OTC 3625

THE INFLUENCE OF WAVES ON THE LOW-FREQUENCY HYDRODYNAM l C COEFF l C l ENTS OF MOORED VESSELS

by J.E.W. Wichers and M.F. van Sluljs, Netherlands Ship Model Basin

0 Copyright 1979. Offshore Technology Conference

This paper was presented a t the 11th Annual OTC in Houston, Tex.. April 30-May 3. 1979. The material is subject to correction by the author. Permission to copy is restricted to an abstract of not more than 3% words.

I ABSTRACT -

Vessels moored at sea will undergo low-frequen- cy motions due to wind, current and waves. Computer programs are being developed to predict theoretical- ly the motions of such vessels, particularly the motions in the horizontal plane, which consist of a high-frequency part (oscillations with periods corresponding to those of waves) and a low-frequen- cy part. In these theoretical approaches only po- tential damping is taken into account; viscous damping, which substantially influences the low- frequency phenomena, is-not included. Moreover, the damping of mooredstructures in resonance con- ditions seems to be influenced by additional damp- ing caused by the waves. Both viscous and wave damping have to be d&emined.qerimentally. At the Netherlands Ship Model Basin model tests have been carried out to determine the low-frequency hydrodynamic coefficients of a VLCC and a LNG car- rier in surge direction in calmwater and in reg- ular waves. The measured data are compared with theoretical results obtained using a computer pro- gram based on three-dimensional linear potential theory.

INTRODUCTION . . . . . . . . . . . . . . . . . . . - . .- . . . . . .

A vessel moored at sea is subjected to forces that tend to shift it from the.desired position. For a given vessel and position in the horizontal plane, the motions depend both on the mooring sys-. tern and on the external forces acting on the vessel. The forces on the vessel caused by an irregular sea are of an irregular nature and may be split into two parts :

- first-order oscillatory forces with wave fre- quency, and - second-order, slowly varying forces with fre- quencies much lower than the wave frequency.

The first-order oscillatory wave forces on a vessel cause the well-known ship motions, whose frequencies equal. the frequencies present in the spectrum of the irregular waves. These are the

........

References and illustrations at end of paper.

translatory motions surge, sway and heave and the angular motions roll, pitch and yaw. In general, the first-order wave forces are proportional to the wave height, as are the ensuing motions. The magnitude of the translatory motions is in the order of the height of the waves.

The second-order wave forces, better known as the wave drifting forces, have been shown to be roportional to the square of the wave height (ref.

711 9 [21 and [31 ) . These forces, though small in magnitude, are the cause of the low-frequency, large-amplitude, hosizontal. motions sometimes ob- served during model tests and on prototype vessels moored at sea. An example of these low-frequency, large-amplitude surge motions, in irregular head seas as measured on a model of a 125,000 cu m LNG carrier moored with an ideal linear mooring system, is shown in Fig. 1. The results are given as full scale values. This paper concerns only the low- frequency longitudinal surge phenomena in still water and in head waves of a VLCC and a LEG carries moored in relatively deep water.

In order to calculate theoretically the low- frequency surge motions of a moored vessel, the equations of motion have to be drawn up and solved. The low-frequency (oscillatory) surge motions of the moored vessel will be excitated by the wave drift forces. Wave drift forces are originated by wave group trains, which occur as a result of dis- persion of a number of regular waves. Incase these waves have different wave heights, frequencies,. and random phase angles, an irregular wave group train results, while by dispersion of two regular waves having a small deviation in frequency a regular wave group train will pccur . . The wave drift forces, consisting of a constant and an oscillating part, are a function of the square of the amplitude of the wave group envelope and the corresponding wave frequency. For a theoreti- cal approach of the calculation of the constant and oscillating wave drift forces, see ref. [3] .

The wave drift forces are supposed to be the right-hand terms in the equation of the low-fre- quency motion in the longitudinal ( X ) direction:

(m + axx)% + b= .k.+ Cx.x = ~ ~ ( t ) . . . . (l_)_ -

In this equation C, rep~esents the spring constant of the mooring system. To solve the equation the added mass of2he vessel a, and the hydrodynamic damping bxx have to be known. Both coefficients are dependent on the frequency of motion.

These coefficients are normally determined from extinction tests or oscillation tests in still water, though the added mass of the vessel can be calculated in a theoretical way for the still water condition by means oT a three-dimensional linear source technique.

The damping of the vessel in still water con- sists of the following parts:

- potential damping (energy loss due to the radi- ated waves caused by the oscillating vessel); - viscous damging (energy Loss caused by friction).

