819. Nonlinear vibration of fluid-conveying doublewalled carbon nanotubes under random material property

Transcription

1 89. Nonlnear vbraton of flud-conveyng doublealled carbon nanotubes under random materal property Ta-Png Chang Natonal Kaohsung Frst Unversty of Scence and Technology, Kaohsung, Taan, R. O. C. E-mal: (Receved July ; accepted September ) Abstract. In ths study, one performs the stochastc dynamc analyss of nonlnear vbraton of the flud-conveyng double-alled carbon nanotubes (DWCNTs) by consderng the effects of the geometrc nonlnearty and the nonlnearty of van der Waals (vdw) force. Based on the Hamlton s prncple, the nonlnear governng equatons of the flud-conveyng DWCNTs are derved. In order to truly descrbe the random materal propertes of the DWCNTs, the Young s modulus of elastcty of the DWCNTs s assumed as stochastc th respect to the poston. By adoptng the perturbaton technque, the nonlnear governng equatons of the flud-conveyng can be decomposed nto to sets of nonlnear dfferental equatons nvolvng the mean value of the dsplacement and the frst varaton of the dsplacement separately. Then one uses the harmonc balance method n conjuncton th Galerkn s method to solve the nonlnear dfferental equatons successvely. Some statstcal dynamc response of the DWCNTs such as the mean values and standard devatons of the ampltude of the dsplacement are calculated. It s concluded that the mean value and standard devaton of the ampltude of the dsplacement ncrease nonlnearly th the ncrease of the frequences for both cases of couplng beteen longtudnal dsplacement and transverse dsplacement and uncouplng beteen them. Hoever, the coeffcents of varaton of the ampltude of the dsplacement reman almost constant and stay thn certan range th respect to the frequency. The calculated stochastc dynamc response plays an mportant role n estmatng the structural relablty of the DWCNTs. Keyords: nonlnear vbraton, flud-loaded double-alled carbon nanotubes, random materal propertes, Galerkn s method. Introducton A landmark paper regardng Carbon nanotubes (CNTs) by Ijma [] has attracted orldde attenton due to ther potental use n the felds of chemstry, physcs, nano-engneerng, electrcal engneerng, materals scence, renforced composte structures and constructon engneerng. Carbon nanotubes (CNTs) are used for a varety of technologcal and bomedcal applcatons ncludng nanocontaners for gas storage and nanoppes conveyng fluds [-]. The sngle-elastc beam model [5-6] ere dely adopted to study the dynamc behavors of fludconveyng sngle-alled carbon nanotubes (SWCNTs) and mult-alled carbon nanotubes (MWCNTs). Normally speakng, the beam models mentoned above are lnear; hoever, the vdw forces n the nterlay space of MWCNTs are essentally nonlnear. Furthermore, the slender ratos are normally large f the beam models are adopted, that s, the large deformaton ll occur. Therefore, t s qute essental to consder to types of nonlnear factors, namely, the geometrc nonlnearty and the nonlnearty of vdw force n nvestgatng the dynamc behavors of flud-conveyng MWCNTs. Salvetat et al. [7] measured the flexural Young s modulus and shear modulus usng AFM test on clamped clamped nanoropes, gettng values th 5 % of error. Informaton related to statstcal dstrbutons of expermental data s also rare, and the mportant study from Krshnan et al. [8] provdes one of the fe examples avalable of hstogram dstrbuton of the flexural Young s modulus derved from 7 CNTs. The Young s modulus as estmated observng free-standng vbratons at room temperature usng transmsson electro-mcroscope (TEM), th a mean value of. TPa -. Tpa / +.6 TPa. VIBROENGINEERING. JOURNA OF VIBROENGINEERING. SEPTEMBER. VOUME, ISSUE. ISSN

