The low-frequency surge motion hydrodynamic coefficients for a semi-submersible platform

CHINA SHIP SCÍENTÍFIÇ RESEARCH CENTER

The Low-F re4uen cy Surgé Móti on Hydrodya4 c

Còef

ficients for a Semisubwersible Platform

Cáo Jin-Zhong,

Fu Yu-Fei,

Feng Yue

October 1985

CSSRC Report

English version-85008

P. 0

.

BOX 116, W.UXI, JIANGSU

CHINA

TEC}INISCH UIVERSIThIT L ScheöpShydmmechanica chIet Mekeiweg Z 2628 CD De!ft TeLO16-78ß73- Fax 015-78183$

Contents page

bstract .1

Nomenclature i

Introduction 2

Determination of the low-frequency surge .'otion hydrod- .5 naniic coefficiènt - 11 Model tést Çonclusion - 14 Acknowledgment 21 References

21

The, Lo-requency Surge Mòtion }Tydrodynaiñic Coefficients f r a Seinisubinersible Platform'

by

Cao Jin-Zhàng, Fu Yu-Fei,

FengYue

CSSRC Chiná

Abstráct

In this paper, the low-freqùency surge motion, hydrodynainic coefficients

for a semisubmersible platform vere studied experimentally. The model tests Were carried out in still vatér,-regular waves and irregular waveso

The model test results show that waves have significant effects on the low-frequency srge motion hydrodynamic coefficients, not only on the

low-frequency damping, but also on the low-low-frequency added mass for the seinisub-mersible platform. .

It is found that the average hydrodynamic coefficients in irregular waves can becalculated o the basis of the test resúlta in regular waves.

N omenc 1 ature

virtual mass pertaining to .low-fréqutcy sUrge motion

a ad4ed mass pertaining

to lowfrequency surge motion

B damping coefficient pertaining to low-frequency surge motion

Ca added mass coefficient pertaining to low-frequency surge motion c0= coefficient of restoring force.

wave excitation fOrce mean Wave dtift force

g = acceleration due to gravity

L length f the semisubmesible

m mass of thé sèmnisubrnersibie

n = order number of samples

t = time

t = initial time of the n-th sample

x displacement along x-axis

= amplitude of surge

absolute value of the amplitude of k-th wävelet of surge motion

= ratio of extinction

= logarithmic decrement

c = phase angle

Ca wave amplitude

p = nondimensional dinping coefficient

u = extinction coefficient

t

= sample time

(*) wave frequency

= natural frequency pertaining to low-frequency surge motion

= undaiuped natural frequency pertaining to low-frequency surge motion

Superscript (1) ( ) = first order ( )(2) = second order

C)

= derivative with respect to time

( )' = nondimensional coefficient rationalized with respect to wave

height Subscript ( ) = in still water ( in waves C = induced by waves Introduction

Semisubmersible platforms have found wide application in ocean

production platform, accommodation and service platform etc.

Therefore

seinisubmersible platforms occupy an important place in the exploitation of

marine natural resources.

Seinisubmersible platforms used to be operated in moored conditions.

On

the continental shelf, where water depth is less than 200 meters, most

of the

seinisubmersible platforms are moored by chains and anchors.

The

wooring.sys-tern provides restoring forces for the sen,isubmersihle platform and keeps it

in position with desired accuracy.

The stationkeeping stability, and motion

characteristics of the semisubmersible platform are closely related to the

operation efficiency, economics and safety etc.

The seinisubinersible platform moored at sea is exposed to action of

the

complex environmental conditions.

The main factors are waves, winds and

cur-rents.

The forces acting on the semisubmersible platform caused by

environ-ment are irregular in nature and may he split into three parts:

stationary

drift force, high-frequency oscillatory force and low-frequency oscillatory

force.

The stationary drift force is the wean value of the environmental force,

it causes a stationary excursion.

The high-frequency oscillatory force

per-tains to the first-order wave excitation force, whose frequencies are equal

to the frequencies present in the wave spectrum.

The motions caused by this

force are called high-frequency motions or first-order motion response.

They

are similar to the osil].atory motions of a freely

floating body in waves.

The low-frequency oscillatory force con8its of the second order

oscil-latory wave drift force, the slowly varying wind force and current force.

Although the magnitude of thse forces may be rather small in comparison with

the first order wave force, the low-frequency surge motion due to it is

domi-nant, because the force frequency is closed to the natural frequency of the

moored semnisubmersible platform, whose mass is generally large, while the

restoring force provided by the mooring system is generally small, as aresult

the natural frequency of surge motion of the moóred semnisubmrersib.le is quite

ir-regular waves.

