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MODELLING OF WAVE LOADS ON

MOORED SYSTEMS IN SPREAD SEAS

R. Eatock Taylor

Department of Engineering Science

University of Oxford, UK. and

K.L Mitchell

Brown & Root Vickers Ltd

Wimbledon, London, UK

ABSTRACT

Design of the moorings for floating production and compliant systems

is strongly influenced by low frequency wave drift forces which excite

resonant responses. The paper discusses the modelling of these

forces by

second order theory, involving the use of directional quadratic transfer functions. Theoretical predictions using

such functions are compared with

the results of experiments in bidirectional and fully spread seas.

Simulated force spectra are also compared with spectra estimated

from

experiments in directional seas. Conclusions are drawn concerning the

influence of directionality, and uncertainties in the modelling

of drift forces.

1. INTRODUCTION

Design of the moorings for floating production and compliant systems

requires knowledge of eivironmental load effects due

to winds, ocean

currents and waves. Critical aspects are the static offset (hence mooring

line tension) due to these effects, and dynamic responses In the low

frequency modes of surge, sway and yaw of the moored platform. These

dynamic responses may be associated with instabilities driven predominantly

by quasistatic winds and currents; or by dynamic load effects associated

with the waves. The work described here forms part of an investigation of

the latter.

The resonant frequencies of floating systems in the horizontal modes

are generally much lower than the frequencies at which there is significant

energy in ocean waves. At full scale, typical resonant periods may lie in

the range of 1 to 5 minutes. This leads to the type of behaviour

illustrated in Figure 1, which shows time histories of wave elevation and

surge response of a turret moored tanker in spread seas (The results have

been scaled from model tests performed at a scale of 1:81 in a wave basin,

where no winds or currents were present).

In this example the waves had a

characteristic period of lOs, whereas the surge response period is seen to

be about 94s. The problem for the designer,

concerned with the ultimate

strength and the fatigue life of the mooring lines, is to predict

the

statistics of the line tensions under these conditions And the part of

that problem with which we are concerned here is prediction of the low

frequency wave forces causing such drift responses.

The underlying hypothesis is that the loading causing large

low

frequency motions and tensions depends on terms

proportional to the square

of the wave elevation process. Such terms are omitted in

first order

theory, which then can be developed using a frequency domain approach based

on linear transfer functions between loads and

waves. The analogous

frequency domain procedure for the second order theory employs Quadratic

Transfer Functions (QTF's). It is clearly more complex than linear theory,

and there are several practical difficulties in its appli:atlon; but recent

years have seen much work in this area, and a methodology for long

crested

random seas is now reasonably well established. The secofld order frequency

domain analysis reported by

Langley1is

characteristic of the approach

that has been taken by several investigators.

Notwithstanding this development of a rational theory, several

difficulties have delayed Its widespread implementation in professional

practice. These include uncertainties in the calculation of

second order

hydrodynamic forces; and problems in performing appropriate experiments and

analysing the resulting data. The present investigation has been directed

at shedding some light on theme two aspects; and at meeting the requirement

for extending application of the theory to study the influence of

Birectional spreading in the waves. The latter problem has also been

considered recently by Pinkster2,

Dalzell3,

Haeda, Morooka and

3

Some of the work has reviously been reported by Eatock Taylor, Hung

and Mitchell6. There we summarised some of the theoretical considerations

underlying the extension to directionally spread seas, and provided

calculations illustrating the influence of various approximations in the

hydrodynamic analysis. The present paper attempts to provide experimental

evidence confirming the underlying hypothesis of the wave drift force analysis, and offers some limited comparisons between theory and experiments in directionally spread seas. The latter were performed at

small scale, and are subject to several sources of uncertainty - but we

believe such data to be extremely rare at the present time.

The paper is organised as follows. The next Section provides a brief

recapitulation of the theoretical background to the QTF corresponding to

unidirectional and bidirectional waves; and. describes experiments and the

resulting data leading to comparison of theoretical and experimentally

estimated QTF's. Section 3 is concerned with theoretical simulation and

experimental estimation of drift force spectra: first in crossing (le

bidirectional) seas; and secondly in fully spread seas. The final Section

lists some brief conclusions.

