An actuator comprised of a rigid substrate and two parallel clamped-clamped microbeams is modeled under the influence of electrostatic loading. The problem is considered under the context of nonlinear Euler's mechanics, where the actuating system is described by coupled integrodifferential equations with relevant boundary conditions. Galerkin-based discretization is utilized to obtain a reduced-order model, which is solved numerically. Actuators with different gap sizes between electrode and beams are investigated. The obtained results are compared to simulations gotten by the finite-element commercial software ANSYS.
One of the basic and most common MEMS devices is the parallel-plate electrostatic actuator. A clear advantage of the parallel-plate electrostatic actuators is their capability of generating high force. One drawback of these actuators, however, is the low deflection they can perform due to the gap size between the parallel plates and the induced pull-in instability caused by system nonlinearities. The main source of these nonlinearities is the fact that the electrostatic produced force is inversely proportional to the squared value of the gap distance between the two electrodes. Inclusive analyses of the pull-in instability can be found in published papers by Gupta et al. [
In order to put limitations on the instability domain, an intuitive solution is to decrease the gap distance between the parallel plates. Two additional actions can also help: (i) reduction of the rigidity of the parallel plates and (ii) increase the areas of the electrostatic surfaces. There have been several attempts to increase the stable range of travel of parallel-plate actuators. Early attempts include the use of curved electrodes [
A unique approach of intensifying the electrostatic force and increasing the out-of-plane deflection was to utilize more than one parallel-plate electrostatic device built in parallel fashion. Abbaspour-Sani and Afrang [
Inspired by these works, the current work is intended to provide an analytical solution for the deflection of a parallel double actuator comprised of a fixed electrostatic substrate and two electrostatic parallel layers. A continuum model based on Euler’s beam is used to describe the two clamped-clamped parallel microbeams under the influence of applied voltage. The presented model includes the effect of geometric nonlinearity due to midplane stretching. The quasi-static response of a single actuator (made of a substrate and a single electrostatic layer) is compared with that of double actuators (made of a substrate and two electrostatic layers). Then, a finite-element numerical solution is obtained by utilizing ANSYS, and a comparison between both solutions is presented.
In this section, we present the structural-electrical model of single-microbeam as well as double-microbeam actuators under electrostatic loading due to parallel-plate effect. Figure
Schematic of an electrostatically actuated clamped-clamped single microbeam.
Based on the above assumptions, the equation of motion and associated boundary conditions of the microbeam shown in Figure
Schematic of electrostatically actuated clamped-clamped doublemicrobeams.
Figure
For convenience, we introduce the following nondimensional variables:
In nondimensional forms, (
Equations (
After substituting (
The same procedure described in the above paragraph is applied to system of two equations describing the deflection of each microbeam in the double-microbeam actuator system. First, (
At this point, we substitute (
Equations (
The commercial software ANSYS
ANSYS finite-element model of double-microbeam actuator.
Representative cases are presented in the sequel for a single-microbeam actuator and a double-microbeam actuator. Let us assume that each microbeam has the geometric properties of
In order to ensure the convergence of the ROM, we calculate the maximum static deflection of the upper microbeam (
Variation of the maximum static deflection of the upper beam of the double-microbeam actuator for different applied DC voltages.
Next, we move on to compare the maximum static defection of the single-microbeam actuator with that of the double-microbeam actuator. Figure
Comparison of the static response of the single-microbeam actuator and the upper beam of the double-microbeam actuator. The pull-in voltage values for single and double-microbeams are 18 and 168 Volts, respectively.
The analytical results obtained from the reduced-order model are then compared with the results obtained from finite element ANSYS simulation. Figure
Comparison between ROM results and ANSYS FEM results.
Hence, ANSYS can be utilized to predict response of a triple-microbeams actuator. Keeping the same total gap thickness, Figure
Maximum deflection versus applied DC voltage for different number of beam-electrode configuration.
Finally, we investigate the effect varying the gap thicknesses on the maximum static deflection of the double-microbeam- actuator. Figure
Effect of the gap sizes on the static response of the single-microbeam and double-microbeam actuators.
The paper presented a MEMS actuator comprising of an electrostatic substrate-single microbeam and substrate-double microbeams. Different arrangements of clamped-clamped microbeam-electrode combinations were presented in this study. Reduced-order model based on Galerkin method and FEM analysis were used to calculate maximum transverse deflections and obtain pull-in voltages. It is found that it is possible to stimulate the response of the system by adding more layers of micro-beam-electrode combinations. The effects of spacing between microbeams and/or electrodes are found to be of a great importance on the transverse response of the system as well as on the pull-in voltage value. The investigation provides a ground for implementing the technique of using multiple layers of microbeam-electrode combination whenever larger response is desired.
The authors greatly appreciate the support received from King Fahd University of Petroleum and Minerals (KFUPM) through its Deanship of Scientific Research (DSR).