A semi-analytical model for squeeze-film damping including rarefaction in a MEMS torsion mirror with complex geometry

Abstract

A semi-analytical approach is presented to model the effects of complicated boundary conditions and rarefaction on the squeeze-film damping dependent quality factor in a double-gimballed MEMS torsion mirror. To compute squeeze-film damping in a rectangular torsion mirror with simple boundaries, compact models derived by solving the conventional Reynolds equation with zero pressure boundary conditions on the edges of the plate are generally used. These models are not applicable if the air-gap thickness is comparable to the length of the plate. To extend the validity of the existing models in devices with large air-gap thickness and complicated boundaries, we present a procedure that requires the computation of the effective length of the structure and uses this length for the computation of damping in all flow regimes using a modified effective viscosity model. The effective length is computed by comparing the damping obtained from a numerical solution of Navier-Stokes equations with that obtained from a Reynolds-equation-based compact model. To capture the effect of rarefaction in different flow regimes, we use two different approaches: the effective viscosity approach which is valid for continuum, slip, transition and molecular flow regimes, and an approach based on the free molecular model which is valid only in a molecular flow regime. We show that the effective length obtained for complicated structures in the continuum regime may still be used to capture the rarefaction effect in the slip, transition and molecular regimes. On comparing different empirical models based on the effective viscosity approach with experimental results, we find some anomaly in the region between the molecular regime and the intrinsic regime where non-fluid damping dominates. To improve modelling in the rarified regimes, we modify the best model among the existing models by minimizing error obtained with respect to the experimental results. We find that the proposed model captures the rarefaction effect not only in the slip, transition and molecular regimes but also couples well with the non-fluid damping in the intrinsic regime and captures the transition to purely intrinsic losses.

On the modified Reynolds equation model for the prediction of squeeze-film gas damping in a low vacuum

“…For this reason, extensive research efforts have been devoted to the development of modeling approaches and tools for the accurate prediction of squeeze-film damping. Some relevant work can be found in (Andrews et al 1993;Pan et al 1998;Hutcherson and Ye 2004;Gallis and Torczynski 2004;Nayfeh and Younis 2004;Veijola et al 1995Veijola et al , 2005Bao and Yang 2007;Pandey and Pratap 2008;Sujilen et al 2009;Hong and Ye 2010;Leung et al 2010;Bidkar et al 2009;Guo and Alexeenko 2009;Veijola 2004;Lee et al 2009) and references cited in a review article by Bao and Yang (2007). Depending on the ambient pressure level and the characteristic length of the resonators, gas between the resonant structure and the fixed substrate could be in a rarefied flow regime and therefore be governed by fundamentally different physics.…”

“…A popular approach for the modeling of squeeze-film damping of a plate resonator is to use Reynolds equation coupled with an effective viscosity (Nayfeh and Younis 2004;Veijola et al 1995;Pandey and Pratap 2008;Lee et al 2009;Li and Hughes 2000;Li 1999). A fundamental assumption inherent in this approach is that rarefaction effect can be accurately accounted for via a modified viscosity model that depends only on the Knudsen number.…”

“…For highly rarefied cases particularly when the gas is in the free-molecule regime, such an assumption does not have any theoretical ground and has not been validated rigorously. Nevertheless, several viscosity models have been proposed in the literature and have been applied to the highly rarefied situations (Nayfeh and Younis 2004;Veijola et al 1995;Pandey and Pratap 2008;Li and Hughes 2000;Li 1999). Most of these models were developed based on Fukui and Kaneko's work (Fukui and Kaneko 1990) in which the effective viscosity was obtained from solving the linearized BGK model equation of a Poiseuille flow.…”

“…Most of these models were developed based on Fukui and Kaneko's work (Fukui and Kaneko 1990) in which the effective viscosity was obtained from solving the linearized BGK model equation of a Poiseuille flow. In a recent study (Pandey and Pratap 2008), a comparison of several effective viscosity models was conducted and it was concluded that Li's model (Li 1999) fitted the best with the numerical solutions of the BGK model equation. The application of Li's model to the modeling of a simple torsion mirror, however, did not produce a good prediction of the quality factor when compared with the experimental data (Pandey and Pratap 2008).…”

A Macro Model of Squeeze-Film Air Damping in the Free-Molecule Regime

“…One is based on the modification of continuum theories. Examples in this category include, but are not limited to, the modification of Reynolds' equation by employing the effective viscosity obtained either from a linearized Bhatnagar-Gross-Krook solution of the Poiseuille flow 12,14,23 or from experimental data 24,25 via curve fitting, by using a pressure dependent effective flow rate coefficient 13 and a modified pressure boundary condition with coefficients extracted from the NavierStokes slip-jump simulations and the direct simulation Monte Carlo ͑MC͒ simulations. 15 Most of these methods have demonstrated their accuracy in the slip regime ͑Kn Յ 0.1͒, i.e., the near continuum regime.…”