Dynamic behavior of micro-resonator under alternating current voltage

This paper investigates the dynamic behavior of a micro-resonator under various levels of Alternating Current (AC) voltage, without a biased Direct Current voltage. The governing equations are developed in the framework of Euler–Bernoulli beam theory, accounting for the effects of damping, fringing field, and mid-plane stretching using von Karman nonlinear strain. The steady-state frequency response of the micro-resonator is derived from the governing equations by the method of multiple scales. The transient response is also derived by the long-time integration. The results of our work reveal that the applied AC voltage and the mid-plane stretching (quantified by a stretching parameter) determine the characteristic feature of the dynamic behavior of the micro-resonator, such as the dynamic pull-in, the frequency response of linear or hardening characteristic. A design diagram in terms of AC voltage amplitude and stretching parameter is developed to show the domains of the different dynamic behavior characteristics. Our results also reveal the significant effects of damping and boundary conditions on the dynamic behavior and the design diagram of the micro-resonator.     

Size-dependent behaviour of electrically actuated microcantilever-based MEMS

In this paper, the nonlinear size-dependent static and dynamic behaviours of a microelectromechanical system under an electric excitation are investigated. A microcantilever is considered for the modelling of the deformable electrode of the MEMS. The governing equation of motion is derived based on the modified couple stress theory (MCST), a non-classical model capable of capturing small-size effects. With the aid of a high-dimensional Galerkin scheme, the nonlinear partial differential equation governing the motion of the deformable electrode is converted into a reduced-order model of the system. Then, the pseudo-arclength continuation technique is used to solve the governing equations. In order to investigate the static behaviour and static pull-in instabilities, the system is excited only by the electrostatic actuation (i.e., a DC voltage). The results obtained for the static pull-in instability predicted by both the classical theory and MCST are compared. In the second stage of analysis, the nonlinear dynamic behaviour of the deformable electrode due to the AC harmonic actuation is investigated around the deflected configuration, incorporating size dependence.     

介电弹性体驱动单元的动态特性分析

针对不同电压载荷情况下介电弹性体驱动单元的动态特性,考虑材料非线性、介电常数变化和惯性效应等因素的影响,从热力学能量转化的角度得到了基于Ogden应变能函数的介电弹性体驱动单元的二阶常微分运动方程,并分析了系统的动态响应及模型几何尺寸的影响。结果表明:介电弹性体驱动单元动态响应的振幅由电场强度决定。在恒定场强下,随着电场强度幅值的增大,驱动单元的振幅将随之增大、振动频率将随之减小;在简谐场强下,随着电场强度频率的增大,驱动单元将发生共振,且共振频率将随电场强度的增大而减小。     

On the nonlinear dynamics of a piezoelectrically tuned micro-resonator based on non-classical elasticity theories

Size dependent static and dynamic behavior of a fully clamped micro beam under electrostatic and piezoelectric actuations is investigated. The microbeam is modeled under the assumptions of Euler–Bernoulli beam theory. Viscous damping and nonlinearities due to electrostatic actuation and mid-plane stretching are considered. Residual stress and fringing field effect are taken into account as well. Governing equation of motion is derived using Hamilton’s principle along with the strain gradient theory (SGT), which is a non-classical continuum theory capable of taking size effect of elastic materials into account. Reduced order model of the partial differential equations of the system is obtained using Galerkin method. Static deflection, pull-in voltage and the primary resonance of the microbeam are examined and the effect of piezoelectric voltage and its polarization on the size dependent static and dynamic response is studied. It is found that the piezoelectric voltage can effectively change the flexural rigidity of the system which in turn affects the pull-in instability regime. The effect of material length scale parameter is examined by comparing the results of the SGT with the modified couple stress (MCST) and classical theory (CT), both of which are special cases of the former. Comparison demonstrates that the CT underestimates the stiffness and consequently the pull-in voltage and overestimates the amplitude of periodic solutions. The difference between the results of classical and non-classical theories becomes more and more as the dimensions of the system gets close to the length scale parameter. Non-classical theories predict more realistic behaviors for the micro system. The results of this paper can be used in designing microbeam based MEMS devices.     

