Eihab M. Abdel-Rahman

A reduced-order model for electrically actuated microbeam-based MEMS

We present an analytical approach and a reduced-order model (macromodel) to investigate the behavior of electrically actuated microbeam-based MEMS. The macromodel provides an effective and accurate design tool for this class of MEMS devices. The macromodel is obtained by discretizing the distributed-parameter system using a Galerkin procedure into a finite-degree-of-freedom system consisting of ordinary-differential equations in time. The macromodel accounts for moderately large deflections, dynamic loads, and the coupling between the mechanical and electrical forces. It accounts for linear and nonlinear elastic restoring forces and the nonlinear electric forces generated by the capacitors. A new technique is developed to represent the electric force in the equations of motion. The new approach allows the use of few linear-undamped mode shapes of a microbeam in its straight position as basis functions in a Galerkin procedure. The macromodel is validated by comparing its results with experimental results and finite-element solutions available in the literature. Our approach shows attractive features compared to finite-element softwares used in the literature. It is robust over the whole device operation range up to the instability limit of the device (i.e., pull-in). Moreover, it has low computational cost and allows for an easier understanding of the influence of the various design parameters. As a result, it can be of significant benefit to the development of MEMS design software.

Characterization of the mechanical behavior of an electrically actuated microbeam

We present a nonlinear model of electrically actuated microbeams accounting for the electrostatic forcing of the air gap capacitor, the restoring force of the microbeam and the axial load applied to the microbeam. The boundary-value problem describing the static deflection of the microbeam under the electrostatic force due to a dc polarization voltage is solved numerically. The eigenvalue problem describing the vibration of the microbeam around its statically deflected position is solved numerically for the natural frequencies and mode shapes. Comparison of results generated by our model to the experimental results shows excellent agreement, thus verifying the model. Our results show that failure to account for mid-plane stretching in the microbeam restoring force leads to an underestimation of the stability limits. It also shows that the ratio of the width of the air gap to the microbeam thickness can be tuned to extend the domain of the linear relationship between the dc polarization voltage and the fundamental natural frequency. This fact and the ability of the nonlinear model to accurately predict the natural frequencies for any dc polarization voltage allow designers to use a wider range of dc polarization voltages in resonators.

Dynamics and Control of Cranes: A Review:

We review crane models available in the literature, classify them, and discuss their applications and limitations. A generalized formulation of the most widely used crane model is analyzed using the method of multiple scales. We also review crane control strategies in the literature, classify them, and discuss their applications and limitations. In conclusion, we recommend appropriate models and control criteria for various crane applications and suggest directions for further work.

A wideband vibration-based energy harvester

We present a new architecture for wideband vibration-based micro-power generators (MPGs). It replaces a linear oscillator with a piecewise-linear oscillator as the energy harvesting element of the MPG. A prototype of an electromagnetic MPG designed accordingly is analyzed analytically, numerically and experimentally. We find that the new architecture increases the bandwidth of the MPG during a frequency up-sweep, while maintaining the same bandwidth in a down-sweep. Closed-form expressions for the response of the new MPG as well as the up-sweep bandwidth are presented and validated experimentally. Simulations show that under random-frequency excitations, the new MPG collects more energy than the traditional MPG.

Secondary resonances of electrically actuated resonant microsensors

We investigate the response of a microbeam-based resonant sensor to superharmonic and subharmonic electric actuations using a model that incorporates the nonlinearities associated with moderately large displacements and electric forces. The method of multiple scales is used, in each case, to obtain two first-order nonlinear ordinary-differential equations that describe the modulation of the amplitude and phase of the response and its stability. We present typical frequency–response and force–response curves demonstrating, in both cases, the coexistence of multivalued solutions. The solution corresponding to a superharmonic excitation consists of three branches, which meet at two saddle-node bifurcation points. The solution corresponding to a subharmonic excitation consists of two branches meeting a branch of trivial solutions at two pitchfork bifurcation points. One of these bifurcation points is supercritical and the other is subcritical. The results provide an analytical tool to predict the microsensor response to superharmonic and subharmonic excitations, specifically the locations of sudden jumps and regions of hysteretic behavior, thereby enabling designers to safely use these frequencies as measurement signals. They also allow designers to examine the impact of various design parameters on the device behavior.

A Reduced-order Model for Electrically Actuated Microplates

We present a reduced-order model for electrically actuated microplate-based MEMS. The model accounts for the electric force nonlinearity and the mid-plane stretching of the plate. The linear undamped vibration modes are found numerically using the hierarchical finite-element method. These mode shapes are used in a Galerkin approximation to reduce the partial-differential equations of motion and associated boundary conditions into a finite-dimensional system of nonlinearly coupled second-order ordinary-differential equations. The model is validated by comparing its results with those obtained experimentally and those obtained by solving the distributed-parameter system. The model is used to calculate the deflection of the microplate under dc voltages and study the pull-in phenomenon. The natural frequencies and mode shapes around these deflected positions of the microplate are calculated by solving the linear eigenvalue problem. The effects of various design parameters on both the static and dynamic characteristics of microplates are studied. The reduced-order model provides an effective and accurate design tool, useful in design optimization and determination of the stable operation range of MEMS devices.

Modeling and design of variable-geometry electrostatic microactuators

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.

A two-dimensional dynamic anatomical model of the human knee joint.

The objective of this study is to develop a two-dimensional dynamic model of the knee joint to simulate its response under sudden impact. The knee joint is modeled as two rigid bodies, representing a fixed femur and a moving tibia, connected by 10 nonlinear springs representing the different fibers of the anterior and posterior cruciate ligaments, the medial and lateral collateral ligaments, and the posterior part of the capsule. In the analysis, the joint profiles were represented by polynomials. Model equations include three nonlinear differential equations of motion and three nonlinear algebraic equations representing the geometric constraints. A single point contact was assumed to exist at all times. Numerical solutions were obtained by applying Newmark constant-average-acceleration scheme of differential approximation to transform the motion equations into a set of nonlinear simultaneous algebraic equations. The equations reduced thus to six nonlinear algebraic equations in six unknowns. The Newton-Raphson iteration technique was then used to obtain the solution. Knee response was determined under sudden rectangular pulsing posterior forces applied to the tibia and having different amplitudes and durations. The results indicate that increasing pulse amplitude and/or duration produced a decrease in the magnitude of the tibio-femoral contact force, indicating thus a reduction in the joint stiffness.(ABSTRACT TRUNCATED AT 250 WORDS)

A Design Procedure for Wideband Micropower Generators

We developed a design procedure for wideband electromagnetic micropower generators (WMPGs) based on piecewise-linear oscillators. We find that the dominant factors in the performance of this class of WMPGs are the stiffness ratio of the oscillator and the velocity of the moving structure at the point of impact with the stopper. We also find that designing these WMPGs requires additional steps beyond those required in the design of regular MPGs. The additional steps match the output power and bandwidth of the WMPG to the probability density function of environmental vibrations. While these steps add complexity to the design of WMPGs, they are shown to significantly increase harvested energy.

Dynamic analysis of variable-geometry electrostatic microactuators

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.