A coupled two degree of freedom pull-in model for micromirrors under capillary force

The current paper presents a two degree of freedom model for the problem of micromirrors under capillary force. The principal of minimum potential energy is employed for finding the equilibrium equations governing the deflection and the rotation of the micromirror. Then, using the implicit function theorem, a coupled bending–torsion model is presented for pull-in characteristics of micromirrors under capillary force and the concept of instability mode is introduced. It is observed that with increasing ratio of bending and torsion stiffness, the dominant instability mode changes from bending mode to the torsion mode. In order to verify the accuracy of the coupled model, static behavior of a group of micromirrors is investigated both analytically using the presented model and numerically using the commercial finite element software ANSYS. It is observed that results of the coupled model match well with the results of finite element simulations, but they both deviate considerably from the results of the pure torsion model. The presented coupled model can be used for safe and stable design of micromirrors under capillary force.     

Nonlinear nonlocal pull-in instability of boron nitride nanoswitches

In the present study, nonlinear pull-in instability of boron nitride nanoswitches (BNNSs) subjected to electrostatic and van der Waals (vdW) forces is investigated. Based on Euler–Bernoulli beam theory, von Kármán geometric nonlinearity, nonlocal piezoelasticity theory and the principle of virtual work, the governing equations are obtained. The differential quadrature method is employed to discretize the nonlinear governing equations, which are then solved by a direct iterative method to obtain the nonlinear pull-in and pull-out voltages for cantilever and fixed–fixed boundary conditions. A detailed parametric study is conducted to elucidate the influences of nonlocal parameter, vdW force, fringing field, beam length and gap distance on the behavior of the pull-in instability voltage. Numerical results indicate that the magnitude of the pull-in voltage increases with increase in the gap distance. Furthermore, as the effective gap distance increases, the pull-out voltage tends toward the electrostatically pull-out voltage.     

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。     

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.     

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.     

滑条式静磁微执行器的pull-in现象分析

从滑条式静磁微执行器的模型出发,应用静磁微执行器的pull—in方程,分析了滑条式静磁微执行器的pull-in现象,用磁动势控制模型和力学平衡两种方式得出了执行器的吸合位置及吸合时的磁动势.对于特定的滑条式磁微执行器结构,pull-in位置为初始气隙宽度的1／3处．算例计算结果表明,该静磁微执行器的pull-in磁动势仅为3．84A．     

一类静电驱动微结构谐振传感器的吸合不稳定性研究及控制

本文以一类静电驱动微结构谐振传感器为研究对象,基于安全域思想研究了系统直流偏置电压和交流激     

The influence of the intermolecular surface forces on the static deflection and pull-in instability of the micro/nano cantilever gyroscopes

In this paper, the effects of van der Waals and Casimir forces on the static deflection and pull-in instability of a micro/nano cantilever gyroscope with proof mass at its end are investigated. The micro/nano gyroscope is subjected to coupled bending motions which are related by base rotation and nonlinearities due to the geometry and the inertial terms. It is actuated and detected by capacitance plates which are placed on the proof mass. The extended Hamilton principle is used to find the equations governing the static behavior of the clamp-free micro/nano gyroscopes under electrostatic, Casimir and van der Waals forces. The equations of static motion are discritized by Galerkin’s decomposition method. The nonlinear equilibrium equations are solved analytically using homotopy perturbation method (HPM). The static response of the micro/nano gyroscopes to variations in the DC voltage across the drive and sense electrodes is obtained and the effects of different parameters on pull-in instability are investigated. The presented results can be used for accurate estimations of the instability and performance of the micro/nano gyroscopes.     

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。     

Stability analysis of a capacitive micro-resonator with embedded pre-strained SMA wires

This paper presents a study on the effects of the SMA wires’ characteristics on tuning the stability of a capacitive micro-resonator. In the proposed model, pre-strained SMA wires have been embedded in a double clamped resonant microbeam which is actuated electrostatically. The governing equations of the system have been introduced and then an eigen-value problem has been adopted to investigate stability. Galerkin-based numerical methods have been used to solve the governing equation of motion for obtaining the motion trajectories of the beam. The effects of the number of SMA wires, their diameter, pre-strain and temperature on the pull-in instability and frequency response of the resonator have been shown. Critical values of recovery stress and SMA temperature for avoiding instability, with and without applying DC voltage have been obtained. The results have shown that by changing each of the SMA parameters, one can reach a needed magnitude of recovery stress as well as transmitted longitudinal force to the host beam, and consequently control and tune the stability and resonance frequency of the micro-resonator.

