Gaussian wave packets - A new approach to seakeeping tests of ocean structures

June 1985

TECUNISCHE uNrVERsITr BERLIN INSTITUX FUR SCHIFFS- UND MEERESTECRNIK

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Gaussian Wave Packets - A New Approach to Seakeeping-Tests of Ocean Structures

by

Prof. Dr.-Ing. GO.nther F. Clause

and

Dipl.-Ing. Jan Bergmann

Report-No. TUB/ISM 85/9 CONTENTS Abtract Introduction Theoretical background Application of Gaussian Wávê Packets for Seakeeping Analysis

Seaisubmersible RS 35 Articulated Tower Oil skimming system

Conclusions

References

GPUSSIflN WPVE PPCKETS - P New Ppproach To Seakeeping

- Tests Of Ocean Structures

by Gunther F. Ct'auss and Jan Dergmann Institut für Schi-ffs- und Meerestechnik Technische Uni'versltät BerLin

PBSTRPCT

The nsient wave train technique: has been widely

used in testing ocean structures.. The method present-ed here is baspresent-ed n a special Gauss-modulated ampli-tude spectrum. These wave groups of Limited: length can be superimposed,, the actual surface elevation being a'

fi.incti'on of packet characteristics and. initial time

Lag. '

The application of this technique i's demonstrated in three cases presenting a semisubmers'ibLe an articu Lated tower and a floating oil skiosier. It i's shown

that a particular-problem needs a taiLored: wave packet.

containing sufficient energy all over the- relevant' frequency' range. These packets are' designed by

super-imposing indi'vi'dua,L Gaussian wave groups.

The new sealceepi'ng test technique has the following advantages

- the wave train is- weLL defined at any Location of the tank

- the wave elevation and spectrum can be

st;andar-ized or easily adapted to any specific problem - the duration of the test is very short;

ref:l'ac-tions (beach)' do no't interfere the results

- the results show high 'resolUtion and are in good agreement with reguLar wave- test data

Suninarizi'ng, the Gaussian wave packet method is a:

highly versatile technique yielding precise and highly resouked results in short time.

INTRODUCTION

During the Last decades-, offshore technology has be-come one of the most vital fields

of

engineering. P great, variety of new structures - floating,, articulat-ed and' fixed - have been' designed These ocean en-gineering Struclures show a large diversity of shapes

Depending on operational conditions - to out,, 'tran-sit, flooding, survival, etc. - they operate- at many different draughts.

Their analysis requires careful' theoretical and

exper-imental investigations. Ps offshore operations are

supposed to survive severe environmental conditions, appropriate model tests' have to be performed in waves.-

-Consnonly these investigations are carried out in wave

tanks or basins 'using regular waves or Irregular sea spectrai Both methods -are time consuming. With 'reou-lar waves a great number of individual' runs are

peaks o transfer functions are lost. With irreguiar sea tests the statistical anaLysis requires at least

20 minutes to Limit excessive scattering of the

resut Is.

Of course these procedures have to be repeated for

every modification such as different draughts or

orientations of the structure in the wave field. For

these reasons a "quick Look-technique for the

analysis and evaluation of offshore structures is

highly' desirable.

The technique presented in this paper is based on the

generation of short wave trains with an energy spec-trüm covering the relevant frequency range.

THEORETICaL BPCKGROUND

Fundamental investigations using transi'en water waves have been ptihtished by Davis and Zarnick (1). They use the deep water impuLse response function

9f2

wfr).V

cost--T)

as input to the wave 'generator. 'This driving signaL

generates a wave train of increasing wave length. -Due to higher celerity of longer wave components, the wave train concentrates 'and converges to its highest

ampli-tude at distance x from the wave generator. II is

difficult1 however, to -control, the shape of the wave

train as well as the appropriate spectrum. Further-more, the transfer function of the wave generator

in-ttuences the resuLts.

Takezawa et at. (2-] improve theabove method. Starting

-from 'the actual wave spectrum they derived the- input

signal of the wave generator using inverse Fourier

transformation. In this' procedure the transfer func-tion of Linear deep water

waves-(w)

exp (-iaIwIAx/gJ

and the transfer -function of the wave maker

H5tWJ /H(w)Iexp iitw)j

are taken' into consideration. This method allows

ex-cellent control of spectral characteristics.. The

con-trot of the shape of the wave train , however, is

rather poor.

