Time-Varying drift forces in cross seas: measurements and theory

Maritime Research InstituteNetherlands

2, Haagsteeg; P.O. Box 28

6700 M 4fagerthgen. The Netheflands

Telephone +31837093911 Telex 45148 nsmb nl Telefax +31 8370 93245

Delit University of Technology

Ship Hydromechanics Laboratory

Library

Mekelweg2, 2628 CD DeItt

The Netheriands

f Phone: +31 152786873 - Fax: +31 15 2781836

Z52242 - Paper ISOPE-92-05-174

TIME-VARYING DRIFT FORCES IN

CROSS SEAS: MEASUREMENTS AND

THEORY

By: Jan O. de Kat

March 1992.

L

ABSI1AC7

This paper presents a novél method to determine

experimentally both mean and time-varying

drift

forces., This method is applied to measure the low

Ire-quency _{forces in cross-sea conditions.. £ brief review}

is given of the theory for determining the second

or-der forces acting on a moored vessel _{in directional}

seas. Phereas in previous experimental work only mean

drift

forces could he determined, the present study shows _{that time-varying drift forces can be measured}

with _{reasonable accuracy.. surge and sway forces and}

yaw moments are measñred for a 200,000 DWT tanker

sub-.jected _{té cross waves in 82.5 n water depth. Compari'}

son between measured and theoretical values is

reason-able.

KEY PORDS: _{Drift forces, directional seas, sécond}

or-der effects, Kalman tutes, _{experiments, tanker}

1.. IN1'R0DUCTI0r

A _{vessel aoored in seas with directional}

spread-ing _{vU.l be subjected to mean and low frequency} wave-induced

drift forces

The lev f reqùency behavior can

have an important influence on the design of the

moor-ing _{system.. In a mooring analysis, one typically}

con-siders long-crested seas, for which an _extensive

amount _{of iodai test results and validated analytical}

tools _{exist.. Relatively littlè work has been carried}

out with respect

to drift

forces in short-crested

seas. So far, experimental wòrk has ben limited

wort-ly to cross seas, i..e.. the còdbination of two long-crested wave systems coming from different directions.

Aside_results,

from facilitating

the interpretation

of test

cross seas do occur gùité frequently _{in a}

typical ocean environment.

Pinkster _{(1985) derived expressions for the}

sec-ond _{order drift forcés in directional seas and per'}

TIMZ-VASYING DRIfl FORCES IN CROSS SEASz HEASIJREENTS AND THEORY

JanO. deKat

Maritime Research Institute Netherlands (MARIN) Kageningén, The Netherlands

formed _{tests to measure the meén drift forces in}

reg-ular Cross waves with equal frequencies and also in irregular cross seas.. Experiments with a spring-moored vessel in cross waves with equal frequencies vere.

re-potted by Krokstad (1990), who also investigated

re-spense and wave 4ft damping effects in thode

condi-tions Nwogu and Isaacscn (1991) conducted tests in

both long-crested and multi-directional

seas,

illus-trating that overall low frequency rotions are reduced

by wave directionality. Other _theoretical

considera-tions concerning drift forces In directional seas have

been presented by Kim and The (1989) and Viaje (1988).

This work is an extension of the research

re-ported _{by Pinkster (1985).. For cross sea coéditions}

with equAl frequencies for both wave components,

it

was _{shown that}

_{for a particular realization}

_{of the} wave process

the mean drift

_{force consists of the} indepeddent _{contributions of the individual wave}

com-ponents and an interaction term _{The agreement between}

theory _{and measurements vas reasonable. The objective}

of the present work is (1) to show that time-varying

drift

forces can be measured with a reasonable degree of accuracy; (2) to compare measured drift forces with

computed values in crosS wave conditions..

A review is given of the expréssions for the sec-ond order drift forces in directional seas.. The _focus of _{this study will be on the slowly vérying part of}

the _{excitation forces, and in particular on the surge}

drift force.. To determine possible interaction effects between directions, the elementary case of two regular

waves _{from perpendicular directions is cons.dered} experimentally.

