ECTION FOR RECTANGULAR DESIGN MATRICES

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1 Pceedngs f ICD3 The Sevenh Inenana Cnfeence n xac Desgn Wcese June 7-8, 3 ICD-3-7 DESIGN PRMETER SELECTION FOR RECTNGULR DESIGN MTRICES Efén M. Benavdes efen.en@up.es Depaen f Ppusn and Fud Mechancs Unvesdad Pécnca de Madd Pza. Cadena Csnes, 3 84, Madd, Span Jan B. Rdíguez an.dguez@aan.c an Cae Capez, 8 Madd, Span BSTRCT Desgn aces ha ae deved f phsca aws ae, n genea, ecangua aces wh a age nube f desgn paaees han funcna eueens. Ths pape expes se agebac ppees f such aces and uses he n de fnd a dagna suae ax, whch s he dea desgn eued b he Independence and Infan xs. Based n hese ppees, a easue f he dsance he dea desgn s ppsed. Uncuped, decuped and cuped desgn aces ae exped. Fna, a ue f seecng he bes desgn paaees f achevng a suae desgn ax s ppsed. Kewds: desgn ax, adusen decns, dea desgn, dagnazan hee. INTRODUCTION xac Desgn [Suh, 99; ] pvdes a sd sucue f aheaca chaacezng he desgn ax asscaed wh he bes desgn. In addn xac Desgn, he desgn aces ae subec he aws f ageba and us be deved f phsca aws. Hence, a a fs gance, xac Desgn, geba and Phscs ae he s ha he engnee has f achevng he bes desgn. On ne hand, phscs s a gd aheaca faewk wh a fxed se f phsca aws. Na, he nube f euans deved f he aws f phscs s uch, uch, we han he nube f vaabes ha us be used f descbng a deened sun f a desgn pbe. Hence, he desgn euans ha ae used b he desgnes have a f paaees be exped, and a uesn abu wha ae he bes paaees be seeced n fs pace appeas. F educng he pac f hs esuce-cnsung ask, engnees eue a cen f dng ha seecn as uck as pssbe. On he he hand, he ageba s a gd aheaca faewk ha aws he desgne exac nfan abu he sucue f he desgn ax. In hs case, he dffcu uesn be sved s hw exac he eued nfan. Ths pape ppses a cen f hs pupse. The cen pesened f seecng desgn paaees w be funded n xac Desgn The, and n geba, akng n accun ha he aces deved f phsca cnsdeans ae ecangua aces. hugh xac Desgn esabshes a genea pcedue f banng an dea desgn, he aheaca eanshps ha ae ebedded n he dea desgn cann awas be peened as a phsca sun devce. In genea, he esu s a desgn ha us sasf funcna eueens and ha have wh desgn paaees. Hweve, xac Desgn esabshes ha n desgn paaees us be seeced as ue desgn paaees and he he us be dscaded fzen. Whu addna nfan, hee ae a age nube f pssbes f hs seecn, bu s expeced ha n ne se f desgn paaees w be he bes. Ne ha he nube f pssbes s gven b he cbnaa nube N!/ ( )!/! whch nceases when nceases. F hs easn, f he bes se f desgn paaees s n seeced a he ve en f wng he desgn ax he cs deved f a ae ean cud be huge. s sad, he a f hs pape s ppse a cen f akng hs ask ease. The pape s sucued as fws. Fs, he desgn euans ae pesened n he faewk f a desgn envnen. Secnd, a aheaca chaacezan f he bes desgn s gven b usng he baned desgn euans and xac Desgn. Thd, he agebac ppees f a ecangua desgn ax ae pesened. Fuh, based n hese ppees a cen f seecng he bes se f paaees s ppsed. Then, he cen s used f cpang uncuped, decuped and cuped desgns. Fna, an exape shwng hw he cen dscads a desgn paaee s pesened. TRNSFER FUNCTION ND DESIGN MTRIX In engneeng desgn pbes, s cn fnd a gea vae f needs, specfcans eueens ha can be descbed as vaabes whse vaue us beng an awed ange. F exape, we can hnk n he psn f a gven pa, he cncenan f an addve, he epeaue f an nfaed sens, ec. In addn, f a gea vae f specfcans, hs awed ange can be denfed wh an neva. Thus, a age nube f engneeng needs specfcans can be defned b usng n w vaues: he nu awed vaue and he axu awed vaue. Suppse ha f a gven desgn pbe hee ae needs ha can be specfed b a se f awed nevas ha defne he hpe-vue f accepan D as: Cpgh 3 b ICD3

