On the slow motions of tankers moored to single point mooring systems

AB It w i a ti Sc sh mo is of Du an ca In sp in mo ap me se In tu fr th cl t.e P.e pa

ARCHIEF

24 JUU 1978

OFFSHORE TECHNOLOGY CONFERENCE 6200 Ncrth Central Expressway

Dallas, Texas 75206

Technische Hogeschool

DIf

ein NUMBER O T C 2548

i PAPER

On the Slovì Motions of Tankers Moored to Single

Point Mooring Systems

By

J. E. W. Wichers, Netherlands Ship Model Basin

ThIS PAPER IS SUBJECT To CORRECTION ©Copyright 1976

Offshore Technc/ogy Conference on behalf of the American InstituteofMining, MeaI!urgical, and Petroleum

Engineers, Inc. (Society of Mining Engineers, The Metallurgical Society and Society of Petroleum Engineers),

American Association of Petroleum Geologists, American Instifutc of Chemical Engineers, American Society of Civil Engineers, American Society of Mechanical Engineers, Institute of Electrical and Electronics En-gin cers, Marine Technology Society, Society of Exploration Geophysicists, and Society of Naval Architects and Marine Enciineors.

This paper was prepared for presentation at the Eighth Annual Offshore Technology Conference, Houston, Tex., May 3-6, 1976. Permission to copy ¡s restricted to an abstract of not more than 300 words. Illustrations may not be copied. Such use of_{an abstract should contain conspicuous acknowledgment of where and by}

whom the paper is presented.

¿

7L"-s

i

9

presented.A procedure will be shown to

obtain practical values of added mass and dampin _{to calculate the nature of the} stability and the natural _{freguencies of} the system. To have insight _{in the} sensi--tivity of the various system parameters on the stability and the natural fre--quencies, results of calculations will be shown. Trends following from the calcula-tions are compared with results of model tests. The results of the tests and the

calculations show a reasonable _agreement.

INTRODUCTION

An SPM system occupied by a tanker will

generally he designed for a maximum operational condition with regard to the sea state. The sea state, at which the down-time starts (disconnectinc bow haw-ser and floating hoses) will he deter-mined for an important part by the force

level in the bow hawser.

The behaviour of a tanker moored to an SPN system is greatly determined by the slow motions of the tanker in the hori-zonta]. plane. Due to this slow motion

(fish-tailing and galloping) behaviour

ST RA CT

is known that in steady current and

nd. in combination with irregular waves

tanker will not take up a steady posi-on in the horizposi-ontal plane. Both full ale observations and model tests have own that the behaviour of a tanker cred to a single point mooring system

greatly determined by the slow motions

the tanker in the horizontal plane.

e to these slow motions (fish-tailing

B galloping) cf the tanker high loads n occur in the bow hawser.

order to be able to solve the general

'4 mooring problem of the slowly vary-g driftinvary-g behaviour of the object Dred by means of a bow hawser, it is oarent that first a number of funda-rital aspects will have to be studied parately.

this paper the results of a study On dynamic stability and the natural quencies of the modes of motions of

tanker in the horizontal plane, in-.ding the effect of the system parame-rs in steady current and wind will be Eerences and illustrations at and of

284 ON THE SLOW MOTIONS OF TANKERS MOORED TO SINGLE POINT MOORING SYSTEMS OTC 2548

of the tanker a slowly varying force occur in the bow hawser. Superimposed on the slowly varying force in the hawser are the forces caused by buoy and tanker motions with wave frequency. The

combi-nation of these forces can lead to hioh

peak loads in the bow hawser force.

The phenomena occurring while the tanker is moored, are complicated and will be iniuenced by a lot of parameters, for instance: the size of the tanker and loading condition, the elasticity pro-perties of the buoy and bow hawser, the length of the hawser, the forces on the vessel exerted by wind, current, waves, astern propulsion and the underkeel

clearance.

Some model tests have been carried out to illustrate the slow motion behaviour in the horizontal plane of a tanker in ballasted and fully loaded condition moored to an SPM system in steady wind

and current and in irregular waves.

The slow motion behaviour of the bal-lasted tanker is shown in Figure 1 and

of the fully loaded tanker in Figure 2.

The weather conditions are given in the

Figures.

In Figures 1 and 2 the positions of the tanker are shown at the time-points that the maximum bow hawser forces occur. The time-points at which the peak values occur are inscribed in the Figures. For some oscillations the registration of the force in the bow hawser as func-tion of time are shown in Figure 3. The spring characteristics of the hawsers

were non-linear.

In comparison with the hawser loads in-duced by the ballasted tanker, the loads in the hawser of the fully loaded tanker can be lower, while also for the fully loaded tanker the amplitude and the fre-quency of the slow motions are smaller.

In ref. [i] it is shown that the slowly

oscillating character of the surge

mo-tion of a moored vessel in head waves

(one-dimensional direction) is in

reso-nance when wave groups are present which encounter the vessel with a period in the vicinity of the natural period of

the mooring system. In ref. j2j it was

found that in one and the same sea state a vessel moored in head seas by means of a linear mooring system, using different values for the mooring stiffness, the low frequency surge motions were in each case of a resonant nature, which demon-strated that the low frequency wave ex-citation in irregular waves covers the whole range of normal natural frequencies

of a moored vessel.

Both examples dealt with vessels having one degree of freedom only. A vessel moored to an SPM, however, has three

degrees of freedom for horizontal moti-ons namely surge, sway and yaw.

In solving the general SPM mooring pro-blem it is in first ins,tance necessary

to know the state of the dynamic stabi-lity and secondly the frequency range of

the natural frequencies of these modes of the motions in the horizontal plane

of the tanker in steady conditions since knowledge of these aspects can give

in-sight in the nature of the low frequency motions of the vessel about its mean

position in the horizontal plane.

