Statistics of high and low frequency motions of a moored tanker

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Scheepshydromechanin

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7

A 88014

OTC No. 6178

STATISTICS OF HIGH AND LOW

FREQUENCY MOTIONS OF A MOORED

TANKER

J.A. Pinkster

.February 1989..

_J

Copyright 1989. Offshore Technology Conference

ThiS paper was presented at the 21st Annual OTC in Houston, Texas, May 1-4, 1989.

This paper was.selected for presentation by the OTC Program Committee following revieW of information contained in an abstract submitted by the author(s). Contents of the paper. as presented, have not been reviewed by the Offshore Technology Conference and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Offshore Technology Conference or its officers. Permission to copy is restricted to an abstract of not More than 300 Words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented.

ABSTRACT

The separation of the total motions,,of a

tanker moored in head seas into wave frequency and

low frequency components with the purpose of de-veloping simplified simulation computations: is

discussed. The relationship between these motion

components is clarified using results of long du-ration model tests in irregular head seas. Results

of time domain simulations using a simple model

are used to illustrate some of the problems facing the designer of a moorinusystem

INTRODUCTION

Rational, design of mooring system; for perma-nently moored floating production and/or storage

systems requires accurate and statistically reli-able data on the 'motions of the vessel and the

' forces in the mooring system.

Tanker-based FPS systems moored in high seas carry out wave frequency motions and superimposed,

low frequency horizontal motions induced by mean and slowly varying environmental forces such as

the second order wave drift forces or wind gusts. Low frequency motions are generally resonant due

to the fact that the excitation frequencies coin-cide with the natural frequency resulting from the

mass of the vessel and the stiffness Of the

moor-ing system. Such motion components can be large

partly due to the relatively low system damping. The combined high and low frequency motions of the 'Vessel combined vith the hydrodynamic forces acting on the components of the system excite forces in the Mooring system. The design of the

mooring system and its components requires insight References and illustrations at end Of paper.

in the frequency content, distribution and ex-tremes of the mooring system loads. Data on the system loads can be obtained directly from model tests carried out with a scale model of the vessel

and the mooring system in model basins in which

all .relevant environmental effects can be

gen-erated on scale. This generally requires model

basins in which it is possible. to generate simul-taneously current', irregular waves and wind. Model tests of- this kind are generally carried Out in a

later stage of the design process and nowadays are often carried out to confirm previoUs design

deci-sions made on the basis of existing data for

sim-ilar systems or based on computations-.

In the earlier stages of the design process

the motions of the vessel and the associated

moor-ing loads can be estimated based on computations.

The phenomenon involved are however very complex

and mathematical models which describe in detail

the major effects of the environment, the response properties of the vessel and of the mooring system

can become very time consuming to develop and too difficult to use for all but the most experienced

(ref. [1]).

For

a detailed insight in the. system behaviour for general cases

such

models are how-ever necessary. For preliminary design purposes

simpler models Vhich require less experience in

use and less computational polder are needed. One

of the Most important simplifications is based on

the knowledge that in many practical cases the

mooting system design is governed by the survival head sea condition in which all environmental effects are parallel and from ahead. This

assump-tion has proved to be one which has reduced the

problems associated with the mathematical

model-ling by an order of magnitude. In terms of the

applicability of this case for the design of the mooring system this assumption has also been

con-firmed in many caset. Only for mooring systems

sensitive to lateral loading such as the SALS and

2 STATISTICS OF HIGH AND LOW EREQUENCY MOTIONS OF A MOORED TANKER OTC 617

to consider more complex mathematical models in-volving not only the surge motions of the vessel

but also the sway and yaw motions. In this paper

we will,

only

consider the head sea case. This is

always the governing case for, for instance,

tur-ret- moored vessels. In this 'case the motions of

interest are the surge , heave and pitch motions .

of the tanker.

