N a .. n o s .. c a l e .. S .. y .. s t e .. m .. s .. M .. M .. T .. A bullet .. I SSN : .. 2299 hyphen 3290 \ centerlineR e s e a r c{ hN A a r\ tquad ic le bulletn o .. s D\ Oquad I : 1c 0 period a l e 2\ 4quad 7 8 slashS \ nquad .. s my m\ tquad hyphens 2 t 0 1e 3\ hyphenquad m 0 0\quad 0 eight-bullets \quad N aM n\ oquad .. M MM T\ Aquad bulletT V\quad o l A Na nos cale S y ste m s M M T A • I SSN : 2299 - 3290 period$ \ bullet 2 bullet$ 2 0\quad 1 3 bulletI SSN 1 2 4 : hyphen\quad 1 42299 4 − 3290 } R e s e a r c h A r t ic le • DOI:10.2478/n smmt −2013 − 000eight − bullet N a n o M M T A • V o l .2 • 20 13 • 124 − 144 M .. o .. d .. e l i .. n g .. t h .. e .. t i .. p hyphen s .. a .. m p .. e .. i .. n t .. e .. r a .. c t i .. o .. n .. i .. n .. a .. t o .. m .. i c .. f o \ centerlineMm .. ic or .. o{ResearchArticle .. d s .. c eli .. o .. py .. ng w i t .. T thi m .. o .. $ e s ..\ bullet h ti.. e .. n$ .. p-s k\quad .. o ..DOI:10.2478/n b .. a e .. a mp .. m .. t .. eh e .. oi nt\quad s m m t $ − 2 0 1Abstra 3 − ct 0 0 0 eight −bullet $ Na n o \quad M M T A $ \ bullet $ V o l $ . 2 \ bullet 2 0 1 3 eractioninatomicfoA\ matbullet r i-x .. framewor1 2 k i 4 sde v− elop1 .. df o4 r s i 4 n $ lea d-n} .. m u-l ts .. J u l i o R C laeysse to the power of 1 comma 2 * comma Teresa .. Ts ukaza to microscopywitTimoshenkobeamtheothe power of 1 dagger sub comma Letic a-i \noindentAbstram i cro ct hyphenM \quad c an to lever\quad s T-id mo\quad shen oe be l m i mo\quad d l-e sn .. g ou\ squad eina it to h ..\ Tonequad te t-o\ toquad the powert i \ ofquad 1 ddaggerp − s sub\quad commaa Dan\quad e a-lm n sub p \ Tquad o lf e \quad i \quad n t \quad e \quad r a \quad c t i \quad o \quad n \quad i \quad n \quad a \quad t o \quad m \quad i c \quad f o 1,2∗ 1 toA the mat power r i − ofx 1 Sframewor k i sde v elop df o r s i n lea d − n m u − l ts J u l i o R C laeysse , Teresa Ts ukaza †, \ centerlineLeticfo rcea − mi icroscop{m \quad y openiparenthesis c r \quad AFo M\quad periods T\ hquad y a recoc \ nsidequad edo f sub\quad sub ecpy ttog\quad .... hlinew i t \quad T i m \quad o \quad s \quad h \quad e \quad n \quad k \quad o \quad b \quad e \quad a \quad m \quad t \quad h e \quad o } 1‡ 1§ mEquation: i cro - c an quencyt lever s toT the− i powermo shen of o eralbe m sub mo d sampl − e fos e r subou s aneina toit theo power Tone of t t c− i subo, ntDan eraa e a sub− ln lyT too lf the power of in sub s to the power of load o nside \noindent Abstra ct r efo din rce from m icroscop r mcrobea y ( AF to M .i T to h the y a powerreco nside of edc t-ifsub o nec period ttog S urfa e eff r sub e tsa rec from so fsupp ort d a ndc a n i-t l e to san dbou nda y co n dt i-o sub e-v to the power of n s or m j sub o de t he from ms period to i ng .. 1 do Su l A v from Brazi to Ins titut eo to the power of fMathem Bent atic oGon s \noindent A mat r $ i−x $ \quad framewor k i sde v elop \quad dforsinlea $d−n $ \quad m $ u−l $ t s \quad J u l i o R C ccedilla to the powerBrazi of commafMathem a l-v to the power of, U sub e to theU nive powerrsda of nivedeF subedera comma-sdo 950 toRioG thea power−r of rsda sub 0 comma 915 to the power of $ l a e y1 s s edoSulAv ˆ{ 1 , 2 ∗BentaticoGons } , $ Teresa \alquad− v Ts $ ukaza ˆ{ 1 }\−9 dagger { , g}−$re, L e t i c $ a−i $ deF sub 9 sub hyphenInstituteo 9 to the power of ede ra subccedilla zero-zero sube commacomma to− thes9500 power,9159 of dzero o sub−zero P, rtoAP rtoA tole the power of RioG sub le to the power of a-r quencyeralsampfoerc iin loadonsideredinrmcrobea thems. sub g-r e comma an nteraaly s ct−ion sofsupportdandcani−tlen ing i .Surfaeeffretsarecsandboundaycondti−o e−v sormjode \noindentte ns i ve ..m us i i cro .. mad− eocan f a di t s levert-r ibut sdm ..$T a− rxi fu $ d-n moshen a .. me n ot-a bemmod 2 .. Mecha nica $ lEnginee l−e $ r s n-i\ gquad Graduaou t-e s Pro eina r-g sub $ i { a mt comma}$ o U\quad niver s-iTone t $ t−o ˆ{ 1 \ddagger } { , }$ Dan e $ a−l n { T }$ o $ l f ˆ{ 1 \S }$ subte d ns adeF i ve ed us i mad eo f a di s t − r ibut dm a rx fu d − n a me n t − a 2 Mecha nica lEnginee r n − i g Gradua t sponse− e Pro convor − g ua t-hm , ioU niver t to thes − powerid adeF of ed e a nan to the power of a s sub l-l sub dt to the power of ow st od oabs o to the power of et sub b to the power \noindent fo rce m icroscop y (e AFM−r . Th y a reco nside ed $ f o { sub }$ ec ttog \ h f i l l $ \ r u l e {3em}{0.4 pt } $ sponse t−hea nanasl−l owstodoabsoet n miee f−orc edte−rsponusesthrbound P ortodoRioAGr n−agred,ed of e-r n miconvou o n hypheniot o-h todt the power of eb e sub mogeneoon−o−hmogeneo to the power of f-o rc to the poweraryc of ed t e-r sponl−e us e st hr bound sub aryc to the power of o oS u ,Ru PortoRS to, the poweraS ofa do− Rior men A too the− t powerL ei e , of 4 5 Gr , 900 sub 5 0l-e - 1 n-a gre to the power of d comma to the power of ed RS comma to the power of oS sub Bra l to \ begin { aBral l i gz n ∗} thed ipower t i on ofls u subTran z s to ie nthe s a power e ide n oft − commaifi ed Rur − aSf a-ro m men in ial o-tv − La eil e ue comma sofper 4 5 comma 900 5 0 hyphen 1 \ tagd i∗{ t$ i on quency l sub s .. ˆ{ Trane r a s l ie n{ ssamp a e ide} n t-if o fi ed{ r-fe o} m inr ial}ˆ v-a{ c l ue} sofper{ an } i ˆ{ in } { nt { era { a }} { l y }}ˆ{ load } { s } o nsidehline r e dinˆ{ r mcrobea } { i ˆ{ c t−i o n } . 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J aSu n u a l r y 2 A 0 1 v ˆ{ Brazi } { Ins t i t u t eo }ˆ{ fMathem } Bent a t i c { oGon } s { c c e d i l l a t }ˆ{ , } a l−v ˆ{ U }ˆ{ nive } { e }ˆ{ rsda } { comma−s 950 } { 0 , 915 }ˆ{ deF } { 9 }ˆ{ ede nenttal resp second onse su s rfacetEigenanalys ma t r i subis f or hyphen de t − suber mn e orde ngfrequ sub ffect en to the power of x sub respons rma sar t r sub e i to the power of etha x sub n flue t charac differeraand} ma{ nal t− r f i o rt9 mod sub} { eshape tizero nt ts sdonerz−zero aleq e w he{ tht u, at heu}} sub seˆ{ o n fd to afu then o− power}d a{ mP of es Received n rtoA period 24}ˆ I{ ts JRioG o a b n se u a} ved r{ y 2froml 0 e 1 } atuˆ{ ra alfreque−r } { ncg− yr a to i e to the , power} of s o lui on s o-f xrespons etha es aturalfrequencya Revised pri 20 1 \endtal { a l i g n ∗} ma t r i rma t r x tcharacdiffere nt terzaleq u @ n.Itsobseved s Acce 25ptedA adamsecond Accesurface to the power−e oforde Revisedffect 25 ptedsar Aei i revisenflue to the power of pn r ialfort 2 0 subti df or to thehe powern of 1 m 2 5 Ai oluionso p r l 2− 0fadam i revisedfor mnanosca 2 5 A p r l l e 2 .. 0 S imul a ton s a e p erorm ed .. o .. a .. b hyphen to the power of s-e g .. me s−e \noindentnanoscafree hyphen l ete S fre imul ns .. abea i ton ve man s a\ equad dwp erorm i ius h ed a mic i o\quad ro hyphen amadeo b − c a ng ft-i a l me e di er r s e-b $t am− ir sub $ a ctu ibut te dm \quad a rx fu $ d−n $ a \quad me n $ t−a 2 $ \quad Mecha nica lEnginee r $ n−i $ gGradua $ t−e $ Pro $ r−g { a }$ m,Univer $s−i { d }$ adeF ed freea p - frei ezoe bea e sub man lect dw r-ii i h claye a mic ra ro minat - c a n et d− inoi l e n er e r se i − b am ia ctu te aKeywords p i ezoe electr−i claye ra minat e d ino n e s i \noindentKeywordsAtom c f orc$ e sponse m icroscop{ yconvo bullet nanosca u } l-et m−h .. a{ teri o a s-l{ ..t an}} dstrˆ{ ucte a } nan ˆ{ a s { l−l }}ˆ{ ow } { dt } s t od { oabs } oAtom ˆbullet{ et c f chem orc}ˆ e{ m i-ce icroscop− ar l} slash{ y •b bnanosca i} o$ lo ngcal − ls$e ensom mi r a{ s ter bulleto a s n− nanoml −an a dstr co hn− ucth i} gˆ bullet{ e microsc e }ˆ{ f−o rc } { mogeneo }ˆ{ ed t e−r } spon{ us } e•Timoshenchem s ti − hrc ka obeam{ l /bound b i o lo gca}ˆ ls{ ensoo } r s{• arycnanom a} c hnPorto i g • microsc ˆ{ do Rio } A ˆ{ Gr } { l−e } n−a{ gre }ˆ{ d } , ˆ{ ed }$ RS $ , ˆ{ oS }ˆ{ u } { Bra l Timoshen}ˆ{ , k obeam Ru } { z }$ aS $ a−r $ men $ o−t$ Leie,45,90050 − 1 PACS : 7 period 79 L h .. 6 2 period 2 5 period-hyphen sub g comma 8 1 period 0 7 period hyphen b comma 8 7 period 8 5 period fkcomma46period70 PACSMSC : : 747 . H7955 L h 74 ..62 H.25 1period 0 comma− hyphen 35 E 0g, 581 comma.07.− b 3 5, 87 L.85 3 zero-comma.fk, 46.70 35 L \noindentMSCcopyright : 74 H Versid 55 i 74 t-at i H.. on1 s 0 p , z 35 $ .. E o l 0 period 5{ , 3 5s L o}$3zero\quad− commaTrans35 L ie nsae iden $t−i f i $ ed $ r−f$ omin ial $v−a $ lcopyright ue1 period s o f p ..eVersi r I ntrot − ..a ductios p z o . o 1I . n t h I is ntro w o r k ductio comma w e det er m i-n e .. dy n am icr e s p on e .. sofTimo s h n ko m ic ro hyphen can t le v r beam m o d e ls .. o u s .. in t h es I n t h is\ begin wp o ro r k b{ ,a e w l .. i e gt-e det n ∗} c er h m nii q− un ..e o fAFM dy n amperiod icr e Th s p sAF on e M te sofTimo c h n q-i s h u n e ko s m a-l ic l ro .. -w can t o t .. le ob v r ai beam nim m ge o s d o e f ls su ra o ce u s .. to in p t o h g es r a h y a tt h e .. a to m i \prin ro u l a b e n{ e 3em o ninvasivt}{− e0.4c h ptem ni}q a u nn e o comma fAFM . f Th r-o sAF m a M w te i de c h var n q i− tyi ..u o e sf sa amp− l l l e o w n t a o s cal ob e ai r nim o m ge an s gs o f ro su m ra st ce o 1 to 0 m p o .. ro ns period .. Con d uc \endgins r a{ uah l lay i g a ti n tt∗} n h gs e a m a tople m s i oin f s a ur no fa ninvasiv e .. s r em u ctu a nn es e c , a f nr ..− bo m ec a on w sid i de er var e dini ty b o o t f h s aiamp r a l en o d n .. a i s qui cal de r env o m i-r an o gs n .. me t s period c sub t-I s pre d ecero s m or st o 1 0 m ro ns . Con d uc \noindentinsst u y la l u ti s n .. gsnent profil a m ple resp e r s t o h f onse sa ur tmag fas e nifie\quad s rd-comma u ctu$ es t c gr a{ ne aE e b r ec t igenan a on n sid 1 zero-multiply er e din aly b o t h sub s ai} r zero-comma$ a n di s f i qui or t dhe de env ve ri $− tic tr− aoe .. n ˆ s{ u me rt ac} eo{ fr a ..}$ a-s mn m p\ equad a nd rngfrequ ecor e en d ..t s he.c ..t− mIs opre i o d n ece o s or \noindent and ma t r i \quad mod eshape s sdon e w \quad tht heu se o f afu $n−d $ a m \quad Received 24 \quad J a n u a r y 2 0 1 stst y y l u l us s o profil n .. pe r hot t h o a gr tmag a p nifie h c pad − pcomma e r-periodgr e T a e rt t F-A a n ..1zero M i ..− ammultiply 0 subzero e c− hcomma a n ct a he l s ve y r te tic m a .. f s or u ser ac si eo n fg f o c e .. t t h e nano N .. ewt o nlevela ba e− s m p e a nd r ecor e d he m o i o n o \noindent $ t a l { second { su r f a c e }}$ ma t r $ i { − { e } orde }ˆ{ x { respons }} { f f e c t } rma { sar }$ stsa y ml u p s leo n a nd p a hot .. v o e gr r a y p t-i h nc pa y t p p e lessr − 1period 0 n mT ra e d t iuF s− ..A thaM ti ism o am u n0 e-tech da on nc a a l micr s y te o m f abr f ic or a se ed si c n an g f ti o l c v e e 1 parenleft-zero 0 mu m closing parenthesistt t $ h er nano ˆ{ period NethaTh ewt} s-i o{ nlevel ..e a llow b i e } x { n f l ue } t charac { di f f e r e } { n a l f o r t } { t i }$ nt $ t e rsa zAFM m{ pa le to l e a qb nd e} .. a{ c aphe v ea r} bl$ y et o− u fii mn $ ay @ tgi p ˆ g{< f ees10 a turn} m{ e-s ran ..d} iu w stn .. a tha .. . mti sm agnific I o u n ats ioe − nt ofgr od o e n b a a t micr e rse one-t o f abr vedˆ h-zero ic a ed{ to catu the an ti power l v r e of 6alfreque times period ncS in ce y it .. inv a } { i ˆ{ s } eno1 i-tparenleftl o u n i b y−on Bzero i n ni0 sµ gm o)− .f Th s adam− i a} llowAcce ˆ{ Revised } 25{ pted } A { i r e v i s e }ˆ{ p r i 2 0 }ˆ{ 1 } { df 6 orAFMone-a}$ to ml-bracketright b e 2 5 c A ap p a bl ropen e l o fi2 square m 0 a gi bracket g f e a tur i he a− ..s sunw e t rg o a ne s m e eralagnific d ea elopio n ofgr m e e to a tthe e r powerone − ofth n-t− zero s period× . S in ce it* .. inv E hyphen en i − mait o n l b : y julio B i nat ni m g a t period ufrg s period n−t \noindentonedagger− al ..− Ebracketrightnanosca hyphen mai l[ i e l h :\ a ..quad teresa sunS e period rg imul o ne mat sa eton eral at gm d s e b elop a sub e m ap ie erorm periods . c-o ed \quad o \quad a \quad b $ − ˆ{ s−e }$ g \quad me ∗ddaggerE - mai .. lE : hyphen julio @ maim a l t: . .. ufrg ltonett s . o-period mat at gm l sub a i period-l c \noindent f r e e − f r e \quad bea mandw i i h a mic ro − c a n $ t−i $ l e er r $ e−b $ am $ i { a }$ ctu te †S ..E E - maihyphen l : mai teresa l : d .an mat itolf@ o period-cgm bai.c p−o at gm a i period-l c ‡timesE - N mai a noMMTAl : ltonett timeso Vo− period l periodmat 2 times@ gm 20 1laiperiod 3 times−lc 1 24 hyphen 1 44 times 1 24 \noindent§ E - maia l : p d an i ezoeitolf o period $ e {− clp e c t@ gm r a− i }period$ claye− l c ra minat e d ino n e s i · N a noMMTA · Vo l .2 · 2013 · 124 − 144 · 124 \noindent Keywords

\noindent Atom c f orc em icroscop y $ \ bullet $ nanosca $ l−e $ m \quad a t e r a $ s−l $ \quad an d s t r uct

\noindent $ \ bullet $ chem $ i−c$ al/biologca ls ensors $ \ bullet $ nanomachn i g $ \ bullet $ microsc

\noindent Timoshen k obeam

\noindent PACS : 7 . 79 Lh \quad $6 2 . 2 5 period−hyphen { g } , 8 1 . 0 7 . − $ b $ , 8 7 . 8 5 . f k , 46 . 70 $

\noindent MSC : 74 H 55 74 \quad H10,35E05,35L $3 zero−comma 35 $ L

\noindent $ copyright $ Versi $ t−a $ \quad s p z \quad o . o

\noindent 1 . \quad I ntro \quad ductio

\ centerline { Inthiswork ,wedeterm $i−n $ e \quad dynam icr e s p on e \quad sofTimo s h n ko m ic ro − can t le v r beammo d e ls \quad o u s \quad in t h es }

\noindent p ro b e \quad $ t−e $ c h ni q u \quad ofAFM . ThsAFMte c h n $ q−i $ u e s $ a−l $ l \quad w t o \quad ob ai nim ge s o f su ra ce \quad topograhyatthe \quad a to m i in ano ninvasiv emann e , f $ r−o$ mawi devar i ty \quad o f sampl eonas cal e romangs romst o10m \quad ro ns . \quad Con d uc

\noindent ins u la ti ngs ample s o f s ur fa e \quad s ructu es can \quad bec on sid er e dinbo th ai r and \quad i qui d env $ i−r $ o n \quad me t s $ . c { t−I s }$ pre d ece s or

\noindent s t y l u s \quad profil e r t h a tmag nifie $ d−comma$ greaer tan $1 zero−multiply { zero−comma }$ t he ve r t i c a \quad s u r ac eo f a \quad $ a−s$ mpeandr ecor ed \quad he \quad m o i o n o

\noindent s t y l u s o n \quad photographcpape $r−period $ Te t $F−A $ \quad M i \quad am $ 0 { e c h }$ ancalsytem \quad forse singfoce \quad t t h e nano N \quad ewt o nlevel b e

\noindent sa m p l e a nd a \quad v e r y $ t−i $ n y t p $ < 1 0$ nmradius \quad tha ti smoun $e−t $ dona micr o f abr ic a ed c an ti l ve $1 parenleft −zero 0 \mu $ m ) . Th $ s−i $ \quad a llow

\noindent AFM to b e \quad capableo fimagigfeatur $e−s $ \quad w t \quad a \quad magnific a io n ofgr e a t e r $ one−t h−zero ˆ{ 6 }\times . $ S in ce i t \quad inv en $ i−t$ onbyBinnig

\noindent $ one−a l−bracketright [ $ i h a \quad sun e rg o ne s e eral de elopm $ e ˆ{ n−t }$ s .

\noindent $ ∗ $ \quad E − mai l : julio $@$ mat . ufrg s .

\noindent $ \dagger $ \quad E − mai l : \quad teresa . mat $@$ gm $b { a i . c−o }$

\noindent $ \ddagger $ \quad E − mai l : \quad l t o n e t t $ o−period $ mat $@$ gm $ l { a i period −l c }$

\noindent \S \quad E − mai l : d an itolf o $ period −c$ p $@$ gmai $period−l $ c

\noindent $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 24 $ M o d e l i n g t h .. e t i p hyphen s a m .. p l e i n t e r a c t i o n i n a t o .. m i c f o r c e .. m i c r o s c o p y .. ellipsis \noindentA t y p icaM l AFMo d e .. cl on i sis n ..g so t f h a\ sequad n sit ..e e t .. i mi p cro− hyphens a m c\ aquad nti l v-epleinteractioninato r .. i .. ha .. mo u n ed .. h a .. p t ip .. ct ng as\ foquad rce ..m e i c f o r c e \quad m i c r o s c o p y \quad Modelingth etip-sam pleinteractioninato micforce microscopy ellipsis $ es l y l iste p s m i s t h $ a .. move s t h es a m p e .. r t e sen s o r i n o r de .. to p o be th e .. sa m p l e s u f ce comma a d e e c ion .. s e nso r sy t em A t y p ica l AFM c on sis so f a se n sit e mi cro - c a nti l v − e r i ha mo u n ed h a p t ip ca nt i lev e .. de fl e ct o .. comma a f eed hyphen ba c k sy s em w h ic h r e gul a te .. t he .. o r c ei n te r c to n an d .. a co n t ro ll er .. el c t ron i ct ng as fo rce e s y ste m t h a move s t h es a m p e r t e sen s o r i n o r de to p o be th e sa m p l e s u f ce , Atsy st y p ica lAFM \quad c on s i s \quad so f a se n s i t \quad e \quad mi cro − c a n t i l $ v−e $ r \quad i \quad ha \quad mo u n ed \quad h a \quad p t ip \quad ct ng as fo rce \quad e a d e e c ion s e nso r sy t em ca nt i lev e de fl e ct o , a f eed - ba c k sy s em w h ic h r e gul a te t he o r c sr y e-one s t e parenright-c m t h a \ period-oquad move rds m s ovem t h ent es s comma amp c e o\ ntquad r l t hertesensor fe ed ba ck l o p .. nd inorde se n d s h-t e\ ..quad m easto u e p d o da be a to th .. a e co\quad m p utsampl er p ro e s esuf ce ,adeecion \quad s e nso r sy tem ei n te r c to n an d a co n t ro ll er el c t ron i sy st r e − oneparenright − cperiod − o rds m ovem ent s , c o nt r l t sica .. g nt u n i t open l e v parenthesis e \quad de $ f l $ e ct o \quad , a f eed − ba ck sy semwh ic h r e gul a te \quad t he \quad orc einte rctonand \quad acont ro ll er \quad el c t ron i sy st he fe ed ba ck l o p nd se n d s h − t e m eas u e d da a to a co m p ut er p ro e s si g u n t ( I n te rms of t he ca rI n $ te e− rmsone of t he parenright ca n t i leve− rc sta te period o f m− oto$ io n d rdsmovement u r i ng m eas u re ments , c comma o nt t rh e l two t hebas icfe t edbay pes of ck AFM l modes op \ aquad re : ..nd sta se t ic n mode d s $ h−t $ n t i leve r sta te o f m ot io n d u r i ng m eas u re ment , t h e two bas ic t y pes of AFM modes a re : sta t ic mode ( co e \openquad parenthesismeas u co e nta dda ct comma a to \ fquad r ict ioa n comput o r la te ra l er fo rce pro closing e s parenthesis si \quad ag nd u d n y na t (m ic mode open parenthesis no n hyphen co nta ct nta ct , f r ict io n o r la te ra l fo rce ) a nd d y na m ic mode ( no n - co nta ct , ta p p in g o r sem i - co nta ct , a co u st ic , commaI n te ta p rms p in of g o t r sem hei ca hyphen n t co i nta leve ct comma r sta a te co ou st fmot ic comma io .. ndu p iezoe rle ct i r ngmeas ic comma u re ment , t h e two bas ic t y pes ofAFMmodes a re : \quad sta t i c mode (p iezoe co nta le ct ctr ic , ,e f− rtwol ict− bracketright io no reT la−ctr te−he ra−os− lytat fo−ar rce−ie−c )e− anddynamicty−t p − parenrightbracketleft mode (− nonc a l− yofco l n gt nta h ct , ta pp in g o r sem i − co nta ct , a cou st ic , \quad p iezoe le ct r ic , e-two l-bracketright sub e T-c t r-h e-o s-y ta t-a r-i sub e-c sub comma, c e-t y-t sub c p-parenright bracketleft-c a l yof l n gt h 1 25 hyphen 4 5 0 mu m 125 − 450 µ m , width 28 − 45µ m , t h ic kness 1 − 8µ m , reso na nt fre q u e ncy 1 2 - 300 KHz , s p r in g co nsta n t 0 . comma$ e− widthtwo 28 l hyphen−bracketright 45 mu m comma{ e t h Tic− knessc 1t hyphen r−h 8 mu e− mo comma s−y reso ta na nt fret−a q u e r ncy−i { e−c { , } e−t y−t { c }}} p−p a r e n r i g h t 1 - 48 N / m a nd t ip p ro be he ig ht 17µ m a nd t i p ra d i u s less t h a n 1 0 nm . The asso - c ia te d le n gth s ca le s bracketleft1 2 hyphen− 300c$ KHz alyoflngth comma s p r in g co nsta $1 n t 0 period 25 1− hyphen4 48 5 N slash 0 m\mu a nd$ t ip m, p ro widthbe he ig ht $ 1 28 7 mu− m a nd45 t i p\ ramu d i$ u s m, less t t h h a n ic 1 kness associated a re s ufficie nt l y s ma l l to ca l l t h e a p p l ica b i l it y o f c la ss ica l co nt i n u u m mode ls i nto q u est io n . 0$ nm 1 period− ..8 The\ assomu hyphen$ m , reso na nt fre qu e ncy Fig 1 . Schematic of an Atomic Force M i croscope operation 1c 2 ia− te300KHz d le n gth s , ca s le p s r associated in g co a re nsta s ufficie n t nt 0 l y . s 1 ma− l48N/mandt l to ca l l t h e a p p l ipprobehe ica b i l it y o f c la ig ss icaht l co $1 nt i n 7 u u m\mu mode$ ls mandt i nto q u est ipradius io less than10nm. \quad The asso − The g eo metry a nd t he mate r ia l o f t he ca nt i le ve r both co nt r i b u te to t he p ro pe rt ie s t hat ma ke a ca nt i leve nc period ia te d le ngth s ca le s associated a re s ufficie nt l y smal l to ca l l t he app l ica b i l it yo f c la ss ica l co nt i nuummode ls i nto qu est io n . r s u ita b le fo r a n y pa rt ic u la r im a g in g modes . B o t h s i l ico n a n d s i l ico n n it r id e m ic ro - ca nt i le ve rs a Fig 1 period Schematic of an Atomic Force M i croscope operation re com m e rc ia l l y ava i la b le b u t refl ect ive ba ck s u rfa ce co a t in g is u sed fo r a b ette r fe ed ba ck . N ew ge ne ra t io \ centerlineThe g eo metry{ Fig a nd 1 t. he Schematic mate r ia l o of f t an he ca Atomic nt i le ve Force r both M co i nt croscope r i b u te to operation t he p ro pe} rt ie s t hat ma ke a ca nt i leve r s u ita b le ns of n a no bea ms have i nc l u d ed p iezoe le ct r ic three − ma − period te r − T h − ia − e ls l − n s − ll − i fo r a n y pa rt ic u la r im a g in g modes period .. B o t h s i l ico n a n d s i l ico n n it r id eo− mcu ic−c ro−la hyphen cao− nty i le ve rs a re com m e rc ia l l y a t − ot − f m − h d − r a t t he m ic r o - b ea m with t he ro le of se nso rs a n d / o r a ct u a to rs [ avaTheg i la b eo le bmetryac u− ts a nda−e t he mate r ia l o f t he ca nt i le ve r both co nt r i bu te to t he p ro pe rt ie s t hatmake a ca nt i leve r s u ita b le mate r ia ls la y e rs wil l mod ify mate r ia l p ro pe rt ie s betwee n ne ig h b o r i ng la y e rs . Active b ea ms fo r AFM have be forefl r ect anypa ive ba ck s rt u rfa ic ce u co la a t r in ima g is u gsed in fo r gmodes a b ette r . fe\ edquad ba ckBoths period N ew gei lne iconands ra t io ns of n a no i bea l ms iconnit have i nc l u ridemic d ed p iezoe le ct ro r− ic ca nt i le ve rs a recomme rc ia l l yava i la b le but e n s u bj ect to a va r iet y o f t ip - sa m p le i nte ra ct io n t y pes mode ls a nd t h e y ca n be fo rm u la te d as a seco nd - o rd reflthree-m ect a-period ive ba te ckr-T h-is u a-e rfa ls l-n ce sub co o-c a u-c-l t in a s-l g l-i is sub u o-y sed a t-o fo t-f r sub a b a c-s ette m-h r sub fe a-e ed d-r ba a t ck t he . m New ic r o ge hyphen ne ra b ea t m io with ns t he of ro n le a of no se bea ms have i nc l ud ed p iezoe le ct r ic e r matrix d iffe re nt ia l eq u a t io n s u bj ect to bo u n da r y co nd it io ns a n d com pat i b i l it y co n d it io ns fo r t ra nsve nso$ rs three a n d− slashm o a r− aperiod ct u a to $ rs open te square$ r−T bracket h−i a−e $ l s $ l−n { o−c u−c−l a } s−l l−i { o−y }$ a $ t−o rs a l vi b ra t io ns i n ne ig h bo r in g seg ments seven − four − wbracketright − endash − h − period e − t−fmate{ ra ia ls c la−s y e} rs wilm−h l mod{ a ify−e mate} rd− iar$ l p ro atthemicro pe rt ie s betwee n ne ig− hbeamwith b o r i ng la y e rst periodheen− roT hActive− leev− ofeeu b− ea ser ms nso fo r rs AFM and/o have be e n r s au bj ct ect ua to rs [ h e − aF − m − A u M − l m − no − i b − h [ or a s a p a fo m f toao− avi va−fn r− ietth− yg o f t ip hyphen sa m p letcomma i nte ra−i− cta− iosp− nsan t− yn− pesa mode ls ac− ndmra t− ho− e y cae n−ni be−an fo− rmm u la te d as a seco nd hyphen o rd e r matrix d iffe re or he i − m al ntmate ia l r iac ls la y e rs wil l mod ify mate r ia l p ro pe rt ie s betwee n ne ig h b o r i ng la y e rs . Active b ea ms fo rAFMhave be e n s u bj ect a nd b io lo g ica l se nso rs i n co n ne ct io n w it h a nd s u rfa ce a n d t h e rm a l effects , ma ke t hat t h e effects o toavareq u a t io n iet s u bj yo ect f to t bo ip u n− dasamp r y co nd le it io i ns nte a n ra d com ct pat io i n b ti l y it y pes co n mode d it io ls ns foa rnd t ra t nsve h e rs y a ca l vi n b ra be t io fo ns rmu i n ne la ig h te bo d r in as g seg a seco nd − o rd e r matrix d iffe re nt ia l f t ra nsve rs e s hea r defo rm a t io n a n d rota r y in e rt ia o n t h e fre q u e ncy b e s ig n ifica nt . With s ma l le mentsequa t io n s u bj ect to bounda r y co nd it io ns andcompat i b i l it y cond it io ns fo r t ra nsve rs a l vi b ra t io ns i nne ig hbo r in g seg ments r va l u es o f t h e ra t io o f t h e p ro be le n gth to its t h ic kness , t h e Tim os he n ko be a m t he o r y is a b le to p $seven-four-w seven−four bracketright-endash-h-period−w bracketright −endash sub e− n-Th−period h-e v-e e u-r{ e e-h sub n−T a o-v h i-f−e n-t h-g v−e e-a F-m-A e u u− M-lr } sube− th comma-i{ a sub o− hyphenv i− a-sf p-s n a−t n-n-a h−g } re d ict t he f req u e n c ies of flexu ra l vib ra t io ns o f t he h ig he r eight − mo − period d e − As − sh − wi − e t m-ne−a o-i subF−m− c-mA $r a-o u hyphen $ M− b-hl sub{ t e-n i-a comma n-m− openi { square − bracketa−s or p− as s a p a a fo n m−n ..−a f or}} he i-mm−n sub c o al−i { c−m r a−o − } b−h { e−n s − hh − r t − h s − l t i − sff − i f − c t − ah − s A − to − F w − Md − c − r a n − tt − h i−aa nd n b− iomi− lo}ug g−c ica[$ l seu− orasapafom nsoer−r rs i n co n ne ct ion n−\ze wquad− ited h−ss af nd or sr u− heor rfa−e ce $ a i− nm d t{e h−e c rm}$ a l effectsa l comma ma ke t hat t h e effects o f t ra nsve rs e s hea r i e − l s − bracketleft a er e g m e , defo rmev− ane t− ioan− nrs a n d rota r y in ec rt ia o n t h e fre q u e ncy b e s ig n ifica nt period .. With s ma l le r va l u es o f t h e ra t io o f t h e p ro be le n gth s − nineone − u − commar − zerof − bracketright / te ns io n effects m u st b e ta ke n in to a cco u nt [ o l andbto its t io h ic lo kness g ica comma lse t h enso Tim rs os he i n ko co be nne aperiod m t− ct heace o io r y nw is a b it le tohand p re d ict s u t he rfa f req ce u eand n c ies t of h .. flexuerma ra l vib l effects ra t io ns o ,make f t he h ig t he hat r t h e effects o f t ra nsve rs e s hea r I n t h is wo r k , we s ha l l d is c u ss AFM Timoshenko m ic r o - ca nt i le ve rs mode ls t hat ca n i nvo lve s ma rt mate r deformaeight-m o-period t io d nand e-A s-s h-w rota i-e t s-h r y h-r in sub e i-u rt g-c ia t-h on sub t u-e he r-r s-l fre t i-s que ff-i sub ncybe n-z e-e d-s s s f-cig sub n ifica r-o r-e t-a nt h-s . sub\quad e-e A-tWithsmal o-F w-M d-c-r le a n-t r va l ues o f the ra t io o f thepro be le ngth ia ls o r t-hto i e-l its sub t e h v-n ic e-a kness n-r s s-bracketleft , t heTimos sub c a er henko e g m e comma be amt he o r y is ab le to p re d ict t he f req uenc ies of \quad flexu ra l vib ra t io ns o f t he h ig he r ge n e ra l d evices fo r describ i ng t i p - sa m p le in te ra ct io n fo rces . These mode ls a re fo rm u la te d as seco nd $s-nine eight one-u-comma−m o−period r-zero f-bracketright $ d $ e− subA period-a s−s ce h− slashw te i− nse io $ n effects t $ ms− uh st b h e− tar ke{ ni in−u to a gcco−c u} nt opent−h square{ u− brackete r− or ..} l s−l $ - o rd e r evo l u t io n s y ste m s s u bj e ct to i n it ia l , fo rc in g a nd bo u n da r y data . I ts mathematica l st u d y wil tI $ n ti− hs is wo f fr− ki comma{ n− wez s ha e− le l d is d c− us ss AFM s } Timoshenkof−c { r− mo ic r or− hyphene } t ca−a nt i le h− ves rs{ modee−e ls} t hatA− cat n i onvo−F lve sw− maM rt mate d−c− rr ia $ ls o a r $ n−t l m a ke a n exte ns ive u s e of d ist r i b u ted matrix im p u ls e res po nse o r in it ia l - va l u e G re e n matrix res p o nse . t−hge $ n e ira l $ d evices e−l fo{ re describ v−n i ng t e i− pa hyphen n−r sa m s p} le ins− tebracketleft ra ct io n fo rces{ periodc }$ .. These a er mode e g lsm a e re , fo rm u la te d as seco nd hyphen o rd e r Th is a l lows to cha ra cte r ize t ra ns ie n ts a n d fo rced res po nses of a va r iet y o f AFM mode ls . The vib ra t evo l u t io n io n m od es fo r ge ne ra l t i p - sa m p le i nte ra ct io n will b e exp l ic it l y fo rm u la te d in te rms of a fu nd a menta l \ centerlines y ste m s s{ u$ bj s e− ctnine to i n it one ia− l commau−comma fo rc in r− gzero a nd bo uf− nbracketright da r y data period{ ..period I ts mathematica−a ce l} st u/$ d y wil te l m ns a ke io a n n exte effectsmu ns ive u s e ofst d be ta ken in to a cco u nt [ o \quad l } matrix re s po nse of a seco nd - o rd e r o rd in a r y matrix d iffe re nt ia l eq u a t io n where t he co r re s po nd i ng st iffn ist r i b u ted ess m a t r ix co efficient de p e nds u po n t h e f req u e nc y . Th is m a t r ix re s po nse ca n b e d ete rm in ed i n c lo sed \ hspacematrix∗{\ im pf i u l lsl } eI res n po t nse h is o r woin it r ia k l hyphen , we vas hal u e l G l re d e n is matrix c u res ssAFMTimoshenkomic p o nse period .. Th is a l lows r to o cha− ca ra cte nt r iize le t ra ve ns ie rs n modets a n d ls fo rced t hat ca n i nvo lve s ma rt mate r ia ls o r f − oneo − onetwo − r − onem − bracketright i n te rms of a s ca la r so l u t io n t h a t ha s a com p le te l y osci l la to r y res po nses of a va r iet y o f AFM .. mode ls period .. The vib ra t io n m od es fo r ge ne ra l t i p hyphen sa m p le i nte ra ct io n will b e exp l ic it l y be havio r b e y o nd a c r it ica l f req u e n c y va l u e [ gefo n rm e u ra la tel d in evices te rms of fo a fu r nd describ a menta i l matrix ng t re i sp po− nsesamp of a seco le nd in hyphen te ra o rd ct e r io o rd n in fo a r rces y matrix . \quad d iffe reThese nt ia l eqmode u a t ls io na where re fo t he rm u la te d as seco nd − order evo lut ion Th is wo r k is o rg a n ized as fo l lows . I n se ct io n 2 , we fo rm u la te t he Timoshen ko b ea m model i n a m a t r ix syco r stems re s po nd subj i ng st iffn e ess ct m to a t i r ixn co it efficient ia l de , fop e nds rc u in po gandbounda n t h e f req u e nc y period rydata Th is m . a\quad t r ix reI s ts po nse mathematica ca n b e d ete l rm st in ud ed i n y c wil lma ke a n exte ns ive u s e of d ist r i bu ted fo rm . I n sectio n 3 , t he d y na m ic r es po nse of t h e matrix model s u bj e ct to t i p - sa m p le i nte ra ct io ns a n d lomatrix sed im p u ls e res po nse o r in it ia l − va l u eGre e n matrix res p o nse . \quad Th is a l lows to cha ra cte r ize t ra ns ie n ts a nd fo rced exte r na l fo rc in g is g ive n in te rms o f t he d ist r ib u ted matrix im p u ls e res po nse . E ig e na na l y s is is d is c u sse resf-one po o-one nses two-r-one of a va m-bracketright r iet y o i n f te AFM rms\ ofquad a s camode la r so l ls u. t io\quad n t h aThe t ha vib s a com ra p t le io te l nmod y osci l la es to rfo y be r havio ge ne r b ra e y l o ndt i a c p r− it icasamp l f le i nte ra ct io n will be exp l ic it l y d i n se ct io n 4 b y so lv in g a n o n - co nse rva t ive reqfo u rm e n uc y la va l te u e d open in square te rms bracket of a fu nd a menta l matrix re s po nse of a seco nd − o rd e r o rd in a r y matrix d iffe re nt ia l eq ua t io n where t he · N a noMMTA · Vo l .2 · 2013 · 124 − 144 · 125 coTh r is re wo s r k pond is o rg a i n ng ized st as foiffn l lows essma period .. t I n r se ix ct ioco n 2efficient comma we fo de rm p u e la nds te t he upon Timoshen t h ko e b eaf reqm model u e i ncn a my a . t Th r ix isma fo rm period t r .. ix re s po nse ca nb e d ete rm in ed i n c lo sed I n$ f−one o−one two−r−one m−bracketright $ i n te rms of a s ca la r so l u t io n t ha t ha s acomp le te l y osci l la to r ybe havio r beyonda c r it ica l f req uencyva l ue [ sectio n 3 comma t he d y na m ic r es po nse of t h e matrix model s u bj e ct to t i p hyphen sa m p le i nte ra ct io ns a n d exte r na l fo rc in g is g Thive n is in wo r k is o rg a n ized as fo l lows . \quad I n se ct io n 2 , we fo rmu la te t he Timoshen ko b eammodel i n ama t r ix fo rm . \quad I n sectiote rms o n f 3t he , d t ist he r ib d u y ted namic matrix im r p es u ls po e res nse po of nse t period h e E matrix ig e na namodel l y s is s is u d bj is c e u ct sse d to i n t se i ct p io− nsamp 4 b y so lv le in i g a nte n o nra hyphen ct io co ns and exte r na l fo rc in g is g ive n in nsete rva rmso t ive f t hed ist r ib u ted matrix impu ls e res po nse . Eig e nana l y s is is d is cu sse d i n se ct io n4by so lv in ganon − co nse rva t ive times N a noMMTA times Vo l period 2 times 20 1 3 times 1 24 hyphen 1 44 times 1 25 \ hspace ∗{\ f i l l } $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 25 $ J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo seco nd - o rd e r m a t r ix d iffe re n t ia l s y ste m t hat d e p e nds no n l i nea r l y u po n t h e e ig e nva l u e . The cases of m ic ro - ca nt i leve rs invo lvi ng a n e la st ic a p p e nd a t t he fre e o r l in e a r ize d b o u nda r y co nd it io ns a re d is c u ssed in te rms o f a fu nd a menta l r es po nse t hat g ive n in c lo sed fo rm . I n se ct io n 5 , t h e Ga le r k in method is u sed to o bta in red u ced - o rde r fo rce d models . Fo rced res po nses a re a p p rox im a ted b y co nce nt ra ted res po nses in vo lvi ng co nvo l u t io n with a co n ce n t ra te d im p u ls e res po nse a nd a lo ca l ized res po nse a t t h e e nd o f t h e be a m d u e to bo u n da r y co nd it io ns . F in a l l y , in se ct io n 6 we d is c u ss t h e matrix methodolog y with moda l a na l y s is fo r t he case o f ha rm o n ic i n p u ts as we l l as fo r mod u la ted l in ea r p ie cewise in p u ts w it h com p os ite m ic ro - ca n t i le ve r bea ms . 2 . Transversal v ib ration s of AFM u s i ng th e Timoshenko model AFM was d eve lo pe d fo r p ro d u c in g h ig h - reso l u t io n im a ges of s u rfa ce st r u ct u res . The AFM t i p h as a ve rt ica l reso lu t io n o n t h e o rd e r o f 1n.mathm.math o r be low , a nd it ca n detect low - a m p l it u de v ib ra t io ns co r re s po n d i ng to h ig h f req u e n c ie s . N owda y s it is a ls o u se d to p ro b e p ro pe r t ie s t h ro u g h i nte ra ct io ns b etwe e n t he t ip a n d t he sa m p le a nd to mod ify s u rfa ces . Th is i nte ra ct io n p ro cess has le a d AFM to be u se d i n s ma rt mate r ia l te ch no lo g y , ch em ica l / b io lo g ica l se nso rs , t r ib o lo g y a − sevenone − one − n − commacomma − bracketright − d − three − four n M − ao − no − de − ma − ln − ch − gia−nin−ng−d,a − im − mu − oa − ng − t o t − ah − ner−Asfi − Me l d − ms−i[ r o t r u ct u r a s − ly st e m i s ac o mp l ex one − two − tone − a − five − bracketrightcomma − six − sk − bracketright − T − periodbracketleft − hT − e h h a − mr−ii − na − eu−s−tsr−iz−oed−th a n − tt − pl − ii−e e o f − r en a fu n d am en t lreso l u t − i on l − hyphen i mi t p a − r a mt−e er [ device d e pe nds o n t he a ccu ra te ext ra ct o f sta t ic b e nd i ng a nd re so n a nt fre q u e ncy . Test measu re m e nts a nd t h eo ret ica l st u d ies have s hown t h a t t he vi b ra t io n be h avio u r of m ic rost r u ct u re s a t t he na n osca le is s ig n ifica nt l y s ize a nd pa ra metric de p e nd e n t . As t he s t r u ct u ra l s ize de c reases towa rd t he na nosca le r eg im e , s u rfa ce effe cts m u st be ta ke n i nto a cco u n t nine − bracketleft]. T h − is d e p e nd e nce h as m t − o i v a te d h e u se o − f i − s z e − hyphen d e pe nd e t − n co n − t i n u u m h e r − o i e s n mod e − l i n g m cr o t − s r u c tu r e s − colon co u l − p e

i r−i−r s − one − one − two − twotwo − one − t − seven − eight − zero − twonine−one−re−comma−comma−bracketright−bracketrightf−comma−comma−ss−s−u−n−sr − t − r − o − nh − c − f − n − a − ac−i−i−l−e−to−n−o−o−en−r−cg−a−y−e−abracketleft − a − l − nlf − eo − d − y e − gg−ma − u − nl − t − dh − ae − t − ei−do−oy−n−r[

two − bracketleftbracketright − comma − comma − bracketrightthree − bracketright, a m o ng o t he s . I n t h is work , we u se t he Tim os h e n ko be a m - t y p e st r u ct u re fo r in co r po ra t in g s h ea r fo rm a t io n a n d ro ta r y i ne rt ia effects . Th is a l lows to co ns id e r m ic ro be a ms with s ma l l le ngth - to - t h ic kness as p ect ra t io . Th ic k be a ms have re la t ive l y h ig h t ra nsve rs e s he a r m od u l u s a n d t he effects o f r ota r y i ne rt ia a nd t ra nsve rs e s h ea r defo rm a t io n m u st be u se d in t he d y n a m ic a na l y s is o f s u ch be a ms . The Timoshen ko mode l co r re cts t he c la ss ica l b ea m t h eo r y with first - o rde r s hea r defo rm a t io n effe cts . Also , p iezoe le ct r ic s he a r i ng co efficie nts ca n be co ns id e red i n t he co nst it u t ive re la t io ns o n ce s hea r

two−dtwo−e−fourcomma−f −fiveo−bracketrightrT −ma−hti−is−om−n a−od−lse−or −in−es−du−tc−se−oa−nn−te−e le a − ct − ss−ru − icd − pii−s−tp−on − la−sco−em−fs−en−mtl − bracketleft l d for m a − t i on s a nd l in ea r e la st ic is o t ro p ic mate r ia l b e h avio r . Contin u u m - base d fo rm a l is m s fo r na nosca le have be e n p ro posed t ha t in c l u de t he effect of s u rfa ce p ro pe r t ies o n t he mecha n ica l be havio r . H e re we s ha l l co ns id e r La p la ce - Yo u ng s u rfa ce e − twol − sixbracketright−aperiod−sticity a nd re s id u a l s u rfa ce te ns io n a d a pted to so l id ma te r ia ls [ r micro−cant ilever L,width b.math, 2h.math and ensity a of thebeam . W e le t I W e 2b.mathh.math3 co nsidebethe t h em omentof inertiahavingofthelecrossngthsectio nare athAic=kness2b.mathh.math,w.math(t.mathcomma.mathx.math)massthedfl exuralaredeflection o f theρbea m, = 3 ψ(t.mathcomma.mathx.math) t he ro ta t io n a n g le o f c ro ss s ect io n o f t h e b ea m , f.math(t.mathcomma.mathx.math) a t ra nsve rs e d y na m ic lo a d a nd q.math(t.mathcomma.mathx.math) a mome nt lo a d . The two−gtwo−o−sevencomma−v−eight e r n in g e q u a t io ns a re g ive n b y [

