Hydrodynamic Coefficients to Calculate the Motions of Cutter Dredgers in the Time Domain by Cummins Equations

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Hydromeóhanic Coefficients to Calculate

the Motions of Catter Dredgers

in the Time Domain by Ciumnins Equations

by Ir. J.M.J. Journée

Report No. 968

March 1993

DOIft University of Technology

Shop Hydromechanics Laboratory Mekelweg 2

2628 CD Deift TheNetherlands Phone 015 - 786882



The software package DREDMO/4.O of the "Deif t Hydraulics;" predicts the behaviour of seagoing cutter dregders in near-shore conditions,

which can be important with respect to the cotxgtruction of the

dredger and the assesment of the downtime.

The motion behaviour of the seagoing dredger has been descrIbed by

non.- ]!inear Cummins equations, whióh have to be solved in the time domain. The required input data on hydromechanical coefficients,

retardation functions and frequency domain wave loads can be deri-ved now with a new prë-processing program. This new program, based on the ship motions program SEAWAY/4 10 and called SEAWAY-D/4 lo, creates the hydromechanical input data file with .a minimum risk on human input errors and it makes DREDMO/4. O much móre accessible

for less-specialist users.,

The present report describes the underlying hydromechanical theory of the new SEAWÀY-D/4.1O release.. Also, comparisons with frequency

domain results have been given.



i. Introduction i

Equations of Motion .. . 3

Frequency Domain Calculations 4

Time Domain Calculations ...Il

Viscous Damping 23

5-i. Experïmentai Roil Damping Data ...24

52. Empirical Roll Damping Data 28

Cömparative Simulations 35

7.. Conclusions and Remarks


T 38




1,. Introduction

The prediction of the behaviour of cutter dregders in near-shore conditions can be important with respect to the construction of the dredger and the assesment of downtime.. To be able to make downtime predictions, the "Delf t Hydraulics" together with the "Laboratory of Soil Movement" and the "Deif t Shiphydromechanics Laboratory",

both of the "Deif t University of Technology", developed the compu-ter program DREDMO [13] in the early 80's.

The behaviour of the seagoing dredger has been described by non-linear so-called Cuinmins equations, which have to be solved in the

time domain. These Cunimins equations require hydromechanical

coef-ficients, retardation functions and wave load series as an input,

together with geometrical and operational data on the ship and the

operational working method.

The required input data on the hydromechanical coefficients, the retardation functions and the frequency domain wave loads had to be derived from model experiments or from calculation results of a

suitable ship motion computer program. However,, the creation of this hydromechanical input data file appeared to be very much time

consuming. The DREDMO-program and the pre.- and post-processing pro-j

grams were running on main frames, but a specialist was required to

operate the software.

In 1984, the "Deift Hydraulics" decided to develop a PC version of the program DREDMO, to promote the use of their commercial DREDMO


In the late 80's the author had developed program SEAWAY [6,8.].

SEAWAY is a frequency domain ship motions computer program, based

on the ordinary an the modified strip theory, to calculate the wave-induced motions and the resulting mechanic loads of monohull

ships or barges, moving forward with six degrees of freedom in a

seaway. When not taking into account interaction effects between the two individual hulls, also these calculations can be carried out for symmetrical twin-hull ships, such as semi-submersibles and catamarans. Also local twin-hull sections, as for instance appear

by cutter dredgers, are permitted.

The potential coefficients are calculated for an infinite water depth., but the wave potential has been defined for a restricted water depth. This holds that the program is fairly suitable for

restricted water depths, with keel clearances down to about half

the ship's draught, too.

Also, added resistances due to waves and shearing forces and ben-ding and torsional moments can be calculated. Linearised springs,

to calculate the behaviour of anchored ships, and free surface

anti-rolling tanks can be included.

Several ideal wave spectra definitions have been used and an input

of wave spectra definitions is possible too.

Computed data have been validated with results of other computer programs and with a lot of experimental data. Based on these vali-dation studies and on the user's experiences, obtained durIng an

intensive use of the program by the author, students, institutes and industrial users for over four years, it is expected that the

program.isf ree of_significant, errors ._At the present, SEAWAY or a

derivative of this program is licensed by 32 users. i.



The program, written in the. standard FORTRAN/77 language, can be

used on any IBM-compatible AT Personal Computer equipped with a

mathematical co-processor. HOwever, because ail system-related

parts have been assembled in only one separate subroutine, the

program. can be made suitable for other computer systems easily. A demo-diskette of SEAWAY/4.iO can be obtained, free of charge.

In 1990, on behalf of the "Laboratory of Soil Movement" of the

"Deif t University of Technology", the author used the relevant

parts of the program SEAWAY/3.0O [6], to create a pre-processing

program for DREDMO. The required formats of the hydromechanical

input data of the DREDMO-program, had been defined by the

"Labora-tory of Soil Movement" for this.

This pre-processing program, called SEAWAY-D, creates the hydrome-chanical input data file for DREDMO with a minimum risk on human input errors and makes DREDMO more accessible for less-specialist students. The use of this pre-processing program S'EAWAY-D makes no very high demands on the shiphydromechanical knowledge of the user

of the program DREDMO.

in 1991, the "Laboratory of Soil Movement" had completed their work on DREDMO with the delivery of PC pre- and post-processing programs and a user-interface, which permits less-specialists to work with

the DREDMO package [131 too.

in 1992, on behalf of the "Delf t Hydraulics", the release 3.00 of the pre-processing program SEAWAY-D (10] and an input control pro-gram SEAWAY-H, to check the input data of the offsets of the under water geometry of the ship, have been added to the new commercial

DREDMO/4.0 software package of the "Deif t Hydraulics!'.

In.1993, a fully revised release 4.10 of SEAWAY-D, based on program

SEAWAY-4.i0 [8], has beenmade available,. Also, the input control

program of the huliform SEAWAY-H has been modified.

This new release of SEAWAY-D includes the use of twin-hull cross sections, the N-Parameter Close-Fit Conformal Mapping Method and: (non-)linear viscous roll damping coefficients. Special attentior has been paid to longitudinal jumps in the cross sections and to

fully-submerged cross sections. Also, improved definitions of the hydrodynamic potential masses at an infinite frequency and the wave

loads have been added. Finally, many numerical routines have been


From now, program SEAWAY-D will follow all new developments of the parent program SEAWAY. New releases of SEAWAY-D will be submitted

to the users free of charge.

This report contains a brief survey of the underlying hydromechani-cal theory of DREDMO and SEAWAY-D, with references to the relevant literature. A validation of the calculated results of SEAWAY-D has been given. For all strip-theory algorithms, special reference is

given to [9].