SIMPLIFIED MODEL OF VESSEL AND MOORING SYSTEM BEHAVIOUR

A complicating factor in setting up a simpli-fied mathematical model of the horizontal motions

of the vessel is formed by the fact that the

mo-tions are carried out in tvo.distinctly different frequency regimes, i.e. at wave frequencies and at frequencies corresponding to the natural surge period. To all intents and purposes the vertical motions of tankers only consist of wave frequency motion components. In integrating the equations of motion the time step is governed by the highest

frequencies present it the notions.' When consider-ing wave.frequency components this results in time

steps of the order of 0.5 to 1.0 s. If it is

de-sired to carry out simulation computations for durations which give results with sufficient sta-tistical reliability, the total simulation time is governed by the number of oscillations of the low frequency horizontal motions. Combined with the

small time step of the simulations as required by

the vave frequency motion components, this leads

to a very large number of time steps in the simu-lation with correspondingly large computational

loads. This is an undesirable situation

in

the

preliminary design stage. An obvious. approach to this problem is to view the wave frequency motions

and the low frequency motions as separate pro-cesses which can be analyzed independently. Wave

frequency motions are computed based on standard linear hydrodynamic theory, often in the frequency

domain, while the low frequency Motions are com-puted it the time domain. The extreme in the total

motion is often determined by adding the extreme by frequency motion to the extreme wave frequency

motion. This is a practical approach _by means of

which it is possible to assess the extreme loads in the mooring system.

This method has, however, shortcomings from

the 'point of view of obtaining statistical data on the motions and mooring forces. Based on this sim-plification it is not possible to assign proba-bility levels to the extreme mooring forces or

motions. The question also arises whether it is

permissible to view the wave frequency motions and

the low frequency motions as independent

pro-cesses. Even though these motions take place in

different frequency regites, both are derived from

the wave elevations and as

such

Must be related. The separation into independent processes does

however have its attractions so that it is of interest to investigate the validity of this assumption.

MODEL TEST WITH A 300 kDVT TANKER

In order to evaluate the independence of the wave frequency and low frequency motions Of a

tanker in head seas, long duration model tests

previously _{carried out at MARIN with a scale 1:80}

model of a fully loaded 300 kDVT tanker vere

ana-lyzed. The vessel was moored by a spring system with linear and with non-linear restoring

charac-teristics simulating a SALS and a Turret type mooring.

The tests were carried out for a duration of 12 hours full scale in a North Sea survival condi-tion with the following particulars:

Significant_vave height: 13.0 m Mean wave period : 12.0 s

The main particulars of the vessel are given

in Table 1. The test set-up is shown in Figure 1. The wave spectrum is shown in Figure 2 and the restoring characteristics of the mooring system are shown

in

Figure 3.

The surge motions of the vessel were measured

by means of an optical tracking device consisting of a point light source on the model being tracked

by a basin-fixed tracking system. In Figure 4

re-sults of the measurements in terms of the wave

elevation record and the associated surge motion

record for the linearly and non-linearly mooted vessel are shown. This figure shows that the surge

motions are dominated by the low frequency motion

components. This does not mean however, that the

wave frequency motions can be neglected. For

in-stance, when considering such aspects as dynamic magnification effects in chain forces, the combi-nation of low frequency motions which give the

static offset to the mooring system and the wave frequency motion component which induce additional

dynamic _{magnification effects, is of great}

impor-tance.

In order to determine whether, as a first approximation, the high and low, frequency motion

components can be regarded as independent quan-tities the following analyses were carried out.

The total surge motion record was filtered to

yield separate time records of. the wave frequency

and low frequency motion components. The cut-off frequency of the filter was Chosen based on the

spectrum of the total motions. In Figure 5 the

high and low frequency surge motion components are

shown along with the wave elevation record. The dependence of the wave frequency motions on the

by frequency

Motions (or the reverse) can be

viewed in different ways. One way is to determine the coherence between the two components. Since

the _{motion components cover non-overlapping}

fre-quency bands this will Indicate that there is no coherence. A second way of analyzing the

depen-dence is to determine the coherence between the

low frequency motions and the low frequency part

of the square of the high frequency motions. The background to this method is the knowledge that

the by frequency motions are caused by second

order low frequency wave drift forces. These

forces _{are related to the square of the wave}

ele-vation record and hence one may expect that the

by frequency motions viii show some coherence in relation _{to the low frequency part, of the square} of the wave frequency surge motions.

