Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran

Transcription

1 Journal of Sold Mecanc Vol., o. (8) pp. - Vbraton naly of Magneto-Electro-Elatc Tmoeno Mcro Beam Ung Surface Stre Effect and Modfed Stran Gradent Teory under Movng ano-partcle M. Moammadmer *, H. Moammad Hooye Department of Sold Mecanc, Faculty of Mecancal Engneerng, Unverty of Kaan, Kaan, Iran Receved September 7; accepted ovember 7 BSTRCT In t artcle, te free vbraton analy of magneto-electro-elatc (MEE) Tmoeno mcro beam model baed on urface tre effect and modfed tran gradent teory (MSGT) under movng nano-partcle preented. Te governng equaton of moton ung Hamlton prncple are derved and tee equaton are olved ung dfferental quadrature metod (DQM). Te effect of dmenonle electrc potental, dmenonle magnetc parameter, materal lengt cale parameter, eternal electrc voltage, eternal magnetc parameter, lenderne rato, temperature cange, urface tre effect, two parameter of elatc foundaton on te dmenonle natural frequency are nvetgated. It own tat te effect of electrc potental and magnetc parameter multaneouly ncreae te dmenonle natural frequency. On te oter and, wt conderng two parameter, te tffne of MEE Tmoeno mcro beam model ncreae. It can be een tat te dmenonle natural frequency of mcro tructure ncreae by MSGT more tan modfed couple tre teory (MCST) and clacal teory (CT). It found tat by ncreang te ma of nanopartcle, te dmenonle natural frequency of ytem decreae. Te reult of t tudy can be employed to degn and manufacture mcrodevce to prevent reonance penomenon or a a enor to control te dynamc tablty of mcro tructure. 8 IU, ra Branc. ll rgt reerved. Keyword: Vbraton analy; Movng nano-partcle; Tmoeno mcro beam model; Surface tre effect, MSGT; Magneto-electro-elatc loadng; DQM. ITRODUCTIO M GETO-ELECTRO-ELSTIC (MEE) compote materal wt pezoelectrc and pezomagnetc pae can be utlzed a materal provdng energy converon among magnetc, electrc and mecancal energe []. Tee materal are appled n varou applcaton uc a tructural ealt montorng, vbraton control, enor and actuator applcaton, robotc, medcal ntrument and energy arvetng [-5]. In recent year, tatc, buclng, and vbraton analy of MEE materal a found gnfcance among many reearcer. Bangala and Ganean [6] tuded te free vbraton of functonally graded (FG) non-omogeneou MEE cylndrcal ell by ung * Correpondng autor. Tel.: ; Fa: E-mal addre: mmoammadmer@aanu.ac.r (M.Moammadmer). 8 IU, ra Branc. ll rgt reerved.

