Frequency response of primary resonance of electrostatically actuated CNT cantilevers

This paper deals with electrostatically actuated carbon nanotube (CNT) cantilever over a parallel ground plate. Three forces act on the CNTs cantilever, namely electrostatic, van der Waals, and damping. The van der Waals force is significant for values of 50 nm or less of the gap between the CNT and the ground plate. As both forces electrostatic and van der Waals are nonlinear, and the CNTs electrostatic actuation is given by AC voltage, the CNT undergoes nonlinear parametric dynamics. The methods of multiple scales and reduced order model (ROM) are used to investigate the system under soft AC near half natural frequency of the CNT and weak nonlinearities. The frequency–amplitude response and damping, voltage, and van der Waals effects on the response are reported. It is showed that only five terms ROM predicts and accurately predicts the pull-in instability and the saddle-node bifurcation, respectively.     

Effects of van der Waals force on nonlinear vibration of electromechanical integrated electrostatic harmonic actuator

An electromechanical integrated electrostatic harmonic actuator is promising for the miniaturization of electromechanical devices. As the dimensions of the actuator decrease, the effects of the van der Waals force become obvious. In this study, by considering the nonlinearity of electrostatic and van der Waals forces, nonlinear vibration equations of the flexible ring of an electrostatic harmonic actuator are deduced. Using these equations, the nonlinear free vibration and nonlinear forced response of the actuator are investigated. The effects of the van der Waals force on the nonlinear vibration of the flexible ring are analyzed. Results show that these effects of the van der Waals force are relatively obvious under some conditions and should be considered.     

Rippling effect on the structural response of electrostatically actuated single-walled carbon nanotube based NEMS actuators

Several nonlinear phenomena have shown to have significant effect on the electromechanical performance of single-walled carbon nanotube (SWCNT) based nanoelectromechanical (NEMS) devices. To name few: the van der Waals forces, the Casimir forces, the tip charge concentration and the rippling phenomenon. Some of these effects have been take care of in previous investigation; however, some have been disregarded in the mechanical models suggested for simulation of the SWCNT based structures. In this paper, the influence of rippling deformation on the vibration characteristics of SWCNT based actuators is investigated using a nonlinear Euler-Bernoulli beam theory that incorporates the effect of rippling deformation using an improved function including some correcting terms for the SWCNT curvature (rippling deformation). The influence of the Casimir and the van der Waals attraction forces are considered in the proposed model as well as the size-dependent behavior assuming the so-called Eringen nonlocal elasticity theory. The dynamic response of CNT is investigated based on time history and phase portrait plots of the CNT based nano-actuator. It is shown that the rippling deformation can significantly decrease the static as well as the dynamic pull-in voltage of the SWCNT based actuator. The rippling deformation of SWCNT decreases the dynamic pull-in time as well. Effect of various factors such as the DC actuation load and the Casimir attractive forces on the dynamic stability and the pull-in characteristics of the nano-actuator are examined. Results of the present study are beneficial to accurate design and fabrication of electromechanical CNT based actuators. Comparison between the obtained results and those reported in the literature by experiments and molecular dynamics, verifies the integrity of the present numerical analysis.     

Nonlinear analysis of a SWCNT over a bundle of nanotubes

The deformation of a single wall carbon nanotube (SWCNT) interacting with a curved bundle of nanotubes is analyzed. The SWCNT is modeled as a straight elastic inextensible beam based on small deformation. The bundle of nanotubes is assumed rigid and the interaction is due to the van der Waals forces. An analytical solution is obtained using a bilinear approximation to the van der Waals forces. The analytical results are in good agreement with the results of two numerical methods. The results indicate that the SWCNT remains near the curved bundle provided that its curvature is below a critical value. For curvatures above this critical value the SWCNT breaks contact with the curved bundle and nearly returns to its straight position. A parameter study shows that the critical curvature depends on the stiffness of the SWCNT and the absolute minimum energy associated with the van der Waals forces but it is independent of the SWCNT's length in general. An analytical estimate of the critical curvature is developed. The results of this study may be applicable to composites of nanotubes where separation phenomena are suspected to occur.     

Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges

This study investigates the size-dependent quasistatic response of a nonlinear viscoelastic microelectromechanical system (MEMS) under an electric actuation. To have this problem in view, the deformable electrode of the MEMS is modelled using cantilever and doubly-clamped viscoelastic microbeams. The modified couple stress theory in conjunction with Bernoulli–Euler beam theory are used for mathematical modeling of the size-dependent instability of microsystems in the framework of linear viscoelastic theory. Simultaneous effect of electrostatic actuation including fringing field, residual stress, mid-plane stretching and Casimir and van der Waals intermolecular forces are considered in the theoretical model. A single element of the standard linear solid element is used to simulate the viscoelastic behavior. Based on the extended Hamilton’s variational principle, the nonlinear governing integro-differential equation and boundary conditions are derived. Thereafter, a new generalized differential-integral quadrature solution for the nonlinear quasistatic response of electrically actuated viscoelastic micro/nanobeams under two different boundary conditions; doubly-clamped microbridge and clamped-free microcantilever. The developed model is verified and a good agreement is obtained. Finally, a comprehensive study is conducted to investigate the effects of various parameters such as material relaxation time, durable modulus, material length scale parameter, Casimir force, van der Waals force, initial gap and beam length on the pull-in response of viscoelastic microbridges and microcantilevers in the framework of viscoelasticity.     

DEM analysis of the effect of electrostatic interaction on particle mixing for carrier-based dry powder inhaler formulations

Particle interactions play a significant role in controlling the performance of dry powder inhalers (DPIs), which mainly arise through van der Waals potentials, electrostatic interactions, and capillary forces. Our aim is to investigate the influence of electrostatic charge on the performance of DPIs as a basis for improving the formulation of the particle ingredients. The mixing process of carrier and active pharmaceutical ingredient (API) particles in a vibrating container is investigated using a discrete element method (DEM). The number of API particles attaching to the carrier particle (i.e., contact number) increases with increasing charge and decreases with increasing container size. The contact number decreases with increasing vibrational velocity amplitude and frequency. Moreover, a mechanism governed by the electrostatic force is proposed for the mixing process. This mechanism is different from that previously proposed for the mixing process governed by van der Waals forces, indicating that long-range and short-range adhesive forces can result in different mixing behaviours.     

Non-linear dynamic instability of a double-sided nano-bridge considering centrifugal force and rarefied gas flow

Double-sided electromechanical nano-bridges can potentially be used as angular speed sensors and accelerometers in rotary systems such as turbine blades and vacuum pumps. In such applications, the influences of the centrifugal force and rarefied flow should be considered in the analysis. In the present study, the non-linear dynamic pull-in instability of a double-sided nano-bridge is investigated incorporating the effects of angular velocity and rarefied gas damping. The non-linear governing equation of the nanostructure is derived using Euler-beam model and Hamilton׳s principle including the dispersion forces. The strain gradient elasticity theory is used for modeling the size-dependent behavior of the system. The reduced order method is also implemented to discretize and solve the partial differential equation of motion. The influences of damping, centrifugal force, length scale parameters, van der Waals force and Casimir attraction on the dynamic pull-in voltage are studied. It is found that the dispersion and centrifugal forces decrease the pull-in voltage of a nano-bridge. Dynamic response of the nano-bridge is investigated by plotting time history and phase portrait of the system. The validity of the proposed method is confirmed by comparing the results from the present study with the experimental and numerical results reported in the literature.     

Nonlinear vibrations of a single-walled carbon nanotube for delivering of nanoparticles

The capability of carbon nanotubes (CNTs) in efficient transporting of drug molecules into the biological cells has been the focus of attention of various scientific disciplines during the past decade. From applied mechanics points of view, translocation of a nanoparticle inside the pore of a CNT would result in vibrations. The true understanding of the interactive forces between the moving nanoparticle and the inner surface of the CNT is a vital step in factual realization of such vibrations. Herein, by employing the nonlocal Rayleigh beam theory, nonlinear vibrations of single-walled carbon nanotubes (SWCNTs) as nanoparticle delivery nanodevices are studied. The existing van der Waals interactional forces between the constitutive atoms of the nanoparticle and those of the SWCNT, frictional force, and both longitudinal and transverse inertial effects of the moving nanoparticle are taken into account in the proposed model. The nonlinear-nonlocal governing equations are explicitly obtained and then numerically solved using Galerkin method and a finite difference scheme in the space and time domains, respectively. The roles of the velocity and mass weight of the nanoparticle, small-scale effect, slenderness ratio, and vdW force on the maximum longitudinal and transverse displacements as well as the maximum nonlocal axial force and bending moment within the SWCNT are examined. In general, the obtained results reveal that the nonlinear analysis should be performed when the nanotube structure is traversed by a moving nanoparticle with high levels of the mass weight and velocity.     