The amplitude of low-frequency surge

motion may be many tibies

larger than the high-frequency surge

amplitude, causing high peak mooring

loads.

Therefore it is necessary to study

the low-frequency surge motion

of

the moored seipisubtrersihle platform.

The physical model of the low-frequency surge

motion maybe cnsidered

as à mass-damper-spring system.

It is known that

the rezonance amplitude

of such a system is determined by

the dping effect, and the

natural

frequ-ency of the system is dependent. on the added mass. Therefore a good

knowle-dge. about the low-frequency surge

motion hydrodyiIaTn.c coefficients

is very

important to determine the low-frequency surge motion amplitude and the

mooring load of the moored seniisuhirersible platform.

It is necessary to know not

only the low-frequency surge

motion hydrody

nic coefficients in -still watero hut

also in waves.

The hydrodynamic

coef-ficients in waves are more important, because

the Iraximuin

surge motion

ampli-tudes and the peak value mooring loads always

occur in rough sea.

Wichers

[1] [21 [3) [4]

has studied the low-frequency

hydrodynardc

-coefficients of two tankers moored at head sea.

!e found that the damping

of the low-frequency sur-ge motion of

moored tanker in waves changed

substantial-ly, while the added mass remained the same as

.iñ still water.

- . -,

Chkrabarti (5] has carried out model experiments

with two- different

tankers and a bárge, it indeated that the low-frequency

hydrodynamic

dnip-iñg in waves is dependent on the -ship shape.

Saito et ai [61 has also investigated the

low-frequency hydrodynamic

coef-ficits oUa ship-shape floating body.

Up to now, published results low-frequency hydrôdynaniic coefficients of

a moored, . seniistibmersibie

have not been found in open literature.

The present paper aims to analyse

the low-fréqúency hydrodynamic.

coef--ficients. -of a semisubme-r'ible moored

in s.tfli water, in regular waves and in

irregular vavés, to determine the wave effects on

them by model testing and

to provide some basic data fOr calculation of' low-frequency surge motion.ad

mooring load of a semiuhtnersihle platform

moored in waves.

Determination of the low-frequency surge

motion hydrody!lamic coefficients

In still water

The low-frequency surge motion

hydrodynawic coefficients of the

semi-subersible platform in still water can be

tested and determined by the

curve of extinction.

Assuming the system to be linear and the

hydrodynai'tc

coefficients to be constantT with titre, then

the motion eouation-inay be written

as:

A

+ B

+ C X = O

o. o o

in which:

A

virtual mass pertaining to low-frequency surge motjon in. still water

B0 = dampin,g coefficient pertaining to

low-frequency surge motion in still

water

'

c. = coefficient of restoring force

x

= displacement along x-axis,

the dot indicates the derivative with respect

to time. .

. . .

The virtual mass consists of the mass of.the

semisuhimmersible platform

m andthe low-frequency, sUrge

motion added' mass in still vater a0, namely

The standard fôrni of thé motion equation

is

.

in which:

'VC, = = low-frequency surge motion eitinctiofl cóeffcient in still

waér

(4)

"CO '"

(m.3

= undairped natural frequency pertaining to low-frequecy surge motion

The solution of this linear differential. equationwith constant

coeffici-ents will be -

-X cos + C ) (6)

in which:

Xa tnaxiÌwn amplitude of the lw_frequency surge motion

t time

natural fre4uency pertaining to low-frequency surge motion

e = phase angle

The non-ditiensional damping coefficient for low-frequency surge motion is

defined by the following formula

(7)

oo

The relation between the low-frequency surge natural frequency and the tndamped natural frequency is given

o

W

xô =L)xoI P' - 2

it is clear, if the damping is negligtblly small (i.e p,<l ) then

, but with the increasing of dampiig, the difference between

to and

cf0

will be larger and larger. The restoring force of the system is assumed to be proportional to the surge displacement, and is easy to

determine on the basis of measuring the static characteristicof the moox

ing system.

Once the restoring force coefficient and the undamped natural frequency of the systen, are known, the lowfrequency surge motiön added mass of the

semisubmersible platfOrm in still water can be calculated by the following

formula C

a = n,

_° Wo_xo o (.9)

The added mass coefficient in still water is defined as

a

C o

ao

in.

For a weakly demped system, one can consider (à d can deter-mine C. fron, the corresponding, curve of extinction ininediately when the

damping is strong, it is necessary to take into accourtof the damping ef-fect and to determine C according formula (8). The way to determine the

nondiwensional damping .oeffic:ient is described below.

