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 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.

Servo-fluid-elastic modeling of contactless levitated adaptive secondary mirrors

“…7, highlights the different nature of the fluid dynamic action at increasing frequencies. As expected [2,15], the fluid film provides a dominant damping effect roughly below 4 kHz, while an increasingly elastic contribution prevails at higher frequencies. Such a damping loss paves the way to possible spillover instabilities, profoundly affecting a controlled mirror response.…”

Microelectromechanical systems (MEMS)

“…simulations. A repetitive pattern of the pressure distribution around each hole in perforated plates is also observed by Mohite et al [9] that identified independent pressure cells of circular geometry, analytically solved by a one-dimensional Reynolds equation in polar coordinates. The complex pressure obtained is used to identify the stiffness and damping offered by the pressure cell, and then added up separately to extract global dynamic parameters.…”

Squeeze-film effects in MEMS devices with perforated plates for small amplitude vibration

“…However, the conventional equation can be modified to be applicable for perforated plate. As most papers did (Bao et al 2003;Homentcovschi and Miles 2004;Mohite et al 2005), the uniformly perforated plate with mass holes can be divided into cells. As shown in Fig.…”

“…According to the Table 1 by Mohite et al (2005) and for the comparisons between different analytical results and ANSYS simulation results, a typical onedimensional rectangular perforated plate with square holes is considered. The plate thickness h = 5 lm and the initial air gap d 0 = 1 lm.…”

“…Recent work by Homentcovschi and Miles (2004) and Mohite et al (2005) gave analytical approximations and simulations to the squeeze-film effects of perforated plates. Homentcovschi considered the perforated plate as infinite, and divided the plate into mass pressure cells.…”