In this paper the procedure of calcula-tion of the stability and the natural

frequencies are for small deviations (sway, surge and yaw) from the equi-libium positions of the tanker in steady

wind and current. By means of a devel-oped computer program the effect of the

system parameters were calculated. The system parameters were:

- length of the bow hawser

- spring characteristics of the

hawser-buoy arrangement

-- astern propulsion

The combination of the tankers and wind and current condition including the various system parameters are given in

Table III.

EXTERNAL FORCES DUE TO STEADY WIND AND

CURRENT

By, as it were, holding the tanker at

the bow fairlead, but allowing it to ro-tate around this location, the equili-briuin position of the tanker relative to

the environment and the direction and the magnitude of the bow hawser force induced by the steady wind and current

forces on the tanker, were calculated.

For the calculations use is made of the current and wind force computer program

of which a description is given in ref. The sign convention of the total

exter-nal current and wind forces on the tan-ker expressed by the transverse force

the longitudinal force X and the moment are given in Figure 4.

The equilibrium position with regard to. the current direction _{will be found}

if the moment with regard to the

fair-lead A corresponds to:

OTC 2548

J.E.W. WICJ-IERS

in which a = distance between the fair-lead and the centre of

gra-vity

It may be noted that the _{magnitude of X} does not influence the equilibrium posi-tion relative to the environment.

The magnitude and the _{direction of the}

bow hawser will be _{determined by X(p)}

and

The total environmental _{forces X,} and N _{will only be influenced}

by the

deviaion in the yaw direction _{() since}

they are not dependent on the x and y co-ordinates of the vessel _{in the} horizon-tal plane.

The variation in the _{forces and moment}

due to a small angular _{displacement A}

of the vessel will be: 3X -.A ₍₂₎ AFyu - .A1) (3) 3N = .A ₍₄₎

EXTERNAL FORCES DUE TO THE BOW HAWSER

There will be interaction _{between the}

tanker and the hawser _{force, when the} tanker will have small _{deviations out of} her equilibrium _{position. For reason of}

simplicity all calculations _{on the}

haw-ser force and direction have _{to be}

con-sidered in the horizontal plane through

the fairlead A. For the determination of the variation of the _{force and direction}

of the hawser due to _{the small}

devia-tions of fairlead A _{an earth-bound}

co-ordinate system is _{chosen, of which the} origin corresponds _{to the vertical centr} line of the buoy in unoccupied rest con-dition. The definitions _{and sign} conven-tions are given in Figure _4.

In the equilibrium _{position the}

quanti-ties will be denoted by the indices 0,

while in the position _{after the}

devia-tion the quantities will have the indices

1.

The interaction _{due to the djslcement}

of the fairlead _{can be split up into a}

variation in magnitude _{of the hawser}

force (elongation) and _{in a variation in}

the direction of the force.

The variation in the magnitude of the

force in the bow hawser _{will be:}

AFb = F1 - F0 = 3 F0 in which Al = l - 10 31 ₃₁ = _3x .Ax + A 2

2½

while 10 = (x0 +y0 3lo Xo

5T

= cos 310 y0 = sin so: AFb = (cos

OeA+5iO.AyA).. (5)

The variation in the bow hawser direc-t.iori will be:

A0 =

-ap0 =

.i--_ Ax+ --;-.AyA

Yo X0 sin 2 10 - cos I)0 10 + 2 10 n 10 .AxAf cos 10 .Ay (6) Equation (5) and (6) _{represent the} vari-ation in magnitude

and direction of the bow hawser force caused by small positive displacements of the _fairlead

AXA ard y. in the earth-bound

co-ordinate axis. pressed in terms of the surge, sway and

yaw displacement of and _{about the centre}

of gravity

of the tanker

(see Figure D)

the fairlead displacements _Axa and Ay become:

Ax _{= - Ax}

AYa = Ay

-Substituting in equation _{(5) and (6) we} have: 3F0 AFb =-(cos0.Ax+sin0.Ay+sin0.Atp.a)--y-(7)

sin0

cosij0 = lo Ax -Ay 285 cosVJ0 .a (8) O in which _arctg si so -YO 10 X0

The external restoring _{forces of the bow}

hawser, indicated by the indices

to the variation of the _position fairlead, will be:

¿Fb= (Fo+AFb) cos ( _{+Ap-Aip) -F0cosp0. . (9)}

EFb= (FO+AFb) sin(0+A0-Ap) -F0sinp0. ₍₁₀₎

¿Mb = ¿FYb.a ₍₁₁₎

Due to the displacements _{Ax, ¿y and ¿}

the total external variation in forces

and moments will be:

¿F = ¿F + ¿F (12) x xb xu ¿F_y = ¿F + ¿F ₍₁₃₎ yb yu AM _{= ¿Fb.a + ¿Nu} (14) Substituting equation (2) , (3) _{, (4)} (9) , (10) and (11) in equation (12) , (13)

and (14) will result _{in the following}

restoring forces: F0 2 F0 ₂ ¿F

_{= -(---.cos i +.sjn}

)Ax +

o0.si0

+

-T--

r_)c0sPo.sinh10 -o o

-

F0sin0 - u}A = = - cxx.Axcxy.AY-cx.AP ₍₁₅₎ aF0 AF =

_{(- sin20+ -.cos2i0)Ay}

₊ F - _{)cosU0.sin0}Ax +} F0 sin2p0 F0 2

-{(-1--.a.

