non-linear response of moored floating structures in random waves and its stochastic analysis - Part 2. Comparison between simulations and statistical predictions and a full scale measured data

Nonlinear Response of

_{Moored Floating Structures}

in Random Waves and its

_{Stochastic Analysis}

Part 2 Comparison between Simulations

_{and Statistical Ptedictions and}

a Full Scale Measured Data

Shunji KATO 1

Masakatsu SAlTO 2

Satoru TAKASE

January 26, 1993

'Ship Research Institute

2SlLip Research Institute 3Hirosliinìa University TECHNISCHE UNIVERSITEIT Laboratoriwn voor Scheepshydromechafllca Arctdef Mekelweg 2,2628 CD Deift TeL 015-786373' Fa3c015-781838

Abstract

This paper is concerned vit1i a siirnilatjon an(1 statistical prediction of total second _order responses, iiicludiug slow dtift; rnoions caused by waves and wiud, of a full scale floating ofEsliore structure "POSEIDON". The at-sea experiment was carried out from June, 1986

to July, 1990 at the Japan Sea. Firstly, in order_{tó study the hydrodyiiainjc and restoring}

force tharacterjstics from full scale free_{decaying test data, a nw ana1ysis method. which is} based Oil time series fitting method using _{a notilinear optimization technique, is developed.}

Sccoii.Ily, in OtdCr to investigae Uie SCCOII(.1 _{order force chatactiistics and the contril)ution}

of wind fluctuations to slow drift motions, Not only the cross bi-spcctral analysis _{of inot.iou} aud .vaves but also tile multi-input analysis of motion, waves, intantaneoiis _{wave power} and wind fluctuatjoiis is carried out. Furthermore, with. respect to the seëond orck'rforces,

a comparison between analyzed results and uumeric _{ones calculated from the potential}

theory is Ifla(lC.

On a basis of these investigations, a comparison 1)ctwecu a measured time history of slow

drift motion and simulations has been carried out.

Relating to statistical estimates of time PDF(Probability (lemisity function) and the

ex-treme response, a new prediction method is developed to take account of both second order wave forces and varying wind loads.

At-sea measured sample data, the statistical prediction based on the Rayleigh.

distribu-tion, i.e. the so-called Cartwrzght-Longv.et-Hjgqi7m.9'_{estimates amicI the results Obtained from}

the present method are compared. Main results arc as follows:

The 1)res(11t method analyzing free _{decaying (lata is effective to get the drag coefficient}

depending on the K-C number ( Kcuiegun-Carpenter number). In this case, the drag

coefficient of surge and _{sway motions can be described as sum of a constant term and} a 1<-C (1e1)endeut term, which is inversely proportional _{to the K-C number. Amid the}

constant term of the full-scale structure is _{as saine as that of model while the K-C} (lcl)endcnt term of the füll-scale structure is larger than that of modäl.

In. order to simulate slow drift motions, not only in-line wind fluctuations bitt also

transverse OIICS should lc taken into account even though the rucan wind direction

is head. As a wind spectriun _{representing wind fluctuations, a spectrum form with}

significante low frequency _{power compared with the well-known Davenport and Hino}

3) The I)robablliy distribution estimated by the presnt method agree very well with the

measiucd one And as foi the estimation of cxtrcmc response, it is confirmed that the Cartwright-Longuct-Higgins' estimate sIgnificantly underestimates the measured results while the estimateS by the present method overestimates slightly them

Chapter 1

Introduction

Diuing tuìe past (ICCtde or so, a fairly large ulluhll)er of papers have I)CCfl reported relating

t() slow (111ff foices and iesponses of offshoie_{stnictuies The great inajoiity of these papers}

have l)CCII devoted to theoretical and model experimental researdies. Ï'lie paper dealing with the analysis results of atsea experimental _{(lata is few.}

VVc carried out at-sea exi)erinlellt using a prototy1 _{floating iílatform " POSEIDON"} from September, 198G to .July, 199O. One _{of the most important rescatch themes in this} project _{as a study of Predicting maximum excursions of slow drift motions and móoring}

) 1('PS.

This paper presents the analysis _{results of at-sea ('xperijiICIital (lata.}

In order to investigate the hydrodynamic and restoring force characteristics of the struc-titres, full scale free decaying tests have been carried _{out. For analyzing this data, a new} uietlio.l, which is based on time series fitting method using a nQnhinear optimization tech-muque, is developed. By using this mnethöd, the hydrodynamic _{coefficients, especially the}

(.lrag coefficient and the s.tiffucss coefficients _{of full-scale structure. have been found out.}

Furthermore the scale effect of drag coefficient is stu(Iied through a comparison between model test results and the analysis ones of at-sea data.

For investigating the second order force _{characteristics and the coitribution of wind}

fluctuations to slow drift motions of the fuir-scale structure, both the cross bi-spectral analysis of motion and waves and the multi-input analysis of motion, _{waves, instautaneoùs} wave power and wind fluctuations has been carried out. With respect to the second order forces, a comparison between analyzed _{results and numerical ones calculated from the} potential tlieoiy is also done.

On the basis of these investigations, we coiuipare a measured time history of slow drift

motioui with sunulatious.

Relating to statistical estimates of tite PDF(Probal)ility density function) and the

ex-treme response, a new prediction method is developed to take account of both second order

At-sea nicasured sample data, the statistictl l)re(liction based ou the Rayleigh

thstri-bution, i.e. the so-called Cartwright-Longuet-Higgins' estimates and the. results obtained from the present method are compared.

The conclusions are summarized as follows.

The present method developed newly to analyze free decaying data is effective to get the drag coefficient depending on the K-C number ( Keulegan-Carpenter number ₎

Jim this case, the drag coefficient of surge and sway motions can be described as simm

of a constant term and a K-C dependent term, which is inversely proportional to the K-C number. And the constant terni of the fnil-scale structure is as saine as that of

model while the K-C dependent term of the full-scale structure is larger than that. of

model.

In order to simulat;e slow drift motions, not only in-line wind fluctuations but also transverse ones should be taken into accountcvcu though the mean wind direction is head. As a wind spectrum re)reseIiting wind fluctuations, a spectrum fornì with significant low frejuency power conipared itIi the well-known Davenport and Humo

spectra, e.g. spectrum forms suggeste(l by Oclii-Shin and the uthpr, should be used.

The Pob

the Cartwright-Longiiet-Higgins' estimate significantly underestimates the measured results while the estimates by the present method overestimates slightly them.

Chapter 2

Qutline of At-sea Experiment

Concerning the outline of this experiment, _{we have reported to a number of papers. The} details aie referred to other papers, e.g. S. Ohfnatsu et a1'. In this paper, we shall describe

it 1)riefly

2.1

_{Floating test structure "POSEIDON"}

The test structure used is named "POSEIDON", whicli means Platform for Oceaíi Space

Exploitation. Its structiue are shown in Fig.2;1. It consists of twelve legs with footings

which support the uppei structure, which is mainly composed of the boc type giideis around four sides. The iiistrumentation house is ananged on the upper deck for power

supply and data acquisition.

2.2

Location, oftest area

The location of test

PicÑ dure, soiitliwestcrn part of the Tapan Sea as shown in Fig 2 2 The watei depth is about 41 in. We have constantly _{severe sea conditiofis duc to the strong seasonal wind} in winter. Tue POSEIDON wa.s coiitructed at Naruto, Tokushinia Prefecture. _{After an}

inclining and free decaying test were carried out there, it was towed by a tag boat from

Naruto port to Yura port, about 800 miles over 9 clays, and installed in the test area of

Yura port in July, 1986.

