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1 Pirrr. & Wr P.. 8. Gomrilly onlinr Fir Orr Sr ormion Tory or Gnrl nioropi Sll. n 9 /SE/SCE/HS/SC Srr Srrl ynmi n ril Conrn: 6 /SE/HS pi Srr Conrn on-rminii ppro Conrn 9 Gomr Spr Form liiiplinry ign Opimizion Spili Conrn [ 8-86] mrin ni o roni n roni. O:./ Plir' PF lo non Vrion o ror in o pli rion i ill:./ in o pliion ror in Explor riol r PF-omn Uniriy o riol - Explor riol r Gnrl rig Ti omn i m ill in orn i plir polii. Pl i only pli rion ing rrn o. Fll rm o r ill: p://.riol../pr/o/r-rm
2 9 /SE/SCE/HS/SC Srr Srrl ynmi n ril Conrn <r>6 7 - pril 8 Smrg 8-86 Gomrilly onlinr Fir Orr Sr ormion Tory or Gnrl nioropi Sll lro Pirrr n Pl. Wr Uniriy o riol riol U S8 T onlo y STO UVESTY on g p://r.i.org O:./ gnrliz ir orr r ormion ory or nioropi mlilyr ll i prn. inl o gomrilly nonlinr ormion n gnrl iniil rr. T liiy qion r xpr in orogonl rilinr oorin lying on ll mil r n n i ormlion rn o o prilrly il or nlyi o rr orm ing ir plmn mnring niq. nol xprion or in mrix i prn in i rlionip n ll p n in oiin i iglig. omnlr orogonl rilinr oorin r poiion or o poin on mil r poiion or o n rirry poin E F G r mri nor lmn l or mé oiin norml rii o rr o mil r r rln pr ni lng i r omponn nonlinr rin omponn nonlinr longion o o lin lmn ing or ormion irion oinin o oorin irion nonlinr r ormion ng o ngl n o lin lmn ing or ormion irion oinin o oorin irion linr longion o o lin lmn ing or ormion irion oinin o oorin irion linr r ormion n o lin lmn ing or ormion irion oinin o oorin irion omponn o rl o iplmn il iplmn iplmn o mil r o ll roion o norml o rrn r rnorm in rrr o lmin oorin irion r orrion or in mrix oiin in mrix oiin P Sn p. o rop Enginring Uniriy o riol U n mmr r p. o rop Enginring Uniriy o riol U mmr mrin ni o roni n roni Copyrig 8 y lro Pirrr. Pli y mrin ni o roni n roni n. i prmiion.
3 in mrix oiin δ irl riion o inmi nrgy δ U irl riion o rin nrgy δ V irl riion o ponil o ppli or m inri onlo y STO UVESTY on g p://r.i.org O:./ nroion HS or prn ir-orr r ormion ory or mlilyr nioropi in ll; i i on T or o y or l pl. i rrn im o rr lop y or o ll o gnrl p lo inling o gomrilly nonlinr ormion ri y oozilo. On o mo rmrl r o ompoi mril i y llo nginr o ign no only rr lo i onin mril. Prly o ir xlln pii in r i nny o m o mimi ll non ior o ioropi mril. i oming inringly imporn or nol ppliion o xploi pilii ompoi lmin or y ir inring rrl iiny or y ring nol nionliy. For inn pr m rom nymmri ing qn n rrly y my inro rl rrl opling n on ooling-on rom r o room mprr y lop inrnl r n rp. onl pnomn or gr pilii or nol onp o in mrging rr il li li iloring n morping rr. n orr o xploi pilii i i rilly imporn o lly nrn rrl ior o mril n o xmin ll or o nioropy. T im o i ppr i o gr nrning o ign mril o oin ilor rrl rpon o gnrl ll. For ll ron rrn or mp o lop nol mol riing ll-li o-imnionl rr. Prilr