i
Introduction
Wave conditions determine to a large extent the feasibility of a mooring system. To ensure the safety of a floating offshore system, one requirement is that the maximum anchor line loads in survival conditions be below a specified level. These loads are dominated by the vessel or platform dynamics, in particular by the low-frequency component of the motions.
The second-order drift forces acting on a given vessel strongly depend on the spectral characteristics and on the realization of the wave process. The mean and slowly varying drift forces can decrease as the spectral peak period increases for constant significant wave height. This will influence the mooring line and turret loads. Also, the possible importance of spectrâl shape as regards low-frequency dynamics is assessed. Anchor chains can be a major source of low-frequency damping, de-pending on the peak period and spectral shape, among others. To investigate the dynamic behavior of a turret-moored tanker, extensive use is made of a recent simulation model, which couples the tanker dynamics to the actual anchor chain dynamics. This method has evolved over the past few years with the following history. Wichers and Huijsmans (1990) show that for a turret-moored tanker, the influence of the interaction forces between the anchor chains and tanker on the low-fre-quency motions cannot be neglected. Huijsmans and Wichers
(1991) present an approximate method to incorporäte the in-teraction forces due to the high and low-frequency motions of a turret-moored tanker. Dercksen et al. (1992), and Dercksen and Wichers (1992) present an integrated method, which ac-counts for the combined high and low-frequency dynamics of tanker and anchor chains.
As an example, we consider a typical, but nonexistent turret mooring system in 82.5 m water depth. The 200,000 DWT tanker, the particulars of which are given in Table 1, is moored
Contributed by the OMAE Division and presented at the 12th International Symposium and Exhibit on Offshore Mechanicsand Arctic Engineering, Olas-gow. Scotland,Juae 20-24, 1993. of Ties AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the OMAE Division, March 7, 1993; revised manuscript received October 4, 1993. Associate Technical Editor: R. C. Ertekin.
Journal of Offshore Mechanics and ArctiC Engineering
ThCI*ISCHE U1IVERSITIT
Laboratorium voor Scheepshydromechardca Archief Mekeiweg 2,2828 CD De!ft ToL:015'786873.Fwc 015-781838The Influence of Wave Spectrum
Formulation onthe Dynamics of a
Turret-Moored Tanker
The. objective of this paper is to provide physical insights into the response of a turret-moored tanker in a severe sea state as a function of spectral wave character-istics. For one sea state various realizations of the wave process are applied to investigate the effect on maximum turret and anchor chain forces. Use is made of a validated mathematical model that accounts for the high and low-frequency dy-namics of coupled tanker and anchor chains. Its predictions are compared with a simpler method and conclusions are drawn with respect tò the significance of peak wave period, spectrum type, and chain damping.
to the seabed with six anchor chains. The layout of the system is shown schematically in Fig. 1, and anchor chain data are given in Table 2. The theoretical and measured (in conjunction with other model tests) static load excursion curves of the total mooring system and most heavily loaded anchor chains are
Table i Particulars of 200-RDWT tankers
Fig. i Layout of turret.moored tanker
FEBRUARY 1994, VoI. 116 Ii
Quantity Symbol Unit Magnitude
Length between perpendiculars t. in 310.00
Breadth B in 47.17
Depth H in 29.70
Draft T in 18.90
Displacement volume y 234994
Transverse inetacentric height iÇ, in 5.78
F GiAIN 2 3
J. O. de Kat
A. Dercksen
Maritime Research Institute Netherlands, Wageningen, The Netherlands
TARRUI LOADEIG COESTlON 100 % T
FX 11PJIETICALCJ C*4A11 aol FORIc*LC.J
o FXTtTDASJ o CHM4 OtSI FCRM
CHAIN 4 FOR1CA.CJ 04AE1 4 FORcGA&l
+
LOAD EXCURSION CURVES - TEST NO. 7000 07I07IO2l
Fig. 2 Static load-excursioncharacteriStins of mooring system and most heavily loaded anchorchains
shown in Fig. 2. Thenatural surge period at the mean position is around 110 s(w=0.O57 rad/s).
To assess the motions and mooring loadsfor the foregoing system, two computationalapproaches are considered:
1 Method 1: An approximate method to estimate the
low-frequency motionsand turret loads.
2 Method 11: An integratedmethod, taking into account the
high and wfrequenCY dynamics of the cou-pled tanker and anchor chains.