2 Desgn Paaee Seecn f Recangua Desgn Maces The Sevenh Inenana Cnfeence n xac Desgn Wcese June 7-8, 3 D [, ] [, ]... [, ] () hen we can esabsh he success cndn f he desgn pcess as (,,..., ) D and he fa cndn as D. In addn, he vaabes,,..., asscaed wh he needs specfcans ae cnsdeed be a se f funcna eueens such as ae defned b Suh [99]: he funcna eueens ae he saes se f ndependen eueens ha cpee chaaceze he desgn becves f a specfc need. Because a desgn sun us be peened n he phsca dan [Suh, 99; ], he desgn euans us eae he funcna eueens a se f phsca paaees. Ths se f phsca paaees has ncude a he phsca cnsans (such as aea ppees), descpve paaees (such as geeca densns), and peana paaees (such as ana speeds, epeaues, and vages). The desgne has n easn f n usng a hese vaabes n he pcess f seekng an adeuae desgn pn. F hs pn f vew, a hese vaabes can be cnsdeed as desgn paaees. I s neesng ne ha, defned n hs wa, he nube f desgn paaees s na age han he nube f funcna eueens be sasfed. Le (wh ) be he nube f desgn paaees. In addn, as has been agued f he funcna eueens, suppse ha he desgn paaees can be defned b he neva whee he can be esabshed. Suppse ha f a gven desgn sun hee ae desgn paaees ha can be specfed b a se f awed nevas ha defne he hpe-vue f vaan C as: C [, ] [, ]... [, ] () hen we can esabsh he desgn ange as (,,..., ) C. I s usefu defne he cene f he hpe-vues D and C as he fwng vecs:,,..., (3),,..., (4) The engnee peens he aws f phscs ha eae he vec f funcna eueens he vec f desgn paaees n he fwng funcn: f : C (5) Ths s he ap ha ansfes he decsns adped b he desgne n he space C (paaees f desgn) he space D (funcna eueens). F hs easn can be cnsdeed a ansfe funcn. Funcn f w be cnsdeed a dffeenabe funcn, and hence, b appng he Ta hee, we can we: ( ) J( )( )... (6) The sucue f Es. (), () and (6) advses he fwng changes f vaabe [Benavdes, ]: ;,,..., x ;,,..., s a esu f hese changes f vaabe, he hpe-vues D and C ansf especve : (7) (8) E [,] [,]... [,] (9) E [,] [,]... [,] () The subsun f Es. (7) and (8) n E. (6) eads : ( x) () x... () In hs expessn, he ax s a ecangua ax f sze : f( ) J () (3) Ths expessn f an eeen f he desgn ax was deduced b Benavdes [] and gves a ana wa f banng densness desgn aces. 3 IDEL DESIGN The cndns x E and E assue ha he axu devan f he funcna eueen can be wen as: () (4) ax () (5) n The subsacn and he addn f Es. (4) and (5) ead especve : ax n (6) ax n () (7) Ineua (6) shws ha n a he desgn aces pduce an accepabe desgn. Indeed, he escn ha he hpe-vue f accepance pses ve he eeens f he desgn ax s even e exgen. Ths new escn ces f he neuaes (4) and (5) and can be cndensed n he fwng neua: Page: /6 Cpgh 3 b ICD3