ρ A w.matht.matht.math − κ GA w.mathx.mathx.math + κ GA ψx.math = f.math(t.mathcomma.mathx.math)comma.math

(1)

ρ I ψt.matht.math− EI ψx.mathx.math−κ GA (w.mathx.math−ψ) = q.math(t.mathcomma.mathx.math)comma.math where

κGA = κGA − (τu.math + τb.math)b.mathcomma.math (2) 2 EI = (EI + 2b.mathh.math Es.math)comma.math (3) J u l i o R period Claeyssen comma Teresa Ts u kazan comma Leticia Tonetto comma Dan i e l a Tol fo \ centerlineseco nd hyphen{Ju o rd l e i r om R a t . r ix Claeyssen d iffe re n t ,ia Teresal s y ste m Ts t hat u kazan d e p e nds , Leticia no n l i nea Tonetto r l y u po , n Dan t h e i e ig e e l nva a l Tol u e period fo } .. The cases of m ic ro a re t he effective c u rva t u re effe ct a nd flex u ra l r ig id it y , res p ect ive l y . H e re τ a nd τ d e n ote t hyphen ca nt i leve rs u.math b.math he u p pe r a nd lowe r s u rfa ces res id u a l te ns io ns a nd E b e i ng a s u rfa ce e la st ic modu lu s . The b o u \noindentinvo lvi ngseco a n e la nd st− ic aorde p p e nd rmat a t t he fre r eixd o r l in iffe e a rs.math ize rent d b o u ia nda l r ysystemt co nd it io ns a hatdependsnon re d is c u ssed in te rms o f la fu i nd near a menta l yuponthee l r es po nse ig enva l ue . \quad The cases of m ic ro − ca nt i leve rs nd a r y co nd it io ns a re t h ose of a ca nt i leve r be a m o r s u bj e ct to ba la nce o f t h e mo me nt a n d s h ea r a t t he f invot hat lvi g ive ngane n in c lo sed la fo strm period ic appe .. I n se nda ct io n t 5 t comma he fre t h e e Ga o le r r l k in in method e a r is ize u sed dbounda to o bta in red r u y ced co hyphen nd it o rde io r ns fo rce a re d models d is cu ssed in te rmso f a fu ndamenta l r es po nse re e e n d x.math = L . I n t h is wo r k , we s h a l l ass u me t h a t fo r a u n ifo rm b ea m t he in vo lve d coefficie nts a re periodt hat .. Fo g rcedive n in c lo sed fo rm . \quad I n se ct io n 5 , t h eGa le r k in method is u sed to o bta in red u ced − o rde r fo rce d models . \quad Fo rced co nsta n t . resres po nses nses a re a a re p p a rox p im p a rox ted im b y aco tednce nt b ra y ted co res nce po ntnses ra in vo ted lvi resng co po nvo nses l u t io in n with vo lvi a co ngn ce co n t nvo ra te ld im u tp u io ls e n res with po nse a aco nd n ce n t ra te d im p u ls e res po nse a nd · N a noMMTA · Vo l .2 · 2013 · 124 − 144 · 126 aa lo lo ca ca l ized l ized res po res nse apo t t nse h e e a nd t o the f t h e be endo a m d u f e tothe bo u n beamdue da r y co nd it io to ns bounda period .. F in r a lycond l y comma it in seio ct ns io n . 6\ wequad d isFin c u ss t a h le l y , in se ct io n6wed is cu ss the matrix matrixmethodolog y with moda l a na l y s is fo r t he case o f harmo n ic i npu ts as we l l as fo r modu la ted l in ea r p ie cewise in pu ts w it h commethodolog p os ite y with m ic moda ro l− a naca l ny s t is ifo ler t he ve case r beams o f ha rm . o n ic i n p u ts as we l l as fo r mod u la ted l in ea r p ie cewise in p u ts w it h com p os ite m ic ro hyphen ca n t i le ve r bea ms period \noindent2 period ..2 Transversal . \quad vTransversal ib ration s of AFM v ib u s ration i ng th e s Timoshenko of AFM u model s i ng th e Timoshenko model AFM was d eve lo pe d fo r p ro d u c in g h ig h hyphen reso l u t io n im a ges of s u rfa ce st r u ct u res period .. The AFM t i p h as a ve rt ica l AFMwasreso lu t d io eve n o n lo t h pe e o d rd fo e r or f p 1 n.math ro du m.math c in o g r h be ig low h comma− reso a nd l it u ca t n iodetect nima low hyphen ges a of m s p l u it rfa u de vce ib st ra t r io u ns ct co r u re res s po .n d\quad i ng TheAFMt i p h as a ve rt ica l toreso h ig h lu f req t u io e n non c ie s period t he o rd e r o f $1 n.math m.math$ o r be low , and it ca n detect low − amp l it udevib ra t io nsco r re spond i ngtohighf requenc ie s . NowdayN owda y s it it is a is ls o a u ls se d ou to p se ro b d e top ro p pe ro r t bep ie s t h ro u pe g h r i nte t raie ct s io t ns h b ro etwe ugh e n t he i t nte ip a nra d tct he io sa m nsbetwe p le a nd to en mod tify he s u t rfa ip ces and t he samp le and tomod ify s u rfa ces . periodTh is i nte ra ct io np ro cess has le adAFMto beu se d i n sma rt mate r ia l te ch no lo gy , chem ica l /b io lo g ica l se nso rs , t r ib o lo gy $Th a− isseven i nte ra ct one io n− pone ro− cessn−comma has le a d commaAFM to−bracketright be u se d i n s ma−d− rtthree mate− r iafo url te ch $ no n lo $g y M− commaa o ch−n em ica o−d l slash e− bm io lo a g− ical l se n− nsoc rscomma h−g { i ta− rn ib o lo i g y n−n g−d , } a−i m−m u−o a−n g−t $ o $ t−a h−n { e r−A s } f i −M { e }$ l $ d−m { s−i } [$a-seven rotructura one-one-n-comma comma-bracketright-d-three-four $s−l { y }$ st emi n M-a sacompl o-n o-d e-m a-l ex n-c h-g sub i a-n i n-n g-d comma a-i m-m u-o a-n g-t o t-a h-n sub e r-A$ s onefi-M− subtwo− e lt d-m one sub− s-ia− openf i v e − squarebracketright bracket r o t r commau ct u− r asix s-l− subs y k st− ebracketright m i s ac o mp− lT ex−period bracketleft −h T−e $ h h $ a−m { r−i } i−none-two-t a−e one-a-five-bracketright{ u−s−t s r− comma-six-si ˆ{ z−o } k-bracketright-T-periode d−t h }$ bracketleft-h a $ n−t T-e t h−p h a-m l− subi r-i{ i-ni− a-ee } sub$ u-s-t e o s r-i $ to f− ther $ power ena of z-o fu e ndamen d-t h t lreso l u a$ n-t t− t-pi $ l-i sub on i-e $ e l o− f-rhyphen$ en a fu n d i am mi en t t lresop $a l u− t-ir on $ l-hyphen a \quad i mi$ t p m a-r ˆ{ a ..t− me to}$ the erpower [ of t-e er open square bracket devicedevice d d e epe pe nds nds o n t on he a ccut he ra ate extccu ra ra ct o te f sta ext t ic ra b e ct nd io ng f a sta nd re t so ic n a b nt e fre nd q u i e ng ncy a period nd re .. Test so measuna nt re m fre e nts qu a nd e t ncy h eo . ret\quad ica l Test measu re me nts a nd t h eo ret ica l stst ud u d ies ies have have s hown s hown t h a t t t he ha vi b t ra t t he io n vi be bh avio ra ut r io of m n ic be rost h r avio u ct u u re r s a ofmic t t he na rost n osca r le u is s ct ig nu ifica re nt s la y t s ize t hea nd nan pa ra osca metric le is s ig n ifica nt l y s ize a nd pa ra metric dede p p e nd nd e en t n period t . \ ..quad As t heAs s t r he u ct s u t ra r l s u ize ct de u c reases ra l towa s ize rd t de he c na reases nosca le rtowa eg im rd e comma t he sna u rfa nosca ce effe le cts r m eg u st im be e ta , ke s n u i nto rfa a ce effe ctsmu st be ta ke n i nto a cco un t cco$ u nine n t −bracketleft ] . $ \quad T $ h−i { s }$ \quad dependencehasm $t−o$ ivatedheuse $ onine-bracketleft−f i−s $ closing z $ square e−hyphen bracket $ period d e pe.. T nd h-i sube $s .. t− dn e p $ e nd\quad e nceco h as m $ t-o n−t$ i v a te inuumhe d h e u se o-f i-s z e-hyphen $r−o $ d e i pe e nd s e\ t-nquad .. con n-t mod i n$ u e− ul$ m h er-o ingmcro i e s .. n mod e-l $t i n− gs$ m cr ructureo t-s r u c tu r e s-colon $s−colon .. co u l-p $ e \quad co u $ l−p $ e Equation: s-one-one-two-two two-one-t-seven-eight-zero-two sub nine-one-r e-comma-comma-bracketright-bracketright f-comma-comma-s s-s-u-n-s r-t-r- o-n\ begin h-c-f-n-a-a{ a l i g nsub∗} c-i-i-l-e-t o-n-o-o-e n-r-c g-a-y-e-a bracketleft-a-l-n l f-e o-d-y to the power of r-i-r e-g sub g-m a-u-n l-t-d h-a e-t-e sub i-d o-o y-n-r open square\ tag ∗{ bracket$ s−one .. i−one−two−two two−one−t−seven−eight −zero−two { nine−one−r e−comma−comma−bracketright −bracketright f−commatwo-bracketleft−comma− bracketright-comma-comma-bracketrights s−s−u−n−s } r−t−r−o−n h− three-bracketrightc−f−n−a−a { commac−i−i− al m−e o−t ng o t o− hen− so period−o−e n−r−c g−a−y−e−a } bracketleft −a−l−n lI n f− te h is work o−d− commay ˆ{ r we−i u−r se} t hee Tim−g os{ hg e−m n ko} bea− au m−n hyphen l−t t− yd p e st h− ra u cte u− ret− foe r in{ coi− rd po ra o− to in g sy− hn ea−r r fo} rm[ a $ t} ioi n a n d ro ta r y i ne\end rt ia{ a l i g n ∗} effects period .. Th is a l lows to co ns id e r m ic ro be a ms with s ma l l le ngth hyphen to hyphen t h ic kness as p ect ra t io period .. Th ic k be a ms\noindent have re la t$ ive two l y−bracketleft bracketright −comma−comma−bracketright { three −bracketright } ,$ amongothes. h ig h t ra nsve rs e s he a r m od u l u s a n d t he effects o f r ota r y i ne rt ia a nd t ra nsve rs e s h ea r defo rm a t io n m u st be u se d in t he I nd y t n h a ism ic work a na l , y weu s is o f se s u t ch heTim be a ms period os h ..e The n ko Timoshen be am ko− modetype l co r re st cts ructure t he c la ss ica fo l b rea inm t co h eo rpora r y with first t ingshea hyphen o rde r r s format ionandro ta ry i ne rt ia heae f rf e c t s . \quad Th is a l lows to co ns id e rm ic ro be a ms with s ma l l le ngth − to − t h ic kness as p ect ra t io . \quad Th ic k be ams have re la t ive l y hdefo ig rmh t a t ra io n nsve effe cts rs period e s heaAlso comma rmodu p iezoe l le u ct s r ic and s he a t r i he ng co effects efficie nts o ca f n r be ota co ns r id y e ired ne i n rt t he ia co andnst it u t t raive re nsve la t io rs ns e o n s ce hea s r deforma t io nmu st beu se d in t he headynamic r ana lys is of suchbeams. \quad The Timoshen ko mode l co r re cts t he c la ss ica l b eam t h eo r y with first − o rde r s hea r defotwo-d rma two-e-four t io comma-f-five n effe cts sub . o-bracketright Also , p iezoe r T-m a-h le t ct i-i sub r ic s-o s m-n he a-o a d-l r subi ng s e-o co r-i efficie sub n-e s-d nts sub ca u-t n c-s be e-o co a-n ns n-t id e-e ele a-c red t-s i sub n ts-r he co nst it u t ive re la t io ns o n ce s hea r u-i$ sub two c− d-pd sub two i i-s-t−e− p-ofo ur n-l sub comma a-s c o-e−f− m-ff i v s-e e n-m{ o t− l-bracketleftbracketright l d .. for r m .. T a-t−m i on sa−h t i−i { s−o m−n }} a−o d−l { s e−oa nd} lr in− eai r{ e lan− ste ic} is os− td ro p{ icu mate−t r ia c− ls b e h e avio−o } r perioda−n .. Contin n−t u eu− me hyphen $ l e base $ ad− foc rm at− ls is m{ s fos− rr na} noscau−i le have{ c be} e nd− pp ro posed{ i ti− has− tt p−o } n−l { a−s c o−e m−f s−e n−m t } l−bracketleft $ l d \quad f o r m \quad $ a−t $ i on s andin c l l u inde t ea he effect r e laof sst u rfa ic ceis p ro o pe t r rop t ies oic n t matehe mecha r ia n ica l l behavio be havio r period r . ..\ Hquad e re weContin s ha l l u co u ns m id− e rbase La p lad ce fo hyphen rm a Yo l uis ng ms s u fo r na nosca le have be e n p ro posed t ha t rfain ce c l u de t he effect of s u rfa ce p ro pe r t ies o n t he mechan ica l be havio r . \quad He re wes ha l l co ns id e rLap la ce − You ng s u rfa ce $e-two e−two l-six sub l− bracketright-as i x { bracketright period-s t ic− ita y a periodnd re s id−s u a l t s u rfa i c ce te i ns t io n y a} d$ a pted and to re so l s id id ma uate r ia l ls su open rfa square ce bracket te ns io nadapted to so l idmate r ia ls [ le t I We = 2 b.math h.math to the power of 3 divided by 3 co ns id e b sub e t he to the power of r t h e sub m sub ome nt to the power of m ic ro hyphen\noindent ca subl eo f t to I the $ power We of{ n= t sub\ f rin a ec { rt2 ia to b.maththe power of h.math i leve r having ˆ{ 3 o}}{ f t he3 le}} c$ r oss co ngth $ sub ns sectio id sub e n to{ theb } powerˆ{ r of} L{ commae t width he }$ subt h a re $ sub e a{ tom the}ˆ power{ m of i b.mathc ro comma− t hca sub} A{ icome sub = kness nt }ˆ sub{ n 2 b.math t } h.math{ o sub f } commaˆ{ i to l the e v e power r of} 2{ h.mathin sub e w.math r t sub i a open} having { o parenthesisf t he t.math} comma.mathl e { c rto the oss power} ofngth a n d sub{ s x.math e c t i o closing}ˆ{ L parenthesis , width mass t} sub{ n he} d{ suba fl sub re exu}ˆ ra{ b l to. math the power , of} e{ nsa it} yt a{ reh d} { A } subi c e{ fle= to} thekness power of{ a sub2 ct b.math io to the power h.math of o f}ˆ sub{ 2 n to the h . math power} of{ t h, e} sub{ ow. sub math f to} theˆ{ powera nof be d sub} t{ he( to the t.math power of a comma.math m rho sub b } { x . math sub) } eamass to the{ powert } { ofhe period} d sub{ m commaf l }ˆ{ toe the power ns of i t We y } { exu ra l } a re { d }ˆ{ a } { e f l e }ˆ{ o f } { ct i o psi}ˆ{ opent parenthesis h e } t.math{ n } comma.math{ o }ˆ{ be x.math} { closingf }ˆ{ parenthesisa m } t{ het ro ta he t io}\ n a nrho g le o{ f cb ro}ˆ ss{ s. ect} io{ nea o f t}ˆ h{ eWe b ea} m{ commam f.math, }$ open parenthesis t.math comma.math x.math closing parenthesis a t ra nsve rs e d y na m ic lo a d a nd q.math open parenthesis t.math comma.math x.math closing\noindent parenthesis$ \ apsi mome( nt lo t.math a d period The comma.math x.math ) $ t he ro ta t io nang le o f c ro ss s ect io no f thebeam $ ,two-g f.math to the power ( of two-o-seven t.math comma-v-eight comma.math e r x.math n in g e q u ) a t $ io nsa at re ra g ive nsve n brs y open e dynamic square bracket lo adand $q.math ( t.math comma.mathhline x.math ) $ a mome nt lo a d . The $rho two A− w.mathg ˆ{ two sub−o t.math−seven t.math comma minus−v kappa−e i g h GA t } w.math$ erningequationsaregivenby[ sub x.math x.math plus kappa GA psi sub x.math = f.math open parenthesis t.math comma.math x.math closing parenthesis comma.math \ [ open\ r u l parenthesis e {3em}{0.4 1 closing pt }\ parenthesis] hline rho I psi t.math t.math minus EI psi sub x.math x.math minus kappa GA open parenthesis w.math sub x.math minus psi closing parenthesis = q.math open parenthesis t.math comma.math x.math closing parenthesis comma.math \ centerlinewhere { $ \rho $ A $ w. math { t.math t.math } − \kappa $ GA $ w. math { x.math x.math } + \kappa $ GAhline $ \ Equation:psi { x open . math parenthesis} = 2 f.math closing parenthesis ( t.math .. kappa GAcomma.math = kappa GA minusx.math open parenthesis ) comma.math tau sub u.math $ } plus tau sub b.math closing parenthesis b.math comma.math Equation: open parenthesis 3 closing parenthesis .. overbar EI = open parenthesis E I plus 2 b.math h.math to the power of\ begin 2 E sub{ a s.math l i g n ∗} closing parenthesis comma.math (a re 1 t he )effective\\\ r c u u l e rva{3em t u}{ re0.4 effe pt ct} a nd .. flex u ra l r ig id it y comma res p ect ive l y period H e re tau sub u.math a nd tau sub b.math d e n ote t\end he u{ pa lpe i g r n a∗} nd lowe r s u rfa ces res id u a l te ns io ns a nd E sub s.math b e i ng a s u rfa ce e la st ic modu lu s period .. The b o u nd a r y co nd it io ns a re t h ose of a ca nt i leve r\noindent be a m o r $ \rho $ I $ \ psi t.math t.math − $ EI $ \ psi { x.math x.math } − \kappa $ GA $ ( w.math { x . math } −s u \ bjpsi e ct to) ba la= nce q.math o f t h e mo ( me nt t.math a n d s h ea comma.math r a t t he f re e e x.math n d x.math ) = L comma.math period I n t h is $ wo r k comma we s h a l l ass u me t h a t fo r awhere u n ifo rm b ea m t he in vo lve d coefficie nts a re co nsta n t period \ begintimes{ Na l a i g noMMTA n ∗} times Vo l period 2 times 20 1 3 times 1 24 hyphen 1 44 times 1 26 \ r u l e {3em}{0.4 pt }\\\kappa GA = \kappa GA − ( \tau { u . math } + \tau { b . math } ) b . math comma . math \ tag ∗{$ ( 2 ) $}\\\ overline {\}{ EI } = ( E I + 2 b.math h.mathˆ{ 2 } E { s . math } ) comma . math \ tag ∗{$ ( 3 ) $} \end{ a l i g n ∗}

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K 1 v . =\ Fquad comma.mathMatha−e r open ie− xf parenthesisk ormu 4 l closing ato parenthesis} ix M v¨+ K v = F comma.math (4) where where \ centerlineEquation: open{ $ parenthesis one−T h 5− closingp a r e n r parenthesis i g h t { c ..−e v} = Rown−c 1 $ w.math o $ open b−u parenthesis p−e t.math l−w $ comma.math e ttT $ x.math n−i−e closing $ m parenthesis $ o−a Row s−s 2{ psih a−e } s−n { e−k } o−o n−m $ d $ hyphen−o d−o r−e ˆ{ d−l }$ e rd $ f f −i $ e $ r−e t−n $ i a l e q u a open parenthesis w.math t.math(t.mathcomma.mathx.math comma.math x.math closing)  parenthesis . comma.math f.math F(t.mathcomma.mathx.math = Row 1 f.math open parenthesis)  t.math comma.math x.math closing $v i−=o $ n wi h m a $ r−t { i x }$comma.math c oe ffic F ie= nt s } comma.math parenthesis Rowψ(t.mathcomma.mathx.math 2 q.math open parenthesis t.math) comma.math x.math closingq.math parenthesis(t.mathcomma.mathx.math . comma.math ) M = Row 1 rho A 0 Row 2 0 rho I . comma.math .. K = E partialdiff to the power of 2 divided by partialdiff x.math(5) to the power of 2 plus N partialdiff Mdivided $ \ byddot partialdiff{v} +$ x.math Kv plus R $=$ comma.math F $comma.math open parenthesis 6 closing( 4 parenthesis )$ where   with ρA 0 2 M = comma.math K = E ∂ + N ∂ + R comma.math (6) E = Row 1 minus kappa GA 0 Row0 2 0ρI hline minus EI . comma.math∂x.math N =2 parenlefttp-parenleftbt∂x.math 0 kappa minus GA kappa GA from 0 to hline parenrightbt-parenrighttp\ begin { a l i g n ∗} comma.math .. 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( mathφi.math ˆ(w.math{ i . math (φ}0 { i( .t.math matht.mathcomma.math}}ˆ{ ( t.mathx.mathx.math comma.math)) )comma.math} { ( C.kali t.math1comma.mathi.math comma.math= (E41}} { w. mathF41 { w. mathG41 ˆ{ i . math H}41{ i . math)comma.mathC.kali}}ˆ{ ( 2comma.mathi.math = (E42 F42 G42 H42 ) comma.mathi.math = 1comma.mathperiod.mathperiod.mathperiod.mathcomma.mathN−1period.math open parenthesis sub i.math closing parenthesis to the poweri.math of 1i.math to the powercomma.math of open parenthesis i.math closing parenthesis sub psi sub psi to the power oft.math prime from comma.math i.math to i.math} { to( the t.math power of prime comma.math sub open parenthesis}} x . math t.math ˆ{ comma.mathx . math { toi . the math power}ˆ{ ofx open . math parenthesis{ i . math t.math}} comma.math{ x . math to{ i . math }}} { i . math }ˆ{ ) { ) }} { ) { ) }} · N a noMMTA · Vo l .2 · 2013 · 124 − 144 · 127 +the ˆ power{ + ˆ of{ psi+ sub} { psi+ to}} theF power ˆ{ F of ˆ prime{ ( from i . i.mathmath to )i.math} { toF the ˆ{ powerF ˆ{1 of{ prime( }ˆ sub{ 1 open} { parenthesisi . math t.math ) }} comma.math{2{ ( }ˆ{ to1 the} power{ i . math of open parenthesis) }}} {3{ t.math( }ˆ comma.math{ 1 } { i .x.math math sub ) i.math}}}} to{ the4 power 1 }ˆ of{\ x.mathpsi sub{\ i.mathpsi fromˆ{ i x.math . math sub} { i.mathi . math to x.math}}ˆ{ sub( i.math t.math sub closing comma.math parenthesis subx . math closing{ parenthesisi . math } to the) power} { ( of closing t.math parenthesis comma.math sub closing parenthesis x.math { comma.mathi . math } = to) the}} power{\ psi of ={\ frompsi = toˆ{ = Ei . sub math 42 to} { thei power . math of}}ˆ{ ( Et.math sub E sub comma.math 32 open parenthesis x.math sub i.math{ i closing . math parenthesis} ) } { to( the power t.math of E sub comma.math 22 open parenthesis x.math sub i.math{ i . math closing} parenthesis) }} + to ˆ the{ + power ˆ{ + of} 1 { + }} Gopen ˆ{ parenthesisG ˆ{ ( sub i . i.math math closing ) } parenthesis{ G ˆ{ G to ˆ{ the1{ power( }ˆ{ of 21 to} the{ i power . math of open ) parenthesis}} {2{ ( i.math}ˆ{ 1 closing} { i parenthesis . math )sub}}} w.math{3{ sub( } w.mathˆ{ 1 } sub{ i . math i.math) }}}} plus{ 14 to the 1 } powerˆ{ w. of math i.math ˆ{\ plusprime 1 to the} power{\prime of w.mathˆ{ w. sub math w.math{ subi . math i.math}} plus{ 1i . to math the power}}ˆ{ of( i.math t.math plus 1 open comma.math parenthesis} to{ the( powert.math of open comma.math parenthesis from}} { openw. math parenthesis ˆ{\ toprime open parenthesis} {\prime subˆ t.math{ w. math sub t.math{ i . sub math comma.math}} { i . math to the}} powerˆ{ ( of comma.math t.math to comma.math the power of } { ( t.matht.math sub t.math comma.math sub comma.math}} x . math to the ˆ power{ x . math of comma.math{ i . math x.math}ˆ{ subx . math i.math to{ thei . math power of}} x.math{ x . math sub i.math{ i from . math x.math}}} sub{ i i.math . math to} x.mathˆ{ ) { ) }} { ) { ) }} +sub ˆ i.math{ + ˆ{ sub+ closing} { + parenthesis}} H ˆ{ subH closing ˆ{ ( parenthesis i . math to the ) } power{ H of ˆ{ closingH ˆ{ parenthesis1{ ( }ˆ{ sub1 } closing{ i . parenthesis math ) plus}} { to2{ the( power}ˆ{ of1 plus} { fromi . math plus to plus) }}} F sub{3{ 42( to} theˆ{ power1 } { of Fi . sub math F sub ) 32}}}} open parenthesis{ 4 1 } subˆ{\ i.mathpsi closingˆ{\prime parenthesis} {\ topsi the powerˆ{\prime of F sub} 22ˆ{ openi . math parenthesis} { i sub . math i.math}}ˆ closing{ ( parenthesist.math to comma.math the power of} 1 open{ ( parenthesis t.math sub comma.math i.math closing parenthesis}} {\ psi toˆ the{\ powerprime of} 2 to{\ thepsi powerˆ{\ ofprime open parenthesis}ˆ{ i . math i.math} closing{ i . math parenthesis}}ˆ{ ( subt.math psi sub psi comma.math sub i.math to} the{ ( power t.math of i.math sub comma.math plus to the power}} x of . math plus sub ˆ{ 1x to . math the power{ i of . math 1 to the}ˆ power{ x . math of psi sub{ psii . math sub i.math}} { tox the . math power{ i . math }}} { i . math }ˆ{ ) { ) } commaof i.math . math sub plus} { to) the{ power) }} of plus= ˆ sub{ = 1 to ˆ{ the= power} { = of}} 1 openE parenthesis ˆ{ E ˆ{ ( to the i power. math of open ) } parenthesis{ E ˆ{ E from ˆ{1 open{ ( parenthesis}ˆ{ 2 } to{ openi . math parenthesis ) }} {22{ ( } { i . math sub) }}} t.math{32 sub{ ( t.math} { i sub . math comma.math ) }}}} to the{ 42 power}ˆ{ ofw. comma.math math { w. to math the power ˆ{ i of . math t.math sub + t.math 1 } sub{ i comma.math . math + to the 1 }}} power{ ofw. comma.math math { w. math ˆ{ i . math +x.math 1 sub} { i.mathi . math to the + power 1 of}}} x.math( sub ˆ{ i.math( ˆ{ from( } x.math{ ( }} subˆ{ i.matht . math to x.math{ t . math sub i.math}ˆ{ comma sub closing . math parenthesis} { comma sub . closing math parenthesis}} { t . math to the{ t . math }ˆ{ comma . math } { comma . math }} powerx . math of closing ˆ{ x . math parenthesis{ i sub . math closing}ˆ{ parenthesisx . math plus{ i to . math the power}} { ofx plus . math from plus{ i .to math plus G}}} sub{ 42ito . math the power}ˆ{ of) G{ sub) G}} sub{ 32) open{ ) parenthesis}} + ˆ{ + ˆ{ + } { + }} Fsub ˆ{ i.mathF ˆ{ closing( parenthesisi . math to ) the} { powerF ˆ{ ofF G ˆ{ sub1{ 22( open}ˆ{ parenthesis2 } { i . math sub i.math ) closing}} {22 parenthesis{ ( } { i to . math the power ) of}}} 1 open{32{ parenthesis( } { isub . math i.math ) }}}} { 42 }ˆ{\ psi {\ psi ˆ{ i . math } { i . math }ˆ{ + } { + }}ˆ{ 1 } { 1 }} {\ psi {\ psi ˆ{ i . math } { i . math }ˆ{ + } { + }}ˆ{ 1 } { 1 }} closing( ˆ{ ( parenthesis ˆ{ ( } { to( the}} powerˆ{ t of. math 2 to the{ powert . math of open}ˆ{ parenthesiscomma . math i.math} { closingcomma parenthesis . math }} sub{ i.matht . math plus{ 1 fromt . math w.math}ˆ{ subcomma i.math . math plus} 1 to{ thecomma . math }} powerx . math of prime ˆ{ x .to math w.math{ toi .the math power}ˆ of{ primex . math to the{ poweri . math of i.math}} { plusx . math 1 from w.math{ i . math sub i.math}}} { plusi . math 1 to the}ˆ power{ ) of{ prime) }} to{ w.math) { to) the}} power+ ˆ{ + ˆ{ + } { + }} Gof prime ˆ{ G open ˆ{ ( parenthesis i . math to the ) power} { ofG open ˆ{ G parenthesis ˆ{1{ ( } fromˆ{ open2 } { parenthesisi . math to open ) }} parenthesis{22{ ( sub} { t.mathi . math sub t.math ) }}} sub{32 comma.math{ ( } { toi .the math power ) }}}} { 42 }ˆ{ i . math +of comma.math 1 ˆ{ w. math to the ˆ{\ powerprime of t.math} { subi . math t.math sub + comma.math 1 }} { w. to math the power ˆ{\prime of comma.math}}} { i x.math . math sub + i.math 1 to ˆ{ thew. power math of ˆ{\ x.mathprime sub} i.math{ i . math +from x.math1 }} { subw. i.mathmath ˆ to{\ x.mathprime sub}}} i.math( sub ˆ{ closing( ˆ{ ( parenthesis} { ( }} subˆ{ closingt . math parenthesis{ t . math to the}ˆ power{ comma of closing . math parenthesis} { comma sub . math closing}} parenthesis{ t . math { t . math }ˆ{ comma . math } { comma . math }} plusx . math to the ˆ{ powerx . math of plus{ fromi . math plus to}ˆ plus{ x . H math sub 42{ toi the . math power}} of{ Hx sub . math H sub{ 32iopen . math parenthesis}}} { i sub . math i.math}ˆ{ closing) { parenthesis) }} { ) to the{ ) power}} of+ H ˆ{ + ˆ{ + } { + }} Hsub ˆ{ 22H open ˆ{ parenthesis( i . math sub i.math ) } closing{ H ˆ parenthesis{ H ˆ{1{ to( the}ˆ{ power2 } of{ 1i open . math parenthesis ) }} sub{22 i.math{ ( } closing{ i . mathparenthesis ) to}}} the{ power32{ ( of} 2 to{ thei . math power of ) }}}} { 42 }ˆ{ i . math +open parenthesis1 ˆ{\ psi i.mathˆ{\ closingprime parenthesis} { i . math sub i.math + plus 1 }} 1 from{\ psi subˆ{\ i.mathprime plus}}} 1 to the{ i power . math of prime + to 1 psi ˆ{\ to thepsi powerˆ{\ ofprime prime to} { thei power . math of +i.math 1 plus}} 1{\ frompsi psiˆ sub{\prime i.math plus}}} 1 to( the ˆ{ power( ˆ{ of( prime} { to( psi}}ˆ to{ thet . math power of{ primet . math open}ˆ parenthesis{ comma . tomath the power} { comma of open . math parenthesis}} { fromt . math open { t . math }ˆ{ comma . math } { comma . math }} parenthesisx . math ˆ{ tox open . math parenthesis{ i . math sub t.math}ˆ{ x sub . math t.math{ subi . math comma.math}} { x to . math the power{ ofi . comma.mathmath }}} { toi the . math power}ˆ of{ t.math) { sub) }ˆ t.math{ comma sub . comma.math math } { comma . math }} { ) { ) }ˆ{ comma . math } { period .math }}\] to the power of comma.math x.math sub i.math to the power of x.math sub i.math from x.math sub i.math to x.math sub i.math sub closing parenthesis sub closing parenthesis sub period.math to the power of comma.math to the power of closing parenthesis sub closing parenthesis sub comma.math to the power\ centerline of comma.math{ I n matrix te rms } I n matrix te rms \ [C.kali C. k a sub l i 1{ comma.math1 comma.math i.math w sub i.math i.math open} w parenthesis{ i . math t.math} comma.math( t.math x.math comma.math sub i.math closing x.math parenthesis{ i . = math C.kali} sub) 2 comma.math = C. k a l i { 2 comma.mathi.math w sub i.math i.math plus 1 open} w parenthesis{ i . math t.math + comma.math 1 } ( x.math t.math sub i.math comma.math closing parenthesis x.math comma.math{ i . math i.math} =) 1 comma.math comma.math period.math i.math =period.math 1 comma.math period.math comma.math period.math N minus period.math 1 comma.math period.math comma.math N − 1 comma . math \ ] where w sub i.math open parenthesis t.math comma.math x.math closing parenthesis = parenlefttp-parenleftex-parenleftex-parenleftbt phi sub i.math open parenthesis\ centerline from{where w.math} sub i.math to the power of prime open parenthesis to w.math sub i.math open parenthesis phi sub prime i.math open parenthesis t.math sub comma.math to the power of t.math comma.math x.math to the power of t.math sub comma.math to the power of t.math comma.math sub x.math\ [ w closing{ i . math parenthesis} ( closing t.math parenthesis comma.math to the power of x.math x.math sub ) x.math = sub ( closing\phi parenthesis{ i . math to the} power( ˆ{ ofw. closing math parenthesis ˆ{\prime parenrightbt-} { i . math } parenrightex-parenrightex-parenrighttp( } { w. math { i . math } ( }{\ comma.mathphi }ˆ{ C.kalit . math sub 1 ˆ comma.math{ t.math i.math comma.math = parenlefttp-parenleftex-parenleftex-parenleftex-parenleftbt} { comma . math }} {\prime { i . math } E sub( 4 1t .to math the power ˆ{ t.math of E sub E sub comma.math 3 open parenthesis} { comma sub i.math . math closing} x parenthesis . math }ˆ{ tox the . math power{ of 1x .to math the power}ˆ{ of) E} sub{ 2) open}} { parenthesisx . math sub ) i.math{ ) }} closing) comma.mathparenthesis to the C.kali power of 1 to{ the1 power comma.math of 1 open parenthesis i.math sub} i.math= closing ( E parenthesis ˆ{ E ˆ{ to( the power i . math of 1 to ) the} power{ E of ˆ{ openE ˆ parenthesis{1{ ( }ˆ i.math{ 1 } closing{ i . math parenthesis) }} {2{ F( sub}ˆ{ 4 11 to} the{ poweri . math of F sub ) F}}} sub 3{3 open{ ( parenthesis}ˆ{ 1 } sub{ i i.math . math closing ) parenthesis}}}} { 4 to the 1 } powerF ofˆ{ 1F to theˆ{ power( i of . math F sub 2 open ) } parenthesis{ F ˆ{ F ˆ{1{ ( }ˆ{ 1 } { i . math sub) }} i.math{2{ closing( }ˆ{ parenthesis1 } { i . to math the power ) }}} of 1 to{3 the{ ( power}ˆ{ of1 1} open{ i parenthesis. math ) sub}}}} i.math{ 4 closing 1 parenthesis} G ˆ{ G to ˆ the{ ( power i . of math 1 to the ) power} { ofG open ˆ{ G ˆ{1{ ( }ˆ{ 1 } { i . math parenthesis) }} {2{ i.math( }ˆ{ closing1 } { parenthesisi . math G sub ) }}} 4 1 to{ the3{ power( }ˆ{ of1 G sub} { Gi sub . math 3 open parenthesis ) }}}} { sub4 i.math 1 } closingH ˆ{ parenthesisH ˆ{ ( to i the . math power of ) 1 to} the{ H power ˆ{ H ˆ{1{ ( }ˆ{ 1 } { i . math of) G}} sub{2 2{ open( } parenthesisˆ{ 1 } { subi . math i.math closing ) }}} parenthesis{3{ ( } toˆ{ the1 power} { ofi . 1 math to the power ) }}}} of 1 open{ 4 parenthesis 1 } ) sub comma.mathi.math closing parenthesis C.kali to{ the2 power comma of . math 1i .to math the power} = of open ( parenthesis E ˆ{ E ˆi.math{ ( closing i . math parenthesis ) } H{ subE ˆ4{ 1 toE the ˆ{1 power{ ( } ofˆ{ H sub2 } H{ subi . 3 math open parenthesis ) }} {22 sub{ i.math( } { closingi . math parenthesis ) }}} to the{32{ ( } { i . math power) }}}} of 1{ to42 the} powerF ˆ of{ HF sub ˆ{ 2( open i parenthesis . math ) sub} i.math{ F ˆ closing{ F ˆ{ parenthesis1 2 { to( the} { poweri . math of 1 to the ) power}} {22 of{ 1 open( } { parenthesisi . math sub ) i.math}}} { closing32{ ( } { i . math parenthesis) }}}} { to42 the} powerG ˆ{ ofG 1 to ˆ{ the( power i . math of open parenthesis ) } { G i.math ˆ{ G closing ˆ{1{ ( parenthesis}ˆ{ 2 } parenrightbt-parenrightex-parenrightex-parenrightex-parenrighttp{ i . math ) }} {22{ ( } { i . math ) }}} {32{ ( } { i . math comma.math) }}}} { 42 C.kali} subH ˆ 2{ comma.mathH ˆ{ ( i.mathi . math = parenlefttp-parenleftex-parenleftex-parenleftex-parenleftbt ) } { H ˆ{ H ˆ{1{ ( }ˆ{ 2 } { i . math ) }} E sub{22 42{ to( the} { poweri . math of E sub ) E}}} sub 32{32 open{ ( } { i . math parenthesis) }}}} { sub42 i.math} ) closing comma.math parenthesis to i.math the power =of E sub 1 22 comma.math open parenthesis period.math sub i.math closing parenthesisperiod.math to the power period.math of 1 open parenthesis comma.math sub Ni.math− closing1 parenthesis period.math to the\ power] of 2 to the power of open parenthesis i.math closing parenthesis F sub 42 to the power of F sub F sub 32 open parenthesis sub i.math closing parenthesis to the power of F sub 22 open parenthesis sub i.math closing parenthesis to the power of 1 2 open parenthesis sub i.math closing parenthesis to the power of open parenthesis i.math closing parenthesis G sub 42 to the power of G sub G sub 32 open parenthesis sub i.math\ hspace closing∗{\ f parenthesis i l l } $ \cdot to the$ power N a of noMMTA G sub 22 open $ \cdot parenthesis$ Vo sub l i.math $ . closing 2 parenthesis\cdot to20 the power 1 3 of 1 open\cdot parenthesis1 24 sub i.math− 1 closing 44 parenthesis\cdot 1 to the 27 power $ of 2 to the power of open parenthesis i.math closing parenthesis H sub 42 to the power of H sub H sub 32 open parenthesis sub i.math closing parenthesis to the power of H sub 22 open parenthesis sub i.math closing parenthesis to the power of 1 open parenthesis sub i.math closing parenthesis to the power of 2 to the power of open parenthesis i.math closing parenthesis parenrightbt-parenrightex-parenrightex-parenrightex-parenrighttp comma.math i.math = 1 comma.math period.math period.math period.math comma.math N minus 1 period.math times N a noMMTA times Vo l period 2 times 20 1 3 times 1 24 hyphen 1 44 times 1 27 J u l i o R period Claeyssen comma Teresa Ts u kazan comma Leticia Tonetto comma Dan i e l a Tol fo \ centerlineFig 2 period{ MuJ u l t l is i pan o R m i. cro Claeyssen hyphen beam , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo } J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo Thus fo r a m u lt is pa n Timoshenko m ic ro hyphen b ea m comma i nc l u d in g p ro po r t io na l da m p in g C = a.math M plus b.math K comma Fig 2 . Mu l t is pan m i cro - beam we\noindent have t h eFig seco nd 2 hyphen. Mu l t is panmi cro − beam Thus fo r a m u lt is pa n Timoshenko m ic ro - b ea m , i nc l u d in g p ro po r t io na l da m p in g C = a.math M o rd e r b lo ck matrix d iffe re nt ia l eq u a t io ns +b.math K , we have t h e seco nd - \ hspaceM dieresis-v∗{\ f i open l l }Thus parenthesis fo r t.math a mu comma.math lt is pa nx.math Timoshenko closing parenthesis m ic ro plus− beam, C v-dotaccent i ncopen l parenthesis udin gpropor t.math comma.math t io x.math na l dampinclosing gC o rd e r b lo ck matrix d iffe re nt ia l eq u a t io ns parenthesis$= a.math$ plus Kv open M parenthesis$+ b.math$ t.math comma.math K , we have x.math t closing h e seco parenthesis nd − = F comma.math 0 less x.math less L comma.math t.math greater 0 M v¨(t.mathcomma.mathx.math)+ C v˙(t.mathcomma.mathx.math)+ Kv (t.mathcomma.mathx.math) = F comma.math 0 < comma.math open parenthesis 8 closing parenthesis \noindentx.math < Locomma.math rd e r b lo t.math ck matrix > 0comma.math d iffe re(8) ntwhere ia l eq ua t io ns where   v.math1   v = Row 1 v.math sub 1 Row 2 v.math sub 2 Row 3 period Row 4 period Row 5 periodM Row1 60 v.math sub N . comma.math M = Row 1 M sub 1 0 Row M $ \ddot{v} ( t.math comma.math v.math x.math2  ) +$ C $ \dot{v} ( t.math comma.math x.math ) 2 period Row 3 period Row 4 period Row 5 0 M sub N . comma.math .. K = Row 1 K sub 1. 0 Row 2 period Row 3 period Row 4 period Row 5 0 K sub N +$ Kv $ ( t.math comma.math x.math.  ) =$ F $comma.math 0 < x . math < $ L $ comma.math t.math . comma.math .. F = Row 1 F sub 1 Rowv 2= F sub 2 Row 3 period comma.math Row 4 periodM = Row 5 period. Row 6comma.math F sub N . comma.mathK open parenthesis 9 closing > 0 comma.math ( 8 )$  .    parenthesis    .  where  .  with 0 MN v.mathN M sub j = Row 1 rho sub j.math A sub j.math 0 Row 2 0 rho sub j.math I sub j.math . comma.math K sub j = parenlefttp-parenleftex-parenleftbt \ hspace ∗{\ f i l l }v $ = \ l e f t (\ begin { arrayF1 }{ c} v . math { 1 }\\ v . math { 2 }\\ . \\ . \\ . \\ v . math { N }\end{ array }\ right ) minus subK minus1 0 sub kappa sub j.math G sub j.math A sub j.math to the power of hline sub partialdiff sub x.math to the power of partialdiff to the power comma.math $ M $ = \ l e f t (\ begin { array F2 }{cc } M { 1 } & 0 \\ . \\ . \\ . \\ 0 & M { N }\end{ array }\ right ) of kappa sub. j.math G sub j.math A sub j.math to the power of hline sub partialdiff sub x.math to the power of partialdiff to the power of 2 sub 2 minus E comma . math $ \quad K $ = \ l e f t (\ begin. { array }{ cc } K { 1 } & 0 \\ . \\ . \\ . \\ 0 & K { N }\end{ array }\ right ) sub= j.math . x.math from comma.math kappa sub j.mathF to= I sub j.math comma.math to the power of(9) two-line sub partialdiff x.math to the power of kappa sub j.math sub partialdiff    .  commasub 2 to . math the. power $ \ ofquad G subF j.math $ = A\ subl e f plus t (\ tobegin the power{array of}{ j.mathc} F to{ the1 power}\\ ofF hline{ sub2 }\\ partialdiff. \\ to the. \\ power. of\\ partialdiffF { N G}\ subend j.math{ array A sub}\ right )    .  comma.mathj.math parenrighttp-parenrightex-parenrightbt0 K ( 9 ) $ comma.math  v.math sub j.math = Row 1 w.math sub j.math Row 2 psi sub j.math . comma.math N F s u bj e ct to g ive n in it ia l co nd it io ns v openN parenthesis 0 comma.math x.math closing parenthesis = r sub o.math open parenthesis x.math closing with parenthesis\noindent comma.mathwith v sub t.math open parenthesis 0 comma.math x.math closing parenthesis = r sub 1 open parenthesis x.math closing parenthesis comma b o u nda r y a n d com p a t i b i l it y co n d it io ns \ [M { j } = \ l e f t (\ begin { array}{ cc }\rho { j . math } A { j∂ .2 math } & 0 \\ 0 & \rho { j . math } I { j . math }\end{ array }\ right )   Equation: open parenthesis 1 0 closing parenthesis .. B.kali-one w sub 1κj.math = n.mathGj.math subAj.math 0 comma.mathx.math B.kali2 sub N w sub N = n.math sub L comma.math ρj.mathAj.math 0 ∂ κj.math w.mathj.math commaMj = . math K { j } = ( − {comma.math − }ˆ{\kappa Kj ={ (j− . math } G { j . math } A { j . math∂ ˆ{\ r u l e {3em}{−0.4Ej.math pt }}x.math{\ partialG A }}ˆ{\ partialGj.mathˆ{ A2j.math}} ){comma.mathx . math } { v.math2 }}j.math{\kappa= { j . math } comma.math Equation: open parenthesis 1 1 closing parenthesis .. C.kali sub 1 comma.math− i.math w sub i open parenthesisκj.math t.mathGj.math comma.mathAj.math x.mathx.math sub i.math closing j.math 0 ρj.mathIj.math ∂ κ 2 j.math ψj.math G { j . math } A { j . math ˆ{\ r u l e {3em}{0.4 pt }} {\ partial }}ˆ{\ partial } { x . math }} − E { j . math } x . mathj.math ˆ{\∂kappa {∂ j . math∂ }} { I ˆ{\kappa { j . math {\ partial }}ˆ{ G { j . math } I two−line + parenthesis = C.kali sub 2 comma.math i.math w sub i plus 1 open parenthesis t.math comma.math x.math sub i.math closing parenthesis comma.math j.math ∂x.math Ai.math} { =2 1}} comma.math{ j . math period.math ˆ{ two−l period.mathi n e } {\ partial period.math comma.mathx . math }}ˆ N{ minusj . math 1 period.math ˆ{\ r u l e {3em}{0.4 pt }} {\ partial }} { + }ˆ{\ partial }} G { j . math } A { j . math } ) comma.math v.math { j . math } = \ l e f t (\ begin { array }{ c} w. math { j . math }\\ s2 u bj period e ct to 3 g period ive n in .. it AFMia l co nd hyphen it io ns t v i(0 pcomma.mathx.math hyphen canti l ever i) nte = r ractio.math ons(x.math)comma.mathvt.math(0comma.mathx.math) = \ psi { j . math }\end{ array }\ right ) comma . math \ ] r1The(x.math t i p) i, nteb o ura nda ct io r y n a with n d com t he p sa a mt i pb lei l h it as y co b een d n it u io s ns u a l l y m od e le d as be in g s u bj ect to s p r in gs o r das h hyphen s p r i ngs o r eight-a three-t-comma zero-t sub a-bracketright che d mass fo r no rm a l a nd la te ra l inte ra ct io n a nd to a n exte r na l exc ita t io n of t he base open square bracket \noindent subj e ct tog ivenin it ia l cond it io nsv $( 0 comma.math x.math ) = r { o . math } Fo r i nsta n ce comma whe n t he t i p of le ngth h.math a nd mass m.math is s u bj e ct to no rm a l k.math sub N a n d la te ra l s p r in gs k.math B.kali − onew1 = n.math0comma.math B.kaliN wN = n.mathLcomma.math sub( L x.math a n d vis co ) u s comma.math v { t . math } ( 0 comma.math x.math ) = r (10){ 1 } ( x . math ) , $ boundaryandcompat i b i l it ycond it io ns C.kalida m1 pcomma.mathi.math e rs c.math subw Ni(t.mathcomma.mathx.math comma c.math sub L commai.math t he) = momC.kali e nt2comma.mathi.math a nd s h ea r cow ndi+1 it(t.mathcomma.mathx.math io ns a t t he f ree e n d a rei.math g ive)comma.math n b y i.math = 1comma.mathperiod.mathperiod.mathperiod.mathcomma.mathN − 1period.math kappa GA parenleftbigg partialdiff w.math divided by partialdiff x.math minus psi parenrightbigg = minus k.math(11) sub N minus c.math sub N partialdiff w.math\ begin divided{ a l i g n by∗} partialdiff t.math minus m.math partialdiff to the power of 2 w.math divided by partialdiff t.math to the power of 2 sub vextendsingle- vextendsingle-vextendsingleB.2 . k 3 a l. i −one AFM - wt i p{ - canti1 sub} l x.math=ever i n nte . = math ractiL to theons{ power0 } ofcomma.math comma.math Equation: B.kali open{ N parenthesis} w { 1N 2} closing= parenthesis n . math ..{ EL I} partialdiffcomma psi . math divided\ tag ∗{$ ( by1 partialdiffThe 0 t )i p x.math$ i}\\ nte raC. = ct k minus io a ln i with k.math{ t1 he sa sub comma.math m L p leh.math h as b to ee the n u i.math power s u a l ofl y} 2 m psiw od minus e{ le di as c.math} be( in subg s t.math u L bj h.math ect to s to comma.math p the r in power gs o r das of 2 h partialdiff -x.math psi{ dividedi . math by} partialdiff) = C. k a l i { 2 comma.math i.math } w { i + 1 } ( t.math comma.math x.math { i . math } ) comma . math t.maths p r i minus ngs o r m.matheight − c.mathathree to− t the− commazero power of 2− subta− hlinebracketright to theche power d mass of partialdiff fo r no rm a sub l a partialdiff nd la te ra lt.math inte ra to ctio the n apower nd of 2 to the power of 2 sub psi vextendsingle-vextendsingle-vextendsinglei.mathto a n exte = r na l 1 exc ita comma.math t io n of t he base period.math sub [ x.math = L to period.math the power of comma.math period.math comma.math N − 1 period.math \ tag ∗{$ ( 1 1 ) $} whereFo r ic.math nsta n cede , notes whe n t t h he e t d i ista p of nce le ngth betweeh.math n t ha e nd lowe mass r edm.math ge o f tis he s u ca bj nt e ct i leve to no r rm a n a d l tk.math he ce ntN roa n id d laof te t h ra e l c ro ss se ct io n period \end{ a l i g n ∗} sI p n r ain g gs e nk.math e ra lL setta n i d ng vis comma co u s se da pa m rap e te rs dc.math bo u ndN , a c.math r y coL n, t d he it momio ns ea nt t t a h nd e es h nds ea x.math r co nd it = io 0 ns comma.math a t t he f ree L e of t h e m ic ro hyphen b ea m ca n ben d g a ive re g n ive as n b y \noindenta.math sub2 1 .1 w.math 3 . \quad sub 1AFM open− parenthesist i p − t.mathcanti comma.math l ever i 0 closingnte racti parenthesis ons plus a.math sub 1 2 psi 1 open parenthesis t.math comma.math 0 closing parenthesis plus b.math sub 1 1 w.math sub 1 to the power of prime open parenthesis t.math comma.math 0 closing parenthesis plus b.math sub Thet i p i nte ra ct io∂w.math nwith t he samp le hasbeenusua∂w.math ∂ l2w.math l ymode le das be in g su bj ect to spr in gs o r dash − s p r i ngs o r 1 2 psi sub 1 to the powerκGA of prime( open− parenthesisψ) = −k.math t.mathN − comma.mathc.mathN 0 closing− m.math parenthesis = n.math sub 1 1 comma.math open parenthesis 1 3 closing $ eight −a three −t−comma∂x.math zero−t { a−bracketright∂t.math}$ che d mass∂t.math fo2 r| norma l a nd la te ra l inte ra ct io na nd to a n exte r na l exc ita t io n of t he base [ parenthesis a.math sub 2 1 w.math sub 1 open parenthesis t.math comma.math 0 closing parenthesisx.math plus=L a.mathcomma.math sub 22 psi 1 open parenthesis t.math ∂ comma.math∂ψ 0 closing parenthesis plus2 b.math sub 2 1 w.math2 ∂ψ sub 1 to the power of prime2 open2ψ parenthesis t.math comma.math 0 closing parenthesis plus EI = −k.math h.math ψ − c.math h.math − m.mathc.math | comma.math b.mathFo r∂x.math i sub nsta 22 psi n sub ce 1 , to when theL power t he of prime t i p openL of parenthesis le ngth∂t.math t.math $ h.math comma.math $ a nd 0 closing mass∂t.math parenthesis $m.math2 x.math =L n.math$ is sub s u 1 bj2 semicolon e ct to norma l $ k.math { N }$ andtimes la N a te noMMTA ra l s times p r Vo in l period gs $k.math 2 times 20 1{ 3 timesL }$ 1 24 andvis hyphen 1 44 cous times 1 28 (12) damp e rs $ c.math { N } , c . math { L } , $ t hemome ntandshea r cond it io nsa t t he f ree enda re g ive nby where c.math de notes t h e d ista nce betwee n t h e lowe r ed ge o f t he ca nt i leve r a n d t he ce nt ro id of t h e c ro ss se \ beginct io n{ . a l i g n ∗} I n a g e n e ra\ lkappa sett i ng , seGA pa ra ( te d\ bof r u a cnd{\ a rpartial y co n d it iow. ns amath t t h e}{\ e ndspartialx.math = 0comma.mathx . math } −L of \tpsi h e m ic) ro - b = ea m− ca n bek . g math ive n as{ N } − c . math { N }\ f r a c {\ partial w. math }{\ partial t . math } − m. math \ f r a c {\ partial ˆ{ 2 } w. math }{\ partial t . math ˆ{ 2 }} {\arrowvert { x . math = L ˆ{ comma . math }}}\\ EI \ f r a c {\ partial \ psi }{\ partial x . math } = − k . math { L } h . math ˆ{ 2 } 0 0 \ psia.math−11w.mathc . math1(t.mathcomma.math{ L } h . math0) + ˆa.math{ 2 }\12ψ1(f rt.mathcomma.math a c {\ partial 0)\ psi + b.math}{\11partialw.math1(t.mathcomma.matht . math } − 0)m.math + b.math12ψ c.math1(t.mathcomma.math ˆ{ 2 {\ r0) u =l en.math{3em}{110.4comma.math pt }}ˆ{\ partial }}ˆ{ 2 {\ psi }} {\ partial t . math ˆ{ 2 }}\arrowvert { x . math = L ˆ{ comma . math }}\ tag ∗{$ ( 1 2 ) $} (13) \end{ a l i g n ∗} 0 0 a.math21w.math1(t.mathcomma.math0) + a.math22ψ1(t.mathcomma.math0) + b.math21w.math1(t.mathcomma.math0) + b.math22ψ1(t.mathcomma.math0) = n.math12;

