Pull-in Value Research Articles

FRACTAL PULL-IN MOTION OF ELECTROSTATIC MEMS RESONATORS BY THE VARIATIONAL ITERATION METHOD

The dynamic pull-in instability of a microstructure is a vast research field and its analysis is of great significance for ensuring the effective operation and reliability of micro-electromechanical systems (MEMS). A fractal modification for the traditional MEMS system is suggested to be closer to the real state as a practical application in the air with impurities or humidity. In this paper, we establish a fractal model for a class of electrostatically driven microstructure resonant sensors and find the phenomenon of pull-in instability caused by DC bias voltage and AC excitation voltage. The variational iteration method has been extended to obtain approximate analytical solutions and the pull-in threshold value for the fractal MEMS system. The result obtained from this method shows good agreement with the numerical solution. The simple and efficient operability is demonstrated through theoretical analysis and results comparisons.

Stabilization of Electrostatically Actuated Micro-pipe Conveying Fluid Using Parametric Excitation

This paper investigates the parametric excitation of a micro-pipe conveying fluid suspended between two symmetric electrodes. Electrostatically actuated micro-pipes may become unstable when the exciting voltage is greater than the pull-in value. It is demonstrated that the parametric excitation of a micro-pipe by periodic (ac) voltages may have a stabilizing effect and permit an increase of the steady (dc) component of the actuation voltage beyond the pull-in value. Mathieu type equation of the system is obtained by applying Taylor series expansion and Galerkin method to the nonlinear partial differential equation of motion. Floquet theory is used to extract the transition curves and stability margins in physical parameters space (Vdc-Vac). In addition, the stability margins are plotted in flow velocity and excitation amplitude space (u-Vac space). The results depict that the micro-pipe remains stable even if the flow velocity is more than the critical value for a certain dc voltage. For instance, in absence of the (ac) component, it is shown that pull-in voltages associated to critical velocities 3 and 6 are 14.06 and 5.4 volt, respectively. However, transition curves show that superimposing an (ac) component with forcing frequency Ω=10 increases the pull-in voltage beyond these values. Furthermore, for the present pull-in voltages the critical velocities 3 and 6 could be increases with imposing some (ac) component. These results are discussed in detail in simulation results section where the transion curves are ploted quantitatively.

Instability threshold of rippled carbon nanotube nanotweezers in the low vacuum gas flow incorporating Dirichlet and Neumann modes of Casimir energy

The aim of this research work is to address the influences of dispersion forces and rippled configuration on the instability threshold of carbon nanotube (CNT) based nanotweezers. To this end, the Dirichlet and Neumann modes of Casimir force arisen from the electric and magnetic energies is developed for cylinder–cylinder geometry. Moreover, the CNTs rippling deformation which experimentally revealed is included in the Euler-Bernoulli beam model to modify the governing equations. The differential quadrature method (DQM) in conjunction with the 4th-order Runge-Kutta algorithm is employed to numerically simulate the non-linear partial differential equations. It is interestingly demonstrated that these phenomena remarkably affect the electromechanical behavior of nanotweezers fabricated from CNTs. By taking the rippling configuration and Casimir attraction between tubes into account, the pull-in voltage decreases. On the other hand, when the gas damping effect due to low vacuum environment is taken into consideration, the pull-in value increases. The accuracy of the present modeling is compared with those experimentally published in the literature, giving excellent results.

Pull-In Effect of Suspended Microchannel Resonator Sensor Subjected to Electrostatic Actuation.

In this article, the pull-in instability and dynamic characteristics of electrostatically actuated suspended microchannel resonators are studied. A theoretical model is presented to describe the pull-in effect of suspended microchannel resonators by considering the electrostatic field and the internal fluid. The results indicate that the system is subjected to both the pull-in instability and the flutter. The former is induced by the applied voltage which exceeds the pull-in value while the latter occurs as the velocity of steady flow get closer to the critical velocity. The statically and dynamically stable regions are presented by thoroughly studying the two forms of instability. It is demonstrated that the steady flow can remarkably extend the dynamic stable range of pull-in while the applied voltage slightly decreases the critical velocity. It is also shown that the dc voltage and the steady flow can adjust the resonant frequency while the ac voltage can modulate the vibrational amplitude of the resonator.

