Offshore Hydromechanics

Offshore Hydromechanics

J.M.J. Journée and J.A.Pinkster

July 17, 1997

Contents

i 1 2 3

introduction

Potential. flow theory

Linear wave induced motions and loads on floating strudtures

1 2

14

3.1 Principle of equations of motion . . 14

3.2 Hydrodynamic coefficients 22

3.2.1 Experimental determination . 25

3.2.2 Theoretical determination 34

3.3 Response in reguilar waves , . . 41

3.4 Behaviour of offshore structures 42

.3 5 Behaviour of ships in regular waves 42

3.5.1 Motions . . 43

3.5.2 Shear forces and bending moments . . 52

4 The panel Method' or Singularity Method

. 61

4.1 introduction. 61

4.2 Mathematical tools . 61

42.1 Elémentary potential solutions . 62

4.2.2 Green's identities 64

4.2.3 A more general formulation of the integral equations 66 4 3 Body in movement in a unlimited flow 67

4.3.1 Integral equation 68

4.3.2 Boundary conditions ,

4.3.3 Finding the Velocity .

68 69

4.3.4 Finding the Potential . i

..

. 69

4.4 Body in movement in a limited flów 72 4.5 The panel method in engineering: analysis of offshore structures . 73

4.5.1

General remarks...

,

4.5.2 More details about the' way to create the. gii'd shape

73 74 4.6 Some results of computation . .. , 75

5

Second order drift fOrces

₇₉

5..1 Characteristic' behaviour of floating'structures . . 79

.1.i The behaviour of a bow-hawser moored vessel in wind and current 79 5.1.2' The behaviour of a large concrete structure. under tow. 82 5.1.3 Low frequency horizontal motions of permanently moored tankers

5,1.4 Motions and mooring forces of semi'- submersibles. 88 5.1.5 The vertical motions of ships in shallow water, long waves. 91

5,. 1.6 Behaviour of a jetty-moored tanker 94

5.2 Historical development concerning second order wave forces 99

5.3 Theory 100

5.3.1 Second order wave forces on a body floating in waves 'IOu

5.3.2 Co-ordinate systems 101

5.3.3 Motions of a point on the body 1101

5.3.4 Fluid motions 103

5.3.5 Body boundary conditions 1103

5.3.6 Boundary conditions at infinity 105

5.3.7 Pressure in a point withjn the fluid 105

5.4 Second order wave forces 106

5.4.1 Second order wave moment 109

55

Quadratic transfer functions for the mean and low frequency wave drift force109

5.5.1 Approximation for the contribution of the sec-ond order potential . 110

55.2 Wave drift forces in regular wave groups 113 5.5.3 Frequency domain calculation of the mean and low frequency drift

force 114

5.6 Some results of calculations of wave drift forces 115

56.1 Mean drift forces in regular waves 115.

5.6.2 Components of the mean wave drift forces 118

5.7 Irregular waves 1121

5.8 Final remarks 127

5.9 Nomencláture 1129

6 Non linear problems

132

6.1 Added resistances . . . . 133

6.2 Simple frequency domain method to approximate the low frequency hori-zontal motions of a moored vessel in irregular head or beam seas 139

7 'Responses in irregular waves

142

7.1 Wave spectra and statistics 143

7.1.1 Quick Review of Spectra and Statistics 143

7.1.2 S.tandard sea spectra 143

7.2 Response spectra and statistics 147

Chapter 1

Introduction

Knowledge about wave induced loads and motions of ships and offshore

structures is important both in design and. operational studies. The

significant wave height (the mean of the highest one-third of the waves) can be larger than 2rn for 60% of the time in hostile areas like the North Sea. Wave heights higher than 30 m can ocdur The mean wave period

can be from i 5 to 20 s in extreme weather situauons and it is seldom

below 4 s.

Environmental loads due to current and wind are also

inaportant. Extreme wind velocities of 40 to 45 m s

' have to be used in

the design ofoffthore structures inthe. North Sea.

Fig. i

.i shows five examples óf offshore structures. Two of them, the

jacket type and the gravity platfórrn, penetrate the sea floor At present, fixed structures have been built for water depths up to about 300 m. Two of the structures, the semi-submersible and the floating production ship,

are free-floating. The tension leg platform (TLP) is restrained 'from.

oscillating vertically by tethers, which are vertical anchorlines that are

tensioned by the platform buoyancy being larger than the platform

weight. Both the ship and the semi-submersible are kept in position by a spread mooring system. An alternative would be to use thrusters and a

dynamic positioning system. Pipes (risers) are used as conneçtions

between equipment on the sea floor aM the platform.

Ships serve a large variety of purposes. Examples are transportation of

goods and passengers, naval operations, drilling, marine operations,

fishing, sport and leisure activities. Fig. 1 .2 shows three types of ships: a

monohull, a SWATH and a S;ES. The monohull is exemplified by

a

LNG (liquid natural gas) carrier with spherical tanks SWATH stands

for small-wacerplane-area, twin-hull ship and consists of two

fully-submerged hulls that are connected to the above water structure by one or several thin struts. Between the hulls there may be fitted fins or foils

as in Fig.

1.2. SES (surface effect ship) is an air-cushion supported

high-speed vehicle where the air-cushion is enclosed

on the sides by

rigid sidewalls and on the bow and stern by compliant seals. By high

Speed we mean high Froude number (Fn). This is defined

as Fn

U/(Lg)i(U = ship speed, L = ship length, g = acceleration of gravity). A ship is considered a high-speed marine vehicle when Fn> O.5. From a

2 INTRODUCTION

Fig. 1. 1. Five types of offshore structures. From left to right we have, jacket, gravity platform, semi-submersible, floating production ship, tension leg platform (TLP). (Partly based on a figure provided by Veritec A/S.) SWAT k Sidewall Cushion beam Stern seat LNG CARRIER Rudder or ventral fin We deck Bow seal Lift fan SES

Fig. 1.2. Three types of ships. SWATH (small-waterplane area, twin-hull. ship), LNG (liquid natural gas) carrier, SES (surface effect ship).

DE.FINfTFONIS OFMOTI:ONS 3

hydrodynamical view point one can distinguish between ships at zero, normal and high speed. SWATH concepts have been designed for both normal and high-speed applications.

Most of the applications presented: in the main text will deal with ships

at zero or normal speed and with

offshore structures. Applications, to hgh-speed marine. vehicles will; be given by exercises. We will discuss both wave-induced lads and motions, with motions being the result of integrated hydrodynamic loads on the structure. In the introduction we will give a survey of important wave load and seakeeping problems for ships and offshore structures. Before doing that we need to define the motions.

DEFINITIONS OF MOTIONS

Motions of floating structures can be divided into wave-frequency

motion, high-frequency motion, slow-drift motion and mean drift. The oscillatory rigid-body translatory motions are referred to as surge, sway

and heave, with heave being the vertical motion

see Fig. 1.3). The. oscillatory angular motions are referred to. as. roll, pitch and yaw, with

yaw being rotation about a

vertical, axis. For a ship, surge is the

longitudinal: motion and roll is the angular motion about the longitudinal axis.

The wave-frequency motion is mainly linearly-excited motion in the wave-frequency range of significant wave energy. High-frequency

mo-_Fig. 1.3. Definition of rigid-body motion modes. Exemplified for a deep concrete floater.

Table 1 .1. Resonant Ireave oscillations of s/zips, offshore structures and high speedvehicles

Vessel:

Rough estimate: \/(L/1.5), where L is ship length in metres.

