Any response quantity, x(t), may then be described by a second-order Volterra series representation as follows:

Transcription

1 ir gap assessen for foaing srres in rando seas L. Mane Deparen of Civi ngineering Universiy of Teas a sin, sin, TX 787 ane@ai.eas.ed bsra Firs- and seond-order iniden and diffraed wave effes are sdied o deerine he infene of he oions of a sei-sbersibe on he insananeos air gap. Saisis of he air gap response are esiaed wih differen odeing asspions. In hese sdies, a singe fied poin is sdied one a he geoeri ener of he sei-sbersibe. e, a oparison of he air gap a differen oaions is sdied by eaining response saisis a differen fied poins for he sei-sbersibe. These inde oaions ose o ons of he for-oned sei-sbersibe. nayia prediions, inding firs- and seond-order diffraed wave effes, are opared wih wave an easreens a severa oaions. In pariar, air gap response saisis sh as he roo-ean-sqare and he -hor edian eree are opared. Bagrond The air gap response, and poenia de ipa, of oean srres nder rando waves is of onsiderabe ineres. ir gap odeing is pariary opiaed in he ase of foaers bease of heir arge voe, and he resing effes of wave diffraion and radiaion. These give rise o wo disin effes: goba fores and resing oions are signifiany affeed by diffraion effes; and oa wave eevaion odeing an aso be onsideraby infened by diffraion, pariary a oaions nderneah he de and/or near a aor on. Boh effes are iporan in air gap prediion: we need o now how high he waves rise ie above, and how ow he de ransaes veriay de o ne heave and pih a a given poin o ee he waves. Moreover, effes and are orreaed in ie, as hey res fro he sae nderying iniden wave eiaion proess. We fos here on anayia diffraion odes of air gap response, and is resing sohasi nare and neria prediions nder rando wave eiaion. enion is fosed on a sei-sbersibe pafor, for whih boh sow-drif oions heave/pih and diffraion effes are poeniay signifian. Varios effes are sdied separaey: e.g., wave fores on a fied oed-down srre, he effe of srra oions on air gap response, and finay, he effe of differen oa wave eevaion odes in sep. For referene, a opee seond-order diffraion ode is foraed and sdied. Response Saisis: Moens and rees In odeing foaing srres, i is oon o epoy Voerra series o desribe he response op of hese noninear syses. The noninear syse is defined in ers of firs- and seond-order ransfer fnions. For foaing srres, hese ransfer fnions are obained fro firs- and seond-order wave diffraion anaysis progras sh as WMIT e.g., WMIT, 995. We sar by defining a sea srfae eevaion, η, in ers of a s of sinsoida oponens a disin freqenies and a wave sper, S η ω as foows: η a os ω θ Re ep iω ; S ω ω ep iθ ny response qaniy,, ay hen be desribed by a seond-order Voerra series represenaion as foows:

2 where, -, and are he firs-order, seond-order differene-freqeny and seond-order s-freqeny onribions, respeivey, o he response. We an wrie eah of hese oponens in ers of ransfer fnions. Ths, we have: i H i H * ; ]; ep[ Re ep Re ω ω ω We wi haraerize he physia response ode and esiae response erees sing he firs for saisia oens of he response. One an hen epress in ers of ay independen sandard Gassian proesses,. Ths, we have: ; The oeffiiens and are obained by soving an eigenvae probe invoving he ransfer fnions and inp power spera densiies see Ka and Seiger, 97. To represen he for saisia oens, we wi epoy he ean, sandard deviaion, and he diensioness oeffiiens of sewness and rosis : ]/ [ ; ]/ [ ] [ ]; [ 5 In ers of and, hese firs for oens ay be given as foows: 8 8 ; 8 6 ; 6 One hese oens are fond, he response proess ay be reaed o a sandard Gassian proess sing a Herie ransforaion ode Winersein, 998: ] [ κ 7 where, and κ are oeffiiens ha an be esiaed in ers of and. The p-fraie eree response in a seasae draion T an hen be esiaed fro q. 7 aing as he orresponding Gassian p-fraie eree: / 0 a, ] n/ [n p T p ν 8 where ν 0 is he average response freqeny. ir Gap Response We sar by defining he ne wave eevaion, η T, wih respe o a fied origin. Then, if a a fied poin of ineres,,y, δ denoes he ne veria dispaeen of he srre, he reaive wave eevaion r easred wih respe o he oving srre ay be given by: where ; 5 y r T ξ ξ ξ δ δ η 9 ipying ha δ is given in ers of he heave ξ, ro ξ, and pih ξ 5, oions. The avaiabe air gap a is he differene beween he si-waer air gap and r: 0 r a a 0

