Progress in computer simulations of SPM moored vessels

OTe 5175

Progress in Computer Simulations of SPM Moored Vessels

by J.E.W. Wichers.

MARIN

Copyright 1986 Offshore Technology Conference

-This paper wasprEfsented at the 18th Annual OTe in ,Houston, Texas, May 5-8, _1986. The material is subject to correction by the author. Permission to copy is restricted to an abstract of not mOJe than: 300 word§. _ _ __

ABSTRACT

In the last ten years computer simulation pro-grams for SPM moored vessels have been developed _by

several authors. At . present the application of such

programs, if at all, is limited to some preliminary

calculations in the very _early design stage. The

reasons for the reluctance to apply such computation

methods in the more final stage of the design is

clearly due to the failure of reliable input data

and uncertainties concerning the procedures to

describe the governing physical phenomena more

accurately.

In this paper experimental and theoretical

re-sults for the input and the methodologies involved

in simulations of the behaviour of and the mooring

forces induced by a large tan1<er 1D.00red in wind,

irregular waves and current will_be presented. INTRODUCTION

A vessel moored in the open sea can perform

high and low frequency motions. The low frequency

oscillations can be an important problem. The large

horizontal excursions that occur can cause large

forces in the mooring system.

As is schematically indicated in Fig. 1, in the

computation procedure distinction can be made in a

low frequency and a high frequency part of the

equa-tions of motion of a moored vessel. Nowadays

rea-sonably reliable computations can be carried out

concerning the high frequency input for the

equa-tions of motion. For the low frequency part,

how-ever, uncertainties exist concerning the input of

the hydrodynamic reaction forces and the wave drift

force excitation in current. Since, in general, an

SPM system is a weakly damped system, reasonance

peaks in the modes of motions in the horizontal

plane can often be clearly distinguished. Therefore

in order to predict the motion responses correctly,

attention must be paid not only to the excitation

but also to the damping.

References and illustrations at end of paper.

In this _paper results will be presented on

input data and computation procedures in order to

clarify the mentioned uncertain terms in the

equa-tions of motion of the low frequency part.

In order to obtain the low frequency

hydrody-namic damping an extensive model test program has

been carried out. . A review of the experiments is

shown in Fig. 2. Some results will be given in this

paper. To show the computation procedures examples

for 1-DOF and 3-DOF will be highlighted.

For the 1-DOF of a linearly moored 200 kDWT

tanker the following computation procedures will be

dealt with:

- wave drift force and wave drift damping in still

water;

- wave drift force and wave drift damping in

cur-rent;

- viscous surge damping in still water and in

cur-rent;

- solution of the equations of motion in the

fre-quency domain.

For the 3-DOF case a formulation of the coupled equations of motion for tanker motions in the hori-zontal plane (in still water and in a current field)

has been experimentally determined. In the past

de-scriptions of the equations of motion for the low

frequency tanker motions in the horizontal plane

have been given by many authors, ref. [1] through

[11]. To evaluate the present formulation of the

equations of motion the results have been compared

with the descriptions as given by Molin [6] and

Obokata [10]. Finally, time domain simulations with

the present formulation have been carried out with a

200 kDWT tanker in fully loaded and in ballasted

condition, while moored by means of a non-linear

hawser. The tanker has been exposed to current and

wind. Model tests under the same conditions have

been carried out to check the reliability of the

computation procedure.

2 PROGRESS IN COMPUTER SIMULATIONS OF SPM MOORED VESSELS OTC 5175

ONE DEGREE OF FREEDOM

_{For the wave exciting forces the following}

expansion is used:

The motions of a moored tanker in irregular

head waves consist of small amplitude high frequency

surge, heave and pitch motions and large amplitude

low frequency surge motions, see [12]. The high

fre-quency motions are related to the individual wave

frequency components in the wave train. The low

frequency surge motion is concentrated around the

natural frequency of the moored vessel.

To study the motions of a structure use has

been made of two different systems of axes as

in-dicated in Fig. 3; the system of axes Ox(1)x(3) is

fixed in space, with the Ox(1 )-axis in the still

water surface and the Ox(3)-axis coinciding with the

vertical axis GX3 of the ship-fixed system of axes

Gx_lx

3, at rest. We shall assume that the surge,

heave and pitch motions can be decoupled into the

following form:

• • • • • • • • (3)

where F(1) is the first order

wave(2~xciting

force

oscillating at wave frequencies. F is the low

frequency second order wave drift force on a

float-ing body in irregular waves, oscillating at wave

group frequencies, disregarding the force

oscilla-tions at the double of the wave frequencies. The

first order wave exciting forces are only dependent

on the incoming wave height, wave period, water

depth and geometry of the body. These first order

wave forces can be calculated by a linear Z-~

dif-fraction program, see [13]. The wave force F 2 on a

stationary floating body in waves is calculated by a

pressure integration technique as developed by

Pinkster [14]. In his study it is assumed that the

floating body only performs small amplitude high

frequency motions around the mean position.

Follow-ing the condition of the mentioned computations the

first order wave exciting forces and the second

or-der wave drift forces should be written as follows:

For the wave exciting loads we shall therefore assume:

As mentioned earlier, in reality however, the vessel

performs small amplitude high frequency motions

while travelling with large amplitude low frequency surge oscillations through the wave field.

Due to the low frequency oscillating speed not

only the pressures on the hull but also the

oscilla-tion frequency of the vessel will be affected. A

similar phenomenon can be observed in the case of

wave resistance at forward speed. The speed effect

may influence both the wave exciting loads and the

high frequency fluid reactive forces.

(7) (6) (4) (5) O,t)

0,~(1)

,t)

F(l)( (2) .(2)

_{xl ,Xl '}t) F(2)( (2)xl ,Xl.(2),~(1) t), F(1) (xi 2) F(2)(xi 2 )

with E and n being small parameters, viz.:

- E relates to the wave steepness;

- n considers the ratio between the two time

scales of the motions: the II frequency range

of the natural frequency of the system and the

W frequency range of the wave spectrum

fre-quencies. Further:

xi 1), x(1) and

x~

1) relate to the wave frequency

surge,

~eave

and pitch motions;

(2) (2) (2)

x

llf' x3lf and x5lf stand for the large amplitude

low frequency second order surge, heave and pitch motions;

x(2) x(2) and x(2) represent the second order

m6gf~ns3gf which5~be

frequency range is twice the wave frequency range.

(1) 2( (2) (2) 2t )

X

_s

(t/n,t)

+

E: x_S1f(t)

+

xShf(n,t)

• • • • • • • • • (1)

Of the second order motions only the low

fre-quency part will be considered and will be denoted

as x(2). Following the expansion of the motions and

substitution into Newton's law we obtain for the

surge direction:

It was pointed out by Wichers and Huijsmans

[15] that the complete equation of the low frequency

motion of a floating body moored in irregular head

waves can be written as follows:

M(,,(2)

+ ..