2 89. NONINEAR VIBRATION OF FUID-CONVEYING DOUBE-WAED CARBON NANOTUBES UNDER RANDOM MATERIA PROPERTY. Uncertanty s also assocated to the equvalent atomstc-contnuum models adopted extensvely n partcular by the engneerng and materals scence communtes. Therefore, to be realstc, the Young s modulus of elastcty of carbon nanotube (CNTs) should be consdered as stochastc th respect to the poston to actually descrbe the random property of the CNTs under certan condtons. In the present study, e nvestgate the stochastc dynamc behavors of nonlnear vbraton of the double-alled carbon nanotubes (DWCNTs) conveyng flud by consderng the effects of the geometrc nonlnearty and the nonlnearty of van der Waals (vdw) force. Based on the Hamlton s prncple, the nonlnear governng equatons of the flud-conveyng doublealled carbon nanotubes are formulated. To dfferent cases of nonlnearty are consdered; case one s to nclude the couplng beteen the longtudnal dsplacement and transverse dsplacement of the DWCNTs, on the other hand, case to s to neglect the couplng beteen them. The Young s modulus of elastcty of the DWCNTs s consdered as stochastc th respect to the poston to actually characterze the random materal propertes of the DWCNTs. Nonlnear beam model for flud-loaded DWCNTs Fg.. Flud-loaded double-alled carbon nanotubes In Fg., the double-alled carbon nanotubes (DWCNTs) are modeled as a double-tube ppe hch s composed of the nner tube of radus R and the outer tube of radus R. The thckness of each tube s h, the length s, and Young s modulus of elastcty s E. It s noted that the Young s modulus of elastcty E s assumed as stochastc th respect to the poston to actually descrbe the random materal property of the DWCNTs. The nternal flud s assumed to flo steadly through the nner tube th a constant velocty U. Besdes, the boundary condtons of the DWCNTs are assumed as clamped at both ends. Based on the theory of Euler-Bernoull beam and a nonlnear stran-dsplacement relatonshp of Von Karman type, the dsplacement feld and stran-dsplacement relaton can be rtten as follos: u x z t u x t z (,, ) (, ) = x (,, ) = (, ) x z t x t u ε = + x x () here x s the axal coordnate, t s tme, u and denote the total dsplacements of the th tube along the x coordnate drectons, u and defne the axal and transverse dsplacements of the th tube on the neutral axs, ε the correspondng total stran, and the subscrpt = and =. Notce that tube s the nner tube hle tube s the outer tube. The potental energy V stored n a DWCNTs and the vrtual knetc energy T n the DWCNTs as ell as the flud nsde the DWCNTs are ndvdually rtten as follos: 96 VIBROENGINEERING. JOURNA OF VIBROENGINEERING. SEPTEMBER. VOUME, ISSUE. ISSN 9-876

3 89. NONINEAR VIBRATION OF FUID-CONVEYING DOUBE-WAED CARBON NANOTUBES UNDER RANDOM MATERIA PROPERTY. u u V = E( x) + z da A x x x + E( x) z da + dx A x x x () ρ t u ɺ u ɺ T = z da+ z da dx A + t x t A + t x t u + ρ cos f U θ + U snθ A + z dadx f t t x t () here θ = x, I and m are the moment of nerta and the mass of the th tube per unt length; ρ s the mass densty of the beam materal; t ρ f s the mass densty of the flud nsde A = R + h R tube ; π ( ) tube and tube, respectvely, and A = R + h R and π ( ) Af are the cross-sectonal areas of = π R s the cross-sectonal areas of the flud passage n tube. Based on Hamlton s prncple, the varatonal form of the equatons of moton for the DWCNTs can be gven by: t t ( δ δ δ ) V T Ψ dt = () here the vrtual ork due to the vdw nteracton and the nteracton beteen tube and the flong flud s gven by: δ x θ δ δ Ψ= x θ δ cos + ( ) P MU dx P dx sn MU dx u (5) P s the nonlnear vdw force per unt length n the nterlayer of the DWCNTs. The nterlayer potental per unt area Π ( δ) can be expressed n terms the nterlayer spacng δ as follos: ( δ) K δ. δ Π = δ δ (6) the vdw force P s then obtaned by consderng the loest-order nonlnear term n Taylor expanson of U, hch s rtten as [9]: Π P = R ( Π ) + ( ) = c ( ) c ( ) δ δ δ δ + δ 6 δ δ= δ δ= δ (7) here: δ δ K = mev/atom; = ; δ =. nm s the equlbrum nterlayer spacng. Π c = R ; δ δ= δ Π = R 6 δ δ= δ c and VIBROENGINEERING. JOURNA OF VIBROENGINEERING. SEPTEMBER. VOUME, ISSUE. ISSN