The generation, of a specific wave profile i's the mai'n objective of the method of Funke and Mansard (3]. 'In their method the spectral characteristics are not well

controlled. Furthermore, the use of a zig-zag wave

profile seems 'to be unrealistic.

'Further' applications of the transient wave, technique,

have been developed for the generation of freak waves (see Kjeidsea (4]). ThIs procedure is based on

non-linear theory. Therefore. control of the wave spectrum is not essential.

Gaussian wave 'packets allow an excellent- control of

both shape of the wave train and its spectrum. These specific types of transient waves have first been dis-cussed by Coulson (5]. P fundamental description is presented by Kinsman (6].

-5-The Gaussian wave packet Is composed of an infinite number of superimposed harmonic components., i.e.

faKexp(i(Kw)JdK

-00

with a Gauss-shaped amplitude spectrum

o(kJ

.exp((k-ko)2/2s21

The standard deviation s is the form factor of the

wave packet.

The integral can be solved if the wave treqency

w (k)=. Vgkth(kdt

is expanded into a ayior series

This results in the function, of the Gaussian wave

train profile

ofr.t)oao/9,

expl-f j-,p(x Atl'Jexpfflkox-woS.fr, lx

All')

P typicaL Gaussian wave' group is shown in Fig. 1.. Note

that at s=O the expression degenerates to a simple

harmonic wave.

Of course the above approximation according to the

Taylor series expansion leads to some restrictions re-garding' the form factor s. These problems ,and

Limita-tions are amply discussed by Bergmann (7).

- 'the damping factor

B-.i

_dk'jk.k0- C0

_sh(2ldI

_2shi2kdiV (sh(Zkodl_{2k5d sh(21dI}2k0d )J

with the phase velocity

ç

Th (k0dI

Ps the central wave number kO is related to the dom-inating wave period TO, i.e.

2"

To

-C0K0 }lgK0lh1K0d'

the Gaussian wave packet will be characterized by this period TO, the water depth d and the form factor. s.

-6-The above expression of the Gaussian wave train con-tains Imaginary' components even In the amplitude term. For the discussion of 'the propagation characteristics

of' such packets it is necessary to separate the

com-plex function Into It real and imaginary parts.

Lengthy algebraic operations (see Bergmann (7)) resuL.t

in

{a(x;t)J

Oof.,g2

exp1--

f.5 22

ix Al)2)

.C('K0X_W0(, otg( s2Bt

l.sB2I2

tx Al)')

This. shows thal the Gaussian 'wave packet is defined with three. terms which depend on the. parameters of the Taylor series expansion

- 'the group velocity

AJ-1

=.2(t. 2k0d /

dkIk=ko 2 sh(2kod)

The damping term

has its maximum (Xi=i) at the concentration point tO. The amplitudes increase up to this- Location as the Long wave components overtake the shorter ones. Beyond the culmination point the amplitudes decrease again as theLong waves are now driving away due to their

higher celerity. With increasing values of the form factor s the damping term deteriorates rapidly with time. Re the forth .fac.tor s is also the standard devia-lion of the ampLitude spectrum, the total wave energy is distributed over a larger frequency range. Thus a large form factor implies a wide spectrum with many wave components of different Lengths resulting in a quick dispersion of the packet (see Fig. 2). To

dis-cuss the damping term Xl in detail, the Influence of

the dominating, period and the water depth on the

second significant parameter B the damping factor -has to be anaiyzed Fig. 3 shows the rapid decline of the damping factor B to high negative values as water depth and wave period increase. From Fig. 4 it can be

seen that these effects influence the damping term

significantly. Low dominating periods result in small damping terms at all water depths. With Larger

periods, i.e. Longer waves., the damping term

in-creases, predominantLy in deeper waters.

X2 exp

1-f

_.

ix At)'J

yields the envelope of the Gaussian wave packet, which Is propagating with the group velocity P. The modula-tion funcmodula-tion governs the shape of the packet and has the characteristic of a distorted Gauss-distribution. Its skewness and variance in the time domain depend on form factor s, group velocity R, damping factor :B as welt as on time and Location respectively. These

ef-fects- are also demonstrated in Flgs 2 and 4, showing

the -envelopes of 6ausian wave packets atvariaus

po-sitions.