     

《薄壁Timoshenko梁弯扭耦合振动的动态有限元法》

通过直接求解单对称均匀薄壁Timoshenko梁单元弯扭耦合振动的运动微分方程,推导了其精确的动态刚度矩阵。在本文研究中考虑了弯扭耦合、翘曲刚度、转动惯量和剪切变形的影响。针对某弯扭耦合的薄壁梁算例,应用本文推导的动态刚度矩阵,采用自动Muller法和结合频率扫描法的二分法求解频率特征方程,计算了该薄壁梁的固有特性,并讨论了翘曲刚度、剪切变形和转动惯量对该弯扭耦合薄壁梁的固有频率和模态形状的影响。数值结果验证了本文方法的精确性和有效性,并指出随着模态阶次的增加,剪切变形、转动惯量和翘曲刚度对薄壁梁的固有特性的影响更加显著。     

Coupled mechanism and pull-in instability of several probe-membrane assemblies subjected to electrostatic force

The mathematical model of a coupled probe-membrane system subjected to electrostatic force is constructed. It is different than the conventional nano/micro actuator, which is constructed by two independent fixed/mobile conducting electrodes. The formula of pull-in voltage for different membranes is presented. The effects of several parameters on the pull-in instability are studied. The analytical method for the coupled vibration is proposed. The effects of the probe tips and several differential boundary conditions of membranes on the coupled characteristics are investigated. The coupled characteristic mechanism is clearly described.     

Coupled bending–torsional dynamic stiffness matrix for axially loaded beam elements

Analytical expressions for the coupled bending–torsional dynamic stiffness matrix elements of an axially loaded uniform beam element are derived in an exact sense by solving the governing differential equations of motion of the beam. The influence of axial force on the coupled bending–torsional frequencies of a cantilever and hinged–hinged beam of thin-walled section is demonstrated by numerical results. Application of the developed theory includes coupled bending–torsional frequency and mode calculations of helicopter, turbine and propeller blades, plane and space frames, and grillages consisting of axially loaded beam elements with non-coincident mass centre and shear centre.     

非对称Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀Bernoulli-Euler薄壁梁单元自由振动的控制运动微分方程,推导了其精确的动态传递矩阵.采用Bernoulli-Euler弯扭耦合梁理论,假定梁横截面没有任何对称性,考虑了薄壁梁在两个方向的弯曲振动及翘曲刚度的影响.动态传递矩阵可以用于计算非对称薄壁梁及其集合体的精确固有频率和模态形状.针对具体的算例,给出了各种边界条件下固有频率的数值结果并与文献中已有的结果进行了比较,还讨论了翘曲刚度对固有频率和模态形状的影响,结果表明如果忽略翘曲刚度的影响,可能得到毫无意义的结果.     

Fabrication of piezoelectric ceramic micro-actuator and its reliability for hard disk drives

A new U-type micro-actuator for precisely positioning a magnetic head in high-density hard disk drives was proposed and developed. The micro-actuator is composed of a U-type stainless steel substrate and two piezoelectric ceramic elements. Using a high-d31 piezoelectric coefficient PMN-PZT ceramic plate and adopting reactive ion etching process fabricate the piezoelectric elements. Reliability against temperature was investigated to ensure the practical application to the drive products. The U-type substrate attached to each side via piezoelectric elements also was simulated by the finite-element method and practically measured by a laser Doppler vibrometer in order to testify the driving mechanics of it. The micro-actuator coupled with two piezoelectric elements featured large displacement of 0.875 microm and high-resonance frequency over 22 kHz. The novel piezoelectric micro-actuators then possess a useful compromise performance to displacement, resonance frequency, and generative force. The results reveal that the new design concept provides a valuable alternative for multilayer piezoelectric micro-actuators.     

