一种微机电非线性耦合系统奇点稳定性研究

工程中许多微机电系统都采用电容驱动原理，这类结构实际上存在着强烈的静电和机械两个物理场的非线性耦合，因此系统的动态特性比较复杂。本文基于一种扭转微镜系统，通过数值分析方法，研究其非线性动态特性，根据理论分析和数值计算证明该系统在相平面中存在两个奇点，一个是稳定中心，一个是鞍点，且两个奇点位置均随施加电压的变化而逐渐靠近，从而得出系统的静态分叉点；同时分析了有阻尼和无阻尼时电压的变化对相轨道的影响，以及阻尼对吸合电压的影响，吸合电压随阻尼的增大而提高，这些研究结果不仅对扭转微镜的设计和应用提供了理论和方法，而且对用电容驱动的微机电系统的设计亦有参考价值。     

Reprogrammable logic-memory device of a mechanical resonator

From the viewpoint of application of nonlinear dynamics, we report multifunctional operation in a single microelectromechanical system (MEMS) resonator. This paper addresses a reprogrammable logic-memory device that uses a nonlinear MEMS resonator with multi-states. In order to develop the reprogrammable logic-memory device, we discuss the nonlinear dynamics of the MEMS resonator with and without control input as logic and memory operations. Through the experiments and numerical simulations, we realize the reprogrammable logic function that consists of OR/AND gate by adjusting the excitation amplitude and the memory function by storing logic information in the single nonlinear MEMS resonator.     

Mechanical measurements of adhesion in microcantilevers: Transitions in geometry and cyclic energy changes

Adhesion between initially separated components is a critical issue in microelectromechanical systems (MEMS), as it plays an important role in determining device reliability and the forces (and energy) required for successful operation. In this paper we outline a new approach for characterizing adhesion using microfabricated MEMS cantilevers, wherein transitions between adhered geometries and the corresponding energy changes are quantified using an instrumented nanoindenter. The use of an instrumented mechanical probe offers an important advantage over other techniques, in that the measured load-displacement response can be used to directly quantify energy changes during changes in adhered geometry and cyclic loading. In addition, the adhered portion of the system can be determined from the mechanical response of the beam, without having to view the system optically as required via interferometric techniques. Experimental results are presented which detail the transitions from free-standing cantilevers to arc-shaped and to s-shaped configurations. Measurements of the energy changes that occur under cyclic loading are also presented. The experiments reveal interesting adhesion behaviors suggested by vastly different experiments reported elsewhere, namely unstable transitions from one adhered geometry to another. The results are interpreted in the contexts of beam theory and fracture mechanics models, which can be used to infer interfacial adhesion energy.     

High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point

This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems in oscillatory form expressed in the physical basis, so that the technique is directly applicable to mechanical problems discretised by the finite element method. Two nonlinear mappings, respectively related to displacement and velocity, are introduced, and the link between the two is made explicit at arbitrary order of expansion, under the assumption that the damping matrix is diagonalised by the conservative linear eigenvectors. The same development is performed on the reduced-order dynamics which is computed at generic order following different styles of parametrisation. More specifically, three different styles are introduced and commented: the graph style, the complex normal form style and the real normal form style. These developments allow making better connections with earlier works using these parametrisation methods. The technique is then applied to three different examples. A clamped-clamped arch with increasing curvature is first used to show an example of a system with a softening behaviour turning to hardening at larger amplitudes, which can be replicated with a single mode reduction. Secondly, the case of a cantilever beam is investigated. It is shown that invariant manifold of the first mode shows a folding point at large amplitudes. This exemplifies the failure of the graph style due to the folding point on a real structure, whereas the normal form style is able to pass over the folding. Finally, a MEMS (Micro Electro Mechanical System) micromirror undergoing large rotations is used to show the importance of using high-order expansions on an industrial example.

     

Energy harvesting in a Mathieu–van der Pol–Duffing MEMS device using time delay

This paper investigates quasi-periodic vibration-based energy harvesting in a delayed nonlinear MEMS device consisting of a delayed Mathieu–van der Pol–Duffing type oscillator coupled to a delayed piezoelectric coupling mechanism. We use the multiple scales method to approximate the quasi-periodic response and the related power output near the principal parametric resonance. The effect of time delay on the energy harvesting performance is studied. It is shown that for appropriate combination of time delay parameters, there exists an optimum range of excitation frequency beyond the resonance where quasi-periodic vibration-based energy harvesting is maximum. Numerical simulations are performed to confirm the analytical predictions.     

