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.     

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.     

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.     

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.     

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.     

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.     

Design of a nonsingular adaptive fuzzy backstepping controller for electrostatically actuated microplates

This paper adopts some alternative strategies to design a nonlinear controller for double electrostatically actuated microplates. The novel design is carried out to solve the singularity problem reported in many articles due to the use of the Taylor expansion to simplify the electrostatic force. The nonlinear governing partial differential equation is converted to the modal equation using the Galerkin method. Then, based on the Lyapunov stability criterion, a fuzzy backstepping controller facilitated by prescribed performance functions is applied to the non-affine system to extend the travel range beyond the pull-in region and capture the structural and nonstructural uncertainties that exist in the practical systems. The present work also aims to bring satisfactory transient and steady-state performance indices to the system. Moreover, unknown time-varying delays as the indispensable part of practical systems are considered in the proposed control scheme to suppress the delays occurring in the measurement of the states by constructing Lyapunov–Krasovskii function. The accuracy of the modal equation in both the static and dynamic analysis is verified through a meshless method as a direct solution of the partial differential equation. The proposed controller guarantees that all the closed-loop signals are semi-globally, uniformly ultimately bounded, and the error evolves within the decaying prescribed bounds. Finally, the proposed controller demonstrates its feasibility to extend the travel range within and beyond the pull-in range despite the unknown uncertainties and time-varying delays which exist in the system.     

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-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.     

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.     

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.     

On the partial differential equations of electrostatic MEMS devices III: Refined touchdown behavior

This paper is a continuation of [N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal. 38 (2007) 1423-1449] and [N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, NoDEA Nonlinear Differential Equations Appl. (2008), in press], where we analyzed nonlinear parabolic problem on a bounded domain Ω of RN with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at −1. Here u is modeled to describe dynamic deflection of the elastic membrane. When a voltage—represented here by λ—is applied, the membrane deflects towards the ground plate and a snap-through (touchdown) must occur when it exceeds a certain critical value λ^∗ (pull-in voltage), creating a so-called “pull-in instability” which greatly affects the design of many devices. In an effort to achieve better MEMS design, the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f(x). In this work, some a priori estimates of touchdown behavior are established, based on which the refined touchdown profiles are obtained by adapting self-similar method and center manifold analysis. Applying various analytical and numerical techniques, some properties of touchdown set—such as compactness, location and shape—are also discussed for different classes of varying permittivity profiles.     

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.     

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.     

具有转向点的二阶非线性系统Robin边值问题的奇摄动

本文研究一类具有转向点的二阶非线性系统 Robin边值问题的奇摄动 ,在适当的假设条件下 ,利用微分不等式方法证明了解的存在性 ,并得到了解的按分量的渐近估计式 .     

返航备降航班高风险频发子集搜索模型

返航备降航班是一种保证航空安全的有效措施,但返航备降航班会带来成本损失,所以对返航备降业务规律进行分析,从而重点监控,在确保安全生产的基础上,有效降低返航备降航班的经济损失。我们可以把确定返航备降航班发生的密集时间段描述为一个多属性空间搜索问题,在许多实践领域存在该类问题,比如商业银行对贷款客户的信用评级、绩效考核等。在多属性组成的多维空间中,存在许多正常点和不正常点,通过建立高风险频发子集搜索模型和算法确定不正常点比较密集的多属性范围,使得在该空间范围内不正常点的比例较高并且总的点的数量达到一定的水平。本文以航空公司安全生产为例,其模型结果在实际生产的运用中取得良好效果,在一年内避免30余班次返航备降。     

Aircraft proximity termination conditions for 3D turn centric modes

International airspace design is undergoing significant change that requires formal and rigorous mathematical specifications to assure the safety of flight operations. Aircraft proximity management is one such area. The Point of Closest Approach (PCA) between aircraft flightpaths is the position along a flightpath at which the minima in relative range occurs. To date, PCA has been estimated primarily based on the assumption of a linear extrapolation of the velocity vectors of each aircraft involved, however, this assumption is limiting and aircraft in turning flight must also be considered. A generalised geometric and vector construction for the determination of PCA is presented. A solution based on a characterization of Fermat’s method for stationary points is presented that results in a complex transcendental equation. By casting the equation in a determinant structure a co-linearity condition is revealed between three unique 2D points. A novel aspect is that one of these points is a fixed reference point that lies on either the vector between the aircraft turn centres or on one of its extensions providing a reference to determine the location of the PCA. The analytic method can be readily applied in a laboratory test environment or in an automated guidance context. The rigorous proof enables a higher confidence in achieving dependable operations in a safety critical context and in the adequacy of test strategies when developing algorithms for aircraft avionics.     

Nonlinear behavior of a nano-scale beam considering length scale-parameter

Size dependent behavior of materials arises for a structure when the characteristic size such as thickness or diameter is close to its internal length-scale parameter. In these cases, ignoring this behavior in modeling may leads to incorrect results. In this paper, strong effects of size dependence on static and dynamic behavior of electro-statically actuated nano-beams have been studied. The fixed points of the Aluminum nano-beams have been determined and shown that for a given DC voltage, there is a considerable difference between the calculated fixed points using classic beam theory and modified couple stress theory. In addition, it has been also shown that ignoring couple stress theory results in an order of magnitude error in calculated static and dynamic pull-in voltages. Some previous studies have applied the classic beam theory in their models and introduced a considerable hypothetical value of residual stress to justify the discrepancies between experimental and theoretical results.     

A new analysis of reciprocated beam bending in electrostatic comb drives using a semi-analytical approach

This paper investigates a new modeling and analysis of the voltage induced reciprocated beam bending effects in the unit cell of a planar, variable gap type, capacitive comb drive structures. A semi-analytical approach has been efficiently formulated to solve this coupled electromechanical problem under a steady state condition. The effect of fringe field is also incorporated to improvise on the accuracy of the solution. Additionally, an energy-based method as well as a finite element (FE) based model has been constructed to simulate and validate the semi-analytical approach. The results show that the reciprocated bending of the beams has significant effects on the performance parameters like displacement, capacitance and pull-in characteristics in the comb drive systems. This effect is also seen to vary with applied voltage and stiffness of the combs and springs. Further, a comparison with conventional lumped model shows considerable difference in the estimated values for different parameters indicating a more practical prediction through the proposed approach. Finally, a set of design guidelines is discussed to reduce this bending, so that its effects on the performance of comb drive systems can be minimized.