Nonlinear Electrostatic Force Research Articles

Enhancement of static and dynamic travel range of electrostatically actuated microbeams using hybrid simulated annealing

This study focuses on the enhancement of travel range of electrostatically driven microbeams in static and dynamic mode using hybrid simulated annealing optimization. Continuous, parametric functions are presented for redistribution of width and thickness profiles of the microbeams. The beam model includes nonlinear electrostatic force with fringing field effect, flexure and midplane stretching. For structural analysis, an energy based technique is used for extracting the pull-in displacement and pull-in voltage. The accuracy of this model is established by validating the results with 3-D finite element simulations. Constraints having engineering importance are implemented through penalty approach. The optimization results show substantial improvement in pull-in displacement of microbeams compared to conventional prismatic microbeam.

Design of Direct Exponential Observers for Fault Detection of Nonlinear MEMS Tunable Capacitor

In this paper a novel method is proposed for construction of an exponential observer for nonlinear system. The presented method is based on direct solution of dynamic error without any linearzing of nonlinear terms. Necessary and sufficient conditions for construction of direct observer are presented. Stability of the observer is checked using Lyapunov theorem. Also the ability of this observer is checked with implementing of observer for fault detection of micro tunable capacitor subjected to nonlinear electrostatic force.

Mechanical behavior analysis on electrostatically actuated rectangular microplates

Microplates are widely used in various MEMS devices based on electrostatic actuation such as MEMS switches, micro pumps and capacitive micromachined ultrasonic transducers (CMUTs). Accurate predictions for the mechanical behavior of the microplate under electrostatic force are important not only for the design and optimization of these electrostatic devices but also for their operation. This paper presents a novel reduced-order model for electrostatically actuated rectangular and square microplates with a new method to treat the nonlinear electrostatic force. The model was developed using Galerkin method which turned the partial-differential equation governing the microplates into an ordinary equation system. Using this model and cosine-like deflection functions, explicit expressions were established for the deflection and pull-in voltage of the rectangular and square microplates. The theoretical results were well validated with the finite element method simulations and experimental data of literature. The expressions for the deflection analysis are able to predict the deflection up to the pull-in position with an error less than 5.0%. The expressions for the pull-in voltage analysis can determine the pull-in voltages with errors less than 1.0%. Additionally, the method to calculate the capacitance variation of the electrostatically actuated microplates was proposed. These theoretical analyses are helpful for design and optimization of electrostatically actuated microdevices.

Large scale electromechanical transistor with application in mass sensing

Nanomechanical transistor (NMT) has evolved from the single electron transistor, a device that operates by shuttling electrons with a self-excited central conductor. The unfavoured aspects of the NMT are the complexity of the fabrication process and its signal processing unit, which could potentially be overcome by designing much larger devices. This paper reports a new design of large scale electromechanical transistor (LSEMT), still taking advantage of the principle of shuttling electrons. However, because of the large size, nonlinear electrostatic forces induced by the transistor itself are not sufficient to drive the mechanical member into vibration—an external force has to be used. In this paper, a LSEMT device is modelled, and its new application in mass sensing is postulated using two coupled mechanical cantilevers, with one of them being embedded in the transistor. The sensor is capable of detecting added mass using the eigenstate shifts method by reading the change of electrical current from the transistor, which has much higher sensitivity than conventional eigenfrequency shift approach used in classical cantilever based mass sensors. Numerical simulations are conducted to investigate the performance of the mass sensor.

Research on Nonlinear Dynamics of Drive Mode inZ-Axis Silicon Microgyroscope

A novel drive method is proposed to increase the vibration amplitude of drive mode of silicon vibratory gyroscopes based on the nonlinear dynamics research made in this paper. A nonlinear dynamic equation, considering the nonlinear mechanical-elastic and electrostatic force, was firstly deduced according to the simplified structure schematic of drive mode. Then, experiments were made to characterize the nonlinear amplitude-frequency response characteristic as well as its dependence on drive voltage. Besides, by solving the nondimensional nonlinear dynamics using Matlab and Simulink, the influence of initial status of drive mode, that is, the displacement and the velocity, on the stable vibration amplitude was presented. The amplitude-frequency characteristic was also analyzed by simulation. The result of simulation shows good agreement with that of experiments. When nonlinearity is present in the amplitude-frequency characteristic, there is a wide flat region with larger amplitude than the peak amplitude obtained in the linear characteristic. Finally, a novel drive method was presented to excite the nonlinear vibration of large amplitude in the downwards-sweeping characteristic curve. It is numerically proved that this merit can be utilized to enlarge the drive mode’s vibration amplitude so as to enhance silicon gyroscopes’ performance.

