FREE VIBRATION AND BENDING ANALYSES OF CANTILEVER MICROTUBULES BASED ON NONLOCAL CONTINUUM MODEL

Transcription

1 Mhmicl nd Compuionl Applicions Vol. 15 o. pp Associion for Scinific Rsrch FREE VIBRATIO AD BEDIG AALYSES OF CATILEVER MICROTUBULES BASED O OLOCAL COTIUUM MODEL Ömr Civlk Çiğdm Dmir nd Bkir Akgöz Akdniz Univrsiy Civil Enginring Dprmn Anly-TURKIYE civlk@yhoo.com Absrc- Th quions of moion nd bnding of Eulr-Brnoulli bm r formuld ug h nonlocl lsiciy hory for cnilvr microubuls MTs. Th mhod of diffrnil qudrur DQ hs bn usd for numricl modling. Th siz ffc is kn ino considrion ug h Eringn s non-locl lsiciy hory. Frquncis nd dflcions of MTs r obind. umricl rsuls r prsnd o sho h ffc of smll-scl ffc on bnding nd vibrion of MTs. Ky Words: Microubuls onlocl lsiciy hory Vibrion Dflcion. 1. ITRODUCTIO Microubuls MTs microfilmns nd inrmdi filmns r h min componns of cyosklon. MTs r proins orgnizd in nork h is inrconncd ih microfilmns nd inrmdi filmns o form h cyosklon srucurs [1]. Th mchnicl propris of microubuls ply n imporn rol in procsss such s cll division nd inrcllulr rnspor []. Thr hv bn numbr of primnl nd mhmicl sudis in h ls f dcds dling ih h mchnicl propris of MTs [3]. Microubuls r h mos rigid ons of h cyoskll filmns nd hv h mos compl srucur. Bnding vibrion nd buckling nlyss of microubuls hv bn rcnly invsigd ug h coninuum modl [3-6]. Much nion hs bn dvod lso o h mchnicl bhvior of micro/nno srucurs such s nnobms nnorods nd nnoubs [7-15]. In his sudy h govrning quions for h bm modl for cnilvr MTs r drivd for vibrion nd bnding nlysis. Th rsuls for sic loding nd vibrion r obind by ug h DQ mhod.. OLOCAL COTIUUM MECHAICS FOR BEAMS In h clssicl lsiciy h srss s of ny body poin is rld o h srin s h sm poin. mly h consiuiv quions of clssicl mcroscopic lsiciy r lgbric rlionships bn h srss nd srin componns. Bu his hory dos no conflic h omic hory of lic dynmics nd primnl obsrvion of phonon disprsion. As sd by Eringn [16] h linr hory of nonlocl lsiciy lds o s of ingropril diffrnil quions for h displcmns fild for homognous isoropic bodis. According o h nonlocl lsiciy hory of Eringn h srss ny rfrnc poin in h body dpnds no only on h srins his poin bu lso on srins ll poins of h body. This

2 9 Ö. Civlk Ç. Dmir nd B Akgöz dfiniion of h Eringn s nonlocl lsiciy is bsd on h omic hory of lic dynmics nd som primnl obsrvions on phonon disprsion. In his hory h fundmnl quions involv spil ingrls hich rprsn ighd vrgs of h conribuions of rld srin nsor h rld poin in h body. Thus h hory inroducs h smll lngh scl ffc hrough spil ingrl consiuiv rlion. For homognous nd isoropic lsic solids h linr hory of nonlocl lsiciy is dscribd by h folloing quions [16]: σ ul σkl l + ρ fl 1 α χ τ dv kl kl V τ λε δ + µε 3 kl mm kl kl 1 u k ul ε kl + l k hr σkl is h nonlocl srss nsor ρ is h mss dnsiy of h body f l is h body or pplid forc dnsiy ul is h displcmn vcor rfrnc poin in h body τ kl is h clssicl Cuchy or locl srss nsor ny poin in h body ε kl is h linr srin nsor poin in h body dnos h im V is h volum occupid by h lsic body α is h disnc in Euclidn form λ nd µ r h Lm consns. Th non-locl krnl α is dfind s h impc of h srin h poin on h srss h poin in h lsic body. Th vlu of χ dpnds on h rio / l hich is mril consn. Th vlu dpnds on h inrnl grnulr disnc lic prmr disnc bn C-C bonds s molculr dimrs nd rnl chrcrisic lnghs crck lngh or v lngh nd is consn ppropri o ch mril for djug h modl o mch rlibl rsuls by primns or som ohr horis. If α ks on Grn funcion of linr diffrnil opror givn s [16] δ R α 5 h nonlocl consiuiv rlion givn by Eq. is rducd o h diffrnil quion R σ τ 6 kl kl