Z. QUADRATIC TRANSFER FUNCTIONS FOR WAVE DRIFT FORCES

.l Summary of Theory

It is convenient first to consider two sinusoidal waves of frequency

direction of propagation ai and phase 4 relative to some datum; and

splitude A, where i-1,2. The low frequency second order wave force on a

ody in direction k may then be expressed as

(2) 1 2 1 2

kW

A1 H11(11-1) + A2H22(2,-w2)

+A1A2Re [Hi2(i,_2)exp1i((w1- w2)t +

12J]

ere H1j(w_w) is the quadratic transfer function for the force due to

nit amplitude waves of frequencies w and in directions i and j. The

orce is in direction k (k=l,2 or 3 for surge, sway and yaw), but the

ubscript k Is here omitted from the QTF for simplicity. Pinkster2uses

of

H12 1-w2) - 2((P12 _{- °12m)}

(2)

with a. and denoting the

wave directions instead of and a.2. The

advantage of the form given in Equation (1) is

that the QTF thereby defined

is the double Fourier transform of a quadratic impulse

response function,

arising in the Volterra serles representation of the second order

process3'6)

We next use the QTF to describe the force in directional

random

seas. We assume that the waves

are constituted by a sum of N

unidirectional waves, having one sided spectral density functions

of wave

elevation Gaaj(w), for i-1 to N.

The mean drift force in direction k is

then found to be given by

E [fk(t) j -

L1

H

-w)G (w) d;

,aa i

and the single sided drift force spectrum is

Gff(w) -2

lH1(w-P)I2 G

(ri) G

(lw-ui) dp

aal aaj

o

This form in terms of one sided spectra is based cn

the assumption

that the frequencies w relevant to low frequency

drift forcing are much

lover than wave frequencies .i; hence there is no contribution

of any

significance to the integral in Equation

(4) when u<w. One observes that

the drift force spectrum at frequency w is made up of

contributions from

all pairs of wave components whose frequencies sum to w.

This is also observed from the Fourier component

jf drift force at

frequency w, which in spread seas can be written

with the obvious notation that A.(i) is the Fourier transform of the wave

elevation a.(t) in direction i. We nay then take the inverse transform of

Equation (5) to obtain the time history of low frequency drift force in

directional seas when N individual directional components have been

distinguished as time series. The latter may be resolved using beam

forming techniques7.

Arising from Equations (4) and (5) is the concept of using sum and

difference frequencies to identify the dependence of the drift force on the

QTF. Thus it is useful to define

21.w1-w2, Q2=w1+w2. (6)

The second order force at frequency w is then determined by the line

in the bifrequency plane defined by axes w1and w2. It is noted that

in the general formulation the frequencies may be both positive and

negative (c.f. Equation (5)), and in that region of the bifrequency plane

corresponding to slowly varying drift forces, one of these is usually

negative (c.f. the mean drift component in Equation (3)). The frequency

Q2- w of the slowly varing component is then given by the sum of a positive

and negative 01F frequency argument. These ideas are illustrated in the

results below.

2.2 Results for a Tanker.

An extensive set of theoretical and experimental results has been

obtained7'8, from which the following are typical. The experiments were

conducted in the 9m square directional wave basin at Herlot-Watt

University, using a system of mooring lines attached to springs to react

the low frequency drift forces. The natural frequency of the system was

designed to lie between the frequency range of relevance to low frequency

second order forcing and that of the higher wave frequency motions.

Although the low frequency forces transmitted through the springs to force

transducers In each mooring line are then a good approximation to the wave

drift forces on a vessel responding freely at wave frequencies there are

(i) The measured force can

contain substantial force components at wave frequencies;

(il) in random seas the frequency ranges of relevant drifz forces

and wave frequency forces can overlap, leaving no

lnermediate

region in which the resonant frequency can safely lis.

These problems have been minimised by adjusting the natursl

frequency so

that the force component in the mooring lines due to

the resonant response

is minimal; and by filtering the resulting time series to

remove all but

the low frequency signals.

An additional source of complexity In these experiments

is the

estimation of the OTF from the measured

data. It is relatively

straightforward to estimate single points on the QTF

bifequency plane from

tests in regular and biharmonic waves, but a large number of tests

would be

required to define even a small area of the QTF. The alternative

explored

here is to estimate the QTF by the application of bi-spectral analysis

techniques to elevation and force time histories measured

from tests in

random waves. Various methods for estimating the QTF from random wave

tests have been examined in considerable detail by

Sincock9,

from which

the "direct' method was adopted here. This is based on the

relation

* *

LimT

E [F(o1+ e) A1(1)A9(,)1

(7)

E[ A1)

2 _{Et 1A2(e2)}12i

where F and A1 are the finite Fourier transforms of the low frequency

force

and wave records over an intevral T, * designates complex

conjugate, and

the expectation operator E Implies averaging across

different realisatlons.

