Study of a vibration problem for a perforated plate with Fourier boundary conditions
type: Item
instance of: scholarly article
zbMATH Open document ID: 0977.35016
author: J.-M. Sac-Épée
author: J. Saint Jean Paulin
publication date: 2002-01-06T00:00:00Z
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 74K20
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35L55
Mathematics Subject Classification ID: 74H45
zbMATH DE Number: 1619395
zbMATH Keywords: thin plate periodically perforated by cylindrical holes
zbMATH Keywords: asymptotic behaviour
MaRDI profile type: Publication
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title: Study of a vibration problem for a perforated plate with Fourier boundary conditions
review text: The authors consider a thin plate periodically perforated by cylindrical holes, the axis of which are perpendicular to the middle plane of the domain. The size of the period is \(\varepsilon\), the thickness of the plate is \(e>0\). The vibration problem of the plate has the form NEWLINE\[NEWLINEe^2\dfrac {\partial^2 u_i^{e\varepsilon}}{\partial t^2} - \dfrac \partial{\partial x_j}(a_{ijkh}\dfrac {\partial u_k^{e\varepsilon}} {\partial x_h})=F_i^{e\varepsilon}\quad \text{in} \;\Omega_{e\varepsilon}\times (0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINEu_i^{e\varepsilon}=0\;\text{on} \Gamma_e^0\times (0,T),\quad a_{i3kh}\dfrac{\partial u_k^{e\varepsilon}}{\partial x_h}n_3=0 \quad \text{on} \Gamma_{e\varepsilon}^{\pm}\times (0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINEa_{i\alpha kh}\dfrac{\partial u_k^{e\varepsilon}}{\partial x_h}n_\alpha+\lambda e^2\varepsilon u_i^{e\varepsilon}+\gamma e^2 \varepsilon \dfrac {\partial u_i^{e\varepsilon}}{\partial t}=G_i^{e\varepsilon}\quad \text{on} \partial S_{e\varepsilon}\times (0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINE u_i^{e\varepsilon}(0)=u^{\varepsilon 0}, \;\dfrac {\partial u_i^{e\varepsilon}}{\partial t}(0)=u^{\varepsilon 1} \quad \text{in} \Omega_{e\varepsilon}.NEWLINE\]NEWLINE Here \(\Omega_{e\varepsilon}=\omega_\varepsilon\times (-e/2,e/2)\) is the perforated cylindrical plate, \(s_\varepsilon=\omega\setminus \omega_\varepsilon,\;\Gamma_e^0=\partial \omega\times (-\varepsilon,\varepsilon),\;\Gamma_{e\varepsilon}^{\pm}=\partial \omega_\varepsilon \times \{\pm e/2\},\;S_{e\varepsilon}=s_\varepsilon \times (-e/2,e/2).\;\) The asymptotic behaviour of this plate, when the two parameters \(e, \varepsilon\) tend to zero are studied. Several possibilities are considered.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00045].
reviewed by: Igor Bock
description: scientific article; zbMATH DE number 1619395