Bistability of curved microbeams actuated by fringing electrostatic fields

In this work we investigate the feasibility of two-directional switching of an initially curved or pre-buckled electrostatically actuated microbeams using a single electrode fabricated from the same structural layer. The distributed electrostatic force, which is engendered by the asymmetry of the fringing fields in the deformed state, acts in the direction opposite to the deflection of the beam and can be effectively viewed as a reaction of a nonlinear elastic foundation with stiffness parameterized by the voltage. The reduced order model was built using the Galerkin decomposition with linear undamped modes of a straight beam as base functions and verified using the results of the numerical solution of the differential equation. The electrostatic force was approximated by means of fitting the results of three-dimensional numerical solution of the electrostatic problem. Static stability analysis reveals that the presence of the restoring electrostatic force may result in the suppression of the snap-through instability as well as in the appearance of additional stable configurations associated with higher buckling modes of the beam that are not observed in “mechanically” loaded structures. We show that two-directional switching of a pre-buckled beam between two stable configurations cannot be achieved using quasistatic loading. Furthermore, we show that switching is both associated with the dynamic snap-through mechanism and possible within certain interval of actuation voltages. Using a single-degree-of-freedom (lumped) model, estimation voltage boundaries are obtained. Theoretical results illustrate the feasibility of the suggested operational principle as an efficient mechanism in the arena of non-volatile mechanical memory devices.     

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.     

Suppression of pull-in in a microstructure actuated by mechanical shocks and electrostatic forces

This work investigates the effect of a high-frequency voltage (HFV) on the pull-in instability in a microstructure actuated by mechanical shocks and electrostatic forces. The microstructure is modelled as a single-degree-of-freedom mass-spring-damper system. The method of direct partition of motion is used to split the fast and slow dynamics. Analysis of steady-state solutions of the slow dynamic allows the investigation of the influence of the HFV on the pull-in. The results show that adding HFV rigidifies the system, creates new stable equilibria and suppresses the pull-in instability for adequate high-frequency voltages. To illustrate the applicability of the result, a specific capacitive microelectromechanical system consisting of a clamped-clamped microbeam is considered.     

Energy-efficient full-range oscillation analysis of parallel-plate electrostatically actuated MEMS resonators

Nonlinear Dynamics - This paper aims to report a novel route to chaotic bursting, i.e., the route via bifurcation delay and chaos crisis, based on the parametrically driven Lorenz system. A...     

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.     

Dynamics of a cantilever arm actuated by a nonlinear electrical circuit

The idea in this paper is to present some analytical and numerical results on the investigation of the dynamics of a nonlinear electromechanical system including a cantilever robot arm manipulator, harmonically actuated through an electric circuit. We use the method of harmonic balance to derive oscillatory solutions. Forced vibrations are analyzed showing that numerical results are in agreement with those obtained analytically for the stationary response. The system presents various types of nonlinear behaviors including chaos.     

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.     

Input-shaping control of nonlinear MEMS

We develop a new technique for preshaping input commands to control microelectromechanical systems (MEMS). In general, MEMS are excited using an electrostatic field which is a nonlinear function of the states and the input voltage. Due to the nonlinearity, the frequency of the device response to a step input depends on the input magnitude. Therefore, traditional shaping techniques which are based on linear theory fail to provide good performance over the whole input range. The technique we propose combines the equations describing the static response of the device, an energy balance argument, and an approximate nonlinear analytical solution of the device response to preshape the voltage commands. As an example, we consider set-point stabilization of an electrostatically actuated torsional micromirror. The shaped commands are applied to drive the micromirror to a desired tilt angle with zero residual vibrations. Simulations show that fast mirror switching operation with almost zero overshoot can be realized using this technique. The proposed methodology accounts for the energy of the significant higher modes and can be used to shape input commands applied to other nonlinear micro- and macro-systems.     