Actually, however, the system is moored in waves, so attention has also to be given to the influence of the waves on these low-frequency hydrodynamic coefficients. To determine the influ- ence of waves on the low-frequency coefficients a model test program has been carried out, consisting of extinction tests in the longitudinal (X) direc- tion of vessels moored in linear spring systems in still water and in regular waves. Prom the results of this study it turns out that the total damping coefficient can be split up into the following components :

- potential damping; - viscous damping; - wave damping.

In this paper the results arediscussed, whereas also an example is given of a possible approach to estimate the effect of wave damping on the resonance surge motions of a moored VLCC in regular wave groups.

TEST SET-UP _ - P

.- -

The motion extinction tests in still water and in regular waves were carried out in the Sea- keeping Laboratory of the Netherlands Ship Model Basin having a water depthof 2.5 m, a length of 100 m, while the breadth was reduced to 6 m. Use was made of models of a fully loaded 200,000 DWT VLCC (scale 1 : 82.5) with a block coefficient of 0.85 and a fully loaded 125,000 cu m LNG carrier (scale 1 :TO) with a block coefficient of 0.75. The particulars of the vessels are given in Table 1. The body plans of both vessels are shown in Fig. 2. -

The vessels were moored in linear mooring systems, of which the spring constants are given below:

125,000 cu m LNG carrier _ 200,000 DWT VLCC 1 1 C = X 38.67 t0n.m-, 53.72 t~n.m-~

180.81 t0n.m- 251.15 t0n.m-, - 754.15 t0n.m-

The set-up of the experimental mooring arrange- ment is shown in Fig. 4. The decaying surge motions were measured in the centre of gravity (c.o.G.) by means of an optical tracking device, which has no

mechanical interaction with the model and a high degree of accuracy. A review of the tests in still water and in waves is given in Tables 2, 3 and 4.

DETERMINATION OF THE HYDRODYNAMIC COEFFICIEPTTS AT LOW-FREQrnCIES

Ex'.i!~2i~e-t_zs4s-i~~stii~-~a4f r For the extinction tests in still water the .

low-frequency decaying surge motion can be described by the following equation:

& = - a .%-boxx.k-C .X . . . . . . . ( 2 ) xx X

in which:

a = added mass coefficient corresponding to the XX natural low-frequency wx of the moored ves-

sel (assumed to be constant) = damping coefficient in still water corre- sponding to the natural low-frequency ox of the moored vessel (assumed to be constant)

C, = spring constant of the mooring system.

The solution of this linear differential equa- tion with constant coefficients of the low-frequen- cy surge motion will be:

boxx 2(m + a;uO

t

x = e (c1 COS wxt + C2 sin w t) X

. . . . . . . . . . . . . . . . . . . . . ( 3 )

in which:

W = X 1- while :

Cl and C2 depend on the starting position of the vessel.

Further, cox is real because the condition:

2 [2(."!~nl] (m L.am, is satisfied for the low-frequency motion due to the relatively small values of the damping coeffi- cient .

From equation (3) it can be seen that the am- plitude of the decay curve consists of:

- a decreasing eqonential function; - a sinusoidal oscillation. The total mass or vTrtual mass of the vessel

is further denoted:

m = m + a xx XX

The decrease of the subsequent amplitudes of the decay curve, X and x,+~, will be: n

I

boxx - - t 2mxx -~ - X X .e

- A = - - n = X b n+l . oxx - - ( t +%)

X . e - " wx n . : ....... -. . . . . .... ..- .

boxx i T . . . . . . . . . . .... - - . . . -.

' W m xx X = e-6 = e . . . . . . . . . . ( 4 )

i n which 6 i s cal led the logarithmic decrement.

. . . . . . . . . .... ... boxx 2 rho,- - - .- -

6 = - S m xx wx "xx V- . .

- (-1

Because t h e s t i l l water damping coeff ic ient boxx i s small fo r the low-frequency motions, the loga- rithmic decrement can be simplified according:

6 = b . 71 ~- ~ OXX

o r t h e - s t i l l watef-damping coeff ic ient can be de- termined from:

- - c- -

X XX = h - . . . . . . . . . . . . . . b o ~ i~ . . . . . ; .( 5 )

while the v i r t u a l mass coeff ic ient m can be de- . - xx termined from:

Cx m = - - ~ ~ ~ p - " . . . " ' . . . " " ' ( 4 ) XX .... -.

