Pull-In Retarding in Nonlinear Nanoelectromechanical Resonators Under Superharmonic Excitation

International audienceIn order to compensate for the loss of performance when scaling resonant sensors down to NEMS, a complete analytical model, including all main sources of nonlinearities, is presented as a predictive tool for the dynamic behavior of clamped-clamped nanoresonators electrostatically actuated. The nonlinear dynamics of such NEMS under superharmonic resonance of an order half their fundamental natural frequencies is investigated. It is shown that the critical amplitude has the same dependence on the quality factor Q and the thickness h as the case of the primary resonance. Finally, a way to retard the pull-in by decreasing the AC voltage is proposed in order to enhance the performance of NEMS resonators

Forced large amplitude periodic vibrations of non-linear Mathieu resonators for microgyroscope applications

International audienceThis paper describes a comprehensive non-linear multiphysics model based on the Euler-Bernoulli beam equation that remains valid up to large displacements in the case of electrostatically actuated Mathieu resonators. This purely analytical model takes into account the fringing field effects and is used to track the periodic motions of the sensing parts in resonant microgyroscopes. Several parametric analyses are presented in order to investigate the effect of the proof mass frequency on the bifurcation topology. The model shows that the optimal sensitivity is reached for resonant microgyroscopes designed with sensing frequency four times faster than the actuation one

Dynamics of a Close-Loop Controlled MEMS Resonator

The dynamics of a close-loop electrostatic MEMS resonator, proposed as a platform for ultra sensitive mass sensors, is investigated. The parameter space of the resonator actuation voltage is investigated to determine the optimal operating regions. Bifurcation diagrams of the resonator response are obtained at five different actuation voltage levels. The resonator exhibits bi-stability with two coexisting stable equilibrium points located inside a lower and an upper potential wells. Steady-state chaotic attractors develop inside each of the potential wells and around both wells. The optimal region in the parameter space for mass sensing purposes is determined. In that region, steady-state chaotic attractors develop and spend most of the time in the safe lower well while occasionally visiting the upper well. The robustness of the chaotic attractors in that region is demonstrated by studying their basins of attraction. Further, regions of large dynamic amplification are also identified in the parameter space. In these regions, the resonator can be used as an efficient long-stroke actuator

