Non-linear normal modes of a fixed-moored offshore catenary riser

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1 No-ler orl odes of fed-oored offshore cer er Crlos E.N. Mzzll Césr Tr Sches Depre of Srcrl d Geoechcl Egeerg Escol Polécc Ue of São Plo Brzl E-l: cezz@sp.r E-l: cesr.sches@gl.co ewords: No-ler orl ode cer er offshore egeerg. SUMMAR. Offshore cer e re sed he eploo of deep-wer ol d gs felds sch s hose log he Brzl Sohes cos. The re seced o seere dcl lods sch s hose reled o sde flow of flds der hgh pressre eerl flow of se crres d posed oo he flog prodco plfor de o se wes. I hs pper he o-ler orl odes (NNM s) of fed-oored cer er re ddressed eglecg he oo of he och dow po (TDP) he och dow zoe (TDZ) where he e ees he seed e kg o cco he ol of he flog prodco plfor he op ed. The NNM s re sefl for he redced-order odellg of he er-fld sse grel fclg he forced respose lss der cople dc lodg sch s h whch cses ore-dced ro (VIV). NNM s of cer e wh og TDP he lred ee sded [] cosderg he er s cred e lhogh he oorg effec hs o ee ke o cco. Here fe-elee odel wll e sed sed for he oored er lhogh he TDP wll e kep fed. I fre work oh he og TDP d he oorg wll e cosdered. A copol ool [] sed o he r fold procedre proposed Shw d Perre [4] hs ee sed o o he NNM s for D fres sg geoercll o-ler e heor p o cc e. INTRODUCTION I he ps wo e Brzl hs dscoered he lrges ol deposs he cor's hsor d he world s os prosg felds sce he dscoeres de shg zkhs he er. Ths hs p Brzl well o s w o ecog or prodcer he fre echologcl d fcl hrdles wll he o e oercoe f. The deposs coss of pre-sl resero. The gges hdrocro cclos were fod he Sos Bs's Pre-Sl Pole d re loced lr-deep we elow ler of sl h soe res s hcker h ee. I ddo oshore d offshore eploro felds he red owhsdg he growg glol ded for eerg d he oll ol prces. Ths scero hs led he operg copes o focs o deep d lr-deep wer eploro rgg forh ew coce o redcg he eploro coss hrogh dced echologcl deelopes. Coseqel he offshore eploro dsr hs cresed s eo o ew rels of reserch sch s o-ler dcs of e for deep d lr-deep wer order o defe he os ecoocl d ppropre solo for ech reseror. If sse eploro hs so fr ee cople d dedg c fro ow o ew of he pcog deelopes deepwer d lr deepwer wll e ee ore chllegg.

2 The offshore dsr hs lred eg o eplore wer dephs he ls of he crre echolog d hs pls o ccess dephs oer 5. I s deelopg sse prodco sses prepro for lr deepwer prodco whch clde ore flele e d hh eroel codos. Uder hese crcsces o-ler effecs wll crese her flece oer he er glol dcs. Moreoer he edec owrds ore flele d lgher sse srcres wll oos p he oerll o-ler ehor. The oece of hs pper s o prese ercl sd o he o-ler orl odes of fed-oored deepwer cer er resorg o he r fold pproch o deere he free ro oo. A copol odel s sogh sg he fe elee ehod. The eploed fe elees cosder geoercl o-leres. The ssoced eqos of oo ke o cosdero o-leres p o hrd order ledg o hrd- d forh-order eso. These o-ler eqos llow for oh cle d e ehor. I hs work he ol of he flog prodco (FPU) he op ed s ke o cosdero sg sprg o odel he sffess of he oorg sse d cocered ss s he FPU s horzol er. The lerl coc srog sorce of o-leres he ochdow zoe (TDZ) hs ee egleced here. Ised he cer er hs ee odelled wh fed ed s llowg for roo he TDP. Frher dces regrdg ercl pproch for he TDZ re der deelope sce cosderle progrg effor s reqred o ke o cosdero he coc prole TDZ. I s eleed h o-ler odes pl por role he ccre srcrl represeo of deep-wer e odels wh sll er of degrees of freedo. A free-hgg cer s ddressed s cse sd cosderg he f o-ler orl ode of ro. The le gol s o deelop ore coprehese low-desol odel of he dcs of er rodcg o-ler odels o cpre he copled dcs of he fld-srcre sse. NONLINEAR MODES Ths pper shold e regrded s l reserch o ele he o-lerl fld-srcre dcs of e whch wll e odelled s ple fres wh geoerc o-leres de o he coplg of gel d el dsplcees. For he e eg erl ler wll e ssed. Too d D effecs s well s he FPU oo d o-leres de o he lerl coc he TDZ wll e egleced. Alhogh erl resoce coe o pl hs reqrg he cosdero of he soclled o-ler l-odes he pper wll cocere o he o-ler orl odes. I fre works he o-ler odes eher orl or l wll e sed o proec he forced dcs of e oo low-desol phse spces hs geerg relle odels wh few degrees of freedo sll keepg he essel ehor of he srcre der dffere se lodg codos sch s he cses of hgh-freqec VIV d low-freqec drfg.. The FEM Forlo Tpcll he eqos of oo of geerl degree-of-freedo fe-elee odel of elsc ple fre wh geoerc o-leres der free ros red: M & p s D p& s U r () where p re he geerlsed coordes d he rces of ss M eqle dpg D d elsc force ecor U deped o he geerlsed coordes d eloces s follows: r

