The statistical properties of low frequency motions of non-linearly moored tankers

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OTC 5457

'The Statistical Properties of Low-Frequency Motions of

Nonlinearly Moored Tankers

by J.A. Pinkster and J.E.W. Wichers, Maritime Research Inst. Netherlands

317 11115/1111113r _wooled= war geneml1611111.1110101011

MOW

Makshing 2. 1023 CD Delft

Tet: 015- 7116873

This paper was presented at the 19th Annual OTC In Houston, Texas, April 27-30.1987. The material is subject to correction by the author. Permisilon to copy Is restricted to an abstract of not more than 309 words.

_

ABSTRACT

The ,motions and mooring loads of

storage/pro-duction tanker's are dominated by large amplitude,

low frequency components related to wave drift

forces. In the design stage use is often Made of

time domain simulation and/or model tests to

deter-mine those quantities. In view of the relatively long periods associated with _low _{frequency Motions} and forces, long simulation or model test times are required in order to reduce statistical variance of

the results.

In this paper a theoretical expression is given for the itatistical variance of these quantities for the case of a linearly moored tanker in irregular head seas. This expression is compared with results of model tests and time domain simulations. For a non-linearly moored vessel, model tests and time

domain simulations are used to quantify the statis-tical variance. The results indicate that the main elements governing the statistical variance of the

data are the natural period of the moored vessel,

the system damping and the simulation/model teat duration.

INTRODUCTION

The dramatic fall in the oil prices and the

resultant need to reduce oil production costs has resulted in an increase in the interest for the ap-plication of floaters as IOW coat floating produc-tion/storage systems. Such units, both tanker based and semi-submersible based, have been in operation for several years, see ref. [1], [2] and [3]. The experience gained through the operation of these

units, although not, always positive

_in

all respects

'(see ref. [4]) has strengthened the conviction that

taking into account in 4 proper manner

all

those aspects Which have a significant bearing on the system. performance at the design stage, tenable,

cost effective systems can be designed and built. One of the major components in the design of

floating production systems is the Mooring System by

means of which the unit is kept on station.

References and ilinatkatiens at end of paper.

The mooring system can be a critical factor in

the design in those locations where environmental

conditions

lead to high t000ring loads. Such condi-tions are met, for instance,

in

the northern North

Sea and east of Canada.

In general, mooring systems are designed based on two major load cases, iie. extreme loads

asso-ciated with extreme environmental conditions and

fatigue, loads. In All casei attention Is mainly

focussed on the dynamic parts of the tooting loads.

' For most offshore, locations, wave loads are the

major source of the dynamic loads in the mooring'

system. In particular, the large low frequency hori-zontal motions and mooring loads associated with law frequency second order wave drift forces form a dew-inapt part of the system behaviour and mooring loads under extreme wave condition. The origins And

char-acterititics of second order wave drift forces in

irregular waves have been the subject of study for

some time, see for instance ref, [5J. [6] and [7].

Methods to compute wave drift forces have been developed which, for many practical cases, can be used for an assessment of environmental loads

_in

the

preliminary design Stage. The motion response of a

moored 'vessel to low frequency drift forces has also

been the subject of research. Also in this area sig-nificant progress has been made toward methods to predict the Minions and mooring loads under realis-tic environmental conditions, see ref. [8], [9] and [10].

Most of the aforementioned studies' are

con-cerned with the physical zspecta of the wave loads and the motion response. Such knowledge 14

fundamen-tal to a proper understanding of the processes as

involved. However, for the design of a 'mooring

sys-tem which is dominated by dynamic effects of a

random nature such as wave drift forces, knowledge

of the statistical properties of, the Motion and mooring loads, is also essential. General studies

have been carried Out with respect to the statisti-cal properties' of the low frequency motion response

Of a vessel under the influence, of low' frequency second order wave drift forces; see ref. [11], (12J

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318

The subject is greatly complicated due to the complex statistical properties of the drift force. excitation Which in many cases is compounded by

non-linearities in the mooring system properties or hydrodynamic reaction forces on the Vessel. As a

result, for practical design cases, very little use can be made of the theoretical expression feOr the

statistical properties of low frequency Motions and

mooring loads.

Recourse is taken to time domain simulation

computations incorporating, as much as possible, the

relevant non-linearities in excitation forces and

system response And the output of such simulations subjected to standard statistical analyses. In some specific cases, which Will be treated in this paper, useful results can also be derived from frequency

domain calculations. For the final design model

tests involving modelling of environmental

condi-tions (waves, wind and current) and the mooring system are often used to generate design loads.

An important aspect of both time domain simula-tion and model tests iS the amount of information contained in the output in terms of the duration of either tests or computations. Prior to carrying out such simulations on model tests, the (full scale) duration must be decided upon, in view

_of,

on the one hand, the statistical reliability of the results

and, on the other 'hand, the costs.

In order to obtain statistically reliable re-sults on the dynamic components of a given signal, rules of thumb involving the number of oscillations

can be applied. .Depending on the data required

(mean, r.m.s., spectral density, distribution

func-tion, distribution of peaks, etc.) the number of.

os-cillations in the motions can range from 10 0 200.

Since the IOW frequency output

is

predominantly

about the natural period of the moored vessel this can require simulation or model test durations

cor-responding to many hours full scale.

An important item in this respect is an assess-ment of the necessary duration of either simulation or model testa in view of the required statistical reliability of some quantity related to the motions or mooring forces. In this paper this particular

problem Will be elucidated and for the case of a

tanker moored in irregular head teas by means of a mooring system with linear restoring

characteris-tics, theoretical expressions will be developed

which highlight the main factors invOlved. The

re-sults predicted. by these expressions will be

com-pared with results Of model tests and time domain simulation computations for the case of a linearly

moored 300 kTDW tanker in irregular head waves.

Results Of model tests and time domain simulation calculations are also compared for the case of the

same tanker with a mooring system with non=linear restoring characteristics.

THEORY

The low freguency_surge motions of a

linearly moored vessel

A tanker moored in irregular head waves carries Out wave frequency motions and large low frequency surge motions induced by drift forces. In this paper we will only consider the low frequency motion

com-ponents.

