Electrostatic Nonlinearities Research Articles

Nonlinearity Reduction in Disk Resonator Gyroscopes Based on the Vibration Amplification Effect

This article reports a novel method to reduce the electrostatic and capacitive nonlinearities in a disk resonator gyroscope (DRG) and to further improve its ultimate oscillation amplitude for performances improvement. The DRG consists of multiple rings, in the working mode, the outer rings always have larger deformation than the inner rings, which is called the vibration amplification effect (VAE). Based on this effect, we propose a method to reduce the nonlinearity in a DRG by optimizing the electrode arrangement, that is, by using the inner electrodes for driving and the outer electrodes for sensing. To verify this method, a DRG with eight layers of driving electrodes was designed and fabricated. The influence of the electrode positions on the nonlinearity of the DRG was theoretically analyzed and experimentally tested. The results show that by using the nonlinearity reduction method, the nonlinearity of the DRG can be reduced and the ultimate oscillation amplitude can be increased to 49% of the capacitive gap, which is much larger than the oscillation amplitude of traditional DRGs. Meanwhile, the sensitivity of the DRG can be improved by 100% and the angle random walk and bias instability can be reduced by 53% and 63%, respectively.

Dynamics of a parallel-plate electrostatic actuator in viscous dielectric media

Understanding the dynamics of a parallel-plate electrostatic actuator, that is, the complex temporal interactions between the mechanical and electrostatic nonlinearities will help optimize the actuator performance for micromanipulation applications. In this paper, we analyze the response of the actuator in a clamped–clamped configuration immersed in viscous dielectric media. We model the actuator as a continuous system by deriving a reduced-order model, and solving it by employing the Galerkin method and linear undamped mode shapes for a clamped–clamped beam. Our model incorporates the inertial loading effect and squeeze film damping by the media, nonlinear mid-plane stretching forces in the beam electrode of the actuator, and nonlinear contact force during the physical contact of the beam electrode with the stationary electrode. The model is utilized to study the actuator dynamics over a broad range, three orders of magnitude of viscosity and two orders of magnitude of relative permittivity of the media. We report three main observations: (1) the actuator response near the vicinity of the pull-in instability region is sensitive to the external forces such as electrostatic force and viscous damping force; (2) the actuator at lower actuation voltages and/or higher actuation frequencies can be approximated to be a linear system; and (3) the high relative permittivity of a media amplifies the generated force, resulting in pull-in to occur at lower actuation voltage amplitudes, and the high dynamic viscosity of a media increases the pull-in time.

Mechanical Behaviors Research and the Structural Design of a Bipolar Electrostatic Actuation Microbeam Resonator.

A class of bipolar electrostatically actuated micro-resonators is presented in this paper. Two parametric equations are proposed for changing the microbeam shape of the upper and lower sections. The mechanical properties of a micro-resonator can be enhanced by optimizing the two section parameters. The electrostatic force nonlinearity, neutral surface tension, and neutral surface bending are considered in the model. First, the theoretical results are verified with finite element results from COMSOL Multiphysics simulations. The influence of section variation on the electrostatic force, pull-in behaviors and safe working area of the micro-resonator are studied. Moreover, the impact of residual stress on pull-in voltage is discussed. The multi-scale method (MMS) is used to further study the vibration of the microbeam near equilibrium, and the relationship between the two section parameters of the microbeam under linear vibration was determined. The vibration amplitude and resonance frequency are investigated when the two section parameters satisfy the linear vibration. In order to research dynamic analysis under the case of large amplitude. The Simulink dynamics simulation was used to study the influence of section variation on the response frequency. It is found that electrostatic softening increases as the vibration amplitude increases. If the nonlinearity initially shows hardening behavior, the frequency response will shift from hardening to softening as the amplitude increases. The position of softening-hardening transition point decreases with the increase of residual stress. The relationship between DC voltage, section parameters, and softening-hardening transition points is presented. The accuracy of the results is verified using theoretical, numerical, and finite element methods.