nion n gin o rlionip n rr n in oiin. Sll rr n ily in nginring ppliion. T lirr or riy o ori moling o gnrl liiy prolm n prilr ign prpo. E ory or nlyi n lop ring rom ommon poin nmly inini qion o li qilirim. Hor y my ir grly pning on irn prpo-rin mpion n pproximion. Frrmor pi ililiy o g riy o ppr i o y o mo ll rl rrl pnomn lirr lmo xlily ppli o nlyi o ll o pril n ommon in nginring. Tror mo pli or n onrn piilly i nr p ylinr pr on or gnrlly i ll i mll in o ri o rr rio. mr o nr i ypoi o rr on in i on ngligil. - T prn or l i gnrliz ir orr r ormion ory or nioropi mlilyr ll. n n mp o gnrl poil mol ino on ll nioropy gnrl ll gomry nonlinr ormion n rnr r ormion. T liiy qion r xpr in orogonl rilinr oorin lying on ll mil r. nol xprion or in mrix i prn. i lo on mny o opling rm r rongly pnn on p o rr.. Toril lopmn n olloing ion oril lopmn ling rom gorning il qion o nlyil olion nmly lo-iplmn qion or ll rr r prn. Ul mpion r ollo: inr li ior o mril. T rnr norml ir r no long. T in irion norml r i ngligil ompr o or r in m irion. T o-iro ypoi i rlx o o ir i r rig n norml o mil pln or ormion rmin rig no longr norml o pln r ormion. mrin ni o roni n roni
4 . Gomry o Cr Sr i m mil r o ll rr i ri y orogonl rilinr oorin ym r n r oorin riing poiion on mil r n i oorin in in irion. Ti ing poin on mil r n on n rirry poiion r ri rpily y or r n. T mri propri o r r omplly ri y ir nmnl orm. rmin lng o n lmn o mil r r E F r G T oiin rprn lmn o r mri nor n r in onlo y STO UVESTY on g p://r.i.org O:./ r r E r r F r r G n rilinr oorin ym qnii E n F r ll l or n F o inilly ql o zro. Similrly n o ll mé oiin imilr mning or poin rog in. Proi n no norml rii o rr o mil r n ing o pring orml on n ri irnil lmn o r n olm rpily or mil r or r n V.. Srin-iplmn lion T non-linr rin omponn nr ypoi o mll rli ormion r in in rilinr oorin mrin ni o roni n roni
5 mrin ni o roni n roni T xprion in Eq. r non-linr ominion o o lmn lly ri oninm ormion nr ypoi o mll iplmn n roion i.. in lil linr ory o liiy in i. i on in rl or 6 linr omponn o ormion in orogonl rilinr oorin r ri ing oring o ypoi ri ginning o ion r iplmn n r m o 6 Siing Eq. 6 ino Eq. nl linr rin omponn o pr ino rm pning on iplmn n roion o mil r onlo y STO UVESTY on g p://r.i.org O:./6.8-86
6 mrin ni o roni n roni 7 Hrin prrip n rr o in-r n o-o-r omponn o linr ormion rpily. Siing Eq. 7 ino Eq. olloing xprion or nonlinr rin omponn r oin 8 onlo y STO UVESTY on g p://r.i.org O:./6.8-86
7 mrin ni o roni n roni 6 C. Sr ln T r rln ing on ll lmn r oin y ingring r omponn or in ; y r r i r orrion or o j irpny n r riion o rnr r n i n impo.. Conii lion Sppo ll rr i ompo o lyr. For lyr onii l i For onnin in Eq. rin omponn or i pli in o pr in i prrip n rpily mning o linr n nonlinr. Similrly r rln r prn m o o or orrponing rpily o iri or n momn rling rom linr n nonlinr rin. So or xmpl ill m o n. y mn o Eq. on n ri 6 6 onlo y STO UVESTY on g p://r.i.org O:./6.8-86
