Vibration Of Double-walled Carbon Nanotubes Research Articles

A nonlocal rod model for axial vibration of double-walled carbon nanotubes including axial van der Waals force effects

In this study, the axial vibration of double-walled carbon nanotubes (DWCNTs) is studied using nonlocal elasticity theory. The nonlocal constitutive equations of Eringen are used in the formulations. The van der Waals forces are considered in the axial direction.An interaction coefficient is obtained in terms of the previously proposed van der Waals coefficient and effective thickness of the single-walled carbon nanotubes. The effect of various parameters like geometry of nanotubes, van der Waals forces and nonlocal parameters on the axial vibration of DWCNTs is discussed. Some mode shapes are also depicted in order to show the relative motion of the nested tubes. The van der Waals forces have strong effects on the axial vibration characteristics of DWCNTs. The present results can be used in the continuum modeling of nanoscale linear motors and oscillators.

Nonlocal Flügge Shell Model for Vibrations of Double-Walled Carbon Nanotubes With Different Boundary Conditions

In this paper, the vibrational behavior of double-walled carbon nanotubes (DWCNTs) is studied by a nonlocal elastic shell model. The nonlocal continuum model accounting for the small scale effects encompasses its classical continuum counterpart as a particular case. Based upon the constitutive equations of nonlocal elasticity, the displacement field equations coupled by van der Waals forces are derived. The set of governing equations of motion are then numerically solved by a novel method emerged from incorporating the radial point interpolation approximation within the framework of the generalized differential quadrature method. The present analysis provides the possibility of considering different combinations of layerwise boundary conditions. The influences of small scale factor, layerwise boundary conditions and geometrical parameters on the mechanical behavior of DWCNTs are fully investigated. Explicit expressions for the nonlocal frequencies of DWCNTs with all edges simply supported are also analytically obtained by a nonlocal elastic beam model. Some new intertube resonant frequencies and the corresponding noncoaxial vibrational modes are identified due to incorporating circumferential modes into the shell model. A shift in noncoaxial mode numbers, not predictable by the beam model, is also observed when the radius of DWCNTs is varied. The results generated also provide valuable information concerning the applicability of the beam model and new noncoaxial modes affecting the physical properties of nested nanotubes.

Nonlinear nonlocal vibration of embedded DWCNT conveying fluid using shell model

In this work nonlinear vibration of double-walled carbon nanotube (DWCNT) embedded in an elastic medium and subjected to an axial fluid flow (incompressible and non-viscose) is investigated. The elastic medium is simulated using Pasternak foundation in which adjacent layer interactions are assumed to have been coupled by van der Waals (VdW) force. The higher-order equation of motion is derived using Hamilton's principle and nonlocal-nonlinear shell theory. Galerkin and averaging methods are adopted to solve the higher-order governing equations. Elastic medium, small scale parameter, velocity and fluid density are taken into account to calculate the effects of axial and circumferential wave numbers in this study. Results reveal that increasing circumferential wave number, leads to enhanced nonlinearity. Critical flow velocities of DWCNT are inversely related to the non-local parameter (e0a), so that increase in the later lead to reduced critical flow velocities.

Thermal vibration of double-walled carbon nanotubes predicted via double-Euler-beam model and molecular dynamics

The paper presents how to study the thermal vibration of a double-walled carbon nanotube (DWCNT) by using a model of double-Euler beams, together with the law of energy equipartition, with the energy of van der Waals interaction between layers taken into consideration. The basic finding of the study is the relation, derived via the model of double-Euler beams and the law of energy equipartition, between the temperature and the root-of-mean-squared (RMS) amplitude of the thermal vibration at any cross section of the DWCNT. The molecular dynamics simulations of thermal vibration of the DWCNT in argon atmosphere show that the model of double-Euler beams can predict the RMS amplitude of the thermal vibration of the DWCNT reasonably well.

Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory

The nonlinear free vibration of double-walled carbon nanotubes based on the nonlocal elasticity theory is studied in this paper. The nonlinear equations of motion of the double-walled carbon nanotubes are derived by using Euler beam theory and Hamilton principle, with considering the von Kármán type geometric nonlinearity and the nonlinear van der Waals forces. The surrounding elastic medium is formulated as the Winkler model. The harmonic balance method and Davidon–Fletcher–Powell method are utilized for the analysis and simulation of the nonlinear vibration. The simulation results show that the nonlocal parameter, aspect ratio and surrounding elastic medium play more important roles in the nonlinear noncoaxial vibration than those in the coaxial vibration of the double-walled carbon nanotubes. The noncoaxial vibration amplitudes of only considering nonlinear van der Waals forces are larger than those of considering both geometric nonlinearity and nonlinear van der Waals forces.

