1

Reprinted: 05-12-2000Website: www.shipmotions.nl

Report 968, March 1993,Delft University of Technology,Ship Hydromechanics Laboratory,Mekelweg 2, 2628 CD Delft,The Netherlands.

Hydromechanic Coefficients for CalculatingTime Domain Motions of Cutter Suction Dredges

by Cummins Equations

J.M.J. Journée

Delft University of Technology

Abstract

Software package DREDMO of the Delft Hydraulics predicts the behaviour of seagoingcutter dredges in near-shore conditions, which behaviour can be important with respect to theconstruction of the dredge and the assessment of downtime. The motion behaviour of theseagoing dredge has been described by non-linear Cummins Equations, which have to besolved in the time domain.The required input data on hydromechanic coefficients, retardation functions and frequencydomain wave load series can be derived with a newly developed pre-processing programSEWAY-D. This program, based on the frequency domain ship motions program SEAWAY,creates the hydromechanic input data file for DREDMO with a minimum risk on human inputerrors and makes DREDMO much more accessible for less-specialist users.This report describes the underlying hydromechanic theory of the new SEAWAY-D andDREDMO releases. Also, comparisons with frequency domain results have been given.

1 Introduction

The prediction of the behaviour of cutterdredges in near-shore conditions can beimportant with respect to the constructionof the dredge and the assessment ofdowntime. To be able to make downtimepredictions, the Delft Hydraulics togetherwith the Laboratory of Soil Movement and

the Ship Hydromechanics Laboratory(both are laboratories of the DelftUniversity of Technology) developed thecomputer code DREDMO [13] in the early80’s. The behaviour of the seagoingdredge has been described by non-linearso-called Cummins Equations, which haveto be solved in the time domain. Theseequations require hydromechanic

2

coefficients, retardation functions andwave load series as input, together withgeometric and operational data of the shipand the operational working method.The required input data – hydromechaniccoefficients, retardation functions andfrequency domain wave loads - have to bederived from model experiments or fromcalculated data by a suitable ship motioncomputer program. However, the creationof this hydromechanic input data fileappeared to be very much time consuming.Besides this, DREDMO and its pre- andpost-processing programs were running onmain frames, and specialists were requiredto operate the software.

In 1984, Delft Hydraulics decided todevelop a PC version of DREDMO, topromote the use of their commercialpackage.

In the late 80’s the author had developedcomputer code SEAWAY [6,8], afrequency domain ship motions programbased on the strip theory, for calculatingthe wave-induced motions and resultingmechanic loads of mono-hull ships movingforward with six degrees of freedom in aseaway.The potential coefficients and wave loadsare calculated for infinite and finite waterdepths. Added resistance, shearing forcesand bending and torsion moments can becalculated too. Linear(ised) springs andfree surface anti-rolling tanks, bilge keelsand other anti-rolling devices can beincluded. Several wave spectra definitionscan been used.

Computed data have been validated withresults of other computer programs andexperimental data. Based on thesevalidation studies - and experiences,obtained during an intensive use of theprogram by the author, students and over30 institutes and industrial users - it isexpected that this program is free ofsignificant errors.

In 1990, the author used relevant parts ofSEAWAY to create a pre-processingprogram for DREDMO. The Laboratory ofSoil Movement had defined the formats ofthe hydromechanic input data ofDREDMO. This pre-processing program iscalled SEAWAY-D and makes DREDMOmore accessible for less-specialist users.

In 1991, the Laboratory of Soil Movementhad completed their work on DREDMOwith the delivery of PC pre- and post-processing programs and a user-interface,which minimises the risk on human inputerrors; see [13].

In 1992, an input control programSEAWAY-H, to check the input data ofthe offsets of the under water geometry ofthe ship, have been added to theDREDMO software package.

Since 1993, DREDMO calculations can becarried out too for ships with localsymmetric twin-hull sections - as forinstance appear at cutter suction dredges -provided that interaction effects betweenindividual sections are not accounted for.Special attention has been paid tolongitudinal jumps in the cross sections,fully submerged cross sections and(non)linear viscous roll dampingcoefficients. Improved definitions of thehydrodynamic potential masses at aninfinite frequency and the wave loads havebeen added. Finally, many numericalroutines have been improved.

All strip-theory algorithms are described in[9]. A user manual of SEAWAY-D, withan example of the input and the outputfiles, has been given in [11].This report contains a survey of thehydromechanic part of the underlyingtheory of DREDMO and SEAWAY-D anda validation of the calculated results.A separately developed “stripped” versionof DREDMO, called SEAWAY-T, hasbeen used here to verify all algorithms inthe time domain.

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2 Equations of Motion

The co-ordinate systems are defined inFigure 1.

Figure 1 Co-ordinate System andDefinitions

Three right-handed co-ordinate systemshave been defined:

1. ),,( bbb zyxG connected to the ship,with G at the ship’s centre of gravity,

bx in the ship’s centre line in theforward direction, by in the ship’s portside direction and bz in the upwarddirection

2. ),,( 000 zyxS fixed in space, with S inthe still water surface, 0x in thedirection of the wave propagation and

0z in the upward direction.

3. ),,( zyxO or ),,( 321 xxxO movingwith the ship’s speed, with O above orbelow the average position of theship’s centre of gravity, x or 1xparallel to still water surface, y or 2xparallel to still water surface and z or

3x in the upward direction. Theangular motions of the body about thebody axes are denoted by: φ , θ and ψor 4x , 5x and 6x .

3 Frequency Domain Calculations

Based on Newton’s second law ofdynamics, the equations of motion of afloating object in a seaway are given by:

∑=

=⋅6

1,

jijji FxM &&

for i = 1, 2, … 6

in which:

jiM , 6x6 matrix of solid mass andinertia of the body

jx&& acceleration of the body indirection j

iF sum of forces or moments acting indirection i

When defining a linear system with simpleharmonic wave exciting forces andmoments, defined by:

( ))(cos)(),( ωεωωωia ii www tFtF +⋅=

The resulting simple harmonic displace-ments, velocities and accelerations are:

( )( )

( )txtx

txtx

txtx

ajj

ajj

ajj

ωωωω

ωωωω

ωωω

cos)(),(

sin)(),(

cos)(),(

2 ⋅⋅−=

⋅⋅−=

⋅=

&&

&

The hydromechanic forces and momentsiF , acting on the free floating object in

waves, consist of:

- linear hydrodynamic reaction forcesand moments expressed in terms withthe hydrodynamic mass and dampingcoefficients:

),()(),()( ,, txbtxa jjijji ωωωω &&& ⋅−⋅−

- linear hydrostatic restoring forces andmoments expressed in a term with aspring coefficient:

),(, txc jji ω⋅−

4

With this, the linear equations of motionbecome:

( ))(cos)(),(

),()(),()(

,

6

1,,

6

1,

ωεωωω

ωωωω

ia i wwjji

jjjijji

jjji

tFtxc

txbtxa

xM

+⋅+⋅−

⋅−⋅−

=⋅

∑

∑

=

=

&&&

&&

for i = 1, 2, … 6

So:

( )

( ))(cos)(

),(),()(

),()(6

1 ,,

,,

ωεωω

ωωω

ωω

ia i ww

j jjijji

jjiji

tF

txctxb

txaM

+⋅=

=

⋅+⋅+

⋅+∑

= &

&&

for i = 1, 2, … 6

The hydromechanic coefficients )(, ωjia ,

)(, ωjib and jic , and the wave load com-

ponents )(ωa iwF can be calculated with

existing 2-D or 3-D techniques.For this, strip theory programs - like forinstance the program SEAWAY [3] - canbe used. According to the strip theory, thetotal hydromechanic coefficients and waveloads of the ship can be found easily byintegrating the cross-sectional values overthe ship length.

