Size-dependent modal interactions of a piezoelectrically laminated microarch resonator with 3:1 internal resonance

The nonlinear interactions of a microarch resonator with 3:1 internal resonance are studied. The microarch is subjected to a combination of direct current(DC)and alternating current(AC) electric voltages. Thin piezoelectric layers are thoroughly bonded on the top and bottom surfaces of the microarch. The piezoelectric actuation is not only used to modulate the stiffness and resonance frequency of the resonator but also to provide the suitable linear frequency ratio for the activation of the inte...     

An investigation of the static and dynamic behavior of electrically actuated rectangular microplates

We present an investigation of the static and dynamic behavior of the nonlinear von-Karman plates when actuated by the nonlinear electrostatic forces. The investigation is based on a reduced order model developed using the Galerkin method, which rely on modeshapes and in-plane shape functions extracted using a finite element method. In this study, a fully clamped microplate is considered. We investigate the static behavior and the effect of different non-dimensional design parameters. The static results are validated by comparison with the results calculated by a finite element model. The forced-vibration response of the plate is then investigated when the plate is excited by a harmonic AC load superimposed to a DC load. The dynamic behavior is examined near the primary and secondary (superharmonic and subharmonic) resonances. The microplate shows a strong hardening behavior due to the cubic nonlinearity of mid-plane stretching. However, the behavior switches to softening as the DC load is increased. Finally, near-square plates are studied to understand the effect of geometric imperfections of microplates.     

The dynamic behavior of MEMS arch resonators actuated electrically

In this paper, we investigate the dynamic behavior of clamped-clamped micromachined arches when actuated by a small DC electrostatic load superimposed to an AC harmonic load. A Galerkin-based reduced-order model is derived and utilized to simulate the static behavior and the eigenvalue problem under the DC load actuation. The natural frequencies and mode shapes of the arch are calculated for various values of DC voltages and initial rises. In addition, the dynamic behavior of the arch under the actuation of a DC load superimposed to an AC harmonic load is investigated. A perturbation method, the method of multiple scales, is used to obtain analytically the forced vibration response of the arch due to DC and small AC loads. Results of the perturbation method are compared with those obtained by numerically integrating the reduced-order model equations. The non-linear resonance frequency and the effective non-linearity of the arch are calculated as a function of the initial rise and the DC and AC loads. The results show locally softening-type behavior for the resonance frequency for all DC and AC loads as well as the initial rise of the arch.     

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).     

Size effect on the static behavior of electrostatically actuated microbeams

We present a new analytical model for electrostatically actuated microbeams to explore the size effect by using the modified couple stress theory and the minimum total potential energy principle. A material length scale parameter is introduced to represent the size-dependent characteristics of microbeams. This model also accounts for the nonlinearities associated with the mid-plane stretching force and the electrostatical force. Numerical analysis for microbeams with clamped-clamped and cantilevered conditions has been performed. It is found that the intensity of size effect is closely associated with the thickness of the microbeam,and smaller beam thickness displays stronger size effect and hence yields smaller deffection and larger pull-in voltage. When the beam thickness is comparable to the material length scale parameter,the size effect is significant and the present theoretical model including the material length scale parameter is adequate for predicting the static behavior of microbeam-based MEMS.     

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.     

Non-lane-discipline-based car-following model incorporating the electronic throttle dynamics under connected environment

We investigate the nonlinear dynamics of a system of generalized Duffing-type MEMS resonator in the frame of simple analog electronic circuit. A mathematical model formed for the proposed generalized Duffing-type MEMS oscillator in which nonlinearities arising out of two different sources such as mid-plane stretching and electrostatic force can lead to variety of nonlinear phenomena such as period-doubling route, transient chaos and homo-/heteroclinic oscillations. These phenomena were confirmed through detailed numerical investigations such as phase portraits, bifurcation diagram, Poincaré map, Lyapunov exponent spectrum and finite-time Lyapunov exponent. The analog circuit realization for the Duffing-type MEMS resonator is constructed. The numerically simulated results are confirmed in the laboratory experimental observations which are closely matched with each other. The experimentally observed chaotic attractor confirmed through FFT spectrum, 0–1 test and Poincaré cross section. In addition, the robustness of the signal strength is confirmed through signal-to-noise ratio.     

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.     