Vibration of double-walled carbon nanotube based nanomechanical sensor with initial axial stress

This paper studies free vibration of double-walled carbon nanotubes (DWCNTs) based nanomechanical sensor under initial axial stress. A bridged DWCNT carrying a nanoparticle at any position of the outer tube is modeled as two clamped nonlocal Euler–Bernoulli beams, and the interaction between two tubes is governed by van der Waals force. For comparison, a bridged single-walled carbon nanotube (SWCNT) based nanomechanical sensor is considered in a similar way. Using the transfer function method, the critical buckling stress and natural frequencies of these nanomechanical sensors are computed. Under small initial stress, the effects of the attached nanoparticle, and the small scale parameter on the frequency shift are discussed. The obtained results show that when the mass of the nanoparticle increases or its location is closed to the beam center, the natural frequency decreases, but frequency shift increases. Initial tensile stress increases the natural frequency, while initial compressive stress decreases the natural frequency. In comparison with SWCNT sensor, DWCNT is more stable, but less sensitive for smaller attached mass. Obtained results are helpful to the design of DWCNT-based resonator as nanomechanical sensor.

Effects of Initial Axial Stress on the Vibration of Double-Walled Carbon Nanotubes under Temperature Field

The paper studies the linear vibrating mechanical behaviors of double-walled carbon nanotubes (DWCNTs) with the initial axial stress under the temperature field. The DWCNTs are modeled as double elastic shells coupled together through vdW interaction between inner and outer nanotubes. Based on the model, the relation between the amplitudes and the frequencies of the tubes is achieved. The results show that the initial axial stress could crucially affect the existence of the natural and intertube frequencies. Meanwhile, it is also concluded that the temperature change is significant for natural frequency, but weakly affects intertube frequency and amplitude ratios.

Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics

In the present study, the free vibration response of double-walled carbon nanotubes (DWCNTs) is investigated. Eringen's nonlocal elasticity equations are incorporated into the classical Donnell shell theory accounting for small scale effects. The Rayleigh–Ritz technique is applied to consider different sets of boundary conditions. The displacements are represented as functions of polynomial series to implement the Rayleigh–Ritz method to the governing differential equations of nonlocal shell model and obtain the natural frequencies of DWCNTs relevant to different values of nonlocal parameter and aspect ratio. To extract the proper values of nonlocal parameter, molecular dynamics (MD) simulations are employed for various armchair and zigzag DWCNTs, the results of which are matched with those of nonlocal continuum model through a nonlinear least square fitting procedure. It is found that the present nonlocal elastic shell model with its appropriate values of nonlocal parameter has the capability to predict the free vibration behavior of DWCNTs, which is comparable with the results of MD simulations.

Radial breathing vibration of double-walled carbon nanotubes subjected to pressure

A theoretical vibrational analysis of the radial breathing mode (RBM) of double-walled carbon nanotubes (DWCNTs) subjected to pressure is presented based on an elastic continuum model. The results agree with reported experimental results obtained under different conditions. Frequencies of the RBM in DWCNTs subjected to increasing pressure depend strongly on circumferential wave numbers, but weakly on the aspect ratio and axial half-wave numbers. For the inner and outer tubes of DWCNTs, the frequency of the RBM increases obviously as the pressure increases under different conditions. The range of variation is smaller for the inner tube than the outer tube.

Effects of Initial Axial Stress on Nonlinear Vibration of Double-Walled Carbon Nanotubes Under Temperature Field

The paper studies the nonlinear vibrating mechanical behaviors of double-walled carbon nanotubes (DWCNTs) with the initial axial stress under the temperature field. The DWCNTs are modeled as double elastic shells coupled together through vdW interaction between inner and outer nanotubes. Based on the model, the relation between the amplitudes and the frequencies of the tubes is achieved. The results show that the initial axial stress could crucially affect the topological forms of the amplitude-frequency curve, whereas it is insensitive to the amplitude ratios in DWCNTs. Meanwhile, it is also concluded that the temperature change is significant for natural frequency, but weakly affects intertube frequency and amplitude ratios as well as amplitude-frequency curve.

