Abstract
In this study, the influences of the squeezefilm damping effect on the dynamic behavior of the microelectromechanical electrostatic actuators are investigated by the hybrid numerical scheme comprising the differential transformation method and the finite difference method. There are two types of actuators which including the circular microplate and the clampedclamped microbeam, relatively. The analyses take account of the axial stress effect, the residual stress and the fringing field effect within the micro actuators and explore the dynamic response of the plate/beam as a function of the magnitude of the AC driving voltage. The effectiveness of a combined DC/AC loading scheme in driving the micro actuators are examined. It is shown that the use of an AC actuating voltage in addition to the DC driving voltage provides an effective means of tuning the dynamic response of the micro actuators. Therefore, the results show that the hybrid method provides an accurate and computationallyefficient means of analyzing the nonlinear behavior of the microbeam structures used in many of today’s MEMSbased actuator systems.
1. Introduction
Microelectromechanical systems (MEMS) devices have applications in many engineering fields such as communications [1], automotive industry [2] and chemical [3], and so on. Such devices are generally actuated using either electrostatic [4], electromagnetic [5], thermal or piezoelectric [6] techniques. Of these various techniques, electrostatic actuation schemes are the most commonly employed due to their fast response, low power consumption, reliability and their batch fabrications [7]. Actually, in the electrostatic actuation of a microstructure system, the electrostatic force is produced from the voltages of two electrodes. If the electrostatic force is greater than the elastic restoring force of the microstructure system, this represents an unstable phenomenon, and the two electrodes attract and come into contact with each other suddenly [8]. The critical value of the voltage is defined as the pullin voltage, which has a tremendous influence on the electrostatic device. For example, the electrostatic device is regarded as a digital actuator when the operation voltage is greater than the pullin voltage and the upper electrode can be attracted to the fixed bottom electrode very quickly; therefore, the pullin voltage limits the operation range of the actuator. The pullin behavior phenomenon, however, can be used in the design of such components as switches and relays. Hence, the pullin voltage is a very important parameter in the design of microelectromechanical devices.
Accordingly, many researchers have proposed more sophisticated techniques for modeling the nonlinear behavior of electrostatic actuators which including the microplate and the microbeam. Chao et al. [9] proposed a novel computational method for predicting the static pullin event between two microplates actuated by a distributed electrostatic force. Chen et al. [10] used a hybrid differential transformation / finite difference method to analyze the nonlinear dynamic behavior of a clampedclamped microbeam in the absence of a squeezefilm damping effect. Younis et al. [11] presented a continuous reducedorder model for predicting the static pullin voltage and position of electrically actuated MEMS microbeams. Taking into account geometric nonlinearity and residual stresses, Vogl and Nayfeh [12] presented an analytical continuous reducedorder model (macromodel) for an electrically actuated clamped circular plate with a uniform residual biaxial plane stress consideration for actuated, and than using a Galerkin approach method to discretize the microstructure.
To increase the actuation efficiency and detection sensitivity of MEMSbased devices, the separation distance between the two electrodes must be minimized and the overlapping area between them maximized. Under such conditions, a squeezefilm damping effect occurs between the two electrodes as the upper electrode deforms [13]. The Squeezefilm damping can be modeled using the Reynolds equation, which is derived from the Navier–Stokes equations and the continuity equation. The main assumption is that the gas in the gap can be treated as a continuum. The validity of this assumption depends on the socalled Knudsen number (${K}_{n}$) which defined as the ratio of the mean free path of the air particals to the film thickness. Based on the Knudsen number, the air flows can be divided into four regimes: continuum flow when $Kn<$ 0.01, slip flow when 0.01 $<{K}_{n}<$ 0.1, transitional flow when 0.1 $<Kn<$ 10 and free molecular flow when $Kn>$ 10. Krylov and Maimon [14] have studied the transient dynamics of an electrically actuated microbeam considering the electrostatic forces, squeezefilm damping, and rotational inertia of a mass carried by the microbeam.
Dynamic analysis currently MEMS devices are being positively developed for various widely applications of industries. Younis [15] examined the dynamic behavior of microbeams subject to combined DC / AC loading and derived analytical expressions for the microbeam motion under primary resonance conditions. Chen et al. [10] demonstrated that the hybrid differential transformation and finite difference method provides a precise and computationallyefficient means of analyzing the nonlinear dynamic behavior of fixedfixed microbeams.
In the present study, the hybrid numerical scheme comprising the differential transformation method and the finite difference method is applied to analyze the dynamic behavior of the circular microplate / the clampedclamped microbeam actuated by pure DC or combined DC / AC loading schemes. The analyses take account of the electrostatic coupling effect, the fringing field effect, the residual stress, the nonlinear electrostatic force, the axial stress effect and squeezefilm damping effect.