Chain and tanker behavior, according to Method II, have been validated usingmodel tests, as is alsoillustrated by Derck-sen and Wichers (1992). To restrict thenumber of parameters, waves and steadywind are applied, and current is excluded from the present case. Waves and wind arecolinearlY directed. Method lis applied to obtain long-term statistics and to
com-pare its predictions with results fromMethod II
After a brief discussion of these methods, the following topics are addressed:
validation of Method 11 using model test results; influence of spectral period and shape on mooring loads using Method I and Method II;
dependence of long-term statistics on spectralpeak period and shape usingMethods I and II;
influence of anchor chaindamping;
influence of individual time histories of waveelevation.
2 TheoretiCal Analysis
2.1 Approximate Method for owFreqUeflCY Motions (Method I). Here we consider only the lowfrequeflCY
surge behavior in head seas with parallel windand current. The six
8! Vol. 116, FEBRUARY 1994
mooring lines are represented by a horizontalnonlinear spring. The equation of motion to be solved in the time domain is given by Eq. (1)
(M+aii)i±Bii*
+k(x)x=+2(t) ++F
The total (linearized) slow drift damping is B11 B'anker + Bwincj +B00 +B1500i8
B50k = viscous damping
of tanker in current or
calm water
= wind damping coefficient
B30 = wave
drift dampingBmooring = dampinginduced by anchor lines
and
TURRET EXCURSION 04 U
M = mass
of tankerau added mass at frequency ¡L
(2) mean wave drift force
= 2S' S (u')
[T(w, )]2 dX2(t) = time-varying wave drift force 2
ç
II
P¡L
=8l S(w)S(W+)
1T1w+',w+
dJo
L\
2 2T(,w) = quadratic
transfer function of wave drift force¡L natural surge frequency
k (xJ = mooring stiffness
F = mean current
force;E; = mean windforce
A description of the simplified computation ofthe slow drift force is given by De KatandWichers (1991), where is assumed to be zero in the determination of the wave group spectral density. The quadratictransfer function, T, isobtained from 3-D diffractiOn analysis.With the àbsexice of more exact
danp-ing models, all low-frequenCYdamping contributions areused in a linearized format.An analysis based onEq. (1) typically consists of 20 sim-ulations of 3 h duration each, i.e., 20 realizationS of a given sea state are simulated using the theoretical wavespectrum. This provides sufficieñt statistical information on the low-frequency surge motion and mooring force.Associated anchor line forces can be obtained frOm the static load curve.
2.2 Integrated Method for Combined High and Low-Fre-quency Motions (Method U). Instead of estimating the
low-frequency chain damping, time domain simulations can be carried out using a direct method. The chainturret-moored tanker is simulated by applying the correct couplingbetween the high and low-frequency vessel motions.This approachhas the advantage that anymooring system can be used without having to estimate the low-frequency chaindamping.
To predict the motions of a moored tanker inirregular head seas in combination with its mooring system, theequations of motion for all elements are solved. The equation,of low-fre-quency surge motion for the tanker is given by
Transactions of theASME (I)
Table 2 AnChOr chain particulars
Uni t sagnitude QuantitY 6 Number of chains L n 700 Length d inch 5.0 Chain diameter Chalntable diameter weight 5bmerged
weight in air
n w w EA m kg/in kg/SI N 7.22 321 369 1. 392E9Stiffness
Pre-tensiOfl angle a P degtf
60 40.83 Pre-teflsioflX TURRET p.q ZTURRE1 XX TURRET F CHN 4 1T9
Fig. 3 Sample time histories of meastired and computed (Method Il) quantities
(M+a11)ï+B115. =21+X2(t) Xmoonng(t)+Fc+Fw (2)
where
Bu = + BWjnd +
B85
Xmoorip.g(t) = horizontal mooring force in surge direction The quadratic transfer functions and resulting drift forces are based on 3-D potential theory.
The mooring system is modeled by means of the lumped mass method, where each anchor leg is represented by a large number of discrete elements. For each node j in the discreti-zation, the governing equation of motion in a global system of coordinates is the following:
(LA1+ [a(t)])x=F(t)
(3)where
EAJ = the inertia matrix
[a(t)J the time-dependent added inertia matrix
x,
(t) =
the global acceleration vectorF(t) =
the nodal force vector, comprising tension, weight, fluid forces and seabed reacting forcesThe fluid forces originate from element motions and water particle motions due to current and waves. The Morison for-mulation and relative motion concept are applied. Seabed re-action is modeled by a system of linear, critically damped springs.
