Mohammad I. Younis

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.

A STUDY OF THE NONLINEAR RESPONSE OF A RESONANT MICRO-BEAM TO AN ELECTRIC ACTUATION

An investigation into the response of a resonant microbeam to anelectric actuation is presented. A nonlinear model is used to accountfor the mid-plane stretching, a DC electrostatic force, and an ACharmonic force. Design parameters are included in the model by lumpingthem into nondimensional parameters. A perturbation method, the methodof multiple scales, is used to obtain two first-order nonlinearordinary-differential equations that describe the modulation of theamplitude and phase of the response and its stability. The model and theresults obtained by the perturbation analysis are validated by comparingthem with published experimental results. The case of three-to-oneinternal resonance is treated.The effect of the design parameters on the dynamic responses isdiscussed. The results show that increasing the axial force improves thelinear characteristics of the resonance frequency and decreases theundesirable frequency shift produced by the nonlinearities. In contrast,increasing the mid-plane stretching has the reverse effect. Moreover,the DC electrostatic load is found to affect the qualitative andquantitative nature of the frequency-response curves, resulting ineither a softening or a hardening behavior. The results also show thatan inaccurate representation of the system nonlinearities may lead to anerroneous prediction of the frequency response.

A NEW APPROACH TO THE MODELING AND SIMULATION OF FLEXIBLE MICROSTRUCTURES UNDER THE EFFECT OF SQUEEZE-FILM DAMPING

We present a new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping. Our approach utilizes the compressible Reynolds equation coupled with the equation governing the plate deflection. The model accounts for the electrostatic forcing of the capacitor airgap, the restoring force of the microplate and the applied in-plane loads. It also accounts for the slip condition of the flow at very low pressures. Perturbation methods are used to derive an analytical expression for the pressure distribution in terms of the structural mode shapes. This expression is substituted into the plate equation, which is solved in turn using a finite-element method for the structural mode shapes, the pressure distributions, the natural frequencies and the quality factors. We apply the new approach to a variety of rectangular and circular plates and present the final expressions for the pressure distributions and quality factors. Our theoretically calculated quality factors are in excellent agreement with available experimental data and hence our methodology can be used to simulate accurately the dynamics of flexible microstructures and predict their quality factors under a wide range of gas pressures. Because the pressure distribution is related analytically to the deflection, the dimension of the coupled structural-fluidic problem and hence the number of global variables needed to describe the dynamics of the system is reduced considerably. Consequently, the new approach can be significant to the development of computationally efficient CAD tools for microelectromechanical systems.

Dynamics of MEMS resonators under superharmonic and subharmonic excitations

We present an analysis and simulations for the dynamics of electrically actuated microbeams under secondary resonance excitations. The presented model and methodology enable simulation of the transient and steady-state dynamics of microbeams undergoing small or large motions. The microbeams are excited by a dc electrostatic force and an ac harmonic force with a frequency tuned near twice their fundamental natural frequencies (subharmonic excitation of order one-half) or half their fundamental natural frequencies (superharmonic excitation of order two). In the case of superharmonic excitation, we present results showing the effect of varying the dc bias, the damping and the ac excitation amplitude on the frequency–response curves. In the case of subharmonic excitation, we show that, once the subharmonic resonance is activated, all frequency–response curves reach pull-in, regardless of the magnitude of the ac forcing. We conclude that the quality factor has a limited influence on the frequency response in this case. This result and the fact that the frequency–response curves have very steep passband-to-stopband transitions make the combination of a dc voltage and a subharmonic excitation of order one-half a promising candidate for designing improved high-sensitive RF MEMS filters. For both excitation methods, we show that the dynamic pull-in instability can occur at an electric load much lower than a purely dc voltage and of the same order of magnitude as that in the case of primary-resonance excitation.

Modeling and simulations of thermoelastic damping in microplates

We present a model and analytical expressions for the quality factors of microplates of general shapes and boundary conditions due to thermoelastic damping. We solve the heat equation for the thermal flow across the microplate, and hence decouple the thermal equation from the plate equation. We utilize a perturbation method to derive an analytical expression for the quality factor of a microplate under electrostatic loading and residual stresses in terms of its structural mode shapes. For the special case of no electrostatic and in-plane loadings, we derive a simple analytical expression for the quality factor, which is independent of the mode shapes. We verify the model by specializing it to a microbeam and comparing the resulting quality factor to that obtained using microbeam models in the literature. We present several results for various modes of rectangular microplates with various boundary conditions.

