Vibration Of Double-walled Carbon Nanotubes Research Articles

Nonlinear Vibration of Double-Walled Carbon Nanotubes Subjected to Mechanical Impact and Embedded on Winkler-Pasternak Foundation.

This study was devoted to an investigation on the dynamics of double-walled carbon nanotubes (DWCNTs) under the influence of Winkler-Pasternak foundation near the primary resonance. Two Euler-Bernoulli beams embedded on nonlinear foundation, interacting through van der Waals forces, subjected to mechanical impact are considered. By means of Hamilton's principle, Eringen's nonlocal elastic theory, and taking into account the moving nanoparticles, the Galerkin-Bubnov method is applied and accordingly, governing partial differential equations are reduced to two differential equations with variable coefficients. The nonlinear damped and forced vibration is studied using the optimal auxiliary functions method (OAFM). An explicit and very accurate analytical solution is obtained by means of OAFM without considering simplifying hypotheses. An accurate analysis is for the first time reported considering the cumulated effects of nonlinearities simultaneously induced by the Winkler-Pasternak foundation, the curvature of beams and van der Waals force, and also the effect of discontinuities marked by the presence of the Dirac function. Finally, a stability analysis of the considered model is developed by means of the homotopy perturbation method (HPM) using the condition of existence of the two frequencies. It was shown that an increasing of some constitutive parameters substantially reduces the area of stability, all these being of much help in guiding the design of advanced nanoelectromechanical devices, in which nanotubes act as basic elements.

Applicability and Limitations of Ru’s Formulation for Vibration Modelling of Double-Walled Carbon Nanotubes

In this paper, a comparison is conducted between two different formulations of the van der Waals interaction coefficient between layers, as applied to the vibrations of double-walled carbon nanotubes (DWCNTs); specifically, the evaluation of the natural frequencies is achieved through Ru’s and He’s formulations. The actual discrete DWCNT is modelled by means of a couple of concentric equivalent continuous thin cylindrical shells, where Donnell shell theory is adopted to obtain strain-displacement relationships. In order to take into account the chirality effect of DWCNT, an anisotropic elastic shell model is considered. Simply supported boundary conditions are imposed and the Rayleigh–Ritz method is used to obtain approximate natural frequencies and mode shapes. A parametric analysis considering different values of diameters and numbers of waves along longitudinal and circumferential directions is performed by adopting Ru’s and He’s formulations. From the comparisons, it is evident that Ru’s formulation provides unsatisfactory results for relatively low values of diameters and relatively high numbers of circumferential waves with respect to the more accurate He’s formulation. This behaviour is observed for every number of longitudinal half-waves. Therefore, Ru’s formulation cannot be used for the vibration modelling of DWCNTs in a large range of diameters and wavenumbers.

Comments on “Axial vibration of double-walled carbon nanotubes using double-nanorod model with van der Waals force under Pasternak medium and magnetic effects [Vietnam Journal of Mechanics, Vol.44, No.1 (2022), pp. 29-43

First of all Sentilkumar has to be congratulated for his paper [1]. In the paper, the author presented the vibration of double walled carbon nanotubes. He claimed that the equations of motion given in [2] are incorrect. The aim of this communication is to clarify this issue.

Nonlocal anisotropic elastic shell model for vibrations of double-walled carbon nanotubes under nonlinear van der Waals interaction forces

In this paper, a novel nonlocal anisotropic elastic shell model is developed to investigate the nonlinear vibrations of double-walled carbon nanotubes (DWCNTs) in the framework of Sanders–Koiter shell theory. Van der Waals interaction forces between the two concentric single-walled carbon nanotubes (SWCNTs) composing a DWCNT are modelled via Lennard-Jones potential and He’s formulation. In the linear vibration analysis, the displacement field of each SWCNT is expanded by means of a double mixed series in terms of Chebyshev orthogonal polynomials along the longitudinal direction and harmonic functions along the circumferential direction, and Rayleigh–Ritz method is considered to get approximate natural frequencies and modal shapes. In the nonlinear vibration analysis, the three displacements of each SWCNT are re-expanded by means of the approximate eigenfunctions derived in the linear analysis, and an energy approach based on Lagrange equations is adopted to obtain a set of nonlinear ordinary differential equations of motion, which is then solved numerically. Molecular dynamics simulations are performed in order to calibrate the proper value of nonlocal parameter to be inserted in the constitutive equations of the proposed elastic continuum model. A simplified linear distribution of van der Waals interaction forces is initially adopted to analyse the nonlinear vibrations of DWCNTs, obtaining a hardening nonlinear behaviour. By considering a more realistic nonlinear distribution of van der Waals interaction forces, a stronger hardening nonlinear behaviour is found.