The "User Manual" of the present SEAWAY-D/4.10release, with an


2. EquationB of Motion

The coordinate systems are defined in the figure below.

Figure 2-A. Coordinate System and Definitions


Three right-handed coordinate systems are defined.:

G-(xb,yb,zb).: connected to the..ship,

with G at the .ship'.s centre of gravity, xb in the ship's. centre line,,

Yb in the Ship's port side direction and

Zb in the upward direction

O-(x0.,y0,z0): fixed in space.,

with O in the still water surface,

x0 in the ship's forward direction at t=O,

Yo in the ship's port side direct!on at t=O and z0 in the ship's. upward direction at .t=O

The axes X0 and Yo are. lying in the stili, water

surface. G- (x,y,z) or

G- (x1,x21,x3) : moving with the ship's displacement,

with G at the. ship's centre of gravity,

x or x1 parallel to still watér surface

y or x2 parallel to still. water surface z or x3 in, the upward direction

The angular motions' of the body about .the body

axes' are denoted by.:


3. Frequency Domain Calculations

Based on Newton's second law of dynamics, the equations of motion

of a floating object in a seaway are given by:


Mj,j.xj Fj for i = 1, 2, .... 6

in which:


: 6x6 matrix of solid mass and inertia of the body

Xj : acceleration of the body in direction j

sum of forces or moments acting in direction i

When defining a linear system with simple harmonic wave exciting

forces. and moments, defined by: FwÏ (a), t) = Fwai(w)

COS[wt+ (w)]

then the resulting simple harmonic motions are:


Xaj (w) CoB(wt)

wxa1(w) sin(wt)

?Caj (w) COB (wi:)

The hydromechanic forces and moments F, acting on the free

float-ing object in waves, consist of:

linear hydrodynamic reaction forces and moments expreSsed in

terms with the hydrod'namic mass and damping coefficients:

-aj,j (w)

i(w,t) -bjj


linear hydrostatic restoring forces and moments expressed in a term with. a spring coefficient:

With this,, the linear equations of motion become:


J1 [-aj,j (w) 'xj (w,t) -b,1

+Fwai (w). Cos [wt+Ej C


k (w,t,)







(w) Xj (w,t)

+Cj,j Xj


Fwaj(w) cog [wt+(w)]

for i = 1, 2, .. 6

The hydrodynarnic coefficients aij(w) and b1 (w) and the wave load

components Fwaj(w) and Ej(W) cati be calcuia'ted with the available two- or three-dimensional techniques.

So for this the results of strip theory programs, like for instance

the program SEAWAY [8], can be used. According to the strip theozy,

the total hydromechanic coefficients and wave loads of the ship can

be found easily by integrating the cross sectional values over the

ship length.

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 have shown 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 sometimes

even lower.

The strip theory is based upon 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

ex-perimental results or by empirical formulas.

The strip theory is based upon 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, inaccu-racies can be expected.. Also, the strip theory does not distinguish betweèn alternative above water hullf orme.

Nevertheless these limitations for zero forward speed., generally

the strip theory appears to be a successful and practical theory

for the calculation of the wave induced motions of a ship.

For the determination of the two-dimensional potential coefficients.

for sway, heave and roll motions of not fully submerged ship-like cross sections, these cross sections can be mapped conf ormally to the unit circle, by the so-called 'Two-Parameter Lewis Transforma-tìon. Also the N-Parameter Close-Fit Conformai Mapping Method has

been included here..

The advantage of conformal mapping is that the velocity potential

of the fluid around an arbitrary shape of a cross section in a

com-plex piane can be derived from the more. convenient circular cross section in another complex plane. In this manner hyrodynamic prob-lems can be solved directly with the coefficients of the mapping

function, as reported by Tàsai [15,16].

The advantäge -of mking "use of the -Two--Pa-ameter- Lewis Conformal

Mapping Methodis that the frequency-depending potential coef f i-ciente are a fu.ctlon of the breadth, the draught and the area of

the cross section, only.



Another method is the Frank Method '[2]:, also suitable for fully

submerged cross sections. This method determines the velocity

potential of a floating or a submerged oscillating cylinder of

infinite length by the integrai equation method u'tilising the

Green's function, which represents a puisating source below the

free surface.

To avoid so-called "irregular frequencies" in the operatibnal

fre-quency range of not fully submerged cross sections, each Frank

sec-tion will be closed automatically at the free surface with a few

extra points. This results into a shift of these irregular f requen-cies towards a higher frequency region.

The two-dimensional pitch and yaw coefficients follow from the

heave and sway moments, respectively.

Finally, a method based on work published by Kaplan and Jacobs [121 and a longitudinal strip' method' has been used for the determination of the two-dimensional potential coefficient's for the surge motion. At the following pages, the hydrornechanic coefficients and the wave

loads for zero forward speed are given' as they can be derived from

the two-dimensional values, defined in a coordinate, system with the origin O in the waterline..

The symbols', used here, are:

solid mass and inertia coefficents of the body

mj,j' (w) sectional hydrodynamic. mass coefficient

(w) sectional hydrodynainic damping coefficient

Fwj' (w) sectional wave exciting force or moment


(w) sectional Froudè-Krilov force or moment

Wj*(,w) equïv'alent sectional orbital acceleration


equivalent sectional orbitai velocity


sectional half breadth of waterline

Xb longitudinal distance of cross section to centre of gravity, positive forwards

0G vertical distance of waterline to centre of gravity, positive upwards

BG vertical distance of centre of buoyancy to centre. of gravity, positive upward's

V volume of displacement

radius of gyration in air for roll k radius of gyration in air for pitch kzz radius of. gyration in air for yaw

density of' water


The solid mass and the potential mass coefficients are given by: M1,1 = M. M113 = O M1,5 = O M2,2 = M =

M24 = O

M2,6 = O

M3, =M =

M3,5, = O M4,4 i

M46 = O

M55 = I=




b -BG.a'1,5 M6,6 = 'zz = a6, 6 =



= M1,

aj,i = aj,j

The remaïnng, mass coefficient's are zero.



a11 =

a,3 = O

a1,5 -BG: a2,2 = a2,4 = Jm2,4 d.Xb +OGa2,2 a2,,6 = Jm2,2..xb.dxb



j'm3,3'dxb a3,5 = -jm3,3


= kpV

a414 fm4,4 dXb L L

a46 = Jm42

bdXb +OGa2,6


The potential damping and the spr±ng coefficients are given by: b1,1 = Jrli,l.dxb C1,1 = O 8 b113. = O cl,3 = O b.,5 = -BGb111 c1,5 = O I '



C22 = O

L b2,4 = lcb +0Gb2,2 c2,4 = O L b2,6


c216 = O L

b33. = Jn33.dxb

L c33, b35,



C35 =

-2.p.g.Jyw'.xb. b4,4 = Jn44.dxb +0G.Jn4,2 b.