The solution of equation of motion given by formula (6) represents a harmonic oscillation with amplitudes decaying in an expotential way. Let

II

and be absolute values of two;iubsequent amplitudes (see Fig.l), the extinction ratio is defined as

"Oli i(a)

e xo 6

where 6 = ln IXk! ln IXk+l! (12)

is called the logarithmic decrement.

The low-frequency surge motion damping coefficient can be represented as

4Ç2

_{+ 62} (1.3)

as

1

Cx=

o o o

The left hand side of equation (15) is _{the same as that in still water, but}

on the _{rïght hand side of the equation the first order and second order} wavè

exciting forces F1 , F appear. _{The. firstorder wave exciting force is} causal of the surge _{otion at wave frequency.}

The second order wave forces

are assumed to consist of three parts, i.e. _{tbe,me wave drift force}

the second order wave force related t _{the low-frequency surge motion el.ocity} -

L

and the second order _{wave, force related to the low-frequency surge}

0F

motion acceleration

- x,

then

F2

(2)

(2)

(16)

In order to obtain the extinction _{curve of surge motion, we eliminate}

the first order respense and the státionary. excUrsion from the _{total motion} in waves by a low-pass ìfilter, _{after which the motion equation takes}

the form

P x + B1x + C X = O

J. -. J_ o

in which

virtual mass of the low-frequency _{surge motion in regular vaves}

total damping coefficient .òf the low-frequency _{surge motion in regular}

waves

6/it ₍₁₄₎

Therefore _{, having the extinction curve of the low-frequency surge}

mo-tion the nondimensional damping coefficient _{may he 4etermined.}

In regular vàves

The low-frequency surge motion equation _{n regular waves may he written}

(1.5)

and

a F2

A1 = A0 + a

aF2

B1 = B0 + a

and

a F2

can be considered as low-frequency surge ir'oton added

ax

mass induced by regular waves and low-frequency surge motion damping induced by regular waves respectively and denoted by a and

(2o) u1 UI

=

-B C

The standard form of equation (17) is

+ 2v1+

(.f2 x = 0 (22)

in which B

v =

_!

= total low-frequency surge motion extinction coefficient in

2A (23)

1 regular waves

(1)0 _{= (C0}

i

_A1) mmdamped natural frequency pertaining to

low-1 _{frequency surge motion in regular waves} (24)

The total low-frequency surge motion nondimensioTtal damping coefficient in

regular waves is defined by

wo X

-9

(21) (25)

i

(18) (10)

Assume to consist of two components, the firstone 4s the nondiinensional

damping coefficient in still water p0, and the second one is the

nondimen-siona]. damping coefficient induced by waves p , then

pl = Po + PC (26)

The total low-frequency surge motion nonditnensional damping cofficient

in waves p1 and the total low-frequency surge motion added mass in waves a1

can be determined from the filtered low-frequency surge motion dacay curve,

consequently the nondimensional damping coefficient caused by waves u and

the added mass caused by waves a can be determined.

As assumed , - . are propotional to the square of the wave

ax

height, then we can define the nondimensional coefficients rationalized with

respect to wave height as follows:

t fl.1L 2

(27) a

Which is the low-frequency surge motion nondimensional damping coefficient

induced by waves rationalized with respect to wave height, and a = m t O.1L 2 a lo -(2 R)

which is the low-frequency surge motion nondimensional added mass

coeffi-cient induced by waves rationalized with reèpect to wave height. Tere, T

is the length of the semisubmersible and is the wave amplitude.

In irregular waves

In irregular waves, amplitudes and freciuencies of waves are changing

with time , so strictly speaking, low-frequency surge moton hydrodynamic coefficients are also time dependant. However, for engineering calculation purposes, we can assume the hydrodynamic coefficients in irregular waves to

of thé hydrodynamic coefficients áre related to the wave spectrum.

The average hydrodynamic coefficients in irregular waves can he model

tested and determined by the random decrement techniquell.]. TFe role of

the random decrement technique is to help us to obtain curves of free

extinc-tion from the surge moextinc-tion time history in irregular waves. Once the curve

of free extinction is obtained, the hydrodynariic coefficients can l'e

deter-mined by the same method outlined in the case of regular waves.