3 Desgn Paaee Seecn f Recangua Desgn Maces The Sevenh Inenana Cnfeence n xac Desgn Wcese June 7-8, 3 n (), () (8) The ange whee hs neua s sasfed eaches a axu when he fwng cndns ae acheved: () (9),,..., () Ne ha hs s a aheaca fuan f he Infan x ha saes ha he bes desgn us have a nu vaue f he nfan cnen,.e., a axu vaue f he pbab f success [Suh, 99]. Ne as ha cndn (9) cnves he neua (8) n he neua (6). On he he hand, he endenc gven n () eads he fwng endences [see E. (3)]: f ( ) () () (3) Ne ha he endenc gven b (3) s he aheaca fuan f he Ca 6 gven b Suh [99]. Hweve, he endenc gven b () cnadcs he endenc gven b (3) because due he heach f he desgn pcess he desgn paaees f ne eve bece he funcna eueens f he fwng eve [Suh, 99]. Hence, when s cnsdeed a desgn paaee f he fs eve, shud be as w as pssbe [see E. ()]; and when s cnsdeed a funcna eueen f he secnd eve, shud be as age as pssbe [see E. (3)]. F hs easn, he endenc gven b () w ake an accepabe sun n he fwng eve f he heach f desgn pssbe. In addn, he desgne n an eve f he heach wans ge a fuan f he funcna eueens ha fuf he cndn (3). Theefe, s an becve f he desgne ncease as uch as pssbe he nevas f accepance f bh he funcna eueens and he desgn paaees. Ths aws us we ha he fwng endenc us be bseved dung he desgn pcess: (4) Snce he fs funcna eueen s fxed b he cuse, he cndn (3) cann be cpee sasfed, bu he desgne has be ceave enugh f achevng he cndn (4). If we assue ha we have ceaed he bes desgn, whch n hs case s he ne ha nceases as uch as pssbe he engh f he accepance nevas f he nex sep, we can cncude ha he fwng endenc s a necessa chaacesc f he bes desgn: (5) On he he hand, he endenc gven b () cann epesen a ea phsca devce. In effec, f a he devaves n he desgn ax ae ze, hee wud n be an eanshp beween he funcna eueens and he desgn paaees. F hs easn a eas ne devave cann be ze: f ( ) The cndns (5) and (6) ead Ke (6) (7) f se. Ths cnadcs he cndn (), and hence he neua (8) cann be fufed. Thus, he desgne us seek ha he cndn () hds f he a nube f eeens n ne w f he desgn ax. On he he hand, he desgne us ban he cndn (7) f a eas ne eeen f he w, bu hs fac s fbdden b neua (8). In addn, E. (9) us be psed b he desgne n E. (8), and hence he axu awabe vaue n he gh hand sde f ha neua s. Pung a hs nfan gehe (.e.,, f as a he eeens, and f a eas ne eeen) and aken n accun he Independence x (and, f necessa, peung ws and peung cuns) we ban he fwng fuan f he desgn ax f he bes desgn (.e., he desgn ax f he dea desgn): () (8) 4 QUNTITTIVE STUDY OF THE DESIGN MTRIX (9) In genea, he desgn ax baned b he desgnes dung he ceave pcess s n he dea ne. S, s cnvenen fnd a genea pcedue cnve he nndea desgn n an dea desgn. genea descpn f he agebac ppees f hs ax can be fund n Benavdes []. Ths secn pvdes he nu eued ageba f dng hs ask. Le us esabsh a se f funcna eueens as a vec n usng s cdnaes n he cannca bass. Le us esabsh a sun chaacezed b a se f desgn paaees ha can be vaed ndependen. s seen n he pevus secn, he desgn paaees can be denfed usng he cdnaes f he vec x. s dscussed n he pevus secns, hds. In addn, he ank f he desgn ax us be [see E. (9)] and hence, s w vecs a,..., a us be nea ndependen. F he sae easn, he vec se a,..., a s a bass f. Ths se f vecs can be wen n ax nan as, whch s nvebe. Thus, I ( ) hds. Theefe, he cun vecs n he ax ( ) ae a cbnan f desgn paaees ha enabe us va he funcna eueens Cpgh 3 b ICD3 Page: 3/6

