Strip theory algorithms

Zr. J.M.J. Journée RéportvNo. 815 December. 1988

Deift University of Technology

Ship Hydromechanics Laboratory Mekelweg 2

2628 CD Deift The Netherlands Phone 015-786882

by: Ir. J.M.J. Journée

Report No. 815

Deift University of Technology

Ship Hydromechanics Laboratory

Summary

This report describes n detail the theoretic backgrounds and the

algorithms of the (ordinary) strip theory method _{to calculate the}

hydroinechanic coefficients, the wave loads, the motions, the added

resistance and the structural.loads of a ship, sailing in regular

and irregular waves coming from any direction

For the wave exciting forces. and moments an alternative approach

has been used, which gives in particular for the roll motions

better results.

This report aims to be an aid for those who want to write or study

a computer program, based on the strip-theory method, to calculate

the behaviour of a ship in a seaway.

Contents page Introduction ₁ Equations of Motions ₃ 2.1. Definitions ₄ 2.2. Potential Theory 6 2.3. Equations of Euler ₁₈

2.4. Strip Theory Method ₂₀

Two-dimensional Potential Mass and Damping ₂₃

Damping of Surge 27 Damping of Sway 31. Damping of Heave 47 Damping of Roll ₆₁ Damping of Pitch 78 Damping of Yaw 79 Viscous Damping ₈₀

4.1. Viscous Damping of Surge ₈₁

4.2. Viscous Damping of Roil ₈₂

Hydromechanic Forces and Moments ₉₆

5.1. _{Hydrorne'chanic Forces of Surge}

. 97

5.2. _{Hydrome.chanic Forces of Sway 100}

5.3. _{Hydromechanic Forces of Heave 104}

5.4 Hydromechanic Moments of Roil ₁₀₇

5.5. _{Hydromechanic Moments of Pitch 111}

5.6. _{Hydromechanic Moments of Yaw} . . . 114

Wave Exciting Forces and Moments ₁₁₇

6.1. _{Wave Exciting Forces of Surge 121}

6.2 Wave Exciting Forces of Sway 125

6.3. _{Wave Exciting Forces of Heave 129}

6.4. _{Wave Exciting Moments o:f Roll 134}

6.5. _{Wave Exciting Moments of Pitch 140}

6--6. _{Wa-v-eE-x-c-i-t-i-ngMo men ts_o-f_a.w}

_...-.

.

3.1. _{Potential Mass and} 3.2. Potential Mass and 3.3. Potential Mass and 3.4. Potential Mass and 3.5. Potential Mass and 3.6. Potential Mass and

page

Transferfunctions of Ship Motions 143

Shear Forces and Bending and Torsional Moments 149

8.1. _{Still Water Loads} . . _.. 153

8.2. _{Lateral Dynamic Loads 154}

83.

Vertical Dynamic Loads ₁₅₇

8.4. Torsional Dynamic Loads 160

Added Resistance due to Waves ₁₆₃

9.1. _{Radiated Wave Energy Method 164}

9.2. _{Integrated Pressure Method 166}

Wave Energy Spectra ₁₆₉

Response Spectra and Statistics ₁₇₃

11.1. Displacements ₁₇₇

11.2. Velocities ₁₇₈

ll.3 Accelerations ₁₇₉

11.4. Vertical Relative Motions ₁₈₀

11.5. Shipping Green Water ₁₈₁

11.6. Slamming ₁₈₃

11.7. Structural Loads ₁₈₅

1l.8. Added Resistance ₁₈₇

1. Introduction

Thi,s report describes the algorithms and theoretic backgrounds of a

computer program, based on the "ordinary striptheory'!, for the calculation of the wave-induced motions and loads _{of a ship, moving}

forward in a seaway with six degrees of freedom.

The purpos.e of the report is to be an aid for those who want to

write or study shipmotion computer _{programs which are based on the}

strip - theory.

The information given here i.s based on the referenced reports taken

from the open literature and the _{common kno.wledg,e on the state of}

art of the research in ship hydromechanics,

However, some approaches of the author _{have been. included with} respect to the use of the equivalent relative velocities and

accelerations of the waterparticles in the exciting _{wave forces and}

moments and ideas on restricted waterdepth pro;blems, additional damping, dynamic loads and calculation routines.

In this introduct:ion, a short description of each _{of the chapters} in the report is given.

Chapter 2 gives the definitions of the axis _{systems, the velocity} potentials and the ship motions.

Based on Lecture Notes of Gerritsma _{[1987], a general description} of the derivation of the velocity _{potentials is given.}

Th.e equations of motions are given with mass and inertia terms and

hydromechanic forces and moments in the left _{hand side and wave} exciting forces and moments in the right hand _side.

The principal assumptions _{are a linear relation between forces and}

motions and the. validity of obtaining the total. forces by a simple integration over the ship length of the _{two-dimensional crOss}

sectional, forces. This includes _{a speed effect as havE b:een defined}

by Korvin-Kroukôvsky and Jacobs [1957] for _{heavE and pitch mot1ons.} This approach is called the "ordinary _{strip-theory method". If one}

s familiar to this approach, other definitions of the _speed

depen-ding sectional hydromechani.c forces, _{as for instance given by Tasai}

[1969] , c.an be included easily.

Chapter 3 describes the determination of the _{two-dimensional}

poten-tial hydrodynamic mass and damping for the _{six modes of motions in} deep water.

The ship's cross sections are conformally mapped to unit circles by the so-called Lewis transformation. Boundary values _{of the validity}

of the transformation are given, _{as for instance presented by von} Kerczek and Tuck [1969].

According to the reports of Urseil [19.49], Tasai [1959], _Tasai

[1961] and de. Jong [1973], a source _{or a doublet and a distribution}

of multipoles are used for the calculation of the _hydrodynamic

forces and moments acting on these ship-like cross sections, oscillating in the free suface of a fluid.

Only for the surge mode a different approach _{is used. Use has been}

made here of work from Kaplan and Jacobs [1960] _{and Sargent and} Kaplan [1974].

Chapter 4 gives some corrections on thehydrodynamic damping due to

viscous effects.

The surge damping coefficient is corrected for viscous effects by

an empirical method, based on a simple still water resistance curve

as published by Troost [1955].

The analysis of free rolling experiments and _{an empirical method,}

published by Ikeda, Himeno and Tanaka [1978], _{to determine a}

viscous correction of the roll damping coeffi.cient are described.

Chapter 5 describes the determination of the hydromechani.c forces

and moments in the left hand side of the equations of _{motions of}

the sailing ship in deep water, as proposed by Korvin-Kroukovsky

and Jacobs [1.957]

Chapter 6 de!scrb.es the wave exciting forces and moments in the right hand side of the equations of motions. of the sailing ship in

water with an arbitrary depth, using the relative motion _concept and the approach of Korvin-Kroukovsky and Jacobs _{[1957]. However,}

the hydrodynamic coefficients used here _{are valid for deep water}

only.

Chapter 7 gives the solutions of the equations of motions.

The transferfunctions o:f displacements, rotations, velocities,

accelerations and vertical relative displacements _{are described.} The use of a wave potential valid for _{an arbitrary waterdepth makes} the calculation method, described here, suitable _{for keel} c1earan-ces down to about 40 until 50percent of the ship's _draught.

Chapter 8 describes the determination of _{the tranferfunctions of} the lateral and vertical shear forces and bending _{moments and the} the torsional moments in _{a way as presented by Fukuda [19621 for}

the vertical mode. Still water phenomena _{are described too.}

Cha.pter 9 describes two methods to calculate the _{tranferfunctions}

of the mean added resistance of _{a sh.i.p in waves The first one is a} radiated energy method, as published by Cerritsma and Beukelman

[1972] . The second one is an integrated pressure method, as

published by Boese [1970]

Chapter 10 gives three examples of normalised _{wave energy spectra,}

as often used in ship motion calculations: _{a somewhat wide spectrum}

of Neumann, an average spectrum of _{Brets:chneider and a more narro.w}

spectrum, the mean JONSWAP spectrum.