SWATH (small

Natural heave period:

Restoring force: Dominating excitation

nicchanisrn around tite

natural heave period:

SES (Surface effect

ship)

<Is

Air compressibility Linear wave forces

due to high encounter

frequency between ship and waves

TLP (tension leg platform) 2-4 s Elasticity of tethers Non-linear sum frequency wave forces Monohull ship Catamaran 4-16 s" Waerplanc area

Linear wave forces

Semi-submersible

>20s

Waterplane area

Swell (long waves)

waterplane area twin hull ship)

>20s

Waterplane area Linear wave forces due

to low encounter frequency between ship and waves

Important damping: 'Ride Control' Viscous effects Wave radiation Viscous effects

TRADItIONAL SHIP PROBLEMS

tion is significant for TLPs and is often referred to as 'ringing' and

'springing' and is due to resonance oscillations in heave, pitch and roll of the platform. The restoring forces for the TLP are due to tethers and the

mass forces due tó the platform. The natural periods of these motion

modes are typically 2-4 s which are less than most wave periods. They

are excited by non-lineai wave effects.

' Ringing ' is associated wii± transient effects and ' springing ' is steady-state oscillations.

SimiJar non-linear effects cause slow drift and mean motions in waves

and current. Wind will also induce slow drift and mean motion. Slow

drift motion arises from resonance oscillations. For a moored structure it

ours 'in surge,. sway and yaw. The restoring forces are due to the

mooring system and the mass forces due to the structure. Typical

resonance periods are. of the order of i to 2. minutes for conventionally moored systems.

Heave is an important response variable for many structures Table

1,. 1 illustrates the range of the natural heave periods .of different types of marine, structures.. These: include SE'S., TLPs, monohull ships, catamar-ans, SWATH ships and semi-submersibles The table indicates how the natural heave oscillations can be excited. For instance for the SES-hiill 'it occurs due to: high encounter frequency between the ship and the waves, while for the' SWATH it óccurs due to low encounter frequency between

the ship and the waves. The table also shows what types of restoring

forces can caùse heave, resonance. För the SES it is the compressibility

effect öf the; air in the cushion. For the mono'hu'li ship, catamaran,

SWATH and semi-submersible 'it is due tó change. in buoyancy forces. This. is reläted directly to the waterp'làne area of the vessels. Finally we.

see in' Table 1. 1 either the most important physical source of natural

heave damping or how one artificially increases the damping by control systems.

For the SES it is the heave accelerations and not the heave motions

that are importänt. If no 'ride control' is used, acceleration values of 1 .5g can occur in relatively calm sea. If the natural heave period is 0.5 s, it

means the. hea.ve amplitude is O.,1 m.

A semi-submersible is designed to avoid resonance heave motion and the maximum heave motion in severe sea states will be less than half the maximum wave amplitude.

TRADITIONAL SHIP PROBLEMS

Examples of important seakeeping and wave load problems for ships. are illustrated in Fig. 1.4. In particular, vertical accelerations and relative vertical motions between the ship and the waves are important responses.

Accelerations. determine loads on cargo and equipment and are an

important reason for seasickness. The relative vertical motions can be

6 INTRODUCTION

on deck. (Slamming means impact between the ship and the water..) For a ship it is important to avoid slamming as well as water on deck because of the. resulting local damage of the structures.

Rolling may be a problem. from an operational point of view of fishing vessels, crane vessels, passenger ships and naval vessels. Means to reduce

Local motions

Slamming

Effect of breaking waves

Accelerations

Water on deck

Liquid sloshing in tanks

Wave bending moments and shear forces

Fig. 1.4. Examples of important seakeeping and wave load problems for

TRADITIONAL SHIP PROBLEMS 7

the rolling of a ship are therefore of interest. Examples are bilge keels, anti-roll tanks and active fins. For smaller ships, rolling in combination with either wind, vater on deck r motion of the cargo can cause the ship (o capsize. Another important reason for capsizing of smaller ships 'is breaking waves. Several accidents off the Norwegian coast have been ex1ained by breaking waves. Following sea can cause different critical capsizing situations. If the wave profik is stationary relative to the ship, the ship may be statically unstable in roll relative to the waterline defined

by the wave proWe. The ship may also lose its directional

stability in

following waves. This can happen when the frequency

of encounter between the ship and the waves is small. The result is an aItered course

relative to the waves. This situation is called 'broaching' and is most

critical with respect to capsizing of ships with small static stability. Liquid sloshmg in tanks may be a problem for bulkships, combination

ships oilbulkore (ORO), liquid natural gas (LNG)

carriers and tankers loading at offshore terminals. There are two reasons why the fluid motion in a tank can be violent. One is that a natural period for the fluid motion in the tank is in a period domain where there is significant ship motion. The other reason is that there is often little damping connected with fluid' motion in a tank. If the excitation period is close 'to a natural period for the tank motion, a strong amplification of the fluid rotion in a tank 'will

occur Liquid sloshing can cause

high local pressures as well as large total forces. Both effects may be important in design.

For larger ships,, wave-induced bending moments, shear forces and torsional moments are important. More specific problems are whipping

and springing. Whipping is transient elastic vibration of the ship hull

girder caused for instance by slamming. Springing is steady-state elastic vibratiön caused by the waves and is of special importantce for larger oceangoing ships and Great Lake carriers. Springing is due to both linear

and non-linear excitation mechanisms. The linear exciting

forces are associated with waves of small wavelengths relative to the ship length.

Sh'ip motions and sea loads can influence the ship speed significantly

due to voluntary and involuntary speed reduction. Voluntary

speed

redu'ction means that the ship master reduces the speed due to heavy

slamming, water on deck or large' accelerations.

Involuntary speed reduction is the result of added, resistance of the ship due to waves and

wind and changes in the propeller efficiency due to waves. The

importance 'of involuntary speed reduction is exemplified in Fig. 1.5. It shows the results of computer calculations for a container ship at a given

señ state. The significant wave height H is 8.25 m.. The waves are

assumed longcrested with different pro,pagation directions relative to the ship. The ship has a length of 185 rn. The actual speed at constant engine power is given for different wave headings together with the design speed in still water at the same engine power. For instance in h'ead seas che ship

INTRO D U.CTIO,N

speed is 8 knots (4.1 m s1) compared to 16,2 knots (8.3

_{m s') in still}

water. Depending on the wave direction, the actual ship speed may be

lower than that shown in Fig.

1.5. This, is due to voluntary speed

reduction. Information like this may be used to choose optimum ship

routes based on relevant criteria like the lowest fuel consumption or the shortest time of voyage.

Criteria for acceptablè levels of ship motions have been discussed in the Nordic co-operative project 'Seakeeping performance of ships' (NORDFORSK, 1987). Considerations have been given to hull safety operation of equipment, cargo safety, personnel safety and efficiency. General operabi:lity limiting criteria for ships

are given in Table 1.2..

Criteria with regard to accelerations and roll for special types of work and for passenger comfort are given in Table 1.3.. The limiting criteria for fast small craft are only indicative of trends. A fast small craft is defined as a vessel under about 35 metres in length with speed in excess of 30 knots.

A reason why the vertical acceleration level for fast small craft is

set higher than. for merchant ships and naval vessels, is that personnel can tolerate higher vertical acceleration when the frequency of oscillation is high.

OFFSHORE STRUCTURE PROBLEMS

For drilling operations heave motion is a limiting factor The

reason is that the vertical motion of the risers has to be compensated and there are

limits to how much the motion can be compensated. An example of

a

Foilowing sea

Fig. 1.5. Effect of added ship resistance due to waves and wind (invoLuntary

speed reduction). Ship length = 185 m. (H = significant wave height).

OFFSHORE STRUCTURE PROBLEMS 9 Table 1.2. General aperabiliiv limiting criteria fcr ships (NOkDFORSK, 1987)

a The limiting criterion for lengths between 100 and 330 m varies almost linearly

between the values L = 100 m and 330 m, where L is the length of the ship. b.The limiting criterion for lengths between 100 and 300 m varies linearly

between the valuesL =' 100 m and. 300 m.