2 M. Moammadmer and H. Moammad Hooye te em-analytcal fnte element approac. Tey llutrated tat te pezoelectrc effect a te tendency of tffenng te ell and ence ncreae te tructural natural frequency but magnetc effect a a negatve nfluence on te ytem frequency and reduce te natural frequency. Lang and Xuewu [7] analyzed te buclng and vbraton analy of FG magneto-electro-termo-elatc crcular cylndrcal ell. Ter reult repreented tat nfluence of pezo-magnetc contant on te value of te crtcal termal buclng load more tan tat of pezoelectrc contant. Razav and Sootar [8] nvetgated te nonlnear free vbraton of MEE rectangular plate. In ter wor, t oberved tat lengt-to-tcne rato a neglgble effect on te nonlnear frequency rato n comparng wt te effect of de rato. Ke et al. [9] employed te free vbraton of ze-dependent MEE nanoplate baed on te nonlocal elatcty teory. Tey obtaned tat te natural frequency qute entve to te mecancal loadng, electrc loadng and magnetc loadng, wle t nentve to te termal loadng. L et al. [] analyzed te buclng and free vbraton of MEE nanoplate under Paterna foundaton baed on Mndln plate teory. Ter reult depcted tat te buclng load and vbraton frequency decreae lnearly wt ncreang of electrc potental, prng and ear coeffcent of te Paterna foundaton, and ncreae wt an ncreae n te magnetc potental. Sootar and Razav [] tuded lnear and nonlnear free vbraton of a multlayered MEE doubly-curved ell on elatc foundaton. Tey owed tat negatve electrc potental and potve magnetc potental ncreae te fundamental natural frequence of MEE doubly curved ell, werea te potve electrc potental and negatve magnetc potental ave nvere effect on te natural frequency of MEE doubly-curved ell. Tey [] alo nvetgated large ampltude free vbraton of ymmetrcally lamnated MEE rectangular plate on Paterna type foundaton. Moammadmer et al. [3] nvetgated electro-elatc analy of a andwc tc plate conderng FG core and compote pezoelectrc layer on Paterna foundaton ung trd-order ear deformaton teory (TSDT). Ter reult owed tat te dmenonle natural frequency and crtcal buclng load decreae wt ncreang of te power law nde, and vce vera for dmenonle deflecton and electrcal potental functon, becaue of te andwc tc plate wt conderng FG core become more fleble; wle tee reult are revere for tcne rato. nar et al. [] decrbed nonlnear forced vbraton of magneto-electro-termo-elatc ano beam baed on Erngen nonlocal elatcty teory. Ter reearc owed tat te effect of eternal magnetc potental and electrc voltage are dependent on ter gn. Xn and Hu [5] derved em-analytcal oluton for free vbraton of arbtrary layered MEE beam ung te tate pace approac (SS) and dcrete ngular convoluton (DSC) algortm. Ter reult revealed tat by ncreang tcne natural frequency ncreae for ger mode and dfferent materal parameter affect natural frequency greatly. lo, tey [6] obtaned free vbraton of multlayered MEE plate ung SS-DSC approac wt mply upported boundary condton. Baed on tran gradent teory, Moammadmer et al. [7] nvetgated te free vbraton analy of tapered vcoelatc mcrorod retng on vco-paterna foundaton. Tey aumed te materal properte of mcro-rod a te vco-elatc and modeled ung te Kelvn-Vogt. Ten, tey derved te governng equaton of moton of vcoelatc mcrorod ung Hamlton' prncple and energy metod, and alo olved tee obtaned equaton ung te dfferental quadrature metod (DQM) for dfferent boundary condton. Ter reult owed tat wt an ncreae n te Wnler and Paterna coeffcent, te natural frequency ncreae a well a te obtaned non-dmenonal natural frequence by modfed couple tre teory (MCST) and tran gradent teory (SGT) decreae by ncreang te materal lengt to radu rato. It wa own tat te non-dmenonal frequence ncreae by ncreang te dampng coeffcent for all teore. Moreover, at te pecfed value of dampng coeffcent of te elatc medum, te varaton of non-dmenonal natural frequency appromately moot. In te oter wor, Ramat and Moammadmer [8] preented electro-termo-mecancal vbraton analy of non-unform and nonomogeneou boron ntrde ano rod (BR) embedded n elatc medum. Tey developed te teady tate eat tranfer equaton wtout eternal eat ource for non-omogeneou rod and derved temperature dtrbuton. lo, ung Mawell equaton and nonlocal elatcty teory, tey obtaned te coupled dplacement and electrcal potental equaton and mplemented te DQM to evaluate te natural frequence. Ke and Wang [9] owed te free vbraton of MEE Tmoeno ano beam baed on te nonlocal elatcty teory. Tey depcted tat te natural frequency of nonlocal ano beam alway maller tan tat of te clacal ano beam, and t decreae wt ncreang of te nonlocal parameter. Wang et al. [] analyzed aymmetrc bendng of FG crcular MEE plate of tranverely otropc materal baed on lnear tree-dmenonal elatcty teory. Tey repreented tat te electrc potental and magnetc potental are parabolc-le along te tcne of te plate and t penomenon may be condered n ome appromate metod for bendng of te MEE plate. Rao et al. [] tuded geometrcally nonlnear fnte rotaton ell element for tatc analy of layered MEE coupled compote tructure. Moammadmer et al. [] nvetgated free vbraton of vcoelatc double-bonded polymerc nanocompote plate renforced by functonally graded-ngle walled carbon nanotube (FG-SWCT) embedded n vcoelatc foundaton baed on modfed tran gradent teory (MSGT). Tey defned materal properte of 8 IU, ra Branc

