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Existence and uniqueness theorems for a class of equations of interaction type between a vibrating structure and a uid

Journal Name:

  • Mathematica Æterna

Publication Year:

  • 2012

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Abstract (2. Language): 
The problems of the interaction between a vibrating structure and a uid have been studied by many authors, see for example the inter- esting article [7] and their references. The principal objective of this work is to investigate the solvability of some problems of interaction between structure and uid by a mathematical method based upon the analytic semigroups and fractional powers of operators and which can be applied to wider range of physical situations. In this paper, we de- velop this method on a three-dimensional model of interaction between a vibrating structure and a light uid occupying a bounded domain in IR3. This model was introduced in J. Sound Vibration 177 (1994) [3] by Filippi-Lagarrigue-Mattei for an one-dimensional clamped thin plate, extended by an in nite perfectly rigid bae. Intissar and Jeribi have shown in J. Math. Anal. Appl. (2004) [4] the existence of a Riesz basis of generalized eigenvectors of this one-dimensional model.A two-dimensional model of the vibration and the acoustic radiation of a baed rectangular plate in contact with a dense uid was considered by Mattei in J. Sound Vibration (1996) [9].
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  • English

REFERENCES

References: 

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radiation by a vibrating plate in a light
uid: comparison with the exact
solution, J. Sound Vibration 177 (1994) 259-275
[4] A. Intissar, A. Jeribi, On an Riesz basis of generalized eigenvectors of the
non-self-adjoint problem deduced from a perturbation method for sound
radiation by a vibrating plate in a light
uid, J. Math. Anal. Appl. 292
(2004) 1-16
30 Marie-Therese Aimar and Abdelkader Intissar
[5] A. Intissar, Analyse fonctionnelle et theorie spectrale pour les operateurs
compacts non auto-adjoints, avec exercices et solutions, CEPADUES Editions,
(1997)
[6] M.V. Keldysh , On completeness of the eigenfunctions for certain classes
of nonselfadjoint linear operators, Uspekhi Mat. Nauk 27 (1971) no. 4
(160), 14-41; English transl. in Russian Math. Surveys 27 (1971)
[7] D.P. Kouzov, G.V. Filippenko ,Vibrations of an elastic plate partially
immersed in a liquid,Journal of Applied Mathematics and Mechanics 74
(2010) 617-626
[8] S.G. Krein, Linear Dierential Equations in Banach Space, Providence,
(1971)
[9] P.O. Mattei, A two-dimensional Tchebyche collocation method for study
of the vibration of baed
uid-loaded rectangular plate,J. Sound Vibration
196 (1996) 407-427
[10] S.G. Mikhlin, Variational Methods in Mathematical Physics, Oxford London
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[11] H. Tribel, interpolation Theory, functions Spaces, Dierential Operators,
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