The surge motions of a vessel moored by a sys-tem with linear restoring characteristics, i.e. a

linear system, can be described by the following

equation of motion::

(m + a)* + bit + cx = F(t) ₍₁₎

where:

m = mass of the vessel

a = added mass

b = damping including effect of still water and wave drift damping (see ref. [8])

c = spring coefficient of the linear mooring

system

F(t) = low frequency wave drift forte x = surge motion.

. Added mass And still water viscous damping are,

in 'general, frequency dependent quantities. However,

the resultant motion is, due to the relatively low

damping, generally sufficiently narrow banded to

alb:if,' the Assumption of constant mass and damping

coefficients.

For the purpose of prediction of the surge Mo-tions, added mass and damping values can be obtained

based on calculations and model tests, see ref. [10]. The low frequency wave drift forces can be

calculated based On 3-dimensional diffraction theory

computations, see ref. [6].

The wave drift forces in irregular waves can be

described

in

the frequency domain by the mean value and the spectral density of the oscillatory compo-nents. The distribution function is a complex quan-tity Which in practical cases, is similar to the exponential distribution. This will be treated in

the next section.

The mean drift forces follow from:

f=

2

f

s,'(w) dw 0 Ca where:

Sc = wave spectral density

= mean drift force coefficient

2

,a

= regular wave amplitude

= wave frequency.

The mean drift force coefficient can be

ob-tained from model tests in regular waves or by calculations.

The spectral density Of the wave drift forces

follows from:

SF(u) = 8

f

S(w).S (.1w+u).1T(w,w+u)12 du 0

(3) in Which:

T(w,w+u) quadratic transfer function of the Slowly

Varying drift force

= frequency.

The quadratic transfer function of the drift forces is obtained from theoretical .Computations or from Model tests in irregular waves. This data can also be obtained from tests in regular wave groups consisting of a regular wave with frequency w super-imposed on a regular wave with frequency (w+u), see

ref. [6].

(2) 4

(3)

The frequency domain solution to the eqtatiOn.

of motion fellott fee:6:

- Mean surge displacement.

-x

(4)

= Spectral density of the oscillatory components of

the surge motions.

S(u) °

1

22

N

2 2

SF( P) . .

(5)

(c - (m + a)p j

+ b p

The variance of the surge motion follows from:

2

x0

where:

mx0 ° variance

a

= root-mean square (r.m.s.) value.

The expression for the surge mOtiOn variance

can be simplified by taking into account that

the

system damping is small. In that case the following

result is found:

2

a_X

2bc Fe

I

S (p )

where:

pe =

+ a) = natural surge period of the moored

vessel.

Equation (7)

is very useful for a preliminary

Assessment of the motions. Eased

on

the estimate Of

the mean and variance of the Motion and assuming a

Rayleigh distribution for the peaks and tkOughs, an

estimate can be made of the most probable maximum of

the low frequency surge Motions. A further

simplifi-cation of equation : (7) can be made by assuming that

SF(pe) can be replaced. by SF(p=0).

Tide domain simulation computations can also be

carried out based on equation (1). In such cases the

wave drift force excitation can be

based,

for

in-,.

stance, on the following type of: equation:

N

F(t)

E E

Cj Pij cos((w-wj)t +

i=1. j=1

N

+

2 2

Ci Cj Qij sin((wwj)t +

i=1 j=1

in which:

CI ,

Ci

°

cOj

°

C" C

rip Wij °

amplitude of regular WiVe components of

the irregular wave train

frequency of regular wave components

random phase angle

CO- And quadrature components of the

quadratic transfer, function

(Tij = Pij + i Qij).

The amplitude of the regular wave components

are found in the usual way based on the wave

spec-tral density SOO.

The above representation is the senetal

formu-lation for the wave drift forces.. The computational

effort involved in evaluating equation (8) As

con-siderable.. Other representations 'based an the

appli-cation of

second order impulse response functions

which can make use of an arbitrary wave record as

input can 'also be applied. Again,

the computational

effort involved is considerable, see ref. [14].

SiMOlified model for generating the wave

drift forces

For

the computations

treated in this paper a

simplified Model is used to generate the time

rec-ords of the wave drift fOrces. This model is based

On a .band-Width limited white noise representation

of the low frequency force components With an

expo-nential distribution.

The spectral density of the simplified

excita-tion force is equal to the spectral density Of the

"true" Wave drift farce given by equation (3) for U

being equal to

zero.

In terms Of Mathehatical

ex-pressions the aforementioned description fer

gener-ating the wave drift forte will be:

.aF(A + i)

₍₉₎

in which:

A = ln(rnd(a)) for 0 .< rrid(a) 4 1

(10)

The quantity A represents an exponential

dis-tribution with average of minus one while the

stand-ard deviation amotnts.to One.

For the white noise representation the total

energy of the wave drift force is found from:

mOF =

_SF(u0)

in which:

r/At = Nyquist-frequency being the maximum Observed

freqtency.

The variance of the wave drift force F(t) will

Taking a sample frequency of once every At the

function F in equation (9) can be computed.

This simple model for the drift forces is

ap-plicable due

to the feet that the natural

frequen-cies of the moored vessel is near to zero and the

motion damping idaMall, Bee ref. [15] and [16).

A consequence

of

this method of

representing

the wave drift force excitation is

that comparison

between results of time domain simulations and model

tests can only be based on statistical parameters

and not on A deterministic comparison.

Knowing the wave drift force F(t) the low

fre-quency

surge motions are determined in

the

time

domain by

integration

of

the equation

of

motion

using standard means.

Variance: of results

When carrying Out

time domain simulations or

model tests in order to determine the statistics of

the low frequency Motion, it will be found that when

tests or computations are carried out for a given

duration but for different realizations of the input

wave Spectrum,

the mean,

t.m.di and Maxima of the

surge: Motion Will be different for each

test/simula-tion. This has been pointed out by sevetal authors,

see for instance ref. [17].