The POSEIDON was shckly mooied by six chain hues as shown in Fig 2 3 The foiward (lirectiou was WNW, whicli corresponds to the dominauit direction of seasonal wind and

2.3

Measurements

The POSEIDON has many measuring items. They are shown in Table 2.1. Now, we shall explain briefly the main measuicIncnts in them, i.e. wave, wind, wind pressure, slow drift motion measurements.

2.3.1

Wave measurement

Three ultrasonic wave probes _{were installed as a line array on the bottom of 180 m offshore}

of the POSEIDON (see Fig.2.3). They _{were used to estimate directional, wave spectra.}

And data measured l)y them are used as a magiutude iiiforrnation of unidirectional incident waves required in the following sinuilatiou. At the saine time, we also measured a relative wave level and vertical accelcratjoii at tue centre column of offshore side. Their dataare

used a.s a phase information for sirnulating unidirectional incident _waves.

2.3.2

_{Wind and Wind pressure measùrements}

An ultrasonic type three axes anemometer _{was installed at tue top mast, which is 19.m} height. above sea surface. _{It was used to ineasüre mean and fluctuating wind velocities} in three directions (surge, sway and heave motion direction). _{The response time of this}

anemometer is 0.5 Hz.

-Wind l)reSS.UCS were measured by using difference _{pressure sensors installed on fore &} aft side an(,l starboard & port sidle of_{upper structure (see Fig.24). Each pressure holes are}

led to two semiconductor type difference pressure sensors through tubes.

Concerning mean wind pressures, Fig. 2.5 compares the measured result with the model

test result.

_{Both results are in good agreeinnt. It is found from this figure that the}

pressure coefficient C, is about 1.3. Figure 2.6 indicates a comparison between a measured

\vin(I pressure time history -in surge direction inn.1 the 0110 cst.iiiiatcd froni wmd fluctuation

in surge direction by. means of the following relation

j3 + p

=

pC(U+u 2

Î; .=

p -

pCUu

where p', is the air density, U is the measured mean wind velocity in surge direction. Froni this figure it is found that fluctuating wind pressure and wind velocity have

2.3.3

_{Measurement of slow drift motion}

A new system to measure a slow drift motion at sea was developed after detail preliminary

investigations. It consists of two ultrasonic type tran.smittcrs and three receivers. The

transmitters were installed ou the footings of POSEIDON and receivers were donc ou the sea floor of 41 m water depth as shown in Fig.2.7. Since distances between transmitters and receivers arc measured successively, by using the triangle quadrature method we can

ol)tain tune series of six mode motions. And at, the same time first order motions were measured by three axes servo-type accelerometers and vertical and directional gyroscopes settled ou the centre öf POSEIDON.

Figure 2.8 shows ann example of xnea.sILred surge inutioil time histoiy an,l a (:omparison

1)etweeu its filtered one, raw data of which were measured by this new measuring system,

and the one measured by the servo type horizontal acceicrômeter, which device has a

fiunction removing the angular motion effects. A cut-off frequency used for high_{i)as filter}

was 1/25 Hz. It is found that this new measuring systm has a good accuracy.

2.4

_{Data Acquisition System}

Forty eight items are automatically recorded by a personal computer at every six hours. Que record til3.ie was 34 minutes arid 8 seconds. The sampling interval was 0.5 sec. thus 409G data arc recorded for each items.

In or(ler to record a long duratiomi data, another recording system is aLso used. The

system is that the long duration measurement can be st.arte(I by a cou.imand of telemeter

system. In thi case, the sampling interval was 1.0 sec. The recorded data were store on l)oar(I into the hard. disc of 40 MB.

Chapter 3

Measured results

Figure 3.1 shows time histories of wind velocity, wind J)rcssurc difference and sway motion.

Enviroïimcutaj conditions of this time were that. mcaii wind velocity was 13.8 in/sec, mçan wind direction was 104° in the direction from Fore to Port, significant wave ìeight was 0.3

in, and current speed was less than 0.5 rn/sec.

This situation is a typical case that wind is dominant compared îitli waves and current. From this figure it is fotuid that there exists a very long fluctuation component in sway motioii t,illL(' lustory. And it secijis l;hat the time lustory of such flu1(:tuatjon agrees WPII

with one of wind velocity fluctuation. To _{confirm this fact, a simple coherency analysis was} carried out. Figure 3.2 iiidicates _{au example of ccherency analysis results. As for frequency}

components lower than 0.01 Hz, wind _{pressure fluctuation and slow drift motion of svay lie}

in a linear correlation and the phase of frequency ransfet function between them is about

O degree in such frequency _{range., i.e. the relation between wiid prcssnrc and slow drift} motion of sway has one-to-one correspondence.

3.1

_{Slow drift motion due to wind}

_{and waves}

As a data assumed that _{waves are stationary in statistical sense, a long duration data (9}

hour successive data) shown in Fig.3.3 were used and analyzed to study how much wind and waves contribute to the slow drift motion. Mean wind and mean wave directions were

about head as found from Fig.3.3. Mean values of significant wave height and siguificant nican l)eriod arc 3() in and 7.0 sec. respectively. Figure 3.4 shows spectra of _{waves, surge} aiid sway motions. As a spectral analysis, t11( AR(Auto Regression) model method was

used andI the order of AR model _{was determined on the basis of least Akaike Informnatjoii} Criteria principle.

There exist three peaks in the surge spectrum of Fig.3.4. The first one lies in the vicinity of wave frequency 0.12 Hz the second one does in 0.07 Hz and the last one does in 0.0156

Hz (64 seconds in period). it seems that the last one is not recogiiied in the sway spectrmn. The first spectrum peak in the surge and sway spectra is (lue to waves and the second _one is (lue to coupling motions since they agree with the nttura1 frequencies of roll and pitch

motions.

Figure 3.5 shows time histories of surge, sway and wind pressure differehces. The sway

motion time history and wind prese difference _{one in sway direction are in good}

agree-ment 1)llt the surge motion time history does not accord with the wind _{ptessure difference} one in surge dection. And Fig.3.G shows a sim1)le coherency bctwcen sway motion and sway direction wind pressure differcuce. _{It is fouiid that the slow drift motion ofsway}

OCCUrS CVCII if the iiiean wind velocity direction ics in the _{surge direction and it is caused}

J)y sway direction wind pressure difference, i.e. _{transverse wind fluctuation. Figure 3.7}

shows the wind spectra in surge direction (mean wind velocity direction) and sway di

rection (transverse direction). From this figure, wind spectrum iñ sway direction lias the saine power as one of surge direction. Spectrum peák frequencies of non-dimensional _wind

Sl)('(tril. in i )OtJl (lire(tions shift tOW1r(l low freqitcucy Si(le coznl)are(l with _{oiies of}

represen-tative...spectra, eg. Davenport and Hino spectra, representing wind pcctra over land. And measured spectra are close to a spectrum form suggested by the author. Moreover, the fact that .rjiid spectrum over sea has more significant power in low frequency than that over

laIl(l Si(IC has recently been measured in the wcirl(1. Ochi _{and Shin2 have suggested a new} spectrum ou the 1)aSiS of the meaii Value of main wim.l _{spectra mcasnred over sea surface.}

Aii(.1 the authors 3) have also suggested a new wjnd Spectrum form based on both sonic

1)hysical investigations and wind _{sI)ectra measured at the .Japan Sea. Figure 3.8 shows a}

comparison of Win(1 spectra I)CtWeCfl Och and Shin's fòtnu1a, the _{authors' one and the} other spectrum forms under a representative strong wind, condition. The object data for this coliq)arjsoll is a one hour data which the _{mucan wind velocity was 22.208 rn/sec. Iii this}

figure, heavy solid line indicates the measured spectrum, heavy l)roken line is the estimated 011e b the author, thin lines _{are the representative spectrum forms, cited by Oclii et al.,}

which have ieen proposed by other authors. This figure shows that the measured spectrum agree ver _{well with the estimated .oiic and it is close to the Ochi-Shiu's spectrum in the}

o tuer J)t01)OSC(l spec trum forms.

