Characterization of extensional multi-layer microbeams in pull-in phenomenon and vibrations

This paper presents an analysis of the pull-in instability and vibrational behaviors for a multi-layer microbeam actuated electrostatically. Based on the accurate geometrically nonlinear theory of Euler–Bernoulli beams, a distributed electromechanical model that accounts for finite deformation and residual stress is proposed. The governing differential equations are established in the form similar to those of the single-layer beam theory by re-determination of the neutral axis. These equations, in conjunction with corresponding boundary conditions, are transformed into two two-point boundary value problems. The geometrical nonlinearity is taken into account in static deformation research. The pull-in parameters are obtained using the shooting method through taking the applied voltage as an unknown and the central deflection as a control parameter. The same algorithm is applied to the small-amplitude free vibration around the predeformed bending configuration following an assumed harmonic time mode. The corresponding fundamental natural frequency is presented. The proposed method is validated by comparing several case studies with available published simulations. The influences of pivotal parameters on the pull-in instability behavior and natural frequency are examined, including the length, thickness and residual stress of the microbeam.     

Pull-in analysis of an electrostatically actuated nano-cantilever beam with nonlinearity in curvature and inertia

In this paper, the nonlinear behavior of electrostatically actuated carbon nanotubes (CNTs) is investigated based on a comprehensive model with nonlinearity in curvature, inertia and electrostatic force. The aim of this study is to show when the nonlinear formulation needs to be taken into account and when the linear formulation can simulate the system behavior accurately. The model comprises a cantilevered CNT suspended over a fixed electrode plate from which a DC potential difference is imposed. A relatively large gap between the CNT and the ground plate is considered. The versatile Galerkin method is employed to reduce the nonlinear integro-differential equations of motion to a set of nonlinear ordinary differential equations in time, and then, the reduced equations are solved by direct numerical integration. Dynamic response of the system before and beyond the pull-in voltages and effect of gap to length ratio of the CNT are studied. It is shown that in a large gap to length ratio, when the applied voltage is close to the corresponding pull-in voltage the nonlinear terms have a profound role in the dynamic behavior of the system. Eventually, the contribution of nonlinear terms are examined and it is found that the nonlinear inertia and curvature terms have softening and hardening effects, respectively, whereas the hardening effect of the nonlinear curvature has a major contribution.     

Static deflection and pull-in instability analysis of an electrostatically actuated mirocantilever gyroscope considering geometric nonlinearities

In this paper, a mathematical modeling of a microcantilever gyroscope is presented considering the nonlinearities of the system due to electrostatic forces, fringing field, geometry and the inertial terms. The microgyroscope is actuated and detected by electrostatic methods and subjected to coupled bending oscillations. First a system of two nonlinear integro-differential equations is derived which describes flexural-flexural motion of electrostatically actuated and detected microbeam gyroscopes. Afterward, static deflection and pull-in instability of the microgyroscopes acted upon by DC voltages in both (driving and sensing) directions are studied for different parameters. The model’s predictions are in good agreement with the experimental data found in the literature and finite element simulation. Results show that the nonlinearities become important when pull-in happens.     

Modeling the effect of intermolecular force on the size-dependent pull-in behavior of beam-type NEMS using modified couple stress theory

Experimental observations reveal that the physical response of nanostructures is size-dependent. Herein, modified couple stress theory has been used to study the effect of intermolecular van der Waals force on the size dependent pull-in of nanobridges and nanocantilevers. Three approaches including using differential transformation method, applying numerical method and developing a simple lumped parameter model have been employed to solve the governing equation of the systems. The pull-in parameters i.e. critical tip deflection and instability voltage of the nanostructures have been determined. Effect of the van der Waals attraction and the size dependency and the importance of coupling between them on the pull-in performance have been discussed.     

静电微泵吸合与释放特性研究

为避免静电微泵因驱动电压过高而出现吸合现象,建立描述静电泵膜吸合与释放现象的一维集中质量模型、二维分布式模型和三维有限元模型,对比研究静电致动泵膜在准静态下吸合与释放的循环过程,获得整个循环过程的泵膜挠曲线变化、泵腔体积变形、电容变化等大量的数据与曲线.比较结果表明,二维分布式模型求解效果最好,并利用三维有限元模型和宏...     