\noindent· N a noMMTAwhere· Vo l $.2 c.math· 2013 · 124 $− de144 notes· 128 t h e d ista nce betwee n t h e lowe r ed ge o f t he ca nt i leve r a nd t he ce nt ro id of t h e c ro ss se ct io n .

\ centerline { I nagene ra l sett i ng , se pa ra te dbounda r ycond it io nsa t the e nds $x.math = 0 comma.math$ L of t h emic ro − b eamca n be g ive n as }

\ begin { a l i g n ∗} a . math { 1 1 } w. math { 1 } ( t.math comma.math 0 ) + a.math { 1 2 }\ psi 1 ( t . math comma.math 0 ) + b.math { 1 1 } w. math ˆ{\prime } { 1 } ( t.math comma.math 0 ) + b.math { 1 2 }\ psi ˆ{\prime } { 1 } ( t.math comma.math 0 ) = n.math { 1 1 } comma . math \\ ( 1 3 ) \\ a . math { 2 1 } w. math { 1 } ( t.math comma.math 0 ) + a.math { 22 }\ psi 1 ( t.math comma.math 0 ) + b . math { 2 1 } w. math ˆ{\prime } { 1 } ( t.math comma.math 0 ) + b.math { 22 }\ psi ˆ{\prime } { 1 } ( t.math comma.math 0 ) = n.math { 1 2 } ; \end{ a l i g n ∗}

\noindent $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 28 $ M o d e l i n g t h .. e t i p hyphen s a m .. p l e i n t e r a c t i o n i n a t o .. m i c f o r c e .. m i c r o s c o p y .. ellipsis \noindentFig 3 periodM AFMo d e t i l p hyphen-si n g t am h p\quad l e interae t c t i o-i p − s a m \quad pleinteractioninato \quad m i c f o r c e \quad m i c r o s c o p y \quad Modelingth etip-sam pleinteractioninato micforce microscopy ellipsis $ ep.math l l i p s i1 s 1 $w.math sub N open parenthesis t.math comma.math L closing parenthesis plus p.math 1 2 psi N open parenthesis t.math comma.math L Fig 3 . AFM ti p hyphen − s am p l e intera c t o − i closing parenthesis plus q.math 1 1 w.math sub N to the power of prime0 open parenthesis t.math comma.math0 L closing parenthesis plus q.math 1 2 psi p.math11w.math (t.mathcomma.math L ) + p.math12ψ N (t.mathcomma.math L ) + q.math11w.math (t.mathcomma.math L ) + q.math12ψ (t.mathcomma.math L ) = n.math comma.math N sub\ centerline N to the power{ Fig of 3 prime . AFM open $ parenthesis t { i }$ t.math p $ comma.math hyphen−s$ LN closing ampl parenthesis e intera = n.math ct sub $o 2− 1i comma.math $ N } 21 open parenthesis 1 4 closing parenthesis \ centerline { $p.math 1 1 w.math { N } ( t.math comma.math$ L $) + p.math 1 2 \ psi $ N p.math 2 1 w.math sub N open parenthesis t.math comma.math(14) L closing parenthesis plus p.math 22 psi N open parenthesis t.math comma.math L closing$ ( parenthesis t.math plus comma.math$ q.math 2 1 w.math L sub$ ) N to + the power q.math of prime 1 open 1 parenthesis w.mathˆ t.math{\prime comma.math} { N 0} closing( parenthesis t.math plus comma.math q.math 22 psi $ sub L N$) to the + power q.math of prime open 1 parenthesis 2 \ psi t.mathˆ{\prime comma.math} { N L closing} ( parenthesis t.math = comma.mathn.math sub 22 period.math$0 L $ ) = n.math { 2 1 } comma . math $ p.math21w.mathN (t.mathcomma.math L )+p.math22ψ N (t.mathcomma.math L )+q.math21w.mathN (t.mathcomma.math0)+ } O r in ma0 t ri .. f or q.math22ψN (t.mathcomma.math L ) = n.math22period.math O r in ma t ri f or B.kali v open parenthesis t.math comma.math 0 closing parenthesis = Av open parenthesis t.math comma.math 0 closing parenthesis plus Bv sub x.math open\ begin parenthesis{ a l i g n ∗} t.math comma.math 0 closing parenthesis = n sub 1 comma.math ( 1 4 ) B.kalivB.kali-two(t.mathcomma.math v open parenthesis0) t.math = Av comma.math(t.mathcomma.math L closing0) parenthesis + Bvx.math =(t.mathcomma.math .. Pv open parenthesis0) = t.mathn1comma.math comma.math L closing parenthesis plus Qv sub\end x.math{ a l i g openn ∗} parenthesis t.math comma.math L closing parenthesis = n sub 2 period.math open parenthesis 1 5 closing parenthesis MoB.kali re com− ..two plev ..( desct.mathcomma.math r i pt io s .. o ftL h e) tip = s a mPv pl( et.mathcomma.math f-o rc e i nc ud e .. nL on) +hyphenQvx.math l n( et.mathcomma.math a r s r ac e and c onL t ac f o r ce .. a t t he b u-x.math \noindent $p.math 2 1 w.math { N } ( t.math comma.math$ L $ ) + p.math 22 \ psi $ N $ ( =) n = L-an2period.math (15) t.mathone-dMo re sub com comma.math$ u-six plebracketright-e-period desc r Li pt io $ s ) to o De+ ft h r e j q.mathtip a g s u a in m hyphenpl e 2f − M 1o rc ul eler w.mathˆ i nc hyphen ud e T{\ n o on porprime - l e n v e D-parenleft a} r{ s rN ac} e and hyphen( c on t.math tsub ac fM o T closing comma.math parenthesis comma 0 ) Joh + nsq .r mathon ce hyphen a t 22t Kehe b ndal\upsi− lx.math hyphenˆ{\prime= Rn o bL e−} toa{ theN power} ( of r-t t.math s .. J K u comma.mathsub T $ L $ ) = n.math { 22 } period.math $ O r in ma t r i \quad f or r−t one − du−sixbracketright2 period− 4e period−period ..to The De r Timoshenko j a g u in - M AFM ul ler model - T o por e v D − parenleft −M T ) , Joh ns on - Ke ndal l - R o b e s J K uT The Timoshen ko m ic ro b ea m m od2 e . l 4 fo . r AFM The Timoshenko o pe ra t io nAFM modes model comma ca n be e nco m passed as t he se co n d hyphen o rde r matrix \ [B.kalievoThe lu Timoshent io nmodel v ko ( m ic ro t.math b ea m m od comma.math e l fo r AFM o pe 0 ra t ) io n modes = Av , ca n be ( e nco t.math m passed as comma.math t he se co n d - o 0 rde r ) + Bv { x . math } ( t.mathmatrixM dieresis-v comma.math open parenthesis 0 t.math ) =comma.math n { 1 x.math} comma closing . math parenthesis\ ] plus Kv open parenthesis t.math comma.math x.math closing parenthesis = F comma.math 0 less x.math less L comma.mathevo t.math lu t iogreater n model 0 comma.math M v¨(t.mathcomma.mathx.mathEquation: open)+ parenthesisKv (t.mathcomma.mathx.math 1 6 closing parenthesis ..) = vF opencomma.math parenthesis 00 comma.math< x.math < L B.kalicomma.math sub v sub open t.math parenthesis > 0comma.math t.math comma.math to the power\ hspace of x.math∗{\ f i l closing l } $ B. parenthesis k a l i −two$ = 0 sub v closing $ ( parenthesis t.math v = comma.math$ to the power of o.math L $ open ) parenthesis =$ \quad x.mathPv sub $ (n sub t.math 1 to the power comma.math of closing $ L $ ) + Qv { x . math } ( t.math comma.math$ L $) = n { 2 } period.math ( 1 5 )$ parenthesis comma.mathx.math sub comma.math) v sub t.matho.math open parenthesis)comma.math 0 comma.math x.math closing parenthesis = v sub 1 open parenthesis x.math closingv(0comma.mathB.kali parenthesis comma.mathv (t.mathcomma.math B.kali-two v open= parenthesis 0)v= (x.math t.mathn comma.math1comma.math L closing parenthesisvt.math(0comma.mathx.math = n sub 2 comma.math) = v1(x.math)comma.math \ hspacewhere∗{\ F caf n i l in l } cMo lu de re d com r ive\ nquad exc itap tl e io\ nsquad o r hdesc y d ro r d y i n pt a m io ic das \ mquad p in gofthetip a n d n sub 1 comma.math sample n(16) sub $f 2 in−o$ te ra ct rce io ns incudete rms with t\quad n on − lnear s raceandcontac for ce \quad a t t he b $ u−x.math =$ n $L−a $ he f ree e nd period .. Fo r B.kali − twov(t.mathcomma.mathL) = n2comma.math insta nce comma \ centerline { $ one−d { u−six bracketright −e−period }$ toDer jaguin − M ul l e r − T o por e v $ D−p a r e n l e f t Equation:where F ca open n in parenthesis c lu de d r ive 1 7 n closingexc ita t parenthesis io ns o r h y .. d nro sub d y n1 a = m minus ic da m Cv p open in g a parenthesis n d n1comma.mathn t.math comma.math2 in te ra ct 0 io closing parenthesis minus Dv sub −t.mathns{ teM rmsopen}$ with parenthesis T) t he f , ree Joh t.math e nd nson . comma.math Fo− rKe insta ndal nce 0 closing , l − parenthesisR o b $ minus e ˆ{ Evr− subt } t.math$ s t.math\quad openJ K parenthesis $ u { t.mathT }$ comma.math} 0 closing parenthesis comma.math \ centerlinen sub 2 = minus{2 . Rv 4 open . \quad parenthesisThe Timoshenko t.math comma.math AFM model L closing} parenthesis minus Ov sub t.math open parenthesis t.math comma.math L closing parenthesisn1 = −Cv minus(t.mathcomma.math Sv sub t.math0) t.math− Dv opent.math parenthesis(t.mathcomma.math t.math comma.math0) − Evt.matht.math L closing(t.mathcomma.math parenthesis period.math0)comma.math open parenthesis 1 8 closing parenthesis \ hspaceone-I n-two∗{\ f parenleft-parenrighti l l }The Timoshen comma kom th ic e .. ro g i-v b eeammod n c o di io n e s at l ..fo x.math rAFMo = L wil pe l ra have t io n modes ,(17) ca n be e ncom passed as t he se co n d − o rde r matrix R = Row 1 k.math sub N minus kappa GA Row 2 0 k.math sub L h.math to the power of 2 . comma.math O = Row 1 c.math sub N 0 Row 2 0 c.math n = − Rv (t.mathcomma.math L ) − Ov (t.mathcomma.math L ) − Sv (t.mathcomma.math L sub\ centerline L h.math2 to{ evo the power lu t of io 2 .n comma.math model } .. St.math = Row 1 m.math 0 Row 2 0 m.matht.matht.math c.math to the power of 2 . period.math open parenthesis 1 9 closing )period.math (18) one − In − twoparenleft − parenright, th e g i − v e n c o di io n s at x.math = L wil l have parenthesis     \ centerlinek.math{M $N \ddot{−vκGA} ( t.math comma.mathc.math x.mathN )0 +$ Kv $ ( t.math comma.math x.math ) AlthoR = u g h t he u nfo rced gove r n in gcomma.math eq u a t io nO M= dieresis-v open parenthesis t.math comma.mathcomma.math x.mathS = closing parenthesis plus Kv open 0 k.math h.math2 0 c.math h.math2 =$parenthesis F $ t.math comma.math comma.math 0 L x.math< closingx . math parenthesis< $ = L 0 m $ ig comma.math ht lo o k to be writte t.mathL n in co> nse rva0 t ive comma fo rm . math comma $ t he}  m.math 0  bo u nd a r y co n d it io ns coperiod.math u ld cha n ge s(19) u chAltho ch a u ra g h ct t he e r u int nfo rorced d gove u c in r n g in ext g eq ra u ae t ne io rgn M yv¨ te(t.mathcomma.mathx.math rms in to AFM s y ste m period)+ When u s i ng m 0 m.mathc.math2 od\ begin a l { a l i g n ∗} Kv (t.mathcomma.mathx.math) = 0 m ig ht lo o k to be writte n in co nse rva t ive fo rm , t he va na ( l y s 0 is f ro comma m a m . math ic r o{ hyphenB. k a ca l i nt} i{ lev ve} rˆ be{ ax m. math with bo )u nd} { a r( y co t.mathn d it io ns comma.math } ={ 0 } { ) } v { = }ˆ{ o . math } bo u nd a r y co n d it io ns co u ld cha n ge s u ch ch a ra ct e r int ro d u c in g ext ra e ne rg y te rms in to AFM s y ste m (Row x . math 1 1 0 Row{ n 2 0} 1ˆ{ . v) open comma parenthesis . math t.math} { 1 comma.math} { comma 0 . closing math parenthesis} v { t plus . math Row} 1 0( 0 Row 0 2 0 comma.math 0 . v sub x.math x.math open parenthesis ) = t.math v { 1 } . When u s i ng m od a l comma.math( x.math 0 closing ) comma.math parenthesis =\ 0tag comma.math∗{$ ( open 1 parenthesis6 ) $}\\ 20B. closing k a l i − parenthesistwo v Row ( 1 0 t.math 0 Row 2 0 comma.math minus 1 . v open L parenthesis ) = t.math n { 2 } a na l y s is f ro m a m ic r o - ca nt i le ve r be a m with bo u nd a r y co n d it io ns commacomma.math . math 0 closing parenthesis plus Row 1 0 1 Row 2 1 0 . v sub x.math open parenthesis t.math comma.math L closing parenthesis = 0 comma.math \endtimes{ a l Ni g an ∗} noMMTA times Vo l period 2 times 20 1 3 times 1 24 hyphen 1 44 times 1 29  1 0   0 0  whereFcan in c luv(t.mathcomma.math ded r ive nexc0) + ita t iov ns o(t.mathcomma.math r hyd ro dynamic0) = 0comma.math damp in gand $n { 1 } comma . math 0 1 0 0 x.math n { 2 }$ in te ra ct io ns te rms with t he f ree e nd . \quad Fo r i n s t a nce , (20)  0 0   0 1  v(t.mathcomma.math0) + v (t.mathcomma.mathL) = 0comma.math \ begin { a l i g n0∗}−1 1 0 x.math n { 1 } = − Cv ( t.math comma.math 0 ) − Dv { t . math } ( t.math comma.math 0 ) − Ev { t.math t.math } ( t.math comma.math 0· N )a noMMTA comma.math· Vo l .2\·tag2013∗{· $124 (− 144 1· 129 7 ) $} \end{ a l i g n ∗}

$ n { 2 } = − $ Rv $ ( t.math comma.math$ L $ ) − Ov { t . math } ( t.math comma.math $ L $ ) − Sv { t.math t.math } ( t.math comma.math$ L $) period.math ( 1 8 )$ $ one−I n−two parenleft −parenright , $ th e \quad g $ i−v$ encodiionsat \quad $ x . math = $ L w i l l have

R $ = \ l e f t (\ begin { array }{ cc } k . math { N } & − \kappa GA \\ 0 & k . math { L } h . math ˆ{ 2 }\end{ array }\ right ) comma.math $ O $ = \ l e f t (\ begin { array }{ cc } c . math { N } & 0 \\ 0 & c . math { L } h . math ˆ{ 2 }\end{ array }\ right ) comma . math $ \quad S $ = \ l e f t (\ begin { array }{ cc } m. math & 0 \\ 0 & m.math c.math ˆ{ 2 }\end{ array }\ right ) period.math ( 1 9 )$ Altho u g h t he u nfo rced gove r n in g eq u a t io nM $ \ddot{v} ( t.math comma.math x.math ) +$ Kv $ ( t.math comma.math x.math ) = 0 $ mig ht lo o k to be writte n in co nse rva t ive fo rm , t he

\ hspace ∗{\ f i l l }bound a r y co nd it io ns co u ld cha n ge s u ch ch a ra ct e r int ro du c in g ext ra e ne rg y te rms in toAFMs y stem. Whenu s i ngmod a l

\ centerline {analys is f romamic ro − ca nt i le ve r be amwith bounda r y co nd it io ns }

\ begin { a l i g n ∗} \ l e f t (\ begin { array }{ cc } 1 & 0 \\ 0 & 1 \end{ array }\ right ) v ( t.math comma.math 0 ) + \ l e f t (\ begin { array }{ cc } 0 & 0 \\ 0 & 0 \end{ array }\ right ) v { x . math } ( t.math comma.math 0 ) = 0 comma.math \\ ( 20 ) \\\ l e f t (\ begin { array }{ cc } 0 & 0 \\ 0 & − 1 \end{ array }\ right ) v ( t.math comma.math 0 ) + \ l e f t (\ begin { array }{ cc } 0 & 1 \\ 1 & 0 \end{ array }\ right ) v { x . math } ( t.math comma.math L ) = 0 comma.math \end{ a l i g n ∗}

\ hspace ∗{\ f i l l } $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 29 $ J u l i o R period Claeyssen comma Teresa Ts u kazan comma Leticia Tonetto comma Dan i e l a Tol fo \ centerlinet he AFM t{ iJ p uhyphen l i sa o Rm p . le Claeyssen inte ra ct io n, ca Teresa n b e co Ts ns u id kazan e re d a ,s lo Leticia ca l ized foTonetto rces a t t , he Dan fr ee i e e nd l period a Tol fo } J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo 3 period .. The AFM dynamic response \noindentt he AFM t it p he - sa AFM m p le t inte i rap − ct iosamp n ca n b le e co inte ns id e ra re d ct a s loio ca n l ized ca fonbe rces a cot t he ns fr ee id e ndere . da s lo ca l ized fo rces a t t he fr ee e nd . 3T-one-four . The parenright-parenright-h AFM dynamic response sub o-c-e d-n sub y-e to the power of q-b-n a-e-u a-m-d-t e-i-i sub o-c r-c-n e-r-parenleft b-s-i p-e o n-d s-i e-n t-o f-e t-m h e-s o-T-f sub i h-t-m o-e s m-h sub e-a t-n k-r sub x-o-i m-i o-m p-d sub e-u l-l open parenthesis e T − one − fourparenright − parenright − h d − n q−b−na−e−ua−m−d−te−i−i r − c − ne − r − parenleftb − s − ip − e o n − ds − ie − nt − of − et − m h e − so − T − f m − h k − r m − io − mp − d l − l( e o−c−e y−\enoindentres po nse3 o r . matrix\quado− Gc reeThe n AFM fu nct dynamic io n h open response parenthesis t.math comma.math x.math comma.math xi closing parenthesisih−t−mo− oes f t he associatede−at−n hx omoge−o−i e−u res po nse o r matrix G ree n fu nct io n h (t.mathcomma.mathx.mathcomma.mathξ) o f t he associated h omoge neo u s neo u s in it ia l hyphen bo u n da r y va l u e p ro b le m in it ia l - bo u n da r y va l u e p ro b le m \ centerlineM dieresis-h{ open$ T− parenthesisone−four t.math parenright comma.math−parenright x.math comma.math−h { o− xic− closinge } parenthesisd−n { y− pluse ˆ{ K hq− openb−n parenthesis a−e−u t.math a−m− comma.mathd−t e−i− x.mathi { o−c }}} M h¨(t.mathcomma.mathx.mathcomma.mathξ)+ K h (t.mathcomma.mathx.mathcomma.mathξ) = comma.mathr−c−n e− xir− closingp a r e n parenthesis l e f t b− =s− 0i comma.math p−e $ o0 less $ x.math n−d comma.math s−i e−n xi less t−o L comma.math f−e t− t.mathm $h greater $ e 0−s comma.math o−T−f open{ i parenthesis h−t−m 2 1 o−e 0comma.math0 < x.mathcomma.mathξ < L comma.math t.math > 0comma.math (21) closings } m parenthesis−h { e−a t−n } k−r { x−o−i } m−i o−m p−d { e−u } l−l ( $ e } h (0comma.mathx.mathcomma.mathξ) = 0comma.math M h (0comma.mathx.mathcomma.mathξ) = δ(x.math − ξ) I comma.math h open parenthesis 0 comma.math x.math comma.matht.math xi closing parenthesis = 0 comma.math .. M h sub t.math open parenthesis 0 comma.math x.math\noindent comma.mathres po xi nse closing o parenthesisr matrix =G delta ree open n fu parenthesis nct io n x.math h $ minus (xi t.math closing parenthesis comma.math I comma.math x.math comma.math \ xi ) $ o fAh t open he parenthesis associated t.math h omoge comma.math neo u 0 comma.maths in it ia xi l closing− boundaryva parenthesis plus B h sub l ueprob x.math open parenthesis lem t.math comma.math 0 comma.math Ah(t.mathcomma.math0comma.mathξ) + Bhx.math(t.mathcomma.math0comma.mathξ) = 0comma.math xi closing parenthesis = 0 comma.math \ hspace ∗{\ f i l l }M $ \ddot{h} ( t.math comma.math x.math comma.math \ xi ) +$ Kh $( t.math PPh h open(t.mathcomma.math parenthesis t.mathL comma.mathcomma.mathξ L comma.math)+ Q hx.math xi( closingt.mathcomma.math parenthesis plusL Qcomma.mathξ h sub x.math open) = 0 parenthesiscomma.math t.math comma.math L comma.math comma.mathxi closing parenthesis x.math = 0 comma.math comma.math \ xi ) = 0 comma.math 0 < x.math comma.math \ xi < $ L $ comma . math t .where math0 de> notes0 t he comma.math2 × 2 n u l l matrix ( a 2 n d I 1t he )$2 × 2 id e nt it y ma t r ix . The La p la ce t ra nsfo rm o f h (t.mathcomma.mathx.mathcomma.mathξwhere 0 de notes t he 2 times 2 n u l l matrix) with ares n pe d Ict t he 2 times 2 id e nt it y ma t r ix period The La p la ce t ra nsfo rm o f h open parenthesis t.mathto t im comma.math e wil l be d e x.math noted b comma.math y H (s.mathcomma.mathx.mathcomma.mathξ xi closing parenthesis with res pe ct ) a nd refe r red to as t he matrix t ra nsfe r \ centerlineto t im e wil{h l be $ d e( noted 0 b y comma.math H open parenthesis x.math s.math comma.math comma.math x.math\ xi comma.math) = xi closing 0 comma.math$ parenthesis a nd refe\quad r redM to $ as h t he{ matrixt . math t } (res po0 nse comma.math . Thus x.math comma.math \ xi ) = \ delta ( x . math − \ xi ) $ I $ comma.math $ } ra nsfe r res po nse period Thus (s.math2 M + K ) H (s.mathcomma.mathx.mathcomma.mathξ) = δ(x.math − ξ) I comma.mathparenleftbig0 s.math< x.mathcomma.mathξ to the power of 2 < ML pluscomma.math K parenrightbig(22) H open parenthesis s.math comma.math x.math comma.math xi closing parenthesis = delta\ [ Ah open ( parenthesis t.math x.math comma.math minus xi closing 0 parenthesis comma.math I comma.math\ xi 0 less) x.math + B comma.math h { x . ximath less} L comma.math( t.math open comma.mathparenthesis 22 closing 0 commaA H . math(s.mathcomma.math\ xi ) =0comma.mathξ 0 comma)+ . mathB Hx.math\ ] (s.mathcomma.math0comma.mathξ) = 0comma.math parenthesisP H (s.mathcomma.math L comma.mathξ)+ Q Hx.math(s.mathcomma.math L comma.mathξ) = 0period.math IA t t H u open r ns o parenthesis u t t ha t h s.math(t.math comma.math− τcomma.mathx.mathcomma.mathξ 0 comma.math xi closing parenthesis) a cts a plus i nte B g Hra sub t i ng x.math fa cto open r in La parenthesis g ra ng e s.math0s comma.math 0 comma.math xi closing parenthesis = 0 comma.math \noindenta dj o in t methodP h fo $ r ( t he n t.math o n h om og comma.math e ne o u s eq u a$ t io L n $ comma.math \ xi ) + $ Q $ h { x . math } ( t.math comma.math $ P H open parenthesis s.math comma.math L comma.math xi closing parenthesis plus Q H sub x.math open parenthesisM s.math comma.math L Lcomma.math $ comma xi . math closing parenthesis\ xi ) = =0 period.math 0 comma.math$ wherev¨(t.mathcomma.mathx.math 0 de notes t he)+ $Kv 2 (t.mathcomma.mathx.math\times 2$ nul) l = matrixandItheF (t.mathcomma.mathx.math $2)comma.math\times 0 <2$ id e nt it ymat r ix . TheLap la ce t ra nsfo rmo f h x.mathcomma.mathξI t t u r ns o u t t ha < t hL opencomma.math parenthesis t.math >minus0comma.math tau comma.math(23) x.math comma.math xi closing parenthesis a cts a i nte g ra t i ng fa cto r in La$ ( g ra ng t.math e quoteright comma.math s a dj o in t method x.math fo r t comma.mathhe n o n h om og\ exi ne o u) s eq $ u with a t io n res pe ct M dieresis-v open parenthesis t.math comma.math x.math closing parenthesis plus Kv open parenthesis t.math comma.math x.math closing parenthesis \noindent to t im e wil l be d e noted b yH $ ( s.math comma.math x.math comma.math \ xi ) $ a nd refe r red to as t he matrix t ra nsfe r res po nse . Thus =v F(0 opencomma.mathx.math parenthesis t.math) = v comma.matho.math(x.math x.math)comma.math closing parenthesis v.matht.math comma.math(0comma.mathx.math 0 less x.math) = comma.mathv.math1(x.math xi less)comma.math L comma.math t.math greater 0 comma.math open parenthesis 23 closing parenthesisAv(t.mathcomma.math0) + Bvx.math(t.mathcomma.math0) = n1(t.math)comma.math \ hspaceLine 1∗{\ v openf i l lparenthesis} $ ( 0 s comma.math. math ˆ{ 2 x.math}$ M closing $+$ parenthesis K $ )= v$ sub H o.math $ ( open s.math parenthesis comma.math x.math closing x.mathparenthesis comma.math v.math\ xi ) = \ delta ( x . math − \ xi ) $ I $comma.math 0 < x.math comma.math \ xi < $ L $ comma . math sub t.math openPv parenthesis(t.mathcomma.math 0 comma.mathL ) + x.mathQvx.math closing(t.mathcomma.math parenthesis = v.mathL ) = subn2 1(t.math open parenthesis)period.math x.math closing parenthesis comma.math Line 2 ( 22 ) $ AvM open− two parenthesis− threeparenright t.math− comma.mathu l b − t i p 0− closingy l y i ng parenthesis ( h (t.math plus Bv− subτcomma.mathx.mathcomma.mathξ x.math open parenthesis t.math) comma.matha nd i nte g 0 closing parenthesis = n sub 1 openra t parenthesis i ng b y pa r t.math ts , it t closing u r ns o parenthesis u t t he d y comma.math na m ic re s po nse \ centerline {AH $ ( s.math comma.mathR t.math R L 0 comma.math \ xi ) + $ B $ H { x . math } ( s.math comma.math Pv open parenthesisv (t.mathcomma.mathx.math t.math comma.math) L = closing0 parenthesis0 (ht.math plus(τcomma.mathx.mathcomma.mathξ Qv sub x.math open parenthesis) t.mathMvo.math comma.math(ξ)+ h L closing parenthesis = n sub 20 open comma parenthesis . math t.math\ xi closing) parenthesis = 0 period.math comma.math$ R}t.math R L (τcomma.mathx.mathcomma.mathξ)Mv1(ξ))d.mathξd.mathτ+ 0 0 h (t.math−τcomma.mathx.mathcomma.mathξ) M-two-three sub parenright-u l b-t i p-y l y i ng open parenthesisL .. h open parenthesis t.math minus tau comma.math x.math comma.math xi closing F (τcomma.mathξ)d.mathξd.mathτ+ J ( v comma.math h )| comma.math (24) parenthesis\noindent aPH nd i nte $ g ( ra t i s.math ng b y pa r comma.math ts comma it t $u r nsL o $ u0 tcomma.math t he d y na m ic\ rexi s po nse) + $ Q $ H { x . math } ( s.math comma.math $ Lwhere $ comma J is a te . math rm co nta\ ixi n in g effects) = o f t he0 in it period.math$ ia l - va l u e G re e n fu nct io n with va l u es o f v a t t he bo u nd a r y . v openThe pparenthesis ro ced u re t.math mentio comma.math ned a b ove is x.math re la ted closing to t he parenthesis Riema n n fu = nintegral ct io n sub method 0 to fothe r powerin teg ra of tt.math i ng h integral y p e r b sub o l 0ic to the power of L open parenthesis hItturnsoutthath sub t.math open parenthesis tau comma.math $( x.math t.math comma.math− \tau xi closingcomma.math parenthesis Mv x.math sub o.math comma.math open parenthesis\ xi xi closing) $ parenthesis a cts a plus i hnte g ra t i ng fa cto r in Lag ra nge $ ’eq{ us a} t$ io ns a dj o in t method fo r t henonhomog e neous equa t io n open parenthesis tau comma.math x.math comma.math xi closing parenthesis Mv sub 1 open parenthesis xi closinga−ea parenthesis−m closing parenthesis d.math bracketleft−three−threeone−twobracketright−bracketright, n−he−do−nrl−e−mu−a−la−sta−dy−pp−n xi d.math tau plus integral sub 0 to the power of t.math integral sub 0 to the power of L h open parenthesiss−t t.math minus tau comma.math x.math sc sn i h c o − en − fi − nl−ee − d c f − o − o n c − o w t − i − nt − ro−hl o − f d str i b t − u ed s y ems comma.math\ hspace ∗{\ xif i closing l l }M parenthesis $ \ddot{ Fv} open( parenthesis t.math tau comma.math comma.math xi closing x.math parenthesis ) d.math +$ xi Kv d.math $ ( tau plus t.math J open parenthesis comma.math v comma.math x.math three − vi − three b period − r a t − F i o − on − r s o − an − mo − dg − cr − en − ae − ck − ou−i o − p r h) closing =$ parenthesis F $ ( vextendsingle-vextendsingle t.math comma.math sub 0 x.math to the power ) of comma.math comma.math to the 0 power< ofx.math L open parenthesisns−g comma.math 24 closing\ parenthesisxi < $ L $ comma . math u − ob − nd − la−em−rs−y is re fe r red to as t he d u a l in teg ra l re p re se ntat io n [ t . mathwhere J is> a te0 rm co comma.math nta i n in g effects ( o f 23 t he in )$ it ia l hyphen va l u e G re e n fu nct io n with va l u es o f v a t t he bo u nd a r y period co nd it io ns , t he te rm J v − twoa − fourparenright − n i s − be − he−co−sm − an − es − dparenleft − a v a atio n − s The p ro ced u re mentio ned a b ove is re la ted to t he Riema n n fu n ct io n method fo r in teg ra t i ng h y p e r b o l ic eq u a t io ns \ [o\ cbegin on s ta{ nta l isf g rmn e d ul} aforv a ( se co 0 n d comma.math- rd e r inea rm a t x.mathi − rx ) = v { o . math } ( x.math ) comma.math v.math { t . math } dbracketleft-three-three iffe re n t ia l eq u a t io one-two n . bracketright-bracketright comma n-h e-d o-n sub r l-e-m u-a-l a-s sub t a-d y-p p-n to the power of a-e a-m sc sn i h c o-e n-fi-n( 0 sub l-e comma.math e-d c f-o-o n c-o x.math w t-i-n t-r sub ) o-h = l o-f v.math .. d str i b{ t-u1 ed} s( y to thex.math power of ) s-t ems comma.math \\ I f we co ns id e r a m ic ro - ca nt i leve r b ea m with a t im e d e p e nd e nt b o u nda r y co nd it io n s2(t.math) a t t h Avthree-v ( i-three t.math b period-r comma.math a t-F i o-o n-r s o-a0 n-m ) o-d + g-c r-e Bv n-a{ e-cx . k-o math sub} u-i sub( n t.maths-g o-p r u-o comma.math b-n d-l sub a-e m-r 0 s-y ) is re = fe r red n to{ as1 t} he d( u a l t . math )e fre comma e e nd . , math t he te\ rmend J { a l i g n e d }\ ] inca teg n ra b e l id re e p n re t ified se ntat as io n open square bracket co nd itR iot.math ns comma t he te rm J v-two a-four parenright-n i s-b e-h subT e-c o-s m-a n-e s-d parenleft-a v a atio n-s o c on s ta nt sf rm ul afor a .. se co J = h (t.math − τcomma.mathx.mathcomma.math L )E Q n2(τ)d.mathτcomma.math (25) where n d hyphen rd0 e r inea rm a t i-r sub x \ centerlined iffe re n t{ iaPv l eq $ u a ( t io t.mathn period comma.math$ L $ ) + Qv { x . math } ( t.math comma.math$ L $ ) = n { 2 } ( t.math )−κGA period.math0  $ }  0 1  I f we co ns id e r a m icE ro= hyphen ca nt i leve rcomma.math b ea m with a Q t im= e d e p e nd eperiod.math nt b o u nda r y co nd it io n s(26) sub 2 open parenthesis t.math closing parenthesis a t t h e fre e e nd comma0 t he− teEI rm J 1 0 \noindentca n b e id e$ n M t− ifiedtwo− asthre e { parenright −u }$ l $ b−t $ i $ p−y $ l y i ng ( \quad h $ ( t . math − \tau comma.mathtwoJ =− integralW − four sub x.mathparenright 0 to the−e powerw comma.math− ci of− t.mathan − l − hli open\−xion parenthesis− b)$s−vo−ev and− t.mathrv−ee ih minus− ntegrath − taueac− comma.mathto t− itn ngbypar− he−v x.mathf − ol− comma.matho− tsuri− ,c−te it−on tur L− closing nsout parenthesis thedynamic E to the power re sponse o−e ofdr T− Qof n sub− es 2− openth − parenthesisp n m − taus e closingg − u parenthesisi s − v − le d.math− e n e tau− b comma.mathy−s( o n s ea n open d dis parenthesis tri u t e d 25 closing parenthesis \ hspaceo − twor∗{\− fivef i l lc}va − on $ (− nd − t.mathc e n − it comma.math− n t − a x.mathd − hyphenf )− a = \ int ˆ{ et− . mathee − ff} { 0 }\c −int ˆ{ L } { 0 } ( h { t . math } where i−ri−a−t l−e o−lu−rc−eing−G e−nct−fu−s ( \tau comma.mathi x.math comma.math \ xi ) Mv { o . math } ( \ xi ) + $ h $ ( \tau comma . math asEquation:− ti − on open− n(w parenthesisi pt h v a l 26 ue closing s of v parenthesis a t t he bo u.. nd E a = r Row y . · 1N minus a noMMTA kappa· GAVo l 0. Row2 · 2013 2 0· 124 minus− 144 EI· . comma.math130 Q = Row 1 0 1 Row 2 1 0 . period.mathx.math comma.math \ xi ) Mv { 1 } ( \ xi ) ) d . math \ xi d . math \tau + \ int ˆ{ t . math } { 0 } \ inttwo-W-fourˆ{ L } sub{ 0 parenright-e}$ h $ w-c ( i-a t n-l-l . math i-o n-b− sub s-v \tau o-e v-rcomma.math v-e e h-t h-e sub x.math a c-t o-t n-h comma.math sub e-v f-o sub l-o-u\ xi r i-c-t) $ e-o F n-d $r-o ( f-e s-t\tau h-p to thecomma . math power\ xi of) o-e n d m-s . math e g-u i\ s-v-lxi e-ed n . e-bmath sub y-s\tau open parenthesis+$ J (o n v s ea $comma.math$ n d dis tri u t e d h $ ) \arrowvert ˆ{ L } { 0 ˆ{ comma . math }} (o-two 24 r-five ) $ c a-o n-n d-c e n-i t-n sub i-r i-a-t t-a sub l-e d-hyphen f-a sub o-l u-r c-e in g-G e-e e-ff sub e-n c t-f sub u-s c-a s-t i-o n-n open parenthesis sub w to the power of i i p t h v a l ue s of .. v a t t he bo u nd a r y period \noindenttimes N awhere noMMTA J times is a Vo te l period rm co 2 nta times i 20 n 1 in 3 times g effects 1 24 hyphen o f 1 44t he times in 1 30it ia l − va l ueGre enfu nct io nwith va l ues o f va t t hebounda ry . \ hspace ∗{\ f i l l }The p ro ced u re mentio ned a b ove is re la ted to t he Riema n n fu n ct io n method fo r in teg ra t i ng h y p e r b o l ic eq u a t io ns