A coupled set of nonlinear equations is obtained for the nodal positions in time, and the boundary conditions are used to superimpose the vessel motions. The mooring reaction force on the turret, caused by the chain legs, is updated at each time step. Due to the importan of the high-frequency chain dy-nanlics, the simulation technicue is essentially in the wave frequency range, but time steps may be as small as 0.05 s. The computational procedure consists of two stages: preparation of input, followed by simulation in the time domain. The preparation of input consists of:
generation of wave drift force time history and high-fre-quency turret surge and heave motions (based on measured or theoretical wave elevation time history);
discretization of mooring system.
The tithe-domain simulation comprises two basic steps:.
solve equation of low-frequency surge motion for the tanker, using the mooring force from the previous time step, in the high-frequency time scale;
add the high-frequency turret motions to the actual tañker position, update the mooring forces and proceed with first step.
Model tests have been carried out at scale i to 82.5 with the 200,000 DWT tanker described in Section 1. Using Method II, test conditions were duplicated in the time domain. The time history of the measured, undisturbed wave at the mean location of the turret was used as input to the program. In the program the second-order transfer functions of the tanker forces are applied through Fourier transformation and con-volutiOn integrals. To illustrate the correspondence between model test results and Method II predictions for severe wave conditions, Fig. 3 shows some sample-computed and measured time histories of the surge and heave motions turret fOrces and forces in the most heavily loaded anchor chain, i.e., the chain to the weatherside. Dercksen and Withers (1992) describe these validation tests in detail, and in this paper the method is con-sidered sufficiently validated without any further proof.
In contrast to Method II, Method I requires an estimate of the linearized damping associated with the mooring system. A hybrid method is used to estimate the linear chain damping associated with a particular simulation: the mooring system is modeled as in Method I, but the actual time history of the undisturbed wave is used as input. Hence, it can be considered as a low-frequency versión of Method II. By varying the chain damping contribution in this hybrid model, the response can be "tuned" until close correspondence (with respect to the standard deviation of the surge motion and horizontal turret force) is found with the Method II results. The resulting chain damping is then used as input to Method I.
3
Overview of Computational Conditions
The following parameters are varied to assess the influence of the wave climate on the mooring dynamics:
peak period (T=I2, 14 and 17 s); the significant wave height is kept constant at H5=8 m;
spectral shape (JONSWAP with y = 3.3, and Bretschnei-der formulation);
random generation of the phase angles of the wave com-ponents representing the irregular sea.
The Bretschneider formulation is obtained from the JON-SWAP formulation withy = 1.0. Figure.4 shows the two wave spectra, and the associated group spectra. An overview of the computational test conditions is given jEu Table 3. In the com-putations the following procedure is used for each sea state:
i Run Method II for 3 h real time, using a sufficiently large number of wave components with random phases to avoid repetition of the wave elevatiOn process.
2 Repeat foregoing run with hybrid model for various chain damping values.
3 Run Method I with "tuned" chain damping for 20x3 h real time, i.e., 20 realizations of 3 h duration.
Toinvestigate the influence of the choice of wave component phase angles in Method II, the 3 h JONSWAP sea state with 12 s peak period is repeated 5 times with different random phase angles.
4 Results
The critical parameters that characterize the mooring dy-namics are the following:
Journal of Offshore Mechanics and Arctic Engineering FEBRUARY 1994, Vol.116 I 9 Sample time histories of measured ar computed quantities
-- 10
FREQ(4CYrtd/5
FIg. 4 ComparIson between JONSWAP and Bretschnelder wave and
group spectra (T,,=12.O s, H=8.O m)
Table 3 Overview of computational conditions
surge at the turret location (X-turret);
longitudinal and vertical force acting on the turret (FX and FZ-turrets);
force in most heavily loaded (weather) anchor chain (F-chain), in this case chainno. 4.
For each of these quantities, the mean, standard deviation, and most probable maximum (MPM) value are determined. The MPM value can be regarded as the expected maximum value in a sea state with a given duration, and is defined by the following probability foran ensemble of rea1iations of
a
(linear) system X:
Prob (X >XMPM) = 0.63
The MPM value can be estimated by assuming the maxima of the relevant quantities to fôllow the Weibtill distribution. As an example,, the anchor chain force according to Method H for test nô.5is shown on Weibull paper, together with the
estimated MPM value for 3
h (= 1/total no. of
oscillations), in Fig.5.For Method I the MPMvalue is based on the ensemble of 20 realizations.
