Dynamic analysis of variable-geometry electrostatic microactuators

Abstract

This paper investigates the dynamic behavior of a microbeam-based electrostatic microactuator. The cross-section of the microbeam under consideration varies along its length. A mathematical model, accounting for the system nonlinearities due to mid-plane stretching and electrostatic forcing, is adopted and used to examine the microbeam dynamics. The differential quadrature method (DQM) and finite difference method (FDM) are used to discretize the partial–differential–integral equation and generate frequency-response curves for various microstructure geometries and different voltages. We show that the use of the DQM, with a few grid points, in conjunction with the FDM applied to the space derivatives and time derivatives, respectively, yields excellent convergence of the dynamic solutions. The stability of these solutions is examined using Floquet theory. Results are presented to display the dynamics and the effect of variable geometry on the frequency-response curves of the microstructure. We first demonstrate convergence of the DQM–FDM discretized dynamics model as the number of grid points is varied from 5 to 13, while the number of time steps in one time period is fixed at 100. The proposed DQM–FDM discretized dynamic model is then compared to recently reported models. We show that the shape of the frequency-response curves of the microbeam, excited near its first natural frequency, is very sensitive to the approximations employed in the construction of the model. Finally, we examine the effect of varying the gap size and the microbeam thickness and width on its frequency-response curves for hardening-type and softening-type behaviors.

Dynamic Analysis of Multilayers Based MEMS Resonators

“…To solve for the coupled equations of motion of both the lower and upper microbeams using the ROM, it is necessary to solve for the unknown functions ( ) and ( ) in (11) and then substituting them back into (9) and (10), which will give the deflection of both microbeams. For the method of multiple scales based perturbation technique, we start first by scaling the first-order coupled equations, (29) for both functions B 1 and B 2 .…”

“…Solving nonlinear dynamical behaviors for MEMS devices is fundamental, since it helps in accurately characterizing and designing them to obtain the desired features quickly and effectively. However, the resolution could be sometime cumbersome and many researchers have struggled to find an effective way to tackle this problem [9][10][11][12]. The key techniques in solving such nonlinear problems involve reducing the order of the partial differential governing equation, which sometime is difficult to be solved, into ordinary differential equations, which are much easier to deal with.…”

Comprehensive Analytical Approximations of the Pull-In Characteristics of an Electrostatically Actuated Nanobeam under the Influences of Intermolecular Forces

“…Numerous methods can be used to numerically solve the above nonlinear equation: such as the finite-element method [24], the finite-difference method [25], the shooting method [26,27], the differential-quadrature method [27], etc., which are all considered to be computationally expensive and in some cases unstable since some rely on initial guesses. In this investigation, we propose to use Galerkin-based reduced-order modeling that transforms the above nonlinear governing differential equation into a nonlinear algebraic equation system (considering only static DC load).…”

Structural Behavior of a Multi-Layer Based Microbeam Actuator

“…To solve the nonlinear differential equations governing the structural behavior of the above described MEMS actuators, various methods can be assumed such as the Finite-Element Method [26], the Finite-Difference Method [27], Differential-Quadrature Method [27], the Shooting Method [28,29], etc., which are considered to be computationally expensive and in some cases unstable since some rely on initial guesses. Another powerful technique is the so-called Galerkin expansion discretization which is mainly used to derive Reduced-Order Models (ROM) from distributed (continuous) systems.…”