4 Desgn Paaee Seecn f Recangua Desgn Maces The Sevenh Inenana Cnfeence n xac Desgn Wcese June 7-8, 3 ndependen. The kene f he nea ap s he subspace geneaed wh he cun vecs f he ax B, whch has vef B. Le us defne an ( ) aba ax ( ), and cnsuc he ax X ( ) B (3) Ths ax cnans n s cuns a he cbnans f he nea paaees ha keep he funcna eueens ndependen. F hs easn he ae caed adusen decns [Benavdes, ]. The aba ax can be chsen f enang he nfuence f a desgn paaee ( a nea cbnan f desgn paaees). Because ax has cun vecs, desgne can eve he nfuence f desgn paaees ( specfed decns). Le desgne defne a ax X ' whse cun ( ) vecs ae he decns n he space f he desgn paaees ha he desgne wans eve. The evng f hese decns eues sve he nea sse X' X, whch eads ( ' ) ' ( ) X B X (3) Ne ha, as s eaked n Benavdes [], he ax X ' B ( ) ( ) cud n be nvebe. The subsun f eads X I B( X ' B) X ' ( ) (3) Ths esu es us assue ha hee s a vec e ha epesens a new se f desgn paaees. In effec, f hs s assued, hen he ansfe funcn can be wen as: I B X B X e e... ( ' ) ' ( )... (33) Ne ha n hs euan he desgne has educed he nube f desgn paaees f and has acheved an dea desgn. Ths esu was used b Benavdes [] pve he dagnazan hee ha saes ha he dea desgn awas exss. F he neesng agebac esus, such as he speca decpsn f he desgn ax, pease efe Benavdes []. Ths expessn shws as ha, f he desgne acs n he desgn paaees b fwng he saeg f vang sevea f he a he sae e, as ndcaed b he cun vecs n ax X, s awas pssbe anan he ndependence beween he eueens. B akng he cun vecs f X as a bass, he nea ap akes he f f he dea desgn gven b E. (9). E. (33) shws ha he exsence f he dea desgn ces f he fwng ppe f he desgn ax: X I (34) In addn, E. (34) shws ha a he eevan nfan f banng an dea desgn f a gven (ecangua n) desgn ax s ceced n he ax X defned b E. (3) whch defnes he adusen decns. 5 MESSURE OF THE GOODNESS OF THE DESIGN MTRIX The vec X cecs he eevan nfan f he desgn ax eued f ansfng a genea desgn n an dea ne. E. (34) saes ha he cun vecs f he ax X cec he vaues f he desgn paaees ha ve he funcna eueens he pn., whch s he axu vaue acceped b he cuse. Bu because he dea desgn ax s he den ax, saes as ha each cun vec f X ves ne and n ne funcna eueen f he vaue. he vaue.. F E. (3) we can ban he fwng aces: X ( ) B (35) X X ( ) B B (36) E. (36) shws ha he cndn f he dea desgn s X X I (ne ha when I hds, B as hds). Hweve, n genea, hs cndn cann be eached and hence, s cnvenen defne he ax: E ( ) B B I (37) Ne ha E s a seca ax ha shud be denca he ze ax f he dea desgn. If an eeen n he ax E s n ze, hen he n f he especve cun vec w n be ze. Ths fac aws us cnsuc a ea psve nube ha easues hw uch he ax E devaes f he ze ax. Ths nube s: whee ace( E E) ace( E ) (38) E s gven b he fwng expessn E ( ) I ( ) B B B B ( ) ( BB ) (39) Theefe, he dea desgn ( I ) ees he cndn =. The cacuan f hs devan s ue had because he desgne shud expe a he pssbe vaues f he ax. E. (3) gves he adusen decns f a gven. When s cacuaed wh E. (3) he cacuan f s educed he adusen decns ha esu f evng exsng desgn paaees. In an case, he adusen decns ha pduce he nu vaue f cnsue he new se f desgn paaees ha acheves he dea desgn. Hweve, as s we dscussed b Suh [99], hese new paaees ae n awas feasbe n he ea wd because hee cud have se ans, f exape ceav, ha avd such peenan. When desgn paaees cann be cbned and he adusen decns cann be fwed, a e pacca cen exs. Ths s he ne whee he desgne checks f he cun vecs f X have a axu cpnen wh an absue vaue cse. and he he cpnens eans beween and +. In hs case, he devan funcn gven b E. (38) cud be subsued b: Page: 4/6 Cpgh 3 b ICD3