Table 1.3. Criteria with regard to accelerations and roll

(NORDFORSK, 1987)

Root mean square criterion

heave motion criterion is that the heave amplitude shóu'ld be less than 4 m. It is. therefore important to design' structures. with low heave motion

so tha

it i's

possible to. drill 'in as high a 'percentage of the time as

possible. Serni-submersibles are examples of structures. with very low heave motion in the actual frequency domain.. Roiling. may also be an important motion mode. o evaluate,, for example for operation of crane vessels or for transportation of jackets and' semi-submersibles on ships and barges. Rolling, pitching and accelerations may represent limiting

Merchant ships Naval vessels Fast small craft

Vertical acceleration at forward perpendicular (RM'S-value)

0.275g (L 100m)

0.05g (L 330 m)' O '75» 0.65g.

Vertical acceleration at bridge

(RMS-value) 0.15g 0.2g. 0.275g

Lateral, acceleration at bridge

(RMS-value) 0.12g 0.lg 0.1g

Roll (RMS-value) 6.Odeg 4:.Odeg 4.O'deg

Slamming criteria (Probability)

0.03(Ll00m)

_{0.01(L30Om)ò} 003 0.03

Deck wetness criteria

(Probability) 0.05 0.05 0.05

Vertical

acceleration,

Lateral

acceleration Roll Description

0.20g 0. 10g 6.0° Light manùal work

0. 15g 0.07g 4.00 _{Heavy manual work}

0.10g 005g 3.0° Intellectual work

0.05g 0.04g 2.5° Transit passengers

Io INTRODUCTION

factors for the operation of process equipment on board a

floating production platform.

In the design of mooring systems for offshore structures 'loads due to

current, wind, wave-drift fòrces and wind- and wave-induced motion

are generally of equal importance. There are two important design

parameters. One is the breaking strength of the mooring lines. The other is the flxibi'lity of the riser system which means, in practice, for a rigid rise system that the extreme horizontal offsets of the platform relative to the connection point of the riser to the sea 'floor should be less than say 10% of the water depth.

Wind, current, mean wave drift forces and slowly varying wave drift forces are also important in the design of thrusters and in' station keeping of crane vesseis, diving vessels, supply ships, offshore 'loading tankers and pipelaying vessels'. Interaction of thrusters with other thrusters, the free-surface and structures may 'also be important for dynamic position-ing systems, towposition-ing and marine operations in waves.

Examples of the main objectives of the hydrodynamic analysis of a

tension, leg. platform are,,:to calculate the vertical dynamic. loads on the platform with the purpose of estimating axial forces in the tethers and to calculate the wave elevation in order to evaluate the air gap between the waves and the underside of the platform. The minimum air gap is also an important consideration for other types of platforms,

HYDRODYNAMIC CLASSIFICATION OF'

STRUCTURES

Both viscous effects and potential flow effects may' be important in

determining the wave-induced motions and loads on marine structures.

Included in the potential flow is the wave diffraction and radiation

around the structure. In order to judge when viscous effects or different

types of potential flow effects are important, it is useful to refer to a

simple picture like Fig.

1.6. This drawing is based on results for

horizontal wave forces on a vertical circular cylinder standing on the sea floor and penetrating the free surface. The incident waves are regular. H

is the wave height and A is the wavelength of the incident waves. D

is the cylinder

diameter. The

results are based: on the use, of Morison's equation (see chapter 7) with a mass coefficient of 2 and a drag coefficient of 1'. The linear McCamy .& Fuchs (1954) theory 'has been used in the wave, diffraction regime.

Let us try to use the figure for offshore structures. We will consider a

regular wave of wave height 30 m and wavelength 300m. This

cor-responds to an extreme wave condition. Let us consider wave loads on 'the caisson of a gravity platform where typical cross-sectional dimensions are 100 m. This implies equivalent HID- and A/D-values of 0.3 and 3, respectively. This means that wave diffraction is most important. If we

ENGINEERING TOOLS

II

consider the columns of a semi-submersible, a relevant diameter would be approximately 10 rn. This implies ./D

30, HID = 3, which means

that the hydrodynamic forces are mainly potential flow forces in phase

with the undisturbed local

fluid acceleration. Wave diffraction and

viscous forces are of less significance.

For the lega of a jacket a relevant diameter is approximately 1 m.. This. implies that viscous forces are most important. By viscous forces we do

not mean shear forces, but pressure forces due to separated flow. The

examples above are for an extreme wave condition.. In an operational

wave condition the relative importance of viscous and

potential flow effects are. different. We should bear in mind that Fig.. 1.6 only provides a very rough classification. For instance, resulting forces may be small

dùe to the cancelling out of effects of loads from different parts of the

structure.

ENGINEERING TOOLS

Both numerical calculations, model tests and full-scale trials are used to

assess wave-indúced motions and loads. From an ideal point

of vjew fill-scale tests are desirable but expensive and difficult to perform undçr controlled conditiôns. It mayalso be unrealistic to wait for 'the extreme weather situations to

occur. Model tests

are

therefore needed. A

drawback with model tests is the difficulty of scaling. test results to full scale results when viscous hydrodynamic forces matter. The geometrical dimensions and equipment of the model test facilities may also limit the experimental possibilities.

Due to the rapid development of computers with large memory

capacity and high computational speed, numerical calculations have

=10 .1:1 O WAVE BREAKING L IM IT VISCOUS FORCES MAS S I FORCES WAVE O IF FRACTION =5 A/O

Fig. 1.6. Relative importance of mass, viscous drag and diffraction forces on marine structures.

1.2 INTRODUCTION

played an increasingly important

role in calculating wave-induced morions and loads on ships and. offshore Structures. A significant step in the development started about 1970. For offshore structures it was partly

cónnected with the beginning of offshore oil and gas production and

exploration in the North Sea. However, it' is important to stress that

numerical computer programs are also dependent on the. development of

hydrodynarnic theories. More theoretical research is

still needed, in

particular to increase the knowledge on separated viscous flow

and 'extreme. wave effects on ships and offshore structures.

It is unrealistic to expect that computer' programs will, totally replace model tests in the foreseeable future. The ideal way is to combine model tests and numerical calculations. In some cases computer programs are

not reliable. Model tests often give m'ore confidence than computer

programs when totally new concepts are

tested out

When computer programs have been' validated and the

theoretical

basis of the computer program has been satisfactorily

compared with

experimental results, computer programs offer' an' advantage relative:, to model testS... Computer programs can often be. used. in a more efficient way 'thanl model' tests. to evaluate

different designs in a large vriet' 6f éa

conditions. Häwever, sound judgement of resuits is always important. A basis for this. is physical understanding and practical feeling.

One aim of the book is to provide physical understanding to the reader

and try 'to simplify the problems mathematically. In this way one can

'develop simple tools to ev., aluate 'results .from model tests, full-scale trials or' computer programs.

Chap. ter 2

Potential flow theory

Suppose a rigid body, floating in an ideal fluid with harmonic waves The time-averaged speed of the body is zero in all directions. To get simple notations it is assumed here that the O(x,y, z) system is identical to the S(xo, yo, zo) system. The z axis and z0 axis are positive upwards.

The linear velocity potential of the fluid is splitted into three parts:

(x,y, z,.t,), = ±

_{± Ii}

in which:

= the radiation potential for the oscillatory motion of the body in still water = the incident undisturbed wave potential

= the diffraëtion potential of the waves around the restrained body

Boundary conditions of the velocity potentials

From the definition of a velocity potential follOws the velocity of the waterparticles in the three translational directions:

vx.=-

_Ox

vy=-

vz=-ay

As the fluid is homogeneous and incompressible, the continuity condition:

0v,, 0.v öv,

Ox _Oy .Oz

results into the equation of Laplace for potential flows:

O2 02 O2

Ox2 _0y2 0z2

The pressure in a point HP (.x,y, z) of the fluid is given by the linearised equation of Bernoulli: .