The experimental determination _{of time-varying}

drift

forces poses

a difficult

problem: .the veisél

must _{be free to perform first order motions (as these}

influence _{the second order forces to a large extent),} yet it should not, exhibit any low frequency mottoés.. A novel approach has been instigated _{at MARIN, where use} is made of dynamic positioning techniques _{and a}

the vessel is unrestrained as regards first order mo-tions. The forces required to maintain, the average

position are, equal to the second order drift forces..

A description is given of experimentAl proce-dures the Objective of the tests is to determine the second order

drift

forces acting :fl a 200,000 DWT tanker moored in 82.5 u vater depth The vessel is

subjected to regular cross vares, where the angle

between the wave train directions is 90 degrees and

where the relative heading angle of the vessé.1 is 45 degrees with respect to bOth vave systems. Tests are

carried out for a range of vave frequencies and

ampli-tndes, and the obtaiñed experimental values are

corn-pared with theoretical predictions..

2 SECOND ORDER DRITT FORCES TN DIRECTIONAL SEAS: THEORY. RZVIV

This section presents a brief review of the

the-oretiral de6cription of the mean and slowly varying

drift

forces in directional seas, dee to Finhetet (1985).

2.1. Incident Paves

We asse that the wave elevation at a point can

be described by a double Tourier series, which

charac-terizes the surface elevation as a sum of regular

long-crested waves from various directions:

j! ki

ik 'COS(L:t + !jk) (2.1) where:

wave frequency

Lik random phase angle

N = number of discrete frequency components

N = nimber of directional cOmponents

ik wave amplitude = wave frequency index

k wave direction index..

In a multi-directional sea the amplitudes of the

wave components are found from:

ik 2

4o.$

_(2.2)

where:

directional wave spectrum

Le frequency interval

a direction interval..

After selection of the random phase angles a time

record of the surface elevation can be generated from

eq. (2.1).

2.2.. Second Order Pave Drift Torces

Second order wave fOrcés acting on vessels or

structures in waves can be computed based on direct

integration of second order pressures and forces, see Pinkster (1980)..

In vector notation the general expression for the second order wave forces is as follows:

Y2(t)

-

_f

pg

- rií-

p (1)2 ds + S0 -

_Jf(p(Ï'

_V+)).dS

+ _{x (N} (1)) g + 2))) ₅ So

t

t

úhere:

41) first order relative wave elevation äround the_{waterline PL}

- first

order velocity potential inclnditg

ef-fects of incoming waves, diffracted vavès and

waves generated by the body motions

N - mass matrix of the body in air ( 1)

( 1)

Realizations of Drift Torce Records in Time Domain

Eq. (2..3) Chews that the secondörder torce

consists. of terms involving products of first order

quantities

or ifltegfts

of products

of first order

quantities, and

at'- /et

terms.. Using the discrete

formulation of eq. (2.1) for the waves, thé second

order force in irregular directional seas can be

written as:

T2(t)

i-1 j-1 k=1 1=1'2 E H E

_ik

N N M N cos((w-e)t +

(AikEjj)} +

N .N M X E Z E

t

_ik

1 ikl

sin((e a )t +

i1 j1 k1 11

first order oscillatory linear motion vector

- first order

oscillatory angular motion vector

of the body

second order 'incoming yayO' potential

second order 'diffraction' potential..

(2.3)

where P and Qj are the in-phase and quadrature

quadratiraasfer-nctions based on. eq. (2.3). The quantities and represent the amplitudes of the

varé cbnponêRts, whéf e i and j are the frequency in-dices and k and 1 are the diréction inin-dices.

Each cf the components Of eq.. (2.3) contributes to the total th-phasé and quadtature quadratic

trans-fer functions. This will not be treated further here. 2 The cpributions due to the nonlinear potentials

and ' ar-e approximated using the method given

b Pthkster (1980), making use of the uoñlinear second

order potential for the undisturbed directional waves.

