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Stability of the Equilibrium State of a Rigid Body with Multilayer Ideal Liquid Separated by Elastic Plates

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Ukrainian Mathematical Journal Aims and scope

L. N. Sretenskii’s problem and the problem of oscillation of physical pendulum are generalized to the case of multilayer ideal liquid separated by elastic plates. Under the assumption of positive definiteness of the potential energy (Sretenskii’s problem) and for the modified potential energy (physical pendulum), we obtain the conditions of stability for the equilibrium state in these problems. A more detailed analysis is performed in the presence of a cylindrical cavity with arbitrary cross section. It is shown that, for the stability of the equilibrium state in Sretenskii’s problem, it is necessary and sufficient that a stable equilibrium state of the elastic plates and liquid in the immobile rigid body exist and, in addition, it is sufficient that the heavier liquid be located below the lighter liquid. For the stability of the equilibrium state in the problem of physical pendulum, it is also necessary that the stable equilibrium state of the elastic plates and liquid exist in the immobile rigid body. It is also shown that the unstable equilibrium position of the physical pendulum can be stabilized by the procedure of preliminary tension of the plates.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Sloviansk, Ukraine

    Yu. M. Kononov

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Correspondence to Yu. M. Kononov.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 10, pp. 1342–1354, October, 2021. Ukrainian DOI: https://doi.org/10.37863/umzh.v73i10.6840.

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Kononov, Y.M. Stability of the Equilibrium State of a Rigid Body with Multilayer Ideal Liquid Separated by Elastic Plates. Ukr Math J 73, 1551–1565 (2022). https://doi.org/10.1007/s11253-022-02013-5

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  • DOI: https://doi.org/10.1007/s11253-022-02013-5

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