1975 - pinkster - low-frequency phenomena associated with moored vessels - old spe jour. - june 15

8/13/2019 1975 - Pinkster - Low-Frequency Phenomena Associated With Moored Vessels - Old SPE Jour. - June 15

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Low-Frequency Phenom ena A ssociated With VesselsMoored at SeaJ A N E T H E R L A N D S S H I P M O D E L B A SI NW A G EN I NG E N T H E N E T H E R L A N D S

ABSTRACT The magnitude of the linear oscillatory motions i sThe in/ luence o j the low /requency waue drifting

force on the motion^ o/ moored vessel s a nd theloads in the mooring syst em i s demonstrated fromresults of model tests in irregular waves. Theorigin o/ the wave di i l t ing force i s discus sed andmethods /or calculating th mean drifting /orce arer ~ u i e u z ~ d . o /aciIitate calculatzon of the low-frequency-wuve drifizcg force on an object inirregular waves, an existing method using the meczndrilting Jorce in regular waves is generalized Theresul ts o j calculat ions using the method introducedin thi s paper a re comparr*dwith previously prtblishecites t reszllts. Ftnally, some remarks ar e add edconcernzng e/jects that have not been ac co mt cd /orin ex is ti ng ca1cu:ation methods.

ves sel m-ured at s ea in s tat ionary condi t ionswith regard to waves, wind, and current i s subjectedto forces that tend to shift i t from the desiredposition. For a given vesse l and position in thehorizontal planc, the motions depend on both themooring system and the external forces acting onthe vessel . In s teady condi :ions, the forces c a u s ~ dby a constant wind and current are constantquantities for a given headirig angle of the vessel. lThe fo rces caused by a statronary irregular se a areof an irr egula r natur e and may be sp li t into twoparts : first-order oscillato:y for ces with wavefrequency, and second-order, slowly varying forceswith frequencies much lower than the wavefrequency.The first-order osci1l ator~ - wave f orces on aves sel cau se the well known ship motions whosefrequencies equal the frequetlcies prcscnt in thespecxrun of he irregular waves. The se are theline.*[ motions of su rge, sway. an d he av e and thethree angular nlctions of roll, pitch, and yaw. Ingeneral, the first-order wave f orce s are proportionalto the wave height , a s are thz ensuing motions.-- .p

Or;r .na manuscr ip t rece ive d in S o c i ~ : ~ ~p r t r ~ l r u r n n g i n e e rso f f l c e ' l a r c h 18 1 9 74 . R e v i s e d m a n u s c r ~ ~ .e c e i v e d o c t . 10 1975.P a p e r .CpE 837; vns f l r s t P r e s e n t e d a t the SPE-AIME EuropeanSprlnt IT--tlng, treld in Amsterdam. The Xerher lands. May 2 9 - 3 0 ,1974. C C o p yr t g ht 1 9 7 5 A m e r ~ c a n I n s t i t ut e of Mlning,Metal lurarcal , and Petrole um Engin eers , I -.

in the order of the height of the waves..The second-order wave forces, perh;tps betterknown a s the wave drifting forces, ha ve beenshown2 to be proportional to the square of r h r

wave height. Th es e forces, though s mall inmagnitude, are the ca us e of t he low-frequency,large-amplitude, horizontal motions sometimesobserved in large ve ss le s moored in i rregular waves.T e s t s run in irregular waves in wave tanks of theNetherlands Ship Model Basi n rev caled a number ofproperties and effects of the low-frequency-wavedrif t ing force that are discussed here -s ing theres ult s of two tes t programs.The first of these programs coflcerns tests runwith the model of a 125,000-cu m LNG carriermoored in head seas with an ideal linear mooringsystem. The se cond program dea ls with300,000-DWT VLCC myored with g rea l i s t i cnonlincar bow hawser to a single-buoy mooring inwaves, wind, and current coming from differentdirections.The resul t s of the te s ts with the L N G carrier areshown in Figs. 1 through 3, while t he res ul ts ofthe tests with the 300,000-DWT VLCC are shownj Fig. 4. All resu l ts ar e given in ful l -scale values .Fig. 1 sho ws the wave trace and t he surg e motionof the L N G carrier to a ba se of time. From thisfigure i t i s seen that the surge motions show n o