The expression of the Gaussian wave packet is

complet-ed by multiplying the envelope function with the

as-cill-ating term

X3

{Kox-wol.

olg(;s2Bt

_{+..- 1.5Z2}

(x A2}

which shows a characteristic phase shift compared with

a simple harmonic wave of the same frequerky WO. The phase shiift has two components. The arctg-term is..

nearly constant as it fades away rapidly from zero at taO to sign(t)Mn/4.. The second term decreases at the

maximum of the modulation function. Thus the -wave com-ponent which coincides with the maximum of the.

modula-tion function is characterized by a wave period close to the dominant wave period.

The three terms of the Gaussian wave packet are now

MuLtipLying the damping term wilh the moduLation

lo

-Remarkable results are obtained if wave

packets

with

different dominant periods are generated. Fig. 8 shows

a 0.8-second: wave train followed by a fas:ter

2-second

wave, packet. Pt 'the culmination posi'ti'on x=0 both wave

groups coincide and penetrate each other. Pt L'a:ter

p0-sitions

the long wave group has clearly overtaken the

short one. Both 'theoretically as well as

experimental-ly

the

Fourier spectra of each individual wave group

are clearLy identifiedeven at the

culmination

posi'-lion.

Ps indi.vidua,l wave groups act independently, they

can

be

arbitrarily combined for the design of wide energy

spectra. The wave train 'in Fig.

Is composed of eight

individual

wave packets. Superposition results In one

big

freak' wave

at

the

culmination

position x0.

Pgain

theoretical and experimental results correspond

quite welt.

PFPLICPTION OF GPUSSIPN WPVE PPCKETS FOR SEPKEEPING

PNPLY5I5

flf Icr presenting the theoretical background, the

tot-Lowing

examples

discuss

the. application of Gaussian:

wave packets for .seakeepi'ng analyses..

Varibus

struc-tures have been investigated in transient wave 'trains'..

The results are compared

with

regular' wave

experi-ments.

In the lot lowing sections the

Gaussian wave

packets

are

characterized

by

the amplitude function aCt) at

various positions

and by

the

accompanying

Fourier

spectra Fa(w). i.e

F0(w) :

f'a(t).exp(_iwt) dl

The spectra. caiculated,uslngFast Fourier

Transforma-tion:

techniques

(FFfl, are supposed to be. invariant.

The area under the Fourier spectrum yields the maximum

amplitude

of the Gaussian wave packet at xt0, which

is also a dIrect measure of the total

energy

of

the

wave train.

Ps discussed earlier, the

generation

of

the

actual

wave

groups requires modification of the input signal.

using the transfer function

of

the

wave

generator.

Fig.

S

shows a sketch a

the wave tank. Fig. S shows

some typical time histories, I.e.. the electrical input

signal of the wave generator and its response, the

mo-tion of the wave plate and the registramo-tion of the

wa-ter surface elevation (middLe).

The

combination

of

electrical/mechanical and hydrociynamic transfer

func-tions

yields

the

complete

transfer function of the

wave generator, including magnitude

and phase

(bot-tom).

These relationships have been determined by

us-ing superimposed Gaussian, wave

packets

and

regular

waves.

Ps an Illustration of propagating wave groups

Fig.

7

shows samples

of

a wave packet with dominant period

T02.O s obtained theoretically (top)

and

experimen-ta1iy

(bottom).. Note the converging and diverging

in-terim packet variations as well

as the

Gauss-shaped

envelope

at

the

culminationpoint x0. Pt alt

päsi-lions the Fourier spectrum is invariant

according

to

Its

shape

and area. The. agreement between theory and

Semi'submersibLe RS 35

The RS 35 is a 37.000 ton' deepwater drilling and

pro-duction system with a variabLe Load capacity' of nearly

15.000 tons (Fig. 10). The basic construction - a

toroidät double-wall hut.l with four columns carrying an integrated' 'deck of modular design - shows exceLlent

motion characteristics and a high safety standard.

Sealceeping tests have been performed at a scale of

1:53. Fig. 11' hows the time histories of a typical

experiment using' Gaussian wave packets. The two top 'registrations present records of the wave train 'in

front and on the beam of the semisubmersible. The

fol-Lowing records show heave,' pitch and surge of the

structure. Note the .pers'isltant resonance heave and

pitch motions. Fig.11 'presents ' also the', related

Fourier spectra. The resulting transfer functions

(maghitude and phase) are plotted in Fig. 12. For corn-parison., regular wave data are also' shown in alt

di-agrams . The high resolution and precision, of the

resuLts obtained by the Gaussian wave technique is' re-'

markabl'e, and this''me'thod can be recommended as a quick Look technique..