4 89. NONINEAR VIBRATION OF FUID-CONVEYING DOUBE-WAED CARBON NANOTUBES UNDER RANDOM MATERIA PROPERTY. By utlzng the Eqs. (-5) and consderng the boundary condtons of the clamped ends, and the assumpton that all varables and dervatves are zero at t= t and t= t, all the terms t nvolvng [] and [] vansh. In addton, consderng the moderate large-ampltude deflecton, t cosθ and snθ are adopted n the follong dervaton. In the present study, the x boundary condtons of the DWCNTs are assumed as clamped, therefore, the follong boundary condtons can be rtten for the axal dsplacement: ( ) ( ) ( ) ( ) u, t = u, t =, u, t = u, t = (8) Furthermore, e deal th to cases n the formulatons. Case one s that the couplng beteen the axal dsplacement u and transverse dsplacement s consdered and case to s that the couplng beteen u and s neglected. Frst of all, the case consderng the couplng beteen u and s nvestgated. By neglectng the rotaton nerta and utlzng Eqs. (-8), after some tedous dervatons e can obtan the coupled nonlnear governng equatons for the free vbraton of DWCNTs conveyng flud as follos: ( E( x)( / x )) E( x) A MU MU I MU dx x x x x x x (9) +( M + m ) +M U MU = c ( ) c ( ) t x t t x x + ( E( x)( / x )) E( x) A + = - ( ) ( ) I m dx c c x t x x () For case to, the formulaton neglectng the couplng beteen u and s nvestgated. Once agan, by neglectng the rotaton nerta and utlzng Eqs. (-8), after some tedous dervatons e can obtan the coupled nonlnear governng equatons for the free vbraton of DWCNTs conveyng flud, for smplcty, the hole dervatons for case to s not presented explctly. In the follong dervatons, only case one s depcted explctly. Case one: consderng the couplng beteen u and In the present study, the Young s modulus of elastcty E(x) s consdered as stochastc th respect to the poston to actually characterze the random propertes of the DWCNTs and t s assumed as Gaussan dstrbuted. Applyng the perturbaton technque on the Young s modulus of elastcty E(x), the follong equatons can be rtten: I E( x) = E ( x) + ε E ( x) + () here E ( x ) s the mean value of the Young s modulus of elastcty E(x), ε s a zero-mean I small parameter, and ε E ( x) s the frst varaton of the Young s modulus of elastcty E(x). Smlarly, the dsplacement ( x), ( x ) of the DWCNTs can be rtten as follos: 96 VIBROENGINEERING. JOURNA OF VIBROENGINEERING. SEPTEMBER. VOUME, ISSUE. ISSN 9-876

5 89. NONINEAR VIBRATION OF FUID-CONVEYING DOUBE-WAED CARBON NANOTUBES UNDER RANDOM MATERIA PROPERTY. ( x) = ( x) + ε ( x) + () I ( x) = ( x) + ε ( x) + () I here ( x), ( x ) are the mean values of dsplacement of the nner and outer tubes separately. Substtutng Eqs. (-) nto Eqs. (9-), e can obtan the follong to coupled equatons based on the zero order of ε: ( ) ( ) MU E A ( ) ( ) MU Ι E MU dx x x x x x x () ( ) + ( M + m ) + MU MU ( )( )( ) = c ( ) + c ( ) t x t t x x ( ) ( ) E A ( ) Ι + = ( ) ( ) E m dx c c x t x x (5) Based on the frst order of ε, e can acheve the follong to coupled equatons: I I I I I ( ) ( E ( / x )) ( ) MU ( ) ( ) ( ) EΙ +Ι + MU dx + dx x x x x x x x x I A I ( ) ( ) E + E dx ( I ) + E dx x x x x x x I I I I MU ( ) ( ) + ( + + M + m ) + MU x x x x x x x t I I I MU ( )( )( ) + ( )( )( ) + ( )( )( ) t x x t x x t x x = c ( ) + c ( ) ( ) I I I I I Ι I ( E ( / x )) E I + I + m ( ) x x t I I A ( ) I E E dx E dx + + x x x x x x I I ( ) I I = c c ( ) ( ) (6) (7) Frst of all, e have to solve, n Eqs. (-5). By applyng the harmonc balance method and Galerkn s method and substtutng = Aφ ( x)sn( ωt) ( =, ) nto Eqs. (-5), after some tedous dervatons the relatonshp beteen the ampltude A and the resonant φ can be acheved as follos: frequency ω of the loest-order mode ( ) x G A + G A + G ( A A ) + G ( A A ) = (8) G A + G A + G ( A A ) + G ( A A ) = (9) 5 6 VIBROENGINEERING. JOURNA OF VIBROENGINEERING. SEPTEMBER. VOUME, ISSUE. ISSN