3 The insananeos ne wave eevaion, η T, in q. 9 is a res of boh he iniden waves ha wod or if he srre were no presen, and he diffraed waves ha arise bease of he presene of he srre ha aers he fow fied. η T η, T η, T η, T η, I η, D ; η, T η, I η, D In q., we see ha in or seond-order ode, we need o represen he ne wave eevaion as ade p of firs- and seond-order effes de o boh iniden and diffraed waves. The firs-order iniden wave η,i is odeed as a saionary Gassian proess, and onsisen vaes of η,d, η,i, and η,d are aaed fro hydrodynai heory. The ehodoogy sing Voerra series odes and oen-based eree esiaion has been ipeened in a pos-proessing roine Ude e a., 996 ha ses firs- and seond-order fore ransfer fnions and added ass and daping. The ransfer fnions obined wih siffness, daping, and ineria properies of he srre provide firs- and seond-order ransfer fnions o any response qaniy,. In oping he air gap response, we need o sianeosy inde boh seond-order s-freqeny effes on he wave srfae, and seond-order differenefreqeny effes on sow drif oions. The air gap response, as desribed by qs. 9 hrogh above, has been ipeened in he foraion based on seond-order Voerra series and oen-based erees odes Mane and Winersein, 998. Pafor Desripion The srre hosen for he neria sdies is he Tro sei-sbersibe. Figre shows a pan view of he pafor ha has for ons and pan diensions, The si-waer air gap is 5 eers and he ean waer deph is 5 eers. Measreens of air gap were ade a seven differen fied poin oaions. hese sae oaions, WMIT diffraion anayses were perfored for waves wih differen headings and wih wave periods ranging fro 7. o 0.0 seonds. The seven fied poins oaions are indiaed on Figre. ernaive Modeing Opions We are ineresed in sdying he infene of aernaive odeing opions in desribing he reaive wave. We wi sdy he air gap a fied poin no. and wi onsider a - hor seasae haraerized by head seas and wih a signifian wave heigh of 0.6 eers and a spera pea period of.5 seonds. To isoae he effes of iniden and diffraed waves of firs- and seond-order, and he effe of ne veria oion, we onsider severa differen ases. Tabe sarizes he response saisis for he Tro sbersibe in eah of he ases defined. Conribions o he response saisis fro he firs-order effes aone and fro he s of firs- and seond-order onribions are inded. Fro he abe, he foowing findings ay be noed as we ove aross he varios ases: Case 0 Ony firs-order iniden waves η,i inded; srre oed down i.e., oions prevened, δ 0. Bease his ase indes ony firs-order iniden waves, he ne wave eevaion is seen o be Gassian. so, he rs response.65 eers is eqa o one-forh of he signifian wave heigh 0.6 eers. The Gassian haraer is aso onfired by he pea faor on he edian eree of.8 see q..8 wih p 0.5, ν 0 T 080 as epeed for a -hor seasae. Case Ony firs- and seond-order iniden waves η,i and η,i inded; srre oed down. Case indes a seond-order Soes iniden wave proess, whih ases he ne wave eevaion o be non-gassian and posiivey sewed. The seondorder proess provides a sa onribion o he oa response: is rs is ony % of ha of he firs-order proess. The pea faor of he oa proess is..

