H Y D R O D Y N A M I C ASPECTS OF SEMI-SUBMERSIBLE P L A T F O R M S
S T E L L I N G E N
I
Om de maximale uitwendige belasting op een lichaam in een onregelmatige
lang-kammige zee te bepalen is het niet korrekt om het lichaam te onderzoeken in een
regelmatige langkammige golf, waarvan de periode en de hoogte gelijk zijn aan de
schijnbare periode en hoogte van een significante golf uit de onregelmatige
langkam-mige zee.
II
Doordat voor oneindig kleine golffrequenties de relatieve bewegingstheorie geldig
blijft, kan worden aangetoond dat voor deze golffrequenties de slingerhoek of
stamp-hoek van een drijvend lichaam niet gelijk behoeft te zijn aan de golfhelling.
Ill
De kleinere bewegingen van een halfondergedompelde 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 lichamen 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 schip ontstaan.
V
In tegenstelling tot de vaart op diep water heeft men als een schip op ondiep water in
een homogene stroom vaart behalve een bepaalde opstuurkoers ook een gemiddelde
roeruitwijking nodig om 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 trainen van stuurlieden biedt grote voordelen
boven de training door waarneming van het manoeuvreren met schepen in de
werke-lijkheid.
VIII
De inzinking van een model tijdens de voortstuwingsproef is groter dan die tijdens de
weerstandsproef De hierdoor ontstane weerstandsverhoging heeft een meetbare
in-vloed op het zoggetal van het schip.
IX
Voor de bepaling van de ankerlijnkrachten van een afgemeerd lichaam in golven
dient een onderzoek in onregelmatige golven uitgevoeid te worden, daar de variatie
in laagfrequente driftkrachten ten gevolge van onregelmatige golven een essentiele
bijdrage levert in het gedrag van het verankerde lichaam.
X
De begrippen 'boven' en 'onder' duiden richtingen aan langs de straal van een
aard-vast bolcoordinaten 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 om ook de positieve en negatieve richting langs de derde as van dit laatste
assenstelsel te definieren.
HYDRODYNAMIC ASPECTS OF
SEMI-SUBMERSIBLE PLATFORMS
PROEFSCHRIFT
TE VERDED.OEN OP WOENSDAO , M A A R T , , " ' T E ' 4 X 1 , "
DOOR
JAN PIETER HOOFT
S C H E E P S B O U W K U N D I G I N G E N I E U R GEBOREN TE BATAVIA b l / L l 0 1 H L L K TECHNlSCHb HOGESCHOOL DELFT / O / i TQ^^^ )i^i^Z
MAN EN ZONEN R V . W A G E N I N G E N-Dit proefschrift is goedgekeurd
door de promotoren
Prof. Ir. J. Gerritsma
Prof. Dr. R. Timman.
CONTENTS
I INTRODUCTION 1 II OSCILLATORY WAVE FORCES ON SMALL BODIES 3
II-l 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-l Introduction 23 III-2 Added mass 24 III-3 Damping 32 IV THE MOTIONS OF A SEMI-SUBMERSIBLE IN WAVES 41
IV-1 Introduction 41 lV-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 DETERMINATION OF THE DIMENSIONS OF A PLATFORM WITH
MINIMUM VERTICAL MOTIONS IN WAVES 73
V-1 Introduction 73 V-2 Platforms with minimum heave at the natural frequency 7 3
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 10 3 III Contribution of the added m a s s of the elements
of the platform to the total added mass 107 IV C o n t r i b u t i o n of a cylinder to the total r e s
-toring force o f the p l a t f o r m 110 V C o n t r i b u t i o n of the w a v e - e x c i t e d force on an
element of the platform t o t h e total w a v e e x
SUMMARY 121 SAMENVATTING 122 REFERENCES 124 NOMENCLATURE 129 ACKNOWLEDGEMENT 132
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 barges, pipelaying barges, storage platforms, production platforms and drilling platforms are built in this way since it is assumed that the motions of this type of construction 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 follow from an optimization technique. The designs are therefore based on assumptions more or less rea-soned at random (see among others Paulling [l-lj and Carrive [l-2j) supplemented with model experiments 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 seml-submerslble platforms follows from the fact that only recently the need was felt for large seml-submersibles on which high demands were made. In the meantime ship researchers on one side had developed the hydrodynsunics on ship behaviour in waves (see Vugts [l-3]) while on the other side the hydraulic experts had developed the hydro-dynamics for complicated fixed constructions in sea (see Wiegel fl-4j and Ward ^1-5]). However, it turned out that these expert knowledges could not be simply adapted for predicting the beha-viour of a semi-submersible platform in waves (see erniong others Paulling [l-l] and Fujii [l-s]) .
In the present study an attempt is made to derive a design method by which the behaviour of seml-submersibles 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-glected in this method. The prediction method to be developed, has been based on the relative motion concept (see Motora fl-v] and Gerritsma ^I~8j) applied to the components of the platform. This means that:
a. The method originates from a potential theory for small bodies (see chapter II) .
b. 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 [l-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 deduced of a semi-submersible platform which from a hydrodynamical point of 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 consideration since hydro-dyncimic aspects are then introduced which are different from the aspects investigated in the present study.
II OSCILLATORY WAVE FORCES ON SMALL BODIES
II-l Introduction
The 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. However, an analytical method to determine the influence of viscous effects on the total force is not yet available.
To predict the wave forces, several approximations have already been given, for instance that given by Morisen [ll"l] • A review of the work carried out experimentally and theoretically is describ-ed in chapter II of Wiegel [l-4]. In addition to that review the papers by Harleman [11-2] and [lI-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 of the lack 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 on a structure in a fluid which is supposed to be incompressible, irrotational and in-viscid. The limitations of this approximation will be deduced 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 Havelock [lI-4] for deep water. It can be adapted for shallow water as given by Flokstra [ll-s] . In the meantime MacCamy
[11-6] 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 [lI-V] . 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 dieimeter 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_oscillatin2 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 subdivided in three components:
* = *j^ + $2 + *3j in which
*, = velocity potential of incident waves
$- = velocity potential of the waves reflected on the body *-.= velocity potential of the waves generated by the motion
of the body in the direction j .
