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.     

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.     

On the probability of the outcomes in buckling of an elastic beam

A nonlinear time-varying one-degree-of-freedom system, which is used for the modelling of the buckling of a loaded beam in Euler??s problem, is considered. For a slowly changing load, the deterministic approach in this problem fails if the trajectories pass through the separatrix. An expression for the probability of possible outcomes of the evolution of the oscillations is obtained. The analytical and numerical results are compared.     

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.     

Adomian decomposition method used for solving nonlinear pull-in behavior in electrostatic micro-actuators

Nonlinear pull-in behavior for different electrostatic micro-actuators were simulated in this study. The Adomian decomposition method was employed to overcome the difficulty in the nonlinear equation of motion. Because no iteration is required in solving the nonlinear deformation, the decomposition method is one of the most efficient methods for evaluating the unstable pull-in behavior of an electrostatic micro-actuator. To investigate the feasibility of applying the Adomian decomposition method in dealing with the nonlinear deflection equation in the micro-actuator problem, different types of micro-actuators, e.g., fixed-fixed beam actuator and cantilever beam actuator were studied and analyzed. The calculated results agreed well with those from the literature.     

Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation

In this paper, post-buckling and nonlinear vibration analysis of geometrically imperfect beams made of functionally graded materials (FGMs) resting on nonlinear elastic foundation subjected to axial force are studied. The material properties of FGMs are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The assumptions of a small strain and moderate deformation are used. Based on Euler–Bernoulli beam theory and von-Karman geometric nonlinearity, the integral partial differential equation of motion is derived. Then this partial differential equation (PDE) problem, which has quadratic and cubic nonlinearities, is simplified into an ordinary differential equation (ODE) problem by using the Galerkin method. Finally, the governing equation is solved analytically using the variational iteration method (VIM). Some new results for the nonlinear natural frequencies and buckling load of the imperfect functionally graded (FG) beams such as the effects of vibration amplitude, elastic coefficients of foundation, axial force, end supports and material inhomogeneity are presented for future references. Results show that the imperfection has a significant effect on the post-buckling and vibration response of FG beams.     

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.     

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.     

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.     

Application of nonlocal continuum theory to the primary resonance analysis of an axially loaded nano beam under time delay control

The paper is concerned with the nonlinear primary resonance of nano beams with axial load under the velocity time delay control. In order to have a deep insight into the system, the amplitude frequency response curve of the system is firstly obtained using the multiple scales method. The effects of the control gains and time delays on the system stability are then investigated. The analyses illustrated that both delay feedback gain coefficient and velocity time delay control can mitigate the system vibrations properties (e.g. hardening nonlinearity, resonance amplitude and the corresponding width) to an excellent level. The nonlinear primary resonance of nano beam is also discussed with the influences of small scale effect, axial initial load, wave number, Winkler foundation modulus and the ratio of the length to the diameter. This paper establishes the relationship between the time delay controller and vibration properties of a nano beam system which provides the guidance for applying time delayed active control for different types of nano devices in engineering practices.     

Study on asymptotic analytical solutions using HAM for strongly nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass

Presented herein is to establish the asymptotic analytical solutions for the fifth-order Duffing type temporal problem having strongly inertial and static nonlinearities. Such a problem corresponds to the strongly nonlinear vibrations of an elastically restrained beam with a lumped mass. Taking into consideration of the inextensibility condition and using an assumed single mode Lagrangian method, the single-degree-of-freedom ordinary differential equation can be derived from the governing equations of the beam model. Various parameters of the nonlinear unimodal temporal equation stand for different vibration modes of inextensible cantilever beam. By imposing the homotopy analysis method (HAM), we establish the asymptotic analytical approximations for solving the fifth-order nonlinear unimodal temporal problem. Within this research framework, both the frequencies and periodic solutions of the nonlinear unimodal temporal equation can be explicitly and analytically formulated. For verification, numerical comparisons are conducted between the results obtained by the homotopy analysis and numerical integration methods. Illustrative examples are selected to demonstrate the accuracy and correctness of this approach. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions.     

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.     

