Analysis of Cantilever beams in Liquid Media: A case study of a microcantilever

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1 Intrntionl Journl of Enginring Scinc Invntion (IJESI) ISSN (Onlin): 9 67, ISSN (Prin: ǁ PP.57-6 Anlysis of Cntilvr bms in iquid Mdi: A cs study of microcntilvr S. Mnojkumr * nd J. Srinivs Dprtmnt of Mchnicl Enginring, Institut of Tchnology, Rourkl, Indi. ABSTRACT: In modrn r of tomic forc microscop, it cn b usd in micro snsing pplictions in rospc nd fluid-flow nginring. Th micro-snsor in such pplictions ncountrs vrious typs of fluid mdi. Th study of convntionl micro-cntilvrs is not pplicbl in liquids. Th bhvior of th AFM cntilvr in liquid mdi hs bn studid by mny rsrchrs during th pst fiv yrs. Hydrodynmic forcs in th systm r oftn modld s nonlinr functions of th tip displcmnt. On th othr hnd microcntilvrs snsors cn lso b usd for msurmnt of micro scl viscosity, dnsity, nd tmprtur in vionic pplictions by nlyzing frquncy rspons of th cntilvr. In this ppr, micro-cntilvr with its tip oprting in tpping mod is considrd with liquid nvironmnt nd modld using continuous systm dynmics. Th hydrodynmic forcs nd dditionl mss from th liquid r ccountd in th qution of motion. Also th first mod dynmics is considrd for solving qutions of motion using Glrkin s mthod. It is lso shown mthodology to msur fluid dnsity nd viscosity using microcntilvr probs. Kywords: Flow physics, Flxurl vibrtion, Hydrodynmic forc, Modl pproximtion. I. INTRODUCTION Th study of flxurl vibrtions of bms nd plts submrgd in viscous fluid is drwing n incrsd ttntion in mny rsrch filds such s tomic forc microscopy, micromchnicl oscilltors for snsing nd ctutions, micro scl nrgy hrvstrs nd biomimtic propulsions. In ll ths pplictions th stimtion of forcs xrtd by th fluid on th structur is of primry importnc. Such forcs includ distributd lift nd thrust producd by momntum trnsfrrd to th fluid. Ths forcs r rltd to complx flow fild gnrtd by solid body motion which is influncd by inrtil nd viscous phnomnon. A first stimt of distributd lift of thin bm with rctngulr cross sction is givn by Sdr []. In this work, lngth to width rtio ws slctd vry lrg nd is subjctd to low frquncy xcittion, so tht bm is loclly considrd s infinitly long cylindr nd fluid loding is nlyzd using numricl findings bsd on unstdy Stoks flow. Brntto t l. [] xplord th possibilitis of xtrcting nrgy from mchnicl vibrtion using ionic polymr mtl composits in which th hydrodynmic function-xprssions wr proposd ovr som rng of Rnult s numbrs. Aurli t l. [] proposd n xtnsion to tk in to ccount finit mplitud oscilltions for two dimntionl numricl simultions of flow physics inducd by rigid lmin oscillting in viscous fluid. It is dmonstrtd in othr pprs [] tht s th mplitud incrss, th rlvnt nonlinr hydrodynmic dmping would lwys xists. In th prsnt work, w considr th flow inducd by vibrtion of cntilvr bm submrgd in viscous fluid to dtrmin th influnc of prmtrs, such s frquncy nd mplitud of oscilltion, spct rtio on th forcs xrtd by fluid on th structur. An in-pln flxurl vibrtion of th bm modld using clssicl linr bm thory nd is ssumd to b vibrting long its fundmntl mod shp. Th fluid is ssumd to b Nwtonin nd flow is incomprssibl. II. PROBEM STATEMENT. Bm vibrtion in liquids W considrd flxurl vibrtion of cntilvr bm undr hrmonic bs xcittion. t x b th coordint long bm xis with y nd z r th co-ordints long width nd thicknss. Bm is slndr nd composd of homognous nd isotropic mtril. Th clssicl linr Eulr-Brnoulli bm thory givs th qution of motion s: wx, t w x, t K bh F ( x, S( x, F( hyd () x x t 57 P g

