### rId D

### '

_{«Jing}

### Congress 1983

ApriI 19-22, 1983

The dynamic behaviour of a cutter suction dredge in waves, and especially the teduion of motions and forces, receive increasing attention. For this the

availabil-numerical tools

### is

essential.The relft University of Technology took the initiative to start a research project on

iròing workability in cooperation with the Delft Hydraulics Laboratory.

One of the elements of this project was the development of a mathematical model for ccnnputation of the behaviour of the cutter suction dredge in irregular seas, which is descxi1ed in this paper.

It 1swell known that this behaviour is strongly influenced by the non-linear soil re-Zctionon the ¿utter head. For this reason it is not possible any more to conduct the cfUqut4tion in the frequency domain.

eDeftre the motions have to be calculated in the time domain. This makes itnecessary to formilate a set of equations which relates instantaneous values of hydrodynamic

c.es nd motions. Hereto the Cummins approach is used.

### 4

descrition is given of the developed computer programme. Also some computational reu1ts are presented. These clearly demonstrate the importance of the incorporationcf the _{il behaviour in the mathematical model.}

### j

9.

Singapore,

tiani AoaUSingapore 0923

spono d by BHRA Fluid Engineering and Marl ntec S.E.A. (Pte) Ltd.. Singapore.

CtOZT WttI W3tA, which incorporates CEDA- Central Dredging Association, WEDA - Western Dredging Association, EADA - Eastern

45tl

on Oceanic Resources. ESCAP - United Nations Economic & Social Commission forAsia and the Pacific tflternaxionar.Associatjon for Dredging Companies. JAHR - International Association for Hydraulic Research, PIANC - Permanent "teLt1On1 Association for Navigation Congresses

### icft ivrj Techcy

### S

### Lrar?

### Mekelweg 2 - 2628 CD Deift

### The Netherlands

Phqna 31 15 758373 - Fax:31 15 781538

CALCULATION METHOD FOR THE BEHAVIOUR OF A CUTTER SUCTION DREDGE OPERATING IN IRREGULAR WAVES

P.J. Keuning

Deift Hydraulics Laboratory, Deif t, the Netherlands

J.M.J. Journée

### TiI

### PCVr

_{S8-2.--P}

Deift University of Technology, Ship Hydromechanics Laboratory, Deift, the Netherlands

Summary

ÑOÑCLAflTRE

added mass

cross sectional area damping

ships beam drag coefficient inertia coefficient

hydrostatic spring coefficient

cable diameter,. unloaded inertia, fOrce

drag force

unis tangent vector to cable element

unit vector normal' to cablé lu verticàl plane unit vector normal to i and i in horizontal plane

1 = length

I ships ]ength

M. , = mass matrix

K]

N = number

R = hydrodyuarnic force per úflit length on cable

### s,,0

s. distance along cable

= energy dénsity spectrì

t' = tithè

T = cable tension

y = relativé velocity between water and constrüction

y = relative water velocity in s, , direction

### W' '

= nett eight pér 'unit length of cable-z. = degrees of freedom

= curreüt angle of attack

V = volume'

= wave elevation wave amplitude

E unit elongation

c. = phase angle'

= wave phase angle

EF r = phase añgle between wave 'and force

### ki

V ' = poisson ratio

T = time

= density of water

= cable angle in vertical plaue

O cable angle in horizo,ütal plane

w - = angular velocity a(w) = A = b(u) = B = C = cM = Ck D = FD = is = i0 =

.ÌNTRODUCTION

During the last decade the operations of cutter suction dredges have shifted frörn fairly protected waters to nearshore areas. In these areas the dredge is much niot'd exposed to wind-, wave- and current forces, which usually was not anticipated in the design. This results both in an increase in downtime and in higher Loads on the

copnents of the dredge.

Attmpts to improve the design lead to mechanical solutions, such as _{swell-compensation}
othe spud carriage and ladder. In very hostile environments, the spud is replaced by
a _{omplicated anchoring system, the 'christmas tree', to extend the limit of }

work-a14,jty.

