13.42 Design Principles for Ocean Vehicles Prof. A.H. Techet Spring 2005

Transcription

1 13. Dsgn Prncpls fr Ocan Vhcls Radng # 13. Dsgn Prncpls fr Ocan Vhcls Prf. A.H. Tcht Sprng Ocan Wav Spctra 1. Wav nrgy spctra. Rd txt ndcats wav gnratn chanss and blu txt ndcats dapng/rstrng frcs. Th ajrty f can wavs ar wnd gnratd. Othr wav gnratng chanss nclud arthquaks and plantary frcs. Plantary frcs drv tds and caus lng 00, 005 A. H. Tcht 1 Vrsn 3.1, updatd //005

2 13. Dsgn Prncpls fr Ocan Vhcls Radng # prd wavs n th rdr f 1 t hurs. Earthquaks ar th ajr caus f tsunas whch, whl rar, can b catastrphc f th arthquak ccurs nar r n th cast. Wavs als ncuntr frcs that tnd t rstr th t a flat surfac. Fr sall wavlngth (hgh frquncy) wavs surfac tnsn plays a larg rl n dapng ut ths wavs. Th ajrty f wavs ar rstrd by gravty and lngr prd wavs ar dapd by th Crls frc. As wnd bgns t blw (btwn knts) n a cal surfac sall rppls, capllary wavs r cat-paws, tnd t fr. Ths sall wavs ar n th rdr f lss than c. As th wnd bcs strngr wav apltud ncrass and th wavs bc lngr n rdr t satsfy th dsprsn rlatnshp. Ths grwth s drvn by th Brnull ffct, frctnal drag, and sparatn drag n th wav crsts. Wnd ust blw vr lng prds f t and larg dstancs t rach a fully dvlpd sa stat. Whn th phas spd f th wav crst atchs th wnd spd nn-lnar ntractns stp (xcpt frctn) and th phas spd s axzd. Th ltng frquncy f th wavs can b dtrnd by th quatn fr phas spd and th dsprsn rlatnshp: C U = / k = g/ (1) p w g c () U w whr U w s th wnd spd and c s th ltng frquncy. Onc wnd stps vscsty rds th wavs slwly. Th sallst wavlngths dcay th fastst. Sapl spctru shaps ar shwn n fgur. 00, 005 A. H. Tcht Vrsn 3.1, updatd //005

3 13. Dsgn Prncpls fr Ocan Vhcls Radng # Fr a str wth wnd spd, U w, th ffcts f th str can b flt at a dstanc fr th str, R. Th nubr f wav cycls btwn th str and th bsrvatn lcatn s = R/ λ. Th apltud f th wavs dcays as Landau and Lfshtz). γ t whr γ ν ν = k = / g (fr Th dvlpnt f strs can b tabulatd. Ftch s th lngth vr whch th wnd ust blw t hav fully dvlpd sas (gvn n standard ls), and th str duratn, gvn n hurs, s th t th str ust last t rsult n a fully dvlpd sa. Wnd warnngs Baufrt scal Wnd spd (ph) Ftch (ls) Str duratn (hr) sall craft gal hurrcan , 005 A. H. Tcht 3 Vrsn 3.1, updatd //005

4 13. Dsgn Prncpls fr Ocan Vhcls Radng #. Typcal Wav Spctra Rsarchrs hav studyng can wavs hav prpsd svral frulatn fr wav spctra dpndnt n a a nubr f paratrs (such as wnd spd, ftch, r dal frquncy). Ths frulatns ar vry usful n th absnc f asurd data, but thy can b subjct t ggraphcal and sasnal ltatns. Mst can wav spctra tak a standard fr fllwng th athatcal frulatn: S A = (3) + B/ ( ) 5 Th frquncy pak s calld th dal frquncy. Th ara undr th spctru s th zrth nt, M, whch ay b dfnd n trs f th sgnfcant wav hght. Fr a narrw-bandd spctru th sgnfcant wav hght s apprxatly fur ts th squar rt f th zrth nt. Snc th sgnfcant wav hght dpnds n th wnd spd, th spctru culd b frulatd n trs f th wnd spd nstad f th sgnfcant wav hght. Whl crtan spctra can hav r than n pak, t s assud that a sngl str prducs a sngl-pakd spctru and any scnd pak s du t a dstant str that snds wavs t th cnsdrd lcatn. Svral r spcfc thrtcal rprsntatns f wav spctra hav bn dvlpd usng data cllctd by bsrvatn platfrs and satllt data n varus rgns. Ths spctra ar dscussd blw. Whn cnsdrng whch spctru frulatn t s prtant t tak nt accunt th spcfc crtra that wr usd n dvlpng th spctru. Typcally paratrs that nflunc th spctru ar: Ftch ltatns,.. whthr th lcatn w ar cnsdrng has s physcal bundars that d nt prt th wavs t fully dvlp. Whthr th sas ar dvlpng r dcayng Saflr tpgraphy: Dp watr wav spctra ar nvald n shallw watrs, and vc vrsa. It ay als b ncssary t accunt fr wav dffractn. Lcal currnts: Strng currnts ay sgnfcantly pact th wav spctru 00, 005 A. H. Tcht Vrsn 3.1, updatd //005

