FFM. Friction. Sample surface. Fig. 8 (a) Schematic operation of FFM, and (b) twist of the FFM tip, [8]. = 2

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1 A.. Couled Flexural-Torsioal Noliear Viratios of PZT-atuated Miroatilevers: Plaar dyamis of miroatilever eams have ee ivestigated y may researhers. I additio the rolem of viratios of miroatilevers has reeived a osiderale attetio sie it aears i several sietifi ad idustrial aliatios suh as Atomi Fore Mirosoy (AFM Naomehaial atilever sesors (NMCS ad Fritio Fore Mirosoy (FFM. Utilizig miroatilevers rovides the aility to raidly detet small quatities of materials i differet eviromet suh as gas or liquid. The miroatilever eam viratio setrum has ee used to detet a target mass artile attahed eetrially to the eam whih rodues oth flexural ad torsioal viratios. I oth FFM ad AFM a miroatilever eam with a shar ti is utilized (as show i Fig. 8-a for aquirig toograhi iformatio from the surfae through mehaial iteratio etwee miroatilever ti ad samle. While mostly flexural viratios our i AFM FFM may exeriee torsioal viratio i additio to edig. This is due to the fat that the saig diretio i FFM is erediular to eam legth ad the ti of the eam. The resultig fritio fore twists the ti of the eam roduig torsio i the eam as show i Fig. 8-. It is show here that the torsio is ouled with flexure eause of the oliear geometry ad resee of iezoeletri atuator. I additio a idiret method for measurig the fritio fore is reseted. a FFM Sa Fritio Samle surfae Sa Twist Fritio Fig. 8 (a Shemati oeratio of FFM ad ( twist of the FFM ti [8]. As show i Fig. 9 the iezoeletri layer is ot ouded o the etire legth of the eam; therefore the eutral surfae varies for eah setio. For s<l or s>l where iezoeletri is ot attahed the eutral surfae is the geometri eter of the eam (y =0. The ositio of eutral surfae y for the setio where iezoeletri layer is attahed a e otaied as y E = h ( h + h ( E h + E h where h idiates the eam thikess ad the susrits ad deote the eam material ad iezoeletri layer roerties resetively ad w is the eam width. The oulig relatio etwee the stress ad the eletrial field for iezoeletri layer is exressed as P( σ = E ε E d3 h where P( is alied voltage to the iezoeletri material.

2 Piezoeletri layer h h Fixed ed l w w l l Fig. 9 Geometry of the miroatilever eam [8]. A eergy method is utilized here to derive the equatios of motio. Utilizig the Lagragia method the equatios of motio of the system with reset to three variales u (logitudial viratio φ (torsioal viratio ad v (edig viratio a e otaied as [8 9] ( J J φv ( C C φ + ( C φ C φ P t = J ( φ φ = 0 at s = 0 ; φ = 0 at s = l C v P( + J v + EA u + v C + C v Cv P t = mu ( J ( u v + v v C ( u + v Cφ + t + C u = 0 at s = 0 ; u = 0 at s = l ( φ v u v u [ J v φ + J ( v v φ u v u v v v ] = mv 3 + EA u v + v + C u + v P( φ C u v P( v = v = 0 at s = 0 ; v = = 0 at s = l. As see from these equatios there exist oliear terms i the equatios from order two to three. However the oliearities of the seod order are due to resee of iezoeletri layer. Cosiderig the equatios of motio it is oserved the torsioal ad flexural viratios are ouled i two ways; oe is a third order oliearity oulig due to eam geometry ad the other oe is a seod order oliearity due to oth geometry ad eletromehaial oulig of iezoeletri layer. The former oliear terms have ee otaied i revious studies while the latter is a ew oservatio whih is dislosed for the first time here. Havig the equatios of motio a umer of ase studies are osidered ext to study the oulig of flexural ad torsioal viratio i resee of iezoeletri atuator ath ad effet of oliearity due to eam geometry. The iextesiility oditio exresses that the elogatio of the eutral axis durig the viratio is igorale. Cosiderig the iextesiility oditio the goverig equatios o logitudial ad flexural viratios a e redued ito oe equatio osequetly. The the equatios of motio of the system a e reseted i the followig form [8]