+ -1--.a.cos ₊ Yu + F0sinP0 -= - c _{.Ay-c .Ax - c .A} (16) yy yx ¿M = - c _{.a.Ax - c .a.Ay +} yx _yy F _F + _{.a.cos20 +} o _o + F0sinp0)a - =

= -

CjjxAX - c.AY -

. _{. (17)}

In the _{above expressions all the}

the sum the

buoy and the hawser _{are linear, then the}

loaded length 10 will be: 10 = +

in which i = the unloaded _{length of the}

hawser.

However, if the sum of the _elastic

cha-racteristics of the buoy and _{the hawser}

are non-linear, then the elongation _and

the spring constant at _{the level of bow} hawser force F0 have _{to be determined} from the load elongation _{relationship.} As was mentioned before, _{the equilibrium}

direction of the tanker with _{respect to}

the wind and current will _{be independent}

of the magnitude of the _longitudinal

force. In case astern _{propulsion is used}

only the bow hawser direction and the

magnitude of the force in _{the bow hawser}

will change.

EQUATIONS OF MOTIONS

It can be shown that the _linearized equa-tions of motion referred _{to the axes of} the tanker, which due to the small mo-tions regarded _{may be considered as}

an earth-bound system of _{co-ordinates, are} as follows: Surge: m_xx

.+b

_xx.z+c_xx.x+m

.+b

xy

xyxy

= O Sway: m_yx

.+b

_yx

.+c

.x+m .-fh _{.y+ (18)} yx _{yy yy yy} +m

_.+b

_.i+c . = O y,b Yaw:

mIJJX +biJix

.+c

.x+m

.+b

.y+

Y _PY

O

The coupling terms in mass and damping will be neglected. This _{is justified} partly from symmetry _{considerations and}

partly by considering _{the relatively}

small contributions of these terms.

Thus it is assumed that _{for the present}

study m

-m

_-m

_-m

_-m

_-m

-O

xy yx x bx yb and byx = = = = = O system

b, due _{parameters are incorporated. If}

of the _{of the elastic characteristic of}

OTC 2548 _{J.E.Ñ. WICHERS}

Thus the coupling exists only by _means of the spring constants.

The equations of motion ar. thus simpli-fled:

m_xx

.+b

_xx.x+c_xx.x+c_xy.y+c .q = O

= O .. (19) = O

In the following sections the determina-tion of the added mass and damping

coef-ficients are dealt with. The spring

con-stants are already discussed in the pre-vious sections.

DETERMINATION OF ADDED MASS

The quantities in and m represent the

virtual mass, wich is he sum of the mass of the tanker and the added mass.

The yaw moment of inertia in1 of the

tanker about _{he vertical a1s through}

the centre of gravity will be the moment

of the virtual mass. The added masses are

dependent on the motion frequencies _of

the vessel. For the determination of the

added masses use is made of the values

given in ref. 4j . Expecting that the

frequencies of the modes of motion of the slow oscillations of_the tanker will not

exceed 0.07 rad.sec. i _{(period T larger}

than 90 sec.) mean values of a_xx' a and

a1 _{can be estimated and these valu}

we used for the calculations.

In Table II the masses of the tankers _and

the estimated added masses have been given. These values are representative

for the tankers used in this paper, the

particulars of which are given in Table

I. All values are valid for a ratio of water depth to fully loaded draft of 1.6.

DETERMINATION OF DAMPING

The damping is regarded to consist of a

potential damping part and a viscous

part due to current.

Potential damping

The potential damping _{is frequency}

de-pendent. Because the frequency _{range is}

limited, average damping _coefficients

have been determined from ref.

[41 under

the same conditions _{as for the added}

masses. The results have been _presented

in Table II.

Damping due to current

The damping due to _{current can be}

ex-pressed in the following _{terrils. For sake} of completeness also the _{oupling terms}

are mentioned, but they will be _neglected

for the calculations.

b

= - - b

_{- -} . b = __2 xx u ' xy v '

b= - ;b

--j ... (20) b

-

. L . b )q) ₁₂ yy in which:

= longitudinal current force in

equi-librium position

= transverse current force in

equili-brium position

u _{= surge velocity of the tanker}

y _{= sway velocity of the tanker}

r _{= yaw rate of the tanker}

L _{= length between perpendiculars}

The expressions for the above mentioned

partial derivatives will be described

below. The sign convention is _{given in}

Figure 5.

The components of the relative _current

velocities will be: y

=v -u

xr X

(21) y

=v -v

yr _y

The relative current velocity is as

fol-lows: -287 Vr ={(v-u) + (vy_v) 2 (22) 2

The current force component in _{the X and}

Y direction of the tanker will _be

respec-tively: X _{= q.v2(u,v).C(u,v,r)} (23) = (24) in which: q = ½p.L.T

C and C are the drag coefficients in longitudinal and lateral direction. The relative yaw moment will be:

2

-Nc = q.L.v

(u,v).CN(u,v,r) (2D) in which: C = yaw moment coefficient.

This moment, however, will not be used

further.

The relative current angle _{lPr amounts to:}

= arctg()

_V (26)

xr

The expression of the partial _derivatives

c

for u and y tending to zero and using equation (22) , (23) and (26) results in:

BX By (u,v) C r - q.2v (u,v) . .0 (u,v,r)+ oU r Bu X BC (u,v,r) +q.

V2

(u,v).

-r Bu in which: By (u,v) y r i x du _VC X _VC

(YL

BC 31f) V X r xr Bi V U r (Vx BC y - Y Bì 2

CV

C SO: V V BC c 2 X 2 _y x = - 2q.v .-_-.C+g.v v2Bl)c =

The expression of the partial derivatives

B

By using equation (22) , (24) and (26) will lead to:

BY sin

3Y(i)

C

(27)

The expressions of the coupled partial

derivatives can be obtained

in

a similar

way, but they will be neglected in the

present calculations.