In practice of course this relation must be transformed In terms of

discrete Fourier transforms of blocks of N points,

at a sampling interval

t. For the following results we used

131,072 data points sampled at intervals of 0.9s, which were split into 256 blocks of N=512 points; the

expected values were then obtained by averaging over

the 256 blocks.

The experiments were performed on a 1:81 scale

model of a tanker

whose properties are given in Table 1: the results, however, are

quoted at

full scale. Figure 2 shows the experimentally estimated QTF

for surge on

the tanker in long crested bow quartering seas. It has been

7

filtering from the force record all frequency components above 0.167 rad/s;

and by smoothing each ordinate with its eight adjacent ordinates in the QTF

plane. Prior to smoothing the function vas set to zero in the region where

the denominator of Equation (7) vas less than 12% of Its peak value. This

is because estimates in regions where the quadratic input is low are

Inevitably unreliable due to the predominance there of the effects of noise

and statistical variability.

It is difficult to compare experimental results in the form of

Figure 2 with corresponding theoretical predictions; but by taking sections

through the QTF surface similarities and differences between theory and

experiment can be readily identified. Figures 3a and 3b show the theoretical and experimentally estimated surge force QTF's in bow

quartering waves, along the sum frequency sections Q20.0 and 0.109 rad/s

respectively. The corresponding sway force QTF's are shown in Figures 4a and 4b. The theoretical results were calculated using the computer program DYHANA, based on a combined finite element/boundary element numerical scheme0which has been extended to permit evaluation of second order

hydrodynamic forces in bichromatic

vaves6,

Two sets of theoretical

results are shown: those corresponding to the freely floating vessel

originally intended to be tested; and results based on using an increased

roll stiffness, which takes some account of the restraint to roll motions

caused by the attachment of the mooring lines above the centre of rotation

(the first order motions in the other five rigid body modes vere not

observed to be affected). The results in Figures 3a and 3b estimated from

the experimental records have not been subjected to frequency smoothing,

but otherwise they have been obtained in the same way as the data

plotted in Figure 2.

By examination of Figures 3 and 4, one can observe rapid changes in

the theoretical OTF's at the difference frequencies which require first

order data at the roll resonances (0.5 rad/s and 0.627 rad/s for the free

and restrained cases respectively). For the sum frequency Sections 2= 0.0

radis, these difference frequencies equal twice the resonant frequencies.

For the sections 02=0.109 rad/s, the corresponding difference frequencies

are Q1= 0.891 rad/s and 1.109 rad/S when the roll resonanse is at 0.5

corresponding to the restrained case roll resonant frequency of 0.627 radis. These irregularities In the theoretical

QTF's, which are not

distinguishable in the experimental estimates, are probably due to

the fact

that viscous roll damping effects were not included

in the calculations.

The roll motions at resonance were consequently overpredicted,

leading to

inaccuracies In the second order forces which use components

at these local

frequencies.

Further details of the experiments, and an extensive commentary

ort

the results, have been given by Mitchell8. Some of the sources of

discrepancy between theory and experiment

can be attributed to the

aforementioned difficulties of measuring

low frequency forces In random

waves; reflections in the wave basin; and the assumptions behind

the theory

of Ideal flow. Futhermore, the effects of the second order velocity

potential were Ignored for the calculations shown

in Figures 3 and 4.

Despite these various problems, the degree of agreement between these

theoretical and experimental results is thought to be most

promising.

3. DRIFT FORCE SPE(,-rRA IN DIRECTIONAL SEAS

Here we present theoretically simulated results

s.nd experimental

data for a vessel in various directional seas.

The vessel is the same

tanker Investigated in the previous section. The objective

of these

results is specifically to shed light on the modelling of low frequency loading in spread seas.

3.1 Crossing seas

Taking N 2 in Equation (4), we have an expression for the

spectrum

of low frequency force in crossing seas comprising a combination of

(different) long - crested random waves from two directions.

One observes

that the low frequency second order excitation

is larger than the sum of

the excitations due to each unidirectional wave taken independently.

The

mean drift force, however, given by Equation (3) with N = 2, is exactly

equal to the sum of the mean forces in each independent

unidirectional

9

The QTF's employed in these expressions can be expensive to obtain.