W ~ - X

Extinctio_p-4~st_s-i!-regular-x:x:s For the low-frequency extinction t e s t s i n

regular head waves with wave amplitude 5, and fre- quency w the decaying surge.motion X can be des- cribed. by t h e following equation:

~ -~

G = -axx(wxda , - b x x ( ~ x y ca, w)k + . -

hydrodynamic reaction forces

- C .X I I , +

W ) X 1

res tor ing force exci ta t ion force

. . . . . . . . . . . . . . . . . . . . . . . .

i n which:

w = natural frequency of the moored vessel i n X

.- waves - -

Ta = wave amplitude . . . . . . . . . =

w = wave frequency . . . .

a = added mass in-phase with _the low--rreqtcency XX . . . . swge acceleration -. . - .. -

bxx = .dampin_g in-phase with t h e low-frequency surge ve loc i ty

Cx = spring constant in-phasewith the low-fre- quency surge. displ+cement .... - . .

Pd = constant wave .dFift force.

From the measurements of t h e natural frequencies of the moored vessel i n s t i l l w a t e ? and i n waves it was found t h a t f o r the saae spring constant the natural f r e p e n c y i n waves was equal t o the na tura l frequency i n s t i l l water. This means t h a t the added mass coeff ic ient am i n t h e low- frequency surge motion i n combinat ion with high- frequency head waves corresponds t o the added mass coeff ic ient i n s t i l l water condition. From t h e ex- periments it turned out t h a t t h e added mass i n . .

st i l l water was the same as i n waves; however, a l a rge difference was recognized i n the damping.

The damping coeff ic ient bxx i s assumed t o de- pend on wx, r;, and U ; the ' damping force b,,H w i l l be s p l i t up into:

- the s t i l l water damping force boxx(wx)%; - an addi t ional damping force bl,.k, which damping

coeff ic ient i s assumed t o be dependent on ox, ca and U.

Now equation ( 7 ) can be modified as follows:

[m + az(wx)]2 -l- [bo,(ux) + blxx(wx, C,, l]& +

+ C .X = F ~ ( C , , U ) X

i n which:

blxx is cal led the wave damping coefficient.

O r :

m ( U )H + btotXX(wX, , W)? + cX.x =. xx X a

= F ~ ( < , , w) . . . . . . . . . . . . . . . (8)

i n which:

btotxx ebOn + blxx = s t i l l water damping coeffi- c ient plus wave.danrping caeyficient =

= t o t a l damping coeff ic ient .

I f equation (8) i s assumed t o be a l i n e a r dif - f e r e n t i a l equation with constant coeff ic ients , the steady s t a t e solution of the low-frequency surge motion ( ~ ~ ( 5 ~ ) W ) = constant) w i l l be:

btotxx -- x = e -xx (C, COS W t + C2 s i n w t)

X X

. . . . . . . . . . . . . . . . . . . . . ( g )

Now the same procedure i s followed a s was des- cribed f o r the s t i l l water condition. The total. damping b totxx w i l l be: '

Jcx."xx . . . . . . . . . . . 1 6 -

btotxx (10) 71

The v i r t u a l mass coeff ic ient m= can,be obtained from equation ( 6 ) .

If now t h e natural frequency wx of the moored vessel and the logarithmic decrement i n still. water and i n regular head waves a re derived from the ex- periments, the wave damping coeff ic ients can be determined.

DETERMINATION OF THE LOGARITNMIC DECREMENT AND THE NATURAL PERIOD OF--.THE LOW-FREQUENCY MOTION .. ..

According t o equation (4) t h e decay of the amplitudes of the surge motion w i l l be:

X + X n n+l or: xn - xn+, z z 6 2

The expression xn - xn+l i s the var ia t ion of Ax, while :

i s the mean over the amplitudes.

I f - t h e damping i n s t i l l water o r i n waves i s supposed t o be l i n e a r , t h e logarithmic decrement - _ follows from:

The natural frequency wx of the moored vessel can be determined from t h e ext inct ion time t race re-

~ ~

corded on paper s t r i p chart . -~ - -

. .

Logarithmic decrement i n s t i l l water:

- By means of the motion decay curve t h e logarith- mic decrement of each two subsequent c res t am- pl i tudes and of each two subsequent trough am- p l i tudes was determined.

- From these data t h e mean logarithmic decrement was determined by means of the Least squares method. The r e s u l t s of such an analysis are shown i n Fig. 5. L

Logarithmic decrement i n regular waves:

- The motion decay curve was recorded on magnetic tape.

- From t h i s s ignal the high-frequency surge motion was f i l t e r e d out by computer. . .

- From the f i l t e r e d s igna l the mean value was de- termined i n order t o eliminate the influence of t h e constant wave d r i f t force. This procedure. i s shown i n Fig. 7.

- Next, the same procedure was followed as f o r t h e s t i l l water condition. The r e s u l t s of an analysis a r e shown i n Fig. 6.

