Strip Theory Algorithms. Revised Report, 1992

Latest Revise: 24-10-1992. Delft University of Technology Ship Hydromechanics Laboratory Mekelweg 2 2628 CD Delft The Netherlands Tel: + 31 15 786882 Fax: + 31 15 781836 (Revised Report 1992) by Ir. J.M.J. Journée Report No. 912 March 1992

Journee, J.M.J.

Strip theory algorithms : (revised report 1992) / by J.M.J. Journee. - Delft : Delft University of

Technology, Ship Hydromechanics Laboratory. Ill. -(MEMT, ISSN 0925-6555 ; 24) (Report / Ship Hydromechanics Laboratory ; no. 912)

Met lit. opg.

ISBN 90-370-0068-1

Summary

This report describes in detail the theoretical backgrounds and the algorithms of a ship motions computer program, named SEAWAY-DELFT, release 4.00.

This P.C. program, based on the ordinary and the modified strip theory, calculates the wave-induced loads and motions with six degrees of freedom of mono-hull ships and barges, sailing in a seaway. When not taking into account interaction effects between the two individual hulls, these calculations can be carried out for twin-hull ships, such as semi-submersibles or catamarans. Linear springs, to calculate the behaviour of anchored or moored ships, are included too.

Contents

page

Introduction 1

Strip Theory Methods 5

2-1. Definitions 6

2-2. Potential Flow Theory 8

2-3. Equations of Motions 20

2-4. Various Strip Theory Approaches 22

Conformal Mapping Methods 29

3-1. Lewis Two-Parameter Conformal Mapping 31

3-2. Extended Lewis Three-Parameter Conformal Mapping 35 3-3. Close-Fit Multi-Parameter Conformal Mapping 37

Two-Dimensional Hydrodynamic Potential Coefficients 43 4-1. Potential Coefficients for Surge 44 4-2. Potential Coefficients for Sway 47 4-3. Potential Coefficients for Heave 63 4-4. Potential Coefficients for Roll 77 4-5. Potential Coefficients for Pitch 93

4-6. Potential Coefficients for Yaw 94

4-7. Zero and Infinite Frequency Potential Coefficients ₉₅

Viscous Flow Effects ₉₇

5-1. Viscous Damping for Surge ₉₇

5-2. Viscous Damping for Roll ₉₈

Hydromechanic Forces and Moments 113

6-1. Hydromechanic Forces for Surge ₁₁₄

6-2. Hydromechanic Forces for Sway ₁₁₇

6-3. Hydromechanic Forces for Heave 120 6-4. Hydromechanic Moments for Roll 123 6-5. Hydromechanic Moments for Pitch ₁₂₆

6-6. Hydromechanic Moments for Yaw ₁₂₉

Exciting Wave Forces and Moments 135 for Surge 137 for Sway 140 for Heave 143 for Roll 146 for Pitch 150 for Yaw 151

Transfer Functions of Ship Motions 153

Anti-Rolling Devices 157

10-1. Bilge Keels 158

10-2. Passive Free-Surface Tanks 159

10-3. Active Fin Stabilisers 162

10-4. Active Rudder Stabilisers 168

Added Resistances Due to Waves ₁₇₁

11-1. Radiated Energy Method 172

11-2. Integrated Pressure Method 173

Shear Forces and Bending and Torsional Moments 177

12-1. Still Water Loads 183

12-2. Lateral Dynamic Loads 184

12-3. Vertical Dynamic Loads 186

12-4. Torsional Dynamic Loads 189

Statistics in Irregular Waves 191

13-1. Normalised Wave Energy Spectra ₁₉₂ 13-2. Response Spectra and Statistics ₁₉₇

13-3. Shipping Green Water ₂₀₀

13-4. Bow Slamming ₂₀₂

Twin-Hull Ships ₂₀₇

References ₂₁₇

8-1. Exciting Wave Forces 8-2. Exciting Wave Forces 8-3. Exciting Wave Forces 8-4. Exciting Wave Moments 8-5. Exciting Wave Moments 8-6. Exciting Wave Moments

1. Introduction

This report aims at being a guide and an aid for those who want to study the theoretical backgrounds and the algorithms of a ship motions computer program based on the strip theory.

The present report describes in detail the theoretical backgrounds and the algorithms of a six degrees of freedom ship motions

per-sonal computer program, named SEAWAY-DELFT [1992].

This program, based on the ordinary and the modified strip theory, calculates the wave-induced loads and motions with six degrees of freedom of mono-hull ships and barges, sailing in a seaway. When not taking into account interaction effects between the two indi-vidual hulls, these calculations can be carried out for twin-hull ships, such as semi-submersibles and catamarans, too.

In the past a preliminary description of all algorithms, used in strip theory based ship motions calculations, has been given by the author [1988]. The present report contains revised parts of that report with respect to the determination of the conformal mapping coefficients of cross sections and the two-dimensional hydrodynamic potential coefficients. Extensions are given here with the deter-mination of the hydrodynamic coefficients with multi-parameter

described cross sections. Last but not least, modified descriptions of the determination of the wave loads, mechanic loads and slamming phenomena and the algoritms for twin-hull ships have been given. Chapter 1, this introduction, gives a short survey of the contents of the succeeding chapters in this report.

Chapter 2 gives a general description of the various strip theory approaches.

Based on Lecture Notes of Gerritsma [1987], a general description of the potential flow theory is given. The derivations of the hydromechanic forces and moments, the wave potential and the wave

and diffraction forces and moments have been described.

The equations of motion are given with solid mass and inertia terms and hydromechanic forces and moments in the left hand side and the 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 for all motions a forward speed effect caused by _the potential mass, as it has been defined by Korvin-Kroukovsky and Jacobs [1957] for the heave and pitch motions. This _{approach is} called the "Ordinary Strip Theory Method". Also _{an inclusion of the} forward speed effect caused by the potential damping _{too, as for} instance given by Tasai [1969], is given. This approach _{is called} the "Modified Strip Theory Method".

The inclusion of so-called "End-Terms" has been _described. Chapter 3 describes several conformal mapping methods.

For the determination of the two-dimensional _hydrodynamic poten-tial coefficients for sway, heave and roll _{motions of ship-like} cross sections, these cross sections are conformally mapped _{to the} unit circle. 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 man-ner hydrodynamic problems can be solved directly with the coeffi-cients of the mapping function.

The close-fit multi-parameter conformal mapping method is given. A very simple and straight on iterative least squares method, used to determine these conformal mapping coefficients, has been described. Two special cases of multi-parameter conformal mapping have been described too: the well known classic Lewis transformation [1929] with two parameters and an Extended-Lewis transformation with three parameters, as given by Athanassoulis and Loukakis [1985].

Chapter 4 describes the determination of the two-dimensional poten-tial mass and damping coefficients for the six modes of motions in an infinite deep fluid.

The principle of the calculation of these potential coefficients is based on work of Ursell [1949] for circular cylinders.

Starting from the velocity potentials and the conjugate stream

functions of the fluid with an infinite depth as have been given by Tasai [1959],[1960],[1961] and de Jong [1973] and using the multi-parameter conformal mapping technique, the calculation routines of the two-dimensional hydrodynamic potential coefficients of ship-like cross sections are given for the sway, heave and roll motions. For the determination of the velocity potentials and the conjugate stream functions itself, reference is given to de Jong [1973]. Because of using the strip theory approach here, the pitch and yaw coefficients follow from the moment about the ship's centre of gravity of the heave and sway coefficients, respectively.

Approximations have been given for the surge coefficients.