Potential flow theory 3

p = p-- - pgz

or:

_{-;:;-- +g( = -}

p

ut p

At the free surface of the fluid,, so for z = C(x, y, z, t), the pressure p is constant.

Because of the linearisation, the vertical velocity of a waterparticle in the free surface becomes: dz

dt - 9z

¿C dx + th dy + th dz

ôxdt

aydt

ôzdt

oc

With this, the boundary condition at the free surface can be written as:

02«

_a

---l--; = o

for: z = O

This boundary condition at the free surface applies for the sum of the velocity potentials as well as for each individual component.

The boundary condition on the bottom, following from the definition of the velocity potential, is' given by:

for: z --h

The boundary condition at the surface of the rigid body plays an important role too. The velocity of a waterparticle in' a point at the surface of the body is equal to the velocity of this point of the body itself.

The outward normal velocity v in a point P(x, y, z) at the surface of the body (positive in the direction of the fluid) is given by:

= v(x, y, z, t)

Because of the linearised prcblem, this can be written as:

0c1 6

Potential flow theory 4

with the (generalised) direction-cosines on the surface of the body:

fi cos(n, x) 12 = cos(n, y) f = cos(n, z) = ycos(n., z) - z cos(n, y) z cos(n, x) - x cos(n, z) 16 = z cos(n, y)

_{- y}

cos(n, z)

Finally the radiation condition states that when the distance R of a waterparticle to the oscillating body tends to infinity, the potential value tends to zero:

1im=O

R+c

Forces and moments

The forces Pand moments follow from an integration of the pressure on the submerged surface of the body:

P=

_JJ(P.il)ds

JJp.(x f)dS

The pressure follows, according to the linearised equation of Bernoulli, from the velocity potentials by:

p = p-- - pgz

f Or

w

Öd\

=PI--+

_at

The hydromechanic forces P and moments can be splitted up into four parts:

Potential flow theory 5

or:

jjJ(&I)ra:wott)d).ds

M=pjJ(+w+d+gz)(xn)ds

in which 1 and M3 are the hydrostatic parts.

Hydromechanic forces and moments

The radiation potential cI belongs to the oscillation of the body in still water. It can be written:

6

= ,y, ,t)

j=1

= j(X,y,Z)

v(t)

in which v3(t) is the oscillatory velocity in direction j.

The normal velocity on the surface of the body can be written as:

a6

The generlised direction-cosines are given by:

aj

f,

-

_on

With this the radiation terms in the hydrornechanic forces and moments are: ¡

¡(ar\

-.

F=pjj -ä)fldS

Potential flow theory 6

and:

M'=

(3r)

(x îi)dS

Jj

(a6)

(x ñ)dS

The components of these radiation forces and moments are defined by:

= (Xri,Xr2,Xra)

(Xr, Xr5, Xr6)

with:

=

jJ

(36)

for: k 1,...6

In this expression çb and çbk are not time-depending, so the expression reduces to:

Xrk _>JXrkj

for: k=1,...6

with:

Xrk = dvi

JJ&IkdS

This radiation load Xrkj in the direction k is caused by a forced harmonic oscillation of the body in the direction j.

Suppose a motion:

si = Sa,e

The velocity and acceleration of the oscillation are:

Sj =

Potential flow theory ₇ The hydromechanjc force can be splitted into a force in phase with the acceleration and a force in phase with the velocity:

Xrkj

Mk,

-=

21k[kj + isOwNk3 ju,t

= (_3ajW2

On

So in case of an oscillation of the body in the direction j with a velocity potential j, the hydrodynamic mass and damping (coupling)coefficients are defined by:

Mk =

_e{PJJ3ds}

_{Nk =}

_{_m{PwJJc/Pds}}

In case of an oscillation of the body in the direction k with a velocity potential çk, the hydrodynamic mass and damping (coupling) _{coefficients are defined by:}

-Tt'jk = Je

_¿J

bkdS}

_{Nk =}

-

{PJJds

Suppose two velocity potentials _{j. and q and use Green's second theorema for these} potentials:

JJJ(j.v2k_k.v2j)dv*

_V.

₌₁₁

s.

(

In these expressions S' is a closed surface, with a volume V, consisting of the wall of _a vertical circular cylinder with a very large radius and inside this cylinder the seabottom, the watersurface and the wetted surface of the_{floating body; see figure 2.1.}

The Laplace operator is given by:

02 02 02

Ox2

Oy2'ô2

So according to the equation of Laplace, it can be written:

= V2ç = O

*

dS

Potential flow theory 8

Figure 2.1: Boundary conditions

This results into:

=

_L

kdS*

The boundary condition on the wall of the cylinder, the radiation condition, is:

Um = O

R-The boundary condition on the seabottom is:

ôn

for: z = h

The boundary condition at the free surface:

---; = O

for: z= O

becomes for = e_iWt:

_2+gO for:z=O

or with k = for deep water:

ôz

for:z=0

So for the free surface of the fluid can be written:

aÇbk ôÇbk

Oz ôn and k

85

Potential flöw theory 9

When taking these boundary conditions into account, the integral equation over the sur-face S* reduces to:

jJ

&Io!s

=

JJ

kdS

in which S is the wetted surface of the body only. This means also that:

Mi,, = Mk and Njk =

Because of the symmetry of a ship some coefficients are zero. See also Timman and Newman (1962) for the forward speed effects.

The two matrices with the existing hydrodynamic coefficients are given below.

Wave potential

The velocity potential , of the harmonic waves has to fulfil three boundary conditions:

- the equation of Laplace:

O

+ + D2

Ox Dy Dz2

the boundary condition on the bottom:

Dz

for: z = h

the dynamic boundary condition at the free surface, which follòws from the linearized equation of Bernoulli:

ocIw

+g(=O

for: z = O Hydrodynamic mass matrix:

Hydrodynamic damping matrix: f M1 M3i o M51

'o

O M22 O M42 O M62 IN11 O O N22 N31 O O N42 N51 O O N62

M3

O M15 O

\

O M24 O M26

M3

O M35 O O M44 O M46 M3 O M55 O O M64 O M66j N13 O N15 O O N24 O N26 N33 O N35 O O N44 O

N6

N53 O N55 O O N64 O N66

Potential flow theory LO

With this the corresponding wave potential, depending on the waterdepth h, is given by the relation:

g cosh k(4 + z)

Ca sin(t - k cos ¡L - ky sin ¡)

cI

= cosh(kh)

The dispersion relation follows from the kinematic boundary condition at the free surface:

w2 _{= kgtanh(kh)}

With a known definition of the wave potential, the pressures in the fluid and the orbital motions of the waterparticles can be obtained.

Some examples of the effect of the waterdepth on the paths of individual waterparticles under a wave are presented in figure 2.2.

and Depth 2 m Depth 10 m )

a coshk(h-1-zL

Depthloom Surf ace I

Figure 2.2: Orbit shapes under a loo metre wave

(Lloyd, 1989)

When calculating the hydromechanic forces and the wave exciting forces on a ship, it is assumed:

XXb

_yyb

zz

In case of a forward ship speed, the wave frequency has to be replaced by the frequency of encounter of th wavesWe.

