We analyze the frequency equation of natural vibrations of a heavy ideal two-layer incompressible liquid placed in a rigid circular cylindrical vessel with elastic bases in the form of fixed thin circular plates. We consider the cases of axially symmetric (longitudinal) and asymmetric (transverse) vibrations of the liquid and plates and different limiting cases (degeneration of the plates into membranes, absolutely rigid plates, and the absence of the top plate). It is shown that the frequency spectrum of coupled asymmetric vibrations of the elastic bases and two-layer ideal liquid consists of three sets of frequencies corresponding to the vibrations of the top and bottom elastic bases and to the vibrations of the inner interface of liquids. The frequency spectrum of coupled axially symmetric vibrations consists of the same three sets as for the asymmetric vibrations and an additional frequency of vibration of the liquid column as a single whole. As an example, we perform the analytic and numerical investigations of the frequency spectrum of a homogeneous liquid with free surface and elastic bottom in the form of a membrane.
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References
L. V. Dokuchaev, Nonlinear Dynamics of Aircrafts with Deformed Elements [in Russian], Mashinostroenie, Moscow (1987).
V. A. Trotsenko and R. I. Bogun, “Transverse vibrations of a fluid in a long cylindrical container with membrane or elastic plate on the free surface,” Nelin. Kolyv., 12, No. 3, 379–404 (2009); English translation: Nonlin. Oscillat., 12, No. 3, 392–416 (2009).
R. I. Bohun and V. A. Trotsenko, “Free oscillations of a fluid in a cylindrical container with arbitrary axisymmetric bottom and elastic elements on the free surface of the fluid,” Nelin. Kolyvannya, 13, No. 4, 461–482 (2010); English translation: Nonlin. Oscillat., 13, No. 4, 493–514 (2011).
Yu. N. Kononov and Yu. A. Dzhukha, “Influence of overloading on the axially symmetric vibrations of a membrane placed on the free liquid surface in a cylindrical vessel,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, 14, No. 2 (2017), pp. 32–41.
A. A. Pozhalostin, “Free vibrations of liquid in a rigid circular cylindrical vessel with elastic plane bottom,” Izv. Vyssh. Uchebn. Zaved., Ser. Aviats. Tekh., No. 4, 25–32 (1963).
M. P. Petrenko, “Natural vibrations of a liquid with free surface and the elastic bottom of a cylindrical cavity,” Prikl. Mekh., 5, No. 6, 44–50 (1969).
M. P. Petrenko, “Forced vibrations of a liquid and the elastic bottom of a cylindrical tank,” Prikl. Mekh., 6, No. 6, 127–131 (1970).
A. Yu. Karnaukh and N. K. Didok, “Natural vibrations of the elastic bottom of a cylindrical vessel and liquid with free surface,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, 20 (2010), pp. 102–108.
Yu. N. Kononov, V. P. Shevchenko, and Yu. A. Dzhukha, “Axially symmetric vibrations of a two-layer ideal liquid with free surface in a rigid cylindrical vessel with elastic bottom,” Visn. Don. Nats. Univ., Ser. A, No. 1-2, 116–125 (2015).
Ngo Zui Kan, “Motion of immiscible liquids in a vessel with plane elastic bottom,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 5, 151–163 (1979).
V. D. Lakiza, “Investigation of dynamic processes in a rigid cylindrical vessel with elastic bottom partially filled with liquid,” Prikl. Mekh., 42, No. 11, 114–120 (2006).
N. K. Didok and Yu. N. Kononov, “Dynamics and stability of vibrations of a cylindrical vessel with ideal liquid and elastic bases,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, 27 (2013), pp. 122–131.
N. D. Kopachevskii, S. N. Krein, and Ngo Zui Kan, Operator Methods in Linear Hydrodynamics: Evolutionary and Spectral Problems [in Russian], Nauka, Moscow (1989).
Yu. N. Kononov, V. F. Rusakov, and Yu. A. Dzhukha, “Axisymmetric vibrations of elastic bases and ideal liquid in a rigid cylindrical vessel,” Visn. Zaporiz. Nats. Univ., Ser. Fiz.-Mat. Nauk, No. 2, 105–114 (2015).
Yu. N. Kononov and Yu. A. Dzhukha, “Axisymmetric vibrations of the elastic bases and ideal liquid in a rigid ring cylindrical vessel,” Visn. Zaporiz. Nats. Univ., Ser. Fiz.-Mat. Nauk, No. 1, 103–115 (2016).
Yu. N. Kononov, V. P. Shevchenko, and Yu. O. Dzhukha, “Axisymmetric vibrations of elastic ring bases and a two-layer ideal liquid in a rigid ring cylindrical vessel,” Mat. Met. Fiz.-Mekh. Polya, 60, No. 1, 85–95 (2017).
D. A. Goncharov, “Dynamics of two-layer liquid separated by an elastic membrane with regard for the forces of surface tension ,” Nauka Obrazov., No. 11 (2013). http://technomag.bmstu.ru/doc/619258.html.
A. A. Pozhalostin and D. A. Goncharov, “Free axisymmetric vibrations of a two-layer liquid with rigid separator between layers in the presence of the forces of surface tension ,” Inzh. Zh. Nauka Innovats., No. 12 (2013); http://engjournal.ru/ catalog/eng/teormach/1147.html.
D. A. Goncharov and A. A. Pozhalostin, “Symmetric vibrations of a liquid in a vessel with a separator and an elastic bottom,” J. Phys.: Conf. Series, Vol. 991 (2018). http://iopscience.iop.org/article/10.1088/1742-6596/991/1/012027/pdf.
A. A. Pozhalostin and D. A. Goncharov, “Free axisymmetric oscillations of a two-layer liquid with an elastic separator between layers,” Russ. Aeronaut., 58, No. 1, 37–41 (2015).
P. Tong, “Liquid motion in a circular cylindrical container with a flexible bottom,” AIAA J., 5, No. 10, 1842–1848 (1967).
H. F. Bauer, S. Chang, and J. T. S.Wang, “Nonlinear liquid motion in a longitudinally excited container with elastic bottom,” AIAA J., 9, No. 12, 2333–2339 (1971).
M. Chiba, “Nonlinear hydroelastic vibration of a cylindrical tank with an elastic bottom, containing liquid. Part II: Linear axisymmetric vibration analysis,” J. Fluids Structures, 7, No. 1, 57–73 (1993).
M. Chiba, “Axisymmetric free hydroelastic vibration of a flexural bottom plate in a cylindrical tank supported on an elastic foundation,” J. Sound Vibrat., 169, No. 3, 387–394 (1994).
H. F. Bauer, “Coupled frequencies of a liquid in a circular cylindrical container with elastic liquid surface cover,” J. Sound Vibrat., 180, No. 5, 689–704 (1995).
S. Tariverdilo, M. Shahmardani, J. Mirzapour, and R. Shabani, “Asymmetric free vibration of circular plate in contact with incompressible fluid,” Appl. Math. Model., 37, No. 1-2, 228–239 (2013).
K.-H. Jeong, “Free vibration of two identical circular plates coupled with bounded fluid,” J. Sound Vibrat., 260, No. 4, 653–670 (2003).
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Translated from Neliniini Kolyvannya, Vol. 21, No. 4, pp. 496–513, October–December, 2018.
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Kononov, Y.M., Dzhukha, Y.O. Vibrations of Two-Layer Ideal Liquid in a Rigid Cylindrical Vessel with Elastic Bases. J Math Sci 246, 365–383 (2020). https://doi.org/10.1007/s10958-020-04745-w
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DOI: https://doi.org/10.1007/s10958-020-04745-w