In still water the velocity potential * . due to an oscillating body depends on the direction of oscillation j , the amplitude of oscillation s and the frequency oo.
a
*3j = -aj *3j «'"^ ("-!) with:
d s.
V . = ( J. ^) = iu s . 33 dt max a]
On the hull of the body the following condition has to be satis-fied (see the boundary condition in equation 11-11):
3(t)
3n -j with:
^ = V. . n (II-2)
n being the normal to the body surface, positive in the out-ward direction
and
loot
V. = V • e
One therefore finds:
^ = Vaj • ^j • -'"' ("-3)
with:
f. = cos (n ,
s.)
The combination, of equation (II-l) and (II-3) results in:
3*3.
The hydrodynamic force on the body amounts to:
^k hydr.= -
^[
P-
h ^^ ..
^"-5)
in which:
3*3
P - Pi = - P
- ^
= - ip« $3j
in which p, is the static pressure on the hull.
The total force on an oscillating body can be split into:
d2s,
F, . . = m, 6, . — r ^ + Fu u .:. (II-6)
k tot.
k
k] , 2 k hydr.
in which:
m. = mass or moment of inertia of body
= 0 if k / j
F, , , = hydrodynamic force or moment
The hydrodynamic force can be split into three parts:
d^s. d s.
F, , , = a, . . —
Tp-
+ b, . .
-rr'-
+ c. . . s. (II-7)
k hydr. k] , 2 k] dt k] 3 ,
in which the following definitions are used:
a, . = added mass
by-
= damping coefficient
c, . = spring constant = restoring coefficient
Combining equations (IX-5) and (II-7) one finds: ipu) v^j e // ^^ . - ^ _ d S =
Re p /^/ * 3 . . f^ . d S = a,^. im up ;^/ * 3 , . f^ . d S = -b,^.
The spring force c, . s. follows from the integration of the stat-J*3 3
ic pressure (p.) force over the body after a displacement s.. 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:
$ = 't^ + $2 (II-9)
$-= velocity potential of the waves reflected on the obstacle
The functions $, and $- satisfy the Seime free surface condition and the boundary condition on the hull as the function $_. which has been discussed in the aforegoing. These conditions follow from the assumption that the fluid under consideration has a bound-ary surface 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 surface were given by an equation x, <.\ = 0 it follows that the condition
(x,y,z,t;
f^ = u x^ + V Xy + w x^ + X^. = 0 (11-10)
whould hold on S. From the fact that u = * , v = $ and w = 0.,
x y z and the fact that (Xv' X » X ) is a normal vector to the surface
in consideration, it follows that equation (11-10) can be written in the form:
3$
g^ = V . n (11-11)
on a moving surface, which reduces to:
^ = 0 (11-12)
dn
on a fixed surface.
The functions $_ 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 [ll-12] ) . The radiation condition usually re-quires the waves at infinity to be progressing outwards and impos-es an uniquenimpos-ess which would not otherwise be primpos-esent (see sec-tion 13 page 475 of Wehausen [lI-J.2]).
The velocity potential of the undisturbed Incident wave is known everywhere by the equation:
$ = - l i i l i a g i w t g - l K S ( j j _ ^ 3 ) 1 (0 in which: _ cosh K(h-^) ^1 cosh Kh 5 = wave amplitude h = waterdepth
u = wave frequency = 2IT/T T = wave period
K = wave number = 2Tr/X X = wave l e n g t h
5 = coordinate in the direction of wave propagation C = coordinate in the vertical downward direction On the hull one finds according to equation (11-12):
3$
S*-^ = - S*-^ (11-S*-^4) The generalized force equals:
F]^ = - ;/ p f^ d S =
= iup ;; (0^ + 'i>2) ^k "^ ^ (11-15)
o
According to equation (I1-4) one finds:
f
= !l3]i
3(|>3k
which means that f^ equals — 7 — in the case that the body oscilla-J^ dn
tes in still water in the k-direction.
Substituting equation (II-4) into equation (11-15) :
Fj^ = iup ;/ [<p^ + (i.^) - g ^ d S (11-16)
Using Green's theorem and equation (11-14) one finds:
3((i 3 $ , 3il>
'^ '2^^^ = 'I *3k 3ir ^ ^ = - 4^ *3k 3ir ^ 2 ("-^^'
Combining equations (11-16) and 11-17) one now finds:^k ^kl " '^k2
^kl= ^"P ^g-^ *1 \ ^ S F. _= iup ;/ 4,,. ^ d s
•k2 ^"^ 'g' '<'3k 3n
(11-18)
In this equation, the force F, , due to the pressure variations 3$1 itl
p -r—- Is called the undisturbed wave pressure force (Froude-Kri-d t
loff force) .
The variations of the force F,-, due to the pressure variations
3*T '^^
p T—*• of the disturbances, are split in two parts which are called d t
the inertia forces F, _ (in phase with F. ,) and the damping forces F,_, (out of phase with F. . ) .
The force F, _ due to the disturbance of the incident wave can be k2 rewritten by:
Fk2 = i"P -^Z % *3k fm ^ S (11-19)
since: 3$ 3s 3s 3s aTT- = V, ^ + V, ^ + V, ^ (11-20) dn 1 dn 2 dn 3 dn in which: 30, 3s v„ = T ^ and T-Sl = f„ m 3s dn m mThe total wave excited force now is determined by:
F]^ = iup /; $^ f^ d S - iup ;/ v^ (fj,^ ^m ^ ^ (11-21) o o
The_wave_excited_f2rce_on_a_relatiyely_small_body
If the height and the length of the body are small relative to the wave length one may assume that v in equation (11-21) has the same value all over the body.