The basic concept of the rdoth decrement method is based on the fact

that the irregular, surge motion response of the semisubmersihie due to ir-regular waves may be viewed as cdnsisting of two pàrts.: I) the deterv'inisti.e

part (impulse and/or step) and 2) the random part (assumed to have a

station-ary average). _{Fy averaging, enough -sainles of the same random response the}

random part of the respOnse will he averaged out, leaving the deterministic

part of the response standing Out 'only. In order to ohtan the curve of free decay due toa step response, the samples must be taken starting always with a constant level. In Fig. 2 if x(t) is the random response, the curve

of free decay due to a step response will he

x(r) = _x(t1

_{+ T)}

₍₂₉₎

N

n=1

in which

t

= sample time

N totál number of samples

n = order number of samples

t initial tinte òf n-tb sample

Model Test

In ordér to detérmine low-frequency surge motion hydrodyfiamic. coeffici-ents of the senìisubmersible a series of model tests was carried out, the ex-periments composed of surge motion extinction tests in still water, in

regu-lar waves id measuring the time históry of surge response in irregular

waves.

li-General description o

the model tests

The model tests vere

carried out in the

seakeeping tank of CSSRC.

The

tank is 69m in length,

46m in width, 4m in depth,

equipped with pneumatic

type of wave-akerso A rotating bridge

is mounted over the tank, a

towing

carriage hanging down from

the bridge serves, as a

measuring platform.

Thé setrisubinersible

platform model was of a six-column twin hull type,

its sketch is given in

Fig.3.

During the testing, the

model vas moored at

the center of 'the tank

by a pair of' linear spiral springs on the fore

and

aft sides of the model as

shown in Fig.4.

These two springs were

identical,

with stiffness equal to

1.615 kg/rn for each.

Thus the restoring coefficient

of thé mooring system was

equal to '3.23 kg/rn.

The springs vere pretensioned

so that they

would never go slack during

the test.

Fhe surge rnotioI of the

model was measured by a

six_dereeflf-freed0m

servo-actuated motion apparatus and recorded by a magnetic tape recorder

and an oscillograph.

Thesurge motion signal

of tests in waves vas

trans-ipitted into two channels, one was

récorded. immediately

including the

high-frequency and low-high-frequency surge motion together:, another one vas

passed

through a f jite

and only the low-frequency surge

motion was recorded.

The wave height. was

measured by an ultrasonic wave

probe end recorded

sinmultmineouslY wjth the surge

mot-io1i.

The wavé probe was mounted on

the

right corner of the carriage

in front of' the model.

For extinction tests

in .5till

ater and in regular waves,

the model

was pulled away

in x direction from its

equilibiuni position and. then

re-leased.

Recordings were made 'for the

entire slow oscillation process.

Only head seas were tested.

IT order to investigate

effects of the wave' frequency

and wave height

on low-frequency,

hydrodynamiC coefficients, the wave

lengths for regular

'waves were

íanged from 2 w ,to-l4.5 m, and three wave

heights were tested.

In order to obtain enough samples the duration of testing in irregular

waves was kept for 40 tnjn. The wave spectrum is shown in Fig. 5. The sig-nificant wave height was equal to 92 nm. The peak frequency was equal to

4.91 rad/sec. The spectrum width coefficient was 0.82.

Analysis and summary of results

An example of the free decay curve recorded during the test in still

water is given in Fig. & The nondimensional damping coeffiéient and added

nAs5 in still water were computed on the basis of the curve of free

extinc-tiop from Eqs. (9), (10), (12), (13). The results are given below:

nondimensional damping coefficient in still water = 0.1025

nondiinensional added mass coefficient .in still water C = 0.337

- ao

Fig. 7 shows an example of test recording in regular waves. There are three

cutves on it, the first one is the trace of regular wave, the second one is the trace of total surge motion, the third one is the low-frequency surge extinction curve after filtering, from which the hydrodynamic coefficients

vere derinined. First the total hydrodynaniic coefficients were determined,

then substracting out the values obtained in still water, the effects induced

by wate's ere obtained. Finally we can calculate the low-frequency surge

wotiopnondimensiona1 hydrodynainic coefficients induced by waves. Fig. 8

p1othe low-frequency surge motion nondiutensional added mass coefficient

in1uL b

waves and Fig. 9 plots the low-frequency surge motion

nondimen-stonai

daing coefficient induced by waves against nondimensional frequency

isZJ-,

in which W is the wave frequency and g is acceleration due to

gr*flt

and Fig.9 show clearly that waves have significant effects on the

lowr

surge motion hydrodynamic coefficients, not only on the

low-f encydamping, but also on the low-frequency added mass for the

semisub-mesibIe platform. In this respect, there is some difference between the

present

test on

a semisubmeraible platform and those perfrcnned on vessels

stu4ied by other authors [1) (5) [6)

-It is shown in Fig.8 and Fig.9 that wave effects increase with increas-ing wave frequency in the tested range. For nondimensional frequen&es less

than 1.5, the scatter range in Fig.9 gets grow and the vavé induced damping sometimes becomes negative. These phenomenona are presently under invest4-gation and shall be presented in a future paper.