The effort can be minimised by only evaluating them in the areas of the

bifrequency plane where there is significant input of power. The

contributions from the various terms in Equation (4) depend therefore on

the various products of the wave elevation spectra. If the spectra of the

two unidirectional waves comprising a crossing sea have little overlap on

the frequency axis (eg. In the case of swell from one direction and wind

driven waves from the other), the contribution from interaction between the

two directions will be small In the low sum frequency region of the

bifrequency plane. Under these conditions the low frequency excitation

could be taken as the sum of the drift forces due to the waves approaching

from each direction Independently.

Two other factors can help to limit, the bifrequency region over

which the OTF's are required. If low frequency resonant response is

critical, then a set of sum frequency lines in the bifrequency plane which

span the natural frequency of the moored vessel needs to be considered, but

sum frequencies outside this region can be disregarded. When the frequency

difference between the peaks of the two unidirectional spectra equals the

resonant frequency, the interactive terms (ij) in Equation (4) have their

greatest Influence on forcing at resonance. The second aspect to consider

is that there is little point in evaluating the QTF's for sum frequencies

higher than the lowest frequency of significant wave forcing.

These principles have been applied in the simulation of drift forces

on the tanker in a crossing sea with waves approaching from equal angles of

160 on either side of the bow. The wave components were defined by ISSC

spectra, having significant wave heights / characteristic periods of Srn/12s

from _160, and 6m/lOs from 160 respectively. The QTF's were evaluated over

the grid of points shown In Figure 5. Linear interpolation of real and

Imaginary points of the QTF's was used to obtain intermediate values. The

spectra of surge and sway drift forces on the tanker, evaluated using

Equation (4), are designated closed form solutions in Figure 6. The

continuous lines, designated estimated spectra, were obtained by performing

spectral analysis on 8192 point time series simulations of drift force

(averaging 16 separate estimates and smoothing over 5 adjacent frequency

enerated by colouring white noise, followed by inverse

trarsformation of

quation (5) for the case N - 2.

It may be observed from Figure 6 that there is reasonable

agreement

between the underlying trends of the closed form solutions

and the spectra

estimated from the simulated drift force time series.

The irregularity of

the latter is associated with the statistical

variability of the estimates

based on a limited set of data. The results also

demonstrate the important

contribution made by the

interactive components of Equation (4) in this

case - at some frequencies these constitute up

to 50% of the total low

frequency force spectra in this crossing sea State.

3.2 Fully spread seas

We now consider the behaviour of the tanker in multi-directional

seas where N>2. Our starting point is a continuous

directional spectrum,

which for the theoretical calculations is discretised

into N representative

unidirectional spectra. The number of

TF's required to evaluate the force

spectrum in each mode using Equation (4) is N(N+l)/2 (taking account of the

symmetries of the

QTF's6.

We used N = 9 for the

following theoretical

results. To provide flexibility in simulating

forces due to spread seas

generated in the wave tank, it is convenient to

choose equally spaced

directions, and here we used four directions at

16.80 intervals on either

side of the predominant wave direction (head seas).

Figure 7 shows simulated surge

force results for a series of spread

seas represented by a spreading function of the form cos

2s over

-it/2<8<a12. In each case the point wave spectrum was defined

by the ISSC

formulation with significant wave height 6m and characteristic period

lOs.

It is clear from the figure that over a substantial part

of the frequency

range the low frequency surge drift force increases with increased spreading of the seas. This is related to the

fact, which we have observed

in both our experimental and theoretical data,

that for waves approaching

the vessel from directions other than ahead the magnitudes of the surge

force QTF in some areas of the bifrequency plane are greater

than the

11

Low frequency force spectra vere also estimated from experiments In

spread seas, and compared with simulations in nominally the same wave

conditions. Figure 8 shows the estimated directional spectrum generated in

the tank during one sequence of tests (spread sea A). This was estimated

using the Maximum Likelihood Method (MLM), based on wave elevation time

series from an array of seven probes. In this as in all the experiments the

waves were measured during a separate run without the vessel in the tank:

the pseudo random wave generation signals were repeatable, and by means of

a reference probe could be aligned with the time histories of forces

measured In the separate tests. All records of the wave elevations and

forces consisted of 32,768 data points sampled at an Interval of 0.9s. The

cross power spectral densities required for the MUM analysis wereaveraged

over 64 estImates, and each spectral ordinate was averaged along the

frequency axis with nine adjacent estimates..

