Nonlinear dynamics of a resonant gas sensor

We develop a mathematical model for a resonant gas sensor made up of an microplate electrostatically actuated and attached to the end of a cantilever microbeam. The model considers the microbeam as a continuous medium, the plate as a rigid body, and the electrostatic force as a nonlinear function of the displacement and the voltage applied underneath the microplate. We derive closed-form solutions to the static and eigenvalue problems associated with the microsystem. The Galerkin method is used to discretize the distributed-parameter model and, thus, approximate it by a set of nonlinear ordinary-differential equations that describe the microsystem dynamics. By comparing the exact solution to that associated with the reduced-order model, we show that using the first mode shape alone is sufficient to approximate the static behavior. We employ the Finite Difference Method (FDM) to discretize the orbits of motion and solve the resulting nonlinear algebraic equations for the limit cycles. The stability of these cycles is determined by combining the FDM discretization with Floquet theory. We investigate the basin of attraction of bounded motion for two cases: unforced and damped, and forced and damped systems. In order to detect the lower limit of the forcing at which homoclinic points appear, we conduct a Melnikov analysis. We show the presence of a homoclinic point for a loading case and hence entanglement of the stable and unstable manifolds and non-smoothness of the boundary of the basin of attraction of bounded motion.     

Frequency response of primary resonance of electrostatically actuated CNT cantilevers

This paper deals with electrostatically actuated carbon nanotube (CNT) cantilever over a parallel ground plate. Three forces act on the CNTs cantilever, namely electrostatic, van der Waals, and damping. The van der Waals force is significant for values of 50 nm or less of the gap between the CNT and the ground plate. As both forces electrostatic and van der Waals are nonlinear, and the CNTs electrostatic actuation is given by AC voltage, the CNT undergoes nonlinear parametric dynamics. The methods of multiple scales and reduced order model (ROM) are used to investigate the system under soft AC near half natural frequency of the CNT and weak nonlinearities. The frequency–amplitude response and damping, voltage, and van der Waals effects on the response are reported. It is showed that only five terms ROM predicts and accurately predicts the pull-in instability and the saddle-node bifurcation, respectively.     

Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation

A nonlinear analysis of an energy harvester consisting of a multilayered cantilever beam with a tip mass is performed. The model takes into account geometric, inertia, and piezoelectric nonlinearities. A combination of the Galerkin technique, the extended Hamilton principle, and the Gauss law is used to derive a reduced-order model of the harvester. The method of multiple scales is used to determine analytical expressions for the tip deflection, output voltage, and harvested power near the first global natural frequency. The results show that one- or two-mode approximations are not sufficient to produce accurate estimates of the voltage and harvested power. A parametric study is performed to investigate the effects of the nonlinear piezoelectric coefficients and the excitation amplitude on the system response. The effective nonlinearity may be of the hardening or softening type, depending on the relative magnitudes of the different nonlinearities.     

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.     

The oscillatory behavior, static and dynamic analyses of a micro/nano gyroscope considering geometric nonlinearities and intermolecular forces

The nonlinear dynamic and static deflection of a micro/nano gyroscope under DC voltages and base rotation are investigated. The gyroscope undertakes two cou- pled bending motions along the drive and sense directions and subjected to electrostatic actuations and intermolecular forces. The nonlinear governing equations of motion for the system with the effect of electrostatic force, intermolecular tractions and base rotation are derived using extended Hamilton principle. Under constant voltage, the gyroscope finds the preformed shape. First, the deflection of the rnicro/nano gyroscope under electrostatic forces is obtained by static and dynamic analyses. Furthermore, the static and dynamic in- stability of the system are investigated. Afterward the oscillatory behavior of the pre-deformed micro/nano gyroscope around equilibrium is studied. The effects of intermolecular and nonlinear parameters on the static and dynamic de- flection, natural frequencies and instability of the micro/nano gyroscope are studied. The presented model can be used to exactly determine static and the dynamic behavior of vibratory micro/nano gyroscopes.     

