On the stability of the equilibrium state of a rigid body with a multilayer ideal fluid separated by elastic plates

  • Yu. M. Kononov Inst. Appl. Math. and Mech. Nat. Acad. Sci. Ukraine, Sloviansk

Abstract

UDC 531.36:531.38:533.6.013.42

L. N. Stretensky’s problem and the problem of physical pendulum oscillations are generalized to the case of a multilayer ideal fluid separated by elastic plates. Assuming the positive definiteness of the potential energy (L. N. Stretensky’s problem) and changed potential energy (the physical pendulum), we obtain the conditions of stability for the equilibrium state in these problems. A more detailed study is performed in the case of a cylindrical cavity of arbitrary cross-section. We show that for the stability of the equilibrium state in L. N. Stretensky’s problem it is necessary and sufficient that there exists a stable equilibrium state of the elastic plates and liquid in the stationary rigid body and it is sufficient that a heavier liquid is located below a lighter one. For the stability of the equilibrium state in the problem of physical pendulum oscillations, it is also necessary that there is a stable equilibrium state of the elastic plates and liquid in the stationary rigid body.
It is also shown that we can use pre-tensioning of plates to stabilize an unstable equilibrium position of the physical pendulum.

References

L. N. Sretenskyi, Kolebanye zhydkosty v podvyzhnom sosude, Yzv. AN SSSR, № 10, 1483 – 1494 (1951).

N. N. Moyseev, Zadacha o malykh kolebanyiakh otkrytoho sosuda s zhydkostiu pod deistvyem upruhoi syly, Ukr. mat. zhurn.,4, № 2, 168 – 173 (1952).

N. N. Moyseev, Dvyzhenye tverdoho tela, ymeiushcheho polost, chastychno zapolnennuiu ydealnoi kapelnoi zhydkostiu, Dokl. AN SSSR, 85, № 4, 719 – 722 (1952).

N. N. Moyseev, Zadacha o dvyzhenyy tverdoho tela, soderzhashcheho zhydkye massy, ymeiushchye svobodnuiu poverkhnost, Mat. sb., 32 (74), vyp. 1, 61 – 96 (1953).

H. K. Pozharytskyi, Zadacha mynymuma v zadache ob ustoichyvosty ravnovesyia tverdoho tela s chastychnym zhydkym napolnenyem, Prykl. matematyka y mekhanyka, 26, vyp. 4,593 – 605 (1962).

N. N. Moyseev, V. V. Rumiantsev, Dynamyka tela s polostiamy, soderzhashchymy zhydkost, Nauka, Moskva (1965).

H. K. Pozharytskyi, V. V. Rumiantsev, Zadacha mynymuma v voprose ob ustoichyvosty dvyzhenyia tverdoho tela s polostiu, zapolnennoi zhydkostiu, Prykl. matematyka y mekhanyka, 27, vyp.1, 11 – 26 (1963).

N. K. Dydok, Poperechnye kolebanyia tsylyndrycheskoho sosuda s upruhym dnom, soderzhashcheho zhydkost so svobodnoi poverkhnostiu, Trudy Yn-ta prykl. matematyky y mekhanyky NAN Ukrainy, 22, 71 – 80 (2011).

Yu. N. Kononov, N. K. Dydok, Zadacha Sretenskoho dlia tsylyndrycheskoho sosuda s ydealnoi zhydkostiu y upruhymy osnovanyiamy, Mekhanyka tverdoho tela, vyp.40, 210 – 220 (2010).

N. K. Dydok, Yu. N. Kononov, Dynamyka y ustoichyvost kolebanyi tsylyndrycheskoho rezervuara s ydealnoi zhydkostiu y upruhymy osnovanyiamy, Trudy Yn-ta prykl. matematyky y mekhanyky NAN Ukrayny, 27, 122 – 131 (2013).

Yu. N. Kononov, O kolebanyy fyzycheskoho maiatnyka s mnohosloinoi ydealnoi zhydostiu, Zb. prats In-tu matematyky NAN Ukrainy, 12, № 5, 73 – 89 (2015). DOI: https://doi.org/10.7588/worllitetoda.89.3-4.0012

S. H. Mykhlyn, Varyatsyonnye metodyv matematycheskoi fyzyke, Nauka, Moskva (1957).

Yu. S. Pashkova, Malye kolebanyia systemy ydealnykh zhydkostei, razdelennykh membrannymy perehorodkamy, Symferopol (1992), Dep. v HKPB Ukrayny 02.10.92.

V. P. Shevchenko, Yu. N. Kononov, Ob ustoichyvosty upruhykh plastynok, razdeliaiushchykh mnohosloinuiu zhydkost, Aktualnye aspekty fyzyko-mekhanycheskykh yssledovanyi. Mekhanyka, Naukova dumka, Kyev (2007), s. 348 – 361.

L. V. Dokuchaev, Nelyneinaia dynamyka letatelnykh apparatov s deformyruemymy elementamy, Mashynostroenye, Moskva (1987).

Yu. M. Kononov, V. P. Shevchenko, Y. O. Dzhukha, Axially symmetric oscillations of elastic annular bases and a perfect two-layer liquid in a rigid annular cylindrical reservoir, J. Math. Sci., 240, № 1, 98 – 112 (2019). DOI: https://doi.org/10.1007/s10958-019-04338-2

Y. M. Kononov, Y. O. Dzhukha, Vibrations of two layer ideal liquid in a rigid cylindrical vessel with elastic bases, J. Math. Sci., 246, № 3, 365 – 383 (2020). DOI: https://doi.org/10.1007/s10958-020-04745-w

H. N. Mykyshev, B. Y. Rabynovych, Dynamyka tverdoho tela s polostiamy, chastychno zapolnennymy zhydkostiu, Mashynostroenye, Moskva (1968).

Published
11.10.2021
How to Cite
Kononov, Y. M. “On the Stability of the Equilibrium State of a Rigid Body With a Multilayer Ideal Fluid Separated by Elastic Plates”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 10, Oct. 2021, pp. 1342-54, doi:10.37863/umzh.v73i10.6840.
Section
Research articles