A compact squeeze-film model including inertia, compressibility, and rarefaction effects for perforated 3-D MEMS structures

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 frequency-dependent 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,, 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.