As a result of substantial growth of these near shore activities, _{the }
beha-viour of cutter suction dredges in waves, and especially methods to reduce its motions
ad forces, are receiving increasing attention from both designers and contractors.
Frthis reason knowledge of the motions of the dredge and the forces on the cutter,
spiidpole mooring system etc. are becoming more and more essential.

ece the need to develop a mathematical model for the behaviour of a cutter suction
drdge, operating in irregular waves became apparent, in particular _{to:}

- vestigate improvements on the design

- r.edict downtime.

Th Deift University of Technology took the initiative to start _{a research project in}
elIaboration with the Delft Hydraulics Laboratory.

TSprbject comprised the following main elements.

areview and verification of existing mathematical techniques for computation of wave çforces on the dredge.

ÇThis study was completed in 1979 (Ref. 1).

h _{development of an analytical description of the soil reaction}

forces on an

oscil-lating cutter.

An extensive research programme is being performed at the Deift University of Tech-nology. The first results are presented at the WODA Congress in 1983 (Raf. 2). e. development of a mathematical model simulating the dynamic behaviour of a cutter

suction dredge in waves.

This resulted in the DREDMO progra=e, which will be described in detail in this

paper.

The project was financially supported by the VACB (Dutch Dredging Contractors

### Associa-don).

2. _{THE COMPUTATION OF THE BEHAVIOUR OF THE DREDGE}

2.1 General

The computation of the behaviour of a floating construction _{in a seaway is}
yell established in the last decades, using the frequency-domain formulation.

The equation of motion, based on Newtons law of dynamics, is given _{by:}

= TOT (1)

in which: M : (6x6) matrix of inertia of the body x :

(6) acceleration vector of the body in its six degrees of freedom TOT forces (moments) vector, containing all forces acting on the body. In case of a cutter suction dredge, the most important contributions to the vector

### rTOT

are:the wave exciting forces

- the hydrodynamic reaction forces - the hydrostatic restoring forces

For simple harmonic motions the: hydrodynamic reaction forces are conventionally ex-pressed in terms of the added mass and damping coefficients a(w) and b(w)

The. Equations (1) become:

6 6 E - w2M. ..x. = E j=1 1(3 3 j=1 2 . i

### .a1(w).x.

_{-}

wbk..(w).x. - C...x.j
+ ### Fk(w).sin(wt+)

+ Fk(ext.) kThis f ormu.at-ion of the hydrodynamic reaction forces _{can only be used in the}

frequency domain, since ak _{and bkj depend on the frequency of motion w. The response}
of the body to irregular waves is äetermined, using linear _{response amplitude operator}
between motion- and wave-amplitude

Wichers computed the behaviour of the dredge using this linear approach (Ref _{3)} _{}
Lad-der and spudpole are represented as bodies without inertia, which exerte restoring
forces on the barge.

As a result of the use of the formulation in the frequency domain, _{àny syste:}
influencing the behaviour of the floating body, such as spudpole, ladder and mooring
system may only have a linear relation with the displacement, velocity _{or acceleration}
of the body _{However, for the cutter suction dredge there are several complications,}
which perish this linearity assumption, the most important one being the soil reaction
forces on the cutter, which are known to be strongly non-linear. In order _{to }
incorpo-rate these non-linear effects in the dredge behaviour, it _{is necessary to formulate}
the equations of motion in the time domain, which relates _{instantaneous values of force}
and motions For the description of the hydrodynaniic reaction forces, due _{to time }
va-rying ship motions, use is made of the formulation as given by Cummins (Ref _{4)} _{}
Con-sidering the floating object to be a linear system between the _{input (velocity) and}
output (hydrodynamic reaction force), the hydrodynamic reaòtion force

k becomes:

Fk

=

### _jt

Ç.(t-r)i.'(r)dr

in which: io, . : added mäss tensor

1(3.

retardation function.

The values of mj and Kkj(t) can be derived using the frequency dependènt added mass and dèmping coefficient a(w), b(w) (Ref. 5).

Kki (t)

## :

Jbk. (w)cos Wt d(2)

mkj = ak. (w) + - J

K. (r) sin wt dr

2. .2 Equations of,

### mot ion

t une### domain

Substituting the derived expressions for the hydrodynamic _{reaction forces}

(2),

### the equations of motion for the

dredge become:+

### _J

### (tT)i(TT

Ckjxj(t t)k =

### The contributions to the external forces on the

barge, Fk(t), can be divided in:a. forces due. to the moor-ing system:

- mooring lines

- spud pole FSP

b, exciting forces

- wave forces Fil

- current forces Fe

- wind forces FWd e. other' forces, among which viscous damping FVS

Hence:

### Fk(t) = 'F(t) + F(t) + F(t) + F(t) + F(t) + Fwd(t) + F(t)

All of these forces may be non-linear functions of time and position of the barge, as will be discussed néxt.