Dynamic analysis of a novel wide-tunable microbeam resonator with a sliding free-of-charge electrode

In this paper, a MEMS-based resonator with a novel effective stiffness tunability is presented. The performance of the proposed resonator is based on the transversal vibration of the two porous cantilever microbeams with a rectangular microplate at the end of the structure. The microplate as a free-of-charge slider electrode is in contact with two other fixed substrate electrodes via the thin layer of dielectric material. Applying a constant DC voltage to the two fixed electrodes leads to the movement of free electrons in the slider and eventually to the formation of two series capacitors. As a result, the slider meets a nonlinear electrostatic force proportional to the square of the applied DC voltage. It will act as a nonlinear spring with a tunable stiffness during the oscillation of the resonator. The coupled nonlinear equations governing the longitudinal and transversal vibration of the resonator are extracted in the presence of the nonlinear voltage-sliding spring. Its steady-state solution is obtained based on a physically based learning method that makes it possible to obtain frequency response for the first harmony as well as for the higher harmonies and to predict primary and secondary resonances in different harmonies of the response. The effect of the applied tuning DC voltage, the geometrical parameters of the resonator, and the cantilever's porosity on the dynamic response of the resonator are investigated. It is shown that the tuning stiffness of this voltage-sliding spring provides a highly effective solution to realize an extreme tunable range. In the end, a modified tunable structure is introduced in which the folded beams are replaced with common ones. The modified resonator by making the nonlinear behavior of the resonator least can improve its performance significantly.

     

Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters

A global nonlinear distributed-parameter model for a piezoelectric energy harvester under parametric excitation is developed. The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geometric, inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler–Lagrange principle and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient to evaluate the performance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence on the harvester’s behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that, for small values of the quadratic damping, there is an overhang associated with a subcritical pitchfork bifurcation.     

Dynamic behavior analysis of cantilever-type nano-mechanical electrostatic actuator

An investigation is performed into the nonlinear pull-in behavior of a cantilever-type nano-mechanical electrostatic actuator. In performing the analysis, the actuator is modeled as an Euler–Bernoulli beam and the influence of surface effects, the fringing field effect and the Casimir force effect are taken into explicit account. In general, analyzing the dynamic behavior of nanoscale electrostatic devices is challenging due to the nonlinear coupling of the electrostatic force and Casimir force. In the present study, this problem is resolved by using a hybrid computational scheme comprising the differential transformation method and the finite difference approximation technique. The feasibility of the proposed approach is demonstrated by the two cantilever-type micro-beams when actuated by a DC voltage. The numerical results show that the present results for the pull-in voltage deviate by no more than 1.47% from those presented in the literature using a different scheme. In addition, it is shown that surface effects play a significant role in determining the static deflection and pull-in voltage of the cantilever beam nano-beam. In general, the results confirm that the hybrid differential transformation/finite difference approximation method provides an accurate and computationally efficient means of simulating the nonlinear electrostatic behavior of nanostructure systems.     

Active boundary control of vibrating marine riser with constrained input in three-dimensional space

In this paper, the effects of initial deflection on the static and dynamic behaviors of circular capacitive transducers are experimentally investigated. The obtained results are in good agreement with numerical simulations. It is shown that the initial deflection has a major impact on the static response of the resonator by shifting the pull-in voltage, and on its dynamic response by increasing the resonance frequency and modifying the bifurcation topology from softening to hardening behavior. Moreover, the dynamic behavior of the microplate may display nonlinear periodic and quasiperiodic responses due to geometric and electrostatic nonlinearities.

     

Bifurcation and pull-in voltages of primary resonance of electrostatically actuated SWCNT cantilevers to include van der Waals effect

In this work the voltage response of primary resonance of electrostatically actuated single wall carbon nano tubes (SWCNT) cantilevers over a parallel ground plate is investigated. Three forces act on the SWCNT cantilever, namely electrostatic, van der Waals and damping. While the damping is linear, the electrostatic and van der Waals forces are nonlinear. Moreover, the electrostatic force is also parametric since it is given by AC voltage. Under these forces the dynamics of the SWCNT is nonlinear parametric. The van der Waals force is significant for values less than 50 nm of the gap between the SWCNT and the ground substrate. Reduced order model method (ROM) is used to investigate the system under soft excitation and weak nonlinearities. The voltage-amplitude response and influences of parameters are reported for primary resonance (AC near half natural frequency).     