Chapter 5 gives some corrections on the hydrodynamic 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 model experiments and an empirical method published by Ikeda, Himeno and Tanaka [1978], to determine a

viscous correction of the roll damping coefficients are described

in detail.

Chapter 6 describes the determination of the hydromechanic forces and moments in the left hand side of the six equations of motion of a sailing ship in deep water for both the ordinary and the modified strip theory method.

Chapter 7 describes the inclusion of linear spring _{terms in case of} anchored or moored ships.

Chapter 8 describes the wave exciting forces and moments in the right hand side of the six equations of motion of _{a sailing ship} in water with an arbitrarily depth, using the relative motion con-cept and both the ordinary and the modified strip theory method. However the hydrodynamic coefficients used here _{are valid for deep} water only.

Chapter 9 describes the solutions of the equations of motion.

The determination of the frequency characteristics _{of the absolute} displacements, rotations, velocities and accelerations _{and the} ver-tical relative displacements. The use of a wave potential valid for any arbitrarily waterdepth makes the calculation method, described here, suitable for ships sailing with keel _{clearances down to about} 50 percent of the ship's draught.

Chapter 10 describes some anti-rolling devices.

A description is given of an inclusion of bilge keels, passive free-surface tanks as defined by Van den Bosch and Vughts [1966], active fin stabilisers and active rudder stabilisers.

Chapter 11 describes two methods to determine the transfer func-tions of the added resistances due to waves.

The first method is a radiated wave energy method, as published by Gerritsma and Beukelman [1972]. The second method is an integrated pressure method, as published by Boese [1970].

Chapter 12 describes the determination of the frequency charac-teristics of the lateral and vertical shear forces and bending moments and the torsional moments in a way as presented by Fukuda

[1962] for the vertical mode. Still water phenomena are described too.

Chapter 13 describes the statistics in irregular waves, by using the superposition principle.

Three examples of normalised wave spectra are given: the somewhat wide wave spectrum of Neumann, an average wave spectrum of

Bret-schneider and the more narrow Mean JONSWAP wave spectrum.

A description is given of the calculation procedure of the energy spectra and the statistics of the ship motions for six degrees of freedom, the added resistances, the vertical relative motions and the mechanic loads on the ship in waves coming from any direction. For the calculation of the probability of exceeding a threshold value by the motions, the Rayleigh probability density function has been used.

The static and dynamic swell up of the waves, of importance when calculating the probability of shipping green water, are defined according to Tasaki [1963].

Bow slamming phenomena are defined by both the relative bow velo-city criterium of Ochi [1964] and the peak bottom impact pressure criterium of Conolly [1974].

Chapter 14 describes the additions to the algorithms in case of twin-hull ships, such as semi-submersibles and catamarans. For interaction effects between the two individual hulls will not be accounted here.

Chapter 15 gives all the references to the literature on which the computer program SEAWAY-DELFT is based.

2. Strip Theory Methods

The ship is considered to be a rigid body. Only the external loads on the underwater part of the ship are considered and the effect of the above water environment is fully neglected. The fluid is consi-dered to be ideal: homogeneous, incompressible, free of surface tension, irrotational and without viscosity. It is assumed that the problem of the motions of a floating body in waves is linear or can

be linearised.

The incorporation of seakeeping theories in ship design is discus-sed by Faltinsen and Svensen [1990]. An overview of seakeeping theories for ships is presented and it is concluded that, never-theless some limitations, strip theories are the most successful and practical theories for the calculation of the wave induced motions of the ship, at least in an early design stage of a ship.

The strip theory is a slender body theory, so one should expect less accurate predictions for ships with low length to breadth ratios. However, experiments showed that the strip theory appears to be remarkably effective for predicting the motions of ships with length to breadth ratios down to about 3.0, or even lower.

The strip theory is based on the potential flow theory. This holds that viscous effects are neglected, which can deliver serious

problems when predicting roll motions at resonance frequencies. In practice, viscous roll damping effects can be accounted for by

empirical formulas.

Because of the way that the forced motion problems _{are solved} generally in the strip theory, substantial disagreements can be

found between the calculated results and the experimental data of the wave loads at low frequencies of encounter in following waves.

In practice, these "near zero frequency of encounter problems" _can be solved by forcing the wave loads to go to zero artificially.

For high-speed vessels and for large ship motions, _{as appear in} extreme seastates, the strip theory can deliver less accurate results. Then the so-called "end-terms" can be important.

The strip theory accounts for the interaction with _{the forward} speed in a very simple way. The effect of the steady _{wave system} around the ship is neglected and the free surface _{conditions are} simplified, so that the unsteady waves generated by the ship are propagating in directions perpendicular to the _{centre plane of the} ship. In reality the wave systems around the ship _{are far more} complex. For high-speed vessels, unsteady divergent _{wave systems} become important. This effect is neglected in the _{strip theory.} The strip theory is based on linearity. This _{means that the ship} motions are supposed to be small, relative _{to the cross sectional} dimensions of the ship. Only hydrodynamic effects of _{the hull below} the still water level are accounted for. So when _{parts of the ship} go out of or into the water or when green water is shipped,

inaccuracies can be expected. Also, the strip theory _{does not} distinguish between alternative above _{water hullforms.}

Because of the added resistance of _{a ship due to the waves is} proportional to the relative motions squared, _{its inaccuracy will} be gained strongly by inaccuracies in the _{predicted motions.}

Nevertheless, seakeeping prediction methods _{based upon the strip} theory provide a sufficiently good basis _{for optimisation studies} at an early design stage of the ship. At _{a more detailed design} stage, it can be considered to carry out additional _model experi-ments to investigate for instance added resistance _{or extreme event} phenomena, such as shipping green water and slamming.

2-1. Definitions

Figure 2-A. Coordinate Systems

A right-handed coordinate system S-(x0,y0,z0) is fixed in space. The (x0,y0)-plane lies in the still water surface, xo is directed as the wave propagation and zo is directed upwards.

Another right-handed coordinate system 0-(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 in the upward direction. The ship is supposed to carry out oscillations around this moving 0-(x,y,z) coordinate

system. The origin 0 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-(xb,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 port side direction and zb upwards.

In still water the (xb,yb)-plane is parallel to the still water

surface.

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

j-a.cos(wt

-kxo)

A z,

in which:

a = wave amplitude k = 2w/A = wave number

A = wave length

w = circular wave frequency

t _{= time}

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

c = w/k

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

xo

Vt-cosp +x-cosp +y.sinp

From the relation between the frequency of encounter _{we and the} wave frequency w:

we = w -kV.cosp

follows:

= -a-cos(wet -kx-cosp -ky.sinp)

The resulting six possible shipmotions in the _{0-(x,y,z) system are} defined as three translations of the ship's _{centre of gravity along} the x-, y- and z-axes and three rotations about _them.

These harmonic displacements are indicated by:

surge: x = xa .cos(wet -FE).)

sway: y = ya -cos(wet

-i-ey)

heave: z = za .cos(wet

_-1-ez)

roll: cp cpa .cos(wet _-FE(14)

pitch: O = Oa -cos(wet

-1-e0)

yaw:

0 =

_{0a .cos(wet -1-Elp)}

The phase lags of these motions _{are related to the harmonic wave} elevation at the origin of the 0-(x,y,x) _{system, the average} posi-tion of the ship's centre of gravity:

wave: = a .cos(wet)

The harmonic velocities and accelerations in the 0-(x,y,z) system are found by taking the derivatives of the _{displacements, for}

instance:

surge displacement: x = _{xa .cos(wet +,q)}

surge velocity: _{x = -we .xa .sin(wet -i-c,.)} surge acceleration: _{x = -we2.xa .cos(wet -i-c3.)}

2-2. Potential Flow Theory

For the general description of the potential flow theory, _{use has} been made of Lecture Notes of Gerritsma [1987].