On the dynamic pull-in instability in a mass-spring model of electrostatically actuated MEMS devices

In this work we study the mass-spring system(1)x¨+αx˙+x=−λ(1+x)2, which is a simplified model for an electrostatically actuated MEMS device. The static pull-in value is λ⁎=427, which corresponds to the largest value of λ for which there exists at least one stationary solution. For λ>λ⁎ there are no stationary solutions and x(t) achieves the value −1 in finite time: touchdown occurs. The maximal displacement achieved by a stationary solution, known as the pull-in distance, is equal to −13 in this model. Assuming that the motion starts from rest, we establish the existence of a dynamic pull-in value λd⁎(α)∈(0,λ⁎), defined for α∈[0,∞), which is a threshold in the sense that x(t) approaches a stable stationary solution as t→∞ for 0<λ<λd⁎(α), while touchdown occurs for λ>λd⁎(α). This dynamic pull-in value is a continuous, strictly increasing function of α and limα→∞⁡λd⁎(α)=λ⁎. A similar result is obtained for initial conditions of the form x(0)∈(−13,1), x˙(0)=0.

Dynamic and Static Pull-in instability of electrostatically actuated nano/micro membranes under the effects of Casimir force and squeezed film damping

In the current study, the effects of Casimir force and squeeze film damping on pull-in instability and dynamic behavior of electrostatically actuated nano and micro electromechanical systems are investigated separately. Linear elastic membrane theory is used to model the static and dynamic behavior of the system for strip, annular and disk geometries. Squeeze film damping is modeled using nonlinear Reynolds equation. Both equation of motion and nonlinear Reynolds equation are first nondimensionalized, and then discretized and solved by means of finite element method. Static pullin analysis is performed and validated by previous researches, and then dynamic pull-in values are investigated and compared with static pull-in parameters. In the next step, the effect of squeeze film damping, ambient pressure and Casimir force on the system dynamics is studied. Results show significant effect of Casimir force and squeeze film damping on the system behavior which is considerable for fabrication and design

An experimental and theoretical investigation of electrostatically coupled cantilever microbeams

We present an experimental and theoretical investigation of the static and dynamic behavior of electrostatically coupled laterally actuated silicon microbeams. The coupled beam resonators are composed of two almost identical flexible cantilever beams forming the two sides of a capacitor. The experimental and theoretical analysis of the coupled system is carried out and compared against the results of beams actuated with fixed electrodes individually. The pull-in characteristics of the electrostatically coupled beams are studied, including the pull-in time. The dynamics of the coupled dual beams are explored via frequency sweeps around the neighborhood of the natural frequencies of the system for different input voltages. Good agreement is reported among the simulation results and the experimental data. The results show considerable drop in the pull-in values as compared to single microbeam resonators. The dynamics of the coupled beam resonators are demonstrated as a way to increase the bandwidth of the resonator near primary resonance as well as a way to introduce increased frequency shift, which can be promising for resonant sensing applications. Moreover the dynamic pull-in characteristics are also studied and proposed as a way to sense the shift in resonance frequency.

A numerical study of the pull-in instability in some free boundary models for MEMS

In this work we numerically compute the bifurcation curve for stationary solutions of the free boundary problem for MEMS in one space dimension. It has a single turning point, as in the case of the small aspect ratio limit. We also find a threshold for the existence of global in time solutions of the evolution equation for the MEMS in the form of either a heat or a damped wave equation. This threshold is what we term the dynamical pull-in value: it separates the stable operating regime from the touchdown regime. The numerical calculations show that the dynamical threshold values for the heat equation coincide with the static values. For the damped wave equation the dynamical threshold values are smaller than the static values. This result is in qualitative agreement with those reported for the model of the MEMS based on a simplified mass-spring system, as studied in the engineering literature. In the case of the damped wave equation, we also show that the aspect ratio of the device is more important than the inertia in the determination of the pull-in value.