5 13. Dsgn Prncpls fr Ocan Vhcls Radng # Prsnc f swlls: Swlls ar wavs that rsult fr dstant strs that travl a sgnfcant dstanc and arrv ftn at an angl that dffrs fr th wnd drctn. If w us a spradng functn t crrct a undrctnal spctru t wll nt accunt fr th prsnc f swll. It s als prtant whn asurng wavs that th cpnnt that rsults fr swll b accuntd fr sparatly. Th Prsn-Mskwtz spctru (quatn 11) was dvlpd fr fully dvlpd sas n th rthrn Atlantc Ocan gnratd by lcal wnds. S 81. g = () ( g/ ζ ) ( ) 3 5 whr ζ s th sgnfcant wav hght, ζ 13 / H = M, (5) and s th dal frquncy, = 0. g/ ζ. (6) Ths spctru s dvlpd undr th fllwng cndtns: undrctnal sas, rth Atlantc Ocan, fully dvlpd lcal wnd gnratn wth unltd ftch. Th st crtcal f ths assuptns s th fully dvlpd assuptn. Fr t s pssbl t achv a largr hav rspns fr a platfr fr a dvlpng sa, vn thugh th sgnfcant wav hght ay b sallr that that f a fully dvlpd sa, snc th dal frquncy s hghr and hav tns tnd t hav hghr natural frquncs. In th cas f a rllng shp th dcayng sa ght xct a largr rll tn snc th natural frquncy f rll tnds t b rlatvly lw. In rdr t vrc th ltatn f fully dvlpd sas, a tw paratr spctru was dvlpd. Ths spctru s th Brtschndr spctru (quatn 1). Th B-S spctru 00, 005 A. H. Tcht 5 Vrsn 3.1, updatd //005

6 13. Dsgn Prncpls fr Ocan Vhcls Radng # rplacd th Prsn-Mskwtz spctru as th ITTC standard. S ( / ) ( ) = ζ (7) 5 +. whr agan, ζ s th sgnfcant wav hght, ζ 13 / H = M, (8) If satsfs quatn 13 thn quatn 1 rducs t quatn 11. By allwng th usr t spcfy th dal frquncy and sgnfcant wav hght, ths spctru can b usd fr sa stats f varyng svrty fr dvlpng t dcayng. Th Och Spctru (quatn 16) s a thr paratr spctru that allws th usr t spcfy th sgnfcant wav hght, th dal frquncy, and th stpnss f th spctru pak. S 1 λ λ + 1 ζ + λ + 1 ( ) = xp λ + 1 Γ( λ) (9) whr Γ ( λ) s th gaa functn, and λ s th paratr that cntrls th spctru stpnss. Fr λ = 1, quatn 16 rducs t quatn 1. Th Och spctru s ltd n that t als cnsdrs nly undrctnal sas and unltd ftch, but th dsgnd can nw spcfy th spctrus svrty (ζ ), th stat f dvlpnt (pak frquncy ) and slat th prtant frquncy rang by dctatng th spctru wdth (λ ). Th ablty t dctat λ allws th dsgnr t accunt fr swll fr a dstant str. Th JOSWAP spctru (quatn 17) was dvlpd by th Jnt rth Sa Wav Prjct fr th ltd ftch rth Sa and s usd xtnsvly by th ffshr ndustry. Ths spctru s sgnfcant bcaus t was dvlpd takng nt cnsdratn th grwth f wavs vr a ltd ftch and wav attnuatn n shallw watr. Ovr,000 spctra wr asurd and a last squars thd was usd t btan th spctral frulatn 00, 005 A. H. Tcht 6 Vrsn 3.1, updatd //005

