Nonlinear Pull-in Instability Research Articles

A semi-analytical study on the nonlinear pull-in instability of FGM nanoactuators

In this paper, a new semi-analytical solution for estimating the pull-in parameters of electrically actuated functionally graded (FG) nanobeams is proposed. All the bulk and surface material properties of the FG nanoactuator vary continuously in thickness direction according to power law distribution. Here, the modified couple stress theory (MCST) and Gurtin-Murdoch surface elasticity theory (SET) are jointly employed to capture the size effects of the nanoscale beam in the context of Euler-Bernoulli beam theory. According to the MCST and SET and accounting for the mid-plane stretching, axial residual stress, electrostatic actuation, fringing field, and dispersion (Casimir or/and van der Waals) forces, the nonlinear nonclassical equation of motion and boundary conditions are obtained derived using Hamilton principle. The proposed semi-analytical solution is derived by employing Galerkin method in conjunction with the Particle Swarm Optimization (PSO) method. The proposed solution approach is validated with the available literature. The freestanding behavior of nanoactuators is also investigated. A parametric study is conducted to illustrate the effects of different material and geometrical parameters on the pull-in response of cantilever and doubly-clamped FG nanoactuators. This model and proposed solution are helpful especially in mechanical design of micro/nanoactuators made of FGMs.

Resonant nano-electromechanical systems from 2D materials

Intriguing mechanical properties of two-dimensional (2D) atomically thin materials make them well suit for a new class of resonant NEMS. Resonant 2D NEMS show high fundamental frequencies, outstanding tenability and broad dynamic range due to the ultrahigh strength, ultralow mass density and ultrahigh strain stretchability. Recent work toward exploring fabrication and transduction techniques, device performances and the nonlinear dynamic characteristics of 2D NEMS are summed up. Nonlinear dynamic behaviors including nonlinear mode coupling, nonlinear damping and nonlinear pull-in instability are reported in 2D NEMS.

Nonlinear Pull-in Instability of Rectangular Nanoplates Based on the Positive and Negative Second-Order Strain Gradient Theories with Various Edge Supports

Abstract Based on the positive and negative second-order strain gradient theories along with Kirchhoff thin plate theory and von Kármán hypothesis, the pull-in instability of rectangular nanoplate is analytically investigated in the present article. For this purpose, governing models are extracted under intermolecular, electrostatic, hydrostatic, and thermal forces. The Galerkin method is formally exerted for converting the governing equation into an ordinary differential equation. Then, the homotopy analysis method is implemented as a well-designed technique to acquire the analytical approximations for analyzing the effects of disparate parameters on the nonlinear pull-in behavior. As an outcome, the impacts of nonlinear forces on nondimensional fundamental frequency, the voltage of pull-in, and softening and hardening effects are examined comparatively.

Size-dependent nonlinear pull-in instability of the clamped cylindrical thin micro-/nanoshell based on the non-classical theories

In recent decades, non-classical continuum theories have been commonly utilized to model material discontinuities in micro-/nanoscales. In the present study, modified strain gradient theory (MSGT) together with the fully clamped cylindrical thin-shell model has been adopted to investigate the size-dependent nonlinear pull-in instability of thin cylindrical nano-/microshell. The modified strain gradient theory in cylindrical coordinates and virtual work principle has been applied to derive the constitutive size-dependent equations of motion. In this study, the linear strains have been utilized to establish a system of three-coupled partial differential equations. Further, effect of van der Waals force has been included in the nonlinear governing equations of the systems, and extended Kantorovich method has been used to solve the nonlinear differential equations that in turn lead to achieving the pull-in parameters of the microshell. From theoretical point of view, effect of van der Waals attraction, the size dependency, and the importance of coupling between them on the instability performance have been discussed and the pull-in parameters, i.e., critical mid-deflection and instability voltage have been determined. It was found that comparable pull-in parameter of MSGT to those obtained by the classic theory, and modified couple stress theory depends only on the length scale of the microshell.