6 89. NONINEAR VIBRATION OF FUID-CONVEYING DOUBE-WAED CARBON NANOTUBES UNDER RANDOM MATERIA PROPERTY. here: π ω ( ) ω M λ λ φ ( ) sn( ω ) ( E A + MU ) λ dφ ( ) x π ω φ ( ) sn ( ω ) G = M + m + U E I x dx t dt G = dx x dx t dt + dx dφ ( ) x π ω φ( ) sn( ω ) MU λ + dx x dx t dt dx π ω φ ( ) sn ( ω ) π ω φ ( ) sn ( ω ) π ω λ ω φ ( ) sn( ω ) ( ) λ φ x π ω φ ( ) sn ( ω ) G = c x dx t dt G = c x dx t dt 5 G = E I m x dx t dt E A G6 = dx x dx t dt dx here φ ( x) s the frst vbraton mode of the correspondng lnear system, hch can be expressed as follos for the clamped-clamped boundary condtons as follos: snh( ) -sn( ) ( x) = cosh( x) -cos( x) - λ λ φ λ λ snh( λ x) -sn( λ x), λ =.7 ( ) cosh( λ) -cos( λ) () 966 After solvng coupled Eqs. (8-9) for the ampltudes A, A, e can obtan, readly. Substtutng, nto Eqs. (6-7), and adoptng the same technque for solvng,, VIBROENGINEERING. JOURNA OF VIBROENGINEERING. SEPTEMBER. VOUME, ISSUE. ISSN I I fnally e can obtan, thout any dffcultes except the dervatons are somehat lengthy. Numercal examples and dscusson In the numercal computatons, the clamped-clamped boundary condtons are consdered for the DWCNTs conveyng flud. The nner and the outer tubes are assumed to have the same Young s modulus, the same thckness and the same mass densty. The numercal values of the parameters are adopted as follos: Mean value of Young s modulus E = Tpa, tube thckness h=. nm, mass densty ρ = Kg/m, the mass densty of ater flo s ρ f = Kg/m, the nner radus.7 nm R = and the outer radus R =. nm and mean = square values of ε s assumed as E ε.. The relatons of the mean value of ampltude versus frequency are depcted n Fg.. The mean value of the ampltude ncreases th the ncrease of the frequences for both cases of couplng beteen longtudnal dsplacement u and transverse dsplacement and uncouplng beteen them. It s certanly reasonable that the relaton beteen the mean value of the ampltude and the frequency s nonlnear; n addton, the mean value of the ampltude of the outer tube s larger than that of the nner tube. Furthermore, t s noted that the mean value of the ampltude th consderng the couplng s larger than that th uncouplng for the fxed frequency. In Fg., the standard devaton of the ampltude s plotted th respect to the frequency for both couplng and uncouplng cases. As t can be seen

7 89. NONINEAR VIBRATION OF FUID-CONVEYING DOUBE-WAED CARBON NANOTUBES UNDER RANDOM MATERIA PROPERTY. from the fgure that the standard devaton of the ampltude ncreases nonlnearly th the ncrease of the frequences, and t s noted that the standard devaton of the ampltude of the outer tube s larger than that of the nner tube. In Fg., the coeffcent of varaton of the ampltude s depcted th respect to the frequency. It s nterestng to notce that the coeffcents of varaton of the ampltude of the nner and outer tubes for both couplng and uncouplng cases are around. to Mean value of ampltude (nm) A, couplng beteen u and A, couplng beteen u and A, uncouplng beteen u and A, uncouplng beteen u and Frequency (Hz) x Fg.. Mean value of ampltude versus frequency Standard devaton of ampltude (nm).5... SD of A, couplng beteen u and SD of A, couplng beteen u and. SD of A, uncouplng beteen u and SD of A, uncouplng beteen u and Frequency (Hz) x Fg.. Standard devaton of ampltude versus frequency Coeffcent of varaton (COV) of ampltude COV of A, couplng beteen u and COV of A, couplng beteen u and COV of A, uncouplng beteen u and COV of A, uncouplng beteen u and Frequency (Hz) x Fg.. Coeffcent of varaton of ampltude versus frequency VIBROENGINEERING. JOURNA OF VIBROENGINEERING. SEPTEMBER. VOUME, ISSUE. ISSN