The strip theory is a slender body theory,so one should expect less accuratepredictions for ships with low length tobreadth ratios. However, experiments haveshown that the strip theory appears to beremarkably effective for predicting themotions of ships with length to breadthratios down to about 3.0 or sometimeseven lower.The strip theory is based upon the potentialflow theory. This holds that viscous effectsare neglected, which can deliver seriousproblems when predicting roll motions atresonance frequencies. In practice, viscousroll damping effects can be accounted for

by experimental results or by empiricalformulas.The strip theory is based upon linearity.This means that the ship motions aresupposed to be small, relative to the cross-sectional dimensions of the ship. Onlyhydrodynamic effects of the hull below thestill water level are accounted for. Sowhen parts of the ship go out of or into thewater or when green water is shipped,inaccuracies can be expected. Also, thestrip theory does not distinguish betweenalternative above water hull forms.Nevertheless these limitations for zeroforward speed, generally the strip theoryappears to be a successful and practicaltheory for the calculation of the waveinduced motions of a ship.

For the determination of the two-dimensional potential coefficients forsway, heave and roll motions of not fullysubmerged ship-like cross sections, the so-called 2-Parameter Lewis Transformationcan map cross-sections conformally to theunit circle. Also the N-Parameter Close-FitConformal Mapping Method can be usedfor this.

The advantage of conformal mapping isthat the velocity potential of the fluidaround an arbitrary shape of a crosssection in a complex plane can be derivedfrom the more convenient circular crosssection in another complex plane. In thismanner hydrodynamic problems can besolved directly with the coefficients of themapping function, as reported by Tasai[15,16].The advantage of making use of the 2-Parameter Lewis Conformal MappingMethod is that the frequency-dependingpotential coefficients are a function of thebreadth, the draught and the area of thecross section, only.

Another method is the Frank Method [2],also suitable for fully submerged crosssections. This method determines thevelocity potential of a floating or a

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submerged oscillating cylinder of infinitelength by the integral equation methodutilising the Green’s function, whichrepresents a pulsating source below thefree surface.To avoid so-called “irregular frequencies”in the operational frequency range of notfully submerged cross sections, each Franksection will be closed automatically at thefree surface with a few extra points. Thisresults into a shift of these irregularfrequencies towards a higher frequencyregion.

The two-dimensional pitch and yawcoefficients follow from the heave andsway moments, respectively.

Finally, a method based on work publishedby Kaplan and Jacobs [12] and alongitudinal strip method has been usedfor the determination of the two-dimensional potential coefficients for thesurge motion.

At the following sections, thehydromechanic coefficients and the waveloads for zero forward speed are given asthey can be derived from the two-dimensional values, defined in a co-ordinate system with the origin O in thewaterline.

The symbols, used here, are:

jiM , solid mass and inertiacoefficients of the body

)(', ωjim sectional hydrodynamic

mass coefficient)(', ωjin sectional hydrodynamic

damping coefficient)(' ω

iwF sectional wave excitingforce or moment

)(' ωiwFK sectional Froude-Krylov

force or moment)(* ωζ

iw&& equivalent orbital

acceleration)(* ωζ

iw& equivalent orbital velocity

'wy sectional half breadth of

waterlinebx longitudinal distance of

cross-section to centre ofgravity, positive forwards

OG vertical distance ofwaterline to centre ofgravity, positive upwards

BG vertical distance of centreof buoyancy to centre ofgravity, positive upwards

∇ volume of displacementxxk radius of gyration in air for

rollyyk radius of gyration in air for

pitchzzk radius of gyration in air for

yawρ density of waterg acceleration of gravity

6

The solid mass coefficients are given by:

∇⋅⋅==

∇⋅⋅==

∇⋅⋅==

∇⋅==

∇⋅==

∇⋅==

ρ

ρ

ρ

ρ

ρ

ρ

26,6

25,5

24,4

3,3

2,2

1,1

zzzz

yyyy

xxxx

kIM

kIM

kIM

MM

MM

MM

The remaining solid mass coefficients arezero.

The potential mass coefficients are givenby:

jiij

Lbb

Lbb

Lbb

Lb

Lb

Lbb

Lb

Lbb

Lb

Lb

Lb

aa

dxxma

aBGdxxma

aOGdxxma

aOG

dxmOGdxma

dxxma

dxma

dxxma

aOGdxma

dxma

aBGa

dxma

,,

2'2,26,6

5,12'

3,35,5

6.2'

2,46,4

4,2

'2,4

'4,44,4

'3,35,3

'3,33,3

'2,26,2

2,2'

4,24,2

'2,22,2

1,15,1

'1,11,1

=

⋅⋅=

⋅−⋅⋅=

⋅+⋅⋅=

⋅+

⋅⋅+⋅=

⋅⋅−=

⋅=

⋅⋅=

⋅+⋅=

⋅=

⋅−=

⋅=

∫

∫

∫

∫∫

∫

∫

∫

∫

∫

∫

The remaining potential mass coefficientsare zero.

The potential mass coefficients are givenby:

jiij

Lbb

Lbb

Lbb

Lb

Lb

Lbb

Lb

Lbb

Lb

Lb

Lb

bb

dxxnb

bBGdxxnb

bOGdxxnb

bOG

dxnOGdxnb

dxxnb

dxnb

dxxnb

bOGdxnb

dxnb

bBGb

dxnb

,,

2'2,26,6

5,12'

3,35,5

6.2'

2,46,4

4,2

'2,4

'4,44,4

'3,35,3

'3,33,3

'2,26,2

2,2'

4,24,2

'2,22,2

1,15,1

'1,11,1

=

⋅⋅=

⋅−⋅⋅=

⋅+⋅⋅=

⋅+

⋅⋅+⋅=

⋅⋅−=

⋅=

⋅⋅=

⋅+⋅=

⋅=

⋅−=

⋅=

∫

∫

∫

∫∫

∫

∫

∫

∫

∫

∫

The remaining potential dampingcoefficients are zero.

The spring coefficients are given by:

jiij

L

Lbbw

Lbw

cc

GMgc

GMgc

dxxygc

dxygc

,,

5,5

4,4

'5,3

'3,3

2

2

=

⋅∇⋅⋅=

⋅∇⋅⋅=

⋅⋅⋅⋅⋅−=

⋅⋅⋅⋅=

∫

∫

ρ

ρ

ρ

ρ

The remaining spring coefficients are zero.

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The wave loads are given by:

bww

Lbww

bwww

Lbww

w

www

Lbww

www

Lbww

www

Lbww

www

Lbww

xFF

dxFF

xFBGFF

dxFF

OGF

FKnmF

dxFF

FKnmF

dxFF

FKnmF

dxFF

FKnmF

dxFF

i

ii

⋅=

⋅=

⋅−⋅−=

⋅=

⋅+

+⋅+⋅=

⋅=

+⋅+⋅=

⋅=

+⋅+⋅=

⋅=

+⋅+⋅=

⋅=

∫

∫

∫

∫

∫

∫

''

'

'''

'

'

'4

*'2,4

*'2,4

'

'

'3

*'3,3

*'3,3

'

'

'2

*'2,2

*'2,2

'

'

'1

*'1,1

*'1,1

'

'

26

66

315

55

2

224

44

333

33

222

22

11

:with

:with

:with

:with

:with

:with

ζζ

ζζ

ζζ

ζζ

&&&

&&&

&&&

&&&

These formulations of the hydrodynamicexciting and reaction forces and momentscan only be used in the frequency domain,since jia , and jib , both depend on thefrequency of motion ω only and theexciting wave loads have a linear relationwith the wave amplitude. In irregularwaves the response of the body can bedetermined by using the superpositionprinciple, so using linear responseamplitude operators between motion andwave amplitudes.

In the following figures, an example hasbeen given of the hydrodynamic potentialmass and damping and the wave loads forroll in the frequency domain.