Nonlinear Pull-In Instability of Strain Gradient Microplates Made of Functionally Graded Materials

In this paper, the size-dependent nonlinear pull-in behavior of rectangular microplates made from functionally graded materials (FGMs) subjected to electrostatic actuation is numerically studied using a novel approach. The small scale effects are taken into account according to Mindlin’s first-order strain gradient theory (SGT). The plate model is formulated based on the first-order shear deformation theory (FSDT) using the virtual work principle. The size-dependent relations are derived in general form, which can be reduced to those based on different elasticity theories, including the modified strain gradient, modified couple stress and classical theories (MSGT, MCST and CT). The solution of the problem is arrived at by employing an efficient matrix-based method called the variational differential quadrature (VDQ). First, the quadratic form of the energy functional including the size effects is obtained. Then, it is discretized by the VDQ method using a set of matrix differential and integral operators. Finally, the achieved discretized nonlinear equations are solved by the pseudo arc-length continuation method. In the numerical results, the effects of material length scale parameters, side length-to-thickness ratio and FGM’s material gradient index on the nonlinear pull-in instability of microplates with different boundary conditions are investigated. A comparison is also made between the predictions by the MSGT, MCST and CT.

Nonlinear pull-in instability of suspended graphene-based sensors

Applying an external electric field on a graphene surface is an important way to improve the molecular adsorption capability of graphene, thus pull-in instability of suspended graphene sensors becomes a critical issue. Incorporating residual built-in strains, fringing fields and intermolecular forces, an electromechanical model is developed to characterize the nonlinear pull-in behaviors of suspended graphene-based sensors. The obtained results of pull-in voltages agree well with the reported experimental data. Moreover, the fracture failure of graphene sensors is initially compared to the pull-in failure. To avoid pull-in instability and fracture failure, critical formulas of axial pre-stress for zigzag-oriented and armchair-oriented graphene sensors are derived. It is demonstrated that axial pre-stress is an effective and controllable way to improve the pull-in stability of graphene sensors.

Grain size and nanoscale effects on the nonlinear pull-in instability and vibrations of electrostatic actuators made of nanocrystalline material

Presented herein is the study of grain size, grain surface energy and small scale effects on the nonlinear pull-in instability and free vibration of electrostatic nanoscale actuators made of nanocrystalline silicon (Nc-Si). A Mori–Tanaka micromechanical model is utilized to calculate the effective material properties of Nc-Si considering material structure inhomogeneity, grain size and grain surface energy. The small-scale effect is also taken into account using Mindlin’s strain gradient theory. Governing equations are derived in the discretized weak form using the variational differential quadrature method based on the third-order shear defamation beam theory in conjunction with the von Kármán hypothesis. The electrostatic actuation is modeled considering the fringing field effects based upon the parallel plate approximation. Moreover, the Casimir force effect is considered. The pseudo arc-length continuation technique is used to obtain the applied voltage-deflection curve of Nc-Si actuators. Then, a time-dependent small disturbance around the deflected configuration is assumed to solve the free vibration problem. By performing a numerical study, the influences of various factors such as length scale parameter, volume fraction of the inclusion phase, density ratio, average inclusion radius and Casimir force on the pull-in instability and free vibration of Nc-Si actuators are investigated.

EM radiation from electrostatic nonlinear pull‐in instability of MEMS

Electrodynamics theory tells that EM waves are generated from accelerated charged particles. Conventional radiators such as antennas are understood to emit waves because of the current's distribution and accumulation in the capacitor, where electrons are accelerated. In MEMS mechanisms, charges can be accelerated by the mechanical motion, which suggests that an MEMS structure can also be a radiator. The electrodynamics to theoretically analyse the EM wave radiation from MEMS is used. During the electrostatic nonlinear pull-in instability MEMS, where the electrostatic gap has become narrower than the 2/3 of the initial gap, the distributed charges are accelerated at a very high rate so as to generate significant power radiation in the form of EM waves. The MEMS's mechanical accelerations are analysed to calculate the transient radiations in both spatial and frequency domains.

