TRANSIENT RESPONSE OF A THREE LAYER SANDWICH PLATE WITH ELECTRORHEOLOGICAL CORE AND ORTHOTROPIC FACES

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Carmel Gilbert
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1 RANSIEN RESPONSE OF A HREE LAYER SANDWICH PLAE WIH ELECRORHEOLOGICAL CORE AND ORHOROPIC FACES Jall Rapaand and Lotfolla Palaan Dpartmnt of Mcancal Engnrng Frdos Unrst of Masad Iran jrap@frdos.m.ac.r ABSRAC prsnt ork s as nstgaton of t transnt rspons of a rctanglar tr lar sandc plat t Elctro-rologcal (ER) cor and ortotropc fac sts. ER flds c ar a class of smart matrals bt rrsbl cangs n tr mcancal proprts n t ar sbjctd to an lctrc fld. B applng an lctrc fld to t cor lar s cangd t dnamc caractrstcs of t strctr t ER lar can b sd for bratons spprssng and rdcng t sttlng tm of t sandc plat sbjctd to a transnt ctaton. In ts std an FE modl of t strctr s dlopd and a modfd drct ntgraton algortm s sd to smlat t mpls rspons of t proposd sandc plat. drd fnt lmnt modl and t solton procdr ar programmd as a cod. Snc t baor of lctro-rologcal matrals ar dffrnt n t pr-ld and post-ld rgons t s ncssar to dntf t stat of t ER matral trogot t solton tm. Accordng to nmrcal rslts c ar compatbl t pros prmntal obsratons t prodcd sar stans n t cor s gratr tan t rqrd stran for tng t cor matral from ts pr-ld rgm. Hnc t cor s modld as a Bngam plastc matral. Basd on t proposd modl ffcts of cangs n t appld lctrc fld on t strctr sttlng tm and ts natral frqncs ar rprsntd for dffrnt cor tcknsss.. INRODUCION Elctrorologcal (ER) flds c ar a knd of smart matrals t controllabl rologcal proprts. Wn an lctrc fld s appld to ts flds t rspond b formng can-lk strctrs c rslts n an nancmnt of tr rologcal proprts ncldng scost ld strss and sar modls b sral ordrs of magntd. ER cltcs [ ] and jornal barngs [] ar som ampls for tns applcatons of ER flds. Bt t cptonal caractrstc of ER flds s tr tpcal rspons tm c s of t ordr of a f mllsconds. In fact ts nrnt fatr of ts flds as t man rason for trmndos rsarc actts n dlopmnt of tr tns applcatons n sstms t controllabl dampng [4 5]. aros matmatcal modls a bn ntrodcd b rsarcrs to prsnt t dnamc baor of ER flds n t pr-ld and t post-ld stags [6-8]. Eprmntal nstgatons mpl tat f a small ctaton gan appld to an ER fld t fld dos not pass ts pr-ld stag and tradtonal lnar braton tors can b modfd to dscrb t braton caractrstcs of t sstm. On t otr and gr ctaton gans old b abl to cas t ER fld ntrng ts ld rgon and consqntl mak t sstm rspons nonlnar. Bngam modl as bn domnantl sd b rsarcrs for dscrbng t post-ld baor of ER flds [9 ]. Accordng to ts modl t ER tratd sstm lds a non-armonc dsplacmnt rspons ndr a snsodal ctaton forc. Hang tndd t applcatons of ER flds t braton [] and t dnamc stablt [] problms for bams and plats t ER cor r nstgatd. In sc strctrs applng an lctrc fld to lastc lars cangs t mcancal proprts