RECORD 0 an i R r l W U L R SE


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o s c i l l a t i o n w i t h w a v e f r e q u e n c y . A n a l y s i s o f t h es u r g e m o ti on r e v e a l e d t h a t i t s m e a n p e r io d w a sv ery c lo se t o t h e n a tu ra l su rg e p e r io d of t h em o o red v es se l i n s t i l l wa t e r . T h i s p h eno m en o na p p e a r e d t o h o l d f o r t h e e n t i r e r a n g e o f s t i f f n e s so f t h e m oo ri n g s y s t e m t e s t e d , a s i s s h o w n in F i g .2. In t h i s f i g u re t h e m ean p er io d o f t h e su rg e

M E A S U R E D n an I R R E G U L A R S E A

N A T U R A L SURGE P E R I O DM E A N SURGE P E R I O D

n a t u r a l surge pertoe In secondsFIG. lEAN PER IOD OF SURGE MOTION IN

IRREG ULAR WAVES FOR AN LNG CARRIER.

m o ti on d u r i n g t e s t s i n t h e s a m e i r re g u l a r w a v e s i ss h o w n t o b a s e of t h e n a t u r a l s u r g e pe r i o d i n s t i l lw a t e r. T h e s e r e s u l t s i n d i c a t e t w o p r o p e r ti e s . F i r s t ,t h e m o o r e d v e s s e l c o n s t i t u t e s a m a s s - s p r i n g s y s t e mwith l i t t l e d am p in g , an d sec o n d , t h e wav e d r i f t i n gf o r c e i n i rr e g u l a r w a v e s c o r lt a in s a w i d e r a n g e o ff r eq u en c i es .F r om t h e t e s t s w i t h t h e L N G c a r r i e r , Fig 3s h o w s t h e s i g n i fi c a n t v a l u e o f t h e t o t a l f o r c eb e t w e e n t h e v e s s e l a n d t he m o o ri ng s y s t e m f o rt e s t s i n t h e s a m e i r r e g u ia r w a v e u s i n g d i f f e r e ntv a l u e s o f t h e s t i f f n e s s of t h e m o o r in g sy s t em .T h e s e r e s u l t s i n d i c a t e t h a t, f or t h e s e a c o n d i ti o nt es t ed , t h e re a p p e ar s t o b e an o p t im u m v a lu e oft h e s t i f f n e s s o f t h e m o o r in g s y s t e m t ha c r e s u l t s i nt h e l o w e s t t o t a l m o o ri ng f o rc e . T h e m e a n v a l u e oft h e t o t a l m o o ri ng f o r c e t h a t i s t h e m e a n v a l u e o fth e d r i f t i n g fo rce am o u n ted t o ab o u t 6 t o n s fo r a l lv a l u e s of t h e s t i f f n e s s o f t he m o or i ng s y s t e m . T h er e s u l t s i n F i g . s h o w t ha c i n i r r e g u la r w a v e s t h em o o r in g fo rces i n d u ce d b y t h e w av e d r i f t i n gp h e n o m e n a c a n r e a c h v a l u e s f a r g r e a t e r th a n t h em ean v a lu e . F ro m th e r es u l t s sh o wn in F ig . 2 i two u ld ap p ea r t h a t t h e l o w- f r eq uen cy d r i f t i n g fo rcec a n i n d u c e r e s o n a n c e f or t h e s u r g e m o t io n w h i c h ,i n t u r n , c a n l e a d t o h ig h f o r c e s in t h e m o o r in gs y s t e m .T h e r e s u l t s o f t e s t s w i th t h e m o d e l of a300,000-DWT VLCC s h o w t h e I n f l u e n c e o f t h e

nelgnt 2 6 0 mm o r P od B l l K

IRREGULAR EAI\

EA SU R ED rn an IRREGULAR SEA

WAVE HEIGHT sn metres

5 1 150rcstoronp c o e tt l c xc n t c R n t o n n i t

f G. IGNIFICANT DOUBLE AhlPLITUDE O F TH EF OR C E I N TH E MOORING SYSTEM O F AN LNG

CARRIER.

o I I I 15 100 I59 XIlam s ronds

FIG. 4 HE YA XING MOTION AND FO RC E IN TH EBOW HAWSER O F A VLCC MOORED TO A SINGLE-

BOUY MOORING.