01 course, the technique does' not depend on specific

characteristics of the wave train itself. This Is

shown' 'in the next diagrams . 'Fig. 13 presents resuLts

of seakeepi'ng tests with the semisubmersible being: pa-sitioned at three different Locations.. The same drIv ing. s:ignal of the wave generator and consequently the same wave train was used in alt three tests. With the

first ,run the semi:submerslble is positioned i'n the converging range of the wave train (left columun'). fluring the second run the structure 'was Located at the concentration point (middle column) and: with the third

run the platform is exposed to the wave train at its

diverging range (right column'). The Upper set of

records shows the time his'tories' of the wave train in front and' on. beam of the semisubmersible as we'll as

12

-its heave, pitch and surge response's. The middle set of plots represents the five Fourier spectra of (he above records, respectively. Finally, the lower set of diagrams shows the heave, pitch and surge transfer functions compared with. regular wave test data. Gen-erally it can be stated that the agreement of the

,di'f-ferent test results is excellent. However, it is

recomended to test structures i'n the converging phase of Gaussian wave ,pacet's.

In the foilowing example wave packets' of limited width are used for seakeeping tests. Fig. 14shows different Gaussian wave trains and their effects on heave, pitch

and surge 'motions of the semisubmersible RS 35. The first four packets are characterized by the dominant periods

T01.0

5; 1.5 s,; 1.9 5; 3.0 s. The Last wave train is a superposition o'f the above wave packets.

Fig. 15 shows the Fourier spectra of the five di:f_ ferent wave trains and the subsequent heave: motion spectra. Fig. 16 presents the results 'accordi'ng to the wide variety of wave spectra, i.e. the heave., pitch and surge transfer functions. Of course, reliable data can only be expected if the frequency range contains significant wave energy. Thus, any of the above wave trains: yields segments of the entire transfer func-tions. The superposition of all wave packets,,

howev-er, results into the most 'accurate and resolved

13

-Prticutated tower

Similar experience, has been gained with experiments of

an articulated tower structure Fig,. 17 shows a tower which has been investigated at a scale of 1:100. The

wave train - a combination of individual Gaussian wave packets - is plotted at two positions in front and on

the beam of the structure. (Fig. 18). In this test horizontal force and pitch angle of the oscillating motion has been measured. Fig. 13 ShOws corresponding transfer functions in compari'sàn. with regular wave

data.

Pgain these experiments, require wave packets with an energy dis;tr'ibuti'on 'covering the relevant range of transfer function. For narrow spectra' transfer

func-tions at the low and high frequency edges show consid-erable scatter as the response spect:rurn is related to

small and dubious values of the wave spectrum. For

these reasons a particular problem needs a tailored. wave train combining several standard Gaussian wave groups;. Of course1 if there. is enough energy alL, over the relevant frequency range, the. transfer functions do not depend on the. type of the wave train used.

Oil ski'minq system

High data density over a sufficiently large frequency range Is extremly important when testing complicated

multi-body structures.. This; wiLl be., exemplified for an;

oil skimmer with movable floater/flap systems.. Fig. 20 shows the skimmer, which is a self floating U-shaped vessel with a buoyant tank system On both sidee and at i;ts aft end. By controlled ballasti'ng of the particu-lar compartments, the draft and trim of the unit can be adjusted. Between the sidewal'ls are two horizontal flaps hinged to the bottom structure. Both flaps are. combined with buoyant side tanks which keep the lead-ing edge just below the water surface. . In the seaway

these floater/flap systems follow the wave profile and

14

-undercut the water surface in a welt defined manner. The front separation hap, which takes the heavy wave

load first., is backed up with a compensating wing

which counteracts the severe .flap rotation and

smoothes its motion.

flfter passing the front flap system, the separated

oil/water mixture proceeds to the rear part of the

vessel. There the rear seperati'On flap skims ofi the

oil and transfers it to the oil sump. The accompanying water isdiverted downwards to a duct. From the oil

sump the collected sludge is pumped into settling

tanks on board of a ship or a pontoon. Fig. 20 illuS-trates the dynamic behaviour of the oi:l skimmer and the integrated fLap seperation system and shows the

relevant motions i waves. The eflicient operation of the oil skimmer is based on the controlled

undercut-ting of the fluid surface. With long waves the pon-toon Itself follows the wave profile.. With shorter wives the floater/flap systems react very quickly to

any variation of the oil/water surface,..