3. EXTERNAL FORCES I ,Mooring lines

Cables undersea, connected to a flóatingobject.meet _{several extérnal forces}

e.g.:

- forces caused by the mass and displacement of the cable - hydrodytiamic loads due to cable motions, waves and current - friction forces by the bedding of the cable at the seabed.

The dynamic pa±t in the cable force increaseÉ with the amplitude and frequency of
oscil-lation at the point of suspension _{However at small amplitudes and motions}

at normal wave frequencies this contribution into the total cable force is small (Ref 6) Hence

in case of the cutter suction dredge the dynamic part of the cable force _{may be }

ne-glected.

When neglecting the second order terms, the static cable equations and
bound-ary conditions result in a set of coupled, non-linear first order differential _{}
equa-tions (unelastic cäble) (Réf. 7). See Fig. 2.

dT/ds W.sin - R

Tcos .(d8/ds) = -R0

T(d/ds) = Wcos R

'

dx/ds cos .cos Q

dy/ds = cos . sin O

dz/ds = sinc

The non-lineár cable elast-icity can be taken into _{account by adding the elongation of}
the cable to the length of the unloaded cable _{The elastic behaviour of the cable}

is

approximated' by a parabolic relation between tension and unit elongation.

For the hydromechanicforces _{the formulation as given by Wilson (Ref..}

8) is used, which results in: R = 0.5 p.D.c .ir.v . ¡V

I

= 0.5 R0 = 0.5 in which: D = D(1-Vc)

When the lower end of the cable lies o _{the sea bottom, an adaption is madé for the}
apparent point of anchoring ànd apparent cable length.

The set of implicit Equations (4) can be solved for the boundary conditions
specified at the two ends of the cable _{For a given set of initial conditions these}
equations are numerically integrated using a fourth order Runge-Kutta _{method} _{The }
so-lution is obtained iteratively, using a "hooting method".

The irnknown initial conditions at S = SA are estimated and frOm these the boundary conditions at S = SB are calculated and ëompared with the actual conditions A Newton Raphson iteration ptocedure is used to generate new initial conditions., until the boundary conditions at S.= SB are satisfied (Ref. 9).

It was found that for relative high pretensioning of the cables, the
hydro-mechanic forces are of minor importance under normal external conditions _{If this is}
the case the direct-ion and. magnitude of the cable force is calculated with
two-dimen-sional cable equations, neglecting hydromechanic forces _{This implies that Rs}

e = O

and O = O in Equation (4).

-In the DREDMO programme the directio _{and magnitude of the cable. force is}
calculated in advance for a number of possible positions of the point of suspension of
each cable around its position at t = O In the time domain computation the calculatiot
of the instantaneous values for each cable is performed by a simple Lagrange
interpola-tion procedure between thesè points.

3.2 The cutter ladder

- The ladder imposes boundary conditions on the motions of the barge by means

of the hinge coupling between both bodies and the hoisting wires Due to its large in-ertia and freedom of motion relative to the barge, the cutter ladder is incorporated as a separate body Of the system.

Using Newtons law, the equations of. motion become:

in which the superscript .9. indicates that the válues _{are related to the ladder.}

The external foce (t) contains several contiibutions, i.e..

- soil reaction forces on the cutter FiS

wave and hydrodynamic reaction forces F -

-- current forces

- hoisting wires forces F h

- forces in the coupling with the barge F

underwater weight of the ladder. FLZ

Forces acting On a rotating cutter which is subjected to _{an oscillating }
mo-tion, such as imposed by the motions of the barge in a seaway, will depend on a large
amount of parameters, such as:

- soil characteristics

- imposed motion, e.g. surge, sway or heave - cutter characteristics

- direction of. sway - etc.