Suppose the rigid body is floating in _{an ideal fluid with harmonic} waves. The time-averaged speed of the body is zero in all

direc-tions. To get simple notations it is assumed here that the 0-(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:

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

(1'r -1-- 4)w + I'd

in which:

.1)r the radiation potential for the oscillatory

motion of the body in still water

(1), = the incident wave potential

(Dd _{the diffraction potential of the waves about the}

restrained body

Boundary Conditions of the Velocity Potentials

From the definition of a velocity potential (I) follows the velocity

of the waterparticles in the three translational _directions:

v =vz =

Y _ay

az

As the fluid is homogeneous and incompressible, _{the continuity}

condition:

avx av avz

+ --I + 0

ax ay az

results into the equation of Laplace:

a2(1. a2(1) a24,

A.1. + _{+ = 0}

ax2 ay2 az2

The pressure in a point P(x,y,z) is _{given by the linearised} Bernouilli equation: at Or: al) + ,g. = at -p.g.z

At the free surface of the fluid, _{so for z = (x,y,z,t), the} pres-sure p is constant. vx = ax P

-P

The boundary condition on the bottom follows _{from the definition of} the velocity potential and is given by:

a(1.

az

= 0 for z -h

Because of the linearisation, the vertical velocity of a water-particle in the free surface becomes:

dz a(1)

,a-=

,

dt az at

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

a21, aci)

+ g 0 _{for z = 0}

at2 az

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:

a.t.

an vn(x,y,z,t)

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

6

an vn(x,y,z,t) _{j=1 J}

with the generalised direction-cosines _{on the surface of the body:}

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:

lim ] 0

R.

L fl cos(n,x) f2 cos(n,y) f3 = cos(n,z) f4 y.cos(n,z) -z.cos(n,y) f5 = z-cos(n,x) -x.cos(n,z) f6 = x-cos(n,y) -y.cos(n,x)

Forces and Moments

The force T and moment R follow from an integration of the pressure on the submerged surface of the body:

.i.

-

4 f

(pE)

.dS

S

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

P = -P at -p-g-z Or: .1-

acpr 8 a(Dd

1 = L at _{at at} -I

The hydromechanic force T and moment R can be splitted up into four

parts: F - Fr + Fw + Fd + Fs _ M - Mr + Mw 4- Md 4- Ms -p-g-z 'f' = P .f

f[

aq, atr + a(1. at +

8d

at + g.z

LE

-dS S _ M = P

f

at at at + g-z E) .dS

sf

[ a.1)r atpw

"d

]-(F x + -

-in which Fs and Ms are the hydrostatic parts.

=

-H.

P *(r-x nT .dS

Hydromechanic Forces and Moments

The radiation potential (Dr belongs to the oscillation of the body in still water.

It can be written:

6 6

= E (1)i(x,y,z,t) E yoi(x,y,z) .vi(t)

j=1 j=1

in which:

v(t) = oscillatory velocity in direction j

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

a

r 6

6

a(p-"r

_E .

an an L I j=1 an

n

So the generalised direction-cosines are given by:

f j

an

With this the radiation terms in the hydromechanic force and moment are: Fr p

.1 f

[ar

I E dS at

_pif[a6

.Z (p..v 1.E .dS

at j=1 and: 17Ir P

-fsf

[ atr 1.(y x 17) .dS

ff[

a6

E çov ]-(Y x

) dS at j=1

The components of this force and moment are defined by: (Xr1,Xr2,Xr3)

17fr (Xr4,Xr5,Xr6)

Or: a 6 Xrk = P .f

f

[

-

E p..v. ].fk .dS_{J J} at j=1 Xrk _{= P} -fsf

[

a 6 E pi-vi ]. at j=1 an ds

In this expression pi and pk are not time-depending, so the expression reduces to:

6 Xrk

_j1

2 Xrkj = with: dv i

4k

Xrkj = _---1-P

-f f

40

-dS

dt 3 an S

This last force in the direction k is caused by a forced harmonic oscillation of the body in the direction j.

Suppose a motion:

sai . e-iwt

So the velocity and acceleration of the oscillation are:

s -J Si = Vi dvi si -dt Xrkj for k = 1, ...6 -iw-sai .e-iwt -w2-saj .e-iwt for k 1, ...6 for k = 1, ...6

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

-Mkj'sj -Nkj'sj

= _{sai-w2-Mki +i-sai.w-Nki I .e-iwt}

=

-sai.w2P

f fpi.

_{dS -e-iwt}

J an

So in case of an oscillation of the body in the direction j with a

velocity potential (pi, the hydrodynamic mass (coupling) coefficient

is defined by:

a(Pk

Mkj = -Re [ p.f

f

cp, dS ]

' an

S

and the hydrodynamic damping (coupling) coefficient by:

a9k

Nkj = -im [

P-w.ff

ds ]

S

'''.i. an

In case of an oscillation of the body in the direction k with a

velocity potential (pk, _{the hydrodynamic mass (coupling) coefficient}

is defined by:

acpi

mjk = -Re [

p-I f

q)k. ---j- dS ]

an

S

and the hydrodynamic damping (coupling) coefficient by:

a(pi

NJ-k = -Im [ p.w.f

f

(pk- --L- dS ]

an S

Suppose two velocity potentials (pj _{and (pk and use Green's second} theorema for these potentials:

(

Li-f Li-f Li-f

[ (pj-A99k - a9k _]-dS* Pk (pk-A(pi ]-dV* =

f f

[ (pj an an v _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} water-surface and the wetted water-surface of the _{floating body.}

The Laplace operator is given by:

a2 a2 a2

A + +

ax2 ay2 _az2

So according to the equation of Laplace:

A(pi

This results into:

J

"k

dS* = çok -J dS* aSoi an an

s*

Figure 2-2-A. Boundary conditions.

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

lim

= 0

R=.

The boundary condition on the seabottom is:

a rp

for z = -h an

The boundary condition at the free surface:

82,1, at2 az = O for z = 0 becomes for = _w2.cp g _{for z =} a z

s*

o R

w2

or with k = for deep water:

g

4

k-9

= for z 0

az

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

k-cpk =

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

J I

wi an dS =

i iic

awi

ço. --.1- dS an

in which S is the wetted surface of the body only.

awk a(Pk

az an

and

k=

...i

__I

4-

=

__I

4-az an

This means also that:

Mjk Mkj

Nil, = Nki

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

below. are given _ M11 O _M13 O _M15 -O O _M22 0 _M24 0 _M26

Hydrodynamic mass matrix: M31 0 M33 0 M35 0

O _M42 0 _M44 0 M46 M51 0 _M53 0 _M55 0 0 _M62 0 _M64 0 _M66 -Nll o N13 o N15 0 O N22 O _N24 O _N26

Hydrodynamic damping matrix: N31 0 N33 0 N35 0

O _N42 O _N44 O

N46

N51 0 N53 0 N55 0

O _N62 O _N64 0

Wave Potential 4,,,,,

The velocity potential .1) of the harmonic waves has to fulfil three

boundary conditions:

- the equation of Laplace:

324) a24, a2(I)

A24,1,7w

w

+ +

az2w o

ax2 ay2

With this the corresponding wave potential, depending on the

water-depth h, is given by the relation:

g cosh k(h+z)

cl)w ',..sin(wt -kx.cosp -ky.sinp)

cosh kh

The dispersion relation follows from the kinematic boundary condition at the free surface and is defined by:

w2

g-k tanh kh

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

x .c.-- xb

Y ""--- Yb

z--,-- zb

In case of a forward ship speed, the wave frequency w has to be replaced by the frequency of encounter of the waves we.