319

OTC 5457 PINKSTERNICHERS

be:

aF2 = E[F2(t)] - (E[F(0]/2

(12)

or:

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4 THE STATISTICAL PROPERTIES OF LOW FREQUENC

320

Y MOTIONS OF NON-LINEARLY MOORED TANKERS OTC 5457

The major cause of the variation

in

the output results with respect to quantities As Mean and

r.m.s. is pointed to as being the limited duration

of the simulation, it being argued that for very

long duration, mean and r.m.s. will converge to the true steady state values. It has been demonstrated by Sebastiani [18] that this effect is indeed

pres-ent. Due to the Very low natural frequency of a

moored vessel in some cases records of several hours are needed to reduce the variations in mean and

r.m.s. to acceptable levels.

In the design stage it is necessary to have

some indication of the Model test/simulation dura-tion which will achieve the desired convergence in

this respect. In the following, we

will

deduce

ex-pressions by means of which the duration can be

estimated. For this We make the assumption that, by

approximation, the low frequency surge motions are normally distributed about the mean value. Results of model testa Will be preaented in support of this

assumption.

As a measure, for the degree of convergence of the oUtpUt motions for a given test/simulation dura-tion, we will make use of the r.m.s. of the variance of the surge motion. We will deduce the approximate expression for this quantity and compare this with results found from model tests and time domain

simu-lations.

Starting point is an expression derived by Tucker [17] for. the variance of the variance of a

normally distributed process.:

a2=11fS0)

du X ax2 T o Where:

T = Oration of the record

S(u)

spectral density of the motion record.

Making use of equation (5) for Sr(p) and apply-ing the low dampapply-ing assumption mentioned earlier we obtain the following result for the r.m.s. (square root of the Variance) of the variance:

a.V1(1

+ 62) riT 2

_{T 6e}

L2 x in which: 6 21R-1177177,T coefficient. non-dimensional damping . .

. ... (14)

SF ( )12 . (15) e

The part in squared brackets is recognized as

the steady state value of the variance given in equation (7).

Dividing by the steady state variance and

As-suming that 6 << 1 results

in

the following

assess-ment for the non-dimensional r.m.s. Of the motion

variance:

a' ₂ = . (16)

ax VT ue

From this expression we may see which factors are

_of

importance with respect to the r.m.s. of the vari-ance. We see that, for a given vessel/mooring Sys-tem, which determines the natural frequency pe and

the non-dimensional damping 6, only the test/simula-tion duratest/simula-tion T will influence this quantity. It is

seen that given a particular requirement: with re-spect to the r.m.s. of the motion variance, a system With low damping and low natural period will require

longer test periods T.

The Mentioned theories on the low frequency mo-tions will be compared with results of model testa/ simulations for the case of a 300 kTDW tanker moored in irregular head seas by means

of

a mooring system

with linear restoring characteristics.

The low freguency motion's of a non-linearly moored vessel

In the general case of a vessel moored at sea by means of, for instance, a spited of chains, the

relationship between the surge displacement And the horizontal. restoring forces will be nonlinear. The

non-linearity

is

generally such that the system is

"sae

for small displacements, while at larger off-sets the system becomes "Stiff" as the chain system

becomes more taut.

The equation of motion for the low frequency

surge motion in head seas may be formulated as

fol-lows:

(m + a)R + bi + fr.(x) = F(t) . (17)

where fr(x) is the non-linear restoring force func-tion. Other quantities are as described in the

pre-vioUs section.

For the general case, no simple theoretical

expressions can be given for the surge motion yell,-ancet., Time domain simulation computations must be

resorted to in order to determine the motion time

record frdm which statistical data can then be derived.

In this paper, time domain simulation computa-tions were carried out for the non-linearly moored vessel using the simplified wave excitation model

as

described in the previous section. The results will be compared with the results of Model testa for the

case of the same tanker moored in irregular head

waves by means of 4 mooring System with non-linear

restoring characteristics.

MODEL TESTS

The model and test set-u2

The model tests were carried out with a 1 to

110 scale model of a 300 kDWT tanker in fully leaded

condition. The Main particulars of the full scale vessel are given

in

Table 1. A body plan

is

shown in Fig. 1. The water depth for the model tests

corre-spond to 110 m full scale. The tests were carried

out in the Wave and Current Laboratory of MARIN.

This facility measures 60 t by 40 M.

The model tests were carried out with the tank-er for both a linear and a nonlinear mooring sys-tem. The test set-up is shown in Fig. 2. For the linear spring system the surge restoring force coef-ficient amounted to 15.5 tf/m full scale. The static

load-deflection curve for the non-linear mooring

system is shown in Fig. 3.

The Vessel surge motion was measured by means of an optical tracking system and recorded on a PD?

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OTC 5457 _{PINKSTER/WICHBRS}

5

Wave condition

The following irregular head sea conditions were applied to the tanker:

Significant wave height: m

Mean wave period : 12.0 s

Wave spectrum : Pierson-Moskowitz

One 12-hour realization (full scale) was generated

to these specifications.

The wave spectrum was determined from the un-disturbed wave elevation and is shown

in

Fig. 4.

Also in this figure the distribution of the wave

elevation samples

is

compared to the normal

distri-bution. In Fig. 5 the spectrum of the square of the

measured wave envelope

is compered

With the theoret-ical values found based on the random wave model

(see ref. [7]). In Fig. 6 the distribution Of this quantity is Competed with the corresponding theoret-ical distribution. As can be seen, the wave envelope squared process, which is of great relevance for the low frequency drift forces, corresponds well with

predictions based on the random Wave model. Test results

Time domain plots of the measured surge motions of the linearly and non-linearly moored tanker are shown in Figs. 7 and 8 respectively. The time plots

concern the first 7, (full scale) of the total 12 hours test duration. As can be seen the motions are almost completely dominated by the low frequency components. The irregular signal of the motion was

recorded on magnetic tape, low-pass filtered and subjected to statistical analyses.

The statistical analyses performed on the low

frequency surge motions yield the following data: Mean Value: n°N -

_{x °}

1 E xn

N01

Root-mean square value: a

a E n°N (x n - x) x N n°1 - 2

Maximum value: A MAX. +

Highest peak to zero value.