In this way, wind _{S1)ectrluhl over sea is (lifferemit from one over buid and the former has} a significant low frequency power compared with flic latter. And even if the mean wind velocity direction is tI'ie direction of surge, wind spectrum in sway direction is significant and it has the saine power as one in surge direction.

Furthermore, we should note that the transverse wind fluctuation, _{i.e. rying wind}

velocity in sway direction, possibly causes a slow drift motion iii the sway direction. While froni Fig.3.5 it SCCS that the slow drift motion in the surge direction is caused by 1)0th waves and wind fluctuation.

Hence, in order to investigate how munch each cnviromncnt,al excitations contribute to the surge slow drift motion, _{we carried out a frequency analysis 1)aSCd ou a multiple input}

analysis.

The concept and analysis J)rocedure of multiple input analysis have been shown by Tick4

and the others. Yarrlano1kchi _{applied it to the analysis of transverse stress of ship in} random waves.

The concept of multi-ml)ut analysis is as follows:

If we assiune that the surge mothin process can be expressed as a linear output process y(t) of many environmental excitation processes x(t), eq x1 is the surface elevation, x2

is the instantaneous wave power and x3 is the wind velocity fluctuation, in the following

equation:

y(t)

> f h(r)x(t - r)dr,

theii the auto correlation and spectrum of the output process are given by:

J?,1,,(r)

_{f li(c)h(ß)Rj(a}

_{ß + r)ckdß}

i1j1

n n

S(f) =

H(f)H;(f)s12(f)

i=lj=i

where H(f) is the coniplex conjugate fiuiction of the

_{reponse functionH(f). And the}

cross spcctrufn to the input xj is

J

ST,j(f)

> H(f)S(j)

where

R(T)

inputs respectiirely

Here the second order forcing process f(f(t) can exactly be expressed as

f

f.fg(ri,r2)x

(t - ri)xi(t - r)dr1dr2

However, it becomes equal to the following equation in sense of stochastic equivalence (see

Kate et al.6)):

fd=fh2(r)x(t_r)dr

This implies that the second order forcing process fd can be represeilteci by a linear response

to instantaneous wave 1)ower, i.e. x2 = x. (Moreover, all inputs processes x are stationary and their mean values are eliminated in advance.)

Now, by putting x(t) [z1 (t), x2(t),z3(t)1 and

H(f) = [H[(f),H2(f), H3(f)],

(3.2)

f)

(i,j = 1.2,3)

,Jy,J(f)

EcI.(3.3) is represented as follows:

S?flJ(f) = H(f)S7(f)H*I(f)

(7)

where H'(f) means a complex conJugate adjoiñt function of H(f).

And (lefihling

S,(f) = S(f) H'(f),

from Eq.(37) a three dimensional frequency response function is given as:

H'(f) = S(f) .S2.(f)

Furthermore, in the case of one-input/single output system, a siwple coherency function representing the dcgree of, linearity between input and output eau l*e defineçi. J. the same

way, a coherency function can also he defflied in the case of multi-input/single-output

system. It is called the multiple coherency, which is rc'prcseiitcd in the following form:

(f) EiXf)

₍₃₉

= s(f)H(f)SYZ(f)

And by using the ConCept. of conditional spectriun proposed by Tick'., the partial coherency is given by:

(3.10)

where the partial co1iereny ftrnction '5'7jij _{means the coherency between the output y(t)} fliif i au 1u1)ut x (t) after all of tue liiiear colTelatioul I)Lrt between j( t,) and the inputs

x1(t)(i

_{j) were removed. Ami Sj(f), Sj(f) and S}

and the conditional cross spectrum respectively.

Using such a definition of coherency functions, the frequency response function of the

output y(t) to an input x(t) can be represented in the following form:

H1(f) =

(i,j = 1,2,3)

Figure 3.9 shows the gain, which corresponds to s frequency response function, the

partial coherciìcies and tue multiple coherency betweemi .sl.lrgc motion process and external

excitations. In this figure there exist two frequency regions incicatiiig high values of the

niultiple (oherem1cy. The one region is over 0.1 Hz and the other one is under 0.01 Hz. Siiice

S(f) =

S(f)

the partial coherency between surface elevation and surge motion shows a high value in the former region, the surge response over (.1 Hz is caused by wave surface elevation. While,

it is considered that two excitations contribute to the _{surge iesponse in low frequencies} under 0 01 Hz Wind excitation contributes to the _{surge response in very low frequencies}

undei 0 005 Hz and instantaucous wave _{l)oWer, which coiresponds to wave drift excitations} because square of wave height is in proportion towave drift force, does t the surge motion

u frequency range from 0.005 lz to 0.01 Hz

Chapter 4

A comparison between

a time

domain simulation and a measured

time history

41

_{Determination of hydrodynamic force and}

stiff-ness coefficients

4.1.1

_{Deterrninatjon by Numerical calculation}

The added mass and _{wave radiation (lamping coeffiçicuts are calculated using the three}

(limensiollal source distribution method. In the COflhI)UtatioIl the nican wetted surface of tue 1)od _{Is appro.imatcd by 640 facets In oidcr to solve tin motion equation in time domain,} the so-called memory effect fuiictión is needed. But it is noteasy, because it is irpossib1e to

ll( imlah num n illy the w we i idiation d unping 0V r iithnit fieqw iuis, whi li is ii' ( d d

to get tue memory effect function. Thus we used the approximate calculation method.

Nammicl, we extrapolated the wave radiation (laniping, whnh had l)CCfl obtained in some frequency tange, l)y using the splinc function, found a frequency w0 which the extrapolated

alue becomes equal to 7ero, and calculated the finite mtcgi al o' ci w0 w O instead of

infinite integral.

In order to check the accuracy of its al)proximnatc calculation, we investigated through a

Comparison l)CtWCCU calculation results and theoretical asymptotic ones.

Takagi and Saito7 has shown theoreticallyan a.sym)t()tic behavior of the mc'mnory cffçct

fuuct.ion for a half submerged sphere. Comparisons between their results and time prcseiit calculation ones has been carried out (see the reference 1G]) and then it lias been confirmed that 1)0th results are in agreement although a slight deformation to the calculated memory effect function is ol,serVc(i. Thus, it may be considered fioni prictical point of view that the

¡)resent approximate calculation method is accurate enough to get _{memory effect functions}

since in, general radiatioii damping forces exponentially decrease svith increasing oscillation

frequency; e.g. Kato et _{However we shoul(.l note that the added mass is slightly}

ulo(lificd by the truncation effect. _{In this case calculations were carried out}

_{up to the}

((1uatiou.

