Study of a vibration problem for a perforated plate with Fourier boundary conditions (Q2725530)

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].