2 of t cor lar and consqntl nflncs t dnamc proprts of t strctr. Lnar scolastc modls c a bn dl sd b pros rsarcrs n modlng of ER fld-basd strctrs can onl dscrb t pr-ld baor of ts matrals [9 ]. Usng ts tors for prssng t sar modls and t loss factor of ER flds a grot of som ndrd prcnts n natral frqncs of a sandc strctr ar prdctd b ncrasng t strngt of t appld lctrc fld [4]. Hor som prmntal alatons bt no sgnfcant cang n rsonant frqncs of sc a sstm []. orgn of ts dscrpancs can b t mc lor sar modls and ld strss of aalabl ER flds c can asl cas t matral racs ts ld pont n comparson t otr scolastc dampng matrals sc as rbbr. rfor scolastc modlng of ER flds n dampng lar tratmnt applcatons t rlatl g ctaton gans dos not sm to b approprat d to t stnc of t grat nconsstnc t prmntal rslts. In contrast to most of t arlst stds on armonc analss of sandc strctrs t ER cor lar for c Kn scolastc modls r sd Rapaand and Palaan [5] proposd a fnt lmnt modl of a tr lar bam strctr basd on t Bngam plastc modl c as capabl of smlatng t post-ld baor of ER flds n a sandc bam. In tr dnamc modl a drct tm-ntgraton-basd algortm ad bn appld to modl t transnt rspons of a cantlr sandc bam. In t prsnt ork and n contn of or pros std [5] t transnt rspons of a cantlr sandc sll strctr t an ER cor and ortotropc fac sts sbjctd to an mpls load as an arbtrar ctaton s nstgatd. modfd drct ntgraton mtod as bn sd n alaton of t nflnc of t appld lctrc fld on t strctr sttlng tm and t natral frqncs of t strctr for dffrnt tcknsss of t ER cor.. BINGHAM MODEL FOR HE POS-YIELD BEHAIOR OF ER FLUIDS pcal ER flds ar collodal flds of t partcl dsprson tp btng t caractrstcs of Bngam flds n tr post-ld stag. An ER fld s cangd from a nar Ntonan fld n c ts partcls ar fr to mo n t absnc of lctrc fld to t Bngam plastc baor n c ts partcls ar algnd n a can b applng t lctrc fld. ld strss of ts matrals ars t t applcaton of an lctrc fld accordng to t follong qaton [6] : τ ( E) η & τ ( E) () p p E τ p (Pa) and η p (Pa.s) ar t sar strss and t basc scos coffcnt of ts fld. (/s) E (/mm) and τ E (Pa) ar t sar rat t lctrc fld and t flctatng strss rspctl. Consqntl n ts flds ar sd n an ER dc t can b confgrd as a mcancal mpdanc ang a arabl frcton dampr.. DESCRIPION OF HE PROBLEM Consdr a rctanglar tr-lar sandc sll as son n Fgr n c an ER lar s rstrctd b to ortotropc lars a constranng lar at top and a bas lar at bottom. Matmatcal modl of ts strctr s constrctd basd on t follong assmptons: - ER cor bas as a Bngam plastc matral. - Rotar nrta and sar dformaton of t lastc lars ar nglgbl. - No normal strss can b prodcd n t cor lar. 4- No slppng occrs btn t ngbor lars.

3 5- transrs dsplacmnts of all ponts on a cross-scton ar qal. 6- matral ln n t cor lar old b stragt aftr dformaton (as son b lns B-C and B / - C / n Fgr and smlarl for - plan). Fgr. rctanglar sandc sll t ER cor and ortotropc fac sts. Accordng to t stran-dsplacmnt rlatons of t ortotropc lars and t gomtrcal caractrstcs and proprts of t lars t lastc strans of t constranng and t bas lars (ndd b and rspctl) can b prssd as follos. { } [ ]{ } L ε ε ε () { } [ ]{ } L ε ε ε () In ts rlatons and ar t dsplacmnts of t md-plan of t t lar n t and drctons rspctl and s t dstanc of t md-gt of lar. Also dnots t transrs dsplacmnt of t lars. Consdrng t gomtr t follong rlatonsp for t sar strans of t cor lar can b obtand. (4) Accordng to Fgr t dsplacmnt gradnts n Eqaton (4) can b rplacd b t follong qalnt prssons for t cor sar stans n trms of t dsplacmnts of t lastc lars. (5)

4 Fgr. Confgraton of t lars bfor and aftr dformaton. (6) Sbstttng t abo qaton nto Eqaton (4) t sar strans of t ER lar can b rttn as: { } { } [ ]{ } Q d d (7) r ( ). d (8) 4. FE MODELING OF HE PROBLEM Consdr a for-nod mlt-lar sll lmnt as son n Fgr n c t transrs dsplacmnts of all lars ar qal. Eac nod as 7 dgrs of frdom. Accordngl t n-plan and t transrs dsplacmnts of t sandc plat can b rttn n trms of a nodal dsplacmnt ctor and a sap fncton matr r ts componnts ar common sll sap fnctons at ts cornrs. { } [ ] )} ( { ) ( ) ( t q N t (9) total nrg of t lmnt π can b rttn: U π () r and ar t kntc and t potntal nrgs of t lmnt rspctl. Also U dnots t ork don b trnal forcs. Wl t fac sts ar assmd to a no sar dformaton t stran nrg of t lastc lars s as follos.