求解静电驱动微梁变形的数值与解析方法

给出了一种静电力作用下微悬臂梁控制方程的数值求解方法,提出了求解该模型的近似解析式和修正解析式,并采用模态叠加法计算微梁的动响应。结果表明,修正解析式、数值算法和动响应方法计算的微梁变形结果是一致的。同时也揭示了微梁振动响应周期与驱动电压成正比、动态临界电压低于静态临界电压等相关特性。     

Electromechanical Behavior of Microbeams with Piezoelectric and Electrostatic Actuation

In this paper we present an analysis of a novel microelectromechanical (MEM) cantilever switch that incorporates piezoelectric layers bonded to the microbeam. We have developed a model of a microbeam that can be actuated by applying voltage to the electrostatic areas and piezoelectric layers. The nonlinear governing equations have been derived and linearized using a step-by-step linearization method and the linear system of equations has been solved by means of the generalized differential quadrature method. The results for a simple micro cantilever MEM switch are in good agreement with other simulation results and demonstrate the feasibility of the concept and the effectiveness of the proposed numerical methods. The numerical results show that the proposed model can be actuated electrostatically and piezoelectrically in three different types of actuation sequences, and we show that, as expected, the switching voltages can be reduced due to the high equivalent piezoelectric coefficient.     

介电型电活性聚合物圆柱驱动器特性

介电型电活性聚合物(Electroactive polymer, EAP)驱动器的动、静态特性是其合理使用及优化设计的重要依据。构建圆柱形驱动器几何模型描述其几何变形,结合EAP膜机电耦合方程推导驱动器轴向线性运动的动力学方程。驱动器的电压-位移数值计算结果显示内外层EAP膜柔性电极涂覆方式会影响驱动器位移。基于驱动器的电压位移关系,对其主要失效形式进行讨论。将基于neo-Hookean弹簧的1个时间常数黏弹性模型修正为具有2个时间常数的模型,并将2种模型运用于阶跃和周期电压激励下的驱动器动态响应研究。理论计算与试验结果的对比分析表明,2个时间常数的黏弹性模型对驱动器的动态位移描述更为准确。在周期激励电压下,提高频率会显著减小驱动器位移幅值,驱动器的“激励死区”会减小其振动中心偏移及振动幅度。     

Estimation of oscillation period/switching time for electrostatically actuated microbeam type switches

The undamped dynamic response of step-voltage driven parallel-plates, cantilever, and fixed-fixed type electromechanical switches is numerically investigated. In each case, application of energy technique yields the threshold values of the amplitude and the applied voltage beyond which the oscillatory motion of the movable electrode ceases to exist. These critical values are identified as the dynamic pull-in parameters of the corresponding microactuator model. For all three microactuator configurations, empirical expressions for the switching time and oscillation period are developed. These empirical relations are applicable over a wide range of applied voltage, and the estimates obtained using the proposed empirical relations correlate very well with the previously published results. Furthermore, the phase portraits of these actuators have been thoroughly investigated in order to examine the role of static pull-in point in a dynamic setting and also to propose the design rules to build faster microswitches.     

Analysis of thermoelastic damping in microresonators by considering the stretching effect

In this paper, the previous theory for obtaining the quality factor of thermoelastic damping in microresonators has been modified. Here, the stretching term for the neutral axis of bending is considered, which has been neglected in previous work. The Taylor expansion of electric actuation has described the static position of microbeam, which is more accurate than previous work that extends it about the straight position of a microbeam. Also, the theory of this work may be used for two layered microresonators without this assumption that the length of two layered electrically actuated microresonators may be obtained, for the first time. To obtain this complete theory firstly the equations of motion coupled with thermal effects are obtained in a two layered microbeam where the second layer is deposited on a part of the first layer length. Then, the quality factor of thermoelastic damping of the system, i.e. the half of the ratio of the real part of the free vibration resonance frequency to its imaginary part, has been obtained by solving the free vibration equation using the strained parameter perturbation method. It has been shown that neglecting the stretching of neutral axis of bending and expanding the electrostatic actuation about the straight position of the microbeam considered in previous researches cause an error that is not negligible for the voltage larger than 30% of the pull-in voltage. Also, it is shown that the quality factor of thermoelastic damping in two layered microresonators depends on the value of length, thickness and the kind of material of the deposited layer.     