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w a v e d r i f t i n g f o r c e c o m b i n e d w i t h w i n d a n d c u r r e n t .T h e r e s u l t s ( F i g . 4) s h o w t h a t t h e y a w m o t i o n o ft h e V L C C c a u s e d b y c he i rr e g u l ar w a v e s c o m i n gf ro m t h e p o r t b o w a r e a l m o s t e n t i r e l y o f t h el o w - f r e q u e n c y t y p e . $ h e n t h e VLCC s w i n g s t op o r t ( i n c r e a s i n g y n w a n g l e ) t h e a n g l e of a t t a c k t ot h e c u r r e n t i n c r e a s e s , w h i c h r e s u l t s i n h i g h e rc u r r en t f o r c e s . T h i s i s r e fl e c te d b y t h e i n c r e a s ei n t h e f o r c e i n t h e b o w h a w s e r , a l s o s h o w n i nF i g . 4 I t i s s e e n t h a t t h e b o w h a w s e r f o rc ec o n t a i n s m o re c o m p o n e n t s w i th w a v e f r e q u en c y .T h i s i s c a u s e d b y t h e m o t i on s o f t h e b u oy .

T h e r e s u l t s o f t h e s e t e s t s s h o w t ha t th em a g n i t u d e o f t h e l o w - fr e q u e n c y f o r c e s a n d m o t i o n sc a n b e s u c h t h a t t h e y c o n ~ p l c t e l y d o m i n a t e t h eb e n a v i o r o f m o o re d v e s s e l s .T h e e x a m p l e s g i v e n h e r e i n d i c a t e t h e i m p o r t an c et h a t m u s t b e a t t a c h e d t o t h e l o w - f r e q u e n c y d r if t i ngf o r c e i n t h e d e s i g n o f s y s t e m s f o r m o o r i n g l a r g ev e s s e l s a t s e a . h l e th o d s fo r d e t e r m in i n g t h el o w - f r e q u e n c y d ri ft in ; f o r c e f o r t h e p u r p o s e ofd e s i g n i n g s u c h m o or in g s y s t e m s a r e d i s c u s s e d .m e t h o d o f c a l c u l a t i o n c o m p l e m e n t a r y to a ne x i s t i n g m e t h od i s i n tr o d u c e d a n d t h e r e s u l t s o fc a l c u l a t i o n s a r e c o m p a r ed w it h e x p e r i m e n t a l r e s u l t s .

O RIG LU O F T H E V A V E D R I F T I N GF O R C E O N LV O B J E C TA r i g i d o b j e c t , e i t h e r f r e e - f l o a t i n g o r f i x e d , w i l l

d i s t u r b t h e n a c u ra l p a t t e r n o f t h e w a v e s t h a ts u r r o un d i t . O n m e e t i n g t h e o b j e c t , t h e i n c o m i ngw a v e s u7ill b e p a r t l y r e f l e c t e d a n d s c a t t e r e d . I f t h eo b j e c t i s f lo a ti n g f re e ? , i t s n ~ c i l l a t o r ym o t i o n sa l s o w i ll c a u s e w a v e s t o b e t r a n s m i t t ed ra d i a ll yo u t w a r d . T h e c o m p l e x w a v e p a t t e r n a r o u nd a v e s s e la t s e a , t h e re f or e , i s a m i x t u r e o f t h e o r i g i n a l ,u n d i s tu r b e d w a v e s , t h e r e i l ec r e d a n d s c a t t e r e dw a v e s , a n d t h e w a v e s g e c e r a t e d by t h e o s c i ll a t i n gv e s s e l .:n c a l c u l a t i n g t h e m e a n d r i f ti n g f o r ce o n a v e s s e l