3 M D U r M D s M D p& D p& p () p p M s p p p p p p p whch M M M D D D d r s re sse coss h defe he secod hrd d forh-order eso dced (). Ese s coeo for so s eploed.. Ler d No-ler Modes Drg odl oo he phse recores of dscresed ler sse re cofed o wo-desol egeples ch he se w s he phse recor of oe-degree-offreedo sse wh geerlsed coorde res cofed o he ple &. De o hs rce proper sch egeples re r folds of he dcl sse. I o-ler sses he r folds re o loger ples d he oos whose recores re cofed o he re clled o-ler orl odes. I geerl here re r folds ech oe correspodg o dffere ode; hese folds co he eqlr po d hs po re ge o he correspodg egeples of he leed sse. Sch opologcl chrceo of odl oo sggess he so-clled rfold procedre o deere orl odes whch Shw d Perre [4] proposed d ppled o sses of few degrees of freedo. Sores d Mzzll [] eeded he procedre o fll fe-elee odels of ple fres. A lere echqe o ele o-ler orl odes of fe-elee odels ws proposed Mzzll d Brcho Neo [5]. Sch lere echqe s sed o he ehod of lple scles. To hdle cses of copled odl oos of o-ler sses de o erl resoce he lple-scles procedre hs ee sccessfll eeded Brcho Neo d Mzzll [6]. I hs cse he esg forced ro kes plce r fold eedded he phse spce whose deso s wce he er of he orl odes h erc. Ths fold cos sle eqlr po d s ge here o he s-egespce of he leed sse whch chrcees he correspodg ler odes. The l-ode c e locll descred ler coo of he ler odes. O hs fold he sse ehes lke M-degree-of-freedo oscllor where M s he er of copled orl odes.. Ir Mfold Procedre Here he fdel seps of he r-fold procedre re followed hg d s pplco o fe-elee odels of e. Irodcg he oo p d p& & sse () c e wre f-order for s & & f ( ) s. ()

4 Power-seres epsos for he fcos ( ) f he eghorhood of he eqlr po re rodced (4): ( ) p p p p p p p p R N L H G F E C B f (4) where p p p N L H G F E C B d p R p re kow coss. If drg odl oo he recor of he solo he phse-spce s resrced o wo-desol srfce he s e possle o epress ech geerlsed dsplcee or eloc s fco of wo of he for sce k d k les he eghorhood of he eqlr po. B ssg he epressos ed here s odl reloshps ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( (5) () we rre ( ) ( ) ( ) f f k f k (6) whch s o-ler sse of prl dfferel eqos hg he fcos s kows h e s dffcl o sole s he orgl eqos () or (). Howeer f we look for pproe solo hese fcos c lso e wre s power-seres epsos ) ( ) ( (7) where 9 re coss o e deered. Now f we sse (7) d (4) (6) sse of o-ler polol eqos hg he s d s s kows s fored. I geerl here re solos o hs sse ech oe correspodg o dffere se of odl reloshps (5).e. dffere r fold. Moreoer ssg of hese solos (7) d he reslg epressos (5) he k-h eqo ()-(4) clled he odl oscllor eqo chrcees he dcs of he

