Surface and nonlocal effects on response of linear and nonlinear NEMS devices

Nonlocal and surface effects become important for nanoscale devices. To model these effects on frequency response of linear and nonlinear nanobeam subjected to electrostatic excitation, we use Eringen’s nonlocal elastic theory and surface elastic theory proposed by Gurtin and Murdoch to modify the governing equation. Subsequently, we apply Galerkin’s method with exact mode shape including nonlocal and surface effects to get static and dynamic modal equations. After validating the procedure with the available results, we analyze the variation of pull-in voltage and frequency resonance by varying surface and nonlocal parameters. To do frequency analysis of nonlinear system, we solve nonlinear dynamic equation using the method of multiple scale. We found that the frequency response of nonlinear system reduces for fixed excitation as the surface and nonlocal effects increase. Also, we found that the nature of nonlinearity can be tuned from hardening to softening by increasing the nonlocal effects.     

Bending and vibration of an electrostatically actuated circular microplate in presence of Casimir force

The pull-in instability and the vibration for a prestressed circular electrostatically actuated microplate are investigated in consideration of the Casimir force. Based on von Kármán’s nonlinear bending theory of thin plates, the governing equations for the whole analysis are decomposed into two two-point boundary value problems. For static deformation of the plate, the geometric nonlinearity is involved and the pull-in parameters are obtained by using the shooting method through taking the applied voltage or Casimir parameter as an unknown. This algorithm is also used to study the small amplitude free vibration about the predeformed bending configuration following an assumed harmonic time mode, and the variation of the prestress and Casimir parameters dependent fundamental natural frequency with the applied voltage is presented. Several case studies are compared with available published simulations to confirm the proposed method. The influences of various parameters, such as the initial gap-thickness ratio, Casimir effect, prestress on the pull-in instability behavior and the natural frequency are examined.     

Dynamic characteristic analysis of an electrostatically-actuated circular nanoplate subject to surface effects

This study investigates the influence of surface effect on the nonlinear behavior of an electrostatically actuated circular nanoplate. The Casimir force, surface effects, and the electrostatic force are modelled. In performing the analysis, the nonlinear governing equation of a circular nanoplate is solved using a hybrid computational scheme combining a differential transformation and finite differences. The method is used to model systems found in previous literature using different methods, producing consistent results, thus verifying that it is suitable for treatment of the nonlinear electrostatic coupling phenomenon. The obtained results from numerical methods demonstrated that the relationship between the thickness, radius, and gap size of a circular nanoplate, and its pull-in voltage, is scale-dependent. The model exhibits size-dependent behavior, showing that surface effects significantly influence the dynamic response of circular nanoplate actuators. Moreover, the influence of surface stress on the pull-in voltage of circular nanoplate is found to be more significant than the influence of surface elastic modulus. Finally, the effects of actuation voltage, excitation frequency, and surface effects on the dynamic behavior of the nanoplate are examined through use of phase portraits. Overall, the results show that the using hybrid method here presented is a suitable technique for analyzing nonlinear behavior characteristic of circular nanoplates.     

Size effect on pull-in behavior of electrostatically actuated microbeams based on a modified couple stress theory

A variety of micro-scale experiments have demonstrated that the mechanical property of some metals and polymers on the order of micron scale are size dependence. Taking into account the size effect on the mechanical property of materials and the inherent nonlinear property of electrostatic force, the static pull-in behavior of an electrostatically actuated Bernoulli–Euler microbeam is analyzed on the basis of a modified couple stress theory. The approximate analytical solutions to the pull-in voltage and pull-in displacement of the microbeam are derived by using the Rayleigh–Ritz method. The results show that the normalized pull-in voltage of the microbeam increases by a factor of 3.1 as the microbeam thickness equals to the material length scale parameter and exhibits size effect remarkably. However, the size effect on the pull-in voltage is almost diminishing as the microbeam thickness is far greater than the material length scale parameter. The normalized pull-in displacement of the microbeam exhibits size independence and equals to 0.448 and 0.398 for the cantilever beam and clamped–clamped beam, respectively.     

