Vibration Of Multi-walled Carbon Nanotubes Research Articles

Nonlinear finite element analysis of vibration of multi-walled carbon nanotubes with geometric imperfection resting on elastic foundations in a thermal–magnetic environment

This paper presents nonlinear finite element analysis of the systems of partial differential equations (PDEs) that govern the vibration dynamics and stability responses of slightly curved Multi-walled carbon nanotubes (MWCNTs) by means of nonlocal elasticity principle. The MWCNTs under investigation rest on Pasternak and Winkler foundations and operate in a magneto–thermal environment. With the aid of the nonlinear finite element analysis (FEA) of the developed governing equations, the effects of primary geometric imperfection, nonlocal parameter, magnetic field, temperature, elastic foundation parameters, number of walls, and boundary conditions on the nonlinear vibration of the CNTs are investigated. The simulated and presented results show how frequency ratio reduces with an increase in the number of elastic foundation parameters, strength of magnetic field term, temperature, tube wall numbers and ratio of curvature radius to length of the slightly curved nanotubes under investigation. Similar trend of results is recorded for the different boundary conditions considered. However, frequency ratio is highest for MWCNTs with clamped-pinned boundary conditions and lowest for the case of clamped supports at both ends. Furthermore, it is established that the vibration of quadruple MWCNTs approximates that of linear vibration even as the foundations parameters are augmented. Such result can be used to restrains chaos in vibration of CNTs. It is hoped that the results from this study will help in the design of MWCNTs for various applications.

Nonlinear Vibration of Multi-Walled Carbon Nanotubes with Initial Curvature Resting on Elastic Foundations in a Nonlinear Thermo-Magnetic Environment

The present work focuses on nonlinear dynamics models of multi-walled carbon nanotubes with initial curvature resting on Winkler-Pasternak elastic foundations in a nonlinear thermomagnetic environment using nonlocal elasticity theory. The derived systems of nonlinear vibration models are solved with the aid of the Galerkin decomposition and the homotopy perturbation method. Effects of temperature, magnetic field, multi-layer, and other thermomechanical parameters on the dynamic responses of the slightly curved multi-walled carbon nanotubes are investigated and discussed. As the temperature increases, the frequency ratio decreases as the linear natural frequency of the system increases. The results reveal that the frequency ratios decrease as the number of nanotube walls, temperature, spring constants, magnetic field strength, and the ratio of the radius of curvature to the length of the slightly curved nanotubes increase. These trends are the same for all the boundary conditions considered. However, clamped-simple and clamped-clamped supported multi-walled nanotube have the highest and lowest frequency ratio, respectively. Also, from the parametric studies to control nonlinear vibration of the carbon nanotubes, it is shown that quadruple-walled carbon nanotubes can be taken as pure linear vibration even at any value of linear Winkler and Pasternak constants. Therefore, this can be used for the restraining of the chaos vibration in the objective structure. These research findings will assist the designers and manufacturers in developing multi-walled carbon nanotubes for various structural, electrical, mechanical, and biological applications, especially in the areas of designing nanoelectronics, nanodevices, nanomechanical systems, nanobiological devices, and nanocomposites, and particularly when they are subjected to thermal loads, magnetic fields and elastic foundations.

Nonlocal Anisotropic Shell Model of Linear Vibrations of Multi-walled Carbon Nanotubes

Nonlocal Anisotropic Shell Model of Linear Vibrations of Multi-walled Carbon Nanotubes

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.

Variational principles for buckling and vibration of MWCNTs modeled by strain gradient theory

Variational principles for the buckling and vibration of multi-walled carbon nanotubes (MWCNTs) are established with the aid of the semi-inverse method. They are used to derive the natural and geometric boundary conditions coupled by small scale parameters. Hamilton’s principle and Rayleigh’s quotient for the buckling and vibration of the MWCNTs are given. The Rayleigh-Ritz method is used to study the buckling and vibration of the single-walled carbon nanotubes (SWCNTs) and double-walled carbon nanotubes (DWCNTs) with three typical boundary conditions. The numerical results reveal that the small scale parameter, aspect ratio, and boundary conditions have a profound effect on the buckling and vibration of the SWCNTs and DWCNTs.

