Complex nonlinear oscillations in electrostatically actuated microstructures

In this paper, new nonlinear dynamic properties of electrostatically actuated microstructures [referred to as electrostatic microelectromechanical systems (MEMS)] observed under superharmonic excitations are presented using numerical simulations. Application of a large dc bias (close to the pull-in voltage of the device) is found to bring the device to a nonlinear state. This nonlinear state (referred to as "dc-symmetry breaking") can be clearly observed from the characteristic change in the phase-plot of the device. Once a steady nonlinear state is reached, application of an ac signal at the Mth superharmonic frequency with an amplitude around "ac-symmetry breaking" gives rise to M oscillations per period or M-cycles in the MEM device. "ac-symmetry breaking" can also be observed by a characteristic change in the phase-plot of the device. On further increasing the ac voltage, a period doubling sequence takes place resulting in the formation of 2/sup n/M-cycles in the system at the Mth superharmonic frequency. An interesting chaotic transition (banded chaos) is observed during the period doubling bifurcations. The nonlinear nature of the electrostatic force acting on the MEM device is found to be responsible for the reported observations. The significance of the mechanical and the fluidic nonlinearities is also studied.     

Identification of nonlinear dynamic models of electrostatically actuated MEMS

This paper focuses on the identification of nonlinear dynamic models for physical systems such as electrostatically actuated micro-electro-mechanical systems (MEMS). The proposed approach consists in transforming, by means of suitable global operations, the input–output differential model in such a way that the new equivalent formulation is well adapted to the identification problem, thanks to the following properties: first, the linearity with respect to the parameters to be identified is preserved, second, the continuous dependence on noise measurements is restored. Consequently, a simple least-square resolution can be used, in such a way that some of the difficulties classically encountered with identification methods are by-passed. The method is implemented on real measurement data from a physical system.     

Shape optimization of electrostatically actuated microbeams for extending static and dynamic operating ranges

We present the systematic development and application of a generic shape optimization methodology to enhance the static and dynamic pull-in ranges of electrostatically actuated microbeams. Energy-based techniques are used to extract static and dynamic pull-in parameters of the distributed electromechanical model that accounts for the fringing field capacitance, moderately large deflections, and coupling between the mechanical and electric forces. Versatile parametric width functions are used to characterize the nonprismatic geometries of cantilever and fixed–fixed microbeams, and parameters of the proposed width functions are optimized using the Nelder–Mead method of function minimization together with a penalty method to enforce the constraints. We consider several test cases in order to fully demonstrate the utility of the proposed methodology. Results indicate that an increase in the travel range of as much as 20% can be obtained using our optimization approach. In case of fixed–fixed microbeams, this enhancement in the travel range is found to be dependent on the extent of geometric nonlinearity. We present the optimal shapes of these microbeams, that easily lend themselves to microfabrication, which exhibit the improved pull-in response in both static and dynamic regimes. For a set of representative cases, the enhanced travel range in both static and dynamic modes of actuation is positively verified by 3D finite element analyses performed on the referential and optimized geometries.     

Fabrication and dynamic analysis of the electrostatically actuated MEMS variable capacitor

Future microwave networks require miniature high-performance tunable elements such as switches, inductors, and capacitors. In this paper, high performance variable capacitor was fabricated by simple microelectromechanical systems (MEMS) technology. The capacitance and quality (Q) factor at 1 GHz are 0.792 pF and 51.6. The pull-in voltage is 13.5 V and the tuning ratio of the capacitor is more than 1.31:1. A reduced-order model for the dynamic characteristics of the capacitor is established based on the equilibrium among the plate flexibility.     

