Investigation of the nonlinear static and dynamic behaviour of rectangular microplates under electrostatic actuation

We present an investigation of the static and dynamic behavior of the nonlinear von-Karman plates when actuated by the nonlinear electrostatic forces. The investigation is based on a reduced order model developed using the Galerkin method, which rely on modeshapes and in-plane shape functions extracted using a finite element method. In this study, a fully clamped microplate is considered. We investigate the static behavior and the results are validated by comparison with the results calculated by a finite element model. The forcedvibration response of the plate is then investigated when the plate is excited by a harmonic AC load superimposed to a DC load. The dynamic behavior is examined near the primary resonance. The microplate shows a strong hardening behavior due to the cubic nonlinearity of mid-plane stretching. However, the behavior switches to softening as the DC load is increased.


Introduction
Micro-Electro Mechanical Systems (MEMS) commonly use electrically actuated flexible micro-structures, such as microbeams and microplates [1,2].Examples include pressure, mass and gas sensors, micropumps, microjets, micromirror, and microphones [3,4].Modeling accurately the mechanical behavior of such structures under the applied electric force is required to predict the response prior to the experimental testing and actual use of the device.Accurate models can guide the design engineer through the design process; reducing the design time on one hand and on the other hand can help to improve the existing devices.It is common to study the mechanical behavior of MEMS using linear theory [2]; which is applicable only for small deflections.Since in MEMS, structures often undergo large deflection, linear theory becomes inaccurate.Common modeling approaches include lumped mass models and Finite Element Method (FEM) [4,5].Lumped mass models give rough estimate of the response only.FEM based software tools are accurate but computationally expensive, especially when it comes to study the nonlinear dynamic behavior.On the other hand Reduced Order Models (ROM) based on the Galerkin approach have got popularity during the last decades because of their accuracy and low computational cost [6][7][8][9][10].They have the capability to reveal the effect of different design parameters very conveniently.A common way to model the plate structures in MEMS devices is using the linear plate theory [11][12][13], which is correct when out of plane deflection is small.But in cases when the transverse deflection is comparable to the thickness of the plate, a strong geometric nonlinearity is present due to mid-plane stretching making the predications of linear theories erroneous.Most of the works reported in literature, have been mainly focused on the static and linear vibration behavior of microplates.Almost no work has been presented about the nonlinear behavior of microplates when actuated by large electrostatic loading or when undergoing large motion.Understanding such behaviors is fundamental to the development of the next generation microplates-based MEMS devices.The objective of this paper is to investigate the nonlinear forced vibration behavior of electrically actuated fully clamped microplates.Towards this, we develop a reduced order model based on the von Karman equations of the plate, in which all three equations of the plate motion are discretized using the Galerkin method.For validation purpose, we compare the static results computed by the reduced order model with the results calculated using the FEM software COMSOL [14].Further the reduced order model is employed to investigate the dynamic behavior of the plates under large vibration amplitude.

Problem Formulation
We consider a fully clamped rectangular microplate as shown in figure 1 and adopt the von-Karman equations of motion [15,16].We ignore the in-plane inertia since the in-plane natural frequencies are much higher than the transverse natural frequencies.Hence, the in-plane inertia has negligible effect on the transverse motion.Dropping the in plane inertia terms, the governing equations of the plate motion can be written as where , uvand w are the displacements along the , xy and z direction, respectively, 4  is the bi-harmonic operator, ,  is the mass density per unit area, c is the damping coefficient, and  and E are Poisson's ratio and modulus of elasticity, respectively.The first term on the right side of equation (3) i.e.
is the applied electric pressure in the transverse direction while ij N is the applied force per unit length on the edge perpendicular to th i axis in the th j direction.
For convenience, we introduce the non-dimensional variables (denoted by hats); Substituting equation (4) into equations ( 1)-( 3) and dropping the hats for convenience we get the following equations: The parameters appearing in equations ( 5)-( 7) are b a where

Reduced Order Model
To develop the reduced order model for the governing equations ( 5)-( 7), we assume the solution for the transverse and in-plane displacements as follows 1 ( ) ( , ) where ( , ) Since the linear eigenvalue problem governing the transversal linear modeshapes of the fully clamped plate cannot be solved analytically, we resort to the finite element method for this purpose.We use the commercial software COMSOL [14] to obtain the modeshapes of the plate.The in-plane displacement shape functions are also obtained by FEM while the plate is deflected by a uniform transverse pressure.