Figure 3-A Potential Mass of Roll

Figure 3-B Potential Damping of Roll

Figure Wave Moment of Roll

4 Time Domain Calculations

As a result of the formulation in thefrequency domain, any system influencingthe behaviour of the floating body shouldhave a linear relation with thedisplacement, the velocity and theacceleration of the body. However, in a lotof cases there are several complicationswhich perish this linear assumption, forinstance the non- linear viscous damping,

8

forces and moments due to currents, wind,anchoring, etc.To include these non-linear effects in thevessel behaviour, it is necessary toformulate the equations of motion in thetime domain, which relates instantaneousvalues of forces, moments and motions.For the description of the hydromechanicreaction forces and moments, due to timevarying ship motions, use has been madeof the classic formulation given byCummins [1] with simple frequencydomain solutions of Ogilvie [14].

4.1 Cummins Equations

The floating object is considered to be alinear system, with translational androtational velocities as input and reactionforces and moments of the surroundingwater as output.The object is supposed to be at rest at time

0tt = . Then during a short time t∆ animpulsive displacement x∆ , with aconstant velocity V , is given to the object.So:

tVx ∆⋅=∆

During this impulsive displacement, thewater particles will start to move. Whenassuming that the fluid is rotation-free, avelocity potential Φ , linear proportional toV , can be defined:

Ψ⋅=Φ V for: tttt ∆+<< 00

in which Ψ is the normalised velocitypotential.

After this impulsive displacement x∆ , thewater particles are still moving. Becausethe system is assumed to be linear, themotions of the fluid, described by thevelocity potential Φ , are proportional tothe impulsive displacement x∆ .So:

x∆⋅=Φ χ for: ttt ∆+> 0

In here, χ is a normalised velocitypotential.

The impulsive displacement x∆ during theperiod ),( 00 ttt ∆+ does not influence themotions of the fluid during this periodonly, but also further on in time.This holds that the motions during theperiod ),( 00 ttt ∆+ are influenced also bythe motions before this period.

When the object performs an arbitrarily intime varying motion, this motion can beconsidered as a succession of smallimpulsive displacements.Then the resulting total velocity potential

)(tΦ during the period ),( ttt nn ∆+becomes:

[ ]

∆⋅⋅∆++

+Ψ⋅=Φ

∑

∑

=−−

=

n

kkjknknj

jjnj

tVttt

Vt

1,

6

1,

),(

)(

χ

In here:

n number of time steps

nt tnt ∆⋅+= 0

knt − ( ) tknt ∆⋅−+= 0

njV , j th velocity component during

period ),( ttt nn ∆+

kjV , j th velocity component during

period ),( ttt knkn ∆+−−

jΨ normalised velocity potentialcaused by a displacement indirection j during period

),( ttt nn ∆+

jχ normalised velocity potentialcaused by a displacement indirection j during period

),( ttt knkn ∆+−−

9

Letting t∆ go to zero, yields:

∑ ∫= ∞−

⋅⋅−+Ψ⋅=

=Φ6

1

)()()(

)(

jj

t

jjj dxttx

t

τττχ &&

where )(tx j& is the j th velocity com-ponent at time t .

The pressure in the fluid follows from thelinearised equation of Bernoulli:

tp

∂Φ∂

⋅−= ρ

An integration of these pressures over thewetted surface S of the floating objectprovides the expression for thehydrodynamic reaction forces andmoments iF .With in as the generalised directionalcosine, iF becomes:

⋅⋅

⋅⋅

∂

−∂+

+

⋅⋅Ψ⋅⋅=

⋅⋅−=

∫∫∫

∑ ∫∫

∫∫

∞−

=

τττχ

ρ

dxdSnt

t

dSntx

dSnpF

jS

ij

t

j Sijj

Sii

)()(

)(6

1

&

&&

When defining:

∫∫

∫∫

⋅⋅∂

−∂⋅=

⋅⋅Ψ⋅=

Si

jji

Sijji

dSnt

ttB

dSnA

)()(,

,

τχρ

ρ

the hydrodynamic forces and momentsbecome:

( )∑ ∫= ∞−

⋅⋅−+⋅=

=6

1,, )()(

jj

t

jijji

i

dxtBtxA

F

τττ &&&

for i = 1, 2, … 6

Together with the linear restoring springterms jiji xC ,, ⋅ and the linear external

loads )(tX i , Newton’s second law ofdynamics provides the linear equations ofmotion in the time domain:

( )

)()()()(

)(

,,

6

1,,

tXtxCdxtB

txAM

ijjij

t

ji

jjjiji

=⋅+⋅⋅−+

+⋅+

∫

∑

∞−

=

τττ &

&&

for i = 1, 2, … 6

in which:

)(tx j&& translational or rotationalacceleration in direction at time t

)(tx j& translational or rotational velocityin direction j at time t

)(tx j translational or rotationaldisplacement in direction at time t

jiM , solid mass or inertia coefficient

jiA , hydrodynamic mass coefficient

jiB , retardation function

jiC , spring coefficient

)(tX i external load in direction iat time t

When replacing in the damping part τ byτ−t and changing the integration

boundaries, this part can be written in amore convenient form:

( )

)()()()(

)(

,0

,

6

1,,

tXtxCdtxB

txAM

ijjijji

jjjiji

=⋅+⋅−⋅+

+⋅+

∫

∑∞

=

τττ &

&&

for i = 1, 2, … 6

10

Referring to the classic work on thissubject by Cummins [1], these sixequations of motion are called here the"Cummins Equations".

4.2 Hydromechanic Coefficients

The linear restoring spring coefficients3,3C , 5,3C , 4,4C , 3,5C and 5,5C can be

determined easily from the under watergeometry and the centre of gravity of thefloating object. Generally, the other jiC , -values are zero.

To determine jiA , and jiB , , the velocity

potentials jΨ and jχ have to be found,which is very complex.A much easier method to determine jiA ,

and jiB , can be obtained by making use ofthe hydrodynamic mass and damping datafound with existing 2-D or 3-D potentialtheory based computer programs in thefrequency domain. Relative simplerelations can be found between jiA , , jiB ,

and the calculated data of thehydrodynamic mass and damping in thefrequency domain.

For this, the floating object is supposed tocarry out a harmonic oscillation jx withamplitude 1.0 in the direction j :

( )tx j ωcos1⋅=

A substitution in the Cummins Equationsprovides:

( ) ( )

( )

( ))(

cos

sin)(

cos

,

0,

,,2

tX

tC

dtB

tAM

i

ji

ji

jiji

=

⋅+

⋅−⋅⋅−

⋅+⋅−

∫∞

ω

τωτωτω

ωω

for i = 1, 2, ... 6

This results into:

( ) ( )

( ) ( )

( ))(

cos

sincos)(

cossin)(

1

,

0,

0,

,,2

tX

tC

tdB

tdB

AM

i

ji

ji

ji

jiji

=

⋅+

⋅

⋅⋅⋅−

⋅

⋅⋅⋅

⋅−+⋅−

∫

∫∞

∞

ω

ωτωττω

ωτωττ

ωω

for i = 1, 2, ... 6

In the classic frequency domaindescription these equations of motion arepresented by:

( ) ( )

( ))(

cos

sin)(

cos)(

,

,

,,2

tX

tc

tb

taM

i

ji

ji

jiji

=

⋅+

⋅⋅−

⋅+⋅−

ω

ωωω

ωωω

for i = 1, 2, ... 6

In here:

)(, ωjia frequency dependinghydrodynamic masscoefficient

)(, ωjib frequency dependinghydrodynamic dampingcoefficient

jic , restoring spring termcoefficient

When comparing these time domain andfrequency domain equations - both withlinear terms as published by Ogilvie [14] -it is found:

( )

( )

jiji

jiji

jijiji

Cc

dBb

dBAa

,,

0,,

0,,,

cos)()(

sin)(1

)(

=

⋅⋅=

⋅⋅⋅−=

∫

∫∞

∞

τωττω

τωττω

ω

11

After a Fourier re-transformation, thedamping term provides the retardationfunction:

( ) ωωτωπ

τ dbB jiji ⋅⋅⋅= ∫∞

cos)(2

)(0

,,

Then the mass term follows from:

( ) τωττω

ω dBaA jijiji ⋅⋅⋅+= ∫∞

sin)(1

)(0

,,,

This mass expression is valid for any valueof ω , so also for ∞=ω , which provides:

)(,, ∞== ωjiji aA

4.3 Addition of Non-Linearities

So far, these equations of motion arelinear. But non-linear contributions can beadded now to ( )tX i easily.For instance, non-linear viscous rolldamping contributions can be added to 4Xby:

( ) φφ && ⋅⋅−=∆ 24,44 a

bX

Also it is possible to include non-linearspring terms, by considering the stabilitymoment as an external load and shifting itscontribution to the right hand side of theequation of motion, for instance:

φφρ φ sin)(

0

4

4,4

⋅⋅∇⋅⋅−=∆

=

GNgX

C

in which )(φφGN is the transverse meta-centric height at arbitrarily heeling angles.