This complicated system has been tested at a model

scale of 1:5. Fig. 21 shows the wave train and two as-sociated response functions, i.e. the pitch motion, of

the skimmer and' the angle of the front flap system. One 'single SO-second test yields the highly ,resoluted

transfer functions of Fig. 22. Compared with 'data of

regular wave tests it is evident that the Gaussian

wave packe.t method' results in much more detailed in-formation at Less expense.

CONCLU5 IONS

Detailed investigations with Gaussian 'wave packets have proved the high versatility of this technique in the seakeeping analysis. Ps this procedure is based on

linear theory any superpositions of wave groups are

applicable. Thus, tailored wave spectra can be

designed for specific experiments providing sufficient energy aLL over the relevant frequency range. In this

paper it is demonstrated that the Gaussian wave train method yields accurate and highly resolved resUlts in

seakeepi'ng tests. Ps the required duration of the

ex-periments can be reduced drastically by using wave

packets near the concentration poin.t this, technique may also be applicable in maneuvering tests.

In the future further programs are. considered to In-' vestgate the statistical characteristics of groups of wave packets. This ?roceduce allows the. presel'ect'ion.

of group factors and may lead to a better unders:tand-ing of the damage behaviour of ocean structures..

REFERENCES

(1] 'Davis,, M.C., Zarnick, E.E.:

Testing Ship Models in Transient Waves

5th Symp.. on Naval Prchi:tec.ture, 1964

Takezawa, S. et al.:

fldvanced Experiment Technique for Testing Ship Models in Transient Water Waves, part 1 and 2 11th Symp. on Naval Hydrodynamics,, 1976

Mansard, E'.PcO., Funke, E.R'..:,

P New Ppproch to Transient Wave Generation Proc. Coast. Eng. Conf. 1982

KjeLdsen, S.P.:

2- and 3-dimensional Deterministic Freak Waves Proc. Coastal Eng. 'Conf. 1982

(5'] Coutson, CP.:

Wave5, a Mathematical Pccount 'of the Consnoh

Types.of Wave Motion

Oliver and Boyd, Intersc. PubI. Inc., N.Y.. 1949

(6] Kinsman, B.:

Wind Waves, their Generation and Propagation

on, the Ocean Surface.

Prentice-Hall Inc., Englewood Cliffs, N.J

1965

(7]: _{Bergmann, J.:}

Gauss'sche Wellenpakete - Em Verfahren zur

Pnalyse des Seegangsverhaltens' meeres'tech-'ni'scher Konst rukt ionen

Ph.-D.-thesis, to 'be published at the Ihsti-lute of Naval flrchitecture and Ocean Engineer-ing., Technical' University Berlin in 1985

PO(NOWLEDGEMENTS

This 'paper presents some results of a research.

proj'ect funded by the Federal Deartment of Research and Development (BMFT).. The authors wish to express their gratitude for the. gen-erous support. .They are especially indepted to

Dr. C. ös'tergaard and Dr. T. Schell'in of Ger-manischer Lloyd, Hamburg who stimuLated many valuable discussions and 'helped to 'complete

the manuscript. Thanks also to. Di'. H.

Kalden-hoff and his colleagues 'at the Institute of

Civil Engineering:, Technical University of

Berlin for the support of' this project.

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0-E

l.a

0.5

a

1.

0

O.0O

l.00

0.50

0.00

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x:Om

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_;5:2Z

tx-At)2)

cos1Kor-o,f 4.;

atg(-s28f)

+

1.sB1

tx-At)2)

Fig. 1:

Definition of the Gaussian wave packet

-20

20

60

80

Fig. 2:

Damping term, rnoduL.ation function and

the resulting wave packet. envelope at

various positions

- influence of the form factors

4.. 44

E

U

0.00

4 44

U

1.00

E

0.50

b

0.00

i.00

0.50

0.00

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0.50

0.00

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Group velocity and damping factor as

function of dominating perLod TO and

water depth d

- --

₂

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Fig. 4:

Damping term, modulation function and

the resulting wave packet envelope at

various positions

- influence of the depth dependent

Fig. S:

Wave tank with double hinged plate and beach

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input signol

Fig. 6:

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Input signal and the resutting wave

- the effect of

transfer functions

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Fig. 7:

Propagation of Gaussian wave packets aid

their related Fourier spectra

- comparison of theory and experiment.