Applicability and limitations of Donnell shell theory for vibration modelling of double-walled carbon nanotubes

In this paper, the comparison is conducted between two shell theories as applied to the vibrations of double-walled carbon nanotubes (DWCNTs); specifically, the evaluation of the natural frequencies is conducted via Donnell and Sanders shell theories. The discrete DWCNTs are modelled by means of couples of concentric continuous circular cylindrical shells, where van der Waals interaction forces between the two shells are modelled via He’s formulation. In order to take into account the chirality of DWCNTs, an anisotropic elastic shell model is used. Simply supported, clamped and free boundary conditions are applied and Rayleigh–Ritz method is adopted to obtain approximate natural frequencies and mode shapes. At the beginning, comparisons with experimental and molecular dynamics data are made, from which it is confirmed that anisotropic elastic shell model is more accurate than isotropic one and it is obtained that Sanders shell theory is more accurate than Donnell one. Then, a parametric analysis considering different values of aspect ratio and numbers of waves along the longitudinal and circumferential directions is carried out in the framework of the anisotropic elastic shell model. From these simulations, it is found that Donnell shell theory yields unsatisfactory results for relatively low numbers of longitudinal and circumferential waves, and for relatively high values of aspect ratio, with respect to Sanders shell theory. Therefore Donnell shell theory cannot be used for vibration modelling of DWCNTs in a large range of longitudinal and circumferential wavenumbers and aspect ratios.

Axial vibration of double-walled carbon nanotubes using double-nanorod model with van der Waals force under Pasternak medium and magnetic effects

The present study investigates the axial vibration of double-walled nanotubes. Using the nanorod continuum model with the van der Waals effect, the vibrational frequencies are studied. Aydogdu (Journal of Vibration and Control, Vol. 21, Issue 16, (2015), 3132-3154) proposed a reliable model for the study of axial vibration in a double-walled nanotube. This model provided a detailed investigation of axial vibration using van der Waals effects. But sometimes, the wrong equation might lead to erroneous scientific results. The incorrect term for axial vibration in the double-walled nanotube model is taken care of in the present study for the correct scientific inferences. Effectively, the axial vibrational frequencies appear without decoupling the continuum model as for primary and secondary nanotubes. The semi-analytical method estimates the axial vibrational frequencies of the double-walled nanotube as a coupled model. Two different boundary conditions like clamped-clamped and clamped-free support, are considered in this computation. The Pasternak medium support and magnetic effects influence the vibrational frequencies of the first and second nanotube for the first time. The Pasternak constant and magnetic parameters don't vary with the length of the nanotube for axial vibration. It means that still more understanding requires in modeling the Pasternak medium and magnetic force for the double-nanotube to model axial vibration.

Coaxial vibrations of electrostatically actuated DWCNT resonators: Amplitude–voltage​ response of parametric resonance

This paper investigates the amplitude–voltage response of parametric resonance of coaxial vibrations of double-walled carbon nanotubes (DWCNTs) under electrostatic actuation. The system under investigation consists of a DWCNT parallel to a ground plate and under AC voltage. This voltage produces a nonlinear electrostatic force leading the DWCNT into vibrations. There is a nonlinear intertube van der Waals force between the two coaxial carbon-nanotubes. In coaxial vibration, the two concentric nanotubes move together synchronously. The AC frequency is near the fundamental coaxial natural frequency of the DWCNT. This leads to parametric resonance. The case of small damping, soft electrostatic actuation, and DWCNTs with a high length to diameter ratio (Euler–Bernoulli beam model), is considered. Modal analysis is performed to decouple the equations of free vibrations in their linear part. The solution of the nonlinear problem is then found in terms of modal coordinates. Reduced Order Models (ROMs) using from one to five modes of vibration are used for investigation. Three methods are used to solve these models, (1) the Method of Multiple Scales used to solve the ROM using one mode of vibration, (2) continuation and bifurcation analysis of the five modes of vibration (5T) Reduced Order Model (ROM) using AUTO-07P, and (3) numerical integration of 5T ROM using Matlab. All models and methods are in excellent agreement for amplitudes lower than 0.4 of the gap, while at amplitudes larger than 0.4 only 5T-ROM provide reliable results. The effects of detuning frequency and damping on the amplitude–voltage response of DWCNTs under electrostatic actuation are reported. The importance of the results in this paper are the effect of damping and detuning frequency on the subcritical and supercritical bifurcations, as they define the voltage intervals DWCNT reaches nonzero steady-state amplitudes.