Pull-in parameters of cantilever type nanomechanical switches in presence of Casimir force

In this paper, the effect of the Casimir force on pull-in parameters of cantilever type nanomechanical switches is investigated by using a distributed parameter model. In modeling of the electrostatic force, the fringing field effect is taken into account. The model is nonlinear due to the inherent nonlinearity of the Casimir and electrostatic forces. The nonlinear differential equation of the model is transformed into the integral form by using the Green’s function of the cantilever beam. The integral equation is solved analytically by assuming an appropriate shape function for the beam deflection. The pull-in parameters of the switch are computed in three cases including nanoactuators without applied voltages, microswitches, and the general case of nanocantilevers. Nanoactuators without applied voltages are studied to determine the detachment length and the minimum initial gap of freestanding nanocantilevers, which are the basic design parameters for NEMS switches. The pull-in parameters of microswitches are investigated as a special case of our study by neglecting the Casimir effect and the results are verified through comparison with other works published in the literature. The general case of nanocantilevers is studied considering coexistence of the electrostatic and Casimir forces. The results of the distributed parameter model are compared with the lumped parameter model.     

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.     

Numerical investigation into nonlinear dynamic behavior of electrically-actuated clamped–clamped micro-beam with squeeze-film damping effect

A numerical investigation is performed into the nonlinear dynamic behavior of a clamped–clamped micro-beam actuated by a combined DC/AC voltage and subject to a squeeze-film damping effect. An analytical model based on a nonlinear deflection equation and a linearized Reynolds equation is proposed to describe the deflection of the micro-beam under the effects of the electrostatic actuating force. The deflection of the micro-beam is investigated under various actuating conditions by solving the analytical model using a hybrid numerical scheme comprising the differential transformation method and the finite difference approximation method. It is shown that the numerical results for the dynamic pull-in voltage of the clamped–clamped micro-beam deviate by no more than 2.04% from those presented in the literature based on the conventional finite difference scheme. The effects of the AC voltage amplitude, excitation frequency, residual stress, and ambient pressure on the center-point displacement of the micro-beam are systematically explored. Moreover, the actuation conditions which ensure the stability of the micro-beam are identified by means of phase portraits. Overall, the results presented in this study confirm that the hybrid numerical method provides an accurate means of analyzing the complex nonlinear behavior of common electrostatically-actuated microstructures.     

Pull-in instability and free vibration of electrically actuated poly-SiGe graded micro-beams with a curved ground electrode

This paper investigates the pull-in instability and free vibration of functionally graded poly-SiGe micro-beams under combined electrostatic force, intermolecular force and axial residual stress, with an emphasis on the effects of ground electrode shape, position-dependent material composition, and geometrically nonlinear deformation of the micro-beam. The differential quadrature (DQ) method is employed to solve the nonlinear differential governing equations to obtain the pull-in voltage and vibration frequencies of the clamped poly-SiGe micro-beams. The present analysis is validated through direct comparisons with published experimental results and excellent agreement has been achieved. A parametric study is conducted to investigate the effects of material composition, ground electrode shape, axial residual stress and geometrical nonlinearity on the pull-in voltage and frequency characteristics.     

Vibrational analysis of electrostatically actuated microstructures considering nonlinear effects

The vibrational behavior of electrostatically actuated microstructures subjected to nonlinear squeeze film damping and in-plane forces is investigated. First-Order Shear Deformation Theory is used to model dynamical system by means of finite element method, while finite difference method is applied to solve the nonlinear Reynolds equation of squeeze film damping simultaneously. Vibrational analysis of microplates is performed by solving eigenvalue problem, after validating the model by pull-in phenomenon and transient behavior. In addition, considering nonlinear squeeze film damping and step-input actuations, response frequencies of microplates are calculated. Effect of ambient pressure and in-plane forces on dynamic pull-in phenomenon is also studied. Results for simplified models are verified and are in good agreement with the published literature. This investigation can reveal nonlinear vibrational behavior of microstructures.     

Thermal and surface effects on the pull-in characteristics of circular nanoplate NEMS actuator based on nonlocal elasticity theory

This paper aims to investigate the coupling influences of thermal loading and surface effects on pull-in instability of electrically actuated circular nanoplate based on Eringen's nonlocal elasticity theory, where the electrostatic force and thermally corrected Casimir force are considered. By utilizing the Kirchhoff plate theory, the nonlinear equilibrium equation of axisymmetric circular nanoplate with variable coefficients and clamped boundary conditions is derived and analytically solved. The results describe the influences of surface effect and thermal loading on pull-in displacements and pull-in voltages of nanoplate under thermal corrected Casimir force. It is seen that the surface effect becomes significant at the pull-in state with the decrease of nanoplate thicknesses, and the residual surface tension exerts a greater influence on the pull-in behavior compared to the surface elastic modulus. In addition, it is found that temperature change plays a great role in the pull-in phenomenon; when the temperature change grows, the circular nanoplate without applied voltage is also led to collapse.     