Coupled bending–torsional dynamic stiffness matrix for beam elements

Explicit expressions for the coupled bending–torsional dynamic stiffness matrix of a uniform beam element are derived in an exact sense by solving the governing differential equation of the beam. Implementation of the derived dynamic stiffness matrix in a space frame computer program is discussed with particular reference to an established algorithm to enable vibration analysis of coupled systems to be made. The application of the theory is demonstrated by an illustrative example wherein the results for a cantilever beam with a substantial amount of coupling between bending and torsion are highlighted. The correctness of the theory is confirmed to a high degree of accuracy by computed results and numerical checks.     

The restrained torsional response of open section carbon fibre composite beams

The analysis procedure outlined in this paper essentially makes use of the existing isotropic theories of torsion suitability modified to account for the non isotropic nature of typical carbon fibre composite material.

The warping and St Venant torsional stiffnesses of the beams are determined using the appropriate equivalent engineering elastic constants of the composite material which correspond to the membrane and bending modes of action respectively.

The differential equation governing the constrained torsional equilibrium of the open section beams is solved exactly in the paper for Z and channel sections with some emphasis being given to the influence of ply stacking sequence.

Theoretical results are presented in graphical form and these depict the variations in warping displacement, warping shear flow and longitudinal or axial constraint force intensity with applied torque for the cantilever beam condition with torque applied at the free end.

The paper also gives details of finite element studies of the composite beams and of an experimental programme of work pertaining solely to the behaviour of composite Z beams.

Comparisons between theory, finite element and experiment are presented and these are seen to give exceptionally close agreement.

It is clearly indicated that fibre orientation significantly influences the restrained torsional behaviour of thin-walled open-section composite beams. It is also clear that the use of the appropriate equivalent engineering elastic material constants in the theory is able to closely predict actual behaviour.     

空间薄壁截面梁非线性有限单元模型

基于Timoshenko梁理论和Vlasov薄壁杆件理论,通过设置单元内部节点并对弯曲转角和翘曲角采取独立插值的方法,建立了可考虑横向剪切变形和扭转剪切变形及其耦合作用、弯扭耦合、以及二次剪应力影响的空间薄壁梁非线性有限元模型。以更新的拉格朗日格式描述的几何非线性应变推得几何刚度矩阵。同时考虑了材料非线性,假定材料为理想塑性体,服从Von Mises屈服准则和Prandtle-Reuss增量关系,采用有限分割法,由数值积分得到空间薄壁梁的弹塑性刚度矩阵。算例表明该文所建梁单元模型具有良好的精度,适用于空间薄壁结构的有限元分析。     

A nanoscanning platform for bio-engineering: an in-plane probe with switchable stiffness

Scanning probe microscopy (SPM) has been one of the most important tools to image and, hopefully, to manipulate bio-structures at micro/nanoscales. However, the current out-of-plane cantilever design makes it very difficult to extend the spectrum of the current SPM technology to meet many new functionalities arising from bio-engineering applications. An in-plane scanning probe concept is developed to accommodate the new functional requirements. It is designed to have a single-strand multi-walled carbon nanotube (CNT) tip assembled at the end of the probe, a built-in actuator and a tip deflection sensor, all in the same plane. The coplanar design is compatible with most of the standard MEMS processes and would facilitate the assembly of a carbon nanotube tip to the micromachined probe. The in-plane design features a switchable stiffness which adapts the scanning probe's stiffness to the changing surface hardness of the sample. This paper describes how the variable stiffness is accomplished by engaging or disengaging electrostatically actuated clutches, in addition to the discussions on many possible benefits of the in-plane scanning platform.     

Dynamic stiffness analysis for axial-lateral-torsional vibration

The dynamic stiffness method enables one to model an infinite number of natural modes by means of a finite number of degrees of freedom. The method is extended to analyse the lateral-torsional vibrations of thin-wall columns under the influence of the constant axial force. The constant in-plane moment weakens the flexural mode so that at certain critical applied moments, the flexural mode buckles as the fundamental flexural frequency reaches zero. However, the constant in-plane moment hardens the torsional mode so that the torsional mode never buckles. When both torsion and flexure are considered, the interaction of the convexity of flexural characteristic curves and the concavity torsional characteristic curves becomes very complex. Since the dynamic stiffness is exact in the classical sense, the interaction can be studied easily. Numerical examples are given to show the complexity of the characteristic diagrams.