Journal of Applied and Computational Mechanics
The Structural Synthesis of Non-fractionated, Three-degree-of-freedom Planetary Gear Mechanisms
Document Type : Research Paper
Authors
Department of Mechanical Engineering, University of Al-Qadisiyah, Diwaniyah, 58001, Iraq
Abstract
Planetary gear trains (PGTs) with one or more degrees of freedom (DOFs) have numerous uses in PGT-based mechanisms. The majority of the currently available synthesis methods have focused on 1-DOF PGTs, with only a few investigations on multi-DOF PGT synthesis. The method for synthesizing 7-link 3-DOF PGMs is outlined. All possible link assortments are produced, labeled spanning trees are generated, and potential geared graphs are constructed. The guidelines for including geared edges and how to synthesize geared graphs are outlined. Vertex-degree arrays are generated to validate the geared graphs. Isomorphic geared graphs are identified by comparing the isomorphic identification numbers of geared graphs with the same spanning tree. Fractionated geared graphs are identified using the reachability matrix method. The new method has a straightforward algorithm. In contrast to what is reported in the literature, the results of the synthesis of 7-link 3-DOF PGMs show that there are seven non-fractionated mechanisms. MATLAB programs are used to acquire the vertex-degree arrays.
Keywords
Main Subjects
Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
[1] Li, M., Xie, L.Y., Ding, L.J., Load sharing analysis and reliability prediction for planetary gear train of helicopter, Mech. Mach. Theory, 115, 2017, 97–113.
[2] Ding, H.F., Huang, P., Zi, B., Kecskemethy, A., Automatic synthesis of kinematic structures of mechanisms and robots especially for those with complex structures, Appl. Math. Model., 36, 2012, 6122-6131.
[3] Xie, T.L., Hu, J.B., Peng, Z.X., Liu, C.W., Synthesis of seven-speed planetary gear trains for heavy-duty commercial vehicle, Mech. Mach. Theory, 90, 2015, 230–239.
[4] Du, M., Yang, L., A basis for the computer-aided design of the topological structure of planetary gear trains, Mech. Mach. Theory, 145, 2020, 103690.
[5] Tsai, L.W., Mechanism design: Enumeration of Kinematic Structures According to Function, CRC Press, 2000.
[6] Pennestri, 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, 92–106.
[7] 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, 1987, 329–336.
[8] Kim, J.U., Kwak, B.M., Application of edge permutation group to structural synthesis of epicyclic gear trains, Mech. Mach. Theory, 25, 1990, 563–574.
[9] Shin, J.K., Krishnamurthy, S., Standard code technique in the enumeration of epicyclic gear trains, Mech. Mach. Theory, 28, 1993, 347–355.
[10] Hsu, C.H., Lam, K.T., Yin, Y.L., Automatic synthesis of displacement graphs for planetary gear trains, Math. Comput. Model., 19, 1994, 67–81.
[11] Hsu, C.H., Hsu, J.J., An efficient methodology for the structural synthesis of geared kinematic chains, Mech. Mach. Theory, 32, 1997, 957–973.
[12] Del Castillo, J.M., Enumeration of 1-DOF planetary gear train graphs based on functional constraints, ASME J. Mech. Des., 124, 2002, 723–732.
[13] Rao, Y.V.D., Rao, A.C., Generation of epicyclic gear trains of one degree of freedom, ASME J. Mech. Des., 130, 2008, 052604.
[14] Kamesh, V.V., Rao, K.M., Rao, A.B.S., Detection of degenerate structure in single degree-of-freedom planetary gear trains, ASME J. Mech. Des., 139, 2017, 083302.
[15] Yang, W.J., Ding, H.F., Zi, B., Zhang, D., New graph representation for planetary gear trains, ASME J. Mech. Des., 140, 2018, 012303.
[16] 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, 2019, 043301.
[17] 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.
[18] Shanmukhasundaram, V.R., Rao, Y.V.D., Regalla, S.P., Algorithms for detection of degenerate structure in epicyclic gear trains using graph theory, J. Braz. Soc. Mech. Sci. Eng., 41, 2019, 496.
[19] Buchsbaum, F., Freudenstein F., Synthesis of kinematic structure of geared kinematic chains and other mechanisms, J. Mech., 5, 1970, 357–392.
[20] Freudenstein, F., An application of Boolean algebra to the motion of epicyclic drives, J. Eng. Ind., 93, 1971, 176–182.
[21] Ravisankar, R., Mruthyunjaya, T., Computerized synthesis of the structure of geared kinematic chains, Mech. Mach. Theory, 20, 1985, 367–387.
[22] Cui, R., Ye, Z., Sun, L., Zheng, G., Wu, C., Synthesis method for planetary gear trains without using rotation graphs, Proc. IMechE Part C: J. Mech. Eng. Sci., 236, 2022, 972-983.