The resulting relations between iotions .f the cutter and the soil reaction forces wit].
be strongly non-linear _{In the DR.EDMO programme any desired relation between motions}
and resulting forces can be specified _{Little is known however about the actual}

rela-tions as function of the relevant parameters _{The first results out of the extensive}
research programme at the Deift University of Technology ou the soil reaction
charac-teristics on an oscillating cutter are given by.De Koning _{et ai. (Ref. 2).}

For most cutter ladders the. schematizaton of the body to a clOsed,
cilin-drical construction will be accéptable. If the _{diater/wavelength ratio does not ex}
ceed a value of about 0 15 and the waveheight/diameter

ratio is less than 1, the dif-fraction forces become negligible and the well known Morison formulation for the hydro-dynamic loads can be used.

-FiW

= FM + FD

o

+

### C4

Pf-.v.JvI]d9.. . (6)In the mentioned parameter region the inertia fOrces _{are predominant (FM » FD). The}
contribution of the acceleration of the ladder _{to the relative acceleration between}

6 E j=1

ladder and water will be small

comparedwjth the orbital accelerations and hence this hydrodynamic reaction force will be neglected.

The local wave elevation at the ladder location is obtained _{from the wave elevation}

at

### the centre. of gravity of the barge taking into account its speed of

_{propagation.}

Determination of the orbital acceleration in an irregularseaway _{is treated in the}
same way as described for the wave forces on the barge. See Chapter _{3.4.}

If the ladder geometry is such that

the wavediffraction forces on the ladder are pre-dominant, these forces are calculated in the frequency doai.n using

_{athree-dinsial}

linear potential diffraction theory. Forces in an irregular _{sea are generated as }de-scribed for the barge. Hydrodynamic

reaction forces, if applicable, are in this case
kept constant _{The in this way created time series of wave forces}

serve as input for the time domain calculation.

The Equations of motion (5) now become:

j!1

### (t)

=### F(t)

+### F(t)

+### F(t)

### F(t)

### F(t-)

### + F(:)

As a result of -the hinge coupling between

ladder and barge this set of equations can be reduced to one degree of freedom.

This results in the implicit equation . ..

### i 2

_{Lw}

_{ic}

### X5 =

### X., X, x5, x,

_{F}, F

### 3

In order to solve this second

order-differential equation with a -finite difference

met-hod, it is reduced to a set of first

òrder differential equations. To solve this
set of stiff differential equations, _{use is made of the 'ttheta.methodt (Jf}

11).Thjs

leads to a set of non-linear _{eqúations in- the unknown variables}

x(t) and u(t) =i5(t).
This is solved with the Newton Raphson iteration method _{Because of its reduced}

size

(2x2), the required inversion of the Jacobian matrix _{can be done analytically}

The ne-cessary initial prediction vector for the

iteration process is delivered by a second
order Adams Basforth method, (Ref. _{11).}

Having solved the equations of motion for the cutter -ladder, _{the- reaction}
forces on the barge can be calculated.,

3.3 The spud pole

-The loads acting on a spud pole are caused by (see Fig. 4):

forces and moments in the spud keeper

-- soil reaction forces

- hydrodynatnic forces mass -and buoyancy forces.

:rhe dynamic behaviour of the spud pole is neglected because _{of its small mass-stiffness}
ìatio. Friction forces between spud pole and spud keepers _{are taken into account.}

For the reaction forces in the spud

keepers the freedom of motion of that part of the spud pole, which has

penetrated into the sea bed is very important _{}
How-ever little is known on the soil _{reaction forces on oscillating poles,}

which have a
small ratio betwEen penetration _{depth and pole diameter.}

IÏi case of a penetration depth of less than about 3 _{metres a pinned situation}

is
sup-òsed _{For larger penetration depths the}

pole is supposed to be partially clamped For this condition use is made of computational methods, which

are originally intended for he. calculation of the soil reaction fOrces on mooring dolphins.

The hydrodynamic loading is

determined, using Morison's formula, as given in

quation (6). For typical spud

pole dimensions the contribution of the drag force

pre-dminates (FD > FM) If applicable the current velocity

is added vectorially to the PlDital velocity

This calculation method tends to somewhat overestimate the total
f1uid loading _{The resulting forces and moments in the spud keeper}

are calculated and

### 1Çransferred

to the centre of gravity of the barge_{(Ref. 9, 10).}

3.4 Wave exciting forces

Time series of the wave exciting forces on the barge are required as input for the DREDMO programme.