3 Fr Vibrion nd Bnding Anlyss of Cnilvr Microubuls 91 Furhrmor h ingro-pril diffrnil quion givn by Eq. 1 is lso rducd o h folloing pril diffrnil quion τ R f ρu& 7 kl l + l k Eringn 3 proposd nonlocl modl for his linr diffrnil opror givn s hr R 1 8 is h Lplcin. Consqunly h consiuiv rlions cn b rin s [ ] σ kl τ kl GOVERIG EQUATIOS FOR MTs I is ccpd h som mchnicl bhvior such s vibrion bnding nd buckling of h bm-lik micro srucurs bsd on nonlocl lsiciy hory r nirly diffrn from hir counrprs bsd on h clssicl mcro bm hory [-1]. Thus h hory bsd on siz dpndn nonlocl lsiciy hory could srv s mor rsonbl nd propr pproch in mchnicl modling of micro nd nno sizd componns of nno mchnicl dvics [11-3]. Th nonlocl hory of lsiciy proposd by Eringn [16] hs bn idly usd in h ps fiv yrs in mny nno mchnicl problms including dislocion crck v propgion vibrion nlysis of nnobms nnoubs crbon nnoubs nd microubuls. For MTs in on dimnsionl cs h nonlocl consiuiv rlions cn b rin s blo [6] σ Eε 1 σ hr σ is h il srss ε is h il srin E is h Young modulus. Assum h h rnsvrs displcmn of bm long y il is is in rms of spil coordin nd im vribl. For rnsvrs vibrion of MTs h quilibrium condiions of Eulr-Brnoulli bm cn b rin s M ρa 11 According o h linr hory of Eulr-Brnoulli bm h srin-displcmns nd h momn r givn by ε y M yσda 113 A

4 Ö. Civlk Ç. Dmir nd B Akgöz 9 In his sg muliplying by y on boh sids of Eq. 1 nd ingring ovr h crossscion r of h bm obin da Ey da y yda A A A ε σ σ 1 Afr som mhmicl mnipulions hv + M M 15 By diffrniing his quion ih rspc o h vribl ic obin + M M 16 Finlly by subsiuing Eq. 11 ino Eq. 16 obin h folloing govrning nonlocl quion for vibrion of MTs bsd on Eulr-Brnoulli bm hory [6] + ρa ρa 17 I is cly sn from h Eq. 17 h h locl Eulr-Brnoulli bm hory is obind hn h prmr is s idniclly o zro. In his cs nonlocl bnding momn by ug hrmonic funcion for cn b rin s ω L ρa L M 18 hr ω is h ngulr frquncy. If considr h Eulr-Brnoulli bm subjcd o disribud lod h gnrl nonlocl quion for his cs is givn blo q d q d q d d + 19 by ug h rlion bn h lod nd dflcion h is q Subsiuing h Eq. ino Eq. 19 obin

5 Fr Vibrion nd Bnding Anlyss of Cnilvr Microubuls 93 6 d d + q q 1 6 d d Clmpd-fr boundry condiions r considrd. Ths r: For clmpd C nd For fr F nd L nd d / d V nd M b. DIFFERETIAL QUADRATURE DQ METHOD Diffrnil qudrur DQ mhod is rlivly n numricl chniqu in pplid mchnics. Th mhod of DQ cn yild ccur soluions ih rlivly fr grid poins. I hs bn lso succssfully mployd for diffrn solid nd fluid mchnics problms [17-1]. Unlik h DQ h uss h polynomil funcions such s por funcions Lgrng inrpold nd Lgndr polynomils s h s funcions hrmonic diffrnil qudrur mhod uss hrmonic or rigonomric funcions s h s funcions. Shu nd Xu [17] proposd n plici mns of obining h ighing cofficins for h hrmonic DQ. f is pproimd by Fourir sris pnsion in h form / kπ kπ f c + ck cos + d k k 1 L L 3 nd h Lgrng inrpold rigonomric polynomils r kn s hk π k π k 1 π k + 1 π k k 1 π k k + 1 π π k π for k 1... According o h HDQ h ighing cofficins of h firs-ordr drivivs A for i j cn b obind by ug h folloing formul: A π P i P j [ i j / ] π ; i j hr

