Geodesic
On small oscillations of three joined pendulums with cavities filled with homogeneous ideal fluids
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 260-299

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We study initial boundary value problem on small motions (and normal oscillations) of hydromechanics system consists of three joined pendulums connected with each other by the spherical hinges and filled with homogeneous ideal fluids. We consider two different cases: conservative systems (without any friction forces) and weak dissipative system (friction forces in some hinges are proportional to difference between angular velocities). Using theory of operators acting in Hilbert space we formulate the problem as a Cauchy problem for differential-operator equation of first order, formulate theorem on strong solvability of the problem on the finite time segment. Corresponding spectral problem has a discrete real spectrum (conservative case) or spectrum situated in the strip along the real axis (dissipative case). For the first case we prove new variational principles, and power asymptotic of the eigenvalues with property of orthogonal basis of eigen elements. For the second case we find some estimates of eigenvalues and Abel-Lidsii basis property for the corresponding system of root elements.
Keywords: boundary value problem, self-adjoint operator, Hilbert space, discrete spectrum, eigenvalues asymptotic. 
@article{SEMR_2020_17_a83,
     author = {V. I. Voytitsky and N. D. Kopachevsky},
     title = {On small oscillations of three joined pendulums with cavities filled with homogeneous ideal fluids},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {260--299},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a83/}
}
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V. I. Voytitsky; N. D. Kopachevsky. On small oscillations of three joined pendulums with cavities filled with homogeneous ideal fluids. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 260-299. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a83/