The result 's of the measurements are given i n Tables 2, 3 and 4.

DETERMINATION OF THE ADDED MASS AND DAMPING BY POTENTIAL THFORY --. -.

For t h e s t i l l water condition t h e low-frequen-

cy added mass and damping can be calculated by means of a three-dimensional l i i lear source technique pro-- gram, of which the theory is summarized for t h e present purposes below.

The o s c i l l a t i n g motion of t h e s t ruc ture i n t h e surge ( X ) direct ion is given by:

i n which:

5 is the amplitude of motion i n the X-direction.

The flow f i e l d can be described by a first order 'velocity potent ia l :

i n which:

= time independent pa r t of the veloci ty po ten t ia l due t o unit-velocity i n X-direction.

The po ten t ia l function i s solved nmeri- c a l l y i n such a way t h a t a11 t h e boundary condi- t ions a re f u l f i l l e d . A description of t h e solut ion of the po ten t ia l function i s given i n r e f . [ h ] .

Acco-rding t o Bernoulli 's theorem t h e l inear- ized hydrodynamic pressure is given by:

The o s c i l l a t i n g hydrodynamic force i n surge direc- t i o n due t o surge motion i s :

i n which:

So = surface of the body n = direct ion cosine:

X

The added mass and damping coeff ic ient % and bxxp respectively a re obtained from t h e in- phase and in-quadrature par t of the hydrodynm-ic reaction force:

A review of t h e r e s u l t s of t h e calculations and the model Cests i s given i n Table 2 and ~ i g s . - 8 and 9.

DISCUS6IDN OF THE RESULTS

From t h e r e s u l t s of the measurements of t h e natural frequencies i n surge direct ion of the moored vessel i n s t i l l water and i n regular head waves, it can be concluded t h a t t h e natural. rrequen- c ies did not change. This means t h a t t h e added mass coeff ic ient axx f o r the Low-frequency surge motion i n head waves equals the added mass coeff ic ient i n

s t i l l water condition. The non-dimensional added mass coeff ic ients a re shown i n Fig. 8. The numeri- c a l values a re given i n Table 2. These r e s u l t s show t h a t the coeff ic ients a re ra ther constant over the frequency range investigated and t h a t the magnitude amounts t o approximately 5 percent of the mass of the vessel .

The added maSs de?i.ved from the measurements corresponds wel l -with the calculated added mass by means of the l i n e a r po ten t ia l theory.

Damping of surge motion a$ low-frequencies PI-----------------------^

S t i l l - + a t e r damping coeff ic ient boxx

I n Fig. g the measured and calculated damping coeff ic ients are given. For the calculated damping use has been made of the three-dimensional l inear po ten t ia l theory. The damping coeff ic ients derived from the measurements were scaled up-to prototype values according t o Froude's law of simili tude.

The damping-coefficient i r i s $ i l L water condi- . .

t i o n consists of the following par ts :

= b (po ten t ia l p a r t ) + boxx xxp - - - - ~~ . -

(viscous .par t ) + bxxv

From the r e s u l t s it can be seen t h a t fo r the low- frequency range (U, < 0.1 rad.secyl) the po ten t ia l pa r t i s small or negligible. It can fur ther be con- cluded t h a t a t the low surge ve loc i t i es the viscous damping (see Fig. 5 ) o r the combination of viscous and po ten t ia l damping may be considered as a Lin- ear function. In Fig. 10 the nondimensional damp- i n g coeff ic ient i s s h ~ m , . .

Damping coeff ic ient i n regular waves-b t otxx

The total-damping coeff ic ient of the Low- frequency natural surge motions of the vessels i n regu lm waves seems t o be a superposition of the following damping terms:

= b (po ten t ia l p a r t ) + btotxx xxp

+ bxxv (viscous p a r t ) +

+ blxx (wave influence) -- = - - boxx ( s t i l l water damping) +

+ blxx (wave damping) -

From the t e s t r e s u l t s it can be concluded t h a t be- cause of the smalb ve loc i t i es of t h e low-frequency surge motion i n regular waves the damping coeffi- c ient may be considered as a l i n e a r function (see Fig. 6 ) .

The r e s u l t s of the wave damping blxx (= btotxx - bo,)for regular waves with dif ferentwave helghts and wave frequencies f o r bosh the -@G c a r r i e r and t h e VLCC a r e given i n the Figs. 1 1 and 12.

Considering the wave damping, t h e following conclusions can be drawn:

- The wave &amping has a s ignif icant .value f o r wave f r e q u e n c i ~ s ranging between 0.3 t o 1.3 rad.secyl ( ~ i g s . 11 and 12).