Analytical Study for the Nonlinear Vibrations of Multiwalled Carbon Nanotubes using Homotopy Analysis Method

In this article, the homotopy analysis method (HAM) is implemented for obtaining semi-analytical solutions to the problem of the nonlinear vibrations of multiwalled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the Van der Waals inter layer forces. The amplitude-frequency curves for large-amplitude vibrations of single-walled, double-walled and triple-walled carbon nanotubes are obtained. The influence of changes in material constants of the surr ounding elastic medium and the effect of changes in nanotube geometrical parameters on the vibration characteristics are studied by co mparing the results with those from the previous work. Series solutions of the problem under consideration are developed by means of HAM and the recurrence relations are given explicitly. The obtained numerical results show the rapid convergence of the series constructed by the proposed method to the exact solution. Test problems have been considered to ensure that HAM is accurate and effi cient compared with the Adomian decomposition method.

On the influence of the shear deformation and boundary conditions on the transverse vibration of multi-walled carbon nanotubes

The vibrational properties of multi-walled carbon nanotubes (MWCNTs) are studied, especially the influence of the shear deformation and the boundary conditions on the non-coaxial frequencies. A high order continuum beam model is proposed, which can be applied to study the transverse vibrations of MWCNTs, including those that could have initial deformations due to defects or external actions. Using the model it was found that the non-coaxial intertube frequencies are independent of the shear deformation and the boundary conditions. This result is important considering that the concentric structure of MWCNTs is a crucial geometrical characteristic and a non-coaxial vibration would affect their electronic and optical properties.

Rigorous van der Waals effect on vibration characteristics of multi-walled carbon nanotubes under a transverse magnetic field

An analytical method is presented to investigate rigorous van der Waals interaction effect on vibration characteristics of multi-walled carbon nanotubes embedded in matrix under a transverse magnetic field. Each of the concentric tubes of multiwall carbon nanotubes is considered as an individual elastic shell and coupled with any two walls through a rigorous van der Waals interaction being dependent on the change of interlayer spacing and the radii of tubes. Results show that the rigorous van der Waals interaction effect makes the lowest magneto-vibration frequency of multi-walled carbon nanotubes decrease and the highest magneto-vibration frequency increase. The effect of rigorous van der Waals interaction on magneto-elastic vibrations of multi-walled carbon nanotubes is dependent on the transverse magnetic strength and the matrix constrained stiffness.

Nonlinear vibrations of embedded multi-walled carbon nanotubes using a variational approach

The present work deals with the problem of the nonlinear vibrations of multi-walled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals interlayer force. The variational iteration method (VIM) is adopted to obtain the amplitude–frequency curves for large-amplitude vibrations of single-, double- and triple-walled carbon nanotubes. The influences of changes in material constants of the surrounding elastic medium and the geometric parameters on the vibration characteristics of multi-walled carbon nanotubes are investigated. The results from the VIM solution are compared and shown to be in excellent agreement with the available solutions from the open literature. The capability of the present analytical technique is clarified in terms of numerical accuracy as well as computational efficiency.

Effects of dimensional parameters and various boundary conditions on axisymmetric vibrations of multi-walled carbon nanotubes using a continuum model

Axisymmetric vibrations of multi-walled carbon nanotubes (MWCNTs) with finite length are investigated in this paper. A multi-walled carbon nanotube is modeled with multiple elastic isotropic shells. Based on a continuum model and considering van der Waals forces between tubes, a two-dimensional finite element model is developed to obtain axisymmetric natural frequencies and mode shapes of MWCNTs. First, the axisymmetric vibrational characteristics of single-walled carbon nanotubes are investigated, and then, they are compared with those of MWCNTs. The effects of van der Waals forces on radial vibration of MWCNTs are also explained. Moreover, influences of end conditions on radial and axial natural frequencies of carbon nanotubes (CNTs) with wide range of length-to-thickness ratio are studied by considering the free-free (F-F) and simply supported (S-S) end conditions. Besides, it is focused on dependence of axisymmetric mode frequencies on dimensional parameters such as length, diameter, as well as number of layers of MWCNTs. Through this, explicit expressions are found for calculating the radial breathing mode and longitudinal mode frequencies of MWCNTs.