Modeling and experimental characterization of squeeze film effects in nonlinear capacitive circular microplates

Abstract In this article, an original approach to model the squeeze film effects in capacitive circular microplates is developed. The nonlinear von Karman plate theory is used while taking into consideration the electrostatic and geometric nonlinearities of the clamped edge microplate. The fluid underneath the plate is modeled using the nonlinear Reynolds equation with a corrected effective dynamic viscosity due to size effect. The strongly coupled system of equations is solved using the Differential Quadrature Method (DQM) by discretizing the structural and the fluid domains into a set of grid points. The linear effects of the squeeze film on the microplate have been investigated based on the complex eigenfrequencies of the multiphysical problem. It is shown that the air film can alter the resonance frequencies by adding stiffness as well as damping to the system. The model has been validated numerically with respect to a Finite Element Model (FEM) implemented in ANSYS and experimentally on a fabricated circular microplates. The nonlinear effects of the squeeze film have been studied by determining the steady state solution of the system using the finite difference method (FDM) coupled with the arclength continuation technique. It is shown that the decrease of the static pressure shifts the resonance frequency and leads to an increase of the vibration amplitude due to the reduction of the damping coefficient, while the increase in the pressure enlarges the bistability domain. The developed model can be exploited as an effective tool to predict the nonlinear dynamic behavior of microplates under the effect of air film for the design of Capacitive Micromachined Ultrasonic Transducers (CMUTs).

Investigations of the Effects of Geometric Imperfections on the Nonlinear Static and Dynamic Behavior of Capacitive Micomachined Ultrasonic Transducers.

In order to investigate the effects of geometric imperfections on the static and dynamic behavior of capacitive micomachined ultrasonic transducers (CMUTs), the governing equations of motion of a circular microplate with initial defection have been derived using the von Kármán plate theory while taking into account the mechanical and electrostatic nonlinearities. The partial differential equations are discretized using the differential quadrature method (DQM) and the resulting coupled nonlinear ordinary differential equations (ODEs) are solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). It is shown that the initial deflection has an impact on the static behavior of the CMUT by increasing its pull-in voltage up to 45%. Moreover, the dynamic behavior is affected by the initial deflection, enabling an increase in the resonance frequencies and the bistability domain and leading to a change of the frequency response from softening to hardening. This model allows MEMS designers to predict the nonlinear behavior of imperfect CMUT and tune its bifurcation topology in order to enhance its performances in terms of bandwidth and generated acoustic power while driving the microplate up to 80% beyond its critical amplitude.

Vibration Identification of Folded-MEMS Comb Drive Resonators.

Natural frequency and frequency response are two important indicators for the performances of resonant microelectromechanical systems (MEMS) devices. This paper analytically and numerically investigates the vibration identification of the primary resonance of one type of folded-MEMS comb drive resonator. The governing equation of motion, considering structure and electrostatic nonlinearities, is firstly introduced. To overcome the shortcoming of frequency assumption in the literature, an improved theoretical solution procedure combined with the method of multiple scales and the homotopy concept is applied for primary resonance solutions in which frequency shift due to DC voltage is thoroughly considered. Through theoretical predictions and numerical results via the finite difference method and fourth-order Runge-Kutta simulation, we find that the primary frequency response actually includes low and high-energy branches when AC excitation is small enough. As AC excitation increases to a certain value, both branches intersect with each other. Then, based on the variation properties of frequency response branches, hardening and softening bending, and the ideal estimation of dynamic pull-in instability, a zoning diagram depicting extreme vibration amplitude versus DC voltage is then obtained that separates the dynamic response into five regions. Excellent agreements between the theoretical predictions and simulation results illustrate the effectiveness of the analyses.

Static and DC dynamic pull-in analysis of curled microcantilevers with a compliant support

In the present work, we propose an analytical model for the static and DC dynamic pull-in analysis of electrostatically actuated, initially curled microcantilever beams. The analytical model of the initially deformed microcantilever beam incorporates the effects of flexible boundary conditions, fringing fields, and full-order electrostatic nonlinearity. The problem is formulated based on the Euler–Bernoulli theory with slender beams. The pull-in parameters are extracted by incorporating a single mode in the setting of the Galerkin based reduced order model. The effect of the flexible support on the natural frequencies and mode shapes is presented. The parabolic and linear configurations of initial deformation of the beam are assumed for analysis. The accuracy of the present model for static pull-in is verified with the previously reported experimental results. An excellent agreement of the present results with experimental results from the literature indicates the sufficient accuracy of the present model.

Analytical Modeling and Experimental Verification of Nonlinear Mode Coupling in a Decoupled Tuning Fork Microresonator

This paper provides the analytical and experimental studies into the nonlinear mode coupling in a tuning fork microresonator, with electrostatic actuation at forced and internal resonance frequencies. The analytical investigation leads to modal equations that take into account kinematic and electrostatic nonlinearities. The equations of motion are derived using Lagrange’s energy method. The theoretical resonance curves are determined by using the two-variable expansion perturbation technique near simultaneous subharmonic and internal resonances. The two desired mode shapes of the device are tailored to establish an approximate two-to-one frequency ratio between their natural frequencies. A sample device design is fabricated in a commercial foundry process. The influence of AC driving levels and a detuning parameter, due to inevitable fabrication imperfections, on the system dynamics are investigated theoretically and experimentally. It is shown that the deviation from ideal internal resonance condition separates the region of the nonlinear vibration into two distinct excitation frequencies. The model and simulated results obtained by the perturbation analysis are validated by comparing them with the experimental results. [2017–0297]