5 s { } [ E ]{ } d { } [ E ]{ }d ε ε ε ε () total stran nrg of ac lar can b rttn as: { ( t)} [ D ( ) ]{ q( t) } { ( t)} [ D ( ) ]{ q( t) } Wr [ D ()] [ L][ N() ] and [ D ()] [ L][ N() ] n c ε () ε () s. rfor { q} [ D ] [ E ][ D ]{ q} d { q} [ D ] [ E ][ D ]{ q}d (4) Fgr. A tr lar sll lmnt t 4 nods and 7-DOF pr nod. E υ E υ υ υ υ υ E E [ E ] (5) υ υ υ υ G E and E ar t ortotropc lastct modls υ and υ ar t ortotropc Posson s ratos and G s t sar modls. As dscrbd bfor accordng to Bngam plastc modl t sar strss of a partcl tp ER fld cold b prssd as Eqaton (). rfor t total nrg lost b t ER cor lar can b rttn as: E r { } { τ E) } d { } η { }d E ( p & (6) Sbsttton of Eqaton () nto Eqaton (7) rslts n: and ts { ( t) } [ Q][ N]{ q} [ R]{ q} {& ( t) } [ R]{ q& } (7) (8) E r E { q} [ R] { ( E) } d { q} [ R] η [ R]{ q& }d τ (9) Nglctng t rotar nrta of t lastc lars tr kntc nrg d to tr nplan and tr transrs dsplacmnts can b rttn as: { } ρ { & } d { } ρ { & }d () p

6 { q} [ N] [ N]{ q& } d { q} [ N] [ N]{ q& }d ρ ρ () & On t otr and t kntc nrg of t cor lar nclds an nrg trm assocatd t transrs dsplacmnt and anotr nrg trm assocatd t rotaton of lmnts. c { } ρ { & } d { } [ J ]{ & }d () r [J ] s t mass momnt of nrta matr for a dfnt olm of ER cor. Its dmnsons ar dfnd b t lmnt lngt and dt and t cor tcknss. rfor c { q} [ N] [ H] ρ [ H][ N]{ q& } d { q} [ R] [ J ][ R]{ q& }d () n c H [ ]. Fnall t ork don b trnal forcs f p can b rttn as follos: { } { f p} { q} [ N] { f } W (4) p Accordng to t Hamlton s prncpl for δπ t gornng qaton of moton for a dampd sandc sll lmnt can b prssd n t follong form n c [ ]{ q& } [ C ]{ q& } [ K ]{ q} { F } { F } & (5) M ER t. [ M ] [ N] ρ[ N] d [ N] [ H] [ H][ N] ρ [ R] [ J ][ R] d [ N] ρ[ N]d [ C ] [ R] p[ R]d [ K ] [ D] [ E][ D] d [ D] [ E][ D]d { FER } [ R] τ E ( E) { sgn( & )}d { F } [ N] { f } d (6) η (7) (8) (9) () t. p sgn(& n Eqaton (9) s appard bcas of t natr of ER cor lar c prodcs onl a rsstant forc aganst t moton. It sold b notd tat { )} 5. DIREC INEGRAION MEHOD Drct ntgraton or plct ntgraton mtod s an ffct gnral algortm for solng dnamc problms. In ts mtod t total solton tm s ddd nto sral ntrals. In ac ntral t procdr prforms a complt FE analss. Sbsttton of qalnt dffrnc formlas for loct and acclraton ctors nto t qaton of moton rslts n a rcrs formla tat spcfs t dsplacmnt ctor at tm t t t rspct to dsplacmnt als at tm t and tm t- t [7]. ( t) [ M ] [ C] { U t ( t) t { f ( t) } [ K ] [ M ] { U } [ M ] [ C] { U } ( t) t t } t t t ()