Nonlinear behavior of capacitive micro-beams based on strain gradient theory

This paper studies the size dependent behavior of materials in MEMS structures. This behavior becomes noticeable for a structure when the characteristic size such as thickness or diameter is close to its internal length-scale parameter and is insignificant for the high ratio of the characteristic size to the length-scale parameter, which is the case of the silicon base micro-beams. However, in some types of micro-beams like gold or nickel bases, the size dependent effect cannot be overlooked. In such cases, ignoring this behavior in modeling will lead to incorrect results. Some previous researchers have applied classic beam theory on their models and imposed a considerable hypothetical value of residual stress to match their theoretical results with the experimental ones. The equilibrium positions or fixed points of the gold and nickel micro-beams are obtained and shown that for a given DC voltage, there is a considerable difference between the obtained fixed points using classic beam theory, modified couple stress theory, and modified strain gradient theory. In addition, it is shown that the calculated static and dynamic pull-in voltages using higher order theories are much closer to the experimental results and are higher several times than those obtained by classic beam theory.     

Fluid-solid interaction in electrostatically actuated carbon nanotubes

This paper deals with investigation of fluid flow on static and dynamic behaviors of carbon nanotubes under electrostatic actuation. The effects of various fluid parameters including fluid viscosity, velocity, pressure and mass ratio on the deflection and pull-in behaviors of the cantilever and doubly clamped carbon nanotubes are studied. Furthermore, the effects of temperature variation on the static and dynamic pull-in voltages of the doubly clamped carbon nanotubes are reported. The results reveal that altering the fluid parameters significantly changes the mechanical and pull-in behaviors. Hence, the proposed system can be applied properly as a nano fluidic sensor to sense the various parameters of the fluid.     

On the electromechanical modelling of a resonating nano-cantilever-based transducer

An electromechanical model for a transducer based on a lateral resonating cantilever is described. The on-plane vibrations of the cantilever are excited electrostatically by applying DC and AC voltages from a driver electrode placed closely parallel to the cantilever. The model predicts the static deflection and the frequency response of the oscillation amplitude for different voltage polarization conditions. For the electrostatic force calculation the model takes into account the real deflection shape of the cantilever and the contribution to the cantilever-driver capacitance of the fringing field. Both the static and dynamic predictions have been validated experimentally by measuring the deflection of the cantilever by means of an optical microscope.     

A modeling and analysis of spring-shaped torsion micromirrors for low-voltage applications

An electrostatic torsion micromirror is designed using the optimized spring-shaped torsion beams and U-shaped groove supporters. The main advantages of this design will be a reduction in the pull-in voltage for low-voltage applications and the function as a switch of the instability mode by adjusting the effective bending stiffness of the torsion beam. In this design, a theoretical model is developed to demonstrate the static characteristics of the electrostatic torsion micromirror. The pull-in effect is investigated specifically to predict the pull-in voltage, pull-in angle, and pull-in displacement. These parameters depend significantly on the electrode size and position, position of the groove, and ratio of the bending and torsion effect of the torsion beam. In addition, the effective torsion and bending stiffness model is provided using the energy method with the objectives to optimize the spring-shaped geometries of the torsion beams and to decrease the pull-in voltage below 2 V. The U-shaped groove supporters are applied in the theoretical model to adjust the effective bending stiffness of the torsion beam and to switch the instability mode of the torsion micromirror. The theoretical analysis is consistent with the numerical simulation using MEMCAD and experimental results measured by an optical projection method.     

A bulk-flow analysis of static and dynamic characteristics of floating ring seals

The bulk-flow model was used to calculate the static and the dynamic characteristics of floating ring seals because the solutions based on the numerical integration of the complete Navier–Stokes equations can be very time consuming. A steepest-descent method is used to find the seal's equilibrium position efficiently. The bulk-flow model is governed by three partial differential equations on eccentric working conditions. A finite-difference scheme has been used to solve the nonlinear governing equations. Compared to Nelson and Nguyen's fast fourier transform method, this scheme has better consistency. Perturbation analysis of the flow variables yields a set of zeroth and first-order equations. The SIMPLE algorithm is used to integrate the system of bulk-flow equations. Comparisons of the numerical predictions (lock-up eccentricity ratio, leakage flow rate and rotordynamic coefficients) with Ha's results, which were formulated using the Fourier series, and experimental data are presented subsequently.     