热电耦合场中压电粘弹性微梁的静力坍塌分析

基于粘弹性理论、欧拉-伯努利梁理论以及压电理论,在考虑微结构蠕变现象的情况下,建立了热电耦合场中压电粘弹性微梁的静力坍塌模型及控制方程,研究了压电粘弹性微梁的静力坍塌临界条件,计算和分析了压电层控制电压和温度等参数对微梁结构静力坍塌的影响。研究结果表明:持久坍塌电压可作为压电粘弹性微梁静力坍塌失稳的临界条件,并且温度和压电层控制电压均会对坍塌失稳的临界值造成影响。该文的研究可为MEMS中梁式微结构的设计与优化提供理论参考。     

Stability of a micropolar fluid layer heated from below

The stability of a layer of micropolar fluid heated from below is studied employing a linear theory as well as an energy method. It is proved that the principle of exchange of stability holds and the critical Rayleigh number is obtained. It is observed that the micropolar fluid layer heated from below is more stable as compared with the classical viscous fluid. The energy method is then used to study the stability under finite disturbances. A variational method is applied to obtain the sharp stability limit. It is found that no subcritical instability region exists and the critical Rayleigh number as derived from the energy method is identical to that of the linear limit.     

时滞位置反馈对一类静电微传感器的吸合不稳定的控制研究

针对微结构在静电力作用下的吸合不稳定问题,考虑一类典型的单自由度静电驱动微传感器振动系统,将时滞位置反馈施加在系统的直流偏置电压上,研究引起微结构的动态吸合和吸合不稳定的系统参数条件,以及时滞反馈对吸合不稳定的抑制机理。运用Melnikov函数法得到时滞受控系统中引起结构吸合不稳定的交流电压的临界幅值。并基于时滞受控系统的安全域随控制参数的演变,定量上研究时滞反馈对吸合不稳定的控制。数值结果和理论分析均表明：在正的反馈增益系数和较小的时滞量下,时滞位置反馈能够有效地抑制静电驱动微结构的吸合不稳定现象。     

Stochastic analysis of dynamic characteristics and pull-in instability of FGM micro-switches with uncertain parameters in thermal environments

In this paper, the stochastic vibration characteristics of a functionally graded material micro-switch with random material properties near the pull-in instability are investigated. The uncertainties of the material properties and randomness of the composition of constituents due to the fabrication process are considered in this study. The micro-switch is modeled as a micro-beam and is under the effects of electrostatic and Casimir forces. The properties of the constituent materials are temperature dependent, and the system undergoes a change in temperature. The governing equations of motion of the Euler–Bernoulli micro-beam are derived based on modified couple stress theory and are solved using the state space form finite difference method. The statistics of the dynamic characteristics are obtained based on Monte Carlo simulation method. The effects of Casimir force, the length scale parameter, temperature dependencies of the material properties, the temperature change, the applied voltage and the volume fraction index on stochastic properties of the first natural frequency, the second natural frequency, the pull-in gap, and the pull-in voltage are studied in detail.     

Pull-in instability of carbon nanotube-reinforced nano-switches considering scale,surface and thermal effects

In this paper, an analytical method is presented to investigate the effect of surface characteristic and temperature change on the pull-in instability of electrically actuated nano-switches reinforced by carbon nanotubes (CNTs) based on Eringen's nonlocal beam theory. An extremely nonlinear fourth-order governing equation for the doubly clamped nano-switches made of CNTs/Si composites nanobeam is derived and solved by using the principle of virtual work, where Van der Waals force as atomic interactions and Casimir force as macro effects of quantum field fluctuation of vacuum are combined as an electrostatic force with fringing field effects. The results show that both the pull-in voltage and pull-in deflection of CNTs/Si composite nanobeam increase with the increase of CNTs volume ratio but decrease with the increase of temperature change. The coupling influences of small scale parameter, geometric behavior, surface characteristic and thermal effect on the pull-in instability of electrostatically actuated CNTs/Si nanobeam are detailedly discussed.     

Dynamic analysis of carbon nanotubes under electrostatic actuation using modified couple stress theory

Modified couple stress theory is a size-dependent theorem capturing the micro/nanoscale effects influencing the mechanical behaviors of the micro- and nanostructures. In this paper, it is applied to investigate the nonlinear vibration of carbon nanotubes under step DC voltage. The vibration, natural frequencies and dynamic pull-in characteristics of the carbon nanotubes are studied in detail. Moreover, the effects of various boundary conditions and geometries are scrutinized on the dynamic characteristics. The results reveal that application of this theory leads to the higher values of the natural frequencies and dynamic pull-in voltages.     