Minimum value: A MAX.

-Highest zero to trough value.

Maximum double amplitude: 2A MAX.

This is the maximum crest to trough value. Significant peak value: A 1/3 +

This is the mean of the highest one-third crest

to zero values.

Significant trough value: A 1/3

-This is the mean of the highest one-third zero

trough values.

Significant double amplitude: 2A 1/3

This is the mean of the highest one-third crest

to trough values.

Number of oscillations: NO

This is the total number of oscillations in the

record.

The results of the statistical analysis on the low frequency signal for the total test duration of

12 hours full scale are presented in Table 2. (N is number of samples)

to

SIMULATIONS

Input_data for computations

The still water viscous damping and the virtual mass of the linearly mooted tanker have been deter-mined by means of a motion decay test

_in

calm Watat. The following results were Obtained.:

The natural period

of

the system Te 315 s (ue ° 0.01995 rad/s).

The linear. (still water) Viscous damping

coeffi-cient was be ° 79.6 tfsnCl.

- The Virtual mass (m

+ a)

corresponded to 31,940

tfs2m-1.

Using the non-dimensional wave drift damping coefficient obtained from ref. [10] and the Wave spectrum given

in

Fig. 4, the mean wave drift

damp-ing has been computed accorddamp-ing to (see tel. [8]):

B12

(W)

2

f

S Aw,% dw

0

° wave drift damping coefficient.

damping bw amounted to 51.0

The total surge damping of the vessel exposed

to the specified wave specttua ecinalled: b 29.6 + 51.0 ° 80.6 tfam-1

and the non-dimensional damping coefficient becomes:

6 - 0.0519

The mean and the spectral density of the Wave drift forces were determined according to equations

(2) and (3) respectively, _using the MARIN Wave-Wind-Current (WWC) program. The following results were

found:

-173.6 tf

Sp(U*0) ° 206,073 tf2s

Results of frequency domain. computations

The folldwing results of the frequency domain

computations were found: .

The mean Surge motion is found based on equation

(4) and amounted to: - -175.6

°

15.5 -11.33 m

- The steady state value of the variance given

in

equation (7) amounted to:

2 x

a °

x 2 * 80.6 * while the steady

16.1 m.

Results Of time domain computations

Using the sate input data for the wave drift

force, the hydrodynamic damping and the Virtual Mass for both the linearly and non-linearly moored tank

in which:

B1 (w) 2 'a

The mean Wave tfsm

ran

15.5 * 206,073 ° 259.2 m2

state r.m.s. value amounted to ox

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THE STATISTICAL PROPERTIES OF LOW FREQUENC

322

Y MOTIONS OF NON-LINEARLY MOORED TANKERS OTC 5457

er, time domain Computations have been carried out.

Time domain plots of the computed low frequency

surge Motions of the linearly and non-linearly

moored tanker are shown in Figs. 9 and 10. The time plots concern the first 7f hours. (real time) of the total 12 hours simulation. The statistical analyses

of the simulation were restricted to the mean, r.m.a., maximum and minimum Value only..

The results

of

the statistical analysis of the low frequency signal for the total simulation time

of 12 hours real time are presented in Table 5.

COMPARISON OF RESULTS OF MODEL TESTS AND SIMULATIONS

Linear system

Gaussian distribution.

In Fig. 11 the distribution function of the

computed and measured low frequency surge motions are compared with the normal (Gaussian) distribution

function. The computed results Are given for five

different simulations of each 74 hours real time.

The measured results involve the full 12,-hour

rec-ord.

The correlation shows that the Gaussian

assump-tion fits well for both the simulations and the experiment.

Rayleigh distribution.

In Fig. 12 the distribution of the crests and troughs of the computed and measured low frequency surge motions are compared with the Rayleigh distri-bution. The theoretical distribution is, based on a narrow spectrum (c.0). The computed and measured data

concern

the full 12-hours. Again a pod

corre-lation is Observed.

Root-mean square of the motion variance.

In order to check the theoretical expressions on the motion variance and the r.m.s. of the motion

variance given by equations

(7)

and (16) the 12-hour record of the model test was subdivided in intervals

of 4 hour and 1f hour. These intervals Were analyzed with respect to the mean, r.m.s., maximum and mini-mum values of the motions. Although, strictly speak-ing, consecutive time intervals are not independent samples of the same stochastic process, we have

as-sumed that this will not affect results;

significant-ly. The results are given in Table 3 for subdivision

of

f

hour and in Table 4 for subdivision of

if

hour.

The same procedure was applied to the 12-hour record of the simulation. The results are given in Tables 6 and 7 for the subdivision of

_f

hour and 14 hour respectively.

Simulations Were carried out in order to inves-tigate the theory for longer durations of 3 hours

and 71 hours. The results are given in Tables 8 and

9.

From the results in these tables the mean r.m.s. value and the rim.s, of the motion variance were calculated. The results are shown in the Tables 10 and 11. The r.m.s. of the variance is presented in non-dimensional form, i.e. divided by the mean variance.

The results of these tables indicate a good correlation between the theory, the time domain

sim-ulation and the model tests and clearly show the

converging process on the reliability of the statis-tics for longer test/simulation duration. Further it

can be concluded from the results, that for a lin-early moored tanker In irregular head waves

equa-tions (7) and (16) can be used to estimate the

motion Variance and the r.m.s., of the motion

vari-ance respectively. For a good estimate of these

values knowledge of the correct wave excitation and

motion damping is requited.

Distribution of extremes.

For a designer, however, a certain maximum de-sign value is required. Based on the good,correla-tion with the Gaussian and Rayleigh distribugood,correla-tion of the low frequency motions of a linearly moored tank-er the theory of the extreme values may be applied,

see ref.. [19] and [20]. This theory provides the

probability that a certain maximum value will occur in a particular sea state having a certain duration. The distribution function of the extreme values is

of the following type:

!'max.) . 1 - e-e-Y

For a narrow band spectrum (z..0) the quantity y

Satisfies: fr*max. 12 j - In N ax in which: x mean value r.m.a value x xn

N = nidfiber of oscillations T/Te) T . considered duration of the sea state T natural period of system.