Chapter 11 gives a description of the calculation _{procedure of the} energy spectra and the statistics of the ship's motions for _six

degrees of freedom, the loads on the sh.ip and the added _resistance in uni-directional irregular waves, coming from any direction. The calculation o:f shipping green water is given with an empiric

approximation of Tasaki [1963] for the static and dynamic _{swell up}

at the bow.

The calculation. of slamming is given according to a definition àf the problem as presented by Ochi [1964]

Chapter 12 finally, gives the references of the _{literature used to} prepare this report and a computer program, suitable for _{use on a}

2. Equations of Motions

The ship is considered to be a rigid body. Only the external loads

on the underwaterpart of the ship are considered and the ef:fect of the above water environment is fully neglected.

The fluid is considered to be ideal: homogeneous, incompressible,

free of surface tension, irrotational and without _viscosity. However some empirical adaptions will be made here for viscous effects in surge, roll and pitch motions.

It is assumed that the problem of the motions of _{a floating body in}

The coordinate systems are showed in the next figure.

jzo

Figure 2.1.A. Coordinate system and definitions.

Three right-handed coordinate systems _{are defined:}

S-(x0,y0z0)

fixed in space,

with (x01y0) in the still water surface,

x0 in the direction of the wave propagation

and z0 positive upwards

O-(x,y,z) moving with the constant forward ship speed, with (x,y) in the still water surface,

x in the direction of the ship's speed V

and z positive upwards

C-(xb,yb,zb) connected to the ship,

with C at the ship's center of gravity,

x in the longitudinal forward direction,

Yb in the lateral port side direction, and zb in the upward direction.

In stIll water this ship-fixed system is parallel to the O-(x,y,z) system.

The harmonic elevation of the _{wave surface ç is defined in the} space-fixed coordinate system by:

in which:

wave amplitude k 2ff/A wave number

A = wave length

circular wave frequency

t _time

The wave speed c, defined i.n the direction of _{the x0-axis with an} angle p relative to the ship's speed V, follows from:

c co/k

The righthanded coordinate system 0- (x,y,z) is moving with the ship's speed V, whic.h yields:

x0 = Vt.cosp +xcosp +y.sinp

From the relation between the frequency of _{encounter 0e and the}

wave frequency co:

We

follows:

- kV c os p

c

ca0s(1oet

kxcosp -ky.sinp)

The resulting six. possible shipmotions in the _{0-(x,y,z) system are}

defined as three translations of the ship's _{center of gravity along} the x-, y- and z-axes and three rotations about them.

The.se harmonic displacements are indicated by:

surge: _{x = xa .cos(wet +cxc)} sway: y y cos(wet +yç)

heave: Z _{Za cOS(Wet}

_+Zlc)

roll: _{= 'a}

_0s(et

_+elpç)

pitch: 0 _{0a cos(wet}

+gç)

yaw: _{1'a cos(wt}

+6,ç)

The p:hase lags of these motions _{are related to the harmonic wave} elevation at the origin of the 0- (x,yx) system, the average position of the ship's center of gravity:

wave: _{c = c8 cos(coet)}

The harmonic velocities and accelerations _{in the 0-(x,y,z) system}

are found by taking the derivatives o.f the displacements, _for

instance:

surge displacement: x _{Xa 'cos(coet +exc)}

surge velocity:

_{X =}

_0e

a

5(et +Exc)

surge acceleration: x

_We2Xa

_{cos(coet_+rc)}

2.2. Potential Theory

For a general description of the potential theory, as given in this chapter, use has been made of Lecture Notes of Gerritsma [1987].

Suppose the rigid body is floating in _{an ideal fluid with harmonic}

wave.s. The time-averaged sp.eed 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-(x0,y0.,z0) system.

The linear velocity potential of the fluid is splitted into three

parts:

(x,y,z , t)

in which:

the radiation potential for the oscillatory motion

of the body in still water the incident wave potential

the diffraction potential of the waves about 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:

-8x 8y 8z

As the fluid is homogeneous and incompressible, the _continuity

condition

8v 8v

+

y+-8x ôy

results into the equation of Laplace:

a2 a2 a2

--+---- +.- =

0

8x ôy' 8z'

The pressure in a point P(x,y,z) is given by the _linearised

Bernouilli equation:

p -p.

-pgz

or:

0cx

gç

at

At the free surface of the fluid, so for z

ç(x,yz,t), the

pressure p is constant.

Because of the linearisation, the vertical velocity of a

water-particle in the free surface becomes:

dz aç

dt 8z 8t

With this the boundary condition at the free surface _{can be written}

as:

+ g

-8z

0

forz=0

The boundary condition on the bottom follows from _{the definition of}

the velocity potential and is given by:

= 0

forz=-h

8z

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.} The outward normal velocity in a point P(x,y,z) at the surface of

the body is given by:

an -

v(x,y,z,t)

Because of the linearised problem, this _{can be written as:}

a",

8n

v(x,y,z,t)

=

Evj.f

with the generalised direction-cosines, on the surface of the body: = cos(n,x) - cos(n,y) f3 - cos(n,z)

f4=ycos(n,z) -zcos(n,y)

f5 - z-cos(n,x) -x.cos(n,z) - x.cos(n.,y) .y.cos(n,x) p p

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:

urn r -I

I ' I 0

R=co L J

Forces and Moments

Th'e force F amd moment M follow from an integration of the pressure on the submerged surface of the body:

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

=

p.

-at

r

8r

8w

ô*Zd ₁

-p.

++---J

at at at

The hydromechanic force F and moment M can be splitted up into four

parts: F _{= Fr + F + Fd +} = Mr + w + Hc + ii) dS

-pg.z

in which F' and M5 are the hydrostatic parts. or: F ₌

_p.

fj'

I 8r

aW

aJ

_+gzIñ.dS

I

++

L _{at at}

at

M ₌

_.p.

JJ'

I

_8r

w 8c1c1 x ii) dS

I -+---+

L _a

-+ g.z ].(?

_at -

jsr(

M

..J,ssp

The radiation potential

r belongs to the oscillation of the body in still water. It can be written: 6 6

Ej(x,y,z,t)

-

Ecj(xy1z)

vj(t) j= 1 in which:

vj(t)= oscillatory velocity in direction _j

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

So the generalised direction-cosines _{are given by:}

8q1

8n

With this the radiation terms in the hydromecha.nic force and moment are: and: fj

8r

I 1 S 8 6

-at j=i dS

The components of this force and moment _{are defined by:}

Fr _{(Xrl,Xr2,Xr3)} Mr (Xr4,XrS,Xr6) a

-

- 6 I E L

j_

i

J JI = 6 E. i-i 8q, v 3n ôn 8n Fr

P11 IIi dS

J J L _8t J Mr =

p.ff[!!]

x ii) dS ] x i) dS

or: 3 6 dS Xrk P

JJ [

i i k for subscript k - l,2,...6 dS Xrk - p for subscript k 1,2,.. .6 6 3Pk dS

In this expression _{j and are not time-depending, so the}

expression reduces to:

Xrk =

Xj

for subscript k 1,2,.. .6 with: dvj

₁₁

3'k Xrkj d't (P1.

;;-This last force in the direction k is caused _{by a forced harmonic}

oscillation of the body in the direction j Suppose a motion:

= 5aj

So the velocity and acceleration of the oscillation _are:

Si

= vi

iWSaj

I

_.eiWt

dt

The hydromechanic force _{can be splitted into a force in phase with} the acceleration and a force in phase with _{the velocity:}

Xrkj - -Mkj.sj -Nkj.sj

= [

Sajw .Mkj +iSaiwNkj ]

.eiWt

[

aj"2

.J

$

(Pj

if!iS.