Energy efficient control of electrostatically actuated MEMS

Plenty of Micro-electro-mechanical Systems (MEMS) devices are actuated using electrostatic forces, and specially, parallel-plate actuators are extensively used, due to the simplicity of their design. Nevertheless, parallel-plate actuators have some limitations due to the nonlinearity of the generated force. The dissertation analyzes the dynamics of the lumped electrostatically actuated nonlinear system, in order to obtain insight on its characteristics, define desired performance goals and implement a controller for energy efficient robustly stable actuation of MEMS resonators. In the first part of the dissertation, the modeling of the electromechanical lumped system is developed. From a complete distributed parameters model for MEMS devices which rely on electrostatic actuation, a concentrated parameters simplification is derived to be used for analysis and control design. Based on the model, energy analysis of the pull-in instability is performed. The classic approach is revisited to extend the results to models with a nonlinear springs. Analysis of the effect of dynamics is studied as an important factor for the stability of the system. From this study, the Resonant Pull-in Condition for parallel-plate electrostatically actuated MEMS resonators is defined and experimentally validated. Given the importance of the nonlinear dynamics and its richness in behaviors, Harmonic Balance is chosen as a tool to characterize the steady-state oscillation of the resonators. This characterization leads to the understanding of the key factors for large and stable oscillation of resonators. An important conclusion is reached, Harmonic Balance predicts that any oscillation amplitude is possible for any desired frequency if the appropriate voltage is applied to the resonator. And the energy consumption is dependent on this chosen oscillation frequency. Based on Harmonic Balance results, four main goals are defined for the control strategy: Stable oscillation with large amplitudes of motion; Robust oscillation independently of MEMS imperfections; Pure sinus-like oscillation for high-grade sensing; and Low energy consumption. The second part of the dissertation deals with the controller selection, design and verification. A survey of prior work on MEMS control confirms that existing control approaches cannot provide the desired performance. Consequently, a new three-stage controller is proposed to obtain the desired oscillation with the expected stability and energy efficiency. The controller has three different control loops. The first control loop includes a Robust controller designed using on µ-synthesis, to deal with MEMS resonators uncertainties. The second control loop includes an Internal-Model-Principle Resonant controller, to generate the desired control action to obtain the desired oscillation. And the third control loop handles the energy consumption minimization through an Extremum Seeking Controller, which selects the most efficient working frequency for the desired oscillation. The proposed controller is able to automatically generate the needed control voltage, as predicted by the Harmonic Balance analysis, to operate the parallel-plate electrostatically actuated MEMS resonator at the desired oscillation. Performance verification of stability, robustness, sinus-like oscillation and energy efficiency is carried out through simulation. Finally, the needed steps for a real implementation are analyzed. Independent two-sided actuation for full-range amplitude oscillation is introduced to overcome the limitations of one-sided actuation. And a modification of standard Electromechanical Amplitude Modulation is analyzed and validated for position feedback implementation. With these improvements, a MEMS resonator with the desired specifications for testing the proposed control is designed for fabrication. Based on this design, testing procedure is discussed as future work.Molts microsistemes (MEMS) són actuats amb forces electrostàtiques, i especialment, els actuadors electrostàtics de plaques paral.leles són molt usats, degut a la simplicitat del seu disseny. Tot i això, aquests actuadors tenen limitacions degut a la no-linealitat de les forces generades. La tesi analitza el sistema mecànic no-lineal actuat electrostàticament que forma el MEMS, per tal d'entendre'n les característiques, definir objectius de control de l'oscil.lació, i implementar un controlador robust, estable i eficient energèticament. A la primera part de la tesi es desenvolupa el modelat del sistema electromecànic complert. A partir de la formulació de paràmetres distribuïts aplicada a dispositius MEMS amb actuació electrostàtica, es deriva una formulació de paràmetres concentrats per a l'anàlisi i el disseny del control. Basat en aquest model, s'analitza energèticament la inestabilitat anomenada Pull-in, ampliant els resultats de l'enfocament clàssic al model amb motlles no-lineals. Dins de l'anàlisi, l'evolució dinàmica s'estudia per ser un factor important per a l'estabilitat. D'aquest estudi, la Resonant Pull-in Condition per a actuadors electrostàtics de plaques paral.leles es defineix i es valida experimentalment. Donada la importància de la dinàmica no-lineal del sistema i la seva riquesa de comportaments, s'utilitza Balanç d'Harmònics per tal de caracteritzar les oscil.lacions en estacionari. Aquesta caracterització permet entendre els factors claus per a obtenir oscil.lacions estables i d'amplitud elevada. El Balanç d'Harmònics dóna una conclusió important: qualsevol amplitud d'oscil.lació és possible per a qualsevol freqüència desitjada si s'aplica el voltatge adequat al ressonador. I el consum energètic associat a aquesta oscil.lació depèn de la freqüència triada. Llavors, basat en aquests resultats, quatre objectius es plantegen per a l'estratègia de control: oscil.lació estable amb amplituds elevades; robustesa de l'oscil.lació independentment de les imperfeccions dels MEMS; oscil.lació sinusoïdal sense harmònics per a aplicacions d'alta precisió; i baix consum energètic. La segona part de la tesi tracta la selecció, disseny i verificació dun controlador adequat per a aquests objectius. La revisió dels treballs existents en control de MEMS confirma que cap dels enfocaments actuals permet obtenir els objectius desitjats. En conseqüència, es proposa el disseny d'un nou controlador amb tres etapes per tal d'obtenir l'oscil.lació desitjada amb estabilitat i eficiència energètica. El controlador té tres llaços de control. Al primer llaç, un controlador robust dissenyat amb µ-síntesis gestiona les incertes es dels MEMS. El segon llaç inclou un controlador Ressonant, basat en el Principi del Model Intern, per a generar l'acció de control necessària per a obtenir l'oscil.lació desitjada. I el tercer llaç de control gestiona la minimització de l'energia consumida mitjançant un controlador basat en Extremum Seeking, el qual selecciona la freqüència de treball més eficient energèticament per a l'oscil.lació triada. El controlador proposat és capaç de generar automàticament el voltatge necessari, igual al previst pel Balanç d'Harmònics, per tal d'operar electrostàticament amb plaques paral.leles els ressonadors MEMS. Mitjançant simulació se'n verifica l'estabilitat, robustesa, inexistència d'harmònics i eficiència energètica de l'oscil.lació. Finalment, la implementació real és analitzada. En primer lloc, un nou esquema d'actuació per dos costats amb voltatges independents es proposa per aconseguir l'oscil.lació del ressonador en tot el rang d'amplituds. I en segon lloc, una modificació del sensat amb Modulació d'Amplitud Electromecànica s'utilitza per tancar el llaç de control de posició. Amb aquestes millores, un ressonador MEMS es dissenya per a ser fabricat i validar el control. Basat en aquest disseny, es proposa un procediment de test plantejat com a treball futur.Postprint (published version