(1») = F ((1) (2»)

+

F

M

+

FW

xl xl H! ,Xl

• • • • • • • • (2) in which:

F_H

=

fluid reactive forces

FM

=

restoring force due to the mooring system

FW

=

wave exciting forces.

• • • • • • • • • • (8)

GTC 5175 WICHERS

3

or: (10) (11) • • • (1Z) 2 2 { 1;1 P II + 1;2 PZZ + ~1~2(PI2+PZl) cos (wl-WZ)t + + (El-E_Z)} +

~l1;z(QlZ-QZl)

sin{(wl-wZ)t + (E_l-E₂)}

In order to relate the expression for the drift

force as given in equation (10) to the wave

eleva-tion given by equaeleva-tion (9), we will write the wave

elevation in amplitude modulated form:

in which Pi and Q_i are quadratic transfer

func-tions depenJent on two frequencies being in this

case the frequencies WI and w_Z• Generally P:i.j and

Qi i are computed in such a way that the folLowing

relations exist: The formulation:

in which:

all s added mass at low frequency

B_ll linear damping coefficient

BIll -quadratic damping coefficient

cll spring constant.

aF(Z)(O,O,~(l)

,t)

d. (2)

xl

actually represents the total wave_ drift force. As

can be seen the total wave drift force consists of

the wave drift force and

_<In

~dditiona1 pari. This

additional part is called t1e wave drift damping and

seems to be confirmed also by experiments carried

out by Takagi et 13,1. [16], Ando et al. [17] and

Fa1tinsen et 13,1.

[18].

Before the application of the total drift force

to a vessel moored in waves with and without

cur-rent, first some elucidations of the behaviour of

the drift forces

F(Z)(O,o,~(l)

,t) = F(Z)(t) will be

dealt with, see Pinkster [14].

At first approximation, wave drift forces are a quadratic function oithe amplitude of the incident

waves. The behaviour of the drift forces in waves

can be elucidated by first looking at the general

expression for the drift forces in a wave train

con-sisting of two regular sinusoidal waves with

fre-quencies wI and 00_{2 and amplitude} ~l and ~Z. The wave

elevation is written as: __

(14) • • (13)

in which A( t) represents the envelope of the wave

elevation, 00₀ an arbitrarily chosen fixed frequency

and E(t) a time varying phase angle. It can be shown that the envelope becomes:

The square of the envelope is:

Z

l: 1;i sin(w i t+E i) i=1

~(t)

• • • • • (9)

2

2

A2(t)

=

l: l: 1;i1;j cos{(Wi-Wj)t + (Ei-E_j)}

i=1

j=1

A schematic representation of this wave train is

shown in Fig. 4. Such a wave train will be denoted

by a regular wave group. This type of wave train is

characterized by a periodic variation of the wa,ve

envelope (dotted line). The frequency associated

with the envelope is equal to

wCwZ

being the

dif-ference frequency of the regular wave components.

• (15)

Comparison with equation (10) _{shows that Pii is}

that part of the quadratic transfer function wl1ich

expresses the component of the drift force which is

in-phase with the square of the wave envelope and

Qi' expresses the in-quadrature part of the drift

fo~ces.

A quantity which is a quadratic function of the

wave amplitudes, in this case the wave drift force,

will be as follows:

Z

l: 1;i1;j Pij cos{(Wi-Wj)t + (Ei-E_j)} + j=l

2

l:

~i~j

Q_ij Sin{(Wi-Wj)t + (Ei-E_j)} j=1

Equation (10) shows that the drift force

con-tains several components. The first two are constant parts corresponding to the mean drift force in each of the regular wave components separately. The third

and fourth parts are low frequency varying

compo-nents which arise through the combined presence of

the two regular wave components in the wave group.

From equation (10) it can be seen that the values of

PH and P_ZZ determine the mean value of the drift

force, while PlZ' PZl' Q12 and QZl determine the

slowly varying part of the drift force.

4 PROGRESS IN COMPUTER SINULATI”ONSCl)?SPM MOORED VESSELS _{OTC 5175}

Drift forces in irregular waves _______________________________

In irregular waves the wave drift force is:

&)(t) =

NN

.

z

=

~icj ‘ij

cos{(w~+j)t

+

‘E~-sj)} + i=lj=l

NN

+Z _{~ ~i~j Qij} ain{(ui-uj)t + (St-gj)}

i=l j=l

. . . (16)

Except for the number of frequency components N in

the waves the above expressions are identical to

those used for regular wave groups.

Total wave drift force in waves without current ---_______________________________________ Regular wave groups

Following equations (8) and (10) the total wave

drift force in a regular wave group can be written

as follows: 2 2 = cl ’11 + ‘2 ’22 + 2~1~2 ’12 Cos ““12t+A&12+e12{ ]+ L . . . . . . . . . . . (17) in which: . ‘1 = :$2) ‘11 = T(wl,lil) and: T - T(f.01,fJJ2)= (P;2 + Q12)2+ = 12

-= quadratic transfer function of the amplitude of the drift force

Q12 ~ E12 = arctan

P_1.3

= phase angl~’between the low frequency part of

the second order force relative to the low

frequency part of the square of the wave

ele-vation.

The total wave drift force can be split up Ln a

wave drift force and a wave drift damping force. The

wave drift damping force contains several

compo-nents. The first two terms are derived from the mean wave drift force in each of the regular wave

compon-ents separately. The third term can be derived from

the low frequency part of the second order force.

The derivatives of the quadratic transfer

func-tions of the mean wave drift force T(wi,wi) to the

10W frequency velocity often denoted as the wave

damping coefficients have been experimentally

deter-mined, [15] and [19]. The results derived from a

fully loaded 200 kDWT tanker has been given in Fig.

5. The following nomenclature will be used:

The effect of the dampfng will play an

impor-tant role for the condition that the natural

fre-quency of the moored vessel will correspond to the

frequency of the wave group:

Assuming a linear viscous damping the equation of

low frequency motion can be written as follows:

aT.. 1 1

. . . (19)

The damping term in equation (19) contains a linear

coefficient and a low frequency oscillating

coeffi-cient.

Using the solution:

w m

xl =_ Z ancosn~t+ X bnsinn~t ..(20)

n=l) n=1

it.can be proven that the oscillating coeff~cient

hardly contributes to the motions. Because the low

frequency oscillating coefficient has to be

multi-plied by the velocity, of which the dominant

compo-nent consists of cos pt, the product results in a

double frequency 2P which is beyond the resonance

frequency of the low frequency surge motion, and so

the contribution will be negligible. It must be

mentioned that in principle a constant part will.

remain, which will affect the mean wave drift force slightly.

Neglecting the term with the oscillating

coef-ficient the total wave drift force in a regular wave group will be:

F(2)(t) =

_t

~:1

j:l

‘icj

‘ij

Cos{(Loi-(l)j)t +

. . . (21)

OTC 5175 WICHERS ₅

Irregular waves

-.