This leads to the following expressions for the wave surface and the first order wave

potential in the G(xb,, Yb, Zb) systcm:

C = Ca S(Wet - kxbcosp - kybsinl.L)

=

Potential flow theory 11

Wave and diffraction forces and moments

The wave and diffraction terms in the hydromechanic force and moment are:

Fw+Fd=PJJ(:+_-)flds

$

and:

M+Md

19f

(0w

₊ ( x ñ)dS

For the determination of these wave forces and moments it is supposed that the floating body is restrained at zero forward speed.

Then the boundary condition on the surface of the body reduces to:

On On On

Define now:

y, z, t) = qw(x, y, z) .

y, z, i) = çbd(x,y,z) e"

This results into

Oçb OÇbd

On - - On

With this and the expressions for the generalised direction-cosines it is found for the wave forces and moments on the restrained body in waves:

= _ipe_t ff (

+ çbj) fkdS

_ipe_iwtff(çbw _{+ Çbd) dS}

_{for: k=1,...6}

The potential f the incident waves , is known and the diffraction potential _d has to

be determined.

Green's second theorerna delivers:

OcbddS

Potential flow theOry 12

With:

it is found:

JJdJdS

=

-)ç/

which results into the so-called Haskind relations:

Xwk =

_ipetJ

J

+ dS for: k = 1, ...6

With this the problem of the diffraction potential has been eliminated, because the ex-pression for Xwk is depending on the wave potential çb and the radiation potential k

on'.y.

These Haskind relations are valid fOr a floating body with a zero time-averaged speed in all directions only.

Newman (1962) however, has generalised these Haskind relat;ions for a body with a con-stant forward speed He derived equatrnns which differ oniy slightly from those found by Haskind. According to Newman's approach the wave potential has to be defined in the moving O(x, y, z) system. The radiation potential has to be determined for the constant forward speed case, taking into account an opposite sign.

These Haskind re]ations are very important. They ünderlies the relative motion (displace-ment-velocity-acceleration) hypothesis, used in the strip theory.

The corresponding wave potential at an infinite waterdepth is given by the relations:

g kz

.

kxcosi

-

kysinjt)

_tag

_{6kz eik(xcosI+ysin$)e_iwt} w CaY e ik(xcos+ysin) w

The velocity of the waterparticles in the directión of the outward normal n on the surface of the body is:

fôz

.ôx

_8y.

-;:;--

qk(

-

z(_cos,

₊

smp)

Un

\ôn

Bn 8m

=

qk,(f3

- i(fi cos p +12 sin

thz ôn

s

and

Potential flow theory 13

Then the wave loads are given by:

= ipe"t Jf qfdS

s

+ipe_tkJJWk(f3 i(ficosp+f2sin))dS

for: k = 1,...6

The first term in this expression for the wave färces and moments is the so-called Froude-Krilov force or moment. The second term is caused by the disturbance because of the presence of the body.

Hydrostatic forces and moments

These are given by:

Chapter 3

Linear wave induced motions and

loads on

floating structures

3.1

Principie of equations of motion

A right-handed coordinate system S(xo,Yo, zo) is fixed in space. The (x0, yo)-plane lies in the still water surface, x0 is directed as the wave propagation and z0 is directed upwards.

Another right-handed coordinate system O(x, y, z) is moving forward with :a constant ship speed .V. The directions of the axes are: x in the direction of the forward speed V, y in the lateral port side direction and z upwards The ship is supposed to carry out oscillations around this moving O(x, y,, z) coordinate system. The origin O lies above or under the time-averaged position of the centre of gravity G. The (x, y)-plane lies in the still water surface.

A third right-handed coordinate system G(xò, Yb, zb) is connected to the ship with G at the ship's centre of gravity The directions of the axes are xb in the longitudinal forward direction:, Yb in the lateral pört side direction and Zb upwards. in still water, the (xb, yb)-plane is parallel to the still water surface.

Principle of equations of motion 15

Principle of equations of motièn 16

The harmonic elevation of the wave :surfàce is defined in the space-fixed coordinate system by:

= 'Ca'cos(wt - k ₎

in which:

Ca = wave amplitude. k. = 2ir/.\, = wave number

wave length

'w = circular wave frequency

t

=time

The wave speed c, defined in a direction w.ith an angleji relative to the ship's speed vector. V, follows from:

W

C=-j

The righthanded coordinate system O(x,y,,z) is moving with the ship's speed V, which yields:

xo= Vtcosji+xcosji+ysinji

From the relation between the frequency of encounter We and the, wave frequency w:

We = w - kV cosji

follows:

C = Cacos(wet - kx cos

ji

kysinji)

The resulting six shipmotions in the O(x, y., z) system are defined by three translations of the ship's centre of gravity in the direction of 'the s-, y- and z-axes and three rotations about them:

surge:

s

5a COS(Wet + e)

sway: _Y Ya cos(wet + c)

heave: z Za cos( wet

+ f)

roll: '

= a

COS(Wet +)

pitch: O Oa cos(wet± Eo)

yaw: '

The phase lags f'these motions are related to the harmonic wave elevation at the: origin of the O(x, y, z) system, the average position of the ship's centre of gravity

Principle of equations of motion 17

The harmonic velocities and accelerations in the O-(x,y,z) system are found by taking the derivatives of the displacements, for instance:

roll displacement: = a COS(Wet +

roll velocity:

=

WgÇb sin( Wet + cg)

roll acceleration:

= WÇba cos(wt + c)

=ØQcos(wt+q)

vr

/

Figure 3.2: Harmonic wave and roll signal

The equations of motions. in a space fixed system of a rigid body follow from Newton's law of dynamics.

The vector equations for the translations of and the rotations about the centre of gravity are respectively given by:

d

=(m

in which:

F = resulting external force acting in the centre of gravity

m = mass of the rigid body

U = instantaneous velocity of the centre of grav.ity

M = resulting external möment acting about the centre of gravity H = instantaneous angular momentum about the centre of gravity

t

=tirne

The total mass of the body and its distribution over the body is considered to be constant with time. This assumption is normally valid during a time which islarge relative to the period of the motions.

Principle of equations of motion 18

When assuming small motions, symmetry of the body and the axes z, y and z to be principal axes, it can be written for the motions of a ship.:

in which:

p = dénsity of water

V = volume of displacement of the ship = solid mass moment of inertia of the ship

Xh1, Xh2, Xh3 = hydrornechanic forces in the x-, y- and z-direction respectively Xh4, Xh5, Xh6 = hydromechanic moments about the x-, y- and z-axis respectively

X1, X2, X,,3

exciting wave forces in the x-, y- and z-direction respectively

When the distribution of the solid mass of the ship is unknown, the moments of inertia in here are often expressed in the radii of inertia and the ship's mass:

'xx = kxx2 . pV with: kxx

0.35 B - 0.45 . B

I

= k,2. pV

with:

O.22 L - 0.28 L

I

pV with: lç 0.22 . .L - 0.28 . L

'xz = 'zx = 0.0

Generally, in the full load condition of normal ships the radii of inertia _{and k2 are} smaller than in the ballast condition.

X4,, X,,5,, X6 = exciting wave moments about the z-, y- and z-axis respectively

The solid mass matrix is given below.

(pV

O O O O O

O pV O O O O

Solid mass matrix: O O pV O O O

O O O lxx O _'xz O O O O

I

O

\

O O

O I

O Surge: (pV

= pV

= Xh1 + Xw1 Sway: (pV. = pV = Xh2 + Xw2 Heave: (pV. = pV = Xh3 + Xw3 Roll:

=IçbI

_{= Xh4 + Xw4} Pitch:

(i

. = lxx Ö = Xh5 + Xw5 Yaw:

(JzzJZx.)

