Application of an electrostatically actuated cantilevered carbon nanotube with an attached mass as a bio-mass sensor

In the present paper, another latent capability of SWCNT as a mass sensor is investigated. The relationship between the resonant frequency, dynamic pull-in voltage at the resonance frequency shift, and the attached mass is established by using the nonlocal Euler–Bernoulli beam theory. Using this relationship, a general closed-form nonlinear sensor-equation has been derived for the detection of the mass attached to the SWCNT. The aim of this study and present model is to show the sensitivity of the Cantilevered SWCNT to the values and positions of attached mass. Moreover, the results indicate that by increasing the value of attached mass and considering a single non-local scaling parameter (e₀), the values of dynamic pull-in voltage at the resonance frequency shift are decreased. Because of the small scaling parameter (e₀), the mass sensitivity of carbon nanotube increases, when the position of the attached mass is in the tip of a Cantilevered SWCNT length. The authority and the accuracy of these formulas are examined with other pull-in sensor equations in literatures. The results demonstrate that the new sensor equation can be applied for CNT-based mass sensors with rational accuracy.     

Effect of Longitudinal Magnetic Field on Vibration Characteristics of Single-Walled Carbon Nanotubes in a Viscoelastic Medium

The effect of longitudinal magnetic field on vibration response of a sing-walled carbon nanotube (SWCNT) embedded in viscoelastic medium is investigated. Based on nonlocal Euler-Bernoulli beam theory, Maxwell’s relations, and Kelvin viscoelastic foundation model, the governing equations of motion for vibration analysis are established. The complex natural frequencies and corresponding mode shapes in closed form for the embedded SWCNT with arbitrary boundary conditions are obtained using transfer function method (TFM). The new analytical expressions for the complex natural frequencies are also derived for certain typical boundary conditions and Kelvin-Voigt model. Numerical results from the model are presented to show the effects of nonlocal parameter, viscoelastic parameter, boundary conditions, aspect ratio, and strength of the magnetic field on vibration characteristics for the embedded SWCNT in longitudinal magnetic field. The results demonstrate the efficiency of the proposed methods for vibration analysis of embedded SWCNTs under magnetic field.     

Free vibration of an embedded single-walled carbon nanotube with various boundary conditions using the RMVT-based nonlocal Timoshenko beam theory and DQ method

The nonlocal Timoshenko beam theories (TBTs), based on the Reissner mixed variation theory (RMVT) and principle of virtual displacement (PVD), are derived for the free vibration analysis of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium and with various boundary conditions. The strong formulations of the nonlocal TBTs are derived using Hamilton's principle, in which Eringen's nonlocal constitutive relations are used to account for the small-scale effect. The interaction between the SWCNT and its surrounding elastic medium is simulated using the Winkler and Pasternak foundation models. The frequency parameters of the embedded SWCNT are obtained using the differential quadrature (DQ) method. In the cases of the SWCNT without foundations, the results of RMVT- and PVD-based nonlocal TBTs converge rapidly, and their convergent solutions closely agree with the exact ones available in the literature. Because the highest order with regard to the derivatives of the field variables used in the RMVT-based nonlocal TBT is lower than that used in its PVD-based counterpart, the former is more efficient than the latter with regard to the execution time. The former is thus both faster and obtains more accurate solutions than the latter for the numerical analysis of the embedded SWCNT.     

Nonlinear size-dependent behaviour of single-walled carbon nanotubes

This paper examines the nonlinear size-dependent behaviour of single-walled carbon nanotubes (SWCNTs) based on the von-Karman nonlinearity and the nonlocal elasticity theory capable of predicting size effects. To this end, based on Hamilton’s principle in the framework of the nonlocal Euler–Bernoulli beam theory, the equation of motion and associated boundary conditions are derived. Then, with the aid of a high-dimensional Galerkin scheme, the nonlinear partial differential equation of motion of the SWCNT is recast into a reduced-order model. The dynamic response of the system is then investigated for two different types of excitation, namely primary and superharmonic excitations. Eventually, the effect of the slenderness ratio, forcing amplitude, and excitation frequency on the motion characteristics of the system is investigated.     

Modeling and active vibration suppression of a single-walled carbon nanotube subjected to a moving harmonic load based on a nonlocal elasticity theory

This paper investigates active vibration suppression of a single-walled carbon nanotube (SWCNT) under the action of a moving harmonic load using Eringen’s nonlocal elasticity theory. The SWCNT is modeled according to the nonlocal Euler–Bernoulli beam theory. A Dirac-delta function is used to describe the position of the moving load along the SWCNT. Next, a linear classical optimal control algorithm with displacement-velocity feedback is used to suppress vibration in the SWCNT with control forces acting as actuators. The effects of a small-scale parameter, slenderness ratio, moving load velocity, and the excitation frequency of a moving load on the dynamic deflection of the SWCNT are examined. Finally, the ability of the control algorithm to suppress the response of the SWCNT under the effects of a moving load with a number of controlled modes and control forces is surveyed.     