In that case the force F.2 (see equation (11-19)) can be trans-formed into:
^k2 iup v^ // $3,^ f^ d S (11-22) Combining this and equation (II-8) one finds;
^k2 k21 •k22 ^k21 "^ ^k22 , iut V ..e^"^b.^ 33 3k (11-23)
since v in equation (11-22) can be substituted by v. in equation (II-8). This follows from the fact that it is assumed that v is
m the undisturbed water velocity which for a small body has the same value everywhere. This condition of a body in an oscillating fluid equals the condition of an oscillating body in a stationary fluid. The total wave excited force on a small body then amounts to:
F = F + F + F
k 'kl k21 ''k22 (11-24)
in which:
F,., = iup // *, f,^ d S = undisturbed pressure force S kl '"" V '1 'k iut F, _, = iu.v ..e k21 33 "jk ?.„ = V ..ei-^t ^ k22 aj jk
= Froude - Kriloff force = inertia force
= damping force
Though this result is complete in order to calculate the wave excited force, it is sometimes simplified in the following way: Suppose that the dimensions of the body in the k-direction are small, then the first part of the wave excited force Fj^, can be written in the following way:
k-direction
upperside (S )
^ k l = ^ up ;/ $^ fj^ d s + ;/ $^ f]^ d s L Sj S2 -I
r * - $ 1
L -^(s. .) -^(s. „) J
= iup // *, - *, I d S' since: f d S = - f d S = d S ' («kl> («k2) If the distance As. is small enough, then:3*1 *i " *i " aT" ^°k
'(^kl^ '(^k2' '"^ ' 3$
In that case the velocity v, = TT-^ is constant over the distance 3s.
As, which leads to: k
Fj^j = iup v^ ;/ As^ d S' =
= iu v^^ e^'^^ m (11-25) in which m = mass of the water displaced by the body.
The total wave excited force then is written by: iut
^k = ^ ^ ^ ^ p (6.^ m^ . a.^) - b.,^ iu] (11-26)
Except for the damping forces the equations of Morrison and O'Brien correspond to equation (11-26) 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 to Havelock [ll-4j the horizontal wave excited force on an infinitely long cylinder is given by equation (11-27). This re-sult was obtained by means of diffraction theory in which the dis-turbance potential is expressed in terms of a source distribution over the surface of the cylinder. The approximation is the same 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: g(D/2)^C. K^(D/2)^ d(KD/2) dJ ( K D / 2 ) „. — = cos ut + -dY,(KD/2) d(KD/2) sin ut dY,(KD/2) rdJ^(KD/:2) -] - rdY^(KD/2)-|
L d(KD/2) J L d(KD/2) J
pg(D/2) C •=— sin (ut + a ) (11-27) in which:J and Y, are Bessel functions of the first and second kind of order 1.
a, = phase angle between longitudinal force and wave =
and = + arc r d J tg i L d(K d J , ( K D / 2 ) d Y J ( K D / 2 ) d(KD/2) d(KD/2) pg(D/2)
5-non-dimensional amplitude of wave excited force =
2 2 K ' ' ( D / 2 ) ^ . /rdJj^(KD/2)-| 2 rdY^(KD/2)-|
V L d(KD/2) J L d(KD/2) J
In particular: X pg(D/2)^C 2w for K D / 2 -»• 0According to Flokstrs [il-5] the W3ve 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 MacC3my [jl-s]). The horizontal oscillating wave force on a cylinder with a length equal to the waterdepth and with a diameter D then amounts to:
4 tanh Kh
pg(D/2) C. , , \ /rdJ, (KD/2) -1 2 [- dY,(KD/2) -i
K (D/2)M/ -J^ + - ^
V L d(KD/2) J L d(KD/2) J
(11-28)
Once the exact solution of the total wave excited force is known, this force can be split in three parts as has been discussed in the previous section on the basis of equation (11-18). As has been discussed previously the first part X (the Froude-Kriloff force) and the second part X (the inertia force) are in phase with each other. Therefore in continuation of equation (11-27) the following subdivision can be made (see Fig. II-l):
Length of cylinder>water.depths co Diameter of cylinderiD Xg^ » undisturbed pressure force Xa2i • inertia force
Xa22* damping force Xa • pg^(D/2)Z
/
A
X ' a 2 i \ ^ a 2 21
^
~ ~ ~
-" ^
V,
Q5 1.0 X D;2Fig. II-l. Components of the wave excited force on a vertical cyl-inder according to Havelock.
(^1 + ^2l' * ^22
<^al ^ ^a21> =°^ "^ * ^ 2 2 ^^" "^ (11-29) in which:
(X , + X -,) = X sin a, al a21 a 1 a22 = X cos c, a 1
In this subdivision the first part due to the undisturbed wave can be C3lcul3ted exactly (see equation 11-18):
3*
^1 = P ^/ 3t^ 'x ^ S
in which the dimensions of the cylinder are not yet restricted.
direction of wave propagation R sin B X + R sin 3$ 3$ h IT X, x, X, = p ; dz ; {-r^ 3 ^ ) R sin g d e o o where: R = radius of cylinder
= - ^—3. cos [(ut - KX ) + K R sin gl
u •- o •>
= _ cos [(ut - KX ) - K R sin 3|
u I- o -'
Thus:
X = 2 pg 5 R sin (ut - K X ) / y,dz /' sin B . C O S ( K R sinB)d6
1 a o ... J. ,^ •^ o o From which it follows that (See Fig. II-2)
X al
pg 5, (D/2)' — ^1 K D / 2 ^ ( K D / 2 )
tanh Kh (11-30)
In the following these parts X , X ^ and X making up the total wave excited force^will be determined according to the approxima-tion given in equaapproxima-tion (11-26) and compared to the exact soluapproxima-tion. According to equation (11-25) the first part then is approximated
by:
in which:
" r^2P 4 D
h ..
/ 5
(C)
d5
X, cos ut
la
• ( C ) la= horizontal acceleration of the water particles (see
equation (IV-32))
TT tanh Kh (11-31)
In Fig. II-2 both the exact solution (see equation (11-30)) and
the approximation (see equation (11-31)) 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 (11-31) when not more
than 4% difference with the exact solution is allowed.
CM
S
Length of cylinder*water depth• infinite Diameter of cylinder-D
exact solution
-
^
0.5 1.0
XD/2
Fig. II-2. Component X . of the wave excited force on a vertical
cylinder caused by the undisturbed incident wave (Froude-Kriloff
force).
so-lution for the second part X 51°^ ^^^ wave excited force (see Fig. II-3).
