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We study the influence of von Kármán nonlinearity, van der Waals force, and thermal stresses on pull-in instability and small vibrations of electrostatically actuated microplates. We use the Galerkin method to develop a tractable reduced-order model for electrostatically actuated clamped rectangular microplates in the presence of van der Waals forces and thermal stresses. More specifically, we reduce the governing two-dimensional nonlinear transient boundary-value problem to a single nonlinear ordinary differential equation. For the static problem, the pull-in voltage and the pull-in displacement are determined by solving a pair of nonlinear algebraic equations. The fundamental vibration frequency corresponding to a deflected configuration of the microplate is determined by solving a linear algebraic equation. The proposed reduced-order model allows for accurately estimating the combined effects of van der Waals force and thermal stresses on the pull-in voltage and the pull-in deflection profile with an extremely limited computational effort.

Microelectromechanical sensors are currently used in automotive electronics, medical equipment, smart portable electronics, hard disk drives, and computer peripherals [

An electrostatically actuated MEMS is generally comprised of a conductive deformable electrode suspended above a rigid grounded electrode [

For a wide class of electrostatic MEMS, the deformable electrode is a flat body whose thickness _{0}_{0} is the initial gap, the electrode can be regarded as a linear elastic membrane. The membrane approximation is valid for ℓ/

With the decrease in electrostatic MEMS dimensions from the micro to the nanoscale additional nanoscale surface forces, such as the Casimir force and the van der Waals force [

MEMS can be subjected to considerable temperature variations during sensing operation, such as in monitoring aircraft condition and distributed satellites communication, as well as during device packaging [

We consider rectangular microplates undergoing large displacements under the combined effect of electrostatic and nanoscale forces. We use the Galerkin method to develop a tractable reduced-order model for electrostatically actuated microplates. The reduced-order model is derived by taking a family of linearly independent kinematically admissible functions as basis functions for the transverse displacement and by decomposing the in-plane displacement vector as the sum of displacements for irrotational and isochoric waves in a 2-D medium. Basis functions for the transverse and the in-plane displacements are related through the nonlinear equation governing the plate in-plane motion. The governing equations of the reduced-order model are derived from the equation governing the transverse motion of the microplate. The model is specialized to the case where a single basis function for the transverse displacement is used to yield manageable solutions. In the static analysis, pull-in parameters are found by solving a system of two nonlinear algebraic equations for the transverse displacement amplitude and the load parameter. The eigenvalue problem corresponding to linear vibrations of the system about its statically deflected position is solved for the fundamental frequency. We show that the fundamental frequency goes to zero as pull-in conditions are approached. The pull-in parameters found from the eigenvalue analysis agree well with those derived from the static analysis. We investigate effects of nanoscale forces and thermal stresses on pull-in parameters and small vibrations of electrostatically actuated microplates.

The rest of the paper is organized as follows. In section 2, we present the governing nonlinear equations of motion for a von Kármán microplate under the simultaneous effects of thermal loading, electrostatic force, and nanoscale forces. We present expressions of the distributed loads due to either the Casimir or the van der Waals forces. In section 3, we introduce a reduced-order model for the considered device that is capable of accurately predicting its dynamics. The derivation of the reduced-order model follows a procedure typically used for studying deformations of thin two-dimensional structures. That is, in-plane inertial effects are neglected, and the resulting equation is solved for in-plane displacements in terms of transverse deflections which are then substituted in the equation governing the evolution of transverse deflection. Once transverse deflections have been computed, in-plane displacements can be found. In section 3, we also briefly outline the technique used to solve equations for the reduced order model. In section 4, we present our results, that include the pull-in parameters and the fundamental frequencies for rectangular microplates. We specialize our results to the case where nanoscale effects can be subsumed into the van der Waals force and we investigate the effect of the van der Waals force on the pull-in instability and the lowest frequency of the predeformed plate. Conclusions are summarized in section 5.