3 Vbraton naly of Magneto-Electro-Elatc Tmoeno 3 vcoelatc nanocompote plate by etended mture rule (EMR), and alo etracted te governng equaton of moton ung Hamlton' prncple and nuodal ear deformaton teory. Ten, tey determned natural frequency of nanocompote plate by aver' and mele metod, and alo, ung mele metod, te effect of varou boundary condton on dmenonle natural frequency. In te oter wor, tey [3] etended modfed tran gradent Reddy rectangular plate teory for baal buclng and bendng analy of double-coupled polymerc nanocompote plate renforced by functonally graded ngle-walled boron ntrde nanotube (FG-SWBT) and FG-SWCT. Kattman and Ray [] nvetgated control of geometrcally nonlnear vbraton of functonally graded MEE plate. Ter reult revealed tat te electro-elatc and te magneto-elatc couplng ave neglgble nfluence on te nonlnear tranent repone of te FG MEE plate wle ncreang te tffenng effect margnally. Lu et al. [5] propoed guded wave propagaton and mode dfferentaton n te layered MEE ollow cylnder. Tey llutrated tat n a low frequency range, radu-tcne rato ave gnfcant effect on te wave caractertc. Sedg and Farjam [6] preented a modfed model for dynamc ntablty of CT baed actuator by conderng rpplng deformaton, tp-carge concentraton and Camr attracton. Tey concluded tat te tp carge concentraton and rpplng penomenon can ubtantally affect te electromecancal performance of actuator fabrcated from cantlever CT. Zare [7] condered pull-n ntablty beavor analy of functonally graded mcro-cantlever under uddenly DC voltage. By employng modern aymptotc approac namely omotopy perturbaton metod wt an aulary term, e obtaned g-order frequency-ampltude relaton, ten te nfluence of materal properte and actuaton voltage on dynamc pull-n beavor are nvetgated. Sedg [8] llutrated te nfluence of mall cale on te pull-n beavor of nonlocal nano brdge conderng urface effect, Camr and van der Waal attracton. He nvetgated te effect of appled voltage and ntermolecular parameter on pull-n ntablty a well a te natural frequency. Furtermore, e condered te nfluence of nonlocal parameter and urface energy on te dynamc pull-n voltage. Te effect of materal lengt cale parameter on te rotaton angle, dmenonle electrc potental, and dmenonle magnetc parameter are taen nto account. lo, te nfluence of dmenonle electrc potental, dmenonle magnetc parameter, materal lengt cale parameter, eternal electrc voltage, eternal magnetc parameter, lenderne rato, temperature cange, urface tre effect, prng Wnler contant, and ear Paterna contant on te dmenonle natural frequency are nvetgated. THE GOVERIG EQUTIOS OF MOTIO FOR MEE TIMOSHEKO MICRO BEM MODEL Te governng equaton of moton ung Hamlton prncple are derved for magneto-electro-elatc (MEE) Tmoeno mcro beam model baed on urface tre effect and modfed tran gradent teory (MSGT) under movng nano-partcle and tee equaton are olved ung dfferental quadrature metod (DQM).. Modfed tran gradent teory Te ze dependent effect a an mportant role at mcro cale. Flec and Hutcnon [9-3] etended and reformulated te clacal couple tre teory and renamed t a te tran gradent elatcty teory (SGET), n wc for omogeneou otropc and ncompreble materal, tree addtonal ger-order materal lengt cale parameter are ntroduced. Lam et al. [3] propoed a modfed tran gradent elatcty teory (MSGT) n wc a new addtonal equlbrum equaton to govern te beavor of ger-order tree, te equlbrum of moment of couple ntroduced, n addton to te clacal equlbrum equaton of force and moment of force. Meanwle, tere are only tree ndependent ger-order materal lengt cale parameter for otropc lnear elatc materal n te preent teory. ccordng to te modfed tran gradent teory (MSGT) propoed by Lam et al. [3], te tran energy of a lnear MEE contnuum occupyng regon wrtten a follow [, 5, 33, and 3]: U ( () () D E ) j j p j j mj j B H dv () were denote te tran, te dlataton gradent tenor, te ymmetrc rotaton gradent () j,, j, j, E, H tenor, te devatorc tretc gradent tenor, te electrc feld and magnetc feld, repectvely, wc are defned by followng form [, 33, and 3]: 8 IU, ra Branc

4 M. Moammadmer and H. Moammad Hooye j ( u, j u j, ) () (3) mm, (),,,,,,, j ( j j j ) j mm m m j mm m m mm, j mj, m () j (, j j, ) (curl(u)) (5) E (6), H (7), In wc u,,, and repreent te dplacement vector, dlataton tran, te nfntemal rotaton vector, mm electrc and magnetc potental repectvely. j and ej are te Knocer and te alternate ymbol repectvely. Te bac equaton for a MEE materal may be epreed a follow: c e E q H T (8) j jl l mj m nj n j D e E d H v T (9) l l m m n n B q d E H T () l l m m n n In wc j, j, D, E, B and H are te tre, tran, electrc dplacement, electrc feld, magnetc nducton and magnetc feld, repectvely. cjl, emj, qnj, m, dn, n, v and are te elatc, pezoelectrc, pezo-magnetc, delectrc contant, magneto-electrc, magnetc, pyroelectrc and pyro-magnetc contant, repectvely. j and () T are te termal modul and temperature cange, repectvely. Te ger-order tree p, j and mj are gven by [33, 3]: p l () l () () () j j m j l (3) j were te ear modulu and l, l, l are ndependent materal lengt cale parameter.. Surface tre effect Te rato of urface to volume at nano and mcro tructure g, terefore te urface tre a mportant role and ould be condered n analyze of tee tructure. Te clacal tre tenor related to te urface can be calculated ung Gurtn and Murdoc teory [35-38]: c e E q H () 3 z 3 z 8 IU, ra Branc

5 Vbraton naly of Magneto-Electro-Elatc Tmoeno 5 were, c,, e3 and q 3 are te redual urface tre, elatc modulu, normal tran, pezoelectrc and pezo magnetc contant n urface layer, repectvely. Te ger-order urface tree a [39]: p p, and () j mj can be epreed l (5) () ( ) j l j (6) m (7) j l j () In wc, j, j and are te urface dlataton gradent tenor, te urface ymmetrc rotaton gradent tenor, te urface devatorc tretc gradent tenor and urface ear modul, repectvely..3 MEE Tmoeno mcro beam model cematc vew of MEE mcro-beam ubjected to an electrc and magnetc potental and a unform temperature wt conderng urface layer rand elatc medum under movng nano-partcle own n Fg.. Te dplacement feld baed on Tmoeno mcro-beam model can be epreed a follow [-]: u (,z, t ) u, t z, t (8) u (,z,t) (9) u (,z,t) w ( t, ) 3 () were u and w are aal and tranvere dplacement for neutral a, repectvely, and te rotaton angle of a tranvere normal about te -a. Fg. cematc vew of MEE mcro-beam wt conderng urface layer and elatc medum under movng mcropartcle. Te electrc and magnetc potental dtrbuton along te tcne drecton of mcro-beam are condered a follow [9], []: zv E (, z, t) co( z) E (, t) z (, z, t) co( z) H (, t) H () 8 IU, ra Branc