(2.4)

n - outwa.rd pointing normal Vector to the hull

S0 - mean wetted hull surface 2)

17

-- We

vili

assume hereafter

that

the quadratic

transfer functions include contribUtions from aB.

com-ponents of eq.. (2.3)..

Mean Value of Realizations of Wave Drift Force Records The time-average drift force ïsfound from eq..

(2.1.) for iaj:

2 N N N

( _(t)

t

k1

3.iik1 cos(Éjk-Ejl) .f

+

sin(&-&1))

(2.5)

It

can be shown that the ensemble average of the

drift force is:

N X

EF2J

F E

t

k

i4k1

which, in continuouS form becómes:

- 2.J ¡

S,,(e,a) P(e,e,,a) da de

00

where P(e,e,a,a) is the quadratic transfer function 6f

the mean drift force in regular waves of frequency e

from wave direction a..

Frequency Domain Representation of the Drift Forces:

Drift.Force..Spectrum . -- -

-It is of interest

to obtain information on the

low frequency drift forces in irregular vares in the

frequency domain. We use eq. (2.4) as starting point to obtain the spectral density of the d;ift forces..

The

derivation df the spctral

density wili not

be given here. As is the case with the wave elevation

in

a point as gien by eq

(2,,l) the second örder

force given by eq. (2.4) is non-ergodic and for a par-ticular realization (choice of c's) the spectral den-sit)' of the force is dependent on the phase angles r.

The average of all realizations (ensemble aver-age) is found tó be:

2it2n

- S f

_¡

.1 S7(e+i.,a) Sr(e,e) 0 0 0 da dP de where: P(w+i,e,OE,p) +

Por the quadratic transfer functions the follow-ing symmetry relationships apply:

- P(e,e+t,$,a) (2.10)

e - Q(s,e+.&,p,a) (2.11)

which is the same as:

- T*(s,+1,a)

(2.12)

where * denotes the complex conjugate. Similar

synüne-try equations apply to the discrété quadratic transfer

fuñctions Q1 and the amplitude

Drift Force .iñ Regular Cross Seas with EqUal

Frequencies

The sieso drift force in irregular

multi-.direc-tiona]. seas is given by eq.. (2.5). Each of the terms in this equation reflects a contribution due

to

in-tétactioñ of a

regUlar wave with frequency e from

rection k with a regular wave with frequency ej from directiOn 1.

Consider the elementary case that ve have one

regular wave with frequency e1 frOm direction 1 énd

one regular wave with the sand frequency from

d.irec-tien 2 Taking into account the symmetry relat.onship

of eqs. (2.10) through (2.12) thé meañ drift force is given br:

F(t)

_ii

p1111 + l2 p1122 +

+ 2 t11 12(P1112 os(r11-s12) +

+ Q11 sin(c11-E12fl (2.13)

The first two components of eq.. (2.. 13) are the

mean forces due to each regular wave independently.

The third coñtribution is dUe to interactiòó effects

of the two regular waves in the mean dríft force. It

is

seen that this component is, besides being a

fun-ction of the quadratic transfér fitiOns P1112 and

Q11.1,, a function of the phase angles t11 and £12 of tEe two regular waves.

Drift Force in. Regular Cross Seas with. Unequal

Frequencies

For regular cross seas with unequal component

frequencies the time-dependent drift fòrce follows from eq. (2.4) and the symmetry relationships used above:

F2(t)

-

C112 _{+ t22} P2222 +

+ 2 t11 t22 [p1212 cos{(w1-m)t + (C11C).) -t

+ Q1212 sin((e-e2)t f (cji..c22))J

Thus

also in this case

_{the méan drift force ja}

composed of the superpoSition of the _{mean forces dUe}

to each independent wave component. The elsie-dependent

part

of the drift force is

caused y'the interaction

between the two intersecting vare _{trains. The} ampli-tude of the slowly varying párt is given by:

2 C ? _{'P ..} fl _{2 16}

a

'

' -

_{11. '22 1212 1212}

The theoretical secOnd order drift _{force can be}

determined by computing the frequency and direction

dependent _{quadratic transfer functions P and}

k1 This can be

achieved through 3-D diiction

cgiculatjons..