Applicability and Limitations of Simplified Elastic Shell Theories for Vibration Modelling of Double-Walled Carbon Nanotubes

The applicability and limitations of simplified models of thin elastic circular cylindrical shells for linear vibrations of double-walled carbon nanotubes (DWCNTs) are considered. The simplified models, which are based on the assumptions of membrane and moment approximate thin-shell theories, are compared with the extended Sanders–Koiter shell theory. Actual discrete DWCNTs are modelled by means of couples of concentric equivalent continuous thin, circular cylindrical shells. Van der Waals interaction forces between the layers are taken into account by adopting He’s model. Simply supported and free–free boundary conditions are applied. The Rayleigh–Ritz method is considered to obtain approximate natural frequencies and mode shapes. Different aspect and thickness ratios, and numbers of waves along longitudinal and circumferential directions, are analysed. In the cases of axisymmetric and beam-like modes, it is proven that membrane shell theory, differently from moment shell theory, provides results with excellent agreement with the extended Sanders–Koiter shell theory. On the other hand, in the case of shell-like modes, it is found that both membrane and moment shell theories provide results reporting acceptable agreement with the extended Sanders–Koiter shell theory only for very limited ranges of geometries and wavenumbers. Conversely, for shell-like modes it is found that a newly developed, simplified shell model, based on the combination of membrane and semi-moment theories, provides results in satisfactory agreement with the extended Sanders–Koiter shell theory in all ranges.

Metamaterial-like vibration of doublewalled carbon nanotubes

Inspirited by the similarity between counter-phase intertube vibration of multiwalled carbon nanotubes and counter-phase internal vibration of locally resonant metamaterials, forced vibration of DWCNTs (doublewalled carbon nanotubes) as a metaelastic material is investigated. It is found that unlike known elastic metamaterials characterized by wavelength-independent effective mass, “effective mass” of a DWCNT depends on wavelength and its “bandgap” (within which effective mass is negative) varies with wavelengths. For the first time in existing literature, our results confirmed that when the applied exciting frequency approaches or exceeds the lowest intertube resonant frequency, the amplitude of forced vibration of a DWCNT is more than one order of magnitude smaller than those driven by the force of same magnitude with an exciting frequency much lower than the lowest intertube resonant frequency. This result suggests that the counter-phase intertube vibration stimulated at terahertz frequencies has a remarkable effect to suppress forced vibration of DWCNTs, and DWCNTs exhibit metamaterial-like vibration behavior in the terahertz range. Similar conclusion is expected for MWCNTs (multiwalled carbon nanotubes) and may provide useful insights into vibration controlling of MWCNTs at terahertz frequencies.

Investigation on mechanical vibration of double-walled carbon nanotubes with inter-tube Van der waals forces

This work represents the study of the vibration response of the double walled carbon nanotubes (DWCNT) for various boundary conditions. The inner and outer carbon nanotubes are modeled as two individual Euler-Bernoulli's elastic beams interacting each other by Van der waals force. Differential transform method (DTM) is used as a numerical method to solve the governing differential equations and associated boundary conditions. The influence of Winkler elastic medium on vibration frequency is also examined and results are interpreted. MATLAB is used as a tool for solving the governing differential equations. The fundamental natural frequencies are validating with those available in literature and observed a good agreement between them.