L i n l i e a d w a v e s , G e r r i t sm a a n d B e u k e l m a n 3d e t e r m i n e d t h e e n e r g y p r e s e n t i n t h e w a v e sp r o g r e s s i n g o u t w a r d fro m t h e v e s s e l b y m e a n s o f as t r i p t h e o r y a n d t h e r e l a t i v e m o t i on c o n c e p t. T h e i rm e t h o d a s s u m e s t h a t t h i s e n e r g y i s c hc w or k d o n eb y t h e i n c o m i n g w a v e s . T h e m e t h od i s s u i t a b l e f o rv e s s e l s w it h a n d w i th o u t f o rw a r d s p e e d , b u t w a ss p e c i f i c a l l y d e v e l o pe d t o d e t e r m in e t h e r e s i s t a n c ei n c r e a s e o f v e s s e l s c ra v el in g i n h e a d w a ve s . T h er e s u l t s o f t h e c a l c u l n t i o n s a g r e e c l o s e l y w i th t h er e s u l t s o f m o d e l t e s t s . noteworthy p a r t o fG e r r i ts m a a n d B e u k e l m a n ' s p a p e r i s t h c e x p er i -m e n t a l v e r i f i c a t i o n of t h e d e p e n d e n c y o f t h e w a v ed r i f ti n g f o r c e o n t h e s q u a r e o f t h e w a v e h e i g h t.T h e m e an w a v e d r i f t i n g f o rc e i n re g u l a r w a v e s o nr e c t an g u l a r c a p t i v e b a r g e s o f i n f i n i t e b r ea d th w a sc a lc u la t ed th e or e t ic a ll y by Evfei a n d ~ l i ~ c k . ~h c i rm e t h o d , l i k e G e r ri c sm a a n d E e u k r l m a n ' s m e t h o d ,d e r i v e s t h e m e a n d r i f t i n g f o r c e f r om I s u r v e y o f t h ew a v e s t r a r e li n g o u t w a rd f ro m t h e o b j e c t . T h i s h a sb e e n s h o w n t o b e a c c e p t a b l e w h cn v i e w i n g t h em e a n d r i f t i ng f o r c e i n r e g u l a r w a v e s .

M E i D R I F T I N G F O R C E INR E G U L A R W A V E SA g e n e r a l e q u a t i o n f o r t h e m e a n d r i f t i n g f o r c e inr e g u l ar w a v e s o n a v e s s e l t h a t i s e i t h e r f r ee - fl o at i ng

or c a p t iv e i sAF f P g C F ~ a . . . . . . ] )

I n E q . 1 t h e d r i f ti n g f o r c e c o e f f i c i e n t , C F q w h i c hm a y b e d e t e r m i n e d e x p e r i m e n t a l l y o r , In s o m ec a s e s , by c a l c u la t i o n, i s g e n er a l ly s o m e f u n ct i ono f t h e f r e q u e nc y o f t h e r e g u l a r w a v e s f o r a g i v e nh e a d i n g a n g l e of t h e v e s s e l . I n F i g . 5, a n e x a m p l eof C F ] f o r s f r ee - fl o at i ng r e c t a n g u l a r b a r g e i sg i v e n . T h e r e s u l t s a r e t a k e n fr om R e f . 5 .

D R I F T IN G F O R C E I N I R R E G U L A R WAVESm e t h o d f o r c a l c u l a t i n g the d r i f t i n g f o r c e i n

i rr eg u l ar w a ve s , s u g g e s t ed by H s u a nd ~ l e n k a r n ~a n d u s e d by R e m e ry a n d erm mans^ t o c a l c u l a t e t h el o w - f re q u e n c y s u r g e m o t i o n s o f a m o o r e d r e c t a n g u l a rb a r g e , a s s u m e s t h a t a n i r r e g u l a r w a v e t r a i n m a yb e t h o u g h t of a s a s u c c e s s i o n o f h a l f - p e r i o d s o fr e g u l a r w a v e s . T h e a m p l i t u d e a n d t h e p e r i o d o fe a c h h a lf - p e r io d i s e q u a l t o t h e z e r o -t o -p e a k o rz e r o - to - t r ou g h v a l u e a n d t h e t i m e b e c w e e n z e r oc r o s s i n g s , r e s p e c t i v e l y , o f t h e i r r e g u l a r w a v e z n i n( s e e F i g . 6 . T h e v a l u e o f t h e w a v e d r i f t in g f o r c ed u r i n g t h e p a s s a g e n f p e a k o r t r o u g h w i tha p p l i t u d e A a n d p e r i o d T i s c a l c u la t e d u s in g E q.1 C F D i s s e l e c t e d a t t h e f r e q u e n c y c o r r es p o nd i ri gt o 7 .