4.1.2

Determination by experiments

At-sea experiment

In order to find out time liydroclynarnic force coefficients of _{a full-scale offshore structure,}

free decaying tests using the POSEIDON were carried out in July, 1990. Table 4.1 shows

the content of the cxperimnciit.

Model tests

Two kimls of model experiment were also carried out. The one is a forc'd oÑcillatiou test,

and the other is a free decyiug test. The model uc(l for mimodel _{experiments is a 1/25.}

model of the full scale structure. Six mooring chain lines are set under the same condition as the at-sea experiment shown in Fig.2.3. The chain weight per unit length in water is. 7.5 gf/mn and the water depth is 1.664m. The mass of time structure excluding mooring chain masses is 31.3 kg and ia1ues of the radius of gyration in pitch and roll motiOns are 400 mm

and 53üixim respectively.

Initial displacemcuts of free decaying tests are 20 cm for surge and 12 cm for sway. Table

4.2 m(licatcs ¡3 values, i.e. the ratio I)ctwec11 Reynokis miuniber and Keulegan-Caipenter

miumber ( I, wlmI)er ), at Ihr at-sea ail(l uio(li'l (lecaying tests. As_{a rcpresriitative l('Ilgt1i} for a definition of Reynolds and Kr nun1l)er, we used 0.12 m, i.e. time diameter of one coliumni

And surge and sway forced oscillation tests were carried out under the condition that ¡3 is Table 4.3 shows the condition of the forced oscillation tests.

Analysis met hod of exp erimuemits

1) Representation of viscous drag force

In general viscous drag forces of surge and sway motions can be represented by

F, =

1cr.

-where S is the projection arca and Cd denotes tue drag coefficient and it is said to be a function of the Reynolds aild K numbers or of ß and It _{number if roughness can be}

neglected.

Thus, ni case that ß is constant, namely the _{os illation pcuod is constant, we can deal}

with such coefficient as a fiinction of. K number alone.

In case of forced oscillation tests under a constant period, the Reynolds and K. numbers and ß are defined as:

Xw0T VT D D Xw9D 1'D V il D2 ¡3

=

where T is the pçriod of forced oscillation, X is the amplitude and X' = V denotes time mnaximuni velocity of oscillation, i' is the fluid viscosity and D i

_{the mean (liamtcr of}

twelve columns; i.e. 3.0 in at sea and 0.12 m iii model.

In case of free decaying tests, since tue fluid phenoiicnon is unsteady,

_{boUì K and R}

uhmuhl)rs change with time. if we assume the free decaying displacemnet as

X P(t) sin(wt); (4.2)

then, using the instantaneous velocity amplitude /12

_{+ wP2, we can define K and R}

niunbers as

T/i)2(t)

+ wP2(t)

1cc =

D

Re=

D/i2(t) ±

where P(t) is the envelope amplitude.

When the variation of period is very small, mninely, if the damping f rce is not so strong, /3 l)econies constant and the drag coefficient

_{may he treated as a function of the timé}

depeiu.lent IL imuniber alone, defined by the al.ove equation.

And assuming that the viscous damping moments of pitch and roll niotions are directly proportional to angular motion velocities because of small initial inclinations, we can (lefifle

damping coefficients of angular mOtions as

N

₌

nt,

= 2(1w0I

v1iere I are the total inertia mnoiiieiit.s uicl1i(huig the adde(I inertiaimioments.

i

Estimation methOds from experimental data

There aie uow a uuInl)cr of techniques available for processiug frcc-.Iecay (iata. This involves fitting a 1)arametric model of the (hUul)illg _{to the (.lata, usually in a least sqnare sense. The}

ijiost rOii.iinoii lIlCtlIO(l is l)a.SC(l OU analyzing the eILveloj)e of tl1( resJ)onse, using .KT7J1OV and

J3oyoliubov rncthoIM). (A generalized forixi of this method _{lias beeii develo1)c(i by Robert .$)).} Recently, regarding the 1)al'alnetric identification _{methods which can lie successfully}

ap-1)11C(1 to tue I)rol)1eIn of e.stiniating the dainpiiig _{1)aralllet('rs, two methods are newly}

devel-Ope(1 i)', Roberts et a1)°. The _{one is the state variable filter method and the other is the} invariant inibcdliug method.

Here, a new identification incthol which is base1 on time series fitting method n.sing a

uoiilinear oI)tizati()I1 technique is (IeVelOpC(iand ap)licd to the measured free-dety data

and the accuracy of this method has been evaluated _{through numerical coml)arison between}

the Roberts' results and the present _{ones ( see Fig. 4.1). Comparisons with experimental}

data have also been carried out.

Tue basic idea of the present method is to identify parametrically each coefficients in a

given model motion equation_{so as to minimize the square error beteen the free decay data}

and a numerical solution of tue model equation. As au optimization technique, We used

the Powell method''), a kind of nonlinear planning method. This mèthod is the method

for Ol)taiiliug time value nhinimniziiig ail evaluation fiiiictiou (l(fi11C(i ui a 1)aramneter _SPtCC.

Each coelliciemuls in the iliodel equation, _{ami liii tial (tlSl)laCCiflCllt all(1 Velocity U1(1 t offset,}

of data arc parameters to be estimated. _{The reasons why we aI)plied this niethod to}

estimate the coefficients in the model equation are that Data available for nalyzing were

sonic data of three 1)eriods at most and the result with good _{accuracy cami not he ol)taine(l} using the usual analysis method in case of such short data.

In order to confirm the _{accuracy and time aVa.ilal)ility of the present method to short}

frre-dccay data, we compared time known cocfflcicnt,s with the _{analyzing results due to the} present niethoci by using a given ¡notion equatiomi with nonlinear clamping. _{The estimation}

error of 1)0th is shown in Fig. 4.2.

This figure shows that time present estimation _{method has a good accuracy even for only}

tl.iree-periocl-dc.cay data.

-While the cstimatjoii method of hydrodynamnic coefficients _{from forced oscillation test}

has already been established. Their coefficients, which are equivalently linearized, can he

(1cternhined from Fourier analysis of forced oscillation reaction force and amplitude.

Hydrodynmic coefficients of surge and sway motions

As a free-decay ¡notion c(luation, firstly, we assunic the following equation:

(M + in)Ï + O.5pCgSÎfTJ + 1CV = O _(4.4)

f) S tue iIUi(E (E(11Siy, S is (diC )rc(t;io1I LI(!fl. an(I (Ai + in) iIl(Iirai.cs 1ie virtual

mass including the a(Idcd mass in. This assiuliption is dcrivc(l, from the following results: . Figure 4.3 shows the added mass coefficients iii low frequencies, obtained from _the

forced oscillation test in model. It presents that the added _{mass coefficients can be}

approximated as constant values at the long periods over 12 sec. (60 sec. at sea). I According to a calculation based on the three dimensional source distribution method

( 640 facets), the iiou-dimensioual wave radiation (lampi.ug coefficient N/pVw lies in the order of OOO1. Thus, it can be neglected.