- The wave damping i s porportional t o t h e square of the wave height ( ~ i g . 13).

- The wave damping i s independent of the spring constant of t h e mooring system (Fig. 14).

In general terms, t h e wave damping force can be considered as a correction on t h e constant wave d r i f t force and i s s imilar t o t h e introduction of forward speed i n t h e increase of res is tance of a ship i n waves, see re?. [5] . Due t o t h e introdue- t i o n of the low-frequency surge motion t h e follow- ing conditions w i l l be changed with regard t o t h e constant wave d r i f t condition:

- the frequency of encounter with t h e incoming waves according to :

i n which: k = wave number = 2s/X k = sinusoidal function

resul t ing i n a sinusoidal frequency modulation; - t h e wave height around t h e ship; - t h e ship motions; - the r e l a t i v e wave height.

EXAMPLE OF THE INFLUENCE OF THE WAVE DAMPING ON THE RESONANCE SURGE MOTION OF A MOORED TAMXER IN REGULAR WAVE GROUPS

For t h e formulation of regular wave groups, use is made of two regular waves with the same am- pl i tude ca and different values fo r the wave fre- quency w and phase angle E:

= ca s i n ( U t - k x + c l ) + 1 1

+ 5, s i n ( w 2 t - k2x + c2) =

[w1-w2 kl -k2 ~1-"21 * cos (-+t - ( - + X + - 2 1

i n which:

A = wave group envelope =

For X = 0 and €1 = E~ = 0 the regular wave group can be described by:

- ctOt = ca[2 + 2 cos(ol-o2) t1 ' . s i n w . t

. . . . . . . . . . . . . . . . . . . . . (12)

Input data ---------- Tanker :

200,000 DWT VLCC fulxy looied; f o r pa r t i cu la r s see Table 1.

I Spring constant of mooring system:

C = 53.72 ton.mW1 X

Total mass of the moored vessel:

From Fig. 8 the added mass coefficient a, can be read, so that the total (virtual) mass of the moored vessel becomes:

2 1 m = 25,388 t0n.sec.m - . ._ m

Natural low-frequency of the system:

When both the spring cons_tant of the mooring sys- %em and the total mass (own mass increased with the mass of the water which is moving with it) of the vessel are known, the natural frequency of the system can,be calculated. For the subject case this frequency is:

w = 0.0461 rad.sec~I X

Wave condition: -

The following wave condition was chosen:

=a = 2 m 1 1 = 0.484 rad.secT, - -

w2 - = 0.438 rad. sec: 1 W = mean wave frequency = 0.461 rad.secS1 - u1 - u2 = wave group frequency = 0.046 rad. sec.

Damping coefficients: .- - - . ..

According to Figs. 9 or 10 the still water damping amounts to:

= 23.37 t0n.sec.m -1 boxx According to Figs. 14 or 15 the wave dam2ing coef- ficient amounts to:

bixx (E = 0.461. rsd.sec,'ll = 3.12 t0n.sec.m- 3 . -

Wave drift force:

The oscillating wave drift force is calculated to be approximately 48 tons, see ref. [3] . Eeso:_aece-sur$eT!2t_ionnwithh?t tli-wates-and-w?xe damping coeff~clent - ------------h------

The differential equation of the low-frequen- cy surge motion can be written as:

m xx .%+box.? = + bIx(W, ca, t)k + CX.x=

= Fd + Fa COS (wl - w2)t

In a wave group the wave damping is assumed to be:

2 blXX = biXX(W) .A (t) =

= b;,(ij))~22sz + cos (al - w2)t] =

- - 2biz(~)ca2 + 2biXX(;)<: cos (ul - w2Jt

The differential equation with non-constant coef- f'icients, now results into:

m xx .X + bo_.k + 2 b ; _ ( ~ ) ~ ~ ~ . i +

+ 2bixx(5)5t cos (ol - 02)t.k + Cx.x =

Fd + F cos (m1 - w2)t a

Using the given input data above equation can be solved in a numerical way.

The magnitude of the resonance surge amplitude xa mounts to 21.89 m, as shown in Fig. 16.

!es~n$!$~-~"rg-~ti?n-~ith-?~1z-~ti1_1-~?t_e1:~d~f!in~ coefficient -----------

The equation of the low-frequency surge motion is :

m .% + boxx.2 + Cx.x = Fd + F cos (m1 - ap)t xx a

with the solution for the maximum surge amplitude:

X = Fa a 2 2 2 2

m {mX - (W, - w2)1 + boXXiw, - w2) 2'

- - Fa boxx(wl - w2)

Using the given input data, the magnitude of the resonance surge amplitude X amounts to L4.55 ILL a

The results of these calculations clearly in- dicate that the influence of wave damping on -&he resonance surge motions is very importazt. This should, however, be checked with model test; results, which will be a subject of a fuxther investigation envisaged.