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FIG. 5 A VE D R I F T I N G F O R C E C O E F F I C I E N T F O RA R E C T A N G U L A R B A R G E .O E C M D of an QOFGULAl SE

F I G . 6 E T E R MI NA T IO N O F T H E W A V E D R I F T I N GF O R C E A C C O R D I N G R E 3 l E R Y A N D H E R M i \N S 5A N D H S U A N D B L E N K A H N .


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B y f a i r in g t h e c a l c u l a t e d v a l u e s , a l o w-f r e q u en c y d r i f t i n g f o r c e s i g n a l i s o b t a i n e d f ro m ag i v e n w a v e r e c o rd . T h i s f o r c e r e c o r d t h e n m a y b eu s e d a s in p u t t o a m a s s - sp r i n g s y s t e m s i m u l a t in gt h e m o o re d v e s s e l , fro m w h i ch t h e m o t i o n s m a y b ec a l c u l a t e d .

T h e a b o v e m e t h o d r e q u i r e s t h a t a r e c o r d o f t h ei r re g u la r v a v e s b e a t ha n d . T h e s u b s e q u e n t m e th o dof c a l c u l ~ t i n g h e t i m e r e c o r d o f t h e l o w - f r e q u e nc yc h if ti n g f o r c e r e q u i r e s a s u b s t a n t i a l a m o u n t o fc a l c u la t i o n t h a t, i n m o s t c a s e s , m u s t b e c a r r i e d o u tb y c o m p u te r . H o w e v e r , t h e m e t h o d h a s l e d t o u s e f u lr e s u lt s , a s i s s h ow n i n R ef . 5. In t h e f o l l o w i n g , am o re g e n e r a l a p p r o a c h u s i n g t h e s a m e a s s u m p t i onu s e d f or t h i s m e t h o d i s i n t ro d u ce d .

A n i r r e g u l a r , l o n g - c r e s t e d s e a m a y b e c o n s i d e r e da s t h e s u m o f a l a r g e n u m b e r o f r e g u l a r w a v e s :N

z t , X ) = Ai C o s f . - c Y ~ x + E ~ ) .i=lEq. 2 a l s o m a y be w r it t en a s

i w h e r ei N NC C A A COS { f i - f . )i= j=

. . . . . . .w a v e e n v e l o p e , 4)a n d

t , ~ )a r c t g

C Ai Sin { f i - f r , ) t aix + E ~ }i

L A; C o s { f i - f r ) t O i X + E i di

. ( 5 )f t h e s p e c t r a l d e n s i t y o f b e i r re g u la r w a v e s i s

n a rr o w , m e a n i n g t h a t t h e h i g h e s t a n d l u w e s tf r e q u e n c i e s p r e s e n t i n t h e w a v e s d i f f e r o n l y s l ip h tl yfr om s o m e c e n t r a l l y c h o s e n f r e q u e n c y l t h e n t h ea m p l i t u d e : I , u a n d p h a s e t , X a r e o f a s l o w l yv a r y i n g i or m s e e F i g . 7 ) .

F r om t h e a b o v e e x p r e s s i o n s , w e m ay d e f i n e t h es l o w l y v a r y i n g f r e q u e n c y / ( t , X :

C A . A . C o s { f i - f . ) ti= j= 1 3

o r , a f t e r r e g r o u p i n g t e r m s ,

L A A C o s { fi-f . ) ti = j= 1 3

I f t h e w a v e a m p l i t u d e A t , x a n d t h e f r e q u e n c y/ I , I r e s l o w l y v a ry i n g q u a n t i t i e s , t h e n w e m a ya p p r o x i m a t e t h e w a v e d r i f t i n g f o r c e f ro m Eq. 1:

T h i s i s t h e a s s u m p t i o n ir. R e f s . 5 a n d G w r i t t e n i na c o n t i n u o u s fo rm . If w e a s s u m e t h a t t h e s q u a r e o fC F D s

t h e n the d r i f ti n g f o r ce b e c o m e s

W VE ENVELOPE A I X

T h i s l e n d s t o FIG. E C O R D O F I R R E G U L . 4 R W A V E S WITH AN A RR OW S P E C T R A L D E NS I T Y .


