Closed-form expressions for the stiffness and the damping coefficients of a squeeze film are derived for MEMS devices with perforated back plates. Two kinds of perforation configurations are considered-staggered and matrix or non-staggered configuration. The analytical solutions are motivated from the observation of repetitive pressure patterns obtained from numerical (FEM) solutions of the compressible Reynolds equation for the two configurations using ANSYS. A single pressure pattern is isolated and further subdivided into circular pressure cells. Circular geometry is used based on observed symmetry. Using suitable boundary conditions, the Reynolds equation is analytically solved over the pressure cells. The complex pressure obtained is used to identify the stiffness and damping offered by the pressure cells. The stiffness and damping forces due to pressure cells within a pattern are added up separately. In turn, the stiffness and damping due to all the patterns are summed up resulting in the stiffness and damping forces due to the entire squeeze film. The damping and spring forces thus obtained analytically are compared with those obtained from the FEM simulations in ANSYS. The match is found to be very good. The regime of validity and limitations of the analytical solutions are assessed in terms of design parameters such as pitch to air gap, hole length to diameter and pitch to hole radius ratios. The analysis neglects inertial effects. Hence, the results are presented for low values of Reynolds number.

We present a comprehensive analytical model of squeeze-film damping in perforated 3-D microelectromechanical system structures. The model includes effects of compressibility, inertia, and rarefaction in the flow between two parallel plates forming the squeeze region, as well as the flow through perforations. The two flows are coupled through a nontrivial frequencydependent pressure boundary condition at the flow entry in the hole. This intermediate pressure is obtained by solving the fluid flow equations in the two regions using the frequency-dependent fluid velocity as the input velocity for the hole. The governing equations are derived by considering an approximate circular pressure cell around a hole, which is representative of the spatially invariant pressure pattern over the interior of the flow domain. A modified Reynolds equation that includes the unsteady inertial term is derived from the Navier-Stokes equation to model the flow in the circular cell. Rarefaction effects in the flow through the air gap and the hole are accounted for by considering the slip boundary conditions. The analytical solution for the net force on a single cell is obtained by solving the Reynolds equation over the annular region of the air gap and supplementing the resulting force with a term corresponding to the loss through the hole. The solution thus obtained is valid over a range of air gap and perforation geometries, as well as a wide range of operating frequencies. We compare the analytical results with extensive simulations carried out using the full 3-D Navier-Stokes equation solver in a commercial simulation package (ANSYS-CFX). We show that the analytical solution performs very well in tracking the net force and the damping force up to a frequency f = 0.8f n (where f n is the first resonance frequency) with a maximum error within 20% for thick perforated cells and within 30% for thin perforated cells. The error increases considerably beyond this frequency. The prediction of the first resonance frequency is within 21% error for various perforation geometries.

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