7 13. Dsgn Prncpls fr Ocan Vhcls Radng # assung cndtns lk nar unfr wnds. S + ( ) 5 ag δ ( ) = γ (10) 5 whr δ ( ) = (11) σ a x. ( 0) = (1) gx x = (13) U σ = 007. ; (1) 009. ; > (1) Th wnd spd n knts s U, x s th ftch n nautcal ls, and th dal frquncy can b fund as ( ) 033. = π 35. /. (15) gu x T rcap, n gnral fr a narrw bandd spctru: + ζ S ( ) d = M = (16) W can accunt fr th ffcts f tw sparat strs by addng th rspctv spctrus: S ( ) = S ( ) + S ( ) (17) W can als crrct fr drctnalty ultplyng th spctru by a spradng functn, 00, 005 A. H. Tcht 7 Vrsn 3.1, updatd //005

8 13. Dsgn Prncpls fr Ocan Vhcls Radng # M ( µ ), + + S ( µ, ) = S ( ) M( µ ) (18) BS whr ( ) M µ sprads th nrgy vr a crtan angl cntand wthn th ntrval [ ππ, ] fr th wnd drctn. Th ntgral f M ( µ ) vr ths ntrval s n. π M( µ ) dµ = 1 (19) π Fr xapl w can chs a spradng functn such that n th ntrval = (0) π M ( µ ) cs µ µ µ < µ < (1) 3. Brtschndr Spctru T rcap, th 15th Intrnatnal Twng Tank Cnfrnc (ITTC) n 1978 rcndd usng a fr f th Brtschndr spctru fr avrag sa cndtns whn a r spcfc apprprat fr f th wav spctru s wll dfnd. Th gnral fr f ths spctru s quatn rfq:spcgn. S A = () + B/ ( ) 5 Th tw paratrs A and B ar dpndnt n th dal frquncy,, and th varanc f th spctru, M rs σ = ( ) =. 00, 005 A. H. Tcht 8 Vrsn 3.1, updatd //005

9 13. Dsgn Prncpls fr Ocan Vhcls Radng # = B ; B = 5 / (3) 5 = = /( ) ; = () Varanc σ A B A σ B If w nralz th frquncy,, by th dal frquncy quatn 30 bcs quatn 33. S( ) 5 ( ) 5 = σ (5) 5 Fr a narrw bandd spctru, ε < 06., th sgnfcant wav hght, ζ 13 / = H = M, whr M s th varanc f th spctru. Fr a wd bandd spctru, ε = 1, St. Dns (1980) shwd that th sgnfcant wav hght was apprxatly, ζ = 3 M. Ths lavs us wth th fnal fr f th Brtschndr Spctru. S( ) 15. ( ). 15 = ζ (6) 5 Th nts f th spctru can b calculatd nurcally. Fr splfcatn th fllwng rlatnshps hav bn gvn (s Prncpls f aval Archtctur vl. III fr furthr dscussn). Th furth nt dvrgs slwly as thus th apprxatn hlps analyss whn calculatn f ths nt s ncssary. M =VARIACE = ( RMS) M = ζ ( ). =. M M 709. M fr 5 < 5 00, 005 A. H. Tcht 9 Vrsn 3.1, updatd //005

10 13. Dsgn Prncpls fr Ocan Vhcls Radng #. 1/ th Hghst Maxa Fr dsgn purpss t s usful t dtrn th ccurrncs f wav apltud axa abv a crtan lvl. Gvn a t-trac f wav hght data fr a buy dplyd n th can w can analyz t t dtrn such nfratn. If w lk at a gvn wav tran, yt (), vr an ntrval n t, T ( T s nt ncssarly a wav prd but r lk a duratn f t whch can cnsst f ultpl prds), thn wthn ths ntrval f t thr ar a nubr f axa, a1, a, a3, a, a, n, whr 1 a / s th valu xcdd by 1/ f th axa. (t: 1 a / s nt a t th 1/ pwr) 3. EXAMPLE: Tak = 3 ; Fr a squnc f twlv asurd wav hghts fnd th 13 / rd (1/ ) hghst wav hght, 13 a / : Masurd wav hghts: Thr ar 1 rcrdd wav axa s thr ar fur wav axa abv th 1/3rd 00, 005 A. H. Tcht 10 Vrsn 3.1, updatd //005

