Fadi M. Alsaleem

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)

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.

Nonlinear Dynamics of MEMS Arches Under Harmonic Electrostatic Actuation

We present an investigation of the nonlinear dynamics of clamped-clamped micromachined arches when actuated by a dc electrostatic load superimposed on an ac harmonic load. The Galerkin method is used to discretize the distributed-parameter model of a shallow arch to obtain a reduced-order model. The static response of the arch due to a dc load actuation is simulated, and the results are validated by comparing them to experimental data. The dynamic response of the arch to a combined dc load and ac harmonic load is studied when excited near its fundamental natural frequency, twice its fundamental natural frequency, and near other higher harmonic modes. The results show a variety of interesting nonlinear phenomena, such as hysteresis, softening behavior, dynamic snap-through, and dynamic pull-in. The results are also shown demonstrating the potential to use microelectromechanical systems (MEMS) arches as bandpass filters and low-powered switches. An experimental work is conducted to test arches realized of curved polysilicon microbeams when excited by dc and ac loads. Experimental data are shown for the softening behavior and the dynamic pull-in of the curved microbeams.

Exploration of New Concepts for Mass Detection in Electrostatically-Actuated Structures Based on Nonlinear Phenomena

This study presents an effort to explore the exploitation of dynamic instabilities and bifurcations in micro-electro-mechanical systems to realize novel methods and functionalities for mass sensing and detection. These instabilities are induced by exciting a microstructure with a nonlinear forcing composed of a dc parallel-plate electrostatic load and an ac harmonic load. The frequency of the ac load is tuned to be near the fundamental natural frequency of the structure (primary resonance) or its multiples (subharmonic resonance). For each excitation method, local bifurcations, such as saddle-node and pitchfork, and global bifurcations, such as the escape phenomenon, may occur. This work aims to explore the utilization of these bifurcations to design novel mass sensors and switches of improved characteristics. One explored concept of a device is a switch triggered by mass threshold. The basic idea of this device is based on the phenomenon of escape from a potential well. This device has the potential of serving as a smart switch that combines the functions of two devices: a sensitive gas/mass sensor and an electromechanical switch. The switch can send a strong electrical signal as a sign of mass detection, which can be used to actuate an alarming system or to activate a defensive or a security system. A second type of explored devices is a mass sensor of amplified response. The basic principle of this device is based on the jump phenomena encountered in pitchfork bifurcations during mass detection. This leads to an amplified response of the excited structure making the sensor more sensitive and its signal easier to be measured. As case studies, these device concepts are first demonstrated by simulations on clamped-clamped and cantilever microbeams. Results are presented using long-time integration for the equations of motion of a reduced-order model. An experimental case study of a capacitive sensor is presented illustrating the proposed concepts. It is concluded that exciting a microstructure at twice its fundamental natural frequency produces the most promising results for mass sensing and detection.

Characterization of the performance of capacitive switches activated by mechanical shock

This paper presents experimental and theoretical investigation of a new concept of switches (triggers) that are actuated at or beyond a specific level of mechanical shock or acceleration. The principle of operation of the switches is based on dynamic pull-in instability induced by the combined interaction between electrostatic and mechanical shock forces. These switches can be tuned to be activated at various shock and acceleration thresholds by adjusting the DC voltage bias. Two commercial off-the-shelf capacitive accelerometers operating in air are tested under mechanical shock and electrostatic loading. A single-degree-of-freedom model accounting for squeeze-film damping, electrostatic forces, and mechanical shock is utilized for the theoretical investigation. Good agreement is found between simulation results and experimental data. Our results indicate that designing these new switches to respond quasi-statically to mechanical shock makes them robust against variations in shock shape and duration. More importantly, quasi-static operation makes the switches insensitive to variations in damping conditions. This can be promising to lower the cost of packaging for these switches since they can operate in atmospheric pressure with no hermetic sealing or costly package required.

Stabilization of electrostatic MEMS resonators using a delayed feedback controller

We present a study for the stabilization of a MEMS resonator actuated with DC and AC voltages using a delayed feedback controller. We show that the delayed feedback controller, with a careful selection of its parameters, can be used to stabilize an originally unstable resonator operating in the dynamic pull-in frequency band. Also, the controller is shown to enhance the stability of the resonator near pull-in, where it experiences a strong fractal behavior. In both cases, the controller shows superior performance in rejecting disturbances. Experimental and theoretical results are presented to demonstrate the capability of the feedback controller to stabilize the performance of a capacitive resonator. Good agreement between simulation and experiment is demonstrated.