7 Dtals of t modfd procdr can b fond n [5 7]. drd fnt lmnt modl and t solton mtod ar programmd as a cod. In t nt scton ffcts of som paramtrs n braton rspons of ts sandc sll ar stdd. 6. EALUAION OF HE RANSIEN MEHOD Ong t lack of t pblsd orks on dnamc rspons of sandc strctrs t ER cor basd on Bngam modl t fnt lmnt modl and corrspondng plct solton algortm r alatd dstnctl. Ganng t confdnc of t corrctnss of t procdr and t comptr program b alatng aros cass ncldng sandc slls and bams t transnt baor of a modl sandc sll t ER cor s obtand and t rslts ar rprsntd n t nt scton. abl. Matral proprts of ortotropc fac sts. E 7 Gpa E 58 Gpa υ. - υ. - ρ 75 kg/m G 6.9 Gpa 7. RESULS AND DISCUSSION For nmrcal analss a tr lar sandc cantlr plat t ortotropc facs s consdrd. fac sts ar mad p of Almnm Od t mcancal proprts lstd n abl. abl ndcats t stctr gomtrs. As an arbtrar ctaton a nt mpls load appld at a fr cornr of t plat s consdrd. partcl tp ER fld sd for ts std conssts of cmcal starc as partcls and slcon ol as bas lqd. τ E (E) s t fld-dpndnt ld strss of t ER fld c s gn b αe β. α and β ar constants of t ER fld. als of α 47 and β. ar dtrmnd prmntall b Jng t al. [8]. abl. Gomtrs and matrals proprts. Plat lngt 5 mm Plat dt mm cknss of t constranng lar mm cknss of t bas lar mm cknss of t ER lar mm Fgr 4. Dpndnc of t strctr natral frqncs on t cor tcknss rgardlss of t appld lctrc fld n mm.

7. EFFECS OF HE CORE HICKNESS AND HE APPLIED ELECRIC FIELD ON NAURAL FREQUENCIES OF HE SYSEM Dpndnc of t frst to natral frqncs of t sstm on tcknss of t cor lar s llstratd n Fgr 4.

8 7. EFFECS OF HE CORE HICKNESS AND HE APPLIED ELECRIC FIELD ON NAURAL FREQUENCIES OF HE SYSEM Dpndnc of t frst to natral frqncs of t sstm on tcknss of t cor lar s llstratd n Fgr 4. Applng aros lls of lctrc fld nto t ER lar c bas as a Bngam fld fond to a no ffct on natral frqncs of t strctr. In fact applng an lctrc fld can onl mpro t dampng tratmnt of t sstm not ts stffnss. Snc t qalnt mass of t sstm gros p b ncrasng t tcknss of t cor lar t natral frqncs of t strctr ar pctd to b dcrasd. 7. EFFECS OF HE ELECRIC FIELD SRENGH ON DYNAMIC BEHAIOR OF HE SRUCURE Inflnc of applng lctrc fld on dnamc rspons of t proposd strctr s prsntd n Fgr 5. In ts cas t gomtrs ar accordng to abl. In ordr to std t ffcts of appld lctrc fld aros oltags ar appld to t facs. graps llstrat t dsplacmnt of t plat cornr on c t trnal forc as bn rtd. (a) (b) (c) (d) Fgr 5. Impls rspons of t strctr for dffrnt appld lctrc flds: (a) ro (b) (c) and (d) K/mm. Accordng to Fgr 5-a applng no lctrc fld dos not a an consdrabl nflnc on braton spprsson tm. As son n Fgr 5 t dampng baor of t sstm mpros b applng gr lls of lctrc fld and ts a sortr sttlng tm s acd. It can b concldd tat t scos dampng componnt of t cor as nglgbl ffct (Fgr 5-a) n comparson t t dampng componnt assocatd t applng lctrc fld (Fgr 5-d). scos dampng forc of ER flds acts smlar to a scos dampr n a bratng sstm. Hor t otr trm of t dampng forc prodcd n t rslt of applng lctrc fld can b confgrd as a colomb dampr accordng to Eqaton

9 (). Snc t lctrc fld-casd dampng forc s domnatng n t sstm a lnar dca of braton ampltd sts nstad of a combnaton of a lnar and an ponntal dca. Fgrs 5-c and 5-d clarl so t colomb dampng baor of t ER cor lar. 7. EFFEC OF HE ELECRIC FIELD SRENGH AND HE CORE HICKNESS ON HE SYSEM SELING IME nrg lost n t cor lar ncrass b rasng t tcknss of t cor lar. Consqntl t qalnt dampng of t strctr s pctd to b mprod. As Fgr 6 llstrats nancmnt of t dampng proprts of t strctr can b acd b ncrasng t tcknss rato of t cor lar. Hor d to t grot of t mass and t mass momnt of nrta of t sstm and also t rdcton of t sar dampng forcs b ncrasng t ER lar tcknss strongr lctrc flds as lss ffct n gr als of t cor tcknss. Fgr 6. Dpndnc of t sstm sttlng tm on t tcknss of t cor lar ( mm). 8. CONCLUSION ransnt rspons of a tr lar cantlr sandc plat t an ER cor sbjctd to an mpls ctaton as stdd. A fnt lmnt modl for t bam as bn constrctd and drct ntgraton algortm as bn sd to smlat t tmdpndnt baor of t strctr. D to t consdrabl lo ld strss of ER flds and tr caractrstcs n t post-ld rgon t Bngam plastc modl as sd n draton of FE modl of sandc bam t ER cor. rslts prsntd rn ndcatd tat t appld lctrc fld and t tcknss of t cor lar a consdrabl nflnc on dampng tratmnt of t strctr. Wl t lctrc fld als r fond to a no consdrabl ffct on t sstm natral frqncs at a spcfd cor tcknss t as obsrd tat t sttlng tm of t strctr dcrass b ncrasng t tcknss of t cor lar. s can b pland b t grot of t nrg lost b t ER lar. 9. REFERENCES. Papadopolos C.A Braks and cltc sng ER flds Jornal of Mcatroncs 8: Nakamra. N. Saga and M. Nakaaa. 4. "arabl scos Control of a Homognos ER Fld Dc Consdrng Its Dnamc Caractrstcs" Jornal of Mcatroncs 4: Dmarogonas A. D. and A. Kollas. 99. Elctro-Rologcal Fld-Controlld Smart Jornal Barngs rbolog ransactons; 5:65-68.

10 4. Hong S.R S.B. Co and S. Han.. "braton Control of a Fram Strctr Usng Elctro-Rologcal Fld Monts" Intrnatonal Jornal of Mcancs Scnc 44: Palaan L. and J. Rapaand. 7. " Dnamc Rspons Analss and braton Control of a Cantlr Bam t a Sq-mod ER Dampr" Jornal of Smart Matrals and Strctrs 6: Stangroom J. E. 98. Elctrorologcal flds" Pscs n cnolog 4: Kt D. J. Wss D. Carlson and A. Non scolastc Proprts of Magnto- and Elctro-Rologcal Flds Jornal of Intllgnt Matral Sstms and Strctrs : Gand F. and W. A. Bllog. 5. On t Pnomnologcal Modlng of Elctrorologcal and Magntorologcal Fld Pr-ld Baor" Jornal of Intllgnt Matral Sstms and Strctrs 6: Kt D. J. Wss D. Carlson and J. P. Coltr. 99. Matral Aspcts of Elctrorologcal Sstms " Jornal of Intllgnt Matral Sstms and Strctrs 4:-4.. Haqng G. L. M. Kng and. B. Cr. 99. "Inflnc of a Locall Appld Elctro-Rologcal Fld Lar on braton of a Smpl Cantlr Bam" Jornal of Intllgnt Matral Sstms and Strctrs 4: Srkanta A. and K. nkatraman.. "braton Control of Sandc Bams Usng Elctro-Rologcal Flds" Jornal of Mcancal Sstms and Sgnal Procssng 7: Y J. Y. L. W. Cn and C. C. Wang. 6. "Dnamc stablt analss of a rctanglar ortotropc sandc plat t an lctro-rologcal fld cor" Jornal of Compost Strctrs 7:-4. Haqng G. and L. M. Kng "braton Caractrstcs of Sandc Bams Partall and Fll ratd t Elctro-Rologcal Fld" Jornal of Intllgnt Matral Sstms and Strctrs 8: Y J. Y L. W. Cn and C. C. Wang. 4. "Dnamc Stablt of a Sandc Bam t a Constrand Lar and Elctrorologcal fld Cor " Jornal of Compost Strctrs 64: Rapaand J. and L. Palaan. 8. " ransnt rspons of sandc bams t lctrorologcal cor" Jornal of Intllgnt Matral Sstms and Strctrs Accptd for pblcaton. 6. Wss K.D. Carlson D. and Coltr J.P. 99. "Matral Aspct of Elctrorologcal Sstms" Jornal of Intllgnt Matral Sstms and Strctrs 4:-. 7. Palaan L. 7. "braton Control of Strctrs Usng Elctrorologcal Flds" Mastr ss Dpartmnt of Mcancal Engnrng Frdos Unrst of Masad Iran. 8. Jng W. J. W.B. Jong S. R. Hong and S. B. Co. 4. "braton Control of a Flbl Bam Strctr Usng Sq-Mod ER Monts" Jornal of sond and braton 7:85-99.

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