Theoretical Prediction of Experimental Jump and Pull-In Dynamics in a MEMS Sensor

The present research study deals with an electrically actuated MEMS device. An experimental investigation is performed, via frequency sweeps in a neighbourhood of the first natural frequency. Resonant behavior is explored, with special attention devoted to jump and pull-in dynamics. A theoretical single degree-of-freedom spring-mass model is derived. Classical numerical simulations are observed to properly predict the main nonlinear features. Nevertheless, some discrepancies arise, which are particularly visible in the resonant branch. They mainly concern the practical range of existence of each attractor and the final outcome after its disappearance. These differences are likely due to disturbances, which are unavoidable in practice, but have not been included in the model. To take disturbances into account, in addition to the classical local investigations, we consider the global dynamics and explore the robustness of the obtained results by performing a dynamical integrity analysis. Our aim is that of developing an applicable confident estimate of the system response. Integrity profiles and integrity charts are built to detect the parameter range where reliability is practically strong and where it becomes weak. Integrity curves exactly follow the experimental data. They inform about the practical range of actuality. We discuss the combined use of integrity charts in the engineering design. Although we refer to a particular case-study, the approach is very general

Nonlinear Dynamic Behavior of a Bi-Axial Torsional MEMS Mirror with Sidewall Electrodes

Nonlinear dynamic responses of a Micro-Electro-Mechanical Systems (MEMS) mirror with sidewall electrodes are presented that are in close agreement with previously-reported experimental data. An analysis of frequency responses reveals softening behavior, and secondary resonances originated from the dominant quadratic nonlinearity. The quadratic nonlinearity is an electromechanical coupling effect caused by the electrostatic force. This effect is reflected in our mathematical model used to simulate the dynamic response of the micro-mirror. The effects of increased forcing and decreased damping on the frequency response are investigated as the mirrors are mostly used in vacuum packages. The results can predict MEMS mirror behaviors in optical devices better than previously-reported models

