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.

We develop a mathematical model for a resonant gas sensor made up of an microplate electrostatically actuated and attached to the end of a cantilever microbeam. The model considers the microbeam as a continuous medium, the plate as a rigid body, and the electrostatic force as a nonlinear function of the displacement and the voltage applied underneath the microplate. We derive closed-form solutions to the static and eigenvalue problems associated with the microsystem. The Galerkin method is used to discretize the distributed-parameter model and, thus, approximate it by a set of nonlinear ordinarydifferential equations that describe the microsystem dynamics. By comparing the exact solution to that associated with the reduced-order model, we show that using the first mode shape alone is sufficient to approximate the static behavior. We employ the Finite Difference Method (FDM) to discretize the orbits of motion and solve the resulting nonlinear algebraic equations for the limit cycles. The stability of these cycles is determined by combining the FDM discretization with Floquet theory. We investigate the basin of attraction of bounded motion for two cases: unforced and damped, and forced and damped systems. In order to detect the lower limit of the forcing at which homoclinic points appear, we conduct a Melnikov analysis. We show the presence of a homoclinic point for a loading case and hence entanglement of the stable and unstable manifolds and non-smoothness of the boundary of the basin of attraction of bounded motion.

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.

We investigate the dynamics and global stability of a beam-based electrostatic microactuator, which is modeled as a first-order approximation of a reduced-order model (ROM) derived using the differential quadrature method (DQM). We show that the ROM dynamics is qualitatively similar to that of a higher-order approximation. We simulate the occurrence of dynamic pull-in for excitations near the first primary resonance using the finite difference method (FDM) and long-time integration. Limit-cycle solutions are obtained using the FDM, the generated frequency- and force-response curves exhibit cyclic-fold, saddle-node, and period-doubling bifurcations. We verify that symmetry breaking is not likely to occur because the orbit is already asymmetric. We identify the basin of attraction of bounded motions using various approximation levels. The simulations reveal that the erosion of the basin of attraction depends heavily on the amplitude and frequency of the AC voltage. We show that smoothness of the boundary of the basin of attraction can be lost and replaced by fractal tongues, which dramatically increase the sensitivity of the microbeam to initial conditions. According to these simulations, the locations of the two fixed points are likely to be disturbed.

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.

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In this paper, we propose the design of an ohmic contact RF microswitch with low voltage actuation, where the upper and lower microplates are displaceable. We develop a mathematical model for the RF microswitch made up of two electrostatically actuated microplates; each microplate is attached to the end of a microcantilever. We assume that the microbeams are flexible and that the microplates are rigid. The electrostatic force applied between the two microplates is a nonlinear function of the displacements and applied voltage. We formulate and solve the static and eigenvalue problems associated with the proposed microsystem. We also examine the dynamic behavior of the microswitch by calculating the limitcycle solutions. We discretize the equations of motion by considering the first few dominant modes in the microsystem dynamics. We show that only two modes are sufficient to accurately simulate the response of the H. Samaali ( ) · S. Choura · M. Masmoudi

A novel control strategy for the simultaneous suppression and confinement of vibrations in flexible structures is proposed. The key idea is to alter the original modes by appropriate feedback forces to allow parts of a flexible structure to reach their steady states at fast rates. It has been demonstrated that the convergence of these parts to zero is improved at the expense of slowing down the settling of the remaining parts to their steady states. The proposed control strategy can be applied in the rapid removal of vibration energy in sensitive parts of a flexible structure for safety or performance reasons. Such parts include communication antennas in light space structures and payloads of flexible robot manipulators. In order to gain insight into the viability of the proposed control strategy, the latter is compared with a classical feedback control strategy under the same input bound constraint. It has been shown that the new control strategy brings the sensitive parts to rest with approximately 50 percent reduction in time.