4.4 Some Numerical Recipes

Many computer programs fail whencalculating )(, ωjib at too high a

frequency. This holds that - whendetermining jiB , - the numericalcalculations can be carried out in a limitedfrequency range Ω≤≤ ω0 only:

( ) ωωτωπ

τ dbB jiji ⋅⋅⋅= ∫Ω

cos)(2

)(0

,,

So, a truncation error )(, τjiB∆ will beintroduced:

( )∫∞

Ω

⋅⋅⋅=∆ ωωτωπ

τ dbB jiji cos)(2

)( ,,

For the uncoupled damping coefficients -so when ji = - this truncation error canbe estimated easily.The relation between the dampingcoefficient )(, ωjib and the amplitude ratioof the radiated waves and the oscillatorymotion )(, ωα ii is given by:

)()( 2,3

2

, ωαω

ρω iiii

gb ⋅

⋅=

From this an approximation can be foundfor the tail of the damping curve:

3,

, )(ω

βω ii

iib =

The value of ii,β follows from thecalculated damping value at the highestfrequency used, Ω=ω . This holds that aconstant amplitude ratio )(, ωα ii issupposed here for Ω>ω .

Then the truncation error becomes:

( )( )

( )

( ) ( ) ( )( )

⋅Ω⋅−

+Ω++

ΩΩ

−

Ω

Ω⋅

⋅=∆

∑∞

=1

2

2

2,

,

!221

ln

sincos)(

n

nn

iiii

nn

B

ττγ

ττ

τ

τπ

τβτ

12

in which ...577215.0=γ (Euler constant)

Studies carried out in the past have showedthat in case of a sufficient high value of Ωthe contribution of iiB ,∆ into iiB , is oftensmall. The potential damping calculationswere based on numerical routines as usedin computer program SEAWAY [8,9]. Inthis program special attention has beenpaid to the potential calculations at veryhigh frequencies. For normal merchantships are 5 radians per second, which canbe reached by the routines in SEAWAY, afairly good value for the maximumfrequency Ω .Thus, the retardation function isapproximated by the numerical solution ofthe integral:

( )∫Ω

⋅⋅⋅=0

,, cos)(2

)( τωτωπ

τ dbB jiji

The damping curve has to be calculated atωN constant frequency intervals ω∆ , so:

Ω=∆⋅ ωωN .

When calculating here the retardationfunctions it assumed that at each frequencyinterval the damping curve is linearlyincreasing or decreasing; see Figure 4.

Figure 4 Integration of Damping Curve

Now the contribution of this interval intojiB , can be calculated analytically. This

holds that – caused by a large ω or a largeτ - the influence of a strongly fluctuating

( )ωτcos at this interval can be taken intoaccount.

Then the numerical integration, at constantfrequency intervals ω∆ , is given by:

( ) ( )[ ]

( )τωτπ

τωτωω

τπτ

ωω

ω

NN

N

nnn

n

ji

b

b

B

sin2

coscos

2)(

11

2,

⋅⋅⋅

+

−⋅∆∆

⋅

⋅⋅

=

∑=

−

in which:

1

1

−

−

−=∆∆=−=∆

nnn

nnn

bbbωωωω

For 0=τ , the value of the retardationfunction can be derived simply from theintegral of the damping:

ωωπ

τ dbB jiji ⋅⋅== ∫Ω

0,, )(

2)0(

Because the potential damping is zero for0=ω , the expression for the damping

term leads for 0=ω , so ( ) 1cos =ωτ , intothe following requirement for theretardation functions:

0)(0

, =⋅∫∞

ττ dB ji

In the equations of motion, the retardationfunction multiplied with the velocityshould be integrated over an infinite time:

τττ dtxB jji ⋅−⋅∫∞

)()(0

, &

However, after a certain time ji,Γ=τ , thefluctuating values of the integral havereached already a very small value.

13

A useful limit value for the correspondingintegration time can be found with:

)0(2

,

1,

ji

N

nn

ji B

b

⋅⋅∆⋅

∆⋅=Γ

∑=

εωπ

ω

with: 010.0≈ε

So, the Cummins Equations, which arestill linear here, are given by:

( )

)()()()(

)(

,0

,

6

1,,

,

tXtxCdtxB

txAM

ijjijji

jjjiji

ji

=⋅+⋅−⋅+

+⋅+

∫

∑Γ

=

τττ &

&&

for i = 1, 2, … 6

The numerical integration can be carriedout with the trapezoid rule or withSimpson’s rule. Because of a relativelysmall time step τ∆ is required to solve theequations of motion numerically, generallythe trapezoid rule is sufficient accurate.

The hydrodynamic mass coefficientfollows from:

)(,, ∞== ωjiji aA

When this mass coefficient is not availablefor an infinite frequency, it can becalculated from a mass coefficient at acertain frequency Ω=ω and theretardation function:

( ) τττ dBaAji

jijiji ⋅Ω⋅⋅Ω

+Ω= ∫Γ

sin)(1

)(,

0,,,

With:

ττ ∆⋅=Γ Nji,

the numerical solution of this integral canbe found in an similar way as for theretardation functions:

( )

( ) ( )( )[ ]

( ) ττττ

τττ

τττ

ττ

τ

∆⋅⋅Ω⋅=−=⋅

⋅Ω

+

∆⋅−⋅Ω−⋅⋅Ω⋅∆

∆⋅

⋅Ω

=⋅Ω⋅

∑

∫

=

Γ

NBB

nnB

dB

Njiji

N

n

n

ji

ji

cos)()0(

1

1sinsin

1

sin)(

,,

1

2

0,

,

in which:

)1()( ,, −−=∆ nBnBB jijin

Similar to this, the numerical solution ofthe frequency depending damping is:

( )

( ) ( )( )[ ]

( )τττ

τττ

τττ

ττ

τ

∆⋅⋅Ω⋅=⋅Ω

+

∆⋅−⋅Ω−⋅⋅Ω⋅

∆∆

⋅

⋅Ω

=⋅Ω⋅=Ω

∑

∫

=

Γ

NB

nnB

dBb

Nji

N

n

n

jiji

ji

sin)(1

1coscos

1cos)()(

,

1

20

,,

,

5 Equations of Motion

Integrating the velocities of the ship’scentre of gravity can derive the path of theship in the ( )000 ,, zyx system of axes:

ψψ

θθ

φφ

ψψψψ

&&

&&

&&&&

&&&&&&

=

=

=

=⋅+⋅=⋅−⋅=

0

0

0

0

0

0

cossinsincos

zzyxyyxx

The Euler equations of motion are writtenin the ( )zyx ,, system of axes:

14

( )( )( )( )( )( ) extwhyyxxzz

extwhxxzzyy

extwhzzyyxx

extwh

extwh

extwh

NNNIII

MMMIII

KKKIII

ZZZyxzM

YYYzxyM

XXXzyxM

++=⋅⋅−−⋅

++=⋅⋅−−⋅

++=⋅⋅−−⋅

++=⋅+⋅−⋅

++=⋅−⋅+⋅

++=⋅+⋅−⋅

θφψ

ψφθ

ψθφ

φθ

φψ

φψ

&&&&

&&&&

&&&&

&&&&&&

&&&&&&

&&&&&&

with in the right hand sides:

subscript h linear hydromechanic loadssubscript w linear wave loadssubscript ext non-linear hydromechanic

loads and (non)linearexternal loads, caused bywind, currents, anchorlines, cutter, etc.