6 6 M. Moammadmer and H. Moammad Hooye were, E (, t) and H (, t) are te varaton of electrc and magnetc potental n te -drecton wc mut atfy te electrc and magnetc boundary condton. V E and H are te eternal electrc voltage and eternal magnetc parameter, repectvely. By ubttutng Eq. (8)- () nto Eq. (), te component of normal and ear tran for bot bul and urface are gven by: u z () w z, z (3) Ung Eq. (3), (), (3), te followng equaton are obtaned: u z () z z (5) By employng Eq. (), (), (3), te non-zero component of devatorc tretc gradent tenor yeld: () () u z 5 5 () w () 333, () () () 8 w () () () 3 3 3, () () () () () ( ) u z 5 5 () () () () () () u z 5 5 () () () w () () () 3 3 3, (6) By ubttutng Eq. (), (3) nto Eq. (5) lead to te followng equaton: w y y y y ( ) (7) Te ger-order tree can be obtaned by ubttutng Eq. () - (7) nto Eq. ()- (3) and (5)-(7) a follow: 8 IU, ra Branc

7 Vbraton naly of Magneto-Electro-Elatc Tmoeno 7 () () u l l z 5 5 () () () () () () u l l z 5 5 () w () 333 l l, 333 l w () () () () () () u l l z 5 5 () () () w () () () l l, l () () ( ) () () () l l, l m y m y m y m y l u p p l z pz p l Z w ) ( (8) (9). Plane tre tate If conder tat mcro-beam under te plane tre tate, te conttutve equaton n one-dmenonal form can be wrtten a follow: V E H c e 3( n( z) E ) q3( n( z) H ) T E H z c z e5 co( z) q5 co( z) E H D e5 z co( z) d co( z) V E H Dz e 3 33( n( z) E ) d 33( n( z) H ) 3T E H B q 5 z d co( z) 5 co( z) V E H B z q3 d 33( n( z) E ) 33( n( z) H ) 3T (3) In wc cj, ej, qj, j, j, dj, j, v and are elatc, pezoelectrc, pezo-magnetc, delectrc contant, termal modul, magneto-electrc, magnetc, pyroelectrc and pyro-magnetc contant, repectvely for MEE mcrobeam under plane tre tate. Tee contant are wrtten n ppend wt detal. Te total tran energy wt conderng te tran gradent teory and te urface effect teory wrtten a follow: b () () U U ( D E ) j j p j j mj j B H dd () () ( j j p j j m j j ) dsd (3) 8 IU, ra Branc

8 8 M. Moammadmer and H. Moammad Hooye b were U and U are te tran energy for te bul and urface, repectvely. By ubttutng Eq. (6), (7), ()- (3), (5)-(7) and ()-(3) nto Eq. (3), te total tran energy obtaned a follow: L b u w u U U { M M Q P P () () () () u P3 P3 M M T T 5 () () () w () M M T 333 T () 8 w () () w 3T 3 3 T 3 3T u 3T 6T T 6 M 5 5 () () () () 3 w Y Y }d L V E E { Dz n z E D co z } dd L H H { B z n z H B co z } dd () M 5 (3) Y Y and oter ger- Te normal tre reultant force,, bendng moment, order reultant force and moment can be epreed n ppend [3-5]. Te total netc energe for MEE mcro-beam contan of bul netc energy can be calculated from: M M, couple moment, b and urface netc energy L b u w ( I I ) ( I I ) d t t t I d, I ds,i z d, I z ds S S Te wor done by te eternal force ncludng te eternal magnetc potental V E and temperature cange T can be calculated a follow: H (33), eternal electrc potental L V eternal ( m e t ) w d (3) q, e V, T m 3 H e 3 E t Te wor due to elatc foundaton can be obtaned a follow: V [( w w ) w ] d (35) L elatc medum w g were w te prng contant of te Wnler type and g repreent te ear contant of te Paterna type. Te wor done due to movng nano-partcle can be wrtten a [6]: 8 IU, ra Branc

9 Vbraton naly of Magneto-Electro-Elatc Tmoeno 9 V Partcle m w L c [( ( p )) ] w d (36) b t were m,, b and are te nano-partcle ma, te locaton of nano-partcle, te wdt of mcro-beam and te c p mpule functon, repectvely. Te Drac-delta functon for te movng nano-partcle defned a follow [6-7]: ( ) g ( ) n n n p p g () ( p )d (37) oterwe were n ( ) denote nt dervatve of Drac-delta functon. Hamlton prncple for MEE mcro-beam ued to derve te governng equaton of moton a follow: t b b U U V eternal V elatc medum V partcle (38) t Subttutng Eq. (3) - (36) nto Eq. (38), te followng equaton of moton can be derved a: u 6 : () () () () P P T T T T u I I 5 5 t w : Q Y Y mc w w ( ww g w ) ( ( mp ))( ) I I b t t : () () () T 333 T3 T 3 w ( ) m e t () () () () M M P3 P3 M M M M Q 5 () () () () () () T 333 T T3 T3 T 3 T () ( ) 6 M M Y Y I I 5 t : E D ( co( z) Dz n( z)) d : H B ( co( z) Bz n( z) ) d (39) Subttutng ppend (.) nto Eq. (39), te governng equaton of moton can be obtaned a follow: 8 IU, ra Branc

