VIBRATION ANALYSIS OF CURVED SINGLE-WALLED CARBON NANOTUBES EMBEDDED IN AN ELASTIC MEDIUM BASED ON NONLOCAL ELASTICITY

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1 VIBRATION ANASIS OF CURVED SINGE-AED CARBON NANOTUBES EMBEDDED IN AN EASTIC MEDIUM BASED ON NONOCA EASTICIT Pym Solni Amir Kssi Dprmn of Mchnicl Enginring Islmic Azd Univrsiy-Smnn Brnch Smnm Irn -mil: Mohmmd M. Thrin Dprmn of Mchnicl Enginring Islmic Azd Univrsiy-Smnn Brnch oung Rsrchrs Club Smnm Irn Anoushirvn Frshidinfr Dprmn of Mchnicl Enginring Frdowsi Univrsiy of Mshhd Mshhd Irn In his ppr h flurl vibrion of h curvd singl-wlld crbon nnoub (SCNT) is invsigd bsd on h nonlocl Eulr-Brnoulli bm modl. Th SCNT is ssumd o b mbddd in n lsic mdium. Boh inklr-yp nd Psrnk-yp foundion modls r uilizd o simul h inrcion of h SCNT wih h surrounding lsic mdium. Thr ypicl boundry condiions nmly clmpd-clmpd clmpd-pinnd nd pinndpinnd r usd o invsig h ffc of h suppord nd condiions. Bsd on h Glrkin mhod soluion for nurl frquncy is obind. According o his sudy h influncs of h mpliud of wvinss h nonlocl ffcs h inklr modulus prmr h Psrnk shr modulus prmr h boundry condiion nd h spc rio r nlyzd nd discussd. I is shown h wvinss in h curvd SCNT cuss n obvious incrs in h nurl frquncy in comprison wih h srigh SCNT spcilly for complin mdium pinnd-pinnd boundry condiion nd shor SCNT. 1. Inroducion Afr h invnion of Crbon nnoubs (CNTs) by Iijim 1 in 1991 considrbl moun of rsrch hs bn conducd o invsig h mchnicl hrml nd lcricl propris of CNTs nd dircd owrd undrsnding h sic nd dynmicl bhviours of crbon nnoubs du o hir normous pplicions which in nnochnology lcronics nd ohr filds of mrils scinc. Thrfor hs propris r good dl mp hs so fr bn dvod o h sudy of h vrious spcs of nnoubs. Sinc conrolld primns nnoscl r difficul nd molculr dynmics simulions rmin pnsiv nd im-consuming coninuum mchnics modls such s Eulr lsic-bm modl nd Timoshnko bm modl hv bn widly usd o sudy h vibrionl bhviour of CNTs. For insnc u l 3 sudid h invsigions of wv nd vibr- ICSV18 Rio d Jniro Brzil 1-1 July 11 1

2 ion propris of singl- or muli-wlld crbon nnoubs bsd on nonlocl Eulr nd Timoshnko bm modls. Rskh nd Khdm obind h mpliud frquncy rspons curvs of h nonlinr vibrion nd h ffcs of h surrounding lsic mdium lso influnc of inrnl moving fluid nd comprssiv il lod on h nonlinr vibrion nd sbiliy of mbddd crbon nnoubs is invsigd. In h ohr ppr h smll-scl ffcs on vibrion chrcrisics of CNTs hv bn sudid using nonlocl coninuum mchnics modl by ng nd Vrdn 5. A quliiv vlidion sudy shows h h rsuls bsd on nonlocl coninuum mchnics r in grmn wih h publishd primnl rpors in his fild. Murmu nd Prdhn 6 implmnd h nonlocl Timoshnko bm hory o invsig h sbiliy rspons of SCNTs mbddd in n lsic mdium. This ppr dmonsrs h criicl buckling lods of SCNT r significnly dpndn on h nonlocl consn nd on h propris of surrounding mdium. In mjoriy of prvious sudis r limid o clssicl bm hory for srigh bms whil som rcn primnl rsuls show h hs iny srucurs r no usully srigh nd rhr hv crin dgr of curvur or wvinss long hm. Joshi l 7 invsigd in h vibrion rspons nlysis of crbon nnoubs wih wvinss rd s hin shll. Also Myoof nd Hww 8 sudid h dynmics of h CNT whn i cs s firs-mod rsonor wih focus on h choic bhviour of curvd crbon nnoub undr hrmonic ciion. In his sudy nonlocl Eulr-Brnoulli bm modl hs bn mployd o invsig h rnsvrs vibrion of h wvy SCNT mbddd in n lsic mdium. Th nurl frquncy for h curvd SCNT is prssd using h Glrkin mhod. Also h ffc of mpliud of curvur on h fundmnl frquncy is discussd. Morovr h vriion of frquncy hs bn considrd bsd on h vrious prmrs such s h surrounding lsic mdium h boundry condiions h spc rio of SCNT nd h nonlocl cofficin.. Modling A curvd SCNT wih lngh mbddd in n lsic mdium wih wo clmpd nds which is dscribd s hollow ub s shown in Fig. 1. Th lsic mdium is simuld by inklr-yp nd Psrnk-yp modls 6. Figur 1. A curvd SCNT mbddd in n lsic mdium wih wo fid nds. Bsd on nonlocl Eulr-Brnoulli bm hory nd using h Hmilon principl h govrning quion of moion for h curvd SCNT cn b prssd s