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Cos i fi-f t - o i - o . ) X + E - E . ) l3 lw h ic h i s t h e s a m e ISF t , x ) =

N N f i f5 P q C A . A . 2i = j= F ?

I t i s s e e n fro m Eq. 1 1 t h a t t h e d r if t in g f o r c e i s a l s oo f a s l o w l y v a r y i n g fo r m, s i n c e i t c o n t a i n sf r e q u e n c i e s t h a t a r e d i f f e r - n c e s o f t h e f r e q u e n c i e si a n d j jT h e m e a n d r i f ti n g f o r c e i n i r r e g u la r w a v e sf o l l o w s by m a k i n g i = 1

If o n l y c o m b i n a t i o n s o f ; n d li a r e u s e d i n Eq 11s u c h t h a t ii ; . , , t h e n E q s . 11 a n d 1 2 g i v e t h el o w - f re q u e n c y o s c i l l a t i n g p o r t io n a n d t h e m e a n ,r e s p e c t i v e l y , o f t h e w a v e d r i f t in a i n r c e .

E q s . l 1 a n d 1 2 a r e n o t i n a fo rm t h a t c a n b e u s e de a s i l y f o r p r a c t i c a l c a s e s . S t r a i g h t -f o r w a r d b u tl e n g t h y m a n i p u la t i o n o f E q s . 1 1 a n d 1 2 g i v e s t h ef o l l ow i n g e x p r e s s i o n s f u r t h e s p c c t r a l d e n s i t y ,S V ) , n f t h e l o w - f r e q u e n c y portion and the rnr-an,I : o f t h e d r i f t i n g f o r c e i n i r r e g u la r w a v e s w i t hs p e c t r a l d e n s i t y S Z / J .

C : ~ + V / ~ ) df-

F r o m E q s . 13 a n d 1 4 i t i s s e e n t h a t t h e m ea n a n dt h e s p e c t r a l d e n s i t y o f t h e d r i f t i n g f o r c e n o w m a yb e c a l c u l a t e d d i r e c tl y f ro m t h e s p e c t r u m o f t h ei r r e g ul a r w a v e s w i t h o u t u s i n g a w a v e r e c or d .

I t s h o u l d b e n o t e d t h a t Eq. 13 g i v r s t h e m e a nv a l u e o f t h e s p e c t r a l d e n s i t y , SF v).

S t r i ct l y s p e a k i n g , t h e s p e c t r a l d c n s i t y o f t h ed r i f t i n g f o r c e g i v e n b y Eq 11 c a n a s s u m e a l a r g er a n g e o f v a l u e s , d e p e n d in g o n t h e c h o i c e o f t h e

r a n d o m p h a s e a n g l e s ci a n d c,. Eq. 1 3 i s a r ri v ed a tb y a v e r a g i n g t h e s p e c t r a l d e n s i t y f or a l l p o s s i b l ec o m b i n a t i o n s o f ci a n d C.. T h e s p e c t r a l d e n s i t ya n d t h e m e a n v a l u e o f tA e d r if t i n g f o r c e m a y b ec a l c u l a t e d f r o m E q s . 1 3 a n d 1 4 u s i n g a w a v es p e c t r u m , S Z / ) , a n d t h e w a v e d r i f t i n g - f o r c ec o e f f i c i e n t , F D l ) .

I n t h e following, t h e lo w - f r e q ue n c y s u r g e m o t i o n so f t h e r e c t a n gu l a r b a rg e t e s t e d b y R e m e r y a n ii ~ e r m a n s ~r e c a l c u l a t e d u s i n g E q s . 1 3 a n d 1 4 . ? h er e s u l t s o f t h e s e c a l c u l a t i o n s a r e c o m p a r e d w it h t h er e s u l t s o f m e a s u r e m e n t s g i v e n b y R e m e r y a n dH e r m a n s .