Application of Thau Observer for Fault Detection of Micro Parallel Plate Capacitor Subjected to Nonlinear Electrostatic Force

This paper investigates the fault detection of a micro parallel plate capacitor subjected to nonlinear electrostatic force. For this end Thau observer, which has good ability in fault detection of nonlinear system has been presented and governing nonlinear dynamic equation of the capacitor has been presented. Upper and lower threshold for fault detection have been obtained. The robustness of the observer to noise and uncertainty and sensitivity to fault has been investigated. Finally, simulation results for fault detection of the capacitor are presented and the ability of the observer in the vicinity of dynamic pull-in voltage has been checked.

The response of a micro-electro-mechanical system (MEMS) cantilever-paddle gas sensor to mechanical shock loads

This work presents an investigation into the effect of nonlinearities on the response of a micro-electro-mechanical system gas sensor under mechanical shock and electrostatic loading. The gas sensor consists of a cantilever microbeam with a rigid microplate (micro-paddle) attached to its tip. A nonlinear Euler-Bernoulli beam theory is used to model the system, accounting for both geometric and inertia nonlinearities, in addition to the nonlinear electrostatic force used to actuate the system. The system of integro-partial-differential equations is discretized using a Galerkin procedure to extract a reduced-order model, which is then used for dynamic simulations of the system responses. The influences of the different components of nonlinearity such as geometric and inertia nonlinearities are examined. The results of the nonlinear model are compared to results obtained from linear beam theory and finite element simulations. For mechanical shock loading, both quasi-static and dynamic responses of a microbeam are considered. The effect of nonlinearity is found to be significant when the deflection of the microbeam exceeds around 30% of its length. The consequence of the large deflection is that the geometric nonlinearity has a much stronger influence on the response in comparison to the inertia nonlinearity. It is also apparent that the effect of the paddle is to enhance the dynamic, as opposed to the quasi-static, response of the microbeam to mechanical shock. For electrostatic actuation, it is found that using a nonlinear beam model to predict the pull-in and the deflection produces a slight improvement over using a linear beam model.

Stability and Dynamic Analysis of the Electrostatic MEMS Actuators

The nonlinear dynamic behavior of the micro actuators is analyzed in this paper. In general, analysis of the electrostatic device is quite difficult and complicated due to the electrostatic coupling effect and the nonlinear electrostatic force. In this study, a hybrid method for the micro-structure system, which combines the differential transformation and finite difference approximation techniques, is used to overcome the nonlinear electrostatic coupling phenomenon. The analysis takes account of the electrostatic coupling effect, the fringing field effect, the residual stress, the nonlinear electrostatic force and squeeze-film damping effect. Finally, the actuating conditions which ensure the stability of the micro actuators are identified by reference to phase portraits.

Dynamic Analysis of Micro Devices with Squeeze-Film Damping Effect Using Hybrid Numerical Scheme

Using traditional methods such as perturbation theory or Galerkin approach method to analyze the dynamic response of electrostatic devices is not easy due to the complexity of the interactions between the electrostatic coupling effect, the fringing field effect, the residual stress, the nonlinear electrostatic force and squeeze-film damping effect. Accordingly, the present study proposes a new approach for analyzing the dynamic response of such devices using a hybrid numerical scheme comprising the differential transformation method and the finite difference method by pure DC or combined DC / AC loading. The validity of the proposed scheme is confirmed by comparing the results obtained for the pull-in voltage of the micro-beam with those presented in the literature derived using a variety of schemes. Overall, the results show that the hybrid numerical scheme provides a suitable means of analyzing the nonlinear dynamic behavior of a wide variety of common electrostatically-actuated microstructures.

Static Analysis of Electrostatically Actuated Micro Cantilever Beam

Electrostatically actuated micro devices experience a fundamental limit on their stable travel range due to a phenomenon called as the pull-in Instability. Accurate determination of pull-in parameters (pull-in displacement and pull-in voltage) is vital in the design of electrostatic micro actuators. A systematic method of analysis of prismatic type of electrostatic beam is discussed in this paper. Using Galerkin method static analysis is carried out. Behaviour of interaction of nonlinear electrostatic force with linear restoring force of the micro cantilever beam is studied. Static analysis using COMSOL multiphysics finite element package is done to validate the results. The results are also compared with existing literature.