For the approximation of b

, as

pre-sented in equation (20) Fiure 6 has been prepared in which the flow directions are shown when the vessel moves with a posi-tive velocity y in sway direction and when she yaws with a positive rate of

turning .

In sway direction the damping per unit

length of vessel is assumed to satisfy: b

.v(x)

while in yaw direction the damping force per unit of vessel length is presented

by:

b

Integration of the "local" damping force

over the length of the tanker and

multi-plied by the lever x, leads to the total

damping moment: b

_yy?

+-L

2

12

Mb _{- L}

-L

dx

= -

1--L h.i

So the yaw damping coefficient will be

approximately: b

ipip 12 yy

The same expressions can be applied for

the damping forces due to wind. However, these contributions will be small rela-tive to the contributions due to current.

This can be seen from equation (27) and

(28) were the speed of the current (wind) appears in the denominator.

Adding the potential damping to the damping due to the current, as given in

equation (20) , (27) , (28) and (29) the

total expression for the damping coeffi-cients will be:

surge:

cost

X(p)

C C C b

=2X().

xx c C y C

ON THE SLOW MOTIONS OF TANKERS MOORED TO SINGLE POINT MOORING SYSTEMS _{OTC 2548}

sin

+b p

V C sway: sin

BY()

b = 2Y (p ) C C C . (30) yy c c v Bi cos

+b p

VC YY yaw: =

In having obtained practical expressions for the determination of small motions

relative to the equilibrium position, the tools are available to study the dynamic stability of the equilibrium position and the natural frequencies of the various modes of motions of the tanker, which will be dealt with in the next section. DYNAMIC STABILITY AND NATURAL FREQUENCIES

In the steady state condition of current

and wind, the tanker will be assumed to be in her equilibrium position (static

stability) . Let us suppose now that she

(29) Cos V (28) C and BC(u,v,r) Bu cos _BX('p) = - 2X(lp). VC + sin VC

:

OTC 2548 _{J.E.W WICHERS}

289

C yx

C

will receive a disturbance inducing small

displacements in x, y and ii) direction.

The disturbance is assumed to be of short

duration. The tanker will be defined to

be dynamically stable in a specific

en-vironmental condition if she after some

times returns to her equilibrium

positi-on. The tanker will be called dynamically

unstable, if she after some times devi-ates further from her equilibrium

posi-tion. To obtain the criteria of dynamic

stability, the general solution of _the

linairized equations of motion (equation

(19)) will be analyzed. Because the

equations are linear, homogeneous and

possess constant coefficients, the

as-sumed solutions are taken as: et

= A1e

y = A2e ₍₃₁₎

=

where o _{is constant yet to be found and} A1, A.) and A3 are constants, which will depend on the initial disturbance.

The tanker will be dynamically stable _in

the steady current and wind condition _if

with increasing time the displacement

from the equilibrium position _(equation (31)) diminishes and approaches _{zero at}

the limit. This means that the exponent

in the assumed solutions of _{equation (31)}

must consist of either negative _real

num-bers or complex numnum-bers with negative

real parts. The complex numbers _give

in-sight in the natural frequencies _{of the} modes of motion of the tanker.

Substituting the assumed solution of

equation (31) in equation (19) , the fol-lowing three equations are obtained:

(m_XX.o2+b_XX.o+c_xx)A +c₁ .A +c .A =0 xy 2 xii) 3

cyx.A +(m1 _yy.o2+b_yy.o+c_yy)A +c₂ .A =0. (32)

y 3

c.Ai+c.A2+ (m

.a2+b

.a+c)A30

If the assumed solutions _{are valid, this}

set of three homogeneous equations _must

he satisfied. Equation (32) _{will be}

satisfied if the determinant of

coeffi-cients of A1 , A2 and A3 is equal to zero (A1 = A2 = A3 0) m

.2+

xx C _e +b .a+c X Xii) xx xx m yy C +b .a+c yii) yy yy

cy

+b

.o+c

lin]) 1)1,1) =0.. (33)

On expansion this determinant _{may be}

arranged as: Ao6+Bcy5+Cci4+Do3+Eo2+Fa+G = O in which:

A=m .m

.m XX _{yy ii)I)} B = b_xx.m_yy.m +b .m .m +b .rn _.m 1/)11) XX XX 1)1]) li)J XX _yy C = c_xx.m_yy.m _b+c_yy.m_XX.m +C .m .m + i]) ii)y xx _yy

+bxx.byy.m ti)+bXX.b,bib .myy+byy.bii)ii).mxx D ₌

_{b(cyy.m+cj1j).myy)+bYY (c.m}

c .m_XX)+b (m .c +m .c +b .b l])ii XX yy yy Xx xx yy +m (c .c E ₌ m_XX (c_yy.c -c .c ) ) ii)Y yl])

c,)+m

_{(cxx.cyy_cyx.cxy) +} +b ii)(byy.cXX±bXX.cyy)+c .b .b ii)ip xx yy F = b_XX(c_yy.c_ì]n])-c .c )+b (c .c -ii)y y _yy xx ii) -c .c )+b (c .c -c .c ii)x xi]) i]nîxx yy yx xy

G = eXX.cyy.ci])i])-cXX.c_y]) .c_i])y+cxy.c ,c +

yi]) ])x

+c_yx.c _y.c_xi])-c_yy .c .c -c .c c Xi]) X i]) yx xy

(35)

From the solution of equation _{(34) six}

roots will be obtained representing _in

general three complex and three _added

complex numbers, each with their real

parts.