The first order components of the wave forces can be obtained from frequency domain calculations Several computational methods are readily available, ranging from strip theory calculations using sectional values derived with two-dimensional poten-tial theories, to three-dimensional linear potenpoten-tial diffraction programmes

For irregular seas the wave forces can be determined using the thus calculated frequen-cy dependent transfer functions between the wave force and wave amplitude

_{(FaIa),}

according to
va
### F(t)

= !,### û

### 'e-.

### (w.)

kl ai### côs(w.t+c+F

The wave amplitudes _{aì aré derived from the energy density spectrum describing the}
desired irregular seastate, by assuming this to consist of a number of (N) regular
wavè-components.
N
E
ai cos(w1t+c )
1=1

### = /2 S

(w.)w. ai i iThe phase angles

### E.

are chosen randomly, while the frequency Intérval wj depends on the frequency intse±f.Th general the low frequency second ordér drift forces may lead to consider-able motion amplification for moored floating objects, if any undamped natural fre-quency of the moored, system lies within this frefre-quency-range. In the case of a cutter suction dredge because of the high stiffness of the mooring system (including spud pole and ladder) this usually tends to be not the case In principle however, the second order drift force contributions can be incorporated in the wave force time series, used

as input for the DREJ»!Ò prógramme.

3.5 Current forces

As dredges are frequently operating in tidal regions or river estuaries, _{}
cur-rent loads ón both barge and ladder are important.

At present however, there are no practical computational methods available _{Therefore}
in the computer programme use is made of formulas, which incorporate empirical coef f_{}

i-cients _{For the barge only the current forces in surge, sway and yaw direction are}

taken into accoùnt, i.e.

FC _{= 0.5}

### V2 T42+B2

_{(, )}

1,2. c 1,2 c

F = 0.6 pv2 T(L2-4-B2).c (c )

6 c 6 c

Here the current velocity v _{is corrected for the motions of the barge. The empirical}
coefficients C1,2&() depend on thé. angle of attack. of the current _{OEc.}

Besides these current loads, the current velocity also influences the wave forces and hydrodynamic reaction forces. However for low current velocitiès these ef-fects.may be neglected.

3.6 Viscous roll damping

The potential part of the total roll damping is included in the retardation
function. The viscous part has a non-linear behaviour and is calculated separately and
is added to the forces in the right hand side of Eq. _{(2).}

4. COMPUTATIONAL SCHEME

After assembling all contributions to the external force vector Fk(t), the
equations of motion for the barge, Eq. _{(3) have to be solved. Because of the so called}
"stiffness" of these equations, mainly caused by the ladder- and spud pole-reaction
forces, much attention has been paid to the numerical solution _{procedure of this set}
àf equations.

The second order differential equations are reduced to a _{set of first order non-ljnear}
differential equations which is solved by using a finite difference _{scheme. Because of}

its unconditional stability use is made of the "theta-method".

To avoid high computational costs this set of equations is solved using a modified Newton-Raphson iteration method (Ref. 11). This implies that the total number of

equa-tions can be reduced and the necessary Jacobian matrix need not be determined for each iteration step and not even for each time step, but can be kept constant during the computations until convergence is no longer obtained.

5. RESULTS

5.1 _{Hydrodynamic reaction forces and wave forces}

The added mass and damping coefficients, required for the _{computation of the}
retardation functions can be calculated with readily available _{computer programmes.}
Computational results of several of these programmes, applied to the geometry of a
cutter suction barge, were verified with model experiments. Oscillation _{tests and wave}
forces measurements were performed at the Deift Hydraulics Laboratory for various
waterdepths and compared with computational results (Ref. 1). _{A fair agreement was}

found between theory and experiments.

5.2 Current forces

The Ship Hydromechanics Laboratory of the Deift University of Technology carried out model experiments on a cutter Suction barge including the ladder, in order to get reliable information on the current forces and moments.

The general outline of barge and ladder _{are given in Fig. 5. The full scale current}
speed range was O to 3 knots.

From these experiments the values for the empirical coefficients c1 _{2 6(c) were }

ob-tained. The values are given in Fig. 5 as a function of the angle ¿f'attack

ac.