C4,4 = pgVGM

L L +OG.b24 b4,6: = 4, 2 Xb b +0G b26

C4,6 =


b55 =

jn3.3.xb2. L b -BGb1.5 5,5 = 2.p.g.Jyw'.x2.dx, b6,6 = 2, 21 V C6,6 = O =


= C,j


The wave loads are givén by:

Fwi = JFwI'

with:. Fwi' = +mii.Vwi* .Inl 'V *

JFW2'.dxb with: Fw2' Fw3 =. +11122 w2* +fl22 +m24 w4.* +n2,4 I . .


, *, with: Fw3

= +m33 V3 +n33

Vw3 +FK3



with: F4' +11142I-..,VW2, 'Fw5 = JFw5' dx with: Fw5' = 'Fwi' BG Fw3.' X Fw6 = JFW6' with: FwG' = Fw2'Xb

These 'f ormulationsE of the hydrodynarnic excIting and reaction forces

and moments can only be used in the frequency domain,, since a j and b both, depend on. the frequency of mot ±on w only and the

ek-citatiiig wave loads have a linear relation with the wave amplitude

In irregular waves the response of the body can be determined by

using the superposition principle., so using linear response ampli-- tüdeampli--öprators bétweên mOt±ön=and waveaniplitudes: . -

-* .vw2


V4 +'FK2,


In the following figures an example has been given of the

hydrody-nainic potential mass and damping and the wave loads for roll in the

f reqi.iency domain. loo 80 50 40 20 r-I 4000 ¿J o O .00 4000 ., 2000 .rI u w O eQ .00


o .00 .10 .20 .30 .40 .50 Frequency in Hz

Figure 3-A. Hydrodyna.mic Potential Mass for Roll

.10 .20 .30 .40 .50

Frequency in Hz


Figure 3-B. Hydrodynamic Potential Damping for Roll 10000

. 8000

.10 .20 .30 .40 .50

Frequency in Hz

Figure 3-C. Transfer Function, of Roll Wave Moment

i 5000 20000 15000 :12000 8000


4 Time Domain Calculations

As a result of the formulation, in the frequency domain, any system

influencing the behaviour of the floating body should have a linear

relation with the displacement, the velocity and the acceleration

of the body. However, in a lot of cases there are several

complica-tions which perish this linear assumption, for instance the non-linear viscous damping, forces and moments due to currents, wind,

anóhoring, etc.

To include these non-linear effects in the, vessel behaviour, it is necessary to formulate the equations of motion in the time domain,

which relates instantaneous values of forces, moments and motions.

For the description of the hydromechanic reaction forces and mo-ments, due to time varying ship motions, use has been made of the classic formulation given by Cummins [i]. Also reference is given

here to Ogilvie [14].

Ciuyniins Equations

The floating object is considered to be a linear system with the translational and rotational velocities as input and the reaction

forces and moments of the surrounding water as output.

The object is supposed to be at rest at time t=t0. Then during, a Short time At an impulsive displacement x, with a constant velo-city V, is given to the object.


Ax = VAt

During this impulsive displacement, the waterparticles wil start to

move When assuming that the fluid is rotation-free, a velocity

potential , linear proportional to V, can be defined:

= VW

for: to < t < t+At

in which is the normalised velocity potential.

After this impulsive displacement Ax, the waterparticles are still moving. Because the system is assumed to be linear, the motions of the fluid, described by the velocity potential , are proportional

to the impulsive displacement Ax.



for: t > t+At

In here x is a normalised velocity potential.

The impulsive displacement Ax during the period (t01t0+At) does not

influence the motions of the fluid during this period only, but

also further on in time.

This holds that the motions during the period (t0,t0+At.) are influ-enced also by the motions before this period.

When the object performs an arbitrarily in time varying motion,

this motion can be corsidered as a succession of small impulsive



Then the resulting total velocity potential tt.) during the period (ta, t+t) 'becomes.: 6 = E V4 j=l in here: n = number of timesteps tn = tO +fl.. t. = t: + (n-k)

V = j -th velocity component during period (t,.t+t.)

j-t'h velocity component during period (.tnk.,tnk+At).

= normalised velocity potential, caused by a displacement in

direction. j during period (t:n,.tn+'t)

Xj = normalised velocity potential caused by a dispiacement in

direction j during, period (t.k,tflk+t) Letting t go to zero, yields.:



cj (t)



(-r) .*j ('r) '.dr

in which:

= j-th velocity component at time t.

The pressure in the fluid follows from the. linearised equation of


p.= -p

integration of these pressures. over the wetted surface .S of t'he

floating object delivers the expression for the hydrodynamic

reac-tion forces and moments F'1.



is the genea1ised dire:ctional cos±ne,, F1 becomes:

= -




j!i [ E fljdS + +


[ xj(tnkitnk+At) Vjk.t i






When defining: Aj,.j



ÔX (t-r)

Bj j (t) 'J J n dS


the hydrodynamic forces and moments become:


F [ Aj,jj,(t)


Xj (r)

dr ]

for i =

Together with the linear restoring spring. terms Cij.xj!! and the

linear external loads !'X(t)", Newton's. second law of dynamics de-livers the linear equations of motions in the time domain:



+C 13 (t:) ] = xi(t) 6 E j, j +Aj, ij (t)


+Cj,j xi('t) J = Xj(t) fori =

Ref ering to' the basic work on this subject by Cuxmnins [li, these

equat ions 'of motions are called the "'Cuins Equationshl!.

in which: for i =

(!t) = translational or rotational acceleration in direction J

at. time t

(t.) = translational or rotational velocity in direction j

at time t


(t) = translational or rotational' in direction J

at time t.

= solid mass or inertia coefficient

Ai,j = hydr.odynaini'c mass coefficient

= retardation function

C'i j spring coefficient

X, (t) = external load, in. direction i at time t

When replacing in the damping part "r" by "t-r" and changing the integration boundaries, this part can be written in a. more



Hydroniechanic COefficients

The linear restoring spring terms C,3, C315, C4,4,, C53 and .C5

. can be determined easily from the underwater geometry and the centre of gravity of the floating object. Genera].ly, the, other

C-values are zero.