10 M. Moammadmer and H. Moammad Hooye u u u u ( ) ( 55 55)( l l ) I I 5 t w : : 3 w 6 E ( ) ( ) ( l 55 l (55 55)) ( E 3 5) 5 8 w w H ( l 55 l ( 55 55) ) ( ) m e t ( Q 5) 5 mc w w ( ww g w ) ( ( mp ))( ) I I b t t : w ( l l )(I3 I 3) ( ) ( ) ( l l l l w E ( l 5 5 l (55 55)) ( E 3 5 E 3 E 3) 5 H ( Q5 Q3 Q3) I I t : E (55 55) (55 55) D D) w E H E 3 E5( ) X Y X 33E Y 33H : H Q w Q E 5( ) Y T H Y 33E T 33H () were te contant of Eq. () are defned n ppend. Te dmenonle parameter for magneto-electro-elatc Tmoeno mcro beam model baed on urface tre effect and MSGT under movng nano-partcle are defned a follow: (U, W) X, (u, w ),,,,, L X T E H t L (,, ) ( I, I ),,(,, ),(I, I ) L I m e t m e t I ( I, I ) (I, I ) (I, I ),(I, I ),(a,a,a,a,a I (d,d ),(, ) (, ),m,(,, ) ( D, D) w L g mc (l,l,l ) w g c I b ˆ ˆ ˆ E E E ˆ ˆ ˆ Q (E,E,E ) (,, ),( Q,Q,Q ) ( ,(X ˆ X X, ) (, ) 33 Xˆ 33 T T Y Y ( T,T ) (, ),(Y,Y ) (, ) ˆ ˆ ˆ ˆ 33 (, 55,,, 55, ),a ) Q Q 3 3,, ) () 8 IU, ra Branc

11 Vbraton naly of Magneto-Electro-Elatc Tmoeno Ung Eq. (), te dmenonle governng equaton of moton can be wrtten a: U : U U U ( a a ) ( a 55 a55 )( ) I I X 5 X W : W 6 (a ) ) (a )) ) X X 5 X X 8 W W ( a55 (a 55 a55 )) ( ) m e t ( ˆ Q 5) 5 X X X W w ( ww g W ) ( mc )( ) I I : 3 3 (a ( a55 55 a55 ( Eˆ W ( )(I33 I 33) ( a ) (a ) 5 X X 3 8 ( W 3 ( ˆ ˆ ˆ a55 (a 55 a55 )) ( E5 E 3 E 3) 3 5 X X ˆ ˆ ˆ 3 ( Q5 Q3 Q3) I I X : X a55 a55 (a 55 a55 ) (a 55 a55 ) d d) ˆ ˆ W ˆ ˆ ˆ ˆ E3 E5( ) X Y X 33 Y 33 X X X X X : ˆ W 3 Q5 ˆ ˆ Y ˆ ˆ T Y 33 T 33 Qˆ ( X X ) X X X () Te dmenonle mply upported (S-S) boundary condton for MEE are condered a follow: W U W, at (X,) X X (3) 3 DIFFERETIL QUDRTURE METHOD In t ecton, dfferental quadrature metod (DQM) utlzed to olve te governng equaton of moton. T tecnque numercal metod wc appromate te patal dervatve of a functon at a partcular amplng pont a a wegted lnear um of te functon value at all amplng pont coen n a pecfed drecton. ccordng to t metod, te functon f and dervatve are appromate a [8-9]: n d f d n C f ( ) () j ( n ) j j were f can be taen a UW,, and,. (n) C j and are te wegtng coeffcent matr and number of grd pont repectvely. To determne te unequally paced poton of te grd pont te Cebyev Gau Lobatto polynomal wa employed a follow [5]: 8 IU, ra Branc