\noindent $ bracketleft −three −thre e one−two bracketright −bracketright , n−h e−d o−n { r } l−e−m u−a−l a−s { t } a−d y−p p−n ˆ{ a−e a−m }$ sc sn i h c $ o−e n−f i −n { l−e } e−d $ c $ f−o−o $ n $ c−o $ w $ t−i−n t−r { o−h l } o−f $ \quad d s t r i b $ t−u $ ed s $ y ˆ{ s−t }$ ems

\noindent $ three −v i−three $ b $ period−r $ a $ t−F $ i $ o−o n−r $ s $ o−a n−m o−d g−c r−e n−a e−c k−o { u−i { n s−g }} o−p $ r $ u−o b−n d−l { a−e m−r s−y }$ is re fe r red to as t hedua l in teg ra l re p re se ntat io n [

\noindent cond it io ns , t he termJ $v−two a−four parenright −n $ i $ s−b e−h { e−c o−s } m−a n−e s−d p a r e n l e f t −a $ v a a t i o $ n−s $ o c on s ta nt sf rmul afor a \quad se co n d − rd e r inearma t $ i−r { x }$

\noindent d iffe rent ia l equat ion.

\ hspace ∗{\ f i l l } I f weco ns id e r amic ro − ca nt i leve r b eamwith a t im e d e p e nd e nt bounda r y co nd it io n $ s { 2 } ( t.math )$ atthe fre eend ,thetermJ

\noindent ca nbe id en t ified as

J $ = \ int ˆ{ t . math } { 0 }$ h $ ( t . math − \tau comma.math x.math comma.math $ L $ ) E ˆ{ T }$ Q $ n { 2 } ( \tau ) d . math \tau comma.math ( 25 ) $ where

\ begin { a l i g n ∗} E = \ l e f t (\ begin { array }{ cc } − \kappa GA & 0 \\ 0 & − EI \end{ array }\ right ) comma . math Q = \ l e f t (\ begin { array }{ cc } 0 & 1 \\ 1 & 0 \end{ array }\ right ) period.math \ tag ∗{$ ( 26 ) $} \end{ a l i g n ∗}

\noindent $ two−W−fo ur { parenright −e } w−c i−a n−l−l i−o n−b { s−v o−e v−r v−e e } h−t h−e { a c−t } o−t n−h { e−v } f−o { l−o−u r i−c−t e−o } n−d r−o f−e s−t h−p ˆ{ o−e }$ n $ m−s $ e $ g−u $ i $ s−v−l e−e $ n $ e−b { y−s } ($ onseanddis triuted

\noindent $ o−two r−f i v e $ c $ a−o n−n d−c $ e $ n−i t−n { i−r i−a−t } t−a { l−e } d−hyphen f−a { o−l u−r c−e in g−G } e−e e−f f { e−n c t−f { u−s }} c−a s−t i−o n−n ( ˆ{ i } { w }$ i $p{ t }$ h v a l ue s o f \quad vat theboundary. $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 30 $ M o d e l i n g t h .. e t i p hyphen s a m .. p l e i n t e r a c t i o n i n a t o .. m i c f o r c e .. m i c r o s c o p y .. ellipsis \noindent3 period 1M period o d .. e Freq l i .. n uenc g t yrespons h \quad e t i p − s a m \quad pleinteractioninato \quad m i c f o r c e \quad m i c r o s c o p y \quad Modelingth etip-sam pleinteractioninato micforce microscopy ellipsis $ eI l n l ip p ra s i c s tic $ e comma wh e nc o m p u ti g t he con v o lu t o nin te .. gr a .. f o .. t he f rc e d res p on e comma w e a c tual 3 . 1 . Freq uenc yrespons Equation: open parenthesis 27 closing parenthesis .. v open parenthesis t.math comma.math x.math closing parenthesis = v sub h.math open parenthesis I n p ra c tic e , wh e nc o m p u ti g t he con v o lu t o nin te gr a f o t he f rc e d res p on e , w e a c tual t.math\ centerline comma.math{3 . x.math 1 . \quad closingFreq parenthesis\quad plusuenc v sub yrespons p.math open} parenthesis t.math comma.math x.math closing parenthesis comma.math where v sub h.math open parenthesis t.math comma.math x.math closing parenthesis is a fr ee .. v i br a ti o nint r o duc d .. b yt h e sy s e m a nd wh\ centerline o s e .. i n ti{ I a l npra v l u e s cr e tic .. a .. e p ,whe i oriunk ncompu.. n o w n period ti I g t t t u he convo lu t o nin te \quad gr a \quad f o \quad t he f rc ed res pone ,weac tual } v(t.mathcomma.mathx.math) = vh.math(t.mathcomma.mathx.math) + vp.math(t.mathcomma.mathx.math)comma.math t hat t he s .. i niti a lv a lu s a e s u pp l e d b y .. he p e rm a n en tr esp v sub n-p.math to the power of(27) parenleft-s to the power of t.math-e comma.math\ begin { a l i g x.math n ∗} closing parenthesis t hat ca n .. b .. det er mi n e d b y ot h r m ea v ( t.math comma.math x.math ) = v { h . math } ( t.math comma.math x.math ) + v { p . math } S inwhere ce tv he .. i m(t.mathcomma.mathx.math p ul s ere sp on e a nd .. t) stis ia m fr eee de r v iva i br iv aec ti o o nint n st r t o t duc e ab d a s b s yt f o h rth e sy ef s er mee a res nd o wh ns o e s sa e n d i th efo rc edr ( t.mathh.math comma.math x.math ) comma.math \ tag ∗{$ ( 27 ) $} ntwo-i ti a l subv l u four-n e s r e parenright-parenleft a p i oriunk h n a o .. w n n u. I lt i tn u i i t a hat l va t hel ue s s at i niti.. t.math a lv a lu= s0 a comma e s u pp t hl e e d in b d y u ce he d p s e y rm ste a m n fre e res p o nse d u e to a pe rm \end{ a l i g n ∗} t.math−e a ne n t re sparenleft po nse v− subs p.math open parenthesis t.math comma.math x.math closing parenthesis ca n be en tr esp vn−p.math comma.mathx.math) t hat ca n b det er mi n e d b y ot h r m ea easS iin ly ce d t ete he r m i m in p ed ul period s ere sp I on t t e u a r nd ns o ut st t i m e de r iva iv ec o n st t t e ab a s s f o rth ef r ee res o ns e sa n d th efo whereEquation: $ v open{ h parenthesis . math } 28( closing t.math parenthesis comma.math .. v sub h.math x.math open parenthesis ) $ is t.math a fr comma.math ee \quad x.mathv i br closing a ti parenthesis o nint r = o minus ducd integral\quad bythe sy s emandwho s e \quad intialvluesre \quad a \quad p i oriunk \quad n o w n . I t t u rc edr two − ifour−nparenright − parenleft h a n u l i n i i a l va l ue s at t.math = 0, t h e in d u ce d s y ste m fre subt 0 hat to the t powerhe s of\quad L h subi 1 niti open parenthesis alvalu t.math saesuppl comma.math x.math edby comma.math\quad hepermanen xi closing parenthesis dotaccent-v tr esp sub$v p.math{ n− openp . math parenthesis}ˆ{ p 0 a r e n l e f t −s ˆ{ t . math−e }} comma.mathe res p o nse d u x.math e to a pe rm a) ne $ n t t re hat s po nseca nvp.math\quad(t.mathcomma.mathx.mathb \quad det er min) ca e n dby be ot h rmea comma.math xi closing parenthesis d.matheas xi minusi ly d ete integral r m in sub ed . 0 I to t tthe u r power ns o u of t L h sub o.math open parenthesis t.math comma.math x.math comma.math xi closing parenthesis v sub p.math open parenthesis 0 comma.math xi closing parenthesis d.math xi comma.math S inwhere ce t he \quad imp ul s ere sp on e and \quad t st ime de r iva iv ec on st t t eaba s s f o rth ef r ee res ons e sandth efo rc edr $h sub two 1− openi { parenthesisfour−n } t.mathparenright comma.mathZ−Lparenleft x.math comma.math $ h a \ xiquad closingnul parenthesis ini phi ialvaluesat open parenthesis xi closing\quad parenthesisZ$L t = . math h open parenthesis = 0 v (t.mathcomma.mathx.math) = − h (t.mathcomma.mathx.mathcomma.mathξ)v ˙ (0comma.mathξ)d.mathξ − h (t.mathcomma.mathx.mathcomma.mathξ)v (0comma.mathξ)d.mathξcomma.math t.math, $h.math the comma.math inducedsy x.math comma.math stemfre xi closing1 e parenthesis res ponsedue M phi open parenthesis toapermanent xi closingp.math parenthesis re sponse comma.math $v h sub{ o.mathp .o.math math open} parenthesis( t . math p.math comma.math x.math ) $ ca n be0 0 t.math comma.math x.math comma.math xi closing parenthesis phi open parenthesis xi closing parenthesis = partialdiff(28) h sub t.math open parenthesis t.math comma.math x.math comma.math xi closing parenthesis divided by partialdiff t.math M phi open parenthesis xi closing parenthesis period.math open\ centerline parenthesis{ eas 29 closing i lydete parenthesis rmin ed . I twhere tur nsout } Hh a1 rm(t.mathcomma.mathx.mathcomma.mathξ o n ic a n d p ie cew is e l in ea r fo rc i) ngφ(ξ a) re = oh f( it.mathcomma.mathx.mathcomma.mathξ nte rest in f req u e ncy a na l y s is period) M Whenφ(ξ)comma.math see ki ng a re s poho.math nse of(t.mathcomma.mathx.mathcomma.mathξ t he sa me t y pe )φ(ξ) = \ begin∂ht.math{ (at.mathcomma.mathx.mathcomma.mathξ l i g n ∗} ) t he t ra nsfe r fu∂t.math n ct io n is int ro d u cedM periodφ(ξ)period.math G ive n t h e ha(29) rmH o a n rm ic o i n pic u a nt d p ie cew is e l in ea r fo rc i ng vaEquation: re{ o fh i . nte math openrest} in parenthesis f( req u t.mathe ncy 30 a closing na l y comma.math s parenthesis is . When see .. ki f.math x.math ng a re open s po parenthesis nse) of = t he sa− t.math me t\ int y comma.math pe ˆ{ L } x.math{ 0 } closingh { parenthesis1 } ( = t.math e.math to the comma.math power of i.mathx.math omega comma.math t.math vt open he t parenthesis ra\ xi nsfe r fu) n x.math ct\ iodot n isclosing{v int} ro{ parenthesis d up .ced math . G ivecomma.math} n( t h e 0 ha rm comma o n ic i. mathn p u t \ xi ) d . math \ xi − \ int ˆ{ L } { 0 } h we{ haveo . math t he} h a rm( o nt.math ic o u t p ucomma.math t res p o nse x.math comma.math \ xi ) v { p . math } ( 0 comma . math \ xi )Equation: d . math open\ xi parenthesiscomma 3 . 1 math closing\ tag parenthesis∗{$ ( .. 28 v sub p.math ) $} open parenthesis t.math comma.math x.math closing parenthesis = e.math to the i.mathωt.math power\end{ ofa l i.math i g n ∗} omegaf.math t.math(t.mathcomma.mathx.math H open parenthesis i.math) = omegae.math closing parenthesisv(x.math v open)comma.math parenthesis x.math closing(30) parenthesis comma.math where \ centerlineH open parenthesis{where i.math} omegawe closing have parenthesis t he h a rm o v.math n ic o u open t p u parenthesis t res p o nse x.math closing parenthesis = integral sub 0 to the power of L H open parenthesis i.math omega comma.math x.math comma.math xi closing parenthesis v open parenthesis xi closing parenthesis d.math xi period.math open parenthesis$ h { 1 32} closing( parenthesis t.math comma.math x.math comma.math \ xi ) \phi ( \ xi ) =$ h $( t.math v (t.mathcomma.mathx.math) = e.mathi.mathωt.mathH(i.mathω)v(x.math)comma.math (31) comma.mathThe ke r ne lp.math H x.math open parenthesis comma.math s.math comma.math\ xi ) x.math $ M comma.math $ \phi xi( closing\ xi parenthesis) comma of t he . math t ra ns fe h r o{ peo ra . math to r H} is t h( e La t.math p la ce t ra comma.math x.math comma.math \ xi ) \phi ( \ xi ) = \ f r a c {\ partial h { t . math } ( t.math comma.math nsfo rm o f t h e im p u ls e re s po nse h open parenthesiswhere t.math comma.math x.math comma.math xi closing parenthesis period I n pa rt ic u la r comma x.math comma.math \ xi ) }{\ partial t . mathL }$ M $ \phi ( \ xi ) period.math ( 29 )$ three-f o four-r a-a c o-a n c-pH o-e n-n-i t t-r a te d fo rce open squareR H bracket x.math = a.math of s p a t ia l a m p lv it u d e v open parenthesis x.math Harmon ic andp ie cew(i.mathω is e)v.math l in( eax.math r fo) = rc0 i nga(i.mathωcomma.mathx.mathcomma.mathξ re o f i nte rest in f req u) e ncy ana l y s is . Whensee ki nga re s po nse of t he samet y pe closing(ξ)d.mathξperiod.math parenthesis = v.math(32) open parenthesis x.math closing parenthesis delta open parenthesis x.math minus a.math closing parenthesis we have t he pe rm a ne n t re s po nse \ centerlineThe ke r ne{ t l H he( ts.mathcomma.mathx.mathcomma.mathξ ra nsfe r fu n ct io n is int) roducedof t he t ra ns fe .Give r o pe ra to ntheharmon r H is t h e La p la ce t ic ra i nput } nsfoEquation: rm o f t open h e im parenthesis p u ls e re 33 s po closing nse h parenthesis(t.mathcomma.mathx.mathcomma.mathξ .. v sub p.math open parenthesis). t.mathI n pa rt comma.math ic u la r , three x.math− f closingo parenthesis = e.math to the power of i.math omega t.math H open parenthesis i.math omega comma.math x.math comma.math a.math closing parenthesis v.math open parenthesis \ beginfour −{ raa l i− g na∗}c o − a n c − po − en − n − i t t − r a te d fo rce [ x.math = a.math of s p a t ia l a m p l it u d e v a.math(x.math closing) = v.math parenthesis(x.math period.math)δ(x.math − a.math) we have t he pe rm a ne n t re s po nse f.mathWith t he ( in it iat.math l va lu es comma.math v sub p.math open x.math parenthesis ) 0 comma.math = e.math xi ˆclosing{ i . math parenthesis\omega = H opent .parenthesis math } i.mathv ( omega x.math comma.math ) xi comma.math \ tag ∗{$ ( comma.math30 ) $} a.math closing parenthesis v.math open parenthesis a.math closing parenthesis comma dotaccent-v sub p.math open parenthesis 0 comma.math \end{ a l i g n ∗} i.mathωt.math xiv closingp.math(t.mathcomma.mathx.math parenthesis = i.math omega) =v sube.math p.math open parenthesisH(i.mathωcomma.mathx.mathcomma.matha.math 0 comma.math xi closing parenthesis comma)v.math t h e(a.math in d u ce)period.math d fre e re s po nse is g ive n b y (33) \ centerlinev sub h.math{wehave open parenthesis t heharmon t.math comma.math ic ou x.math t pu closing t res parenthesis po nse =} minus integral sub 0 to the power of L r open parenthesis t.math With t he in it ia l va lu es vp.math(0comma.mathξ) =comma.mathH (i.mathωcomma.mathξcomma.matha.math x.math comma.math xi comma.math)v.math omega( closinga.math) parenthesis, v˙p.math(0comma.mathξ M H open parenthesis) = i.mathωv i.mathp.math omega(0comma.mathξ comma.math), t a.math h e in d comma.math u ce d fre e re xi s po nse is g ive n b y \ begin { a l i g n ∗} L closingv parenthesis(t.mathcomma.mathx.math v.math open parenthesis) = a.math− R r closing(t.mathcomma.mathx.mathcomma.mathξcomma.mathω parenthesis d.math xi comma.math open parenthesis) M 34H closing(i.mathωcomma.matha.mathcomma.mathξ parenthesis )v.math(a.math)d.mathξcomma.math (34) v {h.mathp . math } ( t.math comma.math0 x.math ) = e.math ˆ{ i . math \omega t . math } H ( i . math wherewhere \omegaEquation:) open v parenthesis ( x.math 35 closing ) parenthesis comma.math .. r = h\ subtag t.math∗{$ ( open 3 parenthesis 1 ) t.math $} comma.math x.math comma.math xi closing parenthesis plus\end i.math{ a l i g omegan ∗} h open parenthesis t.math comma.math x.math comma.math xi closing parenthesis period.math rFo= h rt.math a p u( lst.mathcomma.mathx.mathcomma.mathξ e a m p l it u de ) + i.mathωh(t.mathcomma.mathx.mathcomma.mathξ)period.math \ centerlinev open parenthesis{where x.math} closing parenthesis = v sub o.math open parenthesis H e.math a.math v.math i.math s.math(35) d.math e.math open parenthesis x.math minus L plus b.math closing parenthesis minus H e.math a.math v.math i.math s.math i.math d.math e.math open parenthesis x.math minus L closing\ hspace parenthesis∗{\ f i l l } closingH $ parenthesis ( i . math comma.math\omegaFo r a open p u ls) parenthesis e am v.math p l it 36 u de closing ( parenthesis x.math ) = \ int ˆ{ L } { 0 }$ H $ ( i . math \omegatimesv (x.math N acomma.math noMMTA) = vo.math times( H x.math Voe.matha.mathv.mathi.maths.mathd.mathe.math l period comma.math2 times 20 1 3 times\ xi 1 24 hyphen) $ v 1 44 $( timesx.math ( 1\ xi− 3 1L +)b.math d .)− mathH e.matha.mathv.mathi.maths.mathi.mathd.mathe.math\ xi period.math ( 32 ) $ (x.math− L ))comma.math (36) · N a noMMTA · Vo l .2 · 2013 · 124 − 144 · 131 The ke r ne l H $ ( s.math comma.math x.math comma.math \ xi ) $ of t he t ra ns fe r ope ra to rHis theLap la ce t ra nsformo f theimpu ls e re s po nseh $ ( t.math comma.math x.math comma.math \ xi ) .$ Inparticular, $ three −f $ o $ four−r a−a $ c $ o−a $ n $ c−p o−e n−n−i $ t $ t−r$ atedfo rce $ [ x.math = a.math$ ofspatia lampl itudev $( x.math ) = v.math ( x.math ) \ delta ( x . math − a.math ) $ we have t he pe rma ne n t re s po nse

\ begin { a l i g n ∗} v { p . math } ( t.math comma.math x.math ) = e.math ˆ{ i . math \omega t . math } H ( i . math \omega comma.math x.math comma.math a.math ) v.math ( a.math ) period.math \ tag ∗{$ ( 33 ) $} \end{ a l i g n ∗}

\ centerline {With t he in it ia l va lu es $v { p . math } ( 0 comma . math \ xi ) =$ H $( i.math \omega comma . math \ xi comma.math a.math ) v.math ( a.math ) , \dot{v} { p . math } ( 0 comma . math \ xi ) = i . math \omega v { p . math } ( 0 comma . math \ xi ) ,$ theinducedfre eresponse isgivenby }

$ v { h . math } ( t.math comma.math x.math ) = − \ int ˆ{ L } { 0 }$ r $ ( t.math comma.math x.math comma.math \ xi comma . math \omega )$ MH $( i.math \omega comma.math a.math comma.math \ xi ) v.math ( a.math ) d.math \ xi comma.math ( 34 ) $ where

\ begin { a l i g n ∗} r = h { t . math } ( t.math comma.math x.math comma.math \ xi ) + i . math \omega h ( t . math comma.math x.math comma.math \ xi ) period.math \ tag ∗{$ ( 35 ) $} \end{ a l i g n ∗}

\ centerline {Forapu ls eampl it ude }

v $( x.math ) = v { o . math } ( $ H $ e.math a.math v.math i.math s.math d.math e.math ( x . math − $ L $+ b.math ) − $ H $ e.math a.math v.math i.math s.math i.math d.math e.math ( x . math − $ L $) ) comma.math ( 36 )$ $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 3 1 $ J u l i o R period Claeyssen comma Teresa Ts u kazan comma Leticia Tonetto comma Dan i e l a Tol fo \ centerlinet he pe rm a{J ne u n t l re i so po R nse . t Claeyssen u r ns o u t , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo } J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo Equation: open parenthesis 37 closing parenthesis .. v sub p.math open parenthesis x.math closing parenthesis = e.math to the power of i.math omega t he pe rm a ne n t re s po nse t u r ns o u t t.math\noindent integralt sub he L permanen minus b.math to t the re power s po of nse L H t open u r parenthesis ns ou t i.math omega comma.math x.math comma.math xi closing parenthesis v sub o.math d.math xi period.math \ beginA-two{ a eight-s l i g n ∗} b t-e h-f sub e-o r e-i subZ L comma-n d-b sub y-u e-s u-d b-f sub r-s e-t i t-e sub u r-t sub i-e n-s p-g t n-h sub e-s i w-n i-i sub t-l l-i sub a i.mathωt.math n-lvvp.math v-o{ w-ap( .x.math math l b-u e-e)} = se.math( in open x.math parenthesis ) = e.mathˆH(i.mathωcomma.mathx.mathcomma.mathξ{ i . math \omega t . math )v}\o.mathintd.mathξperiod.mathˆ{ L } { L − b . math } H( i . math \omega comma.mathL x.math−b.math comma.math \ xi ) v { o . math } d . math \ xi period .math \ tag ∗{$ ( v sub h.math open parenthesis t.math comma.math x.math closing parenthesis = minus integral sub 0 to the power(37) of L r open parenthesis t.math comma.math37 ) $} x.math comma.math xi comma.math omega closing parenthesis M H open parenthesis i.math omega comma.math 0 comma.math xi closing parenthesis\end{ a l i g vn ∗} open parenthesis xi closing parenthesis d.math xi comma.math open parenthesis 38 closing parenthesis A − twoeight − s b t − eh − fe−ore−icomma−n d − by−ue − su − db − fr−se−tit−eur−ti−en−sp−g t n − he−s i w − ni − it−ll − iawithn − lv r g-three− ow − toa l theb − powerue − ofe s i-five in ( parenright-v e-period I-n n-a t-s hi e-n parenleft-c a s e .. o f a ti e l i-n e a r e xp n-o e n t-i a l o-f rc i n g \noindent $ A−two eight −s $ b $L t−e h−f { e−o r e−i { comma−n }} d−b { y−u } e−s u−d b−f { r−s Equation:v (t.mathcomma.mathx.math open parenthesis 39 closing) parenthesis = − R r (t.mathcomma.mathx.mathcomma.mathξcomma.mathω .. f.math open parenthesis t.math comma.math x.math) closingM H (i.mathωcomma.math parenthesis = e.math0 x.mathcomma.mathξ p.math) e−th.math i t−e { u } r−t { i−e n0−s p−g }}$ t $ n−h { e−s }$ i $ w−n i−i { t−l } l−i { a } n−l v−o open parenthesis lambda t.math closing parenthesisi open−fiveparenright parenthesis−ve c.math−periodI t.math−n plus d.math closing parenthesis comma.math w−va(ξ $)d.mathξcomma.math l $ b−u e−e $(38) swith in r ( g − three n − at − s hi e − nparenleft − c a s e o fwe a ti have e l i t− hen pe a a r rt e icxp un la− ro soe ln ut t− ioi na l o − f rc i n g Equation: open parenthesis 40 closing parenthesis .. w.math p.math open parenthesis t.math comma.math x.math closing parenthesis = e.math x.math p.math$ v open{ h . parenthesis math } ( lambda t.math t.math closing comma.math parenthesis openx.math parenthesis )t.math = − C plus \ int D closingˆ{ L parenthesis} { 0 }$ comma.math r $ ( t.math comma.math x.math comma.math \ xi comma . math \omega )$ MH $( i.math \omega comma . math 0 comma . math wheref.math(t.mathcomma.mathx.math) = e.mathx.mathp.math(λt.math)(c.matht.math + d.math)comma.math (39) \ xiC .. =) open $ v parenthesis $ ( \ lambdaxi to) the d power . math of 2 M\ xi plus Kcomma.math closing parenthesis ( to the38 power ) $ of minus 1 c.math comma.math open parenthesis 4 1 closing parenthesiswithwe have r t he $ p g− athre rt ic u e la ˆ r{ soi l− ufive t io n parenright −v e−period I−n } n−a t−s $ hi $ e−n p a r e n l e f t −c $ a s e \quad o f a t i e l $ iD−n .. = $ open e a parenthesis r e xp lambda $ n−o to $ the e power n $ of t 2− Mi plus $ aK closing l $ o parenthesis−f $ rc to i the n power g of minus 1 d.math minus 2 lambda open parenthesis lambda to the power of 2 M plus K closing parenthesis to the power of minus 2 M c.math comma.math \ beginwhenevew.mathp.math{ a l ir g lambda n ∗} (t.mathcomma.mathx.math is not a n e ig e nva l u e o) r = nae.mathx.mathp.math t u ra l f req u e ncy(λt.math period )(t.mathC + D)comma.math (40) f.math4 period .. ( Free t.math t ransversev comma.math ibratio ns x.math ) = e.math x.math p.math ( \lambda t.math ) ( c.math t.mathwhereThe se a + rc h of d.math expo ne nt ) ia l so comma.math lu t io ns \ tag ∗{$ ( 39 ) $} 2 −1 \endv open{ a l i parenthesisg n ∗} t.math comma.math x.math closing parenthesisC == e.math (λ toM the+ powerK ) ofc.mathcomma.math lambda t.math v open parenthesis(41) x.math closing parenthesis 2 −1 2 −2 comma.math .. vD open parenthesis= (λ M x.math+ K closing) d.math parenthesis− 2λ(λ =M Row+ K 1 w.math) M openc.mathcomma.math parenthesis x.math closing parenthesis Row 2 psi open parenthesis x.math\noindentwheneve closing r λwehaveis parenthesis not a n e t ig . hepa e comma.math nva l u ert o r ic na open tu u parenthesisla ra l rf req so u el 42 ncy u closing t. io parenthesis n 4of . t h Freee u nfo t rceransverse d Timoshen v ibratio ko m od ns e l \ beginEquation:{ a l i g open n ∗} parenthesis 43 closingThe parenthesis se a rc h of .. expo M nepartialdiff nt ia l so to lu the t io nspower of 2 v divided by partialdiff t.math to the power of 2 plus Kv = 0 w.math p.math ( t.math comma.math x.math ) = e.math x.math w.math p.math(x.math)  ( \lambda t . math ) comma.mathv (t.mathcomma.mathx.math) = e.mathλt.math v (x.math)comma.math v (x.math) = comma.math (42) (one-s t.math u-five parenright-b C + subD comma-j ) comma.math e c-a m-t t-o\ subtag o-u∗{$ t-g ( to the 40 power ) of $} s-e t-n e-o r d-a l t-s e-eψ( p-mx.math a r) a-n e-t sub e d-n h-n sub t-o m-i i-o a-g of t h e u nfo rce d Timoshen ko m od e l to\end the{ powera l i g n of∗} e-l n-s o-e o-l u-u t-s-i n-b sub o-s u n-o d a-t r-h y-e co nd it io ns open parenthesis seco nd hyphen o rd e r d iffe re n t ia l eq u a t io n \noindent where M v to the power of prime prime open parenthesis∂2v x closing parenthesis plus C v to the power of prime open parenthesis x closing parenthesis plus K M + Kv = 0comma.math (43) open parenthesis lambda closing parenthesis v∂t.math open parenthesis2 x closing parenthesis = 0 comma.math open parenthesis 44 closing parenthesis \ hspacewith matrix∗{\ f i co l l efficients}C \quad $ = ( \lambda ˆ{ 2 }$ M $ + $ K $ ) ˆ{ − 1 } c.math comma.math ( 4 1 ) $ s−et−ne−o oneM− =su Row− fiveparenright 1 minus kappa− GAbcomma 0 Row−jec 2− 0am overbar−tt − oo EI.−ut comma.math− g ..r d C− =a Rowl t − 1se 0− overbarep − m kappaa r a − subne GA− ted Row− nh 2− minus kappa GA 0 . comma.math ..n Kt− openom−ii− parenthesisoa−ge−ln−so lambda−eo−lu−ut closing−s−i n − parenthesisbo−sun−o d =a Row− tr − 1hy rho− Ae lambdaco nd it io to ns the ( power of 2 0 Row 2 0 lambda to the power of 2 rho I plus kappa GA . comma.math\ centerlineseco nd - o rd open{ eD r d parenthesis\quad iffe re n t$ ia 45= l eq closing u ( a t ioparenthesis\lambda n ˆ{ 2 }$ M $ + $ K $ ) ˆ{ − 1 } d . math − 2 \lambda ( \lambda ˆ{ 2 }$ 00 0 Mt $ hatM + sat $v ( isfy Kx t)+ $ he )C bo ˆ{ uv n−( x da r)+2 yK} co$ nd( Mλ it) v io $( nsx c.math) = 0comma.math comma.math(44) $ with} matrix co efficients  −κGA 0   0 κ  Equation: open parenthesisM 46 closing= parenthesis .. Acomma.math sub P v sub openC parenthesis= L closingGA parenthesiscomma.math to the powerK of v open parenthesis 0 closing parenthesis\noindent pluswheneve to the power r $ of\ pluslambda B v to$ the0 is power notaneEI of prime Q ig sub enva v to the l ueornaturapower of−κGA prime open0 parenthesis l f requency L closing parenthesis . to the power of open  ρAλ2 0  parenthesis(λ) = 0 closing parenthesis = subcomma.math = n sub n sub(45) 2 to the power of 1 sub period.math to the power of comma.math \noindentWe s h o u4 ld0 . o\ bsequadλ2ρI rve+ tFreeκGA hat if t vis ransverse co u s da m v p in ibratio g fo rces ns a re co ns id e r ed comma t he n t he mat r ix M has to be mod ifie d to in c l u d e a n te hat ig sat e nva isfy l t u he e te bo rm u n period da r y Alsoco nd comma it io ns when co ns id e r i ng lo ca l ized l in ea r ize d t ip hyphen sa m p le i nte ra ct io ns a n d vis co u s da m p i ng\ centerline fo rce a ct i{ ngThe se a rc h of expo ne nt ia l so lu t io ns }

o-four n-three a u-m tic ro ca nt i levev(0) r be a+ m open0 (0) parenthesiscomma.math c la m pe d hyphen fre e closing parenthesis comma t he a b ove e ig e nva lu e p ro b v $ ( t.math comma.mathA ( x.mathL) + Bv Q )0 == n e.math1 ˆperiod.math{\lambda t . math }$ v $ ((46) x.math ) comma.math$ le m wil l mod ify t h e co efficient matricesP v in open parenthesisv (L) n2 \quadone-tv h-five $( sub parenright-e x.math b-w ) sub = i-o\ ll-u-l e f t ( n\ d-bbegin sub{ e-aarray t-r}{ y-hc} c-ow.math s-o n-e d-o ( sub f-i x.math sub t i-c l-o ) a-n\\\ s-mpsi p-parenleft( e x d . math hyphen r ) e\ eend .. b{ earray a m }\ right ) We s h o u ld o bse rve t hat if vis co u s da m p in g fo rces a re co ns id e r ed , t he n t he mat r ix M has to be comma.mathperiod ( 42 ) $ mod ifie d to in c l u d e a n e ig e nva l u e te rm . Also , when co ns id e r i ng lo ca l ized l in ea r ize d t ip - sa oftimes t h N e a unoMMTA nfo rce times d Timoshen Vo l period 2komod times 20 1e 3 l times 1 24 hyphen 1 44 times 1 32 m p le i nte ra ct io ns a n d vis co u s da m p i ng fo rce a ct i ng o − fourn − three a u − m tic ro ca nt i leve \ beginr be a{ ma l ( i g c n la∗} m pe d - fre e ) , t he a b ove e ig e nva lu e p ro b le m wil l mod ify t h e co efficient matrices in (

Mone\−f rth a− c {\fivepartialparenright−ˆe{b −2wi}−ol−vu−}{\lnd −partialbe−at−ry−hc −tos . math− on − ˆ{ed2− o}}f−iti−+cl−oa− Kvns−m p =− parenleft 0 commae d - r . mathe e b\ tag ∗{$ ( 43 ) $} \ende a m{ a . l i g n ∗} · N a noMMTA · Vo l .2 · 2013 · 124 − 144 · 132 \noindent $ one−s u−five parenright −b { comma−j e c−a m−t } t−o { o−u } t−g ˆ{ s−e t−n e−o }$ r $ d−a $ l $ t−s e−e p−m $ a r $ a−n e−t { e } d−n h−n { t−o m−i i−o a−g ˆ{ e−l n−s o−e o−l u−u t−s−i }} n−b { o−s u n−o }$ d $ a−t r−h y−e$ cond it io ns (

\noindent seco nd − orderd iffe rent ia l equat ion

M $ v ˆ{\prime \prime } ($x$) +$C$vˆ{\prime } ( $ x $ ) + $ K $ ( \lambda ) $ v ( x $) = 0 comma.math ( 44 )$ with matrix co efficients

\ hspace ∗{\ f i l l }M $ = \ l e f t (\ begin { array }{ cc } − \kappa GA & 0 \\ 0 & \ overline {\}{ EI }\end{ array }\ right ) comma . math $ \quad C $ = \ l e f t (\ begin { array }{ cc } 0 & \ overline {\}{\kappa } { GA }\\ − \kappa GA & 0 \end{ array }\ right ) comma . math $ \quad K $ ( \lambda ) = \ l e f t (\ begin { array }{ cc }\rho A \lambda ˆ{ 2 } & 0 \\ 0 & \lambda ˆ{ 2 } \rho I + \kappa GA \end{ array }\ right ) comma.math ( 45 )$

\noindent t hat sat isfy t he bo un da r y co nd it io ns

\ begin { a l i g n ∗} A { P v }ˆ{ v ( 0 ) } { (L) } + ˆ{ + } B v ˆ{\prime }{ Q }ˆ{ ( 0 ) } { v ˆ{\prime } ( L) } = { = } n { n ˆ{ 1 } { 2 }}ˆ{ comma . math } { period .math }\ tag ∗{$ ( 46 ) $} \end{ a l i g n ∗}

\noindent Wes hou ld o bse rve t hat if vis co u s damp in g fo rces a re co ns id e r ed , t he n t he mat r ixMhas to bemod ifie d to in c l ud e an e ig e nva l u e te rm . Also , when co ns id e r i ng lo ca l ized l in ea r ize d t ip − samp le i nte ra ct io ns and vis co u s damp i ng fo rce a ct i ng $ o−fo ur n−thre e $ a $ u−m$ tic ro ca nt i leve r beam( c lamped − fre e ) , t he a b ove e ig e nva lu e p ro b le mwil l mod ify t h e co efficient matrices in ( $ one−t h−f i v e { parenright −e } b−w { i−o l−u−l n } d−b { e−a t−r y−h } c−o s−o n−e d−o { f−i { t i−c l−o a−n s−m }} p−parenleft $ e d − r e e \quad b e a m .