Following equation (21) the total wave drift

force in irregular waves without current will yield:

. . . (22)

Total wave drift forces in waves combined with ---current

---Regular waves

According to equation (21) the total wave drift

force on a moored vessel in regular head waves with

wave frequency .U1 and amplitude Cl can be written as follows: F(2)(u1,t) 2 3T11 2. ‘Vll+Za* ‘1 ““”* . (23) 1

Based on the earth-related wave frequencY ~1 the ve-locity dependent wave drift force acting on a vessel

sailing with a low speed U may be determined as is

given below, see [15]:

The frequency of encounter will be:

OJ =lo -#ucosu e 11 . . . (25) in which: 2’11h ~ tanh ~ A1=2 ~ :~ve direc;ion, !J

Regular wave groups

Assuming that the vessel is sailing with the

low speed U and performing low frequency oscillation .

‘1 in a regular wave group the total wave drift

force will yield:

F(2)(o+W2,Ll+:1,t)

=

in which:

* 3T22

T22=T22+~u 1

Equation (26) shows an increase of the constant

parts and of the oscillating part of the wave drift

force caused by the low forward speed U of the

ves-sel. The values 3T la; can be found from Fig. 5.

ii 1

For the derivatives of the oscillating Part of

the drift force to the low frequency velocity no

data exist. Computational developments, however, are

underway [20]. At this stage, it is assumed that the oscillating part of the wave drift force will be

af-fected in a same degree by the forward speed as is

found for the constant part of the wave drift force

and for the approximation the following formulation

has been chosen: aT12 aTll + 3T22 (— ’12 —. —) — ● ..*.. ● (27) a+ a+l a+l ‘11+T22

The formulas mentioned so far have been based

on a vessel sailing with low forward speed U in

com-bination with low frequency oscillations, while the

wave frequencies W1 and U2 are considered with

regard to earth. If the vessel is moored in a

cur-rent field with current speed U (bow directed into

the current) the earth-related wave freauencv should

be transformed into the

we:

Irregular waves

Following equation

force in irregular waves

velocity U will yield:

. .

wave frequency of encounter

(26) the total wave drift

combined with current with

N

F(2)(d =

~~1

j=l~ ~icj T*(Wi,LOj)COS{(~ ‘~.)t +

t iJ 3T(~i,Wi) . } + ~:1 ~; a; ‘1 + (E~-Ej) + ‘ij -1 .,? . . . (28) in which: 3T(0Ji,@) T*(LOi,Wj) = ‘(ui’oj) + a; u 1 T(@i,(IJi)= P((lJi,fJJi)= P ii

while u stands for the frequency transformation

according to:

fl)=o-l- :U

6 PROGRESS IN COMPUTER ELEMULA’TIONSOF SPM MOORED VESSELS OTC 51’75

in which U. is frequency in still water and A. is il=(o. xl<<vc

save length in still water.

It is assumed that both the wave drift force in which: V= = current speed

and the wave drift damping are of potential nature. u = frequency of oscillation

To solve the equation of motion the viscous damping xl = amplitude of the low frequency

mo-has to be determined. This will be dealt with in the tion in surge direction;

next section. then it is plausible that the laminak boundary layer

will be disturbed by the current and that the

bound-Viscous damping in surge direction ary layer will be dominated by the turbulence as is

__________________ _______________

assumed in a steady-state current. Surge damping in still water

The quasi-dynamic calculations were carried out

Radiation again for which the equation of motion will be as

follows: Computations by means of 3-D potential

diffrac-tion theory have shown that the radiated damping can _(M+all(~))fl

be neglected for LOm < 0.5, see [19]. This_mteans

- +PWLT cl(180e) (VC++I)2+

that for the low frequencies the frictional necha- _{+ Cllxl = o} (30)

nism in the boundary layer will dominate the

damp-. damp-. damp-. damp-. damp-. damp-. damp-. damp-. damp-. damp-. damp-. ing.

Comparing the calculated and measured results, Fig.

Laminar boundary layer 6, it can be concluded that for the mentioned

condi-tions the above procedure will be allowed.

Further-Since in the low frequency range the oscillat- more, it can be proven that the damping coefficient

ing amplitudes wf.11 be large, stationary current is ~ to first approximation, linear ‘as is shown

formulation is normally used. By applying this for- below.

inulation, however, it is assumed that the boundary

layer will be turbulent. The computed damping term reads:

The equation of the low frequency motion during

decay can be read as follows: +PWLTC1(180Q) (VC+*1)2=

(M+aIl(P))Yl - *PWL T Cl(vcr);;+ CllXL= O = +PWLT C1(180”) v:+ tPwLT C1(180”) 2VC*I +

. . . (29) -1-higher orders

in which:

Cl(Ycr) = resistance coefficient in current 2F

4cr = relative current direction. =Fc+~ c %1 + higher orders

c As is shown in Fig. 6 the result gives

extreme-ly low damping compared to the measured damping. It

The first order damping coefficient will be:

can be concluded that the quasi-dynamic approach in 2FC

calm water is not permissible. b_lIC .—_v . . . ...*. . (31)

c

From extinction tests in still water it is gen- _{It must be noted that in this case the frequency}

de-erally found that the damping is linearly

propor-tional to the velocity. It may be that the friction

pendency of the damping term has been lost.

is dominated by a laminar flow. Applying the lamlnar _Application

boundary theory according to the NAVIER-STOKES equa-

--.---tions for an oscillating plate (see article 328 of

[21]), it will be found that the damping force is

The theory discussed in the previous sections

linearly proportional to the velocity.

will be applied to a fully loaded 200 kDWT tanker

linearly moored in irregular head waves wtth 2 knots

The surge damping coefficients as determined by

current and without current. The tanker is moored in

82.5 m water depth. The body plan of the tanker is

means of motion decay tests in still water and for

several tanker sizes are presented in Fig. 7. The

given in Fig. 8. The main particulars and input data of the tanker are presented in Table 1.

values for the model damping are scaled to prototype

values according to Froudets law. The damping in

Fig. 7 has been made dimensionless using the

quanti-The matrices of the quadratic transfer function

for the amplitudes of the drift forces are given in

ties as given by the solution of the NAVIER-STOKES _{Table 2,}

equation.

while the quadratic transfer function of

the mean wave drift forces and the mean wave drift

Surge damping In current

damping coefficients are shown in Fig. 9.