12 M. Moammadmer and H. Moammad Hooye L co (5) Ung Eq. () and (3), a et of lnear ordnary dfferental equaton and boundary condton are obtaned tat are wrtten n ppend B. Ung Eq. (B.)- (B.), a et of nonlnear omogenou partally dfferental equaton are obtaned wc n matr form can be epreed a follow: X M X ( K ) (6) K,M are te tffne and ma matrce, repectvely and X were X U, W,,, T T T T T T denote dplacement vector a: (7) General oluton of moton equaton condered a follow: U(, t ) U ( X )e W(, t ) W ( X )e (, t ) ( X )e (, t ) ( X )e (, t ) ( X )e (8) I were L te dmenonle natural frequency and, and denote te fundamental natural frequency and te denty of mcro-beam, repectvely. Subttutng Eq. (6) nto Eq. (7) yeld te lnear egenvalue equaton a follow: M X ( K ) (9) By olvng Eq. (9), te dmenonle natural frequence and ter aocated vbraton mode ape can be etracted. UMERICL RESULTS D DISCUSSIO Te preented reult are baed on te followng data for geometry of MEE mcro-beam [5-5]. lo te materal properte of BTO3 CoFeO can be tated a follow []: l 7.6 m, l, b,l (5) Frt, te effect of te grd pont number on te accuracy and convergence of analy tuded. In t regard te dmenonle fundamental natural frequency veru number of total dcrete grd pont llutrated n Fg.. Tu, te number of grd pont for convergence and acceptable accuracy of te reult elected to be >=5. It ould be noted tat all of repreented reult are baed on Eq. (5), te data of eac Table or Fgure tated under tem. To cec te accuracy of te preent wor, te obtaned reult are compared wt te analytcal oluton gven by nar et al. [53] n Table. T Table gve te dmenonle natural frequence for te mply upported Tmoeno mcro-beam and dfferent teore. good agreement found between te preent reult and toe of analytcal oluton. 8 IU, ra Branc

13 ondmenonal natural frequency Vbraton naly of Magneto-Electro-Elatc Tmoeno Fg. Dmenonle natural frequency veru number of grd pont w g V E H T mc c e3 q3. Table Te comparon between te preent wor and te obtaned reult by nar et al. [53] for mply upported Tmoeno mcrobeam baed on varou ze dependent teore. ( l 5 m,.7,e 7 GPa, 3 g ). 3 m Ceramc materal Stran gradent Modfed couple tre Clacal teory(ct) teory(sgt) l l l l teory(mcst) l l, l l l l l nar [53] Preent wor Fg.3 ow te rotaton angle veru lengt of MEE mcro-beam for varou materal lengt cale parameter baed on te modfed tran gradent teory and mply upported boundary condton. It een tat by ncreang te value of l, te value of te rotaton angle for MEE mcro-beam ncreae. lo, te rotaton angle X ymmetrc and at.5 te value of rotaton angle vare from negatve to potve along lengt of te mcrobeam. L.3.. /l= /l= /l=6 /l= /L Fg.3 Te rotaton angle veru lengt of MEE compote mcrobeam for varou materal lengt cale parameter. V T m c e q w g E H c 3 3 Te effect of materal lengt cale parameter on te dmenonle electrc potental for MEE Tmoeno mcro beam model llutrated n Fg. tat mamum electrc potental occurred at mdlne mply upported MEE mcro-beam and by ncreang te value of l range., te value of dmenonle electrc potental decreae at IU, ra Branc

14 M. Moammadmer and H. Moammad Hooye 6-3 /l= /l= /l=6 /l= /L Fg. Te effect of materal lengt cale parameter on te dmenonle electrc potental for MEE mcro compote Tmoeno beam model. V T m c e q w g E H c 3 3 Fg.5 depct dmenonle magnetc parameter for dfferent materal lengt cale parameter baed on modfed tran gradent model. It oberved tat by ncreang value of l ncreae te dmenonle magnetc parameter /l= /l= /l=6 /l= /L Fg.5 Te effect of materal lengt cale parameter on te dmenonle magnetc parameter for MEE mcro compote Tmoeno beam model. V T m c e q w g E H c 3 3 Fg. 6, 7 and 8 preent te effect of electrc potental and magnetc parameter multaneouly on te dmenonle fundamental natural frequency baed on modfed tran gradent (MSGT), modfed couple tre (MCST) and clacal teore (CT), repectvely. It can be een tat te effect of electrc potental and magnetc parameter multaneouly ncreae te dmenonle natural frequency. On te oter and, wt conderng two parameter, te tffne of MEE Tmoeno mcro beam model ncreae. It can be een tat te tffne of mcro tructure ncreae by MSGT more tan MCST and CT. Moreover, te dmenonle natural frequency baed on MSGT more tan two oter teore , == 6 8 /l Fg.6 Te nfluence of dmenonle electrc potental and magnetc parameter on dmenonle frt natural frequency for MEE compote mcro-beam baed on modfed tran gradent model. V T m c e q w g E H c IU, ra Branc

15 Vbraton naly of Magneto-Electro-Elatc Tmoeno , == 6 8 /l Fg.7 Te nfluence of dmenonle electrc potental and magnetc parameter on dmenonle frt natural frequency for MEE compote mcro-beam baed on modfed couple tre teory. V T m c e q l l, l l w g E H c , == /l Fg.8 Te nfluence of dmenonle electrc potental and magnetc parameter on dmenonle frt natural frequency for MEE compote mcro-beam baed on clacal teory. V T m c e q l l l w g E H c 3 3 Te effect of te eternal electrc voltage and eternal magnetc parameter on te dmenonle fundamental natural frequency of te mcro-beam baed on MSGT llutrated n Fg. 9 and, repectvely. It own from t fgure tat wt ncreang lenderne rato, te dmenonle fundamental natural frequency decreae. lo, te nfluence of te eternal electrc voltage and eternal magnetc parameter on te dmenonle frequency for Tmoeno mcro beam model n ger lenderne rato more. lo, t realzed from Fg. 9 tat te dmenonle natural frequency of te MEE mcro-beam decreae wt ncreang te eternal electrc voltage for tran gradent teory. However, t concluded from Fg. tat by ncreang te eternal magnetc parameter, te dmenonle natural frequency of te MEE mcro-beam ncreae..6.. V E =-(KV) V E =-(KV) V E =(KV) V E =(KV) V E =(KV) L/ Fg.9 Te effect of eternal electrc voltage on te dmenonle fundamental natural frequency baed on MSGT. T m c e q w g H c IU, ra Branc

