Hydrodynamic aspects of semi-submersible platforms

16 JAN. .1973 .

ARCHE

Lab. y.

Scheepsbouwkunc

Technische

Hogeschad

HYDRODYNAMIC ASP1CTS OF

SEMI-SUBMERSIBLE

PLATFORMS

J. P. HOOFT

9:

;

/

STELLINGEN

I

0m de maximale uitwendige belasting op een lichaam in een onregelmatige

lang-kammige zee te bepalen is het niet korrekt orn het lichaam te onderzoeken in een

regelmatige langkainmige golf, waarvan de periode en de hoogte gelijk zijn aan de

schijnbare periode en hoogte van een significante golf nit de onregelmatige

langkam-mige zee.

II

Doordat voor oneindig kleine golifrequenties de relatieve bewegingstheorie geldig

blijft, kan worden aangetoond dat voôr deze golifrequenties de slingerhoek of

stamp-hoek van een drijvend lichaam niet gelijk behoeft te zijn aan de golfhelling.

III

De kleinere bewegingen van een halfóndergedompelde drijvende konstruktie

(be-staande uit ondergedompelde lichamen en half ondergedompelde elementen) ten

opzichte van een enkelrompschip worden veroorzaakt doordat de uitwendige

krach-ten op de ondergedompelde hchamen kunnen worden gekompenseerd door de

krachten op de elementen die door het water steken.

IV

Als een schip in een recht kanaal vaart kan tengevolge van oeverzuiging statische

onstabiliteit van de horizontale bewegingen van het schi, ontstaan.

V

In tegenstelling tot de vaart. op diep water heeft men als een schip op ondiep water in

een homogene stroom vaart behive een bepaalde opstuurkoers ook een gemiddelde

roeruitwijking nodig orn een gewenste rechte koers te kunnen varen.

VI

Voor het meten van een bepaalde zeetoestand moet men rekening houden met de

ver-storing van de golf door het meetinstrument.

VII

Het gebruik van een simulator voor het tramen van stùurlieden biçdt grote voordelen

boyen de training door waarneming van het manoeuvreren met schepen in de

werke-lijkheid.

VIII

De inzinlung van een model tijdens de voortstuwingsproef is groter dan die tij dens de

weerstandsproef. De hierdoor ontstane weerstandsverhoging heeft een meetbare

in-vloed op het zoggetal van het schip.

.Ix

-Voor de bepaling van de ankèrlijnkrachten van een afgemeerd lichaani in golven

dient een onderzoek in onregehnatige golven uitgevoerd te worden, daar de variatie

in Iaagfrequente driltkrachten ten gevolge van onregelmatige golven een essentiéle

bijdrage levert in het gedrag van het verankerde lichaam.

X

De begrippen 'boyen' en 'onder' duiden richtingen aan langs de straal van een

aard-vast bolcoördinaten stelsel. Zowel de begrippen 'voor' en 'achter', als 'rechts' en

'links', zijn aanduidingen voor richtingen langs twee assen van een met de mens

mee-bewegend rechthoekig assenstelsel.

Het is zinvol orn ook de positieve en negatieve richting langs de derde as van dit laatste

assenstelsel te definiéren.

ogeschoo,

ibliotheek van

Onderafd! Ing

Tech r i D uMENTATIE

D AlUM:

C li k E N T A TJJ de

ouwkunde

h

HYDRODYNAMIC ASPECTS OF

SEMI-SUBMERSIBLE PLATFORMS

HYDRODYNAMIC ASPECTS OF

SEMI-SUBMERSIBLE PLATFORMS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE

TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL

DELFT, OP GEZAG VAN DE RECTOR MAGNIFICTJS IR.. H. R. VAN

NAUTA LEMKE, HOOGLERAAR IN DE AFDELING DER

ELEKTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT

TE VERDEDIGEN OP WOENSDAG i MAART 1972 TE 14.00 UUR

DOOR

JAN PIETER HOOFT

SCHEEPSBOUWKUNDIG INGENIEUR

GEBOREN TE BATAVIA

Dit proefschdlt is goedgekeurd

door de promotoren

Prof. Ir. J. Gerritsma

CONTENTS

I INTRODUCTION _i

II OSCILLATORY WAVE FORCES ON SMALL BODIES ₃

II-1 Introduction

3

II-2 Determination of wave forces ₄

II-3 Comparison of the approximated and exact solution for the horizontal wave excited force on a

ver-tical circular cylinder ₁₀

II-4 An experimental verification of the results

ob-tained ₁₈

III HYDRODYNAMIC FORCES ₂₃

III-1 Introduction ₂₃

III-2 Added mass ₂₄

III-3 Damping ₃₂

IV THE MOTIONS OFA SEMI-SUBMERSIBLE IN WAVES 41

IV-1 Intrôduction ₄₁

IV-2 Determination of added mass and hydrostatic forces 42 IV-3 Determination of total wave excited force 52 IV-4 Response of the platform motions to waves ₇₁ V DETERMINATIOÑ OF THE DIMENSIONS OF A PLATFORM WITH

MINE MUM VERTICAL MOTIONS IN WAVES ₇₃

V-1 Introduction ₇₃

V-2 Platforms with minimum heave at the natural frequency 73 V-3 Platforms with minimum heave over a range of

frequen-cies including the natural frequency 79

V-4 The design Of a platform with limited heave motions

over a range frequencies ₈₈

VI REVIEW OF MAIN CONCLUSIONS ₉₆

APPENDIX I Description of the model tests ₉₈

II Derivation of the motion of a non-linearly

damped freely oscillating body ₁₀₃

III Contribution of the added mass of the elements

of the platform to the total added mass 107 IV Contribution of a cylinder to the total

res-toring force of the platform 110

V Contribution of the wave-excited force on an element of the platform to the total wave

SUMMARY 121

SANENVATTING 122

REFERENCES 124

NOMENCLATURE 129

I INTRODUCTION

Semi-submersible platforms consist of submerged bodies connected to the working decks above the water by means of columns or slen-der walls. Derrick , pipelaying , storage platforms, production platforms and drilling platforms are built in this way since it is assumed that the motions of. this type of cónstruction are less than the motions of ship type barges or pontoons.

up to now the advantages of semi-submersible platforms strictly have not been proven since no design method is available in which the dimensions of a platform f011ow from an optimization technique. The designs are therefore based on assumptions more or less rea-soned at randOm (see among others Paulling [t-i] and Carrive [x-2)) supplemented with model experimehts of the original design and With a model of an adapted design when the first results are not

satisfactory.

The lack of a systematic design method for semi-submersible platforms follows from the fact that. only recently the need was felt for large semi-submersibles on which high demands wère made. In the meantime Ship researchers on one side had developed the hydrodynamics on ship behaviour in, waves (see Vugts [I-3]) while on the other side the hydraulic experts had developed the hydro-dynamics for complicated fixed constructions in sea (see Wiegel

[I-4] and Ward [I-5]). er, it turned out that these expert knowiedges could not be simply adapted for predicting the beha-viour of a semi-submersible platform in waves (see among others

Paulling [I-l) and Fujii

_Et-6)).

In the present study an attempt is made to derive a design method by which the behaviour of semi-subinersibles in waves can be predicted. Also the influence will be discussed of details which scientifically can be proven to exist but which will be

ne-gected in this method. The prediction method to be developed, has been based on the relative motion concept (see Motora [I-7) and Gerritsea [I-8]) applied to the components of the platform. This means that:

The method ori, ginates from a potential theory for small bodies (see chapter II).

The influence of viscosity on the platform behaviour is of secondary importance, which will be discussed in chap-ter III.

c. The interaction between several components of the platform is neglected which means that the hydrodynamic properties of one component of the platform are not affected by the existence of neighbouring components (see Laird [I-9]). A discussion about interaction effects will be given in section III-2.