6 9 Ö. Civlk Ç. Dmir nd B Akgöz i j P π i ; for j j 1j i Th ighing cofficins of h scond-ordr drivivs B for i j cn b obind ug folloing formul: i j 1 Aii πcg ; i j B A π p Th ighing cofficins of h firs-ordr nd scond-ordr drivivs A r givn s for i j A p A p ii j 1 j i ; p 1 or ; nd for i By ug DQ discrizion h Eq. 1 ks h form for bnding L D j + qi j 1 j 1 F q 9 j i For vibrion h discr form of govrning quion ks h form: j 1 D W j ω ρaw + ρaω B W 3 i j 1 j Th ighing cofficin of h fourh- nd sih-ordr drivivs D F cn b compud sily from B by D B ik k 1 k 1 B kj ; F D Bkj ik UMERICAL RESULTS In his scion svrl mpls of MTs hving clmpd-fr boundry condiions r mind. For his purpos som numricl rsuls in h forms of grphs r prsnd ug h mhod of DQ. Th mril nd gomric consns of MTs r givn in Tbl 1. Th symbol C-F rprsns h MT hving clmpd dg nd fr dg L. To drmin h ccurcy nd convrgnc of h prsn mhod for dflcion nlysis of MTs numricl primnion s crrid ou by vrying h numbr of grid poins.

7 Fr Vibrion nd Bnding Anlyss of Cnilvr Microubuls 95 Tbl 1. Mril nd gomric vlus of MTs Prmr Vlu E 1 9 /m I m ρ 17 kg/m 3 υ.3 Th obind rsuls of non-dimnsionl sic dflcions nd vibrion compud for diffrn vlus r shon in Tbl. Ec nlyicl soluion [5] is lso givn for comprison. I is imporn o s h h rsuls givn by hs rfrncs [] rld o crbon nnoubs bsd on bm hory. Hovr solvd microubuls problm firsly. So compr only cofficin h rsuls r prsnd s nondimnsionl. Tbl. Comprison of non-dimnsionl mimum dflcion /ql of C-F MTs undr uniformly disribud loding Anlyicl Prsn DQ soluions /L Rsul Rf W E q.1 q.3 q /L Fig. 1. Dflcion of C-F Crbon nnoubs m Eclln grmn hs bn chivd bn h prsn rsuls nd h rsuls obind by nlyicl formul givn by Rddy nd Png []. I is sn from his bl

8 96 Ö. Civlk Ç. Dmir nd B Akgöz h hn h grid poin numbrs rchs 7 h prsn mhod givs ccur prdicions for h dflcions. For frquncy vlus hovr hr r no rsuls for comprison. Figs. 1 nd sho h displcmn long h lngh of MTs for diffrn vlus of lod nd nonlocl prmrs. I cn b sn h h ffc of nonlocl prmr on h dflcion is insignificn. In gnrl h nonlocl prmr rsuls in n incrs in h rnsvrs dflcion of MTs undr uniformly disribud lod. 5 W E E-9 3.E-9 5.E /L Fig.. Dflcion of C-F microubuls for diffrn nonlocl prmr q. /m 6 5 α α. α.3 fhz*1^ Mod numbr Fig.3. Frquncis of C-F MTs for diffrn nonlocl prmr L5 1-6 m; α /L