This leads to the following expressions for the _{wave surface and} the first order wave potential in the G-(xb,yb,zb) system:

',e-cos()et -kxb-cosa -kyb.sinp) and

g _{cosh k(h+zb)}

1), - ',..sin(wet -kxb-cosp -kyb.sinp)

cosh kh

- the boundary condition on the bottom:

al),,

0 _{for z = -h}

az

- the dynamic boundary condition at the free surface,

which follows from the linearized equation of Bernouilli:

a(Dw

+ g- = 0 for z = 0

Wave and Diffraction Forces and Moments

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

"w

al'd 1 -wF + T'd P -1

f

[ + j.n .dS at at s and: at w 34'd iiw + _{= [} +

1.(T.

x TO .dS at at s

For the determination of these wave forces and moments it is supposed that the floating body is restrained with _{zero forward}

speed.

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

+

ad

an an an

Define now:

(x,y,z,t) = çow(x,y,z) -e-iwt

(1)d(x,y,z,t) = (pd(x,y,z) .e-iwt

This results into:

a(pw

ad

= _

an an

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: Xwk -i.p-e-iwt .f

f

(,(pw + wd) _{.fk .dS} S _{for k = 1, ...6} = ₀ Or: Xwk =

-i.p-e-iwt .f fdS

(1,7 +

(i9d) . awk an s _{for k = 1, ...6}

The potential of the incident _{waves q'w is known and the diffraction} potential 9,1 has to be determined.

Green's second theorema delivers:

J f

wd 81,1, an dS =

f 1

4c1

_an dS s s With: awd acpw = an - an it is found:

J f

Wd atl. an dS

-fsf

Wk a(pw an dS S

which results into the so-called Haskind _relations:

Xwk = _i.p.e-iwt awk aww

.J

f

[+

] Wk -dS an an s for k = 1, . 6

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

eliminated, because the expression for Xwk _{is depending on the wave} potential cpw and the radiation potential _{c,ok 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 0-(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.

ac,ow

an

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

g

.

(1)_w = - -ekz ',.sin(wt -kx.cosp -ky-sinp)

w

or:

(Dw _ -1.a...g ekz .ei-k-(x-cosA +y-sinA)

w

and

_ -1-.a.g ekz .ei.k.(x-cosA +y-sinA)

Sow

w

The velocity of the waterparticles in the direction _{of the outward} normal on the surface of the body is:

acpwaz

ax

ay

(pw -k I +i-[ cosp + sinA i

an an an an

This can be written as:

= cpw.k -[ _{f3 +i.(fi.cosA + f2.sinA) ]}

Then the wave loads are given by:

Xwk -i-p-e-iwt

-f f

cpw

-fk dS

s

+i-p.e-iwt.k

-j f

cpw-cpk.[ _{f3 +i-(fi-cosA +f2-sinA) i -dS}

S

for k = 1, ...6

The first term in this expression for _{the wave forces 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:

Xsk

Vg

j f

z -fk dS

S _{for k = 1,}

2-3. Equations of Motions

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

-i [ m ]

dt

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 gravity

M = resulting external moment acting about the _{centre of gravity}

_

H = instantaneous angular momentum about the _{centre of gravity}

t time

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 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 axes, it can be written for the motions of}

a ship: and R = d [ 11 ] dt d Surge: [

p-V-X

] _{= p-V-x} Xhl +Xwl dt d

Sway: [

p-V-Y

] _{= p-V-y}

Xh2 1-Xw2 dt d Heave: [ _{] = p-V-z = Xh3 +Xw3} dt d Roll: _{[ I ''',9} xx -Ixz ',-b ] = Ixx'',9 -Ixz

.0

Xh4 +Xw4 dt d Pitch: [ 1YY-è ] = 1YY W = Xh5 +Xw5 dt d Yaw:

[ iz... , - Izx-c.p ] =

Izz.0 -Izx.P

_{= Xh6 1-Xw6}

Surge: Heave: Pitch: Sway: Roll: Yaw:

This results into the following two sets of three coupled equations of motions: Alta Xwi Xh3 lEw3 Xh5 Xw5 .. p-V-y _Alh2 ..

Ixx-V -Ixz',Ti Alh4

a- ,

Izz-r

-Iz3CP -'-h6

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

The solid mass matrix is given below.

-p-V 0 0 0 0 0

0 p.V 0 0 0 0

0 0 p.V 0 0 0

Solid mass matrix:

0 0 0 Ixx 0 -1_xz 0 0 0 0 I YY 0 0 0 0 -I_zx 0 I_zz in which: P = density of water

V volume of displacement of the ship

Iii = solid mass moment of inertia of the ship Xhl, Xh2, Xh3 hydromechanic forces in the x-,

y-z-direction respectively

and

Xh4, Xh5, Xh6 = hydromechanic moments about the x-, z-axis respectively

y- and

Xwl, Xw2, Xw3 = exciting wave forces in the x-, y-z-direction respectively

and

Xw4, Xw5, Xw6 = exciting wave moments about the x-, z-axis respectively

2-4. Various Strip Theory Approaches

The so-called strip theory solves the three-dimensional problem of the hydromechanic 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 zero-speed case. So each cross section of the ship is con-sidered to be part of an infinitely long cylinder.

Xbi = fXbi'.dxb

X_wj

-TX

_wj- .dxb

in which:

Xbi sectional hydromechanic force or moment Xwi sectional exciting wave force or moment

Two assumptions are made for these loads:

- the hydromechanic 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.

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:

D F * - * s hj * = -11j* I and Dt L Dt

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:

In here: O ]

-V-Dt aat ax and Dt Wj w j w j

According to the "Ordinary Strip Theory Method", as published for

"Modified Strip Theory Method", as published for

In the definitions of the two-dimensional hydromechanic loads, the non-diffraction part RSJ1 is the two-dimensional quasi-static restoring spring term.

In the definitions of the two-dimensional wave loads, the

non-diffractionpart FKJ-is the two-dimensional Froude-Krilov force or

moment which is calculated by an integration of the directional pressure gradient or the directional component of the orbital

acceleration in the undisturbed wave over the cross sectional area

of the hull. Equivalent directional components of the orbital

acceleration and velocity, derived from these Froude-Krilov forces or moments, are used to calculate the diffraction parts of the total wave forces and moments.

According to the

instance by Tasai [1969], these loads become:

D _-i Xhj1 [ (MJJ--, --1---- Njii)-i,'hi' ] +RS-1 D , -i Dt we Dt J we X - = [ ( -- +-- --'). -* ] +F -wj _MJJ Njj wj KJ

instance by Korvin-Kroukovski and Jacobs [1957], the uncoupled two-dimensional potential hydromechanic loads and wave loads in the direction

"j"

are defined by:

D , _ XI,J1 [ M , * ] +Nji --1.1j +RSj I * Dt D

[

M--1--wi* ] +N

--H* -* +

FK-1 Dt JJ JJ wJ J

x

_Wj

-So-called "End-Terms"

From the previous, it is obvious that in the equations of motion longitudial derivatives of the two-dimensional potential mass Mij and damping Niji will appear. From a mathematical point of view, these derivatives have to be determined numerically in such a man-ner that the following relation is fulfilled:

xb(L)+E

f

df(xb) dxb dxb xb(0)-E with: E « xb(L)-xb(0) xb(L)-E

_f

df(xb) +f(0) + dxb -f(L) 0 dxb xb(0)+E and f Mij' or

Figure 2-4-A. Integration of Longitudinal Derivatives

In literature, the terms f(0) and f(L) are called "end-terms",

de-fined by: xb(0)+E df(xb) f(0) + dxb dxb xb(0)-E and xb(L)+E

f

df(xb) f(L) - dxb dxb xb(L)-E

Especially for high speed vessels, these "end-terms" can play an important role. A lot of attention is payed in literature to an inclusion or an exclusion of these terms.