Turbulent boundary layer

To compute the motion, the equation of motion

is solved in the frequency domain. Both the variance

Assuming that during the low frequency motions

and the maximum of the surge motion has been

pre-in surge dLrection the followtng relation exists:

dieted. This approach assumes a Gaussian response

and a linear damping. An example of a Gaussian

dis-tribution of the low frequency surge motions of a

OTC 5175 W ICHERS 7

linearly moored 200 kDWT tanker in irregular head

waves is shown in Fig. 10. In the frequency domain

the variance of the low frequency surge motion will

become:

1 11

and the most probable maximum

to Longuet-Higgins [22] gives:

. . . (32)

.

displacement according

● . ...* -. (33)

in which: b = total damping

Cll = spring constant

N = number of oscillations =

duration of time = natural period of system

‘1 =xlt/cll =

= mean displacement of the tanker

Y-=

It total mean

The mean drift force

comes: m

~ = 2 ~ SC(0) T(u,u) do

o

The mean wave drift damping

cm _B-(u)

While the

SF(P) = 8

force.

in irregular waves

be-. be-. be-. be-. be-. be-. be-. be-. (34)

coefficient will be:

. . . (35)

spectral density of the force becomes:

jmS&+iI) SC(LI))T2(hI+P,LU)do . . (36)

o

In equations (34), (35) and (36):

SC(W) = spectral density of the wave elevation

T(fI),w) = continuous equivalent of the discrete ‘ii = (fJJi,~i) ._

.

B@/~: : continu us equivalent of the discrete

Bl(w)/q9

a

T(Lu+u,fJ))= continuous equivalent of the discrete

quadratic transfer function for the amplitude of the drift force

Tij = T(OJi,Uj)tith~i ~ LOj

v = difference frequency > 0

SF(U)

= spectral density of the drift force

F

= mean value of the drift force

%1

= mean wave drift damping coefficient.

For the computation a-linear spring constant

‘=11= 6.8 tffm was applied. For the wave spectrum a

Pierson-Moskowitz type spectrum with a significant

wave height Cwl,3 = IO m and a mean period ~1 = 12 s

was chosen. The computed spectral densities

accord-ing to equation (36) are shown in Fig. 11.

The results of the computation are shown below:

l.le=~

cll/(M+all)

‘~

6.8/26,145.4= 0.0161radfs

T = 390 S Duration of time = 9000 s N = 9000/390 = 23 oscillations

F=

Fc = %’= 1 ’11 = bllc = b .

sF(~e)=

ox = ‘1 = ‘lmax. . Without current -85.6 tf 29.0 tf/m/s 16.0 tflulls 45.0 tf/nl/s 27,700 tf2s 11.92 m -12.59 m -42.45 m With current (2 k.) -103.8 tf -10*4 tf 25.7 tfjmls 20.2 tflnlls 45*9 tf/m/s 43,000 tf2s 14.71 m -16.79 m -53.63 m

For the 1-DOF case the computation procedure

has been shown to calculate the motions of a

linear-ly moored tanker. In an irregular sea combined with

current the mean wave drift force will Lncrease due

to the current effect. For the oscillating part of

the wave drift force a same degree of increase due

to current effect has been assumed. Due to the low

frequency oscillations the wave drift damp~ng has to be taken into account.

So far the hydrodynamic damping forces and com-putation procedures for the 1-DOF. In the next chap-ter some clarification will be given on the

hydrody-namic damping and computation procedures for the

more complicated 3-DOF.

THREE DEGREES OF FREEDOM

Eguations of low fre~uency motion - ---

---‘lo study the motions of the vessel in the 3-DOF

use has been made of two different systems of axes

as indicated in Fig. 12:

the system of axis 0X(1)X(2) ie fixed to earth;

- the frame GXIX2 ia linked to the vessel in which

the origin corresponds with the centre of gravity

of the vessel.

GXIX2 may be transformed in 0x(1)x(2) by the two

translations xl(t) and x2(t) and the rotation x6(t).

Following equation (1) and taktng only the low

(2)

frequency motion components x, , x\2) and X52) into

account ‘the equations of motion fo$ the sh~p-bound

system of axes will read:

8 PROGRESS IN COMPUTER SIMULATIONS OF SPM MOORED VESSELS OTC 5175 where:

{)

‘6 ( MO M= OM 00 [ 00 D= 00 00

-()[)

(2) ‘1 _‘1 x . (2) ‘2 = ‘2 (2) ‘6 o 0 16 ) . -x 2 +*1 o )

~H = hydrodynamic fluid reactive forces

% = wind forces

~= mooring forces

~= wave drift forces

~T = thrust of the bow and main propellera. Low frequency fluid reactive forces

---Evaluating the low frequency fluid reactive

forces

equati;n~h~f ~~o$only: ‘iii be considered in the

M(~+D@=~ . . . ● .(38)

The relative velocity of the vessel with respect to

the fluid is:

v = (U2 + # cr r r . . . (39) in which: =: ‘r .1 - v= Cos($ - X6) ‘r =x - V sin(+c - X6) V= = c~rren~ veloc$ty 4C = current direction

X6 = yaw angle in global co-ordinates.

Because of the low frequency motions it can be

assumed that the disturbances of the free fluid

surface are negligible. Assuming an ideal_ fluid,

Norrbin [24] derived for the forces exerted on the

vessel: x~~ = X2H = x@ = where or: x~H = . - allur+ . a22vr %j6~6 -. .2 a22vrx6 + a26x6 . allurx6 - a26z6 (a22a11)urvr -. -. -. (; + ur*6) a62 r . . . (40)

akj = added mass coefficient at low frequency,

- allkl - (a22-all)Vc sin($c-x6)ft6+

a22~2+6i-.2

+ a26x6 . . . (41)

X21 = - a~.2%2- a26it6- (a._2-all)vcCos(vc-xb)$j +

. .

- allxIx6 . . . . (42)

x6~ = - a&jx6 -

a62%2

- (a22-all)urvr+ a26&1216

. . . (43)

Equations (41) and (42) lead to the well-known

d’Alembert paradox because the right-hand sides are

equal to zero for 11 = X2 = *6 = O. In equation (43)

the term (a22-a l)u vr is the only arising in an

ideal fluid and o$ten referred to as the

Munk-moment.

In the real fluid, however, viscosity is

in-volved. The viscosity leads to modifications of the

velocity dependent terms and/or introduces

addition-al damping (resistance) terms. Further it may be

assumed that the acceleration dependent terms are

hardly affected by viscosity.

Replacing the destabilizing Munk-moment by the

stationary current moment formu~tion and.n~glecting

the small contributions of a x

direction ~’d6xan~i%~~% ~ ~~~

spectively the x

write equation ~38) by the folfowing formulation

combining expressions for the real and ideal fluid:

(Mi-all)%l = (M+a22)i2*6 + ‘lstat ‘Xldyn

(M+a22)%2 + a26it6 = - (M+aI~);~*6 + xzs~at + ‘2dyn

(16+a66)~6 + a62%2 = ‘6stat+ ‘6dyn

. . . (44) in which: ‘Istat = ~Pw L T cl(~cr) ‘~r ‘2stat = ~pwL T C2(~cr) Vtr X6stat = ~pw L2T C6(~cr) V:r . . . (45)

being the stationary current forceslmoment, while

$ . arctan(-vrf-ur) is relative current direction,

a% the dynamic contribution is assumed to be:

Xldw = - (azz-a~~)vc ‘in(+c-xf_)x6. + ‘~~ ‘2dyn . = - (a22-all)Vc cos(Vc-x6)x6 + X2D ‘6dyn = + *6D . . . (46) which consists of a potential and a viscous part.