To determine A


and B

. the velocity potentials j, and Xj. have

to be found, which is very complex.

A much more easy method to determine Ai and Bij can be. obtained.

by making use of the hydrodynami.c mass' ànd. damping data found by existing two- or three-dimensional potential theory based computer

programs in the frequency domain.

Relative simple relations can be found between A11 and Bj and

the calculated data of the hydrodynamic mass and damping, ri the

frequency domain.

The floating object is supposed to carry out an harmonic

oscilla-tion .with an amplitude i in the direcoscilla-tion j:

Xj =


A substitution in the. Cuimnins equations delivers:


c0s(w't) -w. 1(r)'sin(wt-.wr)dr +Ci,jcos.(wt) =

= Xj(t)

.f or ,i = 1.,, . .6

This results into:


[ i,j - JBi,1r:.sin(wr.dr ]vcOs(wt)'



J.Bj.,j(i).cos(wr).dr ]..sinw.t) +[ C,j ]cos(.wt) = Xi(t)

for i =

In the classic frequency domain dscription. these equations of

motions are presented by:'


+ai,j(w)' ] cos(.wt)

,j(w)' ] 5ïfl(.wt)

4 CjJ

]..cos(wt.). = X1


in here: for i =. i, ..6

aj,j (w) = frequency--dependent hyd-rodynam-ic-mass_--coe.f--f:icient

h, ''('c ....= freiny:d'ependert hydrod3namic 'damping-


A,j =





w J


Wh5n cOmparing the time domain and the frequency domain equations,

both with linear terms as published by Ogilvie [14], it is found:

i aj,j(w) = w (r) sin(wr) dr o (w) =



coS(wr) dr



After a Fourier re-transformation, the damping term delivers the

retardation function,:



(w)c08(wr) &

Then the mass term follows f rom:

,j(r) sin(wr) dr

This expression is valid for any value of w, so also for


which delivers:

= aj,j(w=)

Addition of Non-Linearities

So far, these equations of motions are linear. But non-linear

con-tributions can be added now to X(t) easily..

För instance, non-linear viscous roIl dämping contributions can be

added to X4:




Also it

is possible to include non-iinear spring terms, by consi-dering it as an external load and shifting its contiibution to the

right hand side of the equations of motions, for instance: C4,4 = O

AX4 =-pgV

in which GN (), isthe-t-ransverse-metacantricheight_at_arbitrarily




= 3

Bi,i(r)' =



2 lt



cog (ar),




Some Numerical Receipes

Many computer programs fail when calculating bjj

at. too high

a. frequency. This holds that, when. determining Bj, the numerical

calculations can be carried out in a limited frequency range Owû


So, a truncation error Bij. will be introduced::

2 r


(,r) =- 1b1 1(w)co8(wr')dw

' lt J




For the uncoupled damping coefficients, so. when i=j

tion. errör can be


The relation between the damping coefficient b1 j(LJ)

tude ratio of the radiated waves and.the oscilI.tory

is given by.:

sin (r)

Or + -y

+ ln(ûr)' +


t'runca-and .the

ampli-motion ajj(w.')

From this an approxImation can be found for the tail of the damping




The value of ßj j follows from the calculated damping value. at the.

highest f requeny used, w=O. This holds that it is supposed. here that ai,j(w) is constant for


Then the truncation error becomes:




+ E n=1

2n ('2n)!

in which: -y 0.577215. (Euler constant)

Studies carried Out in the past, have showed that in case. of a

suf-ficient high value of' û the contribution of B1 j into Bi ï is

often Small.. The. potential damping calculations werk. 'based on

'nume-rical routines as used in

computer program SEAWAY [8,91 In this

program special attention

has been paid to the potential

calcula-tions at very high frequencies For normal merchant ships 5 radians .per second., which can be 'reached by the routines in SEAWAY, is a

fairly good value for' the maximum frequency û.



So the retardation function iB approximated by the numerical

solu-tion of the integral:

B,(r) =



The damping curve has to be calculated at N, constant

frequency-intervals w, so: =

When calculating here the retardation functions it assumed that at each frequency- interval the damping curve is linear inceasing or


Bi j(r).


Wn.. i

Figure 4-A. integration of the Damping Curve

Now the contribution of this interval into


can be calculated analytically. This holds that, because of a lar.e w or a large r,

the influence of a strongly fluctuating cos(w.r,) at this interval

will be taken into account.

Then the numerical integration is given by:

2 N




ir r n=1 L 2 +


in which: = W n-1 = = b -b...1 [cos(w.r) - cos(w..1.r)] ]


Bi1 (r=0)

2 6 r E


L. .Jb±j(w).d

Because the potential damping is zero for w=0, the expression for the damping. term leads for w=O, so cos(wr)=1, into the following

requirement for the retardation functions:

1(r)dr = o

In the equations. of motions, the retardation function multipliéd

with. the velocity should be integrated over an infinite time:

(r ) x.j. ('t-



However, after a certain time the fluctuating values of the

integral have reached already a very small, value:.

A useful limit-value for the corresponding integration t:±me can be

found with: N E'. IbnI n=i ir 'co E B . ('O) with: = 0.010

So the Cuinmins equations, which are stili linear here, are. 'given





+C,1x(t) ]

= Xj(t)

for i = :L,.. .6

The numerical integrations can be carried out with the trapezoid rule or with Simpson's rule, with a time. step


Because of a relative small time step M required to solve the equations of

motions numerically, generally the trapezoid rüle is sufficient



For r=O, the value of the retardàtion function can be derived

simply from the integral of the damping:




The hydrodynamic mass coefficient follows from:

Aj, =


When, this mass coefficient is not available for an infinite f

re-quency, it can be calculated from a mass coefficient at a certain frequency and the. retardation function:



=a,(fl) +-



Nr A,r = rj

the numerical solution of this integral can be f òund in an analog

way as for the retardation functions: r1

(r) sïn(ûr) dr =




E -

Isin(û n Ar] -sin[û n-1) Ar)]I +

n=1 Ar L J


r + - i B.i JL f in which: 4B = B±,(fl) -B1(n-i)

Analog, to this, the numerical solution of the frequency-dependent

damping is:


bj,j(í) JBi,j(r).cos()r).dr =

.1 AB

= -

-a [Cos[û.n.A.r] -cos[i' (n-1,) .Ar)]] +




+-& B1,j(r=rNr) sin[12NrAr]

(r=O) -B1 r = r WT



The path of the ship in the xo,Yo , z0 system of axes can be derived by integrating the velocities of the ship's centre of gravity:

The Euler. equations of



(ï -y +è)

M {' +x -


ç ) M (z -x è


- ('-') .

e ..iyy' -




with in the right

subscript h: subscript w: subscript ext:


motion are Written,

= Xh +Xw +Xext h, 'W ext = Zh +Zw +Zext Kh +Kw +Kext = Mh +M +Mext = Nh +Nw +Next hand sides:

i near hydromechani.c loads linear wave loads

non- linear hydromechanic loads and (non) linear external loads, caused by wind, currents, anchor

iines, ciitter, etc..

in the x,y,z system of o =


-rsjnb ro = x. +. cosis ZO z = è0 = =



With the hydromechanic ioads as defined before, the equationS of

motion are defined as given below.