Nonlinear size-dependent pull-in instability and stress analysis of thin plate actuator based on enhanced continuum theories including nonlinear effects and surface energy

In recent years, size dependent continuum theories have been commonly used to simulate material discontinuities in micro/nano-scales. In the present article, modified strain gradient theory (MSGT) and Gurtin–Murdoch surface elasticity, are used to investigate the size-dependent nonlinear pull-in instability and conduct a stress analysis of thin microplates. This paper modifies the classical equations and surface elasticity theory as a non-classical method which is able to account for the surface effect of nanomaterials with sufficient precision and lower calculation cost in comparison to atomic methods. The effects of van der Waals and Casimir forces are incorporated into the nonlinear governing equation of the system. A finite difference method (FDM) is employed to solve the nonlinear differential equation, and the pull-in parameters of the microplate are extracted. The effect of dispersion attractions and size-dependency on instability as well as the importance of the coupling between them is discussed. The most common types of thin-plates, including fully-clamped, simply-supported, and plates with mixed boundary conditions, are investigated. It is found that the significant difference between the pull-in instability parameters in the MSGT and the classical theory for thin microplates is merely due to the consideration of the size effect parameter in the modified strain gradient theory. The effect of surface/interface is also accounted for in the numerical FDM approach. Then, the stress analysis of the microplate is studied numerically by examining the surface/interface effect on the microplate with a square/rectangular section. The results obtained through this numerical method are plotted graphically.

Thermal, size and surface effects on the nonlinear pull-in of small-scale piezoelectric actuators

Electrostatically actuated miniature wires/tubes have many operational applications in the high-tech industries. In this research, the nonlinear pull-in instability of piezoelectric thermal small-scale switches subjected to Coulomb and dissipative forces is analyzed using strain gradient and modified couple stress theories. The discretized governing equation is solved numerically by means of the step-by-step linearization method. The correctness of the formulated model and solution procedure is validated through comparison with experimental and several theoretical results. Herein, the length-scale, surface energy, van der Waals attraction and nonlinear curvature are considered in the present comprehensive model and the thermo-electro-mechanical behavior of cantilever piezo-beams are discussed in detail. It is found that the piezoelectric actuation can be used as a design parameter to control the pull-in phenomenon. The obtained results are applicable in stability analysis, practical design and control of actuated miniature intelligent devices.

Nonlinear pull-in instability of microplates with piezoelectric layers using modified couple stress theory

In this study, the effects of piezoelectric layers, residual stresses and von Karman nonlinearity on the pull-in instability of a microplate are studied by taking into account the size effect of the system. The strain energy is obtained using the modified couple stress theory and the set of nonlinear equations of motion are derived using Hamilton's principle and finite element method (FEM) based on classical plate theory. Newton–Raphson's procedure is used to solve the nonlinear equations. The proposed model is used to study the pull-in voltage, pull-in displacement, and small amplitude vibration about the equilibrium position of microplates with piezoelectric layers. The results are validated by available experimental and theoretical data. They show that the pull-in voltage is affected by the material length scale parameter, but this effect is quite negligible when the ratio of plate thickness to material length scale parameter is larger than 10. It is found that the results obtained by the nonlinear theory are larger than those by the linear theory and their differences are increased by increasing the ratio of initial gap to the plate thickness. Moreover, it is found that applying a small positive voltage to the piezoelectric layers can decrease the maximum pull-in voltage.

Scale-dependent pull-in instability of functionally graded carbon nanotubes-reinforced piezoelectric tuning nano-actuator considering finite temperature and conductivity corrections of Casimir force

A scale-dependent analytical model is presented to solve the nonlinear pull-in instability of functionally graded carbon nanotubes (CNTs) reinforced nano-actuator with piezoelectric layer considering high order-corrected electrostatic pressure and finite temperature and conductivity corrections of Casimir force. Based on Eringen’s nonlocal elasticity theory considering the long range forces among atoms, and geometrical nonlinearity, the electro-thermo-mechanical coupling governing equation of nano-actuator is derived, and solved by utilizing natural mode function and Galerkin’s decomposition method. The higher-order corrected model of electrostatic force with fringing field effect accounting for large gap and geometrical nonlinearity is employed. The results indicate that pull-in voltage decreases with increase of positive piezoelectric effect but increases with increment of negative piezoelectric effect. Pull-in voltage declines as the piezoelectric layer thickness of nano-actuator increases. Casimir force appears in more significant effect on the pull-in voltage of nano-actuator than that of van der Waals force, which shows that the analysis of nanoscale devices cannot neglect the influences of intermolecular forces within sub-micron separations. Finally, the coupling influences of van der Waals force and Capillary force on pull-in voltage are compared.