Approximate solutions to the nonlinear vibrations of multiwalled carbon nanotubes using Adomian decomposition method

This paper applies the Adomian decomposition method (ADM) to the search for the approximate solutions to the problem of the nonlinear vibrations of multiwalled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals inter layer forces. The amplitude–frequency curves for large-amplitude vibrations of single-walled, double-walled and triple-walled carbon nanotubes are obtained. The influence of changes in material constants of the surrounding elastic medium and the effect of changes in nanotube geometrical parameters on the vibration characteristics are studied by comparing the results with those from the open literature. This method needs less work in comparison with the traditional methods and decreases considerable volume of calculation, and it’s powerful mathematical tool for solving wide class of nonlinear differential equations. Special attention is given to prove the convergence of the method. Some examples are given to illustrate the determination approximate solutions of the proposed problem.

Low frequency vibration of multiwall carbon nanotubes with heterogeneous boundaries

Heterogeneous end constraints are imposed on multiwall carbon nanotubes (MWCNTs) by sequentially clamping one end of their originally simply supported constituent tubes. The finite element method is employed to study the vibration of such MWCNTs with an emphasis on the effect of the mixed boundary conditions. The results show that the clamping process constantly enhances the dynamic stiffness of MWCNTs, which leads to substantial frequency increase up to 50% and, in some cases, the transformation of the fundamental vibration mode. In particular, the vibration frequency is always found to be most sensitive to fixing the outermost tubes, showing the critical role of this individual tube in determining the structural stiffness of the whole MWCNTs as a coupled system.

Vibration of multi-walled carbon nanotubes with initial axial force and radial pressure

A study is performed for the vibrational characteristics of multi-walled carbon nanotubes (MWCNTs) with initial axial force and radial pressure based on the Donnell model. In particular, explicit expressions are obtained for double-walled carbon nanotubes. Numerical results show that the intertube frequencies are insensitive to the initial axial stress (axial tensile stress or axial compressive stress), internal pressure or external pressure in MWCNTs, while the natural frequency is sensitive to the initial axial stress, internal pressure or external pressure. Further, the natural frequency increases with increasing axial tensile stress and decreases with increasing axial compressive stress. The frequencies increase with increasing internal pressure and decrease with increasing external pressure. The associated modes are insensitive to the initial axial stress, internal pressure or external pressure in MWCNTs.

Application of HPM to the nonlinear vibrations of multiwalled carbon nanotubes

This work deals with applying the homotopy perturbation method to the problem of the nonlinear vibrations of multiwalled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals interlayer forces. The amplitude–frequency curves for large-amplitude vibrations of single-walled, double-walled, and triple-walled carbon nanotubes are obtained. The influence of changes in material constants of the surrounding elastic medium on the vibration characteristics of multiwalled carbon nanotubes is discussed. The comparison of the generated results with those from the open literature illustrates that the solutions obtained are of very high accuracy and clarifies the capability and the simplicity of this method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

DYNAMICAL PROPERTIES OF MULTI-WALLED CARBON NANOTUBES BASED ON A NONLOCAL ELASTICITY MODEL

This paper investigates the free vibration of multi-walled carbon nanotubes (MWCNTs) with simply supported ends. Based on the nonlocal elasticity theory which allows the effects of small length scale and the more refined van der Waals (vdW) interaction formulas, the equation of motion is first derived and then solved analytically. The results reveal that the effects of the small length scale are significant for small aspect ratios and high radial vibration modes, and are instead insensitive to the number of layers of MWCNTs and weakly dependent on the wall thickness of MWNTs. This finding means that the effects of small length scale on complicated MWCNTs may be simplified to double-walled carbon nanotubes (DWCNTs) or even single-walled carbon nanotubes (SWCNTs).

SMALL SCALE EFFECT ON THE FREE VIBRATION OF MULTI-WALLED CARBON NANOTUBES

This paper is concerned with the free vibration of multi-walled carbon nanotubes (MWCNTs) with simply supported ends. Based on the non-local elasticity theory, Timoshenko beam model with the small scale effect and the van der Waals (vdW) interaction is derived and then solved analytically. The results reveal that the small scale effect is quite significant for small aspect ratios, large scale parameters and high radial vibration modes, whereas it is insensitive to the number of layers of MWCNTs and is weakly-dependent on the wall thickness of MWCNTs.