Finally an optimization technique will be presented using the method for predicting the platform behaviour in waves. As a re-sult of this technique the dimensions will be dduced of à seìui-submersiblé platform which from a hyd±odyñámica] point òf view will be optimum with regard to heave, roll and pitch motions. This means that the horizontal motions which are influenced by the mooring system, will not be taken into conéiderätion since

hydro-dynamic aspects are then introduced which ¡re different from the aspects investigated in the presént Study.

II OSCILLATORY WAVE FORCES ON SMALL BODIES

II-1 Introduction

Thé wave forces on a floating or a fixed structure are influenc-ed by viscous and potential effects. In some cases the potential part of the fOrce can be calculated. Howevér, an analytical method to determine the influence of viscous effects on the total force is not yet available.

To predict the wave forces, 8evera]. approximations have already been given, for instance that given by Moneen [II-i]. A review of the work carried out experimentally and theoretically is describ-ed in chapter II of wiegel [I_4]. In additioñ to that review the papers by Harleman [II-2] and [II-3] should also be mentioned. The results of these approximations sometimes correspond well with measured values.

A drawback of these earlier approximations is the omission of information about the limitations of the approximations. In most cases this was caused by the present lack of knowledge concerning the interaction between viscous and potential effects. The phenom-enon of separation of the fluid around a circular cylinder in waves gives an illustration f the lck of knowledge.

The aim of the present study has been to derive an approximation of the wave excited fOrces, of which the limitations are known. First the potential part of the wave excited forces will be analys-ed. From a comparison with systematic model test results it therp-after will be shown that the influence of viscous effects can be neglected for a range of waves which can be assumed to be

represen-tative for the actual sea conditions.

In the present chapter the study starts with the derivation of an approximate calculation of the wave forces oñ à structure in a fluid which is supposed to be incompressible, irrotational and in-viscid. The limitations of this approximation will be dedüced by comparing the approximated oscillatory wave forces on a vertical

cylinder with the result of exact calculations of the wave excited forces.

The exact solution for a vertical circular cylinder first waS given by Have].ock [II-4] for deep water. It can be adapted for shallow water as given by Flokstra, [II-5]. In the meantime MacCamy

[II-6] had given the exact solution for Shallow water. These theories all describe the wave force on a vertical cylinder

pier-cing the water surface and extending to the sea bottom. For waves which are short relative to the cylinder diameter, an exact solu-tion for a cylinder with a draft less than the waterdepth has been given by Miles [II-1]. However, for waves which are long relative to the cylinder diameter, no exact theory is available. Therefore the, approximate wave force on cylinders with a diameter which is small relative to the wave length, which is of interest in the present case, and with a draft less than the waterdepth can only be compared with model test results.

II-2 Determination of wave forces

Forces on an oácillatin bod in still water

The velocity potential of the water motion due to a moving body in an incompressible, irrotational and inviscid fluid can be

sthdivided in three components:

= velocity potential of incident waves = velocity pctential of the waves reflected

velocity potential of the waves generated òf the body-il the direction j.

In still water the vèlôcity potential _3j due to an body depends on the diréction of oscillation j, the oScillatiôn

8a and the frequency w.

it

$»3j - Vai 43j e with: = l + + in which = maX = iw 5aj

On the hull of the body the fdl-lowing condition has to be satis-f jed (see the boundary condition in equation II-li):

a31

with:

n being the normal to the body surf ace, positive in the out-ward directiòn

on the body by the motion

oscillating amplitudé of

and

= Vai ei)t

One therefore finds:

- Val f1

with:

= c°s ( ,

The combination of equation (II-1) and (II-3) results in:

ìi.

= f

The hydrodynamic force on the body amounts to:

Fk hydr. - _k dS (II-5)

in which:

p-p1=-p--1-ipw31

in which p1 is thé static pressure on the hull.

The total force on ¡n oscillating body can be split into: d2s

F = cS

k tot. mk

dt2 k hydr.

in which:

rn.K mass or moment of inertia of body

kj

=1

if

k=j

=0

if

kj

hydr. = hydrodynamic force or moment

The hydrodynamic force can be split into three parts:

d2s

ds.

F_{k hydr.} = _&ki

1+b +c

s

dt2 kj dt kj i

in which the following definitions are used:

aki = added mass

bkj = damping coefficient

Combining equations (II-5) and (II-7) one finds: iwt a$3k ipu Vai e

_{ft 3j}

d S = aj

et [_2

ak + i bki] or:

Rep J'

3i . d S =aki lin 1k . d S = bk. s J

The spring force c, s. follows from the integratiön of the stat-ic _{pressure (p1) force over the body after a displacement}

A fixed obstacle in waves

When a restrained obstacle is placed in long-crested regular waves, the Velocity potential of the water motion may be written

as:

= 1 + (II-9)

= velocity potential of the incident waves

2= velocity potential of the waves ref leç-ted on the obstacle

The functions and

2 satisfy the saine free surface condition

and the boundar' condition on the hull as the function 3j which has been discussed in the aforegoing. These conditions follów

from the assumption that the fluid under consideration has à bound-ary _{surfäce S, fixed or moving which separates it from some other} medium and which has the property that any particle which is once on the surface remains on it. For example, if this sur-face were

given by an equation x = O it follows that the conditïon (x,y,z,t)

(II-lo)

whould hold on S. From .the fact that u = , y = and w =

--

X

_y

Z

and the fact that (xx, xx5) is a normal vector to the surface

ii consideration, it follows that equation (II-10) can be written

in the form:

on a moving surface, which reduces to:

on a fixed surface. The functions

2 and also satisfy the radIation cOndition

which physically implies that these functions can only have such values that no energy arises from infinity (see section 12 page 471 of Wehausen iII-12]). The radiation condition usually re-quires the waves at infinity to be progressing outwards and

impos-es an uniqueness which would not otherwise be present (see sec-tion 13 page 475 of Wehausen [II-r.12]).

The velocity potential of the undisturbed incident wave is known everywhere by the equation:

=

U1a

et

(II-13) in which: - cosh K(h-) cosh Kh = wave amplitude h = waterdepth u = wave frequency = 21T/T T = wave period K = wave hümber = 21T/À X = wave length

= coordinate in the direction of wave propagation = coordinate in the vertical downward direction

On the hull one finds according to equation (II-12):

=

-The generalized force equals:

=

-k d S =

=iwp 1f

(

+ 2k d S

According to equation (II-4) one finds:

= 3k an

3k

which means that

_k

equals -- in thé casé that the by

oscilla-tes in still water iñ the k-direction.

Substituting equation (II-4) into équation (II-15):

Fk = iwp fl

+

-- d S

(II-16)

USing Grêèn's theore and équation (1±-14) one finds:

3k 4

T

2 d S 3k d S = 3k - d S

combining equations (II-16) and II-17) one now f inds:

Fk=FFk2

Fkl= iwp ff f d S

S

Fk2= iwp _'3k d S

in phis equation, the force F due to the pressure variations

p - is called the undisturbed wave pressure force

(Froude-Kri-lof f force).

The variations of the force due to the pressüre variations of the disturbances, are split in two parts which are called the ihertia fôrces

k2j (in phase with Fkl) and the damping forces Fk22 (out of phase with

The force Fk2 due to the disturbance of the incident wave can be rewrittéñ by: Fk2 = iwp If V 3k m d S (II-19) S

I

since: as as2 as3

= vi. j:- + V2

+ V3

-

(±I-2O) (Ii-17) in which: as vm = and

m

=

The total wave excited force now is deterined by:

Fk= iwp If _k d S - iwp

3k

f d S

(±1-21)

The wave excited fQr on a reatiyely small bgy

If the height and the length of the body are small relative to the wave length one may assume that Vm in equation (II-21) has the Same valúe all over the body.