Analytical modeling of static behavior of electrostatically actuated nano/micromirrors considering van der Waals forces

In this paper,the effect of van der Waals(vdW)force on the pull-in behavior of electrostatically actuatednano/micromirrors is investigated.First,the minimum potential energy principle is utilized to find the equation governing the static behavior of nano/micromirror under electrostatic and vdW forces.Then,the stability of static equilibrium points is analyzed using the energy method.It is foundthat when there exist two equilibrium points,the smaller oneis stable and the larger one is unstable.The effects of different design parameters on the mirror’s pull-in angle andpull-in voltage are studied and it is found that vdW forcecan considerably reduce the stability limit of the mirror.Atthe end,the nonlinear equilibrium equation is solved numerically and analytically using homotopy perturbation method(HPM).It is observed that a sixth order perturbation approximation can precisely model the mirror’s behavior.The results of this paper can be used for stable operation design andsafe fabrication of torsional nano/micro actuators.     

INVESTIGATION OF THE DYNAMIC BUCKLING OF DOUBLEWALLED CARBON NANOTUBE SUBJECTED TO AXIAL PERIODIC DISTURBING FORCES

IntroductionThediscoveryofthefirstcarbonnanotubes[1]hasattractedwideattentionandstimulatedextensivestudies[2 - 5 ].Thestudiesshowedthatthecarbonnanotubesexhibitsuperiormechanical,electronicandchemicalproperties.Onthemechanicalbehavior,thecarbonnanotubespossessexceptionallyhighstrength ,stiffnessandelasticmodulus.Theestimatemodulusofthecarbonnanotubemayreachashighas 1TPa.Itisthelargestofallknownmaterials.Thestrengthorstiffnessishigherthananyknownfiber[3].Thecarbonnanotubeareusedascompositemat…     

Linear instability of ultra-thin liquid films flowing down cylindrical fibre

The stability characteristics of an ultra-thin layer of a viscous liquid flowing down a cylindrical fibre are investigated by a linear theory. The film with the thickness less than 100 nm is driven by an external force and under the influence of the van der Waals forces. The results show that, when the relative film thickness decreases, the curvature of the fibre depresses the development of the linear perturbations, whereas the van der Waals forces promote the instabilities. This competition results in a non-monotonous dependence of the growth rate on the relative film thickness. The critical curves are also obtained to describe the transition from the absolute instability to the convective instability, indicating that the van der Waals forces can enlarge the absolutely unstable region. Furthermore, the surface tension can cause the development of the absolute instability, whereas the external force has an opposite effect.     

Linear instability of ultra-thin liquid films flowing down cylindrical fibre

The stability characteristics of an ultra-thin layer of a viscous liquid flowing down a cylindrical fibre are investigated by a linear theory. The film with the thickness less than 100 nm is driven by an external force and under the influence of the van der Waals forces. The results show that, when the relative film thickness decreases, the curvature of the fibre depresses the development of the linear perturbations, whereas the van der Waals forces promote the instabilities. This competition results in a non-monotonous dependence of the growth rate on the relative film thickness. The critical curves are also obtained to describe the transition from the absolute instability to the convective instability, indicating that the van der Waals forces can enlarge the absolutely unstable region. Furthermore, the surface tension can cause the development of the absolute instability, whereas the external force has an opposite effect.     

Effects of the van der Waals force,squeeze-film damping,and contact bounce on the dynamics of electrostatic microcantilevers before and after pull-in

The operational range of microcantilever beams under electrostatic force can be extended beyond pull-in in the presence of an intermediate dielectric layer. In this paper, a systematic method for deriving dynamic equation of microcantilevers under electrostatic force is presented. This model covers the behavior of the microcantilevers before and after the pull-in including the effects of van der Waals force, squeeze-film damping, and contact bounce. First, a polynomial approximate shape function with a time-dependent variable for each configuration is defined. Using Hamilton’s principle, dynamic equations of microcantilever in all configurations have been derived. Comparison between modeling results and previous experimental data that have been used for validation of the model shows a good agreement.     

Taking into account van der Waals interaction in some problems of elasticity

We obtain the correction due to van der Waals forces to the flexural rigidity of a layer. The cases of a monolayer and a film consisting of several monolayers are considered. The correction due to van der Waals interaction to the force acting on an indentor is obtained. Two versions of determining the elastic forces at the microlevel are considered. Numerical estimates of the parameter values for which the correction is essential are given.     