Based on the computed mean displacements, the steady state r.m.s. of the low frequency surge

mo-tion and a duramo-tion of

f

hour, (N 5.71

oscilla-tions) the theoretical distribution of the extreme values can be computed. For the linear syttem the

result is shown

in

Fig. 15.

Associated with the extreme is the most prob-able maximum xeek.(e.pt.). This can be defined as

follows, see ref'. [21]:

m ax.(m.pr.) x". +

axiritTii

The probability of exceedance is:

P(xmax.(m.Pr.)) _{(1 - e} ) * 100%

. 63.2%

It must be noted that in terms of statistics

this means that of 100 independent recordt of the

low frequent), Surge motinns with a duration of

hour each, 63.2 records will show a maximum value

higher than the Most probable maximud value.

By selecting the values of the maximum negative surge motions (troughs) from Test No. 1 through 24 presented

in

Table 3 and from Run No. 1 through 24

given in Table 6, the distribution function of the extreme measured and computed values can be con-structed, The results are shown in Fig. 15. Taking

into account that only 5.8 oscillations occur per sample of 4 hour and only 24 samples were available,

(7)

Non-linear. szstem

For the non-linear system, deviations from the cumulative distribution function according to Gauss and Rayleigh occur, see Figs. 13 and 14.. In order to estimate the maximum design value for non-linear

systems often model tests will be carried out. Model tests are mostly carried out for a. restricted

dura-tion, e.g.

if

hour.

By means of the distribution function of the troughs and crests obtained from this ₁₄ hour sample

the Most probable maximum

_{P(xwax.(w,pro)}

1/N can

be determined. To longer durations extrapolation procedures can be applied.

Starting from the most probable maximum value, a safety factor, which in broad sense covers

uncer-tainties in the whole. process. -by which loads are found as well as structural Uncertainties, has to be applied in order to arrive at the design value. The uncertainties In the process leading to the loads incorporate, of course, the statistical

Uncertain-ties of the kind discussed in this paper.

For the non-linear system the uncertainties in the statistics as expressed in terms of variance of the variance for the linear system is expected to

have a sate. trend. An examine of the uncertainties.

can be seen in the distribution functions _of _the trough values derived from Test No. 27 and 29 and

shown in Fig. 16.

In order to obtain realistic design values, re._T ducing or eliminating the safety factor for the

uncertainties in the statistics, long duration model tests, or long duration simulations in combination with Model tests can be considered. Having the long duration tests/simulation the distribution of the

extreme values can be established. The result fot

non-linear system based on the extremes which can occur with a duration of 4 hOdr is shown in Fig. 15.

CONCLUSIONS

It has been shown that, for the case of a

tanker moored in irregular head waves by means of a

mooring system with linear restoring properties, it' is possible to estimate the statistical variance of the low frequency surge motions based on estimates of the surge damping, the natural surge period and

the test/simulation duration.

Comparisons between results of model tests and time domain simulation show that for the head sea condition, a simple time domain simulation procedure can be used to produce long duration simulation rec-ords of the surge motion for both linearly and non-linearly moored tankers.

It has been shown that the test/simulation

duration has a significant effect on the statistical

variance of the low frequency surge motions.

In-creasing the duration reduces the statistical

vari-ance. If significantly longer test/simulation

di:re,-tions are used than are used to-day in engineering practice, the consequences with respect to possible

reductions in safety factors should be Considered.

NOMENCLATURE

A . wave .group envelope

a . low frequency added mass . breadth of vessel

B (w) . wave drift damping coefficient

bw bo 11.1.0 P(w1,432), Q(w1

rwi)

4(0

S, T1 Ii Ue o Cs 6 323 OTC 5457 PINKSTER/WICHERS.

total damping coefficient

mean wave drift damping coefficient . low frequency still water damping

coefficient spring constant

. wave drift force

m mean wave drift force.

. length between perpendiculars mass of tanker

area of the spectrum

number of low frequency oscillations or samples or tests/rune

= components of the quadratic transfer

wave spectrum

function dependent on miand U2

. spectral density of the wave drift forte tot frequency P

low frequency surge spectrum

. Mean wave period

draft of the vessel or duration of

test/simulation . surge displacement

mean displacement

parameter

. frequency

natural frequency of system root-mean square Value

. frequency

non-dimensional damping coefficient

. wave elevation

REFERENCES

Poranski, P.F., Gillespie, A.M. and Smulders, L., "The first yoke mooring for a VLCC in the

open ocean", OTC Paper No. 3564, 1979.

Rimery, G.F.M. and Stambodzos, M.A., "Tanker based marginal field production: 8T-years

opera-tional experience", OTC: Paper No. 4036, 1985.

Eppley, D.R. and Van Berkel, P.B., "12 Months' operational, experience with an FPSO. Handling the production from 3 fields offshore Nigeria",

OTC Paper No. 5491, 1987.

Grancini, G., Jovenitti, L.M. and Pastore, P., "Moored tanker behaviour in crossed sea. Field experiences and model tests", Symposium on

De-scription and Modelling of Directional Seasi Technical University of Denmark, Copenhagen, 1984.

Remery, and Hermans, A.J., "The slow

drift oscillations of a moored object in random seas", Society of Petroleum Engiheets Journal,

June 1972.

6, Pinkster, J.A., "Low frequency second order

wave exciting forces on floating structures", MARIN publication No. 650, 1980.

Pinkster, J.A., "Numerical modelling of

di-rectional seas", Symposium On Description and Modelling of Directional Seas, Technical

Uni-versity of Denmark, Copenhagen, 1984.

Webers, J.E.W., "On the low frequency surge

motions of vessels moored in high seas", OTC

Paper No. 4437, 1982.

Wichers, J.E.W. and Huijsmans, "On the

low frequency hydrodynamic damping forces act-ing on offshore moored vessels", OTC Paper No.

4813, 1984.

Wichers, J.E.W., "Progrede in computer

simula-tions of SPM moored vessels", OTC Paper No.