5 Desgn Paaee Seecn f Recangua Desgn Maces The Sevenh Inenana Cnfeence n xac Desgn Wcese June 7-8, 3 D ax x (4) Ths e funcn was ppsed, gehe wh he addna easues f he degee f ndependence, b Benavdes [] f deecng whch ne s he bes se f paaees be seeced n a desgn ax (an dea desgn ees he cndn D ). The cndn D ndcaes f a eas ne desgn paaee has eached s axu ange f vaan. F suans whee he desgn paaees cann be phsca cbned, D f E. (4) s e suabe han f E. (38). 6 PLICTION TO UNCOUPLED, DECOUPLED, ND COUPLED DESIGNS Suh [99] cea defnes uncuped, decuped and cuped desgns. Uncuped and decuped desgns ae hse ha have, especve, a dagna desgn ax, and a angua desgn ax. Fna cuped desgns ae hse ha d n beng he pevus caeges. In hs secn we w cec se spe exapes f hese caeges n de cacuae he aces and e funcns defned pevus. These exapes ae usave and f hs easn ae kep as spe as pssbe: a he cacuans [see Es. (37) and (38)] w be dne f fu-ank ( B =) suae desgn aces and f hee funcna eueens. X E Tabe. Cpasn beween desgns. Uncuped Decuped Cuped 8 D Ths exape shws ha a decuped desgn can be wse, n es f he devan, han a cuped desgn. The easn s ha a decuped desgn can have he adusen decns nea paae. Bu bh, he decuped and he cuped desgns, ae wse han he uncuped desgn, such as he dea desgn eues PLICTION TO THE SELECTION OF DESIGN PRMETERS The fs ppsed exape s a cuped desgn wh he fwng ecangua ax: The an esus f dffeen vaues f he ax ceced n he Tabe. Tabe. Seecn f desgn paaees. X ' X E Desgn ax beces sngua X ' ae D The esus n Tabe shw ha, n he suded case, he na desgn ax des n aw banng an dea desgn b evng desgn paaees. When he DPs can be cbned Cpgh 3 b ICD3 Page: 5/6

6 Desgn Paaee Seecn f Recangua Desgn Maces The Sevenh Inenana Cnfeence n xac Desgn Wcese June 7-8, 3 ban new DPs, he abe shws ha he bes seecn f he desgn paaees s {DP, DP, DP4} ( =9/6), whch eans ha DP3 shud be eved fzen. The esus as shw ha hs pn s bee han n evng an desgn paaees. Hweve, when he DPs cann be cbned, hs s n he bes pn and he bes seecns w be {DP, DP3, DP4} {DP, DP, DP3} ( D =/ / ). I s as neesng ha DP cann be eved: s an essena pa f he desgn because s a ke eeen f ananng he ank f he desgn ax. The secnd ppsed exape s he desgn f a fauce ha us cn he fw ae and he epeaue f a ud fw: {FR, FR}={fw ae, epeaue} and {DP, DP, DP3, DP4, DP5}={pessue, pessue, aea, aea, h epeaue}. The desgn ax f hs pbe s [Benavdes, ]: Ths ax s neesng because epesens a ea devce wh a cuped (seec, f exape, {DP,DP} as he desgn paaees) decuped (seec, f exape, {DP4, DP5}) desgn ax ha cann be uncuped b eans f a saghfwad pcedue. Resus f X, and D ae pesened n Tabe 3 f dffeen vaues f he ax X '. In hs case, he bes seecn f he desgn paaees s {DP3, DP4} f bh cea, nu and D. Ths eans ha: ) because D s nu, he pn f cnng he aeas s bee han cnng he pessues he epeaue; and ) because s nu, he pn f cbnng he aeas s bee f achevng an dea desgn han cbnng he pessues and he epeaue. Uncuped phsca suns, baned b dng hs cbnan f aeas, can be fund n Suh [] and Benavdes []. 8 CONCLUSION I s pssbe deve an ndca, based n he devan f he desgn ax f he dea ne, f he agebac ppees f he desgn ax. Ths ndca aws he desgne seec he bes se f desgn paaees when he desgn ax s n a suae ax. The ndca as esabshes ha ecnfgung he PDs cud be e dffcu f a decuped desgn han f a cuped desgn. 9 REFERENCES [] Benavdes E.M., dvanced Engneeng Desgn: n negaed ppach, Wdhead Pubshng,. ISBN [] Suh N.P., The Pncpes f Desgn, New Yk: Oxfd Unves Pess, 99. ISBN [3] Suh N.P., xac Desgn, New Yk: Oxfd Unves Pess,. ISBN Tabe 3. Seecn f desgn paaees. X ' X 4 D Desgn ax beces sngua Desgn ax beces sngua 98.4 Page: 6/6 Cpgh 3 b ICD3