Bistable nonlinear response of MEMS resonators

This study, for the first time, investigates the nonlinear bistable response of electrically actuated initially imperfect viscoelastic microresonators under DC and AC actuations, while including fringing field effects and retaining both longitudinal and transverse motions. The modified version of the couple stress theory, the Kelvin–Voigt material, and the Meijs–Fokkema electrostatic formula are employed in order to develop a multi-physics model of the electrically actuated initially curved microresonator while accounting for the effects of geometrical nonlinearities, damping nonlinearities, fringing field, small size, and residual axial load. The developed continuous multi-physics model is reduced to set of differential equations of ordinary type, which are nonlinearly coupled, employing a weighted residual method and solved numerically making use of a continuation technique. The numerical simulations results are shown via plots of static deflection and natural frequency versus the DC voltage as well as plots of motion amplitude versus the AC frequency.     

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.

     

Three-dimensional oscillations of suspended cables involving simultaneous internal resonances

The near resonant response of suspended, elastic cables driven by planar excitation is investigated using a three degree-of-freedom model. The model captures the interaction of a symmetric in-plane mode with two out-of-plane modes. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. For particular magnitudes of equilibrium curvature, the natural frequency of the in-plane mode is simultaneously commensurable with the natural frequencies of the two out-of-plane modes in 1:1 and 2:1 ratios. A second nonlinear order perturbation analysis is used to determine the existence and stability of four classes of periodic solutions. The perturbation solutions are compared with results obtained by numerically integrating the equations of motion. Furthermore, numerical simulations demonstrate the existence of quasiperiodic responses.A portion of this work was presented at the 1992 ASME Winter Annual Meeting, Anaheim, CA.     

Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges

This study investigates the size-dependent quasistatic response of a nonlinear viscoelastic microelectromechanical system (MEMS) under an electric actuation. To have this problem in view, the deformable electrode of the MEMS is modelled using cantilever and doubly-clamped viscoelastic microbeams. The modified couple stress theory in conjunction with Bernoulli–Euler beam theory are used for mathematical modeling of the size-dependent instability of microsystems in the framework of linear viscoelastic theory. Simultaneous effect of electrostatic actuation including fringing field, residual stress, mid-plane stretching and Casimir and van der Waals intermolecular forces are considered in the theoretical model. A single element of the standard linear solid element is used to simulate the viscoelastic behavior. Based on the extended Hamilton’s variational principle, the nonlinear governing integro-differential equation and boundary conditions are derived. Thereafter, a new generalized differential-integral quadrature solution for the nonlinear quasistatic response of electrically actuated viscoelastic micro/nanobeams under two different boundary conditions; doubly-clamped microbridge and clamped-free microcantilever. The developed model is verified and a good agreement is obtained. Finally, a comprehensive study is conducted to investigate the effects of various parameters such as material relaxation time, durable modulus, material length scale parameter, Casimir force, van der Waals force, initial gap and beam length on the pull-in response of viscoelastic microbridges and microcantilevers in the framework of viscoelasticity.     

The response of clamped-clamped microbeams under mechanical shock

We present modeling, simulation, and characterization for the dynamic response of clamped-clamped microbeams under mechanical shock. A Galerkin-based reduced-order model is utilized and its results are verified by comparing to finite-element results. The results indicate that the response of a microbeam to mechanical shock is inherently non-linear because of the dominating effect of mid-plane stretching. The effect of the shock pulse shape is investigated. It is concluded that the shape of the shock pulse can result in significant dynamic amplification in the response of the microbeam even in cases where the shock load is considered quasi-static.The combined effect of the electrostatic force and mechanical shock is investigated. The results show that this combined effect can lead to early instability in microelectromechanical systems (MEMS) devices through dynamic pull-in. This could explain some of the reported experimental evidences for the existence of strange modes of failure of MEMS devices under mechanical shock and impact. These failures are characterized by overlaps between moving microstructures and stationary electrodes, which cause electrical shorts. The shock-electrostatic interaction is shown to be promising to design smart MEMS switches triggered at predetermined level of shock and acceleration. Finally, the mechanical shock combined with the packaging effect of MEMS devices is analyzed. A single-degree-of-freedom model representing the motion of the package, which is mounted over a printed circuit board, coupled with the continuous beam model is utilized. Our results reveal that neglecting the effect of the package motion on the response of microbeams can overestimate or underestimate their response. It is concluded that a poor design of the package may result in severe amplification of the shock effect leading to a device failure.     