For input in the DREDMO programme, these coefficients are transformed into polynomial functions, whose coefficients are determined by a least squares method.

5.3 Computational results

The Programme is applied to a conventional cutter suction dredge operating in an exposed area. The influence of the soil characteristics on the motion behaviour and forces in the construction was investigated. Three different soil characteristics were used. To illustrate the influence of non-linear soil reaction forces, one computation was executed using linear soil characteristics.

5.3.1 Input

The main dimensions of the dredge are given in Table 1 together with the hy-drostatic spring rates. The coupling between ladder and barge is assumed to be

conven-tional.

Because of the severe external conditions, the barge is kept on location by the
bow-lines and a 'christmas tree' configuration at the stern of the vessel. The total number
of mooring lines is 5. The ladder is swayed by means of two swing wires. The conf
igura-tion is outlined in Fig. 8. The posiigura-tion of the attachment of the mooring lines on the
vessel and the anchor locations are given in Table 3. The mooring _{line characteristics}
are also given in Table 3.

All computations are performed for the same external conditions. _{The wave condition is}
defined by a Pierson-Moskowitch spectrum with a significant waveheight of 1 .0 ni and a

peak-period of 7.0 s. The wave-spectrum is presented in Fig. 7. The angle of incidence of the waves is 300 from the bow (quartering waves). No current is assumed.

mass mj ae calculated. The values of ae suarized in Table 2. In Fig. 6 a

num-ber of retardation functions is given, i.e. the diagonal of the matrix Kkj.

Because different types of soils are to be simulated, no data can yet be used at this point from Ref.. .3. TherefOre here oniy an approximated, dynamic soil behaviour is used to demonstrate its influence on the motions of the dredge.

.1±1 situatiOns where the cuttér is actually cutting in all directions, use can be made Qf the assumption that the specific cutting energy A3, is constant., so:

A

-sp P

with:

### M

shaft torquew = angular velocity of the cutter

P = cutting prodúction.

The soil react-ion force is now calculated using the assumption that.:

M w

### cc

-: Ck

ihere: R = cutter rad-ius

ck_ constant1 function of tiie wear of the cutter.teeth and shape of the cut

profile..

The soil reaction forces are approximated by:

### Fj(t)

= ck.fI(Ah(t)),(s(t).d,(t)),

(v(t)'\

### fV(t)\

vh(t)j kVh(t)) (8)

with: Vh = swing velocity

s = penetration depth in axial dirêction = penetration aepth in radial directiOn VP = penetration velocity, axial diredtion

V = penétration veloëity, rádial direction.

All these variables are a function of swing direction of the cutter, type of cutter and
type of soil The used (mean) values of the important variables are summarized _{in Table}
4 for different type of soils1 i.e. packed sand and soft rock.

5.3.2 Results

The DREDMO prôgrâmme produces time series of the motions of the barge and cutter ladder and of the forces in mooring lines, spud pole keeper, cutter head, side

swing wires, hoisting wires and the coupling between ladder and barge _{All these time}
series are plotted and/Or statistically and spectrally analysed.

The results of the cpmputer runs given here are primarily intended to
demon-strate the capacilities of the programme _{Because the used soil characteristics are}
chosen rather arbitrarily, no definite conclusions should be drawn from the behaviour
of the dredge in the two types of soil _{However some interesting features can be }

ob-served _{The motions of the centre of gravity of the barge are given} _{in Table 5}

The results show an increase in surge motion for test 3, mainly caused by larger motion
amplitudes in the negative x1-direction while no significant differences occur for sway
and heave However because of the chosen formulation of Fj(t) _{in Equation (8), the}
used combination of increasing A5-value with reduced swing velocity in case of soft
rock, although correct as such, also diminishes the. relativé differences in behaviour
of soils between sand and rock in the x1-x3 plane

surge motion the subharmonic behaviour is seen

to be significant.. This is caused by the
non-linear characteristics of the restoring forces, i.e. the soil reaction _{forces. That}
this is actually the case is clearly demonstrated when comparing the _{results with those}
of the test with linear soil characteristics _{(Test 2). See Fig. 10. No significant}

sub-harmonic response occurs. These observed phenomena of subsub-harmonic motions _{are more }
ge-nerally known in connection with moored ships (Ref. 12). In Table 6 _{the forces on the}
cutter are given. As mentioned before, the

used formulation for the soil behaviour
tends to underestimate the differences. The maximum penetration velocity _{in axial }
di-rection is also given. Here _{a pronounced difference can be observed.}

In Fig. 11 sample time series for the soil _{reaction forces F(t) in}

Test 3 are given.
This illustrates the non-linear responses. It can be seen that for this _{condition the}
cutter temporarily looses contact with the soil.