xb(L)-FE _xb(L)-e f df(xb) J dxb xb(°)-E xb(0)-1-e xb(L)-1-6 xb(L)-E

J

df(xb) x.13

2Axb

`-" ₌ . 2.f f(xb)-xb.dxb -i _dxb xb(C)-E xb(0)+E

In case of an exclusion of "end-terms" in the hydrodynamic loads, the integration of the derivatives has to be carried out in the

region:

xb(0)-e xb xb(L)+6

However, all numerical integrations will be carried out in the

region:

xb(0)+e xb xb(L)-6

Then the integral, the longitudinal static moment and the longitu-dinal moment of inertia of the derivatives are given by:

xb-dxb -f f(xb)-dxb xb(L)-FE J r df(xb) dxb 0 dxb xb(3)-E

xb(L)-E

J

df(xb) dxb i dxb xb(0)+E

In case of an inclusion of "end-terms" in the hydrodynamic loads, the integration of the derivatives has to be carried out in the

region:

xb(0)+E 5_ xb xb(L)-E

All numerical integrations will be carried out in this region too.

Then the integral, the longitudinal static moment and the longitu-dinal moment of inertia of the derivatives are given by:

= 0 -f(0) +f(L) xb(L)-6 xb(L)+E f df(xb) xb.dxb -f f(xb)-dxb -f(0)-xb(0) +f(L).xb(L) J dxb xb(0)+E _xb(°)-E xb(L)-E = _

f

f().dxb -f(0)-xb(0) +f(L).xb(L) xb(L)-E xb(L)+E r df(xb) xb'-dxb = -2.f f(xb)-xb-dxb -f(0).xb(0)2 J dxb xb(0)+E _xb(°)-E +f(L)-xb(L)2 xb(L)-E = -2.f f(xb)-xb-dxb -f(0)-xb(0)2 xb(0)+E +f(L)-xb(L)2

Mark that these expressions are valid for the integrations of the potential coefficients over the full ship length only. They _{can not} be used for calculating local hydromechanic loads.

Also for the wave loads, these expressions can not be used, because there these derivatives are multiplied with xb-depending orbital motions. In case of an exclusion of "end-terms" in the left hand side of the equations of motion, extra "end-terms" at xb(0) and xb(L) have to be introduced in the wave loads in the right hand

side of the equations of motion.

So, the expressions exclusion and inclusion of the "end-terms are

Hydrodynamic Potential Coefficients

The two-dimensional potential hydrodynamic coefficients 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], and Tasai [1959, 1961]. 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 "Close-Fit Conformal Mapping

Method".

Another suitable method to determine the two-dimensional potential hydrodynamic coefficients for sway, heave and roll is the "Frank Close-Fit Method" [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 followed by the author [1990]. An equivalent longitudinal cross section has been defined. For each frequency, the two-dimensional potential hydrodynamic sway coefficient of this equivalent cross section is

translated to two-dimensional potential hydrodynamic surge coeffi-cients by an empiric method, which is based on theoretical results of three-dimensional calculations.

By one of these methods, for each cross section the two-dimensional hydrodynamic coefficients have to be obtained. The determination of these two-dimensional potential coefficients by a conformal mapping method will be described further on.

3. Conformal Mapping Methods

For the determination of the two-dimensional added mass and damping in the sway, heave and roll mode of the motions of ship-like cross sections, these cross sections can be conformally mapped to the unit circle.

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

function.

The general transformation formula is given by:

N r

z = Ms La2n-1.-(2n-1)]

n=0 with:

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

= iea-e-i° = plane of the unit circle

Ms = scale factor ail = +1

a2n-1 = conformal mapping coefficients (n=1,...,N) N = number of parameters Bs r is cons Cant rl c - plane

0 is

cons tant 0 z - plane

r is

cons tant

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

So: N

x= -Ms

.E [(-1)n. a2n-l'e-(2n-1)a. sin[(2n-1)0]] n=0 N F Y - +Ms 'E_n=0 L(-1)11-a2n-l'a-(2n-l)a.cos[(2n-1)0]]

When putting a-0, the contour of the, by conformal mapping, mathe-matically described cross section is expressed as follows:

N F X0 = -M 'E L(-1)11-a2n-1. sin[(2n-1)0]] n=0 N F YO = +Ms 'E L(-1)n'a2n-1' cos[(2n-1)0]] n=0

The breadth on the waterline of the, by conformal mapping mathe-matically described, cross section is defined by:

130 -

2-Ms.Aa b0 Ms -2.Aa do - Ms.Ab N F with: Aa = nE0= La2n-11

The draught of the, by conformal mapping mathematically described, cross section is defined by:

N F

with: Ab

-nE=0 L(-1)11-8-2n-l]

The calculation routines of the Close-Fit multi-parameter _conformal mapping coefficients and the two-dimensional hydrodynamic

poten-tial coefficients, as will be described further on, require the derivatives of the coordinates of the contour of the cross section to 0: dxo N - -Ms .E [(-1)n.(2n-1) _{'a2n-1 .cos[(2n-1)0]]} dO n=0 dyo N -Ms -E _{[(-1)n.(2n-1).a2n_i.sin[(2n-1)0]]} dO n=0

3-1. Lewis Two-Parameter Conformal Mapping

A very simple and in a lot of cases also a more or less realistic transformation of the cross sectional hullform will be obtained with N=2 in the transformation formula, the well known Lewis

transformation [1929].

A extended description of the representation of ship hullforms by Lewis two-parameter conformal mapping is given by von Kerczek and Tuck [1969].

The two-parameter Lewis transformation is defined by:

Z

Ms -[a1Ç +a1

-

+a3.-3

I

in which:

+1

The conformal mapping coefficients al and a3 are called the Lewis coefficients, while Ms is the scale factor.

So:

x Ms .[ ea.sinO +al.e-a.sine -a3-e-3a-sin3O ] y Ms -[ ea.cos0 -a1.e-a.cos0 +a3-e-3a.cos30 ]

Putting a=0, the contour of this so-called Lewis form is expressed

as follows:

xo = Ms .[ (1 +a1).sinO -a3-sin30 ]

yo Ms .[ (1 -a1)-cos0 +a3.cos30 ]

with the scale factor:

Ms B5/2 1 +al +a3 Or Ms Ds 1 -al +a3 and:

Bs = sectional breadth on the waterline Ds - sectional draught

Now the coefficients al and a3 and the scale factor Ms will be

determined in such a manner that the sectional breadth, _{draught and} area of the mathematically described and the actual _{cross section} of the ship are equal.

