Structural Synthesis of a Family of Planar 8-Link Kinematic Chains for Linkages with Multiple Joints and the Most Complex Ternary Link

Structural synthesis of closed kinematic chains to create various mechanisms is the first and most difficult stage of creative design of complex machines due to the large variance of possible structural solutions. In this paper, the authors examine the problem of structural synthesis of a family of eight-link kinematic chains with multiple joints of various types and the most complex three-joint link in order to create multi-loop multiple-joint mechanisms with one degree of freedom. To solve this problem, a synthesis technique is proposed based on the search for all integer solutions of a generalized structural mathematical model of plane linkage mechanisms and the identification of all structurally nonisomorphic kinematic chains using a two-column P-matrix. As the result of the structural synthesis, a family of eight-link multiple joint kinematic chains is obtained, which contains seven new kinematic structures. Examples of creating 1-DOF mechanisms with multiple joints based on the obtained new structures are presented. They confirm the effectiveness of using the structural synthesis procedure and analysis of complex mechanisms with multiple joints in various areas of modern engineering (precise guiding mechanisms, automatic lines, technological machines, robots, manipulators, etc.).

References

[1] Assur L.V. Issledovaniye ploskikh sterzhnevykh mekhanizmov s tochki zreniya ikh struktury i klassifikatsii [The study of flat bar mechanisms in terms of their structure and classification]. Moscow, AN SSSR publ., 1952. 529 p.

[2] Kozhevnikov S.N. Osnovaniya strukturnogo sinteza mekhanizmov [The foundations of structural synthesis of mechanisms]. Kiyev, Naukova dumka publ., 1979. 232 p.

[3] Artobolevskiy I.I. Mekhanizmy v sovremennoy tekhnike. Spravochnik. Tom 1. Sharnirno-rychazhnyye mekhanizmy [Mechanisms in modern technology. Directory. Volume 1. Articulated Linkage]. Moscow, Nauka publ., 1979. 495 p.

[4] Kraynev A.F. Mekhanika (iskusstvo postroyeniya) mashin. Fundamental’nyy slovar’ [Mechanics (the art of building) machines. Fundamental Dictionary]. Moscow, Mashinostroyeniye publ., 2000. 904 p.

[5] Timofeyev G.A. Teoriya mekhanizmov i mekhanika mashin [Theory of mechanisms and mecha¬nics of machines]. Moscow, Yurayt publ., 2019. 368 p.

[6] Smelyagin A.I. Struktura mekhanizmov i mashin [The structure of mechanisms and machines]. Novosibirsk, NSTU publ., 2001. 286 p.

[7] Uicker J.J., Pennock G.R., Shigley J.E. Theory of Mechanisms. New York, Oxford University Press, 2016. 976 p.

[8] Ceccarelli M. Fundamentals of Mechanics of Robotic Manipulations. Springer Science & Business Media, 2004. 312 p.

[9] Babichev D., Evgrafov A. Structural-kinematic synthesis method for (planar) link. Advances in Mechanism and Machine Science. Mechanism and Machine Science. Switzerland, Springer, 2019, pp. 2937–2953.

[10] Markovets A.V., Polotebnov V.O. Synthesis of mechanisms of material handling mechanism with a toothed bar straight line section of the movement. The News of higher educational institutions. Technology of Light Industry, 2018, no. 1, vol. 38, pp. 117–121 (in Russ.).

[11] Sukhikh R.D. Structural synthesis of mechanisms for a given number of links. Pt. 3. Sbornik nauchnykh trudov. Raschet, proyektirovaniye i konstruirovaniye zheleznodorozhnykh mashin [Collection of scientific papers. Calculation, design and construction of railway cars]. Sankt-Petersburg, PSUPS publ., 2003, pp. 3–31 (in Russ.).

[12] Dvornikov L.T. Structural analysis of mechanisms. Teoriya mekhanizmov i mashin, 2004, no. 2, vol. 2, pp. 3–17 (in Russ.).

[13] Glazunov V.A., Koliskor A.Sh., Kraynev A.F. Prostranstvennyye mekhanizmy parallel’noy struktury [Spatial mechanisms of parallel structure]. Moscow, Nauka publ., 1991. 95 p.

[14] Peysakh E.E., Nesterov V.A. Sistema proyektirovaniya ploskikh rychazhnykh mekhanizmov [System design flat lever mechanisms]. Moscow, Mashinostroyeniye publ., 1988. 232 p.

[15] Romantsev A.A. On the creation of block diagrams of planar hinged groups of links. Teoriya mekhanizmov i mashin, 2014, no. 1(23), vol. 12, pp. 81–90 (in Russ.).

[16] Crossley F.R.E. The permutations of kinematic chains of eight members or less from graph theoretic viewpoint. Developments in Mechanisms. Oxford, Pergamon Press, 1965, vol. 2, pp. 467–486.