Electromechanical coupled nonlinear dynamics for microbeams

In this paper, an electromechanical coupled nonlinear dynamic equation of a microbeam under an electrostatic force is presented. Using the nonlinear dynamic equations and perturbation method, we investigated nonlinear free vibrations, forced responses far from and near to natural frequency, respectively. Nonlinear natural frequencies and vibrating amplitudes of the electromechanical coupled microbeam are dependent on the mechanical and electric parameters. Compared with linear forced responses, the obvious shift of the mean dynamic response occurs. Under certain condition, the jump phenomenon will occur. The studies can be used to design parameters of the microbeam and remove undesirable dynamic behavior such as jump phenomenon, etc.     

功能梯度旋转圆板的热弹性固有振动

在考虑热因素及旋转运动条件下,针对金属－陶瓷功能梯度圆板的固有振动问题进行研究．给出随温度变化且材料组分沿厚度方向按幂律分布的材料物性参数,依据热弹性理论得到圆板的能量关系式．基于哈密顿原理建立旋转金属－陶瓷功能梯度圆板热弹性动力学方程．采用伽辽金法得到边界约束下圆板的自由振动方程,确定了静挠度及固有振动频率．基于数值计算,得到系统固有频率值随体积分数指数、转速和温度等参量的变化曲线,讨论了静挠度变化规律及动力系统的奇点稳定性问题．结果表明,固有频率随体积分数指数、材料表面温度以及转速的增加而减小．     

Mechanical behaviors of electrostatic microresonators with initial offset imperfection: qualitative analysis via time-varying capacitors

This paper investigates analytically and numerically the effect of initial offset imperfection on the mechanical behaviors of microbeam-based resonators. Symmetry breaking of DC actuation, due to different initial offset distances of microbeam to lower and upper electrodes, is concerned. For qualitative analysis, time-varying capacitors are introduced and a lumped parameter model, considering nonlinear electrostatic force and midplane stretching of microbeam, is adopted to examine the system statics and dynamics. The Method of Multiple Scales (MMS) is applied to determine the primary resonance solution under small vibration assumption. Meanwhile, the Finite Difference Method (FDM) combined with Floquet theory is utilized to generate frequency response curves for medium- and large-amplitude vibration simulations. Static bifurcation, phase portrait and Hamiltonian function are firstly investigated to examine the system inherent behaviors. Besides, basins of attraction are briefly depicted to grasp the effects of initial offset and AC excitation on the system global dynamics. Then, variation of equivalent natural frequency versus DC voltage is analyzed. Results show that initial offset may induce complex frequency rebound phenomenon as well as a separate frequency branch under secondary pull-in condition. In what follows, emergences of softening, linear and hardening vibration are classified through discussing a key parameter obtained from the frequency response equation. New linear behavior induced by initial offset imperfection is found, which exhibits much higher sensitivity to DC voltage. Medium- and large-amplitude in-well motions are also investigated, indicating the existence of alternations of softening and hardening behaviors. Finally, lumped parameters are deduced via Galerkin procedure, and case studies are provided to illustrate the effectiveness of the whole analysis.     

静载荷对夹层圆板超谐波共振的影响

研究静载荷作用下夹层圆板的超谐波共振问题.基于Hoff型夹层板理论,给出了静载荷作用下夹层圆板的非线性动力学方程.应用Galerkin法推导了静载荷作用下夹层圆板的轴对称非线性振动方程.运用多尺度法分别对系统的三次超谐波问题和二次超谐波问题进行了求解,并依据Lyapunov稳定性理论得到了系统稳态运动的稳定性判据.通过...     

中心集中力作用下圆薄板的固有频率—载荷特征关系

本文讨论了中心集中静载作用下圆薄板在非线性弯曲静平衡构形附近的微幅自由振动,其静平衡问题采用问题的精确解在此基础上,用伽辽金法获得了其最低固有频率——载荷特征关系.所得结果可供这类谐振弹性元件设计中参考.     