The half beam to draught ratio is given by: Ho -in which: a3 -B5/2 1 +al +a3 Ds 1 -al +a3 As 1 ..a.12 -3a32 as -B5 D5 4 (1 +a3)2 -a12 c1'a32 -1-c2.a3 4-c3 - ° C2 = 2 (c1 -3) Cl Ho -1 al - (a3 +1) Ho +1

An integration of the Lewis form results into the sectional area coefficient:

Putting al, derived from the expression for Ho, into the expression for us delivers a quadratic equation for a3:

i

4-us ] 4-us ]

Ho

r _, 1 2

cl = [ 3 + + [ 1

-71- L Ho +1 J

C3 = cl -4

The solutions for a3 and al will become: -c1 +3 +(9

The other solution of a3 in the quadratic equation is:

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. -c1 +3 -(9 -2-c1)1/2

a3

Boundaries of Lewis Forms

In some cases the Lewis transformation can give more or less un-acceptable results. In the next the different regions of Ho and as

Ho > 1.0 and [ 37r 2 Ho ] < as < 32 r 1

i

.1 L 2 -HO 3n 1 [ 3 +

i

32

4H0 J

37r <

as

<-32

Bulbous and not-tunneled forms are bounded by:

31f F

Ho ]

3/T 1

i

Ho <

1.0 and < < [

3 +

32 L 4 as 32

4H0 i

Tunneled and not-bulbous forms are bounded by:

< a s [ E 3 Ho 1 4

i

3 1 ] 4H0 -I 37r F

+Q]

32 L 4 -I 1

+ Ho +

-Ho 1

+ Ho +

Ho ] ] are defined.

Re-entrant forms are bounded by:

37r for Ho and: < 1.0 :

as <

32

[ 2

Ho ]

3n 1 for Ho > 1.0 : [ 2

as <

_{32 Ho ]}

Convential forms are bounded by:

Combined bulbous and tunneled

37T

forms

1

are bounded by:

71-< 3 [ ₁₀ for Ho 1.0 32 E + _{4H0 ]} < a s

< -

₃₂ and: 3n 3 + E

Ho i

7C [ 10 for

Ho > 1.0

32 J < aS <

Non-symmetric forms are bounded by:

0 < Ho < .

and

a>

___it

S F

lo

1

+ H 0 +

1 Ho 32 L 37r for

Ho < 1.0

32 and: for

Ho > 1.0

3n 32

These boundaries are showed together in the next figure.

us

2.0 1.5 o for Ho > 1.0

371-7r

1 [ 2 - H < CI s

< -

[

10 + Ho +

-32 32 Ho 37r 32 [ 2 1 < [ 10 + Ho + ] as Ho -I 32 Ho

If a value of us 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 or aft cross sections of a ship, are avoided when the following requirements are

ful-filled:

B5/2 > -rDs and Ds > y.B5/2 with for ionstance ry=0.01.

non-symmetric bulbou. 1111P4.... conventional tunneled re-entrant I 1 o 0.5 10 1.5 20 2.5 3.0 Ho

Figure 3-1-A. Ranges of Half Breadth to Draught Ratio Ho and Area Coefficient as for

Two-Parameter Lewis Forms

Boundaries of Two-Parameter Lewis Hullforms

Not-accepted forms of 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 us are:

for Ho < 1.0

3-2. Extended-Lewis Three-Parameter Conformal Mapping

Somewhat better results will be obtained by taking into account also the first order moments of half the cross section about the

xo-

and yo-axes, as proposed by Reed and Nowacki [1974].

These two additions to the Lewis formulation have been simplified by Athanassoulis and Loukakis [1985], by taking into account the vertical position of the centroid of the cross section.

This has been done by extending the Lewis transformation to N=3 in the general transformation formula.

The three-parameter Extended-Lewis transformation is defined by:

So:

x = Ms .[ ea-sine +al.e-a.sine -a3-e-3a.sin30 +a5.e-5a.sin50 ]

y = Ms -[ ea.cosO -al-e-a-cose +a3-e-3a.cos30 -a5.e-5a.cos50 ]

Putting a=0, the contour of this mathematical form is expressed as

follows:

xo = Ms .[ (1 +a1).sin0 -a3.sin30 +a5.sin5,9 ]

Yo = Ms -[ (1 -a1) cose +a3.cos30 -a5.cos50 ]

with the scale factor:

Bs/2 Ds

MS

1 +al +a3 +a5 Or MS 1 -al +a3 -a5 and:

Bs = sectional breadth on the waterline Ds sectional draught

Now the coefficients al, a3 and a5 and the scale factor Ms can be

determined in such a manner that, except the sectional breadth, draught and area, also the centroids of the mathematically

des-cribed cross section and the actual cross section of the ship _have an equal position.

The half beam to draught ratio is given by:

Z = Ms -[ a_i_... _i_a1.-1 +a3.-3 ±,a5.-5

]

in which:

a-1 = +1

H0

B5/2 1 +al +a3 +a5 Ds 1 -al +a3 -a5

An integration of the mathematical form results into the sectional area coefficient: As n _ 1 -.2.12 _3,32 ..5a52 as _

B -D_{s s} 4 (1 +a3)2 -(al +a5)2

For the relative distance of the centroid to the keelpoint a more complex expression has been obtained by Athanassoulis and Loukakis

[1985]: KB K = = 1 Ds in which: 1] 1 F 1-2k 1-2k 1-2k _1-2k _ _{+ +}

4 L 3-2.(+i+j+k) 1-2.(+i-j+k) 1-2.(+i+j-k) 1-2.(-i+j+k) I

The following requirements should be fulfilled when also bulbous cross sections are allowed:

- re-entrant forms are avoided when both the following

require-ments are fulfilled: 1 -al -3-a3 -5-a5 > 0

and:

1 +al -3-a3 +5-a5 > 0

- existence of a point of selfintersection is avoided when both

the following requirements are fulfilled: 2

,lhS

2 _,..-In

9-a3 ..--.a5 ,.,_-.a3-a5 +20.H0.a5 >0

and:

9.a32 +145.a52 -10.a.3-a5 -20-H0.a5 > 0

3 3 3

i0 j0 k0

F E E E _{LAijk.a2i-l'a2j-1.a2k-11}

===

3 r

Hous Z

L(a2i..1)3] i=0

Taking these restrictions into account, the _{equations above can be} solved in an iterative manner.

3-3. Close-Fit Multi-Parameter Conformal Mapping

A much more accurate transformation of the cross sectional hullform can be obtained with a greater number of parameters N.

A very simple and straight on iterative least squares method of the author to determine the Close-Fit conformal mapping coefficients will be described here shortly.

The scale factor Ms and the conformal mapping coefficients a2n_i,

with a maximum value of n varying from N=2 until N=10, have been determined succesfully from the offsets of various cross sections in such a manner that the mean squares of the deviations of the offsets of the actual cross section from the mathematically des-cribed cross section is minimized.

The general transformation formula is given by:

N F

Ms n=0'E La2n-1'-(2n-1)] in which:

= +1

Then the contour of the cross section is given by:

N

x0= -Ms- E

[(-1)n- _{a2n-1 -sin[(2n-1)9]]} n=0 N

YO =+M5

E [(-1)11'a2n-1. cos[(2n-1)0]] n=O

with the scale factor:

Bs/2 or Ms s N [a2n-l] n=0 Ds N E n=0 [(-1)n.a2n-1]

The procedure starts with initial values for

(M5'a2n-1).

The initial values of Ms, al and _{a3 are be obtained with the Lewis} method as has been described before, while the initial _{values of a5} until a2N_i_ are set to zero.

With these (M5-a2n-1) values, for each offset a 01-value is deter-mined in such a manner that the actual offset (xi,yi) _{lies on the} normal of the contour of the mathematically described _{cross section}

in (x01,y0i).

Now Bi has to be determined.

A function of 0i, which has to be _{zero, can be defined by the} perpendicular distance of the offset (xi,yi) _{to the normal in}

(x0i,y0i) on the contour of the conformally mapped cross section. F(0i) = (x1-x0i)-cos(ai) -(yi-yoi)-sin(ai)

The direction of the outward normal in _{point (x01,y0i) of the} contour of the mathematically described _{cross section and the yo} axis is defined by the angle ai:

and:

-AYO -AYO -dy0/d0

sin(a)

1/2

As0 ( Ax02 AY02 )1/2

With a certain starting value for Oi, an increment A0i and a for this purpose modified Regula-Falsi method, a Oi-value is calcu-lated that satisfies to the requirement:

A more favourable function F(0i) is defined by distance of the offset (xi,yi) to the normal of the contour of the actual cross section through (x01,y01).