CONCLUSIONS

In the present study model experimental re- sults have been shown about the influence of waves on the low-frequency hydrodynamic coefficients of a moored vessel in surge direction.

Concerning the low-frequency surge mass coef- ficients in waves no deviation is found with regard to the still water low-frequency mass coefficients. The low-frequency surge damping coefficient, how- ever, shows a significant wave influence.

Waves seem to contribute an additional damping superimposed on the still water low-frequency damp- ing. Because this wave damping appears to be pro- portional to the square of the wave height, the contribution of the wave damping to the total damp- ing cannot be disregarded. The low-frequency surge damping coefficient can be considered as a super- position of the following components:

- Potential damping coefficient; this contribution is negligible.

- fiscous still water damping coefficient; this con- tribution is dominant.

- Wave damping coefficient; this contribution is dependent on the wave height and the wave period and independent of the spring constant of the mooring system, while the magnitude of the wave damping is of the same order as the viscous still water damping coefficient.

In order to further explain the damping and its components of moored vessels under motion resonance

conditions, additional physical and theoretical investigations are required..

NOMENCLATURE . m - - -- .-

A = wave group envelope

a = added mass in X-direction XX -

a = potential added mass in X-direction =P at = non-dimensional added mass xx -

B = breadth of vessel

= damping in still water boxx b = potential damping in still water xxp

= viscous damping in still water bxxv

= wave damping blxx

= total dmping in waves btotxx

bAxx = non-dimensional damping in still water

bLp = non-dimensional potential damping in still water

= non-dinensional viscous damping in still b&v water

bixx = wave damping in regular head waves per uhit wave amplitude

bTk.X = non-dimensional wave damping in regular head waves

C~ = block coefficient - -

Cx = spring constant in X-direction

. -.

Fa = amplitude of the oscillating wave force - A

Fa = constant wave drift force p

g = acceleration due to gravity

m = mass of the vessel

m = virtual mass of the vessel xx n = normal vector - -

' = generalized direction cosine nx P = water pressure

k = wave number = 2a/A

k = longitudinal gyradius of the vessel 39- L = length between perpend%culars

T = draft of the vessel or wave period

If X = natural period of the moored vessel in X-direction

X' Q

= roll period of the vessel

= pitch period of the vessel

Tz = heave period of the vessel

t = time

X = linear surge displacement or co-ordinate

k, 2 = first and second derivatives of displace- ment with respect to time

6 = logarithmic decrement

E = phase angle

'a = wave amplitude

E = amplitude of surge motion

W = 2 r /T = wave frequency

U = frequency of wave encounter

W = natural frequency of moored vessel in X X-direction

P = specific mass of water

4 = velocity potential

A = displacement X = wave length

V = displacement volume

REFERENCES

l . Maruo, H.: "The Drift of a Body Floating on Waves", Journal of Ship Research, Vol. 4, December 1960.

2. Remery, G.F.M. and Hermans, A.J.: "The Slow Drift Oscillations of a Moored Object in Random seas", Soc. Pet. Eng. J., June 1972.

3. Pinkster, d.A. and Hooft , J.P. : "Low-Frequency- Drifting Forces on Moored Structures in Waves", Proceedings 5th International Ocean Development Conference, Tokyo, September 25-29, 1978.

4. Oortmerssen, G. van: "The Motions of a Ship in Shallow Water", Ocean Engineering, Tiol- 3 , 1976

5. Salvesen, N.: "Added Resistance of Ships in Waves", Journal Hydronautics, Vol. 12, No. 1 , January 1978.

W

Designation

Length between perpendiculars

Breadth

Draft, even keel

Displacement volume

Metacentric height

Centre of gravity above keel

Longitudinal radius of gyration

Block coefficient

Midship section coefficient

Waterline coefficient

Roll period

Pitch period .

Heave period

MEASURED

125,000 cu m LWG carrier

273.00

42.00

11.50

98,740

1-00

16.70

65.52

0.750

0.991

0.805

32.0

8.8

9.8

Cx

t0n.m-'

Bmbol

L

B

T

V - GM

KG

k YY

C~

CM

Cw T9

To

4%

200,000 D!iT VLCC

310.00

47.20

18.90

234,994

5.78

13.32

77.47

0.850

0.995

0.890

14.5

10.2

11.4

Unit

m

m

m

m3

m

m

m

- - - sec.

sec.

sec.