16 6 M. Moammadmer and H. Moammad Hooye H =-.(K) H =-.(K) H =(K) H =.(K) H =.(K) L/ Fg. Te effect of eternal magnetc parameter on te dmenonle fundamental natural frequency baed on MSGT. V T m c e q w g E c 3 3 Fg. (a) and (b) ow te effect of two parameter elatc foundaton on te dmenonle electrc potental baed on MSGT. It een tat wt ncreang elatc foundaton, te electrc potental decreae. 6-3 g =() 6-3 w =(Pa) g =5() g =5() g =35() w =.5e(Pa) w =3e(Pa) w =3.5e(Pa) /L /L (a) (b) Fg. a) Te effect of ear Paterna contant on te dmenonle electrc potental baed on MSGT. V T m c e q w E H c 3 3 b) Te effect of prng Wnler contant on te dmenonle electrc potental baed on MSGT. V T m c e q g E H c 3 3 Fg. (a) and (b) depct te effect of two parameter elatc foundaton on magnetc parameter baed on MSGT. It concluded tat te magnetc parameter ncreae wt an ncreae n te elatc foundaton. 6-3 g =() g =5() g =5() g =35() w =(Pa) w =.5e(Pa) w =3e(Pa) w =3.5e(Pa) /L /L (a) (b) Fg. a) Te effect of ear Paterna contant on te dmenonle magnetc parameter baed on MSGT. V T m c e q w E H c 3 3 b) Te effect of prng Wnler contant on te dmenonle magnetc parameter baed on MSGT. V T m c e q g E H c IU, ra Branc

17 Vbraton naly of Magneto-Electro-Elatc Tmoeno 7 Fg. 3 llutrate te effect of nano partcle ma on dmenonle natural frequency baed on MSGT. It L aumed tat te nano partcle placed at P. It found tat by ncreang te ma of nano-partcle, te dmenonle natural frequency of ytem decreae..6.. m c =(Kg) m c =e-(kg) m c =e-(kg) m c =3e-(Kg) m c =e-(kg) L/ Fg.3 Te effect of nano partcle ma on dmenonle natural frequency baed on MSGT. V T c e q g w E H c 3 3 Table. ow te effect of temperature cange on dmenonle frt tree natural frequence baed on varou ze dependent teore uc a MSGT, MCST and CT. It can be een tat wt an ncreae n te temperature cange, dmenonle natural frequency of mcro beam model decreae. lo wt ncreang temperature cange te tffne of mcro-beam decreae. ccordng to Gorbanpour ran et al. [5], te mecancal and electrcal properte for urface tre effect are defned a follow: g nc 5.6e 6, c. and e3.6, q3 3e 5 m m m m (5) Ung te data of Eq. (5), Table 3. ow te urface tre effect on dmenonle frt tree natural frequence of MEE mcro-beam for dfferent value of lenderne rato ( L ) baed on MSGT, MCST and CT. It concluded tat by conderng te urface tre effect, te dmenonle natural frequency of ytem decreae. Table Te effect of temperature cange T ( C) on te dmenonle frt tree natural frequency baed on varou ze dependent teore. g w V E H mc c e3 q3. Varou teore MSGT MCST CT mode T ( C) IU, ra Branc