. I Figure 4.4 shows the static mooring force characteristic in surge motion, obtained

from the present time series fitting method when it is assmned that the nonlinear

clamping force is expressed by a quadratic velocity model with _{a constant coefficient} and the nonlinear restoring force is represented by a linear-plus-cubic displacement

model. This figure shows that the restoring force can be approximated as a linear

fol( ( And Fig 4 5 .Jiow oluI)rLrisoli 1)( tWe( Ii IIIC1.SU1( (i fn (l( ( tying Unie history

and simulation one. This .siinulation is obtained nuder the situation that the daini)-ing coefficient Cd is constant. Both results do ut agree. According to the approxi-mate solution for the drag coefficient of a oscillation cylinder tinder the assumption that. the flow around the cylinder is laminar flow a(i two dimensional, the so-called

Wang's solution 12, _{it is said that the drag coefficient is in inverse proportion to}

the Kculegan-Caipenter number. Thus it is expected that the damping coefficient

includes a compoíicnt inversely proportional t

_{the K numbcr}

In Eq.(4.4), all unknown paraiiieters arc five. But since the nonlinear optimizatioii

mUCthO(i used to estimate them depends strongly _{OU the initial values for the evaluation}

function having many iniuima( i.e. the square error has many minima if there exist many

nonlinear terms including unknown parameters.), firstly we detcrunned the undamped

imatural frequency wo

₌

ol)tained the parameters A, B and C , i.e. the damping coefficient

Figures 4.6 an:l 4.7 show the relations of C,. to K. ulunl)crs for the surge and sway iiiotions. The thick lics are the results of the at-sea free decay tests, the thin ones are

those of the model tests and the marks indicate the results from the forced oscillation tests of the model.

It seems that C is inversely proportional to K ou the whole and that it approaches a

constant value with an. increase of _{This tendeùcr appears in the resúlts presented by}

Kziwshzta et ai.'3

Regarding the difference between model and at-sea tests, the damping _{coefficient of}

the Pr0t0tYl)C is larger than one of the lno(iel in low K. nuunher.s while 1)0th coefficients

dependence of the damping coefficient to simulate the surge and sway motions including shott-period-small-amplitude motions due to waves.

Iii practice, wO can treat the K depeiideiicc of he damping _{coefficient as a}

linear-plus-quadratic velocity 1110(101 with constant cocfficient.s because if the _{component proportional}

to K number i

_{± C2 ( where C1 and C are}

constant values) from the definition of _{numbets (Kr= TIxI/D), that is,} 0.5pC,jS:i:i:J = 0.5pSCi: + 0.5pSC2i:l

the first tenu of the right hand side in the above equation _{represents a linear damping}

tenu.

Figtire 4.8 shows the cornparisoii between a ruiea.sured free decay data and the rêsuit

simulated by this conventional model, where C1 and C2 _{are estimated from the least square}

fits of Figs. 4.6 and 4.7.

This figure shouvs that; the couveuitiouial simulation 1110(101_{is suital le.}

4) Hydrodynainic coefficients of Roll and Pitch

motions

Figures 4.9 aiuci 4.10 show the extinction_{curves of roll and pitch motions. The marks are}

the results obtained in the following wäy:

Let x,, be sequential peak values(amplitudes) of damping _{curve. If we assume that the} decaying motion can be represented by:

Nrt _2irt

X = X0 exp[ -] sin(

where T0 is the natural period, I the virtual inertia moment, and N the equivalent

lin-earized damping coefficient. Then ifwe plot

-

-

the damping js constant, from Eq.(4.5) we get:

-NeT0

- Xn+I 1

('xI)[

Thus, using tue least square uiethod, tho mininnun eiror estimate of the inclination E can 1)0 obtaine(l. The natural Period 1J is ol)taine(l froni the mean of zero-up-crossing periods

an(1 zcrO-(.lown-crossing l)(riods. Then the virtual mass and equivalent (lam)ing coefficient

are given by:

TK

4.2

Ne

TKlog(E))

where K is the restoring moment coefficient.

(4.5)

(4.6)

(4.7) (4.8)

This method is called the extinction cui-vc fitting method and it has been well used.

Iii these figures, circles are the at-sea frC( (lecay r('Slllt.S and the other ixiark indicates the model free decay results. _{Furthcrinorc the lines indicate thc results duc to the present} method, i.e. time series fitting method.

It is found that both analysis results _{are iii good agreement amici the model damping}

coefficients of roll amid pitch uiotious _{are greater than the at-sea ones.}

5) Final results

The at-sea results relating to the hydrodynamic _{force and stiffness coefficients excluding}

wave radiation damping forces are summarized in Table _4.4.

As for heave motion, the damping force of a quadratic velocity model with a constant C1 equal to 2.0 and the restoring force due _{to statiC Water pressure re taken into account.} And the added mass and _{wave radiation damping coefficients are obtained from calculation} based on the three dimensional source distribution method (a number of facets is 640).

4.2

_{External forces}

4.2.1

Wind loads

Not only steady wind loads but also wind fluctuation loads should be taken imito account

as mentioned )rcviously.

The wind loads are estimated from the _{measured wind pressures as follows:}

=

= Lp2(t)S2

¿p(t), (i = 1, 2) arc the measured wind_{pressure difference time histories, and S, (i 1,2)}

are the projection area for each directions.

In simulation, the wind 1oad are taken into account only for surge and sway motions.

4.2.2

Wavé loads

-First oidcr forces and second order slow drift ones arc taken into account to simulate the

motion time histories. And we assume that the effect duc to second order potential in

the second order forces and the nonlinearity of incident waves can be neglected, and the incident wave system can be dealt with as an unidirectioual wave system.

The incident wave tune lustories used for simulation arc synthcsl7ed using wave data from

from the relative wave height meter aud the ac(:elorneter on tue (leck. The former clata are

1IS('(1 ilS (UI L1I11)1itIlde inforinatioji and the latter ari' (1011e fl.S L 1)haSC iflfolillatiofl.

Furthermore we tried o analyze a phsc diffcreucc of both wave data. As a result, we

could get. the phase difference corresponding to the (listance 194 m. This value was close to the value 180 ni measured by (livers at a calmsea.

As found from Fig.3.5, the main direction of incident waves is some degree starboard direction. This is obvious from Fig.3.7 because the sway spectra have a significant power due to waves. However, the accuracy of direcioual waves data from wave height meters

installed like a line array on the sea bottom

_{is not so good and the objective of this}

simulation is to confirm whether the slow drift surge and sway motions can he simulated under a rough environmental condition. Tuas, we (lare to consider the main direction of wave and wind head.

'We shall investigate the characteristics öf slow drift force at sea.

Tue amplitude of QTF(Quadratic transfer function) ofsurge slow drift force is shown in

Fig4.11, which is obtained by numerical calculations. In this figure, 11 means the nican

wave frequency of two wave components and 112 denotes the cliffefence wave frequency of thcuj.

This figure shows that there exists a peak in the vicinity of Q 1.7 rad/sec. axid the

amplitude of QTF cliamiges slòwly around such frequency.

In order t.o comnl)are this numerical result with the measured one at. sea, we have carried

out a cross bi-spectral analysis. The analysis procedure followed the Daizell's work''. Ou

the analysis, t.heie are two Problems in this case. The one is if waves and wind fluctuation are mutually independent in statistical sense ami the other is if wind fluctuation is Gaussian distributed. However, as shown in Appendix A,we assume a mutual independence between wave and wind fluctuation processes auch a Gau.s.sianity of wind fluctuation process.