In that case the force Fk2 (see equation (II-19)) can be trans-formed into:

'k2 = iwp Vm J. 'p3k find S (II-22)

Combining this and equation (II-B) one finds:

k2 = k21 + iwt -Fk2licAi.vi.e

.ak

ict Fk22= vaj.e

bJk

k-direction = iwp _!

k d S = undisturbed pressure force S

Froude - Xriloff force F = iw.v .ei0 _{a. = inertia force}

k2l aj jk

- iwt

-- va..e . b - amp ng orce

Though this result is complete in order to calculate the wave excited force, it is sometimes simplified in the following way: Suppose. thät the dimensions of the body in the k-direction are small, then the first part of the wave excited force Fkl can be written in the following way

\

_upperside

\

J

underside (S2)

/

since v in equation (II-221 can be substituted by Vj in ecluation (II-8). This follows from the fact that it is assumed that _{Vm is} the undistúíbed water velocity which for a small body has the same value everywhere. This condition of a body in an oscillating f luid equaÏs the condition of an oscillating body in a stationary fluid. _{he total wave excited force on a small body then amounts} to:

I

(II-23)

= Fkl _k21 + Fk22 (II-24)

=iwp [ffkas+f.fkds] =

=iwpff

1dS'

s L _{(Ski) (Sk2)} I since:

dS=-f

dS=dS'

ki k2

If the distance is small enough, then:

-i, s- k

ki ' k2'

In that case the velocity Tk .5_À is constant over the distance which leads to:

= iwp Vk if d S' =

=iwvake0tm

(II-25)

in which m = mass of the water displaced by the body. The total wave excited force then is written by:

y. ₂

= iw [ jk fltj

+ajk)

bi 1w] (II-26)

Except fOr the damping forces the equations of Morrison and O'Brien correspond to equation (II-26) from which it follows that their formulae are only applicable for bodies which are. small rel-ative to the. wave length.

II-3 Comparison of the approximated and exact solution for. the horizontal wave excited .force,on a vertical circular cylinder According toHavelock [lI-4] the horizontal wave excited force on an infinitely long cylinder is given by equation (II-27). This re-suit was obtained by means of diffractioñ theory in which the dis-turbance potential is expressed in terms of a source distribution over the surface of thecylinder., The ápproxúiation .18 the saine as that used in determining the waves produced by, 'a moving cylinder. The source strength at any point is taken to be determined by the horizontal velocity 'in the primary motion and by the gradient of the surface at that point. The obstacle is then replaced by a dis-tribution of sources over the vertical. Elaboration leads to a

horizontal oscillating wave force on an infinitely long cylinder with diameter D:

in which:

J1 and Y1 are Bessel functiòns of the first and second kind of order 1. phase angle between longitudinal force and wave =

= + arctg 1d J1(KD/2) d Y1(KD/2)] L d(KD/2) d(KD/2) dJ (KD/2) dY (KD/2) 2 -

coswt+

X K. (D/2) d(KD/2) d(KD/2) g(D/2)2 [dJl(KD/2) 2 [dYl(KD/2) 2 d(KD/2) d(KD/2) and X -Xa og(D/2)2 In particular: X a. pg(D/2) Ca -kiL a1 2

sin (oet + a1)

sin wt

non-dimensional amplitude of wave excited force

4 1 K2(D/2)2 \ /J-dJ1(KD/2)-J 2 1dY1(KD/211 2

V L

d(icD/2) J L d(KD/2)

J

for KD/2 O (II-27)

According to Flokstra [Il-s] the wave excited force on a vertical cylinder that extends to the bottom of a shallow sea equals the deep water value multiplied by tanh Kh (see also MacCamy [II-6]). The horizontal oscillating wave force on a cylinder with a length equal to the waterdepth and with a diameter D then amounts to:

the previous section on the basis of equation (II-18). As has been discussed. previously the first, part X (thé Froude-Krilof f force) and the Second part X21 (the inertia force) are in phase with each other. Therefore in continuation of equatión (lI-27) the

following subdivision can be made (see Fig. II-1):

4 2 4 tanh ICh, /1dJ1(KD/2) 12 %II I +

V L

d(KD/2) J Once the exact solution of the total wave this force can be split in three parts as

to

XDI2.

Fg. II-l. Components of the wave excitèd foröe inder ccording' to Havelock.

X = (X1 + X21) + X22 = a1 + X21) cos w. + X22

Sifl

ut in which:

f

dY1(KD/2)

12

L d(KD/2)

J

excited forcè Is known, has been discussed in

on a. vértical cyl-(II-29) Length )L21.

X-of cyIinderIwater.depth Diameter ô? cyuìnderD undisturbed pressure Inertia fortS damping force w force X a. pg(D/2) a

direction. of wave propagation where: R = radius of cylinder = cöS [cct - cx ) + icR sin 8] xl (A) O = a [(wt KX0) - KR sin Thus: h ir

X 2 pg Ca R sin (wt -Kx0) f .i1dz r sin ß.cos(icR sinß)dB

1 ₀

.0

From which it follows that (see Fig. II-2): X

al 2

tanh ich (II-3D)

pg Ca (D/2)2 icD/2 1(icD/2)

In the following these parts X1, X21 and X22 ,making up the tötal wave excited force,will be determined according to the

approxinia-tiongivenin equation (II-26) and compared to the exact solution. According to equation (II-25) the first part then is approximated

(X1 + X21) = Xa sin a1

X

x cosa

a22 a J.

In this subdivision the first part due to the undisturbed wave can be calculated exactly (see equation II-18):

X1 = p f1 d S

in which the dimensions of the cylinder are not yet restricte4.

xl. ixo X2

'-I

,\

h ir z x

by: h X1 = p . D f d = X cos wt la o in which:

= horizontal acceleration of the watèr particles (see

In Fig. II-2 both the exact solution (see equation (II-30)) and the approximation (see equation (II-3].)) of the first part of the horizontal wave excited force in a vertical circular cylinder in infinite deep water have been plotted. From this Figure one finds that up to KD/2 = 0.6 the non-dimensional first part of the hori-zontal force can be approximated by equation (II-3].) when not more than 4% difference with the exact solution is allowed.

04

-5

e 4 2 oo

Length of cylinder water depth infinite

Diameter of cylinder D

exact solution

Cpproximation

OES tO

1(D12

Fig. II-2. Component Xa1 of the wave excited force on a vertical cylinder caused by the undisturbed incident wave (Froude-Krïlof f force).

Combining equations (rI-29) and (II-30> will give the exact so-equation (IV-32))

lution for the second part Xa2iOf the wave excited force (see Fig. II-3).

)fl.D12

Fig. II-3. Component Xa21 of the wave excited force on a vertical cylinder _{This component (inertia force) is caused by the} disturb-ances of the incident wavé on the cyìnder and is in phase with the undisturbed pressure component Xal (see Fig,. II-l).

In

order to calculate the forcé X21 according to the approximation given. in equation (XI-23), it is necessary to assume that the di-mensioris of the cylinder- aré small relative' to the wave length. Therefore it lé aésumed that Kb/2 O. The length of the cylinder, hpever, is not small relative to the wave length. tn'ôrder to

-solve this prôblernuàe is màde öf the "strip theory". It then is assumed that the hydrodynamic properties of each strip of the cyl-inder such -as. added mass and dàmpiég are not influenöed by the water motion along the neighboüring strip; see equation(5.1.9) öf Vugts II-3]). One now findé with the aid of equation (II-3) in

which a,=&n; a=a=O:

h

X21 8m _{(C) dC}

Length of cylinder. watér depth. 03 Diameter of cylinder. D

- - deduced from Havelock

6 approximatéd

4

2

in whiôh is the horizontAl acceleration of the water parti-cles (see equation (IV-32)), thus:

X21 = Am. cas (ut - ac0) f d

by which: Xa21 2 - lT C tanh (Kh) (II-32) pg a '2)

in which Cm is the coefficient of added mass (Am = added mass of a small strip of the cylinder = Cm p . D2).

