Normal oscillations of a pendulum with a cavity partially filled with an ideal incompressible fluid

V. I. Voytitsky ; N. D. Kopachevskii

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 1, Tome 190 (2021), pp. 34-49.

Résumé

We consider a linear initial-boundary-value problem generated by the problem of small motions of a spatial pendulum with a cavity partially filled with a homogeneous incompressible fluid, in the case where the moment of friction forces in the spherical hinge is proportional to the angular velocity. We propose an operator interpretation of the problem and prove a theorem on the strong solvability of the Cauchy problem on a finite time interval. For the corresponding spectral problem, the discreteness of the spectrum and its localization in a strip are proved, power asymptotics of eigenvalues are found, and the summability of the system of eigenvectors is established by the Abel–Lidsky method.

Détail

Keywords: initial-boundary-value problem, Hilbert space, self-adjoint linear operator, discrete spectrum, Abel–Lidsky basis property.

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