So in case of an oscillation of the body in the direction _{j with a} velocity potential pj , the hydrodynamic. mass (coupling) coefficient

is defined by:

N.kj - -Im

[

p.w.JJ

ôn

In case of an oscillation of the body in the directionk with a velocity potential q', the hydrodynamic mass (coupling) coefficient

is defined by:

Nik =

S

and the hydrodynamic damping (coupling) coefficient _by:

S

and the hydrodynamic damping (coupling) coefficient _by:

=

+-8x2 8y2 3z2 8Pk 8q [ _j ân dS ] dS ] I I I ôçoj [ _8n dS ]

Suppose two velocity potentials _{pj and}

k and use Green's second theorema for these potentials

[ 'j 'k

- 9k'j

] .dV* = [ Pj k 844 1

*

-'

]

dS

8n

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 water-surface and the wetted water-surface of the floating body.

The Laplace operator is given by:

So according to the equation of Laplace:

0 Mkj = -Re I ( P J

I

.1

I

I (Pj dS ] 8n Mik - -Re

This results into:

Pk dS*

dS*

8n

Figure 2.2.A. Boundary conditions.

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

urn

R= [ q ]

The boundary condition on the seabottom is:

arp

=0

forz=-h

8n

The boundary C:ondition at the free surface:

--+g =0

forz-0

ôz

becomes for ' = - iwt.

2 aco -w

+ g - -

0 for z = 0 0 R

kp_-.zO

tp-.o

R..-o.o

or with k - for deep water:

g

8

k.q,

= -

forz=0

8z

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

k.qk

8z 'än and _{âz 8n}

When taking these boundary conditions into _{account, the integral}

equation over the surface S reduces to:

Jf

(Pj. dS

- P dS

S S

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

Mjk Mkj

- Nkj

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.

Hydrodynamic mass matrix:

Hydrodynamic damping matrix:

0 0 M51 0 0 N22 0 N42 0 M62 0 M33 0 M53 0 0 0 0 N64 N11 0 N13 0 N.51 0 N53 0 M15 0 0 N26 M35 0 0 M46 M55 0 0 M66 N15 0 0 N26 N35 0 0 _N46 N55 0 0 N66 N22 N24 0 ₀ N31 0 N33 0 0 N42 0 N.44 0 N62 0 N64

Wave Potential _w

The velocity potential _{of the harmonic waves has to fulfil three}

boundary conditions:

- the equation of Laplace:

a2 a2

ax 8y 8z

- the boundary condition on the bottom:

__L _{0 for z = -h}

ôz

- the dynamic boundary condition at the free surface,

which follows from the linearized equation of Bernouilli:

at + g.ç = 0 for z 0

With this the corresponding wave potential, depending _{on the}

waterdepth h, is given by the relation: cosh k(h+z)

casin(wt -kxcos

-k.ysin)

-cosh kh

The dispersion relation follows fromthe kinematic _boundary

condition at the free surface and is defined by:

- gk tanh kh

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

X _{xb Y Yb} z _Zb

In case of a forward ship speed, the _{wave frequency has to be} replaced by the frequency of encounter of th,e waves

This leads to the following expressions for the wave surface and

the wave potential in the _{G-(xb,yb,zb) system:}

c

ca0s(oet

kxb.cosFi -ky.siflji)

and

= - cosh k(h+zb)

ca5(wet -kxb.cO.Sp -kyb.sinp)

Wave and Diffraction Forces and Moments

The wave and diffraction terms in the hydromechanic force and moment are: or: F + Fd -and: + Fd

8w

3d

-

__

ön 8n 8n Xwk -i.p. p.

ff

[!

r r -J J L S

-0

-iwt

JJ

+

8d

at Define now:

(x,Y,z,t) (Pw()C,Y,Z) e1.Wt

d(x,Y,z,t) d(x,Y,z)

elWt

This results into:

8n 8n

With this and the expressions fo:r the _{generalised direction cosines} it is found for the wave forces and _{moments on the réstraind body}

in waves:

Xwk

i.p.e)t.

JJ

('

+ 'd)

k dS

for subscript k - 1,2,...6

For the determination of these wave forces and moments it is

supposed that the floating body is restrained with _{zero forward}

spe.ed.

Then the boundary condition on the surface of the body _{reduces to:}

8q,k

dS

8n

for subscript k - 1,2,.. 6

The potential of the incident waves c°w is known and the diffraction potential _d has to be determined.

+ dS

rr

8k

J J cocI

dS On

S

which results into the so-called Haskind relations:

r r J J

'k -

On S dS t

I I

I

OPk 3Pw J .1 L + S dS for subscript k 1,2,. ..6

With this the problem of the diffraction potential _{has been}

eliminated, because the expression for Xwk is depending on thewave

potential _w and the radiation potential _{Pk only.}

These Haskind relations are valid for _{a floating body with a zero}

time-averaged speed in all directions only.

Newman [1965] however, has generalised these _{Haskind relations for} a body with a constant forward speed. He derived equations which

differ only 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 relations are very important They _{underlies the}

relative motion (displacement-velocity-acceleration) _hypothesis, used in the strip-theory, see chapter 2.4.

Green's second theorema delivers:

S S

on dS On dS With: an an it is found: Xwk -i.p

e1-The corresponding wave potentia1 _{at an infinite waterdepth is given} by the relation:

-

_{cancot -kxcosp -kysinp)}

-i.c.g

e e -(4,

-i.c.g

e e

(x.cosp

+y.sin)

C',

The velocity of the waterparticles in the direction _{of the outward}

normal on the surface of the body is:

a _{3z 8y}

L -.

c.osp sinp

This can be written as:

=

Pwk

_[

f3 +i( f1cos1.j +

f2sinp )

]

Then the wave loads are given by:

Xwk =

JJ

Pw k dS

+i.p.e1t.k.

_dS

for subscript k - 1,2,. . .6

The first term in this expression for the _{wave forces and moments}

is the so-called Froude-Krj].ov force or moment. The second terni is caused by the disturbance because of the _{presence of the body.}

Hydrostatic Forces and Moments These are given by:

-

p.

z dS _{for subscript k}

= 1,2,..

_.6

or:

The equatio:ns of motions in a space fi:xed system of a rigid body follow from Newton's law of dynamics.

The vector equations for the translations of and the rotations

about the center of gravity are respectively given by:

F

i,n which:

F = resulting externa,i force acting in the center of. gravity

rn = mass of the rigi.d body

U _{instantaneous velocity of the 'center of gravity}

M _{resultinig external moment acting about the center of gravity}

H = instantaneous angular momentum about the center of gravity t = t1rne

The total mass of the body and its distribution in the _{body is}

c:on'sidered to be constant with time _{This assumption is normally}

valid during a time which is large relative _{to the period of the}

motions..

When assuming small motions, symmetry of the body _{and the axes x, y}

and z. to be principal axe:s, _{it can be written for a ship:}

Surge: Sway: Heave: Roll: Pitch.: Yaw: d dt dt

d[

d[

dt d dt d dt

d[

dt In. [. I

yy

_I Izz. and d t

pV

x _{= Xh + Xw} p.V _{'h +}

pV

Zh + _Zw 'xx = + iyy . = Nh _{+ M} izz = Nh + N =

Ti

pVz ]

= 'xx = d dt

in which: p-V

'xx, 'yy' 'zz

Xh, Zh

total mass of the ship

= mass moments of inertia of the ship

about the x-, y- and z-axis respectively

hydrornechanic forces

in the x-, y- and z-direction respectively

Kh, Mh, Nb - hydromechanic moments

about the x-, y- and z-axis respectively

xw,

zw - wave excitation forces

in the x-, y- and z-direction respectively

K, Mw, N

= wave excitation moments

about the x-, y- and z-axis respectively

This results into the following two sets of three coupled equations of motions:

Surge: p-V x _{Xh xw indicated by motion 1} Heave: p-V - z Zh indicated by motion 3

Pitch: Iv,, - 8 Mh indicated by motion 5

and:

Sway: p-V y _{h = w indicated by motion 2}

Roll: ço _{Rh Kw indicated by motion 4}

Yaw: _{= N indicated by motion 6}

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

2.4. Strip Theory Method

The so-called strip theory solves the three-dimensional _{problem of} the hydromechanic and _{wave exciting 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 zero-speed case. So each cross section of the ship is

considered to be part of an infinitely long cylinder.