This couldalso give an explanation for the enlarged surge amplitudes _{in Test}
3. Because the cutter temporarily looses _{contact with the soil. The non-linear}

charac-teristics of this restoring force change, which

is not the case in test 1. This results in different motion behaviour.

6. CONCLUSIONS

The DREDMO programme simulates the dynamic _{behaviour of a cutter suction}
dredge in waves. In order to be able _{to incorporate non-linear forces}

acting on the
system the equations of motion are formulated and solved in time domain. _{In particular}
the non-linear soil characteristics have

a pronounced influence on the behaviour of the
dredge, as demonstrated by the computational _{results. From these results it is}

also

apparent that this dynamic behaviour depends

on the type of soil. Knowledge of the soil
reaction forces on an oscillating cutter is thus important. This, however, _{is still}
very much a research subject. The DREDMO _{prograimne is a usefull tool both for design}
studies and for down time assesments.

However, the accuracy obtained for down time
cal-culations very much depends on the availability of data on the particular soil _{}

charac-teristics.

### ACKNOWLEDGETS

The authors want to express their

gratefullness to F.C. Vis and A.W.J. Koster of the Deif t Hydraulics Laboratory for

their contributions to the development of this computer programme.

REFERENCES

Keuni.ng, J.A. and Beukelman, W.: "Hydrodynamic

coefficients of rectangular
barges in shallow water". In Proc. 2nd _{International Conference on Behaviour}

of

Off-shore Structures, (London, U.K.: August 28-31, 1979) Cranfield, _{U.K., BHRA}
Fluid Engineering, 1979.

Koning, J. de and Zwartbol, A. and Miedemá, _{S.: "Soil/cutter head interaction}
under wave conditions". In Proc. World Dredging Congress (Singapore: April 19-22,

1983), Cranfield, U.K., BHRA Fluid Engineering, _{1983.}

Wichers, J.E.W.: "On the forces _{on a cutter suction dredger in waves".}

In Proc.

9th WODCON conference (Vancouver, 1980).

Cuxnmins, W.E.: "The impulse response function _{and ship motions". Schifftechnjk}
B.D. 47, 9, 1962, pp. 101-109.

Ogilvie, T.F.: "Recent progress towards _{the understanding and prediction of}

ship

motions". In Proc. 5th Symposium of Naval _{Hydrodynamics (Bergen, 1964).}
Sluys, M.F. van and Blok, J.J.: "The dynamic _{behaviour of mooring lines". In}
Proc. Ocean Technology Conference (Houston; _{U.S.A.: 1977) OTC 2881.}

De Zoysa, A.P.K.: "Steady-state analysis _{of undersea cables". Ocean Engineering,}
5, 1978, pp. 209-223.