With the hydromechanic loads as definedbefore, the equations of motion are definedas given below.

Surge motion:( )

extw XXCBA

zCzBzA

xCxBxAzyMxM

+=⋅+⋅+⋅

+⋅+⋅+⋅

+⋅+⋅+⋅+⋅+⋅−⋅+⋅

θθθ

θψ

5,15,15,1

3,13,13,1

1,11,11,1

&&&&&&

&&&

&&&&&&

Sway motion:( )

extw YYCBA

CBA

yCyByAzxMyM

+=⋅+⋅+⋅

+⋅+⋅+⋅

+⋅+⋅+⋅+⋅−⋅⋅+⋅

ψψψ

φφφ

φψ

6,26,26,2

4,24,24,2

2,22,22,2

&&&

&&&&&&

&&&&&&

Heave motion:( )

extw ZZCBA

zCzBzA

xCxBxAyxMzM

+=⋅+⋅+⋅

+⋅+⋅+⋅

+⋅+⋅+⋅+⋅+⋅−⋅+⋅

θθθ

φθ

5,35,35,3

3,33,33,3

1,31,31,3

&&&&&&

&&&

&&&&&&

Roll motion:( )

extw

zzyyxx

KKCBA

CBA

yCyByA

III

+=⋅+⋅+⋅

+⋅+⋅+⋅

+⋅+⋅+⋅

+⋅⋅−−⋅

ψψψ

φφφ

ψθφ

6,46,46,4

4,44,44,4

2,42,42,4

&&&

&&&

&&&

&&&&

Pitch motion:( )

extw

xxzzyy

MMCBA

zCzBzA

xCxBxA

III

+=⋅+⋅+⋅

+⋅+⋅+⋅

+⋅+⋅+⋅

+⋅⋅−−⋅

θθθ

ψφθ

5,55,55,5

3,53,53,5

1,51,51,5

&&&&&&

&&&

&&&&

Yaw motion:( )

extw

xxzzzz

NNCBA

CBA

yCyByA

III

+=⋅+⋅+⋅

+⋅+⋅+⋅

+⋅+⋅+⋅

+⋅⋅−−⋅

ψψψ

φφφ

θφψ

6,66,66,6

4,64,64,6

2,62,62,6

&&&

&&&&&&

&&&&

Some of the coefficients in these sixequations of motion are zero. Afteromitting these coefficients and orderingthe terms, the equations for theaccelerations are as follows.

Surge motion:( )

( )θψ

θ

θ

&&&&

&&

&&&&

⋅−⋅+⋅+

⋅−⋅−

+=

⋅+⋅+

zyM

BxB

XX

AxAM

extw

5,11,1

5,11,1

Sway motion:( )

( )φψ

ψφ

ψφ

&&&&

&&&

&&&&&&

⋅+⋅−⋅+

⋅−⋅−⋅−

+=

⋅+⋅+⋅+

zxM

BByB

YY

AAyAM

extw

6,24,22,2

6,24,22,2

15

Heave motion:( )

( )φθ

θθ

θ

&&&&

&&

&&&&

⋅−⋅+⋅+

⋅−⋅−⋅−⋅−

+=

⋅+⋅+

yxM

CBzCzB

ZZ

AzAM

extw

5,35,33,33,3

5,33,3

Roll motion:( )

( ) ψθ

φψφ

ψφ

&&&&&

&&&&&&

⋅⋅−+

⋅−⋅−⋅−⋅−

+=

⋅+⋅+⋅+

zzyy

extw

xx

II

CBByB

KK

AyAAI

4,46,44,42,4

6,42,44,4

Pitch motion:( )

( ) ψφ

θθ

θ

&&

&&&

&&&&&&

⋅⋅−+

⋅−⋅−⋅−⋅−⋅−

+=

⋅+⋅+⋅+

xxzz

extw

yy

II

CBzCzBxB

MM

zAxAAI

5,55,53,53,51,5

3,51,55,5

Yaw motion:( )

( ) θφ

ψφ

φψ

&&&&&

&&&&&&

⋅⋅−+

⋅−⋅−⋅−

+=

⋅+⋅+⋅+

yyxx

extw

zz

II

BByB

NN

AyAAI

6,64,62,6

4,62,66,6

With known coefficients and right handsides of these equations, the sixaccelerations can be determined by anumerical method as - for instance - thewell-known Runge-Kutta method.Because of sometimes an extreme highstiffness of the cutter dredge system, DelftHydraulics has adapted the numericalsolution method of these equations inDREDMO for this; see [13].

6 Viscous Damping

Sway and YawThe non-linear viscous sway and yawdamping term in the equations of motionfor sway and yaw can be approximated by:

yybv

&& ⋅⋅)2(2,2 and ψψ && ⋅⋅)2(

6,6 vb

with:

50.16121

3)2(6,6

)2(2,2

≈

⋅⋅⋅⋅=

⋅⋅⋅⋅=

D

D

D

C

CdLb

CdLb

v

v

ρ

ρ

RollThe total non-linear roll damping term inthe equation of motion for roll can beexpressed as:

( ) φφφ &&& ⋅⋅+⋅+ )2(4,4

)1(4,44,4 aa

bbb

with:

4,4b linear potential roll dampingcoefficient

)1(4,4 a

b linear(ised) additional roll dampingcoefficient

)2(4,4 a

b non-linear additional roll dampingcoefficient

The linear potential roll-damping coeffi-cient 4,4b can be determined as describedbefore.For time domain calculations a linear aswell as a non-linear roll-dampingcoefficient can be used. However, forfrequency domain calculations anequivalent linear roll-damping coefficienthas to be estimated. This linearised roll-damping coefficient can be found byrequiring that an equivalent linear dampingdissipate an equal amount of energy as thenon-linear damping, so:

16

dtbdtbTT

aa⋅⋅⋅⋅=⋅⋅⋅ ∫∫ φφφφφ

φφ

&&&&&0

)2(4,4

0

)1(4,4

Then the equivalent linear additional roll-damping coefficient )1(

4,4 ab becomes:

)2(4,4

)1(4,4 3

8aa

bb a ⋅⋅⋅⋅

= ωφπ

The additional roll damping coefficients)1(

4,4 ab and )2(

4,4 ab are mainly caused by

viscous effects. Until now it is not possibleto determine these additional coefficientsin a pure theoretical way. They have to beestimated by free rolling modelexperiments or by a semi-empiricalmethod, based on theory and a largenumber of model experiments withsystematic varied ship forms. Thelinear(ised) and the non-linear equations ofpure roll motions, used to analyse freerolling model experiments, are presentedhere. Also, for zero forward ship speed,the algorithms of the empirical method ofIkeda, Himeno and Tanaka [3] are given.

6.1 Experimental Roll Damping

In case of pure free rolling in still water,the linear equation of the roll motion aboutthe centre of gravity G is given by:

( ) ( ) 04,44,44,44,4 =⋅+⋅++⋅+ φφφ cbbaIaxx

&&&

in which:

4,4a potential mass coefficient

4,4b potential damping coefficient

ab 4,4 linear(ised) additional damping

coefficient

4,4c restoring term coefficient

This equation can be rewritten as:

02 20 =⋅+⋅⋅+ φωφνφ &&&

in which 4,4

4,44,42aI

bb

xx

a

+

+=ν is the quotient

between damping and moment of inertia

and 4,4

4,420 aI

c

xx +=ω is the not-damped

natural roll frequency squared.