3 3 d EA P P m EI (1) hr is h il coordin is h im () is h rnsvrs displcmn componn nd () rprsns h curvur of h SCNT. EI A nd m r h bnding rigidiy h crossscion r nd h mss pr uni lngh of h SCNT. P() shows h rnl disribud lod nd is nonlocl prmr illumining h nno-scl ffc on h rspons of h srucur This quion cn b rducd o h sm quion in Rf. 3 whn h SCNT is ssumd wihou curvur. Furhrmor if K =K G = = Eq. (1) will b chngd o h sm quion h suggsd by Myoof nd Hww 8. Th rnl forc P() is rld o inrcion bwn h SCNT nd h surrounding mdium which cn b dscribd by h inklr-yp nd Psrnk-yp modls 6 nd K K P G () K nd K G indic h inklr consn nd Psrnk shr consn of h lsic mdium in h ordr. Subsiuion of Eq. () ino Eq. (1) lds o h pril quion of moion (3) for h fr vibrion of h curvd SCNT G d EA K K m EI (3) In his ppr r sndrd boundry condiions hv bn considrd bm clmpd boh nds or h clmpd clmpd boundry condiion. () A bm clmpd on nd nd simply suppord h ohr nd i.. clmpd pinnd condiion (5) And simply suppord bm boh nds or pinnd pinnd condiion (6) In ddiion h sinusoidl smll ris funcion is inroducd by H sin ) ( (7) hr H is h mpliud of curvur. 3. Soluion Glrkin s mhod is uilizd in ordr o obin n ordinry diffrnil quion (ODE) from Eq. (3). Iniilly Eq. (3) mus bcom dimnsionlss quion. For his purpos inroducing h dimnsionlss quniis s

4 EI w m A K (8) y s k I EI KG H kg n h EI Eq. (3) cn b wrin in h dimnsionlss form w w w w w n k G n (9) w y y 1 y w k w n s d n Th procss of Glrkin mhod is srd by spring h dpndncs of h dflcion of h bm w(ξτ) ino mporl funcion q(τ) nd h fundmnl mod shp φ(ξ) s w( ) ( ). q( ) (1) Th spilly dpndn mod shp sisfis h corrsponding boundry condiions in Eqs. (-6). Hnc h firs mod of shp funcion for clmpd-clmpd bm cn k from ( ) cosh sinh.73 cos sin. 73 (11) Bsis funcion for clmpd-pinnd s will b ( ) cosh sinh3.9 cos sin3. 9 (1) And for pinnd-pinnd my b givn s ( ) sin3. 1 (13) Muliplying quion (9) by h mod shp nd ingring ovr h lngh of h SCNT h govrning ordinry diffrnil quion (1) is obind by ssuming h dimnsionlss quion of curv y( ) hsin q (1) q hr is h nurl frquncy nd hs h vribl vlu for hr boundry condiions. For h clmpd clmpd boundry condiion my b prssd s n kg n k kg k sh n sh n (15) in h clmpd-pinnd s cn b wrin s n kg n k kg k sh n sh n (16) And for pinnd-pinnd w g. Rsuls n kg n k kg k sh n sh (17) n In his sudy h fr vibrion quion of h curvd SCNT hs bn drivd by using h nonlocl Eulr-Brnoulli hory. Th our dimr hicknsss nd oung s modulus of h nno

5 ub r ssumd o b d =3.19 nm c =.137 nm nd E=.7 TP rspcivly b. Th mss dnsiy of SCNT is 3 kg/m 3 wih nonlocl prmr of nm nd spc rio /d =. Also h inklr-consn nd Psrnk-consn r simd h vlus of K =1 MP nd K G =5nN in h ordr 6. Fig. shows h fundmnl frquncy ω gins h curvur mpliud H for hr sndrd boundry condiions. I dmonsrs h wih incrsing h mpliud of wvinss h frquncy incrss. Also h rsuls r complly dpndn on h boundry condiions nd h nurl frquncy is incrsd whil h bnding siffnss of h SCNT riss from pinnd-pinnd o clmpd-clmpd spcilly for h low curvur mpliud. Figur. Th fundmnl frquncy ω gins h curvur mpliud H for hr ypicl boundry condiions. Morovr o s h ffcs of curvur clrly h diffrnc prcn is dfind s prmr h shows h prcn incrmn of frquncy for h curvd SCNT (H nm) comprd wih h srigh nnoub (H= nm). H H nm H nm 1 Diffrnc prcn H nm (18) Crinly diffrnc prcn givs br illusrion for h pur ffcs of h mpliud of curvur. Figs. 3-7 rprsn h diffrnc prcn s funcion of h wvinss mpliud H whil h ffcs of crin prmr such s h siffnss of modl h spc rio nd h nonlocl prmr hv bn vlud in ch figur. Obviously h vriion of fundmnl frquncy is incrsd whn h wvinss gos up in ll h figurs. Figur 3. Th diffrnc prcn gins h curvur mpliud H for clmpd-clmpd wih diffrn vlus of h inklr modulus K. 5