静电驱动振模微泵的理论分析

利用最小能量法和均匀压力载荷下的圆薄膜大变形半解析解相结合的方法,改进了静电驱动柔性振膜微泵的理论分析模型.计算结果表明:考虑圆薄膜的周向应变,利用均匀压力载荷下的圆薄膜大变形半解析解来预测振膜的形状,对微泵的性能有显著的影响,证明了圆薄膜大变形时的周向应变是不可忽略的.同时讨论了介电层厚度、腔体形状和双腔结构对微泵压缩性能的影响.结果表明:采用双腔结构,减小介电层厚度、减小腔体深度、缩小腔体半径,对静电驱动柔性振膜型微泵性能的提高具有积极意义.     

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.     

Mathematical analysis and test of an electrostatically actuated micro-power relay

This paper reports a mathematical and experimental study on a micro-relay for power applications. The relay is microfabricated using UV-LIGA and electrostatically actuated. The relay was uniquely designed with all major components made of metals except two polymer strips which serve as both mechanical connectors and electrical insulators. The utilization of UV-LIGA fabrication technologies helps to achieve high aspect ratio metal structures in the relay, which helps to achieve high power capacity. Both the static and dynamic characteristics of the prototype relay have been studied. The performances of the prototype relay agreed very with those predicted using mathematical models. The prototype relay was found to be able to operate in a control voltage lower than 18 V and carry a current of 5 A. The prototype relay operated over 10⁶ cycles without any degradation. In addition, the response time of a relay was 3.2–3.5 ms in the “on” operation and 7.9–14.3 ms during the “off” operation.     

Investigation of the internal stress effects on static and dynamic characteristics of an electrostatically actuated beam for MEMS and NEMS application

Internal stress is often encountered in fixed–fixed beam based devices with micron or sub-micron length scales during device fabrication or operation. In this paper, we have investigated the effects of internal stress on static and dynamic characteristics of an electrostatically actuated cylindrical beam. The beam has been modelled using Euler–Bernoulli theory including the nonlinearities due to beam stretching and electrostatic forcing. The analysis has been carried out by solving the governing differential equations using a Galerkin based multi-modal reduced order modelling technique. A standard collocation based numerical scheme has also been used to confirm the results of the reduced order method. Our study shows that internal stress significantly influences the static and dynamic characteristics of the beam. We also find that, when compressive internal stress is high, it is important to include higher modes in the reduced order model. A design technique to achieve high resonant frequency stability under temperature variation, for electrostatically actuated beam oscillators, has also been proposed as a result of this investigation.     

基于模态分析的静电驱动圆薄板宏模型建立方法

为了研究静电致动圆板静态和动态特性,在模态分析法的基础上,利用多维非线性函数的Levenberg-Marquardt拟合方法,将板的动能、弹性能和电容写成以模态坐标表示的解析式.结合Hamilton原理,导出静电致动微圆板的动力学特性方程的宏模型.以此为基础研究了板的静态特性及其三角波、方波信号激励下的动态响应.结果表明,模态分析方法能够考虑到残余应力的影响,所建立的动态宏模型不仅大大地减少了计算费用,而且具有足够的仿真精度.     

Stability analysis of an electrostatically actuated out of plane MEMS structure

In the present research, stability and static analyses of microelectromechanical systems microstructure were investigated by presenting an out-of-plane structure for a lumped mass. The presented model consists of two stationary electrodes in the same plane along with a flexible electrode above and in the middle of the two electrodes. The nonlinear electrostatic force was valuated via numerical methods implemented in COMSOL software where three-dimensional simulations were performed for different gaps. The obtained numerical results were compared to those of previous research works, indicating a good agreement. Continuing with the research, curves of electrostatic and spring forces were demonstrated for different scenarios, with the intersection points (i.e., equilibrium points) further plotted. Also drawn were plots of deflection versus voltage for different cases and phase and time history curves for different values of applied voltage followed by introducing and explaining pull-in and pull-out snap-through voltages in the system for a specific design. It is worth noting that, at voltages between the pull-in and pull-out snap-through voltages, the system was in bi-stable state. Based on the obtained results, it was observed that the gap between the two electrodes and the applied voltage play significant roles in the number and type of the equilibrium points of the system.