Comparison of generalized differential quadrature and Galerkin methods for the analysis of micro-electro-mechanical coupled systems

This paper presents a comprehensive comparison study between the generalized differential quadrature (GDQ) and the well-known global Galerkin method for analysis of pull-in behavior of nonlinear micro-electro-mechanical coupled systems. The nonlinear governing integro-differential equation for double clamped MEMS devices which was derived using variational principle by the authors [Sadeghian H, Rezazadeh G, Osterberg PM. Application of the generalized differential quadrature method to the study of pull-in phenomena of MEMS switches. J Microelectromech Syst 2007;16(6):1334–40] is discretized by applying Galerkin and GDQ methods. The divergence instability or pull-in phenomenon is analyzed. Obtained results are compared with the results of the pervious works. The Galerkin method is implemented with effect of number of used shape functions. Different types of trail functions on calculated pull-in voltage are examined.Furthermore, compare to one term and two terms truncation Galerkin method, it is observed that the GDQ with small number of grid points (non-uniform) performs accurate results for nonlinear micro-electro-mechanical coupled behavior which requires a large number of grid points at high-order approximation.     

Size-dependent dynamic analysis of rectangular nanoplates in the presence of electrostatic,Casimir and thermal forces

In the present study by considering the small-scale effects, the dynamic instability of fully clamped and simply supported nanoplates is studied in the attendance of electrostatic, Casimir as well as thermal forces. To this end, by applying the nonlocal elasticity theory of Eringen along with the classical plate theory, the dynamic equilibrium equation of nanoplates is obtained by incorporating the in-plane thermal and transverse intermolecular distributed loads. The solution of the obtained nonlinear governing equation is done using the Galerkin method and the dynamic pull-in instability voltage of the nanoplates is compared with the available experimental results. Finally, the simultaneous effects of thermal force as well as nonlocal parameter on the dynamic response of nanoplates are examined in the presence of Casimir force in detail.     

Analytical bounds for the electromechanical buckling of a compressed nanocantilever

An analytical approach is presented for the accurate definition of lower and upper bounds for the pull-in voltage and tip displacement of a micro- or nanocantilever beam subject to compressive axial load, electrostatic actuation and intermolecular surface forces. The problem is formulated as a nonlinear two-point boundary value problem and has been transformed into an equivalent nonlinear integral equation. Initially, new analytical estimates are found for the beam deflection, which are then employed for assessing novel and accurate bounds from both sides for the pull-in parameters, taking into account for the effects of the compressive axial load. The analytical predictions are found to closely agree with the numerical results provided by the shooting method. The effects of surface elasticity and residual stresses, which are of significant importance when the physical dimensions of structures descend to nanosize, are also included in the proposed approach.     

Nonlinear vibrations and chaos in electrostatic torsional actuators

Electrostatic torsional micro-mirrors have wide spread use in different industries for diverse purposes. This paper investigates the development of superharmonics and chaotic responses in electrostatic torsional micro-mirrors near the pull-in condition. Appearance of nonlinear phenomena is investigated in models accounting for and disregarding the coupling of torsional and flexural deflections. Analysis of the system response to step and harmonic excitation reveals the appearance of DC and AC symmetry breaking. Increasing the amplitude of harmonic excitation, the response in the form of distinct superharmonics changes to a broad band response, where there is loss of periodicity and the response becomes chaotic. Accounting for flexural deflections in coupled model reduces the voltage thresholds corresponding to symmetry breaking and chaotic responses. It is also shown that damping has a regularizing effect and introduction of damping changes the chaotic undamped response into quasi-periodic one.     

Nonlinear vibrations of piezoelectric microcantilevers for biologically-induced surface stress sensing

In this paper, nonlinear vibrations of a piezoelectrically-driven microcantilever beam in presence of a biological monolayer are investigated and the corresponding equations of motion are derived and simulated. A part of the microcantilever beam surface is covered by a piezoelectric layer, which acts as an actuator. Inextensibility condition and the coupling between electrical and mechanical properties in piezoelectric materials are considered in the bending vibrations of the beam. The adsorbed biological layer is considered to be a monolayer and its adsorption induced surface stress is formulated from the molecular viewpoint. The nonlinear terms in the governing equations of motion of the beam appear in the quadratic form due to the presence of the piezoelectric layer, and the cubic form due to geometry of the beam and the adsorbed biological layer. Through extensive numerical simulations, it is demonstrated that the nonlinear effect of piezoelectric layer is significant in the microcantilever resonance sensing range. It is also shown that the effect of intermolecular attraction–repulsion on the surface stress is less dominant than other sources of surface stress (e.g., the electrostatic forces). Finally, it is observed that piezoelectrically-actuated microcantilever provides the ability of indirect measurement of vibrations and frequency response characteristics, instead of using bulky laser sensor.     