8 89. NONINEAR VIBRATION OF FUID-CONVEYING DOUBE-WAED CARBON NANOTUBES UNDER RANDOM MATERIA PROPERTY. Conclusons In the present study, e nvestgate the stochastc dynamc behavors of nonlnear vbraton of the double-alled carbon nanotubes (DWCNTs) conveyng flud by consderng the effects of the geometrc nonlnearty and the nonlnearty of van der Waals (vdw) force. Based on the Hamlton s prncple, the nonlnear governng equatons of the flud-conveyng double-alled carbon nanotubes are formulated. To dfferent cases of nonlnearty are consdered; case one s to nclude the couplng beteen the longtudnal dsplacement and transverse dsplacement of the DWCNTs, on the other hand, case to s to neglect the couplng beteen them. The Young s modulus of elastcty of the DWCNTs s consdered as stochastc th respect to the poston to actually characterze the random materal propertes of the DWCNTs. By usng the perturbaton technque, the nonlnear governng equatons of the flud-conveyng double-alled carbon nanotubes can be decomposed nto to sets of nonlnear dfferental equatons nvolvng the mean value of the dsplacement and the frst varaton of the dsplacement separately. Then the harmonc balance method and Galerkn s method are used to solve the nonlnear dfferental equatons successvely. Some statstcal dynamc response of the DWCNTs such as the mean values and standard devatons of the ampltude of the dsplacement are computed. It can be concluded that the mean value and standard devaton of the ampltude of the dsplacement ncrease nonlnearly th the ncrease of the frequences for both cases of couplng beteen longtudnal dsplacement and transverse dsplacement and uncouplng beteen them. Hoever, the coeffcents of varaton (COV) of the ampltude of the dsplacement reman almost constant and stay thn certan range th respect to the frequency. It s noted that the computed stochastc dynamc response plays an mportant role n estmatng the structural relablty of the DWCNTs. Acknoledgements Ths research as partally supported by the Natonal Scence Councl n Taan through Grant No. NSC-99--E-7-. The author s grateful for the fnancal support. References [] Ijjma S. Helcal mcrotubules of graphtc carbon. Nature, Vol. 5, 99, p [] Gadd G. E. et. al. The orld s smallest gas cylnders? Scence, Vol. 77, 997, p [] Che G. et. al. Carbon nanotubule membranes for electrochemcal energy storage and producton. Nature, Vol. 9, 998, p [] Karlsson A. et. al. Netorks of nanotubes and contaners. Nature, Vol. 9,, p [5] Yoon J., Ru C. Q., Moduchosk A. Flo-nduced flutter nstablty of cantlever CNTs. Int. J. Solds Struct., Vol., 6, p [6] Wang., N Q. On vbraton and nstablty of carbon nanotubes conveyng flud. Comput. Mater. Sc., Vol., 8, p [7] Salvetat J. P. et. al. Elastc and shear modul of sngle-alled carbon nanotube ropes. Phys. Rev. ett., Vol. 8, No. 5, 999, p [8] Krshnan A. et. al. Young's modulus of sngle-alled nanotubes. Phys. Rev. B, Vol. 58, No., 998, p. -9. [9] Xu K. Y., Guo X. N., Ru C. Q. Vbraton of a double-alled carbon nanotube aroused by nonlnear ntertube van der Waals forces. J. Appl. Phys., Vol. 99, 6, p VIBROENGINEERING. JOURNA OF VIBROENGINEERING. SEPTEMBER. VOUME, ISSUE. ISSN 9-876