On the nonlinear resonances and dynamic pull-in of electrostatically actuated resonators

We present modeling, analysis and experimental investigation for nonlinear resonances and the dynamic pull-in instability in electrostatically actuated resonators. These phenomena are induced by exciting a microstructure with nonlinear forcing composed of a dc parallel-plate electrostatic load superimposed on an ac harmonic load. Nonlinear phenomena are investigated experimentally and theoretically including primary resonance, superharmonic and subharmonic resonances, dynamic pull-in and the escape-from-potential-well phenomenon. As a case study, a capacitive sensor made up of two cantilever beams with a proof mass attached to their tips is studied. A nonlinear spring‐mass‐damper model is utilized accounting for squeeze-film damping and the parallel-plate electrostatic force. Long-time integration and a global dynamic analysis are conducted using a finite-difference method combined with the Floquet theory to capture periodic orbits and analyze their stability. The domains of attraction (basins of attraction) for data points on the frequency‐response curve are calculated numerically. Dover cliff integrity curves are calculated and the erosion of the safe basin of attraction is investigated as the frequency of excitation is swept passing primary resonance and dynamic pull-in. Conclusions are presented regarding the safety and integrity of MEMS resonators based on the simulated basin of attraction and the observed experimental data. (Some figures in this article are in colour only in the electronic version)

Nonlinear Dynamics of Electrically Actuated Carbon Nanotube Resonators

This work presents an investigation of the nonlinear dynamics of carbon nanotubes (CNTs) when actuated by a dc load superimposed to an ac harmonic load. Cantilevered and clamped-clamped CNTs are studied. The carbon nanotube is described by an Euler-Bernoulli beam model that accounts for the geometric nonlinearity and the nonlinear electrostatic force. A reduced-order model based on the Galerkin method is developed and utilized to simulate the static and dynamic responses of the carbon nanotube. The free-vibration problem is solved using both the reduced-order model and by solving directly the coupled in-plane and out-of-plane boundary-value problems governing the motion of the nanotube. Comparison of the results generated by these two methods to published data of a more complicated molecular dynamics model shows good agreement. Dynamic analysis is conducted to explore the nonlinear oscillation of the carbon nanotube near its fundamental natural frequency (primary-resonance) and near one-half, twice, and three times its natural frequency (secondary-resonances). The nonlinear analysis is carried out using a shooting technique to capture periodic orbits combined with the Floquet theory to analyze their stability. The nonlinear resonance frequency of the CNTs is calculated as a function of the ac load. Subharmonic-resonances are found to be activated over a wide range of frequencies, which is a unique property of CNTs. The results show that these resonances can lead to complex nonlinear dynamics phenomena, such as hysteresis, dynamic pull-in, hardening and softening behaviors, and frequency bands with an inevitable escape from a potential well.

Investigation of the response of microstructures under the combined effect of mechanical shock and electrostatic forces.

There is strong experimental evidence for the existence of strange modes of failure of microelectromechanical systems (MEMS) devices under mechanical shock and impact. Such failures have not been explained with conventional models of MEMS. These failures are characterized by overlaps between moving microstructures and stationary electrodes, which cause electrical shorts. This work presents modeling and simulation of MEMS devices under the combination of shock loads and electrostatic actuation, which sheds light on the influence of these forces on the pull-in instability. Our results indicate that the reported strange failures can be attributed to early dynamic pull-in instability. The results show that the combination of a shock load and an electrostatic actuation makes the instability threshold much lower than the threshold predicted, considering the effect of shock alone or electrostatic actuation alone. In this work, a single-degree-of-freedom model is utilized to investigate the effect of the shock-electrostatic interaction on the response of MEMS devices. Then, a reduced-order model is used to demonstrate the effect of this interaction on MEMS devices employing cantilever and clamped-clamped microbeams. The results of the reduced-order model are verified by comparing with finite-element predictions. It is shown that the shock-electrostatic interaction can be used to design smart MEMS switches triggered at a predetermined level of shock and acceleration.

An Experimental and Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated Electrostatically

We present experimental and theoretical investigations of dynamic pull-in of electrostatically actuated resonators. Several experimental data are presented, showing regimes of ac forcing amplitude versus ac frequency, where a resonator is forced to pull in if operated within these regimes. Results are shown for primary and secondary resonance excitations. The influences of the initial conditions of the system, the ac excitation amplitude, the ac frequency, the excitation type, and the sweeping type are investigated. A shooting technique to find periodic motions and a basin-of-attraction analysis are used to predict the limits of the pull-in bands. When compared with the experimental data, the results have shown that the pull-in limits coincide with 30%-40% erosion lines of the safe basin in the case of primary resonance and 5%-15% erosion lines of the safe basin in the case of subharmonic resonance. Bifurcation diagrams have been constructed, which designers can use to establish factors of safety to reliably operate microelectromechanical-systems resonators away from pull-in bands and the danger of pull-in, depending on the expected disturbances and noise in the systems.