For a cylinder this coefficient

Ç

equals unity (see Kennard [III-5)), if the wave frequency is small (u + O).

In Fig. II-3 the approximated value of the second part of the wave excited force calculated according to equation (II-32), has been compared with the exact solution. Again no larger difference than about 4% has been found between the two values when the wave

length is so long that (p1.2 remains less than 0.6.

Normally the damping force X22 for small bodies relative to the wave length, is very small compared with the undisturbed force X1 as well as compared with the inertia force X21. Since the damping force iS 900 out of phase with the inertia force, it has an in-fluence of less than 5% on the total force if it is less than about 30% of the sum of the inertia. förce and undisturbed pressure force, which holds for KD/2 < 0.65. Apart from a few execptions which will be mentioned in section III-3, the damping force therefore will be neglected when the wave excited force is calculated.

when neglecting the. damping force X22 the approximated total wave excited force is found by adding the undisturbed pressure

force X1 from equation (II-30) and th inertia fôrce X21 from equa-tion (1±-32). From 11g. II-4 one finds that up to KD/2 = 0.6 this approximation. differs 4% at most from the exAct Solution.

From this it can be fOund that for bodies of which the diameter is less than one fifth of the wavelength the oscillatory wave force in an incompressible, irrotational and inviscid fluid can be cal-culated by adding the following parts:

Part 1.. The undisturbed pressure force which is thé force that arises from the pressure over the hull in a wave that is not disturbed by the hull.

Part 2. The inertia force which is the force that arises from the acceleration of the added mass of thé hull in a wave that is not disturbed by the hull.

This approximation differs at most 4% with the exact solution in the case that the horizontal force on a vertical circular cylinder which extends to the sea bottom is considered.

6 4 ej o

J

2 o

Length of cyIinder.water depth h Diameter of cyIinderD exact Solution approximation deep water r

14k

1J-11m.

Ir

IA

ri__

05 1.0 l. D12

Fig. II-4. The total horizontal wave excited force xa on a vertic-al circular cylinder.

It should be noted that ónly the diameter has to be restricted while experience shows that the length of the cylinder is of less importance since by cutting up the cylinder in strips the approx-imation still can be maintained.

Another fact that should be noted is that the force n part 2 is caused by the disturbane of the wave due to presence of the hull. Yet for the calculation of the added mass force the accel-eration of the water particles in the undisturbed wave has to be

sed. This conclusion was reached independently by Newman [II-13] and the author [II-14).

II-4 An, experimental verification of the results obtàined

All the exact solutions mentioned above refer to the force ex-e.rted by waves with small amplitude. Also the forces from model test results have only been determined for low waves. The lineari-zation obtained in this way will allow te prediction of maximum förces in irrégular waves by means of spectral density analysis. Earlier studies on the prediction of maximum forces in-irregu-lar waves were based on the principle öf the determination of the maximum force in a regular wave which corresponds' to the maxi-mum wave to be -expected in the irregular wave-train; see for in-stance Bretschneider Ii-8]. In that case the forces due to high wavés had to be analysed by which non-linear effects were intro-duced which sometimes dominated. Due to these non-linear effects the spectral analysis coüld not be adopted and therefore the de-sign:wave criterion was indispensable. A break-through came when the spectral analysis became common practice. After that the stu-dies were- focussed on determining the maximum wave force -to be ex-pected in irregular sea states; see for a review Freudenthal

[Ii-9], in addition to which also the papers by Pierson [liio] and Borgman [ii-ii] should be mentioned.

These earlier studies were carried out in relation to the strength calculation of fixed structures. In the present study, however, the influence of wave forces on the platform motions are studied, which means that the spectra of wave forcés and the significant value of the force are much more predominant on the development of a maxi-mum platform motion than one single maximaxi-mum wave force.

- In Fig. XI-5 a review is given of the model tests carried out.

It- can be expected that the approximations will beäome. inaccurate: 1. when the length of the cylinder becomes small relative to

the diameter of the èylinder (see the line of limitation

-l/D1),

2. when the clearance between the bottom of the cylinder and the sea bottom becomes small (see the line of limitation

= 0.5).

In Appendix I a description is given of the way in which the model tests have been

performéd.-4

2

o

h

Fig. II-S. Review of model tests with a restrained vertical cyl-inder in waves.

In Figs. lI-6, II-7 and II-S the results of the tests are plotted in comparison with the approximated calculation method, which has been discussed in the previous section.

From these Figures it can be conéluded that for a wave-length which is greater than about 5 times the diameter of the body the

approximated horizontal and the approximated vertical wave excit-éd forcés agree well with the measured results when the cylinder diménsions remain within-the limitations indicated in Fig. II5. When the cylinder length is small relative to the éylinder

diame-ter the measured horizontal force will be smaller than the calcu-lated value. When the clearance under the cylinder becomes small relative to the cylinder diameter the measured vertical wave

ex-cited force becomes smaller than the calculated value.

lo

h i

V

approxlrnatIon of thewave excited fOite

A-.-

-4

rpri'

V.

tests with °

for these condit4ons have been performed

*Ith a cylinder dlaeter D.Q30m

IE aloe N .M o loo 00

water depth measured calculated

075m

100 m o

indicates wave period, for

which XD12 0.6 or X-50

I ' 2

wave period in seconds

3

Fig. II-6. Wave excited forces on a vertical cylinder with a length of 0,6 in

(lID

= 2).

41

o

loo

water depth measured calculated

055m a

Q70m A

105m o

indicates wave period for

which tDI2o6 or A5D

-o A OA A o _A 3 2

wave period In seconds

Fig. IX-7. Wave excited forces on a vertical cylinder with a lengtn of 0,42 in (l/D = 1,4).

k

2 X pfl o 100 00

water depth measured calculated

030 m o

040 m A

aeorn

2

wave period in seconds

3

Fig. II-B. Wave excIted forces on a verticàl cylinder with a length of 0.24 ,m (l/D = 0,8).

indicates wave period for

-III HYDODYNAMIC FORCES

III-1 Introduction

According to

equation

(II-7) forces due to a motion of a body in still water cari be split in three parts:

- the force in phase with the acceleration

of the body (added mass force) hydrodynamic

- the force in phase. with the velocity of forces the body (damping force)

- the force in phase with a displacement _] of the body (restoring force, spring I

hydrostatic

I force

force) j

When these forces are linear with the motion, the hydrodynamic coefficients, being the ratio of the hydrodynamic forces and accel-eration, respeótively velocity, are called added mass, respectivi-ly damping coefficient.

Iñ genéral these coefficients are not constant and depend on the frequency of the motion. When viscous effects play an important role the coefficients will, also depend on the amplitude of the motion. In that case òoxnplex methods are requi:red to find the

so-lutipnof.the body motions (see Bellman [III-1]) because the normal relations (see Solodovnikov

_EIII-2J),

which exist for linear sys-teins and,from which additional Information can be obtained, no longer exist.

Whn it is stated that the coef-ficients. depend on the frequency of the motion it is assumed that only harmonic oscillating motions are considered. In that case for each frequency of oscillation different values of the coeffIcients may be found as has been discussed by Ogilvie [III-3]. A description of the mOtions by means of differential equations then is not really possible since it will not be known which value of the coefficient should be used to solve the motion for a force,which changes arbitrarily with the time. Therefore in the case of frequency dependent coefficients the motion will be described by means of response functions to harmonically oscillating forces; these functions are called res-ponse functions in thé frequency domain as will be discussed in section IV-4.

Frequency dependent coefficients can be determined experimental-ly when the bo4y under consiieratiOn is oscillàtèd hàrmonicalexperimental-ly. In case the motion does not change periodically with time, it wIll

become difficult to determine the required coefficients.