(2.14) (2.8) varying part, where the mean component is given by:

The _{drift force ConSists of a mean and a slowly}

F (t) p . _(2.15)

11 1111 '22 2222 (2.9)

(2.6)

3.. EXERIKENTAL PROCEDUB.ES 31 General

ïtodel tests aiid computations were carried out for

a 200,000 DWT tanker (90% loaded) moored in 825m water depth.. The particulars of the tahker are shown

in loble 1, and a body plan is Fig.. 1. The

experimental setup in KASIN's Vaya and Current Basin

is shown schematidafly in Fig. 2. This facility

mea-sures 60 n x 40 n and is equipped with snake type wave generators located on two sides of the baSin. The wave generators can be driven independently to produce reg-ular and irregular cross-sea conditions.. The scale

used in the experiments is i to 82.5.

Table 1 Iain particulars of 200,000 DWT tanker

(.90% loaded) Tanker model- 5612

öde1 scale I to 82.5

The principle of the mooring arrangement is as follows. The tanker is moored using two constant

ten-sion winches connected transversely to the model with

steel wire, and one constant tension winch in longi-tudinal direction.. Pretension is obtained by means of counterweights. The winches are steered through a con-trol system, which checks the low frequency vessel

no-tions against the desired position at each time step

during testing and takes corrective action if neces-sary.

Ieasurenents comprise three earth-fixed ¡notions

(surge, sway and yaw) and three ship-fixed forces (one

longitudinal force, One transverse force forward and

one trañsverse force aft). The forces and yaw moment

that are obtained from the measurements represent the wave drift forces.

3.2. Positioning Control System - laman Filter

The _{displacements of the tanker} model are

re-corded by the control system in a basin-fixed coordi-nate system. In this investigation the center of

grav-ity

is used as the reference point on the model to

determine the excursion of the model with respect to a

reference position in the basin,. The control system

uses thïs information ro determine the required winch

forces.

STATION IO

Fig.. 1 General layout and body plan of tanker

The motion of the tanker model consists of

moza-bimed low frequency and high (. wave) frequency

mo-tions.. The low frequency notions are to be

counter-acted by the winches. The basic principle of the

con-trol

System is to use mathematical models ro estimate

the low frequency motions of the ship and filter the

high frequency oscillations. The estimate is updated

at èvery time step, using the measured surge, sway and

yaw motions.. To predict the motions at the next time

step, the contrOl system has access to-a tine-domain

simtilation program that computes the low frequency

motIons of a moored or dynamically positioned vessel,

taking into account the various hydrodynanic effects

related to waves, current and wind. Given the

calcu-lated drift forces and model position with respect to

the reference position, the required winch forces are

determined and applied.

The ¡boye type of filtering is keovn as Kainan

filtering.

The Kalman- filter

is a set

of equations, which allows an estimator to be updated once a new

ob-servation becomes available, This process is carried

out in two parts. The first step consists of -forming

the optimal pred.ictor of the next observation, given

all

the information currently available. The new ob

servation is then inc5rporated into the estimator of

the state vector, It is assumed th.ar the low frequency

dynamics of the system can be described as follows by the laman filter model:

(3.1)

Y (3.2)

Eq.. (2.17) is the state equation of _{the system}

and _{eq.. (2.18) is the observation equation.. X}

repre-sents deterministic input and and r, are uncorrelated

noise processes, where the covariance matrix is given

by; Designation Symbol Unit liagnitude

ength between perpendiculats L n 310.20

Breadth S n 47.20

Depth D n 24.40

Draft - T n 17.66

Displacement weight 8

t

233,315

Center of gravity above keel KG n 13. 70

Center of buoyancy forward of

section 10 n 7 34

Netacentric height . GM in 5.69

Longitudiñal radius of

gyra-tionizair

-k

yy n 74 41

Transverse radius of gyration

inair

xí n - 16 00

(3)

(4)