Vibration of multi-walled carbon nanotubes by generalized shear deformation theory

In this study, free vibration of simply supported multi-walled carbon nanotubes (CNTs) was investigated by using the generalized shear deformation-beam theory (GSDBT). Parabolic shear deformation theory (PSDT) is used in the specific solutions. Unlike Timoshenko beam theory present theory satisfies zero traction boundary conditions on the upper and lower surface of the structures so there is no need to use a shear correction factor. Free vibration frequencies and amplitude ratios were obtained and results are compared with previous studies. Results showed that significant difference exist between PSDT and Euler beam theory. Present results are slightly higher than the results of Timoshenko beam theory. Shear deformation effects are important especially for higher modes. It is obtained that van der Waals (vdW) forces should be considered for small inner radius.

Vibration of multi-walled carbon nanotubes with initial axial loading

This paper studies the vibrational characteristics of multi-walled carbon nanotubes (MWNTs) with initial axial loading based on Donnell equations. Both the effect of initial axial stress and the effect of the van der Waals forces are discussed in detail. It is shown that the resonant frequencies is related to the tension or compression forms of initial axial stress, which increase with the increase of axial tensile stress, and decrease with the increase of axial compressive stress. Further, the intertube resonant frequencies are insensitive to the initial axial stress in MWNTs, while the natural resonant frequency is sensitive to the initial axial stress, especially for the lower ones. However, the associated vibrational modes are independent of the initial axial stress. In particular, for double-walled carbon nanotubes (DWNTs), the natural resonant frequency of free vibration or vibration with initial axial stress is always between the corresponding one of the isolated inner tube and that of the isolated outer tube. In addition, some of the results obtained are compared with previous results.

The effect of van der Waals interaction modeling on the vibration characteristics of multiwalled carbon nanotubes

An elastic multiple shell model is used for the vibration analysis of multiwalled carbon nanotubes (MWCNTs). The van der Waals (vdW) interaction between any two layers of the MWCNT is modeled as the radius-dependent function. Based on the simplified Donnell shell equations, explicit formulas are obtained for the radial-dominated natural frequencies and mode shapes of double- and triple-walled carbon nanotubes. The natural frequencies are calculated for MWCNT with various radii and number of tubes. The numerical results show that the effect of vdW interaction on the torsionally and longitudinally dominated natural frequencies is very small and can be neglected, and the vdW interaction has only a small influence on the lowest radial-dominated natural frequency, but plays a significant role in the higher radial-dominated natural frequencies for various combinations of m (number of longitudinal) and n (number of circumferential) waves in the mode even for the MWCNTs of small innermost radius. Especially, due to the effect of vdW interaction, there exists a change over in the order of the modes, from radially dominated mode to longitudinally dominated mode, or from longitudinally dominated mode to torsionally dominated mode even when the innermost radius of the MWCNT is very small, indicating that the interlayer vdW interaction plays a significant role in the vibration of MWCNTs with small radius. The effect of the mode order on the natural frequencies of MWCNT is also examined in detail.

Radial breathing vibration of multiwall carbon nanotubes based on a rigorous van der Waals interaction

This paper reports the results of an investigation on radial breathing vibrations of multiwall carbon nanotubes (MWNTs) embedded in an elastic matrix, based on a rigorous van der Waals (vdW) interaction which is dependent on the change of interlayer spacing and the radii of tubes. Each of the concentric tubes of MWNTs is considered an individual elastic shell and coupled with any two tubes through a rigorous vdW interaction force. The matrix surrounding MWNTs is considered a spring in which the spring coefficient is defined by the Winkler model. The results obtained show that the radial breathing frequencies of the first vibration mode increase monotonically as the stiffness of the matrix increases. The matrix surrounding MWNTs not only changes the value of amplitude ratios between any two tube walls of the MWNTs embedded in an elastic matrix, but also changes the vibration modes between the tube walls of the MWNTs from the coaxial vibration to the non-coaxial vibration. The new features and the meaningful numerical results in the present work on radial breathing vibrations of MWNTs embedded in an elastic matrix may be used as a useful reference for Raman scattering of light that can indirectly provide structural data on nanotubes.