8 mrin ni o roni n roni 7 Tn iing Eq. ino Eq. 9 n ingring qn xprion i i poil o oin lmin onii rlion rpor in Eq. - n Eq. -. E. min Sin rix Corrponing o inr Srin T onii qion r or in mor omp orm T lmn o Eq. -6 i r o linr pr o rin omponn n ll ing olloing rlion onlo y STO UVESTY on g p://r.i.org O:./6.8-86
9 mrin ni o roni n roni n ln ln ln Similr oiin i rip n oin imply inrnging rip n. i o igniin in mri rmin ymmri n nlrg o x mrix ym. o ymmry n i y ompoing r rin ino o pr omponn. F. min Sin rix Corrponing o onlinr Srin Similrly i i poil o oin pr o onii qion o nonlinr rin omponn onlo y STO UVESTY on g p://r.i.org O:./6.8-86
10 mrin ni o roni n roni 9 n T oiin o Eq. - i r o nonlinr pr o rin omponn n ll ing olloing rlion onlo y STO UVESTY on g p://r.i.org O:./6.8-86
11 mrin ni o roni n roni g g g g r ln ln ln ln onlo y STO UVESTY on g p://r.i.org O:./6.8-86
12 mrin ni o roni n roni ln ln g ln ln. Eqion o oion T olloing ix qion o qilirim r ll-non n ily p. 6 Ty rl qilirim o mil r n rnr lo q i ppli q. T m inri r ll ing i i ρ i r ρ i m niy. o iniion o r rln l xprion in Eq. onrning rilling qilirim i ly ii n or i ron i i lly no onir in riing irnil qion rling onlo y STO UVESTY on g p://r.i.org O:./6.8-86
13 mrin ni o roni n roni iplmn n ppli lo. i no rilling qilirim i ly ii ing Eq. 9. Hor in riing r rln pproxim xprion or iplmn il n. n monr i irpny ol l o nonzro r rln orrponing o mll rigi oy roion n Eq. 6 i in n o om moiiion. 67 nlyilly i i on y moiying riniplmn rlion in Eq. 6 y rpling o rm io il ing olloing rm i il n ~ n ~ ~ n ~ n Hr rm n i ir omponn o rl o ll mil r iplmn il n n 6 V. C Si On o noli inro in i or i rm i n rin in riion o ll mol. Ti rm i ypilly ngl or rng o ppliiliy o ny ll ory on n pproximion o ll o imnionl rr qniy i i mll i ompr o niy. n olloing ion ri xmpl o ppliion o lop ory ill proi. T xprion o in r prn nion o gomry o ll in i orogonl rilinr ym i.. o norml rii o rr. T nion ror rprn poin o poin mpping n rr iliz omin n in. n or or y llo on o ll nlyilly in o poin iin rr. i lr on pi onirmion o pproximion i i mll ompr o niy y ngling i rm nil lo o ril pi o inormion. n gomry o ll rr n in mrix y inroing opling rm n or ymmri lmin. Ti i rily xplin y impliying xprion in Eq.. For inn onir gnri ll o in n m mril i ioropi. ri xpnion o Eq. yil olloing rlionip: Eqion 7 gi n i o orr o mgni o irn n li lminion ory CT in n on rin prn n lo o i irn pn on ign o. y ompring lr xprion o li in i: 8 onlo y STO UVESTY on g p://r.i.org O:./6.8-86