11 13. Dsgn Prncpls fr Ocan Vhcls Radng # hghst wav hght. Fr ths squnc 13 a / = 6 snc t s th nxt hghst wav axa blw 7. Had ths bn a nfntly lng srs f bsrvd wav hghts and 7 was th lwst valu f th 1/3rd hghst wav axa, thn 6.99 ght bn a bttr statn fr ths valu. Hwvr n ths shrt squnc th answr s 6. Had th squnc bn: [ ] thn 13 a / = 8, snc th fur hghst wav hghts ar all qual t 11. Th prbablty f wav hghts ccurrng abv th 1/th hghst wav s gvn by whr ε / / ( η ) / P( η η ) = (7) ε η 1/ 1/ a 1 ε = = ln 1 1 M + ε (8) Th avrag valu f ALL axa abv a 1/ s calld th 1/ (th) hghst avrag apltud. Ths can b fund usng th frula fr xpctd valu f a varabl: { ( )} 1/ 1/ a = E a a > a (9) Ths s th xpctd valu gvn a s gratr than 1 a / as can b rprsntd as 00, 005 A. H. Tcht 11 Vrsn 3.1, updatd //005

12 13. Dsgn Prncpls fr Ocan Vhcls Radng # (( ) ( )) 1/ 1/ a = a p a = a a > a da 1/ a (30) whr th prbablty, pa = a a > a /, s sply 1 ( 1 / ( = ) ( > )) P a a a a pa = a a > a = ( ) 1/ 1/ Pa > a (31) Kpng th apltud n nn-dnsnal fr w can calculat th th hghst avrag wav hght usng th apprxat pdf gvn n th last radng. 1 1 ε / η / η η 1 dη / η (3) ε Fr a valu f = 3, 1 a / s cnsdrd th sgnfcant wav apltud whr 1/ a σ M = fr ε < 05.. Th sgnfcant wav hght s dfnd as twc th sgnfcant wav apltud, H = a. (33) 13 / 13 / Ths valu s vry cls t that whch a casual bsrvr wuld stat as th wav hght whn watchng th sa. Ths aks th sgnfcant wav hght a vry usful statstc. Maps f th sgnfcant wav hght vr th ntr arth can b sn n satllt ags. Thr ar svral lnks n th curs wbpag that llustrat ths quantty vr th arth surfac. Fr such a satllt cpst ag w can s that th suthrn can s th st tuultuus can and has th hghst sgnfcant wav hght. 00, 005 A. H. Tcht 1 Vrsn 3.1, updatd //005

13 13. Dsgn Prncpls fr Ocan Vhcls Radng # 5. Lng Tr Statstcs Structural dsgn analyss f ffshr structurs vr a lng t, spcfcally th ttal lf span f th structur r T, rqurs knwldg f th shrt tr statstcs f wavs at th syst nstallatn lcatn. Ovr th lf f a platfr r structur thr ar a ttal f strs, n f ths can als nclud a str whch s dscrbd cpltly cal cndtns, slar t th null st n prbablty. Th prbablty f ach str vnt s, P, whch can als b lkd at as th fractn f th ttal lf f th structur vr whch a crtan str,, xsts. Thus th lf f ach str, T s dctatd by quatn 1. T = TP (3) Fr shrt tr statstcs w hav th frquncy f wavs, upcrssngs, xcdng a crtan apltud, a. 1 M 1 1 = = = M T T µ (35) a/ ( M ) a/ ( M) na ( ) π Whr T s th avrag prd f wavs n a str. S t s asy t fnd th ttal nubr f ts th lvl a s xcdd n a str, gvn by as T = na ( ) T = T µ (36) Whr T s th lf f th str, T s avrag prd f th wavs wthn that str. Th ttal nubr f ts lvl bcs: whr a s xcdd durng th lf f th structur thn T P a / ( M ) a = = (37) all T M s th zrth nt f th spctru f str. It s ncssary t ak th assuptn that th spctru s narrw bandd s that w can wrt th zrth nt n 00, 005 A. H. Tcht 13 Vrsn 3.1, updatd //005