Effect of nonlinear electrostatic forces on the dynamic behaviour of a capacitive ring-based Coriolis Vibrating Gyroscope under severe shock

This paper investigates the dynamic behaviour of capacitive ring-based Coriolis Vibrating Gyroscopes (CVGs) under severe shock conditions. A general analytical model is developed for a multi-supported ring resonator by describing the in-plane ring response as a finite sum of modes of a perfect ring and the electrostatic force as a Taylor series expansion. It is shown that the supports can induce mode coupling and that mode coupling occurs when the shock is severe and the electrostatic forces are nonlinear. The influence of electrostatic nonlinearity is investigated by numerically simulating the governing equations of motion. For the severe shock cases investigated, when the electrode gap reduces by ∼60%∼60%, it is found that three ring modes of vibration (1θ,2θ1θ,2θ and 3θ3θ) and a 9th order force expansion are needed to obtain converged results for the global shock behaviour. Numerical results when the 2θ2θ mode is driven at resonance indicate that electrostatic nonlinearity introduces mode coupling which has potential to reduce sensor performance under operating conditions. Under some circumstances it is also found that severe shocks can cause the vibrating response to jump to another stable state with much lower vibration amplitude. This behaviour is mainly a function of shock amplitude and rigid-body motion damping.

Static and Dynamic Mechanical Behaviors of Electrostatic MEMS Resonator with Surface Processing Error.

The micro-electro-mechanical system (MEMS) resonator developed based on surface processing technology usually changes the section shape either due to excessive etching or insufficient etching. In this paper, a section parameter is proposed to describe the microbeam changes in the upper and lower sections. The effect of section change on the mechanical properties is studied analytically and verified through numerical and finite element solutions. A doubly-clamped microbeam-based resonator, which is actuated by an electrode on one side, is investigated. The higher-order model is derived without neglecting the effects of neutral plane stretching and electrostatic nonlinearity. Further, the Galerkin method and Newton–Cotes method are used to reduce the complexity and order of the derived model. First of all, the influence of microbeam shape and gap variation on the static pull-in are studied. Then, the dynamic analysis of the system is investigated. The method of multiple scales (MMS) is applied to determine the response of the system for small amplitude vibrations. The relationship between the microbeam shape and the frequency response is discussed. Results show that the change of section and gap distance can make the vibration soften, harden, and so on. Furthermore, when the amplitude of vibration is large, the frequency response softening effect is weakened by the MMS. If the nonlinearity shows hardening-type behavior at the beginning, with the increase of the amplitude, the frequency response will shift from hardening to softening behavior. The large amplitude in-well motions are studied to investigate the transitions between hardening and softening behaviors. Finally, the finite element analysis using COMSOL software (COMSOL Inc., Stockholm, Sweden) is carried out to verify the theoretical results, and the two results are very close to each other in the stable region.

Synchronization of a micromechanical oscillator in different regimes of electromechanical nonlinearity

In this letter, we investigate the dynamics of injection-locking a nonlinear micromechanical oscillator operating in different regimes of electromechanical nonlinearity to an external tone generated by a secondary oscillator. The micromechanical oscillator exhibits a combination of mechanical and electrostatic nonlinearities that were tuned using a bias voltage to adjust the relative importance of third-order and fifth-order stiffness nonlinearities. While it is well-known that third-order stiffness (Duffing) nonlinearity results in a synchronization range that increases with an oscillator's amplitude, little is known about the impact of other nonlinearities. We show that when using Duffing nonlinearity cancellation, higher order nonlinearities dominate, the synchronization range is smaller but has a greater rate-of-increase with oscillation amplitude. When both mechanical stiffness-hardening and electrostatic stiffness-softening nonlinearities are present, the frequency response follows an “s-curve” and, unlike the other conditions, the synchronization range does not increase monotonically with amplitude but instead reaches a minimum when both nonlinearities have similar magnitude. We develop a nonlinear resonator model and show that this model achieves good quantitative prediction of the measured synchronization range in all nonlinear operating regimes studied.