On the dynamics of a rolling ball actuated by internal point masses

The motion of a rolling ball actuated by internal point masses that move inside the ball’s frame of reference is considered. The equations of motion are derived by applying Euler–Poincaré’s symmetry reduction method in concert with Lagrange–d’Alembert’s principle, which accounts for the presence of the nonholonomic rolling constraint. As a particular example, we consider the case when the masses move along internal rails, or trajectories, of arbitrary shape and fixed within the ball’s frame of reference. Our system of equations can treat most possible methods of actuating the rolling ball with internal moving masses encountered in the literature, such as circular motion of the masses mimicking swinging pendula or straight line motion of the masses mimicking magnets sliding inside linear tubes embedded within a solenoid. Moreover, our method can model arbitrary rail shapes and an arbitrary number of rails such as several ellipses and/or figure eights, which may be important for future designs of rolling ball robots. For further analytical study, we also reduce the system to a single differential equation when the motion is planar, that is, considering the motion of the rolling disk actuated by internal point masses, in which case we show that the results obtained from the variational derivation coincide with those obtained from Newton’s second law. Finally, the equations of motion are solved numerically, illustrating a wealth of complex behaviors exhibited by the system’s dynamics. Our results are relevant to the dynamics of nonholonomic systems containing internal degrees of freedom and to further studies of control of such systems actuated by internal masses.     

Multistability of cantilever MEMS/NEMS switches induced by electrostatic and surface forces

MEMS/NEMS switches are used in a variety of portable electronics and RF telecommunications systems. MEMS/NEMS switches are ideally bi-stable, with one ON and one OFF state and a reliable switching between the two induced by electrical actuation. Presented herein is an exploration into non-ideal behavior, i.e. tri-stability, and parametric sensitivity of a generalization of cantilever MEMS/NEMS switches. The representative system model employs multiphysics features based on Euler–Bernoulli beam theory, parallel plate capacitance for electrostatics, and a Lennard-Jones form of surface interaction. The geometry, material properties, and surface features of the device are condensed into just a few dimensionless quantities, creating a parameter space of low enough dimensionality to provide accessible representations of all system equilibria within physically relevant ranges. Analysis of this system model offers insight regarding conditions necessary for bi- and tri-stability in such systems, which are crucial for informing studies on switching dynamics and various device performance metrics.     

Impact dynamics of MEMS switches

During operation, a MEMS switch is activated by an applied voltage. This causes the switch, often a doped silicon microbeam, to be attracted toward (pulled-into) a substrate. The component–substrate contact completes a circuit and permits the flow of current. Calculations for the minimum voltage required to achieve quasi-static pull-in are well documented. But for these quasi-static pull-in voltages to be meaningful, the voltage would have to be increased gradually until the critical value V_pull-inV_{\mathrm{pull\mbox{-}in}} is reached and the switch closes. Of course, practical considerations might require the switch to cycle on and off quickly, i.e., dynamically. This is particularly true in the case of radio frequency (RF) MEMS switches. In this paper, a model is developed and used to consider the dynamic pull-in characteristics of a clamped-clamped microbeam. This model includes inertial effects, structural and air damping (squeeze-film damping), as well as the impact behavior of the microbeam with the substrate. Parameter combinations leading to various types of behavior (no pull-in, air-bounce, wall bounce, etc.) are clearly identified. In an attempt to ensure fast switch closure and limit bouncing, two new applied voltage profiles are considered.     

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.     

Pull-in instability of a typical electrostatic MEMS resonator and its control by delayed feedback

Pull-in instability of the electrostatic microstructures is a common undesirable phenomenon which implies the loss of reliability of micro-electromechanical systems. Therefore, it is necessary to understand its mechanism and then reduce the phenomenon. In this work, pull-in instability of a typical electrostatic MEMS resonator is discussed in detail. Delayed position feedback and delayed velocity feedback are introduced to suppress pull-in instability, respectively. The thresholds of AC voltage for pull-in instability in the initial system and the controlled systems are obtained analytically by the Melnikov method. The theoretical predictions are in good agreement with the numerical results. It follows that pull-in instability of the MEMS resonator can be ascribed to the homoclinic bifurcation inducing by the AC and DC load. Furthermore, it is found that the controllers are both good strategies to reduce pull-in instability when their gains are positive. The delayed position feedback controller can work well only when the delay is very short and AC voltage is low, while the delayed velocity feedback will be effective under a much higher AC voltage and a wider delay range.     