14 i om lr log r i mll n i non-niorm irn or n rm rm r irn rom zro o orr o n mgnii or rr in i <. For ompoi rr irn i n xp o o m orr o mgni. n olloing ion in mri rling rom Eq. 7 o r prn in omprion i qiln li mri in.. T rr in xmpl r ll m o m rom lyr i mril propri: E 6. 8 GP E. 7 GP G G. GP G. GP ν. onlo y STO UVESTY on g p://r.i.org O:./ n ymmri ly-p i ing qn [ 9 ]. For l omprion i i or noing oring o. n: Proloi o olion > m Figr rprn rr o p i proloi o rolion. Hr δ rprn ril oorin long gnror in orogonl rilinr oorin ym lying on proloi. z [m] 9 y [m] x [m] Figr. Proloi o olion gomry. o ymmry o rr lmn o mrin ni o roni n roni
15 6 6 6 δ δ ; CT onlo y STO UVESTY on g p://r.i.org O:./ r nion o δ only. xp n lmn in Eq. r o orr o n on y rm in Eq. 7 r proporionl o i i.. o irn n grn n r r on rig-n i o Fig.. T iriion o nion o δ i on in Fig.. Figr o r i mll non-ngligil opling n proloi in-r n o-or mnil ior. Ti opling i ri olly rom gomry o rr om o rm in. o proporionliy o in o irn n in o ri rio opling i mximm r i irn i i p. o i l i lo o zro op o rr n n ympoilly o igniin l or lrg l o δ i.. δ > 6. Figr. mrix lmn.. Spril Sll n i r ny gomri on in ippr. C. Cylinril Sll m For ylinril rr o rr on i proporionl o lop rin r ompr i lil l pl lo. T in mri ; CT mrin ni o roni n roni
16 ; CT ; CT onlo y STO UVESTY on g p://r.i.org O:./ ; CT On gin o rr on n i mll r i igniin on.. Conil Sll > m Figr rprn rr o p i rn on. n i xmpl δ oini i ril oorin long gnror in orogonl rilinr oorin ym lying on on. T lmn o opling in mrix r δ ; CT ini in Eq. i prilly popl. From i proporionl o i rm in Eq 7 o rr on z [m] y [m] x [m] Figr. Con gomry. mrin ni o roni n roni
17 For ql in n lol rii o rr on onil ll i lrgr ompr o o proloi. Ti i lrly on y rig-n i o Fig. n Fig. no in ion V- n ill i rr. onlo y STO UVESTY on g p://r.i.org O:./ E. Hyproli Proloi m < For yproli proloi δ n δ rprn orogonl rilinr oorin ym ri y gri lying on r on in Fig.. T xprion or o CT n n r δ δ δ δ ; CT Figr. mrix lmn z [m] x [m] y [m] Figr. Hyproli Proloi gomry. 6 mrin ni o roni n roni
18 T iriion o n i on in Fig. 6 n o igniin l i ir mximm orring nr o rr. ' onlo y STO UVESTY on g p://r.i.org O:./ δ δ δ '' 6 Figr 6. mrix lmn or poii qrn o rr. o olor onor ini prpi only. V. iion o C Sy l For ll o i onir i nol xpion o pr mrix i non-zro n or ymmrilly lmin rr o inrn gomry. n imporn gnrl rl my rom xprion in Eq. 7 n i o iniil gomry on li ior o r r pn on i Gin rr G. Ti qniy i in pro o prinipl rr n i i poii or ynli r proloi o rolion zro or lopl or rl r ylinr on n ngi or nili r yproli proloi. For rr i irn gomri n inil in mgni o lmn o inr or ring G. n mmry rom Eq. 7 ' 6 δ δ '' δ Cnrl poin orrponing o δ δ G > O 7 mrin ni o roni n roni
19 G O G < O. nringly igniin o rm pn only on gomry i G n no on mril in propri. o rr on i omplly pr y G noing lrg or or nili gomri yproli proloi ngi G n mll or ynli rr poii G i zro or pr. Crr on or ylinril ll r inrmi n o prio xmpl my xp o ir zro G l. onlo y STO UVESTY on g p://r.i.org O:./ V. Orll iion i no linr pr o lop mol i in goo grmn i rl rom. ; min irn i xprion o in mrix oiin. n. or rri o ingrion or in rqir o in oiin ion - nmrilly pproximing rin rm. n prn or i pproximion i oi n l o irn nlyil rl. T mol n rr xn o inl o gomrilly nonlinr ormion. Prliminry nmril nlyi o goo gr o oniny i rp o rl prn in. 6. l lo o riion o -% on rin omponn i rp o ir lil l. i no irn my igniinly ling n po-ling pnomn. Fr or ill r i i. V. Conlion Gnrl qion o mlilyr nioropi ll r lop y inling o r ormion iniil rr n gomrilly nonlinr ormion. nol xprion or in mrix n prn in i rlionip n ll p n in oiin n m xplii. oly pr o lmin onii qion riing linr ormion xii ymmril in oiin iin nol x mrix ormlion n og n. T rol o gomry iniil rr or o nioropy n n nlyz. n on o rr igniinly n i mgni pn on ign o Gin rr. Gnrlly lmn o in mrix prilly pn on in/lol ri o rr rio n on Gin rr. nolgmn T or n ning o ir U n HE-SW rog Gr Wrn r grn. rn y J.. ni o min Compoi Pl n Sll: Tory n nlyi n. CC Pr o on Flori. oozilo V. V. Fonion o onlinr Tory o Eliiy or Pliion n. inol Yor 999. Toorni. H. i.. Gnrl Eqion o nioropi Pl n Sll nling Trnr r ormion ory nri n niil Crr E Jornl o Son n Virion Vol. 7 o. pp Cllin C.. Tory o ll Srr. Cmrig Uniriy Pr Cmrig U 98. Gol nizr.. Tory o Eli Tin Sll Prgmon Pr. onon U oozilo V. V. T Tory o Tin Sll Wolr-ooro Pliing Groningn T rln Hoin. C. r.. Compoi ril or irr Srr mrin ni o roni n roni n. Yor Po Y. C. Simpl ning nlyi o min Pl y rg-lion Tory Jornl o Compoi ril Vol. pp onnll. H. m Pl n Sll Gr-Hill Yor mrin ni o roni n roni
20 onlo y STO UVESTY on g p://r.i.org O:./ ioron F.. Sll Tory or-holln ri in ppli mmi n mni. 9 mrm or- Holln 98. Plzoo.. nni S. T. onlinr nlyi o Sll Srr ion ri mrin ni o roni n roni Wingon C 99. Ogn. W. on-inr Eli ormion or Pliion n. inol Yor 98. Pogorlo. V. ning o Sr n Siliy o Sll Trnlion o mmil monogrp. 7 mrin mil Soiy Proin o ln 988. Ci C.-Y. onlinr nlyi o Pl Gr-Hill nrnionl oo Co Yor 98. Cox H.. T ling o Pl n Sll Prgmon Pr Oxor Snr Jr. J.. onlinr ori or in ll rrly o ppli mi Vol. o. 96 pp Snr Jr. J.. n mpro Fir pproximion ory or Tin Sll S T Wir. O. E. n ympoi ory or orly rg lion o nioropi Pl Jornl o Enginring mi Vol. o. 969 pp ir. Smi. in Tori o Eli nioropi Sll oning or Smll Srin n or oion nrnionl Jornl o on-inr ni Vol. o. 988 pp Pi P. F. n y. H. nonlinr ompoi pl ory onlinr ynmi Vol. o pp. 6. Pi P. F. n y. H. nii nonlinr ormlion or pl n ll ori onlinr ynmi Vol. 6 o. 99 pp. 9. Y W. Hog. H. Gomrilly onlinr Sr ormion Tory or Compoi Sll Jornl o ppli ni Vol. 7 o. pp. 9. Co. n Prmrr.. Eiin Higr-Orr Compoi Pl Tory or Gnrl minion Conigrion Jornl Vol. 99 pp Ci C.-Y. Ci. S. E o rnr r on ini ormion o gnrlly lmin llo ll i ol rr ni Soli Sini Vol. 9 o. 996 pp. 7. Ci C.-Y. onlinr nlyi o oly r ymmrilly lmin llo ll i rnglr plnorm ri o ppli ni ngnir ri Vol. 8 o. 988 pp Wr P.. On niil nioropi E in Compoi Srr r /SE/SCE/HS/SC Srr Srrl ynmi n ril Conrn nr Coloro pr mrin ni o roni n roni
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