Figure 9 shows the estimated surge and sway drift force spectra in

this sea state, and also in two other cases (spread sea B and a

unidirectional sea). The calculated results shown in these figures were

based on the theoretical OTF's, and discretisation of the estimated

directional wave spectra (eg. Figure 8) Into nine directions.

These figures lead to several observations. Very obvious are the

peaks at about 0.1 rad/s, which are close to the lowest calculated standing

wave resonant frequency of the wave basin (0.112 rad/s). The resonant

frequencies of the moored vessel itself In surge and sway were estimated

from measurements to be 0.24 rad/s and 0.28 rad/s respectively, and it is

noticeable how the drift forces rise rapidly towards these frequencies.

For these reasons the spectra In Figure 9 which have been estimated from

the experimental measuremmts are not considered to give a reliable

representation of the drift forces on the freely floating vessel, in the

open sea, above a frequency of about 0.075 rad/s.

It should also be noted that the method of discretising the

directional spectra for the simulations increased the spread of the wave

energy (since the lumping occurs at the mid point between directions rather

than at the centroid of the energy lying between those directions). As a

i larger than that estimated from

the experiments; and (in this case) the

calculated surge spectrum would be appear to correspondingly less.

As a final comment on the comparision of theoretical and

experimental results in spread seas, it is appropriate :o emphasise the

complexity of the various analytical and numerical procedures employed In

the calculations, the difficulty in measuring drift forcas on a 'freely'

floating vessel; and the statistical variability inherent i- the estimation

of parameters from tests in random waves.

4. CONCLUDING REP(ARKS

It has been found that the non-linear interaction of waves from

different directions can have a significant influence on the low frequency

loads in mooring lines, at least for the moored tanker system considered here. In some cases this component constituted up

to 50Z of the total

drift force. Futhermore, it has been observed that

the surge drift forces

(and indeed the surge wave frequency forces) can either increase or

decrease as the spread of the sea state is increased. Hence the calculated

or experimentally measured mooring loads in unidirectional seas should

not

be considered a conservative estimate of the loads occurring In a

real sea

having the same point spectral density as the unidirectional sea.

We have performed a series of experiments in crossing seas and fully

spread seas which have tended to support our theoretical simulations. All

the results point to the significant influence of wave directionality.

The

results presented here, however, (both theoretical and experimental)

highlight the difficulties currently associated with predicting

low

frequency loads in real seas. The further stage of predicting responses has

well known additional difficulties related to the modelling of low frequency damping. Considerable further wouk is required, both on

theoretical and numerical aspects, experimental techniques and methods of data analysis. Only then will there be adequate tools, and

scope for

generating sufficient statistical data, to enable soundly based reliability

ACKNOIJLEDGEHENT

This work formed part of a project conducted by the authors

in

the

Department

of

Mechanical

Engineering, University College London.

It was

supported by the Managed Programme on Floating

Production

Systems

(FPS),

jointly

sponsored

by

industry

and

the Science and Engineering Research

Council through Marine Technology Directorate Ltd.

The

experiments

vere

carried

out

in

the

directional

wave basin at Fleriot-tJatt University in

collaboration

with

another

project

in

the

FE'S Programme,

led

by

Mr.B.T.Linfoot.

His help and cooperation are much appreciated.

REFERENCES

LANGLEY,

R.S. :

'Second

order

frequency domain analysis of moored

vessels', Appl. Ocean Res., vol. 9,

pp. 7-18, 1987.

PINKSTER, J.A.: 'Drift forces in directional seas', NSMB Publication

Z50545, 1985.

DALZELL,

J,F.:'Quadratic

response

to

short

crested

seas',

Proceedings of the 16th Symposium on Naval Hydrodynamics,

Berkeley,

1986.

MAEDA, H., NOROOK.A, CF. and KASAHORA, A.: 'Motions of floating type

offshore

structures

in

directional waves' .

In Proceedings of the

5th Offshore Mechanics and Arctic Engineering Symposium, Tokyo, vol.

1,

pp. 94-101, 1986.

NVOGU, O. and ISAACSON, M.: 'Drift motions of a

floating

barge

in

regular

and random multi-directional waves'.

In Proceedings of the

8th Offshore Mechanics and Arctic Engineering Symposium, The

Hague,

vol.

2,

pp. 441-448, 1989.

EATOCK TAYLOR, R.,HUNG ,S.M. and MITCHELL, KL.:

'Advances

in

the

prediction

of

loQ

frequency

drift

behaviour'.

In

BOSS

'88,

Proceedings of the International Conference on Behaviour of Offshore

Structures,

ed.