We consider a rectangular plate-like body of longer side ℓ and shorter side _{1}, _{2}) aligned with the longer and with the shorter sides, respectively. We assume that the initial gap _{0}_{0}

We use the Einstein summation convention, meaning that when an index variable appears twice in a single term we are summing over the range {1,2}; free index also span the range {1,2}. Furthermore, ^{3}^{2})) is the bending stiffness of the plate, _{i}_{e} is the Coulomb force, and _{s}

In the von Kármán plate theory, the components _{ij}

We assume that the strain tensor admits the additive decomposition

Substituting for _{ij}_{t}^{2}).

From an electrical point of view, the system depicted in _{0}_{e} of the Coulomb or the magnitude of the electrostatic force acting on the deformable electrode along its normal is given by the parallel plate approximation [_{0} is the non-dimensional transverse displacement, _{0} is the dielectric constant in vacuum, and _{0}

In ^{−19}J.

van der Waals and Casimir forces between parallel layered metallic surfaces have been extensively studied in the literature, see for example [^{−3} (van der Waals force) to the force distance dependence ^{−}^{4}

The effect of the Casimir force on pull-in parameters of von Kármán microplates has been studied in [_{s}_{vdW} in

We introduce the non-dimensional time _{1} and _{2} are nondimension-alized as _{1} = _{1}/ℓ, _{2} = _{2}

The non-dimensional parameters

We note that

We consider the boundary Γ of Ω to be clamped. The kinematic boundary conditions for a clamped edge are [

Initial conditions are not needed since we either study static deformations of the MEMS, or analyze frequencies of small vibrations around an electrostatically deformed configuration.

A closed-form solution of the initial-boundary-value problem defined in _{i}_{(}_{n}_{)} and _{(}_{p}_{)}_{i}_{(}_{n}_{)} and ξ_{(}_{p}_{)} are the corresponding amplitude parameters or equivalently the mode participation factors. Basis functions are collected into the _{i}

In [

Basis functions for the in-plane displacement can be determined by solving the following linear eigenvalue problem associated with

Therefore,

By integrating over the domain Ω and by applying Green's formulas to transform surface integrals into line integrals,

Additional boundary conditions are provided by

Therefore, basis functions for the in-plane displacement are given by
_{mn}

In order to express the coefficients _{(p)} is defined through

_{(p)} defined in

The reduced-order model is obtained by premultiplying both sides of

From _{mn}_{ij}

_{1}ζ_{2}ζ_{3}ζ^{3}_{e}(ζ)), and the van der Waals force (_{vdW}(ζ)). From

In what follows, we discuss two equivalent methods to extract static pull-in parameters. The first method is based on the solution of the static problem, while the second one is based on the study of small vibrations of the system around its static equilibrium configurations. The method based on the vibration analysis can be used to experimentally determine the pull-in voltage without potentially damaging the MEMS [

At the onset of instability the system's tangent stiffness

Therefore, at the pull-in instability, the reduced-order model satisfies

We solve the problem for λ = 0 to compute the critical value, say _{cr}, of the van der Waals force parameter. When _{cr}_{cr}_{cr} is given by

The effect of the van der Waals force on pull-in parameters λ_{pi} and ║_{PI}║_{∞} is investigated by solving _{cr}]. Here, ║ · ║_{∞} is defined as max_{(x1,x2)∈Ω} | · |. By solving _{PI}:

The lowest positive root of _{PI}. The corresponding non-dimensional pull-in voltage is thereby determined from

The buckling thermal stress parameter _{B}_{B} of the displacement parameter corresponding to buckling instability. The parameter _{B} is obtained as

We determine the lowest frequency of the deflected plate at a given solution (ζ, λ, _{0}

Since the tangent stiffness defined in

Integrals appearing in the governing equations of the reduced-order model, including

For the transverse displacement, we use the following kinematically admissible function in _{0} = 4.73004 is the lowest nonzero root of the transcendental equation cosh ϑ cos ϑ = 1, and the constant _{0} is chosen by normalizing

Since _{1}= 1/2 and x_{2} = _{(p)i}_{(}_{p}_{)} = 0 in _{(}_{p}_{)}_{i}

In [

_{cr}

We note that the critical parameter _{cr} increases rapidly as the plate aspect ratio