Stability and Bifurcation Analysis of an Asymmetrically Electrostatically Actuated Microbeam

This paper deals with the study of bifurcational behavior of a capacitive microbeam actuated by asymmetrically located electrodes in the upper and lower sides of the microbeam. A distributed and a modified two degree of freedom (DOF) mass–spring model have been implemented for the analysis of the microbeam behavior. Fixed or equilibrium points of the microbeam have been obtained and have been shown that with variation of the applied voltage as a control parameter the number of equilibrium points is changed. The stability of the fixed points has been investigated by Jacobian matrix of system in the two DOF mass–spring model. Pull-in or critical values of the applied voltage leading to qualitative changes in the microbeam behavior have been obtained and has been shown that the proposed model has a tendency to a static instability by undergoing a pitchfork bifurcation whereas classic capacitive microbeams cease to have stability by undergoing to a saddle node bifurcation.

Stability Analysis in Parametrically Excited Electrostatic Torsional Micro-actuators

This paper addresses the static and dynamic stabilities of a parametrically excited torsional micro-actuator. The system is composed of a rectangular micro-mirror symmetrically suspended between two electrodes and acted upon by a steady (dc ) while simultaneously superimposed to an (ac ) voltage. First, the stability of the system subjected to a quasi-statically applied (dc ) voltage is investigated, where the pull-in instability, equilibrium positions, and bifurcation points of the system are determined. Then by superimposing an (ac) voltage and extracting a Mathieu type governing equation the effects of (ac ) component on the stability of the system is investigated. By varying excitation parameters (steady (dc) voltage and time-dependent amplitude of (ac ) excitation), transition curves and the stability margins of the micro-mirror are demonstrated. Theoretically obtained margins are checked by means of numerical simulations. The results show that superimposing the harmonic (ac ) component could have a stabilizing effect and allow an increase of the steady (dc ) component beyond the pull-in value. The obtained results could be used in design of micro-actuators.

A Non-Contacted Magnetic Sensor with Direction Memory Based on Biot-Savart Theory

Magnetic sensor with direction memory can be used to control the motion direction. Based on Biot-Savart theory, the magnetic field distribution expression of a bar-type in external space is derived, and the superposition distribution of both inner and outer magnet is directed by the principle of superposition, which can be used to quantitatively describe the magnetic distribution formed by inner and outer magnet, and accurately scope the effective field. According to the operating characteristics of the magnetic reed switch with different value of pull-in and drop-out, by a proper detecting distance to ensure the magnetic field strength value of inner magnet at magnetic reed switch greater than pull-in value and less than drop-out value to make magnetic reed switch maintaining the original state when outer magnet leaving. Meanwhile, by a proper detecting distance to ensure the superinposed magnetic field strength value greater than pull-in value in the forward direction, and less than drop-out value in backward direction. Calculation of response curves show the impacts of magnet size, response intensity and detecting distance variation on the sensor.

Static and dynamic stability modeling of a capacitive FGM micro-beam in presence of temperature changes

Stability of a functionally graded (FG) micro-beam, based on modified couple stress theory (MCST), subjected to nonlinear electrostatic pressure and thermal changes regarding convection and radiation, is the main purpose of this paper. It is assumed that the functionally graded beam, made of metal and ceramic, follows the volume fraction definition and law of mixtures, and its properties change as an exponential function through its thickness. By changing the ceramic constituent percent of the bottom surface, five different types of the micro-beams are investigated. The static pull-in voltages in presence of temperature changes are obtained by using step-by-step linearization method (SSLM) and, by adapting Runge–Kutta approach, the dynamic pull-in voltages are obtained numerically. Though the temperature distribution through the thickness of FG micro-beam (due to its too small measurement) is considered uniform, owing to the different thermal expansions of layers, temperature changes cause deflection in the micro-beam, and consequently affect pull-in values. Hence the profound effects of different material constituent over the pull-in voltages are illustrated and it is graphically displayed that how in some cases neglecting components of the couple stress leads to inaccurate results.

Dynamic pull-in instability of electrostatically actuated beams incorporating Casimir and van der Waals forces

In this study, influences of intermolecular forces on the dynamic pull-in instability of electrostatically actuated beams are investigated. The effects of midplane stretching, electrostatic actuation, fringing fields, and intermolecular forces are considered. The boundary conditions of the beams are clamped—free and clamped—clamped. A finite-element model is developed to discretize the governing equations, and Newmark time discretization is then employed to solve the discretized equations. The static pull-in instability is investigated to validate the model. Finally, dynamic pull-in instability of cantilevers and double-clamped beams are studied considering the Casimir and van der Waals effects. The results indicate that by increasing the Casimir and van der Waals effects, the effect of inertia on pull-in values considerably increases.