Nonlinear dynamic responses of electrostatically actuated microcantilevers containing internal fluid flow

A nonlinear theoretical model for electrostatically actuated microcantilevers containing internal fluid flow is developed in the present study, which takes into account the geometric and electrostatic nonlinearities. A four-degree-of-freedom and eight-dimensional analytical modeling is presented for investigating the stability mechanism and nonlinear dynamic responses near and away from the instability boundaries of the fluid-loaded cantilevered microbeam system. Firstly, the reliability of the theoretical model is examined by comparing the present results with previous experimental and numerical results. It is found that, with the increase in flow velocity, flutter instability, pull-in instability and the combination of both can occur in this dynamical system. It is also found that the instability boundary depends on the initial conditions significantly when the internal fluid is at low flow rate. Nextly, the phase portraits and time histories of the microbeam’s oscillations and bifurcation diagrams are established to show the existence of periodic, chaotic divergence and transient periodic-like motions.

Compensation of nonlinear hardening effect in a nanoelectromechanical torsional resonator

A mechanical resonator based on torsional resonance has been fabricated in our facilities to sense infrared radiation. Actuation and detection are both electrostatic. Reaching phonon noise is highly desirable in order to get low noise uncooled infrared detectors. Therefore a high dynamic range has to be reached to increase as much as possible the overall signal to noise ratio and minimize the contribution of amplitudes noises (thermomechanical or electrical) to frequency noise. The dimensions of resonator body are similar to most clamped-clamped NEMS and our devices are thus also greatly affected by nonlinearities. We present here a nonlinear model for mechanical behavior around fundamental mode resonance taking account of both mechanical and electrostatic nonlinearities. The model correctly reproduces the nonlinear behavior observed for different resonator designs. We experimentally observe on some devices a compensation of hardening effects, allowing a linear operation of torsion angle up to 13°. The model provided in this work allows an engineering strategy in order to design high linear dynamic range for fundamental torsional mode.

Operation of a high quality-factor gyroscope in electromechanical nonlinearities regime

This paper describes the operation of a high quality factor gyroscope in various regimes where electromechanical nonlinearities introduce different forms of amplitude–frequency (A–f) dependence. Experiments are conducted using an epitaxially-encapsulated 2 × 2 mm2 quad-mass gyroscope (QMG) with a quality factor of 85 000. The device exhibits third-order Duffing nonlinearity at low bias voltages (15 V) due to the mechanical nonlinearity in the flexures and at high bias voltages (35 V) due to third-order electrostatic nonlinearity. At intermediate voltages (~26 V), these third-order nonlinearities cancel and the amplitude–frequency dependence is greatly reduced. A model is developed to demonstrate the connection between the electromechanical nonlinearities and the amplitude–frequency dependence, also known as the backbone curve. Gyroscope operation is demonstrated in each nonlinear operating regime and the key performance measures of the gyroscope’s performance, angle random walk (ARW) and bias instability, are measured as a function of drive-mode vibration amplitude. We find that low ARW can be achieved even though the gyroscope’s drive mode exhibits large amplitude–frequency dependence, and that bias instability is largely independent of the operating regime.

On the Nonlinear Dynamics of a Doubly Clamped Microbeam Near Primary Resonance

This work aims to investigate theoretically and experimentally various nonlinear dynamic behaviors of a doubly clamped microbeam near its primary resonance. Mainly, we investigate the transition behavior from hardening, mixed, and then softening behavior. We show in a single frequency–response curve, under a constant voltage load, the transition from hardening to softening behavior demonstrating the dominance of the quadratic electrostatic nonlinearity over the cubic geometric nonlinearity of the beam as the motion amplitudes becomes large, which may lead eventually to dynamic pull-in. The microbeam is fabricated using polyimide as a structural layer coated with nickel from top and chromium and gold layers from the bottom. Frequency sweep tests are conducted for different values of direct current (DC) bias revealing hardening, mixed, and softening behavior of the microbeam. A multimode Galerkin model combined with a shooting technique are implemented to generate the frequency–response curves and to analyze the stability of the periodic motions using the Floquet theory. The simulated curves show a good agreement with the experimental data.

Elaso-plastic behavior of micro strings loaded by distributed electrostatic force

The behavior of a micro string (one-dimensional membrane) made of nonlinear inelastic material and subjected to electrostatic force is analyzed. The present work introduces the coupling between material nonlinearity and the commonly considered geometric and electrostatic nonlinearities. The effects of the material softening behavior on the string response and pull-in instability are studied analytically and numerically. Moreover, the accumulation of irreversible stretching and the residual state of the string after unloading are examined. It is suggested that the residual string elongation can be exploited for various applications, including cold forming and residual stress tuning. Analytical and numerical results show that inelastic irreversible deformation can be achieved in strings of realistic dimensions actuated by feasible voltages.