Thermal effect on wave propagation in double-walled carbon nanotubes embedded in a polymer matrix using nonlocal elasticity

The thermal effect on wave propagation in double-walled carbon nanotubes (DWNTs) embedded in a polymer matrix is studied in this paper via nonlocal elasticity. The small-scale effects on vibration characteristics of carbon nanotubes are explicitly derived through a complete continuum beam model. In example calculations, the mechanical properties of carbon nanotubes and polymer matrix are treated as the functions of temperature change. The research work reveals the significance of the small-scale effect on wave propagation in DWNTs. It is also demonstrated that some properties of transverse vibrations of DWNTs are dependent on the change of temperature.

Vibrational analysis of double-walled carbon nanotubes with inner and outer nanotubes of different lengths

In this study, the Euler–Bernoulli beam model is used to analyze the resonant vibration of double-walled carbon nanotubes (DWCNTs) with inner and outer nanotubes of different lengths. The resonant properties of DWCNTs with different inner and outer nanotube lengths are investigated in detail using this theoretical approach. The resonant vibration is significantly affected by the vibrational modes of the DWCNTs, and by the lengths of the inner and outer nanotubes. For an inner or outer nanotube of constant length, the vibrational frequencies of the DWCNTs increase initially and then decrease as the length of another nanotube increases. A design for nanoelectromechanical devices that operate at various frequencies can be realized by controlling the length of the inner and outer nanotubes of DWCNTs. This investigation may be helpful in applications of carbon nanotubes such as high frequency oscillators, dynamic mechanical analysis and mechanical sensors.

Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part I: theoretical formulations

The current work suggests mathematical models for the vibration of double-walled carbon nanotubes (DWCNTs) subjected to a moving nanoparticle by using nonlocal classical and shear deformable beam theories. The van der Waals interaction forces between atoms of the innermost and outermost tubes are modeled by an elastic layer. The equations of motion are derived for the nonlocal double body Euler–Bernoulli, Timoshenko and higher-order beams connected by a flexible layer under excitation of a moving nanoparticle. Analytical solutions of the problem are provided for the aforementioned nonlocal beam models with simply supported boundary conditions. The dynamical deflections and nonlocal bending moments of the innermost and outermost tubes are then obtained during the courses of excitation and free vibration. Finally, the critical velocities of the moving nanoparticle associated with the nonlocal beam theories are expressed in terms of small-scale effect parameter, geometry, and material properties of DWCNTs.

Free vibration characteristics of double-walled carbon nanotubes embedded in an elastic medium

In this Letter, a theoretical analysis of the resonant vibration of double-walled carbon nanotubes (DWCNTs) and the DWCNTs embedded in an elastic medium is presented based on Euler–Bernoulli beam model and Winkler spring model. The vibration modes of DWCNTs are quite different from those of single-walled carbon nanotubes (SWCNTs). The resonant vibrations of DWCNTs are found to have in-phase and anti-phase modes, in which the deflections of the inner and outer nanotubes occur in the same and opposite directions, respectively. For the vibration of DWCNTs with the same harmonic numbers, the resonant frequencies of anti-phase mode are larger than the ones of in-phase mode. Moreover, influence of the surrounding medium on the resonant vibrations is investigated using the Winkler spring model. The results show that surrounding medium makes a strong impact on the vibration frequencies of in-phase mode, but little on those of anti-phase mode.

Some Modern Problems in Structural Engineering Dynamics

This review paper deals with two problems in structural engineering dynamics; one is deterministic, the other is of stochastic nature. One problem is linear, the other is nonlinear. Authors have a biased preferential view on these problems because of their active involvement in the discussed research topics. Still, these two problems reflect, at least in a small manner, some developments in this vast and fascinating field. The first part deals with deterministic linear vibrations of double-walled carbon nanotubes either in classical or refined setting; the second part is devoted to the nonlinear random vibrations of structures. 1. Fundamental natural frequencies of double-walled carbon nanotubes Nanotechnology became recently a very active field of research. This section deals with evaluation of fundamental natural frequencies of double-walled carbon nanotubes under various boundary conditions. The Bubnov-Galerkin and Petrov-Galerkin methods are applied to derive the expressions for natural frequencies. Apparently for the first time in the literature explicit expressions are obtained for the natural frequencies. These can be useful for the designer to estimate the fundamental frequency in each of two series.

