Rapid CommunicationFundamental natural frequencies of double-walled carbon nanotubes
Introduction
As Qian et al. [1] mention that “the discovery of multi-walled carbon nanotubes (MWCNTs) in 1991 has stimulated ever-broader research activities in science and engineering devoted entirely to carbon nanostructures and their applications. This is due in large part to the combination of their expected structural perfection, small size, low density, high stiffness, high strength (the tensile strength of the outer most shell of MWCNT is approximately 100 time greater that that of aluminum), and excellent electronic properties. As a result, carbon nanotubes (CNT) may find use in a wide range of application in material reinforcement, field emission pane display, chemical sensing, drug delivery, and nanoelectronics.” The state of the art in modeling and simulation of nanostructured materials and systems was given by Gates and Hinkley [2] and Liu et al. [3] inter alia. Vibrations of double-walled carbon nanotubes (DWCNTs) have been considered in several papers. Xu et al. [4], [5] and Ru [6] studied the free vibrations of a DWCNT which composed of two coaxial single-walled CNT interacting each other by the interlayer van der Waals forces. Therefore, the inner and outer CNT are modeled as two individual elastic beams [4], [5], [6]. In these studies the Euler-beam model has been used to derive exact solution for the natural frequencies at various boundary conditions. The results showed that DWCNTs have frequencies in the range of terahertz (THz). Also, in the study of vibration of CNT, Timoshenko beam model has been used for short length-to-diameter ratios which allows for the effect of transverse shear deformation [7], [8]. Likewise, the shell models have been applied recently by He et al. [9], Ru [10] and Wang et al. [11]. Ru [6] stresses that “carbon MWCNTs are different from traditional elastic beams due to their hollow multilayer structure and the associated interlayer van der Waals forces.” The exact solutions lead to the need of solving transcendental equations. It appears that they can be usefully supplemented by simple solutions.
In this paper, approximate solutions are found by using Bubnov–Galerkin [12], [13] and Petrov–Galerkin [14] methods. Explicit formulas of natural frequencies are derived for the DWCNTs at different boundary conditions. Comparison of the results with recent studies shows that the above methods constitute effective alternative techniques to exact solution for studying the vibration properties of CNT.
Section snippets
Analysis
The governing differential equations for free vibration of the DWCNTs readwhere x is the axial coordinate, t the time, wj (x, t) the transverse displacement, Ij the moment of inertia and Aj the cross-sectional area of the jth nanotube; the indexes j=1,2 denote the inner tube and outer tube, respectively.
The exact solutions for various boundary conditions were considered by Xu et al. [4], [5]. Their derivation necessitates numerical
Simply supported DWCNT: exact solution
For the DWCNT that is simply supported at both ends one obtains an exact solution by substitutionwhere ξ=x/L is a non-dimensional axial coordinate and m=1,2,… the number of half-waves in the longitudinal direction as well as the sequence number of the vibrational mode and ω the sought natural frequency. We substitute Eq. (2) into Eq. (1) and demand nontriviality of Y1 and/or Y2. In order to be different from zero the following determinant must
Simply supported DWCNT: Bubnov–Galerkin method
In order to ascertain the accuracy of the approximate solutions we first apply the Bubnov–Galerkin [12], [13] method to the simply supported DWCNT the exact solution for which was reported in preceding section.
We approximate the displacements as follows:whereThe function in Eq. (7) is so called Duncan polynomial [15]. We substitute the expressions (6) into governing differential Eq. (1) having in mind Eq. (2); we then multiply the result of the substitution by
Simply-supported DWCNTs: Petrov–Galerkin Method
In 1940 Petrov [14] suggested a modification to the Bubnov–Galerkin method; he proposed to employ two systems of functions simultaneously; namely, one system of functions is used to approximate the displacement, whereas another set of functions is used for satisfying the orthogonality condition.
We first substitute the coordinate function ϕ(2) in Eq. (13) into governing equations as was done in Eq. (6). The results however are not multiplied by ϕ(2) as in Bubnov–Galerkin method, but by some other
Clamped–clamped DWCNT: Bubnov–Galerkin Method
For this set of boundary conditions, we use so called Filonenko–Borodich [16] trigonometric polynomial as a coordinate function in the context of Bubnov–Galerkin method:
The usual Bubnov–Galerkin procedure yields the equations for D1 and D2:We arrive in the following frequency equation:with roots
Clamped–clamped DWCNT: Petrov–Galerkin method
To contrast the results furnished by the Bubnov–Galerkin method, we compare them with the evaluation by the Petrov–Galerkin method. We employ the substitution function and the multiplicative function .
The Petrov–Galekin procedure result in the equations for D1 and D2:The frequency equation reads
Simply supported–clamped DWCNT
For this set of boundary conditions, again in the context of Bubnov–Galerkin method, we utilize the following coordinate function:The Bubnov–Galerkin procedure yields the equations for D1 and D2:The frequency equation readsThe natural frequencies squared are
Clamped–free DWCNT
In the context Bubnov–Galerkin method we utilize the following Duncan polynomial
The Bubnov–Galerkin procedure yields the equations for D1 and D2The frequency equation readswith roots
Comparison with results of Natsuki et al. [17]
Most recently, Natsuki et al. [17] analyzed free vibration characteristics of DWCNT. Also, in the private communications [18] to the presents authors, he informed on results of calculation of natural frequencies. Specifically, Natsuki et al. [17], [18] adopt the following formula for the van der Waals interaction coefficient c1:whereandwhere σ and ε are the van der Waals radius and the well depth of
Conclusion
In this study we utilize two approximate methods, namely, those of Bubnov–Galerkin, and Petrov–Galerkin to derive explicit expressions for the double-walled carbon nanotubes under various boundary conditions. The attractiveness of the derived results lies in their simplicity. The expressions are not more complicated that the exact analytical formulas for the double-walled carbon nanotubes, that are simply supported at both end. In the cases where the exact solutions are available our solutions
Acknowledgment
I.E. appreciates partial financial support by the J.M. Rubin Foundation at the Florida Atlantic University. D.P. appreciates the partial financial support of Mr. Angelos Langadas Scholarship at the FAU. We appreciate helpful correspondences with Professor C.Q. Ru of University of Alberta, Canada, with Professor K.Y. Xu of Shanghai University, People's Republic of China, and with Professor T. Natsuki of Shinshu University, Japan. This study was conducted as a part of the grant proposal to the
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