Nanoscale mass detection based on vibrating piezoelectric ultrathin films under thermo-electro-mechanical loads

The potential applications of piezoelectric nanofilms (PNFs) and double-piezoelectric-nanofilm (DPNF) systems as nanoelectromechanical mass sensors are examined. The PNFs carrying multiple nanoparticles at arbitrary locations are modeled as rectangular nonlocal plates with attached concentrated masses. Using the nonlocal elasticity theory and Hamilton’s principle, the differential equations of motion are derived for both PNF-based and DPNF-based nanosensors. The influences of small scale, initial stress and temperature change on the frequency shifts of the nanoelectromechanical sensors are taken into consideration. Explicit expressions are derived for the resonance frequencies of the nanosensors by employing the Galerkin method. The present results show that when the value of nonlocal parameter decreases, the frequency shifts of piezoelectric nanosensors increase. Further, the frequency shifts of DPNF-based mass sensors are always greater than those of PNF-based mass sensors. The present work would be helpful in the design of nanoelectromechanical mass sensors using PNFs.     

Longitudinal vibration and stability analysis of carbon nanotubes conveying viscous fluid

Nowadays, carbon nanotubes (CNT) play an important role in practical applications in fluidic devices. To this end, researchers have studied various aspects of vibration analysis of a behavior of CNT conveying fluid. In this paper, based on nonlocal elasticity theory, single-walled carbon nanotube (SWCNT) is simulated. To investigate and analyze the effect of internal fluid flow on the longitudinal vibration and stability of SWCNT, the equation of motion for longitudinal vibration is obtained by using Navier-Stokes equations. In the governing equation of motion, the interaction of fluid-structure, dynamic and fluid flow velocity along the axial coordinate of the nanotube and the nano-scale effect of the structure are considered. To solve the nonlocal longitudinal vibration equation, the approximate Galerkin method is employed and appropriate simply supported boundary conditions are applied. The results show that the axial vibrations of the nanotubesstrongly depend on the small-size effect. In addition, the fluid flowing in nanotube causes a decrease in the natural frequency of the system. It is obvious that the system natural frequencies reach zero at lower critical flow velocities as the wave number increases. Moreover, the critical flow velocity decreases as the nonlocal parameter increases.     

Nonlocal vibration of coupled DLGS systems embedded on Visco-Pasternak foundation

In this paper, vibration analysis of the coupled system of double-layered graphene sheets (CS-DLGSs) embedded in a Visco-Pasternak foundation is carried out using the nonlocal elasticity theory of orthotropic plate. The two DLGSs are coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak foundation. Considering the Von Kármán nonlinear strain-displacement-relations, the motion equations are derived using the Hamilton's principle. Differential quadrature method (DQM) is applied to obtain the frequency ratio for various boundary conditions. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, aspect ratio, graphene sheet's size, boundary conditions and the elastic and viscoelastic medium coefficients on the frequency ratio of CS-DLGSs. In this coupled system, two case of DLGSs vibration are investigated and compared with each other: (1) In-phase vibration (2) Out-of-phase vibration. The results indicate that the frequency ratio of the CS-DLGSs is more than the single-layered graphene sheet (SLGS). The results are in good agreement with the previous researches.     

Wave propagation in fluid-filled single-walled carbon nanotube on analytically nonlocal Euler–Bernoulli beam model

An analytically nonlocal Euler–Bernoulli beam model for the wave propagation in fluid-filled single-walled carbon nanotube (SWCNT) is established. The governing equations with the nonlocal effects are derived on the variational principle, and used in the wave propagation analysis of the SWCNT beam. Compared with the partially nonlocal Euler–Bernoulli beam models used previously, the analytically nonlocal model presented in the present study predicts well the effects of the stiffness enhancement and the wave damping at the high wavenumber or the strong nonlocal effects area for the fluid-filled SWCNT beam. Though the analytical model is less sensitive than the partially nonlocal model when the moving velocity of the internal fluid is high enough, it simulates more of the high-order nonlocal effecting information than the partially nonlocal model does in many cases.     