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Fig. 8:

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Fig. 10:

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- main dimensions

0 -0.00 2.LZ

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time histories

0

ax

300 _I800 aoo Coo aoo COO

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Motion and motion spectra cii

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C

;-

-0.00 -. 0 Is C -'.97

as an U 1..iaI as aDa I p S -U, .214 as - as a', U 1.dIiS

Is

'N as p N 44.4.-U .1.4 as as

Fig. 12:

Transfer functions of the seaway motions of the Semisubme-rsible

-RS 35

-comparison of

-Gaussian wave

packet data and reguLar

wave results

an U as

-Ias,.S .4-

0

C 0 U C I- ClS. 141 Zn as an U 4-. o C 0 .4- L Cl.4- .141 C .0

-

as

-:.N4_; Iflur

:

..

-C

0 1.

Ii.

S 0 0 E

I.,

m Cl .0 Cl

..

U.

Cl

x Gaussian wave packet

regular waves

1 U N as as as

S

0

8 S

I

a

I

I

pitch

n,,_ i.#.i

11g

I'aj

IjI

t

I

1'

I

a a.

I

(

S I, S § S 8 a

.4

-a.

LJ

.

_1,.

Yf

/

/

S

a S

th

uIL

_liii.

S 555

- a.

- a.a a,aj i,wg.s

I#'i-i C/U1

4. 0 0

-'

U S

b I

Fig. la:

Seakeeping tests of the Serni;submers.ibt:e R5 35.

t three

'different locations

a.

a.

a. aIa -- _.

a. aaa

U

a. aaiaa

--U

a. ma

a

-- a

aa

--a. a

a.

aa a

aaai

aa

1

U

005 050 005 _.050 050 ow _.020 T0sWs .1J 050 ₀₀₀ 000 050 000 _-050 aIs 0'5 020 000 T0a1.5s(+) 0 _-05o

4IL

I

050 000 Ta3.OS l) 000

Fig,. 14:

Time histories of seakeeping tests with theSernisubmersibte RS 35

using individual and superimposed wave packets

.2 I 3.70' 000

_-"a

000

4

0 .020 050 020 060 0.05 000 .020 060 0 250 0 250 0 250 time t

(sI

.025

if!IIP1I

L1iI

750 7.65 000 _.165

III1MM

2.25 0 250

E

I-3

21.00

)- To=3.Os

A superposit

iou

o.o

0.00

1.00

(rodsI

(rod/si

Fig. 15:

Fourier spectra of various wave packets

and related heave response

V

I

- 050 0 S S. 000 1.00 LX

a

Y

050 41

trod/si

''

Irod/si

Fig. 16:

Transfer

functions

of the Semisubmersibte

R5 35

- results from seakeeping tests using wave packets

of Limited frequency range

--- > - --X !WS 4. T0.1.St

Y T.'.9.

>- T0.3Os A rpo. U r.9443r0oY.I -->. -

-050 I.00 41

trod/SI

0.00 0.50 1.00

c .167 aQO -387 5.9, aoo -5.9, 97

vrlicol fore.

horizontrl force pitch angle

of oscillation

Fig. 17:

Prticulated tower

d. 150 ii.

1

p

s is,

Fig. 18:

Time histories of a seakeeing test with

I

'3

050 U abl

a"

Mn V IOU/il paw

£

£

Gousslan

regular

.(dilferent

wave packet

_waves

heights)

-'zoo 1.00 a

_*

a N

:f-1

*00 £ £

x

Gaussian

regular

(different

wave waves

heights)

packet

100 000 a a N NM

--.

-

-a"

;.

I

.ISS

a,,

Lb as,

an

Ii/iI

Fig. 19:

Transfer functions of an articulated tower

comparison of Gaussian

wave packet data and regular wave results

0

}

a

Fig. 20:

ERNO-oil skimmer - main dimensions and

mat-ion behaviour in waves

00.0

50

--

-03:

Fig. 21:

Time histories of .seakeepng tests with

an oil skimming system

r--r

z,.

70.0

z

0

0

30

'C 'C 'C

-- x

Gaussian wove

.

regular wove pocket

pocket

'C

x

'C

a

'C )( 11 )C'c

x

'C o 20 o I_fl

-

x

'C 10 0 2.00 3.00 .00

W Ira dtsl

:00

2.00 3.00

W Irod/si

Fig. 22:

Transfer functions a

an oil skimming system

comparison of

Gaussian wave

packet

data

with regular wave results

0 0 E U 0. 0 51