Izz1zx

_{= Xh6 + Xw8}

Principle of equations of motion 19

Becauseof symmetry and asymmetry of motions, this results into two sets of three coupled equations of motions: Surge: pV - Xh1 = X Heave: pV

- Xh3 = X3

Pitch:

I

Ö

-

= X, Sway:

Xh2=X2

Roll:

Xh4 X4

Yaw:

i.4i.4 Xh6 =X6

After the determination of the in and out of phase terms of the hydromechariic and the wave loads, these equations can be solved with a numerical method.

Right hand terms

The so-called strip theory solves the three-dimensional problem of the hydromechaniç and exciting wave forces and moments on the ship by integrating the two-dimensional potential solutions over the ship's length. Interactions between the cross sections are ignored for the zerospeed case. So each cross section of the ship is considered to be part of an infinitely long cylinder, see figure 3.3.

(Lloyd, 1989)

Xh

=/x;

dxb

and = dxb

Figure 3.3 Representation of hul'lform section shape by an infinite cylinder

Principle of equations of motion 20

in which:

X. = sectional hydromechanic force or moment

X,'L,. = sectional exciting wave force or moment

Two assumptions are made for these loads:

the hydrornechanic forces and moments are induced by the harmonic oscillations of the rigid body, moving in the undisturbed surface of the fluid

the wave exciting forces and moments are produced by waves coming in on the restrained ship.

Due to linearity, the hydromechanic loads and the exciting wave loads can be added to obtain the total loads, see figure 3.4.

in here:

DfÔ

a

Dtôt

is an operator, which transforms (xo,yo,z0,t) to

(x,y,z,t

13VJj D

- ¡j

h

,Z* _J.*

'3tUj - _Dt

ltvj

and and D

- Dt

'I'h

Relative to a restrained ship, moving forward in waves, the equivalent displacements, velocities and accelerations in the direction j of a waterparticle in a cross section are defined by:

;.:

_:fi:*

1JJi -

_Dt-'

mOtions in oscillation in restrained in

waves stili water in waves

Figure 3.4: Superposition of hydromechanic loads and wave loads

Relative to an oscillating ship, moving forward in the undisturbed surface of the fluid, the equivalent displacements, velocities and accelerations in the direction j of a waterparticle in a cross section are defined by:

Principie of equations of motion 21

According to the "Ordinary Stri:p Theory",. a more or less intuitive approach as published by Korvin-Kroukovsky and Jacobs (1957) and others, the uncoupled two-dimensional potential hydromechanic loads and' wave loads in the direction j. are defined by:

X =

} + N

± x,

x,,j =

According to the "Modified Strip Theory", as published by Tasai (1969) and others, these loads become:

=

-

+ xsi

= {

(Mj

-

Ivj)

-, /.* i jj

F-In the definitions of the two-dimensional hydromechanic loads, the non-diffraction part X3. is the tw-dimensional quasi-static restoring spring term,generally present for heave, roll and pitch only.

In the definitions of the two-dimensional wave loads, the non-diffraction part Xk. is the two-dimensional Froude-Krilöv force or moment which Th calculated by an integration of the directional pressure gradient in the undisturbed wave over the cross sectional area of the. hull. Equivalent directional components of the orbital acceleration and velocity, derived from these 2-D Froude-Krilov forces, are used to calculate the diffraction parts of the total wave forces and moments.

Hydrodynamic potential coefficients

The two-dimensional potential hydrodynamic coefflcients for sway, heave and roll can be derived by a two-dimensional potential theory for the zero forward speed case, as for instance given by Ursell (1949), Tasai (1959), Tasai (1960) and Tasai (19.61).

This theory can be used after a conformal mapping of the cross sections to the unit circle. The. simplest way here is to use two mapping coefficients derived from the local breadth to draught ratio and the sectional area coefficient, the "Two-Parameter Lewis Transformation Method".

When including information on the vertical' location of the centroid of the cross section, three mapping coefficients can be found, the "Three-Parameter Lewis Transformation Method".

Also a more accurate conformal mapping, based on a least squares method, with up to 10 mapping coefficients can be used, the "Closé-Fit Conformal Mapping Method". Another suitable method to determine the two-dimensional potential hydrodynamic co-efficients for sway, heave and roll is the "Frank Close-Fit Method" (Frank (1967)). This method determines .the velocity potential of a floating or a submerged oscillating cylinder

of infinite length by the integral equation method utilising the Green's function, which represents a pulsating source below the free surface.

For the surge motion, a more or less empiric procedure has f011owed by the author (Joûrnée (1992)). An equivalent longitudinal cross section has been defined. For each frequency, the two-dimensional potential hydrodynanic sway coefficient of this equivalent cross section is translated to two-dimensional potential hydrodynamic surge coefficients by an empiric method, which is based on theoretical results of three-dimensional calçùlations.

3.2

Hydrodyna.mic coefficients

Damping of ship motions is caused by the generated waves which dissipates the energy of the moving ship and by viscous effects such as skinfriction, vortices,, etc.

For sway, heave, pitch and yaw motions the major part of the damping is caused by the wave or potential damping. Then the contribution of the viscous damping can be ignored.

But for roll motions the contributiòn of the viscous damping in the total damping is important. Generally, the wave damping component for röll is relatively small.

Also for surge, the viscous damping can play an important role.

Observe a floating cylinder, carrying out a vertical harmonic oscillation in still water.

Z: Z

sinwt

Hydrodynamic coefficients 23

When supposing a small steepness of the radiated waves the linear equation of the heave motion is given by:

Z Za Sill wt

(pV+A22)+B22.+C22z=F(t)

with: w circular frequency t

=time

p = dénsity of water V = volume of displacement

A2 = potential mass coefficient

B2

= potential damping coefficient C22 = pgA = restoring spring coefficient

g = acceleration of gravity

A waterline area

The work done by the mass, damping and spring force components in this equation per unit of time T (one period of oscillation) is:

f(p + A22)

dt =

_2(T+

A2)w3

J sinwt coswt dt = O

T ₂ T i

fB2

i idt

Za zzW J cos2 wt dt = BzzW2 Za2 JpgA . . dt Za2 pgAwW

J

wt . cos wt . dt

= O

with:

T = 2ir/w = oscillation period

dt = dz = covered distance in dt seconds

It is obvious that only the speed-dependent potential damping term B2 . dissipates

energy. The energy delivered by this damping force is equal to the energy dissipated by the waves:

JB22.dt =

Bzzw2Za2

= 2pgCa2L

Because the wave velocity c = g/w, the right hand side of this equation becomes:

1 ₂ c pg2 Ca2 L

2pg(,,

L

2 2 2w

Then the potential damping coefficient per unit of length is defined by:

B2

pg2 ((a\2

Hydrodynamic coefficients 24

For the roll motions of a ship the wave damping i reltive1y small. The relation with the breadth-draught ratio has been illustrated in figure 3.6.

i

t

2.5

breadth-draught ratio

Figure 3.6: Roll damping

At a breadth-draught ratio of abolit 2.5, the average cross' section of the ship approaches a circular cross section, which has no wave damping.

For very low breadth-draught ratios (paddle-type wave maker in a towing tank) and very high breadth-draught ratios (wave damper of a towing carriage in a towing tank) thewave

3.2.1

Experimental determination

The hydrodynamic coefficients in the equations of motion can be obtained experimentally by decay tests or by forced oscillation tests with ship models.

Free decay tests with a ship model in still water can be carried out for those motions only which have a restoring ability, so the heave, pitch and. roll motions. For these motions the hydrodynarnic mass and damping can be obtained from the decaying motions itself at one frequency only: the natural frequency.

For the other ship motions surge, sway and yaw, this can be done by using external springs. Then, the hydrodynamic coefficients can be found for a restrictèd range of frequencies, by varying the stiffness of these springs.