Nonlocal discrete and continuous modeling of free vibration of stocky ensembles of vertically aligned single-walled carbon nanotubes

Free dynamic analysis of transverse motion of vertically aligned stocky ensembles of single-walled carbon nanotubes is of particular interest. A linear model is developed to take into account the van der Waals forces between adjacent SWCNTs because of their bidirectional transverse displacements. Using Hamilton's principle, the discrete equations of motion of free vibration of the nanostructure are obtained based on the nonlocal Rayleigh, Timoshenko, and higher-order beam theories. The application of such discrete models for frequency analysis of highly populated ensembles would be associated with so much computational effort. To overcome such a problem, some useful nonlocal continuous models are established. The obtained results reveal that the newly developed models can successfully capture the predicted fundamental frequencies of the discrete models. Through various numerical studies, the roles of slenderness ratio, radius of the SWCNT, small-scale parameter, population of the ensemble, and intertube distance on the fundamental flexural frequency of the nanostructure are examined and discussed. The capabilities of the proposed nonlocal continuous models in predicting flexural frequencies of the nanostructure are also addressed.     

Free vibration of non-uniform nanobeams using Rayleigh–Ritz method

In this article, boundary characteristic orthogonal polynomials have been implemented in the Rayleigh–Ritz method to investigate free vibration of non-uniform Euler–Bernoulli nanobeams based on nonlocal elasticity theory. Non-uniform cross section of nanobeams has been considered by taking linear as well as quadratic variations of Young's modulus and density along the space coordinate. Detailed analysis has been reported for all the possible cases of such variations. The objective of the present study is to analyze the effects of nonlocal parameter, boundary condition, length-to-diameter ratio and non-uniform parameter on the frequency parameters. It is found that clamped nanobeams are having highest frequency parameters than other types of boundary conditions for a particular set of parameters. It is also observed that frequency parameters decrease with increase in scaling effect parameter. First four deflection shapes of non-uniform nanobeams have also been incorporated. In this analysis, some of the new results in terms of boundary conditions have also been included.     

Free torsional vibration of nanotubes based on nonlocal stress theory

A new elastic nonlocal stress model and analytical solutions are developed for torsional dynamic behaviors of circular nanorods/nanotubes. Unlike the previous approaches which directly substitute the nonlocal stress into the equations of motion, this new model begins with the derivation of strain energy using the nonlocal stress and by considering the nonlinear history of straining. The variational principle is applied to derive an infinite-order differential nonlocal equation of motion and the corresponding higher-order boundary conditions which contain a nonlocal nanoscale parameter. Subsequently, free torsional vibration of nanorods/nanotubes and axially moving nanorods/nanotubes are investigated in detail. Unlike the previous conclusions of reduced vibration frequency, the solutions indicate that natural frequency for free torsional vibration increases with increasing nonlocal nanoscale. Furthermore, the critical speed for torsional vibration of axially moving nanorods/nanotubes is derived and it is concluded that this critical speed is significantly influenced by the nonlocal nanoscale.     

A nonlocal Levinson beam model for free vibration analysis of zigzag single-walled carbon nanotubes including thermal effects

A nonlocal Levinson beam model is developed to study the free vibrations of a zigzag single-walled carbon nanotube (SWCNT) in thermal environments. The equivalent Young’s modulus and shear modulus for a zigzag SWCNT are derived using an energy-equivalent model. The present study illustrates that the vibration characteristics of an SWCNT are strongly dependent on the temperature change and on the chirality of a zigzag carbon nanotube. The investigation of the chirality and temperature effects on free vibration of carbon nanotubes may be used as a useful reference for the application and the design of nanoelectronic and nanodrive devices, nano-oscillators, and nanosensors, in which carbon nanotubes act as basic elements.     

A hybrid continuum and molecular mechanics model for the axial buckling of chiral single-walled carbon nanotubes

The purpose of this study is to describe the axial buckling behavior of chiral single-walled carbon nanotubes (SWCNTs) using a combined continuum-atomistic approach. To this end, the nonlocal Flugge shell theory is implemented into which the nonlocal elasticity of Eringen incorporated. Molecular mechanics is used in conjunction with density functional theory (DFT) to precisely extract the effective in-plane and bending stiffnesses and Poisson's ratio used in the developed nonlocal Flugge shell model. The Rayleigh-Ritz procedure is employed to analytically solve the problem in the context of calculus of variation. The results generated from the present hybrid model are compared with those from molecular dynamics simulations as a benchmark of good accuracy and excellent agreement is achieved. The influences of small scale factor, commonly used boundary conditions and chirality on the critical buckling load are fully explored. It is indicated that the importance of the small length scale is affected by the type of boundary conditions considered.     

Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions

This paper investigates the thermo-electro-mechanical vibration of the rectangular piezoelectric nanoplate under various boundary conditions based on the nonlocal theory and the Mindlin plate theory. It is assumed that the piezoelectric nanoplate is subjected to a biaxial force, an external electric voltage and a uniform temperature rise. The Hamilton's principle is employed to derive the governing equations and boundary conditions, which are then discretized by using the differential quadrature (DQ) method to determine the natural frequencies and mode shapes. The detailed parametric study is conducted to examine the effect of the nonlocal parameter, thermo-electro-mechanical loadings, boundary conditions, aspect ratio and side-to-thickness ratio on the vibration behaviors.     

Geometrical influence of a deposited particle on the performance of bridged carbon nanotube-based mass detectors

A nonlocal continuum-based model is derived to simulate the dynamic behavior of bridged carbon nanotube-based nano-scale mass detectors. The carbon nanotube (CNT) is modeled as an elastic Euler-Bernoulli beam considering von-Kármán type geometric nonlinearity. In order to achieve better accuracy in characterization of the CNTs, the geometrical properties of an attached nano-scale particle are introduced into the model by its moment of inertia with respect to the central axis of the beam. The inter-atomic long-range interactions within the structure of the CNT are incorporated into the model using Eringen's nonlocal elastic field theory. In this model, the mass can be deposited along an arbitrary length of the CNT. After deriving the full nonlinear equations of motion, the natural frequencies and corresponding mode shapes are extracted based on a linear eigenvalue problem analysis. The results show that the geometry of the attached particle has a significant impact on the dynamic behavior of the CNT-based mechanical resonator, especially, for those with small aspect ratios. The developed model and analysis are beneficial for nano-scale mass identification when a CNT-based mechanical resonator is utilized as a small-scale bio-mass sensor and the deposited particles are those, such as proteins, enzymes, cancer cells, DNA and other nano-scale biological objects with different and complex shapes.     

Surface effect on size-dependent wave propagation in nanoplates via nonlocal elasticity

Within the framework of nonlocal elasticity, the surface layer model is proposed to investigate the wave propagation characteristics in a single-layered nanoplate. The general solutions of nonlocal governing equations are expressed using partial wave technique and the nonclassical boundary conditions are derived. The dispersion relation with the effects of surface and nonlocal small-scale is obtained, and the size-dependent dispersion behaviour is demonstrated. The impacts of surface elasticity, residual surface stress and nonlocal parameter on the dispersion curves of the lowest-order two modes are illustrated. Numerical examples reveal that both the surface effect and nonlocal small-scale effect can obviously decrease the magnitude of phase velocity, and the thinner nanoplate corresponds to the smaller wave velocity and the narrower frequency bandwidth.     

非局域颗粒复合介质的相干完美吸收效应

研究了两束相干光以相同的入射角从左、右两侧分别入射到Au-SiO_2复合介质板时,在不同的体系参数下该复合材料体系发生相干完美吸收的情形.运用有效媒质理论推导出了复合介质的有效介电常数以及有效磁导率;在得到有效电磁参数的基础上进一步推导得到平面波入射复合介质板时的反/透射系数.通过比较分析非局域和局域情况下颗粒复合介质的相干完美吸收现象,发现当颗粒尺寸很小时非局域效应的影响会导致复合介质产生相干完美吸收的入射光的频率范围显著变宽.在进一步的解析计算中,通过调节复合介质板的厚度、入射光波长、金属颗粒体积分数等参数得到了不同情况下产生的相干完美吸收现象,并由此分析非局域情形下对于相干完美吸收现象的调控.     

Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory

The aim of this paper is to study the longitudinal frequency of a cracked nanobeam. The frequency equation of the nanobeam with clamped–clamped and clamped–free boundary conditions is derived based on the nonlocal elasticity theory. According to the equation, it can be found that the effects of the crack parameter, crack location, and nonlocal parameter on the longitudinal frequency of the cracked nanobeam are significant. The frequency decreases with an increase of the crack parameter. However, the increasing nonlocal parameter results in a decrease of the crack effect on the frequency. In addition, when the crack location is near the support, a larger decrease in the frequency can be observed.     

Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium

In the present paper, the sinusoidal shear deformation plate theory (SDPT) is reformulated using the nonlocal differential constitutive relations of Eringen to analyze the bending and vibration of the nanoplates, such as single-layered graphene sheets, resting on two-parameter elastic foundations. The present SDPT is compared with other plate theories. The nanoplates are assumed to be subjected to mechanical and thermal loads. The equations of motion of the nonlocal model are derived including the plate foundation interaction and thermal effects. The governing equations are solved analytically for various boundary conditions. Nonlocal theory is employed to bring out the effect of the nonlocal parameter on the bending and natural frequencies of the nanoplates. The influences of nonlocal parameter, side-to-thickness ratio and elastic foundation moduli on the displacements and vibration frequencies are investigated.