Nonlinear dynamic behavior of a microbeam array subject to parametric actuation at low,medium and large DC-voltages

The dynamic response of parametrically excited microbeam arrays is governed by nonlinear effects which directly influence their performance. To date, most widely used theoretical approaches, although opposite extremes with respect to complexity, are nonlinear lumped-mass and finite-element models. While a lumped-mass approach is useful for a qualitative understanding of the system response it does not resolve the spatio-temporal interaction of the individual elements in the array. Finite-element simulations, on the other hand, are adequate for static analysis, but inadequate for dynamic simulations. A third approach is that of a reduced-order modeling which has gained significant attention for single-element micro-electromechanical systems (MEMS), yet leaves an open amount of fundamental questions when applied to MEMS arrays. In this work, we employ a nonlinear continuum-based model to investigate the dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations focus on the array’s behavior in regions of its internal one-to-one, parametric, and several internal three-to-one and combination resonances, which correspond to low, medium and large DC-voltage inputs, respectively. The nonlinear equations of motion for a two-element system are solved using the asymptotic multiple-scales method for the weakly nonlinear system in the afore mentioned resonance regions, respectively. Analytically obtained results of a two-element system are verified numerically and complemented by a numerical analysis of a three-beam array. The dynamic behavior of the two- and three-beam systems reveal several in- and out-of-phase co-existing periodic and aperiodic solutions. Stability analysis of such co-existing solutions enables construction of a detailed bifurcation structure. This study of small-size microbeam arrays serves for design purposes and the understanding of nonlinear nearest-neighbor interactions of medium- and large-size arrays. Furthermore, the results of this present work motivate future experimental work and can serve as a guideline to investigate the feasibility of new MEMS array applications.     

The effect of time-delayed feedback controller on an electrically actuated resonator

This paper presents a study of the effect of a time-delayed feedback controller on the dynamics of a Microelectromechanical systems (MEMS) capacitor actuated as a resonator by DC and AC voltage loads. A linearization analysis is conducted to determine the stability chart of the linearized system equations as a function of the time delay period and the controller gain. Then the method of multiple-scales is applied to determine the response and stability of the system for small vibration amplitude and voltage loads. It is shown that negative time-delay feedback control gain can lead to unstable responses, even if AC voltage is relatively small compared to the DC voltage. On the other hand, positive time delay can considerably strengthen the system stability even in fractal domains. We also show how the controller can be used to control damping in MEMS, increasing or decreasing, by tuning the gain amplitude and delay period. Agreements among the results of a shooting technique, long-time integration, basin of attraction analysis with the perturbation method are achieved.     

谐波平衡法与复平均法计算强非线性系统稳态响应的区别

近些年，很多学者致力于利用非线性增强振动响应减少的效果或者能量采集器的效率。因而非线性系统的响应值需要从理论计算方面更准确地预测。另外，根据学者已取得的研究成就，非线性能量汇(NES)中存在的立方刚度非线性可以将结构中宽频域的振动能量传递至非线性振子部分。文章将一种由NES和压电能量采集器组成的NES-piezo装置与两自由度主结构耦合连接，系统受谐和激励作用。文章采用谐波平衡法和复平均法分别推导了系统稳态响应，参照数值结果，对比两种近似解析方法在求解强非线性系统稳态响应时的异同。计算结果表明，系统体现较弱非线性时，二者计算结果差异很小；当系统体现强非线性时，复平均法不能准确地呈现系统高阶响应，提高阶数的谐波平衡法能更准确地表示系统响应值。基于谐波平衡法和数值算法，讨论NES-piezo装置对于系统宽频域减振的影响。与仅加入非线性能量汇情况对比，结果表明NES-piezo装置不会恶化宽频域减振效果，并且在第一阶共振频率附近，可以稍微提高结构减振效率。另外，计算结果也表明，采用恰当的NES-piezo装置可实现宽频域范围的结构减振和压电能量采集一体化。此项研究工作为研究不同情形强非线性系统的响应提供了理论方法的指导。另外，研究结果也为宽频域范围的结构减振和压电能量采集一体化提供了理论依据。     