On the secondary resonance of a MEMS resonator: A conceptual study based on shooting and perturbation methods

Nonlinear dynamics of a clamped–clamped capacitive micro-beam resonator subjected to subharmonic excitation of order one-half is studied. The micro-beam resonator is sandwiched with two piezoelectric layers throughout the length, and as a result of piezoelectric actuation a tensile/compressive axial load is induced along the length which is used as a frequency tuning tool. The resonator is subjected to a combination of a bias DC and harmonic AC electrostatic actuations. In order to determine the frequency response subharmonic resonance condition, both perturbation and shooting methods are applied. The stability of the periodic solutions and the bifurcations types are also studied. It is shown that the application of perturbation method imposes some limitations on the order of magnitudes of the terms in the differential equation of the motion; as a result out of the domain where the ordering assumption of the perturbation solution does not hold, some periodic solutions as well as some vital bifurcation points are missed. It is shown that on the frequency domain, the resonator exhibits both softening and hardening behaviors whereas this is not predicted by the perturbation scheme. The effect of DC and AC actuation voltages on the qualitative response of the system is determined. It is shown that based on the polarity of the piezoelectric actuation, the frequency response curves can be shifted both in forward and backward directions which can be used in the design of novel RF MEMS filters/sensors.     

Active vibration control of a dynamical system via negative linear velocity feedback

In this paper, a negative velocity feedback is added to a dynamical system which is represented by second-order nonlinear differential equations having quadratic coupling, quadratic, and cubic nonlinearities. The system describes the vibration of the system subjected to multi-parametric excitation forces. The method of multiple scale perturbation technique is applied to obtain the response equation near the simultaneous internal and super-harmonic resonance case of this system. The stability to the system is investigated applying frequency response equations. The numerical solution and the effects of some parameters on the vibrating system are investigated and reported. The simulation results are achieved using MATLAB 7.0 program. A comparison is made with the available published work.     

Vibration and stability of an aircraft tail under simultaneous primary-combined and internal resonance

This paper adds a negative velocity feedback to the dynamical system of twin-tail aircraft to suppress the vibration. The system is represented by two coupled second-order nonlinear differential equations having both quadratic and cubic nonlinearities. The system describes the vibration of an aircraft tail subjected to both multi-harmonic and multi-tuned excitations. The method of multiple time scale perturbation is adopted to solve the nonlinear differential equations and obtain approximate solutions up to the third order approximations. The stability of the proposed analytic solution near the simultaneous primary, combined and internal resonance is studied and its conditions are determined. The effect of different parameters on the steady state response of the vibrating system is studied and discussed by using frequency response equations. Some different resonance cases are investigated numerically     

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 analysis of electrostatic MEMS filters

We present an analytical model and closed-form expressions describing the response of a tunable MEMS filter made of two electrostatic resonators coupled by a weak microbeam. The model accounts for the filter geometric and electric nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system to produce a reduced-order model. We predict the filter deflection and static pull-in voltage by solving a boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the filter poles (the natural frequencies delineating the filter bandwidth). We found a good agreement between the results obtained using our model and published experimental results. We found that, when the input and output resonators are mismatched, the first mode is localized in the softer resonator whereas the second mode is localized in the stiffer resonator. We demonstrated that mismatch between the resonators can be countermanded by applying different DC voltages to the resonators. As the effective nonlinearities of the filter grow, multi-valued responses appear and distort the filter performance. Once again, we found that the filter can be tuned to operate linearly by choosing a DC voltage that makes the effective nonlinearities vanish.     

Nonlinear oscillations of suspended cables containing a two-to-one internal resonance

The near-resonant response of suspended, elastic cables driven by planar excitation is investigated using a two degree-of-fredom model. The model captures the interaction of a symmetric in-plane mode and an out-of-plane mode with near commensurable natural frequencies in a 2:1 ratio. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. The existence and stability of periodic solutions are investigated using a second order perturbation analysis. The first order analysis shows that suspended cables may exhibit saturation and jump phenomena. The second order analysis, however, reveals that the cubic nonlinearities and higher order corrections disrupt saturation. The stable, steady state solutions for the second order analysis compare favorably with results obtained by numerically integrating the equations of motion.     

Phenomena and modeling of piezoelectric energy harvesting from freely oscillating cylinders

We investigate energy harvesting from vortex-induced vibrations of a freely moving rigid circular cylinder with a piezoelectric transducer attached to its transverse degree of freedom. The power levels that can be generated from these vibrations and variations of these levels with the freestream velocity are determined. A mathematical model that accounts for the coupled lift force, cylinder motion, and harvested voltage is presented. Linear analysis is performed to determine the effect of the electrical load resistance of the transducer on the natural frequency of the cylinder and the onset of synchronization (the shedding frequency is equal to the cylinder oscillating frequency) region. The impact of the nonlinearities on the cylinder response and harvested energy is investigated. The results show that the load resistance shifts the onset of synchronization to higher freestream velocities. For two different system parameters, the results show that the nonlinearities result in a hardening behavior for some values of the load resistance.