Timoshenko beam model for vibrational analysis of double-walled carbon nanotubes bridged on substrate

This paper proposes a continuum three-segment double Timoshenko beam (TSDTB) model to investigate free vibration of double-walled carbon nanotubes (DWCNTs) bridged on a silicon channel. Accurate explicit formulas of the van der Waals (vdW) interactions between each tube as well as between the tubes and a substrate are derived for a better prediction of the vibrational behaviors of the DWCNTs. An analytical modified Fourier series method (MFSM) is developed for the vibrational analysis of the TSDTB model. Numerical results show the good convergence characteristics and accuracy of the MSFM. The overlapped lengths between DWCNTs and the substrate at both ends have almost no effects on the lower natural frequencies of the TSDTB model when the overlapped lengths are long enough. Moreover, the natural frequencies of the TSDTB model are much less than of those of one-segment double Timoshenko beam (OSDTB) model with clamped supported boundary condition, whereas the TSDTB model can be equivalent to the simply supported OSDTB model under certain condition. In addition, the effects of vdW interaction coefficients on the vibrational behaviors of the DWCNTs are also revealed. The obtained results in this work should be greatly helpful in the design and application of CNTs-based nanomechanical resonators.

Characterizing the nonlinear behaviour of double walled carbon nanotube based nano mass sensor

This work deals with the evaluation of nonlinear behaviour of a curved double walled carbon nanotube (DWCNT) when used for mass sensing applications. The DWCNT is considered to be doubly clamped at a source and a drain. To judge the nonlinear behaviour, equations of motion have been derived using Euler beam theory and Hamilton principle, considering nonlinear van der Waals interaction nonlinear oscillations of a double walled carbon nanotube excited harmonically near its primary resonance are considered. The nonlinear free vibration of double-walled carbon nanotubes based on the elasticity theory is studied in this paper. Modelling of the weak van der Waals force of attraction between the inner and outer tubes is represented using a spring element. The equation of motion involves four nonlinear terms due to the curved geometry and the stretching of the central plane. The dynamic response of the double walled carbon nanotube based mass sensor is analyzed in the context of the time response, Poincare maps, and fast Fourier transformation diagrams. The results show the appearance of instability and chaos in the dynamic response as the mass on carbon nanotube is changed. The appearance of regions of periodic, subharmonic, and chaotic behavior is observed to be strongly dependent on mass, inner and outer tubes and the geometric imperfections of double walled carbon nanotube.

Free transverse vibration of double-walled carbon nanotubes embedded in viscoelastic medium

This paper aims to study the vibration characteristics of viscoelastic double-walled carbon nanotubes embedded in a viscoelastic medium. In doing this, the governing equations of the system are derived by combining the Euler–Bernoulli beam theory, nonlocal viscoelastic model and Kelvin viscoelastic foundation model. Subsequently, the transfer function method is employed to solve the governing equations, which enables one to obtain the natural frequencies and the corresponding mode shapes in closed form for the DWCNTs with arbitrary boundary conditions. Here, the developed mechanics model is first compared with the existing techniques available in the literature, where excellent agreement is achieved. Also, a detailed parametric study is conducted to examine the effect of boundary conditions, nonlocal parameter, relaxation time, slenderness ratio, stiffness coefficient and damping coefficient on the vibration response of the DWCNTs.

On the vibration of double-walled carbon nanotubes using molecular structural and cylindrical shell models

The vibrational behavior of double-walled carbon nanotubes is studied by the use of the molecular structural and cylindrical shell models. The spring elements are employed to model the van der Waals interaction. The effects of different parameters such as geometry, chirality, atomic structure and end constraint on the vibration of nanotubes are investigated. Besides, the results of two aforementioned approaches are compared. It is indicated that by increasing the nanotube side length and radius, the computationally efficient cylindrical shell model gives rational results.

Vibration of double-walled carbon nanotubes coupled by temperature-dependent medium under a moving nanoparticle with multi physical fields

ABSTRACTA numerical model on nonlinear vibration of double-walled carbon nanotubes (DWCNTs) subjected to a moving nanoparticle and multi physical fields is proposed. DWCNTs are considered with the kinematic assumption of Euler–Bernoulli beam theory. The surrounding elastic substrate is simulated as Pasternak foundation, which is assumed to be temperature-dependent. Hamilton's principle, incremental harmonic balanced method, Galerkin, and time integration method with direct iteration are employed to establish the equations of motion of zigzag DWCNTs. The study reveals that for the weak van der Waals forces, DWCNTs have the positive and the negative deflections as if it vibrates under a moving nanoparticle.