For the Casimir force, the function _{cr}

In _{cr}] for _{PI} decreases monotonically from its maximum value
_{cr}

The value _{cr}. represents the intersection of the curves with the horizontal axis. With an increase in _{cr}. For a MEMS made of a specified material, _{5} implies that
_{cr}. Thus, _{cr} provides a quantitative indication of the devices' size that can be safely fabricated. This means that reduced deflection ranges are allowable for devices having a large value of _{cr} since _{cr} is inversely proportional to

For _{cr} and _{∞} ⋍ 0.59, and of a rectangular plate with _{∞} ⋍ 0.59. In both cases thermal effects are discarded by selecting _{∞} the maximum magnitude of the van der Waals pressure for the rectangular plate is nearly an order of magnitude higher as compared to that for the square plate; note that _{vdW} in

In _{cr} for a square plate and a rectangular plate with _{cr} increases with decreasing temperature and decreases with increasing temperature. Indeed, from

In the absence of van der Waals force, that is, for _{PI} for a square plate, and a rectangular plate with

In _{0} of the deflected microplate versus A for _{0} corresponding to λ = 0; values of non-dimensional _{0} for _{0} equal 36.108 and 98.592 for the square and the rectangular plate with

The variation of_{0}/_{0} with A is non-mono tonic due to the combined effect of the strain hardening represented by _{3}ζ^{3} and the softening effect introduced by the Coulomb and the van der Waals forces. Indeed, from

Results reported in _{0} = 0 agrees with the λ_{PI}found from

In _{0} corresponding to λ = 0, _{0} = 0 corresponds to the buckling instability of the MEMS plate, and is found with the method explained in section 3.5. For _{B}

For _{0}/_{3}ζ^{3}) and the thermal softening (_{2}ζ). For the case studied, when

We have studied effects of thermal stress and nanoscale forces on pull-in instability and resonant behavior of electrostatically actuated microplates. Mechanical nonlinearities are accounted for by mod-eling the deformable microplate using the von Kármán plate theory. The thermal stress is modeled as a homogenous residual stress depending on the microplate temperature. Nanoscale surface forces are described using the van der Waals force. We have derived a simple and tractable mass spring single degree of freedom model for analyzing the behavior of the considered device. The reduced-order model is derived by using a single basis function for the transverse displacement and eight basis functions to describe strain hardening due to membrane stretching.

Results show that the pull-in voltage and the pull-in deflection are strongly affected by thermal stresses and the van der Waals force. More specifically, as the temperature of the MEMS increases the pull-in voltage decreases. Moreover, the van der Waals force becomes more relevant as the MEMS size is reduced and can potentially lead to the spontaneous collapse of the system in absence of applied voltage.

The authors would like to express their gratitude to the guest editor Dr. S. Mariani for his effort in coordinating this special issue of Sensors on “Modeling, Testing and Reliability Issues in MEMS Engineering”.

In the literature [

Sketch of an electrostatically actuated rectangular microplate.

Variation of _{cr} with the aspect ratio

Variation with

(a) λ_{PI} and (b) ‖_{PI}‖_{∞} versus

(a) λ_{PI} and (b) ‖_{PI} ‖_{∞} versus

For _{cr} and _{∞} ⋍ 0.59, and (b; rectangular plate with _{∞} ⋍ 0.59. Fringe plots of the van der Waals pressure are also displayed.

For _{cr} for (a) square plate and (b) rectangular plate with

For _{PI}for (a) square plate, and (b) rectangular plate with

Normalized fundamental frequency versus A for _{cr} (dashed curve),

Normalized fundamental frequency versus λ for _{cr} (dashed curve),

For the square plate, fundamental frequency versus

For the rectangular plate with

Values of _{0} in

| ||||
---|---|---|---|---|

_{cr} |
_{cr} | |||

1 | 36.1 | 34.2 | 98.6 | 93.4 |

4 | 36.1 | 34.2 | 98.6 | 93.5 |