Surge motion:,

iM. +M



+A11 +BjC +C11x

+A13 ±B13


+A15 +Bj15i +C1.5o = Xr,j +Xext

Sway mot ionj:


+M (+cb +A22' +B22.jr +C22y


+B24q, 24c' i-A26 +B26i +C26'. = ext

Heave motion:

+M (-c

+r) +A3j

+B31-x +C3;1'x.


+C3:3 Z +A35, +335 +C3!5 O = Zext

Roll motion:


(.1yyIzz)'b +A42

+B42r +C4.2y

co +B44 co co +A46 b +B46 ib +C46 b = Kw +Kext

Pitch motion:

'yy - (izzicic'b +A51X +35;1'x +C51x

+A53Z-FB53'Z +C53'z +A55O +Br55.9 +C55.O +Mext

Yaw motion:

- (i-I)


+A62y +B62'y +C62y



Some of the coefficients in these six equations of motion. are

zero After ommitting these coefficients and ordering the terms,

the equations for the aòceierations are as follows.

Surge motion:



= +Xw +Xext



With known

coefficients and righthand sides of theseequations,

the six accelerations can be determined by a numerical method, for

instance the Runge- Kutta method.

Because of periods

withan extreme highs:tiffness of-

a cutter

dred-ger system, an adapted method, developed by the ItDelf t

Hydraulics", has, been used in the program DREDMO/4.O' [13]

Sway motion:

(M+A22) +A24'p +A26 = +Yq


22 Y -B24 rp -B26

+M (-cb


Heave motion:



= +Zq +Zext


-C33 z -B359 C5 O

+M (+c


Roll motion:

( +A4 +A46 = +Kw+Kext

-B42y B44





Pitch motion:


+A51 X


= +Mw +Mext

-B51k -B53



è -0550


Yaw motion:

(Izz+A66) . +A62.r ±A64 = +N +Next

-B62-' -B64



5. Viscous Danuinq

The non-linear vIscous sway and yaw damping can be approximated by:



1:' with: b22(2)

= l/2pLTCd


I'I ì,

with.: b66'(2) =i/6pL3.TCd


Cd 1.50

The total (non-)linear roll damping term in the left hand side of

the equation of motion for roll can be expressed as:

[b44 + b44a')] ' + b44a'2



b44 = linear potential roll, damping coefficient

b44a = linear(ised) additional ròll damping coefficient b44a(2) = non-linear additional roll damping coefficient

The linear potential roll damping coefficient b44 can be determined

as. described before.

For time domain calculations a linear as well as a non- linear roll

damping coefficient can be used.

However, for frequency domain calculations an equivalent linear

roll damping coefficient has to be est±mated. Thi linearised roil

damping coefficient can be found by requiring that an equivalent

linear damping dissipates an equal anount of energy as the

non-linear damping, so: TÇ,



, dt



Then Çhe. equivalent linear additional roll damping coefficient






The additional roil damping coefficients b44a() and b44a2') are

mainly caused by viscous effects.

Until now it is not possible to determine these additional coeff i-cients in a pure theoretical way. They have to be estimated by free roiling model experiments or by a semi-empirical method,, based on

theory and a large number of model experiments with systematic

varied ship forms.

The l±near(ïsed) and the non-linear equations of pure roll motions,

used to analyse free roiling model experiments, are presented here. Also, for zero forward ship speed, the. algorithms of the empirical method of Ikeda, Himeno and Tanaka [3] are given.


= b44a(2)

Pa c


24 5-i.. Experimental Roil DarnpIng Data

in case of pure free roiling in. stili water, the linear equation of the roil motion about the centre of gravity G is given by:

(i, +a44,) + (b44 +b44a) + C44 ço = O,


'a44 = potentia] mass coefficient

'b44 = potential damping coefficient

b44a = iinear(ised) additional, damping coefficient

'C44 = restoring term coefficient = pgV'GM

This equation 'can. be' rewritten as:

+ 2v q' + = O:

in which.:

'i +a44

b44 +'b44a

quotient of, damping. and moment of Inertia


+a unda.mped natural roil frequency squared:

When defining a non-dimensional roll dampIng by':

V k



'the equation of motion can be rewritten as:

+ 2,cw01j, '+ w0q

= O

Then, .the iogarithic decrement .of roil is:




Because the relation:

= 2,

-and the assumption:


it can be written,:


This leads to:




So the non-dimensional total roil damping is given by:


= (b44 +b44a)

2 c44

The non-potential part of the total roll damping coef.filcient

follows from the. average value of



2' C44

'b44a = IC' '



These K-values 'can easily been found when results of free rolling experiments with


model in still, water' are available.



results of these free rolling tests are presented by':

a as a function of a with: 25 and: )'

2 absolute value öf the average of two

successive positive or negative

maximum roll angles

= logIr



L 9, (t+T9,)

= (i.) - Pa(+1) I =

absolute value of' the difference of

two successive positive or negatïve

maximum roll angles


-Then the total non-dimensional roll damping coefficient at the

natural frequency becomes.:

IC = K




log 2 2

These experiments deliver no information on the relation with the

frequency of oscillation.. So it has to be decided to keep the. addi-tional coefficient b44a or the. total coefficient b44+b44a constant.

The successively found values for K, plotted on base of the average

roll amplitude, will often have a non-linear behaviour as

illustra-ted in the' next figure.



experiments fitted by:

Figure 5-i-A. Non-Dimens±bnal Roll Damping doef.fi.cients Determined by Free-Roiling. Experiments

Fora behaviour like this, it will be found,:

K =

K1 +

This holds' that during frequency domain calculations, the damping

tern is depending on the solution for the roll amplitude.

For rectangular barges. (LxBxT), with center of gravity in the

waterline, it is found by Journée [7].:

= 0'. 0013 (B/T)2

IC2 . 0.500 .

