On oscillations of connected pendulums with cavities filled with homogeneous fluids

N. D. Kopachevsky ; V. I. Voytitsky

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 65 (2019) no. 3, pp. 434-512.

Résumé

We consider the problem and normal (eigen) oscillations of the system of three connected (coupled to each other) pendulums with cavities filled with one or several immiscible homogeneous fluids. We study the case of partially dissipative system when the cavity of the first pendulum is completely filled with two ideal fluids, the cavity of the second one is filled with three viscous fluids, and the cavity third one is filled with one ideal fluid. We use methods of functional analysis. We prove the theorem on correct solvability of the initial-boundary value problem on any interval of time. We study the case of eigen oscillations of conservative system where all fluids in cavities of pendulums are ideal and the friction in joints (points of suspension) is not taken into account. We consider in detail three auxiliary problems on small oscillations of single pendulums with three above variants of fluids in cavities.

Bibliographie

[1] M. S. Agranovich, Sobolev Spaces, Their Generalizations, and Elliptic Problems in Domains with Smooth and Lipschitz Boundaries, MTSNMO, M., 2013 (in Russian)

[2] T. Ya. Azizov, I. S. Iokhvidov, Fundamentals of Theory of Linear Operators in Spaces with Indefinite Metrics, Nauka, M., 1986 (in Russian) | MR

[3] T. Ya. Azizov, N. D. Kopachevskiy, Applications of Indefinite Metrics, DIAYPI, Simferopol', 2014 (in Russian)

[4] V. G. Babskiy, N. D. Kopachevskiy, A. D. Myshkis, L. A. Slobozhanin, A. D. Tyuptsov, Hydromechanics of Weightlessness, Nauka, M., 1976 (in Russian)

[5] E. I. Batyr, N. D. Kopachevskiy, “Small motions and normal oscillations in systems of connected gyrostats”, Contemp. Math. Fundam. Directions, 49, 2013, 5–88 (in Russian)

[6] M. Sh. Birman, M. Z. Solomyak, “Asymptotics of spectra of differential equations”, Totals Sci. Tech. Ser. Math. Anal., 14, 1977, 5–52 (in Russian)

[7] V. I. Voytitskiy, “To the problem of small motions of system of three connected pendulums with cavities filled with homogeneous incompressible fluids”, Dynam. Syst., 8:4 (2018), 337–356 (in Russian)

[8] V. I. Voytitskiy, N. D. Kopachevskiy, “On small motions of system of three connected pendulums with cavities filled with immiscible incompressible fluids”, Materials Int. Conf. “Contemporary Methods and Problems of Mathematical Hydrodynamics” (Voronezh, 3–8 May 2018), Voronezh, 2018, 84–91 (in Russian)

[9] V. I. Voytitskiy, N. D. Kopachevskiy, “On small motions of a physical pendulum with cavity filled with a system of homogeneous immiscible fluids”, Matherials Int. Conf. XXIX Crimean Autumnal Mathematical School on Spectral and Evolution Problems, Sec. 1–3, Poliprint, Simferopol', 2018, 58–62 (in Russian)

[10] V. I. Voytitskiy, N. D. Kopachevskiy, “On small motions of a physical pendulum with cavity filled with a system of three homogeneous immiscible viscous fluids”, Tavricheskiy Bull. Inform. Math., 2018, no. 3, 22–45 (in Russian)

[11] I. L. Vulis, M. Z. Solomyak, “Spectral asymptotics of the degenerating Steklov problem”, Bull. LGU, 19 (1973), 148–150 (in Russian) | MR | Zbl

[12] I. L. Vulis, M. Z. Solomyak, “Spectral asymptotics of second-order degenerating elliptic operators”, Bull. Acad. Sci. USSR. Ser. Math., 38:6 (1974), 1362–1392 (in Russian) | MR | Zbl

[13] J. Goldstein, Semigroups of Linear Operators and Applications, Vyshcha shkola, Kiev, 1989 (Russian translation)

[14] N. E. Zhukovskiy, “On motion of solid body with cavities filled with homogeneous drip fluid”, Selected Works, v. 1, Gostekhizdat, M.–L., 1948, 31–52 (in Russian) | MR

[15] Yu. N. Kononov, “On motion of connected solid bodies with cavities containing fluid”, Mech. Solid Body, 30 (2000), 207–216 (in Russian) | Zbl

[16] N. D. Kopachevskiy, “On oscillations of a body with a cavity partially filled with heavy ideal fluid: existence, uniqueness, and stability theorems for strong solutions”, Probl. Dynam. Stability Multidim. Syst., 2:1 (2005), 158–194 (in Russian) | MR