Fig. 10 shows the recording of testing in irregular aves. In

Fig.

11

the curve of free extinction of low-frequency surge mot4on in irregular waves

determined by the random decrement method is given. The average low-f

rerluen-cy hydrodynamic coefficients were determined from it. It is found that the

average low-frequency damping coefficient in irregular waves equals fl.29&,

and the average low-frequency added mass coefficient in irregular waves equals

1.015.

We have also calculated the low-frequency hydrodyriamic coefficients on

the basis of the testing results in regular waves for a regularvaves, whose

wave height was equal to the significant wave height of the tested irregular

wave and the wave frequency was equal to the peak frequency of the tested

ir-regular wave. It vas found that the low-frequency damping coefficient was equal to 0.285 and the low-frequency added mass coefficient was equal to

1.045. These figures are very close to the results of the testing in

irregu-lar waves. It suggests that the results of the testing in regular waves can

be used to calculate the low-frequency hydrodynainic coefficients in irregular

waves, which are useful for concept design dalculations of the smisuhmersible

platform.

Conclusions

The present study show waves have significant effects on the low-f

re-quency surge motion hydrodynamic coefficients of the semisuhmersihle plat-form, not only on the low-frequency surge motion damping coefficient, but

also on the low-frequency strge motion added mass. The effects can be ex-plained as a result of hydrodynamic forces induced by second order waves, these forces are dependant on the low-frequency surge motion velocity and

-acceleration respeçtively. in the tested range the wave effects increase with the increasing of the wave frequency and proportional to the square

of the wavè height. It is Shown that the results of the testing in regular waves can be used to calculate the low-frequency hydrodynamic coefficients

in irregular waves and may be uSéful in assessing design performances of a s.emiSúbinersible plátform.

Fig. i - Sketch of a curve of extinction

Fig. 2 - Sketch of the sampling for ridom decrement technique

J

Fig. 3 - Sketch of the senilsubmersible

-'J

carri age

Fig. 4 - Model set up

peak frequency = 4.9 rad/sec

L) (rad/sec

0

6.14

12.27

18.41

24.54

Fig. 5 - Wave

spectrum

-

17

-t (sec)

I I I .1

I I I I i

10. 20 25

30

35 40

total surge motion i

1

su wave height () 1.0 wave

'k

Fig. 7 - Test recording in regular waves

fi itred

surge lotion

ib

Fig. 8 - Lowu-frequency surge iroton nondiTnensfonøl added ii'ass coefficient induced by waves

-Fig.

9 -

Lfrequeucy surge

òtiOn nondimensiona'l daipin cÀefficientinduced, by waves

wave

filtred surge motion

Fig. 10 - Recording of testing in irregular waves

Fig. 11 - Curve of free extinction determined

by. the random decrement

-Jtcknoviedgment

The authors wishe to acknowledge the help of Mr. Ou Shi-Rao for his

ttbution to the analysis of test data. Peferen ces

1,, Wichers, J.E.W. and van luijs, 1F.: "The Influence of Waves on the

Low-Frequency llydrodynawic Coefficients of Moored Vessels", Offshore

Technology Conference, Paper OTC 3625, 1979.

Wichers, J.E.W.,: "On the Low-FrequencySurge Motions of Vessels

Moor-ed in Bigh Seas", Offshore TechnologyConference, Paper flTC 4437, 1°82.

Wichers, J.F.W. and lTuijsinans, P.M.1T.:

"flu

the Low-Frequency

Ilyiroñy-naniic

tanping Forces Acting on Offshore Moored Vessels", Offshore

'rechnology Conference, Paper OTC 4813, 1QR4.

Lchers, J.F.W: "Written Contributions to the Technical Peort of

' Ocean Engineering Comnittee," ITTC.'84, Cöteherg, Sweden

ciakrabarti, S,,,: "Experiments on Wave Drift Force on a floored Floating

Vessel", Offshore Technology Conference, Papet ()TC 4436, 1982.

Sajto K, et al; "On the Low-Frequency Dat,ping Forces Acting on a

Moor-4ody in Waves"

Journal of the Tansai Society of Naval Architects,

295, l)ec.

i984,

1.

tbr&üm, S.R.:

"Rado Decrement Technique for Modal Identification of tTUCtures " ,

3oura1

of Spacecraft _{td Rockets, Vol.14, No.11, Nov.} 917.,