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Behavior of an elastic plate having a small vibrating rigid support. Asymptotic study of the impedance matrix. (Comportement d’une plaque élastique dont une petite région est rigide et animée d’un mouvement vibratoire. Étude asymptotique de la matrice d’impédance.) (French) Zbl 0834.73041

Summary: This paper consideres a Love-Kirchhoff plate \({\mathcal P}\) which is fixed to a vibrating rigid support along a domain \(\omega _e\). The size of \(\omega _e\) is small so that we set diam \(\omega _e\) = \(\epsilon\) diam \({\mathcal P}\). We study the behaviour of the impedance matrix when \(\epsilon\) vanishes. We give the asymptotic expansion of the solutions of the outer problem together with the fundamental solution of the biharmonic problem and its derivatives. Then we set the inner problem after a dilatation of the domain and we write also asymptotic expansions of its solutions. Finally, in the case of a given rigid motion of \(\omega _e\), we match the outer and inner expansions, and we obtain rational equivalents in \(\ln \varepsilon\) for impedance terms.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
74K20 Plates
35C20 Asymptotic expansions of solutions to PDEs
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References:

[1] Campbell, A.) .- Impédance d’une plaque reliée à un bras rigide suivant un disque de rayon ∈ et comportement quand e tend vers zéro, C.R. Acad. Sci.Paris314, série II (1992), pp. 743-748. · Zbl 0751.73032
[2] Hagedorn, P.) et Kelkel, K.) .- Study of plate impedance, Rapport E.S.A. (avril 1985).
[3] Campbell, A.) .- Matrices d’impédance dans les liaisons de plaques élastiques reliées à des bras rigides, Thèse de Dotorat, Université Paris VI (mai 1989), pp. 1-190.
[4] Van Dyke, M.D.) .- Perturbation methods in fluid mechanics, Academic press, New-York (1964). · Zbl 0136.45001
[5] MAZ’YA, M.G.), Nazarov, S.A.) et Plamenevskii, B.A.) .- The asymptotic behaviour of solutions of elliptic boundary value problems under singular perturbations of the domain, Inst. Prikl. Tbilis. Univ., Tbilisi (1981), en russe; traduit largement en allemand dans “Asymptotische Theorie Elliptischer Randwertaufgaden in singulär gestörten Gebieten” Bd 1 & 2, Berlin, Academic-Verlag (1990-1991 ).
[6] Leguillon, D.) et Sanchez-Palencia, E.) .- Computation of singular solutions in elliptic problems and elasticity, Masson (1987). · Zbl 0647.73010
[7] IL’IN, A.M.) . - Matching of asymptotic expansions of solutions of boundary value problems, Nauka, Moskow (1989), en russe ; traduit en anglais dans American Mathematical Society, Providence, U.S.A., 1992. · Zbl 0754.34002
[8] IL’IN, A.M.) .- A boundary value problem for a second-order elliptic equation in a domain with a narrow slit, II : Domain with a small cavity, Mat. Sb.. 103, n° 145 (1977), pp. 265-284, en russe; traduit en anglais dans Math. USSR Sb.32 (1977). · Zbl 0396.35033
[9] MAZ’YA, M.G.) Nazarov, S.A.) ET Plamenevskii, B.A.) .- Asymptotic expansion of eigenvalues of boundary value problems for the Laplace operator in domains with small holes, Izv. Akad. Nauk. SSSR. Ser. Mat.48 (1984), pp. 347-371; traduit en anglais dans Math. USSR Izv24 (1985). · Zbl 0566.35031
[10] Duvaut, G.) et Lions, J.-L.) .- Les inéquations en mécanique et en physique, Dunod (1972). · Zbl 0298.73001
[11] Courant, R.) et Hilbert, D.) .- Methods of mathematical physics, Interscience (1937).
[12] Petiau, G.) .- La théorie des fonctions de Bessel, Ed. C.N.R.S. (1955). · Zbl 0067.04704
[13] Rektorys, K.) . - Variational methods in mathematical, science and engineering, Reidel publishing company (1980). · Zbl 0481.49002
[14] Hardy, G.H.), Littlewood, J.E.) et Polya, G.) .- Inequalities, Cambridge University Press (1951). · Zbl 0047.05302
[15] KONDRAT’EV, V.A.) . - Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obshch.16 (1967); pp. 209-292, en russe; traduit en anglais dans Trans. Moskow Math. Soc.16 (1967). · Zbl 0194.13405
[16] Pazy, A.) .- Asymptotic expansion of solutions of ordinary differential equations in Hilbert space, Arch. Rat. Mech. Anal.24, n° 2 (1967), pp. 193-218. · Zbl 0147.12303
[17] Nazarov, S.A.) et Plamenevskii, B.A.) .- Elliptic problems in domains with piecewise smooth boundaries, Nauka, Moskow (1991), en russe ; traduit en anglais dans “Walter de Gruyter & Co”, Berlin - New-York (1992). · Zbl 0806.35001
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