Figure 6 shOws X-turretand FX-turret as a function of peak period and spectrum type, where results from both
Methods I and Il are included. FZ-turret and F-chainare shown in Fig. 7, but only for Method II. If required, thesevalues could be
10 IVol. 116, FEBRUARY 1994
TESTNO.5 FcHAIN4
DISTRIBIJTIOP4 OF AMpLrruoEs
NUMBER OF PEARS 152
70 tOO 200 420 700 1000
PEAK VALUES OF FCJ4AIN4IN ITPI
FIg. 5 DlstiibtIofl of maximum anchor
chain forces for most heavily loaded chain and determination of most probable maximum (MPM)value obtained quasi-statically for Method I using the static load curve shown in Fig. 2.
To illustrate the dependence of the low frequency damping caused by the six anchor chains, the "tuned" (linearized) damping values are shown in
Fig. 8 as a function of peak
period and spectrum type. Anoverview of all damping con-tributions is given in Table 4.Test no. j (JONSWAP, T,=12 s sea state) was repeated
5
times with Method II using different time histories of thewave elevation. The resulting mean values, strandard deviations and MPM values are shown in Table5 for the surge motion,turret and anchor chain forces. Thevariability in the MPM values is indicated in Figs. 6 and 7. Figure 9 illuStrates the dependence of the distribution of maximum chain forces on the individual wave realizations.
Figure 10 illustrates that the maximum mooring loads are largely determined by the low-frequency surge motion. The relatively small effect of high-frequency dynamics is typical for a stiff mooring system in head seas.
5 Observations
5.1
Influence of Peak Period. The mean values and
standard deviations are not very sensitive to peak period; they tend to remain constant or decrease with increasing period. The MPM (most probable maximum) values decreasewith increasing period.
The MPM values of the horizontalturret force and anchor chain force show the strongest tendency to decrease with in-creasing period. The rate of decrease according to MethodH is not dependent on spectrum type, while Method Ipredicts a more pronounced decrease in case of the JONSWAP spectrum.
Transactions of theASME
ecO II 41
-tJONSwAP) tBetdv.eMe) .nve IpecEl,,, wove ecec1w,, &,esoctnn oup pscE,a,, JOBIRWAP O,stzdv,etde, 30\
\
I i II;!II
va
Test No. (ej Rs (mj Wave Spectrum V wind fm/a) 1 12.0 8 JONSWAP(y
-3.3) 50 2 14.4 8 JONSWAP (y - 3.3) 50 3 16.8 8 .IONSWAP (y - 3.3) 50 4 12.0 8 Bre t Schneider 50 5 14.4 8 Bretschnejder 50 6 16.8 8 Bretschnejdet 50E
r
ot
X-t,tret JONSWAP ti 13 ISWAVE PERIOD Tp (a)
FX-ItaTet JOPISWAP METHOD I MP 3 Ers) MEAN ST. DCV. I, IS WAVE PERIOD Tp (s)
FIg.6 InI Iueflce of peak period and
gitudinal turret fOrcé; bars indicate
history E >1 t' 4 X METHOD I (LP; 3.20 Irs) MEAN ST. 0EV. MPM X-twrat BRETSOecIOER t3 15 17 WAVE PERIOD Tp (s) FX-twvot BRETSOISIEIOER X 'L X X 2-
¡ -=
I IS It WAVE PERIOD Tp (s) spectrum type on surge and Ion-MPM dependence on wave timeIf the damping from-the anchor chains is considered as a linear damping contribution, this damping value will strongly decrease with increasing peak period. The (generally unknown) dependence on peak period should be taken into account wheñ using a procedure like Method I in a mooring analysis.
5.2 Influence of Spectrum Type. Spectrum formulation does not häve a large influence for the present case: the mean values and standard deviations are approximately the same for both formulations, and the same applies to the MPM values according to Method II. The Method I MPM estimates are lower than for Method II, especially as regards FX-turret. The rate of decrease in linear chain damping with increasing peak period is higher for the JONSWAP formulation.
