Analytical solutions for the stiffness and damping coefficients of squeeze films in MEMS devices with perforated back plates

Abstract

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 fromnumerical (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