In terms of harmonics the motions can be written as follows:

Alt _A2t

X = a1e .cosuj1t+a2e .cosw2t+ A3t +a3e .cosw3t A1t X t y = h1e _.cosw1t+b2.e 2 .cosw2t+ A3t +b3e .cosu3t X

t

_X2t

= c1e 1 .cosi1t+c2.e _.cosw2t+ A3t

+C3e .cosw3t

in which the constant coefficients bn and c _{depend on the initial} distur-bance. From equation (36) _{it will be seen}

that each motion consists of the natural frequencies (Wn) while the effect of the

harmonic of the natural _{frequency on the}

motion will be influenced _{by the decay}

Ant term (e ).

From the mentioned equations and

coeffi-cients it is hardly _{possible to have in}

(34)

(36)

290 ON THE SLOW MOTIONS OF TANKERS MOORED TO SINGLE POINT MOORING

an analytical way insight in the effect

of one of the system parameters e.g. the

length of the bow hawser, astern

propul-sion or the spring constant of the

buoy-hawser arrangement on the natural

fre-quency (wa) and the decay terms (eAt)

To obtain insight in the effect of the

various parameters, calculations have been carried out which will be dealt

with in the next section. CALCULATIONS

The number of variations in tanker con-ditions (sizes and drafts) and sea states

(steady wind and current) were restrict'd,

The tanker conditions and sea states, for which calculations have been carried out,

are given in Table III. In this Table th

parameters (length of bow hawser, spring constant, astern propulsion) and the input data necessary for the calculations on the natural periods and the decay

terms are also presented. The input data

were obtained from the results of the

current-wind force computer program. The

water depth in all cases amounts to 1.6

times the draft of the fully loaded tan-ker.

The weather conditions and the tanker sizes selected, are representative for

this type of mooring system installa-tions.

The range of the bow hawser lengths cover

commonly used lengths. Most of the calcu-lations have been performed with a linear spring characteristic of the buoy-hawser arrangement. For reason of symplicity th spring characteristic was kept constant

for each applied length of hawser. This

means that at increasing length the

dia-meter increases too.

In order to gain insight in the

sensiti-vity of the natural periods and decay terms also calculations for changes in the spring characteristics were carried out with a stiffer spring constant. The same was done with a non-linear elastic

characteristic for the buoy-hawser

ar-rangement. The data on the non-linear elastic characteristics are shown in

Figure 7.

From the results of the calculations on

the natural frequencies and decay terms, the natural frequencies w given as

func-tion of the unloaded bow awser length 1,

have been presented in the Figures 10 and 11. The decay terms have been presented

as motion decay characteristics in

Table IV. The definition of is as

follows: t (37) V

=

-n T n in which:

t = the time that the value of the

aro-plitude has been decreased by 95%

t was determined from:

At

nfl

e = 0.05

t in 0.05

Tn = period of natural frequency w

The motion decay characteristic

indi-cates the effect of the damping on the

appropriate harmonic component as pre-sented in equation (36)

If the value of 'J is zero and t is

finite the appropiate component in

equa-tion (36) will be heavily dampened. How-ever, the tanker can still be unstable dependent on the other two components.

If the value of is infinite and T is

finite, the value of the

appropriatecom-ponent in equation (36) will be infinite. Independent of the nature of the other terms, the tanker will be unstable. DISCUSSION OF RESULTS

The results of some calculations have been corre1ated with the results of a

model test. The periods of the slow mo-tion behaviour of the model were

deter-mined in the neighbourhood of the

equi-libriurn position by means of motion decay

tests. From records made of the force in

the bow hawser and of the angles indica-ting the position of ship and bow hawser as shown in Figure 8, the main frequen-cies of the motions could be discerned.

The results of the calculations and the

results of the model test appear to com-pare reasonable (Figure 9)

The longest natural period is difficult to distinguish partly due to the long period and partly due to its rather

heavily damped nature (see theoretical t/T3 value of calculation A)

Although the results of the tests were encouraging, it must bn borne in mindthat the number of test results was limited

and therefore not conclusive.

From the remaining calculations the fol-lowing trends and conclusioñs can be drawn:

- Direction of wind

the wind direction relative to the cur-rent will have an important influence

both on the natural frequencies and on

the decay characteristics of the slow motions of the ballasted tanker. The

trend exists that the risk for an

un-stable behaviour of the tanker

in-creases at decreasing angle between

the current and wind. - 1èncjth bow hawser

Lengthening of the bow hawser may

re-suit in an unstable behaviour of the tanker in the horizontal plane. This is in agreement with the conclusion of

ref. [511 , in which is described that

at increasing length of the towing line a towed vessel will be unstable in the

horizontal plane.

- Spring characteristic of buoy-hawser

arrangement

From the results of the computations it may be concluded that the trend

exists that neither the various

con-stant spring characteristics nor the non-linear spring characteristic will significantly affect the stability of the equilibrium position of the

bal-lasted tanker in wind and current. - Astern propulsion

The trend exists that astern propulsion generally increases the stability of

the ballasted tanker. - Influence draft

The trend is that (in the steady 5

knots current and 45 knots wind state)

the natural frequencies of the modes of motion of the fully loaded tanker are significantly smaller than the natural frequencies belonging to the ballasted tanker.

The above mentioned tendencies which _were

based on the results

of

_{computations,}

have been confirmed to a large extend _by

model tests on single point mooring

sys-tems performed in the past. CONCLUSIONS

In the present study it is shown that

iith a relatively simple linear model a

good impression can be obtained about _the

stability of the equilibrium position of

a tanker moored to a single point in wind

and current by means of a bow hawser.

Some preliminary model tests have shown the applicability of the method and the reasonable correlation between measured and computed natural periods of the

hori-zontal motions of the ship.

l3oth computations and model tests sho;

that a ship moored to an SP1 system _can

perform slow oscillating motions in the

horizontal plane in wind and current _on1

thus without the presence of a slow

os-cillating external drifting force due

to waves. These slow motions only _occur

when the "equilibrium position" of the

ship in wind and current is unstable. In that case the magnitude of the

oscil-lations can only be determined by taking

into account non-linear damping terms _in the describing equations of motions. _The

introduction of these terms will be one

of the next steps.

