Analysis of small oscillations of a pendulum partially filled with a viscoelastic fluid

Hilal Essaouini; Pierre Capodanno; Hilal Essaouini; Pierre Capodanno

doi:10.3934/cam.2023019

Communications in Analysis and Mechanics

2023, Volume 15, Issue 3: 388-409. doi: 10.3934/cam.2023019

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Analysis of small oscillations of a pendulum partially filled with a viscoelastic fluid

Hilal Essaouini ^{1
,
,},
Pierre Capodanno ²

1.
Department of Physics, Abdelmalek Essaadi University, Faculty of Sciences, Energy Laboratory, 93030 Tetuan, Morocco
2.
Université de Franche-Comté, 2B - Rue des jardins, F- 25000 Besançon, France

Received: 13 March 2023 Revised: 22 June 2023 Accepted: 27 June 2023 Published: 12 July 2023
76A10, 76M22, 76M30, 49R50, 47A75

This paper focuses on the spectral analysis of equations that describe the oscillations of a heavy pendulum partially filled with a homogeneous incompressible viscoelastic fluid. The constitutive equation of the fluid follows the simpler Oldroyd model. By examining the eigenvalues of the linear operator that describes the dynamics of the coupled system, it was demonstrated that under appropriate assumptions the equilibrium configuration remains stable in the linear approximation. Moreover, when the viscosity coefficient is sufficiently large the spectrum comprises three branches of eigenvalues with potential cluster points at $ 0 $, $ \beta $ and $ \infty $ where $ \beta $ represents the viscoelastic parameter of the fluid. These three branches of eigenvalues correspond to frequencies associated with various types of waves.
- viscoelastic fluid,
- normal oscillations,
- variational,
- operatorial and spectral methods
Citation: Hilal Essaouini, Pierre Capodanno. Analysis of small oscillations of a pendulum partially filled with a viscoelastic fluid[J]. Communications in Analysis and Mechanics, 2023, 15(3): 388-409. doi: 10.3934/cam.2023019

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Abstract

This paper focuses on the spectral analysis of equations that describe the oscillations of a heavy pendulum partially filled with a homogeneous incompressible viscoelastic fluid. The constitutive equation of the fluid follows the simpler Oldroyd model. By examining the eigenvalues of the linear operator that describes the dynamics of the coupled system, it was demonstrated that under appropriate assumptions the equilibrium configuration remains stable in the linear approximation. Moreover, when the viscosity coefficient is sufficiently large the spectrum comprises three branches of eigenvalues with potential cluster points at $ 0 $, $ \beta $ and $ \infty $ where $ \beta $ represents the viscoelastic parameter of the fluid. These three branches of eigenvalues correspond to frequencies associated with various types of waves.

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