However, some additions for viscous effects are necessary.

Surge: _{Xh - JXh .dXb + Xh} a = JXW'.dxb L Sway: Heave: Roll: Pitch: Yaw: in which:

Xh' , Yh' , Zh' sectional 2-D potential hydromechanic force Kh' , Mb' , Nh' sectional 2-D potential hydrornechanic moment

Xv',, Y' ,

Z'

sectional wave exciting force

M', N'

- sectional wave exciting moment Xha additional hydromechanic force in surge

due to viscous effects

Kha additional hydromechanic moment in roll due to viscous effects

Mba = additional hydromechanic moment in pitch

due to viscous effects in surge dXb Zh JZh'.dxb Kh - JKh'.dxb + Kha = JMh'.dxb + Mha L Yw JYW, L dxb zw = JZW'.dXb = JKW'.dxb - JMW'.dxb = JNW'.dxb

As ment:ioned before, two assumptions _{are made for these forces and} moments. The hydr'omechanic 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.

Except for the surge motion., the hydromechanic coefficients are

derived by a two-dimensional potential theory for the _{zero forward}

speed case,, used after a conformal mapping with two coefficients of the cross sections to the uni.t circle, the so-called Lewis

transformation.

in cas.e of a failing transformation, like for instance can happen with bulbous sections,, an adaptive procedure is given.

in those cases also a conformal mapping with _{more coefficients can}

be used.

Another method is the Close-Fit method, given _{by Frank [1.967]} . '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. _{However, this method is} not included here.

According to the ordinary strip theory method, _{the forward speed}

effect on the hy'dromec'hanic potential f'orces _{and moments on an}

oscillating cross section is expressed 'as proposed by Korvin-Krouk'o.vsky and Jacobs [19571 for heave and pitch motions:

D Fh = -h-;. [ M .'J ] + N' .V + FS with: D Dt in which:

Fh' - two-dimensional hydromechanic potential force or moment

M' _{two-dimensional potential mass or ine'r'tia coefficient}

N'' two-dimensional potential damping coefficient Vw - the directional component of the velocity of the

waterparticles in still water, relative _{to the cross} sec t ion

FS' _{- two-dimensional restoring, force o,r moment (if present)} V - forward ship speed

Th'e first term is a result of the so-called _{"slender-body theory",,} the second one is a damping term du'e _{to the gener'ated wave's and the} last one is a restoring term.

at

a

- V.

The relative velocity V, follows from:

[ displacement ]

Later on other definitions of the forward speed effect on the

hydromechanic forces are published, see for instance Tasai [1969]. in a so-called "modified strip-theory method" _{a theoretic better} approach is used for the hydromechanic forces:

I

vw]

+FS'

in which Ce is the frequency of encounter.

This strip-theory version is not described _{here. In this stage the}

original one is choo:sen, but o.ther definitions of the hydromechanic

forces can be included easily. Experience learns however that in average the ordinary strip-theory method does not give worser results than modified versions.

It may be noted that for a zero forward speed these _{two versions}

are identical.

Equivalent to the potential hydromechanic forces _{and moments on an}

oscillating cross section in still water, as given by Korvin-Kroukovsky and Jacobs -[1957] , the wave forces and moments on a

restrained cross section in waves can be defined by:

Fh D Dt D Dt in which:

Fw' _{two-dimensional wave exciting force or moment}

= the directional component of an equivalent orbital

velocity of the waterparticles in the undisturbed _wave,

relative to the cross section.

FK' _{= the two-dimensional Fr'oude-Kriiov force or moment;} this is the forc.e or moment on a ship's cros,s section, caused by the undisturbed wave.

The two-dimensional Froude-Krilov force FK' is _{calculated by an} integration of the directional pressure gradient _{in the undisturbed}

wave over the cross sectional area of the hull. [ M'

I N'

D F

.3., Two-Dimensional Potential Mass and Darnping

For the determination of the two-dimensional added _{mass and damping} in the sway, heave and roil mode of the motions of ship-like cross'

sections, these croas sections are conformally mapped' to the unit

circle. An extended description of the representation of ship hulls by conformal mapping, is given by Kerc.zek and Tuck [1969]

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 convenient circular

section in another complex plane. In this way hydrodynamic problems

can be solved di.rect,iy with the coefficients of the mapping function.

The coeff1cjents a1 and a3 are called .the

Figure 3.A. Mapp i1,n.g re ha t ion. between. two comp.lex p.1 anes.

The general transformation. formula is given by:

Z=M

ç

+

nl

a2n

ç_(2nl)

j

in which:

x +iy

complex plane of the ship's cross section

ie.e10

complex plane of the unit circle scale factor

N = maximum number of parameters

A very simple and in most cases also _{a realistic tr'ansformat,io,n. of} the cross sectional hull form will be obtained with, N=2,,, the

generally known Lewis,train,sforma.tion.,

This Lewis-transformation is defined by:

r a'1 a3 Z = . [ .ç

+ - +

j

c ,çJ

So:

x = M [

ea.sjnQ +a1esinQ a3.e-3a.sin39

y=M

ea.cos,6 -a1ecosO +a3.e3a.cos30

Putting aO, the contour of the Lewis-form is expressed as follows.: = M [ (1 +a1).sinO -a3.sin39 ]

Yo - M [ (1 -a1).cosO +a3cos39 ],

with the scale factor:

M=

B5 or M

2.(l +a1 +a3) 1 -a1 +a3

and:

B5 =. sectional breadth on the waterline sectional draught

From this follow the half beam to draught ratio and _{the sectional}

area ratio:

B 1 +aj +a3

2D5

1 -a1 +a3

a A it. 1 -a12 -3a32

B.D5

4 (1 +a3)2 -a12

Putting a1, derived from the expression for H0, into the expresson for a delivers a quadratic equation in _a3:

c1a32

+c2a3 +c3 = 0

with: r

4a ,

r - L + ] + [ 1 -it.

C2 = 2(cj -3)

D5 H0 -i H0 +1 J c3 = c1 -4

The solutions for a3 and a1 will become:

a3 -c1+3 +(9 -2.c1)

H -1

a1

When doin.g this,

a1

and

a3

are determined in such a way that the

sectional breadth-draught ratio and the sectional area coefficient

of the Lewis form and the actual cross section of the ship are

equal. The dimensional values are obtained by the value of the

scale factor M.

The other solution of

a3

in the quadratic equation Is: +3 -(9 2.c1)½

Cl

Lewis forms with this solution are not considered because they are

looped., which means that they intersect themselves at a point within the fourth quadrant.

In some cases the Lewis-transformation _{can give a more or less} unacceptable result. In the next the different regions of _H0 and a

are defined.