A Study for the Effect of the PCB Motion on the Dynamics of MEMS Devices Under Mechanical Shock

We present a theoretical and experimental investigation into the effect of the motion of a printed circuit board (PCB) on the response of microelectromechanical systems (MEMS) devices to shock loading. For the theoretical part, a 2-DOF model is used, where the first degree of freedom accounts for the PCB. The second degree of freedom represents the motion of the MEMS microstructure. Low-g acceleration pulses are applied to the MEMS-PCB assembly base to simulate shock pulses generated from a drop-table test. Simulation data are presented to show the effects of the natural frequency of the PCB, the natural frequency of the microstructure, and the shock pulse duration. Universal 3-D spectra representing the effect of these parameters are presented. It is found that neglecting the PCB effect on the design of MEMS devices under shock loads can lead to undesirable motion of their microstructures. The effects of electrostatic force and squeeze film damping are investigated. It is found that the amplification of motion due to the PCB can cause early pull-in instability for MEMS devices implementing electrostatic forces. The effect of higher order modes of a microbeam is studied through a continuous beam model coupled with a lumped model of the PCB. The limitations of the 2-DOF model are discussed. An experimental investigation is conducted to verify the theoretical results using a capacitive accelerometer. Experimental data for the response of the accelerometer while it is mounted on two representative PCBs due to different low-g shock conditions are shown.

An Investigation Into the Effect of the PCB Motion on the Dynamic Response of MEMS Devices Under Mechanical Shock Loads

We present an investigation into the effect of the motion of a printed circuit board (PCB) on the response of a microelectromechanical system (MEMS) device to shock loads. A two-degrees-of-freedom model is used to model the motion of the PCB and the microstructure, which can be a beam or a plate. The mechanical shock is represented as a single point force impacting the PCB. The effects of the fundamental natural frequency of the PCB, damping, shock pulse duration, electrostatic force, and their interactions are investigated. It is found that neglecting the PCB effect on the modeling of MEMS under shock loads can lead to erroneous predictions of the microstructure motion. Further, contradictory to what is mentioned in literature that a PCB, as a worst-case scenario, transfers the shock pulse to the microstructure without significantly altering its shape or intensity, we show that a poor design of the PCB or the MEMS package may result in severe amplification of the shock effect. This amplification can cause early pull-in instability for MEMS devices employing electrostatic forces.

Integrity Analysis of Electrically Actuated Resonators With Delayed Feedback Controller

In this work, we investigate the stability and integrity of parallel-plate microelectromechanical systems resonators using a delayed feedback controller. Two case studies are investigated: a capacitive sensor made of cantilever beams with a proof mass at their tip and a clamped-clamped microbeam. Dover-cliff integrity curves and basin-of-attraction analysis are used for the stability assessment of the frequency response of the resonators for several scenarios of positive and negative gain in the controller. It is found that in the case of a positive gain, a velocity or a displacement feedback controller can be used to effectively enhance the stability of the resonators. This is confirmed by an increase in the area of the basin of attraction of the resonator and in shifting the Dover-cliff curve to higher values. On the other hand, it is shown that a negative gain can significantly weaken the stability and integrity of the resonators. This can be of useful use in MEMS for actuation applications, such as in the case of capacitive switches, to lower the activation voltage of these devices and to ensure their trigger under all initial conditions.

Dynamic response of an electrostatically actuated microbeam to drop-table test

In this work, we present a theoretical and experimental investigation into the response of an electrostatically actuated microbeam when subjected to drop-table test. For the theoretical part, a reduced-order model based on an Euler-Bernoulli beam model is utilized. The model accounts for the electrostatic bias on the microbeam and the shock pulse of the drop-table test. Simulation results are presented showing the combined effect of electrostatic force and mechanical shock in triggering early pull-in instability of the cantilever microbeams. Dynamic pull-in threshold as a function of the mechanical shock amplitude is shown over wide range of shock spanning hundred of thousand of gs up to zero g. For the experimental part, a micromachined cantilever beam made of gold of length 50 microns is subjected to drop-table tests while being biased by electrostatic loads. Several experimental data are shown demonstrating the phenomenon of collapse due to the combined shock and electrostatic forces. It is demonstrated also that by biasing short and too stiff microbeams with electrostatic voltages, their stiffness is weakened. This lowers their threshold of collapse considerably to the range of acceleration that enables testing them with in-house shock testing equipments, such as drop table tests.