\noindent $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 32 $ M o d e l i n g t h .. e t i p hyphen s a m .. p l e i n t e r a c t i o n i n a t o .. m i c f o r c e .. m i c r o s c o p y .. ellipsis \noindent4 period 1M period o d .. e The l i .. n e igenva g t h l u\quad epro be to t the i power p − s of al-e m \quad pleinteractioninato \quad m i c f o r c e \quad m i c r o s c o p y \quad Modelingth etip-sam pleinteractioninato micforce microscopy ellipsis $ eone-four-I l l i p s i s sub $ one-four-n bracketright-parenright-t sub i-e s-r m i-g e o n i-b n i-y sub t bracketleft-i a lv a lu e s comma t e ge n e al s o l u i o n of the 4 . 1 . The e igenva l u epro bl−e seco nd hyphen o rd e r m a tri x d i ffe r e n t i a l eq u ati o ns open parenthesis one − four − I bracketright − parenright − t e o n i − b n i − y a lv a lu e s , t e ge n e al s o l u i o n of the seco nd - o rd e r m a tri x d i ffe r e n t i a l eq u ati o ns ( one−four−n \ centerlineEquation: open{4i .− parenthesises 1−rmi . −\gquad 47 closingThe \quad parenthesistbracketlefte igenva .. v− openi l u parenthesis epro $ x.math b ˆ{ l closing−e }$ parenthesis} = h sub o.math open parenthesis x.math closing parenthesis v open parenthesis 0 closing parenthesis plus h sub 1 open parenthesis x.math closing parenthesis v to the power of prime open parenthesis 0 \ centerline { $ one−four−I { one−four−n } bracketright −parenright −t { i−e s−r m i−g }$ e o n $ i−b $ n closing parenthesis comma.math 0 v(x.math) = ho.math(x.math)v(0) + h1(x.math)v (0)comma.math (47) $ iwhere−y { t bracketleft −i }$ alvalues , tegene al so lui onof the second − orderma tri xdi ffe rent ial equations( } h sub o.math open parenthesis x.math closing parenthesiswhere = h to the power of prime open parenthesis x.math closing parenthesis M plus h open \ begin { a l i g n ∗} parenthesish x.math(x.math closing) = h0 parenthesis(x.math) M C+ comma.mathh (x.math) hC subcomma.math 1 open parenthesish (x.math x.math) = closingh (x.math parenthesis) M comma.math = h open parenthesis(48) x.math closing parenthesis vo.math ( x.math ) = h { o . math } ( x.math )1 v ( 0 ) + h { 1 } ( x . math ) v ˆ{\prime } Mo comma.math r , i n t he mo open r e p parenthesis ra ct ica l fo 48 rm closing parenthesis (o r 0 comma ) i ncomma t he mo . math r e p\ ratag ct∗{ ica$ l fo( rm 47 ) $} \endEquation:{ a l i g n ∗} open parenthesis 49 closing parenthesis .. v open parenthesis x.math closing parenthesis = h open parenthesis x.math closing parenthesis c 0 sub 1 plus h to the power of primev( openx.math parenthesis) = h(x.math x.math)c1 + closingh (x.math parenthesis)c2comma.math c sub 2 comma.math (49) \ centerlinefo r co nsta{ ntwhere 2 times} 1 vecto rs c sub 1 a n d c sub 2 period H e re h open parenthesis x.math closing parenthesis is t he 2 times 2 matrix so l u t io n fo ro co f t nsta he i nt n it2 × ia1 l vavecto l u rs e pc1 roa b n led mc2. H e re h (x.math) is t he 2 × 2 matrix so l u t io n o f t he i n it ia l va l u e p ro b le m 00 0 $M h h{ too the . math power} of prime( x.math prime open ) parenthesis =M hˆh x closing({\x prime)+ parenthesisC }h ( x( plus)+ x.mathK C h(λ to) h the( )$x power) = M of 0comma.math prime $+$ open h parenthesis $((50) x.math x closing parenthesis )$ C $comma.mathplus K openh { parenthesis1 } ( lambda x.math closing ) parenthesis =$ h h open $( parenthesis x.math x closing )$ parenthesis M $comma.math = 0 comma.math ( open 48 parenthesis )$ 50 closing parenthesis 0 orh open , i parenthesis nt hemor 0 closing epra parenthesish ct(0) = ica = 0comma.math 0 l comma.math form Ah Ah(0) to the = Icomma.math power of prime open parenthesis 0 closing parenthesis = I comma.math where 0 de notes t he 2 times 2 n u l l matrix a nd I t he 2 times 2 i-four d-five sub parenright-e n-period t it y matrix period The matrix co efficients where 0 de notes\ begin t he {2a× l i2 gn n u∗} l l matrix a nd I t he 2 × 2i − fourd − fiveparenright−en−periodtity matrix . The matrix co efficients be in g g ive n as in ( be in g g ive n as in open parenthesis 4 . 2 . Shape modes i n c l osed fo rm v4 period ( x.math2 period .. Shape ) = modes h i n c ( l osed x.math fo rm ) c { 1 } + h ˆ{\prime } ( x . math ) c { 2 } comma . math \ tag ∗{$ ( 49Fo ) r $ a} m ic ro - ca n t i leve r be a m of le ngth L , we have t he c la m ped bo u nd a r y co nd it io n v (0) = 0, t hat is B.kaliFo r a− mone ic ro= hyphenI . B ca y n t i leve r be a m of le ngth L comma we have t he c la m ped bo u nd a r y co nd it io n v open parenthesis 0 closing parenthesis\end{ a l i g =n ∗} 0 comma t hat is B.kali-one = I period .. B y u s in g t he in it ia l va lu es of h four − parenleftx − nine)i − comman − i( t u r n s o u t th t c2 = 0. Th u s we h ave to dete rm i ne λ so t h a t u s in g t he in it ia l va lu es of h four-parenleft x-nine closingv parenthesis(x.math) = i-commah (x.mathcomma.mathλ n-i open parenthesis) tc u r n( s5 ..1 ) o u t th t .. c sub 2 = 0 period Th u\ centerline s we h ave to{ detefo rrm co i ne nsta lambda nt so t$ h 2 a t \times 1$ vecto rs $c { 1 }$ a n d $ c { 2 } .$ Hereh $( x.math ) $sat i s isfies t he t he bo $ u 2 nd a\ rtimes y co nd it io2$ n a t tmatrix he f ree e so n d lutx.math iono= L . f B the y ass u in m i ng it ho ia mogeneo l va u lueproblems b o u n da r } yv co open n d it parenthesis io ns , we have x.math t he closing parenthesis = h open parenthesis x.math comma.math lambda closing parenthesis c .. open parenthesis 5 1 closing parenthesis \ hspace ∗{\ f i l l }M $ h ˆ{\primeno n l\prime in e a r e} ig e($x$) n matrix p ro b le m +$C$hˆ{\prime } ( $ x $ ) + $ K $ ( \lambda sat isfies t he bo u nd a r y co nd it io n a t t he f0 ree e n d x.math = L period .. B y ass u m i ng ho mogeneo u s b o u n da r y co n d it io ns comma )$U h(x(λ) c $)= ( Ph =( L 0comma.mathλ comma.math) + Qh ( (L comma.mathλ 50 )$ )) c = 0period.math (52) From t h is , it t u r wens have o u t t he he cha ra cte r ist ic e q u a t io n no n l in e a r e ig e n matrix p ro b le m \ [hU open ( parenthesis 0 ) lambda = closing 0 comma.math parenthesis c = parenleftbigAhˆ{\prime Ph open} parenthesis( 0 )L comma.math = I lambdacomma.math closing parenthesis\ ] plus Qh to the power of prime open parenthesis L comma.math lambda∆( closingλ) = det( parenthesisU) = 0period.math parenrightbig c = 0 period.math open parenthesis(53) 52 closing parenthesis From t h is comma it t u r ns o u t t he cha ra cte r ist ic e q u a t io n \ centerlineEquation:We s ho u open{ ldwhere o bseparenthesis rve 0 t de hat 53 t notes he closing modes t parenthesis have he t $ he 2 sa .. me Capital\ stimes ha p e Delta , reg2$ aopen rd less parenthesis nul o f t he l co matrixandIlambda n d it io closing ns a t t parenthesis he f the ree e nd= $2 , b determinant u t t \times open parenthesis2 i−fo ur U closing d−f i v e { parenright −e parenthesisn−heperiod e ig e nva = l0 tu period.math e λ id t iffe rs y a} cco$ rd matrix in g to t he. Thebo u nd matrix a r y co co efficie efficients nt matrices P bea n in d Q g. gFo iver a m n ic ro as - ca in nt ( i leve} r b eaWe m ,s t ho h ese u ld m o a bse t r ice rve s ta hatre gt ive he n modesi − twon have− zeroperiod t he sa me− s haparenleft p e commaT reg h a ema rd lessr − oi fx t heU co n(λ d)o it− iofivethree ns a t t he− r f ree e nd comma b u t t he e ig e \ centerline {4 . 2 . \quad Shape modes i n cu− lby− oseda fo rm } nvat lh u− eca−en − cb − ha−erac−de−tt−ee−ri−rs−mti−ic−n d − eq t i c − on − o( p u − t i ng t e unda m e nt l − a mlambda a ri x so d iffel u t rs io a n ccoh rd(x.math in g to). t he bo u nd a r y co efficie nt matrices P a n d Q period Fo r a m ic ro hyphen ca nt i leve r b ea m comma t h ese m a t\ hspace r ice s a∗{\ re gf ive i l l n}Fo r a m i c ro −4ca . 2 . n 1 .t i Computi leve r ng beh amof(x.math) le ngth L , we have t he c lamped bo und a r y co nd it io nv $(i-twoThe 0 n-zero fu n ) da period-parenleft m = e nta 0l res po ,$ nse .. Th thatis h( ..x.math ema r-i) ca x $B.kali n .. b U e opendete− rm parenthesisone i ne d =in c $ lambda lo sed I fo. closing rm\quad as fo parenthesis lB lows y . Expoo-five ne three-r nt ia l t t h-c y p sub a-e n-c b-h sub a-e ra c-d e-t sube vecto t-e e-r r so i-r l s-m u t io t i-i ns sub c-n d-e q to the power of u-b y-a t i c-o n-o open parenthesis p u-t i ng t e .. unda m .. e nt l-a m a ri x v (x.math) = e.mathk.mathx.math\ centerlineso l u tu ioo n−{ husfourf open in parenthesis− gfourparenleft t he x.math in it− closingparenrighte ia l parenthesisva lu− x esi periodt of− s h( $u 6= four 0),−whep a never e n l r e fk.math t x−isnine a ro ot of ) t h e i cha−comma ra cte r ist n− ici po l y ( no $ m ia t l u r n s \quad o u t th t \quad $ c { 2 } = 0 .$ Thuswehavetodetermi ne $ \lambda $ so t h a t } 4 period 2 period 1 period .. Computi ng h open parenthesis4 x.math closing parenthesis The fu n da m e nta l res po nse h openP parenthesis(k.mathcomma.mathλ x.math closing) parenthesis = d.mathe.matht.math ca n b e dete( rmk.math i ne2 dM in c+ lokC sed fo+ rmK as fo l lows period .. Expo ne nt \ hspaceP ∗{\ f i l l }v4− $(j.math x.math ) =$ h $( x.math comma.math \lambda ) $ c \quad ( 5 1 ) ia) l = t y pβ ej.math vectok.math r so l u t io nscomma.math (54) v open parenthesis x.math closing parenthesis = e.math to the power of k.math x.math u o-four f-four parenleft-parenright e-x i t-s open parenthesis .. u\ hspace negationslash-equal∗{\ f i l l } sat 0 closing isfies parenthesis t hebounda comma whe nevej.math r ycond r k.math= 0 it is a io ro ot na of t t h te cha he ra f cte ree r ist end ic po l y$x.math no m ia l =$ L . \quad By ass umi ng ho mogeneo u s b o un da r y co nd it io ns , we have t he 4 where \ centerlineP open parenthesis{non k.math l in e comma.math a r e ig lambda en matrix closing p parenthesis ro b lem = d.math} e.math t.math parenleftbig k.math to the power of 2 M plus kC plus K parenrightbig = sum beta sub j.math k.math to the power of 4 minus j.math comma.math open parenthesis 54 closing parenthesis U $ ( \lambda ) $ c $= ( $ Ph(L $comma.math \lambda ) + Qh ˆ{\prime } ( $ L $ comma.math j.math = 0 2 2 2 2 4 \lambdaβ0 = a.mathb.math) )$m.math ccomma.math $= 0 period.mathβ1 = 0comma.math ( β2 52 = (−a.mathe.mathλ )$ − c.mathλ b.mathm.math − a.math + a.mathm.matha.math)comma.math β3 = 0 β4 = c.mathλ a.math + c.mathλ e.mathcomma.math where (55) FromthEquation: openis , parenthesis it tur 55 nsout closing parenthesis t hecha .. beta ra 0 =cte a.math r ist b.math ic sub equa m.math t comma.math io n beta 1 = 0 comma.math beta 2 = open parenthesis minus a.math e.math lambda to the power of 2 minus c.math lambda to· N the a noMMTA power of 2· Vo b.math l .2 · sub2013 m.math· 124 − minus144 · a.math133 to the power of 2 plus a.math sub\ begin m.math{ a l i a.math g n ∗} closing parenthesis comma.math beta 3 = 0 beta 4 = c.math lambda to the power of 2 a.math plus c.math lambda to the power of 4 e.math\Delta comma.math( \lambda ) = \det ( U ) = 0 period.math \ tag ∗{$ ( 53 ) $} \endtimes{ a l Ni g an ∗} noMMTA times Vo l period 2 times 20 1 3 times 1 24 hyphen 1 44 times 1 33 Wes hou ld o bse rve t hat t he modes have t he sames hap e , reg a rd less o f t he co nd it io ns a t t he f ree e nd , bu t t he e ig e nva l u e $ \lambda $ d iffe rs a cco rd in g to t he bo und a r y co efficie nt matrices Pa ndQ . Fo r amic ro − ca nt i leve rbeam, th esema t r ice s a re g ive n $ i−two n−zero period−p a r e n l e f t $ \quad T h \quad ema $ r−i $ x \quad U $ ( \lambda ) o−f i v e three −r $ t $ h−c { a−e } n−c b−h { a−e ra c−d e−t { t−e e−r i−r s−m } t i−i { c−n }} d−e q ˆ{ u−b y−a }$ t i $ c−o n−o ( $ p $ u−t $ i ng t e \quad unda m \quad e nt $ l−a $ m a r i x solutionh $( x.math ) .$

\ centerline {4 . 2 . 1 . \quad Computi ng h $ ( x.math ) $ }

\ hspace ∗{\ f i l l }The fu ndame nta l res po nse h $ ( x.math ) $ ca nb e dete rm i ned in c lo sed fo rm as fo l lows . \quad Expo ne nt ia l t yp e vecto r so l u t io ns

\ centerline {v $( x.math ) = e.mathˆ{ k.math x.math }$ u $ o−fo ur f−four parenleft −p a r e n r i g h t e−x $ i $ t−s ( $ \quad u $ \not= 0 ) ,$ wheneve r $k.math$ is aro ot of thechara cte r ist ic po l ynomia l }

\ centerline {4 }

\ hspace ∗{\ f i l l }P $ ( k.math comma.math \lambda ) = d.math e.math t.math ( k.mathˆ{ 2 }$ M $+$ kC $+$ K $) = \sum \beta { j . math } k . math ˆ{ 4 − j . math } comma.math ( 54 ) $

\ [ j . math = 0 \ ]

\ centerline {where }

\ begin { a l i g n ∗} \beta 0 = a.math b.math { m. math } comma . math \beta 1 = 0 comma . math \beta 2 = ( − a.math e.math \lambda ˆ{ 2 } − c . math \lambda ˆ{ 2 } b . math { m. math } − a . math ˆ{ 2 } + a . math { m. math } a.math ) comma.math \beta 3 = 0 \beta 4 = c . math \lambda ˆ{ 2 } a.math + c.math \lambda ˆ{ 4 } e.math comma.math \ tag ∗{$ ( 55 ) $} \end{ a l i g n ∗}

\ hspace ∗{\ f i l l } $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 33 $ J u l i o R period Claeyssen comma Teresa Ts u kazan comma Leticia Tonetto comma Dan i e l a Tol fo \ centerlinewith {J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo } J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo a.math = kappa GA comma.math c.math = rho A comma.math e.math = rho I comma.math open parenthesis 56 closing parenthesis with \noindentLine 1 hlinewith Line 2 a.math sub m.math = kappa GA = kappa GA minus open parenthesis tau sub u.math plus tau sub b.math closing parenthesis b.math a.math = κ GA comma.math c.math = ρ A comma.math e.math = ρ I comma.math (56) comma.math Line 3 b.math sub m.math = overbar E sub I = open parenthesis EI plus 2 b.math h.math to the power of 2 E sub s.math closing parenthesis period.math\ hspace ∗{\ f i l l } $ a . math = \kappa $ GA $ comma.math c.math = \rho $ A $ comma.math e.math = \rho $ Ifive-T $comma.math four-h sub e-parenright ( 56 r a-o )$ o-n t b-s o f-e a-parenleft s l-i y .. o b a-i sub n eda f te r w i-r t in g t as a.math = κGA = κGA − (τ + τ )b.mathcomma.math Equation: open parenthesism.math 57 closing parenthesis .. P openu.math parenthesisb.math k.math comma.math lambda closing parenthesis = a.math b.math sub m.math \ [ \ begin { a l i g n e d }\ r u l e {3em}{0.4 pt }\\ 2 open parenthesis k.math to theb.math powerm.math of 4 plus= g.mathEI = to ( theEI power+ 2b.mathh.math of 2 k.math toEs.math the power)period.math of 2 minus overbar r.math 4 sub closing parenthesis comma.math a . math { m. math } = \kappa GA = \kappa GA − ( \tau { u . math } + \tau { b . math } ) b . math fivewhere− T four − he−parenright r a − oo − n t b − s o f − ea − parenleft s l − i y o b a − in eda f te r w i − r t in g t as commaEquation: . math open\\ parenthesis 58 closing parenthesis .. g.math to the power of 2 = overbar g.math 2 m.math plus hline sub s.math to the power of 2 subb comma.math. math { m. overbar math } g.math= 2\ overline m.math ={\}{ minus parenleftbiggE } { I } e.math= ( divided EI by b.math+ 2 sub b.math m.math plus h.mathˆ c.math divided{ 2 } byE a.math{ s . parenrightbigg math } ) P (k.mathcomma.mathλ) = a.mathb.math (k.math4 + g.math2k.math2 − r.math4 (57) lambdaperiod to .math the power\end of{ a 2 l hline i g n e subd }\ s.math] to the powerm.math of 2 = 1 divided by b.math sub m.math open)comma.math parenthesis a.math sub m.math minus a.math closing parenthesiswhere comma.math overbar r.math 4 = minus c.math lambda to the power of 2 parenleftbigg a.math plus e.math lambda to the power of 2 divided by a.math b.math sub m.math parenrightbigg period.math open parenthesis 59 closing parenthesis \noindent $ f i v e −T four−h { e−p a r e n r i g h t }$ r $ a−o o−n $ t $ b−s $ o $ f−e a−parenleft $ s $ l−i $ five-I t2 t-parenright sub u r-a n-r s-e o u2 t t ha t t he ro ots of open parenthesis e.math c.math 2 2 1 y g.math\quad o= b $g.math a−i 2m.math{ n }+$ edafterws.mathcomma.math $i−g.mathr $ t2m.math in g= t− as( + )λ s.math = (a.mathm.math − a.math)comma.math k.math sub 1 = epsilon.alt comma.math k.math sub 2 = minus epsilon.alt comma.mathb.math k.mathm.math suba.math 3 = i.math delta comma.mathb.math k.mathm.math sub 4 = minus i.math delta comma.math (58) \ begin { a l i g n ∗} with a.math + e.mathλ2 PEquation: ( k.math open parenthesis comma.math 60 closing parenthesis\lambda .. epsilon.alt) = = a.math 1 divided by b.math 2 radicalbigg-line{ m. math of minus} ( 2 g.math k . math to ther.math ˆ power{ 44} of = 2+ plus−c.mathλ g2 . radicalBig-line math2( ˆ{ 2 } )period.math a.mathb.math ofk . g.mathmath ˆ to{ the2 } power − of \ overline 4 plus 4 to{\}{ the powerr . math of hline} r.math4 { to) the comma power . of math 4 sub}\ comma.mathtag ∗{$ ( Equation: 57 ) open $} parenthesis 6 1 closing parenthesis .. m.math delta\end{ =a 1 l i divided g n ∗} by 2 radicalbigg-line of 2 g.math to the power of 2 plus 2 radicalBig-line of g.math to the power of 4 plus 4 to the power of hline r.math (59) tofive the power− I t oft − 4parenright sub period.mathur−an−rs−e o u t t ha t t he ro ots of ( \noindentThe re s powhere nse h open parenthesis x.math closing parenthesis i-one s t-bracketright he n g ive n b y t he fo rm u la o b ta i ned i n open square bracket 4 j.math minus 1 Equation: open parenthesis 62 closing parenthesis .. h open parenthesis x.math closing parenthesis = sum sum beta sub i.math k.math = epsilon.altcomma.math k.math = −epsilon.altcomma.math k.math = i.mathδcomma.math k.math = −i.mathδcomma.math d.math\ begin to{1a the l i g power n ∗} of open parenthesis j.math2 minus 1 minus i.math closing parenthesis3 open parenthesis x.math closing4 parenthesis h sub 4 minus j.math comma.mathgwith . math ˆ{ j.math2 } == 1 i.math\ overline = 0 {\}{ g . math } 2 { m. math } + \ r u l e {3em}{0.4 pt } { s . math }ˆ{ 2 } { comma . math }\ overline {\}{ g . math } 2 where{ m. math } = − ( \ f r a c { e . math }{ b . math { m. math }} + \ f r a c { c . math }{ a . math } ) \lambda ˆ{ 2 } \ r u l e {3em}{0.4 pt } { s . math }ˆ{ 2 } = \ f r a c { 1 }{ b . math { m. math }} ( a . math { m. math } − a . math ) Equation: open parenthesis 63 closing1q parenthesis .. d.mathp open parenthesis x.math closing parenthesis = delta se n h open parenthesis epsilon.alt comma . math \ tag ∗{epsilon.alt$ ( 58= ) $}\\\−2g.mathoverline2 + 2{\}{g.mathr .4 math+ 4 } 4r.math = 4 − c . math \lambda(60)ˆ{ 2 } ( \ f r a c { a . math x.math closing parenthesis minus epsilon.alt se n open parenthesis delta x.math closing parenthesiscomma.math divided by a.math b.math sub m.math epsilon.alt delta + e . math \lambda ˆ{ 2 }}{ a.math2 b.math { m. math }} ) period.math \\ ( 59 ) open parenthesis delta to the power of 2 plus1q epsilon.alt to thep power of 2 closing parenthesis sub comma.math \end{ a l i g n ∗} δ = 2g.math2 + 2 g.math4 + 4 r.math4 (61) is t he so l u t io n o f t he i n it ia l va l2 u e p ro b le m period.math beta 0 to the power of d.math to the power of open parenthesis i.math v.math closing parenthesis open parenthesis x.math closing parenthesis plus beta The re s po nse h (x.math)i − one s t − bracketright he n g ive n b y t he fo rm u la o b ta i ned i n [ 2\noindent to the power$ of f d.math i v e −I to $ the t power $ t− ofp aprime r e n r primei g h t open{ u parenthesis r−a x.mathn−r closing s−e }$ parenthesis out plusthat beta thero 4 to the power ots of of d.math ( open parenthesis x.math closing parenthesis = 0 comma.math Equation: open parenthesis 64 closing parenthesis .. d.math open parenthesis 0 closing parenthesis = d.math to\ [ the k . power math of{ prime1 } open= parenthesis epsilon.alt 0 closing parenthesis comma.math = d.math k.math to the power{ 2 } of prime=4 j.math prime− epsilon.alt open− 1 parenthesis 0 comma.math closing parenthesis k.math = 0 comma.math{ 3 } = i . math \ delta comma.mathXX k.math { 4 } = − i . math \ delta comma . math \ ] a.math b.math sub m.mathh(x.math d.math) = to theβ powerd.math of prime(j.math prime−1− openi.math parenthesis)(x.math)h 0 closingcomma.math parenthesis = 1 comma.math(62) a nd t he matrices h sub j.math = h toi.math the power of open parenthesis j.math4− closingj.math parenthesis open parenthesis 0 closing parenthesis sat isfy t he matrix d iffe re n ce e q u a t io n j.math = 1 i.math = 0 \noindent with whereM h sub j.math plus 2 plus C h sub j.math plus 1 plus K h sub j.math = 0 comma.math Equation: open parenthesis 65 closing parenthesis .. h sub 0 = 0 comma.math M h sub 1 = I period.math \ beginB y s{ ua lbst i g n it∗} u t in g va l u es comma we a r r ive to t he c lose d fo rm u la δsenh(epsilon.altx.math) − epsilon.altsen(δx.math) epsilon.altEquation: open parenthesisd.math = \ f( rx.math a c 66{ closing1) =}{ 2 parenthesis}\ sqrt ..{ h − open2 parenthesis g . math x.math ˆ{ 2 closing} + parenthesis 2 \ sqrt = Row{ g .1 math open parenthesis ˆ(63){ 4 } + a.math 4 plus ˆ{\ e.mathr u l e { lambda3em}{0.4 pt }} a.mathb.math epsilon.altδ(δ2 + epsilon.alt2) tor . themath power ˆ{ of4 2}}} closing{ parenthesiscomma . math d.math}\ tag open∗{ parenthesis$ (m.math 60 x.math ) $}\\\ closingdelta parenthesis= minus\ fcomma.math r b.matha c { 1 sub}{ m.math2 }\ sqrt d.math{ 2 to the g power . math of ˆ prime{ 2 prime} + open 2 parenthesis\ sqrtis t he{ sog l. x.math mathu t io n ˆ closing o{ f t4 he} parenthesis i n+ it ia l 4 va ˆ lminus{\ u e pr u roa.math l e b{ le3em m sub}{0.4 m.math pt }} d.mathr . math to the ˆ power{ 4 }}} of prime{ period open parenthesis .math }\ x.mathtag ∗{$ closing ( parenthesis 6 1 ) comma.math $} Row\end 2{ a a.math l i g n ∗} d.math to the power of prime open parenthesis x.math closing parenthesis minus a.math d.math to the power of prime open parenthesis (i.mathv.math) 00 x.math closing parenthesisβ0d.math plus c.math lambda(x.math to the) + powerβ2d.math of 2(x.math d.math) +openβ4d.math parenthesis(x.math x.math) = 0comma.math closing parenthesis comma.math . comma.math hline hline\ centerline {There sponseh $( x.math ) i−one $ s $ t−bracketright $ he n g ive nb y t he fo rmu la o b ta i ned i n [ } d.math(0) = d.math0(0) = d.math00(0) = 0comma.matha.mathb.math d.math00(0) = 1comma.math (64) where a.math = kappa GA comma a.math sub m.math = kappa GA commam.math b.math sub m.math = E I comma c.math = rho A comma e.math = rho I \ begin { a l i g n ∗} perioda nd t he matrices h = h(j.math)(0) sat isfy t he matrix d iffe re n ce e q u a t io n 4 j . math − j.math1 \\ h ( x . math ) = \sum \sum \beta { i . math } d . math ˆ{ ( j . math − 1 − times N a noMMTA times VoM l periodh 2 times+ C 20h 1 3 times+ K 1 24h hyphen= 0 1comma.math 44 times 1 34 i . math ) } ( x . mathj.math ) h+2 { 4 j.math− +1 j . mathj.math} comma . math \ tag ∗{$ ( 62 ) $}\\ j.math = 1 i.math = 0 \end{ a l i g n ∗} h0 = 0comma.mathMh1 = Iperiod.math (65) B y s u bst it u t in g va l u es , we a r r ive to t he c lose d fo rm u la \noindent where  (a.math + e.mathλ2)d.math(x.math) − b.math d.math00(x.math) −a.math d.math0(x.math)comma.math  \ beginh(x.math{ a l) i =g n ∗} m.math m.math comma.math a.mathd.math0(x.math) −a.mathd.math0(x.math) + c.mathλ2d.math(x.math)comma.math d.math ( x.math ) = \ f r a c {\ delta se n h ( epsilon.alt x.math ) − epsilon.alt se n ( \ delta x . math ) }{ a.math b.math { m. math } epsilon.alt \ delta ( \ delta(66) ˆ{ 2 } + epsilon.alt ˆ{ 2 } ) } { comma . math }\ tag ∗{$ ( 63 ) $} \end{ a l i g n ∗} where a.math = κ GA , a.mathm.math = κ GA , b.mathm.math = EI , c.math = ρ A , e.math = ρ I . · N a noMMTA · Vo l .2 · 2013 · 124 − 144 · 134 \noindent is theso lut ionof the init ia lvalueproblem

\ begin { a l i g n ∗} \beta 0 ˆ{ d . math ˆ{ ( i.math v.math ) }} ( x . math ) + \beta 2 ˆ{ d . math ˆ{\prime \prime }} ( x . math ) + \beta 4 ˆ{ d . math } ( x.math ) = 0 comma.math \\ d.math ( 0 ) = d.mathˆ{\prime } ( 0 ) = d.mathˆ{\prime \prime } ( 0 ) = 0 comma.math a.math b.math { m. math } d . math ˆ{\prime \prime } ( 0 ) = 1 comma.math \ tag ∗{$ ( 64 ) $} \end{ a l i g n ∗}

\noindent a nd t he matrices $ h { j . math } = h ˆ{ ( j . math ) } ( 0 )$ sat isfy thematrixd iffe rence equat ion

\ centerline {M $ h { j . math + 2 } + $ C $ h { j . math + 1 } + $ K $ h { j . math } = 0 comma . math $ }

\ begin { a l i g n ∗} h { 0 } = 0 comma.math M h { 1 } = I period.math \ tag ∗{$ ( 65 ) $} \end{ a l i g n ∗}

\noindent By su bst it ut in gva l u es ,wea r r ive to t he c lose d formu la

\ begin { a l i g n ∗} h ( x . math ) = \ l e f t (\ begin { array }{ cc } ( a.math + e.math \lambda ˆ{ 2 } ) d.math ( x.math ) − b . math { m. math } d . math ˆ{\prime \prime } ( x . math ) & − a . math { m. math } d . math ˆ{\prime } ( x.math ) comma.math \\ a.math d.math ˆ{\prime } ( x . math ) & − a.math d.math ˆ{\prime } ( x.math ) + c.math \lambda ˆ{ 2 } d.math ( x.math ) comma.math \end{ array }\ right ) comma . math \ tag ∗{$ ( 66 ) $}\\\ r u l e {3em}{0.4 pt }\ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\noindent where $ a.math = \kappa $ GA $ , a.math { m. math } = \kappa $ GA $ , b.math { m. math } =$ EI $, c.math = \rho $ A $, e.math = \rho $ I . $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 34 $ M o d e l i n g t h .. e t i p hyphen s a m .. p l e i n t e r a c t i o n i n a t o .. m i c f o r c e .. m i c r o s c o p y .. ellipsis \noindent4 period 3M period o d .. e Freq l i .. n uenc g t yeuqt h \quad ao n ..e f r t a i.. suppp − ors e-da m ..\ microquad hyphenpleinteractioninato to the power of e-b \quad m i c f o r c e \quad m i c r o s c o p y \quad Modelingth etip-sam pleinteractioninato micforce microscopy ellipsis $ eFo l l r i pa s s i u s p $ p ort e dTimoshe nk o m o d l comma w eha v e .. he .. b ou n d ar y co n di 4 . 3 . Freq uenc yeuqt ao n f r a supp or e − d micro −e−b u.math open parenthesis t.math comma.math 0 closing parenthesis = 0 comma.math psi sub x.math open parenthesis t.math comma.math 0 closing Fo r a s u p p ort e dTimoshe nk o m o d l , w eha v e he b ou n d ar y co n di parenthesis\ centerline = 0{ comma.math4 . 3 . \quad Equation:Freq open\quad parenthesisuenc yeuqt 67 closing ao parenthesis n \quad ..f u.math r a \quad open parenthesissupp or t.math $ e−d comma.math $ \quad Lmicro closing parenthesis $ − ˆ{ e− =b 0}$ comma.math} psi sub x.math open parenthesis t.math comma.math L closing parenthesis = 0 comma.math one-w h-five sub o a-s e-r co efficient sw rit t n .. n m a t ix f-o r m .. open parenthesis u.math(t.mathcomma.math0) = 0comma.math ψx.math(t.mathcomma.math0) = 0comma.math \ centerlineEquation: open{Fo parenthesis r a s upp 68 closing ort parenthesise dTimoshe .. A.kali nk omo = F.kali d =l Row ,weha 1 1 0 v Row e \ 2quad 0 0 . comma.mathhe \quad bound J.kali = Q.kali ar y = coRow n 1 di 0 0} Row 2 0 1 . u.math(t.mathcomma.mathL) = 0comma.math ψ (t.mathcomma.mathL) = 0comma.math (67) period.math x.math \ beginThe m{ a a l it g r n iz∗} U.kali ca n b e red u ced to a ha lf s ize o ne d u e to t he b o u n da r y co nd it io ns comma t ha t is comma c.math sub 1 2 = 0 comma one − wh − fiveoa−se−r co efficient sw rit t n n m a t ix f − o r m ( c.mathu.math sub 2 1( = 0 t.math period We t comma.math h u s have 0 ) = 0 comma.math \ psi { x . math } ( t.math comma.math 0 ) =t he 0 red comma u ced s. math y ste m\\ u.math ( t.math comma.math L ) = 0 comma.math \ psi { x . math } ( t . math commaU.kali . math sub D c L = 0 comma.math ) = 0 .. c comma1 = Row 0  . math 1 c.math\ tag sub∗{ 1$ 1 Row( 2 67 c.math ) sub $} 0 22 .0 T comma.math open parenthesis 69 closing parenthesis A.kali = F.kali = comma.math J.kali = Q.kali = period.math (68) \endwhere{ a l i g n ∗} 0 0 0 1 Equation: open parenthesis 70 closing parenthesis .. U.kali sub D = Row 1 open parenthesis a.math minus e.math omega to the power of 2 closing parenthesis\ centerlineThe m d.math a t{ r$ iz open oneU.kali−w parenthesisca h n− bf i e v redL e closing u{ cedo to parenthesis a− has lf s e ize− minusr o} ne$ b.math d co u e efficientto sub t he m.math b o u sw n d.math da rit r y to cot nthe nd\ itquad power io nsn of , t m prime ha a t t is prime i x $open f−o parenthesis $ r m \ Lquad closing( } , c.math = 0, c.math = 0. We t h u s have t he red u ced s y ste m parenthesis12 minus d.math21 to the power of prime open parenthesis L closing parenthesis a.math sub m.math Row 2 a.math d.math to the power of prime open\ begin parenthesis{ a l i g n ∗} L closing parenthesis minus a.math d.math to the power of i.mathc.math v.math11 open parenthesis L closing parenthesis minus d.math to the U.kaliD c = 0comma.math c = T comma.math (69) powerA.kali of prime = prime F.kali open parenthesis = \ l e L f t closing(\ begin parenthesis{ array }{ c.mathcc } 1 omega & to 0 the\\ power0c.math & of 222 0 . period.math\end{ array }\ right ) comma.math J.kali = Q. kThe a l i natu = ra l\ fl req e f t u(\ ebegin nc ies lambda{ array =}{ i.mathcc } 0 omega & six-six-c0where\\ 0 a-one-zero & 1 \ parenright-parenright-nend{ array }\ right ) b-o-b period.math sub e-y-r s-o b-u\ tag sub∗{ b-t$ ( sub s-a 68 t-i sub ) $t-i-n} e-u t-d\end n-f{ suba l i g g-r n ∗} o m-t parenleft-e ro t s o f t h e c h a rac te r i st c e q at ion Capital Delta open parenthesis omega2 closing parenthesis = d.math e.math00 t.math open parenthesis0 U.kali sub D closing parenthesis = A s in open Thema t r iz(a.math $U.kali− e.mathω $ canbe)d.math(L) − reducedb.mathm.math tod.math aha(L lf) s ize onedue−d.math to(L) ta.math heboundam.math r ycond it io ns , t ha t is parenthesisU.kaliD = delta L closing parenthesis s in h open parenthesis0 epsilon.alt L closing parenthesis =i.mathv.math 0 comma.math open parenthesis00 7 12 closingperiod.math parenthesis $ , c . math { 1 2 } =a.mathd.math 0 , c . math(L) { 2 1 } =−a.mathd.math 0 .$ Wethushave(L) − d.math (L)c.mathω where (70) tA he = red open u parenthesis ced s y open stem parenthesis minus epsilon.alt to the power of 2 b.math sub m.math a.math plus a.math to the power of 2 minus a.math The natu ra l f req u e nc ies λ = i.mathωsix −e.mathsix − omegaca − one to− thezeroparenright power of 2 closing− parenright parenthesis− nb delta− o to− b the powers − ob of− 4u plus open parenthesist − dn epsilon.alt− f toparenleft the power− ofe 4ro a.math t s o f tb.math h e c h sub a rac m.math te r i st c e q at ion \ hspace ∗{\ f i l l } $ U. k a l i { D }$ c $= 0e−y comma.math$−r b−ts−at−\quadit−i−ne−cu $ = \gl− erom f t −( t\ begin { array }{ c} c . math { 1 1 }\\ plus open parenthesis∆(ω) 2 = c.mathd.mathe.matht.math omega to the power(U.kali of 2) b.math= A s in sub(δ m.mathL ) s in plus h (epsilon.alt 2 a.math subL m.math) = 0comma.math a.math closing(71) parenthesis epsilon.alt to the power c . math { 22 }\end{ array }\ right )$ T $comma.mathD ( 69 )$ of 2 minus c.math omega to the power of 2 a.math pluswhere c.math omega to the power of 4 e.math closing parenthesis delta to the power of 2 plus open parenthesis a.math to the power of 2 minus a.math e.math omega to the power of 2 closing parenthesis epsilon.alt to the power of 4 plus open parenthesis c.math\ centerline omega to{where the power} of 2 a.math minus c.math omega to the power of 4 e.math closing parenthesis epsilon.alt to the power of 2 closing parenthesis ((−epsilon.alt2b.math a.math + a.math2 − a.mathe.mathω2)δ4 + (epsilon.alt4a.mathb.math + (2c.mathω2b.math + 2a.math a.math)epsilon.alt2 − c.mathω2a.math + c.mathω4e.math)δ2 + (a.math2 − a.mathe.mathω2)epsilon.alt4 + (c.mathω2a.math − c.mathω4e.math)epsilon.alt2) dividedA = by delta epsilon.alt a.mathm.math to the power of 2 b.math sub m.math to the power of 2 open parenthesism.math epsilon.alt to the power ofm.math 2 plus delta to them.math \ begin { a l i g n ∗} 2 2 2 2 2 power of 2 closing parenthesis to the power of 2 sub period.math δepsilon.alta.math b.mathm.math(epsilon.alt + δ ) period.math U.We k a ol i bse{ rveD t} hat= fo r\ deltal e f t ( =\ n.mathbegin { piarray divided}{ cc by} L( we ca a .n math o b ta in− omegae . mathsix-f r one-o\omega comma-mˆ{ 2 w-parenleft} ) hd . i math l e o-f r .. ( epsilon.alt L ) = i.math− b . math { m. math } n.mathπ d . math ˆ{\prime \prime } (L)& − d . math ˆ{\prime } m.mathπ( L ) a . math { m. math }\\ a.math d.math ˆ{\prime } We o bse rve t hat fo r δ = L we ca n o b tam.math in ωsix pi− dividedf r one by− Locomma six-w zero-e− mw c T-a− parenleft n u-s s l-eh it-parenleft l e o − f r e r kepsilon.alt .. d-n of = i.math L six − wzero − e c T − a n u − s s l − et − parenleft e r k d − n of (L)&u−bracketleft−bracketleftB− a.math−en−y d.math ˆ{ i.math v.math } (L) − d . math ˆ{\prime \prime } ( L ) c . math f − three − three − three r five − six − seven − ecomma − comma − period −f-three-three-threeq r five-six-seven-ec i comma-comma-period-qe − co − ss − ai − re − da to− thern power− s s g of− u-bracketleft-bracketleftot − c i h − at − ees − df B-e− n-yw − cra i e-c− i o-sc−th s-a− i-re t e-dh − a-rpa n-s−es s− g-oam t-c− o i h-ae t − ca − er − l le d seco nd s pe ct r u m [ h \omega ˆ{ 2 i}\end{ array }\ right )sp period.math e t r um wil la\ ptag ea∗{ r f$ o r ( 70q u e ) n ces $} ab o ve e cl ass c alcr i ca f re q uen c y 2 a.math two − va − sevenbracketright − lu−commatt-e sub−eh e−s s-dg − f-w-rs s a-i− subve − c-tec h-e− nn t h-p− in sub−d[ a-e s-a m-o e t-c a-e r-l le d secoe − ndr s pe ct r u m open squaret − h bracket .. h ωc.math = e.math . \endtwo-v{ a l ia-seven g n ∗} bracketright-l sub4 . 4 u-comma . Freq t-e uency h-s g-seq uati i s-v on e-e fo c-n r a n-i m sub i cro n-d - canti open l square ever bracket sp e t r um .. wil la p ea r f o r e-r q u e n ces .. ab o ve t-h e .. cl ass c alcrFo r it ca he fca re n q t ueni leve c r y Tim .. omega os h e sub n ko c.math mode l to , we the have power t h e of bo 2 u= nd a.math a r y co divided n d it byio ns e.math sub period \ centerline4 period 4 period{The .. natu Freq uencyra l eq f uatireq on u fo e r nc a m ies i cro hyphen$ \lambda canti l ever= i . math \omega six −six −c a−one−zero parenright −parenright −n b−oFo−b r t{ hee ca−yw.math n− tr i} leve(t.mathcomma.maths r− Timo os b− hu e n{ kob− mode0)t = 0{ lcomma.math commas−a } wet have−i ψ( tt.mathcomma.math{ ht e− boi− un nd a e− ru y co}}0) n = dt 0− itcomma.mathd io ns n−f { g−r o m−t } p a r e n l e f t −e$ rotsof thecharacter i stceqation } w.math open parenthesis t.math comma.math 0 closing parenthesis = 0 comma.math psi open parenthesis t.math comma.math 0 closing parenthesis = ψ (t.mathcomma.math L ) = 0comma.math w.math (t.mathcomma.math L )−ψ(t.mathcomma.math 0\ hspace comma.mathx.math∗{\ f i l l } $ \Delta ( \omega ) =x.math d.math e.math t.math ( U.kali { D } ) = $ A s in $ ( L ) = 0comma.math (72) w − onefive − h coefficients writte n i n matrix fo rm ( \ deltapsi sub$ x.math L)sinh open parenthesis $( t.math epsilon.alt$ comma.mathoa−se−r L L closing $) parenthesis = 0 = 0 comma.math comma.math w.math ( sub 7 x.math 1 )$ open parenthesis t.math comma.math L closing parenthesis minus psi open parenthesis t.math comma.math L closing parenthesis = 0 comma.math open parenthesis 72 closing parenthesis \ centerline {where } w-one five-h sub o a-s e-r coefficients 1 writte 0  n i n matrix fo rm open parenthesis 0 0  A.kali = comma.math J.kali = comma.math A.kali = Row 1 1 0 Row 2 0 1 . comma.math0 1 J.kali = Row 1 0 0 Row 20 0 0 0 . comma.math open parenthesis 73 closing parenthesis F.kali = Row 1 0 0 \ [ A = \ f r a c { (( − epsilon.alt ˆ{ 2 } b . math { m. math } a.math + a.math ˆ{ 2 } − a.math e.math Row 2 0 minus 1 . comma.math Q.kali = Row 1 0 1 Row 2 1 0 . period.math (73) \omegatimes Nˆ{ a2 noMMTA} ) times\ delta Vo l periodˆ{ 4 2} times+ 20 ( 1 3 times epsilon.alt 1 24 hyphen ˆ{ 14 44} timesa.math 1 35 b.math { m. math } + ( 2 c . math \omega ˆ{ 2 } b . math { m. math } + 2 a0 . math 0  { m. math } a.math 0 ) 1  epsilon.alt ˆ{ 2 } − c . math \omega ˆ{ 2 } a . math F.kali = comma.math Q.kali = period.math + c . math \omega ˆ{ 4 } e0 . math−1 ) \ delta ˆ{ 2 } +1 0 ( a . math ˆ{ 2 } − a.math e.math \omega ˆ{ 2 } ) epsilon.alt ˆ{ 4 } + ( c . math \omega ˆ{ 2 } a . math − c . math \omega ˆ{ 4 } e.math ) epsilon.alt ˆ{ 2 } ) }{\ delta epsilon.alt a.math ˆ{ 2 } b . math ˆ{ ·2N} a noMMTA{ m. math· Vo} l .(2 · 2013 epsilon.alt· 124 − 144 · ˆ135{ 2 } + \ delta ˆ{ 2 } ) ˆ{ 2 }} { period .math }\ ]

\ centerline {Weo bse rve t hat fo r $ \ delta = \ f r a c { n . math \ pi }{ L }$ wecanobtain $ \omega six −f $ r $ one−o comma−m w−parenleft $ h i l e $o−f $ r \quad $ epsilon.alt = i.math \ f r a c { m. math \ pi }{ L } six −w zero−e $ c $ T−a $ n $ u−s $ s $ l−e t−parenleft $ e r k \quad $ d−n $ o f }