Tuning the primary resonances of a micro resonator,using piezoelectric actuation

Microbeam dynamics is important in MEMS filters and resonators. In this research, the effect of piezoelectric actuation on the resonance frequencies of a piezoelectrically actuated capacitive clamped-clamped microbeam is studied. The microbeam is sandwiched with piezoelectric layers throughout its entire length. The lower piezoelectric layer is exposed to a combination of a DC and a harmonic excitation voltage. The DC electrostatic voltage is applied to prevent the doubling of the excitation frequency. The traditional resonators are tuned using DC electrostatic actuation, which tunes the resonance frequency only in backward direction on the frequency domain. The proposed model enables tuning the resonance frequencies in both forward and backward directions. For small amplitudes of harmonic excitation and high enough quality factor, the frequency response curves obtained by the shooting method are validated with those of the multiple time scales technique. Unlike the perturbation technique, which imposes limitation on both the amplitude of the harmonic excitation and the quality factor to be applicable, the shooting method can be applied to capture the periodic attractors regardless of how big the amplitude of harmonic excitation and the quality factor are.     

A stationary free boundary problem modeling electrostatic MEMS

We investigate a free boundary problem describing small deformations in a membrane based model of electrostatically actuated microelectromechanical systems (MEMS). The existence of stationary solutions is established for small voltage values and non-existence is obtained for high voltage values. We give a justification of the widely studied narrow-gap model by showing that steady state solutions of the free boundary problem converge toward stationary solutions of the narrow-gap model when the aspect ratio of the device tends to zero.     

Axial control for nonlinear resonances of electrostatically actuated nanobeam with graphene sensor

The nonlinear resonance response of an electrostatically actuated nanobeam is studied over the near-half natural frequency with an axial capacitor controller. A graphene sensor deformed by the vibrations of the nanobeam is used to produce the voltage signal.The voltage of the vibration graphene sensor is used as a control signal input to a closedloop circuit to mitigate the nonlinear vibration of the nanobeam. An axial control force produced by the axial capacitor controller can transform the frequency-amplitude curves from nonlinear to linear. The necessary and sufficient conditions for guaranteeing the system stability and a saddle-node bifurcation are studied. The numerical simulations are conducted for uniform nanobeams. The nonlinear terms of the vibration system can be transformed into linear ones by applying the critical control voltage to the system. The nonlinear vibration phenomena can be avoided, and the vibration amplitude is mitigated evidently with the axial capacitor controller.     

Dynamic analysis and design of electrically actuated viscoelastic microbeams considering the scale effect

Viscoelastic phenomena widely exist in MEMS materials, which may have certain effects on quasi-static behaviors and transition mechanism of nonlinear jumping phenomena. The static and dynamic behaviors of a doubly clamped viscoelastic microbeam actuated by one sided electrode are investigated in detail, based on a modified couple stress theory. The governing equation of motion is introduced here, which is essentially nonlinear due to its midplane stretching effect and electrostatic force. Through quasi-static analysis, the equilibrium position, pull-in voltage and pull-in location of the system are obtained with differential quadrature method and finite element method. The equivalent geometric nonlinear parameter is presented to explain the influence of the scale effect on the pull-in location. Different from elastic material, there are two kinds of pull-in voltages called as instantaneous pull-in voltage and the durable pull-in voltage in viscoelastic system. Then, Galerkin discretization and the method of multiple scales are applied to determine the response and stability of the system for small vibration amplitude. A new perturbation method to deal with viscoelastic term is presented. Theoretical expressions about the parameter spaces of linear-like vibration, hardening-type vibration and softening-type vibration are then deduced. The influence of viscoelasticity and scale effect on nonlinear dynamic behavior is studied. Results show that the viscoelasticity can reduce the effective elastic modulus and make the system tend to softening-type vibration; the scale effect can increase effective elastic modulus and make the system tend to hardening-type vibration. And most of all, simulation results of case studies are used to realize parameter optimization. Then parameter conditions of linear-like vibration, which is desired for many applications, are obtained. In this paper, the results of multi-physical field coupling simulation are used to verify the theoretical analysis.     