Fig. 2 _{Schematic representation of setup of tanker model in Vave and Current Basin}

cov{]_{

]

The _{objective is to forecast S at the next time}

step, based on previous observutions (Y) _{At each time} instant a number of steps are applied:

Kalman gain;

P ET(H p M +

where rT represents the transpose of H and F.

defined bys

s E{(S

_{- Stlt_i)(st - S1_1))}

State corrector at current time:

-

F (Y - R Stjt_i)

State predictor:

StIt+i P 5tk + C

Error predictor:

Therefore, at tine t+1 the following information

is needed for the required computations: Y _,

P , F, H, R and Q.. The state predictio

_{givén y}

eq.

_{(,,5) is carried out using}

_{the time-domain} simu-lation program mentioned earlier in this section. _This

program, named DESI, does not take wave directional

spreading into account.. Based on the available

infor-mation _{(surge, sway and yaw position, and the forces}

acting on the tanker) an estimate is ¡nade of the (33) forces _{required to maintain the desired tanker}

posi-tion. This information is subsequently relayed _{to the} three controlled winches and updated every second.

Us-ing _{this procedure the low frequency motions are kept}

to a miniininn and the measured forces are equivalent to the second order wave drift forces.

For the present test setup the moasured _response

of the mooring system to a given displacement away

(3.4) from _{the equilibrium position is shown}

_{in Fig. 3 for}

surge and sway. In both _{cases the tanker is brought}

back _{to its equilibrium position without any}

oscilla-tions.

(3..5)

(3.6)

Waves 2250

+Fx

Force transducer (fi

fi

Beach Wsves l35 '..Winch FY-FORE Winch fl-AFT Beach +FY-FOR . Counterweight Counterweight

.',

+FY-AFT

Force transducer (fi)

t-

..

,.

o t

-

4e

¿

}ig. 3 Surge and sway extinction time traces

Table 2 Review of test conditions

4. TEST CONDITIONS ND RESULTS

An overview of the test conditions is shovn in Table 2.. The wave fre4uencies vere chosen to obtain a realistic range: 0.5 - 0.8 radIs.. For three casez

(0.5, 0.6 and 0.-7 .rad/s) the sane frequency vas used

for both di±ectijns to check the behavior found by Piukster (1985) For all cases with cross waves a fre-quency difference of 0.04 rad/s vas applied, and for

jome add.ttional cases also 0.02 rad/s frequency dif-ference weE used. The amplitudes of the wave

éouipo-nears veré cf the same order, but not necessarily

equal to each other..

P.XN

Test No..

Regular wave f rom 225 de;

Smplitude la)

---

8agular wave from

-- ---Wave Freguenny L] Irad/el 135 dóg fragen-ny difference (red/ej PhÌ.ÒÖ between wave ankara (de;] Wave 1'xequency L-) (red/a) .8mplitude La] 659301 3. 0.500 1.58 7 0.500 2.46. 0.00 0 659901 3. 0.500 1,58 7 0.500 1.46 0.00 ₆₀ 660202. 3. 0.500 1.58 7 0.500 1.46 0.00 120 660502. 2. 0.500 1.58 7 0.500 1.46 0.00 180 660801 1 0._500 1.58 '7 0.500 1.46 0.00 24Ö 661101 1 0.500 1,58 '7 0.500 1.46 0._00 300 659201 3. 0.500 1.58 8 0.540 1.24 0,04 0 658001 l.A 0.500 2.46 EA 0.-540 2.06 0.04 0 659101 2 0.575 1,14 10 0.595 1.69 0.02 658101 2Á 0.575 2.-03. lOA 0.595 2.65 0.02 659001 2 0.575 1.14 11 0.615 1.87 0.04 659401 3 0.600 1.46 12 0.600 1.82 0.00 0 660001 3 0..6Ö0 1.46 12 0.600 1.82 0.00 - 60 660301 3 0.600 1.46 12 0.600 1.82 0.00 120 660601 3 b.600 1.46 12 0.600 1.82 0.00 180 560901 3 0..6Ò0 ' 1.46 12 0.600 1.82 0._00 - 240 663.201 3 0.600 1.46 12 0.600 1.82 0.00 300-658901 3 0.600 1.46 13 0.620 1.-75 0.02 0 658803. 3 0.600 1.46 - 14 0.640