The results of the coherence computations are shown in Figure 6 to a, base of the motion frequen-cy. It is seen that the coherence is close to zero over the complete frequency range of interest.

Based on these results it can be, concluded

that, for a first approximation we May assume the vave frequency surge motion components to be

in-dependent _{of the low frequency surge motions}

cow-ponents. It should be noted however, that, it is probable that a greater degree of coherence Will

be _{detected using bi-spectral analysis techniques}

which take into account in a more correct way the quadratic dependence of low frequency motions and wave frequency quantities. Having established that

high and low frequency motion _{components can, at} first approximation, be Viewed as separate inde-pendent processes, the question nov is how to

model the separate: processes and how to obtain the.

total result.

When viewing such aspects as the motional be-haviour of a moored tanker in head seas it viii be clear that the low frequency motion components are

influenced by the mooring system characteristics. Determining these Motion components for the gener-al _{case involving nonlinear mooring system}

char-acteristics requires time domain simulation

Com-putations. For the case of mooring systems with

linear restoring characteristics, the low fre-quency motions can be determined by analytical

means, assuming that the low frequency surge

mo-tion can be described by means of a second order equation of motion with constant coefficients and the by frequency excitation by meant _{of a white} noise spectral shape (tee ref. [2])t

F_

Mean offset: x_m (1)

c

HMS of the slowly varying part of the offset:

x lit

S2bc

where:

S() = wave spectrum

TT(4)

. mean drift force coefficient. -a

' For the case that the mooring system restoring

characteristics are non-linear, time domain

simu-lation computations can be carried out based on

the _{same assumptions with respect to the low}

fre-quency part of the surge excitation. We' have

chosen for a Monte Carlo process to. describe, the

surge _{excitation force. In order to take into}

ac-Count the basic quadratic relationship betveen the

wave _{elevation process, which is normally}

distri-buted, and the slowly varying Wave drift force ex-citation, we have selected a model which results

in _{an exponentially distributed surge force. The}

following expression for the time domain represen-tation of the force complies with

theserequire-ments:

F - oF(B+1) Sign(F) + Fm (4)

in which:

B ln(rnd(a)) 0 < rnd(a) < 1

Fd mean surge drift. force

4-

17-7-F ' At

At = time step of the simulation.

In Figure 7 and Figure 8 the white noise model and the exponential distribution are compared with

'exact' representations which take: into account in

a correct way the quadratic _{relationship between}

the _{wave and the drift force (ref. [3]). The}

com-parisons

show that with respect to the spectral shape differences occur at freqUencies higher than

about 0,5 rad/s. The influences of these .differ-ences _{on the resultant motions is generally small}

due to the fact that the by frequency motions are resonant in nature and hence highly tuned around

the _{natural surge frequency. Depending on the}

ac-tual mooring system stiffness this will lie in the frequency band from 0 - 0.5.tad/s. Results of sim-ulation _{computations compared to model test'}

re-sults given

in

reL(2) confirm this effect. In Figure 9 an example is _{shown of a typical surge}

drift force record at _{obtained using the simple.}

Monte Carlo method. The simplified method reflects

the skewness in the 'true' force but differences

occur. around the 'zero force level. .

The selection of a simplified model for the

wave _{frequency part of the motions hinges on the} effect _{that the mooring system properties have on} these motion components. It is generally assumed

that _{the effect of the mooring system on the wave}

frequency motions is small. _{In Figure 10 and in}

Figure 11 the distributions and amplitude, response

functions of the wave frequency surge motion

com-(5)

in which:

bm total system damping

. mean part of the surge force . mooring system stiffness

spectral,density of the wave drift forces

2 [S (w) . dco

C 2

'a

ponents as obtained from the previously described

model tests vith the linearly and non-linearly moored tanker in high irregular head seas are shown. The comparison confirms that the effect of

the difference in restoring force characteristics of the mooring is negligible. This suggests that

it

is

not necessary to solve the equation of

mo-tion for these components for a different mooring

system. All that is required is a representation of the wave frequency motion component which

con-forms with the particular sea condition and which

is given

in

a form consistent with the

representa-tion

of

the mooring system response to the wave

frequency motion component. For example, for a

linear mooring system with static restoring

coef-ficient c.and with negligible additional dynamic effects in the response to the vessel motions, it

Is sufficient to describe the wave frequency

mo-tions in the frequency domain in terms of its

spectral density. The same quantities are directly derivable for the corresponding mooring force

com-ponent. The frequency domain description can

easily be transferred into a representative time

record which can be added to the low frequency result obtained from time domain simulations.