MEMS动平板的切向静电阻力

由于微机械的表面积与体积之比远大于宏机械,所以微机械中的表面阻力难以忽略,为了改善MEMS器件的性能和可靠性,必须对其影响进行研究.基于能量守衡法,本文建立了光滑平板和正方形、四棱锥两种微凸体粗糙表面平板的切向静电阻力模型,讨论了微小尺度、表面形貌、外加电压以及因流片制造工艺而产生的微凸体、凹坑或孔对两个相对运动的带电平板间的切向静电阻力的影响.分析表明：当平板宽度与两平板之间的距离之比、表面形貌因数和外加电压增大时,切向静电阻力也将随之增加;表面形貌因数则与微凸体在平板的总投影面积与平板面积之比成正比,随相对表面粗糙度增加而非线性增加.     

Frequency locking in a forced Mathieu–van der Pol–Duffing system

Optically actuated radio frequency microelectromechanical system (MEMS) devices are seen to self-oscillate or vibrate under illumination of sufficient strength (Aubin, Pandey, Zehnder, Rand, Craighead, Zalalutdinov, Parpia (Appl. Phys. Lett. 83, 3281–3283, 2003)). These oscillations can be frequency locked to a periodic forcing, applied through an inertial drive at the forcing frequency, or subharmonically via a parametric drive, hence providing tunability. In a previous work~(Aubin, Zalalutdinov, Alan, Reichenbach, Rand, Zehnder, Parpia, Craighead (IEEE/ASME J. Micromech. Syst. 13, 1018–1026, 2004)), this MEMS device was modeled by a three-dimensional system of coupled thermo-mechanical equations requiring experimental observations and careful finite element simulations to obtain the model parameters. The resulting system of equations is relatively computationally expensive to solve, which could impede its usage in a complex network of such resonators. In this paper, we present a simpler model that shows similar behavior to the MEMS device. We investigate the dynamics of a Mathieu–van der Pol–Duffing equation, which is forced both parametrically and nonparametrically. It is shown that the steady-state response can consist of either 1:1 frequency locking, or 2:1 subharmonic locking, or quasiperiodic motion. The system displays hysteresis when the forcing frequency is slowly varied. We use perturbations to obtain a slow flow, which is then studied using the bifurcation software package AUTO.     

具有组合非线性阻尼的非线性能量阱振动抑制响应分析

非线性能量阱是一种振动能量吸收装置,其在结构振动抑制中具有十分重要的作用.文章对具有组合非线性阻尼非线性能量阱的系统进行振动抑制相关的分析.首先对具有组合非线性阻尼非线性能量阱的系统进行理论模型的描述,对系统模型的运动方程利用复变量平均法进行推导,得到系统的慢变方程.其次对系统的慢变方程运用多尺度法进行强调制响应的分析,通过对系统进行慢不变流形和相轨迹的研究,描述系统强调制响应发生的条件基础.此外,还利用一维映射对系统进行分析,揭示外激励幅值对强调制响应存在时频率失谐系数取值区间的影响规律.最后利用能量谱、时间响应和庞加莱映射对耦合组合非线性阻尼非线性能量阱系统进行了振动抑制的相关研究,揭示组合非线性阻尼的非线性能量阱不同阻尼比、阻尼和刚度对其振动抑制效果的影响规律,得出组合非线性阻尼非线性能量阱和主结构响应存在一致性的现象,并验证所提出的组合非线性阻尼非线性能量阱模型具有较好的振动抑制能力.     

Application of piezoelectric actuation to regularize the chaotic response of an electrostatically actuated micro-beam

The impetus of this study is to investigate the nonlinear chaotic dynamics of a clamped–clamped micro-beam exposed to simultaneous electrostatic and piezoelectric actuation. The micro-beam is sandwiched with piezoelectric layers throughout its length. The combined DC and AC electrostatic actuation is imposed on the micro-beam through two upper and lower electrodes. The piezoelectric layers are actuated via a DC electric voltage applied in the direction of the height of the piezoelectric layers, which produces an axial force proportional to the applied DC voltage. The governing differential equation of the motion is derived using Hamiltonian principle and discretized to a nonlinear Duffing type ODE using Galerkin method. The governing ODE is numerically integrated to get the response of the system in terms of the governing parameters. The results show that the response of the system is greatly affected by the amounts of DC and AC electrostatic voltages applied to the upper and lower electrodes. The results show that the response of the system can be highly nonlinear and in some regions chaotic. Evaluating the K–S entropy of the system, based on several initial conditions given to the system, the chaotic response is distinguished from the periodic or quasiperiodic ones. The main objective is to passively control the chaotic response by applying an appropriate DC voltage to the piezoelectric layers.     

