A Nw High-Pcision Mod Acclation Mthod fo Calculating Fquncy Rspons of Non-Classically Dapd Systs Jingfang Shn School of Scinc Huazhong Agicultual Univsity Wuhan, China shningfang6@6.co Png Wang School of Scinc Huazhong Agicultual Univsity Wuhan, China wangpng505@foxail.co Abstact h odal tuncation pobl of non-classically dapd systs is constantly ncountd in th dynaic analysis of ngining. h psnt study is dsignd to calculat th fquncy spons functions of th nonclassically daping systs accuatly on account of th Nuann xpansion thoy and th fquncy shifting tchniqu. Considing th fist and th scond t influnc of th Nuann xpansion quations in th fquncy spons analysis of th viscolastic systs, w could coct th odal tuncation pobl of odl displacnt thod. h popty givn in th study shows that this cocting thod can duc th high-od ods that can t b calculatd to th low-od ods that a asi to b coputd. And th poposd thod can also solv th pobl causing by th singulaity of stiffnss atix. h sult of cas givn in th aticl shows that it can ipov th accuacy of haonic spons ffctivly copad odl displacnt. Kywods-haonic spons analysis; fquncy shifting tchniqu; odl displacnt thod I. INRODUCION In any ngining pobls, dynaic analysis, vibation contol, stuctual dsign and daag dtction is always an ipotant pat of it. hus, th dsign of th algoith and o contol duing th pocss to calculat th fquncy spons function plays an ipotant ol. h dsign nds to b quic and accuat fo calculating th syst fquncy spons function and has a gat pactical significanc. With th aount of dgs of th ods considd in dynaic spons analysis incasing, th pocss of coputing all th fquncis spons functions can b xtaodinaily ti consuption. Howv, in fact, th only ods considd in th fquncis spons analysis a th ods locatd in th ang of fquncis of intst. Unfotunatly, sinc th thod nglct th contibution of th high-od ods and th viscolastic ods, th will b so o xisting in th odal tuncation. hus, any odifid thods a poposd to solv th o accounting in th odal tuncation. In cnt dcads, lots of studis hav bn don cntd on th odl duction by using dinsionality duction tchniqus in any sach ointations. Mod displacnt thod is th ost basic thod to solv statically indtinat stuctus fo its sipl calculating pocss and accuat calculation sults. In addition, odl supposition thods also hav an xtnsivly us in stuctual fild. Sinc 9th cntuy, odl duction tchniqu is hot spot in th coputing of fquncy spons functions and th stuctual dynaics spons analysis, th ost coon thod is od supposition thod (MSM) that was psntd by Rayligh []. Howv, this thod can hav so ipovnts of th oiginal MSM by using diffnt vctos in th pocdu of Nuann xpansion []. Caig and Bapton [3] also gav a thod to incas th accuacy by analyzing th nonlina dynaic stability of an actual lag-scal oto-baing syst, which is calld th fixdintfac duction thod fo th fixing bounday of od of syst. Basd on th f vibation ods and th availabl ods of th ngining stuctu, th od displacnt thod [4] hav bn poposd by psnting th displacnt in a haonic way. But this condition will not b always satisfid, so this ind of will not b suitabl fo th focd syst. Mod acclation thod (MAM) is put fowad to solv this pobl by considing th supposition of th availabl ods and th f vibation ods. hfo, th MAM is a static coction thod bcaus of zo fquncy. h xpints showd that od acclation thod can ally nhanc th accuacy of th fquncy spons and siplify calculation of th FRF. But in th al situation, ths constaint conditions will not b always satisfid. It s ans that th o of odal tuncation still xists. o solv this pobl, any scholas stuggl fo it ya aft ya, and hav achivd gatifying succsss. Fo xapl, Maio and Giuspp [5] poposd a odifid thod fo dynaic fquncy spons analysis of th systs in th fnc. And th nuical applications a also showing that th poposd thod can ipov calculating fficincy. Ctainly, so oth coctions thod can also hav a good pfoanc in ipoving th accuacy of dynaic spons including dynaic coction thod [6], high-pcision odal supposition thods, slf-adapting supposition thod, coction continuous systs thods and so on. With th widly us of non-viscous daping to analyz chanical systs and dynaic fquncy spons calculating. h calculating of fquncy spons of nonviscously dapd syst has bco incasingly ipotant. o nhanc th accuacy of th fquncy spons functions 5
atix, th aticls poposd a thod, which tis to stiat th influnc of th ods that usd to b ispctiv and consid th nonviscously systs by taing th fist on o two ts of Nuan xpansion into considation. It s cla that with th nub of ods usd in th odal analysis of viscolastic syst incasing, th odal tuncation o will accuulat gadually. A thod is psnt to solv this pobl by considing th low od and th fis ts contibution in Nuann xpansion. As to th non-popotionally systs, a thod basing on th hybid xpansion is poposd to coput th spons functions of th systs. his study is dvisd to coput th haonic sponss of th availabl ods accuatly. Fo th popty obtaind in th study on account of th Nuann xpansion tho and th fquncy shifting tchniqu, it s vidnt that th high ods fquncy spons function can xpss as quations consisting of th low availabl ods and syst atics. W can us this popty to siplify th high odal tuncation o. Ctainly, w can us this thod to ipov th accuacy of fquncy sponss functions by dividing th fquncy ang into sval subfquncy angs of intst and slcting diffnt valus fo p sub-fquncy ang. II. BACKGROUND OF HORY h quations of otion fo a lina non-viscously dapd syst with zo Initialization, obys th govning quation Mu ( t) Cu ( t) Ku( t) f ( t) N N wh M, C and K R a, th ass daping and stiffnss atics, f (t) is th focing vcto. u (t) is th displacnt vcto. In th snsitivity analysis of dapd systs, u(t) can also b calld as th spons vcto. In od to o aptly dscib th phnonon of oy of solid atial o hystsis ffct, in 874, Boltzann put fowad Boltzann s supposition pincipl of lina viscolastic atials. Lat, in 98, Volta giv th thoy of hysttic o oy in viscolastic hditay atials. So th daping foc can b xpssd as t fd ( t) g( t ) u ( t) d 0 wh gt () is a atix of nl function. Diffnt placs and aas hav diffnt choics of nl function. Ctainly, th thotically how to choos nl function ains unsolvd.. h quations of otion of a lina non-classically dapd syst with zo initial condition is t ut () Mu( t) Kv g( t ) dt Ku( t) f ( t) 0 wh K v which is th daping cofficint atix. gt () is a nl function that has diffnt nas in th diffnt placs. If th loading function is haonic, that is N f ( t) Fh ( s)xp( st) with s iw and Fh R, th stadystat fquncy spons will also b haonic, i.. u( t) Uh( s)xp( st). aing th placs of ut () and f() t in (3), w can obtain ( s M sg( s) K) Uh( s) Fh D( s) Uh( s) Fh H G( s) K L[ g( t)] and L[] V dnots th Laplac tansfo, w now that Gs () can also b xpssd as n c G() s s K v H c and a th laxation paats. And fo th dynaic stiffnss atix, it can b xpssd as D( s) s M sg( s) K h accuat stady-stat fquncy spons that w want to gt can b acquid by utilizing th dict fquncy spons thod. Fo th chaactistic quation dt s M sg( s) K 0 h ignvalu a th oots of it. And wh dnots th th ignvcto and can b wittn in anoth way ( M G( ) K) 0 In addition, asytic-atics pobl ay also ais fo using th stat-spac appoachs. Howv, ths noal ods basd on thos appoachs still hav so o whn coputing th fquncy spons functions paticulaly fo high-dinsionality dapd systs. Futho, w can avoid th convgnc pobl by considing itativ statgy. h coplx FRF atix and th spons vcto can b obtaind by U h Fh Hs () Uh () s ( s ) ( s ) 53
wh Ds () s s D( s) D( s) and s ( ) M G s s s s And this situation is suitabl fo th idal situation that ignvalus a spaatd o non-patd. Fo th coplxity of th non-viscously dapd systs, th odl will b psntd by a lag nub of diffnt quations. It s ans that th odal-tuncation o still xists. o solv this pobl, w intoduc th odal tuncation o. Givn that th fquncy fo th to L th of intst can b coputd, th o of odal tuncation of th odal displacnt thod can b obtaind by () s Fh Fh ( s ) ( s ) L Fo th invs atix, Nuann xpansion can b xpssd in th following way 3 ( IN A) IN A A A N N H A R and I on bhalf of th unit atix. Givn that (9) ts convg condition, th pow-sis xpansion can tnd to th xact sult. h FRF atix psntd can b wittn into th atix fo as H( s) U ( si ) U wh =diag[,,, ], U [,,, ] and =diag[,,, ]. Lt s s and using th Nuann xpansion, th fquncy spons function atix can b xpssd by H( s) U s ( I ) U wh is a coplx fquncy shift constant. h dynaic stiffnss atix Ds () givn in (6), can also b xpssd as Lt D s s M s G s M ( ) ( ) ( )( ( ) ) ( K G( s) M ) K( s) K G( s) M G( s) G( s) M Copaing () and (3), lt s 0( s ), w can gt III. U ( I ) U li K( s) s ( K G( ) M ) A MHOD O IMPROV H ACCURACY OF MOD ACCLRAION MHOD Basd on th f vibation ods and availabl of th stuctu, th od displacnt thod hav bn psntd by using a ti-haonic psntation fo th displacnt of th unfocd syst. But this condition will not b always satisfid, so this ind of will not b suitabl fo th focd syst. Mod acclation thod (MAM) is poposd to duc th odal tuncation o by considing th ffct of high ods. Fo th (6), w can s that this pobl of singula pobl of stiffnss atix hav bn ovco whil incoing th fquncy shift constant. Substituting Ksand () Gs () in (5), th quation can b wittn in th anth way as U ( I ) U ( K G( ) M ) ( G( ) M ) ( K G( ) M ) By using th Nuann xpansion tho and lt s s, th FRF atix givn in (3) can b altnativly xpssd as s Hs () ( ) Whn,, considing th contibution of th fist and scond t of th Nuann xpansion of th high ods, (8) can b psntd in th following way by utilizing th low availabl ods H( s) ( K G( ) M ) ( ) ( ) s H ( s) ( K G( ) M ) ( G( ) M ) ( K G( ) M ) Assuing th fquncy ang fo L th to L th of intst can b calculatd, th spons can b coputd pcisly in th following way 54
L Fh Uh ( s) ( s) L ( s ) h sa pocss as (3), using Nuann xpansion, w can obtain tuncation o of odl displacnt thod, it s no doubt that th o can b ducd in this way. Fo th singulaity of stiffnss atix, th sults in th nub of ts that w can us a ly th fist and th scond on. hus, th spons in (9) can b calculatd can b xpssd by ( s) ( s ) GMAM 0 Fh Fh ( ) ( ) L () L Fh Uh ( s) ( s) ( s ) = L Fh ( s ) () Fo th (), it consids th fist t of th ight-hand quation. Whn 0, th quation can b xpssd as () s Fh Fh ( ) ( ) () L hn giv a fquncy shift valu to (), w can obtain ( ) ( K G( ) M ) L ( G( ) M ) ( K G( ) M ) ( ) (3) W can s that (3) can duc th high ods to th low ods. hat is to say, w can us this quation to iplnt dinsionlity duction. Whn, th abov quation will b wittn in this way Using th upp bound of th th coponnt of th o vcto, w can obtain th blow quation H F F h h s ( ) L ( ) () psnts th th lnt of th vcto. Copaing to th o of gnalizd od acclation thod, w can now that th o is bcoing sall by considing th influnc of th scond t of th Nuann xpansion of th odl displacnt o. IV. XAMPL AND DISCUSS In this psnt pat, on cas of th haonic focd vibation of non-classically dapd syst is shown, which is a fou DOF nonviscously syst with f-f bounday condition []. () s Fh Fh ( ) ( ) L L Fh Fh ( s ) ( ) L ( ) (4) Figu. Fou DOF non-classically dapd syst with f-f bounday condition In od to coput fficintly, w a th fist t and th scond t. hat is Fh Fh ( ) ( ) L F F s s h h ( ) ( ) ( ) L ( ) (5) h syst atics of th nonviscously dapd od, shown in Fig., a M, K and G. It s obvious that th ngy dissipation is not unifoly distibutd in th whol syst. hat is say, th syst is a non-classically dapd syst. Suppos th intsting fquncy ang is -8 ad/s. Accoding to th psnt tho in [], th fquncy shift valu is 0i. Fou lastic ods a covd in th fquncy ang of intst. In th quation, all pats can b coputd. By considing th influnc of th scond t of th 55
Plas s Stp 9 fo oding pints of you pap. Rpints ay b odd using th fo povidd as <pint.doc> o <pint.pdf>. ACKNOWLDGMN his wo was suppotd by th Fundantal Rsach Funds fo th Cntal Univsitis (6607JC04), National univsity studnts innovation poct (05300073, 060504076) and High school univsity athatics taching sach and dvlopnt cnt poct (CMC060408). hans to Png Wang who is th cosponding autho. Figu. h FRF of th od in th fquncy ang 0-30 ad/s Fig. shows that th FRF of th scond DOF in od ov th fquncy ang of intst. Sinc th fquncy shift valu 0i, it s vidnt that th odal tuncation o causd by th odl displacnt thod can b ducd whn th considing fquncy is locatd in th fquncy ang of intst. Fo xapl, in 4-30 ad/s, th cocting thod psnting in this study can hav a btt accuacy that th gnalizd acclation thod poposd by Li t al. in []. hat is say th sults hav a btt pfoanc whn th fquncy tnds to th fquncy shift. V. CONCLUSIONS You ust subit th I lctonic Copyight Fo (CF) p Stp 7 of th CPS autho it s wb pag. his fo Must b Subittd in Od to Publish You Pap. RFRNCS [] J.W.S. Rayligh, h hoy of Sound, Dov Publications, Nw Yo, 945. [].L. Wilson, M-W. Yuan, J.M. Dicns, Dynaic analysis by dict supposition of Ritz vctos, athqua ngining & Stuctual Dynaics, 98. [3] R.R. Caig J., M.C.C. Bapton, Coupling of substuctus fo dynaic analyss, 968. [4] B. Bsslin, U. aba, A. Lutowsa, N. van d Wouw, H. Nii, D.J. Rixn, M.. Hochstnbach, W.H.A. Schilds, A copaison of odl duction tchniqus fo stuctual dynaics, nuical athatics and systs and contol, 03. [5] M. Di Paola, G. Failla, A coction thod fo dynaic analysis of lina systs, Coput. Stuct. 8 (004) 7 6. [6] G. Boino, G. Muscolino, Mod-supposition thods in dynaic analysis of classically and non-classically dapd lina systs, athqua ngining & Stuctual Dynaics 4 (986) 705 77. 56