     

Mixed frequency excitation of an electrostatically actuated resonator

We investigate experimentally and theoretically the dynamics of a capacitive resonator under mixed frequency excitation of two AC harmonic signals. The resonator is composed of a proof mass suspended by two cantilever beams. Experimental measurements are conducted using a laser Doppler vibrometer to reveal the interesting dynamics of the system when subjected to two-source excitation. A nonlinear single-degree-of-freedom model is used for the theoretical investigation. The results reveal combination resonances of additive and subtractive type, which are shown to be promising to increase the bandwidth of the resonator near primary resonance frequency. Our results also demonstrate the ability to shift the combination resonances to much lower or much higher frequency ranges. We also demonstrate the dynamic pull-in instability under mixed frequency excitation.     

Integrated magnetic sensing of electrostatically actuated thin-film microbridges

The movement of electrostatically actuated microbridges is measured by sensing the field of a permanent magnet, deposited and patterned on top of the microbridge, with a spin valve magnetic sensor fabricated beside the bridge at the level of the substrate. The spin valve sensor is sensitive to the position of the magnet and thus to the position of the bridge. The thin-film microbridges are fabricated using thin-film technology and surface micromachining at low temperatures (/spl les/100/spl deg/C) on glass substrates. The bridges are electrostatically actuated by applying a voltage between the bridges and a gate counter electrode placed beneath them. The deflection of the bridge is at the same time characterized optically by focusing a laser on the structure and monitoring the position of the reflected beam with a photodetector. A comparison of the bridge position sensing using the optical and magnetic methods is made. The absolute movement of the structures is measured with a precision close to 0.1 /spl Aring/ using the integrated magnetic sensor. The deflection of the electrostatically actuated structures is studied as a function of the applied gate voltage and length of the bridges. The experimental results show qualitative agreement with an electromechanical model.     

Closed-form empirical relations to predict the static pull-in parameters of electrostatically actuated microcantilevers having linear width variation

We develop novel closed-form empirical relations to estimate the static pull-in parameters of electrostatically actuated tapered width microcantilever beams. A computationally efficient single degree-of-freedom model is employed in the setting of Ritz energy technique to extract the static pull-in parameters of the distributed electromechanical model that takes into account the effects of fringing field capacitance. The accuracy of this single-dof model together with the variable-width equivalent of the Palmer’s fringing model is established through a comparison with 3D finite element simulations. A unique surface fitting model is proposed to characterize the variations of both the pull-in displacement and pull-in voltage, over a realistically wide range of system parameters. Optimum coefficients of the proposed surface fitting model are obtained using nonlinear regression analysis. Empirical estimates of pull-in parameters are validated against finite element simulations, and available experimental and numerical data. An excellent agreement indicates that the proposed relationships are sufficiently accurate to be safely used for the electromechanical design of tapered microcantilever beams.     

Experimental characterization of electrostatically actuated in-plane bending of microcantilevers

Experimental validation of numerical models developed by the authors to predict the static behaviour of microelectrostatic actuators is described. Cantilever microbeams, currently used in connection with RF-MEMS and micro-scale material testing were analysed. A set of microcantilevers, bending in the plane of the wafer, i.e. in the same plane as the profiling system’s target, was tested. This differs from the popular case of out-of-plane microbeams, usually studied in the literature. Geometry nonlinearity caused by large deflection of the microbeam was investigated and nonlinear coupled formulation of electromechanical equilibrium was performed. Coupled-field analysis was implemented using the Finite Element Method (FEM), to predict displacements and pull-in voltage measured by Fogale Zoomsurf 3D, subsequently plotting the displacement-versus-voltage curve to complete model validation. FEM nonlinear analysis, based on iterative approach with mesh morphing, and FEM non-incremental approach, including a special element proposed by the authors, are compared to the linear solution and to experimental results. Geometry nonlinearity appears relevant in microbeam modelling and requires a nonlinear solution of the coupled problem. Investigative work, which compared the results of 2D and 3D models to experimental data, revealed that some three dimensional effects are significant in model validation, but the 2D approach may be effective in predicting static behaviour provided that at least a microbeam thickness equivalent is adopted.     