6 Th diffrnc prcn is highly snsiiv o h siffnss of modl du o h foundion nd boundry condiions. Figs. nd 5 dpic h influnc of mdium on h vibrion of curvd SCNT. Th rsuls show h s h mdium siffnss cusd by inklr-consn K nd Psrnk-consn K G incrs h diffrnc prcn dcrss. In ddiion Fig. 5 indics h h boundry condiions hv h considrbl ffcs on h diffrnc prcn. Th illusrion dmonsrs h wih dcrsing h siffnss of SCNT from clmpd-clmpd o pinndpinnd h ffcs of wvinss on vibrionl frquncy incrs. Figur. Th diffrnc prcn gins h curvur mpliud H for clmpd-clmpd wih diffrn vlus of h Psrnk modulus K G. Figur 5. Th diffrnc prcn gins h curvur mpliud H wih diffrn yps of boundry condiions. Fig. 6 shows h impornc of h spc rio /d in h nurl frquncy nd ssocid diffrnc prcn. According o his figur h ris of frquncy wih curvur mpliud is found o b significnly dpndn on h spc rio i.. for /d = h diffrnc prcn incrss spr hn /d =8. I mns for long nnoub h diffrnc of frquncy bwn h srigh SCNT nd curvd SCNT is modrly rducd. Finlly o invsig h ffc of nonlocl hory h diffrnc prcn is plod s funcion of h curvur mpliud H nd h nonlocl prmr in Fig. 7. As i shows by incrs- 6

7 ing h nonlocl ffc h diffrnc prcn incrss for vry fid mpliud H lhough his ffc is ngligibl Figur 6. Th diffrnc prcn gins h curvur mpliud H for clmpd-clmpd wih diffrn vlus of h spc rio /d. Figur 7. Th diffrnc prcn gins h curvur mpliud H for clmpd-clmpd wih diffrn vlus of h nonlocl prmr. 5. Conclusion A nonlocl coninuum modl hs bn dvlopd o nlyz h ffc of wvinss on h curvd singl-wlld crbon nnoub. Th surrounding lsic mdium is simuld s h inklr nd Psrnk modls. Th Glrkin mhod is mployd o solv h govrning quion of moion. Th rsuls show h influnc of h curvur h siffnss of mdium round h SCNT h boundry condiions h spc rio of SCNT nd h nonlocl prmr on h nurl frquncy. Dild rsuls dmonsr h incrsing of h mpliud of curvur cuss h fundmnl frquncy o incrs. Furhrmor wih n incrs in h nonlocl consn s h siffnss of modl du o h boundry condiions or o h foundion nd lngh of SCNT dcrs h frquncy is obind highr vlus for curvd SCNT in comprison srigh SCNT. 7

8 REFERENCES 1. Iijim S. Hlicl microubuls of grphiic crbon. Nur (638) () Ebbsn T. Crbon nnoubs: prprion nd propris. CRC Prss: Nw ork 1997; (b) Gup S. S.; Bosco F. G.; Br R. C. ll hicknss nd lsic moduli of singlwlld crbon nnoubs from frquncis of il orsionl nd innsionl mods of vibrion. Compuionl Mrils Scinc 1 7 () ; (c) Thosnson E. T.; Rn Z.; Chou T.-. Advncs in h scinc nd chnology of crbon nnoubs nd hir composis: rviw. Composis Scinc nd Tchnology 1 61 (13) u P.; H. P.; u C.; Zhng P. Q. Applicion of nonlocl bm modls for crbon nnoubs. Inrnionl Journl of Solids nd Srucurs 7 (16) Rskh M.; Khdm S. E. Nonlinr vibrion nd sbiliy nlysis of illy lodd mbddd crbon nnoubs convying fluid. Journl of Physics D: Applid Physics 9 (13). 5. ng Q.; Vrdn V. K. Vibrion of crbon nnoubs sudid using nonlocl coninuum mchnics. Smr Mrils nd Srucurs 6 15 () Murmu T.; Prdhn S. C. Buckling nlysis of singl-wlld crbon nnoub mbddd in n lsic mdium bsd on nonlocl lsiciy nd Timoshnko bm hory nd using DQM. Physic E: ow-dimnsionl Sysms nd Nnosrucurs 9 1 (7) Joshi A..; Bhngr A.; Shrm S. C.; Hrsh S. P. Vibrory nlysis of doubly clmpd wvy singl wlld crbon nnoub bsd nno mchnicl snsors. Inrnionl Journl of Enginring Scinc nd Tchnology 1 (5) Myoof F. N.; Hww M. A. Choic bhvior of curvd crbon nnoub undr hrmonic ciion. Chos Solions & Frcls 9 (3)