Modeling and simulation of a novel vertical actuator based on electrowetting on dielectric

A novel vertical actuator based on electrowetting on dielectric （EWOD） was designed, analyzed and simulated. Modeling results indicated that the vertical driving force of the actuator obeyed a second order polynomial of applied voltage, which was verified by Covent_ware 2006. As a resuit, the vertical driving force of the EWOD actuator with a 1.1 nL droplet and a 1.75 μm thick polymer was about 0.5 μN under an applied voltage 100V which was comparable to that of the electrostatic actuators. Moreover, the noise from plane forces we analyzed and simulated was very low. Therefore, we made a conclusion that the EWOD actuator can be used in MEMS transducer.     

Spectrographic analysis for the modal testing of nonlinear aeroelastic systems

The spectrograph is a signal-processing tool often used for the frequency domain analysis of time-varying signals. When the signal to be analyzed is a function of time, the spectrograph represents the frequency content of the signal as a sequence of power spectra that change with time. In this paper, the usefulness of the technique is demonstrated in its application to the analysis of the time history response of a nonlinear aeroelastic system. The aeroelastic system is modelled analytically as a two-dimensional, rigid airfoil section free to move in both the bending and pitching directions and possessing a rigid flap. The airfoil is mounted by torsional and translational springs attached at the elastic axis, and the flap is used to provide the forcing input to the system. The nonlinear system is obtained by introducing a freeplay type of nonlinearity in the pitch degree-of-freedom restoring moment. The airfoil is immersed in an aerodynamic flow environment, modelled using incompressible thin airfoil theory for unsteady oscillatory motion. The equations of motion are solved using a fourth-order Runge–Kutta numerical integration technique to provide time-history solutions of the response of the airfoil in the pitch and plunge directions. Time-histories are obtained for the nonlinear responses of the linear and nonlinear aeroelastic systems to a sine-sweep input. The time-histories are analyzed using the spectrographic technique, and the frequency content of the response is plotted directly as a function of the input frequency. Results show that the combination of the sine-sweep input with the spectrographic analysis permits a unique insight into the behavior of the nonlinear system with a minimum of testing. It is shown that the frequency of the nonlinear system response is a function of the input frequency and one other characteristic frequency that can be associated with the limit cycle oscillations of the same nonlinear system subject to a transient input.     

Nonlinear instabilities in a vibratory gyroscope subjected to angular speed fluctuations

Nonlinear instabilities in a single-axis vibrating MEMS gyroscope that is subjected to periodic fluctuations in input angular rates are investigated. For the purpose of characterizing the bifurcation behavior associated with the steady-state, when the angular rate input is subject to small intensity periodic fluctuations, dynamic behavior of periodically perturbed nonlinear gyroscopic systems is studied in detail. An asymptotic approach based on the method of averaging has been employed for this purpose, and closed-form conditions for the frequency response due to parametric resonances have been obtained. This behavior has been illustrated via amplitude-frequency plots.     

斜拉索振动的模糊半主动控制研究

考虑拉索垂度及抗弯刚度的影响,得出了索-阻尼器系统振动偏微分方程;用中心差分法将偏微分方程在空间内离散,导出了系统的面内振动常微分方程组;提出了使用MR阻尼器(Magnetorheological Damper)作为控制设备,模糊集为基础的半主动控制算法,并运用提出的算法对索-阻尼器系统进行了振动控制分析。本文方法的优势在于算法自身的鲁棒性、处理非线性问题的能力强和不需要结构的精确数学模型,算法需要的输入变量少,可以解决实际工程中斜拉索的振动响应信息难以测量的困难。模糊算法的输出直接控制MR阻尼器的输入电压。与LQR-Clipped算法不同,MR阻尼器的输入电压可以是零与最大值之间的任意值。本文以实际斜拉桥拉索为例,分析了拉索的振动控制效果,结果表明本文提出的模糊半主动控制算法,使MR阻尼器的功能得到了更好的发挥,比MR被动控制效果好,且可以减小控制力。