And we note that a QTF obtained from such analysis is just the QTF ofsurge response,

G2. Tliu.s the transform from it to the QTF of second order fqrce G may be given by:

G(wi,w2)

-

where H,. is the linear transfer function of _{s1irge motion to the external force, which is}

obtained from the free oscillation test in still water if the hydrodynamic force coefficients

do not change in waves ami it. may be given by:

-Hj(w)=

It - (M + in)w2 ± iNew

Figure 4.12 a) shows the Q1F of second order force at 112 = 0, i.e. frequency character-istics of steady drift force. Figure 4.12 b) indicates the amplitudes of the QTF of second

order force vs. the difference frequencies, Q2. Moreover, the results of model tests are also slown in Fig.4.12 a). Black circles are the results obtained from the 1/14.3 model tests in

regular waves and the broken line is the ones obtained from tire _{numerical values taking} ac-count of the effect of viscous thift force, which lias been presented in the _{previous paper6).}

The holl7ontal aus indicates the nican ficucncy of different two wave components In

Fig.4.12 b) there exist five theoretical hics for each Q2, hut the_{difference between them is}

too small to be shown in the figure.

-From tlni.s figure the ïiumerical linie l)a.sc(l _{on the potential theory is mincir lower than tire}

exi)crirnental results, but both results have the same tendency. That is, the amplitudes

of QTF of second oider force vs _{the difference wave frequency increase gradually with}

au increase of the mean wave frequency. And concerning steady drift force, the broken

line, i.e. the corrected numerical result taking into _{account the effect of Viscous drift force,} agrees with riot only model test results but- also at-sea experiineirtal ones. This means that in order to estimate the QTF of second order force of the floating structure consisting of many circular slender legs,, we should take into account not only the potential drift force

luit also the viscous One.

4.3

_{Simulation model}

(T +

±

- r)Tdr ±

+

NJijIXI ± (aki +

= F(t)

+

F2(t) ± F,"

\vlierc -

-- --. _{; Mass matrix} -

-in(oo) ; Added mass matrix at = ao

-

-N'; Viscous damping coefficient matrix.

aL amid _{bkl; Stiffness matrix due to static water pressures and mooring line force}

respectively.

I; Memory effect function matrix

- F,l),(2)

First and second order force column vector. A double summation method is ilse(l to get time lustorics of such force.

FLU'; Wind force column vector. Only the components in surge and sway motion directions arc taken into account.

Furthermore, as a simulation condition, since we assume that the incident wave system is

unidirectional and its main direction amici the _{mean wind direction are head, only the three}

mode motions, i.e. surge, heave pitch môtiou.s occur. Dut we used a five mode coupling

iiiotlon equation excluding yaw liiotion in order to take a(:colult. Of :ouiing effects 1)CtWCPII

the sway motion due to wind and other motiOn modes.

As an estimate of the QTF of second order force, iu(lging from the difference between the analyzed results of at-sea experimental (lata and time mmunierically calculated _{ones, we}

used al)011t twice Values Of the caldulate(i QTF.

4.4

Results and DIscussion

Figure 4.13 shows a coml)arison between the _{simulated tinte history and measured one.} Figure 4;14 shoWs the spectra of both results;

Both results are in rough agreement but, as for the pha.se of slow drift motion, they

are slightly different. And the simulation result of sway motion does not include the wave frequency component. This reason is that since it is assiuncI that tile incident wave system

is unidirectional and both the. main _{wave and the mucan wind directions are head. the effect} of waves to the sway motion is not taken into accouiul. This discrepancy is however not so jitiportatit, i n'rai ise it is clear that the sway slow drift niotion dite to wind is (lominant compared with that, duc to waves.

This result shows that estimates of each coefficients of motion equations and the external

forces in tile present. simulation model _{are good, although they are rough.}

Amid relating to the problem whether _{or not the wave drift damping, which is ai a(ldCd}

damping force due to drifting of structure in_{waves and an important clamping term, should}

be taken imito account, the. present result seems to show that it is no uiced to be takeim into

accoiuut, 1)ut we can not judge it from the present result only.

Ou a basis of the siniujation 1110(1cl obtained_{iii this section, we shall discuss about tue}

Chapter .5

Statistical prediction of

_{slow drIft.}

motion at

sea

It is said that ?DF(probability density function) of slow dñft inotiéns apart from the Gau Sitli PDF, the extreme statistics can not be estimated from the Well-known linear theory,

which has been represented by Cartwright-Longuet-Higgins15), _{and that, the contribution of}

wind to slow drift motions is much significant.

Vc llave already (1CVCIÓ1)Cd _{t UCW statistical estimatiofi method taking into acconnt not} only the eIlicL of S('COii(E order WaVes 1)111; also the wIII(1 fluctuation effect (see Appendix

A). In this section, we apply this method to estimate the response PDF and the expected extreme values of slow drift motions and _{compare the estimated results with the at-sea} nicasured ones. Furthermore, we also study the Wind fluctuation effects to the responses PDF and the extreme statistics.

Figure 5.1 shows an example of the instantaneous PDFs of surge and sway motions

imicluding slow drift motions, where the results _{are normalized as the mean is zero and the}

variance is unity.

This figure reveals that the PDF .of_{surge motion is asymmetry with respect to time mean}

\r]je and that its, tail broadens _{toward the drift direction tiime to waves. The solid line of}

this figure indicates the estimation result by the present estimation method represented in Appendix A. This result agrees with the measured one very well.

And the instantaneous PDF of _{sway motion has alsó the same tendency as the one 01} uigc motion But the asyiumctry of the PDF is less than that of the surgi PDF

Figure 5.2 shows the maxima PDF of_{surge motion, where plus direction of surge motion'}

corresponds to th( drift ditection due to _{WtVeS. The I)eakS are (lefifled as the maxinmuin}

al)solute value in the peaks among a mean up-crossing timmie.

This figure shows that the measured maxima PDF _{differs from time Rayleigh PDF}

ob-tained fiom the linear theoiv mncl agices with the present estimate

The lines in this figure mean the expectation, the broken _{hue is the estimation result by} Longuet-Higgins', the solid line is the one l)y the present method and the dash-dotted line is the mean value of observed data. The marks _{are all sample values. It is found that the} dash-dottedlinc is approximately 1.3 times greater than _{the Longuet-Higgins' line while the} line estimated by the present method gives an upper limit of observed extreme data.

And almost of the extreme values of the sway motion distributes around the Longuet-Higgins' line. This reason can be accounted for as follows

As a cause of the slow drift. motion of_{sway motion, the Wind tlil(:tllatiofl is muore (loluliflaIlt}

than waves, as found froiui Fig.4.13. And it has been _{shown by one of the authors that} the wind fluctuation process is _{a quasi-Gaussian process and wide-banded. Thus, it is} expected that the mucan value of extreme values of sway iiiotion is as same as or less than

tue Longue t-Higgins' line.

This fact implies that the wind fluctuation has the effect moderating the asymmetry of the PDF of slow drift motion.

Figures 5.4 and 5.5 show the effect of wind fluctuation _{to the instantaneous PDF awl} the ('xtrdul(' statistics of slow drift motion. The solid hue is the resilts of slow (hilt motion

due to waves alone, and the broken line is those ciime _{to 1)0th waves and wind. It is obvious}

that the wind fluctuations contribute to the relaxatiou.of asymmetry of instantaneous PDF and to a decline of extreme responses.

Chapter 6

Conclusion

T11(' (:o11c1U.siofls iIi. this »iper _{are Ñ1uI1Inariz(1 as follows:}

i ) A iicw I)kra1I1('tcr idciitificatioii lLlctlIo(l is (levclol)('(l to uiaIyzc _{short frce-dccayiuig (lata.}

It; is c1fccive t-,() get hc drag coeflicicul (lc)('I1(liug OIl the IÇ-C uumbcr (

.Keulegan-Carpenter number ). Iii _{1iis ca.se, the (Irag (:ocffic:icu of surge and sway motions cati}

be (kscribc(l as SitU! of a conStant t-erixi an(l a K-C dcpendeiit term, which is inver.ely l)roportiOflaj to tite K- inmiber. And flic constant tenu of the full-scale structure is as saine a that of model while the K-C dependent term of the full-scale structurè is larger than that of model.