(8)

TABLE 3-SIGNIFICANT QUANTITIES OF THE MEASURED UNDISTURBED WAVE HEIGHT AND THE

LOW FREQUENCY SURGE MOTIONS

Duration of each

test

= 1/2 hour'

TABLE 4-SIGNIFICANT QUANTITIES OF THE MEASURED UNDISTURBED WAVE HEIGHT AND THE

LOW-FREQUENCY SURGE MOTIONS

Duration of each test . 1 1/2 hair

TABLE 5-SIGNIFICANT QUANTITIES OF THE COMPUTED LOW FREQUENCY

SURGE MOTIONS

Duration of each simulation = 12 hOurs

.325

Test No.

undisturbed wave height . Low frequency surge motion

Linear system Non-linear system

r.m.s. A MAX. + A MAX. - mean r.m.s. A MAX. + A MAX. - mean r.m.s. A MAX. + A MAX.

-1 3.07 14.33 -9.42 -10.90 17.28 34.58 -45.64 -5.95 22.42 45.09 -45.62 2 3.08 16.50 -9.29 -8.85 16.15 32.94 -41.56 -6.51 20.33 44.78 -45.04 3 3.18 10.24 -10.07 -10.45 21.56 34.48 -53.22 -12.43 13.62 16.13 -43.37 4 3.01 13.27 -8.13 -9.89 17.03 27.59 -46.85 -9.92 15.88 22.96 -39.65 5 3.12 12.29 -10.32 -11.40 18.01 24.21 -50.81 -9.73 17.83 32.84 -47.73 6 3.14 11.84 -9.55 -10.09 22.15 39.49 -49.98 -8.32 18.98 36.82 -45.56 7 3.17 14.59 -10.44 -12.60 38.08 54.94 -76.75 -4.42 26.60 47.66 -51.24 8 3.21 12.33 -9.74 -9.99 15.81 20.11 -38.57 -11.59 13.14 23.07 -39.26 9 3.27 11.39 -9.84 -10.66 14.00 26.30 -38.82 -12.94 15.37 25.73 -37.40 10 3.14 11.21 -8.56 -11.64 21.04 29.76 -53.07 -8.96 16.77 30.98 -43.46 11 3.34 11.77 -9.91 -13.28 15.57 20.10 -40.49 -16.56 10.71 10.77 -35.92 12 3.15 11.19 -9.22 -9.31 13.17 12.99 -36.59 -9.80 12.66 12.98 -36.88 13 3.05 12.54 -10.77 -10.38 13.76 13.23 -43.29 -9.87 16.20 17.16 -42.94 14 3.16 11.73 -8.32 -11.37 12.91 16.83 -35.82 -12.87 14.00 19.45 -37.32 15 3.22 11.72 -10.09 -12.01 11.98 16.13 -36.33 -15.63 7.81 2.78 -28.64 16 3.19 12.61 -8.32 -10.58 15.94 17.36 -49.49 -10.97 12.21 21.93 -42.88 17 3.25 11.87 -9.04 -13.13 14.04 16.14 -37.95 -14.28 14.48 19.57 -38.92 18 3.01 10.19 -8.88 -8.25 17.78 20.66 -44.73 -5.25 20.12 32.14 -43.24 19 3.21 15.28 -10.37 -12.59 15.48 26.22 -45.85 -14.66 13.57 17.47 -38.09 20 3.18 11.93 -7.13 -11.25 13.04 15.96 -44.47 -13.09 13.39 16.94 -39.40 21 3.34 12.01 -8.98 -12.49 15.16 16.33 -42.95 -13.15 14.82 17.56 -40.47 22 3.11 12.78 -8.79 -8.47 16.17 32.23 -38.36 -10.94 14.84 32.81 -36.07 23 3.19 13.10 -10.15 -12.42 15.66 27.62 -45.52 -12.83 14.01 23.06 -41.48 24 3.38 12.15 -9.09 -11.75 22.32 28.86 -61.09 -12.11 14.90 22.63 -42.52

-Low frequency surge motion

Te st Undisturbed wave height

Linear system

. Non-linear system

No.

r.m.s. A MAX. + A MAX. - mean r.m.s

-A M-AX. + A MAX. - mean r.m.s. A MAX. + A MAX.

-25 3.12 16.50 -10.07 -9.92 18.49 34.58 -53.22 -8.30 19.37 45.09 -45.62 26 3.10 13.27 -10.32 -10.43 19.18 39.49 -50.81 -9.33 17.60 36.82 -47.73 27 3.23 14.59 -10.44 -11.07 25.17 54.94 -76.75 -9.64 19.64 47.66 -51.24 28 3.22 11.77 -9.91 -11.40 16.98 29.76 -53.07 -11.77 14.03 30.98 -43.46 29 3.15 12.54 -10.77 -11.24 12.91 16.83 -43.29 -12.78 13.36 19.45 -42.94 90 3.16 12.61 -9.04 -10.65 16.11 2066. -49.49 -10.16 16.37 32.14 -43.24 31 3.25 15.28 -10.37 -12.12 14.61 26.22 -45.85 -13.64 13.96 17.56 -40.47 32 3.24 13.10 -10.15 -10.93 18.32 32.23 -61.09 -12.01 14.55 32.81 -42.52

mean r.m.s.. A MAX. + A MAX. - mean r.M.s. A MAX. + A MAX.

(9)

TABLE 6-SIGNIFICA4T QUANTITIES OF THE COMPUTED LOW-FREQUENCY

Duration of each simulation 1/2 hour

TABLE 7-SIGNIFICANT QUANTITIES OF THE COMPUTED LOW-FREQUENCY SURGE MOTIONS

Duration of each simulation . 1 1/2 hour

TABLE 8-SIGNIFICANT QUANTITIES OF THE COMPUTED LOW-FREQUENCY SURGE _M

Duration of each simulation 3 hours

826 Run

No.

Linear system _{Non-linear system}

mean r.m.s. A MAX. + A MAX. - mean r.m.s. A MAX. + A MAX.