When the equilibrium position of the

tan-ker in wind and current is stable, the

slow motions will be damped after sorne

time unless an oscillating external forc

works on the ship. This force will be present in irregular waves and is called

the wave drifting force.

To introduce estimates of these wave

drifting forces in the right hand sides

of the equations of motions will also be

a subject for further investigation. NOMENCLATURE

= drag coefficient in longitudinal

direction

= drag coefficient in transverse

direction

= mean bow hawser force

= length between perpendiculars

yaw moment

either draft or period longitudinal force = transverse force

a _{= distance between the fairlead and}

the centre of gravity of the tan-ker

= added mass in i-direction result-ing from j-motion

= damping coefficient in i-direction

resulting from j-motion

= spring coefficient in i-direction resulting from j-motion

g _{= gravitational constant}

= unloaded length of the hawser-huoY

arrangement from the fairlead to the vertical line through the

centre of the buoy in zero posi-t ion

OTC 2548 _{J.E.W. WICHERS}

291 C X C y F0 L N T X y ij c.

10 m m.. J-J r t t n u V V C V. 1 X y A X p a)

= loaded length of the hawser-buoy

arrangement from the fairlead to

the vertical line through the

centre of the buoy in zero

posi-tion when the tanker is in the

equilibrium position = mass tanker

= total mass in i-direction re-sulting from j-motion

= yaw rate of tanker = time

= the time that the value of the amplitude has been decreased by

95%

= velocity tanker in surge direc-tion

= velocity tanker in sway directior = current velocity

= current velocity component in

i-direct i on = co-ordinate axis = co-ordinate axis = measured angle = measured angle = small displacement

= real part of complex number = motion decay characteristic = specific density of sea water = angle between current and ship

in equilibrium position

= angle in horizontal plane betweer longitudinal axis of tanker and

direction of hawser

= yaw angle tanker

= frequency of oscillation INDICES A = fairlead b = bow hawser c = due to current i,j,k = directions n

=1,2,3

O _{= indicates quantities in the}

equilibrium position

i = indicates quantities out of the

equilibrium position p = due to potential

r = relative

u = due to current and wind

REFERENCES

Hermans, A.J. and Remery, G.F.M.: "Resonance of moored objects in wave trains"

Conference of Coastal Engineering, Washington, 1970

Pinkster, J.A.:

"Low frequency phenomena associated

with vessels moored at sea"

SPE Paper No. 4837

Amsterdam 1974

Remery, G.F.M. and van Oortmerssen,G

"The mean wave, wind and. current

forces on offshore structures and

their role in design of mooring sys-tems"

OTC Paper No. 1741, May 1973 Van Oortmerssen, G.:

"The motions of a ship in shallow water"

Chesapeake Section of SNAME

November 19, 1975

Strandbgen, A.G., Schoenherr, K.E.

and Kobayashi, F.M.:

"The dynamic stability on course of towed ships"

TABLE i - PARTICULARS OF TANKERS

Designation Symbol Unit 220 KDWT tanker

350 KDWT tanker Length between perpendiculars m 316.00 363.50

Beam B m 48.70 57.88

Depth D ni 26.20 28.54

Area superstructure lateral AL m2 615.00 875.00

Area superstructure transverse AT m2 430.00 589.00

FULLY LOADED

Draft T ni 19.30 21.80

Displacement volume V m3 252,325 387,809 Centre of gravity before

section 10 (midships) f ni 9.61 7.55

Longitudinal gyradi.us k ni 79.05 90.88 Distance between centre of

gravit.y and fairlean a IR

iO.O

iio.zu

Water depth/Draft Wd/T - 1.60 1.61

BALLAST DRAFT: 43 PERCENT OF FULLY LOADED DRAFT

Draft T ni . 8.30 9.37

Displacement volume V ni3 _{103,718 156,000}

Centre of gravity before

sectìon 10 (:nidsnips) f 13.61 10.70

Longitudinal gyr.dius m 79.05 90.88

r)istance between centre of

gravity and fairlcad ni 146.39 173.05

T Confirnatton tests

x Sea state 1: 1

knot current perpendicular to 45 knot wind

Sea state 2: _{internal angle between 1 knot}

current and 45 knot wind amour.ts to 45 _de5rees Sea state 3: _{S knot current perpendicular to 45 knot}

wood

TABLE 3 - REVIEW OF CALCULATIONS AND MODEL TESTS

Designation _{Symbol Juli} 220 KDMT tanker f 310 5101T camer

100% T 43% T I

100% T 43% T

Mass tanker _n

tor..sm1 26,364 10,837 40,520 16,300

Added mass coefficient sway' _a/m

-1.6 1.0 1.6 1.0

Added mass sway _ayy

ton.s?01 42,160 10,637 64,830 16,300

Total mass sway

ton.s?rs 68,544 21,674 105,350 32,600

Added mass coefficient surge2 _a5/m

-0.08 0.05 0.08 0.05

Added nass surge _a5

ton.sm1 2,110 540 3,240 825

rotai mass surge _m

ton.s.rs 28,474 11,379 43,760 17,115

Moment of inertia of mass cf tanker _I,

tOn.s?m 1.65 x 10 6.76 s IO7 3.35 s IO 1.35 X 10

Added nass coefficoent yaw2 _a/(l.V.L2)