In order to simulate slow drift motions, not ouiy in-line wind fluètuations _{hut aLso}

t.ransv('rsr O1i(5 should I>C taketi imito uTOUflt CV('1i though the nieaii Wili(l (lirPctiOit

is licad As a wind spcctriun iepicseutiug wind fit« tuation',, a spectrum form with

significant low frequency power compared with the well-known Davenport and lino spectra, e.g. spectrum forms suggested by Ochi-Shiri. and the author, should he used.

I _{latiug to statistical pu dictions of thc PDF and the extreme r(spons, t nCW}

method has been developed to take into account not only second dtcler wave forces i)ut also wind mean and fluctuating loads;

Tue probability distribution estimated by the _{present thethod agree very well with}

the measured one. And as for tlie estimation of extreme response, it is confirmed that

the Caïtwiiqht-Lori.guct-Hiqqjris' estimate significantly underestiinat.es the uicasured result.s while flic estimate by the present method overestimates slightly them.

As for wind effects to slow drift motion statistics, wind fluctuations contril)utc to tite relaxation of asymmetry of instantaneous PDF and to a decline of extreme response

s.

This work ha leen supporte(l in part i)y the _{special coor(lmation fund for T.T.R.D.} (

Transport Technology Research and Development _{) of the Ministry of Transport of the} Japanese Government.

The authors are grateful to acknowledge Dr. moue, who is Head of Ocean Engineering Division in Ship Research Institute for his _{support and also appreciate the assistance in} the analyses and drawing graphs of Mr., H.Yoshimoto and Mr. H. Sato, who are -research

staff in Ocean Engineering Division.

REFERENCES

1) OhmatsuS. et _{Experiment, of Floating. Platform "POSEIDON", Proc.}

of 8th OMA E Syrnposi?irn.

2): Ochi M. K. and Shuri Y. S. (1988): Wind Turbulent Spectra for Design Consideration of Offshore Structures, OTC 5736.

Kato S. (1991): Wind, Turbulent Spectrum over water at the .Tapan Sea. (to he

ap-- peared)

-Tick, L.J.(1963): Conditional Spectra, Linear Systems and Coherency, Proc. of the Symposium on Time Series Analysis, John Wiley _{& Sous, Newyork.}

Yamauouti,Y.(1969): Ou the Application of the Multiple Input Analysis to the Study of Ship's Behavior and _{an Approach to the Non-linearity of Responses, Journal of}

SNA.J Vol.125.

G Kato S. ami Kinoshit,c T. (1990): Nonlinear response of moored floating structures in rall(lom waves and its stochastic analysis, Part. 1. Theory and riiodel experinients,

Paper of SRI, No27 Vol.4.

T Takagi. M and Saito K.: On the Description of Nou-lannonic Wave Problems in the Frequency Domain (ist, 2nd, 3rd, 4th, 5th, 6th and 7th reports), J. Kansai Soc.

N.A., Vol. 's. 182, 184, 187, 188, 191, 192.

Bogoliubov, N. N. and Mitropoisky, Y. A., Asjrn.ptotic Methods in the Theory of

Non-lm.ear O.scillations Gordon and Breach, New York, 1961.

Roberts, .J. B. (1985): Estimation of Nonlinear Ship ftoll _{Damping from Free-Decay}

Pata, Journal of Ship Research, Vol.29, No.2.

.1.B. Roberts, A. Koiuitzcris, and P.1. Gawthrop (1991): Parametric Identification

Nonno, H.(1978): Nonlinear Mathematical Programming, NITIKAGIREN press (in Japanese).

Wang, C. Y.(1968): On High-Frequency Oscillating Viscous Flows, JourL. of Fluid

Mech., No.32.

Kinoshite, T.. and Takaiwa, K.(1990): A Mathematical Model for Slow Drift

Mo-tion of A Vessél Moored in Waves Determined by OscillaMo-tion Tests in Regular Wave

Trains,J?eport of the Thstitute of Industrial Science' The University of Tokyo, Vol.35,

No;5.

DaIzell, J.F.(1974): Cross-Bi-Spe:tral Analysis : Application to Ship R.esistade in

Waves, J.S,R., Vol.18.

Cartwright, D.& nd Loiguct-Higgius, M.S. (195G): The Statisticäl Distributiou of

'the Maxima of a Random Function, Proc. of thè Royal Society, Vol.237;

S. Nato, T. Kinoshita and S. Takase (1990): Statistical Theory of. Total Second Order

Responses of Moored Vessels in Random Seas, Applied Ocean Research, VoL 12.

T. Tsuda (19690): Montecarlo Method dud Sirrthlation, Baifukan Press, Tokyo. (in .lapaiiese)

T. Ozaki (1985): Non-linear Time Series Models and Dynamical Systems, Handbook of Statistics, Vol.5.

Appendix A

Nonlinear Statistical Prediction

Theory of Slow Drift Motions due to

Wind and Waves

A.i

_{Estimation of Instantaneous Probability}

iJen-sity Function of Slow Drift Motion due

_{to Waves}

Slow drift motion process of a moored Hoaing structure subjected to a Gaussian random excitation at sorne fixed time can be exprssed as:

X(t) = x' +X2

- r)dr

.,1

and the nonlinear second order terni as:

x2

=

/

j92(ri,r2)((t_ rj)((t

(A.)

In above equations, «t) deiiotcs the surface elevation which !s a stationary Gaussian ran-(ioni vatiable with a zero mean. The kernel g1 is a linear impulse response function. The

kernel _{92 is analogous to the linear impulse response function and is called the quadratic}

impulse response function. And we assume that they are continuous and absolutely

inte-gral)le, tlieii they l)OSSCSS a Fourier transform a.s

According to the Iac & Siegert theory (e.g. see the reference [161), Eq.(A.1) represents

as follows:

00 00

where W is a set of independent Gaussian random variables of zôro. mean value and unit variance. The ) are cigdnvalucs which satisfy:

fI(wi,w2)i(w2)(1w2 = (A.5)

The parameters c, which represent the linear response, can be determined by:

where * indicates a complex conjugate and S is a two-sided wave spectrum.. In equation (A.5) is a set of orthogonal cigenfunctious which satisfies:

= fiw)V(;w

(A.)

f(wciw={

:;

(Al)

aùd kernel fiuictiou K(wi, 2.) is a Hermite kernel deflued by:

K(wi,w2) ₌ JS<(wi)S(w2)G2(w1,

w2)

Collecting terms with the same sign ou the egenva1ues, the response process is obtained

as a sum of two random variables Z1 (for positive eigeiivalucs) and Z2 (for _negative

eigeiivahics),. each given by a positive definite _{representation. 'T'cn we can represent each}

PDF of two variables as series expansion in terms of generalized Laguerre polynomials. If

the expansion is truncated after the first term, the Gamma PDFs. with thte parameters approximating the true PDFs of the variables Z1 andZ2 can be obtained. The

param-ctets are determined from a comparison between the _{true and the resulting approximate} characteristic function as:

4À+3E.Ajc

1-(9.X+c2)2

2(2\+c)