With forced oscillation tests instill water the hydrodynamic mass and damping coéfficients can be obtained at any frequency of oscillation from the measured exciting loads. Also the coupling coefficients between the motions can be obtained.

Experimental determination 26

Free decay tests

In case of pure free heaving in still water, the linear equation of the uncoupled heave motion of the centre of gravity G is given by:

This equation can be rewritten as:

+ 2v + w02 z = O

in which the damping coefficient and the undamped natural frequency are defined by.:

2v=

pV+A

and w02 =

When defining a non-dimensional heave damping coefficient by:

V

IC =

-wo

the equation of motion is given by:

-I-2,cwo +wo2 z = O

Suppose that this system is deflected to some initial vertical displacementZa in still water and then released.

The solution of the equation of this decay motion becomes:

u

Z =_{Zae_t (COswt + sinwt}

Wz

Then the logarithmic decrement of heave is:

f

z(t)

Because of the relation w2 = w02 - u2 for the natural frequency and the assumption

u2 wo2 it can be written w w0, which leads to:

w0T

wT = 2ir

Then, the non-dimensional heave damping is given by:

i _I z(t) w0

IC = -lOg1

)J

=

2C vT = ICWOTZ = loge czz

pV+A

Experimental determination 27

These ,-vaiues can easily been found when resiïlts of decay tests with a model in still. water are available.

ct

z0e-v1t

T

za2 z=z0e Vt

(coswt

+ sinW t) 03

Figure 3.7: Determination of logarithmic decrement

Mind you that this damping coefficient is determined here by assuming an uncoupled heave motion. Strictly, this damping coefficient is not valid for the actui coupled motions..

Often the results of these free heave decay tests are presented by

A7

as a function of Za Za

with the absolute, value of the average of two successive positive or negative maximum heave displacements, given by:

Zaj + Zaj+i

Za

2

and the absolute value of the difference of the average of two successive positive or negative maximum displacements, given by:

= Za, - Zai+i

Then the non-dimensionaI heave damping coefficient 'becomes:

i

Í2+

K =

Experimental determination 28

To avoid spreading caused by a zero shift of the measuring signal, double amplitudes can be used too:

Zai - Za.+i + Zai+2 i+3

4

and the non-dimensional heave damping coefficient becomes:

i

I Z - Zaj+i

Ic=1ogÇ

2ir

t.Z2 - Za.+3

The decay coefficient i can therefore be estimated from the decaying oscillation by deter-minig the ratio between any pair of successive Edouble) amplitudes. When the damping is very small and the oscillation decays very slowly, several estimates of the decay can be obtained from a single record. The method is not really practical when u is much greater than about 0.2 and is in any case strictly valid only for small values of u, which is generally the case.

With the previous, the potential mass and damping at the natural frequency can be obtained:

A=---pV

and B- 2KC

wo

The decay tests deliver no information about the relation of the potential coefficients with the frequency of oscillation.

For roll motions the successively found values for ic, plotted on base of the average roll amplitude, will often have a non-linear behaviour as illustrated in figure 3.8.

A behaviour like this can be desribed by:

K = K1 + k2Ça + K3q5

This holds that during frequency domain calculations, the damping term is depending on the solution for the roll amplitude. With a certain wave amplitude, this problem can be solved in an iterative manner.

Experimental determination 29 .04 C w C.) w o o .02 D) C o-E D o t-.01

Product crrier, V =0 knots

first expernient positive angles

O first experiment, negative angles

I

secondexperiment. positive angles D secofld'experiment,negative angles

meanli near and quadratic damping rneanlinear and cubic damping

s

D,

_, -d z / o LU i 2 3 5

mean roll 'amplitude (deg)

Figure 3.8: Non-linear behaviour of roll damping coefficients

From free decay experiments with rectangular barge models at an even keel condition and with the centre of gravity in the waterline, it is found by Journée (1991) that the roll damping coefficients can 'be approximated by:

2

K1

= 0.00130 ()

K2 = 0.500

Experimental determination 30

Forced oscillation tests

By means of forced oscillation tests with ship models, the relation between the potential coefficients and the frequency of oscillation can be found.

The model is mounted on two vertical struts, spaced equally about the centre of gravity. If the struts are oscillated in unison, so that the strut motions aft and fòrward are the sane, the model executes a sinusoidal heave motion. If the strut motions are opposite, so a phase lag of 180 degrees, the model executes a sinusoidal pitch motion. All other motions are restrained and the forces necessary to impose the heave or pitch' oscillation are measured by transducers at the ends of the struts and recorded on suitable apparatus. To avoid internal stresses, the aft transducer is fitted with a swinging link or mounted on .a horizontal sled.

A scheme of the experimental set-up for forced heave, and pitch oscillations is given in figure 3.9.

Figure 3.9: Forced heave oscillation experiment

During the forced heave or pitch oscillations the vertical motions of the transducer aft and forward are ZA(t) and zp'(t) respectively, with equal amplitudes Za.

Then the motions of the model are defined by:

heave oscillations: Z(t' = ZA(t) ZF(t) = Za Sfl

1ZA(t) _ZF(t).) Za .

pitçh oscillations: 9(1) = arctan

smwt = °a SIn wi

I. 2r _J

r

During both type of oscillations, the heave fOrces and pitch moments can b.e obtained from the measured forces' in the transducer aft FA(I) and the transducer forward FF(t):

heave force:

_{F(t) = FA(I + FF(t)}

_{Fa si'n(wt + cFi)} pitch moment: M0(t) = r(F41)

-

FF(t)) = Ma sin(wi + cMi)

Experimental determinaton 31

The linear equations during the 'heave osçiliations are given by

ai +'b +'CZ = Fa5fl(t +'Fz)

d + e. ± fz = Ma sin(wt ±

The components of the force and' moment which are in phase with the heave motion are associated with the' inertia and stiffness coefficients, while the quadrature components are associated with damping.

With:

Z = Z SinWt Z = ZaWcOSWt

= zw2sillwt

-we obtain:

Za(_aw2

+

Sfl wt + ZabW COS wt = Fa COS Fz SjflWt + Fa sin Fz cos

Za(_dw2 + f) sill t + ZaeW COS wt = Ma COS Mz 5fl Wt + Ma sin 'CMz COS

which delivers: w c =

_{C = pg4}

f -

CO5 Ej

d=

_{Ma Sill EMz} Za 0z

1=

Cez:pgStu

To obtain the stiffness coefficients cand f, use has to be made of A (area of the waterline) and S (first order moment of the waterline), which can be obtained from the geometry of the ship nadel.

Also it is possible to obtain th stiffness coefficients from static experim&its:

F

M

z=O and

=O dèlivers:

c=

'and

f=-Z Z

C-a = PV H Za 2 w k_{5fl 0Fz}

b

-

,1? ._Za

-I-JZZ-Experimental determination 32

The linear equations during the pitch oscillations are given by:

aO + bO + cO = Fa sin(ut ±

dO + cO + f0 = M 5jii (ut ± fMO)

The components of the force and moment which are in phase with the pitch motion are associated with the inertia and stiffness coefficients, while the quadrature components are associated with damping.

With:

O = Oa5flWt 0 = OaWcOsWt O = Oaw2sinwt

we obtain:

°a(_aw2

+

smut + °a-' coswt Fa cos E1a'e sin ut + Fa Sfl FO COs.Wt °a(_dw2 + f) Sill ut + OaeW coswt = Ma cos MO sin ut + Ma sin Ej cos ut which delivers:

e -

cos F9

A9=

a

u

sin

b=

_B9=

Oa

u

c=

Cg=pgS

f -

cos Mg

d=I±A09

sin EMO B99= a w

f

CeOPgVGML

To obtain the stiffness coefficients c and f, use have been made of S, (first order moment of the waterline) and GML (longitudinal metacentric height), which can be obtained from the geometry of the ship model.