5 correspodg ode. Dels of he procedre s oled re oded here for re c e fod Sores d Mzzll [] where s lso show h he solo of he o-ler polol eqo eoed oe c e oded proded he egeles d egeeco of he leed sse re kow. NUMERICAL RESULTS Fgre depcs plr seel cer er (SCR). The geoercl d echcl properes of he SCR re see Tle. Tpcl les of rl perods of oscllo of 88 o se-sele plfor oored 8 wer deph re of order 5s ch lrger h he f (ler) rl perod of he cer er of order of s (Fgre preses ercl solos for hree dffere les of l rgd d copo wh WB lcl o pproo; o crre; θ L 7 ; fro Pesce d Mrs [7]).e. he er dcs e cosdered qs-sc wh he hpohess of fed-ed he och dow po qe cceple [7] les l sd. A frher locl correco TDZ e he ppled e.g. odr-ler echqe [8]. og s odls Rser legh Cross-seco re Cross-seco oe of er Rser eerl deer Rser hckess Il eso ( he op) Il eso ( he oo) E. N/ L 8 A. I D. e T N 5 T 6.94 N Rser ss per legh 8kg/ (wer sde dded ss) Rser wegh per legh p 77N/ Tle : Mechcl d geoercl er properes. θ L H L Θ(s) Fgre : Mechcl d geoercl er prole (llsro).

6 Fgre : Nrl freqeces of SCR. Fgre shows he eploed fe-elee odel d he odr codos for he fedoored SCR. A sprg s plced he er op wh he eo of odellg he sffess of he oorg sse. The sprg sffess s 9 kn/. Also cocered ss M56 o s se he er op order o llow for he horzol er of he FPU pls dded-ss effecs. Fgre 4 por he SCR fe elee odel wh he erg of he degrees of freedo. The odl dsplcee s chose s he ffeh degree of freedo. The fe elee odel cosss of 6 o-ler srgh e elees. Alhogh s ol -ple lss he hdlg of hrd- d forh-order eso reqres cosderle copol effor. To clcle he o-ler odes of hs odel ook 7 ho of processg e sg.6ghz processor d GB of RAM eor. Usg he r fold pproch he o-ler odl oscllor for he f ode s sogh s follows: & &. 4646E E- 84. E E-6. 67E- (8) -. 89E E-6. 47E-4. 54E-5 where (he ffeh degree of freedo Fgre 4) d re he odl dsplcee d he odl eloc ( & ) respecel. Eqo (8) e rewre elg he los-ll e s follows: & & (9) Fgres 5 e 6 show respecel he e-hsor respose d he phse porr oed egrg eqo (9). The Rge- Mehod of forh order ws eploed o egre (9). The l codos re () 5 d ().

7 8 6 4 ( ( ) Fgre : Bodr codos ( ) Fgre 4: The erg of he degrees of freedo. The ffeh degree of freedo s chose s he odl dsplcee. ( ) Lkewse Fgres 7 e 8 depc respecel e-hsor respose d phse porr for he followg l codos: () d (). Oserg Fgres 5 o 8 s ocele h lhogh he ler respose prll grees wh he o-ler oe here re rerkle dffereces ewee he. I s oeworh he serc phse recores wh respec o he odl-eloc s he o-ler respose (he odl dsplcee s -64 for he o-ler respose opposo o -5 for he ler oe Fgre 6). I Fgre 8 he o-ler odl dsplcee s -7 whle he ler oe s -. Sch resls re reled o he SCR sc eqlr cofgro h leds o serc sffess. The ler epls he o-serc ehor osered he Fgres 5 o 8.