When defining a non-dimensional rolldamping coefficient by:

0ων

κ =

the equation of motion for roll can berewritten as:

02 200 =⋅+⋅⋅⋅+ φωφωκφ &&&

Then, the logarithmic decrement of roll is:

+=

⋅⋅=⋅

)()(

ln

0

φ

φφ

φφ

ωκν

Ttt

TT

Because of the relation 220

2 νωωφ −=

and the assumption that 20

2 ων << , it canbe written 0ωωφ ≈ .This leads to:

πωω φφφ ⋅=⋅≈⋅ 20 TT

So, the non-dimensional total roll dampingis given by:

( )( )

( )4,4

04,44,4 2

ln2

1

cbb

Ttt

a ⋅⋅+=

+⋅

⋅=

ω

ωω

πκ

φ

17

The non-potential part of the total roll-damping coefficient follows from theaverage value of κ by:

4,40

4,44,4

2b

cb

a−

⋅⋅=

ωκ

When data on free-rolling experimentswith a model in still water are available,these κ -values can easily been found.

Often the results of these free rolling testsare presented by:

a

a

φφ∆

as function of aφ ,

with aφ as the absolute value of theaverage of two successive positive ornegative maximum roll angles:

2)1()( ++

=ii aa

a

φφφ

and aφ∆ as the absolute value of thedifference of two successive positive ornegative maximum roll angles:

)1()( +−=∆ ii aaa φφφ

Then the total non-dimensional naturalfrequency becomes:

∆−

∆+

⋅⋅

=

a

a

a

a

φφ

φφ

πκ

2

2

ln2

1

These experiments deliver no informationon the relation with the frequency ofoscillation. So, it has to be decided to keepthe additional coefficient

ab 4,4 or the total

coefficient a

bb 4,44,4 + constant.

The successively found values for κ ,plotted on base of the average rollamplitude, will often have a non-linearbehaviour as illustrated in Figure 5.For behaviour like this, it will be found:

aφκκκ ⋅+= 21

Figure 5 Free Rolling Data

This holds that during frequency domaincalculations, the damping term isdepending on the solution for the rollamplitude.For rectangular barges (LxBxd ) with thecentre of gravity in the water line, it isfound by Journée [7]:

50.0

0013.0

2

2

1

≈

⋅≈

κ

κdB

Then the total roll-damping term becomes:

( ) φω

φκκφω

κ && ⋅

⋅

⋅⋅+=⋅

⋅

⋅0

4,421

0

4,4 22 cca

The linear additional roll-damping coeffi-cient becomes:

4,40

4,41

)1(4,4

2b

cb

a−

⋅⋅=

ωκ

But for the non-linear additional roll-damping coefficient, a quasi-quadraticdamping coefficient is found:

18

φ

φω

κ&ac

ba

⋅⋅

⋅=0

4,42

)2(4,4

2

Because this roll-damping "coefficient"includes ),( tωφ& in the denominator, it

varies strongly with time.

An equivalent non-linear damping termcan be found by requiring that theequivalent quadratic damping term willdissipate an equal amount of energy as thequasi-quadratic damping term, so:

∫

∫

⋅⋅⋅⋅⋅

⋅=

=⋅⋅⋅⋅

φ

φ

φφφω

κ

φφφ

T

a

T

dtc

dtba

00

4,42

0

)2(4,4

2 &&

&&&

Then the equivalent quadratic additionalroll-damping coefficient )2(

4,4 ab becomes:

20

4,42

)2(44

28

3ω

πκ

cb

a

⋅⋅

⋅⋅=

With this, the roll-damping term based onexperimentally determined κ -values, asgiven in Figure 5, becomes:

( )φφ

ωπ

κφω

κ

φφφ

&&&

&&&

⋅⋅⋅

⋅⋅

⋅+⋅⋅

⋅=

=⋅⋅+⋅+

20

4,42

0

4,41

)2(44

)1(444,4

28

32 cc

bbbaa

So far, pure roll motions with one degreeof freedom are considered in the equationsof motion. Coupling effects between theroll motion and the other motions are nottaken into account. This can be done in aniterative way.

Experimental or empirical values of 1κ

and 2κ provide starting values for )1(4,4 a

b

and )2(4,4 a

b . With these coefficients, a free-rolling experiment with all degrees of

freedom can be simulated in the timedomain. An analyse of this simulated rollmotion, as being a linear pure roll motionwith one degree of freedom, delivers newvalues for )1(

4,4 ab and )2(

4,4 ab . This

procedure has to be repeated until asuitable convergence has been reached. Aninclusion of the natural frequency 0ω inthis iterative procedure provides also areliable value for the estimated solid massmoment of inertia coefficient xxI .However, this procedure is not included inDREDMO yet.

6.2 Empirical Roll Damping

Because of the additional part of the rolldamping is significantly influenced by theviscosity of the fluid, it is not possible tocalculate the total roll damping in a puretheoretical way. Besides this, experimentsshowed also a non-linear (about quadratic)behaviour of the additional parts of the rolldamping.As mentioned before, the total non-linearroll damping term in the left-hand side ofthe equation of motion for roll can beexpressed as:

( ) φφφ &&& ⋅⋅+⋅+ )2(44

)1(444,4 aa

bbb

For the estimation of the additional partsof the roll damping, use has been made ofwork published by Ikeda, Himeno andTanaka [3]. Their empirical method iscalled here the “Ikeda Method”.At zero forward speed, this Ikeda methodestimates the following components of theadditional roll-damping coefficient of aship:

)2(4,4

)2(4,4

)2(4,4

)2(4,4

)1(4,4 0

kefa

a

bbbb

b

++=

=

with:

19

)2(4,4 f

b non-linear friction damping)2(

4,4 eb non-linear eddy damping

)2(4,4 k

b non-linear bilge keel damping

Ikeda, Himeno and Tanaka claim fairlygood agreements between their predictionmethod and experimental results.They conclude that the method can be usedsafely for ordinary ship forms. But forunusual ship forms, very full ship formsand ships with a large breadth to draughtratio the method should not be alwayssufficiently accurate.Even a few cross sections with a largebreadth to draught ratio can result in anextremely large eddy-making componentof the roll damping. So, always judge thecomponents of this damping.

In the description of the Ikeda method, thenomenclature of Ikeda is maintained hereas far as possible:

ρ density of waterν kinematic viscosity of waterg acceleration of gravityω circular roll frequency

aφ roll amplitude

nR Reynolds numberL length of the shipB breadth of the shipD average draught of the ship

BC block coefficient

fS hull surface area

OG distance of centre of gravity abovestill water level

sB sectional breadth on the water line

sD sectional draught

sσ sectional area coefficient

0H sectional half breadth to draughtratio

1a sectional Lewis coefficient

3a sectional Lewis coefficient

sM sectional Lewis scale factor

fr average distance between roll axisand hull surface

kh height of the bilge keels

kL length of the bilge keels

kr distance between roll axis and bilgekeel

kf correction for increase of flowvelocity at the bilge

pC pressure coefficient

ml lever of the moment

br local radius of the bilge circle

For numerical reasons two restrictionshave to be made during the sectionalcalculations:- if 999.0>sσ then 999.0=sσ

- if ssDOG σ⋅−< then ssDOG σ⋅−=

6.2.1 Frictional Damping, )2(4,4 f

b

Kato deduced semi-empirical formulas forthe frictional roll-damping from experi-mental results of circular cylinders, whollyimmersed in the fluid.An effective Reynolds number for the rollmotion was defined by:

( )ν

ωφ ⋅⋅⋅=

2512.0 afn

rR

In here, for ship forms the averagedistance between the roll axis and the hullsurface fr can be approximated by:

( )π

OGL

SC

r

fB

f

⋅+⋅⋅+=

0.2145.0887.0

with a wetted hull surface area fS ,approximated by:

( )BCDLS Bf ⋅+⋅⋅= 70.1

20

When eliminating the temperature ofwater, the kinematic viscosity can beexpressed in the density of water with thefollowing relation in the kg-m-s system:- fresh water:

( )( ) sm 100007424.0

10003924.0442.11022

6

−⋅+

−⋅+=⋅

ρ

ρν

- sea water:

( )( ) sm 100002602.0

10001039.0063.11022

6

−⋅+

−⋅+=⋅

ρ

ρν

Kato expressed the skin friction coefficientas:

114.0500.0 014.0328.1 −− ⋅+⋅= nnf RRC

The first part in this expression representsthe laminar flow case. The second part hasbeen ignored by Ikeda, but has beenincluded here.