CM O
JP
Length of cylinders water depth- co Diameter of cylinder* D ^>^ approximated .
\
V\
0.5 1.0 XD/2Fig. II-3. Component X . of the wave excited force on a vertical cylinder. This component (inertia force) is caused by the disturb-ances of the incident wave on the cylinder and is in phase with the undisturbed pressure component X ^ (see Fig. II-l) .
In order to calculate the force X2J according to the approximation given in equation (11-23), it is necessary to assume that the di-mensions of the cylinder are small relative to the wave length. Therefore it is assumed that K D / 2 ->• 0. The length of the cylinder, however, is not small relative to the wave length. In order to solve this problem use is made of the "strip theory". It then is assumed that the hydrodynamic properties of each strip of the cyl-inder such as added mass and dcimping are not influenced by the water motion along the neighbouring strip; see equation(5.1.9) of Vugts [1-3]). One now finds with the aid of equation (11-23) in which a„„ = Am; a = a = 0: XX ' °yx ZX
in which $,,> is the horizontal acceleration of the water parti-cles (see equation (IV-32)), thus:
h Xjj^ = Am. Kg 5^ cos (ut - KX^^) / ^ dK.
o
by which X ^ii _ = n C tanh (Kh) (11-32) Z m pg Ca (D/2)in which C is the coefficient of added mass (Am = added mass of m IT ?
a small strip of the cylinder = C p i D'') . m 4
For a cylinder this coefficient C equals unity (see Kennard [111-5] ) , if the wave frequency is small (u ->• 0) .
In Fig. II-3 the approximated value of the second part of the wave excited force calculated according to equation (11-32), 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 K D / 2 remains less than 0.6.
Normally the damping force X-, for small bodies relative to the wave length, is very small compared with the undisturbed force X as well as compared with the inertia force X_ . Since the damping force is 90 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 force and undisturbed pressure force, which holds for K D / 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 X the approximated total wave excited force is found by adding the undisturbed pressure force X from equation (11-30) and the inertia force X-, from equa-tion (11-32) . From Fig. II-4 one finds that up to K D / 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 psrts;
Part 1. The undisturbed pressure force which is the 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 the 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.
Length of cylinders water depth =h
Diameter of cylinders 0
exact solution approximation
XD/2
Fig. II-4. The total horizontal wave excited force X on a vertic-al circular cylinder.
It should be noted that only 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 in part 2 is caused by the disturbance 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
used. This conclusion was reached independently by Newman [ll-l3] and the author [11-143.
II-4 An experimental verification of the results obtained All the exact solutions mentioned above refer to the force ex-erted 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 the prediction of maximum forces in irregular waves by means of spectral density analysis. Earlier studies on the prediction of maximum forces in irregu-lar waves were based on the principle of 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 [11-8]. In that case the forces due to high waves had to be analysed by which non-linear effects were intro-duced which sometimes dominated. Due to these non-linear effects the spectral analysis could 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
[11-9] , in addition to which also the papers by Pierson [ll-io] and Borgman [ll-ll] 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 forces 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. II-5 a review is given of the model tests carried out. It can be expected that the approximations will become inaccurate:
1. when the length of the cylinder becomes small relative to the diameter of the cylinder (see the line of limitation 1/D = 1 ) ,
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 performed.
I h
h - I Q5
tor ttiese ccxiditions tests have been performed with a cylinder with diameter D>Q30m
Fig. II-5. Review of model tests with a restrained vertical cyl-inder in waves.
In Figs. II-6, II-7 and II-8 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 concluded 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-ed forces agree well with the measurexcit-ed results when the cylinder dimensions remain within the limitations indicated in Fig. II-5. When the cylinder length is small relative to the cylinder 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.
D«0.30m I •neom
200
S-E
T—<r-<r-r-<r-r—^—v-T-T-r-water depth measured 075 m 4
1JOO m o
indicates wave period for which XD/2-06 o r X - 5 D calculated 0 100 S-IE s • ... JZ N
^
^
""^'•-.^
nl o^
/\
/ •A
>''^
--::;==::
1 2 wave period in secondsFig. II-6. Wave excited forces on a vertical cylinder with a length of 0,6 m (1/D = 2 ) .
2 0 0 y E
x"U
0 100 » £ D = 0 3 0 m I • 0 4 2 mwater depth measured 0 5 5 m a Q70 m 4 1 0 5 m o indicates wave period for which X D / 2 - 0 6 or X - 5D calculated
^
V.
o 0i ^ ^ ? ^ ~ ^
~ ° 4 1^
/0
^
U'
::^^^'
4 2ggs^
p & 3 «l awave period in seconds
Fig. II-7. Wave excited forces on a vertical cylinder with a length of 0,42 m (1/D = 1,4).
2 0 0 I-IE u
n
^ % a nl a O 100 D > Q 3 0 m I • Q24 m w a t e r depth measured 0 3 0 m o 0 4 0 m 4 0 6 0 m o indicates wave period for which XD/2 - 0 6 or X - 5D calculated ^ V ^"^^^-^.r^"**"-*
o " 0 **—^:z:z
/i
^
f
4"^r^^'
8 = : ^ -nl nwave period in seconds
Fig. II-8. Wave excited forces on a vertical cylinder with a length of 0,24 m (1/D = 0,8) .
hydrodynamic forces
hydrostatic force III HYDRODYNAMIC FORCES
III-l Introduction
According to equation (II-7) forces due to a motion of a body in still water can be split in three parts:
- the force in phase with the acceleration of the body (added mass force)
- the force in phase with the velocity of ^the body (damping force)
- the force in phase with a displacement of the body (restoring force, spring force)
When these forces are linear with the motion, the hydrodynamic coefficients, being the ratio of the hydrodynamic forces and accel-eration, respectively velocity, are called added mass, respectivi-ly damping coefficient.
In general 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 complex methods are required to find the so-lution of. the body motions (see Bellman [lII-l] ) because the normal relations (see Solodovnikov rill-2j ) , which exist for linear sys-tems and from which additional information can be obtained, no longer exist.
