INVESTIGATION ON MECHANICAL VIBRATION OF DOUBLE-WALLED CARBON NANOTUBES ON WINKLER FOUNDATION WITH LENGTH EFFECTS VIA DTM

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1 Rasayan J. Chem., 0(), 8-87(07) Vol. 0 No April - June 07 ISSN: e-issn: CODEN: RJCABP INVESTIGATION ON MECHANICAL VIBRATION OF ON WINKLER FOUNDATION WITH LENGTH EFFECTS VIA DTM B. Ravi Kumar * an K. Palaksha Rey School of Mechanical Engineering, SASTRA University, Thanjavur-630,Tamilnau (Inia) * ravikumar@mech.sastra.eu ABSTRACT In the present stuy, vibration response of ouble alle carbon nanotubes embee in a Winkler founation is investigate using ifferential transform metho (DTM). Simply supporte, clampe - Hinge an clampe clampe bounary conitions are employe to analyze the natural frequencies. The effect of length of nanotubes an Winkler elastic founation moulus on the funamental natural frequencies for the Euler- Bernoulli beam is investigate an tabulate. The governing ifferential equations are solve by using ifferential transform metho. The ifferential transform metho is applicable for nanotubes an gives high accuracy results. Keyors: Aerospace, Winkler, DTM, DWCNT, MATLAB, MEMS. RASĀYAN. All rights reserve INTRODUCTION Nanoscience an nanotechnology have mae a principal contribution to the introuction of small-scale structures an evices. Iijima s iscovery paper on multi-alle carbon nanotubes in 99 pulle the attention of scientific researcher s communities in physics, chemistry an engineering. Some of the potential applications of nanoros an nanobeams are image technology, microelectro-mechanical systems (MEMS) an nanoelectromechanical systems (NEMS).,3 In Aerospace inustries, There is a much nee for ne materials hich exhibit improve mechanical properties i.e, high strength to eight ratio, high thermal stability an high corrosion resistance. Carbon nanotubes (CNTs) exhibit novel properties of having tensile strength in Tera pascal range, thermally stable up to 800 C an vibration ithstaning capabilities in the range of Tera Hertz. The structure of a single alle carbon nanotube can be visualize by packaging a single-atom-thick layer of graphite calle graphene into a flaless cyliner. -5 By no, there are three methos to analyze the mechanical properties of CNT. They are spectroscopy experiment, computational simulation an theoretical analysis. Hoever, technically these are ifficult experiments of vibrational properties because of the small iameter of the nanotubes, an plenty of time ill be spent on the molecular ynamics (MD) simulation. Theoretical analyses using elastic continuum moels have shon the avancement on the taken cost an the spent time of researches compare ith experiment operation an MD simulation methos, an they can preict some results hich cannot be achieve by the other to methos as of no. 6 0 In the present ork, DTM has been use to analyze the vibration of carbon nanotubes embee on a Winkler founation. Zhou evelope ifferential transformation metho for solving both linear an nonlinear initial value problems in electric circuit analysis. Later on Chen an Ho use this metho to Eigen value problems. EXPERIMENTAL Differential Transform Metho The ifferential transform metho is a semi-analytical metho base on the Taylor series expansion. In this technique, certain transformation rules are use in converting the governing ifferential equations an the bounary conitions of the system in to a set of algebraic equations in terms of the ifferential