Both the viscous stationary current forces/

I

moment and the viscous part of the dynamic

contri-butions can be determined by model tests. PMM model tests

In order to determine the dynamic contributions

Planar Motion Mechanism (PMM)

~~$ ;’ h%v%ynb~~T ~a%nied out. For the PMM tests a

OTC 5175 WICHERS 9

The tests were carried out at scale 1 to 82.5

on basis of Froude’s law of similitude. A body plan

and the main particulars of the tanker are given in

Fig. 8 and Table 1 respectively. The tests were

ca~-ried out in the Shallow Water Basin. The basin

mea-sures 16 m by 220 m and is provided with a towing

carriage. The oscillator was connected to the

_car-riage. By running the carriage, current @S simu-.

lated. By means of ship-bound force transducers the

vessel was connected to the oscillator allowing

pitch, roll and heave motions. Sway, yaw and

com-bined sway-yaw motions were performed for zero

speed, 2 knots and 4 knots current speed. Although

the stroke was kept constant the frequency of the

motions were changed during the tests. While the

sway and combined sway-yaw oscillation tests were

carried out for a restricted number of current

angles, the yaw tests were performed for sector

steps of 45 degrees.

Besides the oscillation tests steady current

force/moment measurements were also carried out. The

steady current resistance coefficients are shown in

Figs. 13 and 14. Based on the measured steady

resis-tance forces/moment and the low frequency viscous

reactive forces/moment the equations of motion in

calm water and in current will be dealt with in the

next sections.

Equations of motion in calm water

---In Figs. 15 and 16 some results of oscillation

tests for the sway and yaw mode of motions in calm

water have been presented. The data concern the

damping (resistance) coefficients.

For sway direction the damping coefficients B22

and B62 have been obtained as fOllows:

=- $ PwT-B22(Fp-AP) *21*21 . . . . (47)

.- + PWT B62(FP2-AP2) :21i2\ . . . . (48)

while in yaw direction the coefficients B26 and B66

can be determined as follows:

X26=- ~PwT 1 6PWT .-— X66 = -+PWT . -~PwT in which! X22, ’62, ’26, AP FP B26(FP3+AP3)-*6]%6] . . . . (49) B66(FP4+AP4) ;61:61 . . . . (50)

X66 = measured forces/moment,

in-qua-drature with the applied dis-placements

.=.aft perpendicular = fore perpendicular.

The results show that the values B22 and B62 are

frequency independent and correspond to a

Keulegan-Carpenter number:

KC = (~2T)/B= (211A2)/B= 4.

Faltinsen et al. [18] show that for other KC

numbers at the same low range the transverse

resis-tance coefficients for tanker-shaped cross sections

should be approximately constant. Assuming a strip

type approach for the yaw mode of motion using local

transverse resistance coefficients the same will

hold true for the coefficients B66 and B26.

In Spite Of the faCt that B22=B66 and ‘62SB26,

but because _{B 2=B66 + ‘~2WB26 a non-homogenous}

distribution o~the transverse resistance

coeffi-cient over the length of the vessel must exist.

Com-puting the resistance coefficient distribution over

the length of the vessel applying the by Sharma et

al., [25] and [26], suggested high order

polynomin-als failed. A simplified method assuming constant

resistance coefficients over section 20-18, section

18-4, section 4-2 and section 2-O and a strip type

approach as is indicated by equations (47) through

(50) was applied. The four computed resistance

coefficients together with the faired curve giving

the C(k) as function of the longitudinal position

along the tanker centre line is presented in Fig.

17.

Applying the findings

cous fluid reactive forces

ing a decoupling with the

be written as follows:

to the low frequency

vis-in calm water and

assum-surge, equation (44) can

(M+all)%l = (M+a22)%2 + a26k6 = (I+a66)~6 + a62y2 = in which: x~sw = - . ’11 ‘1 FP (M+a22) ~2+6+x~SW - (M+all) ‘~*6+x2SW ‘6sW . . . (51)

Reviewing the values of the damping

coeffi-cients for calm water it can be concluded that

application of the values of the stationary current

resistance coefficients may lead to an

underestima-tion of the damping forces. Equations of motion in current -

---Referring to equations (44) and (46) the

dynam-ic contribution was read as:

‘ldyn = - (a22-all) cV sin(*c-x6)*6 + ‘lD

‘2dyn = - (a22-a~~)Vc coe.(~c-xb)f$j + ‘2D

‘6dyn = + X6D

1 J I I

potential parts viscous

10 PRCGRESS T-RCOMPUTER SllJULATIONSOF SPM MOORED VESSELS OTC 5175

For the viscous parts the following

formula-tions have been used in the past and were based on

local cross flow principles:

1. Local transverse velocity times local transverse

velocity (1979) [4]:

- vcr]vcrl]dl

X6~ = *PW T c2(900) ~ [(Vcr-*6L) ]Vcr-~6~I +

- V=r\Vcrl]$? d;

.,0 . . . ..* (52)

2. Local transverse velocity times local total

ve-locity (1980) [6] :

r

[(vcr-*#){

(vcr-:6~)2

+

‘~D= *pw Tc2(90”) ~ + Ujr}$ - vcrvcr]dg X6D = ~Pw T C2(900) Y [(vcr-~6~){(vcr-~6g)2 + + U:r}$ - vcrv~] % d% .,.0... .*.. (53)

3. Local total velocity times local total velocity

(1983) [IO] : x2D=0 (vcr-

k62)2+

U:r} + . . . (54) in which: _{‘cr = - ‘r} v cr = - ‘r

The present formulation for the viscous pares

in the dynamic contribution have been derived from

model tests and can be read in the following form: XID = f(L,T,Vc,Vc,xl,?$,x2,f2,x6,~6,Fp#)

‘2D = f(L,T,Vc,Yc,xl,*1~x2,+2, ~, b,x ; FP,AP)

X6D = f(L,T,Vc,~c,xl,:l,x2,~2,x6,x6,“ FP,AP)

● . . . (55)

Concerning the potential parts in equation (46)

the present formulation corresponds to Obokata [IO]

but deviates from Molin [6]. The formulation as

giv-en by Molin seems to be:

‘ldyn (potential part) = allVc sin($c-x6) ;6

X2dYn (potential part) = - azzvc cos(4c-x6) *6 X6dYn (potential part) = - a26Vc cos(Vc-x6) *6

. . . (56)

The computed results of XldYn> X2dyn and ‘6dYn

ac-cording to the formulations given in [IO] and [6]

were compared with results of oscillation tests for

the yaw mode of motion in current.