Stochastic finite element method for analyzing static and dynamic pull-in of microsystems

Electro–mechanical sensors and actuators are a specific type of microsystems. The electrostatic pull-in value is one of the defining characteristics for these devices. Because the material and geometrical properties of micro fabricated systems are often very uncertain, this pull-in value can be subject to considerable variations. Therefore it is important to be able to estimate how uncertainty of mechanical properties propagates to the uncertainty of pull-in values. In this work the required design sensitivities of static and dynamic pull-in are derived. These sensitivities are used to perform a perturbation–based stochastic FEM analysis of an electromechanical device. This stochastic analysis consists of an uncertainty analysis and a reliability analysis. This stochastic analysis is validated by an expensive crude Monte Carlo computation.

Design methodology for variable capacitors

A theoretical and systematic methodology has been developed to design variable capacitors based on the parallel plate and two-degree-of-freedom (DOF) torsional actuator. For easy design of a variable capacitor without any time-consuming numerical analysis or simulation, closed-form expressions for the capacitance of parallel plate actuators and torsional microactuators are obtained as functions of applied voltage and geometry. A theoretical capacitance of the parallel plate capacitor is obtained in a closed-form. Model and formulation of the capacitance of two-DOF variable capacitors are presented and the capacitance, height and angle of the capacitor are numerically solved to give their pull-in value in simple expressions. The theoretically and numerically solved capacitances are in good agreement within an error of 4.3% with those from experiments using surface-micromachined variable capacitors. As such, the derived capacitance and suggested methodology can be used for design and analysis of the parallel plate and the two-DOF variable capacitor.

Stabilization of electrostatically actuated microstructures using parametric excitation

Electrostatically actuated microstructures are inherently nonlinear and can become unstable. Pull-in instability is encountered as a basic instability mechanism. We demonstrate that the parametric excitation of a microstructure by periodic (ac) voltages may have a stabilizing effect and permits an increase of the steady (dc) component of the actuation voltage beyond the pull-in value. An elastic string as well as a cantilever beam are considered in order to illustrate the influence of fast-scale excitation on the slow-scale behavior. The main conclusions about the stability are drawn using the simplest model of a parametrically excited system described by Mathieu and Hill's equations. Theoretical results are verified by numerical analysis of microstructure subject to nonlinear electrostatic forces and performed by using Galerkin decomposition with undamped linear modes as base functions. The parametric stabilization of a cantilever beam is demonstrated experimentally.

Pull-in Dynamics of an Elastic Beam Actuated by Continuously Distributed Electrostatic Force

A detailed study of the transient nonlinear dynamics of an electrically actuated micron scale beam is presented. A model developed using the Galerkin procedure with normal modes as a basis accounts for the distributed nonlinear electrostatic forces, nonlinear squeezed film damping, and rotational inertia of a mass carried by the beam. Special attention is paid to the dynamics of the beam near instability points. Results generated by the model and confirmed experimentally show that nonlinear damping leads to shrinkage of the spatial region where stable motion is realizable. The voltage that causes dynamic instability, in turn, approaches the static pull-in value.

Pull-in study for round double-gimbaled electrostatic torsion actuators

This paper reports an analytical study of the pull-in effect in round, double-gimbaled, electrostatic torsion actuators with buried, variable length electrodes, designed for optical cross-connect applications. We find that the fractional tilt at pull-in for the inner round plate in this system depends only on the ratio of the length of the buried electrode to the radius of the plate. The fractional tilt at pull-in for the outer support ring depends only on the ratio of the length of the buried electrode to the outer radius of the ring and the ratio of the ring's inner and outer radii. Expressions for the pull-in voltage are determined in both cases. General relationships are also derived relating the applied voltage to the resulting tilt angle, both normalized by their pull-in values. Calculated results are verified by comparison with finite element MEMCAD simulations, with fractional difference smaller than 4% for torsion-mode dominant systems.

Pull-In Time and Range of Any Order Generalized PLL

The general pull-in range and pull-in time expressions are derived versus the relevant system parameters and the initial detuning for any order loop with any type of phase detector by the quasistationary approach. Even though the pull-in time expression becomes more accurate as the loop's order becomes larger, it is correct also for second-and third-order loops provided that the initial detuning, supposed large, is approaching the pull-in range value.