A numerical study of squeeze-film damping in MEMS-based structures including rarefaction effects

Abstract

In a variety of MEMS applications, the thin film of fluid responsible of squeeze-film damping results to be rarefied and, thus, not suitable to be modeled though the classical Navier-Stokes equation. The simplest way to consider fluid rarefaction is the introduction of a slight modification into its ordinary formulation, by substituting the standard fluid viscosity with an effective viscosity term. In the present paper, some squeeze-film damping problems of both parallel and torsion plates at decreasing pressure are studied by numerical solving a full 3D Navier-Stokes equation, where the effective viscosity is computed according to proper expressions already included in the literature. Furthermore, the same expressions for the effective viscosity are implemented within known analytical models, still derived from the Navier-Stokes equation. In all the considered cases, the numerical results are shown to be very promising, providing comparable or even better agreement with the experimental data than the corresponding analytical results, even at low air pressure.Thus, unlike what is usually agreed in the literature, the effective viscosity approach can be efficiently applied at low pressure regimes, especially when this is combined with a finite element analysis (FEA). SOMMARIO.In molte applicazioni MEMS, il sottile strato di fluido responsabile del fenomeno dello squeezefilm damping risulta essere rarefatto e, quindi, non modellabile mediante l'equazione classica di Navier-Stokes. Il modo più semplice per tener conto della rarefattezza del fluido consiste nell'introduzione di una piccola modifica all'interno della sua formulazione ordinaria, ovvero nella sostituzione della viscosità del fluido con una viscosità effettiva. Nel presente lavoro, vengono presi in considerazione alcuni problemi di squeeze-film damping, riguardanti casi in cui le piastre coinvolte sono dotate di moto normale traslatorio e casi in cui esse sono dotate, invece, di moto torsionale. Tali problemi vengono risolti mediante una modellazione numerica 3D dell'equazione di Navier-Stokes, in cui la viscosità effettiva viene calcolata mediante apposite espressioni già note in letterature. Inoltre, queste stesse espressioni sono utilizzate all'interno di modelli analitici già conosciuti, derivati anch'essi dall'equazione di Navier-Stokes. In tutti i casi qui considerati, i risultati numerici sono molto promettenti, in quanto essi sono caratterizzati da uno scostamento dai dati sperimentali uguale o inferiore a quello fornito dai corrispettivi risultati analitici, anche a basse pressioni. Quindi, nonostante quanto sia usualmente asserito in letteratura, il metodo della viscosità effettiva può essere efficientemente adottato anche a bassi regimi di pressione, specie se in combinazione con analisi numeriche agli elementi finiti (FEA).

A Low-G Silicon Inertial Micro-Switch with Enhanced Contact Effect Using Squeeze-Film Damping

“…In the device structure, the air-gap height ( h ) is at least 35 μm. As a result, we have K n < 0.01, and the fluid between the proof mass and substrate can be considered as a continuum [ 22 ]. The behavior of the continuous fluid can be governed by the linearized Reynolds equation, based on the hypotheses of small amplitude displacement of the proof mass and small squeeze number [ 23 ], which is applicable for this study.…”

Squeeze-Film Air Damping of a Five-Axis Electrostatic Bearing for Rotary Micromotors

“…Gas flowage in the cavity can be characterized by the Knudsen number $K_{\mathrm{n}}=\frac{\lambda }{d}$ where λ is the mean free path of the molecules, and d is the gas film depth in normal direction of gas surface [20,25]. …”

“…The advantage of using the finite-element method lies in the ability to model arbitrary device geometries and boundary conditions. In the case where the rotor moves normally with respect to the top glass wafer along the z -axis, the squeeze film damping coefficient is evaluated as [20] $b_z=\frac{true\int p(x,y)\mathrm{d}x\mathrm{d}y}{v_z}$ where p ( x , y ) is the pressure field over the moving rotor.…”

Simulation Study of Inertial Micro-Switch as Influenced by Squeeze-Film Damping and Applied Acceleration Load

“…For the gas film between the proof mass and substrate, the gap height . As such, we have , and thus the fluid is in the continuum regime [ 18 ]. The behavior of the continuous fluid is generally governed by the well-known Reynolds equation [ 19 ].…”