The oscillation tests shown were performed for

a Yaw angle amplitude ‘6a = 0.3 rad and an

oscilla-ting period T = 269 s. The results of the dynamic

contribution (for ~ = -0.007 rad/s) as function of

the relative curre$~ angle has been plotted in the

Figs. 18, 19 and 20. Ap~lication

--

---In order to check the computation procedures on

the damping forces on the slow motions of an SPM

moored vessel, some computer simulations have been

carried out and compared with model tests. The model

tests concern a bow hawser moored tanker in current

(fully loaded) and in a co-linear current and wind

field (ballasted). The bow hawser was connected to a

fixed point. The load-elongation characteristic of

the hawser is given in Fig. 21. The tanker model was

the same as was mentioned earlier. The SPM model

tests were carried out in the Wave and Current

Ba-sin, measuring 40 m by 60 m and 1 m water depth. The

wind was generated by a battery of wind fans. The

current was pumped from one side to the other side

of the basin.

By means of the present formulation the results

of the simulation are shown in Figs. 22 and 23 and

compared with the results of the model tests. For

the stationary parts in equation (45) the resistance coefficients were used as was measured (see Figs. 14

and 15). For the wind forces reference is made to

[27].

Comparing the results of the computations and

model tests a good correlation was found.

CONCLUSIONS

For a vessel performing low frequency motions

in an irregular sea the total wave drift force can

be split up in a wave drift force and a mean wave

drift damping force. For a vessel moored in

irregu-lar seas combined with current the wave drift force

will increase due to the current. Due to the low

frequency oscillations the total drift force

con-sists of the increased wave drift force and a mean

wave drift damping force.

The hydrodynamic viscous fluid reaction forces

acting on a tanker performing slow oscillating

mo-tions in calm water or in a current field must be

carefully considered. Excessive underestimation can

be made by applying the steady current resistance

formulation for the still water case.

Considering the yaw oscillating motion of a

vessel in current the dynamic viscous contribution

in yaw and sway direction due to the cross flow

principle based on the total local velocity deviates

to some extent from the present finding.

Contribu-tions in surge direction seem to be caused for an

OTC 5175 ‘–-WICHERS 11

In this first phase of the study the computer

simulations give good correlation with the model

tests. In a next phase a sensitivity analysls of the

simulations and a complete description of the

pres-ent simulation will be dealt with.

NOMENCLATURE A AP ai “ B’ Iii ‘i Bij b b .. ~~J Cij F(l; ~(2, 7 FP G g H h I k L f.. M N P . . . . . . . . . . . . . . s . = . . P(ui,uj) = T(fJJi,Uj)= ? T’ : T . t . u . v . x . ;i . x(i) = ‘IN . Eij =

wave group envelope

position of aft perpendicular added mass

breadth of the vessel

wave drift damping coefficient mean wave drift damping coefficient viscous damping coefficient in still water

total damping potential damping

resistance coefficient in current restoring coefficient

high frequency force low frequency force mean wave drift force

position of fore perpendicular centre of gravity

acceleration of gravity depth of vessel

water depth

moment of inertia of the vessel wave number

“length of the vessel between perpendicul-ars

co-ordinate in length direction of the vessel with regard to G

mass of vessel number

position of fai.rlead

in phase part of the quadratic wave

;; f; fo;ce tranafer functioni’

Qij = in-quadrature part of the quadratic wave drift force transfer function wetted surface

spectral density of wave spectrum

spectral density of wave drift wave force spectrum

Ti . = amplitude of the quadratic wave dr~ft force transfer function

mean wave period of wave spectrum draft of the vessel

natural period time

forward speed current speed force/moment vector

linear/rotational displacement in i di-rection (vessel bound)

linear displacement in i direction (earth bound)

N-th amplitude of the decaying surge motion

first and second derivatives of displace-ment with respect to time

high frequency motion vector low frequency motion vector small parameter

phase angle between the low frequency part of the wave drift force and the low

. . . . . . . . . . . . . . . Indices: A= a . c . D . H= i= ij = M . n . r . T . w=

frequency part of the square of the wave elevation

wave elevation

significant wave height small parameter dynamic viscosity = V.PW wave length low frequency wave direction kinematic viscosity = 1.18831 * 10-6 m2s-1

;:;:;;i:f$z:$ty ‘f ‘ea ‘ater ‘

specific density of air = 0.00013 tfm-4s2 standard deviation angle frequency frequency of encounter due to wind amplitude current

viscous part of dynamic contribution due to fluid reaction

direction of motion or number

in i direction due to motion in j direc-tion or numeradirec-tion

due to mooring system number

relative

due to thrusters

due to wave drift forces

1. 2. 3. 4. 5. 6. 7. 8. 9.

Owen, D.G. and Linfoot, B.L.: “The Development

of Mathematical Models of Single-Point Mooring

Installations”, OTC paper 2490, Houston, 1976.

Wtchers, J.E.W.: “On the Slow Motions of

Tank-ers Moored to Single Point Mooring Systems”,

OTC paper 2548, Houston, 1976.

Muga, B.J. and Freeman, M.A.: “Computer

Simula-tion of Single Point Moorings”, OTC paper 2829,

Houston, 1977.

Withers, J.E.W.: “Slowly Oscillating Mooring

Forces in Single Point Mooring Systems”, BOSS

’79, London, August 1979.

Faltinsen, O.M., Kjaerland, O. et al.:

“Hydro-dynamic Analysis of Tankers at Single Point

Mooring Systems”, BOSS ’79, London, August

1979.

Molin, B. and Bureau, G.: “A Simulation Model

for the Dynamic Behaviour of Tankers Moored to

SPM”, International Symposium on Ocean

Engi-neering and Ship Handling, Cothenburg, 1980.

Sorheim, H.R.: “Analysis of Motion in Single

Point Mooring Systems”, Modelling

Identifica-tion and Control, Vol. 1, No. 3, 1980.

Ractliffe, A.T. and Clarke, D.: “Development of

a Comprehensive Simulation Model of a Single

Point Mooring System”, The Royal Institution of

Naval Architects, Paper 9, London, 1980.

Natvig, B.J. and Berta, M.: “Comprehensive

Dy-namic Analysis of Offshore Loading Concepts”,

12 PROGRESS IN COMPUTER SIMULATIONS OF SPM MOORED VESSELS OTC 5175 10. 110 12. 13. 14. 15. 16. 17* 18.

Obokata, J.: “Mathematical Approximation of the

Slow Oscillation of a Ship Moored to Single

Point Moorings”, Marintec Offshore China

Con-ference, Shanghai, October 23-26, 1983.

Withers, J.E.V. and Van den Boom, H.J.J.:

“Simulation of the Behaviour of SPM Moored

Vessels in Irregular Waves, Wind and Current”,

Deep Water Offshore Technology, Malta, 1983.

