Geodesic
Strong constraints in the dynamics of systems with geometric singularities
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 3, pp. 53-67
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The dynamics of holonomic mechanical systems with geometric singularities of the configuration space, such as branch points, is considered. Classical methods for deriving equations of motion are not applicable in neighborhoods of singular points because there are no generalized coordinates. A new method for analyzing the dynamics of systems with singularities is proposed. Some holonomic (rigid) constraints are replaced by elastic ones (springs). As a result, the singularity disappears, but the number of degrees of freedom of the system increases. With an unlimited increase in spring stiffness, the trajectory of a system with elastic constraints should deviate less and less from the configuration space for the original system with holonomic constraints. A hypothesis has been put forward about the motion of a mechanical system whose configuration space could be represented as a union of two smooth manifolds. The limit transition for the spring stiffness is considered using a specific example. For this purpose, a singular pendulum with a spring is constructed. This two-degree mechanical system can be explicitly parameterized, which simplifies its analytical and numerical modeling. In numerical experiments, the motion of the system is consistent with the hypothesis.
Keywords: constraint realization, constraint reactions, manifolds with singularities, singular point, holonomic constraints
Mots-clés : Lagrange multipliers. 
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S. Bur'yan. Strong constraints in the dynamics of systems with geometric singularities. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 3, pp. 53-67. http://geodesic.mathdoc.fr/item/TIMM_2024_30_3_a4/

[1] Penleve P., Lektsii o trenii, per. s frants., Gostekhizdat, Moskva, 1954, 316 pp.

[2] Mukharlyamov R.G., Deressa C.T., “Dynamic equations of controlled Mechanical system with redundant holonomic constraints”, Vestn. Kazan. Tekhnol. Univ., 17:11 (2014), 236–242

[3] Wojtyra M., Fra̧czek J., “Solvability of reactions in rigid multibody systems with redundant nonholonomic constraints”, Multibody Syst Dyn., 30 (2013), 153–171 | DOI | MR

[4] Flores P., Pereira R., Machado M., Seabra E., “Investigation on the Baumgarte stabilization method for dynamic analysis of constrained multibody systems”, Proc. 2nd Eur. Conf. on Mechanism Science (EUCOMES 08) (Cassino, Italy, Sept. 17-20, 2008), Springer-Verlag, Dordrecht, 2009, 305–312 | DOI | Zbl

[5] Zhuravlev V.F., “Ponyatie svyazi v analiticheskoi mekhanike”, Nelineinaya dinamika, 8:4 (2012), 853–860

[6] Arnold V.I., Kozlov V.V., Neishtadt A.I., Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki, VINITI, Moskva, 1985, 304 pp.

[7] Rubin H., Ungar P., “Motion under a strong constraining force”, Communications on pure and applied mathematics, 10 (1957), 65–87 | DOI | MR | Zbl

[8] Kozlov V.V., Neishtadt A.I., “O realizatsii golonomnykh svyazei”, Prikl. matematika i mekhanika, 54:5 (1990), 858–861 | MR | Zbl

[9] Takens F., “Motion under influence of a strong constraining force”, Global theory of Dynamics Systems, Springer-Verlag, Berlin, 1980, 425–445 | DOI | MR

[10] Karapetyan A.V., “O realizatsii negolonomnykh svyazei i ustoichivost keltskikh kamnei”, Prikl. matematika i mekhanika, 45:1 (1981), 42–51 | MR | Zbl

[11] Zegzhda S.A., Soltakhanov Sh.S., Yushkov M.P., Uravneniya dvizheniya negolonomnykh sistem i variatsionnye printsipy mekhaniki. Novyi klass zadach upravleniya, Fizmatlit, Moskva, 2005, 272 pp.

[12] Polyakhov N.N., Zegzhda S.A., Yushkov M.P., Tovstik P.E., Soltakhanov Sh.Kh., Filippov S.B., Petrova V.I.,, Teoreticheskaya i prikladnaya mekhanika, V 2 t., v. I, Obschie voprosy teoreticheskoi mekhaniki, Izd-vo S.-Peterb. un-ta, SPb., 2022, 560 pp.

[13] Zhuravlev V.F., Osnovy teoreticheskoi mekhaniki, Fizmatlit, Moskva, 2001, 320 pp. | MR

[14] Buryan S.N., “Osobennosti dvizheniya mayatnika s singulyarnym konfiguratsionnym prostranstvom”, Vestn. SPbGU. Matematika. Mekhanika. Astronomiya, 4(62):4 (2017), 541–551 | DOI | MR

[15] Burian S.N., Kalnitsky V.S., “On the motion of one-dimensional double pendulum”, AIP Conf. Proc., 1959, 2018, 030004 | DOI

[16] Buryan S.N., “Sily reaktsii singulyarnogo mayatnika”, Vestn. SPbGU. Matematika. Mekhanika. Astronomiya, 9 (67):2 (2022), 278–293 | DOI | MR