\ centerline { $ f−three −three −three $ r $ five −six −seven−e comma−comma−period−q ˆ{ u−bracketleft −bracketleft B−e n−y }$ c i $ e−c o−s s−a i−r e−d a−r n−s $ s $ g−o t−c $ i $ h−a t−e { e } s−d f−w−r a−i { c−t } h−e $ t $ h−p { a−e } s−a m−o $ e $ t−c a−e r−l $ le dseco nds pe ct rum[ \quad h }

\ centerline { $ two−v a−seven bracketright −l { u−comma t−e h−s } g−s $ i $ s−v e−e c−n n−i { n−d } [ $ sp e t r um \quad wil lapear for $e−r $ q u e n ces \quad ab o ve $ t−h $ e \quad cl ass c alcr i ca f re q uen c y \quad $ \omega ˆ{ 2 } { c . math } = \ f r a c { a . math }{ e . math } { . }$ }

\ centerline {4 . 4 . \quad Freq uency eq uati on fo r am i cro − c a n t i l ever }

\ centerline {Fo r t he ca n t i leve r Tim os h e nkomode l , we have t h e bounda r y co nd it io ns }

\ [ w.math ( t.math comma.math 0 ) = 0 comma.math \ psi ( t.math comma.math 0 ) = 0 comma . math \ ]

$ \ psi { x . math } ( t.math comma.math $ L $ ) = 0 comma.math w.math { x . math } ( t.math comma.math $ L $ ) − \ psi ( t.math comma.math$ L $) = 0 comma.math ( 72 )$ $ w−one f i v e −h { o a−s e−r }$ coefficients writte n i n matrix fo rm (

\ begin { a l i g n ∗} A. k a l i = \ l e f t (\ begin { array }{ cc } 1 & 0 \\ 0 & 1 \end{ array }\ right ) comma.math J.kali = \ l e f t (\ begin { array }{ cc } 0 & 0 \\ 0 & 0 \end{ array }\ right ) comma . math \\ ( 73 ) \\ F . k a l i = \ l e f t (\ begin { array }{ cc } 0 & 0 \\ 0 & − 1 \end{ array }\ right ) comma.math Q.kali = \ l e f t (\ begin { array }{ cc } 0 & 1 \\ 1 & 0 \end{ array }\ right ) period .math \end{ a l i g n ∗}

\ hspace ∗{\ f i l l } $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 35 $ J u l i o R period Claeyssen comma Teresa Ts u kazan comma Leticia Tonetto comma Dan i e l a Tol fo \ centerlineB y fo l lowing{J t u h l e s i a o me R reaso . Claeyssen n i ng as befo , re Teresa comma Ts we ou b kazan ta i n t ,h a Leticia t d u e to Tonetto t he b o u , nda Dan r y i co e nd l it a io Tol ns comma fo } t he matrix U.kali ca n J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo b e red u ced B y fo l lowing t h e s a me reaso n i ng as befo re , we o b ta i n t h a t d u e to t he b o u nda r y co nd it io ns , t he matrix \noindentto a ha lf sBy ize o fo n e l comma lowing t h at t h is e comma s ame c.math reaso sub 2 n 1 i = ng 0 comma as befo c.math re sub , weob22 = 0 period ta i We n t t h ha u s h t ave du t he e red to u cet hed s ybounda ste m r y co nd it io ns , t he matrix U.kali ca n b e red u ced $U.kaliU.kali sub $ D cac = n 0 comma.math b e red u .. ced c = Row 1 c.math sub 1 1 Row 2 c.math sub 1 2 . T comma.math open parenthesis 74 closing parenthesis to a ha lf s ize o n e , t h a t is , c.math = 0, c.math = 0. We t h u s h ave t he red u ce d s y ste m where 21  22  c.math11 \noindentU.kali toahac = 0comma.math lf s izeonec = , thatT iscomma.math $ , c.math(74) where{ 2 1 } = 0 , c . math { 22 } = 0 .$ Wethushavethereducedsystem Equation:D open parenthesis 75 closing parenthesisc.math .. U.kali sub D = Row 1 a.math d.math to the power of prime open parenthesis L closing parenthesis minus a.math d.math to the power of prime prime open12 parenthesis L closing parenthesis plus lambda to the power of 2 c.math d.math to the power of prime$ U. open k a l i parenthesis{ D }$ L closingc $= parenthesis 0 comma.math$ Row 2 minus b.math\quad sub m.mathc $ = d.math\ l e f tot (\ thebegin power{ array of prime}{ c prime} c . mathopen parenthesis{ 1 1 L}\\ closingc parenthesis . math { 1  a.mathd.math0(L) −a.mathd.math00(L) + λ2c.mathd.math0(L)  plus2 U.kali}\ lambdaend{=array to the}\ powerright of)$ 2 e.math T $comma.math d.math to the power ( of prime 74 open )$ parenthesis L closing parenthesis a.math d.math to the power of prime primeperiod.math open D −b.math d.math00(L) + λ2e.mathd.math0(L) a.mathd.math00(L) − λ2c.mathd.math(L) − a.math d.math00(L) parenthesiswhere L closing parenthesism.math minus lambda to the power of 2 c.math d.math open parenthesis L closing parenthesis minusm.math a.math sub m.math d.math to the power of prime prime open parenthesis L closing parenthesis . period.math (75) \ beginThe natu{ a l i rag n l∗} f req u e nc ies lambda = i.math omega six-six-c a-one-zero parenright-parenright-n b-o-b sub e-y-r s-o b-u sub b-t sub s-a t-i sub t-i-n e-u The natu ra l f req u e nc ies λ = i.mathωsix − six − ca − one − zeroparenright − parenright − nb − o − b s − ob − t-dU. n-f k a sub l i g-r{ D o m-t} parenleft-e= \ l e f t ..(\ robegin t s o{ farray t h e c}{ h acc rac} a.math te r i st c e q d.math at ion ˆ{\prime } (L)&e−y−r − a.math d.math ˆ{\prime u t − dn − f parenleft − e ro t s o f t h e c h a rac te r i st c e q at ion \primeEquation:b−ts−at}−it−(i open−ne−u L parenthesis ) +g− 76rom closing\−lambdat parenthesisˆ{ 2 } .. Capitalc.math Delta d.math open parenthesis ˆ{\prime omega} closing(L) parenthesis\\ = − minusb omega . math to{ them. power math of} 4 c.mathd . math ˆ{\prime e.math\prime open} parenthesis( L )d.math + to the\lambda power ofˆ{ prime2 } opene.math parenthesis d.math x.math ˆ closing{\prime parenthesis} ( closing L parenthesis ) & a.math to the power d.mathˆ of 2 plus{\ omegaprime to the\prime } (L) − \lambda ˆ{ 2 } c.math d.math ( L ) − a . math { m. math } d . math ˆ{\prime \prime } power of 2 open4 square bracket a.math0 c.math2 d.math2 open parenthesis x.math closing parenthesis00 d.math to the power of prime prime open parenthesis0 000 2 0 2 00 2 (L)∆(ω) = −ω\c.mathe.mathend{ array }\(d.mathright )(x.math period.math)) + ω [a.mathc.mathd.math\ tag ∗{$ ( 75(x.math ) $)d.math} (x.math) − (a.mathe.math + c.mathb.mathm.math)d.math (x.math)d.math (x.math)] + (a.math − a.matha.mathm.math)(d.math (x.math)) − a.mathb.mathm.math(d.math (x.math)) period.math x.math closing parenthesis minus open parenthesis a.math e.math plus c.math b.math sub m.math closing parenthesis(76) d.math to the power of prime open parenthesis\end{ a l i g x.mathn ∗} closing parenthesis d.math to the power of prime prime prime open parenthesis x.math closing parenthesis closing square bracket plus openF − parenthesisfour e − a.mathaf − o to thel − powermli − ofu 2r minus− st − a.matho r t − a.mathc − aa − subes m.math− n t i − closingth − l parenthesisbe a m open d escrib parenthesis ed b y t d.math he to the power of prime open \noindentob−Ther natu rao−wr l f req uc e nc ies $ \lambda = i . math ee−ver\omega six −six −c a−one−zero parenright −parenright −n parenthesisTim os he x.math n ko mode closing l with parenthesis s u rfa ce effects closing , F parenthesis ig u re s ize to de the p powere nd e nof ce 2 iminus n t he a.math natu ra b.math l f req u sub e n m.math c y o f Timoshen open parenthesis d.math to the power of b−o−b { e−y−r } s−o b−u { b−t { s−a } t−i { t−i−n e−u }} t−d n−f { g−r o m−t } p a r e n l e f t −e $ \quad rotsof thecharac te r i st ceqat ion primeko c prime la ss ica open l model parenthesis a n d Timoshenko x.math closing mode parenthesis l in c lu d closing i ng s u parenthesis rfa ce effects to , the The power so l uof t 2 io period.math ns b ased o n c la ss ica l TimoshenF-four sub ko obe b-r a m e-a t heo f-o subr y a o-w nd Timoshenr l-m l i-u ko sub bea c r-sm t t-o h eo r t-c-ar y in a-e c lu s-n d i t ng i-t s h-l u rfa sub ce e effects e-v e ar bere a de m n d ote escrib d b y ed TB b ay n t dhe Tim os he n ko mode l with s u \ begin { a l i g n ∗} rfaTMB ce effects , res p comma ect ive F l y ig . u re The n a t u ra l f req u e nc ies a re n o rm a l ized to fu n da m e nta l f req u e ncy o f ca nt \Delta ( \omega ) = − \omega ˆ{ 4 } c.math e.math ( d.math ˆ{\prime } ( x . math ) ) ˆ{ 2 } is leve ize r detwo p e− ndEseven e n ce− i nu tl hef natu− eo ra− r l fr reqt − uB e n c y o f Timoshenm − l kol ce − laib ss− icap l model a n d TimoshenkoI n − eperiod mode− l int h c lu d i ng s u rfa ce effects comma + \omega ˆ{ 2 } [ a.math c.mathh−er−ens d.math−ou−a ( x.mathu−ea )−rp− d.mathˆmo−period {\prime \prime } ( x . math ) − i −TheT s so− lhfi u t− ioe ns b ased oi n− ca la− ssrr ica− al l− Timoshene co ns id ko e redbe a t hem tpa heo ra mete r y a rs nd u Timoshent i l ized i n ko [ abea nd m t y t pes h eo of r s y u in rfa c ce lu c d r iy ng s u rfa ce effects a re ( a.math e.mathg−mur−at−e + c.math b.math { m. math } ) d . math ˆ{\prime } ( x.math ) d.mathˆ{\prime stade l n o oter ie nd ta by t io TB n da eten d rm TMB in e comma t he s u resrfa ce p ecte la ive st ic l coy period nsta nts .. . The Fo n a r at nu ara no l f d req ic a u l ue mnc i ies na aA rel.math n o rm( aYo l izedu ng to0s fu n da m e nta l f req u e ncy o f \prime \prime } ( x.math ) ] + ( a.mathˆ{ 2 } − a.math a.math { m. math } ) ( d . math ˆ{\prime } camod nt i u leve l u rs E = 70 G Pa , Po iss o n 0s ra t io ν = 0period.math3 a nd ρ = 2700 kg /m3) a re co ns id e re d two t y pes ( x . math ) ) ˆ{ 2 } − a.math b.math { m. math } ( d . math ˆ{\prime \prime } ( x . math ) ) ˆ{ 2 } oftwo-E c r y sta seven-u l lo g lra f-e p ho-r ic r d t-B i re sub ct io h-e n r-e n s-o u-a m-l l e-i b-p sub u-e a-r p-m sub o-period I n-e period-t h i-T s-h fi-e sub g-m u r-a t-e i-a-r r-a l-e co period .math \ tag ∗{$ ( 76 ) $} ns id e redA t hel.math pa ra[10] mete : E rs u t i= l− ized7period.math i n open square9253 N bracket / m a nd τ = 0period.math5689 N / m comma.math \end{ a l i g n ∗} s.math a nd t y pes ofA s ul.math rfa ce[111] c r y : E stas.math l o r= ie 5 nperiod.math ta t io n d1882 ete rmN in/ m e at ndhe sτ u= rfa 0period.math ce e la st ic918 co nstaN / m nts . period .. Fo r a n a no d ic a l u m i na A l.math openFig parenthesis 4 . I n fl Youence u ng of quoteright s u r face effects s and s ize dependence on t he normal ized f u ndamental natu ral f req uency of t he m i cro - canti l \noindent $ F−fo ur { o b−r } e−a f−o { o−w r } l−m l { i−u } { c } r−s t−o $ r $ t−c−a a−e s−n $ evermod fo r u 2 l h u= s E0.2 =L, 70 G Pa comma Po iss o n quoteright s ra t io nu = 0 period.math 3 a nd rho = 2700 kg slash m to the power of 3 closing parenthesis t $ i−t h−l { e e−v e r }$ be amd escrib ed b y t he Tim os he n ko mode l with s u rfa ce effects , F ig u re a re co ns id e re d two t y pes of c r y sta l lo g rab p= h 0 ic.4 L and κ = 5/6 s ize de p e nd e n ce i n t he natu ra l f req u e n c y o f Timoshen ko c la ss ica l model a n d Timoshenko mode l in c lu d i ng s u rfa ce effects , I dn i− refourw ct io n− t h e − ea − F − c o−u e e − v ha t f − o r b eam le n gt h o nt h o rder o f n an o me t The so l u t io ns b ased oin− ng crb la−e ss ica l Timoshen ko be amt heo r y a nd Timoshen ko beamt h eo r y in c lu d i ng s u rfa ce effects a re eA t o l.math m openc − i squarerons , t bracket he d − 1i 0ffe closing re n c esquare be t wee bracket n natu : E ra sub l f reqs.math u e nc = ieminus s is a 7 p period.math pa re nt a nd 9253 b y i N ncre slash as min ag tnd he tau = 0 period.math 5689 N slash de n ote db yTBa ndTMB , res p ect ive l y . \quad Thena t u ra l f req ue nc ies a re norma l ized to fu ndame nta l f req ue ncy o f ca nt i leve r mle comma.math ngth of t h e m ic ro be a m , t he res u lts te n d to Timoshen ko c la ss ica l two − th − sevenbracketright−eo−fr−oy−r. T h$A− two l.mathm −E opens − sevenob square− am−u− bracketee $− lam 1− $1b 1 f− closingeh − i o square−r $ bracket r $u − : t E−waB sub− pp{ s.math−h−soe− =ot 5 r−− period.mathbe− r n s 1−o 882 N u− slashad −} mtim− a− ndhnl− tau $e[ = l 0 $period.math e−i b− 9p 1 8{ Nu slash−e m a−r p−m {i−ois−−periodc }}$ I $ n−es−e perioda−mv−−plt−i $o−yr h $ i−T s−h f i −es−ed{−erg−−periodvOm u−e r−a t−e } i−a−r r−a l−e $ co ns id e red t hepa ra mete rs u t i l ized i n [ periodo b s er v a o n is t ha t h e n a t r a l fre q u e ncy o f vi b ra t io n o f TB be a ms is i nde pe nde nt of t he b ea m le n gth and t y pes of s u rfa ce c r y sta l o r ie n ta t io nd ete rmin e t he s u rfa ce e la st ic co nsta nts . \quad Forananod ic a l umi naA whFig i le 4 fo period r TB I t nh flis uenceis not o of ccu s u r r , facet hat effects is , t he and s u s rfa ize ce dependence effects a re on s ig t n he ifica normal nt o n ized ly i f n u n ndamental a nosca le . natu ral f req uency of t he m i cro hyphen canti l $ l.math ($ Young $’{ s }$ ever· N fo a rnoMMTA 2 h = 0 period· Vo l 2.2 L· 2013 comma· 124 − 144 · 136 b = 0 period 4 L and kappa = 5 slash 6 \noindentI n-four w-tmodulusE h e-e a-F-c sub i n-g $= to the 70$ power of GPa,Poiss o-u r b-e e e-v .. haon t f-o $’ r{ b eams }$ .. le ra n gt t h i o o nt h$ ..\nu o rder= o f n an0 o me period.math t e t o .. m c-i rons 3$ comma and t$ he\rho d-i ffe re= n c e 2700$ be t wee n kg $/ mˆ{ 3 } ) $ a re co ns id e redtwo types of c ry sta l lo g raph ic dnatu i re ra ctl f req i o u n e nc ie s is a p pa re nt a nd b y i ncre as in g t he le ngth of t h e m ic ro be a m comma t he res u lts te n d to Timoshen ko c la ss ica l two-t h-seven sub bracketright-e o-f r-o y-r sub period T h-m sub i-i sub s-c s-o b-a m-e e-a m-b sub s-e h-i sub a-m v-p l-i sub o-y r u-w a-p p-s o-o t-b-r sub\ centerline s-e d-e r-period{A$l.math v O-e d-t i-h n-e [ open 1 square 0 bracket ] : o b s E er v{ as o . n math is t ha} t h= e ..− n a t7 r a l period.math 9253 $ N/ma nd $ \tau =fre 0 q u e period.math ncy o f vi b ra t io 5689 n o f TB $ beN/m a ms is $ i nde comma.math pe nde nt of $ t he} b ea m le n gth wh i le fo r TB t h is is not o ccu r comma t hat is comma t he s u rfa ce \ centerlineeffects a re s{A$l.math ig n ifica nt o n ly i [ n n a 1 nosca 1 le period 1 ] : E { s . math } = 5 period.math 1 882$ N/mand $ \tau =times 0 N period.math a noMMTA times 9 Vo l period 1 8$ 2 times N/m. 20 1 3 times} 1 24 hyphen 1 44 times 1 36 \noindent Fig 4 . I n $ fl $ uence of s u r face effects and s ize dependence on t he normal ized f u ndamental natu ral f req uency of t hem i cro − canti l ever fo r 2 h $ = 0 . 2 $ L ,

\ centerline {b$= 0 . 4$Land$ \kappa = 5 / 6 $ }

\noindent I $ n−fo ur w−t $ h $ e−e a−F−c { i n−g ˆ{ o−u } r b−e }$ e $ e−v $ \quad ha t $ f−o $ r b eam \quad l e n gt h o nt h \quad o rder o f nanomet e t o \quad m $ c−i$ rons ,the $d−i $ ffe rencebetween natu ra l f req u e nc ie s is a p pa re nt a ndby i ncre as in g t he le ngth of t h emic ro be am , t he res u lts te nd to Timoshen ko c la ss ica l $ two−t h−seven { bracketright −e o−f r−o y−r { . }}$ T $ h−m { i−i { s−c }} s−o b−a m−e e−a m−b { s−e } h−i { a−m v−p l−i { o−y r }} u−w a−p p−s o−o t−b−r { s−e d−e r−period v O−e } d−t i−h n−e [$ observaonisthathe \quad n a t r a l fre qu e ncy o f vi b ra t io no fTBbe ams is i nde pe nde nt of t heb eamle n gthwh i le fo rTBt h is is not o ccu r , t hat is , t he s u rfa ce effects a re s ig n ifica nt o n ly i nn a nosca le .

\noindent $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 36 $ M o d e l i n g t h .. e t i p hyphen s a m .. p l e i n t e r a c t i o n i n a t o .. m i c f o r c e .. m i c r o s c o p y .. ellipsis \noindent5 period ..M Moda o d .. e a l .. pproximi n g t a hto\ noquad f .. dye to t the i p power− s of a n-a m \ mquad icresponspleinteractioninato \quad m i c f o r c e \quad m i c r o s c o p y \quad Modelingth etip-sam pleinteractioninato micforce microscopy ellipsis $ etwo-T l l i p s h-eight i s $ sub bracketright-e G-a sub n-a l e-b e r k-u s-i dm e-o r-t sub h d-o d te open square bracket rm in i g a p pr o im te d yn a m ic e s p o n 5 . Moda a pproxim a to no f dyn−a m icrespons se s of h-t sub e .. A F M .. mi c-r o hyphen can il e e-v sub r two − T h − eight G − a l e − b e r k − us − i dm e − or − t d − o d te [ rm in i g a p pr o im te d yn a m ic e s p o n se s of h − t A F M mi c − r o - can il e e − v bracketright−e \ centerlineb-twon−a e-four{ parenright-a5 . \quad mModa d-e sub\quad e s-a c-cah \ r-tquad u-i a-bpproxim sub e-l d-l a bto n-y no t-e f h-d\quad sub e T-o$ dy sub ˆ i{ fi-mn− d-o-na }$ s h-a me icresponssub n-e n k-a} sub p-o p-m-r o-o x-d subr h b − twoe − fourparenright − a m dm-e− e l period F-i-oa sub− b r o-nd − m-ol b openn − paranthesisRowyt − eh − d T − 1o h Row 2 t .h ..− una a mk e− na t a lp − m − ro − ox − d l .F − i − o ( e un a m e n t a l es−ac−cr−tu−i e−l e ifi−md−o−ns n−en p−o m−e ro−nm−o t \ centerlinematrix res p{ o$ nse two h open−T parenthesis h−e i g h t t.math{ bracketright comma.math− x.mathe } comma.mathG−a { n−a xi} closing$ l parenthesis $ e−b $ period e r Fo $ r k t− hu is comma s−i $ we dm first in $ t e rod−o u ce r− tt { h } matrix res p o nse h (t.mathcomma.mathx.mathcomma.mathξ). Fo r t h is , we first in t rod u ce t he b lo ck matrix hed−o$ b lo ck dte matrix [rmin igapproimtedynamic esponse s of $h−t { e }$ \quad AFM \quad mi $ c−r $ o −Equation:can i l open e $ parenthesis e−v { 77r } closing$ } parenthesis .. V open parenthesis x.math closing parenthesis = open parenthesis v sub 1 open parenthesis x.math closing parenthesis v sub 2 open parenthesis x.math closing parenthesis period.math period.math period.math v sub n.math open parenthesis x.math closing V(x.math) = (v1(x.math) v2(x.math) period.mathperiod.mathperiod.math vn.math(x.math))comma.math (77) parenthesis\ centerline closing{ $ parenthesis b−two comma.math e−four parenright −a $ m $ d−e { e s−a c−c r−t u−i } a−b { e−l } d−l $ b $ n−y t−e h−d { e } T−o { i f i −m d−o−n s } h−a { n−e n } k−a { p−o } p−m−r o−o x−d { m−e }$ l $ . whose co l u m ns a re t he first n.math five-c a-one parenright-nd−en−if t− c-inu− o-lg sub e-r r-v e-e s-r p-e i-o g-n to the power of d-e n-i f-n sub u-gv− ne c-t o-t i t-o whose co l u m ns a re t he first n.mathfive − ca − oneparenright − n t c − io − le−rr−ve−es−rp − ei − og − n n c − to − t i t − oh − ns − em − parenleft i c r - c a n i − tle r ei en v a l e s, th ati s , Fh-n−i− s-eo m-parenleft{ r o− in c r hyphen m−o } c a(\ nbegin i-t le to{ array the power}{ c} ofh v-e\\ r eit en\ vend a l{ earray s comma}e$ th ati\quad s commaun a m e n t a l } Equation: open parenthesis 78 closing parenthesis .. v sub j.math open parenthesis x.math closing parenthesis = Row 1 w.math sub j.math open \ centerline { matrix res p o nse h $ ( t.math comma.math x.math comma.math \ xi ) . $ For t h is ,we first in t rodu ce t heb lo ck matrix } parenthesis x.math closing parenthesisw.math Rowj.math 2 psi(x.math sub j.math) open parenthesis x.math closing parenthesis . = h open parenthesis x.math comma.math omega vj.math(x.math) = = h(x.mathcomma.mathωj.math)cj.mathcomma.math (78) sub j.math closing parenthesis c subψ j.mathj.math( comma.mathx.math) \ beginwhere{ a c l sub i g n j.math∗} five-i s parenright-o w-b sub t i-a-t h-i sub n ed b y find in g a no nze ro so l u t io n of open parenthesis lambda sub j.math = i.math V ( x.math )where =c (five v −{ i 1s parenright} ( x .− mathow − b ) v ed{ b2 y} find( in g a x.math no nze ro so ) l u t ioperiod.math n of period.math period.math omega sub j.math period S i nce tj.math h e AFM m ic ro hyphen ca nt i le veti− ra modes−th−in s ha re v ( t{ heλj.mathn n . math o rm= i.mathω a} l m( odj.math e x.math p ro. S pe i nce rt y t ) comma h e AFM ) we m comma.math ca ic nro ass - ca u nt m i e le\ t vetag hat r∗{ modes t he$ (y s have ha 77re bee n ) no $} rm a l ized w it h res pe ct to t h e mass matrix M period The n\end we t{ hea l i n g on rm∗} a l m od e p ro pe rt y , we ca n ass u m e t hat t he y have bee n no rm a l ized w it h res pe ct to t h e mass matrixco nsM id e. r The t h n e we o b co te ns nt id io e n r o t hf a e no b a te p p nt rox io n im o f a a te n are p s p po rox nse im a te re s po nse \ centerlinen.math {whose co l umns a re t he first $ n.math five −c a−one parenright −n $ t $ c−i o−l { e−r r−vv open e− parenthesise s−r } t.mathp−e comma.math i−o g− x.mathn ˆ{ closingd−en.math parenthesis n−i f−n period.math{ u−g }} =$ sum n p.math $ c− subt j.math o−t $open i parenthesis $ t−o t.math h−n closing s−e parenthesis m−p a r e n l e f t $ i c r − c a n $ i−t l e ˆ{ v−e }$ reienvaleP $s { , }$ th a t i s , } v sub j.math openv ( parenthesist.mathcomma.mathx.math x.math closing parenthesis)period.math == V openp.math parenthesisj.math(t.math x.math)vj.math closing(x.math parenthesis) = V P(x.math open parenthesis) P t.math closing parenthesis comma.math(t.math)comma.math open parenthesis(79) 79 closing parenthesis \ beginj.math{ a = l i 1g n ∗} v { j . math } ( x . math ) = \ l e f t (\ begin { array }{ c} w. math { j . math } ( x . math ) \\\ psi { j . math } two-o f-three t-parenright h e AFM m ic ro hyphen caj.math nt i le= ve 1 r Timoshen ko m od e l open parenthesis (Fo x r . mathdete rm i ) n i\end ng t{ hearray t im}\ e aright m p l) itu = d es h P to ( the power x.math of T open comma.math parenthesis t.math\omega closing{ parenthesisj . math } =) open c parenthesis{ j . math p.math} comma 1 open . math \ tag ∗{$ ( parenthesis78 ) $} t.mathtwo closing− of parenthesis− threet − parenright p.math 2 openh e AFM parenthesis m ic ro t.math - ca nt i closing le ve r Timoshen parenthesis ko period.mathm od e l ( period.math period.math p.math n.math open T parenthesis\endFo r{ detea l i rmg t.mathn i∗} n i ng closing t he t im parenthesis e a m p l it u closing d es P parenthesis(t.math) = comma-seven-four (p.math1(t.math) seven-parenright-w-commap.math2(t.math) period.mathperiod.mathperiod.math p-e r-n-s-e hyphen-u-t o-b-m sub p.mathn.math parenleft-s u-t(t.math l-i ))comma− seven − fourseven − parenright − w − commap − er − n − s − ehyphen − u − to − b − mparenleft−su−tl−i ( t sub t-t sub i-u p-t sub l-e open parenthesis t−ti−up−tl−e \ hspacehet reshe resu∗{\ lt u in ltf g i in matrix l l g} matrixwhere d iffe d re iffe $ n t c re ia nl{ eq t iaj u . a lmath eqt io u n a} b t y ioVf in v b e(µ− y)Ti Va $ open nd ints parenthesis eg $ ra parenright te i n mu o r declosing r− too a parenthesis p w p− lb y t he{ tonot therm i a power− la m−t od of e Tp h− ai nd{ intn eg}} ra$ te i edby n o r de r find to a in gano nze ro so l u t io n of p$ pro ( l pe yt r\ hetlambda y no . rm I ta t{ l u mj rod . ns math e p ro} pe= r t y iperiod . math .. I t\ tomega u r ns { j . math } . $ S i nce t heAFMmic ro − ca nt i le ve r modes s ha re o u t t he n.math d im e ns io na l so y u ste t t m he n.math d im e ns io na l s y ste m t heEquation: n o rm open a lparenthesis mod e 80p closingro pe parenthesis rt y , we .. P-dieresis ca n ass open ume parenthesis t hat t.math t he closing y have parenthesis bee n norm plus Capital a l Omega ized w tothe it powerh res of pe 2 P ct open to t h e mass matrixM. The nwe parenthesisco ns id t.math e r closingtheob parenthesis te nt = fio comma.math no f anapproxima te re s po nse ¨ 2 where P (t.math) + Ω P(t.math) = fcomma.math (80) \ [Capital n . math Omega\ ] to the power of 2 = Row 1 omega sub 1 to the power of 2 0 times 0 Row 2 0 omega sub 2 to the power of 2 times 0 Row 3 times times where times times Row 4 02 0 times omega sub N to the power of 2 . comma.math .. f = integral sub 0 to the power of L V open parenthesis mu closing parenthesis ω1 0 · 0 to the power of T Fd mu2 period.math open parenthesis 8 1 closing parenthesis \ hspace2 ∗{\ f i l l }0vω $2 (· 0 t.math comma.mathR L x.mathT ) period.math{ = }\sum p . math { j . math } ( t . math TheΩ a b= ove s y ste m is s u bj ect comma.math to t he in it ia lf co= nd0 itV io ns(µ) Fd µperiod.math (81) The a b ove s y ste m is s ) v { j . math ····} ( x.math ) =$ V $( x.math )$ P $( t.math ) comma.math ( 79 )$ P open parenthesis0 0 0 closing· ω parenthesis2 = integral sub 0 to the power of L V open parenthesis mu closing parenthesis to the power of T v sub o.math open parenthesis mu closing parenthesisN d.math mu comma.math P-dotaccent open parenthesis 0 closing parenthesis = integral sub 0 to the power of L V \ [u bj j . ect math to t he = in it ia 1 l\ co] nd it io ns open parenthesis muR L closing parenthesisT to the power of T v sub 1 open parenthesisR L muT closing parenthesis d.math mu period.math open parenthesis 82 P (0) = V (µ) vo.math(µ)d.mathµcomma.math P˙ (0) = V (µ) v1(µ)d.mathµperiod.math (82) closing parenthesis 0 0 T −eight−eightzero−two−hu−parenright−parenrights−w−ct−i−an−t−he−he−s−he−ow−lr−i−un−t−it−i l− T-eight-eight zero-two-h sub u-parenright-parenright s-w-c t-i-a n-t-h sub e-h e-s-h e-o w-l sub r-i-u n-t-i t-io−t sub−ti− o-t-tn−e i-n-e l-o c-f-a s-o-parenleft n d i o-i \ centerlineoc − f − as −{o −$parenleft two−on f d− ithreo − ei n s t(−parenright $ h e AFMm ic ro − ca nt i le ve r Timoshen komod e l ( } n s open parenthesis t.math P (t.math) = h (t.math) P (0) + h˙ (t.math) P (0) + R h (t.math − ν)( f )d.mathνcomma.math (83) P open parenthesis t.math closing parenthesis = h open parenthesis0 t.math closing parenthesis P open parenthesis 0 closing parenthesis plus h-dotaccent w − eightzero − h d u − ha − et − e h − ta − h d eco u p le d ch a ra ct e r o f ( open\noindent parenthesisFor t.mathcomma determi closing−erw−e parenthesis, n i ngt P heopen timeamp parenthesiso e 0−t closing l it parenthesis udes plus $Pˆ integral{ T } sub( 0 to the t.math power of ) t.math = h open ( parenthesis p.math t.math 1 ( minust.math nu closing ) parenthesis p.math open 2 parenthesis ( t.math f closing ) parenthesis period.math d.math nu period.math comma.math open period.math parenthesis 83 closing p.math parenthesis n.math ( t.math ) ) comma−seven−fo ur seven−parenright −w−comma p−e r−n0−s−e hyphen−u−t o−b−m { p a r e n l e f t −s u−t l−i { t−t { i−u } w-eight zero-h sub comma-e r w-e sub comma d u-h a-e0 t-e sub o h-t a-h sub e-t d0 eco u p le d ch a ra ct e r o f open parenthesis sin(Ωt.math) sin(ω1t.math) (sinω2t.math) sin( ·ωN t.math) p−t {h(lt.math−e }}}) = ( $ = ( · · · · )period.math (84) Equation: open parenthesis 84 closing parenthesis .. h open0 parenthesis· ·· · t.math closing parenthesis = s i n open parenthesis Capital Omega t.math Ω ω 0· · ωN closingt he parenthesis res u lt divided in g matrixd by Capital iffe Omega0 re1 = nt parenlefttp-parenleftex-parenleftex-parenleftex-parenleftbt ia lω2 equa t io nbyV $ ( \mu ) sine ˆ{ T open}$ parenthesis andint omega eg ra sub te 1 t.mathi norder toapp l ythenorma lmodeproper ty . \quad I t t u r ns closing parenthesis divided by omega sub 0 to the power of 0 from times· toN 1 a open noMMTA parenthesis· Vo l sine.2 · 2013 to the· 124 power− 144 of 0· omega137 sub 2 t.math closing parenthesis divided\ centerline by times{ou from t 0 tto he omega $n.math$ sub 2 times sub dime times times ns io sub na times l s from y stem times to} times times sub times s in open parenthesis to the power of 0 to the power of 0 times omega sub N t.math closing parenthesis divided by omega sub N parenrightbt-parenrightex-parenrightex-parenrightex-parenrighttp period.math\ begin { a l i g n ∗} \ddottimes{P N} a noMMTA( t . times math Vo l ) period + 2 times\Omega 20 1ˆ 3{ times2 } 1 24P hyphen ( 1 t.math 44 times 1 )37 = f comma.math \ tag ∗{$ ( 80 ) $} \end{ a l i g n ∗}

\ centerline {where }

$ \Omega ˆ{ 2 } = \ l e f t (\ begin { array }{ cccc }\omega ˆ{ 2 } { 1 } & 0 & \cdot & 0 \\ 0 & \omega ˆ{ 2 } { 2 } & \cdot & 0 \\\cdot & \cdot & \cdot & \cdot \\ 0 & 0 & \cdot & \omega ˆ{ 2 } { N }\end{ array }\ right ) comma . math $ \quad f $ = \ int ˆ{ L } { 0 }$ V $ ( \mu ) ˆ{ T }$ Fd $ \mu period.math ( 8 1 )$ Theab ove s y stemis s u bj ect to t he in it ia l co nd it io ns

P $ ( 0 ) = \ int ˆ{ L } { 0 }$ V $ ( \mu ) ˆ{ T } v { o . math } ( \mu ) d . math \mu comma . math \dot{P} ( 0 ) = \ int ˆ{ L } { 0 }$ V $ ( \mu ) ˆ{ T } v { 1 } ( \mu ) d . math \mu period .math ( 82 ) $ $ T−eight −e i g h t zero−two−h { u−parenright −parenright s−w−c } t−i−a n−t−h { e−h } e−s−h e−o w−l { r−i−u n−t−i t−i { o−t−t i−n−e }} l−o c−f−a s−o−parenleft $ nd i $o−i $ n s (

P $( t.math ) =$ h $( t.math )$ P $( 0 ) + \dot{h} ( t.math )$P$( 0 ) + \ int ˆ{ t . math } { 0 }$ h $ ( t . math − \nu ) ($ f $) d.math \nu comma.math ( 83 ) $ $ w−e i g h t zero−h { comma−e r w−e { , }}$ d $ u−h a−e t−e { o } h−t a−h { e−t }$ decoupledchara cterof (

\ begin { a l i g n ∗} h ( t . math ) = \ f r a c { s i n ( \Omega t . math ) }{\Omega } = ( \ f r a c {\ sin ( \omega { 1 } t . math ) }{\omega { 0 ˆ{ 0 ˆ{\cdot } { 1 }}}}\ f r a c { ( {\ sin }ˆ{ 0 }\omega { 2 } t . math ) }{\cdot ˆ{ 0 } {\omega { 2 }}} \cdot {\cdot }\cdot {\cdot ˆ{\cdot } {\cdot }}\cdot {\cdot }\ f r a c { s in ( ˆ{ 0 ˆ{ 0 }}\cdot{\omega } { N } t . math ) }{\omega { N }} ) period.math \ tag ∗{$ ( 84 ) $} \end{ a l i g n ∗}

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Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo closing parenthesis = V open parenthesis x.math closing parenthesis P open parenthesis t.math closing parenthesis o-two four-f parenright-parenleft comma B − eightthree − yi − su − n b h − s t e − ita−ut−p \i centerlinep − ng − rx −{ parenleft$ B−e i g hi m t a te three d dy n− ay m icr i−es− s p u− onn e $ v b(t.mathcomma.mathx.math $ h−s $ t $ e−i {) =tV a(x.math−u t)−Pp }(t.math$ i)o $− ptwofour−n g−rfparenright x−p a r e− n lparenleft, e f t $ w e ha e w e ha e L L imateddynamicrv (t.mathcomma.mathx.math $e)−s = $R pV on(x.math e \quad) h v(t.math $ ()VT t.math(µ)v (µ comma.math)d.mathµ + R V x.math(x.math) )h =$ V $ ( x.math v open parenthesis t.math comma.math x.math0 closing parenthesis = integralo.math sub 0 to the power of0 L V open parenthesis x.math closing parenthesis )$ P $(T t.math )R t.math o−two four−f parenright −T parenleft , $ we ha e } h( opent.math parenthesis)V (µ)v1( t.mathµ)d.mathµ closing+ 0 parenthesisV (x.math V to the) h power(t.math of T− openν)V ( parenthesisµ) f d.mathνperiod.math mu closing parenthesis(85) Co v nse sub q o.mathu e open parenthesis mu closing parenthesisnt l y , we d.math o b ta i nmu t h plus e s pectra integral l a sub p p 0 rox to im the a tpower io n fo of r L t he V inopen it ia parenthesis l va l u e G x.math re e n matrix closing re parenthesis s po nse h open parenthesis t.math closing parenthesis Vv to $ the ( power t.math of T open comma.math parenthesis mu closingx.math parenthesis ) = v sub\ int 1 openˆ{ parenthesisL } { 0 mu}$ closing V $( parenthesis x.math d.math )$ mu plus h integral $( sub t.math 0 to the power ) Vˆ{ T } of( t.math\mu V open) parenthesis v { o . math x.math} closing( parenthesis\mu ) h d open . math parenthesis\mu t.math+ minus\ int nuˆ{ closingL } { parenthesis0 }$ V V to $( the power x.math of T open )$ parenthesis h $( mu t . math ) V ˆ{ T } ( \mu ) v { 1 } ( \mu ) d . mathj.math=1\mu + \ int ˆ{ t . math } { 0 }$ V $ ( closing parenthesis f d.math nu period.math open parenthesis 85sin closing(Ωt.math parenthesis) T X s.mathi.mathn.math(ωj.matht.math) T x.mathh(t.mathcomma.mathξcomma.mathµ )$ h $( t.math ) period.math− \nu =)V( Vξ) ˆ{ T } ( V \(µmu) = ) $ f $ d . math \nu period.mathvj.math(ξ) (vj.math 85(µ)comma.math Co nse q u e nt l y comma we o b ta i n t h e s pectra l a p p rox imΩ a t io n fo r t he in it ia l va l u e G reω ej.math n matrix re s po nse ) $ N Equation: open parenthesis 86 closing parenthesis .. h open parenthesis t.math comma.math xi comma.math mu(86) closing parenthesis period.math = V openConse parenthesis que xi nt closing l y parenthesis , weob s ta i n openi n tparenthesis he s pectra Capital Omegal app t.math rox closing ima parenthesis t io n fo divided r t he by Capitalin it Omegaia l va V to l theueGre power of en T matrix re s po nse opena nd parenthesis fo r t he t ra mu nsfe closing r m a parenthesis t r ix fu nct = iosum n from j.math = 1 to N s.math i.math n.math open parenthesis omega sub j.math t.math closing parenthesis divided\ begin by{ a omegal i g n ∗} sub j.math v sub j.mathH open(s.mathcomma.mathξcomma.mathµ parenthesis xi closing parenthesis v) subperiod.math j.math to= the powerV (ξ)( ofs.math T open2 I parenthesis mu closing parenthesis h ( t.math comma.math T \ xi comma . math \mu ) period.math{ = } V( \ xi ) \ f r a c { s i n ( comma.math2 −1 T Pj.math=1 vj.math(ξ)vj.math(µ) \Omega+Ω ) V t( .µ math) = N ) }{\s.mathOmega2+ω}2 V ˆ{ T } ( (87)\mu ) = \sum ˆ{ j . math = 1 } { N }\ f r a c { s.math i.math a nd fo r t he t ra nsfe r m a t r ix fuj.math nct io nperiod.math n .We mathH o open bse rve parenthesis ( t h\ aomega t when s.math t he{ pj comma.math ro. math b e d e} flet ct xi . mathio comma.math n is co ns ) id}{\ e mu redomega closing d u e o nparenthesis{ l yj t. math o t he period.math in}} te rav ct{ io nj =t . math ip.. - V sa open m} p le( parenthesis fo rce\ xi xi) closing v ˆ{ parenthesisT } { j . open math } parenthesis( n2 a\mu t t he s.math e) nd x.math comma to the= . power mathL of\ tag 2 I∗{ plus$ ( Capital 86 Omega ) $} to the power of 2 closing parenthesis to the power of minus 1 V to the power of T open \endtwo{−aof l i g− nfivet∗} − parenrighth − t m − oi − bc − ta − ro − i nc a − th − nt − e i l − a i − w − xm − parenthesis mu closing parenthesis =o sum−e from j.math = 1 to N v sub j.math open parenthesisp−evp− xier− closingro−comma parenthesis v sub j.math to the power of T open parenthesisec − aa − mued − closingnu − rs parenthesis− ee − s( dividedons e by s.math to the power of 2 plus omega sub j.math to the power of 2 sub period.math open parenthesis 87 closing \noindent and fo r t he t ra nsfe rmaR t.math t r ix fu nct io n T parenthesis v (t.mathcomma.mathξ) ∼ = 0 h (t.math − τcomma.mathx.mathcomma.math L )E Q n (τcomma.mathWe o bse rve tL h a) td.mathτperiod.math when t he p ro b e d e(88) fle ct io n is co ns id e red d u e o n l y t o t he in te ra ct io n t ip hyphen sa m p le fo rce n sub 2 a t t he e nd\ hspace6 x.math . N∗{\ = umerical Lf i l l }H s $ imu ( l atio s.math ns comma.math \ xi comma . math \mu ) period.math{ = }$ \quad V $ ( \ xi )two-oI ( n t f-five h s is . math sect t-parenright io ˆ n{ , we2 s} h-t ha$sub l lI co o-e ns$ m-o+ id e ri-b t\Omega he c-t e a-r ig e o-iˆ nva{ nc2 lu a-t e} p h-n ro) b t-e ˆ le{ i m− l-a fo sub r a1 frep-e} e v -V p-e f ree ˆ{ r-r bT isub -} seg o-comma mented( \mu Timoshenko i-w-x m-e) c-a = bea a-e\ d-nsum u-rˆ s-e{ j e-s . math open parenthesis = 1 } { N }\ f r a c { v { j . math } ons( m e a\ ndxi ) v ˆ{ T } { j . math } ( \mu ) }{ s . math ˆ{ 2 } + \omega ˆ{ 2 } { j . math }} { period .math } ( 87tv he open o ) b $ te parenthesis n t io n of fo t.math rce d res comma.math p o nses fo r xi a closingTimoshen parenthesis ko m ic ro thicksim - ca n t i leve= integral r b ea m sub with 0 to a the p ie power zo e le of ct t.math r ic la y h e open r parenthesis t.math minus tau comma.matha bove it . x.math comma.math L closing parenthesis E to the power of T Q n open parenthesis tau comma.mathThe L closing parenthesis d.math tau period.math\noindentcom p u ta opentWeo io ns parenthesis we bse re p rve e rfo 88t rm closing ha ed in t exa parenthesis when ct ra t io hep n a l a ro r it bede hmetic u s i fle ng t cthe s io y m n b o is l ic co co m ns p u id ta e t io red n la due ng u a on l y t o t he in te ra ct io n t ip − samp le fo rce $ge n6 Maple period{ 2 . ..} Expa$ N umerical attheend ns io ns s imu l atio $x.mathns =$ L weI n re t t h r isun sect cated io nwith comma a s ma we l ls n ha u l m l bco e ns r N ido e fr te t he rms e ,ig u e s nva u al lu l y ebetwee p ro b len 5 m a fo nd r 1 a 0 fre . e hyphen f ree b i hyphen seg mented Timoshenko bea m a nd \noindent6t . he 1 . o b B te i n$ - tsegmented two io n− ofo fo frce f ree− df i- res v f e ree p o Timoshenko nsest−parenright fo r a Timoshen beam h ko−t m ic{ roo− hyphene } m ca−o n t i ileve−b r bc ea−t m with a− tor a p o− iei zo $ e le nc ct r ic$ la a− yt e r a h− boven it t period−e $ i $ l−a { p−e v p−e r−r { o−comma }} i−w−x m−e c−a a−e d−n u−r sn−−et e−s ( $ ons e three − In − eightbracketleft − bracketright.... The ,i − t wa sco si de re d t h ei e na n al y s is f or a f e e - fe e Eu e r - B er n o u l i − l b i - s egm e e d be a m . B y u s n g t h sa me dcomone p− ua tata t ,io as ns g ive we n re in p Ta e brfo le rm ed in exa ct ra t io n a l a r it hmetic u s i ng t he s y m b o l ic co m p u ta t io n la ng u a ge Maple period Expa ns\ hspaceTable io ns ∗{\1.threef i l l−}Geightv $ (− e − t.mathbracketrightperiod comma.math− o metrical\ xi d i mensions) \ andsim material{ = p}\ ropeint rt i es ofˆ{ beamt . math[ } { 0 }$ h $ ( t . math − \tau comma.mathwe re t r u n cated x.math with a scomma.math ma l l n u m b $P e ro r L p N e ort $ f ies te) of rms bea E comma m ˆ{ e lT em u} e$ sn uts Q a l n l y betwee$ ( n\ 5tau a nd 1comma.math 0 period $ L $ ) d.math \tau period .math (6 period 88 1) period $ .. B i hyphen segmented f ree hyphen f ree Timoshenko beam three-I n-eight bracketleft-bracketright sub comma i-t .. wa sco si de re d t h .. ei e na n al y s is f or .. a f e e hyphen fe e Eu e r hyphen B er n o u l i-l \noindent 6 . \quad N umerical s imu l atio ns .. b i hyphen s egm e to the powerPa of ra n-t m e ete d r be ( u a n m it period) S y B m y bo u l s N n u g m t e h r .. ica sa l va me l u es d one-a ta comma as g ive n in Ta b le \ hspaceTable 1∗{\ periodf i l l three-G} Inth eight-e-bracketright is sect io n period-o ,wesha metrical l d l i mensionsco ns id and e material r t hee p rope ig rt enva i es ofbeam lu eprob open square lemfo bracket r a fre e − f r e e b i − seg mented Timoshenko bea m a nd P ro p e rt ies of bea m e l em e n ts \noindent t he o b te n t io n of fo rce d res p o nses fo r a Timoshen kom ic ro − can t i leve r beamwith to ap ie zo e le ct r ic la y e r a bove it . \ h f i l l The Lenhline g t h of fi rst s eg m e n t (m.mathm.math) l.math1 0comma.math254 Len g t h of secon d seg m e n t (m.mathm.math) l.math2 0comma.math140 ThPa ick ra n m ess ete of r openfi rst parenthesis s eg m e n t u n(m.mathm.math it closing parenthesis) t.math .. S1 y m0comma.math bo l N u m e01905 r ica lTh va ick l u n es ess of secon d seg m e n t \noindent compu ta t io ns we re p e rfo rmed in exa ct ra t io na l a r it hmetic u s i ng t he s ymbo l ic compu ta t io n la ngua ge Maple . Expa ns io ns (m.mathm.mathhline ) t.math2 0comma.math549 Len g t h of fi rst s eg m e n t open parenthesis m.math m.math closing parenthesis l.math sub 1 0 comma.math 254 \noindentLen g t h ofwe secon re td seg r un m e n catedW t open idth( withm.mathm.math parenthesis a smal m.math) w.math l m.math numbe0 closingcomma.math rNo parenthesis2545 f te l.math rms sub , usua 2 0 comma.math l l ybetween5and10 1 40 . YouTh n ick g Mod n ess u l of u sfi ( rst GPa s )eg mE e n t71 opencomma.math parenthesis7 De m.math n s it y m.math ( K g.math/m.math closing parenthesis3) ρ t.math2830 sub 1 0 comma.math 0 1 905 \noindentTh ick n ess6 of . secon 1 . \ dquad seg mB e n i t− opensegmented parenthesis f m.math ree − m.mathf ree closing Timoshenko parenthesis beam t.math sub 2 0 comma.math 549 Width open parenthesis m.math m.math closing parenthesis w.math 0 comma.math 2545 \ centerlineYou n g Mod{ u$ l uthree s open−I parenthesis n−eight GPa closing bracketleft parenthesis−bracketright .. E .. 7 1 comma.math{ , } 7 i−t $ \quad wa sco si de re d t h \quad ei enanal ys is f or \quad a f e e − f e e Eu e r − B er n o u l $ iDe−Wel n $ s ha it ve\ yquad opens im u parenthesisb la i ted− ts h e egm K e ig g.math e $n a e na slash ˆ l{ yn m.maths− ist of} a$ f to ree edbeam.Byusngth the - f power ree Timoshe of 3 closing n ko b parenthesis i - se gme nte rho d 2830be\quad a m bsa y u mes i ng} t h e matrixhline b as is five − ge−twoparenright−n e ra ted b y a fu nd a menta l matr ix r es po nse i n t h e st u d y o f t he e ig e nva l \noindentuWe e p haro b ve le sd m im ( $ u la one ted−a$ t h e e ta ig e , n as a na g l ivey s is ninTab of a f ree hyphen le f ree Timoshe n ko b i hyphen se gme nte d be a m b y u s i ng t h e matrix b as is five-g to the power of e-two parenright-nThe co r eres ra p ted o n b d yi ng a fu bo nd u n a da menta r y co l nd matr it io ix ns r es po nse i n t h e st u d y o f t he e ig e nva l u e p ro b le m open parenthesis\noindent Table $1 . three −G eight −e−bracketright period −o $ metrical d i mensions and material p rope rt i es of beam [