18 8 M. Moammadmer and H. Moammad Hooye Table3 Surface tre effect on te dmenonle frt tree natural frequence for dfferent value of lenderne rato baed on varou ze dependent teore. g w V E H T mc, l m. varou teore MSGT MCST CT mode Wt neglectng urface effect Wt conderng urface effect Wt neglectng urface effect Wt conderng urface effect Wt neglectng urface effect Wt conderng urface effect L L L COCLUSIOS In t reearc, te free vbraton analy of magneto-electro-elatc (MEE) Tmoeno mcro beam model baed on urface tre effect and modfed tran gradent teory (MSGT) under movng nano-partcle wa nvetgated. Te governng equaton of moton ung Hamlton prncple were derved and tee equaton were olved ung dfferental quadrature metod (DQM). Te effect of materal lengt cale parameter on te rotaton angle, dmenonle electrc potental, and dmenonle magnetc parameter were taen nto account. lo, te nfluence of dmenonle electrc potental, dmenonle magnetc parameter, materal lengt cale parameter, eternal electrc voltage, eternal magnetc parameter, lenderne rato, temperature cange, urface tre effect, prng Wnler contant, ear Paterna contant on te dmenonle natural frequency were condered. Te reult of t tudy are lted a follow:. Te number of grd pont for convergence and acceptable accuracy of te obtaned reult elected to be >=5.. It oberved from te reult tat wt an ncreae n te value of l, te value of te rotaton angle for X MEE mcro-beam ncreae. lo, te rotaton angle ymmetrc and at.5 te value of rotaton L angle vare from negatve to potve along lengt of te mcro-beam. 3. It can be een tat te effect of electrc potental and magnetc parameter multaneouly ncreae te dmenonle natural frequency. On te oter and, wt conderng two parameter, te tffne of MEE Tmoeno mcro beam model ncreae. It can be een tat te tffne of mcro tructure ncreae by MSGT more tan MCST and CT. Moreover, te dmenonle natural frequency baed on MSGT more tan two oter teore.. It own tat wt ncreang lenderne rato, te dmenonle fundamental natural frequency decreae. lo, te nfluence of te eternal electrc voltage and eternal magnetc parameter on te dmenonle frequency for Tmoeno mcro beam model n ger lenderne rato more. lo, t found tat te dmenonle natural frequency of te MEE mcro-beam decreae wt ncreang te eternal electrc voltage baed on MSGT. However, t concluded tat by ncreang te eternal magnetc parameter, te dmenonle natural frequency of te MEE mcro-beam ncreae. 5. It een tat wt ncreang elatc foundaton, te electrc potental decreae, wle te tffne of mcro-beam a well a natural frequency ncreae. 6. It concluded tat te electrc potental ncreae wt an ncreae n te elatc foundaton. 7. It found tat by ncreang te ma of nano-partcle, te dmenonle natural frequency of ytem decreae. 8 IU, ra Branc

19 Vbraton naly of Magneto-Electro-Elatc Tmoeno 9 8. It een tat wt ncreang of te temperature cange, te dmenonle natural frequency of mcro beam model decreae. Moreover wt ncreang temperature cange te tffne of mcro-beam decreae. 9. It concluded tat by conderng te urface tre effect, te dmenonle natural frequency of ytem decreae. CKOWLEDGEMETS Te autor would le to tan te referee for ter valuable comment. Moreover, te autor are grateful to te Iranan anotecnology Development Commttee for ter fnancal upport and alo grateful to te Unverty of Kaan for upportng t wor by Grant no /. PPEDIX Te contant n Eq. (3) are wrtten a follow: c c e c q c c c c e e e e q q ,, 3 3, 5 5, 3 3 c33 c33 c33 e e q q d d d d , 33 33,, 33 33,, c33 c33 c33 q c e q q,,, c33 c33 c33 (.) In Eq. (3), te normal tre reultant force,, oter ger-order reultant force and moment can be epreed a follow [3-5]:, bendng moment M M, couple moment, Y Y and d, ds,m z d,m z ds, Q d, S S z P P P P d, ds, Y m yd, Y m yds, P S 3 P z P 3 P z () () M () () M zpd, M zp d, M () z () ds () () () () T T () () () () T T () () () () () () M T 3 3 d, T 3 3 ds, () z () d () () () () M T3 3 T3 3 () () () T () T (.) Te contant n Eq.() are defned a te followng form: c, c S, c,, S,I I, I I , b, S (b ), D c I, D c I, E,Q e, q co( z)d S X, Y, T, d, co ( z)d, X 33, Y 33, T33 33, d 33, 33[ n( z ) z ] d E,Q e, q n( z ) z d, E,Q e, q n( z ) z ds, (.3) 8 IU, ra Branc

20 M. Moammadmer and H. Moammad Hooye PPEDIX B Ung Eq. () and (3), a et of lnear ordnary dfferental equaton are obtaned a follow: ( a a ) C U ( a a )( ) C U () () U I I () 3 () C W (a C (a ) ) 6 ( ( 5 (3) () a55 (a ˆ 55 a55 )) C E5) C 8 ( a ( a () a )) C W 55 () ˆ () C W Q5 C ( ) ( ) m e t W w ( ww g W ) ( mc )( ) I I ( )(I I ) ( ) (a ( a55 a () 3 ( ) C a C W ) 6 ( ( ˆ 3 () () () () 3 5 W C () (a a ) (a a ) d d ) C (3) 3 () a55 (a 55 a ˆ ˆ ˆ 55)) C W E5 E 3 E 3) C ( Qˆ Qˆ Q ) C I I Eˆ C Eˆ ( C () () ˆ ˆ Y X Xˆ C C Yˆ () () () 3 ˆ Q5 W C Qˆ C ( C ) () () C C 33 T 33 Y ˆ Tˆ Yˆ ˆ ) (B.) nd for boundary condton, one can be wrtten a te followng form: W U W, at X X X W U W, at X X X (B.) It ould be noted tat ubcrpt and return to grd pont number. REFERECES [] Sun K.H., Km Y.Y.,, Layout degn optmzaton for magneto-electro-elatc lamnate compote for mamzed energy converon under mecancal loadng, Smart Materal and Structure 9: IU, ra Branc

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