2 Anlysis of Cntilvr bms in iquid Mdi: A cs study of microcntilvr Ebh whr, K, b nd h r width nd thicknss, Mss dnsity of cntilvr, w ( x, Bm dflction, F( F sin( Hrmonic bs xcittion, S(x,=-B w x, t t is th dmping forc, ngth of bm, F hyd (x, dscribs hydrodynmic ction xrtd on th bm by th ncompssing fluid. Th ffct of liquid viscosity cn b tkn cr by simpl modl. Rsrchrs hv pproximtd th hydrodynmic forcs to b in proportion to th cntilvr cclrtion nd vlocity s: dw d w Fhyd x, t c () dt dt Whr, c dditionl hydrodynmic dmping cofficint= b liq nd dditionl mss dnsity liq. Hr, is vibrting frquncy of th cntilvr, is kinmtic liqb b viscosity of liquid, liq is dnsity of th liquid.. Solution mthodology In ordr to solv th dynmic qutions in continuous form, th Glrkin s pproximtion mthod is M mployd. Hr w considrd w(x,= i ( x) qi ( whr M is numbr of mods usd, i (x) is i th i normlizd modl function. As first mod domints, oftn w(x, is pproximtd s (x)q (. Hr, = (x) is obtind from th boundry conditions of th bm. Th mod shp function (x) is multiplid on both sids of th diffrntil q.() nd th rsultnt qution is intgrtd long th cntilvr lngth. i.. q K dx ( bh ) q dx B c q dx F q t dx x ( ) sin () III. NUMERICA EXAMPE In ordr to illustrt th mthodology, microcntilvr bm with nno-tip usd in AFM snsing [5] subjctd to hrmonic bs xcittion is considrd s shown in Fig.. Svrl rlir works dmonstrtd th oprtion of such bms in liquid mdi. Song nd Bhushn [6] usd finit lmnt modl to know frquncy nd trnsint rspons nlysis of cntilvrs in tpping mod oprting in ir s wll s liquid. Korym t l. [7] showd tht th frquncy rspons bhvior of micro cntilvr in liquid is compltly diffrnt from tht in ir nd studid th influnc of mchnicl proprtis of th liquid lik viscosity nd dnsity on frquncy rspons nlysis. Vncur t l.[8] nlyzd chrctristics of rsonnt cntilvr in viscous liquids using rctngulr cntilvrs gomtris in pur wtr, glycrol nd thnol solution with diffrnt concntrtion. His study rsults cn b usd in rsonnt cntilvrs s biochmicl snsors in liquid nvironmnts. Fig. Micro-cntilvr bm undr considrtion 58 P g

3 Anlysis of Cntilvr bms in iquid Mdi: A cs study of microcntilvr In ddition to th hydrodynmic nd hrmonic forcs, th systm is subjctd to n tomic intrction forc f ID ( in microscopic lvl. Th gnrl mod shp function is obtind from th following boundry conditions: w(, x At x =, w(, =, nd w( w( w( At x = K, nd K m f ( ID x x x Hr, f ID (=-k ts w( is linrizd tip-smpl intrction forc, with contct stiffnss f ( k ts = ID = w( E * HR, if ( z w( ) z R( z ), if ( z w( ) whr, H is Hmkr constnt, z is quilibrium distnc btwn cntilvr nd smpl, R is quivlnt tiprdius, E * =[(- t )/E t +(- s )/E s ] - is ffctiv lstic modulus, is intrtomic distnc nd m is quivlnt tip mss ddd. Th frquncy qution nd ignfunction cn b obtind from bov four boundry conditions s follows [9] EI kts m A A whr. Th normlizd mod shp is EI ( x) (cos cosh)(sin x sinh x) (sin sinh )(cosx coshx) N whr N (sin cosh cossinh ) sin cosh cos sinh EI cos cosh (8) (9) Th computtions r prformd with MATAB symbolic logic progrm, which cn rsolv th qutions into ordinry diffrntil form in trms of q. IV. RESUT AND DISCUSSION Tbl shows th dt considrd for nlysis. Tbl. Prmtrs of simultion for th AFM cntilvr [6] Cntilvr lngth () µm Cntilvr width (b) µm Cntilvr thicknss ( 7.7 µm Cntilvr mss dnsity () 7 Kg/m Cntilvr Young s Modulus (E) GP Qulity fctor in ir (Q) 9 iquid dnsity( liq ) Kg/m iquid viscosity(). - Kg/m Cntilvr ngl() 5 Numbr of lmnts(n) Tip lngth(l) µm Tip rdiud(r) nm Hmrkr constnt (H).96-9 J Intrmolculr distnc ( ).8 nm Effctiv lstic modulus (E * ). GP Effctiv lstic modulus (G * ). GP () (5) (6) (7) Th ffct of quivlnt linr intrction stiffnss shown in Fig.. kˆ ts k ts / k, whr k=al n on nturl frquncis is s 59 P g