, and integrating over the plate domain, we get a system of differential algebraic equations (DAEs) for the time dependent coefficients.Equations ( 5) and ( 6) generate algebraic equations while equation ( 7) results in ordinary differential equations (ODEs).The resulting system of DAEs is solved for the unknown time dependent coefficients () Toward this, we solve equations ( 5) and ( 6) for () u qt and () v qt in terms of () i qt and then substitute the results into equation ( 7).This equation is then integrated in time using Runge Kutta method.These coefficients are substituted back into equations ( 9)-( 11) to get the displacements w , u and v .

Static Analysis
To calculate the static deflection of the microplate under a DC load, we drop the time derivatives in the reduced order model and the time dependent unknown coefficients () i qt, () u qt, and () v qtare replaced by constant coefficients i q , u q , and v q .This results in a system of nonlinear algebraic equations, which is numerically solved for i q , u q and v q .Then equation ( 9) is used to find the transversal deflection. .We note that the static results converge by retaining at least four transverse modes in the model.One can note that the deflection curve is limited by the pull-in instability, where the slope of the curve approaches infinity.
Next we validate the reduced order model by comparing its results with the results obtained by FEM software COMSOL [14] for different aspect ratios  .Figure 3 shows the maximum deflection max W at the center of the microplate calculated by both models.We use rectangular microplates of four different aspect ratios of 1

Dynamic Analysis
In this section we investigate the dynamic response of the square microplate in the neighborhood of primary, resonances of the fundamental mode.We analyze the dynamics of the microplate by generating frequency response curves.The Runge Kutta method is used to perform long time numerical integration to solve the system of DAEs.The stable steady solution is captured after making sure that the transient response is no longer contributing to the response.Figure 4 shows the time history of the dynamic response of the microplate when actuated at  To investigate the dynamic response at primary resonance, we fix 1 near primary resonance, which for the linear plate is near 36.Our choice of a constant value of Q means that we assumed negligible effect of squeeze-film damping, which reasonable assumption is assuming that the microplate is placed inside a vacuum chamber and is operated at reduced pressure.Otherwise, squeeze-film damping can have strong effect on the dynamics of microplates and needs to be modeled using Reynolds equation and a structural-fluidic model [17,18].The microplate exhibits strong hardening effect due to the cubic nonlinearity, which comes into play due to midplane stretching.The hysteresis in the curves is captured by performing forward and backward frequency sweeps. .Further we notice that there exist multiple stable solutions over some range of frequency and amplitude jumps from higher to lower or lower to higher values depending on the type of frequency sweep., respectively.A gap between the two stable solutions starts to emerge in the case of 0.4V ac V  by the forward and backward frequency sweeps.This gap might indicate that the microplate becomes unstable at that actuating voltage and pulls on the stationary electrode.This kind of instability is called dynamic pull-in instability, which usually occurs at a lower DC load superimposed to a harmonic load [17,[19][20][21].Another possibility exists, especially in the case of 0.4V ac V  , that this gap is created due to the numerical divergence of the time integration scheme, due to its inability to find suitable initial conditions that lead to

Conclusions
In this article we developed a reduced order model for the investigation of the static as well as the dynamic behavior of electrically actuated rectangular microplates.First convergence of the static results with the number of modeshapes retained in the ROM has been studied.We found that four modes are sufficient for convergence.The dynamic behavior of the microplate has been investigated near primary resonance using long time numerical integration.We captured the stable solutions using forward and backward frequency sweeps.The microplate actuated near primary resonance shows a strong hardening behavior due to the cubic nonlinearity, which comes into play due to mid-plane stretching.Increasing ac V further widens the gap between the two stable solutions captured by forward and backward frequency sweeps and pull-in instability occurs at a lower vibration amplitude.
for the in-plane displacements u and v , while ()

Fig. 2 :
Fig. 2: Convergence of the static response with the number of transverse modes retained in the reduced order model.Variation of the maximum non-dimensional deflection max W Results are shown for various values of dc V , until the pull in instability, for 1.5   .It shows excellent agreement among the results calculated by the ROM model and the FEM model.

Fig. 3 :
Fig. 3: Comparison of the maximum deflection max W at the center of the microplate, calculated by the reduced order with the results obtained from FEM for various values of dc V until the pull-in instability, for a rectangular microplate of aspect ratio 1.5   .
static pull-in voltage for the microplate is near 14 Volts.We investigate the nonlinear dynamic behavior of the microplate near primary resonance, a small harmonic ac V is superimposed to a

Fig. 5 :
Fig. 5: Maximum non-dimensional deflection max W at the center of the microplate against the actuating frequency  when actuated at 3V dc V  and various values of

Figure
Figure6shows the frequency response curves of the microplate when actuated at 7V

Table 1 :
Dimensional specifications of electrically actuated microplates used in the model validation.