The co0 r res p o n d i ng bo u n da r y co nd it io ns 0 \ centerlineEEquation:1I1ψ1(t.mathcomma.math open{Pro parenthesis p e rt0) 89 =ies closing 0comma.math of beameparenthesis κ1 lG .. eme1A E1 sub[w.math n 1 tsI sub1(}t.mathcomma.math 1 psi sub 1 to the0) power− ψ1( oft.mathcomma.math prime open parenthesis0)] = 0t.mathcomma.math comma.math 0 closing parenthesis = 0 comma.math kappa sub 1 G sub 1 A sub 1 open square bracket w.math sub 1 to the power of prime open(89) parenthesis t.math comma.math 0\ [ closing\ r u lparenthesis e {3em}{0.4 minus pt }\ psi] 1 open parenthesis t.math comma.math 0 closing parenthesis closing square bracket = 0 comma.math E I ψ0 (t.mathcomma.math L ) = 0comma.math κ G A [w.math0 (t.mathcomma.math L )−ψ2(t.mathcomma.math E2 sub2 2 2 I sub 2 psi sub 2 to the power of prime open2 parenthesis2 2 t.math2 comma.math L closing parenthesis = 0 comma.math kappa sub 2 G sub 2 A L )] = 0comma.math · N a noMMTA · Vo l .2 · 2013 · 124 − 144 · 138 sub 2 open square bracket w.math sub 2 to the power of prime open parenthesis t.math comma.math L closing parenthesis minus psi 2 open parenthesis t.math\ centerline comma.math{Paramete L closing parenthesis r ( un closing it ) square\quad bracketSymbo = 0 comma.math lNume r ica l va l ues } times N a noMMTA times Vo l period 2 times 20 1 3 times 1 24 hyphen 1 44 times 1 38 \ [ \ r u l e {3em}{0.4 pt }\ ]

\noindent Lengthof $ fi $ rst s egment $( m.math m.math ) l.math { 1 } 0 comma.math 254 $ Len g t h of secon d segme n t $ ( m.math m.math ) l.math { 2 } 0 comma.math 1 40 $ Thick n ess of $ fi $ rst s egment $ ( m.math m.math ) t.math { 1 } 0 comma.math 0 1 905$ Th ick n ess of secon d segme n t $ ( m.math m.math ) t.math { 2 } 0 comma.math 549 $

\ [ Width ( m.math m.math ) w.math 0 comma.math 2545 \ ]

\noindent You n g Mod u l u s ( GPa ) \quad E \quad $7 1 comma.math 7$ Den s it y (K $g.math / m.mathˆ{ 3 } ) \rho 2830 $

\ [ \ r u l e {3em}{0.4 pt }\ ]

Wehave simu la ted the e ig enana l y s is of a f ree − f ree Timoshe n ko b i − se gme nte d be amb y u s i ng t h e matrix b as is $ f i v e −g ˆ{ e−two parenright −n }$ e ra tedbya fu ndamenta l matr ix r es po nse i nthe st udyo f t he e ig e nva l ueprob lem(

\ centerline {The co r res pond i ng bounda r y co nd it io ns }

\ begin { a l i g n ∗} E { 1 } I { 1 }\ psi ˆ{\prime } { 1 } ( t.math comma.math 0 ) = 0 comma.math \kappa { 1 } G { 1 } A { 1 } [ w. math ˆ{\prime } { 1 } ( t.math comma.math 0 ) − \ psi 1 ( t.math comma.math 0 ) ] = 0 comma.math \ tag ∗{$ ( 89 ) $} \end{ a l i g n ∗}

\noindent $ E { 2 } I { 2 }\ psi ˆ{\prime } { 2 } ( t.math comma.math $ L $ ) = 0 comma.math \kappa { 2 } G { 2 } A { 2 } [ w. math ˆ{\prime } { 2 } ( t.math comma.math $ L $ ) − \ psi 2 ( t.math comma.math$ L $ ) ] = 0 comma.math$ $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 38 $ M o d e l i n g t h .. e t i p hyphen s a m .. p l e i n t e r a c t i o n i n a t o .. m i c f o r c e .. m i c r o s c o p y .. ellipsis \noindentca n b e wM ritt o e d ni e nm l a i tr n x g fo tm h \quad e t i p − s a m \quad pleinteractioninato \quad m i c f o r c e \quad m i c r o s c o p y \quad Modelingth etip-sam pleinteractioninato micforce microscopy ellipsis $ eEquation: l l i p s i s open $ parenthesis 90 closing parenthesis ..Row 1 0 0 Row 2 0 minus kappa sub 1 G sub 1 A sub 1 . Row 1 w.math sub 1 open parenthesis ca n b e w ritt e ni nm a tr x fo m t.math comma.math 0 closing parenthesis Row 2 psi 1 open parenthesis t.math comma.math 0 closing parenthesis . plus Row 1 0 E sub 1 I sub 1 Row 2 kappa\ centerline sub 1 G sub{ ca 1 nb A sub ew 1 0 . ritt Row 1 e w.math ninma sub tr1 to x the fom power} of prime open parenthesis t.math comma.math 0 closing parenthesis Row 2 psi sub 1 to the power0 of 0 prime open  parenthesisw.math (t.mathcomma.math t.math comma.math0) 0 closing parenthesis0 E .I = Row  1w.math 0 Row 20 ( 0t.mathcomma.math . comma.math Equation:0)  open 0 parenthesis 9 1 closing \ begin { a l i g n ∗} 1 + 1 1 1 = comma.math parenthesis0 −κ ..RowG A 1 0 0 Rowψ 21( 0t.mathcomma.math minus kappa sub 20) G sub 2 A subκ G 2 .A Row 10 w.math subψ 20 open(t.mathcomma.math parenthesis t.math0) comma.math0 L closing parenthesis \ l e f t (\ begin1 1 {1array }{ cc } 0 & 0 \\ 0 & − \kappa1 1 1{ 1 } G { 1 }1 A { 1 }\end{ array }\ right ) \ l e f t (\ begin { array }{ c} w. math { 1 } Row 2 psi 2 open parenthesis t.math comma.math L closing parenthesis . plus Row 1 0 E sub 2 I sub 2 Row 2 kappa(90) sub 2 G sub 2 A sub 2 0 . Row 1 w.math( t.math sub 2 to the comma.math power of prime 0 open ) parenthesis\\\ psi t.math1 comma.math ( t.math L closing comma.math parenthesis Row 0 2 psi ) \ subend 2{ toarray the}\ powerright of prime) + open\ l e parenthesis f t (\ begin { array }{ cc } 0  0 0   w.math (t.mathcomma.mathL)   0 E I   w.math0 (t.mathcomma.mathL)   0  &Et.math comma.math{ 1 } I L{ closing1 }\\\ parenthesis2 kappa . ={ Row1 } 1 0G Row{ 21+ 0} . period.mathA { 1 }2 &2 0 \end{ array2 }\ right ) \ l e f t (\ begin= { arrayperiod.math}{ c} w. math ˆ{\prime } { 1 } 0 −κ G A ψ2(t.mathcomma.mathL) κ G A 0 ψ0 (t.mathcomma.mathL) 0 (The t.math co m2 p2 a2 ticomma.math bil t yc on d it io 0 s .. ) a-x.math\\\ psi = l.mathˆ{\ subprime 1 a2 re}2 {2 1 } ( t.math2 comma.math 0 ) \end{ array }\ right ) = (91) \ l eLine f t (\ 1begin w.math{ array sub 1}{ openc} parenthesis0 \\ 0 \ t.mathend{ array comma.math}\ right l.math) comma sub 1. math closing\ tag parenthesis∗{$ ( = 90w.math ) sub $}\\\ 2 openl e f t parenthesis(\ begin { t.matharray }{ comma.mathcc } 0 & l.math0 \\ sub0 1 & closing− parenthesis \kappa comma.math{ 2 } G Line{ 2 2} psiA 1 open{ 2 parenthesis}\end{ array t.math}\ right comma.math) \ l e fl.math t (\ begin sub 1{ closingarray }{ parenthesisc} w. math = psi{ 2 open2 } parenthesis( t . math The co m p a ti bil t yc on d it io s a − x.math = l.math1 a re commat.math . comma.math math L l.math ) \\\ sub 1psi closing2 parenthesis ( t.math comma.math comma.math Line 3 psi sub 1 L to the ) power\end{ ofarray prime}\ openright parenthesis) + \ t.mathl e f t (\ comma.mathbegin { array l.math}{ cc sub} 0 &E1 closing{ parenthesis2 } I ={ alpha2 }\\\ sub 1kappa psi sub 2{ to2 the} powerG { of2 prime} A open{ parenthesis2 } & 0 t.math\end{ array comma.math}\ right l.math) \ l sub e f t 1(\ closingbegin parenthesis{ array }{ c comma.math} w. math ˆ{\prime } { 2 } ( t.math comma.math L ) \\\ psi ˆ{\prime } { 2 } ( t.math comma.math L ) \end{ array }\ right ) = Line 4 w.math sub 1 open parenthesis t.math comma.math l.math sub 1 closingw.math parenthesis1(t.mathcomma.mathl.math minus psi 1 open parenthesis1) = t.mathw.math comma.math2(t.mathcomma.mathl.math l.math sub 1)comma.math 1\ l closing e f t (\ parenthesisbegin { array = beta}{ c} 10 open\\ parenthesis0 \end{ array w.math}\ subright 2 open) period.mathparenthesis t.math\ tag comma.math∗{$ ( 9 l.math 1 sub ) 1 $ closing} parenthesis minus psi 2 open ψ1(t.mathcomma.mathl.math1) = ψ2(t.mathcomma.mathl.math1)comma.math parenthesis\end{ a l i g t.mathn ∗} comma.math l.math sub 1 closing parenthesis closing parenthesis comma.math ψ0 (t.mathcomma.mathl.math ) = α ψ0 (t.mathcomma.mathl.math )comma.math where alpha sub 1 = E sub 2 I sub 2 slash E sub 1 I sub 1 a nd beta 1 = kappa sub 21 G sub 2 A sub 2 slash kappa1 sub 1 G sub1 2 1 A sub 1 period I n ma 1 t\ centerline riw.math .. f or1 m(t.mathcomma.mathl.math comma{The w compa e hav ti bil1) − ψ t1( yct.mathcomma.mathl.math ond it io s \quad1)$ = a−βx.math1(w.math2 =(t.mathcomma.mathl.math l.math { 1 }$ a1) − reψ2(} t.mathcomma.mathl.math1))comma.math Equation: open parenthesis 92 closing parenthesis ..Row 1 1 0 0 0 Row 2 0 1 0 0 Row 3 0 0 0 1 Row 4 0 minus 1 1 0 . Row 1 w.math sub 1 open where α = E I /E I a nd β1 = κ G A /κ G A . I n ma t ri f or m , w e hav parenthesis\ [ \ begin { t.matha l i g n e comma.math d } w. math1 l.math2 {2 1 1 sub}1 1( closing t.math parenthesis2 2 2 comma.math1 Row1 1 2 psi 1 open l.math parenthesis{ 1 t.math} ) comma.math = w. math x.math{ sub2 } i.math( closing t.math parenthesis comma.math Rowl . math 3 w.math{ 1 to} the) power comma of prime . math 1 open\\ parenthesis t.math comma.math l.math sub 1 closing parenthesis Row 4 psi sub 1 to the power of prime open \ psi 1 ( t.math comma.math l.math { 1 } ) = \ psi 2 ( t.math comma.math l.math { 1 } parenthesis 1 0 t.math 0 0 comma.math  w.math l.math(t.mathcomma.mathl.math sub 1 closing parenthesis) . = Row 1 11 0 0 0 0 Row 0 2 0 01 0 0 Row 3w.math 0 0 0 alpha(t.mathcomma.mathl.math sub 1 Row 4 0 minus beta)  sub 1 beta ) comma . math \\ 1 1 2 1 sub 10 0 . 1 Row 0 1 0w.math subψ1( 2t.mathcomma.mathx.math open parenthesis t.math comma.math) l.math0 sub 1 1 closing 0 parenthesis 0 Rowψ2( 2t.mathcomma.mathl.math psi 2 open parenthesis t.math) comma.math \ psi ˆ{\prime }{ 1 } ( t.math comma.mathi.math  = l.math { 1 } ) = \alpha { 1 }\ psi ˆ{\prime1 } comma.math{ 2 } l.math 0 sub 0 1 closing 0 1  parenthesis w.math Row01(t.mathcomma.mathl.math 3 w.math to the power of prime)  2 open 0 parenthesis 0 0 t.mathα  comma.math w.math0 l.math2(t.mathcomma.mathl.math sub 1 closing parenthesis)  Row 4 psi ( t.math comma.math  l.math { 1 } ) comma1  . math\\ 1   1  sub 20 to− the1 power 1 0 of prime openψ0 ( parenthesist.mathcomma.mathl.math t.math comma.math) l.math sub0 1 closing−β β parenthesis0 . comma.mathψ0 (t.mathcomma.mathl.math ) w. math { 1 } ( t.math1 comma.math l.math1 { 1 } ) 1−1 \ psi 1 (2 t.math comma.math1 l.math { 1 } o r C.kali sub 1 w sub 1 open parenthesis t.math comma.math l.math sub 1 closing parenthesis = C.kali sub(92) 2 w sub 2 open parenthesis t.math comma.math) = \beta l.math sub1 1 closing ( w. parenthesis math { where2 } ( t.math comma.math l.math { 1 } ) − \ psi 2 ( t.math comma.math l . math { 1 } ) ) comma . math \end{ a l i g n e d }\ ] Equation: openo r C.kali parenthesis1w1(t.mathcomma.mathl.math 93 closing parenthesis .. C.kali1) = C.kali sub 12 comma.mathw2(t.mathcomma.mathl.math i.math = Row 1 11 0) 0where 0 Row 2 0 1 0 0 Row 3 0 0 0 1 Row 4 0 minus 1 1 0 . comma.math C.kali sub 2 comma.math i.math = Row 1 1 0 0 0 Row 2 0 1 0 0 Row 3 0 0 0 alpha sub 1 Row 4 0 minus beta 1 beta 1 0 . comma.math W sub j.math open parenthesis t.math comma.math x.math closing parenthesis = parenlefttp-parenleftex-parenleftex-parenleftex-parenleftbt  psi sub j.math open parenthesis\ centerline from{where w.math sub $ \ j.mathalpha1 to 0{ the1 0 power} 0 = of prime E { open2 } parenthesisI { 2 to} w.math/E sub{ j.math1 } 1 openI 0 parenthesis{ 1 0}$ 0 apsi nd sub prime $ \beta j.math open1 =parenthesis\kappa { 2 } 0 G { 2 } A { 2 } / \kappa0 1 0{ 01 } G { 1 } A { 1 } . $ I n ma t0 r i 1\quad 0f 0 or m , w e hav } w.math ( t.matht.mathcomma.math x.math )) t.math sub comma.math to the power of t.math comma.math to the power of t.math sub comma.math to the power of t.math comma.math x.math to j.math comma.math x.mathx.math C.kali1comma.mathi.math =   comma.math C.kali2comma.mathi.math =   comma.math Wj.math(t.mathcomma.mathx.math) = (ψj.math(w.math (ψ0j.math(t.matht.mathcomma.math x.math )) )period.math the power of x.math from x.math to0 x.math 0 0 sub 1  closing parenthesis to the power of closing parenthesis 0 0 to the 0 powerα1  of closing parenthesis to the power of j.math comma.math closing\ begin parenthesis{ a l i g n ∗} parenrightbt-parenrightex-parenrightex-parenrightex-parenrighttp0 −1 1 0 period.math 0 −β1 β1 0 \ l ethree-I f t (\ begin sub eight-n-two{ array }{ T-wcccc sub} 1 o-a-f & e-b-r 0 l &e-a-h 0 anv & e u-t 0 l-h\\ r-o0 sub & r-hyphen 1 & B-e 0 t-e & c-n 0 a\\ u-l-o0 a-l& n-l i-d 0 b & e-e(93) x-a0 m-p& 1e m-i\\ o-m0 d-e & e-n− t-l-a1 to the& power1 & of 0 l-parenleft\end{ array E v B-a}\ right u-l-T) parenright-comma\ l e f t (\ begin { esarray o bt ai}{ nec} dw. i n math open square{ 1 bracket} ( t.math comma.math l.math { 1 } ) \\\ psi 1three (− Ieight t.math−n−twoT comma.math− wo−a−f e − b − x.mathr l e − a −{h anvi . math e u −}tl −)hr\\− or−w.hyphenB math− ˆet{\−ecprime− n a u}{− l −1 oa} − (ln − li t.math− d comma.math l.math { 1 } t hos e o bta in e d in t h is wo r k with a Tim os hl− eparenleft n ko mode l open parenthesis TBT closing parenthesis a nd b y a p p l y i ng s im i la r methodology fo) b r\\\ me − uex ltpsi is− am pˆ a{\− np Eprimee um le− r hyphenio} −{md1 −} ee −(nt − t.mathl − a comma.mathE v B − au − l.mathl − T parenright{ 1 } − comma) \endes{ array o bt ai}\ neright d i n ) = \ l e f t (\ begin { array }{ cccc } 1 &[B-three t0 hos & e o nine-e 0 bta & in period-r e 0d in\\ t h n is0 o-W wo & r u k l-e 1with l i& ab-b Tim 0 sub os & s-e h e 0 e-a n ko\\ v-m-r mode0 e-s l & ( open TBT 0square ) & a nd 0 b bracket y & a p\ palpha tha l y i t ng .. she{ im r1 i equ la}\\ r enc methodology e0 s o & b ta− fo ned r \..beta f o rt he{ ..1 i-T} mo& sk\beta en ho ..{ 1 } & m0 m o\end du ltlar{ isarray e p c a l-o n E}\ s u erright le o-t r - exB) p−\ rthreeninel ime f t e( nt\ begin al− eperiod{ array− r}{n co}−w.W mathu l − e{l i 2b −} bs−(ee−av t.math−m−re−s[ tha comma.math t he r equ enc l.math e s o { 1 } ) \\\ psi 2 (bo ta five-n t.math ned es f period o rt comma.math he Thei co− T r resmo p sk l.matho en nd ho in g m{ o u1 d lt lar} is epa) c nl\\ s− hao sw. p er e math mo − odt ˆex es{\ pfo rprime r im t ra e nt nsve al}{ ors2five a} l d− is(n pes la t.math . ceme The co nt r a res n comma.math d p rota o t io n a re l.math i l lu st ra{ t ed1 in} F) \\ ig\ psind u re inˆ g{\ m uprime lt is pa} n s{ ha2 p} e m( od es t.mathfo r t ra nsve comma.math rs a l d is p la ceme l.math nt a n d rota{ 1 t io} n a) re\ iend l lu{ starray ra t ed}\ inright F ig u) re comma . math \ tag ∗{$ ( 92 ) $Table} 2 period Natu ral f reqTable uencies 2 . ofNatu a b ral i hyphen f req uencies segmented of a b i - fsegmented ree hyphen f ree -f f ree ree beam . period \endhline{ a l i g n ∗} Freq period Th eo retica l Expe ri m e n ta l Th is work Th is work \ centerline {o r $ C. k a l i { 1 } w { 1 } ( t.math comma.math l.math { 1 } ) = C. k a l i { 2 } w { 2 } Line 1 open parenthesis Hz closingFreq .parenthesis Th eo retica open l Expe parenthesis ri m e n ta l 1Th closing is work parenthesis Th is work open parenthesis 2 closing parenthesis open parenthesis 3 closing parenthesis( t.math Line 2 comma.math hline l.math { 1 } ) $ where } 1 st .. 292 286 minus 29 1 292 period.math 42 29(Hz 1 period.math) (1) (2) 77 (3) \ begin2 n d{ ..a l 1 i g1 n 8∗} 1 1 1 59 minus 1 1 65 1 1 8 1 period.math 28 1 1 67 period.math 89 C.3 k rd a l .. i 1{ 8041 1 759 comma.math minus 1 771 1 804 i.math period.math} = 0\ 1l 1 e 775 f t (\ period.mathbegin { array 94 }{ cccc } 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 01hline st &292 0 & 286 0−291 & 1292\\period.math0 & 42− 2911period.math & 1 &77 02 n\ dend1181{ array 1159}\ right−1165) 1181 comma.mathperiod.math28 C.kali 1167period.math{ 289 comma.math i.math } = 3three-parenleft rd\ l e f1804 t (\ begin 1759 1{ eight-parenright−array1771}{ 1804ccccperiod.math} sub1 : & closing01 0 square 1775& 0period.math bracket & 0 E-R\\94 sub0 e-u & l-f 1 sub & e-e 0 r-r & e-hyphen 0 \\ B-n0 c-e & r-e 0 n-bracketleft & 0 & o\ ualpha l i b e a{ m1 th}\\ e o r0 y &closing− parenthesis \beta 1 & \beta 1 & 0 \end{ array }\ right ) comma . math W { j . math } ( t.math comma.math x.math )parenleft-three = ( \ psi to the{ powerj . math of two-eight} ( ˆ closing{ w. math parenthesis ˆ{\prime : closing} square{ j . math bracket} Refe( } re{ n cew. openmath square{ j bracket. math } ( }{\ psi }ˆ{ t . math ˆ{ t . math comma . math } { comma . math }} {\prime{ j . math } ( t . math ˆ{ t.math comma.math } { comma . math }} x . math ˆ{ x . math ˆ{ x . math } { x . math }}ˆ{ ) ˆ{ ) }} { ) ˆ{ ) }} threethree-parenleft− parenleft to1eight the power− parenright of three-nine]E − closingR l parenthesis− f : closing square bracketn − bracketleft E-R sub e-uo u l-fl i b sub e a e-e m th r-r e e-hyphen o r y ) B-n c-e r-e n-bracketleft sub o ) period.math \ tag ∗{$ ( 93: ) $e−}u e−er−re−hyphenB−nc−er−e u lparenleft i b e a m− ththree e o rtwo y− closingeight) :]parenthesisRefe re n ce [ three − parenleftthree−nine) :]E − R l − f n − \end{ a l i g n ∗} e−u e−er−re−hyphenB−nc−er−e bracketlefthline ou l i b e a m th e o r y ) times N a noMMTA times Vo l period 2 times 20 1 3 times 1 24 hyphen 1 44 times 1 39 \noindent $ three −I { eight −n−two } T−w { o−a−f } e−b−r $ l $ e−a−h $ anv e $ u−t l−h r−o { r−hyphen B−e t−e } c−n $ a $ u−l−o a−l n−l i−d $ b $ e−e x−a m−p $ e $ m−i o−m d−e e−n t−l−a ˆ{ l−p a r e n l e f t }$ E v $ B−a u−l−T parenright −comma$ es o bt ai ne· N d a noMMTA i n [ · Vo l .2 · 2013 · 124 − 144 · 139 t hos e o bta in e d in t h is wo r k with aTim os h e n komode l (TBT) a ndby a pp l y i ng s im i la r methodology fo rmu lt is p anEu le r − $ B−thre e nine−e period−r $ n $ o−W $ u $ l−e $ l i $ b−b { s−e e−a v−m−r e−s } [ $ tha t \quad he r equ enc e s o b ta ned \quad f o r t he \quad $ i−T $ mo sk en ho \quad m o d l a r e c $ l−o $ s er $ o−t$ expriment al o $ f i v e −n$ es . Theco r res pond in gmu lt is pan s hapemod es fo r t ra nsve rs a l d is p la ceme nt and rota t io na re i l lu st ra t ed inFig u re

\ centerline { Table 2 . Natu ral f req uencies of a b i − segmented f ree − f r e e beam . }

\ [ \ r u l e {3em}{0.4 pt }\ ]

\ centerline {Freq . Th eo retica l Expe ri m e n ta l Th is work Th is work }

\ [ \ begin { a l i g n e d } (Hz)(1)(2)(3) \\ \ r u l e {3em}{0.4 pt }\end{ a l i g n e d }\ ]

\noindent 1 s t \quad $ 292 286 − 29 1 292 period.math 42 29 1 period.math 77$ 2 n d \quad $ 1 1 8 1 1 1 59 − 1 1 65 1 1 8 1 period.math 28 1 1 67 period.math 89 $ 3 rd \quad $ 1 804 1 759 − 1 771 1 804 period.math 0 1 1 775 period.math 94$

\ [ \ r u l e {3em}{0.4 pt }\ ]

\noindent $ three −parenleft 1 eight −p a r e n r i g h t { : } ]E−R { e−u } l−f { e−e r−r e−hyphen B−n c−e r−e } n−bracketleft $ ou l i be amth e o r y ) $ p a r e n l e f t −thre e ˆ{ two−e i g h t } ) : ]$ Reference[ $ three −p a r e n l e f t ˆ{ three −nine } ):]E−R { e−u } l−f { e−e r−r e−hyphen B−n c−e r−e } n−bracketleft { o u }$ libeamtheory)

\ [ \ r u l e {3em}{0.4 pt }\ ]

\ hspace ∗{\ f i l l } $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 39 $ J u l i o R period Claeyssen comma Teresa Ts u kazan comma Leticia Tonetto comma Dan i e l a Tol fo \noindentFig 5 periodJ Fu i l rst i t oh ree R matrix. Claeyssen s hape modes , Teresa of a b i Ts hyphen u kazan seg mented , Leticia f ree hyphen Tonetto f ree T , i moshenko Dan i e beam l a period Tol fo Left : t ransve rsal de fl ection J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo Fig 5 . F i rst t h ree matrix s hape modes of a b i - seg mented componentFig 5 . w.math F i rst open t parenthesis h ree matrix x.math s closing hape parenthesis modes of period a b iRig− htseg : rotat mented io n f ree − f ree T i moshenko beam . Left : t ransve rsal de f ree - f ree T i moshenko beam . Left : t ransve rsal de fl ection component w.math(x.math). Rig ht : rotat io n $ flcomponent $ ection psi open component parenthesis $w.math x.math closing ( parenthesis x.math period ) . $ Rig ht : rotat io n 6 period 2 period .. Forced AFM m i cro hyphen canti l eve r beam with p i ezoelectric l ayer componentψ(x.math). \ [A component Timoshen ko m\ psi ic ro hyphen( ca x . ntmath i le ve r ) b ea . m\ a] ct u a ted b y a p ie zo e le ct r ic la y e r la m in a ted o n o ne s id e o f t he b ea m was 6s-four . 2 . t-zero Forced sub AFM bracketright-u m i cro - canti period-d l eve sub r beam i T-e with d-hp i-e i nezoelectric o-bracketleft l ayer ve r .. ng .. quat i o ns .. n l-c u d ed v sc o us .. d a mpi ng a d t-h e mo m en t aA t Timoshenh-t ef r ee ko e nm d ic i-s ro .. - ca su nt b ie le t-c ve .. r ob ..ea a m n a ct u a ted b y a p ie zo e le ct r ic la y e r la m in a ted o n o ne s id e o f t \noindenthea b p ea p l m ie was d6 vo . lta 2 ge . to\quad p iezoForced e le ct r ic AFMm la y e r i period cro ....− Thecanti e q ul a eve t io ns r abeam nd bo with u nd a p r iy co ezoelectric nd it io ns we re l esta ayer b l is h ed fo r a Tim os he n ko sm− fourt ic ro hyphen− zerobracketright ca n t i leve−u rperiod w it h− ad laiT − med in− ahi ted− e pn iezoo − ebracketleft le ct r ic la yve e r r h ngaving quatle ngth i o nsL comma n l − t hc u ic d kness ed v sc h.math o to the power of p.math a n d width\ hspaceus b.math d∗{\ a mpif a-six ing l l a} s-parenrightA d t Timoshen− h e mo i nm koF en ig t m ua t reich open− rot −ef parenthesis rca ee e nt n d ii − les vesu rbeama b e t − c o ct a uan tedbyap ie zo e le ct r ic la ye r lamin a ted onone s id eo f t hebeamwas aFig p p 6l ie period d vo lta Schematic ge to p iezo of beam e le ct with r ic la p y i e ezoelectric r . The e actuq u a ator t io ns a nd bo u nd a r y co nd it io ns we re esta b l is h ed fo r \noindentaB Tim y i os nco he r n po$ ko s ra−fo t i ur ng t he t− bzero o u nd{ a rbracketright y co nd it io n d−u e} to pperiod ie zo e− led ct{ r ici it y T a− te t} h e fred−h e e nd i− ae s a$ co n n ce $ nt o− rabracketleft te d fo rc in g in $ to ve t he r \quad ng \quad quat i o ns \quad n $m lmodel− icc$ ro - comma ca udedvscous n t i levewe ca r w n it d h escribe a la m\ tinquad h a is ted la pdampingad te iezo r as e lea fo ct rcer ic dla da y e m r $t h ped aving−h$ Timoshenko le ngth emomentatL , m t h ic ic ro kness hyphenh.math ca $h np.math t− it$ levea nr efreeendmodeld width period I n matrix $i− fos rm $ u\ laquad t io nsu b e comma$b.matha t−c we $ − have\sixsquad− parenrighto \quad ai n n F ig u re ( FigM dieresis-v6 . Schematic plus C of v-dotaccent beam with p i ezoelectricplus K v actu = F ator comma.math open parenthesis 94 closing parenthesis \noindentwhereB y i ncoapp r po ra tl i ngie t dvo he b o ulta nd a ge r y top co nd it iezo io n d e u e le to pct ie zor e ic le ct la r ic ye it y ra t . t h\ eh fre f i l e l eThe nd a s e a coqua n ce nt t ra io ns a nd bound a r y co nd it io ns we re esta b l is h ed fo r aTim os he n ko tev d = fo Row rc in 1 gw.math in to t he open parenthesis t.math comma.math x.math closing parenthesis Row 2 psi open parenthesis t.math comma.math x.math closing parenthesis\noindentmodel , we . cam comma.math n i d c escribe ro − tca hF is = nla Row te t r i as1 0leve a Row fo rce rw 2 d k.math da it m ped ha sub Timoshenko 1 lamin delta sub m a 1ic opented ro - ca pparenthesis n iezo t i leve e r model x.math le ct . I closing r n matrix ic parenthesisla fo y rm e u rla Vth open aving parenthesis le ngthL t.math , closingt h ic kness parenthesis$io h .n math , we have . ˆ comma.math{ p . math } open$ and parenthesis width 95 closing $b.math parenthesis a−s i x s−parenright $ i nF ig u re ( M = Row 1 M sub 1 1 0 Row 2 0 M sub 22 . comma.math .. K = Row 1M Kv¨ sub+ C 1 1v˙+ KK subv 1= 2 RowF comma.math 2 K sub 2 1 K(94) sub 22 . comma.math .. C = Row 1 c.math\noindentwhere sub 1 0Fig Row 6 2 . 0 c.math Schematic sub 2 . of period.math beam with p i ezoelectric actu ator H e re \ hspacetimes N∗{\ a noMMTAf i l l }By times i nco Vo l rpora period 2 times t i 20 ngt 1 3 times heboundarycond 1 24 hyphen 1 44 times 1 40 it io ndue top ie zo e le ct r ic it yat the fre eendasaconce nt ra tedfo rc in g in to t he  w.math(t.mathcomma.mathx.math)   0  v = comma.math F = comma.math \noindent modelψ(t.mathcomma.mathx.math , we ca n d escribe) t h is la te r as a fok.math rce1δ1 d(x.math damped)V (t.math Timoshenkom) ic ro − ca n t i leve r model . I n matrix fo rmu la t io n , we have (95) \ hspace ∗{\ f i l l }M $ \ddot{v} + $ C $ \dot{v} +$ Kv $=$ F $comma.math ( 94 )$  M 0   K K   c.math 0  M = 11 comma.math K = 11 12 comma.math C = 1 period.math \noindent0 whereM22 K21 K22 0 c.math2 H e re · N a noMMTA · Vo l .2 · 2013 · 124 − 144 · 140 \ begin { a l i g n ∗} v = \ l e f t (\ begin { array }{ c} w.math ( t.math comma.math x.math ) \\\ psi ( t.math comma.math x . math ) \end{ array }\ right ) comma . math F = \ l e f t (\ begin { array }{ c} 0 \\ k . math { 1 }\ delta { 1 } ( x.math ) V ( t.math ) \end{ array }\ right ) comma . math \\ ( 95 ) \end{ a l i g n ∗}

\ centerline {M $ = \ l e f t (\ begin { array }{ cc } M { 1 1 } & 0 \\ 0 & M { 22 }\end{ array }\ right ) comma . math $ \quad K $ = \ l e f t (\ begin { array }{ cc } K { 1 1 } &K { 1 2 }\\ K { 2 1 } &K { 22 }\end{ array }\ right ) comma . math $ \quad C $ = \ l e f t (\ begin { array }{ cc } c . math { 1 } & 0 \\ 0 & c . math { 2 }\end{ array }\ right ) period.math $ }