When it is stated that the coefficients 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 [jlI-33 . 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 harmonic3lly oscillsting forces; these functions are called res-ponse functions in the frequency domain as will be discussed in section IV-4.
Frequency dependent coefficients can be determined experimental-ly when the body under consideration is oscillated h3rmonic3lexperimental-ly. In case the motion does not change periodically with time, it will
become difficult to determine the required coefficients.
It should be noted that the hydrodynamic forces have to be mea-sured when the body is in motion, as it may be possible that a different physical aspect is studied when the hydrodynamic coef-ficients are determined from fixed conditions in moving water. However, from the preceding chapter it will be clear that when the hydrodynamic coefficients are known from tests in still water, the hydrodynamic forces in long waves can be determined; see also Lebreton [lll-4j . As was concluded in section II-3 the wave length in this context should be larger than 5 times the diameter of the body.
III-2 Added mass
Fy^l^_§y^5!?E3S^_^2^if §
When the body is fully submerged in an unbounded fluid, the added mass is mainly determined by the area perpendicular to the direc-tion of oscilladirec-tion (projected area). This is most obvious for
cyl-D
^
/
direction of oscillation
added mass a^^ • C „ Tl p(D/2r
^
Q I — 1 1 1 1 1
L-0 5 1L-0 15 J.
D
Fig. III-l. Added mass of rectangular cylinders according to Kennard.
inders (see Kennard [lll-s] in which 3n extensive list of referen-ces is given) which vary from a plane lamins, to circular and
ellip-2
tic cylinders of which the added mass is p IT a per unit length; 2a being the diameter of the cylinder perpendicular to the direc-tion of acceleradirec-tion. However, for cylinders of which the cross section is different from an elliptic form the added mass does change (see for a rectangular cross section Fig. III-l), as follows from the review by Kennard [111-5J.
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 srbitrary 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 a,. x sin a while its direction is perpendicular to the longitudinal axis of the cylinder. The added mass in the direction X therefore amounts to:
^xx = ^d- ^^" ( I I I - l )
Direction of oscillation Acceleration x sina perpendicular ] to the cylinder
Force a^ x sin O due to the acceleration of the added mass
a^ of the cylinder
Added mass a,^, follows from -:4 • a^ sirra
Added massawy follows from -s- c-a.. sina cosa
y * X d
in which:
a. = added mass of a cylinder when moved in a direction per-pendicular to 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:
^yx ^ " ^d ^^" * '^°^ " (III-2)
When the body has a three dimensional form instead of being cyl-indrical, the added mass will assume a different value. In this case the added mass is not only influenced by the projected area in the direction of oscillation but also by the length of the body in the direction of oscillation as indicated in Fig. III-3, in / which also the added masses of a disk (2b/D=0) and of a sphere
(2b/D=l) are given.
^n
\vj
\
^
\iy
\
direction of oscillation x added mass a ^ = C^. A pTt(D/2)^ O 5 10 15 b D/2As 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:
2-a •T p TT a b c (III-3) in which:
AX.
a = a b c ;. / - 5 o y(a''+X)3 (b'=+X).(c^+X)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 same degree when the body lies at a distance A from a wall as when it lies at a distance 2 A from a neighbouring body of the same form. According to page 389 of Kennard [lll-s] the added mass of a circular cylinder with a diameter D moving perpendicular to the line connecting the centers of both cylinders
t2
1.1
1.0
HO 1
1 V—y direction of oscillation 12A
j J ( . ( added mass of one body 1 1 V l y ' x x • •< »xxu
^xxu ' "^'^"^ mass of one body in infinite fluid
\
\
\ \
\
\
^cylinder ^sphere^
A D/2Fig. III-4. Influence of distance 2A between two bodies on the added mass, according to Kennard.
amounts to:
2 2
^ix = P '^ (7) (1 + ^°^^2 + • • • • ) (III-4)
XX /: 2A''
a' = added mass per unit length
D = diameter of cylinder
A = distance between centre of cylinder and a fixed
infi-nite wall.
iao
Added mass a' of a cylinder per unit length at frequency (0 = 0156 a' 100 Q7
\
y independent of \ direction of \oscillation\
z a'.m' 1 • j (-/
^
> i 2 ^ ' z ' w w c t o n oscillation 1 I D/2Added mass a of a sphere at frequency U)s0
0 6 _a m
as
jhonzontal oscillation \.vertical oscillation^ \
^
;:::;-;;__.
8 ' Z [ 2 32^
-
^
a
.2p,(D/2pJ
- "Section oscillationy
D/2Fig. III-5. Influence of depth of submergence on the added mass
according to Yamamoto.
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,
iSfiH®DE?_2f_£E5§_§HE-S2S
When the body is floating or near the free surface the added mass will 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 [111-6] (see Fig. XII-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 (u) that when taking into account the body diameter
direction of oscillation
added mass of sphere • C „ p .^ TC(D/2)^
=
P\
J ^
• • • • >\ ,
^
\ \
0 as 1.0 15Fig. III-6. Influence of frequency of oscillation on the added mass of a sphere according to Yamamoto.
(D) , the value u \J~~ has to be smaller than about 0.8.
In Fig. III-7 and III-8 the results of model tests (see for a description Appendix I) with vertical cylinders are plotted in combination with the theoretically approximated added mass of an infinitely long cylinder in an unbounded fluid. From these Figures the conclusion can be drawn that the model test results also show no influence of the frequency if the frequency is small (u \ / ^ ^
< 0.8). ^ The model test results also indicate that the added mass
coeffi-cient C^ can be taken unity (C^^^ = 1) under the following restric-tions : direction of oscillation 1 Cm 1 Cm 1 Cm
Cm > ratio between added mass and water displaced by cylinder
Cfff : 1 If cylinder lies in unlimited fluid
model test results cylinder length o 0 24 m & 042m o 1
1 i
& i-.04 & ^ A L1-06
^ • & • ^ • 0 7 6 3 0 5 -10Fig. III-7. Added mass determined from oscillation tests with a vertical cylinder with diameter 0,3 m.
a. The length of the cylinder shall be long enough relative to the diameter (1/D > abt 1.5).
b. When the length of the cylinder is small relative to the diameter, the length has to be long enough relative to the waterdepth (dependent on the ratio between bottom clearance and cylinder diameter).