Then a- is defined by:

xi+1 - xi-1

cos(ai)

1/2

(xi+1 - xi-1)2 1- (y+1

y1)2

and

+Axo _Ax0 +dx0/d0

cos(a-)

1/2 1/2

As0 ( Ax02 AYO2 )

-(y+1

Yi-1) sin(ai)

1/2

(xi+1 - xi-1)2 + (Yi+1 Y1-1)2

These two approaches have been shown in the figure below.

o

(dx0/d0)2 + (dy0/d0)2 I

(dx0/d0)2 + (dy0/d0)2

-/

Foi)

Figure 3-3-A. Close-Fit Conformal Mapping Definitions

With this Oi-value, the numerical value of the square of the devi-ation of (xi,yi) from (x0i,y0i) is calculated:

ei = (xi-xoi)2 + (yiy.002

After doing this for all IwL+1 offsets, the numerical value of the sum of these squares of deviations is known:

1WL

E = E

ei

i=0

Then new values of (Ms-a2_1) have to be obtained in such a manner

that E reaches its minimum value. This means that the derivatives of this equation to the coefficients (Ms-a2_1) are zero.

So: aE a(ms-a2j_i) 0 for j = 0, ... N IWL = E [ _{x-1.sin[(2j-1)0i] -y1.cos[(2j-1)01]} ] i =0 for j = 0, ... N

The sum of the squares as:

of these deviations can also be expressed

_

IWL N 2

E = E

i=0

[ x1- +

n0

E=

[(-1) IMn-_{--s-a2n-1) .sin[(2n-1)0i]]} ] +

-+

[

Yi

-N E

n=0 [(-1)11-1Ms'a2n-1) .cos[(2n-1)0i]] ] 21

From these requirements follow N+1 equations:

Iwi.. N E

i=0

-sin[(2j-1)0i] [

.E

n=0 [(-1)n- tHs-a2n-11. sin[(2n-1)0i]] +

N,

-cos[(2j-1)0i]

.E

n=0 L(-1)n- (Ms-a2n-1). cos[(2n-1)0i]] ]

N E [n=0 (Ms'a2n-1)] = B5/2 N E rL (-1)n.(ms-a2._0] = Ds n=0

So, the re-transformed cross sections will have an equal breadth and draught as the actual cross sections.

These N+1 equations are rewritten as:

IWL I (-1)11.(Ms-a2n-1) .1 [cos[(2j-2n).0i]] n=0 i=0 = E _{-x1.sin[(2j-1)01] +yi-cos[(21-1)Bi]} i =0 for j = 0, ... N

In case of cross sections with a very low breadth to draught ratio, for instance bulbous cross sections, the solution of these equa-tions can result into a negative breadth on the waterline. To avoid this, the intersection of the hullform with the waterline

(breadth) has to be fixed.

To avoid numerical problems for cross sections with a very large breadth to draught ratio, the intersection of the hullform with the centre line (draught) has to be fixed too.

This is done here by replacing the last two equations by the equations for the breadth at the waterline and the draught, as demonstrated below. IWL (-1)11.(M5'a2n-1) 'E [cos[(2j-2n)-Oi]] n=0 i=0 IWL = E [ -xi-sin[(2j-1)Oi] +yi.cos[(2j-l)] i=0 for j 0 ... N-2

These N+1 equations can be solved by a numerical method, as for instance given by de Zwaan [1977].

So new values for (Ms-a2n-1) have been obtained.

These new values are used instead of the initial values to obtain new Oi-values of the IwL+1 offsets again, etc.

This procedure will be repeated several times and stops when the difference between the numerical E-values of two subsequent cal-culations becomes less than a certain threshold value AE, which depends on the dimensions of the cross section; for instance:

.[b 2 l_d 211/2 1

AE = (IwL+1) .[ 0.00005 _{max J}

j

in which:

bmax = maximum half breadth of the cross section

dmax maximum height of the cross section

In this iterative manner, the N+1 values for (MS-a2_1) have been

obtained.

Because of the definition a_1=+1 the scale factor Ms is equal to the final solution of the first coefficient (n=0).

The N other coefficients a2n_i have been found by dividing the final solutions of (Ms-a2n-1) through this M5-value.

Reference is also given here to a report of de Jong [1973].

In this report several other, suitable but more complex, methods are described to determine the scale factor Ms and the conformal mapping coefficients a2n_i from the offsets of a cross section.

Attention has to be paid to divergence in the calculation routines and re-entrant forms.

In these cases the number N will be increased until _{the divergenge} or re-entrance vanish. In the most worse case a "maximum" value of N will be attained without succes. Then _{can be switched to Lewis} coefficients with an area coefficient of the _{cross section set to} the nearest border of the valid area for Lewis _{coefficients, as has} been explained before.

For Close-Fit multi-parameter conformal mapping of a cross section, it is adviced to select the offsets at approximately _{equal mutual} circumferential lengths, eventually with somewhat more dense

offsets at sharp corners.

An example has been given here for the midship cross section of the S-175 containership design:

The calculated data of the Lewis two-parameter and the Close-Fit N-parameter conformal mapping of this midship section are tabled

below.

For the least squares method in the conformal mapping method, 33

new offsets at equidistant length intervals on the contour of this cross section have been determined by a second degree interpolation

routine.

At the last line the RMS-values have been given of the deviations of the 33 equidistant points on the contour of this cross section.

Ds-y1 (m) xi (M) 0.000 0.000 0.135 4.950 0.270 9.900 0.500 10.960 1.000 11.740 2.000 12.440 3.050 12.700 6.000 12.700 9.000 12.700 Lewis Confonma( Napping

Close Fit N-Parameter Conformal Napping

N (-) 2 2 3 4 5 6 7 8 9 10 2-N-1 (-) 3 3 5 7 9 11 13 15 17 19 Ns (m) 12.2400 12.2457 12.2841 12.3193 12.3186 12.3183 12.3191 12.3190 12.3195 12.3194 a_1 (-) +1.0000 +1.0000 +1.0000 .1.0000 +1.0000 +1.0000 +1.0000 +1.0000 +1.0000 +1.0000 al (-) +0.1511 +0.1511 +0.1640 +0.1634 +0.1631 +0.1633 +0.1633 +0.1632 +0.1632 +0.1632 a3 (-) -0.1136 -0.1140 -0.1167 -0.1245 _{-0.1246 -0.1243 -0.1244 -0.1245 -0.1245 -0.1245} a5 (-) 0 0 -0.0134 -0.0133 -0.0105 -0.0108 -0.0108 _{-0.0108 -0.0107 -0.0107} a7 (-) 0 0 0 +0.0053 +0.0054 +0.0031 +0.0030 +0.0032 +0.0031 +0.0030 a9 (-) 0 0 0 0 -0.0024 -0.0023 -0.0024 -0.0026 -0.0029 _-0.0029 all (-) ° 0 0 0 0 +0.0021 +0.0022 +0.0012 +0.0014 +0.0015 a13 (-) 0 0 0 0 0 0 +0.0002 +0.0002 +0.0021 +0.0020 a15 (-) 0 0 0 0 0 0 0 +0.0009 +0.0007 +0.0000 au (-) 0 0 0 0 0 0 0 0 -0.0016 -0.0015 a19 (-) ° 0 0 0 0 0 0 0 0 +0.0006 DIS (m) 0.181 0.180 0.076 0.039 0.027 0.019 0.018 0.017 0.009 0.008 B. = 25.40 m

O= 9.00m

4. Two-Dimensional Hydrodynamic Potential Coefficients

The determination of these two-dimensional potential coefficients by a conformal mapping method will be described here. During the ship motions calculations three different coordinate systems, as shown in figure 2-A, will be used. The two-dimensional hydrodynamic potential coefficients have been defined here with respect to the 0-(x,y,z) coordinate system for the moving ship in still water.