Equivalent area nonlinear static and dynamic analysis of electrostatically actuated microstructures

Nonlinear dynamic investigation of electrostatically actuated micro-electro-mechanical-system (MEMS) microcantilever structures is presented. The nonlinear analysis aims to better quantify, than the linear model, the instability threshold associated with electrostatically actuated MEMS structures, where the pull-in voltage of the microcantilever is determined using a phase portrait analysis of the microsystem. The microcantilever is modeled as a lumped mass-spring system. The nonlinear electrostatic force is incorporated into the lumped microsystem through an equivalent area of the microcantilever for a given electrostatic potential. Electro-mechanical force balance plots are obtained for various electrostatic potentials from which the static equilibrium positions of the microcantilever are obtained and the respective conservative energy values are determined. Subsequently, phase portrait plots are obtained for the corresponding energy values from which the pull-in voltage is estimated for the microsystem. This pull-in voltage value is in good agreement with the previously published results for the same geometric and material parameters. The results obtained for linear electrostatic models are also presented for comparison.

Symmetry breaking in an initially curved micro beam loaded by a distributed electrostatic force

The asymmetric buckling of a shallow initially curved stress-free micro beam subjected to distributed nonlinear deflection-dependent electrostatic force is studied. In order to highlight the symmetry breaking phenomenon and the approach to its analysis, the subsidiary simplified problem of a curved beam attached to a linearly elastic foundation, and subjected to uniformly distributed “mechanical” load, which is independent of deflections, is addressed first. The analysis is based on a two degrees of freedom reduced order (RO) model resulting from the Galerkin decomposition with linear undamped eigenmodes of a straight beam used as the base functions. Simple approximate expressions are derived defining the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric limit point buckling curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the curve. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict non-symmetric buckling in electrostatically actuated bistable micro beams.

Nonlinear Dynamic Behavior Analysis of Microelectrostatic Actuator Based on a Continuous Model Under Electrostatic Loading

Analyzing the dynamic behavior of microelectrostatic devices is problematic due to the complexity of the interactions between the electrostatic coupling effect, the fringing field effect, the residual stress, the tensile stress, and the nonlinear electrostatic force. In this study, this problem is resolved by modeling the electrostatic system using a continuous model and solving the resulting governing equation of motion using a hybrid scheme comprising the differential transformation method and the finite difference method. The feasibility of the proposed approach is demonstrated by modeling the dynamic responses of two fixed-fixed microbeams when actuated by a dc voltage. It is shown that the numerical results for the pull-in voltage deviate by no more than 1.74% from those presented in the literature. The hybrid scheme is then applied to examine the nonlinear behavior of one clamped microbeam actuated by a combined dc/ac scheme. The beam displacement is analyzed as a function of both the magnitude and the frequency of the ac voltage. Finally, the actuating conditions, which ensure the stability of the microbeam, are identified by reference to phase portraits and Poincaré maps. Overall, the results presented in this study show that the hybrid differential transformation and finite difference method provides a suitable means of analyzing a wide variety of common electrostatically actuated microstructures.

Hofmeister effects: interplay of hydration, nonelectrostatic potentials, and ion size

The classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of colloids, and corresponding theories of electrolytes, are unable to explain ion specific forces between colloidal particles quantitatively. The same is true generally, for surfactant aggregates, lipids, proteins, for zeta and membrane potentials and in adsorption phenomena. Even with fitting parameters the theory is not predictive. The classical theories of interactions begin with continuum solvent electrostatic (double layer) forces. Extensions to include surface hydration are taken care of with concepts like inner and outer Helmholtz planes, and "dressed" ion sizes. The opposing quantum mechanical attractive forces (variously termed van der Waals, Hamaker, Lifshitz, dispersion, nonelectrostatic forces) are treated separately from electrostatic forces. The ansatz that separates electrostatic and quantum forces can be shown to be thermodynamically inconsistent. Hofmeister or specific ion effects usually show up above ≈10(-2) molar salt. Parameters to accommodate these in terms of hydration and ion size had to be invoked, specific to each case. Ionic dispersion forces, between ions and solvent, for ion-ion and ion-surface interactions are not explicit in classical theories that use "effective" potentials. It can be shown that the missing ionic quantum fluctuation forces have a large role to play in specific ion effects, and in hydration. In a consistent predictive theory they have to be included at the same level as the nonlinear electrostatic forces that form the skeletal framework of standard theory. This poses a challenge. The challenges go further than academic theory and have implications for the interpretation and meaning of concepts like pH, buffers and membrane potentials, and for their experimental interpretation. In this article we overview recent quantitative developments in our evolving understanding of the theoretical origins of specific ion, or Hofmeister effects. These are demonstrated through an analysis that incorporates nonelectrostatic ion-surface and ion-ion dispersion interactions. This is based on ab initio ionic polarisabilities, and finite ion sizes quantified through recent ab initio work. We underline the central role of ionic polarisabilities and of ion size in the nonelectrostatic interactions that involve ions, solvent molecules and interfaces. Examples of mechanisms through which they operate are discussed in detail. An ab initio hydration model that accounts for polarisabilities of the tightly held hydration shell of "cosmotropic" ions is introduced. It is shown how Hofmeister effects depend on an interplay between specific surface chemistry, surface charge density, pH, buffer, and counterion with polarisabilities and ion size. We also discuss how the most recent theories on surface hydration combined with hydrated nonelectrostatic potentials may predict experimental zeta potentials and hydration forces.

Nonlinear Dynamics of Electrically Actuated Carbon Nanotube Resonators

Abstract This work presents an investigation of the nonlinear dynamics of carbon nanotubes (CNTs) when actuated by a dc load superimposed to an ac harmonic load. Cantilevered and clamped-clamped CNTs are studied. The carbon nanotube is described by an Euler–Bernoulli beam model that accounts for the geometric nonlinearity and the nonlinear electrostatic force. A reduced-order model based on the Galerkin method is developed and utilized to simulate the static and dynamic responses of the carbon nanotube. The free-vibration problem is solved using both the reduced-order model and by solving directly the coupled in-plane and out-of-plane boundary-value problems governing the motion of the nanotube. Comparison of the results generated by these two methods to published data of a more complicated molecular dynamics model shows good agreement. Dynamic analysis is conducted to explore the nonlinear oscillation of the carbon nanotube near its fundamental natural frequency (primary-resonance) and near one-half, twice, and three times its natural frequency (secondary-resonances). The nonlinear analysis is carried out using a shooting technique to capture periodic orbits combined with the Floquet theory to analyze their stability. The nonlinear resonance frequency of the CNTs is calculated as a function of the ac load. Subharmonic-resonances are found to be activated over a wide range of frequencies, which is a unique property of CNTs. The results show that these resonances can lead to complex nonlinear dynamics phenomena, such as hysteresis, dynamic pull-in, hardening and softening behaviors, and frequency bands with an inevitable escape from a potential well.

A Highly Accurate Pull-in Voltage Model for an Atomic Force Microscope Probe

A readily usable closed-form model has been developed to determine the pull-in voltage of an atomic force microscopy (AFM) cantilever probe under electrostatic actuation. The analytical model is derived based on the Euler-Bernoulli beam theory, Taylor series expansion of the nonlinear electrostatic force, and deflection function of the first natural mode of a cantilever beam. The model takes account of the electrostatic force associated with the fringing field capacitances between the cantilever probe and the substrate to predict a more accurate pull-in voltage. The developed closed-form model has been verified by comparing the model predicted values with published experimental results with a maximum deviation of 3.36%. The model has also been compared with a published closed-form model and 3-D electromechanical finite element analysis (FEA) carried out by the authors. The results are found to be in excellent agreement.

