis of Displacement Graphs for Planetary Gear Trains, Math. Comput. Modelling, 19(11), 1994, 67-81.
[28] Rao, Y.V.D., Rao, A.C., Generation of Epicyclic Gear Trains of One Degree of Freedom, ASME J. Mech. Des., 130(5), 2008, 052604.
[29] Kamesh, V.V., Rao, K.M., Rao, A.B.S., Topological Synthesis of Epicyclic Gear Trains Using Vertex Incidence Polynomial, ASME J. Mech. Des., 139(6), 2017, 062304.
[30] Ravisankar, R., Mruthyunjaya, T.S., Computerized Synthesis of the Structure of Geared Kinematic Chains, Mech. Mach. Theory, 20(5), 1985, 367-387.
[31] Shin, J.K., Krishnamurthy, S., Standard Code Technique in the Enumeration of Epicyclic Gear Trains, Mech. Mach. Theory, 28(3), 1993, 347-355.
[32] Hsu, C.H., Hsu, J.J., An Efficient Methodology for the Structural Synthesis of Geared Kinematic Chains, Mech. Mach. Theory, 32(8), 1997, 957-973.
[33] Cui, R., Ye, Z., Sun, L., Zheng, G., Wu, C., Synthesis method for planetary gear trains without using rotation graphs, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 236(2), 2022, 972-983.
[34] Yang, W.J., Ding, H.F., The complete set of one-degree-of-freedom planetary gear trains with up to nine links, ASME J. Mech. Des., 141(4), 2019, 043301.
[35] Prasad Raju Pathapati, V.V.N.R., Rao, A.C., A New Technique Based on Loops to Investigate Displacement Isomorphism in Planetary Gear Trains, ASME J. Mech. Des., 124(4), 2002, 662-675.
[36] Shanmukhasundaram, V.R., Rao, Y.V.D., Regalla, S.P., Review of Structural Synthesis Algorithms for Epicyclic Gear Trains, Springer, Singapore, 2021.
[37] Xu, X., Sun, H., Liu, Y., Dong, P., Matrix-Based Operation Method for Detecting Structural Isomorphism of Planetary Gear Train Structures, ASME J. Mech. Des., 142(6), 2020, 063301
[38] Yang, W.J., Ding, H.F., The perimeter loop-based method for the automatic isomorphism detection in planetary gear trains, ASME J. Mech. Des., 140(12), 2018, 123302.
[39] Hsu, C.H., Displacement Isomorphism of Planetary Gear Trains, Mech. Mach. Theory, 29, 1994, 513–523.