It shòuld be noted that the hydrödynainic forces havè to be mea-sured when the body is in motIon, as it may be possible that a different physical aspect is studied when the hydrodynmic coef-f icients are determined coef-from coef-fixed conditions in moving water. However, from the preceding chapter it will, be clear that whei the hydrodynaxnic coefficients are knowñ from tests in stiÏl water, the hydrodynamic forces in long waves can be determined; see also Lebretön [XII-4]. As was concluded in séction II-3 the wave length

in this context should be larger than, '5 times the diameter òf the body.

II2 Added, mass Fully submer2ed bodies

Whén the body is fully submerqed in an unbounded fluid, the added mass is mainly determined by the area perpendicular to the direc-tionof oscillation (prQjected area). This is most obvious for

cyl-2

00

direction of oscillation

added mass a, Cm1t p(D12f

Fig. III-1. Added mass of rectangular cylinders according to Kennard.

inders, (see Kennard [III-5] in which an extensive list of referen-ces is given) which vary from a plane lamina, to circular and ellip-tic cylinders of which the added mass is p 71 a2 per unit length;

2a being the diameter of the cylinder, perpendicular to the direc-tion of acceleradirec-tion. However, for cylinders of which thé cÊoSS section is different from an elliptic form the added mass does change (see for a rectangular cross section Fig. III-1), as follows from the review by Kennard [III_5].

In addition to the information given in literature about the added mass of cylinders another feature is also used in the pres-ent study. This, is the effect that in an inviscid fluid the added mass of a cylinder oscillating in an arbitrary direction relative to its longitudinal axis can be deduced as indicated by Fig. III-2. The added mass force due to an acceleration in the x-direction amounts to ad. sin a while its direction is perpendicular to the longitudinal axis of the cylinder. The added mass in the direction x therefore amounts to:

a = a . sin2 a

xx d

,1-Acceleration ï sln perpendicular to the cylinder

Added mass follows from ad sln2

Yx

Added mass a follows from -a sin cosa

x d

Fig. III-2. Determination of added mass of an inclIned cylinder.

Direction of scillation

orce

xslnduetothe

acceleration 01 the added mass

in which:

ad = added mass of a cylinder when moved in a dirèction per-pendicular q the cylinder axis.

It will be obvious that the added mass in the y-direction due to the oscillation in the x-direction wIll be:

a= - ad sin

When he body has indrical, the added case the added mass in the direction of in the direction of i 04 iCm 02 o o cos a

a three dihiensional form instead of being cyl-mass will assume a different value. In this is not only -influenced by the projected area oscillaton but also by the length of the body oscillation as indicated in, Fig. IÏI-3, in

direction of oscillation x

added mass a Cm. p1(DI2)3

b

D12

Fig. III-3. Added mass of a spherold.according to Kennard. which also the added masses of a disk. (2b/D=O) and of a sphere

As a most general indication the added mass of an ellipsoid with radii a, b and c along the x-, y- and z-axes, moving in the x-di-rection is given by:

axx = p lT a b c in

which:

dÀ

'V2

(b2+À)(c2+À)

Besides the form of the body., the existence of other bOdies or boundaries will also influence the hydrodynamic coefficients. From the mirror-principle it has been derived that the coefficients are changed to the saine degree when the body lies at. a distance A from a wall as when it lies at a distance 2 A f röm à neighbouring body of the same form. According to page 389 of Kennard [III-5] the added mass of a circular cylinder with a diameter D moving perpendicular to the line connecting the centers òf both cylinders

K t2 1.1

'n

C 0 2

4

D12 6

Fig. III-4. Influence of distance 2A between two bodies on the added mass, according to Kennard.

OdirecUo

of oscillation of one bödy

-mass of- one body in infinite fluid

added added mass

.k.axj

cylinder

amounts to:

D2

= p i

()

(1 + (D,2, +

. ) (III-4)

2A

a = added mass per unit length D = diameter of cylinder

A -= distance between centre of cylinder and a fixed mf i-nite wall.

Added mass a of a cylinder per unit length at frequency WO

al OES a zontal osciiiatin vertical oscillation z D12

Added mass a of a sphere at frequency W O

3

z

D12

Fig. 111-5. Influence of depth of submergence on the added mass according to Yamamoto.

am [i-..

(1.cos2ßXQi] z

[i . I c2LJ

iso

t25 'too

The influence of a neighbouring cylinder (interaction effect) as given by equation (III-4) is plotted in Fig. III-4. In the same way the wall effect on a sphere or the interaction effect between two spheres can be given.

Influence of free surface

When the body is floating or near the free surface the added mass vi].]. be influenced by oscillations of the free surface (waves). For a sphere and a horizontal cylinder the free surface effects have been analysed for deep water by Yamamoto [III-63 (see Fig. III-5)

Besides the effect that the added mass at zero frequency increases when the body reaches the water surface a more serious effect of

the free surface is introduced by the frequency dependency of the added mass (see Fig. III-6). In this study, however, only the zero frequency value will be used. From Fig. III-6 it follows that theoretically this approximation is only allowed for such small frequencies (w) that when taking into account the body diameter

drectic of oscillation

z' D!2.

adde mass of sphere .Ç P ltcob2)3

Fig. III-6. Influence of frequency of oscillation on the added mass of a sphere according to Yainamoto.

aß

z

-vn

(D), the value w \,haé to be sniallér than about

0.8.

In Fig. III-7 and III-8 thé results of model tests (See for a description Appendix I) with vertiöal cylinders are plotted ifi combinatioñ with the theoretically approximated added mass of an infinitely long cylinder in an unbounded fluid. Fr.oin these Figures the conclusion can be drawn that the model test results also show no influence of the frequency if thé frequency is small (w

< 0.8).

The modél test results also Indicate that the added mass coef f

i-dent

can be taken i ity (C = 1) under the following restric-tions: q 2 6

k0.4

o o direction of oscillation

Cm ratio between added mass and water displaced by cylinder

Cm I if cylinderllesIn unlimited fluid

model test results cylinder length

o 024m

£

42m

o

A

0.5 1.0 -- 1.5

Fig. III-1. Added mass determined fròm oscillatiòn tests with a vertical ôylinder with diameter 03 rn

2

*.Qe

A Cm

The length of, the cylinder shall be long enoúgh relative to the diaméter (l/D > abt 1.5).

When the length of thé cylinder is smàll relative to thé diameter, the length has to be long enough relative to the waterdepth (dependent on the ratio between bOttom clearañce and cylinder diameter).

It should be noted that from à point of view öf end effects the requirement lID > 1.5 f Or semi-submerged vertical cylinders has tó be altered to 1/b > 3 for fully submerged cylinders.

directIon of osculation

2

Cm

o

Cm ratio between added mass and water displaced by cylinder

Cm I If cylinder lies in unlimited fluid

model test lesults cylinder len9th

o O.2m L 04m 6 o _e o Q4

Fig. III-8. Added mass determined from oscillation tests with a vertical cylinder with diameter 0,5 in.

A A o o as o G LI_o

5

13 2

i

2 o_e

i

A o o

III-3 Damping

The hjdrodynàxnic damping of a body is influenced as much by po-tential effects as by viscous effects. The damping by popo-tential effects is related to the wave excited forces on the body as follows from the study by Newman 1111-71. This relation will be deduced here for shallow wàter:

a. The velocity potential of the wave generated by the oscilla-ting body in stilt water is defined in equation (1±-1):

iwt = vai +3j e

in whiOh:

Va = amplitude of velocity of oscillation

j = direction of oscillation

with the aid of the radiation condition one finds that at infinity (R + 4) thé potential 3j' amounts to:. iKR 3j = Aj(ß) e cosh K( + d) (III-9)

in which R and are the radius and angle in a system of cylindrical polar coordinates.