Interval static analysis of multi-cracked beams with uncertain size and position of cracks

This paper deals with beams under static loads, in presence of multiple cracks with uncertain parameters. The crack is modelled as a linearly-elastic rotational spring and, following a non-probabilistic approach, both stiffness and position of the spring are taken as uncertain-but-bounded parameters.A novel approach is proposed to compute the bounds of the response. The key idea is a preliminary monotonicity test, which evaluates sensitivity functions of the beam response with respect to the separate variation of every uncertain parameter within the pertinent interval. Next, two alternative procedures calculate lower and upper bounds of the response. If the response is monotonic with respect to all the uncertain parameters, the bounds are calculated by a straightforward sensitivity-based method making use of the sensitivity functions built in the monotonicity test. In contrast, if the response is not monotonic with respect to even one parameter only, the bounds are evaluated via a global optimization technique. The presented approach applies for every response function and the implementation takes advantage of closed analytical forms for all response variables and related sensitivity functions.Numerical results prove efficiency and robustness of the approach, which provides very accurate bounds even for large uncertainties, avoiding the computational effort required by the vertex method and Monte Carlo simulation.     

Nonlinear analysis of buckling,free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment

Employing Euler–Bernoulli beam theory and the physical neutral surface concept, the nonlinear governing equation for the functionally graded material beam with two clamped ends and surface-bonded piezoelectric actuators is derived by the Hamilton’s principle. The thermo-piezoelectric buckling, nonlinear free vibration and dynamic stability for the piezoelectric functionally graded beams, subjected to one-dimensional steady heat conduction in the thickness direction, are studied. The critical buckling loads for the beam are obtained by the existing methods in the analysis of thermo-piezoelectric buckling. The Galerkin’s procedure and elliptic function are adopted to obtain the analytical solution of the nonlinear free vibration, and the incremental harmonic balance method is applied to obtain the principle unstable regions of the piezoelectric functionally graded beam. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and accuracy of the present analysis and solving method. Simultaneously, validation of the results achieved by rule of mixture against those obtained via the Mori–Tanaka scheme is carried out, and excellent agreements are reported. The effects of the thermal load, electric load, and thermal properties of the constituent materials on the thermo-piezoelectric buckling, nonlinear free vibration, and dynamic stability of the piezoelectric functionally graded beam are discussed, and some meaningful conclusions have been drawn.     

A hybrid solution for analyzing nonlinear dynamics of electrostatically-actuated microcantilevers

This paper introduces a closed-form approximation of dynamic response of microcantilevers. The applied load on the system was linearized by Taylor series expansion and to obtain approximate solutions, model of a pure odd-order nonlinear oscillator, subjected to constant excitation was assumed. Pull-in voltage was investigated to analyze the different parameters of the examined microbeam. In order to obtain a comprehensive dynamic model for MEMS devices, before, during and after switching, the pure odd-order nonlinear model was combined with a distributed parameter system and solved after reaching the pull-in voltage. The obtained results demonstrate correct prediction of the static pull-in voltage and also the dynamic deflection of microbeams. By using the same approach, the sensitivity of the pull-in voltage to various geometrical parameters was also investigated. The obtained results indicate that excessive increase in the air gap causes substantial increase in the pull-in voltage; while increasing thickness of microcantilever has even greater effect. It was also observed that for a given thickness of microcantilever, increasing its length beyond a certain amount has no effect on the pull-in voltage.     

A Nonlinear Elastic Beam System with Inelastic Contact Constraints

In this paper we study freely propagating inertial, i.e., unforced, waves, in an elastic beam constrained so that all motion takes place above and on a flat, rigid support surface, subject to a gravitational force and a compressive longitudinal load. Contact between the beam and the support surface is assumed to be completely inelastic. A nonlinear beam model is used, incorporating a quartic extension of the familiar quadratic potential energy functional for the standard Euler--Bernoulli model. After briefly reviewing the rationale for the model and some of its properties, as developed in earlier articles, we present existence and uniqueness results for the constrained system obtained with the use of a ``penalty function' approach involving the addition of a ``uni-directional friction' dissipative term, active only when the constraint is violated, to the unconstrained system.     

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.     

Flow-induced and mechanical stability of cantilever carbon nanotubes subjected to an axial compressive load

In the present study, a modified nonlocal elasticity theory is used for flutter and divergence analyses of the cantilever carbon nanotubes (CNTs) conveying fluid. The CNT is embedded in viscoelastic foundation and is subjected to an axial compressive load acting at the free end. An extreme high-order governing equation as well as higher-order boundary conditions is developed using Hamilton's principle for vibration and stability analysis of the CNT. The numerical solution for flutter and divergence velocities is computed using the extended Galerkin method. The validity of the present analysis is confirmed by comparing with molecular dynamics simulation (MDS) and numerical solutions available in the literature. In the numerical results, the effects of nonlocal parameter, surface effects, viscoelastic foundation and compressive axial load on the stability boundaries of the system are investigated. The results show that the stability boundaries of the CNT are strongly dependent on the small scale coefficient and surface effects.     

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.