[17] Tuttle E.R., Peterson S.W., Titus J.E. Enumeration of basis kinematic chains using the theory of finite groups. ASME Journal of Mechanisms, Transmissions and Automation in Design, 1989, vol. 111(4), pp. 498–503, doi: 10.1115/1.3259028

[18] Hwang W.-M., Hwang Y.-W. Computer-aided structural synthesis of planar kinematic chains with simple joints. Mechanism and Machine Theory, 1992, no. 2, vol. 27, pp. 189–199, doi: 10.1016/0094-114X(92)90008-6

[19] Butcher E.A., Hartman C. Efficient enumeration of planar simple-jointed kinematic chains. Mechanism and Machine Theory, 2005, no. 9, vol. 40, pp. 1030–1050, doi: 10.1016/j.mechmachtheory.2004.12.015

[20] Umnov N.V., Sil’vestrov E.E. The use of homotopy methods in the synthesis of mechanisms. Sbornik dokladov mezhdunarodnoy konferentsii po teorii mekhanizmov i mashin [Collection of reports of the international conference on the theory of mechanisms and machines]. Krasnodar, Kubanskiy STU, 2006, pp. 47–48.

[21] Yan H.-S., Chiu Y.-T. On the number synthesis of kinematic chains. Mechanism and Machine Theory, 2015, no. 9, vol. 89, pp. 128–144, doi: 10.1016/j.mechmachtheory.2014.08.012

[22] Ding H., Hou F., Kecskeméthy A., Huang Z. Synthesis of a complete set of contracted graphs for planar non-fractionated simple-jointed kinematic chains with all possible DOFs. Mechanism and Machine Theory, 2011, no. 11, vol. 46, pp. 1588–1600, doi: 10.1016/j.mechmachtheory.2011.07.012

[23] Chu J., Zou Y. An algorithm for structural synthesis of planar simple and multiple joint kinematic chains. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 228, 2014, pp. 2178–2192, doi: 10.1177/0954406213516306

[24] Ding H., Yang W., Huang P., Kecskeméthy A. Automatic structural synthesis of planar multiple joint kinematic chains. ASME Journal of Mechanical Design, 2013, vol. 135, pp. 091007, doi: 10.1115/1.4024733

[25] Chu J., Cao W. Identification of isomorphism among kinematic chains and inversions using link’s adjacent-chain-table. Mechanism and Machine Theory, 1994, no. 1, vol. 29, pp. 53–58, doi: 10.1016/0094-114X(94)90019-1

[26] Ding H., Huang Z. Isomorphism identification of graphs: Especially for the graphs of kinematic chains. Mechanism and Machine Theory, 2009, no. 1, vol. 44, pp. 122–139, doi: 10.1016/j.mechmachtheory.2008.02.008

[27] Liu J., Yu D. Representations & isomorphism identification of planar kinematic chains with multiple joints based on the converted adjacent matrix. Journal of Mechanical Engineering, 2012, vol. 48, pp. 15–21, doi: 10.3901/JME.2012.05.015

[28] Pozhbelko V.I. A complete theory of structure, structural synthesis and analysis statically determinate mechanical systems on base a new degrees of freedom equation (DOF). Teoriya mekha¬nizmov i mashin, 2013, no. 2(22), vol. 11, pp. 15–37 (in Russ.).

[29] Pozhbelko V.I. Napravlennyy sintez optimal’nykh struktur ploskikh mekhanicheskikh sistem s sovmeshchennymi sharnirami (mekhanizmy, fermy, gruppy Assura, roboty). CH. 1. Izvestiya vysshikh uchebnykh zavedeniy. Mashinostroyeniye, 2012, no. 10, pp. 31–45.

[30] Pozhbelko V. A unified structure theory of multibody open-, closed-, and mixed-loop mechanical systems with simple and multiple joint kinematic chains. Mechanism and Machine Theory, 2016, no. 6, vol. 100, pp. 1–16, doi: 10.1016/j.mechmachtheory.2016.01.001

[31] Pozhbelko V., Ermoshina E. Number structural synthesis and enumeration process of all possible sets of multiple joints for 1-DOF up to 5- loop 12-link mechanisms on base of new mobility equation. Mechanism and Machine Theory, 2015, no. 8, vol. 90, pp. 108–127, doi: 10.1016/j.mechmachtheory.2015.03.006

[32] Pozhbelko V. Advanced Technique of Type Synthesis and Construction of Veritable Complete Atlases of Multiloop F-DOF Generalized Kinematic Chains. Mechanisms and Machine Science, 2019, vol. 59, pp. 207–214, doi: 10.1007/978-3-319-98020-1_24

[33] Pozhbelko V. Type Synthesis Method of Planar and Spherical Mechanisms Using the Universal Structural Table with All Possible Link Assortments. Mechanisms and Machine Science. Switzerland, Springer, 2019, vol. 73, pp. 1517–1526, doi: 10.1007/978-3-030-20131-9_150

[34] Pozhbelko V.I. Method for solving the problem of identifying isomorphism or metaformism in the structural synthesis of complex multi-circuit mechanical systems. Teoriya mekhanizmov i mashin, 2015, no. 1(25), vol. 13. pp. 23–40 (in Russ.), doi: 10.5862/TMM.25.3