One can now deduce a relation between the damping coefficient b. which determines the input of energy of the Oscillating body:

enérgy input = ½ V2 aj

(III-10)

and the function AJ(ß) which determines the energy ràdi.ation at infinity transmittéd by the waves with velocity potential 3:

average work done per unit time

2ir

I, 3$

-p

f

f

Rdzd

o

o

(III-il)

Combining equations CItI-lo) and (III-11) one finds:

b 2. (. +

5l2) f2

A. dB (III-12)

b. The relatiOn between thé wavé excited foröe. and the potential $3 follows from Green's théòrern which states that if

()

is

in which:

- a cosh K(h-z) -iKR cos (ß-) jut

- w

coshh

e e

= wave direction

w

cosh2Kh

4ir pg3 Kh tanh Kh (1 + sinh 2 Kh) 2 Kh

(III-15)

Substituting the functions 03 (see equation (III-9) and into equation (III-14) one finds by evaluating the integral by the method of stationary phase (see Lamb [III-B] page 395):

F.() =

pg

1et

_Kh(1+2Kh)

(III-16)

where F() denotes the exciting force for waves at an angle of incidence ii.

c. By combining the results sub a. (equation (III-12)) and sub b. (equation (III-16)) one finds the relation between the damping and the wave excited force:

2ff

F.

2

b.4 = ai(.') u (III-17)

o

in which:

(XII-18)

In Fig. III-9 the coefficient before the integral in equation (III-17) has been given.

Once the wave excited. force is known from chapter II the relation (III-17) can be compared with an exact solution. As an example the solution for a sphere given by lainamoto [III-6] will be discussed. In this case of a sphere in deep water the rèlation (III-11) can be rewritten as:

any control surface in the fluid outside S (the surface of the body) then silice (velocity potential of undisturbed incident waves) and both satisfy the surface condition, one finds:

- 03

)d S

O (III-13)

which combined with equation (II-18) results in:

=

2pg3

.

(III-19)

in whicil the vertical wave excited forcé follows from equation (II-26)' while neglecting the damping:

z'

= (1 + C)

_{4 i}

p

a3.w2 eZ

(III-20)

a' = radius of sphere

z distance of center of sphere to the. st±ll water surface

K = wavé nümber

Cm coefficient of' added mass (which for the sphere also has been calculated by Yainamoto).

In FigL

ÏII-1O

the euations of the exact solution of the

potent-ial damping and the approxtzitaed potentpotent-ial damping are given. It will be obvious that both solutions are the same as the added mass coefficient of a sphere amounts to ½. Therefore only, one curve of the potential damping as a function of the frequency is given.

id7 water depth h 200m water depth h 25m W3cOsh2lth 4tpg3 ith tanhY.hCi. H . 0.5 ' 1.0 1.5

Wave frequency W Ui red.se

Fig.: III-9. Relation betwèen coefficient y of equation (III-17) and wave frequency w as a function of the waterdepth h.

11 to N C

a

E V

.1

o according to Yamamoto b. Cd7

12!2

-. approximated

b. (

,cjCd7e2Z'Cd2

L . z 1 012 o 13 Dirnen&oflless frequencyw' w

Fig. III-10. Comparison of exact and approximate calculation of potential damping of a sphere.

Except fOr potential damping also damping by the viscosity of the water will exist. The force due to viscus damping is assumed to be proportional to the square of the velocity. The total damp-ing force, bedamp-ing the force in Phase with the velocity theft càn be written as:

= F'dl + Fd2 = b. + jj I I . (III-21)

For the determination of the viscoùs force the drag coéfficiente Cd givénby Hoerner [III_9] for a large variety of body formè have been üsed in this study.

q = l p CD. Sp

(III-22)

in which S _{is thé area of the projection of. the body in}

the di-rection of the vèlocity.

The sOlution of the complete equation of motion:

is quite tedious and ñot relevant.

In thecase that.the damping is small relative to the critical damping (which equals 2 /Tm+a)) ,the damping will influence the amplitudé of the motion only at the natural frequency of oscilla-tiOnaTIerefore the damping will be studied here at the natural

frequency of oscillation. In that case the damping can be deter-mined experimentallyfrom the extinction of thernotion of a freely

vibrating body o

When the viscous damping can be neglected the. motion of a freely oscillating body will be:

b - 2(m+a)

(côs &ut + sin wt) (III-24)

in which:

tI

2

ic

b

Vm+a

4(m+a)2

I/c

V

(III-25) n

Fig. III-il. Influence ot linear damping on the extinction of a freely vibrating system.

b b

- (m.a)uij natural period rn.a virtual mass

b potential damping 0.05 t;.0.10 QiS 4 8 12 t5 1.0 Sn Q o

and:

t = tizne elapsed sihce the time of release = initial amplitude at time of release

From equati.on (III-24) the relation between peak Sn and the next trough

5ni-1 follows from (see Fig III-li):

,rb 2(m+à)w =

- 5n e (III-26)

When, however, the potential damping can be neglected, equation (III-26) changes into equation (III-21). The derivation of this equation is given in Appendix II according to the method of Lindsted (see page 85 of Bellman [Iii-i]):

S+i = -

5n i 1 (III-27) in which: st = n

3m+a5n

Relation (III-27) has been plotted in Fig. III-12.

Now the time between a peak and the next trough follows from:

in which: = m+a

= L [

1

2 (_)2

+ 2 (ti1

1n

m-l-a

Once the damping has been determined from the extinction of the freely oscillating body, one can determine the motion of the body when it is forced to oscillate at the natural frequency.

The amplitude of the forced oscillation at the natural frequency follows from the consideration that the energy input due to the excitation fôrce equals the energy absorbed by the motion of the body platform. From this one f iñds that:

1. In case of potential damping: Fa(wj)

a(W)

= b.w (III-29)

in which b follows from equation (III-17).

2. In case of viscous damping: 5a(w oa i. 2w e

\I

371 F (w.) a

2g

lo S

Fig IIi-12 Influence of viscous damping on the extinction of a freely vibrating system.

)< Fori frequencies largely different from the naturäl frequency the influence of the damping can be neglected. The ranges of frequencies

fôr which the damping has an influence which is less than 10% of the total reaction force follow from:

:w<w V1+2b2_2b1

w>w

\Jl+2b12+2b1

in which: bw. b' -= c = spring constant i 15 (III-30) (III-31) mass of vefocity m.a m.a virtuaI q drag/square

4

! V

yV!+i

r

444

H

In these frequency ranges the total reaction force can be cal-culated without knowledge of the damping. The respOnse function around the natural frequency will be determined in this study by

fairing the response curves, known for the frequency ranges given by equation (IÏI-3i), to the response at the nätural frequency accOrding to equation

(III-29)

or

(III-30).

The influence of the damping on the exciting force has the same tendency as was found for the reaction fOrce. This means that normally the damping force can be néglected with respect to the inertia force or the undistur}ed préssure force except for a frequency range around the frequency for which

th

inèrtià force and the undisturbed pressure forée cancel each other. For the frequency range meant here the damping both from potential and from viscous effects have to be known. The viscous damping normally can be approximated well while the potential damping follows from the relation (III-17) in which the wave exciting forces for other directions also have to be known. When the inertia force and undis-turbed pressure force Oancel each other for one freqùency

indepen-dent of the wave direction, one then finds (neglecting the influence of viscosity) that for instance, the vertical wave excited force

at the frequencyw can be written according to equation

(II-23):

z b b w x

a zz a zz a mz 2

while according to equation (III-11)

z2

= 2i v()

Combining equations

(III-32)

and (III-33) one finds:

i

w

Ca

2rrvij2

_mz

A much more complicated situation arises when the natural

frequency coincides with the frequency for the minimum wave excited force.

In that case the damping dominates both the reaction force and the excited force.