However, in this chapter a deviating axes system, _{as defined by} Tasai and de Jong, is used for the determination of the two-dimensional hydrodynamic potential coefficients for sway, heave and roll motions. This holds for the sway and roll coupling coef-ficients a change of sign. The signs of the uncoupled _{sway, heave} and roll coefficients do not change.

Because of small surge motions, for pitch and roll coefficients it has been assumed: xb-x.

For each cross section, the following two-dimensional _hydrodynamic coefficients have to be obtained:

M11' - two-dimensional potential hydrodynamic mass

coefficient of surge

N11' - two-dimensional potential hydrodynamic damping coefficient of surge

M22' - two-dimensional potential hydrodynamic mass

coefficient of sway

N22' - two-dimensional potential hydrodynamic damping coefficient of sway

M24' - two-dimensional potential hydrodynamic mass

coupling coefficient of roll into sway

N24' - two-dimensional potential hydrodynamic damping

coupling coefficient of roll into sway M33' _{- two-dimensional potential hydrodynamic mass}

coefficient of heave

N33' _{- two-dimensional potential hydrodynamic damping} coefficient of heave

M44' - two-dimensional potential hydrodynamic mass coefficient of roll

N44' - two-dimensional potential hydrodynamic damping coefficient of roll

M42' - two-dimensional potential hydrodynamic mass coupling coefficient of sway into roll

N42' - two-dimensional potential hydrodynamic damping coupling coefficient of sway into roll

The determination of these coefficients _{by a conformal mapping} method will be described here in detail.

4-1. Potential Coefficients for Surge

An equivalent longitudinal section, being 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 CB

By using a Lewis transformation of this equivalent longitudinal section to the unit circle, the two-dimensional potential _{mass M11*} and damping Nil* can be calculated in an analog manner as will be described further on,for the two-dimensional potential _{mass and} damping of sway, M22 and N22 .

With these two-dimensional values, the total potential mass and damping of surge are defined by:

Mll - M11* '13

Nll = N11* 13

These frequency-dependent hydrodynamic coefficients do not include three-dimensional effects. Only the hydrodynamic _{mass coefficient,} of which a large three-dimensional effect is expected, _{will be} adapted here empirically.

According to Tasai [1961] the zero-frequency potential _{mass for} sway can be expressed in Lewis-coefficients:

M22'

(w°)

-When using this formula for surge, the total potential _{mass of} surge is defined by:

M11(w-°) -

M11* (w=0) -B

A frequency-independent total hydrodynamic _{mass coefficient is} estimated empirically by Sargent and Kaplan [1974] _{as a proportion} of the total mass of the ship pV:

M11(S&K) -

a .PV

with:

L - ship length B = breadth

The factor a is depending on the breadth-length _{ratio B/L of the} ship: a 2-a in which: 1-b2 a _ VA" [ Ds 2 2 1-a1+a3 ] -[(1-a1)2 +3-a32] 1+b 1 loger L 1_1, _I -2 -b 1 _{with: b = [} 1 -[

1- i2 1

L 1/2

a=

b3 _

0.50 Mll (S&K) pV 0.40 ! 0.30 0.20 0.10 L 0

1

_....111. o 2 E dAs ]2 dxb N11 _{dA 2} N11

I

_[

_{s- ]}

.dxb dxb L 4 6 L B 8 10

Figure 4-1-A. Surge Hydrodynamic Mass in Proportion to the Ship's Mass as a Function of L/B, According to Sargent and Kaplan [1974]

With this hydrodynamic mass value, a correctionfactor p for three-dimensional effects has been determined:

mil(s8,10

P

M11(w=°)

With this factor the potential mass of surge is defined by:

Mll M11* 'B *)3

The three-dimensional effects for the potential damping of surge are ignored, so:

Nll N11* 'B

To obtain a uniform approach during all ship motions calculations, the cross sectional two-dimensional values of the hydrodynamic mass and damping have to be obtained.

Based on the results of numerical 3-D studies with a Wigley hull-form, a proportionality of both the two-dimensional hydrodynamic mass and damping with the squares of the derivatives of the cross sectional areas As in the xb-direction is assumed:

F dAs i2 L dxb Mll J F dAs 12. dxb Mll L dxb -I

Coupling of Surge into Heave

The coupling effects of surge into heave are ignored, so: M31' O two-dimensional hydrodynamic mass coupling

coefficient of surge into heave

N31' = 0 two-dimensional hydrodynamic damping coupling coefficient of surge into heave

Coupling of Surge into Pitch

Small coupling effects are found for the coupling of surge into

pitch.

In the ship motions 0-(x,y,z) coordinate system these coupling coefficients are defined by:

M51' -M11' b0

N51'

-N11' b0

in which:

two-dimensional hydrodynamic mass coupling coefficient of surge into pitch

two-dimensional hydrodynamic damping coupling coefficient of surge into pitch

b0 = vertical distance from the centroid of the cross section to the waterline, positive upwards

A201)) ₊

ax2 ay2

g

0

Because the sway motion of the fluid is not symmetrical about the y-axis, this velocity potential has the following relation:

.1)(-x,y) -4)(+x,y)

The linearized free surface condition in deep water is expressed

as follows:

w2

aci,

+ 0, for: lxi B5/2 and y 0

g ay

In consequence of the conformal mapping, this results into:

6bN

(I) .Z

[(2n-1) 'a2n-l'e(2n1)a] ±l

_--

_a ₀

aa n-0 ao

n

for: ci>0 and

0-4---2 in which: (4,2 = Ms aa g or:

fi3 = b0/2 non-dimensional frequency squared

This forced lateral oscillation of the cylinder causes a surface disturbance of the fluid.

Because the cylinder is supposed to be infinitely long, the

generated waves will be two-dimensional. These 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.

The amplitudes of these waves decrease strongly with the dis-tance to the cylinder.

- A regular progressive wave system, denoted here by subscript B. These waves dissipate energy. At a distance of a few wave

lengths from the cylinder, the waves on each side can be des-cribed by a single regular wavetrain. The wave amplitude at infinity

na

is proportional to the amplitude of oscillation of the cylinder xa, provided that this amplitude is sufficiently small compared with the radius of the cylinder and the wave length is not much smaller than the diameter of the cylinder.

The two-dimensional velocity potential of the fluid has to fulfil the following three requirements:

1. The velocity potential must satisfy to the equation of Laplace: a2,t,

b0/2

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

84),o(0) . axo

+x

an an

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

Using the stream function Ilf, this boundary condition on the surface of the cylinder (a=0) reduces to:

awo(0) . axo -x ao aa N = -X.Ms .E [(-1)n.(2n-1) _.a2n-1. sin[(2n-1)0]] n=0

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

. N

TO(e)

4-x.14s .E [(-1)n.a2n-1. cos[(2n-1)0]] +C(t)

n=0

in which C(t) is a function of the time only. When defining: YO g(0) 1 N *E [(-1)n.a2n-l cos[(2n-1)0]] n=0 aa

the stream function on the surface of the cylinder is given by: