13. DYNAMIC ANALYSIS USING MODE SUPERPOSITION

Transcription

1 . DYAMI AALYI UIG MODE UPEPOIIO he Mode hes used o Ucoule he Dmc Equlrum Equos eed o Be he Exc Free-Vro Mode hes. EQUAIO O BE OLVED { XE "Mode hes" }{ XE "Mode ueroso Alss" }{ XE "Pece-Wse Ler Lodg" }he dmc force equlrum Equo.4 c e rewre he followg form s se of d secod order dfferel equos: Mu u Ku F J f g. All ossle es of me-deede lodg, cludg wd, wve d sesmc, c e rereseed sum of J sce vecors f, whch re o fuco of me, d J me fucos g. { XE "c odeso" }he umer of dmc degrees of freedom s equl o he umer of lumed msses he ssem. M ulcos dvoce he elmo of ll mssless dslcemes sc codeso efore soluo of Equo.. he sc codeso mehod reduces he umer of dmc equlrum equos o solve; however, c sgfcl crese he des d he dwdh of he codesed sffess mrx. I uldg e

2 - AI AD DYAMI AALYI srucures, whch ech dhrgm hs ol hree lumed msses, hs roch s effecve d s uomcll used uldg lss rogrms. For he dmc soluo of rrr srucurl ssems, however, he elmo of he mssless dslceme s, geerl, o umercll effce. herefore, he moder versos of he AP rogrm do o use sc codeso o re he srseess of he sffess mrx.. AFOMAIO O MODAL EQUAIO he fudmel mhemcl mehod h s used o solve Equo. s he sero of vrles. hs roch ssumes he soluo c e exressed he followg form: u ΦY. Where Φ s d mrx cog sl vecors h re o fuco of me, d Y s vecor cog fucos of me. From Equo., follows h: u Φ Y d u ΦY. d.c { XE "Orhogol odos" }Before soluo, we requre h he sce fucos ssf he followg mss d sffess orhogol codos: Φ M Φ I d Φ KΦ Ω. { XE "Geerlzed Mss" }{ XE "Mss, Geerlzed" }where I s dgol u mrx d Ω s dgol mrx whch he dgol erms re. he erm hs he us of rds er secod d m or m o e free vro frequeces. I should e oed h he fudmels of mhemcs lce o resrcos o hose vecors, oher h he orhogol roeres. I hs ook ech sce fuco vecor,, s lws ormlzed so h he Geerlzed Mss s equl o oe, or M..

3 MODE UPEPOIIO MEHOD - Afer susuo of Equos. o Equo. d he re-mullco Φ, he followg mrx of equos s roduced: J IY dy Ω Y g.4 { XE "Modl Prco Fcors" }where Φ f d re defed s he modl rco fcors for lod fuco. he erm s ssoced wh he h mode. oe h here s oe se of modl rco fcors for ech sl lod codo f. { XE "Dmg Mrx" }{ XE "Dmg:lsscl Dmg" }{ XE "Dmg:Modl Dmg" }{ XE "Modl Dmg" }For ll rel srucures, he mrx d s o dgol; however, o ucoule he modl equos, s ecessr o ssume clsscl dmg where here s o coulg ewee modes. herefore, he dgol erms of he modl dmg re defed : d ζ.5 where ζ s defed s he ro of he dmg mode o he crcl dmg of he mode []. A cl ucouled modl equo for ler srucurl ssems s of he followg form: J ζ g. For hree-dmesol sesmc moo, hs equo c e wre s: ζ u u u.7 x gx g z gz { XE "Erhquke Exco Fcors" }where he hree-drecol modl rco fcors, or hs cse erhquke exco fcors, re defed - M whch s equl o x, or z d s he mode umer. oe h ll mode shes hs ook re ormlzed so h M.

4 -4 AI AD DYAMI AALYI. EPOE DUE O IIIAL ODIIO OLY { XE "Il odos" }Before reseg he soluo of Equo. for vrous es of lodg, s covee o defe ddol coss d fucos h re summrzed le.. hs wll llow m of he equos reseed oher rs of hs ook o e wre comc form. Also, he oo reduces he edum volved he lgerc dervo d verfco of vrous equos. I ddo, wll llow he equos o e form h c e esl rogrmmed d verfed. If he suscr s droed, Equo. c e wre for cl mode s: ξ.8 whch he l modl dslceme d veloc re secfed s resul of revous lodg cg o he srucure. oe h he fucos d gve le. re soluos o Equo.8. { XE "Dmc esose Equos" }le. ummr of oo used Dmc esose Equos

5 MODE UPEPOIIO MEHOD -5 OA D ξ ξ ξ ξ ξ ξ D D FUIO ξ e s e cos D ξ D D D A ξ A D he soluo of Equo.8 c ow e wre he followg comc form: A.9 A hs soluo c e esl verfed ecuse ssfes Equo.8 d he l codos..4 GEEAL OLUIO DUE O ABIAY LOADIG { XE "Arrr Dmc Lodg" }{ XE "Dmc esose Equos" }{ XE "Dmg:umercl Dmg" }{ XE "umercl Dmg" }{ XE "me Icreme" }here re m dffere mehods vlle o solve he cl modl equos. However, he use of he exc soluo for lod, roxmed oloml wh smll me creme, hs ee foud o e he mos ecoomcl d ccure mehod o umercll solve hs equo wh comuer rogrms. I does o hve rolems wh sl, d does o roduce umercl dmg. Becuse mos sesmc groud cceleros re

6 - AI AD DYAMI AALYI defed s ler wh.5 secod ervls, he mehod s exc for hs e of lodg for ll frequeces. Also, f dslcemes re used s he sc u, he lod fuco derved from ler cceleros re cuc fucos wh ech me ervl, s show Aedx J. o smlf he oo, ll lods re dded ogeher o form cl modl equo of he followg form: ζ. where he modl lodg s ece-wse oloml fuco s show Fgure.. oe h he hgher dervves requred he cuc lod fuco c e clculed usg he umercl mehod summrzed Aedx J. herefore, he dfferel equo o e solved, wh he ervl o, s of he followg form for oh ler d cuc lod fucos: ζ.

7 MODE UPEPOIIO MEHOD -7 - ervl - o For ler lodg wh ervl For cuc lodg. wh ervl where d re secfed me Fgure. Modl Lod Fucos From he sc heor of ler dfferel equos, he geerl soluo of Equo. s he sum of homogeeous soluo d rculr soluo d s of he followg form: 4 5. he veloc d ccelero ssoced wh hs soluo re: 4 5..c 5

8 -8 AI AD DYAMI AALYI hese equos re summrzed he followg mrx equo: B I s ow ossle o solve for he coss. he l codos re d. herefore, from Equos. d. 4 D ϖ. he susuo of Equos.,. d.c o Equo. d seg he coeffces of ech oloml erm o e equl roduce he followg four equos: : : : :. hese sx equos, gve Equos. d., c e wre s he followg mrx equo: D ϖ or,.4

9 MODE UPEPOIIO MEHOD -9 herefore,.5 he verso of he uer-rgulr mrx c e formed lcll; or c esl e umercll vered wh he comuer rogrm. Hece, he exc soluo me o of modl equo ecuse of cuc lod wh he me se s he followg: B A. { XE "esose ecrum Alses:umercl Evluo" }{ XE "Duhmel Iegrl" }{ XE "me Icreme" }Equo. s ver smle d owerful recursve relosh. he comlee lgorhm for ler or cuc lodg s summrzed le.. oe h he A mrx s comued ol oce for ech mode. herefore, for ech me creme, roxmel mullcos d ddos re requred. Moder, exesve ersol comuers c comlee oe mullco d oe ddo roxmel - secods. Hece, he comuer me requred o solve ses er secod for 5 secod duro erhquke s roxmel. secods. Or modl equos c e solved oe secod of comuer me. herefore, here s o eed o cosder oher umercl mehods, such s he roxme Fs Fourer rsformo Mehod or he umercl evluo of he Duhmel egrl, o solve hese equos. Becuse of he seed of hs exc ece-wse oloml echque, c lso e used o develo ccure erhquke resose secr usg ver smll mou of comuer me.

10 - AI AD DYAMI AALYI { XE "Algorhms for:oluo of Modl Equos" }le. Hgher-Order ecursve Algorhm for oluo of Modl Equo I. EQUAIO O BE OLVED: ξ II. IIIAL ALULAIO ξ D ξ ξ ξ ξ ξ D D s e D ξ cos e D ξ D D... B..... ϖ D d B A III. EUIVE OLUIO,.. c. A d. d reur o III.

11 MODE UPEPOIIO MEHOD -.5 OLUIO FO PEIODI LOADIG { XE "ecurrece oluo for Arrr Dmc Lodg" }{ XE "Perodc Dmc Lodg" }{ XE "Wve Lodg" }{ XE "Wd Lodg" }he recurrece soluo lgorhm summrzed Equo. s ver effce comuol mehod for rrr, rse, dmc lods wh l codos. I s ossle o use hs sme smle soluo mehod for rrr erodc lodg s show Fgure.. oe h he ol duro of he lodg s from o d he lodg fuco hs he sme mlude d she for ech cl erod. Wd, se wve d cousc forces c roduce hs e of erodc lodg. Also, dmc lve lods o rdges m e of erodc form. F Me Wd Pressure me Fgure. Exmle of Perodc Lodg For cl duro of lodg, umercl soluo for ech mode c e evlued lg Equo. whou l codos. hs soluo s correc ecuse does o hve he correc l codos. herefore, s ecessr for hs soluo o e correced so h he exc soluo z hs he sme dslceme d veloc he egg d ed of ech lodg erod. o ssf he sc dmc equlrum equo, he correcve soluo x mus hve he followg form: x x A x A.7 where he fucos re defed le..

12 - AI AD DYAMI AALYI he ol exc soluo for dslceme d veloc for ech mode c ow e wre s: z x.8 z x.8 o h he exc soluo s erodc, he followg codos mus e ssfed: z z.9 z z.9 he umercl evluo of Equo.4 roduces he followg mrx equo, whch mus e solved for he ukow l codos: A A x A A. x he exc erodc soluo for modl dslcemes d veloces c ow e clculed from Equos.8 d.8. Hece, o ecessr o use frequec dom soluo roch for erodc lodg s suggesed mos ex ooks o srucurl dmcs.. PAIIPAIG MA AIO { XE "Mss Prco os" }{ XE "Prcg Mss os" }everl Buldg odes requre h les 9 erce of he rcg mss s cluded he clculo of resose for ech rcl dreco. hs requreme s sed o u se ccelero rculr dreco d clculg he se sher due o h lod. he sed se soluo for hs cse volves o dmg or elsc forces; herefore, he modl resose equos for u se ccelero he x-dreco c e wre s:. x

13 MODE UPEPOIIO MEHOD - he ode o er forces he x-dreco for h mode re defo: f x Mu M M. x he ressg se sher he x-dreco for mode s he sum of ll ode o x forces. Or: V x - x x I x M. he ol se sher he x-dreco, cludg modes, wll e: x V x.4 For u se ccelero dreco, he exc se sher mus e equl o he sum of ll mss comoes h dreco. herefore, he rcg mss ro s defed s he rcg mss dvded he ol mss h dreco. Or: X Y Z mss mss mss x m m x m z z.5.5.5c { XE "Mss Prco ule" }If ll modes re used, hese ros wll ll e equl o.. I s cler h he 9 erce rco rule s eded o esme he ccurc of soluo for se moo ol. I co e used s error esmor for oher es of lodg, such s o lods or se dslcemes cg o he srucure.

14 -4 AI AD DYAMI AALYI Mos comuer rogrms roduce he coruo of ech mode o hose ros. I ddo, exmo of hose fcors gves he egeer dco of he dreco of he se sher ssoced wh ech mode. For exmle, he gle wh resec o he x-xs of he se sher ssoced wh he frs mode s gve : x θ..7 AI LOAD PAIIPAIO AIO { XE "c Lod Prco os" }For rrr lodg, s useful o deerme f he umer of vecors used s deque o roxme he rue resose of he srucurl ssem. Oe mehod, whch he uhor hs roosed, s o evlue he sc dslcemes usg ruced se of vecors o solve for he resose resulg from sc lod ers. As dced Equo., he lods c e wre s: J F f g.7 Frs, oe solves he scs rolem for he exc dslceme u ssoced wh he lod er f. he, he ol exerl work ssoced wh lod codo s: E f u.8 From Equo., he modl resose, eglecg er d dmg forces, s gve : f.9 From he fudmel defo of he mode sueroso mehod, ruced se of vecors defes he roxme dslceme v s:

15 MODE UPEPOIIO MEHOD -5 v f. he ol exerl work ssoced wh he ruced mode she soluo s: E f f v. A sc lod rco ro r c ow e defed for lod codo s he ro of he sum of he work doe he ruced se of modes o he exerl ol work doe he lod er. Or: r E E L f u. If hs ro s close o., he errors roduced vecor ruco wll e ver smll. However, f hs ro s less h 9 erce, ddol vecors should e used he lss o cure he sc lod resose. I hs ee he exerece of he uhor h he use of exc egevecors s o ccure vecor ss for he dmc lss of srucures sueced o o lods. Wheres, lod-deede vecors, whch re defed he followg cher, lws roduce sc lod rco ro of...8 DYAMI LOAD PAIIPAIO AIO { XE "Dmc Prco os" }I ddo o rcg mss ros d sc lod rco ros, s ossle o clcule dmc lod rco ro for ech lod er. All hree of hese ros re uomcll roduced he AP rogrm. he dmc lod rco ro s sed o he hscl ssumo h ol er forces ress he lod er. osderg ol mss degrees of freedom, he exc ccelero u ecuse of he lod er f s:

16 - AI AD DYAMI AALYI f M u. he veloc of he mss os me s: f M f M u.4 Hece, he ol kec eerg ssoced wh lod er s: E f M f Mu u.5 From Equo., he modl ccelero d veloc, eglecg he mssless degrees of freedom, s gve : d f f f. From he fudmel defo of he mode sueroso mehod, ruced se of vecors defes he roxme veloc v s: f v.7 he ol kec eerg ssoced wh he ruced mode she soluo s: E M Mv v.8 A dmc lod rco ro r c ow e defed for lod codo s he ro of he sum of he kec eerg ssoced wh he ruced se of modes o he ol kec eerg ssoced wh he lod er. Or: E E r f M f.9

17 MODE UPEPOIIO MEHOD -7 he dmc lod rco ro cludes ol lods h re ssoced wh mss degrees of freedom. However, he sc lod rco fcor cludes he effecs of he lods cg he mssless degrees of freedom. A erce dmc lod rco dces h he hgh frequec resose of he srucure s cured. I ddo, for he cses of mss roorol lodg he hree glol drecos, he dmc lod rco ros re decl o he mss rco fcors..9 UMMAY he mode sueroso mehod s ver owerful mehod used o reduce he umer of ukows dmc resose lss. All es of lodg c e ccurel roxmed ece-wse ler or cuc fucos wh smll me creme. Exc soluos exs for hese es of lodg d c e comued wh rvl mou of comuer me for equl me cremes. herefore, here s o eed o rese oher mehods for he umercl evluo of modl equos. o solve for he ler dmc resose of srucures sueced o erodc lodg, s ol ecessr o dd correcve soluo o he rse soluo for cl me erod of lodg. he correcve soluo forces he l codos of cl me erod o e equl o he fl codos he ed of he me erod. Hece, he sme me-dom soluo mehod c e used o solve wd or wve dmc resose rolems srucurl egeerg. Prcg mss fcors c e used o esme he umer of vecors requred elsc sesmc lss where se cceleros re used s he fudmel lodg. he use of mss rco fcors o esme he ccurc of oler sesmc lss c roduce sgfc errors. Ierl oler cocered forces h re equl d oose drecos do o roduce se sher. I ddo, for he cse of secfed se dslcemes, he rcg mss ros do o hve hscl meg. c d dmc rco ros re defed d c e used o esme he umer of vecors requred. I wll ler e show h he use of z vecors,

18 -8 AI AD DYAMI AALYI rher h he exc egevecors, wll roduce vecors h hve sc d dmc rco ros or er erce.