[17] N. D. Kopachevskiy, “Abstract Green formulas for triples of Hilbert spaces and sesquilinear forms”, Contemp. Math. Fundam. Directions, 57, 2015, 71–105 (in Russian)

[18] N. D. Kopachevskiy, Abstract Green Formula and Some Its Applications, Forma, Simferopol', 2016 (in Russian)

[19] N. D. Kopachevskiy, V. I. Voytitskiy, Z. Z. Sitshaeva, “On oscillations of two connected pendulums containing cavities partially filled by incompressible fluids”, Contemp. Math. Fundam. Directions, 63, no. 4, 2017, 627–677 (in Russian) | MR

[20] N. D. Kopachevskiy, S. G. Kreyn, Ngo Zuy Kan, Operator Methods in Nonlinear Hydrodynamics: Evolution and Spectral Problems, Nauka, M., 1989 (in Russian)

[21] S. G. Kreyn, Linear Differential Equations in Banach Space, Nauka, M., 1967 (in Russian)

[22] S. G. Kreyn, N. N. Moiseev, “On oscillations of solid body containing fluid with free boundary”, Appl. Math. Mech., 21:2 (1957), 169–174 (in Russian)

[23] J. L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Mir, M., 1971 (Russian translation)

[24] I. A. Lukovskiy, M. Ya. Barnyak, A. I. Komarenko, Approximate Methods of Solution for Problems of Dynamics of Bounded Volume of Fluid, Naukova dumka, Kiev, 1989 (in Russian)

[25] G. N. Mikishev, B. I. Rabinovich, Dynamics of a Solid Body with Cavities Partially Filled with Fluid, Mashinostroenie, M., 1968 (in Russian)

[26] S. G. Mikhlin, Variational Methods in Mathematical Physics, Nauka, M., 1970 (in Russian) | MR

[27] N. N. Moiseev, A. A. Petrov, Numerical Methods for Computing Eigen Frequencies and Oscillations of Bounded Volume of Fluid, VTs AN SSSR, M., 1966 (in Russian)

[28] N. N. Moiseev, V. V. Rumyantsev, Dynamics of a Body with Cavities Containing Fluid, Nauka, M., 1965 (in Russian)

[29] A. D. Myshkis, V. G. Babskiy, N. D. Kopachevskiy, L. A. Slobozhanin, A. D. Tyuptsov, Methods of Solution of Hydromechanical Problems for Weightlessness Conditions, Naukova dumka, Kiev, 1992 (in Russian)

[30] G. S. Narimanov, L. V. Dokuchaev, I. A. Lukovskiy, Nonlinear Dynamics of an Aircraft with Fluid, Mashinostroenie, M., 1977 (in Russian)

[31] L. S. Pontryagin, “Hermit operators in the space with indefinite metrics”, Bull. Acad. Sci. USSR. Ser. Math., 8:6 (1944), 243–280 (in Russian) | Zbl

[32] I. M. Rapoport, Oscillations of Elastic Casing Partially Filled with Fluid, Mashinostroenie, M., 1967 (in Russian)

[33] S. F. Feshchenko, I. A. Lukovskiy, B. I. Rabinovich, L. V. Dokuchaev, Methods of Determination of Connected Masses of Fluid in Movable Domains, Naukova dumka, Kiev, 1969 (in Russian)

[34] P. V. Kharlamov, “Compound spatial pendulum”, Mech. Solid Body, 4 (1972), 73–82 (in Russian)

[35] F. L. Chernous'ko, Motion of Solid Body with Cavities Filled with Viscous Fluid, VTs AN SSSR, M., 1968 (in Russian)

[36] Gagliargo E., “Caratterizzazioni delle trace sulla frontiera relative ad alaine classi di funzioni in $n$ variabili”, Rend. Semin. Mat. Univ. Padova, 27 (1957), 284–305 | MR

[37] Kopachevsky N. D., Krein S. G., Operator approach to linear problems of hydrodynamics, v. 1, Self-adjoint problems for an ideal fluid, Birkhauser, Basel–Boston–Berlin, 2001 | MR | Zbl

[38] Kopachevsky N. D., Krein S. G., Operator approach to linear problems of hydrodynamics, v. 2, Nonself-adjoint problems for viscous fluids, Birkhauser, Basel–Boston–Berlin, 2003 | MR | Zbl

[39] Myshkis A. D., Babskii V. G., Kopachevsky N. D., Slobozhanin L. A., Tyuptsov A. D., Low-gravity fluid mechanics. mathematical theory of capillary phenomena, Springer, Berlin, 1987 | MR