- ( E À +3 E Ajc

If the slow drift approximation introduced by Naess.is applied, the parameters in Eq.(A.9)

should be replaced by 6

=

_{= 9. Thus the p.d.f.}

The p.d.f. of Z2, as well as that of Z1., _{can be. also approximated by the Gamma PDF}

with three parameters , i.e. 92, z'2 and 62.

f(Ö1,2; Öl,2) J(z ± x -

X exp(_2)

f(Öj,Ò2;j,t2)f'°(zx+1

= (2Òi)/2(2/2r(j/2jt(2/2)'

+

i

(A.il)

A.2

_{Estimation of Instantaneòus Probability}

Den-sity Function of Slöw Drift Motion due

_{to Waves}

and Wind Fluctuatións

As shown in the previous section, the slow drift motion due to wind is caused by a low frequency component of wind fluctuations. It has been shown by the authors 3) that the

wiu(l fluctuation can l)C al)proximated_{as a isotropic turbulence model, i.e. its process can}

be regarded as a wide banded Gaussian process close to white noise process. This implies that the PDFs of wind flucti.iations in surge and sway directions can he expressed in the

following form:

-i

(xm.)2

p(x)

22'

?fl = Sp1H(0)

f

where i = i: surge drectÍon, i 2: sw.y direction. And indicates the wind pressure

differences in surge and sway directions and (w) are their spectra. Pz(X) =

In general, wind and waves are not independent an(l _{an energy transfer from wind to}

waves occurs at the developing pröcess of waves. However, they can be treated_as indepen-dent variables within the inertia sul)range suggested by Kolmogorov, where _{is equilibrium} fioni a viewpoint of eneigy transfer Since the existence of f he inertia subrange has been

confli med by the authors 3), we shall _{assume the statistical independence of the waves and}

wind in this paper Then the I'DF of total slow drift motion due _{to waves and wind can}

be found by convolving each PDFs p and p of slow drift motions due to waves and wind:

=

f p.(z -

A.3

_{Prediction of expectation of maximum}

excur-sion

From a viewpoint of eugineering safety desigfl purpose, we iml)Óse the most severe conditjon

such that the response and its response elocity piocesses are mutualy independent Then, The expectation of inaxinunu excursion in N peaks can be given by

s' E[ZN] .

-

p()}NldZ

(A.1)

P(y)

d fp\(iJ+E[X])

Appendix B

Supplement:Cornparison with

simulations due to Monte-Carlo

method

It. is difficult to get stal)le statistical values by using tijne series. Especially, the statistical variation of the expectation of maximum excursion iii N observations will be. sigiiificantly large even for a linear random system since only one sample value can be. obtained from one time history. In order that. we investigate the variation of statistical values of nonlinear

dynamic system, the Monte-carlo simulations arc the most effective mthód..

As an example of simulation, we shall consider the surge motion including slow drift motion in head waves.

B.1.

_{Monte-Carlo simulation.}

According to Appendix A, the slow drift motion process can be represented by a difference

of two random variables obeying the Gamina PIF if time is fixed. Similarly, a second order forcing process holds this relation. And- it is known that the Gamip.a process

with parameters c and ß satisfies the following dynamical equation, exactly a stochastic

differential equation (SD).

dy(t) = {(c - 0.5)ß

however, a random process (Wiener process) and it is impossible to differentiate with respect. to time over entire space. (Exactly speaking, an integration form with respect to time has a mathematical meaning, but formally, the differential form is well-used.)

We can obtain the SDE of slow drift motion by coupling Eqs.(B.1) and (4J1). To solve its SDE is equivalent to solve the equation dominating

_{the transition?DF (TPDF) of its}

time transition of joint PDF from _{a initial value, p(X, X; X', X'; O).}

We are, however, interested in the expectation of_{maxwum excursion rather than the}

TPDF. Here we consider the method obtaining the _{maximum excursions by means of}

Monte-Carlo simulations.

The process generating the second order forcing process is the Gamma process and the Gamma process can be generated by the Wiener process. In general, the Wiener process can be simulated using the white noise process, ie. Gaussian random numbers.

Iii this paper, in order to generate two indepemlent _{Gaussian random numbers,, we used} the Box-Muller methôcl 17) and the Multiplication Modular metliod'7.

Nextly, in order to generate two Gamma processes with three parameters, we used the

following method, which has been shown l)y _O?aki18.

(/3q)2

=

q11

=

- 0.5)/qi -

/2} ±

where t is the white noise process and can be generated by Gaussian random numbers.

We introduce the slow drift approximation _{suggested by Naess. This approximation}

corresponds to change the Gamma process and its parameters into the following form: (Y -* Li

__

-

Fually, we solve numerically Eci.(4.l1) by inputting instantaneously the external forcing I)roccss expressed by the difference of two Gamina random variabIcs As a solution tech-nique of the differential equation, The Runge-Iutta-Gill method is used. How to give the

initial values is the problem for solving the _{differential equation. However since it can} be show that the solution of the SDE _{considered here is the Markov process and has an}

equilibrium distribution, the result of long duration simulation is independent of the initial

values. We can get the iiumber of _{rncan-up-'crossings and tue extreme values from such}

simulations. By repeating this procedure, we can obtain the distribution of extreme value. Actually, we generated randomly different five simulation data from 20,000 to 120,000 and repeated this procedure 200 times for each one simulation data. The time interval is approximately 1/150 of natural period, i.e. 0.5 sec.

B12

Results

The result of this Simulation is shown in FigB.1. _{This figure expresses that samples}

of the extreme values seethingly distribute around the _{Longuet-Higgins' result hut they}

scatter over a wide raiige with respect to the value of extreme statistics, i.è. the extreme

value/the standard dcvatiou, _{even if the rnunl)er of peaks is 000, corresponding to a}

15-hour simulation. Especially, when the number of peeks is 300, its value distributes over

the range from 2.5 to 5.0. Thus, wO have _{to pay attention to estimate the extreme value} from time domain simulations.

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Pres sure (sway)

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response for wind

(m/(m/S))

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(rn/rn)

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--i

(thin lines : result by Robeits)

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length óf aflalized data (period)

-0.04 0.03 5) -t o. 0.02 0.01 0.0

Q-E

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0.0

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6

9

Kc.

A flOfl ai' eq.

'- _{with srflal ÏÍtiäI vakie}

0N1/(M.rn)

©N/(M.m)

\'

K/(M.m)

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1 2 3 4

length of analized data (period)

flón-Ì1ON eq. (M.m)X.NX.N2X(Xl.KXO

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D mear ebon

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50 .(kN) Q)

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displacement

time

(sec)

(fi)

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Kc

Caracteristics of Cd for surge sotion

forced ceci on teet(modelj

QT:15.c Ø43

DT:lTsec ß:744

¿T=eec ß66

free decayfrig test (at seá)

---state A:wthrteertzthe stete B : without deer t

t 108.8 150.8

time

-

Kc

Caracteristics of Cd for. sway sotion

208.0 250.0

(sec)

o

I

j3

i

o-00

model; èxtfrict$on cxve method at sea ; oxtMctfon cwve method model ; thiie sedAs fitting at sea; thne serles fItting

¿ 3

'2

o-00

Ion-Extinction curve of pitch

(model)

(atsea) 4

(deg)

A modAl ; extinction CtMVO method

o at sea ; exthicllon ctxve method

model;e sacks fitting

- At sea ; time sedes fitting

2 6 8 (model)1O -- i i i ---0 1 2 3

4(atsea)5

-