We model and analyze the deflections and motions of a shaped microbeam in a capacitive-based MEMS device. The model accounts for the system nonlinearities including mid-plane stretching and electrostatic forcing. The differential quadrature method (DQM) is used to discretize the microbeam partial differential equation. It is shown that the use of 11 grid points in the DQM is sufficient to capture the response of the device. It is also observed that, unlike the shooting methods, DQM does not face the problems of system differential equations stiffness and solution sensitivity to the initial guess. The static response to a dc voltage is first determined to investigate the influence of varying the geometric parameters of the device on the range of travel and pull-in voltage. Analytical expressions approximating the range of travel and pull-in voltage, as functions of the capacitor gap size and microbeam width and thickness, are derived. Symmetric and asymmetric spatial distributions of these parameters are considered. For symmetric distribution, an increase (decrease) in the beam width and/or thickness at the middle with respect to those at the endpoints results in an increase (decrease) in the pull-in voltage and a decrease (increase) in the range of travel. An increase (decrease) in the gap size at the middle with respect to those at the endpoints results in an increase (decrease) in the pull-in voltage and an insignificant effect on the range of travel. The dynamic response of the microbeam to a dc voltage is also determined for various distributions of the microbeam width and thickness and the gap size. It is shown that decreasing the microbeam thickness at the middle is the most effective method to reduce the pull-in time.

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In this paper, we consider the plane problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half-space, when the two bodies are pressed together. The graded layer is modeled as a nonhomogeneous medium with an isotropic stress-strain law and over a certain segment of its top surface is subjected to normal tractions while the rest of this surface is free of tractions. Since the contact between the two bodies is assumed to be frictionless, then only compressive normal tractions can be transmitted in the contact area. Using integral transforms, the plane elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact half-length. The global equilibrium condition of the layer is supplemented to solve the problem. The singular integral equation is solved numerically using Chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact half-length that satisfies the global equilibrium condition. The main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact.

We use a discretization technique that combines the differential quadrature method (DQM) and the finite difference method (FDM) for the space and time, respectively, to study the dynamic behavior of a microbeam-based electrostatic microactuator. The adopted mathematical model based on the Euler— Bernoulli beam theory accounts for the system nonlinearities due to mid-plane stretching and electrostatic force. The nonlinear algebraic system obtained by the DQM—FDM is used to investigate the limit-cycle solutions of the microactuator. The stability of these solutions is ascertained using Floquet theory and/or long-time integration. The method is applied for large excitation amplitudes and large quality factors for primary and secondary resonances of the first mode in case of hardening-type and softening-type behaviors. We show that the combined DQM—FDM technique improves convergence of the dynamic solutions. We identify primary, subharmonic, and superharmonic resonances of the microactuator. We observe the occurrence of dynamic pull-in due to subharmonic and superharmonic resonances as the excitation amplitude is increased. Simultaneous resonances of the first and higher modes are identified for large orbits in both primary and secondary resonances.

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