Re-entrant forms are bounded by: a3 3ir if H0 < 1.0 : 0.0 < a < 32 and: if H0 > 1.0 : 0.0 < a < 32

Convential forms are bounded by:

if H0 < 1.0 and: if _{H0 > 1.0} 3ir 32 3ir 32 3,r if H0 < 1.0 : and: 3r if _{H0 > 1.0} : [ [

Bulbous and not-tunneled forms are bounded by:

Tunneled and not-bulbous forms are bounded by:

3r

11

3ir

H0>1.O

and

13+i< a

<-32 1 _4H0 ' ₃₂

Combined bulbous and tunneled forms _{are bounded by:}

[ [

L1

H0 [ 1 2 H0

2Ho]

< a

<-

3ir 32 3r < a

<-32

1I3+ <

a

<-4H0 32 H0-1

3 + 1<

a

<--4 J 32 [ [ 1 4H0 H 3 4 ]

Lj

H0

H0<l.O

and 3,r r H

13

3ir 32 L 4 Ii

a <-

32 [ 2

Ho]

[

10 + H0 +

Ii

10 + HO +

_H0

Non-symmetric forms are bounded by:

or:

These boundaries are showed together in the _{next figure.}

2.0 1.5 1.0 0.5 0 0 0.5 10 1.5 -j

Figure 3.B. _{Ranges of H0 and a for Lewis forms.}

Not-accepted forms for, ships are supposed to be the re-entrant forms and the non-symmetric forms.

So conventional forms,, bulbous forms _{and tunneled forms are} cOnsidered to be valid' forms here.

Then the boundaries for the sectional _{area coefficient a are:}

3ir if H0 < 1.0 :

-

_{[ 2} - H 32

1<

20 2.5 3.0 11 _{.- 1} 0

< -

I

10 + Hn +

-3.2 L '' _H0-1 3ir r 1 1 r 1 if H0 > 1.0 :

-

I 2

- - 1

< a

<

-I 10 + H

±

-32 L

_H0-

₃₂ L _H0

But., _{if a value of a is outside of this}

range it has to be set to

the value of the nearest border of this _{range, to calculate the}

Lewis. coefficients.

Numerical problems, for instance with bulbous sections, are avoided

when the following requirements _{are fulfilled:}

B5/2 > 0.01 .D

> 0.01 .B/2

In the following sub-chapters this conformal mapping method is used

to determine the hydrodynamic coefficients.

bulboua bulbou8 + tunneled tunneled conventional I

I

re-entrant

-I

0<H0<co

and I r 1

10+H0+-a>

32 L _H0

1 -b2 a = b3 log and: in which: L - ship length B = breadth

[

-[ B 12 L ]½

3.1. Potential. Mass and DampinL of Surge

Hydrodynamic Mass

For the determination of the potential hydrodynamic mass of surge

an empiric formula, given by Sargent and Kaplan [1974], can be

used.

A frequency-independent total hydrodynamic mass is estimated as a

proportion of the total mass of the ship pV:

M11 a pV

The factor a is depending on the breadth-length ratio B/L of the

ship: a 2-a with:

1 +b ,

I -2.b

1-ba

Figure 3.1.A. _{Surge hydrodynamic mass in proport:ion to} the ship's mass as a function of L/B.

0.50 pV

I

0.25 0 5 10 L B

To have a uniform approach during all calculations. the _cross

sectional two-dimensional values of the hydrodyn'ami.c mass have to be obtained.

For this, a proportionality with the absolute values _{of the}

derivatives of the cross sectional areas is assumed:

= dA5

Am dxb

in which:

midship sectional area local sectional area longitudinal axis

Hydrodynamic Damping

For the derivation of the two-dimensional pote.ntional hydrodynamic damping of surge work of Kaplan and Jacobs [1960] _{can be used. They}

derived the hydrodynamic damping for a Haskind _{cross sectional form} on the basis of an expanding two-dimensional section, where the

expansion is proportional to dBs,fdxb, the longitudinal rate of

change of the breadth of the waterline. The derivation of the

two-dimensional damping is based on the thin-ship theory in deep water.

Using this approach, the sectional potential _{surge damping i.n deep} water can be determined for the Haskind form _{or the actual}

hullfo.rm: N.11' - p.w. 2 dB dXb

exp(k.zb) dzb

in which: p density of water

= circular frequency of oscillation

k wave number

B8 = sectional breadth on the waterline

= sectional draught

a _{= sectional area coefficient} xb, Yb zb - as defined before This Yb with: N equivalent B.5 1 Haskind form

[]N

was defined by:

a < 1.0) = 2 a (provided that 1-or Am = A.8 = xb =

and

Alteinative Method to Estimate Potential Mass and Damping An alternative method can be obtained by the _{calculation of a}

longitudinal sectional two-dimensional potential mass 1411* and

damping N11*.

An equivalent longitudinal section, to be constant over the ship's breadth B, is defined by:

sectional breadth _{- ship length L} sectional draught _{= midship draught D}

sectional area coefficient block coefficient C.B

Now M11* and Nl1* can be calculated in an analog way as will be described further On f:or the two-dimensional potential mass and damping of away.

With this the total potential _{mass and damping of surge are defined}

by:

*

M11 = M11

B

*

N11 N

To have a uniform approach during al.l calculations _the

two-dimensional cross sectional values of the hydrodynamic _{mass and}

damping have to be obtained.

As defined before, a proportionality with the _{absolute values of}

the derivatives of the cross sectional _{areas is assumed.}

So:

M11

dXb

N11'

N11

2Am

dXb

Coupling of Surge into Heave

Because surge and heave motions are symmetric _{translations, the}

coupling effects o.f surge into heave can be ignored. So:

0 _{two-dimensional hydrodynam.ic mass coupling} coefficient of surge into heave

N31'

= 0 _{two-dimensional hydrodynamic damping coupling} coefficient ofsurge into heave

Coupling of Surge into Pitch

Small coupling effects are assumed for the coupling o,f surge into

pitch:

M51'

= -M11 B0

two-dimensional hydrodynamic mass coupling coefficient of surge into pitch

N51'

= -N11 BO

two-dimensional hydrodynamic damping coupling coefficient of surge into pitch

in which:

BO = vertical distance of the local center of buoyancy

Figure 3.2.A. Axes system for sway oscillations,

as used by Tasai.

The cylinder is forced to carry out a simple harmonic lateral

motion about its initial position with a frequency of oscillation _w and a small amplitude of displacement _xa:

X -

Xa .cos(wt

+6)

in which e is a phase angle.

The lateral velocity and acceleration of the cylinder becomes:

X W X sin(wt

+6)

X -

C&)2Xa .cos(Ct,t +e)

B5/2

3.2. Potential Mass and Damping of Sway

The determination of the hydrodynamic coefficients of a swaying

cross section of a ship in deep and still water at zero forward speed, as described here, is based on work published by Tasai

[1961]

Reference is given here to de Jong [1973] for a description of the derivation of the velocity potentials and conjugate stream

functions.

Suppose an infinite long cylinder in the suface of a fluid, of which a cross section is given in the next figure.

-

+--

82cZ

3x2 8y2

0

The forced lateral motion of the cylinder _{causes a surface}

disturbance of the fluid. Waves travel _{away from}

_the

_{cylinder and a} stationary state is rapidly attained.

Two kinds of waves will be produced:

- a standing wave system, denoted here by subscript A

- _{a regular progressive wave system, denoted here by subscript B.}

The amplitudes of the standing _{wave system decrease strongly with} the distance to the cylinder.

The regular wave system dissipates _{energy. It is known that, at a} distance of a few wave lengths from the cylinder, _{the waves on each} side can be described by a single regular wave-train.

The wave amplitude at infinity _{'1a is proportional to the amplitude}

of oscillation, of the cylnder _{Xa, provided that:}

- the amplitude of oscillation is sufficiently _{small compared}

with the radius of the cylinder

- the wave length is not much smaller than the _{diameter of the}

cylinder.

The cylinder is supposed to be infinitely long, _{so the motion of}

the water and also the velocity potential and the _{conjungate stream}

function will b.e two-dimensional.