Using this, the non-linear roll-dampingcoefficient due to skin friction at zeroforward speed is expressed as:

fff CSrbf

⋅⋅⋅⋅= 34,4 2

1ρ

Ikeda confirmed the use of this formula forthe three-dimensional turbulent boundarylayer over the hull of an oscillatingellipsoid in roll motion.

6.2.2 Eddy-Making Damping, )2(4,4 e

b

At zero forward speed the eddy makingroll damping for the naked hull is mainlycaused by vortices, generated by a two-dimensional separation. From a number ofexperiments with two-dimensionalcylinders it was found that for a naked hullthis component of the roll moment isproportional to the roll velocity squaredand the roll amplitude. This means that the

non-linear roll-damping coefficient doesnot depend on the period parameter but onthe hull form only.When using a simple form for the pressuredistribution on the hull surface it appearsthat the pressure coefficient pC is afunction of the ratio γ of the maximumrelative velocity maxU and the meanvelocity meanU on the hull surface:

mean

max

UU

=γ

The relation between pC and γ wasobtained from experimental roll dampingdata of two-dimensional models.These experimental results are fitted by:

50.10.2435.0 187.0 +⋅−⋅= ⋅−− γγ eeC p

The value of γ around a cross-section isapproximated by the potential flow theoryfor a rotating Lewis-form cylinder in aninfinite fluid.An estimation of the sectional maximumdistance between the roll axis and the hullsurface, maxr , has to be made.Values of )(max ψr have to be calculatedfor:

0.01 == ψψ

and:( )

⋅+⋅

⋅==3

312 4

1arccos5.0

aaa

ψψ

The values of ( )ψmaxr follow from:

( ) ( ) ( ) ( ) ( ) ( ) 2

31

231

max

3coscos1

3sinsin1

)(

ψψ

ψψ

ψ

⋅⋅+⋅−

+⋅⋅−⋅+⋅

=

aa

aaM

r

s

With these two results, maxr and ψ followfrom the conditions:

21

- if )()( 2max1max ψψ rr > then:)( 1maxmax ψrr = and 1ψψ =

- if )()( 2max1max ψψ rr < then:)( 2maxmax ψrr = and 2ψψ =

The relative velocity ratio γ on a cross-section is obtained by:

+⋅

⋅+⋅

⋅

+⋅⋅⋅

⋅=

22max

0

3

2

2

baHM

r

DOG

HD

f

s

sss σ

πγ

with:

( ) ( )( )

( )( ) ( )

( ) ( ) ( )

( )( ) ( )

( ) ( ) ( )

( )25 11065.13

21

312

12

31

31

3

21

312

12

31

31

3

3

31

23

21

41

sin

336

3sin1

5sin2

cos

336

3cos1

5cos2

4cos62cos312

91

sef

a

aaaaa

aa

ab

a

aaaaa

aa

aa

aaa

aaH

σ

ψ

ψ

ψ

ψ

ψ

ψ

ψψ

−⋅⋅−⋅+=

⋅+

⋅⋅++⋅⋅++

⋅⋅−⋅+

⋅⋅⋅−=

⋅+

⋅⋅−+⋅⋅−+

⋅⋅−⋅+

⋅⋅⋅−=

⋅⋅⋅−⋅⋅⋅−⋅⋅+

⋅++=

With this a non-linear sectional eddymaking damping coefficient for zeroforward speed follows from:

⋅−⋅+

⋅−++⋅

⋅−⋅

⋅⋅

⋅⋅⋅=

2

102

11

2

max4'

11

21

s

b

s

b

ss

b

ps

sE

DRf

Hf

Drf

DOG

DRf

CDr

DB ρ

with:

( )

( ) ( )s

s

s

se

f

f

σπ

σπ

σ

σ ⋅⋅−⋅−

⋅−⋅=

−⋅+⋅=

⋅− 255

2

1

sin15.1

cos15.0

0.1420tanh15.0

The term between square brackets

⋅−

s

b

Drf1 is included in the program

listing in the paper of Ikeda et. al. [3], butit does not appear in the formulas given inthe paper. After contacting Ikeda, this termhas been omitted in SEAWAY andSEAWAY-D.

The approximation of the local radius ofthe bilge circle bR is:

• for sb DR < and 2/sb BR < :

( )4

12 0

−−⋅

⋅⋅=π

σ ssb

HDR

• for 10 >H and sb DR > :

sb DR =

• for 10 <H and sb DHR ⋅< 0 :2/sb BR =

For three-dimensional ship forms, the zeroforward speed eddy-making dampingcoefficient is found by integration over theship length:

22

∫ ⋅=L

bE dxBbe

')2(4,4

6.2.3 Bilge Keel Damping, )2(4,4 k

b

The bilge keel component of the non-linear roll-damping coefficient is dividedinto two components:- a component NB due to the normal

force of the bilge keels- a component SB due to the pressure an

the hull surface, created by the bilgekeels.

The normal force component NB of thebilge keel damping can be deduced fromexperimental results of oscillating flatplates. The drag coefficient DC dependson the period parameter or the Keulegan-Carpenter number. Ikeda measured thisnon-linear drag also by carrying out freerolling experiments with an ellipsoid withand without bilge keels.This results in a non-linear sectionaldamping coefficient:

DkkkN CfhrB ⋅⋅⋅= 23'

with:

( )sef

frh

C

k

kak

kD

σ

φπ−⋅−⋅+=

+⋅⋅⋅

⋅=

0.11603.00.1

40.25.22

The local distance between the roll axisand the bilge keel, kr , will be determinedfurther on.

Assuming a pressure distribution on thehull caused by the bilge keels, a non-linearsectional roll-damping coefficient can bedefined:

∫ ⋅⋅⋅⋅⋅=kh

mpkkS dhlCfrB0

22'

21

Ikeda carried out experiments to measurethe pressure on the hull surface created bybilge keels. He found that the coefficient

+pC of the pressure on the front-face of

the bilge keel does not depending on theperiod parameter, while the coefficient

−pC of the pressure on the back-face of

the bilge keel and the length of thenegative pressure region depend on theperiod parameter.

Ikeda defines an equivalent length of aconstant negative pressure region 0S overthe height of the bilge keels, which is fittedto the following empirical formula:

kakk hrfS ⋅+⋅⋅⋅⋅= 95.130.00 φπ

The pressure coefficient on the front-faceof the bilge keel is given by:

20.1=+pC

The pressure coefficient on the back-faceof the bilge keel is given by:

20.15.22 −⋅⋅⋅

⋅−=−

akk

kp fr

hC

φπ

The sectional pressure moment is givenby:

( )+− ⋅+⋅−⋅=⋅⋅∫ pps

h

mp CBCADdhlCk

2

0

with:

( )

( )( ) ( )

( )( )64531

1

232

1

10

34

27843

215.01621

215.03

mmmmmm

mmm

mHm

B

mmmmA

⋅+⋅⋅+⋅−⋅

−⋅⋅−+

⋅−⋅=

−⋅+=

23

while:

( )( ) ( )

( )( ) ( )110

10

210

6

110

10

210

5

104

213

2

1

215.01215.0

0106.0382.00651.0414.0

215.01215.00106.0382.0

0651.0414.0

0.1

mmH

mHmH

m

mmHmH

mH

m

mHm

mmm

DOG

m

DR

m

s

s

b

⋅−⋅⋅−

⋅⋅+−⋅+⋅

=

⋅−⋅⋅−

⋅+⋅−

⋅+⋅

=

−=

−−=

−=

=

For bRS ⋅⋅> π25.00 :

178

10

7

414.0

25.0

mmm

mDS

ms

⋅+=

⋅⋅−= π

For bRS ⋅⋅< π25.00 :

−⋅⋅+=

=

bRS

mmm

m

0178

7

cos1414.0

0.0

The approximation of the local radius ofthe bilge circle, bR , is given before.

