Nonlinear electro-dynamic analysis of micro-actuators: Effect of material nonlinearity

This paper presents a nonlinear dynamic analysis of a micro-actuator made of nonlinear elasticity materials. The theoretical formulations are based on Bernoulli–Euler beam theory and include the effects of mid-plane stretching due to large deformation and material nonlinearity. By employing Linstedt–Poincaré perturbation method, the nonlinear governing equation is transformed into a set of linear differential equations which are then solved using Galerkin’s method. Numerical results show that the linear constitutive relationship used in previous studies is valid for small deformation only whereas for large deformation, the nonlinear elasticity constitutive relationship must be used for accurate analysis. The effects of initial gap and beam length on the nonlinear electro-dynamic behavior of the micro-actuator are also discussed.     

联合Duffing方程和Van der Pol方程的非线性分数阶微分方程

本文研究了Adomian分解方法在非线性分数阶微分方程求解中的应用. 利用Riemann-Liouville分数阶导数和Adomian分解方法, 将Duffing方程和Van der Pol方程联合在一个分数阶方程中,并获得了此方程的解析近似解.     

Application of Laplace decomposition method on semi-infinite domain

In this article, Laplace decomposition method (LDM) is applied to obtain series solutions of classical Blasius equation. The technique is based on the application of Laplace transform to nonlinear Blasius flow equation. The nonlinear term can easily be handled with the help of Adomian polynomials. The results of the present technique have closed agreement with series solutions obtained with the help of Adomian decomposition method (ADM), variational iterative method (VIM) and homotopy perturbation method (HPM).     

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.     

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.     

Finding all real roots of a polynomial by matrix algebra and the Adomian decomposition method

In this paper, we put forth a combined method for calculation of all real zeroes of a polynomial equation through the Adomian decomposition method equipped with a number of developed theorems from matrix algebra. These auxiliary theorems are associated with eigenvalues of matrices and enable convergence of the Adomian decomposition method toward different real roots of the target polynomial equation. To further improve the computational speed of our technique, a nonlinear convergence accelerator known as the Shanks transform has optionally been employed. For the sake of illustration, a number of numerical examples are given.     

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.     

A New Efficient Transform Mechanism with Convergence Analysis of the Space-Fractional Telegraph Equations

This article"s goal is to investigate the space-fractional telegraph equation using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). Using the Banach fixed point theorem, we explore proofs for the existence and uniqueness theorems applying it to a nonlinear differential equation. Using our method, exact solutions of the space-fractional telegraph equation and time-fractional diffusion problems have been obtained. To demonstrate the effectiveness of the suggested scheme, four examples are provided.     

A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell–Whitehead–Segel equation

In this paper, we will carry out a comparative study between the reduced differential transform method and the Adomian decomposition method. This is been achieved by handling the Newell–Whitehead–Segel equation. Two numerical examples have also been carried out to validate and demonstrate efficiency of the two methods. Furthermost, it is shown that the reduced differential transform method has an advantage over the Adomian decomposition method that it takes less time to solve the nonlinear problems without using the Adomian polynomials.     

Near-field and far-field approximations by the Adomian and asymptotic decomposition methods

The Adomian decomposition method and the asymptotic decomposition method give the near-field approximate solution and far-field approximate solution, respectively, for linear and nonlinear differential equations. The Padé approximants give solution continuation of series solutions, but the continuation is usually effective only on some finite domain, and it can not always give the asymptotic behavior as the independent variables approach infinity. We investigate the global approximate solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from the asymptotic decomposition method for linear and nonlinear differential equations. For several examples we find that there exists an overlap between the near-field approximation and the far-field approximation, so we can match them to obtain a global approximate solution. For other nonlinear examples where the series solution from the Adomian decomposition method has a finite convergent domain, we can match the Padé approximant of the near-field approximation with the far-field approximation to obtain a global approximate solution representing the true, entire solution over an infinite domain.     

Adomian decomposition method for nonlinear differential-difference equations

We extend Adomian decomposition method (ADM) to find the approximate solutions for the nonlinear differential-difference equations (NDDEs), such as the discretized mKdV lattice equation, the discretized nonlinear Schrödinger equation and the Toda lattice equation. By comparing the approximate solutions with the exact analytical solutions, we find the extend method for NDDEs is of good accuracy.     

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.     

A note on the use of modified Adomian decomposition method for solving singular boundary value problems of higher-order ordinary differential equations

This paper extend the work [Yahya Qaid Hasan, Liu Ming Zhu. Solving singular boundary value problems of higher-order ordinary differential equations by modified Adomian decomposition method. Commun Nonlinear Sci Numer Simul. doi :10.1016/j.cnsns.2008.09.027] to high order of singular boundary value problems. Solution of these problems is considered by proposed modification of Adomian decomposition method. The proposed method can be applied to linear and nonlinear problems. Some examples are presented to show the ability of the method for linear and non-linear ordinary differential equation.     

Numerical solutions for the damped generalized regularized long-wave equation with a variable coefficient by Adomian decomposition method

Adomian’s decomposition method is proposed to approximate the solutions of the nonlinear damped generalized regularized long-wave (DGRLW) equation with a variable coefficient. The solution of the nonlinear DGRLW equation is calculated in the form of series with computable components. Numerical examples are tested to illustrate the proposed scheme. Moreover, the approximate solution is compared with the exact solution.     

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.     

Adomian solution of a nonlinear quadratic integral equation

We are concerned here with a nonlinear quadratic integral equation (QIE). The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM) applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian’s series solution. Two methods are used to solve these type of equations; ADM and repeated trapezoidal method. The obtained results are compared.     

Numerical solutions of the Laplace’s equation

This paper uses the sinc methods to construct a solution of the Laplace’s equation using two solutions of the heat equation. A numerical approximation is obtained with an exponential accuracy. We also present a reliable algorithm of Adomian decomposition method to construct a numerical solution of the Laplace’s equation in the form a rapidly convergence series and not at grid points. Numerical examples are given and comparisons are made to the sinc solution with the Adomian decomposition method. The comparison shows that the Adomian decomposition method is efficient and easy to use.     

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.     

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.     

The solution of linear and nonlinear systems of Volterra functional equations using Adomian–Pade technique

The purpose of this study is to implement Adomian–Pade (Modified Adomian–Pade) technique, which is a combination of Adomian decomposition method (Modified Adomian decomposition method) and Pade approximation, for solving linear and nonlinear systems of Volterra functional equations. The results obtained by using Adomian–Pade (Modified Adomian–Pade) technique, are compared to those obtained by using Adomian decomposition method (Modified Adomian decomposition method) alone. The numerical results, demonstrate that ADM–PADE (MADM–PADE) technique, gives the approximate solution with faster convergence rate and higher accuracy than using the standard ADM (MADM).