Withers, J.E.W.: “On the Low Frequency Surge

Motions of Vessels Mooring in High Seas”, OTC

paper 4437, 1982.

Van Oortmerssen, G.: “The Motions of a Moored

Ship in Waves”, MARIN publication, Wageningen,

1976.

Pinkster, J.A.: “Low Frequency Second Order

Wave Exciting Forces on Floating Structures”,

MARIN publication, Wageningen, 1980.

Withers, J.E.W. and Huijsmans, R.H.M.: “On the

Low Frequency Hydrodynamic Damping Forces

Act-ing on Offshore Moored Vessels”, OTC paper

4813, 1984.

Takagi, M., Nakamura, S. and Saito, K.: “On the Low Frequency Damping Forces Acting on a Floored

Body in Waves”, written contribution to the

Technical Report of Ocean Engineering

Commit-tee, ITTC ’84, Cothenburg, 1984.

Ando, S. and Kate, S.: “On the Hydrodynamic

Forces at the Low Frequency Motion of Moored

Floating Structures”, 7th Ocean Engineering

symposium, June 13-14, 1984, The Society of

Naval Architects of Ja-pan.

Faltinsen, O.M.. Dahle. L.A. and Sortland. B.:

“Slow Drift Damping and Response of a M~ored

Ship in Irregular Waves”, OMAE, Tokyo, April

1986. 19. 20. 21* 22. 23. 24. 25. 26. 27.

Withers, J.E.W. and Van Sluijs, M.F.: “The

Influence of Waves on the Low Frequency

Hydro-dynamic Coefficients of Moored Vessels”, OTC

paper 3625, 1979.

Huijsmans, R.H.M. and Hermans, A.J.: “A Fast

Algorithm for Computation of 3-D Ship Motions

at Moderate Forward Speed”, 4th International

Conference on NumericaL Ship Hydrodynamics,

Washington, 1985.

Lamb, H.: “Hydrodynamics”, Cambridge, six

edi-tion, 1932.

Cartwright, D.E. and Longuet-Higgins, M.S.:

“The Statistical Distribution of the Maxima of

a Random Function”, Proceedings Royal Society

of London, Ser. A237, No. 1203, 1956.

Newman, J.N.: “Second Order, Slowly Varying

Forces on Vessels in Irregular Waves”,

Inter-national Symposium on the Dynamics of Marine

Vehicles and Structures in Waves, London, 1974.

Norrbin, N.H.: “Theory and Observations on the

Use of a Mathematical Model for a Ship’s

Ma-noeuvring in Deep and Confined Watera”,

Pro-ceedings 8th Symposium on Naval Hydrodynamics,

1970.

Sharma, S.D. and Zimmerman, B. :

“Schr3g-schlepp- und Drehversuche in Vier Quadranten

-Teil 1/2”, Schiff und Hafen/KommandobriickeHeft

10/81, 33 Jahrgang.

Sharma, S.D.: “Schr3gachlepp- und Drehversuche

in Vier Quadranten - Teil 2“, Schiff und Hafen/ KommandobriickeHeft 9/82, 34 Jahrgang.

Remerv. G.F.Ff. and Van Oortmerssen. G.: “The

Mean ‘W~ve, Wind and Current Forces kn Offshore

Structures and their Role in the Design of

Table 1

MAIN PARTICULARSAND INPUT DATA FOR THE 200 kDWT TANKER

Designation

Magnitude

Symbol Unit

Fully loaded Ballasted

Length between perpendiculars L m 310.00 310.00

Breadth B m 47.17 47.17

Depth H m 29.70 29.70

Draft T m 18.90 7.56

Wetted area

s

m2 22,804

Displacementvolume v m3 234,994 88,956

Centre of buoyancy forward of

section 10 m m 6.6 10.46

Centre of gravity above keel m m 13.32 13.32

Metacentricheight m m 5.78 13.94

Transverse radius of gyration _kll m 17.00 17.00

Longitudinalradius of gyration _k22 m 77.47 82.15

Yaw radius of gyration _k66 m 79.30 83.90

.------

---blindarea of superstructure(aft):

- lateral area _‘LS m2 922 922

- transversearea _‘TS m2 853 853

---.---

--Added mass _al1 tfs2m-1 1,592 250

u = O.O79 rad/s _a22 tfs2m-1 25,336 4,293

(water depth 82.5 m) _a26 tfs2’ -78,802 -16,132

a62 tfsz -78,802 -16,132 a66 tfms2 125,300,000 23,200,000 ------------.--- ---Potentialdamping _bll tfsm-l 5.6 1.0 u = 0.079 rad/s _b22 tfsm-1 23.2 3.5 (water depth 82.5 m) b26 tfs -38.9 -5.15 b62 tfs -38.9 -5.15 b66 tfms 1,416 339.0

Table 2

WAVE DRIFT AMPLITUDE QUADRATIC TRANSFER FUNCTIONS WITH ANO WITHOUT CURRENT FOR A 200 kDWT TANKER IN 82.5 m WATER DEPTH

Amplitude quadratic transfer function T(tJJi,tiJ7-wTth6utcurrent

‘+ 0.20 0.24 0.28 0.32 0.36 13,4130.44 0.48 0.52 0.56 0.60 9.64 0.68 0.72 0.76 0.80 U.₁ 0.20 0.25 4.97 8.59 0.24 0.47 3.98 5.61 0.28 0.82 2.28 3.56 0.32 1.40 2.50 2.20 0.36 2.22 2.21 2.45 0.40 0.44 4.49 4.39 4.04 0.48 6.17 4.84 6.31 0.52 9.41 6.98 6.64 0.56 12.16 6.73 6.92 0.60 13.09 7.31 5.46 0.64 13.88 6.64 4.01 D.68 11.95 4.83 3.29 0.72 10.02 6.70 5.82 0.76 8.60 4.57 1,68 0.80 8.92 4.88 9.23

Ampl itude quadratic transfer function T(IJJi,Uj) with 2 kn. current

‘j 0.20 0.24 0.28 0.32 0.36 (3.40 13.44 0.48 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.80 l!+ 1 0.20 0.3 8.0 11.5 0.24 0.5 4.7 8.0 0.28 0.9 3.75 !.3 0.32 1.5 3.1 3.6 0.36 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.80 2.6 3.1 2.8 4.3 5.0 4.8 6.8 6.6 7.0 10.5 8.2 &.9 16.0 10.3 8.3 16.7 9.2 8.5 16.9 9.5 6.8 16.9 8.4 5.0 14.5 6.5 4.3 12.5 7.6 6.0 10.4 9.3 10.1

LON frequency farces

I I I

EXCITATION FORCES OAMPINGFORCES INERTIAFORCES

~1

Wave drift force damping f($, V) 1= Mean current Hydrodynamic viscous damping

Mean wind !+nd damping I

I

Linearized equation STABLE of motion STABILITY =...