The heave response in case of viscous damping then follows from:

g

a(w)

= q a(w2)

(III-32)

(III-33)

which reduces to:.

zJ

a

- -

(III-35)

Ca _'2

which will be the sanie heave response in case the potential damp-i ing döminates.. It will now be

obvious

that when the natural fre-quency, w and the frequency at which the excited force becomes minimum differ a little from each other, then the response at. the natural frequency will still be according to equation (III-35). The difference between the frequencies _{and umj has to be so} small.' that the damping still dominates for more than 80%. In that case the difference between w. and w follows from:

j mj

Vi -.

0.35

b' <

_{< w}

_V

0.35 b' (III-36) bw.

when b' = is small.

IV THE MOTIONS OF A SEMI-SUBMERSIBLE IN WAVES

IV-1 Introduction

In the preceding chapters the hydrodynamically induced forces on a body of which the dimensions are small relative to the wave length, or small relative to a value g/w2, have been discussed. In this connection is the wave frequency or the frequency of oscillation of the body.

In order to determine the hydrodynamic forces on asemi-submer-sible, the underwater

construction

is subdivided in small elements of which the hydrodynamically induced forces are kñown when the disturbances due to neighbouring elements can be neglected.

From the paper by Kennard [III-8] it has been shown in the pre-ceding section. that in an unbounded fluid the variàtiön in the hydrodynainic force due to

interaction

effects is less than 15% if the distance between two elements is more than the diameter of one of the elements.

However, when free surface effects play a role, it might be expected that

interaction

will have a larger effect on the hydro-dynamic fôrces on the elements.

From model test results and calculations, Boreel [iv=i) has shown that the variation in the wave excited forces on a vertical

circu-lar cylinder will be less than 30% when:

the distance between neighbouring cylinder is more than three times the diameter of one of the cylinders.

the wave length is larger than about five times the diam-eter of one of the elements.

An estimate of the influence of interaction effects on the total hydrodynainic forces on a semi-submersible - consisting of submerged bodies and cylinders piercing the water surface - was made by

comparing the sum of the hydrodynamic fôrces on each of the elextients of the platform, with the total hydrodynainic forces determined from model test results.

In the present chapter the formulae for the hydrodynamically in-duced forces on the elements of the platform together with the summation will be given.

First the added masses ai and the spring constants ci of the platform will be determiñed by sunation of the coefficients pf each elementary part of the platform. These same coefficients of

each part also determine the wave excited .force on each part of the platform from which the total force on the platform can be deter-mined as discussed in sections II-3 and. III-3.

Once the exciting and the reaction fOrces are known, the response functiòns of the platform motions to waves can be determined as inicaed in- section IV-4.

When these calculations are compared with model test resultS it is found that the calculations differ 10%- at most from the model test results (see Ref. IV-2).. From this agreement it may be con-cludedthat the derived alculation method provides sufficiently aÒcurate information about the behaviour of a semi-sübmersjble in

a seaway. -

-The 'method therefore provides a reliable means for.designing semi-submersibles from a point of view of Seaworthiness.

--IV-2 Determination of added mass and hydostatic forces

In his section the added mass au and hydrostatic coèfficiénts

c1 for the total platform will be deduced, when for- each part of thé platform these òoefficients are known.

-The platform will be split in several parts of the following nature:

- a. parts which are fülly sùbmèrged cylinders,

- b. parts which are cylinders which -pierce the water surfaCe,

:

_{parts of arbitrary fOrm Of which the hydrodynainic}

charac-teristics can be assumed to be concentrated in one point. Thé axes of reference are defined to be fixed to the platform in such a way -that the z-axi's is the vertical axis through the

center of gravity G. (See Fig. IV-l). The xy piane coincides with the undistuzbed water surface while the x-axis runs parallel tó the horizontal axis of symmetry.

ad a. y_gjbrnerged cylinders

It is assumed thatthe length 1 of the cylinder is large rela-tive to the diameter D of the cylinder. When the pòint A

(x1 ,y1 ,z1)

and B, )are the extreme poiñts of the cylinder, the length

2

of the cylinder follows from (see Fig. IV-2)

dI

) O Cm i cos dr

Fig. IV-2. Descrition of added mass fOrces on a cylinder.

When the body is oscillated in the x-direction a force, of which the components are X,, and and a mment, of which the

compo-nentsareK , M and Ñ. are encountered.

X X X

-The oscillation of the acceleration is harmonical and amounts to:

x = L. sin (amt (IV-2)

The acceleration component ( fros a) which is perpendicular tö thé cylinder and lies in a plane through the cylinder axis parallel to the x-axis (see Fig. IV-2)' mainly introduces the forces on the cylinder,while the forces due to the acceleration

along the cylinder axis can be neglected. The angle a is deter.-stined by:

X2 - X1

-

sina=-Due to the acceleration (X cos a) a hydÍodrnaic fOrce d on an element or strip dr of the cylinder is generated in the direction of this acceleratòn,the index X refers to the direction of the motion that causeß the force. The force d follows from:

d = i da cas a (IV-3)

in which the added mass da is a function of the location

R on the cylinder. (XriYrZr)

X <X <X

X

x +rsjna

i r 2 r i < 'r < 2 'r = y1 + r sin ß (IV-4) z

<Z <Z

z z

+rsiny

i r 2 r i in which:

r = Vri2 +

'r'i + (z-z1)2 and: sin a = (x2-x1)/l < - 900 < ₉₀ sin B = (y2-y1)/1 < - 900 < B < 90° sin y = (z2-z1))i < - 90° < y < 900

Whi]e resolving the farce d onè finds:

d X= A cas2 a

d Y =-A sin ß sin a

X X

dz

sin y sin a

in which:

A

da=ipO C

dr

in which:

= cross sectional area of the cylinder at point R.

Cm added mass coefficient.

(IV-5)

(IV-6)

(IV-7)

Due to the forces d F a moment relative to the center of grav-ity G arises of which the components follow from:

(x01y0,

_'

d K = d Z

'r'o

- d Y(Zr_Zo)

d M = d X

(z-z0) -

d Z (Xr_Xo)

d N

= d Y (x-x) - d X

When substituting (IV-4) and (IV-7) into (IV-8):

d K = A [(z1-z0) sin a sin ß - (y1-y0)ein a sin ï]

dM =A rsiny+A

[(z

-z)cos2cx+

X X X

i

O

(x1-x) sin a sin ï] (IV-9)

d N =-A r sin ß .- A, [(x1-x0) sin a Sin B + (y1-y0)os2 a]

Due to a motion along the y- and z-axis,similar equations for the forces X, Y and Z and the moments K, M and N are found as given .in equation (IV-8) and (IV-9).

Due to a rotation $ about the x-axis1 forces and moments on the cylinder will also be encountered. The rotation $ will introduce an athwart motion y and a ve±tical motion z of some point

of the cylinder resulting in the following équations:

r'r' r

= -

(ZZ0)

_i

=

'r'o

_j (IV-10)

in which it is assumed that the body is rotated harmonically:

If one now wants to know some components of the force due to the rolling motion, use is made of the equion:

dx =dX +dX

y z

d Y = d Yy + d

dz =dZ +dZ

y z

When using equation (IV-10) in combination with equations similar to (IV-7) one finds:

]

in which:

B=

da = _{°r c} dr

While comparing equation (IV-9) and (IV-12) the resemblance is obvióus.