The velocity potential of the fluid must satisfy to the equation of

Laplace:

Because the motion of the fluid is not symmetrical _{about the} y-axis, this velocty potential has the following relation:

(-x,y) - -(+x,y')

The linearized free surfac.e condition in deep water is expressed as

follows: 8'I' el, + 0, g öy g 2 for:

>-

B8 2 and _y

In consequence of the Lewis transformation, _{this free surface} condition can be written as:

ea -a1e

3a3e3a

₈₁₁ 0, for a and 0 - ± l+a1+a3 ₈₉ 2 in which: B

From the definition of the velocity potential _{follows the boundary} condition on the surface of the cylinder for a = 0:

8cX0(0) 8x0 x

8n 8n

in which n is the outward normal of _{the cylinder surface.}

Using the stream function 1', _{this boundary condition on the surface} of the cylinder reduces to:

-8w0(o)

89

Integration results into the following _{requirement for the stream} function on the surface of the cylinder:

B (1-a1).cos9 +a3cos39

x + C*(t)

2

where C*(t) is a function of the time _only

For 9 ½,i, _{so at the surface of the fluid, this requirement for} the stream function reduces to:

W0(½ir) - C*(t)

8x0 B5 (l-a1).sjnO +3a3sin3O

x

=x

For the standing wave system a velocity potential _{and a stream} function satisfying to the Laplace equation, the free surface condition and the non.- symmetrical motion of the fluid has to be found.

The following set of velocity potentials, as given by Tasai [19611,. fulfil these requirements:

g

+EP2m(pA2m(a,e) cos(wt)

.sin(ct)]

with: (PA2,m(a,O) + - e(2m+1)a.cos[(2m+1)9j l+a1+a3 e -2ma 2 m

(2m+2)a _{3a3 .e(2m+4)a}

+ sin[(.2m-i-2)9] - _{sin[(2m+4)Oj I}

2m+2 _2m+4

The set of conjugate stream functIons i.s expressed as:

-lr.w

[+mlP2mAA2m(czPO) cos(wt)

.sin(wt)]

with: t'A2m(a,8) e - 2111a 2m sin [ 2mG] cos [2mG]

3a3.e(2m+4)a

cos[(2m+2)9] - cos[(2m+4)9J 2m+4

In these expressions the magnitudes of the

2m and the Q2m series follow from the boundary conditions _{as will be explained further}

on.

These sets of functions tend to _{zero as a tends to infinity.}

sin[(2m+l)9] + B

l+a1+a3

ai

+

Another requirement is a diverging wave-train for a goes to infinity. It is therefore necessary to add a stream function,

satisfying the free surface condition and the non-symmetrical

motion of the fluid., representing such a train of waves at

infinity.

For this, a function describing a two-dimensional horizontal doublet at the origin 0 is choosen.

The velocity potential of the progressive wave system is given by

Tasa.i [1961] as:

g'1a

[ 9Bc(x,Y)

cos(wt) +9Bs(X,y) sin(wt) ]

with: 'Bci .sin(kixi) k

r

k.cos(iy)

+L'.sin(L'y)

9Bsi = +

e ' cos(kx) - I e 0

g

irw '

The conjugate stream function is given by:

[

Bc(X,)') cos(wt) + Az8(x,y) si.n(wt)

] with: Bc = +

.eky

.cos(kx) Bs = + sin(kixi) + y k (x2-f-y2) Ch.angtng the parameters:

g

irw

+

in which:

for x > 0 :

j =

+1 and for x < 0

:

j

= -1

k - - wave number for deep water

g

Changing the par:ameters: lxi k.(x2+y2) g = lr.c4, [ q'B0(8) .cos(wt) +

_9Bs(,0)

[ c(a,O) cos(wt) + frBs(a,9) sin(wt) ]

V

sin(wt)

I

k.sin(vy) -i#.co;s(iy)

When calculating the integrals in the expressions for _Bs and

in a numerical way, the convergence is very slowly.

Power serie.s expansions, as given by Porter [1960], can be used instead of theàe integrals:

I

kcos(iiy) +vsin( in which:

Q -

7 + log[k.(x2I.y2)½

S-+Ep.sin(n)

n-i x P arctan y Y)

_exdv

Pn

The conjugate stream function is:

r irw

].Ir(a,O) -

+ g '1a

f

ksin(&iy) -v.cos(#y) e YX.d, - [

j

0 + pCos(flfl) n-i

kn.(X2Y2)½n

n(ni)

-+ [

BC(,0) +

P rn-i (a,O)

miQ2m'A2m

[

Bc(a,8) +miP2mbA2m(aPO)

cos(kx) +(S-1r)fsi.n(kx)].e.Y

The summation in these expansions _{converge much faster than the} numeric integration procedure. A suitable maximum value of n should

be choosen.

The total velocity potential and _{stream function to describe the}

waves generated by a swaying cilinder are:

- A +

- '1'A + WB

So the velocity potential i,s expressed by:

PA2m(a,O) ] 7 - 0.5772156.649.. (Euler constant) cos (t) ,8) ]

sln(wt

cos (t)

+ { B t,O)

_{miQ2m2m a,O) ]}

sin(wt)

[ sin(kx) -(S-ir).cos(kx)].eY

irw

Putting a=O, the stream function is equal to the expression, found before, from the boundary condition _{on the Surface of the cylinder:}

with:

rlrW

L

11

2rn

r irw

_j.w

a (½w) B

(l)m+1

AO2m()

=

_l+a1+a3

So C(t) will become: C(t)

=

'BOc() +miP2inbA02m(½1r) ] cos(wt)

IB0s() +miQ2mbA02m(½1r)

_I sin(wt)

-+ BOc(½

Bos(½

= 0

+ C(t)

11

2m ) + E P. rni

)+E

rni a1

_3a3

cos[2mO] + cos[(2m+2)9] - cos[(2m-i-4)9]

2m+2 2m+4

In this expression _{BOc(0) and B0s(0) are the values of bBc(a,O)}

and _{,Bs(a,O) at the surface of the cylinder, so for 0.}

When putting 0 = ½ir, we obtain C(t):

a1 3a3 2m+2

2m+4

AO2m()

AO2m()

cos (wt) sin(wt) 1

j.Wo(9) =

+ r

L 1'BOc(9) +_m=lP A02m(0) I cos

+ 1'BOs(0) _{A02m(0) I} sin(wt) 1 B5 (l-a1).cosO +a3.cos39 + I

1x

+

C(t)

L

_g'a

-' 2 1+a1+a3 ) - - _{cos[(2m-i-l)9]} l+a1+a3 with: AO2m(

+ [

BOg(9) -

BOs()

m=im

[AO2m(9) AO,2m(½

A substitution C(t) in the expression for _{results into the}

following equation of the stream, functions:

+ BOc(9) - ABO(½7r)

m=12m

[AO2m°

-AC.2.m(½1r)] I co.s (wt) )_I ] sin(wt) r irw . B (1-a1).cos9 +a3cos39 L -2

The right h'and side of this equation can be. written as follows:

irw . B (1-a1).cos9 +a3cos39 2 _i±a1+a3 Xa

1tBh(0) sin(w.t +e)

Xa

tBh(0) [ sine cos(wt) +cose sin(wt

a

+ h(9) [ P0 cos(wt)

_{+ Qo sin(wt) ]} in which: h (.9) (l-a1)co.s9 +a3cos39 l+a1+a3 P0 Xa

_eB

_s1t Qo Xa CO5E

This results for any value of 0 in two sets of equations with the

unknown parameters P2m and _Q2m

So:

BOc(0) BOc("') h(0).P0 + E

f2ni(0)P2m

m=l

BOs(0) BOs('r) h(0).P0 + E

_2m(8)Q2m

m= 1

with:

2'm(0) - AO2m(0)

_AO2m()

This can als.o be written as:

1'BOc(0) BOc(½h1)

m020)"2m

BOs(9)

BOs()

₌

_m=02mQ2m

with:

for mO:

f0 (0) = -i-h(0)

for m>O: _{= AO2m(0) +bAO2m(½r)}

The series of these relations _{converges uniformly with an} increasing value of m in the ranges for a and 8:

and

-½ir9+½,r

For practical reasons the maximum value of m is limited to _user However, each 0-value will deliver an equation for the

2m and Q2m series. So at least flser+] values of 0 are required to solve these equations.