### 8.

### Bendenbènder, J.W.: "Three dimensional, boundary value problems for flexible

### cables". Iù Proc. 0cean Technology Conference (Houston, U.S.A.: 1970) OTC 1281

### Boom, E. yan, den: "Mooring forces on a cutter suction dredge in waves".

### (Vér-ankeringkfachten op een snijkopzuiger in zeegang)

### Master thesis, Deif t

### Uni-versity of Technology,. Ship Hydromechanics Laboratory, 1979. (In Dutch).

### 'Marien Technologisch Speü±rerk (MaTS): "Forces on a vertical pile by wavês and

### current"

### (Krachten op een vertikale paal ten gevolge van de cotnbinatie van

### stroom en golven). Netherlands Industrial Couni1 for Oceanology, Report MaTS

### VN-1, novethber 1980. (In Dutch).

### Lambert, J.D.: "Computational methods in ordinary differential equations".

### Lndòn, JOhn Wiley and Sons, 1973.

### Qortmerssen, G. van: "The motions of a moored ship in waves".. Wageningen,

### VèeTrïñíàñ en ZòñenN.V., 1976.

### Táble

1### Main dimensions cutter/suction dredge

### length

### beam

### depth

### draught (1/2 L)

### displacement weight

### ràdii of gyration: k,

### Metacentre height

### position centre of gravity above base

### position centre of gravity from APP

### hydrostatic spring rates C33

### C»

C55

### length cutter ladr

C35,### pOsition hinge relative to CG barge

### mass ladder

### Table 2

### Added mass

_{kj}

7.1 .80
### 17.15

### 5.40

-.### 4.00

### 4290.Ò0

### 6.8

### 22,2

### 21.9

### 4.4

### .5.0

### 37.4

11025 139171### 4257740

### 36331

### 41.5

### 22.5

513 ni ni ni ni### ton

ni ni m ni ni ni### kN'/rad

### kNm/rad

### kNm/rad

### kN/rad

ni -ni### ton.

1 2 3 4 5 6 1 291 . O O O### 3604

0 2 Q### 860

0### -3476

0 55 3 0 0### 8688

. O### -

4Ó165 O .4 0### -3476

0### 263356

0### -

### 33475

5### -3604

' 0### -40165

0### 5116363

0 6 0 55 0### - 33475

0### 2555953

Table 4

Soil characteristics for the test 1, 2, 3

Table 5

Motion amplitudes of centre of gravity barge (in)

Table 6

Soil reaction at cutter Table 3

Mooring line characteristics

max. crest-trough values soil reaction forces

### F1

F 1 2_{3}(kN) (kN) (kN) Test 1 1120 370 1299

_{0.20}Test 3

_{j 1340}400 1502 0.12 max. penetration velocity Vp (m/s)

Remark: in test 3 all minimum soil reaction forces are zero, due to cutter loosing contact with soil.

Line Point of attachment _{Anchor positiOn}

number x .y z x y z
1 -33.6 0 _{- 7.3} _{-233.6} _{0} _{-27.1}
200.2 85021
2 -33.6 0 7.3 - 33.6 -150 -27.1 130.7 85021
3 -33.6 0 - 7.3 - 33.6 +150 -27.1 150.7 85021
6 _{51.6} _{0} _{10.0} _{162.7} _{- 77.8}
-27.1 140.1 102203
7 51.6 0 .10.0 162.7 + 77.8 -27.1 140.1 102203
packed sand
test i
linear (sand)
test 2
soft rock
test 3
Asp kJ/m3 600 600 3000
Vh in/s 0.3 0.3 0.1
cl i _{-} 1
c3 2 _{-} _{2}
R in 0.87 0.87 _{0.75}
Test I 0.25 0.40 _{0.38} _{0.34}
0.54 0.50
Test 3 0.31 0.43 0.37 0.43 0.59 0.48
Significant _{Maximum}
xl x2 x3 xl x2 _{x3}

x,, x6 : earth fixed coordinafe

### x: body fixed

coordina tesFIGURE 1. DEFINITION COORDINATE SYSTEMS

FIGURE 3. LOAD CONDITION CUTTER LADDER

FIGURE 2 LOCAL COORDINATE SYSTEM FOR MOORING LINE

'T

30 60

### FIGURE 5

_{COEFFICIENTS OF CURRENT FORcES}

AND MOMENT L BARGE L LADDER BARGE * LADDER

- POLYNoM, _{APPRQx',.qA TION}

2600 0 ¡ 3 300 2600 260 20X200 zw iso tOOL' 100 000 50

### 00

500 -I.e 9FiGURE 5. RETARDATION FUNCTIONS COTTER SUCTION DREDGE

### 587is

14 10 10 20 22 24 26 28### r

-a

FiGURE 8. MOORING LINE ARRANGEMENT

FOR CHRISTMAS TREE CONFiGURA TION

FIGURE 7. APPUEÒ WAVE SPECTRUM

.E1UM r H. W., r Z0 3 2.0 7.5 a 1.4 7

### - 'ç4

12 1### -Ku

K1Lc

### f

flO.9 Ttu tutu @7 tu tutU @7tU72TT @7 Utu
tUT3
'io
710, 10 T tu _{TItu Q? tu} _{@7}
OUYXTO Q?
2
It,
"Lw

FIGURE 1? TIME SERIES SOIL REACT/ON