For mooring systems vith non-linear restoring characteristics and significant additional dynamic effects such as turret moorings, which require

time domain solutions of the equations describing

the moorings system response, presentation of the wave frequency vessel motions as time records can

be more appropriate. This approach to determining

the forces in the mooring system requires

consid-erable computational effort. For preliminary

de-sign purposes analytical solutions to the mooring

line dynamics problem may be more suitable (see

ref. 14]). Selection of such methods will require an appropriate representation Of the vave frequen-cy motion components.

The total motions and mooring forces are found

by addition of time records of the wave frequency

and low frequency components. The statistics of

the total motions and forces can then be deter-mined and preliminary design data such.as the most

probable maximum mooring. force can be derived.

APPLICATION TO THE LOW FREQUENCY SURGE MOTIONS AND MOORING FORCES

In the folloving two examples of the

applica-tion of the above described approach for the low

frequency components of the mooring force will be

'

The first applies to the case of a sensitivity analysis with respect to the stiffness of a lin-early Moored tanker. The case concerns the 300

kDWT tanker moored in head seas in the

aforemen-tioned survival condition. It

is

required to in-vestigate the effect of

a

20% increase in the

stiffness of the linear mooring system on the

maximum mooring load.

This will be carried out making use of time domain Simulations of the low frequency motions

based on the constant coefficient equation of

motion for surge. The following values were used for OE simulation computations:.

Virtual mass - m = 38940 ifs2/m -Damping coefficient b 80.6 tfs/m Mean drift force Fm = -175.6 tf, Drift force spectral density S 2060073 tf's

Such time domain simulations are carried out

for a limited duration. One of the questions aris-.

ing from this approach concerns the validity of

the answers in view of the statistical reliability

of the results. Due to the sample variance effect

there is the possibility that comparison of re-sults of tvo simulation calculations carried out -for identical environmental conditions for two

different .values of the mooring stiffness may yield incorrect conclusions.

We will investigate the probability of

incor-rect conclusions being drawn by carrying out 10

simulation computations for the' original mooring

stiffness and 10 simulations for the increased

mooring stiffness. The environmental force records

will

be the same thus simulating the case of

re-peated model tests in the same wave train.

Compar-ison of all ten cases with respect to the maximum mooring force

will

reveal the probability level

that an arbitrary realization can lead to

incor-rect conclusions. This process will be repeated

for three simulation durations of 0.5 hour, 1.0

hour and 1.5 hours. The mooring stiffnesses corre-spond to 15.5 tf/m and 18.6 tf/m respectively.

The results of the ten simulations in terms of

the maximum mooring force are given in Table 2. The results

in

the table show that for a test

du-ration of 0.5 hour in six out of ten cases the

maximum mooring force will be reduced when the

mooring system stiffness is increased. However,

for a duration of 1.0 hour and of 1.5 hours, seven out of ten cases indicate that the maximum mooring

force will increase due to the increase in

stiff-ness. The results of this exercise show that care

has to be exercised in drawing conclusions from

simulations or model tests which have a relatively

short duration from the point of view of the phe-nomenon being investigated.

The second example concerns the determination of the design load of a mooring system. The design

load of a mooring system is often based on the

most probable maximum value of the force which

will occur in a selected sea condition for some assumed duration of the particular condition. The

most probable maximum value of a quantity is the

value of the force for which the distribution

function of the extremes of the force it at Its' maximum. For a quantity of Vhich the distribution of the extremes conforms with the Rayleigh

distri-bution, the most probable maximum value is found

by intersecting the distribution of the extremes

at the probability level found from the following equation:

1

p(F) 100% (6)

in which:.