-0,1 0.02 0.1 0.02

Added nass moment _a9

ton.sn 2.90 s iO7 2.16 x 10' 5.35 s 10 0.43 s l0

Total moment of inertia _{n ton.s}

4.55 s 10 8.92 s 1O 8.70 a 1.78 x lO8

Potential dampIng coefficient sway' _b/Cm./i7t)

-0.1 0.05 0.1 0.05

Potential damping sway _b0

ton.s.m1 465 95 666 113

Potential damping coefficient surge2 _b5/lm./j7EI

-0.03 0 0.03 0

Potensial damping surge

ton.s.m1 140 0 _{200 0}

Potensial damping yaw2

ton.s.n C 0 0 0

Ratio water depth to fill loaded draft Wd/T - 1.60 3.73 1.61 3.75

Calco-lation Sea2 state Taro-size Draft in I o.. aoaded draft Length of bow hawse.. i in n 8F O t/m Ae sion in tons 9c 0 00 X, T

Ix

ly e xu 6?

rad. ton _tor/rad.

ton/rad. ton.no/rtd. A 1 220 43 80, 123 _{6.3, 12.6} -1.9 _{0.4110 34.1 - 0.0} 33.5 -0.3 - 11.7 - 1.1 - 160.1 - 605.0 1 220 43 40, 123, 240 - 1.9 0.4110 34.1 - 0.1 33.5 -0.3 - 11.7 - 1.1 - 160.1 - 605.6 1 220 43 40, 123, 240 6.3 _{15 1.9} 0.3650 49.54 - 0.1 33.5 -0.3 - 11.7 - 1.1 - 060.1 - 605.0 1 220 43 40, 123, 240 6.3 30 1.9 0.2812 63.78 - 0.1 33.5 -0.3 - 11.7 - 1.0 - 160.1 - 605.0 A' 11 220220 4343 12340 6.36.3 -- 1.81.9 6.390.37 29.028.1 -- -- -- -- -- -- - -B 1 350 43 40. 80, 123 _{6.3, 12.6} -1.905 0.3850 44.02 - 0.1 43.3 -0.3 - 05.2 0 - 211.5 0 1 350 43 40, 123, 240 6.3 ₁₅ 1.905 0.2293 72.70 - 0.1 43.3 -0.3 - 15.2 0 - 211.5 0 C 2 350 43 8Q123 6.3, 12.6 - 2.495 0.089 39.61 - 0.4 20.81-0.6 - 46.6 4.8 - 222.8 - 12651.0 2 350 43 83l23 _6.3 15 2.495 0.0645 54.56 - 0.4 20.6 -0.6 - 46.6 4.8 - 222.8 - 12691.0 2 350 43 40, 80. 123 6.3 30 2.495 0.0507 69.54 - 0.4 20.8 -0.6 - 46.6 4.8 - 222.6 - 02691.3 0 3 220 100 _6.3 - _{3.0023 -1.4073 127.3} -17.5 160.0 0 -1384.3 11.3 -1568.0 -027678.0 E 3 220 43 6380 6.3 - 2.676 -1.372 163.4 - 7.7 248.0 -6.9 - 747.5 _{26.0 - 795.2 - 24957.0}

At

in e n = 0.05 t - 29

w . 0.05

n

TABLE 4 - MOTION DECAY CHARACTERISTICS _{OF MOTIONS OBTAINED FROM CALCULATIONS} Astern propulsion in tons 0 ₁₅

30 ₀

Spring constant

in - 6.3 6.3 6.3 _12.6

ton/rn Length of

Calculation bow hawser t1/T1 t2/T2 t3/T3 t1/T1 t2/T2 t3/T3 t1/T1 t2/T2 t3/T3 t1/T1 t2/T2 _t3/3

A - 40 3.2 0.85 0.31 3.2 1.16 0.21 4.7 1.16 0.17 33.2 0.83 0.31 80 - - - 3.5 0.40 123 3.9 0.30 0.56 4.3 0.48 0.28 4.6 0.57 0.13 3.6 0.30 0.56 240 4.0 0.23 0.52 4.6 0.31 0.25 7.0 0.39 0 3.6 0.21 0.52 B 40 3.0 1.05 0.35 - - - 7.20 1.10 0.26 4.64 0.33 1.02 80 3.70 0.55 0.47 - - - 5.10 0.46 0.56 123 3.92 0.39 0.54 - - - 8.82 0.57 0.26 - - -240 _{4.15 0.28 0.51} - - - 9.63 0.39 0.18 5.44 0.52 0.26 C 40 47.64 0.45 0.84 64.60 0.77 0.47 78.30 0.96 0.38 59.60 0.47 0.51 80 53.47 2.79 0.18 73.14 0.32 1.06 87.14 0.53 0.59 65.45 0.19 2.56 123 55.70 20.8 0.07 76.30 1.85 0.19 91.30 0.31 0.92 67.55 0.08 15.28 180 57.00 0 78.38 3.36 0.10 93.90 0.21 1.31 68.80 0 240 57.79 0 79.50 7.00 0 95.30 1.65 0.14 69.47 0 D 40 15.55 1.55 0.21 60 16.67 1.31 0.21 80 _{18.27 1.14 0.21} 100 19.37 1.02 0.21 123 20.21 0.91 0.21 E 40 10.08 0.70 0.46 60 7.29 _0.70 0.47 80 5.94 0.70 0.47 100 5.14 0.70 0.48 123 4.55 0.70 0.45 2 -, 2 A_n

50 loo 50 loo W I ND k42 kn. IRREGULAR WAVES 13 ft. CURRENT 0.9kn. EQUILIBRIUM POSITION MEASURED IN CURRENT AND WIND

TN

\NN

EQUILIBRIUM POSITION ASURED IN CURRENT

\

/

Nø 1N EQUILIBRIUM POSITION CALCULATED IN DISTANCE IN m.