Upper and lower bounds for the pull-in parameters of a micro- or nanocantilever on a flexible support

An analytical method is proposed to accurately estimate the pull-in parameters of a micro- or nanocantilever beam elastically constrained by a rotational spring at one end. The system is actuated by electrostatic force and subject to Casimir or van der Waals forces according to the beam size. The deflection of the beam is described by a fourth-order nonlinear boundary value problem, or equivalently in terms of a nonlinear integral equation. New a priori analytical estimates on the solution from both sides are first derived and then lower and upper bounds for the pull-in parameters are obtained, with no need of solving the nonlinear boundary value problem. The lower and upper bounds turn out to be very close each other and in excellent agreement with the numerical results provided by the shooting method. The approach also provides accurate predictions for the pull-in parameters of a freestanding nanoactuator.     

Study of dynamic stability of the double-walled carbon nanotube under axial loading embedded in an elastic medium by the energy method

In this paper, the dynamic stability of single- and double-walled carbon nanotubes (SWCNT and DWCNT) under dynamic axial loading is investigated using the continuum mechanics model and the minimum total energy method. The natural frequencies of the SWCNT and the critical dynamic axial load of the SWCNT and DWCNT are obtained using the Rayleigh-Ritz method. The effects of the elastic medium and the van der Waals forces between the two layers in the DWCNT are taken into account using the Winkler model and Lennard-Jones theory, respectively. The effect of the small length scale is also considered using the Eringen Model. The critical dynamic axial load is increased by inserting an inner carbon nanotube (CNT) into an isolated CNT embedded in an elastic medium.     

范德华力对磁头/硬盘系统气体润滑特性的影响

通过数值方法分析范德华力对磁头承载力的影响以及随飞行姿态的变化规律.结果表明,范德华力主要作用在磁头滑块尾部的凸台上,当飞行高度低于10 nm后,范德华力使得滑块的承载能力明显降低,从而导致飞行高度下降.范德华力主要与飞行高度和俯仰角有关,随着飞行高度和俯仰角减小,范德华力使承载能力和俯仰转矩明显减小.范德华力对侧翻角的影响不敏感,随着侧翻角的增加,范德华力使滑块的承载能力和侧翻转矩略微减小.在磁头设计时,通过增加俯仰角和减小尾部凸台面积可以减小范德华力的影响.     

范德华力对多壁纳米碳管力学性质的影响

用分子动力学方法模拟了多壁纳米碳管在压缩、弯曲变形下力与变形的关系．通过与组成多壁碳管的各单壁碳管的比较分析,揭示了多壁纳米碳管层间范德华力对碳管力学性质的影响．采用Tersoff-Brenner势描述每一单壁纳米碳管内原子间作用,采用Lennard-Jones势描述碳管壁间范德华力．计算结果表明：多壁纳米碳管的比强度明显高于单壁纳米碳管．纳米碳管的半径虽然对杨氏模量影响不大,但对纳米碳管的曲屈行为影响却相当显著。     

Dynamic stability of electrostatic torsional actuators with van der Waals effect

The influence of van der Waals (vdW) force on the stability of electrostatic torsional nano-electro-mechanical systems (NEMS) actuators is analyzed in the paper. The dependence of the critical tilting angle and voltage is investigated on the sizes of structure with the consideration of vdW effects. The pull-in phenomenon without the electrostatic torque is studied, and a critical pull-in gap is derived. A dimensionless equation of motion is presented, and the qualitative analysis of it shows that the equilibrium points of the corresponding autonomous system include center points, stable focus points, and unstable saddle points. The Hopf bifurcation points and fork bifurcation points also exist in the system. The phase portraits connecting these equilibrium points exhibit periodic orbits, heteroclinic orbits, as well as homoclinic orbits.     

A glance on the effects of temperature on axisymmetric dynamic behavior of multiwall carbon nanotubes

In this paper the effects of temperature on theradial breathing modes(RBMs) and radial wave propagation in multiwall carbon nanotubes(MWCNTs) are investigated using a continuum model of multiple elastic isotropicshells.The van der Waals forces between tubes are simulatedas a nonlinear function of interlayer spacing of MWCNTs.The governing equations are solved using a finite elementmethod.A wide range of innermost radius-to-thickness ratioof MWCNTs is considered to enhance the investigation.Thepresented solution is verified by comparing the results withthose reported in the literature.The effects of temperature onthe van der Waals interaction coefficient between layers ofMWCNTs are examined.It is found that the variation of thevan der Waals interaction coefficient at high temperature issensible.Subsequently,variations of RBM frequencies andradial wave propagation in MWCNTs with temperatures upto 1 600 K are illustrated.It is shown that the thick MWCNTs are more sensible to temperature than the thin ones.