Also it is possible. to obtain the stiffness coefficients from static experiments:

F

M

-j-Experimental determination 33

The in and out phase. parts of the measuring signal can be found easily from an integration over a whole number n of periods T of the measuring signals multiplied with coswt and sin wt, respectively.

For the heave oscillatioiïs, this results in:

Fa sin fp2 =

_{J (F'(t) ± FF(t)) coswt dt}

Fa cos

J(F(t) + FF(t))

sinwt dt

Ma Sfl Mz =

J(it)

-

FF(t)) coswt dt

Ma COS Mz =

J(F(t) - FF(t))

sinwt dt

and for the pitch oscillations in:

Fa Sfl FO = -;; J (FÀL(t) ± FF(t)) cos wt dt

Fa cos FO

_J

(t

+

F(t)) 'sinwt

. dt

Ma Sjfl MO = -

J(FA(t)

-

FF(t)) cos wt dt

M COS MO = J(FA(t)

-

F(t)) sinwt

dt

With the measured hydromechanic coefficients, the coupled linear equations of heave and pitch motions in stilli water are given by:

(pV+A)

+Czz +A0Ö +Bo +CoO

O

3.2.2

Theoretical determination

Consider a cylinder, oscillating in or below the free surace of a fluid, The fluid is assumed to be incompressible, inviscid and irrotational, without any effects of surface tension. The water depth was assumed to be infinite and thç motion cf the cross section was taken to be harmonic.

The cylinder is forced to carry out a simple harmonic motion A(m) . cos(wt) with a pre-scribed frequency of oscillation w. The superscript .m may take on the val:ues 2, 3 and 4, denoting swaying, heaving and rolling motions, respectively It is assumed that steady state conditions have been attained.

Take :the x axis to be coincident with the undisturbed free surface and the y axis being the axis of symmetry of the cross section, positive upwards.

A velocity potential has to be found:

(m)(yt)

_y).

e_iwt}

satisfying the fllowing six conditions:

1., Equation of Laplace.

The velocity potential must satisfy to the equation of Laplace: a2(m) a2(m)

V2(m)=.

₊

₌₀

82

ay2

Symmetry or antisymmetry condition.

Because both the sway and t:he roil motion of the fluid are antisymmetriçal and the heave motion is symmetrical, these velocity potentials have the following relation:

(2)(_x,y.

_2(+x,y)

_{for sway}

= +(+x,y)

for heave

= '(+x,y)

for roll Free surface condition.

Thelinearized free surface condition in deep water is expressed as follows: a2(m)

a(rn)

O for: y = O

at2 _ay

on the free surface outsidé the cylinder, in which g is the acceleration of gravity.

Bottom condition.

For deep water, the boundary on the bottom is expressed by:

a(m)

=0

ay Kinematic boundary condition.

Theoretical determination 35

The normal velocity component of the fluid at the surface of the cylinder is equal to the normal component of the forced velocity of the cylinder.

if v, is the component of the forced velocity of the cylinder in the direction f the outgoing unit normal vector n, then a kinematic boundary condition has to be satisfied at themean or rest position of the cylindrical surface:

6. Radiation condition.

At a large distance from the cylinder the disturbed surface of the fluid has :to take the form of a regular progressive .outgoing gravity wave.

The motion amplitudes and velocities are small enough, so that all but the linear terms of the free surface condition, the kinematic boundary conditión on the cylinder and the

Bernöulli equation may be neglected.

With known velocity potentials, the hydrodynarnic pressures on the surface of the cylin-der can be obtained from the linearised equation of Bernoulli. An integration of these pressures in the required direction delivers the hydrodynarnic force or moment.

This force ore moment can also beexpressed in potential mass and damping terms. Còrnparions of the in and out phase parts of these two expressions deliver the potential mass and damping coefficients.

Analytical solution of Ursell

Waves radiate away from the

cylinder r

= Ya cos(wt + ô)

Figure 3.10: Circular cylinder heaving in the free surface

Urseli (}949) made the first step towards solving the general problem of calculating the twö-dimensional flow around a cylinder of arbitrary Shape floating in a free surface. He derived an analytical solution for an oscillating circular cylinder, semi-immersed in a fluid, as shown for heavein figure 3.10.

Theoretical determination 36

The forced oscillation of the cylinder causes a surface disturbance of the flùid. Because the cylinder is supposed to be infinitely long, the generated waves will be two-dimensional. After initiai transients have died away, the oscillating cylinder generates a train of regular waves which radiate away to infinity on either side of the cylinder, serving as a mechanism for the dissipation of energy.

Two kinds of waves will be produced: a standing wave system and a regular progressive wave system.

The standing wave system can be described by an infinite number of pulsating multipoles aligned: with they-axis. The amplitudesof these waves decreasestrongly with the distance to the cylinder.

The progressive wave system can be described by a pulsating source at the origin for the heave motion and by a horizontal doublet at the origin for the antisymmetric sway and roll motion. These waves dissipate energy. At a distance of a few Wave lengths from the cylinder, the waves on each side can be described by a single regular wavetrain. The wave amplitude at infinity is proportional to the amplitude f oscillation of the cylinder, provided that these amplitudes are sufficiently small compared with the radjus of the cylinder and the wave length is not much smaller than the diameter of the cylinder. Ursell defined the stream and the potential functions, satisfying the boundary conditions. Then the potential mass and damping coefficients are obtained by considering the pres sure fluctuations in the surface of the cylinder, obtained with the linearised equat-oii of Bernoulli.

Theory of Tasai

For the determination of the two-dimensional addéd ru ass and damping in the sway, heave and roll mode of the motions of ship-like cross sections, Tasai (1959) mapped these cross sections conformally to the unit circle. Then Ursell's theory was used for a circular cylinder to obtain thç potential coefficients.

The advantage of conformal mapping is that the velocity potential of the fluid around an arbitrary shape of a cross section in a complex plane can be derived from the more conve-nient circular section in another complex plane. In this manner hydrodynamic problems can be solved directly with the coefficients of the mapping function.

Conformal mapping of a cross section to the unit circle

The general transformatiön formula, which requires an intersection of the cross section with the waterline, is given by:

z

M3>{a

-(2n-i') }

Theoretical determination 37

with:

z = x + iy = plane of the ship's cross section

ie'e'0

= plane of the unit circle

M3 = scale factor

a1

=+l

a2fl_.1 = conformal mapping coefficients (n = 1, ..., N)

N

= number of parameters Be

r is

constant r' - plane (r,ø) o is constant r is cons tan t

Figure 3ll: Mapping relation between two planes

From this follows the relation between the coordinates in the z-plane and the variables in the (-plane:

=

_{_Ms{(_1)na2n_ie_(2n_1sin((2n}

-

_1)O)} y =

±Ms{(_1)fla2n_ie_(2n_1cos((2n

-

i)O)}

When putting a=O, the contour of the by conformal mapping mathematical described cross section is fOund.

The scale factor M3 and the conformal mapping coefficients a2fl_l., with a maximum value of n varying from N=2 until N=1O, can be determined succesfully from the offsets of a cross section in such .a manner that the mean squares of the deviations of the actual cross section from the mathematical described cross secticn is minimized. A simple and straight on iterative least squares. method to determine the Close-Fit conformal mapping coefficients is given by Journée (1t992).

A very simple and in a lot of cases also a more or less realist ic transformation of the cross section will be obtained With N = 2 in the transformation formula, the well known Lewis transformation (Lewis (1929).).

The contour of this so-called Lewisform is expressed by:

x0=M3. ((1-i-ai)sinOa3sin3O)

Yò = M3.

((i - ai)cosO +

a3cos3O)