Figure 10 shows parts of the response time histories at the instant where the maximum peak load in chain no. 4 occurs during test no. 4 (T 12 s, Bretschneider). Also shown
are-Journal of Offshore Mechanics and Arctic Engineering
o METHOD S MEAN S 0CV.'-
MPM FZ-tuuTBt JOAP-4
13 IS WAVE PERIOD Tp (s) n o o E ti FZ-t,sret aRETSO*ER bOO Is tWAVE PERIOD Tp (a)
000 '7 Bmoortg 1 o o G + 4 o o X X 'N X
*
Thst No. Bwjfld Itfs/mJ 8tartker (tfs/mJ Bwave (tfs/mJ Bmooring (tfs/tn) 1 5 20 23 62 2 5 20 21 34 3 5 20 14 21 4 5 20 20 48 5 5 20 20 45 6 5 20 - 16 22 O JONS WAP O BRETSc*lElDERTable4 Low-frequency damping contributions (lineàrized)
11 13 15 17
WAVE PERIOD Tp (s)
Ag.8 Unearlzed chain damping as a function of peak period and
spec-trum type
the corresponding time histories for test no i with the JON-SWAP spectrum; the same random phase angles and wave frequencies were used in these simulations. Figure lO suggests that the maximum forces are dominated by the low-frequency motions. Furthermore, for thi particular stretch in time, the JONSWAP sea results in somewhat larger heave motions of
FEBRUARY 1994, Vol. 116111
13 tI '7 I,. 13 15
WAVEPERIOD Tp (s) WAVE PERICO Tp (s) Fig. 7 Influence of peak period and spectrum type on vertical turret
f oróe and Weather chain force; bars IndicateMPMdependence on wave tIme history
N
woeF cheEr JONSWAP FcheErBRETSC**uDER
4...
0.605 0.010 0.050 0.100
e:
i
;
i
30.00 oe 70.60 00.00 DSEIR04UI1ON OF FCI4AiN NO.1 NO.2 NO. S NO.4 NO.5 1It
f
I 12! Vol. 116, FEBRUARY 1994 X 1URRET z1ur 'SII FX IURPET 0F' F044814 (TPJ JONSWAPÇrp =128) BRETSCHNEIDER (Tp 12 s) SECONDSFig. lo Comparison of time histories of responses for JONSWAP and Bretschnelder spectra, using the same frequencies and random phases in generating the wave trains (occurrence of peak chain load during test
no. 4)
Table 5 ComparIson of 5 simulations with Method ii using different
Wave realizations (T,, = 12 s,H1= 8 m, JONSWAP)
Notation Teat NO. Mean St. dcv. MPM
x-turret 1 10.87 7.76 28.59 2 10.1.4 8.90 30.88 3 11.31 7.21 26.48 4 10.58 8.41 28.79 5 12.01 7.09 29.59 TX-turret 1 156.50 187.20 1364.00 (tonfi 2 153.60 204.40 1790.00 3 154.50 164.60 1050.00 4 156.10 196.20 1400.00 5 156.30 162.10 1475.00 YZ-turret 1 250.00 4093 (tonti 2 250.80 42.05 3 249.70 38.05 4 250.90 41.38 5 251.40 34.83 442.60 503.20 418.70 45Ö.80 490.80 Pchain4 1 158.10 159.60 1263.00 (tonfi 2 159.20 172.50 1659.00 3 154.30 139.30 967.80 4 159.70 166.00 1297.00 5 154.90 137.10 1376.00
The high-frequency motions of the turret are dependent on the peak period; the maximum heave velocity response
of
the turret occurs at the longest peak periods. Therefore, one would expect maximum chain damping to occur for the longest peak periods. Again, the results show the opposite, which suggests strong dependence of chain damping on both high and low-frequency tanker motions.The time history of the .wave drift force, which provides the most important time-varying excitation, depends on the peak period, spectrum type, wave realization, and computa-tional method. The spectral density of the wave groups as-sociated with a theoretical JONSWAP and Bretschneider spectrum is shown in Fig. 5. The difference in densities at the
Transactions of the ASME.
70 100 300 400 700 1000
PEAXVAUJES OF F04804 04 tOI
of the same sea state (T,,=12 s, H1=8 m, JONSWAP, Method ii)
FIg. 9 DistrIbution of maximum anchor chain forces for 5 realizations
the turret during the maximum slow drift excursion (at I 100
s); this in turn will cause higher chain damping, which may explain the smaller slow drift surge amplitude for the JON-SWAP case.