The best fit values of _{2m and Q2m are supposed to be those found}

from the nser+l equations by means of a least squares method.

After multiplying both sides of the equations _{with 0, the}

This means that two sets of

_ser-

_{equations, one for 1'2m and one}

for Q2m' are obtained:

'ser E P m=O m ½,r

2m(0)2n()

=J [o

_BOc]2fl(0)9

0 n 0, 1, 2, _tser ½ir nser

_I

2m

2m(9)2n(0)10J [BoSn

BOs(½)]2n(9)9

n - 0, 1, 2, . 1se,r

Now the _{2m and Q2m series can be solved with a numerical method.}

When using the definitions of the first _{terms P0 and Q of these}

solutions, as given before, the amplitude _{rati.o of the radiated}

waves and the forced sway oscillation is given by the following

expression:

with:

] o

'AO2m(9) = + sin[(2m+l)Oi +

The hydrodynarnic pressure on the surface of the cylinder can be obtained from the linearised Bernouilli equation:

8. (9) p(9) -p

8t

Putting aO, the velocity potential _{on the surface of the cylinder}

becomes:

'BOc(0) + E P2

rni PAO2m(0)

I

1 + a + a

r I a1 _3a3

- sin.[2mO] +

sin[(2m+2)9]: sin[(2rn+4)9]

2m 2m+2 2m+4

In this expression _{PB0c(0) and BOs(9) are the values}

,

and q(a,O) at the surface of the cylinder, _{so for a - 0.} So the hydrodynamic pressure on the surface of the cylinder becomes:

p(0)

[

+ [ 'BOs(9)

rn1Q2m021°)

I cos (cot)

[ 'B0c(9) +miP2mPAO2m(0) _I sin(cot)

The two-dimensional hydrodynamic lateral f;orce, acting on the

cylinder in the direction of the x-axis, _{can be found by}

integrating the lateral component of the _{hydrodynarnic pressure on} the surface of the cylinder:

½ir cos (wt) +

[ P(9)

_m1Q2m

A02m(9) ] sin(wt) F in which: dy0 = +2 _I J -B 0 r

p(0)

dyn dO +3:a3sin39 (l-a1)sin0

--dO dO 2

With this the two-dimensional hydrodynamic lateral force due to

sway oscillations can be written as follows:

F with: ½ir

r

(l-a1).sinO +3a3sin39 -- J 9'BOs(9) dO l+a1+a3

- miQ2m'2m

3a3 Q2 1+a1+a3 4 and : ½ir N0 = - PBOc(0) (l-a1).sin9 +3a3sjn39 dO 0 l+a1-4-a3

- m='l2m2m

3a3 i+a1+a3 4 M0 cos(wt) - N0 sin(wt) ] P2 in which: D2m B (..I)m+i [.(l-al).[

+3a3

+ [ 1 + a1 -+ 3a3 ] 4m2 -1 1 (2m+2)2-1 a1 (2m+4)2-1 3a 4m2 (2m+2)2-9 (2m+4)2-9 I

It was found before that:

sin

and cose

Xa.'lreB

Using this, the two-dimensional hydrodynainic _{lateral force can be}

resolved into components in phase and out of phase with the lateral displacement of the cylinder:

F

pgB.5,

[ [MQ + N0.P.0] cos(wt +

) + [M0.P -

oQo] sin(ot +

This hydrodynamic lateral force can also be written as:

= -M22 ₂₂

X

[M22'w2xa]

cos(wt +e) _{+ [N22 W.Xa] si.n(cot +e)}

in which:

two-dimensional hydrodynamic mass coefficient of _sway

N22 _{= two-dimensional hydrodynamic damping coefficient of sway}

When using also the amplitude ratio of _{the radiated waves and the}

forced sway oscillation, found before, _{the two-dimensional}

hydrodynamic mass coefficient of _{sway is given by:}

pB52

M0.Q0 + N0.P0

p02 +

and the two-dimensional hydrodynarnic _{damping coefficient of sway} by:

p'B5 'M0.P0

-'N22 (0

N2.2' - .

B2

4 w 1 p02 +

The energy delivered by the exciting forces should be equal to the

energy radiated b.y the waves.

So: Os C 1

N22'x xdt

- 2. -T05 2 in which:

T05 - period of, oscillation

With the relation fo.r the wave speed c = g/w, follows the relation

between the two-dimensional sway damping coefficient _{and the} amplitude ratio of the radiated waves and the forced sway oscillation:

pg2

_1'?a12

N22

- _W

I_i

With this amplitude ratio the two-dimensional hydrodynamic _damping

coefficient of sway is also given by:

When comparing this expression for N22' with _{the expression found}

before, the. following energy balance relation is found:

M0.P0 - N0.Q0

Coupling of Sway into Roll

In the case of a sway osciliatio.n generally a roll moment is

produced outside of F . The hydrodynamic pressures which act upon the right and left hand side differ from each _other.

The two-dimensional hydrodynamic _{moment acting on the. cylinder in}

the clockwise direction _{can be found by integrating the roll}

component of the hydrodynamic pressure _{on the surface of the} cylinder: dx0 dy0 -i

+ Ycr - I

dO

dO dO J = -2

_J(o).

[

in which:

dx0 +B5 (l+a1).cos9 -3a3cos39

dO 2 l+a1+a3

dy0 _-B (l-a1)sin8 +3a3.sin38

dO 2

With this the two-dimensiona.l hydrodynamic roll moment due to sway oscillations can be written as follows

MR

It [ 'R cos(wt) - XR sin(wt) ]

with:

+

JBOs(9)

a1. (i+a3) .sin[20] -2a3.sin[40]

0 (l+a1+a3)2 dO

+ miQ2mE2m

in which:

(l)m+1

1 2a1(1+a3) 8a3 (l+a1-i-a3)2 L _(2m+l)2-4 + (2m-i-l)2-16

This hydrodynamic roll moment _{can be resolved into components in}

phase and out of phase with the lateral. displacement of the

cylinder in an analog way _{as was done for the hydrodynamic lateral}

force. E2m + ( 8(l-i-a1+a3)2 and: + 8. (l+a1+a3)2 Q2

a3Q4)

XR = + JBOc(9)

a1.(l+a3).sin[2O] -2a3.sin[4O]

0 (l+a1+a3)2

dO

+ m=l2m2m

So: P. B52 . MR 2 11 [

YRQO +

XRf P0]

This hydrodynamic roll moment can also be written as:

MR'

-M42'x -N42'x

in which:

N42

EM42' W2Xal cos(wt +e)

When using also the amplitude ratio of the _{radiated waves and the} forced sway oscillation, found before, the _{two-dimensional}

hydrodynamic mass coupling coefficient of _{sway into, roll is given}

by:

M42

= 2

= two-dimensional hydro.dynamic mass coupling coefficient of sway into roll

= two-dimensional hydrodynamic damping coupling

coefficient of sway into roll

pB53

and the two-dimensional hydrodynamic _{damping coupling coefficient}

of sway into roll by:

- XR.Q0

p02 ₊

Coupling of Sway into Yaw

The coupling of two-dimensional potential hydrodynamic mass and damping of sway into yaw is the moment of the two-dimensional values of sway around the origin of _{the axis system:}

M62' +M22' _{.xb two-dimensional potential hydrodynamic mass}

coupling coefficient of sway into yaw

two-dimensional potential hydrodynamic damping

coupling coefficient of sway into yaw

p.B 2 3

RQ0 + XRPO

p02 + w N62' _+N22'.xb in which:

= longitudinal distance to the origin of the axis system

cos(c,t

+6) + ER0 -

_{XR.Q0] sin(wt +)}

I

[N42'wxa] sin(wt -i-c)

M42'

3.3. Potential Mass and Damping of Heave

The determination of the hydrodynamic coefficients of a heaving

cross section of a ship in deep and still water at zero forward speed, as described here, is based on work published by Ursell

[1949] and Tasai [1959]

Reference is given here to de Jong [1973] for a description of the derivation of the velocity potentials and conjugate stream

functions.