8 5.. () < <.e; -64. < < 5. (s) Fgre 5: Modl rle e hsor. Ler[le] d No-ler[red].. (/s) < < < <. Z Fgre 6: Phse porr of odl rle. Ler[le] d No-ler[red]. ().. () < <.e; -.7e < <.e Fgre 7: Modl rle e hsor. Ler[le] d No-ler[red]. (s). (/s) e < <.e -7. < < 7. Z Fgre 8: Phse porr of odl rle. Ler[le] d No-ler[red]. ()

9 Oher slos cosderg o oorg sse led o (lgercll) sller o-ler odl dsplcees For he se er cosderg fed eds he dsplcee s -8 for () 5 d () d -7 for () d (). The phscl resog ehd hese resls s h he FPU de o s hge er so cosderle pr of he kec eerg whe llowed o oe hs redcg he er odl respose. B sdg eqo (9) oe oce h he e d he oppose sgs.e. he er redces he sse sffess whls he er ehes he oppose w. The wer er depeds o he er pree/geoer d he ge l codos. Therefore s o rl o def he red o hrdeg or sofeg. For l plde of 5 he o-ler perod (.9s) s slghl lrger h he ler oe (.8s). Followg hs edec he o-ler perod creses o 4.9s whe he l plde s se eql o. The e e re resposle for he serc ehor sce he re o fod he solos of sercl sses. I s es o ded her role f oe keeps d h hese e he he se sg (pose hs sd) drg he sse oo. Ths es her coro s o-serc wh respec o he odl dsplcee. I s por o ephsze h relsc les of plde he ee sed hs work order o sress eresg qle respecs of he resls here ddressed. For prccl prposes csoso effor wold e eeded o ds he deeloped procedre o he feld reqrees. 4 CONCLUSIONS Resls po o h o-ler odellg of er srcrl ehor gh e reqred for lrge-plde ro. Rele dffereces gh he pper he odl shpes d phse recores of seel cer e wh copl sppors. Those ler dffereces e por f drg d er fld lods re frher cosdered. I s o e frher esged he possl of o-ler coplg of lower odes whch reqre he cosdero of l-odes. For oh cer d srgh pre-esoed e for whch ler freqeces e show o e erl proporol o he ode er wold o coe s srpe he pperce of : or : erl resoce of lower odes ledg o srog o-ler coplgs. The coplg lss wold lso e rele he cse of lodl eco of hgher odes VIV despe he fc h sch dc rege he plde of oscllos s sll sll of he order of oe or wo dee ol. I s reclled here h he lerl coc TDZ d he oo of he TDP he o ee ke o cco hs sd so h hese effecs shold e looked e. No-ler odes re eleed o pl por role degree-of-freedo odel redco. To ssess how good or d re he qle d qe resls oed wh he redcedorder odel geered fro he NNM s he er respose o dc lods sch s sde flow VIV d FPU drfg s e copred wh hose oed wh he fll fe-elee odel. The f seps hs dreco he lred ee ke [].

10 Refereces [] Mzzll C.E.N. Lec S. Norl ro odes of sleder e o elsc fodo wh lerl coc Proc. II ICTAM Adelde (8). [] Mzzll C.E.N. Sores M.E.S. No-ler orl odes of plr fres dscresed he fe elee ehod Cope & Srcres (). [4] Shw S.W. Perre C. Norl odes for oler ror sses Jorl of Sod d Vro 64 () 85-4 (99). [5] Mzzll C.E.N. Brcho Neo O.G.P. Elo of o-ler orl odes for feelee odels Cope & Srcres Vol 8 () pp (). [6] Brcho Neo O.G.P. Mzzll C.E.N. Elo of l-odes for fe-elee odels: sses ed o : erl resoce Ierol Jorl of Solds d Srcres Vol 4 pp (5). [7] Pesce C.P. Mrs C.A. Sler L.M.. Rser-Sol Ierco: Locl Dcs TDP d Dscsso o he Egele d he VIV Proles Jorl of Offshore Mechcs d Arcc Egeerg 8 pp (6). [8] Pesce C.P. d Mrs C.A. Nercl Copo of Rser Dcs Nercl Modellg Fld-Srcre Ierco Adces Fld Mechcs Seres Ed. S. Chkrr WIT Press Sohpo U Chper 7 pp. 5-9 (5). [9] Sches C.T. Modos ão leres de rção e corole o de e PhD Thess Escol Polécc d Uedde de São Plo ( Porgese) (9). hp://