N number

of

oscillations of the quantity

in

the

assumed duration of the storm cOndition. In general the distribution of the extremes of

a quantity such as the :mooring force- will not be

in accordance with the Rayleigh formulation. In

such cases the probability level can be deduced directly from the distribution of the. extremes by determining the force_ value at the peak of the

distribution. One of the problems associated

with

this approach Is the amount of data, in terms' of

the duration of the record on which the

distribu-tion is based. In most cases only a limited amount

of data is available which Means that the results will always be. inflUenced by finite simple

ef-fects.

For the case in hand we will assume that we

may use equation (6) to determine the probability

level at which the distribution of the extremes must be intersected in order to obtain the most

probable value.

For the 300 kDWT tanker moored by meant of the non-linear mooring system, simulation computations were cattied out to determine: the statistical

var-iance of the Most probable maximum mooring force

for the same sea condition as before It was as-sUmed that the most probable maximum was to be de-termined for a storm duration Of .3 hours, In order to determine the dittribUtion of the extremes from which the most probable value was to be deter.-mined, simulations were 'carried out for durations

of 6 hours, 12 hours and for 18 hours.. The effect

of sample variance was determined by carrying out each siMUlation ten times. ,For each simulation the most probable maximum mooring force vas calculated according to the procedure outlined above. Final-/y, the mean, Maximum, minimum and RMS of the most ptobable maximum mooring force values were

deter-mined from the ten simulations carried out for

each duration value. The results of the

computa-tions

are

Shown in Figure 12. This figure shows that as the test duration is increased, so the RMS Value of the mint probable maximum mooring force decreases thus taking it more. probable that the

resUlts. Obtained from a single. simulation (test) will yield data which is Close to the 'true'

Val-ue.

CONCLUSIONS

In this paper an approach to developing

sim-plified mathematical models of the wave frequency

and low frequency motions and mooring fbrces of

s

tanker moored In head seas has been discussed ; It

has been shown on the

basis

of model test results

that it it permissible from the point of view of obtaining results in the early design stage, to

view low frequency motion components as being sta tistically independent of wave frequency motions

components. This is an important and practical simplification

which

can lead to a significant reduction in the computational effort required to simulate the motions and mooring forces.

Results of simplified low frequency- simula-tions 'were used to highlight some of the pitfalls awaiting the designer when determining the

sen-sitivity of the mooring loads for small changes

in

a Mooring system characteristics and when

deter-Mining

the design loads.

REFERENCES

1. wichers4 J.E.V.: "A Simulation Model for a Single Point Moored Tanker," Doctoral

Disser-tation, University of Delft, June 1988.

1. Pinkster, J.A. and Vichers, "The

Sta-tistical Properties of Low Frequency ?lotions

of Non-Linearly Moored Tankers," Offshore Technology Conference, Paper No. 5457, Houston, 1987.

Pinkster, J.A.: "Low Frequency Second Order Wave Exciting Forces on Floating Structures," $ARIN Publication No. 650, 1980.

Polderdijk, S.A.: "Response of Anchor. Line; to

Excitation at the Top," BOSS'85, Delft, 1985.

OTC 6178/J.A. Pinkster

Table 1

Particulars of tanker

Designation

Symbol

Unit

300 kTDW

VLCC

Length between perpendiculars

L m.

347.6

Breadth

B m

53.57

Draft fore

TF m

21.19

Draft mean'

TM

m

21.19

Draft aft

_TA

m

21.19

Displacement

V

m3

335,415

Centre of gravity above base

KG

m

14.94

Metacentric height

U1

m

7.04.

Centre of buoyancy forward

of station 10

LCB

m

10.56

Longitudinal radius of gyration

kYY

m

86.90

Natural roll period

Table 2 - Influence of mooring stiffness

on maximum mooring forces

Run duration in hour

0.5

1.0

1.5

Run

No.

