Characterization of coupled-domain multi-layer microplates in pull-in phenomenon, vibrations and dynamics

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Abstract

This paper presents a model to analyze pull-in phenomenon, vibrational behavior and dynamics of multi-layer microplates using coupled finite element and finite difference methods (FDM). First-order shear deformation theory (FSDT) is used to model dynamical system using finite element method, while FDM is applied to solve nonlinear Reynolds equation of squeeze film damping. Using this model, pull-in analysis of single- and multi-layer microplates are studied. Vibrational behavior of single- and multi-layer microplates are analyzed to compute resonance frequencies and mode shapes of the system. Also, an algorithm is presented to study dynamics of microplates under the actuation of nonlinear electrostatic force and squeeze film damping. Results for simplified single-layer microplates are validated and in good agreement with the published literature. This investigation can be implemented in the design of multi-layer microplates.

Introduction

Technology of microelectromechanical systems has experienced a lot of progress recently. Their light weight, low cost, small dimensions, low-energy consumption, and durability make them quite useful. Studying the design parameters of microelectromechanical systems by scientific techniques is of great importance; therefore simulation tools are being improved.

Among various methods, electrostatic actuation and sensing is widely used in many MEMS sensors and actuators, including acoustic resonators, RF switches and pressure sensors. In designing microplates, pull-in effect, as an important phenomenon, must be considered. Due to this importance, many researches have been done on this phenomenon. Osterberg [1] achieved several closed-form models for pull-in phenomena in these systems and has compared his findings with experiments.

Damping in microstructures strongly affects their characteristics especially their dynamic behavior. As important sources of damping, squeeze film and thermoelastic damping has extensively been studied in literature. Nayfeh and Younis [2] presented an approach to model squeeze film damping considering in-plane forces and slipping flow by perturbation method. Younis [3] has modeled micro systems in multi-physics field considering thermoelastic effect. Zhang and Meng [4] used harmonic balance method to study the behavior of a micro cantilever influenced by parametric actuation and squeeze film damping.

Some researchers have paid attention to design-parameters study of micro systems. Zhao et al. [5] presented a model for plates considering electrostatic actuation and plate stretching. Using this model, they have discussed about design parameters. Abdullah et al. [6] considered the effect of microbeam dimensions on pull-in voltage.

MEMS devices could be of single-layer type or multi-layer one. Unfortunately, dynamical behavior and vibration analysis of multi-layer microplates are not addressed well in the literature. Rong et al. [7] presented an analytical model to study pull-in phenomenon of multi-layer beams using energy method.

In this paper, pull-in analysis, vibrational behavior and transient dynamics of orthotropic multi-layer microplates are studied. Nine-node elements, with three or five degrees of freedom in each node are considered. The approach utilizes first-order shear deformation theory (FSDT) of multi-layer plates coupled with nonlinear electrostatic actuation and Reynolds equation.

Multi-layer model is used to verify pull-in results with the results published in the literature. Afterwards, vibrational analysis for a single-layer model is performed to compare results with literature.

Using these certifications an analysis for vibrational and dynamic behavior of multi-layer microplates are presented.

Section snippets

Equations of motion

In an n-layer prismatic microplate, there are one conductive and some dielectric layers (Fig. 1). It is assumed that the conductive layer always positioned at the top of the stack. A fixed planar electrode underlies the plate as a ground. The multi-layer microplate is deformable while the substrate plate is rigid. When a voltage applied between the microplate and the substrate plate, an attractive electrostatic force causes the microplate to deform. The length of the plate is l, the width is b,

Finite element model of the plate

Multiplying Eqs. (12a), (12b), (12c), (12d), (12e) with δu0, δv0, δw0, δϕx and δϕy, respectively, and integrating over the domain, an integrated form of governing equations are obtained. Integration by parts, results in the weakened expressions.

Assuming displacements and rotations as [8]: u0(x,y,t)=j=1muj(t)ψj(x,y),v0(x,y,t)=j=1mvj(t)ψj(x,y),w0(x,y,t)=j=1mwj(t)ψj(x,y),ϕx(x,y,t)=j=1mSj1(t)ψj(x,y),ϕy(x,y,t)=j=1mSj2(t)ψj(x,y),where ψi and m are Lagrange interpolation functions and number of

Finite difference model for squeeze film damping

In studying transient behavior of microplates, finite difference method can be used to model squeeze film damping. Depending on the magnitudes of deflection, nonlinear or linearized Reynolds equation can be used. For problems with small deflection, either nonlinear or linearized Reynolds equation and for problems with large deflection, nonlinear Reynolds equation is used.

Reynolds equation has the form:x((dgap-w0)3PPx)+y((dgap-w0)3PPy)=12η((dgap-w0)Pt-Pw0t).

AssumingP(x,y,t)=P0+P¯(x,y,

The pull-in analysis of single- and multi-layer microplates

With increasing applied voltage, the plate will bend further, but the system is stable and there will be an equilibrium state due to the equivalence of the electrostatic force and plate restoring force. However, there is a critical voltage in which, system becomes unstable. In other words, if the applied voltage reaches this critical value or beyond, there will be no static equilibrium in the system. This value is called the pull-in voltage (Vpi) and this phenomenon is called pull-in

Conclusions

Multi-layer microplates subjected to electric field have been modeled. The model is developed to consider in-plane forces, nonlinear electrostatic actuation and nonlinear squeeze film damping using FSDT. Pull-in analysis of single- and multi-layer microplates is performed and results are in good agreement with literature. Also vibrational behavior of single-layer microplates is studied. The model is validated using results of investigation on single-layer microplates reported in the literature.

References (13)

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