Natural Frequencies Of Carbon Nanotubes Research Articles

Small size and rotary inertia effects on the natural frequencies of carbon nanotubes

Abstract In the present paper, rotary inertia and small size effects on the free vibration response of single-walled carbon nanotubes are examined. The equations in motion and associated boundary conditions are obtained by using Hamilton’s principle on the basis of modified couple stress and Rayleigh beam theories. The size effect is taken into account by modified couple stress theory while the rotary inertia effect is considered by Rayleigh beam theory. The resulting equations are analytically solved by implementing Navier’s solution technique for pinned-pinned carbon nanotubes. Influences of slenderness ratio, length scale parameter and rotary inertia on the natural frequencies of single-walled carbon nanotubes are studied in detail.

Study on the effect of small scale on the wave reflection in carbon nanotubes using nonlocal Timoshenko beam theory and wave propagation approach

Abstract This paper presents the wave propagation approach for analyzing the free vibration and wave reflection in carbon nanotubes. The propagation and reflection matrices obtained by the wave propagation method are useful for the analysis of mechanical energy transmission and reflection in micro/nano devices. These matrices can also be used for obtaining the natural frequencies of carbon nanotubes. Firstly, the obtained natural frequencies by this method are compared with the results in the literature. Then, the effect of small scale on the reflected power of an incident wave upon different boundary conditions is studied in details. It was found that the reflected power of an incident wave upon a simply supported boundary is independent of the small scale, incident wave frequency, material and geometrical parameters of the carbon nanotubes while the reflected power of a wave upon clamped and free boundary conditions depends on the small scale, incident wave frequency, material and geometrical parameters.

Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models

Recently, there have been an increasing number of studies on nonlocal theoretical models, of which different kinds of nonlocal elasticity approaches including softening and hardening models have been investigated extensively. In the present work, the longitudinal dynamic behaviors of some common one-dimensional nanostructures (e.g. nanorods/nanotubes) were examined using the hardening nonlocal approach. The hardening nonlocal longitudinal stress was employed and an eigenvalue analysis method was utilized in deriving some dynamic characteristics of the minute structures at nanoscale. The effects of a dimensionless nonlocal small scale parameter at molecular level unavailable in classical rods/tubes were investigated. Subsequently, nonlocal longitudinal vibration responses under various boundary conditions including soft and hard constraints were determined through a numerical method. The correlations between natural frequencies and the nonlocal nanoscale parameter were obtained and discussed in detail. Within the framework of hardening nonlocal stress theory, it concludes for the first time that the longitudinal free vibration frequencies of nanorods/nanotubes are higher than those based on the classical continuum mechanics but they are quite different from the softening nonlocal model. The strengthening effects on nonlocal stiffness of nanorods/nanotubes are observed and they are consistent with some other theoretical approaches or atomistic simulations. In particular, a case study on nonlocal natural frequencies of carbon nanotubes (CNTs) was presented and the results were calculated by atomistic simulations, classical continuum theory, softening and hardening nonlocal models, respectively. The validity of both existing softening and hardening nonlocal models was confirmed in comparison with the atomistic simulations. In addition, a comparative calculation for dimensional natural frequencies with respect to length of CNTs by different methodologies was provided to explain why the softening and hardening nonlocal models are both correct in nonlocal elasticity theory.

Influence of Thermal and Magnetic Field on Vibration of Double Walled Carbon Nanotubes Using Nonlocal Timoshenko Beam Theory

In this paper, the influence of thermal and magnetic field on transverse vibration of double walled carbon nanotubes are investigated using nonlocal Timoshenko beam theory. The governing equation of motion with small scale effects are formulated by considering thermal and Lorentz magnetic force. The dispersion relation is derived for carbon nanotubes with simply supported ends and the numerical simulations are done using MATLAB. Dispersion curves are drawn for both double and single walled carbon nanotubes for different mode and magnetic field strength. The numerical result shows that the strength of magnetic field increases the natural frequencies and the nonlocal parameter decreases the natural frequencies of carbon nanotubes. Further to discuss the accuracy of the present result, the dispersion curves of single walled carbon nanotubes obtained by omitting the magnetic field are compared with the existing literature.

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.

Nonlocal continuum theories of beams for the analysis of carbon nanotubes

The equations of motion of the Euler–Bernoulli and Timoshenko beam theories are reformulated using the nonlocal differential constitutive relations of Eringen [International Journal of Engineering Science 10, 1–16 (1972)]. The equations of motion are then used to evaluate the static bending, vibration, and buckling responses of beams with various boundary conditions. Numerical results are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies of carbon nanotubes.

Torsional structural properties of single-walled carbon nanotubes

In this paper, a dynamic continuum modelling method is developed based on the molecular mechanics as well as on the energy equivalence principle for the single-walled carbon nanotubes subject to torsional vibration. The dynamic continuum modelling method enables one to evaluate torsional structural properties without assuming any fictitious wall thickness of single-walled carbon nanotubes. The present dynamic continuum modelling method is then applied to armchair single-walled carbon nanotubes to evaluate torsional rigidities and mass inertia properties. By using the evaluated torsional structural properties, the natural frequencies of carbon nanotubes are also investigated.