Lateral pull-in instability of electrostatic MEMS transducers employing repulsive force

Nonlinear Dynamics - We report on the lateral pull-in in capacitive MEMS transducers that employ a repulsive electrostatic force. The moving element in this system undergoes motion in two...     

Pull-in voltage analysis of electrostatically actuated MEMS with piezoelectric layers: A size-dependent model

A size-dependent model for electrostatically actuated microbeam-based MEMS (micro-electro-mechanical systems) with piezoelectric layers attached is developed based on a modified couple stress theory. By using Hamilton's principle, the nonlinear differential governing equation and boundary conditions of the MEM structure are derived. In the newly developed model, the residual stresses, fringing-field and axial stress effects are considered for the fixed–fixed microbeam with piezoelectric layers. The results of the present model are compared with those from the classical model. The results show the size effect becomes prominent if the beam dimension is comparable to the material length scale parameter (MLSP). The effects of MLSP, the residual stresses and axial stress on the pull-in voltage are also studied. The study may be helpful to characterize the mechanical and electrostatic properties of small size MEMS, or guide the design of microbeam-based devices for a wide range of potential applications.     

Non-local effects on the non-linear modes of vibration of carbon nanotubes under electrostatic actuation

Carbon nanotubes (CNTs) based NEMS with electrostatic sensing/actuation may be employed as sensors, in situations where it is fundamental to understand their dynamic behaviour. Due to displacements that are large in comparison with the thickness and to the non-linearity of the electrostatic force, these CNT based NEMS operate in the non-linear regime. The knowledge of the modes of vibration of a CNT provides a picture of what one may expect from its dynamic behaviour not only in free, but also in forced vibrations. In this paper, the non-linear modes of vibration of CNTs actuated by electrostatic forces are investigated. For that purpose, a p-version finite element type formulation is implemented, leading to ordinary differential equations of motion in the time domain. The formulation takes into account non-local effects, which influence the inertia and the stiffness of CNTs, as well as the electrostatic actuation. The ordinary differential equations of motion are transformed into algebraic equations of motion via the harmonic balance method (HBM) and then solved by an arc-length continuation method. Several harmonics are considered in the HBM. The importance of non-local effects, combined with the geometrical non-linearity and with the action of the electrostatic force, is analysed. It is found that different combinations of these effects can result in alterations of the natural frequencies, variations in the degrees of softening or hardening, changes in the frequency content of the free vibrations, and alterations in the mode shapes of vibration. It is furthermore found that the small scale, here represented by the non-local theory, has an effect on interactions between the first and higher order modes which are induced by the geometrical and material non-linearities of the system.     

Voltage–amplitude response of alternating current near half natural frequency electrostatically actuated MEMS resonators

This paper uses the Reduced Order Model (ROM) method to investigate the nonlinear-parametric dynamics of electrostatically actuated MEMS cantilever resonators under soft Alternating Current (AC) voltage of frequency near half natural frequency of the resonator. The voltage is between the resonator and a ground plate, and provides a nonlinear parametric actuation for the resonator. Fringe effect and damping forces are included. The resonator is modeled as an Euler–Bernoulli cantilever. Two methods of investigations are compared, Method of Multiple Scales (MMS), and Reduced Order Model. Moreover, the instabilities (bifurcation points) are predicted for both cases, when the voltage is swept up, and when the voltage is swept down. Although MMS and ROM are in good agreement for small amplitudes, MMS fails to accurately predict the behavior of the MEMS resonator for greater amplitudes. Only ROM captures the behavior of the system for large amplitudes. ROM convergence shows that five terms model accurately predicts the steady-states of the resonator for both small and large amplitudes.