2 Vol. 0 No April - June 07 transforms of the original functions. The solution of these algebraic equations provies the results ith a high egree of precision. The ifferential transformation of the i th erivative of the function u(x) is efine as follos: i u( x) U(i) = i i! x= x0 An the ifferential inverse transformation of U(k) is expresse as: 0 () i= 0 u( x) = U(i)( x x ) i In real application function, u(x) is expresse as finite series an equation () can be ritten as: n 0 i (3) i= 0 u( x) = U( k)( x x ) We can use the folloing transformation table for converting a ifferential equation in to the DTM form. Table-: Differential Transformations for Mathematical Equations Original Function Transforme Function y ( x ) = u ( x ) ± v ( x ) Y ( x) = U (i) ± V (i) y ( x ) = λ u ( x ) Y (i) = λ U (i) y n u ( x ) ( x ) = Y(i) = (i+ )(i + )...(i + nu ) (i + n) n The superiority of the DTM is its simplicity an goo precision an it takes less time to solve polynomial series. It is ifferent from the traitional high orer Taylor s series technique, hich nees symbolic computation of the necessary erivatives of the ata functions. The Taylor series metho takes relatively long time for computations involving large orers. With DTM technique, it is also possible to compute near exact solutions for ifferential equations. Mathematical Formulation Using the Euler Bernoulli beam moel, the require governing partial ifferential equation for motion of a beam is given by: 8 ρ A + EI = P(x) Where, x an t are the spatial coorinates. is the eflection of carbon nanotubes an P is the transverse force acte on the nanotubes. E an I are the Young s moulus of elasticity an the mass moment of inertia, respectively. A is the area of cross section an ρ is the ensity of nanotubes. By using the Equation () governing equations for inner an outer nanotube can be ritten as: ρ A + EI = p ρa + EI = p Where, the subscripts an enote the properties of the inner an outer nanotubes respectively. p (j=,) are the pressures exerte on inner an outer nanotubes. The pressure P acting on the inner j () () (5) (6) 8 B. Ravi Kumar an K. Palaksha Rey

3 Vol. 0 No April - June 07 nanotube ue to thevw interaction is given by, p = c( ) (7) Where, c is the Vaner Waals interaction coefficient beteen inner an outer nanotubes. The pressure acting on the surface of outer nanotube ue to presence of the Winkler meium can be given by, p = k (8) Where, negative sign represents that p is acting in opposite irection to the eflection of nanotubes.k is the spring s stiffness. Thus, for the DWCNTs, the pressure of the outer nanotube contacting ith the Winkler meium is given by, p = p c( ) (9) In the moel Van er Waals interaction (vw) coefficient (c) can be obtaine from the interlayer energy potential, given as He XQ et al, 6 6 πε R Rσ 00σ 0 c = H H a 3 9 Where, 3 7 π / m θ m/ ( KCos θ ) 0 m H = ( R + R ),( m = 7,3) an K = RR ( R + R ) (0) Here a = 0. nm (carbon to carbon atom bon length). R, R are the inner an outer raii of ouble alle carbon nanotubes. σ=0.3 nm, ԑ=.967 mev are the vw raius an the ell epth of Lennar- Jones potential respectively as given by Saito et al. 8 Thus, ρ A + EI = c( ) ρa + EI = k c( ) In this simulation, e consier the eflection of ouble alle carbon nanotubes has ifferent vibration moes, (x), j =, for the insie an outsie nanotubes respectively. The isplacements of the W j vibration analysis in ouble alle carbon nanotubes can be given by iωt ( x, t) = W ( x) e (3) j j So equations () an () can be further simplifie as: W Ω W = β ( W W ) X W δ ηω W = β ( W W ) kw (5) X () () () 83 B. Ravi Kumar an K. Palaksha Rey