Alleviation of residual oscillations in electrostatically actuated variable-width microbeams using a feedforward control strategy

This article presents a feedforward control strategy for alleviating the residual oscillations in electrostatically driven nonprismatic microbeams. The two prevalently employed microbeam configurations (fixed-free and fixed-fixed) are analyzed in this work. The governing differential equation is first developed taking into account the effects of electrostatic nonlinearity, viscous energy dissipation, moderately large beam-deflections, and the fringing-fields. Modal superposition method with full-order electrostatic nonlinearity is then used to solve the equation of motion numerically. The mass-normalized mode shapes of the respective nonprismatic configuration are extracted using the differential transform method. Response of the microbeams driven by a multi-step voltage waveform is investigated highlighting the presence of residual oscillations about the anticipated equilibrium positions. A feed-forward control strategy is proposed to alleviate the effect of such residual oscillations; thereby achieving fast switching between the successive equilibrium configurations. Efficacy of the proposed control technique is demonstrated by controlling the motion over various user-defined equilibrium sequences. A parametric study involving variations in the taper parameter is performed to bring out the utility of using nonprismatic geometries in place of conventional prismatic beams. Additionally, an impact of higher vibration modes on the performance of the proposed control strategy is investigated. The present work can find its potential use in the development of open-loop controllers to be used in microelectromechanical systems involving beam-type actuators.

Influence of the Driving Waveform on the Open-Loop Frequency Response of MEMS Resonators With Nonlinear Actuation Schemes

This paper deals with the influence of the shape of the driving waveform on the frequency responses of microelectromechanical systems (MEMS) resonators under nonlinear actuation. Our models show that, at large oscillation amplitudes, these responses are strongly dependent on the shape of the actuation waveform so that the nonlinear frequency response is not the system signature, but the system signature under a specific driving waveform. The case of a MEMS resonator electrostatically driven with a sine-, pulsed-, or square-wave voltage is specifically addressed. Our models and simulations, supported by experimental evidence, predict counterintuitive phenomena resulting from the distortion of the actuation waveform by the displacement-dependent electrostatic nonlinearity. This paper emphasizes that this issue should not be overlooked in order to perform quantitative MEMS characterization in the nonlinear regime. [2015-0283]

Pull-in characteristics of electrically actuated MEMS arches

This paper examines the pull-in characteristics of initially curved microelectromechanical resonators actuated by a DC voltage along with an AC harmonic voltage. The deformable electrode is modeled by means of a clamped–clamped Euler–Bernoulli microarch in which a combination of mechanical, electrostatic, and electrodynamic nonlinearities are taken into account. A high-dimensional reduced-order Galerkin model is built, and the effect of various system parameters on pull-in instabilities is investigated numerically. Specifically, the electroelastostatic deformation as well as pull-in instabilities are analyzed for the cases actuated electrostatically via the DC voltage; the DC voltage deflection characteristics due to the simultaneous presence of mechanical and electrical nonlinearities are presented. The electroelastodynamic deformations are investigated for the system under simultaneous actuation of the DC and AC voltages; the AC frequency–motion as well as the AC amplitude–motion curves is analyzed to highlight the effect of electroelastodynamic nonlinearities.

Mitigation of residual oscillations in electrostatically actuated microbeams using a command-shaping approach

When electrostatically actuated microbeams are driven by an input-waveform comprising multiple voltage steps, the resulting response inherently contains residual oscillations, which may prove detrimental to the device performance and accuracy. In this article, we report the systematic development of a command shaping technique for mitigating such residual oscillations in electrostatically actuated microbeams and achieving fast switching between the successive equilibrium states. Invoking the force balance at a critical point in an oscillation cycle, the proposed technique relies on bringing the actuator to a stagnation state by applying an additional voltage signal of specific amplitude at a predetermined time. The underlying principle of the technique is enunciated for the lumped parallel-plates model of the microactuator, and further extended to the cases of microbeams. The electromechanical model of the microbeam incorporates the effects of full-order electrostatic nonlinearity, moderately large deflections, viscous energy dissipation, and fringing fields. The modal superposition method is employed to obtain the dynamic response of microbeams. Based on a single-mode assumption, the proposed technique lends itself to a simple multistep waveform, which is attractive from the implementation point of view. The applicability of the proposed technique is demonstrated by considering a wide range of parameters involving variations in the extent of geometric nonlinearity, damping, and equilibrium sequences. The impact of higher modes on the stabilized response is exposited, and a command shaping approach based on the multi-mode response of the actuator is suggested. In particular, such an approach is shown to be effective in controlling the motion of the beam in the vicinity of the static pull-in displacement, which is associated with strong electrostatic nonlinearity. The present investigation can find its potential use in the development of an open-loop controller for electrostatically actuated microbeams.