Tx

sec.

125,000 cu m LNG carrier

38.67 104.6 0.060 10318 10742 424 4.11 14.14 180.81 I 8.5 0 . 3 1 I ,0699 I I 3.69 I 2%:;; 913.00 21.7 0.289 10931 5.94

200,000 DWT VLCC

Ox

rad.sec;'

53.72 251.15 754.15

m

ton.sec?m-l

136.2 63.5 36.7

CMCULATELI WITH THREE-DIMENSIONAL DIFFRACTION PROGRAM

"xx

ton.sec?mml

245.54 0.046 0.099 0.171

sad.sec,'

a*

ton.sec?m-l

25388 25625 25791

m

ton.sec?m-l ton.sec?m-l X t0n.sec.m-l

125,000 cu m LUG carrier

* 100 a

XX

m 7

834 1071 1237

0.029 0.057 0.089 0.130 0.180 0.288

borx

t0n.sec.m-l

3.40 4.36 5.04

10318

23.37 41.20 92.58

200,000 DWT VLCC

436 440 447

480 459 534

0.02F 0.049 0.088 0.120 0.204

4.23 4.26 4.33 4.45 4.65 5.18

24554

0.013 0.103 0.408 1.418 4.553 32.111

1422 1433 1461

1618 1493

5.81 5.84 5.95 6.08 6.59

0.033 0.260 1.499 4.154

29.625

TABLE 3 R VIEW AND RESULTS OF MEASUREMENTS ON THE DETERMINATION OF THE !'IAVE DAMPING COEFFICIENTS OF A 125,000 CU M L ~ G CARRIER

TABLE 4 V F ~ ~ E W AND RESULTS OF MEASUREMENTS ON THE DETERMINATION OF THE WAVE DAMPING COEFFICIENTS OF A 200,000 DWT

Cx

t0n.m-l

38.67

180.81

Wx

rad.secf

0.060

0.130

Cx

t0n.m-'

53.72

251.15

754.15

5,

m

2.45 1.75 1.05

2.45 1.75 1.05

2.45 1.75 1.05

1.75 1.05

2.45 1.75 1.05

2.45 1.75 1.05

2.45 1.75 1.05

- 1.75 1.05

U

rad.secyl

0.046

0.099

0.171

T

sec.

19.0

12.6

9.2

6.7

19.0

12.6

9.2

6.7

'a

m

2.89 2.06 1.24

2.89 2.06 1.24

2.89 2.06 1.24

- 2.06 l .2b

2.89 2.06 1.24

2.89 2.06 1.24

- 2.06 1.24

2.89 2.06 1.24

o

rad.eec;'

0.331

0.500

0.683

0.938

0.331

0.500

0.683

0.938

T

sec.

20.9

13.6

10.0

7.3

13.6

10.0

7.3

13.6

6

0.072 0.073 0.071

0.120 0.099 0.079

0.137 0.105 0.083

- 0.089 0.075

0.055 0.056 0.055

0.077 0.070 0.061

0.087 0.073 0.062

- ~~

0.063 0.059

W

rad.sec;'

0.301

0.462

0.628

0.861

0.462

0.628

0.861

0.462

btotxx

t0n.sec.m-l

14.75 14.96 14.55

24.59 20.29 16.19

28.07 21.52 17.01

18.24 15.37

24.43 24.87 24.43

34.20 31.09 27.10

38.64 32.43 27.57

- 27.83 26.03

6

0.063 0.063 0.063

0.133 0.105 0.078

0.174 0.119 0.081

- 0.097 0.077

0.083 0.068 0.058

0.097 0.076 0.061

- 0.063 0.058

0.083 0.076 0.070

boxx

t0n.sec.m-l

14.14

25.10

btotxx

ton.sec.ml

23.37 23.37 23.37

49.33 38.95 28.93

64.55 43.99 30.11

35.98 28.56

67.06 54.94 46.86

78.37 61.41 49.29 -

50.90 46.86

116.43 106.61 98.20

blXX

t0n.sec.m-l

- - -

10.45 6.15 2.05

13.93 7.38 2.87

- 4.10 1.23..

