On~graphs and~structural formulas of~the~mechanisms theory

M. D. Kovalev

Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 3, pp. 42-55.

Résumé

Structural formulas in the theory of mechanisms are called formulas expressing the number of degrees of freedom of the device through the numbers of its links and kinematic pairs. It is known that they are not always fair. Mathematical graph theory helps to understand this phenomenon. The validity of structural formulas in the case of generic frameworks is completely determined by their structure, described by graphs. The paper considers two models of planar frameworks with rotational pairs. The first model is a construction made up of straight rods (levers) bearing hinges at the ends. Such devices are naturally associated with a graph $G$ with vertices corresponding to hinges and edges corresponding to levers. In the theory of mechanisms, it is customary to consider another graph $\mathcal G$, whose vertices correspond to links, and the edges correspond to kinematic pairs. It turns out that the use of the graph $G$ to describe the structure both in the first model and in the second one, which contains all planar constructions with rotational pairs, is preferable to the graph $\mathcal G$. In particular, it allows to give a criterion for the applicability of structural formulas for generic constructions of a given structure.

Détail

Keywords: structure of mechanisms, structural graphs, structural formulas of mechanisms theory, hinged mechanisms, mathematical models of mechanisms.
Mots-clés : dimension of configuration space

Bibliographie

[1] Dvornikov L.T., “Experience of structural synthesis of mechanisms”, Teoriya Mekhanizmov i Mashin, 2:4 (2004), 3–17 (in Russian)

[2] Dvornikov L.T., “To prove the validity of the experience of structural synthesis of mechanisms”, Teoriya Mekhanizmov i Mashin, 4:1 (2006), 44–48 (in Russian) | MR

[3] Peisakh E.E., “To the discussion on the problem of structural synthesis of plane hinge mechanisms”, Teoriya Mekhanizmov i Mashin, 4:1 (2006), 49–54 (in Russian) | MR

[4] Dvornikov L.T., “About fundamental inaccuracies in the research of Prof. Pozhbelko V.I. on the structure of mechanisms”, Teoriya Mekhanizmov i Mashin, 14:3(31) (2016), 145–166 (in Russian)

[5] Pozhbelko V.I., Kuts E.N., “Integer structural synthesis of multi-contour lever mechanisms with complex hinges for various fields of mechanical engineering”, Izv. Vyssh. Uchebn. Zaved. Mashinostroenie, 2021, no. 6, 23–36 (in Russian) | DOI

[6] Peisakh E.E., “About Assur groups, Baranov trusses, Grubler chains, plain hinge mechanisms and their structural synthesis”, Elektron. Zhurn. «Nauka i Obrazovanie» , MGTU, 2007, no. 4 (in Russian)

[7] Dvornikov L.T., SibGIU scientific schools. Theory of the structure of mechanical systems and the practice of its use in the synthesis of complex machines, including mining and metallurgical, Izd. Tsentr SibGIU, Novokuznetsk, 2015 (in Russian)

[8] Peisakh E.E., Nesterov V.A., Design systems for plain lever mechanisms, Mashinostroenie, M., 1988 (in Russian)

[9] Davies T.H., “An extension of Manolescu's classification of planar kinematic chains and mechanisms of mobility $M\geq 1$, using graph theory”, J. Mechanisms, 3 (1968), 87–100 | DOI

[10] Didenko E.V., Development and analysis of planar multi-contour mechanisms based on graph theory, Diss. ... kand. tekhn. nauk, M., 2019 (in Russian)

[11] Kovalev M.D., “Geometric theory of hinged devices”, Russian Acad. Sci. Izv. Math., 44:1 (1995), 43–68 | MR | Zbl

[12] Kovalev M.D., “Questions of geometry of hinge devices and schemes”, Vestn. MGTU. Ser. Mashinostroenie, 2001, no. 4, 33–51 (in Russian)

[13] Kovalev M.D., Geometric questions of kinematics and statics, Lenand, M., 2019 (in Russian)

[14] Kovalev M.D., What is a hinge mechanism? And what did Kempe prove?, Itogi Nauki i Tekh. Ser. Sovremen. Mat. i Ee Prilozh. Temat. Obzory, 179, VINITI RAN, M., 2020, 16–28 (in Russian) | DOI

[15] Tay T.S., “Rigidity of multi-graphs I, linking rigid bodies in n-space”, J. Combin. Theory. Ser. B, 36 (1984), 95–112 | DOI | MR | Zbl

[16] Harary F., Graph Theory, Addison-Wesley Publ, MA, 1969 | MR | Zbl

[17] Benedetti R., Risler J., Real Algebraic and Semi-Algebraic Sets, Hermann, Paris, 1990 | MR | Zbl

[18] Kovalev M.D., “Configuration spaces of hinged mechanisms, and their projections”, Mat. Sbornik, 213:4 (2022), 74–99 (in Russian) | DOI | MR | Zbl

[19] Graver J., Servatius B., Servatius H., Combinatorial Rigidity, Amer. Math. Soc., Providence, 1993 | MR | Zbl

[20] Chebyshev P.L., “On parallelograms”, Trudy Vtorogo S"ezda Rus. Estestvoispytatelei. Otdel Tekh. Prakt. Mekh., 1870, 9–30 (in Russian)

[21] Levitskii N.I., Theory of mechanisms and machines, Nauka, M., 1990 (in Russian)

[22] Pollaczek-Geiringer H., “Über die Gliederung ebener Fuchwerke”, Z. Angew. Math. Mech., 7:1 (1927), 58–72 | DOI

[23] Laman G., “On graphs and rigidity of plane skeletal structures”, J. Engrg. Math., 4 (1970), 331–340 | DOI | MR | Zbl

[24] Tay T.S., “Rigidity of multi-graphs. I: Linking rigid bodies in $n$-space”, J. Combin. Theory. Ser. B, 36:1 (1984), 95–112 | DOI | MR | Zbl

[25] Handbook of Geometric Constraint Systems Principles, Taylor and Francis Group, 2019

[26] Tay T.S., “Linking ($n-2$)-dimensional panels in $n$-space II: $(n-2;2)$-frameworks and body and hinge structures”, Graphs Combin., 5:1 (1989), 245–273 | DOI | MR | Zbl