-1 -11.68 12.98 22.35 -38.02 -13.09 14.32 25.91 -38.19 2 -8.40 9.09 7.74 -28.13 -11.33 11.18 8.87 -31.73 3 -13.81 11.71 6.91 -52.74 -13.12 13.62 16.00 -42.95 4 -9.98 20.77 33.39 -50.71 -7.80 21.87 39.72 -44.21 5 -12.57 7.64 5.12 -30.38 -15.63 9.86 8.05 -32.95 6 -9.89 9.03 12.03 -29.37 -11.88 12.46 16.93 -34.91 7 -11.85 26.20 41.43 -58.32 -9.09 19.66 33.80 -46.85 8 -9.53 20.79 36.28 -6i.43 -9.06 16.71 22.41 -42.48 9 -11.50 24.65 44.43 -74.10 -9.28 19.07 41.72 -51.10 10 -13.56 25.99 36.04 -59.96 -11.13 16.44 27.15 -40.58 11 -9.17 12.59 -19.82 -34.86 -11.96 11.61 13.00 -35.60 12 -10.36 8.98 11.60 -34.50 -12.66 10.70 7.56 -36.39 13 -9.62 13.38 16.43 -42.82 -7.62 17.34 26.01 -43.50 14 -11.32 15.02 13.62 -48.98 -11.52 13.95 16.57 -41.66 15 -10.65 12.05 13.08 -33.63 -11.78 13.25 13.56 -35.73 16 -13.50 15.96 14.78 -45.51 -12.87 15.47 16.51 -39.72 17 -9.06 7.85 11.50 -26.60 -12.13 7.46 2.37 -31.50 18 -9.94 16.27 24.44 -44.49 -8.12 19.32 32.79 -41.29 19 -13.15 21.12 28.16 -53.46 -12.39 15.53 18.54 -40.22 20 -14.07 16.44 21.00 -52.80 -9.99 19.82 44.76 -49.12 21 -9.19 13.40 14.76 -36.20 -9.46 16.62 26.03 -38.84 22 -11.51 16.96 16.53 -41.95 -9.77 15.60 14.54 -40.71 23 -11.93 20.95 29.01 -56.50 -2.61 30.19 60.09 -54.48 24 -13.23 15.44 18.03 -47.63 -14.30 12.47 10.90 -37.91 Run No.

mean r.m.s. A MAX. + AMA)(. :, Mean r,m.s _{A MAX; +} _{A MAX.}

-25 -11.26 11.61 22.35 -52i74 -12.60 13.00 25:91 -42.95 26 40.59 13.82 33.39 -50:71 -11.55 16.06 39.72 -44.21 27 -11.19 24.03 44.43 -74.10 -8.94 18.51 41.72 -51.10 28 29 10.71 -10.74 17.53 13.41 36.04 16.43 -59.96 -48.98 -11.95 -0.56 13.11 15.01 27.15._ . 26.01 -40.58 -43.50 30 40.92 13.97 24.44 -45.51 40.89 14.50 11.57 -41.29 31 41.91 17.88 28.16 -53.46 -10.10 17.90 44.76 -49.12 32 -12.12 17.66 29.01 -56.50 -9.28 21.42 6049 -54.48 Run No. Linear system _ Non-linear system

mean r.m.s A MAX. + A MAX. - mean r.m.s. A MAX. + A MAX.

-33 T10.88 12.75 33.39 -52.74 -12.03 14.61 39.72 -44.21 34 -10.99 21.04 44.43 -74.10 -10.49 16.12 41.72 -51.10 35 -10.82 13.69 24.44 -48.98 -10.83 14.59 29.67 -43.50 36 -11.90 17.76 29.01 -56.50 -9.42 19.81 60.09 -54.48 37 -12.43 18.40 32.88 -73.38 -10.53 17.63 51.65 -54.43 38 -10.85 14.87 25.52 -51.78 -11.55 14.47 34.83 -42.55 39 -10.95 13.37 24.16 -50.02 -11.98 13.79 28.44 -41.66 40 -11.12 12.49 20.26 -47.04 -12.68 12.75 25.23 -41.47

(10)

TABLE 9-SIGNIFICANT QUANTITIES OF THE COMPUTED LOW-FREQUENCY SURGE MOTIONS

Duration of each simulation =7 1/2 hours

TABLE 10-STEADY-STATE AND MEAN VALUES OF THE ROOT-MEAN SQUARE

(LINEAR SYSTEM)

TABLE 11-NONDIMENSIONAL R.M.S. OF THE VARIANCE AS FUNCTION OF THE

TEST/SIMULATION DURATION (LINEAR SYSTEM)

327

Run Linear system Non-linear system

No.

mean r.m.s. A MAX. + A MAX. - mean r.m.s A MAX. + AMAX

-41 -10.84 16.68 44.43 -74.10 -11.10 15.33 41.72 -51.10 42 -11.91 17.35 32.88 -73.38 -10.26 17.89 60.09 -54.48 43 -10.87 13.69 25.52 -51.78 -11.70 14.02 34.83 -42.55 44 -10.83 15.92 35.43 -63.24 -11.72 14.32 37.50 -44.35 45 -10.91 16.97 40.51 -68.50 -10.37 16.41 41.70 -46.47 Duration of test/run Steady state az (m) N 0, t fa )2)1/2 N n=1 % xn, , Number Of tests/runs N _ Simulation (m) __ Measurement (m) -1/2 hour 1 1/2 hour 3 hours 7 1/2 hours 12 hours 16.1 16.1 16.1 16.1 16.1 16.6 16.6 15.8 16.2 16.8 18.0 18.0 -18.1 24 8 8 5 1 1 (la )2 ()2)2 " )1/2 1 Duration of test/run T (s) NUmber of tests/runs N /74717-e (1T E n..1 (-2-2"Ii i

Computed Simulation Measurement

1/2 hour 1800 0.73 0.68 0.79 24

1 1/2 hour 5400 0.42 0.47 0.41 8

3 hours 10800 0.30 0.39 T 8

7 1/2 hours 27000 0.19 0.15 - 5

(11)

Fig. 1flode glen of tenlon. A.P. 25.0 12.5 0.0 40.0 20.0 0.0

Fig. aThal setup.