It should be noted that from a point of view of end effects the requirement 1/D > 1.5 for semi-submerged vertical cylinders has to be altered to 1/D > 3 for fully submerged cylinders.
d i r e c t i o n of oscillation
\ \ \ \ \ \ \ \ \ \ \
Cm ' r a t i o b e t w e e n added mass and
w a t e r displaced by c y l i n d e r Cm • 1 If c y l i n d e r lies m u n l i m i t e d f l u i d model t e s t r e s u l t s c y l i n d e r length o 0.2 m A 0.4 m 1 Cm 1 Cm 1 Cm 6 A , 0 t 1
>
^ . 0 4 6 ' 1 A i 1 i.= 0.6 o A! 2
f
1Tr-°«
Q 5 .1.0 I SFig. III-8. Added mass determined from oscillation tests with a vertical cylinder with diameter 0,5 m.
III-3 Damping
The hydrodynamic damping of a body is influenced as much by po-tential effects as by viscous effects. The damping by popo-tential eff.ects is related to the wave excited forces on the body as follows from the study by Newman [ I I I - T ] . This relation will be deduced here for shallow water:
a. The velocity potential $, of the wave generated by the oscilla-ting body in still water is defined in equation (II-l):
*3j = ^aj *3j «'"' (^"-«) in which:
V = amplitude of velocity of oscillation j = direction of oscillation
With the aid of the radiation condition one finds that at infinity (R -»• ") the potential <ti-. amounts to:
- I K R
*^-i = A.,Q> cosh K ( C + d) (III-9) 33 D(ti) ^
in which R and g are the radius and angle in a system of cylindrical polar coordinates.
One can now deduce a relation between the damping coefficient b.j which determines the input of energy of the oscillating body:
2
energy input = ij v , b. . (III-IO) a] 3 3
and the function A. .„, which determines the energy radi3tion 3t infinity transmitted by the waves with velocity potential * :
average work done per unit time =
2-n =0
= - p ; • ' ' i l ^ K ^^^^ (iii-ii)
o o
Combining equations (III-lO) and (III-ll) one finds:
b £ 0 ^ (1 ^ sinh2Kh, / ' ' A . , ^ , d6 (III-12) 33 2 2Kh ](g)
b. The relation between the wave excited force and the potential *- follows from Green's theorem which states that if S. . is
any control surface in the fluid outside S (the surface of the body) then since 0 (velocity potential of undisturbed incident waves) and *, both satisfy the surface condition, one finds:
3(ti 3$
•''•'' <*i aTT - 'f'-j ^ ) d S = 0 (III-13) S+s„ 1 3n 3 3n
which combined with equation (11-18) results in: F. k
S in which;
3()>,k S$
iup ;; ($^ - ^ — ((,3^ j^) d s (III-14)
$ = £5a cosh K(h-Z) -IKR cos (g-u) iut (TTT-lSl
*1 u cosh Kh ® ® ^^^^ '•^>
wave direction
Substituting the functions ^, (see equation (III-9) and $, into equation (III-14) one finds by evaluating the integral by the method of st3tion3ry phase (see Lamb [lll-sj page 395) :
^J(U)
- I T \ /2Tr .iut - Kh ,, .sinh 2Kh^
^ pg V ~ ^(U)-Cosh Kh<^^ 2 Kh ^
(III-16) where F., , denotes the exciting force for waves at an angle of
incidence y•
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:
2TT F . 2 b. . = V / ( ^3it^)) a u (III-17) -'•' o ^a in which: V = ^ cosh Kh (ITI-181 47T pg3 Kh tanh Kh (1 + sinh2jch < " ^ ^8' ' 2 Kh
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 Yamamoto [lll-e] will be discussed. In this case of a sphere in deep water the relation (III-17) can be rewritten as:
Z 2 3
^zz
-
^r' ; — 3
a 2 pg
(III-19)
in which the vertical wave excited force follows from equation (11-26) while neglecting the damping:
_a = n 4. r 1 4 IT p a^ u^ e"''^ (III-20) m
a = radius of sphere
z = distance of center of sphere to the still water surface K = wave number
C = coefficient of added mass (which for the sphere also has been calculated by Yamamoto).
In Fig. III-IO the equations of the exact solution of the potent-ial damping and the approximated 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 ^i. Therefore only one curve of the potential damping as a function of the frequency is given.
15-10 10-10' 5-10' water depth h« 200 m water depth hi 25m U)-^cosh^Xh 4Tlpg3 Xh tanhXhCI • ^iM^ 2Xh W a v e frequency U in rad.sec.
Fig. III-9. Relation between coefficient v of equation (III-17) and wave frequency u as a function of the waterdepth h.
a-o|cv 5. •«lr)1.0
f
£ 0.5according to Yamamoto b^^ • ^ oj'^e^^''^ approximated b^^ • (1 • C m f ^ w ' ^ e ^ ^ ' " '
»o
zy
D/2• N
0 Q5 1.0 1.5 Dimensionless frequency U)' = U) \ / ^ =-Fig. III-IO. Comparison of exact and approximate calculation of potential damping b of a sphere.
Except for potential damping also damping by the viscosity of the water will exist. The force due to viscous 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 then can be written as:
F- = F + F,. = b.. S. + q.. S. &. (III-21) d dl d2 ]] 3 ^33 3 I 3 I
For the determination of the viscous force the drag coefficients C^ given by Hoerner [111-93 for a large variety of body forms have been used in this study.
q = is P Cp. Sp (III-22)
in which S is the area of the projection of the body in the di-rection of the velocity.
The solution of the complete equation of motion:
is quite tedious and not relevant.
In the case that the damping is small relative to the critical damping (which equals 2 ^c(m+3)),the damping will influence the amplitude of the motion only at the natural frequency of oscilla-"**
tlon.Therefore the damping will be studied here at the natural
frequency of oscillation. In that case the damping can be deter-mined experimentally from the extinction of the motion of a freely vibrating body.