Size-dependent dynamics of a FG Nanobeam near nonlinear resonances induced by heat

This study explores heat-induced nonlinear vibration of a functionally graded (FG) capacitive nanobeam within the framework of nonlocal strain gradient theory (NLSGT). The elastic FG beam, which is firstly deflected by a DC voltage, is driven to vibrate about its deflected position by a periodic heat load. The nano-structure, which consists of a clamped-clamped nanobeam, is modeled assuming Euler–Bernoulli beam assumption which accounts for the nonlinear von-Karman strain and the electrostatic and intermolecular forcing. To simulate the static and dynamic responses, a model reduction procedure is carried out by employing the Galerkin method. The method of Averaging as a regular semi-analytic perturbation method is applied to obtain governing equations of the steady-state responses. With the purpose of establishing the validity of the solution, a Shooting technique in conjunction with the Floquet theory is used to capture the periodic motions and then examine their stability. The nonlinear resonance frequency of the FG nanobeam near its fundamental natural frequency (primary resonance) and near principal parametric resonance is investigated while the emphasis is placed on studying the effect of various parameters including DC voltage, amplitude of the periodic heat source, material index, damping ratio, and small scale parameters. The main objective of this study is to model a miniature structure which can be used as either a sensitive remote temperature sensor or a high-efficiency thermal energy harvester.     

Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies

A modified continuum model of electrically actuated nanobeams is presented by incorporating surface elasticity in this paper. The classical beam theory is adopted to model the bulk, while the bulk stresses along the surfaces of the bulk substrate are required to satisfy the surface balance equations of the continuum surface elasticity. On the basis of this modified beam theory the governing equation of an electrically actuated nanobeam is derived and a powerful technology, analog equation method (AEM) is applied to solve this complex problem. Beams made from two materials: aluminum and silicon are chosen as examples. The numerical results show that the pull-in phenomena in electrically actuated nanobeams are size-dependent. The effects of the surface energies on the static and dynamic responses, pull-in voltage and pull-in time are discussed.     

Nonlinear modeling and analysis of electrically actuated viscoelastic microbeams based on the modified couple stress theory

A nonclassical nonlinear continuum model of electrically actuated viscoelastic microbeams is presented based on the modified couple stress theory to consider the microstructure effect in the framework of viscoelasticity. The nonlinear integral-differential governing equation and related boundary conditions of are derived based on the extended Hamilton's principle and Euler–Bernoulli hypothesis for viscoelastic microbeams with clamped-free, clamped-clamped, simply-supported boundary conditions. The proposed model accounts for system nonlinearities including the axial residual stress, geometric nonlinearity due to midplane stretching, electrical forcing with fringing effect. The behavior of the microbeam is simulated using generalized Maxwell viscoelastic model. A new generalized differential/integral quadrature method is developed to solve the resulting governing equation. The developed model is verified against elastic behavior and a favorable agreement is obtained. Efficiency of the developed model is demonstrated by analyzing the quasistatic pull-in phenomena of electrically actuated viscoelastic microbeams with different boundaries at various material length scale parameters and axial residual stresses in the framework of linear viscoelasticity.     

Suppression of pull-in instability in MEMS using a high-frequency actuation

We study the effect of a high-frequency AC tension on the pull-in instability induced by a DC tension in a microelectromechanical system. The microstructure is modelled as a single-degree-of-freedom system. The method of direct partition of motion is used to split the fast and slow dynamics. Analysis of steady-states of the slow dynamic enables us to depict the effect of the AC voltage on the pull-in. The main result of this paper indicates that it is possible to suppress the electrostatically induced pull-in instability for a range of values of the amplitude and the frequency of the high-frequency AC.     

LINEARIZED OSCILLATION FOR ODD-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

By an associate linear equation, we obtain a linearized oscillation result of certain odd-order nonlinear neutral delay differential equation. The result answers partially an open problem proposed by Gyori and Ladas.