IAN:CLAM SEAS (117111. 21 C0CLE31

NAVE SPECTRUM

ISeemorysd 14%6--12.B.pTi - 11.0

17morostLesol. I P.M.)

s44.

13.0 TI ILO

...

F.P.

IOC ELEVATION WIVE

I.

7499

a , -3.17. 0 16.5 C_{o emu}--10.8. 0. - Z1.9 - 2' 4

In1

..7". ..- -N-.

.--1.,-i

tf 1250

I

1 1 I ? 1000 s 1 1 1 I 750 / 1 I I 560 250 ti 60 _ _ 40 - 0 . metres 40-- - 40 --60

/

Fig. 1-91a9e loe&defleellon come for the Meat end nonlinear mooring system. _{Fig. 4-01ErIbullon funeflen end apeelnan of Int grimly wave.}

1.5

0.0 0.5 .1.0

WAVEPRES1DICVINRFOtS

-10.0 0.0

ELEVFITION IN 11

(12)

4000.0

I

moo.°

51030.0

0.0

TEST NO. 7499

OCR WOO MOM LOW rACOUCNCT PART Or SOUAAtO AVE RECORD

DERIVED THEORETICALLY BASED ON SPECTRUA or monsuoco Ron

40 20 40 20 ° I, 40

a

20 0 -20 0 20 -20 0 20 TEST NO. 7499

DERIVED nom LOWrocourxeyPART or SOuRRED mRvC WJI3R0

OCRIval THEORETICALLY BASED ON SPECTRUA Or REASURED mAvC

I.

i 1

i

I I i r , i l i

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1111M11UMillI1111111121MIN11 NIMIN/iN1111 MINNA UII

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I I T

I

111117.111111MMIIIII1111111111111t=1111MIE1111111111i1U11111111MCW. .

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I

.5

This in hours

6-Reeord of Om nurse motion dolmol horn model lasts (nonlinear $ystem).

329

/C\-'s

\ S

S S \ \ \

\\

\ ...

N...

0.00 0.25 0.50 0.75

GROUP CREOUENCT IN ORD/0

Flo. E.-Spectrum of the is groups.

0.0 50.0

INIt

11-.DMI11butIon lonalon of ems poop.

60 75

This In hours

Fig. 7--Ilecord ofIllsmoos motion dethld front model tabONMEN segforr*

20

limenwiaimensimaimarmlosiromwaisiwimmuurimimmivsnum

I 1111111MIMIIMMIIMIW5W1111111111 Nal11111111111111111MIIVA

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60

_-75

ad 16' ID. 100.0 150.0

(13)

1

411

Pig. 11-Diaribution function of the lawfrespianey surge ashen.

I

ilIIICIIIII411/.=111111=111111WWWINNOGIWIWIIIMM

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l _{2111Ilibrana 111:1111,1211 MUM:11=11=111=ica la}

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lIMINI11 =ma.. 1.111=hommIllIIIIIIIIMITIMAIIITSIIIIIIi=1MIT

Ii'

60

Time in hours

Fig. 10-Roccad of tha ahnutalad knalraciunney mono= guanines, Nyman*

LINEAR SYSTEM 330

I

:2 to 20 10 5 1 0.1 50.0

75

LINEAR

51aulationTest No. 7501Piration 12 hours

T7140177/ (Rayleigh)

o. o

X DOWER 14 11

Fig. 12-0141tOsulhan lunation of the ants end trusts al Use losafregueney saga modem

-100.0 Tia. to 09.13, 041.5 89.0 1113.0 05.0 93.0 05.0 00.0 70.0 60.0 50.0 40.0 30.0 20.0 15.0 10.0 5.0 2.0 Na. 7507 No. 41 No. 42 No. 43 No. 44 No. 45 Maus - Duration 12 hours Cunt ion 7 1/2 hours I Test X Run 0 Run A kin 0 Run 0 Run - Theory w 1.0

es\

3.5 3.7 1.1 -3 -2 -1 0 , \*S' 50.0 0.0_ 50.0 ion

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(14)

26.0 15.11 99.5 99.0 59.0 93.0 58.0 85.0 80.0 d 70.'0

I

15.0 10.0 5.0 2.0 1.0 NON-LINEAR SYSTEM 1

Test No. 75 2 - Duration

12 hours ti Run No. el Run no. 42 A lion No. 43 Run No. 44 a Run No. 45 -Theory (Gam) Ikration 7 3/2 hours -2 /a.

Rs. 13-Diartbution funcain 411 the 1.w4taq010ey mg. meta&

808-LINEAR MUM

9 CFINKCIt 11

R. 111-011411butket function of MO kreetrequomee extreme Wan ORRIN*

2 3 0 1 OS es 1 es ei: 70 3 80 P40so 40 20 10 5 0.1 1 100 99 95 90 BO 70 GO SO 40 30 20 10 5 0.1 0.1 50.0 11.0 -53.0 1209231 INN

Flo. l4---01eidhotIon Mallon otmote end 000950 01 the towdniquener amp motion.

-NON-LINEAR SYSTEM

Test No. 7S12)

Sleulatten Dural on 12 hiurs

CRESTS -50.0 56:6 4.0 100.0 -1CO. 0 4 0

NON-LINEAR SYSTEM - EXTREBE ALOES

modaltosa

t.=01

Oetermined f 2i tests eite ell Ifefe of 5.8

f

_{low frequeniy oscillations}

1201 Stoulation 4..., ,

_\

s -....".... ... N... N _..- -Theory . N.... sN... s.,... \\*.:**>

'N..,

N...

yew , _{Test No. 751}

- 12 hours Ito. 27 _{11/2 hours} No. 29 es 90

- Test

_{--..-- Test} ro so -o3 to 1 so to 5 1 1 1.1 53.0 MO_ -50.0 -tan .. ___

....t Cleiermined from 14, tints

Withan averageof6.0

Ulufrequency oscillations

,,,,Liii.

,,,,,,e, -...\.

\.

_ -i)/°e

Ro. 16-01ohlbellom londlon of the houghs et the lowetnquenee some motions.

331 ON so SO 70 80 50 40 30 ON IC 5 1 1 0.1