The approximation of the local distancebetween the roll axis and the bilge keel,

kr , is given as:

2

2

0

293.00.1

293.0

⋅−+

+

⋅−

⋅=

s

b

s

s

b

sk

DR

DOG

DR

H

Dr

The total bilge keel damping coefficientcan be obtained now by integrating thesum of the sectional roll dampingcoefficients '

NB and 'SB over the length

of the bilge keels:

( )∫ ⋅+=k

k

LbSN dxBBb '')2(

4,4

7 Comparative Simulations

As far as ship motions are concerned,SEAWAY-T is an equivalent ofDREDMO. To check the calculationroutines for the time domain, as used in thepre-processing program SEAWAY-D andin the time domain program SEAWAY-T,comparisons have been made with theresults of the frequency domain programSEAWAY [8] for a number of ship types.An example of the results of thesevalidations is given here for the S-175containership design in deep water, as hasbeen used in the Manual of SEAWAY too,with principal dimensions as given below.

Length between perp., ppL 175.00 mBreadth, B 25.40 mAmidships draught, md 9.50 mTrim by stern, t 0.00 mBlock coefficient, BC 0.57

Metacentric height, GM 0.98 mLongitudinal CoB , ppCoB LL / -1.42 %

Radius of inertia, ppxx Lk / 0.33

Radius of inertia, ppyy Lk / 0.24

24

Radius of inertia, ppzz Lk / 0.24

Height of bilge keels, kh 0.45 mLength of bilge keels, kL 43.75 m

The body plan of this container ship designis given in the Figure 6.

Figure 6 Body Plan S-175 Ship

This S-175 containership design had beensubject of several computer andexperimental studies, co-ordinated by theShipbuilding Research Association ofJapan and the Seakeeping Committee ofthe International Towing Tank Conference[4,5]. Results of these studies have beenused continuously for validating programSEAWAY after each modification duringits development.For this ship, the motions have beencalculated in the frequency domain and inthe time domain at zero forward shipspeed.Additional data, used during the timedomain simulations, are:- maximum frequency of damping

curves: 00.5=Ω rad/s- frequency interval: 05.0=∆ω rad/s- maximum time in retardation

functions: 00.50, =Γ ji s- time interval: 25.0=∆t sThe potential coefficients and thefrequency characteristics of the wave loadsat zero forward speed, calculated bySEAWAY-D, have been input inSEAWAY-T and the calculations havebeen carried out for a regular wave

amplitude of 1.0 meter. Extra attention hasbeen paid here to the roll motions. In bothcalculations, the viscous roll damping hasbeen estimated with the Ikeda method. Inthe frequency domain, the results arelinearised for this wave amplitude of 1.0meter. Because of the relatively small roll-damping at zero forward speed, in thenatural frequency region the initialconditions of the wave loads will occurunstable roll motions in the time domain.Then, a long simulation time is required toobtain stable motions.

The agreements between the amplitudesand the phase lags of the six basic motions,calculated both in the frequency domainand in the time domain, are remarkablygood. Some comparative results of the sixmotion amplitudes of the S-175containership design are given in Table I.

Table I Comparison of Computations for S-175 Ship

25

Comparisons for a rectangular barge (100x 20 x 4 meter), with hoppers (25 x 14meter) fore and aft, are given in Table IIfor the natural roll frequency region. Theexperimental roll damping data were inputhere.

Table II Comparison of Computations for a Barge with Hoppers

Based on these and a lot of othercomparisons between the time domain andthe frequency domain approaches forlinear systems, it may be concluded thatSEAWAY-D + SEAWAY-T has an equalaccuracy as the frequency domainpredictions of these linear motions by theparent program SEAWAY.This conclusion holds that the pre-processing program SEAWAY-D providesreliable results.

8 Conclusions and Remarks

This new release of SEAWAY-D includesthe use of local twin-hull cross-sections,the N-Parameter Close-Fit ConformalMapping Method and (non)linear viscousroll damping coefficients. Special attentionhas been paid to longitudinal jumps in thecross sections and fully submerged cross-sections. Also, improved definitions of thehydrodynamic potential masses at aninfinite frequency and the wave loads havebeen added.

Based on a lot of comparisons, madebetween the time domain and thefrequency domain approaches for

linear(ised) systems, it may be concludedthat the computer codes SEAWAY-D andSEAWAY-T have an equal accuracy asthe frequency domain predictions of theselinear(ised) motions by the parent codeSEAWAY.

This conclusion holds that the pre-processing program SEAWAY-D deliversreliable results. It is advised however tocarry out a similar validation study withthe computer codes SEAWAY-D andDREDMO too.

9 References

[1] W.E. CumminsThe Impulse Response Function and ShipMotions, Symposium on Ship Theory,Institüt für Schiffbau der UniversitätHamburg, Hamburg, Germany, January1962, Schiffstechnik, 9, 101-109, 1962.

[2] W. FrankOscillation of Cylinders in or below theFree Surface of a Fluid, Naval ShipResearch and Development Center,Washington, U.S.A., Report 2375, 1967.

[3] Y. Ikeda, Y. Himeno and N. TanakaA Prediction Method for Ship Rolling,Department of Naval Architecture,University of Osaka Prefecture, Japan,Report 00405, 1978.

[4] I.T.T.C.Proceedings of the 15th InternationalTowing Tank Conference, The Hague, theNetherlands, 1978.

[5] I.T.T.C.Proceedings of the 18th InternationalTowing Tank Conference, Kobe, Japan,1987.

[6] J.M.J. JournéeSEAWAY-Delft, User Manual of Release3.00, Delft University of Technology,Shiphydromechanics Laboratory, Delft,

26

the Netherlands, Report 849, January1990.

[7] J.M.J. JournéeMotions of Rectangular Barges, 10th

International Conference on OffshoreMechanics and Arctic Engineering,Stavanger, Norway, June 1991.

[8] J.M.J. JournéeSEAWAY-Delft, User Manual of Release4.00, Delft University of Technology,Shiphydromechanics Laboratory, Delft,the Netherlands, Report 910, March 1992.

[9] J.M.J. JournéeStrip Theory Algorithms, Revised Report1992, Delft University of Technology,Shiphydromechanics Laboratory, Delft,the Netherlands, Report 912, March 1992.

[10] J.M.J. JournéeSEAWAY-D/3.00, a Pre-processingProgram to Calculate the HydromechanicInput Data for Program DREDMO/4.0,September 1992.

[11] J.M.J. JournéeUser Manual of Program SEAWAY-D/4.10, a Pre-Processing Program ofDREDMO/4.0, Delft University ofTechnology, Shiphydromechanics Labora-tory, Delft, the Netherlands, Report 969,March 1993.

[12] P. Kaplan and W.R. JacobsTwo-Dimensional Damping Coefficientsfrom Thin-Ship Theory, Stevens Instituteof Technology, Davidson Laboratory,Note 586, 1960.

[13] S.A. Miedema, J.M.J. Journée and S. SchuurmansOn the Motions of a Seagoing CutterDredge, a Study in Continuity,Proceedings of 13th World DredgingCongress WODCON’92, Bombay, India,1992, www.shipmotions.nl

[14] T.F. OgilvieRecent Progress Towards the Under-standing and Prediction of Ship Motions,Fifth Symposium on Naval Hydrody-namics, Bergen, Norway, 1964.

[15] F. TasaiOn the Damping Force and Added Mass ofShips Heaving and Pitching, ResearchInstitute for Applied Mechanics, KyushuUniversity, Japan, Vol. VII, No 26, 1959.

[16] F. TasaiHydrodynamic Force and Moment Produ-ced by Swaying and Rolling Oscillation ofCylinders on the Free Surface, ResearchInstitute for Applied Mechanics, KyushuUniversity, Japan, Vol. IX, No 35, 1961.