Ezzizl

I I I -. .-, Excitation forces Slowly varying drift forces f($,v)

I

Frequency/time domain Time domain I

I I

High frequency forces I [ E’c’’a”~” f“+ EEEEC! I Inertia forces EG+EEl~D=EI=l .EEEK1

Ffg.l—Flow diagram5PMsimulation.

Model tests

Surge direction Sway direction Yaw direction

I

I Steady 1 inear motions (stationary current forces)

I

11 Steady rotating motion in calm water

III Steady rotating motion in current

Iq Osc711at0ry motions in calm water

I

EiEEEE+

Oscillatory motions in current Fig.2—Review of model tests performed to determine hydrodynamic

+X3,+X(3)

‘“a i

/

Fig. 3—System of coordinates.

L o

t

Fig. 4—Regular wave group.

2

2

1

0

II = 180 deg. ● Extinction tests @ TVwimg--tesEs \ ● ✏✍

/’

●

b

● 1 I I I I I I 1.0 1.5 2.0 2 ‘m 5

X,N in Still water 22 1-” 0 Measured .,”>.. ● Computed according 20 Q •~ to equation (29) 0 ‘0 9>% 18 K,. “L b% 16 b bb 14 m .:x b. 12 ‘.Q v 11 ‘Oa_~. o? 10 –- 0.0 ‘f,o 9 y o .,0 \\ 0 8 \\

++-++&+

N SURGE DECAY \ Current (2 kn. ) \ 22 Q 4/ 0 Measured “:/ ● Computed according to equation [30) ;; z ‘$. ~b\* 16 — \4\ $. 14 — :,,\ “so\ 12 — “<) 11 — _%\ 10 — 9 — 8 — ‘\ 7 I I I I I I I \\ o 4 8 12 16 N N = number of oscillations

Fig. 6—Surge motion extinction tests with and without current.

3.106

Tanker size:

❑ 200 kDWT (fully loaded) Z50 kOWT (fully loaded) : 55 kDWT (80% loaded) 2 ./’” ,/- //-811 — /0 n~ ,/ .n~ / “ 1 _{u ..} .: ‘ / “ + =$ o 0 5 10 15 20*1 6s/v

Fig. 7—Results of surge damping coefficients determined by means of motion decay tests In still water.

10-6

Body plan

-16 -15-10

8

— / ,0 -— — — —

7

—

—

— — / — — — — — — / . — -. — — —

7

.

—

—

— —

9

—

—

—

— / . -,0’ — — — — / / o-— — — — — — / . — — — — —

G* IO’ 4 SF (11) X2 in tf s 2 0 Vc : Z .kn. [— ———. — ‘c= 0 ‘n” [== Newman approximation [23] FuIT matrix _. Newman approximation [23] Full matrix =-- —--- _--_ --- _—. _ -. \ .\ .—— _ _{——. .}

“\

-

“\

“\.

\.\

:\.\

“\._+ I I I 0.02 0.04 . II in rad/s —.

Fig. 1l—Computed spectral densities of the wave. drift forces.

‘A Vc +(lA h +$c “X2‘+X2 x(2)

I

x(1)

Fig. 12-System of coordinates.

loaded: A 1.03 mls o -0 ● 2.06 mls

1

h[T = 4.37 0 2.57 mls 1 o Ballasted e 1.03 M/s] h/T = 10.9 $~r in degrees

922 666 0.0 C6 -0.0 -0.1 -0.1 /+\ Fully loaded: ‘8 A 8, i, ,/ Ballasted : J ‘? 1 \ \ A 1.03 mrs ● 2.06 m[s o 2.57 m[s 0 1.03 mls h/T = 4.37 hlT = 10.9 ‘\ i \ \\ 0/

O\

1’

\

/’

‘8,

_,0=98

1

I I I I ) 135 .“. ’30 45 0 *~r in degrees

Fig.14–Measured current moment resistance coefficient.

3

Ful 1y 1oaded tanker KC=4 2 -0---o---o 1 -0. I I I 0 0.025 0.05 0.075 6

Fully loaded tanker KC=4

4 - 0----@

----o--662 2

-15—Damping coefficients for the swaymode of motion in stillwater.

Fully loaded tanker

o--- fy--- ’26 2

t

Fully loaded tanker o-__-*__Q.

., . . Calm water 4 I I o Computed 3 - II \ I 2 - ‘o, / % I =..; _{-— —.-—}~ I_{- .—— ——} 1 - I 0 —-––-l-o-11 _I 0 I ! +t I Section: O 2 4 10 G 18 : (AP) (FP) I AP = -161.61 m FP = +148.39 m

Fig. 17—Transverse resistance coefficient as function of the longitudinal position atong the tanker center line.

@ Oscillation tests 200kOl{T ballasted tanker

— Method [101

— Method ~61 v. = 1.03 mis ———- Present -f;nnulation X6, = -0.007 radls .-30 – 20 – o 10 – , OH ‘Idyn ,/ Q, / in tf -lo -20 -30 -W :,.. _{-“’-:::. .--41~ _.Tl_} I I I I I I ,. ‘: , -=,..*. - _{.-, . 90 180 270’} 3 ‘-7tic-i6)in degrees -.

Fig. 18-The dynamic contribution in surge direction caused by oscillating yaw motion-hlurrent. +--._==-—— ; — _{t .-.____} ;...—..=” t a.x Oscillation tests “: - Method [101 ,, .. . .= _{—~- Method [61}

— ---:-y Present formulation ++, .: ’...

200 kDLKb_aJ 1as ted tanker

v = 1.03 m/s .C x6a = -0.007 rad/s bu ,- .. . ,..., ,. . ... +—~.&—– . ₄₀ -20 - ,/-/ ( ‘2dyn o in tf 20 --40 – + \ / \ -60 - b /1 1 /1 \ \ /’ -80 - \ \\x _--- /“”’ I 1 I I I o 90 180 270 I ) o ,. (!IC-X6)indegrees

15,00 10,001 6dyn in tfm 5,00[ o ox Oscillation tests

Method [10] 200 kDWT ballasted tanker

—— Method [6] Vc = 1.03 m/s

o Method [41

———— Present formulation _‘6a = -0.007 rad/s

0 0 0 0 0 0 000 0 0 I

_I

I o 90 180 270 3 (4C-X6) in degrees

Fig. 20—Thedynamic contribution inyaw direction caused byoscillating yaw motionin current. Load in tf 500 250 -/ o /

~o/O

o 0 10 20 30 Elongation in m Fig.21—Bowhawser load-elongation characteristic.

II IJ > E m r. II r-. l.n LO e-w . . . Coml.n NN+ NJ 1 II II II ..-000 ---.NLC ---Xxx . .

E

.r-.lJ .r-: n .

‘-s

(

w

3

x-u c