4 vlocity of th cntilvr displcmnt Cntilvr t th tip q (µm) Nturl frquncy (Hz) Anlysis of Cntilvr bms in iquid Mdi: A cs study of microcntilvr.6 x Normlisd quivlnt intrction stiffnss Fig. Grph of Normlisd quivlnt stiffnss vs. nturl frquncy Hr th dottd lin indicts th nturl frquncy of norml cntilvr in ir without tip mss. It is sn tht vn if intrction stiffnss is zro, th nturl frquncy mismtch with dshd lin is du to th tipmss boundry condition. Using th modl function vilbl ftr solving frquncy qution, th prtil diffrntil is rducd into scond ordr diffrntil qution in trms of vribl q s pr Eq.(). This is solvd with Rung-Kutt s fourth ordr mthod, to study th ffct quivlnt stiffnss on tim rspons. Th viscous dmping rtio considrd in prsnt work is.. Fig. shows th tim history nd phs digrm for th systm with kˆ ts =.. x tim(s) Fig. Vrition of th displcmnt(µm) of systm with rspct to tim (s) Fig. shows th grph of th displcmnt of th cntilvr vs. vlocity of th cntilvr for th systm. x displcmnt of cntilvr x - Fig. Grph of displcmnt vs. vlocity of th cntilvr. 6 P g

5 Anlysis of Cntilvr bms in iquid Mdi: A cs study of microcntilvr IV. CONCUSIONS In this ppr cntilvr bm dynmics using first mod mchnics in liquids ws considrd. Th ffct of hydrodynmic forc xrtd by ncompssing fluid ws studid. Glrkin s pproximtion mthod ws usd to gt th normlizd modl function. Rung-Kutt solvr is usd to solv this scond ordr ordinry diffrntil qution in tim vribl. Th ffct of normlizd quivlnt intrction stiffnss on nturl frquncy is studid. Furthr work is going on. It cn b concludd tht thr is trmndous ffct of hydrodynmic forcs on th modl chrctristics of cntilvrs. APPENDIX Modl function is pproximtd in trms of frquncy prmtr s: x) C cos x C sin x C coshx C sinh x ( Th constnts C to C r obtind from following boundry conditions: At x, w (, ( ) C C At x w (, ( ) C C ( x) C(cos x coshx) C (sin x sinh x) Furthr t x K w ( C cos cosh) C ( sin sinh ) (A) At ( d w Kw ( m k w( t ts x ) dt K C ( sin - sinh) C ( cos - cosh ) m ( )( jt ) k ( ) ts jt K C( sin - sinh) C ( cos - cosh ) ( m kts ) ( C(cos cosh) C (sin sinh ) K ( sin - sinh) ( m kts )(cos cosh) C K ( cos - cosh ) ( m kts )(sin sinh ) C Eliminting C nd C frin qs.(a) nd (A), w gt th frquncy qution (7): jt REFERENCES []. J. E. Sdr, frquncy rspons of cntilvr bm immrsd in viscous fluids with ppliction to tomic forc microscop, journl of pplid physics, vol.8 pp 6-76, 998. []. P. Brntto nd. fortun, S. Grzini nd S. Strzzr, A modl of ionic polymr mtl composits in undrwtr oprtions, smrt mtrils nd structurs, vol. 7, pp. 5-9, 8 []. M. Aurli, M. Msrn, M. Porfir, Non-linr finit mplitud vibrtions of shrpd dgd bms in viscous fluids, Journl of sound nd vibrtion, vol., pp. 6-65, []. G. Flcucci, M. Aurli, S. Ubrtini, nd M. porfir, trnsvrs hrmonic osciltions of lmi in viscous fluid in lttic Boltzmnn study, philosophicl trnsctions of royl socity of ondon, Prt A, vol. 69, pp ,. [5]. N. Jlili, nd K. xminryn, A rviw of tomic forc microscopy imging systms: ppliction to molculr mtrology nd biologicl scincs, Mchtronics, vol., pp.97-95,. [6]. Y. song, nd B. Bhushn, Finit-lmnt vibrtion nlysis of tpping mod tomic forc microscopy in liquid, Ultrmicoscopy, vol. 7, pp. 95-, 7. [7]. M. H. korym, H. Shrhi, nd A. H. Korym, Comprison of frquncy rspons of tomic forc microscopy cntilvrs undr tip smpl intrction in ir nd liquids, Scinti Irnic, vol. 9, pp. 6-,. [8]. C. Vncu, I. Dufour, S. Hinrich, F. Joss nd A hirlmnn, Anlysis of rsonting microcntilvr oprting in liquid nvironmnt, Snsors nd Actutors,A, vol.,pp.-5, 8. [9]. A.F.Pym nd M.Fthipour, Study of th tip mss nd intrction forc ffcts on th frquncy rspons nd mod shps of th AFM cntilvr, Int. J.Adv.Mnuf Tchnology, DOI.7/s7---z,. jt (A) 6 P g