14 13. Dsgn Prncpls fr Ocan Vhcls Radng # trs f th sgnfcant wav hght. Ths splfs th prbl snc th sgnfcant wav hght s th data st ftn avalabl that dscrb sa cndtns vr a crtan t prd. xt w can adjust th quatn fr ttal nubr f upcrssngs t rflct a nubr f upcrssngs xcdng th hght, h whch s sply dfnd as twc th apltud, a. h h ζ T = P (38) T Equatn 5 rprsnts th ttal nubr f upcrssngs past a lvl h vr a t ntrval T. Th ttal nubr f upcrssngs past th an watr lvl, h = 0, s gvn by quatn 6. T = P (39) T Havng dtrnd th quatn fr th ttal nubr f wavs xcdng a lvl h and als th ttal nubr f upcrssngs durng th lf f th structur, w can fnd th prbablty f a wav hght xcdng a dsgn hght h n quatn (0) T h /ζ P h T > = = T P T Ph ( h ) (0) Ths quatn rprsnts th fractn f ttal wavs abv lvl h. Ralzng th lf span f th structur, T, s cnstant w can cancl t ut f quatn (0) and rwrt t n th fr 1 h T > = = Ph ( h ) h/ ζ 1 ( T ) P P (1) 00, 005 A. H. Tcht 1 Vrsn 3.1, updatd //005

15 13. Dsgn Prncpls fr Ocan Vhcls Radng # Th nuratr and dnnatr f quatn (1) ach hav th fr f th xpctd valu frula fr a rand prcss wth rand varabls T and ζ. S w can rwrt th prbablty as 1 T h / ζ E Ph ( > h ) = () E 1 { T } t th subscrpt s drppd snc T s th rand varabl and T s a sngl rand vnt. Thus th xpctd valu f th rand varabl s th su vr all th th rand vnts. Ths can als b xtndd t a rand prcss. W nw hav and quatn fr th prbablty f wavs abv a lvl that s nt plctly dpndnt n th lf span f th structur. In practcal applcatn th avrag wav prd can b nglctd (canclld) and th prbablty apprxatd by Ph ( > h ) E. (3) h / ζ On valuabl calculatn s rlatd t th hundrd yar wav. Ths s th wav that xcds s valu, h 00 1, n th avrag nly nc n vry n hundrd yars. # ( f wavsvr h ) 1 T Ph ( > h 00) = = = () # /. scnds ( f ttalupcrssngs) 100 yars T t: Ths quatn taks nt cnsdratn 100 yars wth th apprprat nubr f lap yars. S nw w hav an quatn fr th prbablty f th hundrd yar wav. h100/ ζ E T = (5) Whr T s th avrag prd f wavs vr all strs. In rdr t slv fr h 100 w ust hav statstcs fr vry rugh strs wth hgh sgnfcant wav hghts. Sallr, r 00, 005 A. H. Tcht 15 Vrsn 3.1, updatd //005

16 13. Dsgn Prncpls fr Ocan Vhcls Radng # frqunt, valus f wav hght d nt play a rl n ths calculatn and t s nly th st frcus strs that cntrbut t ths valu. Thus yu can agn n rdr t cllct ths data bsrvatns durng ajr strs, hurrcans, typhns, tc., ust b prfrd vr t. Undr ths cndtns t s t rsky fr huan bsrvatn and ftn qupnt s lst at sa akng btanng ths challngng. Fr prbablty thry t s pssbl t apprxat th prbablty f xtr vnts by usng a Wbull dstrbutn n th fr x x1 ( x x ) γ 1 > = (6) Px ( x ) whr x 1, x, and γ ar statd fr hstrcal data and x 1 s a thrshld lvl blw whch th statstcs wll nt ffct th xtrs. 6. Encuntr Frquncy Th prncpl tns f a shp r structur n sas ar dfnd as surg, sway, hav, ptch, rll and yaw. A dagra f th crdnat syst s shwn n fgur 1. 00, 005 A. H. Tcht 16 Vrsn 3.1, updatd //005

17 13. Dsgn Prncpls fr Ocan Vhcls Radng #. Mtn f a shp dfnd wth a rght handd crdnat syst wth frward tn as th pstv x-drctn and pstv hav upwards. Wavs ncdnt n th structur r shp can b dscrbd as had sas, fllwng sas, ba sas, r quartrng sas dpndng n th ncdnt drctn. Fgur llustrats th frst thr cass and quartrng sas ar dfnd as ths that apprach th shp fr th thr th prt (lft) r starbard (rght) strn quartr f th shp (btwn 90 and 180 r 180 and 70 ). 5. Incdnt sa dscrptn. Fgur at lft shws had, ba and fllwng sas. Th ncdnt angl, µ, s asurd fr th bw (x-axs s µ = 0 ) cuntrclckws. Th tn f a shp, frward r thrws, affcts th way ncdnt wavs ar vwd by sn abard th vssl. Fr xapl f th shp s akng way n had sas wth a cnstant vlcty, U, thn th wavs wll appar t t th shp at a fastr rat than th actual frquncy f th wavs. Ths nw, r bsrvd, frquncy s trd th ncuntr frquncy,. If th wavs ar ncdnt n th shp at s angl, µ, thn th cpnnt f th spd f th shp n th drctn f wav prpagatn s U = U csµ. Th wav crsts v at th phas spd, Cp = / k and th rlatv spd btwn th shp and th a 00, 005 A. H. Tcht 17 Vrsn 3.1, updatd //005