ì93

0.04 0 658701 4 0.625 - 2.59 16 0645 1.77 0.02 ₀ 658601 4 0.625 1.59 17 0.665 1.70 0.-04 O 659701 5 0 700 1 67 18 0 700 1 71 0 00 0 660101 5 0.700 1.67 18 0.700 1.71 0.00 60 660401 5 0.700 1.67 18 0.700 -1.11 . 0._00 2.20 660701 5 0.70Ö i.67 18 0.700 1,71 0.00 180 661001 5- 0.700 1.67 18 0.700 - 1_71 0.00 240 661301 5 Ó..700 1.67 18 0.700 1,71 0.00 - 300 658501 5 - 0.700 1.67 19 0.740 1.44 0.04 658301 5A Ò..700 2.88 19A 0.740 2.65 0.-04 658401 6 0.800 2.11 21 0.840 1.79 0.04 o 6$820i GA 0.800 2.82 23.1 0.840 2.56 0.Ö4 I)

The above test conditions vere used as input to a

3- diffraction program, which allows one to determine

the first and second erdér forces actin& on a moored structure in long-crested and short-crested seas. Por this purpose the tanker is modeled with a sufficiently

large number of panels. Comparisons are ¿nade for the

surge drift force.

Three forces vere measured; FI, FT-PORE and fl-AU. The surge drift force is equal to PI, thé sway drift force is obtained by surnznatian of the two trans verse FT forces,. and the yaw moment is given by sum-ining the moments of FT about G.

Cross Vaves with Equal Frequencies

In cross-wave conditions vtth equal frequencies the mean drift force is a function of the phase angle betweho the two wave trains. Therefore, for each ¡et

of frequencies the phase angle between the two wave.

generators vas varied fron O to 360 degrees.

The measured surge and sway drift force, as veil

as the yaw moment, are sbovn as a fùnctian of the phase angle in Figs,. 4, 5 and -6 for ¿n .0.5, 0.6 and

0.7 radis, reàpéctively..

It is seen that the measured drift forces and yaw nouent contain a mean part and a part that ts a har-monic function of the phase angle differénce

(c11-c12), as vas observed by Pinkster, (1985).

PMS5AO5 DSTWnBI WAY ATRI I DOa

Fig.. 4 Measured mean drift forces and yaw moment. in cross waves with equal frequency (0.5 rad/s)

iAI1

-

maw

R_ama

mama_ama

_w

n ii0

P14*25 ,am.s 25TWlvEIAyO5,, OEg

Pig. 5 MeaÉured mean drift forces and .yaú moinént in cross waves with equal frequency (0.6 rad/s)

FMA25AIIOLZarwam.WAVI 25' UWVI PI D5O

12

Fig. 6 Hèasured mean drift forces and yaw moment in cross waves with equal frequency (0.7 radis)

12' n

In _{this paper ve vil], focus on the surge}

_drift

force. The ensémble _{average of the Eleúured surge}

dr.f t force is given together with the theoretical

mean _{value in lable 3, ivenby eq.. (-2J4). The} com-parison between experiment and theory i.s shown grapKi-daily in Pig.. 7 and is qüite good for _{the freqüecries}

considered..

lable 3 Ensemble _{average of measured and theoretical}

surge

_{drift force iñ cross}

_{waves with equal} frequencies

Yrequency

(rid/c) Measured

(tf)

Theoretical_C")

20

o

Measured Theoretical

0 5

The theoretical average surge drift force and

ais-plitude are given by eqs (2 15) and (2 16),

respec-tively. _{These quantities are shown in brackets in}

Ta-bles 4 and 5 The same _{results are shown graphacally}

in Figs. 8 and 9,.