'Then the total damping term becomes:

2 c4




The linear additional roll, damping coefficient becomes:



for the non-linear additional roll damping coefficient, quasi-quadratic damping coefficient is found:


2 c44



2 c44

Because of this roll damping "coefficient" includes

denominator, it varies strongly with the time.

An equivalent non-linear damping term can be found by

that the equivalent quadratic damping term dissipates

amount of energy as the quasi-quadratic damping term, so: T97 b44a2) 1971 b44a2') = .97

., dt



.2.c44 Wo2 b44 2.7 T 2' C44 IC2 a 'j? W'0 o

Then the equivalent quadratic additional roll damping cQefficient

b44a(2) becomes:

With this, the damping term 'based on experimental -vaiues, as

given in figure 5-i-A, becomes:

'[b44 +b44aW] + b44a(21



(p + Wo


in the. .;, dt





W.'Q2 1971

So far in the equations of motion, pure roll motions with one

degree of f reedom are observed.. Cöupling effects between the roil motion and the other motions are not taken ±nto account. ThIs can

be done in an iterative way.

Experimental or enpiricai v1es of

and K2 deliver starting

values for b44a"

and b44a2J. With these coefficients, a free

rolling experiment with all degrees of freedom can be simulated in the time domain. An analyse of this simulated roll motion, as being a linear pure roll, motion wit1 one degree of freedom, delivers new

values for b44a(1) and b44a2. This procedure has to be repeated

until a suitable convergenge has been reached.

An inclusion of the natural frequency W0 in this iterative proce-dure delivers also a reliable, value for the estimated solid mass

moment of inertia I.

However, this procedure i's not inciudedi-nthe--DREDMOprog-ram.



an equal


5-2.. EmDirical Roll Damping Data

Because of the additional part of the roil damping is significantly influenced by the viscosity of the fluid, it is not possible to

calculate the total roll damping in. a pure theoretical way. Besides

this, experiments showed also a non-linear (about quadratic)

beha-viour of the äddit'ional parts of the roil damping.

As mentioned before, the total non-linear roll damping term in the

l,ef t hand side of the equation of motion for roll can be expressèd


[b44 + b44aW] i


For the estimation of the additional parts of the roil damping, use

has been made of work published by ikeda, Himeno and Tanaka [.3]. Their empiric method is called here the "Ikeda Method".

At zero forward speed, this Ikeda method estimates the following,

components of the additional roil damping coefficient of a ship:


= o


= b4:4f2 + b44e(2 + b44k2


- b44f(2) non- linear f riction damping

- b44e(2) non- linear eddy damping

- b44,k(2) non-linear bilge keel damping

ikeda,, Himeno and Tanaka claim fairly good agreements between their prediction method and experimental results.

They conclude that the method can be used safely for ordinary ship forms. But for unusual ship forms,, very full ship forms and ships with a large breadth to draught ratio the method should not be

al-ways sufficiently accurate.

Even a few cross sections with a large breadth to draught ratio can

result in an extremely large eddy making component of the roll dam-ing. So, always judge the components of the dampdam-ing.


Nomenclature of Ikeda

in the description of the Ikeda thethod, the nomenclature of. I'keda is maintained here as far as possible:

p = density of water

V = kinematic viscosity of water g. = acceleration, of gravity

w, = circular roll f requncy

P'a = roll amplitude

R, = Reynolds number

L = length of the ship

B: = breádth of the ship

D = average draught of the. ship

'CB' = block coefficient

Sf = hull surface area

0G = distance. of centre of gravity above still water level B8, = .sectionaÏ breadth on the waterline

D5 = sectional draught

u5 = sectional area coefficient

H0 = sectional half breadth to diaught ratio

a1 = sec...iona.1 Lewis coefficient

a3 = sectional Lewis coefficient

= sectional Lewis scale: factor

average . distance. between roll, axis and. hull surface

hk = height of the bilge keels

= length of the bilge keels

rk = distance between. roil axis and bilge keel

= correction for increase of flow velocity at the bilge

= pressure coefficient

= lever of the. moment

= local radius of the bilge circle

For numerical reasons two restrictions have to be made during the

sectional calculations:

- if g,8 .> 0.999 then u.5. = '0.999 - 'if 0G < -D5a5 then 0G. =





with a


hull surface area Sf, approximated by:.

Sf = L (1.7D +C B)

When eliminating the temperature of water, the kinematic viscosity can be expressed into the density of water by the following

rela-tion in the kg-m-s system: - fresh water;



1.442 +

'0.3924 (p -1C00) + 0.074'24 (p



- sea water:

zi106 l.063 + 0.1039 (,p -1025) + 0.02602 (.p i025),2 m2/s

Kato expressed the skin friction coefficient as:

Cf =

1.328R0500 +O.0i4R0-14

The first, part in this expression represents the laminar flow case..

The second part has been ignored by ikeda, but has. been included


Using this,

the non-linear roll


coefficient due to s:kin

friction at zero forward speed is expressed as:




Ikeda confirmed the use of this. formula for the three-dimensional turbulent boundary layer over the hull, of an oscillating ellipsoid

in roll motion.

30 Frictional Roll Damping, b44f (2)

Kato deduced semi-empirical formulas for the frictional roil

damping from experimental results of circular cylinders, wholly

immersed in the fluid.

An effective Reynolds number for the roil motion was defined by:


0..5Ï2 (rfa)2.w

In 'here, for Ship forms the average distance between the roil axis and the hull surface rf can be approximated by:



= 0.5 cos'

= M8.

a1 (i+a3)



The values of rmax(r) follow from:


Rddy Making ROil Damping, b44e(2)

At zero forward speed the eddy making roll damping f or the naked hull is mainly caused by vortices, generated by a two-dimensional

separation. From

a number of

experiments with twodimensional

cylinders it was found that for a naked hull this component of the roil moment is proportional to the roll velocity squared and the

roll amplitude. This means that the non-linear roll damping coef

fi-cient does not depend on the period parameter but on the hull form


When using a simple form for the pressure distribution on the hull surface i.t appears that the pressure coefficient is a function of the ratio y of the maximum relative velocity to the mean

velocity timean on tie hull surface:

i = U / Umea,n

The Cg--y relation was obtained from experimental roil damping data

of two-dimensIonal models.