- - - 9.10 5.99 2.00

13.54 7.33 2.47

- 2.73 0.93

boxx

t0n.sec.m-'

23.37

41.20

92.58

blxx 2

ca ton.~ec.m-~

- -

1.74 2.01 1.86

2.32 2.41 2 ..60

- 1.34 1.12

- - -

1.52 1 .g6 1.81

2.26 2.39 2.24

0.89 0.84

blxx

ton.6ec.m-l

0 o 0

25-96 15.58 5.56

41.18 20.62 6.74

12.61 5.19

25.86 13.74 5.66

37.17 20.21 8.09

9.70 5.66

23.85 14.03 5.62

blxx 2

5, ton.~ec.m-~

0 o 0

3.11 3.67 3.62

4.93 4.86 4.38

- 2.97 3.38

3.10 3.24 3.68

4.45 4.76 5.26

- 2.29 3.68

2.86 3.?l 3.66

RECORD OF AN IRREGULAR SEA L

RECORD OF SURGE MOTION

-10m -

I o o 1 6 0 . 140 d o 250 3h

TIME IN seconds

Fig. 1 - Low-frequency surge motions of a noored LUG oarriar.

10000 125.000 cu m LNG CARRIER

rao.81 Ton m-'

2

0 10 20 30

l0000

2 W

z W

P e

5000

P 3 J

2 r

0 10 20 3 0 STATIC DISPLACEMENT I N m

125.000 w m L N O CARRIER

200P00 DWT V L C C

Fig. 2 - Body plans o f the vessels.

......... ~. I i6.40 m(125.000 cu m LNG CARRIER) 24.68 m (2W.000 D W T YLCC J

Fig. 4 - Test set-up.

Fig. 3 - S t a t i c load tests i n surge direction. -

W, = 0.099 nd.scc?

WAVE: W 0.628 rad.$to:' 5.. 2.89 m

OCREST VALUES

. . X"+ Xn.1 2 I N m

Flg. 5 - Surge motion decay test i n s t i l l water. Fig. 6 - Surge m t i o n decay tes t i n regular waves.

125,000 cu m LNG CARRIER m POTENTIAL THEORY

1 + MEASURED

200,000 DWT VLCC O POTENTIAL THEORY MEASURED

-X 0.1

MEAN

.--.

125,000 cu m LNG CARRIER @ POTEtsTiAL THEORY

~ 2 w , o o o VLCC 0 + MEASURE0 POTEtWAC THEORY 0 MEASURED

Fig. 7 - Procedure o f analysis of ext inct ion wave tes t recording. Fig. 8 - Non-dimensio~l low-frequency surge added mass cmff ic ient . Fig. g - Calculated and measured damping c w f f i c i e n t s i n s t i l l uater.

200,000 DWT VLCC

---- 115,QOO su m L W CARRIER

P- 2OO.WO DWT VLCC

Fig. 10 - Non-dimensional Iw-frequency surge damping coeff ic ient i n a t i l l uater.

125.000 cu m LNG CARRIER

MWRING SPRING CONSTANT

MOORINO SPRINO CONSTANT Cx =180.81 ton.m4 W, . 0.llO rrd.sacll mx, . 10,699ton.rec.~mi box, 25.10 ton.sec. d

MOORING SPRING CONSTANT Cx 5 53.72 t0n.m-l wx 5,346 raa+eef m,, a 25388 tw t .~ to .~m-~ box, = 2 i 3 7 ton.scc. m-'

X.

MOORING SPRING CONSTANT cX .251.15 mn.m-' Wx X 0.099 iaasts? m,, t 25,625 ton.$ec?rn"

R b,, 41.20 ton.*=.. m!

- d W MOORINO SPRING CONSTANT C, ,754.15 tonrn-'

W, o.r t~raates; ' W, * 25,791 tonsec.*m4 box, 92.58 tan sac. m"

Fig. 11 - Wave danping o f the LNG carr ier . Fig. 12 - Have damping o f the ME.

I SPRlNG CONSTANTS MOORING SYSTEM: 1 125,000 cu m LNG CARRIER

C x = 38.67 ton.6 ' I a C,= 53.72 t0n.m" i', = 104.6 sec. T, = 136.2 sec.

200.000 DWT VLCC

C X . 180.81 ton. m-' CX.251.15 ton.mil T X - 48.5 sec. T,= 695 sec. 2 so

----- 125,000 W m LNG CARRIER as C,- 38.67ton.m" + =180.81 t0n.m-'

200,000 DWT VLCC o C,= 53.72 t0n.m-' o =251.1 5 tonm' v ~754.1 S ton.m'l

Fig. 13 - Relation between wave daq ing and wave height.

Rg. 14 - Relation between wave daq ing coeff icient and spring constants o f the mowing system.

---- 125.000 CU m LNG CARRIER

l - 200.WO DWT VLCC

Fig. 15 - Non-dimensional wave damping coeff icient.

Fig. 16 -Resul t o f calculation o f the resonance surge motion of a . - VLCC using the wave damping coeff icient.