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Fundamental natural frequencies of double-walled carbon nanotubes

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Abstract

This study 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.

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 readc1(w2-w1)=EI14w1x4+ρA12w1t2-c1(w2-w1)=EI24w2x4+ρA22w2t2where 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 substitutionw1=Y1sin(mπξ)sin(ωt)w2=Y2sin(mπξ)sin(ωt)where 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 Y12+Y22 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:w1=D1ϕ(1),w2=D2ϕ(1)whereϕ(1)=3ξ5-10ξ3+7ξThe 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:ϕ(4)=1-cos(2πξ)

The usual Bubnov–Galerkin procedure yields the equations for D1 and D2:(3L4ρA1ω¯2-3L4c1-16π4EI1)D1+3L4c1D2=03L4c1D1+(3L4ρA1ω¯2-3L4c1-16π4EI2)D2=0We arrive in the following frequency equation:9L8ρ2A1A2ω¯4+(-233785L4A1EI2-9L8A1c1-233785EI1L4A2-9L8c1A2)ρω¯2+233785L4c1EI2+233785EI1L4c1+2429100E2I1I2=0with rootsω¯1,12=[

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 ϕ(5)=ξ4-2ξ3+ξ2 and the multiplicative function ψ=1-cos(2πξ).

The Petrov–Galekin procedure result in the equations for D1 and D2:(L4π4ρA1ω¯2+45L4ρA1ω¯2-45L4c1-720π4EI1-L4c1π4)D1+(45L4c1+L4c1π4)D2=0(45L4c1+L4c1π4)D1+(L4π4ρA2ω¯2+45L4ρA2ω¯2-45L4c1-720π4EI2-L4c1π4)D2=0The frequency equation reads20280L8ρ2A1A2ω¯4+(-9987000L4A1EI2-20280L8

Simply supported–clamped DWCNT

For this set of boundary conditions, again in the context of Bubnov–Galerkin method, we utilize the following coordinate function:ϕ(6)=2ξ4-3ξ3+ξThe Bubnov–Galerkin procedure yields the equations for D1 and D2:(19L4ρA1ω¯2-19L4c1-4536EI1)D1+19L4c1D2=019L4c1D1+(19L4ρA1ω¯2-19L4c1-4536EI2)D2=0The frequency equation reads361L8ρ2A1A2ω¯4+(-86186L4A1EI2-361L8A1c1-86186EI1L4A2-361L8c1A2)ρω¯2+86186L4c1EI2+86186EI1L4c1+20575296E2I1I2=0The natural frequencies squared areω¯1,12=[19L4A1c1+19L4c1A2+4536A1EI2+

Clamped–free DWCNT

In the context Bubnov–Galerkin method we utilize the following Duncan polynomialϕ(8)=ξ5-103ξ4+103ξ3

The Bubnov–Galerkin procedure yields the equations for D1 and D2(163L4ρA1ω¯2-163L4c1-2970EI1)D1+163L4c1D2=0163L4c1D1+(163L4ρA1ω¯2-163L4c1-2970EI2)D2=0The frequency equation reads26569L8ρ2A1A2ω¯4+(-484110L4A1EI2-26569L8A1c1-484110EI1L4A2-26569L8c1A2)ρω¯2+484110L4c1EI2+484110EI1L4c1+8820900E2I1I2=0with rootsω¯1,12=[163L4A1c1+163L4c1A2+2970A1EI2+2970EI1A2-(26529L8A12c12+53138L8A1c12A2+968220L4A12c1EI2

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:c1=πεR1R2σ6α4[1001σ63H13-1120σ69H17]whereHm=(R1+R2)-m0π/21/(1-Kcos2θ)m/2dθ,(m=7,13)andK=4R1R2/(R1+R2)2where 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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