These experimental results are fitted by:

= 0.435exp(--y) -2.0exp(-0.187-y) +1.50

The value of -y around a cross section is approximated by the

poten-tial flow theory for a rotating Lewis-form cylinder in an infinite


An estimation of the sectional maximum distance between the roll

axis and the hull surface, rmax, has to be made. Values of rmax() have, to be calculated for:

= p1 = 0.0


{(i+a1) sin() -a3sin(3b)}2


With. these two results, rmax and i/' follow from the conditions:

for rmax ( i > (i2): = rmax(i/ii)' and -' = 1/Pi


The relative velocity ratio y on a cross


is obtained by:



2D5 [H0


With this a non-linear sectional eddy making damping coefficient.

for zero forward speed follows f rom:




2. Rb


=1D5 1r1 .f3 E


f1 =O;.5 [1+tanh(2Oa5-i4)] f2 = O.5Ei-cos(',ro5)] -l.5[i-exp(.5-5-a5)]sin2(ira5)


bold printed

term -f


is included in the programlisting in the paper of Ikeda et.aI. ?3], but it does not appear in the formulas given in the paper. in the programs SEAWAY and SEAWAY-D,

this term is ommitted.

The approximation of the local radius of the bilge circle Rb ±8:








ship forms, the zero forward speed eddy

making damping coefficient is found by an, integration over the ship






32 {


(a2b2)11 ]

I fiRi 0G



- D


+ -

-+ 2 for: Rb<Ps and Rb<B/2 for: H0:>i and Rb>Ds

f or:

H0<i and Rb>HOP$

f1 Rb 12

D8 J



i +a1

+9a32 +2a1 (l-3a3) cos(2) -6a3cos(4b)



-2a3cos(5), +a1 (l-a3) cos(3b)


(6-3a1) a3,2 +(a12-3a1)a3 +a12 ]cos()




-2a3, sin(5i) +a1 (l-a3) sin(3)


(6+3a1 )a32 +(a123a1)'a3 +a12

i +





Bilge Keel Rôli Daing,, b44k(2)

The bilge keel component of.the non-linear roll damping coefficient, is divided into two components:

a component BN dúe to the normal force of the bilge keels

- a component B8 due to the pressure on the hull surface, created by the bilge keels.

The normai force component N of the bilge keel damping can be

deduced from experimental result's of oscillating flat plates The

drag coefficient CD depends on the period' parameter or the Keule-gan-Carpenter number. Ikeda measured this non-linear drag also by carrying out free rolling experiments with an ellipsoid with and

without bilge keels.

This results in a non-linear sectional damping coeffic±ent: 'BN' = rk3. hkfk2CD


CD = 22.5 hk +2.40

1t r 'a k

= 1.0 +0.aexp[-'160


The local distance between the roll axis and the bilge keel., rk, wiÏl be determined further on

Assuming a pressure distribution on the, 'hull caused by the biig,e 'keels, anon-linear sectional roll damping coefficient can be



B8 = rk2f k2





Ikeda carried out experiments to measure the pressure: on the 'hull

surface created by bilge keels. 'He found that the coefficient C of

the pressure n the front-face of the. bilge keel does not depen on

the periöd parameter,, while the coeff'cient 'c of the pressure on

the back-face of the 'bilge keel and the length of the negative

pressure region depend on' the period parameter.

Ikeda defines an' equivalent length of a constant negative pressure

region S, over the height of the bilge keels, which is fitted to

the following empirical formula.: = O.3O.?r.fkrk.a +i.9!5.hk

The pressure coefficient on the front-face of the bilge keel is

given by: = 1.20

The pressure coefficient on the back-face of the bilge keel is

given by: -.. -. .

= -22.5 hk

if.. r k 'Pa


m=HQ -m1

0.414.H0 +0.065'ILrn12

-(0.382'H0 +.0.0;10:6)m1

in5 =

(H'0 -0.2i5m1) (1 -O.,2:15.m1')

0.4l4H0 +0.Ó65l'rn12-(0.,382 -i-0.0106'H0)rn1

rn6-(Ho -.0.215m1) . (i -0..215rn1)

rn7 = - 0.,25irrn1

for: S0 > 0.257rRb,



for: S0


m8 = in7 +0'.414m1

for:. S0 > O.25.ir.Rb

= rn7 +1.414m1 [1-cos(SO/Rb.)]

for: S0 <' O2i5:.Rb

'The approximation of the local radius of the bilge circle, Rb, is

given bef ore.

The approximation of the local distance between the roll axis and

the bilge, keel, 'rk' is given as:

Rb '2 ' 0G 'Rb 2

rk = D8

1Ho -0.,293 i


[ .1.0



-L D5

' D D5

The total bilge keel damping coefficient can be' obtained' now by

integrating the sum of the sectional roll damping coefficients BN

and B8

over the length of the bilge keels':



E B +



The sectional pressure moment is given by:.


JCp.lrn.dh = D52 (.AC



A (rn3+m4)rn8



(1 -m1)2 (2rn3 -rn2)

3 ('H'0



(i -0.215m1)

+ m1 (m3m5 +m4m6)

D5 -0G rn2 = D5 xn3

=. 1.0 -l



6. Comparitive Siimilations

To check the calculation routines for the time domain, as used in the pre-processing program SEAWAY-D and in the time domain program SEAWAY-T, comparisons have been made with the results of the

frequ-domain SEAWAY [8] for

ency program a number of ship types.


The body plan of this container ship design is given in thé figure


Figure 6-A. Body Plan of S-175 Container Ship Design [4,51

This S-175 containership design had been subject of several com-puter and experimental studies, coordinated by the Shipbuilding

Research Association of Japan and the Seakeeping Committee of the International Towing Tank Conference [4,5]. Results of these

stu-dies have been used continuously for validating program SEAWAY

after each modification during its development.

An example of the results of these validations is given here for

the S-175 containership design in deep water, as has been used in the manual of SEAWAY too, with dimensions as tabled below.

Length between perp., Lpp Breadth, B 175.000 m 25.400 m Midship draught, Tm 9.500 m Trim by stern, t 0.000 m Block coefficient, CB 0.572 Metacentric height, GM 0.980 m Longitudinal CoB, Lcb/LDp -1.420 Radius of inertia, k/ Radius of inertia, k ¡L 0.328 0.240 Radius of inertia, k/Lpp

Height of bilge keels, hk

0.240 0.450 m



Table 6-I. Cómparison of Calculated Motions in the Frequency Domain (SEAWAY) and in the Time Domain (SEAWAY-T)

of the S-175 Container Ship Design in Regular Bow

Waves (pl'5O°) with. aflXflplitUde of l..O Me.tcr.-at..

Zero Forward Ship Speed.