Nonlinear pull-in instability of carbon nanotubes reinforced nano-actuator with thermally corrected Casimir force and surface effect

An analytical model of investigating the pull-in characteristics of CNTs reinforced nano-actuator with temperature-dependency subjected to coupled electrostatic loading, dispersion forces as van der Waals and thermally corrected Casimir force, and surface stress are derived based on von Karman׳s geometric nonlinearity and surface elasticity. The results illustrate that increment of volume fraction of CNTs enhances the structural stiffness and leads to the increases of pull-in voltage of CNTs reinforced nano-actuator; the growth of temperature increases the axial compression stress and thus decreases the pull-in voltage; and free-standing behavior depends on the characteristic scale of the CNTs reinforced nano-actuator. The doubly-clamped and doubly-supported configuration of nano-actuator are considered and the pull-in voltage of doubly-clamped nano-actuator is larger than that of doubly-supported type. Casimir force with thermal correction significantly depends on the temperature and initial gap between electrodes. The results show the difference between Casimir force with and without thermal correction is small within couples of nanometers gap but the effect of Casimir force on pull-in instability of CNTs reinforced nano-actuator cannot be neglected.

Studying the effects of small scale and Casimir force on the non-linear pull-in instability and vibrations of FGM microswitches under electrostatic actuation

The size-dependent non-linear pull-in instability and free vibration of electrostatically actuated microswitches with the consideration of Casimir force effect are studied using a numerical solution approach. To this end, a non-classical non-linear beam model is developed based on Mindlin׳s strain gradient elasticity and the Timoshenko beam theory. The geometric non-linearity is taken into account according to the von Kármán hypothesis. Also, the microswitches are assumed to be made of functionally graded materials (FGMs). To obtain the size-dependent governing equations and boundary conditions, the virtual work principle is applied. The presented equations can be simply reduced to those on the basis of modified versions of strain gradient and couple stress theories (MSGT and MCST) as well as the classical elasticity theory. For solving the problem, the generalized differential quadrature (GDQ) method and the pseudo arc-length continuation technique are employed. In the numerical results, the influences of different parameters such as length scale parameter, Casimir force, material gradient index and geometrical properties on the pull-in instability and free vibration of actuated microswitches are examined.

Nonlinear pull-in instability and free vibration of micro/nanoscale plates with surface energy – A modified couple stress theory model

Effects of surface energy on the pull-in instability and free vibration of electrostatically actuated micro/nanoscale plates are analyzed based on the modified couple stress theory. A reduced-order model is derived to consider the geometrically nonlinear strain, surface energy, the Casimir force and the material length scale simultaneously. Results show that the pull-in voltage and fundamental frequency of the plate are considerably enhanced by the material length scale, surface energy and geometrically nonlinear deformation. However, these quantities are weakened with the inclusion of Casimir force. The effects of surface energy and the material length scale become more significant if the thickness decreases. In addition, the effects of surface energy and geometrically nonlinear strain on the pull-in voltage are the largest for a square plate.

Influence of surface energy on the non-linear pull-in instability of nano-switches

This paper investigates the pull-in instability of a nano-switch under electrostatic and intermolecular Casimir forces. The analysis is based on the geometrically non-linear Euler–Bernoulli beam theory with consideration of the surface energy. Through differential quadrature method (DQM), the pull-in voltages of the nano-switch are obtained. Results show that the effect of surface energy and geometrically non-linear deformation on the pull-in voltage depends on the length, height and initial gap of the nano-switch. In addition, the effect of intermolecular Casimir force on the pull-in voltage weakens as the initial gap increases.

Nonlinear nonlocal pull-in instability of boron nitride nanoswitches

In the present study, nonlinear pull-in instability of boron nitride nanoswitches (BNNSs) subjected to electrostatic and van der Waals (vdW) forces is investigated. Based on Euler–Bernoulli beam theory, von Karman geometric nonlinearity, nonlocal piezoelasticity theory and the principle of virtual work, the governing equations are obtained. The differential quadrature method is employed to discretize the nonlinear governing equations, which are then solved by a direct iterative method to obtain the nonlinear pull-in and pull-out voltages for cantilever and fixed–fixed boundary conditions. A detailed parametric study is conducted to elucidate the influences of nonlocal parameter, vdW force, fringing field, beam length and gap distance on the behavior of the pull-in instability voltage. Numerical results indicate that the magnitude of the pull-in voltage increases with increase in the gap distance. Furthermore, as the effective gap distance increases, the pull-out voltage tends toward the electrostatically pull-out voltage.