非线性压电效应下压电弯曲执行器的动力分析

研究压电弯曲执行器在强电场作用下的非线性动力行为．考虑电致伸缩和电致弹性的非线性压电效应,导出了压电悬臂执行器变刚度的弯曲振动控制方程．利用非定常振动的渐近理论,讨论了弯曲压电执行器的动力特征．根据目前的非线性模型可以计算压电悬臂执行器的固有共振频率与电场的变化关系．结果表明压电执行器端头挠度谐振幅度随作用电场振幅的增大而增大,以及力学品质因数随电场振幅的增大而减少,并且与实验结果非常吻合．通过数值比较得到在电场频率随时间变化非常缓慢的情况下非定常振动问题可以近似地用定常振动来处理．     

Micromechanical fatigue testing

This paper describes the design, modeling, and experimental test results of a single crystal silicon micromechanical device developed to evaluate fracture and fatigue of silicon based micromechanical devices. The structure is a cantilever beam, 300 microns long, with a large silicon plate and gold inertial mass at the free end. Torquing and sensing electrodes extend over the plate, and with associated electronics, drive the structure at resonance. Fatigue crack propagation is measured by detecting the shift in the natural frequency caused by the extension of a preexisting crack introduced near the fixed end of the cantilever. Experimental data are presented demonstrating time-dependent crack growth in silicon. Crack extensions of 10 to 300 nm have been measured with a resolution of approximately 2.5 nm, and crack tip velocities as low as 2.1×10⁻¹⁴ m/s. It is postulated that static fatigue of the native surface silica layer is the mechanism for crack growth. The methodology established here is generic in concept, permitting sensitive measurement of crack growth in larger fatigue specimens as well.     

On vibration mitigation and energy harvesting of a non-ideal system with autoparametric vibration absorber system

This paper demonstrates that vibration mitigation and energy harvesting can be achieved simultaneously by using of an electricity-generating from autoparametric vibration absorber system (AVAS) and non-ideal system (NIS). The NIS consists of a simple portal frame excited by a small dc motor with eccentric mass, with limited power supply and located on the top. The AVAS consists of a cantilever beam with tip mass parallel coupled to NIS. A piezoelectric material is considered for energy harvesting installed in the base of the AVAS and an electric circuit is connected to the piezoelectric material in order to produce voltage output. Several numerical simulations were carried out focusing on the passage through the resonance of NIS, when the motor rotational frequency is near the portal frame natural frequency and when the non-ideal subsystem frequency is approximately twice the absorber beam frequency (two-to-one internal resonance). The results showed the existence of Sommerfeld effect in NIS and saturation phenomenon in the NIS–AVAS.     

热环境中旋转运动功能梯度圆板的强非线性固有振动

研究热环境中旋转运动功能梯度圆板的非线性固有振动问题．针对金属－陶瓷功能梯度圆板,考虑几何非线性、材料物理属性参数随温度变化以及材料组分沿厚度方向按幂律分布的情况,应用哈密顿原理推得热环境中旋转运动功能梯度圆板的非线性振动微分方程．考虑周边夹支边界条件,利用伽辽金法得到了横向非线性固有振动方程,并确定了静载荷引起的静挠度．用改进的多尺度法求解强非线性方程,得出非线性固有频率表达式．通过算例,分析了旋转运动功能梯度圆板固有频率随转速、温度等参量的变化情况．结果表明,非线性固有频率随金属含量的增加而降低;随转速和圆板厚度的增大而升高;随功能梯度圆板表面温度的升高而降低．     

Elastic instability of a uniformly compressed annular plate with axisymmetric initial deflection

This paper presents a theoretical study of the elastic instability of a uniformly compressed, thin, circular annular plate with axisymmetric initial deflection. The dynamic version of the nonlinear Marguerre plate theory is used, and the linear free vibration problems around the axisymmetric finite deformation of the plate are solved by a finite difference method. By examining the frequency spectrum with various asymmetric modes, the critical compressive load under which the axisymmetric additional deformation of the plate becomes unstable due to the bifurcation buckling is determined, which is found to depend severely on the magnitude of the axisymmetric initial deflection.