Specific features of attenuated light transmission by liquid-crystal twist cells in constant and alternating electric fields

Optical transmission characteristics of dual-frequency nematic liquid crystal (NLC) twist cells with different alignment layers (rubbed polyimide and obliquely deposited cerium dioxide) have been studied in constant and alternating electric fields. It has been established that a change in the optical (twist effect) threshold and dynamic range of attenuated transmission depend both on the boundary conditions (that influence the screening of applied voltage) and on the parameters of the applied electric field. The maximum dynamic range (49.5 dB) has been obtained in the cell with a CeO₂ alignment layer controlled by a constant potential. In the case of an alternating electric field, the dynamic range decreases because of reduced effective voltage.     

Thermoelectromechanical stability of dielectric elastomers undergoing temperature variation

In this paper, the influence of both temperature and deformation on dielectric constant is considered during the establishment of free energy function of dielectric elastomers. A constitutive model of the thermodynamic systems undergoing adiabatic process is derived to study its thermoelectromechanical stability. The relations between different work conjugated parameters of dielectric elastomer are theoretically described, including the relations between nominal electric field and nominal electric displacement, entropy and temperature. Under different temperatures and electric fields, the allowable energy range of dielectric elastomer is calculated. Furthermore, the electric-induced variation of dielectric elastomer’s temperature and entropy is also studied under various principal planar stretch ratios. These simulation results should offer assistances in guiding the design and fabrication of excellent actuators featuring dielectric elastomers.     

粘弹性圆柱壳在轴向恒压下的动力稳定性

基于Timoshenko-Mindlin假设,得到考虑粘弹性的各向同性圆柱壳及纤维增强正交铺设层合圆柱壳在轴向恒压下的动力学方程。文中对两端简支的圆柱壳进行了分析,依Laplace变换,导出动力稳定的特征方程,由Routh-Hurwitz判据建立动力稳定性条件,对两类圆柱壳讨论了横向剪切变形的影响。     

Dynamic analysis of an electrostatically actuated circular micro-plate interacting with compressible fluid

This paper investigates the dynamic behavior of a circular micro-plate interacting with compressible fluid and excited by electrostatic force. Utilizing Kirchhoff’s thin plate theory for the actuating micro-plate and assuming inviscidity for the operating fluid, an eigenvalue problem of the coupled system is derived using Fourier-Bessel expansion. Investigating the change in free vibration properties of the system, a parametric study is done accounting for the variation of physical and geometric properties of the bounded domain. Then considering step input voltages, the response of the coupled system, pull-in time, and pull-in voltages are derived. It is shown that besides the electric permittivity the inertial effect of the contained fluid also changes the transient response significantly. The impact of fluid added mass is observed in the decreased response frequencies and increased pull-in times. In addition, by constructing phase plane diagrams, it is found that the attraction zones of stable fixed points vary for different contained fluids. This could affect the response of the micro-plate qualitatively when there exists an uncertainty in the initial conditions.     

A nonlinear bilayer beam model for an interfacial crack in dielectric bimaterials under mechanical/electrical loading

A bilayer beam model is extended to study the fracture behavior of dielectric interfacial cracks. In this model, a semi-infinite crack with an original opening value is oriented along the interface between two dielectric layers which are under mechanical/electrical loading. Taking into account the effect of the electrostatic traction on the interfacial crack, a nonlinear analytical solution is derived, along with also a developed finite element analysis method where a special constitutive equation for the capacitor element in ANSYS is utilized to simulate the electrostatic tractions. Both the analytical and numerical solutions predict the same results which further show that the elastic and dielectric mismatches can play a significant role in the interfacial cracking behavior under mechanical and electrical loading. Furthermore, the electrostatic tractions may cause hysteresis loops in the curve of crack opening versus applied mechanical displacement or versus applied electric voltage. An applied mechanical load is the driving force for the interfacial cracking, while an applied electric field retards it.     