L s O.02 rada Measured, 4 si 0.02 red/s Theoretical 4 si -0.04 rad/s Meaved

4

0.04 ròd/e Theoretical

80

Pig.. 8 Keäsured _{and theoretical values of meansurge}

drift force

o

04 _{06 08}

si[rad/sJ Fig.. 9 _{Measured and theoretical values}

force amplitude of surge drift MEAN SURGE DRIFT PORC

04 O + o g o 06 r. frawsj

4

= 0.02 rad/e MeasuÑd o .Asi= 0.02 radis ThOoretical £ + _{8 ci = 0.04 radis Theoretlhaj}

SURGE DRIFT FORCE AMPUTUDE

50 £ s 0.8 07 0.9 e1rad]

Fig.. 7 _{amplitude of wean longitudinal drift force in}

regular cross waves with equal frequencies

Cross waves with Unequal Frequencies

In _{the generation of waves with unequal f}

reqiiec-cies _{a bound second order wave travels with the}

_first

order vaies.. Due to the discrepancy _{in boundary}

condi-tions between the wave generators and fluid, free

second _{order waves have been kñòvn te occur in some}

medal basins.. Suth long period _{oscillations can}

influ-ence _{results significantly. Measurements in}

the Wave

and Current gasta, however, have shown that under

typical _{test conditions (is in the present case) free}

second order Úaves do not occur.

The _{measured drift terces contain amean ¡md}

os-cillatory _{component The measured average surge drift} force is shown in Table 4 as a function o wave

ampli-tude _{and frequency, and the drift}

force amplitude is given iñ Table 5 fär the ¡aise _cnditions.:

0.5 _{-45.1 -46.6}

0.6 -79.. 9 -82.7

lable 4 Measured and theoretical average surge drift force in cross waves with unequal frequencies

10543.02 w. 04.. .135 dog - rgo..., ¿ud/O) 111t.d. t.)

-0 7-01 O .750 5,4' 0.500 a u. 0 500 2.52 50.0 110.0 (241.2) 40. 0 441.4) 04.-0 470 2)

lable 5 _{fleasured and theoretical anplitude of surge drift fotce iii}

cross waves

02.4/.) Sog41tud.

with unequal frequencies

10goioz ou. fr...235 dog - 520qwcy ¿.j

0120.2.... 0 04 0 04 0 02 0 02 0 04 0 02 0 04 0 02 0 04 0 04 0 04 0 04 0 04 .I'...cy 3i.jt.do 0.540 0.540 0.555 0.505 0.02.5 0.020 0(40 0.045 0002 5.74ó 0 iII 0.40 ₀₅₄₀ 2 24 3.04 1.05 3.12 1.07 175 1.13 Irs. 2.70 144 2 02 3.70 3.54 0.300 34.0 3.30 (42 i) 0 500 43.0 2 40 (100.5) 100.02.4 -ti 4Th00015.1 - ti) 0 575 07.0 1 14 417 35 5.573 _2.22.0 2 01 (230.5) 0 575 53.0 1054102 i 1m (47 0) 0.000 o 04.. 2 40 ₍₅₅₅ 225 dog 0. (00 ₀₂₀ 2.40 _(10.1) 0 125 i 50 _(45.0)J, O 0. 025 2.50 -SS Q (45 S) dtiZ.z.... 0.04 0.04 0.02 0.03 0.01 0.02 0.04 0.04 0.04 0.04 0.54 Yosqosooy 1144O 0 5401.24 2.000 540 0.035105 0.1552(5 0.0251.07 0.0202.75 0 0401 03 0.4431.75 0 102170 5.7401.44 .2.020.740 0 0102.70 .0 040251 0 .500 30 O 1 5* (39 5) 0 500 3.44 (-123.0)-2.2.4. 0 _(Th00r.tifri- ti_{- ti)} 0. 575 -44:0 1.14 (-0 0) 0 075 -132.0 3.23 (-175.0) 0.575 -55.-0 1.14 (-04.0) 0 050 -12.0.4 fo 1.-41 (.70.0) 225 dog 0 000 -124. 0 2.40 _{(-70 0)} 1.425 1.'9 o osi x_5, o :7,0 1.47 5.700 2 ¡0 0. 005 2.2.2 0.050 i SS -104.0 (-490) -205.0 (-45 0) -343 0 (-50 4) -234 0 -(-044 0) -153.1 -103.0 (-153.0)