Suppose an infinite long cylinder in the suface of a fluid, of

which a cross section is given in the next figure.

cos(wt+6)

z - plane

Figure 3.3.A. Axes system for heave oscillations,

as used by Tasai.

The cylinder is forced to carry out a simple harmonic vertical

motion about its initial position with a frequency of oscillation w and a small amplitude of displacement

_y:

y = y .cos(t +6)

in which 6 is a phase angle.

The vertical velocity and acceleration of the cylinder _becomes:

=

- ya Sifl(wt +6)

80

The linearized free surface condition in deep _{water is expressed as}

follows:

g 8y

for: _lxi and _y 0

The forced vertical motion of the cylinder causes a surface

disturbance of the fluid. Waves travel away from the cylinder and a stationary state is rapidly attained.

Two kinds of waves will be produced:

- a standing wave system,, denoted here by subscript A

- a regular progressive wave system, denoted here by subscript B.

The amplitude.s 'of the standing _{wave system decrease strongly with}

the distanc,e to the cylinder.

The regular wave system dissipates _{energy. it is known that, at a}

distance of a few wave lengths from the cylinder, the _{waves on each}

side can be described by a single regular wave-train.

The wave amplitude at infinity _{a is proportional to the amplitude.}

of oscillation of the cylinder Ya' provided that:

- the amplitude of. oscillation i,s sufficiently small _compared

with the radius of the cylinder

- the wave length is not much smaller than the diameter of the

cylinder

The cylinder is supposed to be infinitely long, _{so the motion of}

the water and also the velocity potential and the _{conjungate stream}

function will be two-dimensional.

The velocity potential of the fluid _{must satisfy to the equation of}

Laplace:

()

= __-;c-+--- 0

OxL

Oy

Because the motion of the fluid i.s symmetrical about the y-axi.s,

this velocity potential has the following relation: (+x,y)

from which follows:

in which n is the outward normal of the cylinder surface.

Using the stream function W,, _{this boundary condition on the surface} of the cylinder reduces to:

-8W.0(0) _8Y0

80 8cr

Integration results into the followin.g requirement for the stream

function on the surface of the cylinder:

'P0(0) _{= -y}

Because of the symmetry of the fluid aboutthe y-axis:

C*(t) _{= 0}

For 0 - ½,r,, so at the surface of the fluid, this requirement for the stream function reduces to:

W0(½ir) =

2

B (l+a1)sin.O -a3sin39

+ C*(t)

2 l+a1+a3

In consequence of the Lewis transformation, this free surface condition can be writtenas:

ea _-a1e

_-3a3e3

-

0, for a and 0 ± -B l+a1+a3 ₈₀ in which: 2 -g 2

= non-dimensional frequency squared

From the definition of the velocity potential follows the boundary condition on the surface of the cylinder for a 0:

' (l+a1).cos9 -3a3cos39

2 l-i-a1+a3

0y0

y

with:

'A2m('0)

a1 e (2m+l)a 2m+1

A2m

For the standing wave system a velocity potential and a stream function satisfying to the Laplace equation, the free surface

condition and the symmetrical motion of the fluid has to be found.

The following set of velocity potentials, as given by Tasai [1959], fulfil these requirements:

g

w

+ E

_{.cos(wt) + E Q2q3(a,O) .sin(wt)]}

m=1

mi

with: = + e2mx.cos[2m9] + 1+a1+a3 = + e2ma.sjn[2m9] + sin[(2m+l)9] - 2m+3 + 2m+1

e(2ml)a

2m- 1 3a3 e (2m+3)a cos[(2m+1)O] - cos[(2m-i-3)9] ] 2m+3

The set of conjugate stream functions is expressed as.:

ga

irw

a,9) cos(wt) + E Q2mbA2m(a,O) sin(ct)]

m= 1

e(2ml)a

2m- 1

3a3.e(2m+3)a

In these expressions the magnitude,s of the _{2m and the Q2m series} follow from the boundary conditions as will be explained further

on.

These sets of functions tend to _{zero as a tends to infinity.}

cos [ (2m-1)O]

sinE (2m-1)O}

Another requirement is a diverging wave-train for a goes to infinity. It is therefore necessary to add a stream function, satisfying the free surface condition and the symmetry about the

y-axis, representing such a train of waves at infinity. For this, a function describing _{a source at the origin 0 is} choosen.

The velocity potential of the progressive _{wave system is given by}

Tasai [1959] as:

irw

[ Bc(X.Y) cos(wt)

+ rp5(x,y)

.sin(wt) ]

with:

PBc = +

ir.eY

cos(kx)

in which:

k - = wave number for deep water

g

Changing the parameters:

g

=

_{[ PBc(')}

cos(wt) + 9?Bs(a,O) sin(wt) ]

The conjugate stream function is given by:

[ Bc(',Y) cos(wt) +

5(x,y)

sin(wt) ] with:

Bc = + sin(klxI)

Bs =

.-ky

cos(kx) +

Changing the parameters:

= g ''a

[ B a,O) co:s(wt) + ,bB5(a,O) sin(cot) ]

sin(kIxI) +

r k.sin(vy) -i#.cos(vy)I ) e

J

0

k.cos(Ly) +vsin(&iy)

e

When calculating the integrals in the expressions for and _Bs

in a numerical way, the convergence is very slowly.

Power series expansions, as given by Porter [1960], can be used instead of these integrals. This has been showed before for the sway case. The summation in these expansions converge much faster

than the numeric integration procedure.

The total velocity potential and stream function to describe the

waves generated by a heaving cilinder are.:

= "A +

So the velocity potential is expressed by:

[

rw

g

+

_{[ p(a,9)}

The conjugate stream function is:

lBc(a,O) +EP2m.7,bA2m(a,9) ] a,9) + m=.L

+E

m= 1 + E Q m= 1 a, 0) A2rn(a,O)

_I

cos (wt) A2m(°'0) ] sin(cc,t) co.s (wt) sin(wt) ].W(cz,e) = + { [ Bs (, 0) 'I

g

+ [

Putting aO, the stream function is equal to the expressio.n, found before, from the boundary condition _{on the surface of the cylinder:}

[ ;:; 1 o(0) + { t'BOc(0) +miP2mbAO2m(0) ]

. [ B0's(0)

miQ2mbAo2mce) ]

sin(,t) with: with: AO2m(6) + sin[2m9] + r I a1 3a3

[ _2m-1 sin[(2m-l)O] + _2m+l .sin[(2m+l)9] - sin[(2m+3)e] 2m+3

In this expression _{BOc(9) BOs(0) are the values of}

and t'Bs(a,O) at the surfac.e of the cylinder, _{so for a 0.}

When putting 0 ½ir, it follows.:

An elimination of:

-i

jy

a

l+a1+a3

+

[ %6BOc() +EP2mlAO2m(½1r) ]

cos(wt)

+ [ 'B0s(½

)-FEQ

m= 1 I irw B8

j.y.-2

11

a1 3a3 2m-1 2m+l 2m+3

AO2m() ]

s.in1

in the expressions for W0(0.) and _{0(½ir) results for any value of 0}

in two sets of equations with the unknown _parameters

2m

and Q2m

irw 1 B (li-a1).sin0 -a3sin30 ' y 2 AO2m(½ )