BOW HAWSER LENGTH 50 m.

\

. EQUILIBRIUM POSITION CALCULATED IN CURRENT AND WIND - DISTANCE IN m. 220 KDWT TANKER 150

Fig. i - Position of a ballasted 220 KDWT tanker at maximum bow hawser

forces. t . IN SEC. PEAK VALUE 1N/0OF BREAKING STRENGTH BOW HAWSER 1 166 9 2 346 17 3 489 14 4 616 13 5 740 33 6 889 11 7 1120 37 6 1327 8 9 1461 40 lo 1750 28

I

- 5 U - 4 N. _{IN SC.}t PEAK VALUE 1N100F BREAKING STRENGTH BOW HAWSER 1 125 8 2 292 19 3 476 18 4 607 16 5 794 17 6 980 15 7 1163 12 8 1370 12 9 1610 20 lo 1880 15

I

- S - 4 50 loo 50 loo 150

50 100 WIND 42 kn.

----IRREGULAR WAVES 13 lt. EQUILIBRIUM POSITION CALCULATED IN CURRENT AND WIND

BOW HAWSER LENGTH 65m.

CURRENT O9kn.

DISTANCE IN ni.

Fig. 2 - Position of fully loaded 220 KDWT tanker at maximum bow _hawser forces. N2. IN SEC. PEAK VALUE IN /. OF BREAKING STRENGTH BOW HAWSER 1 17 7 2 790 8 3 1391 £ 4 1864 6

I

- 3 o _{50 100}

w O z 40 30 20

!i'

:

CURRENT SHIP BOUND CO-ORDINATE AXIS

EXTURNAL FORCES ANO MOMENT EXCERTED ON _THE

TANKER BY cuRRENT AND WIND

BUOY-BOW HAWSER

DIRECTION

EARTH BOUND CO-ORDINATE AXIS

--A FAIRLEAD

BOW HAWSER DIRECTION WITH REGARD TO BUOY ANCHORING

Fig. 4 - Current, wind and bow hawser interaction. 607 - ,-,,-,,- ---

-t.

L 12 _2c2 476 t in seconds

Fig. 5 - Sign convention for forces, moments, displacements, and velocities.

CURRENT VELOCITY IN SWAY DIRECTION

a _{POSITIVE ANGLE BOW HAWSER} ANTI CLOCKWISE

1/2 L

1/2 L

CURRENT VELOCITIES IN YAW DIRECTION

Fig. 6 - Current velocity for determination of yaw-damping _coefficient.

'X POSITIVE ANGLE CURRENT ANTI CLOCKWISE

+-- Ax POSITIVE SURGE FORWARD AY POSITIVE SWAY PORT + y + V + Y..

G

POSITIVE YAW BOW TO PORT

X

y,v

LI) z O I-z O 150 loo 50 o o

r'

D SP L AC E M E N T BUOY F0 10 20

ELONGATION OF BOW HAWSER INCLUDING DISPLACEMENT

OF BUOY IN m.

Fig. 7 - Relationship load-displaCement of buoy-bow hawser arrangement.

30

z z o O 3000 U-0 2U-0U-0U-0 'n w o O w

_I

WIND 60W HAWSER FORCE F0 TOE T 13 Tp CURRENT

MEASURING OF BOW HAWSER FORCE F0, ANGLE G AND ANGLE 13

Fig. 8 - Test set-up for confirmation tests.

220 KDWT TANKER IN BALLAST CONDITION.

i KNOT CURRENT PERPENDICULAR TO 45 KNOTS WIND

HAWSER - BUOY SPRING CONSTANT 6.3 toR/m

MEASURED COMPUTED

0

100 200

HAWSER LENGHT L IN rn.

Fig;9 - Results of confirmation tests.

Ti

300

--(n O z O o w " 3000 z z O I-O O (n w o O 2000

I

I-O (n o O w û--J 1000 z 4000 CALCULATION A WIND 45 KNOTS CURRENT 1 KNOT BALLAST CONDITION T 43/ T3 J

\

I

\

_I T1

/

0 0 100 200

BOW HAWSER LENGHT IN m

4000

3000

2000

1000

BOW HAWSER LENGHT IN m.

4000

3000

2000

1000

O 100 200 00 100 200

BOW HAWSER LENGHI IN m.

Fig. 10 - Natural periods of the modes of motion of a 220 KDWT tanker as function of the hawser length.

CHARACTERISTIC El IO IN TONS

LINEAR 6.3 0 LINEAR 12.6 0 NON-LINEAR SEE FIG.7 O

-

LINEAR 6.3 15 LINEAR 6.3 30 CALCULATION E WIND 45 KNOTS CURRENT 5 KNOTS BALLAST CONDITION

T 43/

À

T2 CALCULATION D WIND 45 KNOTS CURRENT 5 KNOTS FULLY LOADED T 1O0/

-4

n O z O o w n 3000 z O F-O u-O V) LU C) O 2000 u-O V) û O a w a -J 4 a 4000 1000 CALCULATION B WIND 45 KNOTS CURRENT 1 KNOT 2 4000 3000 2000 1000 CALCULATION C WIND 45 KNOTS CURRENT 1 KNOT o I o O 100 200 0 100 200

BOW HAWSER LENGHTINrn. BOW HAWSER LENGHT INm

Fig. 11 - Natural periods of the modes of motion of a 350 KOWT tanker in ballast condition (T=43%) as function of the hawser length.

COMPUTED _{CHARACTERISTIC}HAWSER _Ello IN ton/rn ASTERN PROPULSIONIN TONS

LINEAR LINEAR NON-LINEAR LINEAR LINEAR 6.3 1a6 SEE FIG.7 6.3 63 0 O O 15 30