Influence of the nonlocal parameter on the transverse vibration of double-walled carbon nanotubes

AbstractA high-order nonlocal continuum beam model is proposed, which can be applied to study the transverse vibrations of double-walled carbon nanotubes (DWCNTs), including those that could have initial deformations due to defects or external actions. A beam element is developed adopting Hermite cubic polynomials as shape functions, and mass and elastic stiffness matrix are presented. The influence of the nonlocal parameter on the vibrational properties of DWCNTs is studied. Using the proposed model, it was found that the nonlocal parameter has a strong influence on the natural frequencies.

Vibration of Cantilevered Double-Walled Carbon Nanotubes Predicted by Timoshenko Beam Model and Molecular Dynamics

Vibration of double-walled carbon nanotubes (DWCNTs) with one end fixed and the other end free is studied by using different beam models of continuum mechanics and the molecular dynamics (MD) simulations. The models of the double-Euler beams (DEB) and the double-Timoshenko beams (DTB) of cantilevered case, with the van der Waals interaction between layers taken into consideration, are applied to predict the natural frequencies of DWCNTs. An analytical solution is first obtained for the DTB model with cantilevered boundary condition. The fundamental frequencies obtained by the DEB model and the DTB model are very close, for the relatively long DWCNTs. The MD simulations show that these two models can predict the natural frequencies well. However, the difference between the DEB model and the DTB model becomes obvious, for the vibration of the relatively short DWCNTs. The DTB model can offer a much better prediction than the DEB model when the DWCNT is very short especially for high-order frequencies.

A continuum shell model including van derWaals interaction for free vibrations of double-walled carbon nanotubes

This paper proposes the free vibration analysis of Double-Walled Carbon Nano Tubes (DWCNTs). A continuum elastic three-dimensional shell model is used for natural frequency investigation of simply supported DWCNTs. The 3D shell method is compared with beam analyses to show the applicability limits of 1D beam models. The effect of van der Waals interaction between the two cylinders is shown for different Carbon Nano Tube (CNT) lengths and vibration modes. Results give the van der Waals interaction effect in terms of frequency values. In order to apply the 3D shell continuum model, DWCNTs are defined as two concentric isotropic cylinders (with an equivalent thickness and Young modulus) which can be linked by means of the interlaminar continuity conditions or by means of an infinitesimal fictitious layer which represents the van der Waals interaction.

Nonlocal vibration and instability analysis of embedded DWCNT conveying fluid under magnetic field with slip conditions consideration

In this study, vibration of double-walled carbon nanotubes (DWCNTs) conveying fluid placed in uniform magnetic field is carried out based on nonlocal elasticity theory. DWCNT is embedded in Pasternak foundation and is simulated as a Timoshenko beam (TB) model which includes rotary inertia and transverse shear deformation in the formulation. Considering slip boundary conditions and van der Waals (vdW) forces between inner and the outer nanotubes, the governing equations of motion are discretized and differential quadrature method (DQM) is applied to obtain the frequency of DWCNTs for clamped–clamped boundary condition. The detailed parametric study is conducted, focusing on the remarkable effects of small scale, Knudsen number, elastic medium, magnetic field, density, and velocity of conveying fluid on the stability of DWCNT. Results indicate that considering slip boundary conditions has significant effect on stability of DWCNTs. Also, it is found that trend of figures have good agreement with the previous researches. Results of this investigation could be applied for optimum design of nano/micro mechanical devices for controlling stability of DWCNTs conveying fluid under magnetic fields.

A nonlocal meshless solution for flexural vibrations of double-walled carbon nanotubes

The true understanding of free vibration of double-walled carbon nanotubes (DWCNTs) plays a vital role in optimal design and dynamic control of such nanostructures. This paper is aimed to examine free flexural vibration of lengthy DWCNTs with arbitrary boundary conditions in the framework of nonlocal elasticity theory. The DWCNTs are embedded in an elastic medium and are subjected to initially axial forces. Equivalent continuum structures associated with the innermost and outermost tubes of the DWCNT are considered. The transverse and rotational interactions of the DWCNT with the surrounding elastic medium are also taken into account. The generalized equations of motion of lengthy DWCNTs are established based on the nonlocal Rayleigh beam theory. Seeking an analytical solution to the developed equations, particularly in their general form, is a very problematic task. As an alternative solution, an efficient numerical scheme is proposed. The effects of slenderness ratio, small-scale parameter, lateral and rotational stiffness of the surrounding matrix, and initially axial force on the first five natural frequencies of DWCNTs under different boundary conditions are comprehensively scrutinized.