[23] Kamesh, V.V., Mallikarjuna Rao, K., Rao, S., et al., Topological synthesis of epicyclic gear trains using vertex incidence polynomial, J. Mech. Des., 139, 2017, 139.
[24] Tsai, L.-W., Lin, C.-C., The creation of nonfractionated, two-degree-of-freedom epicyclic gear trains, J. Mech. Transm. Autom., 111, 1989, 524–529.
[25] 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.
[26] Nafeh, H.A., Esmail, E.L., Abdali, S.H., Automatic Structural Synthesis of Planetary Geared Mechanisms using Graph Theory, J. Appl. Comput. Mech., 9, 2023, 384-403.
[27] Shanmukhasundaram, V.R., Rao, Y.V.D., Regalla, S.P., Review of structural synthesis algorithms for epicyclic gear trains, In: D. Sen, S. Mohan, G. Ananthasuresh (eds), Mechanism and Machine Science. Lecture Notes in Mechanical Engineering, Springer, Singapore, 2021.
[28] Xue, H.L., Liu, G., Yang, X.H., A review of graph theory application research in gears, Proc. IMechE Part C: J. Mech. Eng. Sci., 230, 2016, 1697–1714.
[29] 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.
[30] Yang, W., Ding, H., The Perimeter Loop-Based Method for the Automatic Isomorphism Detection in Planetary Gear Trains, J. Mech. Design, 140, 2018, 1–10.
[31] Yang, W., Ding, H., Automatic detection of degenerate planetary gear trains with different degree of freedoms, Appl. Math. Model., 64, 2018, 320–332.
[32] Rai, R.K., Punjabi, S., A new algorithm of links labelling for the isomorphism detection of various kinematic chains using binary code, Mech. Mach. Theory, 131, 2019, 1–32.
[33] Yi, H.J., Wang, J.P., Hu, Y.L., Yang, P., Mechanism isomorphism identification based on artificial fish swarm algorithm, Proc. IMechE Part C: J. Mech. Eng. Sci., 235, 2021, 5421–5433
[34] Zou, Y.H., He, P., An algorithm for identifying the isomorphism of planar multiple joint and gear train kinematic chains, Math. Probl. Eng., 2016, 5310582
[35] Huang, P., Liu, T.T., Ding, H.F., Zhao, Y.Q., Isomorphism identification algorithm and database generation for planar 2–6 DOFs fractionated kinematic chains combined by two or three non-fractionated kinematic chains, Mech. Mach. Theory, 166, 2021, 104520
[36] He, L.Y., Liu, F.X., Sun, L., Wu, C.Y., Isomorphic identification for kinematic chains using variable high-order adjacency link values, J. Mech. Sci. Technol., 33, 2019, 4899–4907.
[37] Rai, R.K., Punjabi, S., Kinematic chains isomorphism identification using link connectivity number and entropy neglecting tolerance and clearance, Mech. Mach. Theory, 123, 2018, 40–65.
[38] Kamesh, V.V., Rao, K.M., Rao, A.B.S., An innovative approach to detect isomorphism in planar and geared kinematic chains using graph theory, J. Mech. Des., 139, 2017, 122301
[39] Hsu, C.-H., Displacement isomorphism of planetary gear trains, Mech. Mach. Theory, 29, 1994, 513–523.
[40] Shin, J.K., Krishnamurthy, S., Standard code technique in the enumeration of epicyclic gear trains, Mech. Mach. Theory, 28, 1993, 347–355.
[41] Mruthyunjaya, T.S., Raghavan, M.R., Structural analysis of kinematic chains and mechanisms based on matrix representation, ASME J. Mech. Des., 101, 1979, 488–494.
[42] Ravisankar, R., Mruthyunjaya, T.S., Computerized synthesis of the structure of geared kinematic chains, Mech. Mach. Theory, 20, 1985, 367–387.
[43] Hsu, C.H., Lam, K.T., A new graph representation for the automatic kinematic analysis of planetary spur-gear trains, ASME J. Mech. Des., 114, 1992, 196–200.
[44] Hsu, C.H., A graph representation for the structural synthesis of geared kinematic chains, J. Franklin Inst., 330, 1993, 913–927.
[45] Salgado, D.R., Castillo, J.M.D., A method for detecting degenerate structures in planetary gear trains, Mech. Mach. Theory, 40, 2005, 948–962.
[46] Vijayabarathi, A., Anjaneyulu, G.S.G.N., Wiener Index of a graph and chemical applications, Int. J. Chem. Tech. Res., 5, 2013, 1847–1853.
[47] Hsu, C.H., Synthesis of kinematic structure of planetary gear trains by admissible graph method, J. Frankl. Inst., 330, 1993, 913–927.
)