Experimental measurements and behavioral modeling of an electrostatically actuated bi-axial micromirror

Recent developments in micro-electro-mechanical systems and micromachining technologies have made possible the development and integration of micromirrors that can be employed for a number of consumer applications such as free space optical switching (Syms in J Lightwave Technol 20(7):1084–1094, 2002; Pan in Micromech Microeng 14(1):129–137, 2004), 2D scanning or image projection (Texas Instruments DLP ). This high level of integration provides many advantages such as reproductivity and improved performance. Obviously, these increased levels of miniaturization raise new characterization and modeling concerns. We present in this paper the design, the fabrication process, the characterization and the behavioral modeling approach of a novel electrostatically actuated bi-axial micromirror. This micro-opto-electro-mechanical system is a result of a joint development between our partners Colibrys and Coventor companies (Colibrys ; Coventor ). In the field of this work, two HDL multi-physics models using Verilog-A language and Matlab have been developed and validated by different experimental measurements.     

Linearization of electrostatically actuated surface micromachined2-D optical scanner

This paper presents an effective method of linearizing the electrostatic transfer characteristics of micromachined two-dimensional (2-D) scanners. The orthogonal scan angles of surface micromachined polysilicon scanner are controlled by using quadrant electrodes for electrostatic actuation. By using a pair of differential voltages over a bias voltage, we could improve the distortion of projected images from 72% to only 13%. A theoretical model has been developed to predict the angle-voltage transfer characteristics of the 2-D scanner. The simulation results agree very well with experimental data. Differential voltage operation has been found to suppress the crosstalk of two orthogonal scan axes by both experiment and theoretically. We have found that a circular mirror is expected to have the lowest angular distortion compared with square mirrors. Perfect grid scanning pattern of small distortion (0.33%) has been successfully obtained by predistorting the driving voltages after calibration     

Numerical solution of static and dynamic nonlinear multi-degree-of-freedom systems

The performance of a dynamic algorithm is demonstrated on a series of nonlinear multi-degree-of-freedom systems. Nonlinear dynamics, large deflections and postbuckling behaviour are dealt with. In highly nonlinear problems of a prevailing static nature it is shown that, due to its clear physical interpretation, the dynamic approach allows easier guessing of initial conditions assuring convergence.     

Nonlocal and strain gradient based model for electrostatically actuated silicon nano-beams

Conventional continuum theory does not account for contributions from length scale effects which are important in modeling of nano-beams. Failure to include size-dependent contributions can lead to underestimates of deflection, stresses, and pull-in voltage of electrostatic actuated micro and nano-beams. This research aims to use nonlocal and strain gradient elasticity theories to study the static behavior of electrically actuated micro- and nano-beams. To solve the boundary value nonlinear differential equations, analogue equation and Gauss–Seidel iteration methods are used. Both clamped-free and clamped–clamped micro- and nano-beams under electrostatical actuation are considered where mid-plane stretching, axial residual stress and fringing field effect are taken into account for clamped–clamped cases. The accuracy of solution is evaluated by comparison of the pull-in voltages of different micro-electro-mechanical systems with those results previously published in the literature. A main drawback of the previous theoretical researches using nonlocal or strain gradient methods was that they don’t account for effects of the size on the Young modulus of the beam and merely they adjust the length scale parameters for small sizes to fit data with experimental results. In the present study, the experimental voltages for static pull-in of different micro- and nano-beams are used to estimate the silicon Young’s modulus, nonlocal and length scale parameters. Using the estimated parameters, pull-in voltages of silicon micro- and nano-beams based on strain gradient and nonlocal theories are compared with respect to each other and also with the experimental and previous theoretical results. The conducted results demonstrate that the predicted pull-in voltages using proposed strain gradient theory will give the best fit with experimental observations.