温度变化对受静电力作用微圆板的影响分析

以考虑温度效应时,受静电力作用的微圆板静力弯曲变形微分方程为基础,该文首先针对轴对称问题的特点,利用级数展开和求极限法则,对圆板静力弯曲变形微分方程进行了改进。改进后的微分方程消除了圆心处的奇异性。其次,利用改进后的圆板静力变形弯曲微分方程,对周边固支圆板受静电力和温度变化作用下的受力变形进行分析。分析过程分为3个阶段：正常模式、过渡模式和接触模式。在数值求解圆板弯曲变形微分方程时,主要将非线性微分方程的求解化成迭代求解两个未知量的问题,一个未知量是形成静电场力的电压,而另一个未知量是圆板的一个边界条件。在实例中,给出了部分计算结果,包括：温度变化对不同阶段圆板受力变形的影响,温度变化对吸合电压的影响,以及不同温度变化下,圆板的几何尺寸对吸合电压的影响,从而可以更好地了解温度变化的作用和影响。     

Nonlocal problems in MEMS device control

Perhaps the most ubiquitous phenomena associated with electrostatically actuated MEMS devices is the `pull-in' voltage instability. In this instability, when applied voltages are increased beyond a certain critical voltage there is no longer a steady-state configuration of the device where mechanical members remain separate. This instability severely restricts the range of stable operation of many devices. Here, a mathematical model of an idealized electrostatically actuated MEMS device is constructed for the purpose of analyzing various schemes proposed for the control of the pull-in instability. This embedding of a device into a control circuit gives rise to a nonlinear and nonlocal elliptic problem which is analyzed through a variety of asymptotic, analytical, and numerical techniques. The pull-in voltage instability is characterized in terms of the bifurcation diagram for the mathematical model. Variations in various capacitive control schemes are shown to give rise to variations in the bifurcation diagram and hence to effect the pull-in voltage and pull-in distance.     

On the dynamic behavior of a piezoelectric-elastic laminate with a crack in the piezoelectric material under electro-mechanical loads

The dynamic behavior of a piezoelectric-elastic laminate with a crack in the piezoelectric material under in-plane steady-state electro-mechanical loads is considered. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which are solved using Chebyshev polynomial expansions. Numerical results are provided to show the variation of both the dynamic stress intensity factors and electric displacement intensity factor with frequencies of the applied electro-mechanical loads. A phenomenon similar to “resonance” is observed when the applied loads act in some specific ranges of frequencies, and both the dynamic stress intensity factors and electric displacement intensity factor may increase significantly, which will lead to the failure of piezoelectric material. The effects of applied electric fields, crack geometry and elastic layer thickness on the phenomenon are also discussed.     

Flexural vibration behavior of piezoelectric heterogeneous bimorph beams

Bending oscillations of piezoelectric bimorph beams are effective sound sources in gases or fluids, and, therefore, of practical interest. On the basis of the piezoelectric constitutive relations and the elastodynamic equations, the differential equation of flexural vibrations of thin rectangular piezoelectric heterogeneous bimorph beams, consisting of a piezoelectric layer glued onto an elastic substrate, is derived. The piezoelectric layer is polarized in thickness direction and can be excited to thickness vibrations by an electric alternating current voltage applied to electrodes covering the upper and lower surface of the layer. This causes an oscillating transverse contraction in the piezoelectric layer but not in the substrate, and, thus, generates flexural vibrations of the beam. The differential equation is solved analytically for beams of finite length with both ends free, one clamped and one free end, as well as for both ends clamped. Their vibration behavior in viscous fluids is considered. For a piezo-ceramic composite layer joined to a steel plate vibrating in air and in water, the analytical results are evaluated numerically as function of frequency     

Mechanical behavior of a FGM micro-beam subjected to a nonlinear electrostatic pressure

The present article studies the mechanical behavior of a FGM micro-beam subjected to a nonlinear electrostatic pressure. The FGM micro-beam is made of metal and ceramic and material properties vary continuously along the beam thickness according to a power-law. The nonlinear equation of dynamic motion of the FGM micro-beam is derived. By solving the equation of the static deflection, equilibrium positions of the micro-beam are determined and shown in the state control space. To study the stability of the fixed points, the trajectories of the beam motion are illustrated in the phase plane for different initial conditions. In order to find the response of the micro-beam to a step DC applied voltage, the nonlinear equation of motion is solved using a Galerkin based reduced order model. Moreover, time histories and phase portraits for different applied voltages are illustrated. The effect of different power law exponent on the stability of the micro-beam is studied.