In order to accurately describe the processes involved in the motion of am embrane loaded by as queeze fluid film, it is necessary to have solutions for both the membrane and the fluid filmw hich account for their strong coupling. The dynamic solution approaches presented here include the effects of the bending stiffness of the membrane (assumed to be ac lamped square plate)a nd the motion of the thin air filmb etween the membrane and the backing wall, assuming realistic boundary conditions at its periphery.T he mode shapes are predicted accurately for both the plate and the air film, and the approach followed includes their strong coupling. Owing to the accuracyofthe method, there are applications in the characterisation of the properties of square microphones (those miniaturised on silicon chip)a sshown at the end of the paper.Actually,the method would be useful for other applications which involverectangular or square plates coupled to resistive and reactive squeeze fluid films, including or not an in-plane tension. PACS no. 43.38.Bs ACTA ACUSTICA UNITED WITH ACUSTICA Le VanSuu et al.:O nthe modelling of clamped plates Vol. 96 (2010) Le VanSuu et al.:O nthe modelling of clamped plates Vol. 96 (2010)

Current electrostatic transducers are designed with aplanar perforated backing electrode while nowadays some authors suggest that anon planar backing wall would enhance their performances. Accurate analytic results for these newshapes of backing electrode are not available until nowwhereas current applications (metrology,miniaturisation)arise which need more refined modelling. Therefore the present paper aims at providing an analytical description of the strong coupling between afl uid layer trapped between an axisymmetrically curved smooth backing wall and acircularvibrating membrane (the thickness of the fluid layer depends on the radial coordinate) using modelling wherein viscosity,h eat conduction, inertia and compressibility of the fluid are considered, beyond realistic boundary conditions (nos lip condition and no temperature variations near the walls). Solutions are givenasthe sum of an expansion on eigenfunctions of the membrane in vacuo and an extended power series (using Frobenius method)toovercome the limitations of astandard electrical equivalent network analysis in the frequencyr ange of interest (upt o1 00 kHz for circular membranes approximately one millimetre in diameter). The results obtained provide tools for optimising the design of such small electrostatic transducers.

The thermo-viscous damping due to the thin fluid film between the membrane and the backing electrode strongly influences both the sensitivity of the condenser microphones in the lower frequency range and the upper limit of the frequency bandwidth. Nowadays, most of the MEMs microphones use a perforated backing electrode while some authors suggest that a continuously curved backing electrode could enhance their performances (among avantages in the design when etching). The present paper provides two kinds of modeling for such microphones with a tapered fluid film: the first one lies on Kirchhoff-network analysis (neglecting cross-coupling between elements) whereas the second one is based upon the direct resolution of the set of basic equations (including heat transfer phenomena). The results are presented and discussed for both models in the cases of flat, parabolic, and stepped shapes backing electrode. Finally, the pressure field in the fluid film, computed (for an axisymmetrical configuration) using the above-mentioned models, is compared to the one computed with a new FEM formulation taking into account both viscous and thermal phenomena in the boundary layers.