简谐电压激励下静电谐波微电机的响应分析

提出了一种基于静电驱动的新型谐波微电机，描述了该微电机的基本工作原理。给出了微电机中静电场力的静态分量、动态分量和简谐电压激励下的载荷。将受迫振动的位移解表示为柔轮自由振动振型的级数形式，进而把按照Donnell壳理论建立的控制微分方程予以化简。利用壳体振型的正交性，确定了简谐电压激励下系统的动态响应，并总结了其规律。最后，分析了系统在电压激励下的共振特性，以及随主要系统参数的变化规律。       

On the Mathematical Modeling of a MEMS-Based Sensor for Simultaneous Measurement of Fluids Viscosity and Density

This paper discusses a novel microelectromechanical sensor for simultaneous measurement of fluids viscosity and density. Its operation is based on a longitudinal piezoelectric actuation of a micro-beam in a laminar fluid field. The proposed sensor consists of a micro-beam and a micro-cylindrical sensing element immersed in a fluid. In order to actuate the sensor longitudinally, the micro-beam is bonded with two piezoelectric layers on its upper and lower surfaces which are subjected to an AC voltage. The coupled governing partial differential equations of the micro-beam and fluid field have been derived. The obtained governing differential equations with time-varying boundary conditions have been transformed to an enhanced form with homogeneous boundary conditions. The enhanced equations have been discretized over the beam and fluid domain using Galerkin based reduced order model. The dynamic response of the sensing element for different piezoelectric actuation voltages and different exciting frequencies has been investigated. The effects of viscosity and density of fluids and geometrical parameters of the sensor on sensing element response have been studied.     

力反馈微加速度计的准静态和阶跃响应模型

对于微机械电容式加速度计,为了检测微小电容变化,必须引入交直流电压驱动信号,然而该静电力的存在显著影响传感器的最大可靠工作范围。对准静态惯性信号和阶跃惯性信号,深入研究了力反馈加速度计的可靠工作范围,建立了不同偏压配置的驱动信号对引起吸合失效的临界加速度信号的影响模型。研究结果表明静电力对阶跃信号的影响要比准静态信号大,动态工作条件下所能承受的最大阶跃加速度信号比准静态信号要小一个数量因子。对于双边驱动的力反馈加速度计,正正偏压配置或正负偏压配置可以最大程度地减小驱动信号引起的传感器吸合失效,从而增大可靠工作范围;而负负偏压配置或负正偏压配置增大了驱动信号效应,从而大大减小了可靠工作范围。     

Atomic force microscope characterization of a resonating nanocantilever

An atomic force microscope (AFM) is used as a nanometer-scale resolution tool for the characterization of the electromechanical behaviour of a resonant cantilever-based mass sensor. The cantilever is actuated electrostatically by applying DC and AC voltages from a driver electrode placed closely parallel to the cantilever. In order to minimize the interaction between AFM probe and the resonating transducer cantilever, the AFM is operated in a dynamic non-contact mode, using oscillation amplitudes corresponding to a low force regime. The dependence of the static cantilever deflection on DC voltage and of the oscillation amplitude on the frequency of the AC voltage is measured by this technique and the results are fitted by a simple non-linear electromechanical model.     

基于双电压驱动下高速开关阀的特性研究

为了提高高速开关阀的响应特性,采用双电压驱动的控制方法。运用AMESim建立高速开关阀的动态仿真模型,仿真分析得出不同控制方法下（单电压驱动、双电压驱动）高速开关阀的动态特性曲线及其特征值,通过试验测试证实了仿真结果的可靠性。结果表明：当高压源为25 V时,双电压驱动下开关阀的吸合时间减少了28.3%,释放时间减少了44.2%。仿真分析基于该高速开关阀在工作因素不变的情况下,不同控制方法对其动态特性的影响,结果进一步证实了双电压驱动控制方法的优越性。