4 Where, ρ Aω L Ω =, cl β =, A η =, I δ =, k EI EI A I kl =, X = E I x L Vol. 0 No April - June 07 Application of DTM In converting the governing equations in DTM formulation, e use Table- an e can conveniently rite the folloing expressions: ( β.( W (i) W ( i)) + rw. (i)) W (i+ ) = (i+ )(i + )(i+ 3)(i + ) (( β / δ ).( W (i) W (i)) + r( η / δ ). W (i) ( k / δ ). W (i)) W (i+ ) = (7) (i+ )(i+ )(i+ 3)(i+ ) Where, ρaω L r =,[ k = 0.,,3. N_max] EI Using the appropriate bounary conitions an transforme bounary conitions for DTM, the recurrence Eq. (6) an (7) can be solve for vibration frequencies for various moes. Bounary Conitions Simply Supporte CNTs For the simply supporte CNTs beam bounary conitions at both ens are efine mathematically as: = 0, = 0, = 0, = 0 (8) Clampe- Clampe CNTs For clampe-clampe CNTs, the bounary conitions at both ens are efine as: = 0, = 0, 0, 0 = = (9) Clampe-Hinge CNTs For clampe-hinge CNTs case, the bounary conitions are efine as: at x=0 = 0, = 0, = 0, = 0 (0) at x=l = 0, = 0, = 0, = 0 () RESULTS AND DISCUSSION Comparison ith Analytical Solutions In this stuy, e consier ouble alle carbon nanotubes mounte in a Winkler meium having iameter of the inner an outer nanotubes of.8 nm an 5.5 nm, respectively. The thickness of each nanotube is taken to be 0.3 nm. The carbon nanotube has a Young s moulus of TPa an the ensity of.3 gm/cm. 3, (6) 8 B. Ravi Kumar an K. Palaksha Rey

5 Vol. 0 No April - June 07 By using the DTM as the numerical metho the natural frequency for DWCNTs has been compute. Results are then compare ith T.Natsukiet al an Xuet al. 5 Very goo agreement is observe ith the T.Natsuki et al an Xu et al results for the simpy supporte DWCNTs. We have taken n= 50 so that the result converges up to four ecimal places, here n is the number of iterations require to converge the result. Table-: Simply supporte (S-S) DWCNTs Funamental frequency in THz Moe ω ω ω 3 ω ω 5 ω 6 Present [DTM] T.Natsukiet al Xu et al From the above Table-, it is clearly observe that initial in-phase moes are matching perfectly an higher moes are slightly iffering. As e observe that there is very little increment in vibration frequency after moe, as vibration eflection ill be in opposite irection of both nanotubes. Ne Results for DWCNTs Table-3 is the ne extene results for the Clampe-Hinge (C-H) an Clampe- Clampe (C-C) bounary conitions. Table 3: Funamental frequency for the DWCNTs in THz Moe ω ω ω 3 ω ω 5 ω 6 C - C C - H It is observe that ouble alle carbon naotubessho very high frequency over Terahertz. Moes to represent the first four vibration moes, hich are in-phase moes, hich means eflection of the inner an outer nanotubes are in the same irection. Moes 5 to 6 are the to vibrations of anti-phase moe, in hich the eflections of the inner an outer nanotubes are in the opposite irections. Influence of variation of Length of DWCNTs on vibration frequencies No if e change the length of nanotubes, vibration frequencies vary as: Fig.-: Variation of vibration frequencies ith length of DWCNTs for Simple supporte bounary conition. Figures-, an 3 sho the variation of the vibrational frequncy ith the length of nanotube for the various moes. Vibration frequnecy ith the first moe in simply-supporte DWCNTs is about B. Ravi Kumar an K. Palaksha Rey

6 Vol. 0 No April - June 07 THz from DTM hich is very close to the available reference result ith length of 0 nm. It is clearly observe that ith the increase of length of CNTs vibrational frequencies ecrease. This effect is occuring ue to the strong influence of phase moes on vibration frequencies. CONCLUSION In this stuy, the vibration investigation of DWCNTs embee in an elastic meium for various bounary conitions like clampe-clampe, simply supporte, an clampe hinge has been presente by using a semi-analytical numerical technique calle the ifferential transform metho in a simple an accurate ay. The solution of the present vibration analysis problem using the DTM involves transformation of the governing equations of motion into algebraic set of equations an solving the transforme equations. Results sho that phase moes ominate on vibration frequencies of CNTs. The analysis presente coul be helpful in the application of CNTs in high-frequency oscillators, ynamic mechanical analysis an micro mechanical sensors. Fig.-: Variation of vibration frequencies ith Length of Clampe-Clampe DWCNTs Fig.-3:Variation of vibration frequencies ith Length of Clampe-Hinge DWCNTs 86 B. Ravi Kumar an K. Palaksha Rey

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