5.3 Influence of Computatioflal Method. Mean values, standard deviations, and MPM values of the surge motion show the same trends for both methods. The mean values and standard deviations for X-turret and FX-turret are in reason-able agreement. The MPM values according to Method I are derived from the eñsemble of 20 realizations of 3 h, where the variability in the individual runs is not shown. Method Il tends to predict somewhat higher MPM values but these include
some high-frequency dynamics.
5.4 Influence of Wave Realization. Table 5 and Fig. 9 suggest that the standard deviations and maxima of the motions and forces are subject to considerable statistical variability. The MPM values of the anchor chain force in Fig 10would correspond approximately to the 1 percent exceedan level. 6
Discussion of Results
The items listed in the following in combination with the nonlinearities in the restoring characteristics affect the response
results:
s The total linearized damping, values depend strongly on the peak period and spectrum type. One would expect that conditions with maximum damping would lead to the lowest loads. The results show the opposite, which implies strong interaction between damping and excitation. It should be noted that chain damping also strongly depends on thepre-tension (and suspended length).
50.00 60.00 s 60.00 60.00 &000 10.00 80.00 oe
natur .1 surge frequency of the system is obvious. Taking into account the quadratictransfer function of the drift force, the dependence øfl peak period and spectrum type becomes ap-parent.
The predominantly low-frequency surge motion and as-sociated moóriñg forces are sensitive to theduration and re-alization of the irregular sea. Pinkster (1992) discusses the "variance of the variance" of low-frequency motions, and Stansberg (1992) presents experimentally observed and theo-retically derived statistical variability of slow drift motions.
7 Conclusions
A validated simulation model is used to compute the mooring dynamics due to the combined high and low frequency motions of a turret-moored tanker. Results are also obtained with a low-freqùency model. Both methods are used to determine the influence of peak periOd and sectra1 shape on the tanker motions, turret forces, and anchOr chain forces.
The following tendencies are observed for the present (stiff) mooring system in heád seas and wind:
Maximum turret and anchor chain forces are sensitive to the peak period and decrease with increasing peak period.
Mooring forces are not very sensitive to wave spectral shape.
o Anchor chains provide the most important contribution to low-frequency damping in combination with high-frequency motions.
Low-frequency (linearized) damping due to the mooring chains is very sensitive to the peak period, and decreases with increasing period.
Maximum chain and turret loads are governed by the low-frequency surge motion of the tanker.
For a stiff mooring system with relatively taut lines, lim-ited dynamic amplification of the low-frequency loads occurs due to high-frequency dynamics of the tanker.
The standard deviation and distribution of maximum
val-ues of sÚre motion and anchor chain force are subject to statistical variàbility, which strongly depends On the actual wave realization, implyiñg that a sufficiently large number of wave records must be used in both numerical simulations and model tests.
References
De Kai, .1. Ö., and Wichers. J. E. W., 1991, "Behavior of a Moored Ship in Unsteady Current, Wind, and Waves." Marine Technology, VoI. 28,No. 5,
Sept., pp. 251-264.
Dercksen, A., Huijsmans, R. 1-I. M.. and Wichers, J. E. W., 1992, "An
Improved Method for Ca1cu1ting the Contribution of Hydrodynaniic Chain Damping on Low Frequency Vessèl'Motions," OTC Paper No. 6967, Houston, TX. May.
Dercksen. A., and Wiçhers, J. E. W.. 1992, "A Discrete Element Method on a Chain Turret Tanker Exposed to Survival Conditions," Proceeding BOSS'92. Mar.
Huijsmans, R. H. M.. àndWichers, J. E. W., 1991, "A ComputationModel
on a Chain-lu ret Moored Tanker iñ Irregülar Seas,"OTC Paper No. 6594.
Hòuston, TX, May.
Pinkster, J. A., 1992, "Statistical Aspects of the Behaviour of Moòred Float. ing Structures Hydrodynamice Computations Model Tests and Reality
Pro-ceedings of MARIN Workshop, Wageningen, The Netherlands,Elsevier Developments in Marine Technology Vol 10 May pp 373-386
Stansberg C T 1992 Basic Staustical Uncertainties in Predicting Extreme Second Order Slow Drift Motion," Proceedings ISOPE-92 Conference, Vol. Ill, San Francisco, CA, June, pp. 526-531.
Wichers, J.. E. W., and Huijsrnans, R. H. M., 1990, "The Contributionof Hydrodynaniic Damping Induced by Mooring Chains on Low Frequency
Mo-tions," OTC Paper No. 6218, Houston, IX, May.