Inorder to determine the components of the moment.due to the rólling motion, use is máde of equations Similar to equation

(IV-11)

X s B [(z1-z0) sin a sin B -

(y1-y0).

sin a sin -r]

2

d Y =-B r sin y - B [(z1-z0) cos B + (y1-y0) sin B sin

d Z = B r sin B + B [(z1Za sin

B giri y +

(y1-y0) cos2y]

dK- +dK +dK

y z

dM =dM +dM

y z

dN=dN i-d

substitution of equatiOn (IV-13) one

equations (IV-4), (IV-7) and (IV-8) into finds:

2 2 2

d K = B _[(y1-y0) cos y + (z1-z0) cos B

+ 2

(y1-y0)

(z1-z0)

sin B sin

yJ

+ Br [2(y1-y0) sin B + 2(z1-Z0) sin y

+ B4r2 [sin2 B + sin2 y]

d

4=

-Br2 sin a sin ß

- Br

[(x1-x0) sin B +

(y1-y0)

sin B [-(x1-x0) sin y +

(z1-z0)

sin 'a]

[-(z1-z0) sin B + (y1-y0) sin y] + - B (x1-x0) (y1-y0)

d M = -Ber sin a sin y

- Br [ sin y(x1-x0) + sin a (z1-z0)] - B [sin B (x1-x0) sin a

[sin y (y1-y0) sin B (z1-z0)] +

B (z1-z0) (x1-x)

By integration Of the forces over e length oÉ the cylinder, the added mass and damping of each cylinder can be determined. T1e formulae för the added mass as determined above are given in

(IV-12)

in Appendix III-A.

The spring constants of these cylinders are all zero.

ad-b. Cylinders Qiercin2 the water surface

Only the submerged part of the cylinder has to be taken into account.

For. that reason the length of the cylinder is determined by the distance between the one eìtrezne point A( at the water

surface and the point B in the

2

For the determination of the. added mass the formulae given in Appendix III-A can be used.

For the determination of the buoyancy coefficients (spring constants) ci the change of displacement as à result of some motion

has to ne analysed. Due to surge x, sway y and yaw no change of displacement will be found..

In order to compensate for the change of displacement due to heàve (z) a vertical force Z as well as a moment K. and M have to act

z z z

on the body. This force and the moments can be deduced in the following way:

When the cross section of the cylinder amounts to O, the water plane of the cylinder amounts to O/sin y, in which sin y is defined

as-:

-sin y

=

From this it follows that:

pg

p gO

K

= gIg y

(y1-y0)

Mz - . z (x1-x)

1n which

(x0,y

,z) are the co-ordinates of the center of gravity, while z is the heave motion of the center of gravity.

In the saine way the vertical restoring force Z and the restoring moments K and M for roll and pitch respectively can be determined. To this end it is asssumed that the two moments of inertia Of the waterplane area relative to the axes which are parallel and per-pendicular to the x-axis and which run through the center of the water plane of the cyliàder, amount to I. and I

xx yy

As a result of the roll mötion the waterplane will be moved

I

according to:

z= O (y1-y)

In. order 'to generate a pure roll motiön a vertical force. is needed:

z = (IV-'16a)

The roll moment due to heave of the water plane amounts to:

K0 = p g I

+ . (y1-y0)2 (IV-16h)

while the pitch moment arnoüntS to:

=

- ° O (y1-y0) (x1-x) (IV-16c)

'In the, same Way. Z0, K and M0 can be determined.

In addition tô these moments, a moment will also be introduced due to the rotation of the. center of buoyancy B around the center of gravity G.

The otal moment fóllows from:

K0=0GM

i

= O t G M0

j

.

(III-17)

in wiich t is the total displacement of the platform and:

= - BC0 ' (IV-18)

in which:

BM0Z

K0 being the moment K0 of the dth cylinder according to equation (IV-16b).

Using the ¡boye given considerations the restoring force'and moment coefficients canbe. deduced. They are given in Appendix

IV.1

adc. Arbitrari 2arts of the làtform

These parts may be spheres, the ends of a cylinder, ellipsoids or' other bodies of arbitrary form. ,

Fo each part of the platform of which the mass and the added mass can be assumed to be coñcentrated.in one point (x1,y1,z1) one has

x-Y-respectively z-direction. Once these added masses are kñown and assuming that:

a

=a

a

=a

=a

=a

=0

y yx yz zx

yz

zy

the coupling added maSseS f011ow from:

a,

0

axe = ae

= a(z1-z0)

a

=a

=-a '

-p. px

xx'110

In the same way the other cross cOupling coefficients _{can be} determined1

The inertia mOments due to the added masses fllow frOm:

=

a(z1_z0)2

+ a(y1-y0)2

a66 = a5(x1-x)

+ a(Z1-z)

a

= a(y1-y0)

+ a(x1-x0)2

While the rotational òoupling coefficients are:

a6

a6 _{= - a(x1-x0)(y1-y0)}

a = a _{- a(x1_x0) (z1-z0)}

= a,8 = - a(y1y) (z1-z)

The total added mass of the Elatform

'The platform has been split into the following parts: a, cylinders which are fully submerged

cylinders which pierce the water surface

parts of arbitrary form. of which the center lies in (xdj,ydl,zdj).

The center of gravity G of the platform lies in (x,y,z).

The displacement followS from:

4 = p g.V

in which:

V = E V

The volume of each cylinder, follows from:

Vd=

f

_Oddr

o

i

(IV-21) (IV-23)

j

(IV-2 0)

j

(IV-22)

in which l is the length of the cylinder between (X1,

dI' zdl) and (xd2,yd2szd2).

The volume Vd of each of the parts of arbitrary form has to be stated.

The virtual mass now can be calculated by ad4lng the total mass (p V) of the platform to the total added mass,which follows from:

= E adji (IV-24)

in which aj of the cylinders are given in Appendix III-A whIle adii of the other parts are' given in Appendix III-B. In Fig.IV-4 the calculated added mass and added mass moments of the Staflo Drilling Plátform as indicated in Fig. IV-3 have been plotted to-qet1er with the results of model tests.

ELEVA11ON AA

Fig. IV-3. Dimen:

STAFLO

IIr&iIEI

Fur

_803m

i

longitudinal metacentric height GM., lasa m

tranaverso metacentric height GM.1. 858 m

radius of gyration - roll 2477 m

- Fitch 2188 m -yaw 'lisp 2852 m displacement volume V 12,700 m3 405m E

t°.1o9 o 2.1ó

io

I

0! 00_C0000 #0° - oo-IDOS 0000 e 40 io6 C '00 000 oJ.J -10 i CALCULATED -MEASURED o ee Lflu el. 0 flflS. - -0 0 000°0CeCO0! o 000 0 C 0 000 0 0 10.106 1x8 0 001 0 e

0'.

00

Fig. IV-4. Added mass coefficients of Staf lo Drilling Platforms.

o o

ao106 20106 20.1O

o -2fr10 20106

o

o to 20 0 10 2.0 10 20

W ¡n rad.sec W In rad.se Win rad.se

o O O0 00 o 10106 20106 ay. .0 e O #0 o e..0

Total restorin coefficients

Thetotal spring coefficients Cjj are given in Appendix IV. In order to calculate the restoring fôrce coefficients the center of

buoyañcy B, has tö bè knöwñ: Xb,ybZb; in wkich: 1S = E Sd sin d + p V

(*d1o)

S, = E Sd _d

+ o

Vd S = E Sd sin + p Vd (zdl_zo) and :

where r is measured along the center line öf the cylinder from extreme point d1,Yd1lZdl

IVi3 Determination of total wave excited fOrce

In this section the total force due tó regular long creste4 waves will be deduced.

ttis assumed that the direction of propagation of the waves makes ànr angle of i degrees relative to the x-axis f the floating body. The system of co-ordinates of the motion of the water particles is' defined by

The center of this system coincides with the center of the System o co-ordinates (x,y,z).

The vertical -àxis and the z-axis also coincide while the e-axis is parallel to the direction of propagation of the waveS.

From thiS it follows that:

sx

s

XbXO

g

T

'b'o = g - ;

z-Z

= X- COB 11 + Y sin u

r =-x sin U + Y COB U

The characteristics of the undisturbed wave can be deduced from the wave potential (see equation II-13):

g = -

--

cos (wt - (IV-27) with:

cosh K(h)

coshith

J

=g

Sz A (IV-2 5) (IV-26)