When the viscous damping can be neglected the motion of a freely oscillating body will be:
b 2(m+a) (cos wt + 2JJ s^" "^) (III-24) in which:
y
m+a 15 1.0 Sn«1 Sn 0.5 4 (m+a)' (III-25)^
\
^
\
• ^ V . ,b'.0.05 -b' = 0.10 ^ . 0 . 1 5 b'- " (m*a)a)j ii)j 3 natural periodb ' potential damping
V vy ^y
^
_
~
^ ^
e
12Fig. III-ll. Influence of linear damping on the extinction of a freely vibrating system.
and:
t = time elapsed since the time of release s. = initial amplitude at time of release
From equation (III-24) the relation between peak s and the next trough s . follows from (see Fig. Ill-ll) :
TT b 2(m+a)u
^n+1 = - ^n « (III-26) When, however, the potential damping can be neglected, equation
(III-26) changes into equation (III-27) . The derivation of this equation is given in Appendix II according to the method of Lindsted (see page 85 of Bellman [lll-l] ) :
«n+l = - "n riij <"I-27)
in which: 4 s = — —*— s
n 3 m+a n
Relation (III-27) has been plotted in Fig. III-12.
Now the time between a peak and the next trough follows from:
!n = JL r 1 1
2 '-I Ll - is2 (-a_)2 + . . J
(III-28) 6 n m+a
in which:
"l = \/i?S
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 st the natural frequency follows from the consideration that the energy input due to the excitation force equals the energy absorbed by the motion of the body platform. From this one finds that:
1. In C3se of potential dcimping: ^a(u.)
s. = ^ .. •' (III-29) a, . b.u.
(<»)^) 3
2. In case of viscous damping: ° a ( u . ) 2u 3 6 Q7 0 i 8 Sn«1 Sn 0 9 3TT F ( u . ) a j _ 2 q ( I I I - 3 0 ) 1.0
/
/
^
/
/ ^
/
X
/
/ q ' . 0 . 0 4/
/
/ q ' . 0 0 2 - ^ 0 0 1/
' '
/
/
^y
^
m»a n . a = virtual mass q s d r a g / s q u a r e of velocity 1\ A
/pfn
/ ^
V ^ N^^'^l
15Fig. III-12. Influence of viscous damping on the extinction of a freely vibrating system.
For frequencies largely different from the natural frequency the influence of the damping can be neglected. The ranges of frequencies for which the damping has an influence which is less than 10% of the total reaction force follow from:
u <
Uj
Y
u > u. Y
1 + 2 b'^ - 2 b 1 + 2 b'^ + 2 b in which: b' = u. = natural frequency c = spring constant (III-31)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 (III-31), to the response at the natural 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 neglected with respect to the inertia force or the undisturbed pressure force except for a frequency range around the frequency for which the ^ nertia force and the undisturbed pressure force cancel each other. For the frequency range meant here the deunping both from potential and from viscous effects have to be known. The viscous daunping 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 cancel each other for one frequency
indepen-dent of the wave direction, one then finds (neglecting the influence of viscosity) that, for instance, the vertical wave excited force at the frequency u can be written according to equation (11-23):
(III-32)
(III-33)
(III-34) A much more complicsted situation arises when the natural
frequency u. coincides with the frequency u . 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:
'^ *a(u^) = ^ ^a(u^)
2 = b q = b C u y_ a zz a zz a mz 2 while according to equation
(III-Z» 2 ^ z = 2- ^ ( ^ )
Combining equations (III-32) and *a _ 1
^a ^TT V y2 -^ ^ 2
•17)
which reduces to:
•T" = r- (III-35)
^a ^2
which will be the same heave response in case the potential damp-ing dominates. It will now be obvious that when the natural fre-quency u. and the frefre-quency u , at which the excited force becomes
J mj
minimum differ a little from each other, then the response at the natursl frequency will still be according to equation (III-35). The difference between the frequencies u. and u . has to be so small that the damping still dominates for more than 80%. In that case the difference between u. and u . follows from:
3 m3 u. N/l - 0.35 b' < u . < u. \/l + 0.35 b' (III-36) 3 V m3 3 V bu . when b' = — ^ is small. c 40
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/u^, have been discussed. In this connection u is the wave frequency or the frequency of oscillation of the body.
In order to determine the hydrodynamic forces on a semi-submer-sible, the underwater construction is subdivided in small elements of which the hydrodynamically induced forces are known when the disturbances due to neighbouring elements can be neglected.
From the paper by Kennard [lII-Sj it has been shown in the pre-ceding section that in an unbounded fluid the variation in the hydrodynamic 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 forces on the elements.
From model test results and calculstions, Boreel [iV-l] has shown that the variation in the wave excited forces on a vertical circu-lar cylinder will be less than 30% when:
a. the distance between neighbouring cylinder is more than three times the dicuneter of one of the cylinders.
b. the wave length is larger than about five times the diaun-eter of one of the elements.
An estimate of the influence of interaction effects on the total hydrodynamic forces on a semi-submersible — consisting of submerged bodies and cylinders piercing the water surface — was made by comparing the sxaa of the hydrodynamic forces on each of the elements of the platform, with the total hydrodynamic forces determined from model test results.
In the present chapter the formulae for the hydiJodynamically in-duced forces on the elements of the platform together with the summation will be given.
First the added masses a.. and the spring constants c.. of the platform will be determined by summation of the coefficients of each elementary part of the platform. These seime 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 functions of the platform motions to waves can be determined as indicated 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-cluded that the derived calculation method provides sufficiently accurate information about the behaviour of a semi-submersible in a seaway.
The method therefore provides a reliable means for designing seml-submersibles from a point of view of seaworthiness.
IV-2 Determination of added mass and hydrostatic forces
In this section the added msss a.. and hydrostatic coefficients c.. for the total platform will be deduced when for each part of the platform these coefficients are known.