2. Differential transformation theory
Differential transformation theory was originally proposed by Zhou [16] in 1986 as a means of solving linear and nonlinear initial value problems in the circuit analysis field. However, later researchers extended its use to the analysis of the mechanical engineering domain [10, 13, 17]. The basic principles of the differential transformation method are introduced below. If $y\left(t\right)$ is an analyzable function in time domain $T$, a definition of the differential transformation of $x$ at $t={t}_{0}$ in the $K$ domain is:
where $k$ belongs to the set of nonnegative integers denoted as the $K$ domain, $Y(k;{t}_{0})$ is the transformed function in the transformation domain, otherwise called the spectrum of $x\left(t\right)$ at $t={t}_{0}$ in the $K$ domain, $M\left(k\right)$ is the weighting factor, and $q\left(t\right)$ is regarded as a kernel corresponding to $y\left(t\right)$. Both $M\left(k\right)$ and $q\left(t\right)$ are nonzero and $q\left(t\right)$ is an analyzable function in time domain $T$. Therefore, the differential inverse transformation of $Y(k;{t}_{0})$ can be described as:
If $M\left(k\right)={H}^{k}/k!$ and $q\left(t\right)=\text{1}$, where $H$ is the time interval. Let ${t}_{0}=\text{0}$; Eq. (1) then becomes:
The differential inverse transformation of $Y\left(k\right)$can then be expressed as below by Eq. (2):
Substituting Eq. (3) into Eq. (4) gives:
Eq. (5) can be derived by Taylor series expansion. Therefore, the main basic operation properties of the differential transform are listed below:
a) Linearity operation:
where $T$ denotes the differential transform and $\alpha $ and $\beta $ can be any real number.
b) Differential operation:
where $T$ denotes the differential transform and n is the order of differentiation [10, 13, 17].
3. Modeling of microelectromechanical electrostatic actuators
3.1. Modeling of clampedclamped microbeam
The analysis performed in this study considers the clampedclamped microbeam shown in Fig. 1. As shown, the upper plate is actuated by a driving voltage $V\left(t\right)={V}_{DC}+{V}_{AC}\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)$, where ${V}_{DC}$ is the DC polarization voltage, ${V}_{AC}$ is the magnitude of AC voltage, and $\omega $ is the excitation frequency. In deriving the governing equation of the clampedclamped microbeam which considers the fringing field effect, the residual stress and the squeezefilm damping effect within the beam. The corresponding governing equation is therefore given by [13]:
Regarding the righthand side of Eq. (1), ${F}_{E}$ represents the excitation force per unit beam length and is given by:
where ${\epsilon}_{0}$, $b$ and $G$ denote the permittivity of free space, the width of the microbeam and the initial gap size between the upper and lower electrodes, respectively. Moreover, $\stackrel{~}{E}$ is the effective beam material modulus. For the case of a wide microbeam (i.e., $b\ge 5h$), $\stackrel{~}{E}$ is equal to the plate modulus, i.e. $E/(1{\nu}^{2})$, where $\nu $ is the Poisson ratio and $h$ is the beam thickness. Conversely, for a narrow beam, $\stackrel{~}{E}$ is equal to the Young’s modulus, $E$. In Eq. (8), $I$ denotes the moment of inertia of the microbeam, ${N}_{0}$ is a constant denoting the residual force acting on the beam and $w$ is the transverse displacement of the beam and varies as a function of both the longitudinal position $x$ and the time $t$, i.e.,$w=w(x,t)$. Note that the residual force is given by ${N}_{0}=\stackrel{~}{\sigma}bh$, where $\stackrel{~}{\sigma}$ is the residual stress and has the form of $\stackrel{~}{\sigma}=\sigma (1\nu )$, in which $\sigma $ is the biaxial residual stress.
Finally, ${F}_{S}$ denotes the force acting on the beam due to the squeezefilm damping effect and is given by:
where $P$ is the absolute pressure in the air gap and ${P}_{a}$ is the ambient pressure. The net pressure can be expressed as $\stackrel{}{P}=P{P}_{a}$. The squeezefilm damping effect in Fig. 1 can be described by the nonlinear Reynolds equation:
Note that${h}_{a}\text{,}$${\rho}_{a}$ and $\mu $ represent the variable distance between the two electrodes (${h}_{a}=Gw$), the density of the air in the air gap, and the effective viscosity of the air in the air gap, respectively.
Fig. 1Schematic illustration of the clampedclamped microbeam
3.2. Modeling of circular microplate
In deriving the governing equation of motion for the circular microplate shown in Fig. 2, the study considers both the residual stress within the plate and the squeezefilm damping effect between the two plates. The governing equation is therefore given as [8]:
where ${\epsilon}_{0}$, $h$ and $G$ are represented the permittivity of free space, the thickness of the circular microplate, and the initial gap height between the upper and lower plates, respectively. In addition, $V\left(t\right)$ is the voltage between the two plates, (i.e., $V\left(t\right)={V}_{DC}+{V}_{AC}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)$), $\rho $ is the density of the micro circular plate, and $w$ is the transverse deflection of the circular microplate at a distance $r$ from the center of the plate. In other words, the symmetry transverse deflection of micro circular plate is irrelevant to polar coordinate $\theta $. Finally, ${T}_{r}$ is the residual stress within the plate, ${P}_{n}$ is the net pressure within the air gap and $D$ is the flexural rigidity of the plate, i.e.:
where $\nu $ and $E$ are the Poisson ratio and Young’s modulus of the upper circular plate, respectively. The squeezefilm damping effect within the air gap is modeled using the linearized compressible gasfilm Reynolds equation [13], which has the form:
where ${h}_{a}$ and $\mu $ represent the variable distance between the two electrodes (i.e., ${h}_{a}=Gw$) and the effective viscosity of the air in the air gap, respectively. The net pressure in the air gap can be expressed as ${P}_{n}=P{P}_{a}$, where $P$ is the absolute pressure in the gap and ${P}_{a}$ is the ambient pressure.