Table 4 and Pig .8 chow that the théoretical and experimental average surge drift forces agree reason-ably véll for frequencies below 0.6 tad/s. The theore-tical average drift forces for freqùeñciès between 0.6

and O.8 radis tend to be significantly smaller than

thé .measüred values. Por the _{tests with 0.7 sTud 0.8}

rad/s the agreement between measùrèments end theory is

significantly better when the wave amplitude is

in-creased (test No. 658301 and 658201).

The results in Table 5 and Fig. 9 show that the theoretical and measured amplitudes of the slowly

varying surge drift f rca are in reasonable agreement. This suggeCt.s that _{interaction effects are properly}

accounted _{for in the theory; Heasured drift force}

amplitudes tend to be 10 to 20% lower than theoretical

values The experimental technique used for measuring tine-dependent

drift

forces is promising. Research

.efforts are continuing to improve the Kalman

filter

and mechanical conzpoüents used in the _{test setup.}

5.. CONCLUSIONS

qave-induced

drift forces in

short-crested seas

are _{investigated theoretically añd experimentally. A}

brief overview is ptesented of the theory for

deter-mining the mean and time-varying _{secoud ôrder forces} acting on a moored _{vessel in waves With directional} spreading.. _{Surge and sway forces and yaw moments were}

measured for a 200,000 DWT tanker _{subjected to cross} waves in 82.5 e water depth..

In the case of regular cross Waves vith equal

cémponent frequencies the measured and theoretical

mean

drift

forcé

aré in

reasonable agreement. Por

cross waves with unequal frequencies _{the theoretical}

amplitudes of 'the slowly varing _{drift force compare}

quite well with the measured values; in some cases

large discrepancies exist _{with respect to the mean} component of the drift force.

&efore _{this study no reliable method existed to}

measure time-dependent

_drift

_{forces Using complex} mithematicá], models and Ka].man filtering _{to position a}

vessel in waves,

it is

_{possible to experimentally}

determine _{slowly varying drift forces with reasonable} accuracy..

A0VLEDGEHETS

I would like to thank LLE.. Huijsmans

for hs

valuable _{assistance in the expex'inents and}

computa-tions, Furthermore I would like to express my

grati-tude to J.A. Pthkster, who _{provided useful advice}

concerning this study..

REFERENCES

Kim, X.H. and Yue, D.LP. (1989), "Slòv1y-Va-ing

Drift _{Forces in Short-Crested Irregular Seas",}

Applied Ocean Research, Vol.. 11, No.. 1, pp. 2-18..

K±okstad, J.R.. (1990), "Drift Fores,Moments and

Cor-responding Responses in Multi-Directional Seas. A

Theoretical and Experieentgl Study", IDEAN, Sym-posium on the Dynadics of Xarine Vehicles énd

Stiuctures in Waves, Brunel University, 24-27

Jude.

Nwogu, 0. and Isaacson, M..

(1991), "fljft

Motions of

a Floating Barge in Random Multi-Directional

Waves", Journal of Offshoreilechanjcs. and Artjc

Engiñeéring, Vol. 113, pp. 37-42.

Pinkster, J.A.. (1980), Lov Frequency Secénd Order

Wave _{Exciting Forces on floating Structures",} MARIN Publication No. 650, Vageningen..

Pinkster, J.A. _{(1985), "Drift Forces in Directional} Seas", Proc. KARflIEC CEINA '85 Shingha.i,Decem-ber 2-8T.

Vinje, _{T. (1988), "On the Effect of Short-Crestedness}

Ön _{the Statistical Distribütion of Slowly}

Vary-ing, Second Order Forces", Applied Ocean;Re