T. Moan, N.

Janbu

and O.

Faltinsen,

Tapir

Publishers, Trondheirn, vol.

2, pp 651-666, 1988.

MITCHELL,

K.L., KNOOP, J. and EATOCK TAYLOR, R.; 'Prediction of low

frequency responses in directional seas,

Final

Report

on

project

A2, Managed Programme on Floating Production Systems, 1989.

MITCHELL, K.: 'Slow drift behaviour of floating structures in

multi-directional

seas',

Thesis

submitted

for

the degree of Doctor of

Philosophy in the University of London, University

College

London,

9. SINCOCK, P.: 'Non-linear compliant systems in irregular seas'

Thesis submitted for the degree of Doctor OL Philosophy In the

University of London, University College London, 1989.

lO. EATOCK TAYLOR, R. and ZIETSMAN, J.: 'Hydrodynamic

loading on

multi-component bodies' , In Behaviour of Offshore Structures, Proceedings

of the 3rd BOSS Conference, ed. C. Chryssostomidis and J.J. Connor,

Hemisphere Publishing Corp., Jashington, vol.1, pp. 424-446, 1982.

Table i Particulars of tanker vessel

Scale Loaded displacement Length L Breadth B Draft T Block coefficient C8

Centre of buoyancy above keel

Centre of buoyancy forward of midship

centre of gravity above keel

Radii of gyration Transverse k xx Longitudinal k yy Vertical k zz 1:81 109,000 Tonnes 254.0 mr 38.4 m 13.0 it. 765 6.7 ni 6.6 ni 8.0 ri 13.4 mm 55.8 m 55.8 r E : -80 179820 251784

I

_f

323740 3957 12 Time/Sec b

If!

323748 3957 12 Tm me/Sec

\

VIM

467676 539640

Figure 1 Behaviour of a turret moored tanker in spread seas

a) wave elevation; b) surge response.

.i9i, 41) '(vr

f

_f

50 E °° 50 79820 251784

u1-.., r.th',

a

15

Figure 2 Magnitude of estimated surge force QTF in bow

quartering waves.

r.ds/s&

Figure 3 Theoretical and experimental surge force QTF in bow

quartering waves a) 0.0 rad/s - real part;

b) û2 0.109 rad/s magnitude.

LEGEND Lt GE U D

R..i p.fl QtF OIT

DfltA.N DINGUA ND...

* Dfl4ANA,..Ir*.d?*,..l DUD GD .1 *0

Figure 4 Theoretical and experimental sway force QTF in bow

quartering waves a) 02 0.0 rad/s - real part;

¶

Caa.4 form sOlution l... txstsraCUr. compoa.nta)

Figure 5 Grid of points used for Figure 7 Spectra of surge drift

QTF calculations, force in unidirectional

and spread seas.

e

øoo O Ut û o Oit O Q 023 030 03

rada/sec:

LEGEND .ticuaL.d ap.00ccIm Closed term solution

Clos.d form soiutio (i... IntsrsçtsVs cOpoa.at..)

Figure 6 Spectra of drift forces in crossing seas a) surge; b) sway.

b

t

F

z

0_00 0 02 Q Q 0 10 020 020 030 030 rads/eec LEGEND EoLIrnat.d .p.otrum fused form solution

000 OUt DIO o it Q 20 0 20 030 030 O 'U

rads/sec

02 04 0$ 02

ii rads/sec

LEGEND LEG E N D

PetaLs c.itruint.d ett5 SUSANA O Unidirectional

Moan SnOt Op.r,Lor lin. n Co,20

Co.l0

O CosO

Units of vertice! nuis ni' ,etod0'

t..

CONTOUR 'LOT 0iret,un aegean.

Figure 8

Estimated directional spectrum.

000000m 0050 0 075 0 lOO O LOS O ISO O £75 O 20! rads/sec

O Ua,duect,ann! se. (measured) o Spread cee A (measured) £ Spend see S (measures) O

Uaidor.auoa.A e.. 4c.Auuiat.d)

O Spread en. A (OOlasilaO.d( O Spread n.. B (a.laui.t.d(

a -t, 00000 025 0050 0075 0 lOO O 125 0 50 U liS 020! rmd.s,'aeor

O Spread ce. A (measured)

Spread sen S (measured) Spread axa A cu.Acul.00.d( Spreco ava B (ca.loruilred)

Figure 9

Spectra of drift forces in spread and unidirectional seas

a) surge; b) sway.

b

LEGUND

O