Some Modern Problems in Structural Engineering Dynamics

This review paper deals with two problems in structural engineering dynamics; one is deterministic, the other is of stochastic nature. One problem is linear, the other is nonlinear. Authors have a biased preferential view on these problems because of their active involvement in the discussed research topics. Still, these two problems reflect, at least in a small manner, some developments in this vast and fascinating field. The first part deals with deterministic linear vibrations of double-walled carbon nanotubes either in classical or refined setting; the second part is devoted to the nonlinear random vibrations of structures.

Free flexural vibration studies of double-walled carbon nanotubes with different boundary conditions and modeled as nonlocal Euler beams via the Galerkin method

In this work, a numerical Galerkin approach which can handle different boundary conditions in a general manner is presented to study the free vibration of double-walled carbon nanotubes (DWCNTs) modeled as nonlocal Euler beams. The effects of small scale, different boundary conditions, and geometrical parameters on the vibration of DWCNTs are examined in detail. It is also observed from the numerical results that the higher modes of DWCNT vibration (where the inner and outer nanotubes vibrate out of phase) are dominated by van der Waals interaction between the inner and outer nanotubes, and small scale effect and boundary conditions are noted to have minimal consequence on the first higher natural frequency mode of a DWCNT.

Analysis of nonlinear vibrations of double-walled carbon nanotubes conveying fluid

Abstract This paper investigates the effect of the geometric nonlinearity and the nonlinearity of van der Waals (vdW) force on the transverse vibration of the double-walled carbon nanotubes conveying fluid and the interaction between two types of nonlinearities. By using the Hamilton’s principle, the nonlinear governing equations of the double-walled carbon nanotubes conveying fluid are deduced. The effects of two types of nonlinearities on the coaxial and noncoaxial vibrations of the double-walled carbon nanotubes conveying fluid are discussed in numerical examples. The results show that the effect of geometric nonlinearity on the amplitude–frequency properties can be neglected if two types of nonlinearities are simultaneously considered. Compared with the uncoupling, the coupling between the longitudinal and transverse vibrations has little effect on the amplitude–frequency properties with considering two types of nonlinearities simultaneously. However, the coupling has significant effect on the amplitude–frequency properties with only considering the geometric nonlinearity.

Vibrations of Double-Walled Carbon Nanotubes With Different Boundary Conditions Between Inner and Outer Tubes

Free vibrations of a double-walled carbon nanotube (DWNT) are studied. The inner and outer carbon nanotubes are modeled as two individual elastic beams interacting each other by van der Waals forces. An original method is proposed to calculate the first seven order resonant frequencies and relative vibrational modes. Detailed results are demonstrated for DWNTs according to the different boundary conditions between inner and outer tubes, such as fixed-free, cantilever-free, fixed-simple and fixed-fixed (reduced form) supported ends. Our results indicate that there is a special invariable frequency for a DWNT that is not affected by different combinations of boundary conditions. All vibrational modes of the DWNT must be coaxial when the resonant frequency is smaller than this frequency. Some noncoaxial vibrations will occur when their resonant frequencies exceed the frequency. Especially, the first noncoaxial resonant frequency is still invariable for all different boundary conditions. A change of resonant frequency for various lengths of DWNTs is discussed in detail. In addition, our model predicts a new coaxial-noncoaxial vibrational mode in fixed-simple supports for inner and outer tubes of a DWNT.

Effects of shear deformation on vibration of doublewalled carbon nanotubes embedded in an elastic medium

In this study, free vibration of simply supported multi-walled carbon nanotubes (CNTs) embedded in an elastic medium was investigated by using the generalized shear deformation-beam theory (GSDBT). The effects of surrounding elastic medium, which is considered as a spring, defined by the Winkler model, and van der Waals forces from adjacent nanotubes are taken into account. Third-order shear deformation (TOSD) theory is used to study free vibration of a multi-walled carbon nanotube embedded in an elastic medium. Unlike Timoshenko beam theory, TOSD 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 for various sides to thickness ratios and elastic medium effects and results are compared with previous studies. The results showed that significant difference exist between TOSD and Euler beam theory. It is also interesting to note that, although frequency parameter is increasing by increasing stiffness of embedded medium, amplitude ratios are insensitive to stiffness of embedded elastic medium.