A Highly Accurate Closed-Form Model for Pull-in Voltage of Circular Diaphragms under Large Deflection

A simple easy to use highly accurate closed-form model to determine the pull-in voltage of electrostatically actuated circular diaphragms subject to large deflection has been developed. The model takes account of the nonlinear stretching of the diaphragm during large deflection, effects of residual stress, bending stress, and the effects of fringing field capacitances. At first a linearized uniform model of the nonlinear non-uniform electrostatic force has been derived that includes the effects of the fringing field capacitances. This model is then used in conjunction with the load-deflection model of a circular diaphragm subject to large deflection to derive the closed-form model for the pull-in voltage based on the condition that at unstable equilibrium, the electrostatic force is just balanced by elastic restoring force. The model has been verified by extensive 3-D electromechanical finite element analysis (FEA) using IntelliSuite. The model is in excellent agreement with 3-D electromechanical FEA with a maximum deviation of 1.6% for a wide range of residual stress and pull-in voltage values. The model leads to an integrated design strategy to optimize the electrical and mechanical design variables for MEMS-based capacitive type sensors having circular diaphragms. The model can be used to determine pull-in voltages of MEMS capacitive type pressure sensors, capacitive micromachined ultrasound transducers (CMUT) for medical diagnostic imaging, MEMS based microphones, touch mode pressure sensors, and other application areas where electrostatically actuated circular diaphragms are used. Keywords: MEMS, electrostatic, sensor, circular diaphragm, pull-in voltage

Analysis of Microelectromechanical Systems Using the Meshless Random Differential Quadrature Method

Several devices of microelectromechanical systems (MEMS) are analyzed in the presented work, using a novel numerical meshless method called the random differential quadrature (RDQ) method. The differential quadrature (DQ) is an effective derivative discretization technique but it requires all the field nodes to be arranged in a collinear manner with a pre-defined pattern. This limitation of the DQ method is overcome in the RDQ method using the interpolation function by the fixed reproducing kernel particle method (fixed RKPM). The RDQ method extends the applicability of the DQ method over a regular as well as an irregular domain discretized by uniform or randomly distributed field nodes. Due to the strong-form nature, RDQ method captures well the local high gradients. These features of the RDQ method enable it to efficiently solve the MEMS problems with different boundary conditions. In the presented work, several MEMS devices that are governed by the nonlinear electrostatic force are analyzed using the RDQ method, and their results are compared with the other simulation results presented in the existing literature. It is seen that the RDQ method effectively and accurately solves the MEMS devices problems.

Application of hybrid differential transformation/finite difference method to nonlinear analysis of micro fixed-fixed beam

Analyzing the dynamic response of electrostatic devices is problematic due to the complexity of the interactions between the electrostatic coupling effect, the fringing field effect and the nonlinear electrostatic force. To resolve this problem, this study presents an efficient computational scheme in which the nonlinear governing equation of the electrostatic device is obtained in accordance with Hamilton’s principle and is then solved using a hybrid differential transformation/finite difference method. The feasibility of the proposed approach is demonstrated by modeling the dynamic responses of two micro fixed-fixed beams with lengths of 250 and 350 μm, respectively. The numerical results show that the pull-in voltage reduces as the beam length increases due to a loss in the structural rigidity. Furthermore, it is shown that the present results for the pull-in voltage deviate by no more than 0.75% from those derived in the literature using a variety of different schemes. Overall, the results presented in this study demonstrate that the proposed hybrid method represents a computationally efficient and precise means of obtaining detailed insights into the nonlinear dynamic behavior of micro fixed-fixed beams and similar micro-electro-mechanical systems (MEMS)-based devices.

Non-contact electrostatic microactuator using slit structures: theory and a preliminary test

A new type of electrostatic microactuator with pull-in and pull-out behaviors has been presented based on nonlinear electrostatic force acting on a slit structure. Theory for the microactuator has also been developed to understand the nonlinear behaviors of the microactuator when a voltage is applied to the microactuator. Using dimensional analysis and finite element analysis (FEA), the nonlinear electrostatic force was investigated and was formulated in a simple closed-form equation whose coefficients are functions of the geometry of the microactuator. Afterwards, a force balance equation of the microactuator was set up and then an equation for the height of the movable structure at a voltage was derived. Theoretical analysis shows that without mechanical contact the microactuator suffers from a pull-out as well as a pull-in phenomenon. When the movable grating is spaced by a distance larger than a minimum distance, the pull-in and pull-out phenomena occur at the corresponding pull-in and pull-out voltages, respectively. In a preliminary experiment using a microactuator made of a 1.5 µm thick polysilicon, the pull-in voltage is measured to be 19 V. As a new type of electrostatic microactuator, microactuators using slit structures have potential applications in micro- and nano-systems such as actuators with a large travel range, micromirrors, variable capacitors, tunable RF filters and microaccelerometers.