9 Fr Vibrion nd Bnding Anlyss of Cnilvr Microubuls 97 Fig.3 shos h vriion of h frquncy vlu ih mod numbr of C-F microubuls for diffrn nonlocl prmr. I is sn h h frquncy vlus for ll h hr nonlocl prmrs α/l incrs s mod numbr incrss. I is lso shon h h frquncy vlus dcrs ih incrg nonlocl prmr. 6. COCLUDIG REMARKS In his ppr fr vibrion nd sic bhvior of MTs r invsigd. Th numricl rsuls sho h h nonlocl prmr is ffcd on sic nd dynmic bhviour of MTs. Th mhod is suibl for h problm considrd du o is gnrliy simpliciy nd ponil for furhr dvlopmn. Alhough no providd hr h mhod is lso usful in providing buckling soluions of MTs ug nonlocl lsiciy hory. Acknoldgmns Th finncil suppor of h Scinific Rsrch Projcs Uni of Akdniz Univrsiy is grfully cknoldgd. 7. REFERECES 1. J. Hord Mchnics of Moor Proins nd h Cyosklon. Sinur Associs Inc Sundrlnd MA.1. D. Bol Mchnics of h cll Cmbridg Univrsiy Prss Cmbridg. 3. C.Y. Wng C.Q. Ru A. Mioduchoski Orhoropic lsic shll modl for buckling of microubuls. Phys Rv. E J.. Rddy S.D. Png onlocl coninuum horis of bms for h nlysis of crbon nnoubs. J Appl Phy J.. Rddy onlocl horis for bnding buckling nd vibrion of bms. In J Eng Scincs K.M. Li Q. Wng Anlysis of v propgion in crbon nnoubs vi lsic shll horis Inrnionl Journl of Enginring Scinc Q. Wng K.M. Li Applicion of nonlocl coninuum mchnics o sic nlysis of micro- nd nno-srucurs. Physics Lrs A Ö. Civlk Ç. Dmir B. Akgöz Sic Anlysis of Singl Wlld Crbon noubs SWCT Bsd On Eringn s onlocl Elsiciy Thory. Inrnionl Journl of Enginring nd Applid Scincs IJEAS Ç. Dmir Ö. Civlk B. Akgöz Fr Vibrion Anlysis of Crbon noubs Bsd On Shr Dformbl Bm Thory By Discr Singulr Convoluion Tchniqu. Mhmicl nd Compuionl Applicions In prss C. Li C.Q. Ru A. Mioduchoski Torsion of h cnrl pir microubuls in ukryoic flgll du bnding-drivn lrl buckling Biochmicl nd Biophysicl Rsrch Communicions

10 98 Ö. Civlk Ç. Dmir nd B Akgöz 11. C.Y. Wng L.C. Zhng Circumfrnil vibrion of microubuls ih long il vlngh Journl of Biomchnics X.S. Qin J.Q. Zhng C.Q. Ru Wv propgion of orhoropic microubuls J Appl Phys B. Akgöz Modling of Crbon nnoubs s Timoshnko nd Eulr bms B.Sc. Thsis in Turkish Akdniz Univrsiy9. 1. Ç. Dmir B. Akgöz Ö. Civlk Fr Vibrion And Bnding Anlysis Of Crbon noubs Ug onlocl Eulr Bm Thory Inrnionl Symposium on Enginring nd Archicurl Scincs of Blkn Cucsus nd Turkic Rpublics Ç. Dmir Ö. Civlk A. Korkmz Frquncis Anlysis Of Tprd Crbon noubs Embddd In To-Prmr Elsic Foundion Inrnionl Symposium on Enginring nd Archicurl Scincs of Blkn Cucsus And Turkic Rpublics A.C. Eringn On diffrnil quions of nonlocl lsiciy nd soluions of scr dislocion nd surfc vs. J. Applid Physics C. Shu H. Xu Eplici compuions of ighing cofficins in h hrmonic diffrnil qudrur. J. Sound Vib Ö. Civlk Applicion of Diffrnil Qudrur DQ nd Hrmonic Diffrnil Qudrur HDQ For Buckling Anlysis of Thin Isoropic Pls nd Elsic Columns. Enginring Srucurs Ö. Civlk M. Ülkr Hrmonic Diffrnil Qudrur HDQ For Aisymmric Bnding Anlysis Of Thin Isoropic Circulr Pls. In. J. Srucurl Enginring nd Mchnics C. H. W. g Y. B. Zho Y. Xing nd G. W. Wi On h Accurcy nd Sbiliy of Vriy of Diffrnil Qudrur Formulions for h Vibrion Anlysis of Bms. Inrnionl Journl of Enginring nd Applid Scincs IJEAS M. Şimşk Sic Anlysis of Funcionlly Grdd Bm undr Uniformly Disribud Lod by Riz Mhod Inrnionl Journl of Enginring nd Applid Scincs IJEAS M. Aydoğdu Ail vibrion of nnorods ih h nonlocl coninuum rod modl. Physic E R. Arn A. Tp Th iniil vlus mhod for buckling nonlocl brs ih pplicion in nnochnology. Europn Journl of Mchnics A/Solids