4 Case Firs-order iniden and diffraed waves η,i and η,d, seond-order iniden waves η,i inded; srre oed down. ddiion of he firs-order diffraed waves in Case has he effe of raising he rs of he firs-order proess by % and he edian eree by 0%. The pea faor of he oa proess response is.7 i.e., he oa proess is ore Gassian han in Case de o he arger reaive onribion of he firs-order effes. Case Firs- and seond-order iniden and diffraed waves η,i, η,d, η,i, and η,d inded; srre oed down. ddiion of seond-order diffraed waves in Case ases a arge inrease in response, os noaby in is eree eves. In pariar, he rs eve hanges ony fro.7 o.50, whie he pea faor grows fro.7 o 5.5. This enhaned pea faor is de o he ared non-gassian behavior predied in his ase: he sewness vae is fond o be 0.5, and he rosis vae,.. Heave Srre s heave oion ξ sdied. gain, we see ha noninear effes here, he effe of differene-freqeny sow-drif oions ony idy infene rs vaes 0.78 inreases ony o 0.79, b ore noaby affe non-gassian behavior sewness vae of 0.. Base Firs- and seond-order iniden and diffraed waves η,i, η,d, η,i and η,d inded; srre peried o ove i.e., δ 0. Finay, he base ase ress predi he reaive wave response for a srre peried o ove. These ress obine or "bes" ode of he wave eevaion inding seond-order diffraion effes as in Case wih or orreaed ode of assoiaed veria oions. The ress show srong non- Gassian behavior: sewness of 0.5, rosis of.55, and pea faor of 6.. In view of heir siiariy wih Case vaes, hese srongy non-gassian effes appear o be de o he presene of seond-order diffraed waves, and are no weaened when srra oions are inded. Of orse, by periing he srre o ove wih he waves, he reaive wave response is reded as opared wih he oed-down srre in Case. nayia Prediions verss Wave Tan Tes Daa We ne opare anayia prediions wih ode es ress. For differen seasaes are onsidered: H s, T p T p.0s. Ress are shown beow for hree fied poins, inding again he pafor id-poin and wo oher oaions nearer o ons. Fro Tabe, we see ha seond-order diffraion is fond o ony oderaey inrease he rs response, b aredy inrease pea faors and erees. Ths, even if seond-order effes in he iniden wave are reained, nonservaive air gap prediions ay res if seond-order diffraion effes are negeed. Pariary srong non-gassian behavior is predied a he pafor idpoin, as opared wih prediions orresponding o oher poins nearer a on. Finay, we opare he ode prediions wih he observed wave an saisis, shown in he fina wo ons of Tabe. We firs noe ha hese observed erees pea faors eah arise fro a singe -hor es, and are herefore raher noisy esiaes of he edian vaes aross any siiar -hor ondiions. Si, hese ress sgges ha ignoring seond-order diffraion effes od ead o nderesiaion of response erees. When seond-order diffraion effes are inded, pea faors are no onger syseaiay nderesiaed b agreeen wih observaions reains iperfe. he pafor idpoin, for insane, he prediions appear "oo noninear." oneheess, hese ress sgges ha noninear diffraion effes an be iporan, and shod be sdied frher. We beieve ha he genera saisia odes presened here, whih esiae erees fro a iied se of saisia oens, offer an effiien approah o assess he ipa of varios noninear odes on eree response eves.

5 Consions ehodoogy has been presened for desribing he air gap response for foaing srres. The iporane of iniden waves reaive o diffraed waves of boh firs and seond order has been sdied. For he Tro sei-sbersibe, a f seond-order anaysis inding iniden and diffraed waves p o seond order is neessary; i is nonservaive o nege he seond-order diffraed waves. In oparing he differen fied poin oaions, iporan non-gassian effes are observed ha differ sighy depending on he proiiy o a on. The air gap a he ener of he pafor ehibied he greaes non-gassian haraer. This non- Gassian haraer is argey a res of seond-order diffraed waves. nowedgeens The ahor graefy anowedges sef disssions wih Dr. Seven. R. Winersein of Sanford Universiy and wih Dr. Gdnd Keiven of ors Hydro who aso provided he daa o ae his sdy possibe. Referenes. Ka, M. and. J. F. Seiger 97. On he Theory of oise in Radio Reeivers wih Sqare Law Deeors, Jorna of ppied Physis, Vo. 8, pp Mane, L. and S. R. Winersein 998. siaion of ir Gap Saisis for Foaing Srres: Reease of TFPOP Version. Coper Progra for Perforing Sohasi Response naysis of Foaing Srres, Tehnia oe T-, Reiabiiy of Marine Srres Progra, Sanford Universiy.. Ude, T. C., S. Kar, and S. R. Winersein 996. TFPOP.: Sohasi Response naysis of Foaing Srres nder Wind, Crren, and Seond-Order Wave Loads, Tehnia Repor RMS-8, Reiabiiy of Marine Srres Progra, Sanford Universiy.. WMIT, WMIT: Radiaion-Diffraion Pane Progra for Wave- Body Ineraion Users Mana, Dep. of Oean ngineering, M.I.T. 5. Winersein, S.R oninear Vibraion Modes for rees and Faige, Jorna of ngineering Mehanis, SC, Vo., o. 0, pp Figre. Pan view of Tro sei-sbersibe showing fied poins sdied.

6 Type Mean Sd. Dev Median Pea Response Sewness Krosis of Rn ree Faor Case 0 TOTL Case s -order TOTL Case s -order TOTL Case s -order TOTL Heave s -order TOTL Base s -order TOTL Tabe ir Gap Response Saisis for Cases invoving differen Modeing Opions Fied Poin a ener Hs Tp naysis w/o nd -order naysis w/ nd -order Wave Tan s Tess PF PF PF Fied Poin 5 on Hs Tp naysis w/o nd -order naysis w/ nd -order Wave Tan s Tess PF PF PF Fied Poin 6 in fron of a on Hs Tp naysis w/o nd -order naysis w/ nd -order Wave Tan s Tess PF PF PF Tabe Response Saisis RMS Response and Pea Faor on Median ree in hors nayia Prediions verss Wave Tan Tes Daa a Fied Poins