T o c a l c u l a t e t h e s u r g e m o ti o n s , a s s u m e t h a t t h em o o r ed b a r g e i s a l i n e a r m a s s - s p ri n g s y s t e m w i t ht h e f o l l o w i n g e q u a t i o n o f m o t i o n :

T h e a m p l i t u d e, ,4,, o f t h e s u r g e r e s p o n s e t o as i n u s o i d a l i n p u t w it h a m p l i t u d e F a n d f r e q l l e n c yv i s

T h e s u r g e r e s p o n s e c ?e r a t o r t h en b e c o m e s

T h e s p e c t r a l d e n s i t y o f t h e l o w - f r e q u en c y s u r g em o t i o n f o l l o w s fr o m

F i n a l ly , t h e m e a n s q u a r e v a l u e , [L:, o f t h e s u r g em o t i o n f o l l o w s by i n t e g r a t i o n o f Eq. 18:

w h e r e SF v) i s t h e s p e c t r a l d e n s i t y of t h el o w - f r e q u e n c y d r i f t i n g f o r c e f r om E q . 13T h e r o o t m e an s q u a r e v a l u e s o f t h e s u r g e m o t i on so f t h e r e c t a n g ul a r b a r ge w e r e c a l c u l a t e d u s i n g t h ef o l l o w i n g v a l u e s . 5

V i r t u al m a s s o f b a r g e \l 1 3 0 . 5 0 t o n s e c 2 / mD a m p i n g c o e f f i c i e n t D = 2 7 0 t o n s e c / m


6/8

Restoring coefficients R 136 68 9 40 29and 20 ton/m

The spectrum, S, /), of the irregular v7a ves i s givenin Fig. 8.

Furthermore, the mean wave drifting force givenby Eq. 14 i s als o given in Ref. 5 The calculatedmean valu es, there fore, ace equivalent. The meanforce amounted to 28 tons. T he measured meanforces were between 22 and 3 tons. Th e resul ts ofthe calculations, showing the root mean square ofthe low-frequency surge motion for the differentvalues of the stiffn ess CR ) of the mooring systemar e given in Fig. 9 -41~ 0 hown in this f igure arethe results of tests and cslculations carried out byRemery and d er man ss using the method of Hsu and~ l e n k a r n . ~

The resul ts of the calculat ions using the spectraldensity givm by Eq. 13 are somewhat higher thanthe test v J u e s and the calculated va lues given inRef. 5 possi ble explanation for the difference i sthat the pr esent method of calculation i s bas ed on

\ an infinite test period, while the results given inl Ref. 5 are based on a finite period correspondingto about 3 5 minutes in reality.An exam ple of the inf lue nce of the length of thete st period on the root mean squ ar e of the surgemotion i s given in the following table.

Te st Peri od Root Mean Square ofPrototype Low-Frequency Surge(minutes) Motion (meters)

35 1.70105 1.96The above result%were taken from te st s carried outin head waves with the model of a larg e LNGcarrier. .4s may be s een , the root mean squ are of

2SPECTRUM of an I R R E GUL AR S E A

S I G N I F I C A N T W A VE HEIGHT 5 mMEAN WAVE PERIOD . JCC

the low-frequency surge motion incre ased by about15 perc ent when increasi ng the tes t period by afactor of three. secon d possible explanation forthe differ ences may be in the linearization a ssume din Eq. 9.

FURT HER DEVELOPXIENTSIn deriving the present method of calculation ir

was assumed that the low-frequency drifting forcein irregular waves could be calculated using themean wave drifting force coeffi cient s for regularwaves.

Using the exact solution for the low-frequencypressure variat ions in an undisturbed irregularwave, Newman8 concluded that this method isvalid provided only low frequencies appr0achir.gzero are considered.

When consi derin g higher freq uenc ies of the wavedrifting force effects &at must be accounted forare (1) influence of low-frequency pressure variationsnot accounted for by the mean drifting force i nregular waves, and 2 ) influence of the dimensionsof the obj ec t rel ati ve to the wave length of :heslowly varying envelo pe of the irregular waves.

When considering the exact theoretical solutionfor the low-frequency drifting force on an object,i t can be shown that che low-frequency pressurevariation acting on the object may be ca lcula tedusing the velocity potential, includingsecond-order contributions:

l ) 2 ) . . . 20)Q @The physi cal im plication of the contribution of thefirst-order potential, @ l ) ,s tha t i t desc r ibes eachregular wave as a perfect s ine form. The contribution

] y R E M E R Y and H E R M A N S 3)o CALCULATED

CALCULATED by P R E S E N T M E T H O D

0 5 050 0 75 100trequcncy on r a a .c

1975 - pinkster - low-frequency phenomena associated with moored vessels - old spe jour. - june 15

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