18 13. Dsgn Prncpls fr Ocan Vhcls Radng # wavs s U = U + C = Ucsµ + / k (7) r a p Thus t appars that th wavs hav a phas spd U r such that Ur = = Ucsµ + (8) k k Usng th dsprsn rlatnshp fr wavs n dp watr w can rwrt th quatn fr ncuntr frquncy as U = cs µ + (9) g fr > 0. In practc w usually hav th ncuntr frquncy snc t s bsrvd and w wuld lk t calculat th actual wav frquncy, s takng quatn (9) w can slv fr frquncy,. g Ucsµ = 1± 1+ (50) U cs µ g Lkng at th quatn 17 w s that thr ar svral pssbl slutns fr. Ths ar dpndnt n th ncdnt angl µ. Fr U cs µ g 1 > (ral valus f ) w can lk at dffrnt ncdnt angls: (1) HEAD SEAS: ncdnt angl btwn π / < µ < π/ and th csn f th angl s pstv (csµ > 0). Th ncuntr frquncy s always pstv fr had sas. () FOLLOWIG SEAS: ncdnt angl btwn π / < µ < 3π/ and csµ < 0. Hr thr ar thr pssbl scnars: 00, 005 A. H. Tcht 18 Vrsn 3.1, updatd //005

19 13. Dsgn Prncpls fr Ocan Vhcls Radng # Ucs µ > C, < 0, shp vrtaks wavs p Ucs µ = C, = 0, shp surfs wavs p Ucs µ < C, > 0, wavs vrtak shp p Th fnal cas, whr th wavs v fastr than th shp, can tnd t caus prbls n cntrllng a vssl, spcally whn th sas ar nt drctly fr bhnd th shp. Ths ffct s gratst n rll and yaw. Whn a shp s vng wth th wavs but vrtakng th t can als appar as f th sas ar apprachng fr th bw, thus th actual ncdnt angl s abguus and culd b µ r µ + π. Th ffct f ncuntr frquncy als changs th bsrvd spctru f th sas. Enrgy ust b prsrvd undr th spctru thus w can lk at th fllwng S( ) d = S( ) d (51) S( ) S( ) = (5) d d W can fnd th drvatv f ncuntr frquncy by th actual frquncy by usng quatn (9). d 1 U cs µ d = + g (53) S th spctru f th ncuntr frquncy bcs S( ) S( ) = (5) + µ 1 g U cs 00, 005 A. H. Tcht 19 Vrsn 3.1, updatd //005

20 13. Dsgn Prncpls fr Ocan Vhcls Radng # Ths spctru has an ntgrabl sngularty at g =. (55) U cs µ At ths valu f th ncuntr frquncy s = g U cs µ. (56) Gvn U, µ, w can lk th tw cass wth ncuntr frquncy = + U cs µ (57) g 1. f µ s n ( π/, π/ ) thn > 0 fr > 0 and > fr > 0. g. f µ s n ( π/ 3, π/ ) thn < 0 fr < 0 and fr > U cs µ. 6. Encuntr frquncy vrsus actual frquncy fr fllwng sas ( csµ < 0) and > 0. 00, 005 A. H. Tcht 0 Vrsn 3.1, updatd //005

21 13. Dsgn Prncpls fr Ocan Vhcls Radng # 7. Encuntr frquncy vrsus actual frquncy fr th thr cndtns: had, ba and fllwng sas. 7. Usful Rfrncs Rad sctn fur f th supplntal nts: Trantafyllu and Chryssstds, (1980) "Envrnnt Dscrptn, Frc Prdctn and Statstcs fr Dsgn Applcatns n Ocan Engnrng" 00, 005 A. H. Tcht 1 Vrsn 3.1, updatd //005