3 ( J J φv ( C C φ + ( C φ C φ P t = J ( φ v φ = 0 at s=0; φ = 0 at s= l s s [ C v P( ] v ( C v mv ( v v + v Cφ + t + C ( φ [ J v φ + J ( v v φ ] = mv l v φ C 0 dsds P( v = v = 0 at s=0; v = = 0 at s=l. Usig the oulig of strethig ad flexure to the iextesiility oditio the two oliear equatios for logitudial ad flexure viratios are omied ito oe. There are ow two orders of oliearities the seod order oliearity is due to resee of iezoeletri layer ad the third order oliear terms are due to geometry ad aears as oliear iertia ad stiffess terms. I order to rodue the ordiary differetial equatios goverig the time futios of equatios of motio these equatios are searated ito ositio ad time omoets usig Galerki aroximatio as m= φ( s = φ ( s = α ( s q ( = m m= v ( s = v ( s = β ( s r ( where α m ad β are the omariso futios (satisfyig oly eam oudary oditios ad ot eessarily the equatios of motio for edig ad torsio of the miroatilever eam ad q are the geeralized time-deedet oordiates. The ordiary differetial equatios goverig the torsio ad flexure viratio are resetively: k = q + k q + k q r + k q r P( 0 m m m m 3m m 4m m = ( k r + k r+ k rpt ( + k r + k rr + rr + k rq + k q Pt ( = k Pt (. 3 5m 6m 7m 8m 9m 0m m m m m where k im are ostats that a e otaied y orthogoality oditio. A exerimetal setu is desiged for the oliear ouled flexural-torsioal viratios of the miroatilever eam. The exerimet aims at fidig the ouled flexural-torsioal viratio frequey. For this urose a DMASP miroatilever eam made y Veeo Istrumets is utilized as deited i Fig. 0. m m 3

4 Fig. 0 (a The setu for exerimet ad ( miroatilever eam [8]. I order to erform the exerimet our state-of-the-art MRI the Miro System Aalyzer (MSA-400 as deited i Fig. -a is used for o-otat measuremet of three-dimesioal motios i the miroatilever. The laser iterferometery measures the out-of-lae viratio of the miroatilever ti. Usig MSA-400 oe a reisely adjust the laser light o the eam ti as show i Fig -. a Miroatilever ih µm ZO Piezoeletri layer Laser eam Fig. (a The Polyte MSA-400 testig devie ad ( the ti of the miroatilever eam [8]. The DMASP miroatilever utilized i the exerimetal tests has a smaller width at the ed whe omared with the rest of the miroatilever (see Fig. -. There is a small ti vertial to the elogatio of the miroatilever at its ed whih is used i AFM aliatios. The iezoeletri layer osists of a 3.5µm ZO layer ad two Ti/Au layers of 0.5µm height o to ad eeath the ZO layer at the ase of eah roe. The miroatilever is istalled uder the MSA-400 o the setu show i Fig. 0 i order to use the laser light for measurig the eam viratio. The miroatilever eam viratio is the measured i resose to a Volt AC voltage alied to the iezoeletri layer as the exitatio soure. As a demostrale examle the exerimetal results for first flexural atural frequey are deited i Fig.. 4

5 a 50 Frequey Resose 0 v(l (d Frequey (KHz First resoae KHz Seod resoae 39.7 KHz Fig. a Exerimetal result ad logarithmi ad liear simulatio results resetively for Volt hir exitatio sigal with first flexural atural frequey highlighted [8]. 5

6 This sutask has resulted i the followig uliatios durig this reort eriod. [8] Mahmoodi S.N. ad Jalili N. Couled Flexural-Torsioal Noliear Viratios of Piezoeletrially-Atuated Miroatilevers aeted edig revisio ASME Joural of Viratio ad Aoustis (May 007. [9] Mahmoodi S.N. ad Jalili N. Couled Flexural-Torsioal Noliear Viratios of Miroatilever Beams Proeedigs of the 4 th Iteratioal SPIE 007 Smart Strutures & Materials Coferee Sa Diego CA (Feruary