(m) (m)


Cm) (deg) (deg) (deg) PROGRAM

0.2 0.843 0.485 0.992 0.173 0.204 0.088 SEAWAY 0.82 0.48 0.99 0.18 0.20 :0.08 SEAWAY-I 0.3 0.795 0.451 0.956 1.128 0.456 0.195 SEAWAY 0.78 0.45 0.95 1.13 0.46 0.20 SEAWAY -T 0.4 0.700 0.397 0.860 1.692 0772 0.282 SEAWAY 0.69 0.40 0.86 1.71 0.77 0.28 SEAWAY-I 0.5 0,. 546 0.293 0.671 0.998 1.071 0.357 SEAWAY 0.54 0.30 0.67 1.01 1.07 0.35 SEAWAY-T 06 0.344 0.161. 0.385 0.837 1.191 0.338 SEAWAY 0.34 0.16 O .38: 0.85 1.20 0.34 SEAWAY-I 0.7 0.140 0.056 O .173 0.558 0.937 0.211 SEAWAY 0.14 0.06 0.1.7 0.56 0.94 0.21 SEAWAYI 0.8 0.005 0.057 0.395 0.150 0.309 0.039 SEAWAY 0.00 0.06 0.40 0.15 0.31 0.04 SEAWAY-I 0.9 0.038 0.035 0.385 0.376 0.392 0.069 SEAWAY 0.04 0.04 0.39 0.38 0.39 0.07 SEAWAY-I 1.0 0.016 0.014 0.101 0.408 0.299 0.046 SEAWAY 0.02 0.01 0.10 0.41 0.30 0.05 SEAWAY-I 1.1 0.009 0.0i0 0.027 0.137 0.031 0.017 SEAWAY 0.01 0.01 0.03 0.14 0.03 0.02 SEAWAY-I 1.2 0.005 0.005 0.002 0.064 0.052 0.014 SEAWAY 0.01 0.01 0.00 0.06 0.05 0.01 SEAWAY-I 1.3 0.006 0.002 0.010 0.031 0.023 0.010 SEAWAY Q.00E 0.00 0.01 0.03 0.02 0.01 SEAWAY-I 1.4 0.004 0.002 0.005 0.030 0.015 0.003 SEAWAY 000 0.00 0.00 0.03 0.02 0.00 SEAWAY-I 1.5 0.002 0.001 0.003 0.016 0.016 O .003 SEAWAY 0.00k 0.00 o oo 0.02 0.02 0.00 SEAWAY-I

Wave Motions cálcuLated by SEAWAY and SEAWAY-I respectiveLy freq.



For this ship, the motions have been calculated in the frequency domain and in the time domain at zero forward ship speed.

Additional data,, used during the time domain simulations, are: - maximum frequency of damping curves = 5.00 rad/s

- frequency interval = 0.05 tad/s

- maximum time in retardation functions r = 50.00 s

- time interval 0.25 s

The potential coefficients and the frequency characteristics of the wave, loads at zero forward speed, calculated by program SEAWAY-D, have been input in program SEAWAY-T and the calculations have been

carried out for a regular wave amplitude of 1.0 meter.

Extra attention has been paid here to the roll motions. In both

calculations, the viscous roll damping has been estimated with the Ikeda method. In the frequency domain, the results are linearised for this wave amplitude of 1.0 meter. Because. of the relatively small roll damping at zero forward speed, in the natural frequency

region the initial conditions of the wave, loads will occur unstable

roll motions in the time domain. Then, a long simulation time is

required to obtain stable motions.

The agreements between the amplitudes ana the phase lags of the six basic motions, calculated both in the frequency domain and in the time domain, are remarkably good.. Some comparative results of the six motion amplitudes of the S- 175 containership design are given

in table 6-I.

Comparisons for a rectangular barge (100 x 20 x 4 meter), with a well (25 x 14 meter) fore and aft, are given in the tàble below for the natural roll frequency region. The experimental roll damping

data are input here.

Table 6-II. Comparison of Calculated Motions in the Frequency Domain (SEAWAY.) and in the Time Domain (SEAWAY-T)

of a Rectangular Barge with a Well Fore and Aft in

Regular Bow Waves (=150°) with an Amplitude. of 1.0 Meter at Zero Forward Ship Speed..

Based on these and a lot of other comparisons between the time

do-main and the frequency dodo-main approachea for. linear systems, . it may

be concluded that the programs SEAWAY-D/SEAWAY-T have an equal

ac-curacy as the freqtency domain predictions of these lInear motions

by the parent program SEAWAY.

This conclusion holds that the pre-processing program SEAWAY-D/4.10

delivers reliable results.

Wave freq.


Motions caLcuLated bySEAWAY and SEAWAY-T respectiveLy

PROGRAM Xa Cm) a Cm) Za Cm) q'5 (deg) 9a (deg) (deg) 03 .0.818 0.451 0.980 0.501 0.449 0.223 SEAWAY 0.81 0.45 0.97 0.50 0.45 0.22 SEAWAY-T 0.4 0.767 0.451 0.937 1.644 0.774 0.375 SEAWAY 0.76 0.45 0.93 1.65 0.77 0.38 .SEAWAY-T 0.5 0.679 0.415 0.846 0665 1.131 0.530 SEAWAY 0.67 0.42 0.84 067 1.13 0.53 SEAWAYT


7. Conclusions and Remarks

This new reIease of S'EAWAY-D includes the use of local twin-hull cross sections, the N-Parameter Close-Fit Conformal Mapping Ñèthod and (non-)lirear viscous roll damping coefficients. Special atten-tion has been paid to longitudinal jumps in the cross secatten-tions and fully-submerged cross sections Also, improved definitions of the

hydrodynainic potential masses at an infinite frequency and the wave loads have been added.

Based on a lot of comparisons, made between the time domain and the frequency domain approaches for iinear(ised) systems, it may be

concluded that the new programs SEAWAY- D/SEAWAY-T have an equal accuracy as the frequency dOmain predictions of these linear(ised)

motions by the parent program SEAWAY.

This conclusion holds that the pre-processing program.SEAWAY-D/4.lO

delivers reliable results.

It is advised to carry out a similar validation study with the pro-grams SEAWAY-D/DREDMO too.


8. References

ciimmins, LE.

The Impulse Response Function and Ship Motions.

Symposium on Ship Theory, Institût fûr Schiffbau der Universi-tât Hamburg, Hamburg, Germany, January 1962.

Schiffstechnik, 9, 101-109, 1962. Frank, W.

Oscillation of Cylinders in or below the Free Surface of a


Naval Ship Research and Development Center, Washington, U.S .A., Report No 2375, 1967.

E3] ikeda Y., Y. Himeno and N. Tanaka.

A Prediction Method for Ship Rolling.

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