The pl3tform will be split in several parts of the following nature: a. parts which are fully submerged cylinders,
b. parts which are cylinders which pierce the water surface, c. parts of arbitr3ry form of which the hydrodyn3mic
charac-teristics can be assumed to be concentrated in one point. The axes of reference are defined to be fixed to the platform in such a way that the z-axls is the verticsl axis through the
Motions Forces
center of gravity G. (see Fig. I V - 1 ) . The x-y plane coincides with the undisturbed water surface while the x-axis runs parallel to the horizontal axis of symmetry.
ad a. FBlly_§!ibmerged_cylinders
It is assumed that the length 1 of the cylinder is large rela-tive to the diameter D of the cylinder. When the point A
(5c, , y I ' ^ 1 '
and B , >are the extreme points of the cylinder, the length ^ 2'^2 2'
of the cylinder follows from (see F i g . IV-2) ,2
1 = \ / ( X 2 - x ^ ) 2 + (y2-yi)^ + (^2"^1>' (IV-1)
'Xo-Sfa-Jo'
dly ' p C^ C^ X cosa dr
Fig. I V - 2 . Description of sdded m3ss forces on 3 cylinder.
When the body is oscillated in the x-direction a force, of which the components are X , Y and Z 3nd a moment, of which the
compo-X X X
nents are K , M and N are encountered.
X X X
The oscillation of the acceleration is harmonical and amounts to:
x^ sin ut (IV-2)
The accelerstion component (x cos a) which is perpendicular to the cylinder and lies in a plane through the cylinder axis P3r3llel 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-jained by:
^2 - ^1
sin a = ^
Due to the acceleration (x cos
a.)
a hydrodyncunic force d F on sn
element or strip dr of the cylinder is generated in the direction of
this acceleration,the index X refers to the direction of the motion that
causes the force. The force d F follows from:
d F = X da cos a (IV-3)
X
in which the added mass da is a function of the location
'*(x^,y^,z^) °" the cylinder.
X < X < X X = X + r sin a
y^ < Yj. < 72
Yj. = y^ + ^ sin
3 (IV-4)
z, < z < z- z = z, + r sin v
1 r 2 r 1 '
in which:
^(x^-x^)^ + (y^-y^)^ + (z^-^i)^ (iv-5)
and:
sin a = (x -X )/l < - 90° < a < 90°
sin B = (y2-yi)/l < - 90° < B < 90° (IV-6)
sin Y = (z -z )/l < - 90°
< y <
90°
While resolving the force d F one finds:
2
d X = A cos
a.
X X
d Y =-A sin B sin a (IV-7)
d Z =-A sin
Ysin a
X X '
in which:
A = x d a = x p O C dr
X '^ r m
in which:
0 = cross sectional ares of the cylinder at point R.
C_ = added mass coefficient.
Due to the forces d F a moment relative to the centner of
grsv-ity G, . arises of which the components follow from:
(x^'y,^'^^) o o o d K = d Z ( y - y ) - d Y ( z - z ) d M = d X (z - z ) - d Z (x -X ) r o r o d N = d Y (Xj.-x^) - d X (yj-'y^) (IV-8)
When s u b s t i t u t i n g (IV-4) and (IV-7) i n t o ( I V - 8 ) :
d K^ = A^
[<^l"^o^ ^^" " ^^" ^ " <yi"yo'*^" " 3^"
Y J
d M = A r sin v + A R z , - z ) cos a +
X X X L 1 o
(x^-x ) sin a sin y] (IV-9)
d N =-A r sin 3 - A [(x -x ) sin a sin B+ (y,-y )=os a "I
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 t about the x-axis,forces and moments on the
cylinder will slso be encountered. The rotstion * will introduce
sn athwart motion y and a vertical motion z of some point
R, 2 ) °f *^^® cylinder resulting in the following equations:
r r r
y =
z =
'(-r-V
5(yr-yo>
in which it is assumed that the body is rotated harmonically
CIV-10)
? = ? sin ut
a
If one now wants to know some components of the force due to the
rolling motion, use is made of the equartlon:
d X^ = d X + d X
$ y z
d Y. = d Y, + d Y,
4 y z
d Z . = d Z + d Z
$ y Z
(IV-11)
When using equation (IV-10) in combination with equations similar
to (IV-7) one finds:
^ ^i = '^i
Q^r^o^ ^^" tx sin B - (y^-y^).
sin a sin yl
1 d Y. =-B^ r sin y - B. [(z -z ) cos B +
(y^-y^) sin B sin y]
d Z, r sin B + B [(z -z ) sin B sin y +
(yj-yg) C O S ^ Y ]
(IV-12)
in which:
B
*
? da = ? p 0 C dr ^ r mWhile comparing equation (IV-9) and (IV-12) the resemblance is obvious.
In order to determine the components of the moment due to the rolling motion, use is made of equations similsr to equation
(IV-11): d K . = d K + d K * y z d M ^ = d M + d M * y z d N * = d N + d N y z
'$
(IV-13)By substitution of equations (IV-4), (IV-7) and (IV-8) into equation (IV-13) one finds:
5 5 0
<^ K^ = B^ [(yryo^ cos
y
+ (z^-Zg) COS 6
+ 2 (y^-yo) (ZI-ZQ^ ^^" ^ ^^" "»•]+ B^r [2(y^-yQ) sin B + 2 ( Z ^ - Z Q ) sin y]
-^ [sin^
d L,
+ B.r •^ B + sin^ y] sin a sin B
- B^r [(x^-x^) sin B + (y^-yg) sin a] - B^ [~(''i~''o^ ^^" ^ ••• ^^l"^o^ ^^^ ^ '
[-(z^-z^) sin B + (yj^'y^j) sin y] + - B,
$
d M, -B^r sin a sin y
t^r^o> ^^r^o*
- B. ,r [ sin y(x -X ) + sin a (z,-z )"| ' *- 1 o 1 o ->
- Bj [sin B (X;I-XQ) - sin a (y^-yo)]
[sin y (yj^-y^) - sin B (z^^-z^)]
By integration of the forces over the length of the cylinder, the added mass and damping of each cylinder can be determined. The formulse for the added mass as determined above are given in
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