\noindent H e re $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 40 $ M o d e l i n g t h .. e t i p hyphen s a m .. p l e i n t e r a c t i o n i n a t o .. m i c f o r c e .. m i c r o s c o p y .. ellipsis \noindentM sub 1 1M = o open d eparenthesis l i n g rho t toh the\quad powere of t p.math i p − h.maths a m to\quad the powerpleinteractioninato of p.math b.math plus rho to the power of b.math\quad h.mathm to i the c f power o r c e \quad m i c r o s c o p y \quad Modelingth etip-sam pleinteractioninato micforce microscopy ellipsis of$ b.math e l l i p s b.math i s $ closing parenthesis comma.math M sub 22 = open parenthesis rho to the power of p.math I to the power of p.math plus rho to the power of b.math I to the power of b.math closing parenthesis comma.math Equation: open parenthesis 96 closing parenthesis .. K sub 2 1 = minus 4 open \ begin { a l i g n ∗} parenthesis kappa to the power of prime p.math from k.math sub 1 = e.math sub 1 3 z.math sub m.math to the power of p.math b.math comma.math to p.math p.math b.math b.math p.math p.math b.math b.math M { 1 1 } = ( \rho ˆ{ p . math } h . math ˆ{ p . math } b . math + \rho ˆ{ b . math } h . math ˆ{ b . math } b . math M11 = (ρ h.math b.math + ρ h.math b.math)comma.math M22 = (ρ I + ρ I )comma.math K sub 1 1 = minus 4 open parenthesis kappa to the power of prime p.math kappa to the power of prime b.math c.math sub 55 to the power of b.math p.math b.math 0p.math 10(1+ν ) ) comma . math M { 22 } =p.math ( \rho ˆ{ p . math } I ˆ{ p . math } + \rho ˆ{ b . math } I ˆ{ b . math } ) comma . math \\ K { 2 0b.math 10(1+ν ) κ = ∂ κ = p.math b.mathb.math) comma.math from open parenthesis0p.mathk.math t.math1=e.math closing13z.math parenthesism.mathb.mathcomma.math comma.math to0 c.mathb.math sub 55b.math to the(t.math power)comma.math of p.math h.math to the power of p.math b.math plus from b.math ∂ 12+11ν = p.math b.math b.math 12+11ν period.math 0b.math h.math ∂ x.math 0b.math b.math b.math K21 = −4(κ 0p.math κ c.math 0b.math b.math h.math b.math) K b.math ∂2 comma.math = −(c.math Ip.math + c.math I p.math κ ) ∂ c.math255b.math p.math + κ c.math h.math b.math)comma.math 1 } = − 4K11 (=−4(κ\kappa ˆ{\prime p . math }ˆ{ k . math 55{ 1 } =p.math e . math { 1u.math 3 } (t.mathz . math)= ˆ{ p . mathκ } {c.mathm. math } 22 h.math b.math) K12 11 11 0p.math p.math line−two + 0p.math p.math 55 u.math open parenthesis t.math closing parenthesis = to c.math sub 55 to the power of p.math h.mathp.math to the power of p.math b.math plus sub V to the55 ∂x.math comma.math ∂x.math2 comma.math 4(κ c.math55 h.math b.math+ ∂x.math 4(κ c.math55 h.math b.math c.math55 h.math b.math+ p.math V b.math comma.math } { K { 1 1 } = − 4 ( \kappa ˆ{\prime p . math }}\c.mathkappa ˆh.math{\primep.mathb.math+b . math } c . math ˆ{ b . math } { 55 }ˆ{ ( power of kappa to the power of prime b.math c.math sub 55 to the power of b.math h.math to the power of b.math b.math55 closing parenthesis partialdiff (96) dividedt.math by partialdiff ) comma.math x.math sub} comma.math{ c . math Kˆ{ subp . math22 from} kappa{ 55 to} theh .power math of ˆ{ primep . math b.math} =b hline . math sub 1 + 2 plus ˆ{ u.math 1 1 nu to the ( power t.math of 1 0 open ) =parenthesis} { c . math 1 plus ˆ nu{ p to . math the power} { of55 b.math} h closing . math parenthesis ˆ{ p . math to} h.mathb . math to the power + } { ofV b.math}ˆ{\ b.mathkappa closingˆ{\prime parenthesisb partialdiff . math } toc the . math power ˆ{ ofb . math } { 55 }}} c.math1 a n d c.math2 a re vis c o u d a m p in gc o ns t an z.math − scomma−m.math is t h e dist anc ebetwe e n t e 2h .divided math ˆ by{ partialdiffb . math } x.mathb . math to the power ) \ off r 2a c sub{\ comma.mathpartial }{\ K subpartial 1 2 sub comma.mathx . math } ={ tocomma the power . math of =} minusK { open22 parenthesis}ˆ{\kappa c.mathˆ{\ subprime 1 b .m math i dd l} e l ne= of\ r h u epie l e {3em z oe}{ l e0.4 tr i ptc l} a yeˆ{ 1 0 ( 1 + \nu ˆ{ b . math } ) } { 1 2 + 1 1 \nu }} { h . math ˆ{ b . math } 1 to thet he power ne u tr of a p.math a xi so I sub f be p.math a m a n plusd − c.mathV (t.math sub) is 1 t1 he to a the p p power li e dv of o b.matht − l ag I et to o the pie z power oe l e of ct b.mathr i − c froml a ykappa to the power of prime p.math = 1b . 0 math open parenthesis ) \ f r a 1 c {\ pluspartial nu to theˆ power{ 2 }}{\ of p.mathpartial closing parenthesisx . math ˆ divided{ 2 }} by{ 1comma 2 plus 1. math 1 nu to} theK power{ 1 of p.math 2 }} { subcomma period.math . math to} 4= open ˆ{ = } T − three − fourperiod − he−ab − nd−eam g eo me t ric a l an d m a t er i a l po p er e s a red e sc r be d i th e T ab e s −parenthesis( c kappa . math to ˆ the{ p power . math of} prime{ 1 p.math 1 } c.mathI { subp . 55 math to the} power+ of c . p.math math ˆ h.math{ b . math to the} power{ 1 of p.math 1 } I b.math ˆ{ b .plus math kappa}ˆ{\ to thekappa powerˆ{\ of prime p . math } = \ f r a c {Table1 3 0. Geometrical ( 1 d + i mensions\nu andˆ{ materialp . math p rope} rt i) es of}{ beam1 e l ements 2 + . 1 1 \nu ˆ{ p . math }} { period .math }} { 4 prime b.math closing parenthesis sub line-twoP sub ro p partialdiff e rt ies of bea x.math m e l emto the e n ts power of partialdiff c.math 2 55 b.math plus to the power of h.math sub 4 open parenthesis( \kappa kappaˆ{\ toprime the powerp of . math prime p.math} c . math c.math ˆ{ subp .55 math to the} power{ 55 of} p.mathh . math h.math ˆ{ top . themath power} ofb . p.math math b.math + }\ tokappa the powerˆ{\ of b.mathprime subb . math }{ ) } { l i n e −two ˆ{\ partial } {\ partial b.mathx . math closing}} c parenthesis . math { sub2 partialdiff} 55{ b . to math the power} { + of hline}ˆ{ h sub . math x.math}ˆ to{ theb . math power of{ comma.mathb . math )to ˆ the{\ powerr u l e of{3em partialdiff}{0.4 pt plus}} kappa{\ topartial the power}}ˆ{\ partial } { x . math ˆ{ comma . math }}} { 4 of( prime\kappa b.mathˆ{\ c.mathprime sub 55p to . math the power} ofc .b.math math ˆ h.math{ p . math to the} power{ 55 of} b.mathh . math b.math ˆ{ closingp . math parenthesis} b . math comma.math} + \kappa ˆ{\prime b . math } c . mathc.math ˆ sub{ b 1 . math a n d c.math} { 55 sub} Pa 2 ah ra re . mmath vis ete c r o ( ˆ u{ n ..b it d. )math a m S p y} in m gcbob.math lo N ns u tm an e r z.math-s ica ) l va comma.math l u sub es comma-m.math\ tag ∗{ is$ t ( h e dist 96 anc ) ebetwe $} e n t e m i dd l e l ne .. of h\end epie{ za oel i g l n e∗} tr i c l a ye t he ne u tr a .. a xi so f be a m a n d-V open parenthesis t.math closing parenthesis is t he a p p li e dv o t-l ag et o pie z oe l e ct r i-c .. l a y \ hspaceT-three-four∗{\ f i period-h l l } $ c sub . math e-a b-n{ sub1 } d-e$ a and m g eo me $ t c.math ric a l an{ d m2 a} t$ er ia a re l po v p i s er ce s o a ured\quad e sc r bed d amp i th .. ein T abgc e o s ns t an $ z.math−s { comma−m. math }$ Len g t h (µm.math) L 1 50 isTable t h 3e period dist Geometrical anc ebetwe d i e mensions n t emi and material dd l e p rope l ne rt\ iquad es of beamofhepie e l ements zoe period l e tr i c l aye P ro p e rt ies of bea m e l em e n ts W idth(µm.math) b.math 30 \ centerlinehline { t he ne u t r a \quad axiso fbeaman $d−V ( t.math )$ istheappliedvo $t−l $ agetopie zoe l ect r $i−c $ \quad l a y } Pa ra m ete r open parenthesis u n it closing parenthesisb.math .. S y m bo l N u m e r ica l vab.math l u es Th i ck n ess (µm.math) h.math 10 Yo u n g Mod u l u s ( GPa ) 1c.math1 73 hline \ centerlineLen g t h open{ $ parenthesis T−three − mufo ur m.math period closing−h parenthesis{ e−a ..} L ..b− 1n 50 { d−e a m }$ geomet ric a l andmat er i a l poper esarede sc rbedi th \quad e T ab e s } Wid th open parenthesis mu m.mathDensity closing(Kg.math/m.math parenthesis b.math3) 30ρb.math 2200 \ centerlineTh i ck n ess{ Table open parenthesis 3 . Geometrical mu m.math dclosing i mensions parenthesis and h.math material to the power p rope of b.math rt i 1 es 0 of beam e l ements . } Poisso n coefficient νb.math 0period.math17 Yo u n g Mod u l u s open parenthesis GPa closing parenthesis 1 c.math sub 1 to the power of b.math 73 \ centerlineDensity open{Pro parenthesis p e rtK g.math ies of slash beame m.math tol eme the power n ts of 3} closing parenthesis rho to the power of b.math 2200 Poisso n coefficient .. nu to the power of b.math 0 period.math 1 7 \ [ \ r u l e {3em}{0.4 pt }\ ] hline Table 4 . Geometrical d i mensions and material p rope rt i es of p iezoel ect r i c e l ement . Table 4 period Geometrical d i mensionsP ro and p e material rt ies of p p iezoe rope lectric rt i es e l of em p e iezoel n t ect r i c e l ement period P ro p e rt ies of p iezoe lectric e l em e n t \ centerlinehline {Paramete r ( un it ) \quad Symbo lNume r ica l va l ues } Pa ra m ete r open parenthesis u n it closing parenthesis .. S y m bo l N u m e r ica l va l u es Pa ra m ete r ( u n it ) S y m bo l N u m e r ica l va l u es \ [ hline\ r u l e {3em}{0.4 pt }\ ] Len g t h open parenthesis mu m.math closing parenthesis l.math sub p.math 1 50 Wid th open parenthesis mu m.math closing parenthesis b.math 30 \ centerline {Len g t h $ ( \mu m. math ) $ \quad L \quad 1 50 } Th i ck n ess open parenthesis mu m.mathLen g t closing h (µm.math parenthesis) l.math h.mathp.math to the150 power of p.math 1 0 Yo u n g Mod u l u s open parenthesis GPa closing parenthesis 1 c.math sub 1 to the power of p.math 7 1 \ [Density Wid open th parenthesis ( \mu K g.mathm.math slash m.math )W idth b.math(µm.math to the power) 30 b.math of\ ] 3 closing30 parenthesis rho to the power of p.math 7700 Poisso n coefficient .. nu to the power of p.math 0 period.math 3 1 p.math p.math hline Th i ck n ess (µm.math) h.math 10 Yo u n g Mod u l u s ( GPa ) 1c.math 1 71 \ beginTable{ 5c e period n t e r } Comparative natu ral f req u encies period Thihline ckness $( \mu m.math ) h.math ˆ{ b . math } 1 0 $ 3 p.math YoungModulusF req period .. Refe re n ce (GPa P rese nDensity $) t work(Kg.math/m.math 1 { c . math }ˆ{) bρ . math }7700{ 1 } 73 $ \end{ c e n t e r } Line 1 open parenthesis KHz closingPoisso parenthesis n coefficient open parenthesisνp.math 0period.math 1 closing parenthesis31 Line 2 hline 1 st .. 547 .. 544 \ [2 Density n d .. 331 4 .. ( 3298 K g.math / m.mathˆ{ 3 } ) \rho ˆ{ b . math } 2200 \ ] 3 rd .. 8833 .. 8797 4 th .. 1 5951 .. 1 62 1 0 Table 5 . Comparative natu ral f req u encies . \ centerlinehline { Poisso n coefficient \quad $ \nu ˆ{ b . math } 0 period.math 1 7$ } four-parenleft 1 zero-parenright sub : closing square bracket Refe re n ce open square bracket \ [ \ r u l e {3em}{0.4 pt }\ ] hline F req . Refe re n ce P rese n t work times N a noMMTA times Vo l period 2 times 20 1 3 times 1 24 hyphen 1 44 times 1 4 1 (KHz) (1) \ centerline { Table 4 . Geometrical d i mensions and material p rope rt i es of p iezoel ect r i c e l ement . }

\ centerline1 st 547{ 544P ro 2 n dp e 331 rt 4 ies 3298 of 3 p rd iezoe 8833 lectric 8797 4 th e l 1 eme 5951 n 1 62t 1} 0

\ [ \ r u l e {3em}{0.4 pt }\ ]

\ centerline {Paramete r ( un it ) \quad Symbo lNume r ica l va l ues }

\ [ \ r u l e {3em}{0.4 pt }\ ]

\ centerline {Len g t h $ ( \mu m.math ) l.math { p . math } 1 50 $ }

\ [ Wid th ( \mu m.math ) b.math 30 \ ]

\ begin { c e n t e r } Thi ckness $( \mu m.math ) h.math ˆ{ p . math } 1 0 $ YoungModulus (GPa $) 1 { c . math }ˆ{ p . math } { 1 } 7 1 $ \end{ c e n t e r }

\ [ Density ( K g.math / m.mathˆ{ 3 } ) \rho ˆ{ p . math } 7700 \ ]

\ centerline { Poisso n coefficient \quad $ \nu ˆ{ p . math } 0 period.math 3 1$ }

\ [ \ r u l e {3em}{0.4 pt }\ ]

\ centerline { Table 5 . Comparative natu ral f req u encies . }

\ [ \ r u l e {3em}{0.4 pt }\ ]

\ centerline {F req . \quad Refe re n ce P rese n t work }

\ [ \ begin { a l i g n e d } ( KHz ) ( 1 ) \\ \ r u l e {3em}{0.4 pt }\end{ a l i g n e d }\ ]

\noindent 1 s t \quad 547 \quad 544 2 n d \quad 331 4 \quad 3298 3 rd \quad 8833 \quad 8797 4 th \quad 1 5951 \quad 1 62 1 0

\ [ \ r u l e {3em}{0.4 pt }\ ]

\ centerline { $ four−parenleft 1 zero−p a r e n r i g h t { : } ] $ Refe re nce [ }

\ [ \ r u l e {3em}{0.4 pt }\ ]

\ hspace ∗{\ f i l l } $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 4 1 $ four − parenleft1zero − parenright:] Refe re n ce [

· N a noMMTA · Vo l .2 · 2013 · 124 − 144 · 141 J u l i o R period Claeyssen comma Teresa Ts u kazan comma Leticia Tonetto comma Dan i e l a Tol fo \ centerlineFig 7 period{ MassJ u nol i rmal o R i zed . Claeyssenmatrix s hape , modes Teresa of a Ts m i ucro kazan hyphen , canti Leticia l ever beam Tonetto with a , p Dan iezoel i ectric e l l a aye Tol r period fo } Left : t ransve rsal de fl J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo ect ion com ponent w.math open parenthesis x.math closing parenthesis period Rig ht : Fig 7 . Mass no rmal i zed matrix s hape modes of a m i cro - canti l ever beam with a p iezoel ectric l aye r . Left : t ransve rsal de fl ect ion \noindentrotat ion componentFig 7 . Mass psi open no parenthesis rmal i zed x.math matrix closing s parenthesis hape modes period of F a i m rst i mode cro :− socanti l id b l l ue ever l i n e beam comma with Second a p mode iezoel : dash ectric hyphen l aye r . Left : t ransve rsal de com ponent w.math(x.math). Rig ht : rotat ion component ψ(x.math). F i rst mode : so l id b l ue l i n e , Second mode : dash - dotted dotted$ fl $ red lect i ne ion comma component Th i rd mode $w.math : dotted b l ack ( l i ne x.math comma Fou ) rt h . mode $ Rig : ht : red l i ne , Th i rd mode : dotted b l ack l i ne , Fou rt h mode : dashed g ray l i ne . rotatdashed ion g ray component l i ne period $ \ psi ( x.math ) . $ F i rst mode : so l id b l ue l i ne , Secondmode : dash − dotted red l i ne , Th i rd mode : dotted b l ack l i ne , Fou rt h mode : dashedLine 1 hline g ray Line l 2 i F ne open . parenthesis t.math comma.math x.math closing parenthesis = c.math o.math l.math open square bracket 0 q.math open parenthesis t.math comma.math x.math closing parenthesis closing square bracket Line 3 hline Line 4 hline Line 5 hline F (t.mathcomma.mathx.math) = c.matho.mathl.math[0 q.math(t.mathcomma.mathx.math)] \ [ \Figbegin 8 period{ a l i gAbove n e d }\ : t ransversalr u l e {3em de}{ fl0.4 ection pt }\\ com ponent w.math open parenthesis t.math comma.math x.math closing parenthesis d ue to a concentrated momentF ( q.math t.math open parenthesis comma.math t.math comma.math x.math x.math ) = closing c.math parenthesis o.math = k.math l.math sub 1 delta [ sub 1 0 open q.math parenthesis ( x.math t.math closing parenthesis comma.math Vx . open math parenthesis ) ] t.math\\ closing parenthesis at t he f ree end of t he m ic ro hyphen canti l ever \andr u l p e { ro3em fi l}{ es0.4 fo r pt several}\\ t i mes period Below : rotat i on component psi open parenthesis t.math comma.math x.math closing parenthesis and pro fi l es fo\ r r u several l e {3em t}{ i mes0.4 periodpt }\\ Fig\timesr u 8 l e . N{3em a noMMTA}{Above0.4 :pt t times}\ ransversalend Vo{ dela period l ifl g nection ed 2}\ times com] ponent 20 1 3 timesw.math 1( 24t.mathcomma.mathx.math hyphen 1 44 times 1 42 ) d ue to a concentrated moment q.math(t.mathcomma.mathx.math) = k.math1δ1(x.math) V (t.math) at t he f ree end of t he m ic ro - canti l ever and p ro fi l es fo r several t i mes . Below : rotat i on component ψ(t.mathcomma.mathx.math) and pro fi l es fo r several t i mes . · \noindentN a noMMTAFig· Vo 8 l ..2 Above· 2013 · 124 : t− ransversal144 · 142 de $ fl $ ection com ponent $ w.math ( t.math comma.math x.math ) $ d ue to a concentrated moment $ q.math ( t.math comma.math x.math ) = k.math { 1 }\ delta { 1 } ( x.math )$ V $( t.math )$ atthe f reeendof themic ro − c a n t i l ever

\noindent and p ro $ fi $ l es fo r several t i mes . Below : rotat i on component $ \ psi ( t.math comma.math x.math ) $ and pro $ fi $ l es fo r several t i mes . $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 42 $ M o d e l i n g t h .. e t i p hyphen s a m .. p l e i n t e r a c t i o n i n a t o .. m i c f o r c e .. m i c r o s c o p y .. ellipsis \noindentLine 1 hlineM Lineo d 2 e F l open i n parenthesis g t h \quad t.mathe comma.math t i p − s x.math a m \quad closingpleinteractioninato parenthesis = c.math o.math l.math open square\ bracketquad m 0 q.math i c f open o r c e \quad m i c r o s c o p y \quad Modelingth etip-sam pleinteractioninato micforce microscopy ellipsis parenthesis$ e l l i p s i t.math s $ comma.math x.math closing parenthesis closing square bracket Line 3 hline Line 4 hline Line 5 hline Fig 9 period Above : .. transversa ld e fle c-t i n o-c m pone w.math-n parenleft-t t.math comma.math x.math closing parenthesis d ue to a co ncentrate dp\ [ \ ubegin l-s emo{ a me l i g q.math-n n e d }\ r t-parenleft u l e {3em}{ t.math0.4 pt comma.math}\\ x.math closing parenthesis = 0 period.math 0 1 V open parenthesis t.math closing parenthesis F (t.mathcomma.mathx.math) = c.matho.mathl.math[0 q.math(t.mathcomma.mathx.math)] openF parenthesis ( t.math H open parenthesis comma.math x.math x.math minus 6 L slash) = 8 closing c.math parenthesis o.math minus H l.math open parenthesis [ 0x.math q.math minus 7 L ( slash t.math 8 closing parenthesis comma.math closingx . math parenthesis ) ] at\\ t h e \fr ree u l e end{3em o ft}{ h0.4 e m pt icr}\\ oca n tlev era nd p r ofi es forse v e r al t-i sub m es period Be lo w : r o ato nco m p o-psi n-parenleft to the power of e-t.math sub\ comma.mathr u l e {3em}{ n-x.math0.4 pt }\\ parenright-t and pro fi le sf o rsever a l t-i m e \T-fiver u l e { h3em e-f r-fi}{0.4 sub pt r-c}\ s-oend t m{ p-fa l i go-a n e r-u d }\ s] o-o sub n-b sub t w-a i t-n h-e h-t-d n-o sub a-s e-t u r-o a t-l h r-e to the power of e-f q-o r-u m e n-u l-c-a e-e-t s-dFig m-a 9 . r-o-eAbove sub : d e-s transversa h-l o w ld ni e nfle Tc ab− t lei n o − c m pone w.math − nparenleft − tt.mathcomma.mathx.math) d ue toi-four a co ncentrate n-seven-zero dp u l − period-bracketlefts emo me q.math aTe− nt h− aparenleftt.mathcomma.mathx.math m-s sub i-s r-n-o o r-hyphen c-m aa) i-t-l = 0 i-z-lperiod.math e-e d-v sub01 V period(t.math erts)( i-hH p o-e sub b e-m-s to the power of a-r\ hspace(x.math r-e d-x∗{\− m-t6fL h i l o-a l/}8)Fig e-t-d− H 9 t-s-h(x.math . Above n-i-e− i n7 :L/ F-c\quad l-i8 ) u-s ) attransversa sub t h e i-r o-e n o ld e fle $ c−t $ i n $ o−c $ mpone $ w.math−n p a r e n l e f t −t t.math comma.math x.math ) $ d ue to a co ncentratee−t.math dp u $ l−s $ emo me $ q.math−n t−parenleft t.math f ree end o ft h e m icr oca n tlev erarota nd p r to ofi r es y forse in e v rt e r ia al at nd− i sm hes ea . Be r i lo n w b : ea r o m ato mode nco m l p io ng− ipsin n fl u− eparenleft n ces t hcomma.math e ro ta t ion n− comx.mathparenright po n e nt of t h− et foand rce pro dfi resle po sf onses rsever d au l et to− ai sm pat e ia l comma.math x.math ) = 0 period.math 0 1$ V $( t.math ) ($e−fq H−or− $(u x.math − 6 $ L Tco−five nce nth rae− tedfr − afi n dr− scs pat−ot iam lp p− ufo ls− ear mo− meu s nto− excitaon−btw t− ioait− nsn h t− hateh a− ret− mdn od−o ua− lase ted−tur with−o a a ht− al rmh r o− ne ic in p u t period $m /7 e periodn 8− ul ..− ) Conclusionsc −−ae −$e − Hts − $dm (− ar x .− matho − ede−−sh−l o7$ w ni n L/8))atthe T ab le i − fourn − seven − zeroperiod − bracketleft aTe hTh a m is− pasi p−s er r− an dd− o resseso r − ahyphenc matrix fo− rmm uaa lai t− iot − nli fo− rz m− icle ro− hyphened − v. erts ca ni t− i leveh p ro models− ebe−m in−s AFMa−r r − ted h− a txm a re− st uh bj e ct to q u ite ge n e ra l \ centerlineot− ipae hyphen− t − dt{ saf− ms ree− phn le end i− ntei − o rae fti ctnF io h ns− emicrcl comma− iu − s ocas ui− rfaro− n ceen tlev effectso rota era ato n r dy ndp inexte e rt r r ia n ofi and l excita ses h ea forse t r io i n ns b v periodea emmode r .. al Altho l i ng$ u ti− n gi hfl we{ havem }$ co ns es id . e Bere d low: a fin ite le r o ato ncomp ngth$u o− e u np nsces i ifo t h rm n e− rop ata r te ion l n e fcom t ˆ po{ ne− et nt . math of t h e} fo{ rcecomma d res po . math nses d} u e ton− ax.math s pat ia l co parenright nce nt ra ted− at n d $ s pat andpro ia l p u ls e $ fi $ le sf o rsever a l $mo tTimos− mei $ nt he excita m n eko t} b io ea ns m t hatm od a re e l m comma od u la t ted h e with ma ta r h ix a rm fo rm o n u ic la in t p io u n t .ca n b e u sed wit h o t he r bea m mode ls period The u se of p ie zo e le ct r ic ma te r ia ls 7 . Conclusions \noindentasTh both is pa a n p$ a e ct rT− a u ddf a i v to resses e r $a n admatrix h a se $ nsofo e− rmf r has u la r m− t iofot i n iva fo{ ted rr m− to icc roi nco - s ca− ro n po t ira leve t te} tr$ hemodels m matrix $ in pAFM t− reatmf t h o ea− nt ta a o re f msr− u u bj lt $ e i ct hyphen s to q $ u s o− po a n{ ben a− msb period{ t .. Iw− na t h is i t−itenwo ge}} r n k e commah ra−e l t ip it h -− is sat p− m rod p posed le n i− nteo t he ra{ ctextea io−s ns , ive s e u− u rfat se ce of u effects fu} nd ar− meno dnta exte $ la r m n a a $t l excitart− ixl res $ t iopo h ns nses . $ s r u− Altho che ˆa{ u s g te− hhef we d haveist q− r ioco b ns u ted r−u matr}$ ix m im e p u $ ls n e− resu po l nse−c−a e−ideof− ett re he d a m s− fin icd ite ro le hyphen m− ngtha u ca n r− ifo nto− rm ie le Timosve{ rd fo he rp ne− rekosd b eaict h m− i ngl m} od fo$ erced l owninTab , t res h e p ma o nsest r ix a fo n rm d le co u la nce t io nt n ra cated n b ematrix u sed wit res h p o o t nses he r fo r dete rm i n i ng modes a nd bea$fre i m− qfomode u e ur nc ls ies . nThe of−seven t u he se m of− ic pzeroie ro zo hyphen e le period ct ca r ic nt ma− ibracketleft le te ve r ia rs ls period as both .. $ Thea n aTe a e ct ig hu e a na to a r l $ma y n s− diss a i senvo{ nsoi lved− rs has t} h m er ot so−n iva l− uo ted t $ io to n oi i nco n $c r lose r−hyphen d fo rm of a c se−m co $ n d aa hyphen $ i−t−l oi− rdpoz− e ral r te da t e mhe−e pmatrix ed d− tv reatm{ e. nt} o$ f m e u r tlt s i - s $ p a i− nh be $ a ms p . $ I o n− te h is{ wob r k e,− itm is− ps ro ˆ{ poseda−r t he}} exter− nse ive u d se−x of fu ndm−t $ h $ o−a e−t−d t−s−h n−aid me− iffee nta $ re l n m i t a ia t$n lr eq{ ix resF u− ac po t} io nses nl with s− ui ch m a ua s− ts rhe ix d{ coefficie isti− rr i b u nts tedo− periode matr n ix ..} im The$ p ucase o ls e o res f a po s unse p po of t rte he d m m ic ic ro ro - ca hyphen nt i le bve ea r fo m r with s u r fa ce effe cts ca n le a d torotap a re d ictto i rng y fo rcedin e res rt p o ia nses and a n d co shea nce nt ra r ted i matrix nbeammode res p o nses fo l r dete i ng rm i i n n i ng $ modes fl $ a nd uen fre q u ces e nc ies the of t ro ta t io ncompone nt of the fo rce d res po nses due to a s pat ia l coheseco m nce ic nd ro nt s - pectruca ra nt i ted le m ve a a rsb ove.nd a The sc r pat e it ig ica e ia n l afre l y q pu s u is e i ncynvo ls lvedperiodemoment t h .. e so Fo l r u t t excita he io n m i nic c ro t lose hyphen io d fons rm ca t of n hat a t se i leveco a n remod rd case - o rd comma e ru da la m it ted was o with bse rved a h t h armo e s ize d n e icpe in pu t . ndep ed nce d in iffe t here n t ia l eq u a t io n with m a t r ix coefficie nts . The case o f a s u p po rte d m ic ro - b ea m with s u r fa ce \ centerlineeffenatu cts ra ca l n f le re{ a7 q d u to. e a\ ncyquad seco o ndfConclusions Timoshe s pectru n m ko a b c ove la} ss a cica r it l modelica l fre a q nd u e Tim ncy .os h Foe n r ko t he model m ic ro a n- ca d tn hat t i leve s u r rfa case ce , effects it was o a re s ig n ifica nt o n l y in bsena rved nosca t h le e period s ize d e S pe im nde u la nce t io in ns t he we natu re p ra e rfo l f re rm q ed u e b ncy y u o s f iTimoshe ng t he n Ga ko le c lar k ss in ica method l model with a nd m Tim ic roos hhyphen e n ko camodel nt i leve r e ig e n fu nct io ns period ThThea n iss d h t paa hat pe p s u e rfa r ce a effects dd resses a re s ig n a ifica matrix nt o n l fo y in rmu na nosca la le t . Sio im n u la fo t io rm ns we ic re ro p e− rfoca rm ed n b t y iu s leve i ng t he r Ga models le inAFMt ha t a re s u bj e ct to qu ite ge n e ra l trmatrix k ip in− method modessamp with a nd m le icf reqi ro nte- u ca e nt n ra c i leveie ct s for io e r ig a ns e b n i fuhyphen , nct s u io sens rfa gme. The ce nted s effects h a fre pe e matrixhyphen a nd modes f ree exte Timoshenkoa nd fr req n u a e b ln ea c excita ie m s we fo r re a t deteb ioi - se rm ns i ne . \ dquad periodAltho .. Fo rce u d g res h p we o have co ns id e re d a fin ite le ngth u n ifo rm nsTimosgme es of nted a he fre nkob e - f ree Timoshenko eammod b ea e m l we , re t dete h ematrm i ne d . r ix Fo fo rce rmud res p ola ns t es ofio a n p ieca zoe nb le ct e r ic u m sed ic ro wit - ca n ho t i t he r beammode ls . Theu se of p ie zo e le ct r ic mate r ia ls aslevep ieboth r bzoe ea le ma ct where n r a ic ctcomm ic u p ro u ahyphen ted to when r ca a s un nd bj t i ect leve a to se r co b nso nce ea m nt r where ra hasmot ted coma nd p u iva ls ted e ha when ted rm o sto n u ic bj i exc ect nco ita to t r co io po ncens a ra nt t t ra hete ted fre t a e hend e nd matrixp . u ls e ha t rm reatm o n ic e exc nt ita o t fmuio ns a lt i − s p a n be a ms . \quad I n t h i s two t he r k , it is p ro posed t he exte ns ive u se of fu· N nd a noMMTA ame nta· Vo lma l .2 · 2013 t r· 124 ix− res144 · po143 nses s u ch a s t he d ist r i bu ted matr ix impu ls e res po nse o ffre t ehe e nd m period i c ro − ca nt i le ve r fo r p re d ict i ng fo rced res p o nses a n d co nce nt ra ted matrix res p o nses fo r dete rm i n i ng modes a nd fretimes qu N a e noMMTA nc ies times of t Vo hemic l period 2 ro times− ca 20 1nt 3 times i le 1 ve 24 hyphen rs . \ 1quad 44 timesThee 1 43 ig ena l ys is i nvo lved the so l ut io n i nc lose d formof a se cond − o rd e r damped d iffe re n t ia l eq ua t io n withma t r ix coefficie nts . \quad The case o f a s up po rte dmic ro − b eamwith s u r fa ce effe cts ca n le a d to a seco nd s pectruma b ove a c r it ica l fre q u e ncy . \quad Fo r t hemic ro − ca n t i leve r case , it was o bse rved t h e s ize d e pe nde nce in t he natu ra l f re q u e ncy o f Timoshe n ko c la ss ica l model a ndTim os h e n ko model a n d t hat s u rfa ce effects a re s ig n ifica nt o n l y in na nosca le . S im u la t io ns we re p e rfo rm ed b y u s i ng t he Ga le r k in method withm ic ro − ca nt i leve r e ig e n fu nct io ns . The s ha pe matrix modes a nd f req u e n c ie s fo r a b i − se gme nted fre e − f ree Timoshenko b ea mwe re dete rm i ne d . \quad Fo rce d res p o ns es of a p ie zoe le ct r icmic ro − ca n t i leve r b eamwhere comp u ted when s u bj ect to co nce nt ra ted a nd p u ls e ha rm o n ic exc ita t io ns a t t he f r e e e nd .

\ hspace ∗{\ f i l l } $ \cdot $ N a noMMTA $ \cdot $ Vo l $ . 2 \cdot 20 1 3 \cdot 1 24 − 1 44 \cdot 1 43 $ J u l i o R period Claeyssen comma Teresa Ts u kazan comma Leticia Tonetto comma Dan i e l a Tol fo \ centerlineAcknowledgments{J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo } J u l i o R . Claeyssen , Teresa Ts u kazan , Leticia Tonetto , Dan i e l a Tol fo AcknowledgmentsWe t ha n ks t he rev iewe rs fo r t h e i r im po rta n t co mments a n d s u gg est io ns period \noindentR e f e r eAcknowledgments n .. c e s We t ha n ks t he rev iewe rs fo r t h e i r im po rta n t co mments a n d s u gg est io ns . open square bracket 1 closing square bracketR e G f .. e Bin r nie gn comma c C e period s F period Q u a t comma .. period G e r b P-r-comma hysics R evi w-e .. L e\ centerline t-t er five-s 6{ commaWet 930ha nendash ks t 933 he .. rev open iewe parenthesis rs fo 1986 r t h e i r im po rta n t co mments a nd s u gg est io ns . } [ 1 ] G Bin ni g , C . F . Q u a t , . G e r b P − r − comma hysics R evi w − e L e t − t er five − s6, 930 – 933 ( 1986 open square bracket 2 closing square bracket .. Y So n g .. B .. B h u sh a .. J-comma ournal of P e sub hysics : Cond ens dM .. at two-e r-zero sub [ 2 ] Y So n g B B h u sh a J − comma ournal of P e : Cond ens dM at two − er − zero 225012 – 22504 1 ( 200 8 comma\ centerline 22501 2{ endashR e f 22504 e r e1 open n \quad parenthesisc e s200} 8 hysics , [ 3 ] N J a l i l i Piezoele ctr ic - Base d Vibrati n C ontr l − colon F m − o M a cr to MicroN ano S cale Sy open square bracket 3 closing square bracket N .. J a l i l i .. Piezoele ctr ic hyphen Base d Vibrati n C ontr .. l-colon F m-o M a cr to .. MicroN ano S . e − S s − r in g er - V e rl a g cale\ centerline Sy periodp−m e-S{ [ sub 1 ] p-m G \ s-rquad in g erBinnig hyphen V ,C.F.Quate rl a g , \quad . G e r b $ P−r−comma $ hysics R evi $ w−e $ \quad L e $ topen−t $ parenthesis er $ 201f i v e 0− closings 6 parenthesis , 930 $ −− 933 \quad ( 1986 } (2010) open square bracket 4 closing square bracket S .. E sl a m i .. N period J a l i l U-comma l tramicroscop 1 1 7 comma 3 1 endash 45 .. open parenthesis 2\ centerline 0 12 closing parenthesis{[[ 4 2 ] S ] \ Equad sl a mY i So N n . J g a\ lquad i l U −Bcomma\quadlB tramicroscop h u sh a1\ 1quad 7 , 3 1 –$ 45 J−comma ( 2 0 12 $ ) ournal of P $ e { h y s i c s } : $ Cond ens dM \quad at $ two−e r−zero { , } 22501 2 $ −− 22504 1 ( 200 8 } [ 5 ] A Sa l e hi - Khopen o jsquare i , S . B bracket as h a s 5 h closing N . squareJ ya bracketJ − li − Ao Saurnal l e of hi hyphen Micromecha Kh o j i ni comma s a d M S periodi − c Broe as ngineer h a s h ..n N− periodoneeight J y− subg,11 a J-lp p i-o ( 2 urnal 8 ) of .. Micromecha [ 6 ] H J i − h−niLi s a a d n M, F i-c . R roe o n ngineerhyphen n-one− g Fo eight-g n g − subcommaC comma, S 1 h1 engp p - open H s parenthesisJ − in − o urnal 2 8 closing of parenthesis Soun dan dVib rat i n − two89, 529 – 550 ( 2006 \ hspace ∗{\ f i l l } [ 3 ] N \quad J a l i l i \quad Piezoele ctr ic − Base d Vibrati n C ontr \quad $ l−colon $ F $ m−o $ open[ 7 ] square V K bracket W 6 closing H o ile s square , B . C bracket o r n e H ..Nanosca J i-h hyphenl − e LiSystem a n comma sMMT F periodV − A Ro ol un hyphen-gme one Fo− nparenleft g-comma C to the power of comma S h eng M a cr to \quad MicroN ano S cale Sy $ . e−S { p−m } s−r $l in g er − V e r l a g hyphenI SS N H 22 s 99 J-i - n-o 329 urnal D Oof : .. 1 Soun 0 . 2478 dan /dVib nsm m rathyphen i n-two− 89t21 comma 9.)143 529– 1 endash7 1 Dece 550 m b .. e openr (201 parenthesis 2006 open square[8]J bracket Hsu 7 closing Hsquare L.Lee,W.Chan bracket V .. KN ..− Wcomma .. H oano ile stech comma n olog B1 period 8 , 28503 C o –r 2850n e .. 8 Nanosca( 200 7 l-e .. System sMMT V-A sub o l u me .. \ [ ( 201 0 ) \ ] one-parenleft[ 9 ] J W I SS I N sr 22 a99 el ach hyphen v il l 329I − ..comma D O : 1ntermo 0 periodlecula 2478 ran slash dSu nsmrfa e m Forc hyphen-t.A − 2sca 1 d em i Pres s , 3 r de d it i o9 n period , ( 2 0 sub 1 [ closing 1 0 ] M parenthesis G ur 1 t 4 i n3 endash, . W 1 7ei 1s Dece− s m ulb e le r r to , F the . L power a r c ofe l− openP comma parenthesis− hilosop 2 0c 1hica lMaga zn e openseven square− A bracket5(5), 193 8 closing– 1 10 9 square ( 199 bracket8 [ 1 1 J] .. H J s.R u ..C H l a .. e L yss period e n quotedblbase L e e commaG W . period C a n aC h h u a al n N-commap a , C sub J u ano tech n olog 1 8 comma 28503 \ centerline { [ 4 ] S \quad E s l a m i \quad N . J a l i l $ U−comma $ l tramicroscop 1 1 7 , 3 1 −− 45 \quad ( 2 0 12 ) } endashA − g 2850comma 8− openp plied parenthesis Nu merica 200 7 lMathem ati three − s0(1), 65 – 78 ( 199 9 [ 1 2 ] J C l a e yss e n , S . C os t aopen− J − squarecomma bracketournal 9 of closing Soun square dan bracket dVib ra tiJ ..n W− ..two I sr96(4 a el− ach5)1 v il53 l– I-comma 105 8 ( 200 sub 6 n[13] te rmo.F L lecula an d ol ran s , FdSu . Gh rfa o e r Forc b e period A-s sub ca d em i .. Pres s\ centerline commal − S − 3comma r de{ d[ itmart 5 i o ] Ma n A comma terial Sa l san open e d hi Structur parenthesis− Khoj1 9 2, 0 i 65028 1 ,S.Bashash ( 2 0 1 0 [ 1 4 ] T\quad F anN g . , W J . Ch$ y a n{ g Ja −} commaJ−l i−o $ urnal o f \quad Micromecha ni s a d M $ i−c $ roe ngineer $ n−one eight −g { , } 1 1 $ p p ( 2 8 ) } ournalopen ofsquare P hysic bracket san dChe 1 0 closing m ist y square e of S ol bracketd − sixs M ..− Gfour .. ur,913 t i n– comma 9 1 8 .. ( period 2003 [ W 1 5 ei ] s-sL C m al ul a leb r r , comma N . P u F g period n , . M L a r c e-P comma-h ilosop to the powereno z ofz c− hicai, S lMaga . V a l znr e− seven-AJo − iu 5− opencomma parenthesisrnal of P 5 hysics closing : Con parenthesis dens dM comma at 1 93e − endashtwozero 1 10− r 9,474208 open parenthesis (2008 [ 1 199 8 \ centerline6open ] H square B ut{ t bracket[ 6 B ] . C H1 a1\ pp closingquad el l ,J squareM . $ K ai bracket− ph S −− ..l −$ Jcomma period Lianurface sub ,F.RonR S C c l ienc a e eRepoyss e n r $hyphen quotedblbases − five−9,g1 G$– 1 period 52Fo n (C 2005 a$ n g− )acomma h [ 1 u al p a C comma ˆ{ , ..}$ C J S u h A-g eng sub− H s $ J−i n−o $ urnal o f \quad Soun dan dVib rat i $ n−two 89 , 529 $ −− 550 \quad ( 2006 } comma-p7 ] M plied As gh Nu a r.. , merica M . K alMathem h r o b a ati i y three-s a n − 0comma open parenthesisM . Ahm a 1 d closing i I n − parenthesisncomma − commaterna tiona 65 endash lJourna 78 lo .. fE open ngine parenthesis 199 9 e openi − squarern − g bracketScie 1c 2− closingfoureight square− e bracket1749 – .. 1 J C 7 l 6 a e yss ( 2 0e 1n 0 comma [ 1 8 ] S S period H a C sh os e tm a-J-comma in e a d , B ournal . Gh of e .. sh Soun l dan dVib ra ti n-two 96 open [ 7 ] V \quad K \quad W \quad Hoile, s ,B.Corne \quad Nanosca $ l−e $ \quad System sMMT $ V−A { o }$ parenthesisa g A − i 4pplied hyphen P 5a closinghysic sL parenthesis et er s − 1nine 53 endash7, 25313 105 (2010 8 open[ 1 parenthesis 9 ] B O 200 n 6 E . Al t h u , E . T a − d m l u me \quad $ one−parenleft $ I SS N 22 99 − 329 \quad DO : 1 0 . 2478 / nsmm $ hyphen−t 2 1 $ Iopen− rn− squarecommaterna brackettiona 1 3 ljourna closing lo square f s oli s bracket a n d structu period sube − four F L anseven d ol− s commas,1243 – F 125 period ( Gh 2 0 1o 0r b [ e20 l-S-comma ] P L u mart,H MaP terial san d Structur 1 9 comma $ 9 . { ) } 1 4 3 $ −− 171Decembe $rˆ{ l } ( 2 0 1 $ 65028. L e .. e open, C . L parenthesisJ − comma 2 0ournal 1 0 of Applie dPh ysi s − nine9(7351 0) 1 – (200r6 [ 2 1 ] H T h a i In terna t ionaopen lJourna square lo bracket fE n gine 14 e closingr − ig square− n bracketScie five .. T− ..ce F− antwo g,56 comma– 64 – W 60 period ( 2 0Ch 1 2 a n [ 22 g J-comma ] S K on ournal g ofS . P Zh hysic o u san dChe m ist y e of S ol d-six \ centerline { [ 8 ] J \quad H s u \quad H \quad L.Lee,W.Chan $N−comma { ano }$ tech n olog 1 8 , 28503 −− 2850 8 ( 200 7 } s-four, Z . sub N i comma,K − period 9 1 3 endashWa n I 9− 1g 8n− ..commaterna open parenthesist iona lJourna 2003 lo f Engine e i − rn − g Scie c − fourseven − e,487 t−i –open 498 square ( 2009 bracket [ 23 ] L 1 . 5 L closing K e square , Y . S bracket . W an g L C , al . aS b.Ki r commap o rnN periodc h I − Pa un− giterna n commat iona period lJourna M lo eno f Engine z z-i esub comma S period V a l r-J o-i \noindent [ 9 ] J \quad W \quad I sraelachv il l $I−comma { n te rmo }$ lecula ran dSu rfa e Forc $ . u-commai − rn − rnalg ofScie P hysicsc − five : Conzero dens− e dM,256 ..– at 267 e-two ( zero-r2 0 1 2 sub [ 24 comma ] J S 474208 choeft openn e , H parenthesis . I rs ch 2008k − S − comma mart A−Masopen terial{ squareca san}$ d bracket Stu d ctur em 1 6is closing−\quadtwo0, square25007Press bracket (2010 ,3rdedit[ H 25 .. ] G B ut W t .. an Bion g periodJournal ,(201 C of a I pp nte el l ll comma− i gen M tM period ateri Kl − aa pS S-l-comma y s − t urface S c ienc eRepo r s-five 9 comma[e msa 1 0 1 ndendash ] St M ruct\quad 1 52r − ..Gtwoe open\quad− parenthesisfourparenleftur t i 2005 n− , closings\6)quad, 226 parenthesis–. 23 W e ( i 2 0 $ 1 2 s− [s$ 26 ] G muller,F.Larc W an g , X . F en g A − comma $epplied−P comma−h i l o s o p ˆ{ c }$ hicaPopen hysic lMaga square sL et er zn brackets e− nine $ 1 seven70 closing, 231904−A square (2007 5 bracket[ ( 27 ] S 5 M A .. bba )As si gh o, n a ,r A comma 1 . R af 93 sa M n $period j a−− i , R K1 . Avaz a 10 h r 9 m o ( bo − a199h i yam 8a n-comma m d M , A period . F Ahm a d i I n-n comma-t sub [ 1 1 ] \quad J $ . { R }$ Cl ae yss en $ quotedblbase $ G.Canahu al pa , \quad C J u $ A−g { comma−p }$ ernaar s tiona hid i lJourna a A − n lop− fEar− nginepcomma e i-r− n-gl ied .. P Scie hysic c-four sL et eight-e er s − subnine comma5(14), 114312 749 endash (2009 1[ 28.. 7 ] 6 .. J open G insber parenthesis g Mechan 2 0i 1− 0c p l i e d Nu \quad merica lMathem ati $ three −s 0 ( 1 ) , 65 $ −− 78 \quad ( 199 9 aopen an square dStru bracket ctur l Vibr 1 8 closinga i − t squareo .J brackets − o h S n .. Wiley H a sh e ( m 20 in 0 e 1 a[29] d comma.T . C H B period u an .. g Journal Gh e sh of l a l gApplie A-i pplied dMecha P nia sub hysic sL et er s-nine 7 comma 253[two 1 3− 2 opens ]8, 579\ parenthesisquad– 584JClaeyssen,S.Cost (2 19 0 1 6 0 1 [ 30 ] U R ab e E . Ke s t $ae , W−J .− Arcomma no $d − S ournal− comma of urface\quad andSoun I nt erfac dan eAn dVib ra ti $ n−two 96 (alysopen 4 s square−−two7 bracket5, 386 – ) 391 1 9 1 closing ( 1999 53 square $ [ 3−− 1 bracket ] R105 B B 8 .. ( GO 200 u.. enth n 6.. eE , period . W Al . L t eh u ..P comma− comma E periodartial T a-d Differen .. m I-ri sub− t n-commaa te rna tiona ljourna lo f s $ [ 1 3 ] . { F }$ Landols ,F.Ghorbe $l−S−comma $ mart Ma terial san d Structur 1 9 , 65028 \quad ( 2 0 1 0 olilEqu s a n a dtio structu s o e-four− fM seven-sathe m ati sub al comma P hy ics 1 an 243 dIn endash t gral E125 qua .. open. o parenthesis− D − io−ns 2−ve 0 1r 0 [open 1 4 square ] \quad bracketT \ 20quad closingFang square ,W.Changbracket .. P L u comma $J sub−comma H .. P $ period ournal L e e comma of P hysic C period san L J-comma dChem ournal ist y of e .. 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