Fig. 2Schematic illustration of the circular microplate
4. Numerical results and discussion
4.1. Clampedclamped microbeam
In this section, the validity of the proposed hybrid numerical scheme was demonstrated by comparing the predicted value of the pullin voltage with that obtained using different scheme in the literature and the computations were performed using MATLAB. Note that in performing the analysis, the material and geometry properties of the microbeam were assigned the values given in Table 1. In performing the comparison, as shown in Table 2, it is seen that the pullin voltage calculated using the proposed method is just 1.4 % lower than that presented in the literature.
Table 1Material and geometry parameters of the clampedclamped microbeam
Parameters / Symbol / Unit  Value 
Young's modulus ($E$) (GPa)  151 
Poisson’s Ratio ($\nu $)  0.3 
Density ($\rho $) (kg/m^{3})  2332 
Thickness of circular plate ($h$) (μm)  1.5 
Initial gap ($G$) (μm)  1.18 
Length of beam ($L$) (μm)  210 
Biaxial residual stress ($\sigma $) (MPa)  6 
Table 2Comparison of present analytical results and literature results for pullin voltages
Analytical results  Deviation  
Hybrid numerical scheme  DQM [18]  $e$ (%)  
Pullin Voltage (V)  27.7  28.1  1.4 
Fig. 3 shows the variation over time of the centerpoint deflection of the microbeam for a constant DC voltage of 27 V and AC voltages in the range of 01.9 V. It is seen that the centerpoint deflection increases nonlinearly with an increasing AC voltage due to the nonlinear of the electrostatic coupling effect. Furthermore, it is observed that for a dimensionless time of less than 60, the DC voltage is unstable with a smaller deflection of the center point. In addition, the results show that for an AC voltage of 1.8 V, the microbeam oscillates in a stable manner about the equilibrium deflection point. However, when the AC voltage is increased to 1.9 V, the microbeam collapses and makes transient contact with the lower electrode.
Fig. 3Variation of dimensionless centerpoint displacement over time for different AC voltages
4.2. Circular microplate
Fig. 4 illustrates the variation of the centerpoint deflection of the microplate over time as a function of the AC voltage. Note that in performing the analysis, the basic material and geometry properties of the microplate is assigned in accordance with the data presented in Table 3. The results show that as the AC voltage increases, the centerpoint deflection of the microplate also increases as a result of the enhanced electrostatic coupling effect. In addition, it can be observed that the center deflection of the microplate increased nonlinearly as the AC voltage increased due to the microplate coupling effect.
Fig. 4Variation of dimensionless centerpoint displacement over time for constant DC voltage and AC voltages ranging from 0.011.5 V
Fig. 5 shows the phase portraits of the circular microplate for a constant DC voltage of 94 V and AC voltages of 5.0 V, 8.0 V,11.5 V and 12.0 V, respectively. The results show that for an AC voltage less than 12.0 V, the system exhibits a stable behavior, i.e., the size of the orbit remains approximately constant over time. However, for an actuating voltage of 12.0 V, the size of the orbit gradually increases until the system becomes unstable and the pullin event occurs.
Table 3Material and geometry parameters of the circular microplate
Parameters / Symbol / Unit  Value 
Young’s modulus ($E$) (GPa)  130 
Poisson’s Ratio ($\nu $)  0.23 
Density ($\rho $) (kg/m^{3})  2330 
Thickness of circular plate ($h$) (μm)  1.0 
Initial gap ($G$) (μm)  1.0 
Radius ($R$) (μm)  45 
Fig. 5Phase portraits for the circular microplate given constant DC voltage of 94 V
5. Conclusions
The present study has analyzed the nonlinear dynamic behavior of the microelectromechanical electrostatic actuators subject to a squeezefilm damping effect by using a hybrid numerical scheme comprising the differential transformation method and the finite difference method. The validity of the proposed scheme has been confirmed by comparing the predicted value of the pullin voltage for the microbeam with the results presented in the literature. Finally, it has been shown that the stability of the circular microplate reduces as the magnitude of the AC voltage increases. Overall, the numerical results presented in this study show that the hybrid numerical scheme provides an accurate and computationallyefficient means of analyzing the nonlinear dynamic behavior of the microelectromechanical electrostatic actuators used in many of today’s MEMSbased systems.
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About this article
The authors gratefully acknowledge the financial support provided to this study by the Ministry of Science and Technology of Taiwan under Grant Number MOST 1042221E018022.