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 Rus. J. Nonlin. Dyn., 2020, Volume 16, Number 2, Pages 309–324 (Mi nd712)

Nonlinear physics and mechanics

Mathematical Study of the Small Oscillations of a Pendulum Completely Filled with a Viscoelastic Fluid

H. Essaouinia, P. Capodannob

a Abdelmalek Essaâdi University, Faculty of Sciences, M2SM ER28/FS/05, 93030 Tetuan, Morocco
b Université de Franche-Comté, 2B — Rue des jardins, 25000 Besancon, France

Abstract: We study the small oscillations of a pendulum completely filled by a viscoelastic fluid, restricting ourselves for the fluid to the simpler Oldroyd model. We establish the equations of motion of the system. Writing them in a suitable form, we obtain an existence and unicity theorem of the solution of the associated evolution problem by means of semigroup theory. Afterwards, we show the existence and symmetry of the spectrum and prove the stability of the system. We show the existence of two sets of positive real eigenvalues, of which the first has infinity, and the second a point of the real axis, as points of accumulation. Finally, we specify the location of the possible nonreal eigenvalues.

Keywords: viscoelastic fluid, small oscillations, variational-operatorial and spectral methods, semigroups

DOI: https://doi.org/10.20537/nd200206

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MSC: 76A10, 76M22, 76M30, 49R05, 47A75
Accepted:30.03.2020
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Citation: H. Essaouini, P. Capodanno, “Mathematical Study of the Small Oscillations of a Pendulum Completely Filled with a Viscoelastic Fluid”, Rus. J. Nonlin. Dyn., 16:2 (2020), 309–324

Citation in format AMSBIB
\Bibitem{EssCap20} \by H.~Essaouini, P.~Capodanno \paper Mathematical Study of the Small Oscillations of a Pendulum Completely Filled with a Viscoelastic Fluid \jour Rus. J. Nonlin. Dyn. \yr 2020 \vol 16 \issue 2 \pages 309--324 \mathnet{http://mi.mathnet.ru/nd712} \crossref{https://doi.org/10.20537/nd200206} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85093900049}