Electrostatic MEMS Bifurcation Sensors

We report experimental evidence of a new instability in electrostatic sensors, dubbed quasi-static pull-in, in two types of micro-sensors operating in ambient air. We find that the underlying mechanism and features of this instability are distinct from those characterizing hitherto known static and dynamic pull-in instabilities. Specifically, the mechanism instigating quasi-static pull-in is a global Shilnikov homoclinic bifurcation where a slow-varying waveform drives the sensor periodically through a saddle-node bifurcation. Based on these findings, we propose a new taxonomy of pull-in instabilities in electrostatic sensors. Experimental evidence of nonlinear chaotic behaviors were observed in an electrostatic MEMS sensor. Period doubling bifurcation (P-2), period three (P-3), and period six (P-6) were observed. A new class of intermittency subsequent to homoclinic bifurcation in addition to the traditional intermittencies of type-I and type-II were demonstrated. Quasiperiodicity and homoclinic tangles leading to chaos were also reported. All of these nonlinear phenomena instigate either banded chaos or full chaos and both are observed in this work. Based on our knowledge, this is the first observation such chaotic behaviors in electrostatic MEMS sensors. All of the experimental observations have been measured optically via a laser Doppler-vibrometer (LDV) in ambient pressure. Also, a new class of intermittencies was found in the oscillations of an electrostatic sensor. These intermittencies involve a dynamic system spending irregular time intervals in the vicinity of the ghost of an orbit before undergoing bursts that are arrested by landing on a larger attractor. Re-injection into the vicinity of the ghost orbit is noise induced. As a control parameter is increased, switching intermittency of type-I leads to a stable periodic orbit, whereas switching intermittency of type-II leads to a chaotic attractor. These significant findings in nonlinear dynamic were used to develop novel MEMS sensors. An electrostatic MEMS gas sensor is demonstrated. It employs a dynamic-bifurcation detection technique. In contrast to traditional gas or chemical sensors that measure (quantify) the concentration of an analyte in analog mode, this class of sensors does not seek to quantify the concentration. Rather, it detects the analyte's concentration in binary mode, reporting ON-state (1) for concentrations above a preset threshold and OFF-state (0) for concentrations below the threshold. The sensing mechanism exploits the qualitative difference between the sensor state before and after the dynamic pull-in bifurcation. Experimental demonstration was carried out using a laser-Doppler vibrometer to measure the sensor response before and after detection. The sensor was able to detect ethanol vapor concentrations as 100\,ppb in dry nitrogen. A closed-form expression for the sensitivity of dynamic bifurcation sensors was derived. It captured the dependence of sensitivity on the sensor dimensions, material properties, and electrostatic field. An analog dynamic bifurcation mass sensor is developed to demonstrate a sensing mechanism that exploits a quantitative change in the sensor state before and after depositing added mass. A polymeric material was deposited on the top surface of the sensor plate to represent added mass. A variation in the frequency and current amplitude were utilized to demarcate the added mass optically and electrically. A chemical sensor was also developed to detect mercury in deionized-water in a fashion of analog detection. A polymeric sensing material that has high selectivity to mercury was utilized to captured mercury molecules in water. The sensor was submerged completely in water with a pre-defined flow-rate. The sensor was excited electrostatically. A variation in the frequency response due to added mass was measured electrically using a lock-in amplifier. A frequency-shift was observed while releasing the mercury to the water

HARMONIC DETECTION OF RESONANCE IN MICRO- AND NANO- CANTILEVERS

Over the past decade there has been an explosion in the study of cantilevered beams on the micron and submicron dimension. The applications and research that involve these structures include state-of-the-art electronic components, sensors, and more recently, studies aimed at elucidating the mechanical properties of cantilevered carbon nanotubes and semiconducting nanowires. In nanoelectro-mechanical systems (NEMS), it is desirable to develop a capacitive readout method involving only two electrodes that are fully compliant with standard CMOS technology. However, the main drawback with this method is the ability to detect resonance in the presence of parasitic capacitance, which is due to the fringing electric fields present between the electrodes (cantilever and the counter electrode). The work presented in this thesis deals with the electrical actuation / detection of mechanical resonance in individual micron and sub-micron sized cantilevers. The aim is to overcome parasitic capacitance which masks the detection of resonance signal in these cantilevers thereby increasing the signal-to-background ratio (SBR). In our method, a silicon microcantilever, or cantilevered multi-walled carbon nanotube (MWNT), is placed close to a counter electrode whose potential is varied at a frequency ω. An electrical signal comes from the flow of charge on and off of the cantilever when ω equals a resonant frequency 0 of the cantilever. Higher harmonics of 0 are measured to overcome the parasitic capacitance. This technique, termed harmonic detection of resonance (HDR), allows detection at frequencies well removed from the driving frequency thereby increasing the SBR by ~3 orders of magnitude. It is shown that HDR allows the detection of resonance even in multi-walled carbon nanotubes, which have diameters on the order of 50 nm. Furthermore, superharmonics inherent to electrostatic actuation, are shown to occur at driving frequencies of ω0/n where n=1,2,3,...