Mooring

stiffness

Mooring

stiffness

Mooring,

stiffness

in tf/m

in tf/m

in tf/m

15.5

18.6

15.5

111.6

15.5

18.6

1

768

990

827

990.

,

828

990

2

827

'

878

717

704

802

828-3

594

564

802

,

828

.

993

954

4

717 : 704

993

i

954

961

1124

5

802

.

828

728

736

735

760.

6

572

570

'967 -1124

728

887'

7

806

778

,

735

,

759

613

806

8

99.5

954

728

.

888

764

718

9

626

736

649

727

820

864

10

728

686

613

Eia

1129

971

OTC 6178/J.A. Pinkster

A.P.

OTC 6178/J.A. Pinkster

40.0

20.0

0.0

0.0

0.5

1.0

WAVE FREQUENCY IN RAD/S

Fig. 2

Spectrum of irregular waves

1.5

WAVE SPECTRUM Measured'

;41c17-- 12.6 m ; Ti - 14.0 s

c0

ThepretLcat. t P.M.) ;4147---, 13.0 m ; Ti -. 12.0 s

c0

J 1 I % I

_i

t % % % % 1 1

:

1 1 I t _ %

.

.

I t t r ; S r

.

_.

.

_.

. .

_.

.

_.

.

_.

_{. .}

.._.

..

. ..." ims,

OTC 6178/J.A. Pinkster

Non-linear mooring

Linear mooring (15.5 tf/m)

Fig. 3

_{Restoring characteristics of mooring ,systems}

1250 / 1 1 1

/

I 1000 / 1 I 1 1 750 / 1 1 1 11

f(X)

i

n tf

500 1 1 1 r r t / 250

/

/

/

/

/

/ 60 40 20

/

/

/

/

//

e

/

,

e

/

/

/

/

/

-250 0 _-20

X in m

-40 -60

/

/

/

/

/

-500

i

/

/

/

/

/

/

/

/

-750

Woe

(m)

Surge15

(m)

Iii

_{Fig. 4}

Surge

(P)

0

Surgel5

(m)

0

Wave frequency part

Low frequency part

Fig. 5

OTC 6178/J.A. Pinkster

1.0 0

Linear

mooring

mooring

Non-linear

---... -_. -..- . : ..2. ...

2:- .--..L.... A:1.

Ak...i

....

....

0 0.1

0.2

0.3

w in red/t

Fig. 6

_{Coherence of squared wave frequency motions and low}

OTC 6178/J.A. Pinkster

0

---

Spectral density, complete expression

Approximation

./...

/

_\

I

s(o)

\

\,1

i\

i

\

F

\

_\

\

\

-I

\

\

_SF(u)

\

\

I 1 I I I I I

I...

....----...." ---4,

\

\ .

\

_\

. ; 1

Approximate range of

surge frequencies

\

_\

..._---.

\

1 1

natural

I

1 1 0

0.03

0.05

010

0.15

p in rad/s

Fig. 7

Spectral density of low frequency surge drift forces on

OTC 6178/J.A. Pinkster

0.01

p(F)

0.005

F in t

Fig. 8

Distribution of drift forces

Il

I'

I'

I'

I

\

1

\

\

\

\

k

Approximation

Complete

expression

----I I 1

\

\

\

_\

I I I

i

1

\

_N

_\

\\

0

-100

_-200

_-300

_-ann

F in

tf

200

100

OTC 6178/J.A. Pinkster

4

Fig. 9

Drift force time record

OTC 6178/J.A. Pinkster

99.9

IS CDC

99

s_

90

80

60

WC°

40

4-

20

0

10 4-) 5 1

0

-3 -2

Linear mooring

--- Non-linear mooring

1 X3 In in 1 2

Fig. 10

_{Distribution of wave frequency}

OTC 6178/J.A. Pinkster

1.5 1.0 0.5 0

Linear mooring

--- Non-linear mooring

Fig. 11

Amplitude response of wave frequency surge motions

0 0.25

0.50

0.75

OTC 6178/J.A. Pinkster

3000

2000

0

max

Fig. 12

Influence of simulation duration on variability of

most probable maximum mooring force in three hours

0 10 . 20