Roll damping by free surface tanks

Report DIo. 13k.

Publication No. 26.

P

nov.bar 1965.

LABORATORIUM VOOR

SCH EEPSBOUWKUNDE

TECHNISCHE HOGESCHOOL DELFT

PING BY

E S FCE

By: Zr J.J. van de

B080h

ß Conten t8 page Liat of aymbole. 2 List of figures.

k

Summary. 6 Introduction. 7 Basic assumptions. 8

The water motion in the tank ad its

effective moment on the ship. 10

3.1 Physical phenomena. 10

3.2 Theoretical results. I i

3.3

The experimental set-up. 12

1e. _{Results of the measurements of K. for a}

simple rectangular tank. ₁₃

General. 13

k,2 Frequency of motion. 1k

L3 The amplitude of oscillation. 15

+.k Water depth. 15

11.5 Tank length. 15

k.6

Tank breadth. '16

k.7 Position of the tank with respect

to the axis of rotation.

16

1i.8

The influence of the topdeck of the tank. 17

5. Modifications of the rectangular tank. 18

r

6.

The rolling motion with and without roll

damping tank. 19

6.1 General. 19

6.2 The ship without tank. 20

6.3

The ship with tank. 22

Design procedure for a passive roll

damping tank. 2k

Example of tank design and the resulting

rolling motion. 28

Slack tanks. ₃₃

Conclusions and remarks,

₃₅

Ó

Ó

List of symbols.

B Ship's breadth.

G Ship's centre of gravity.

Virtual mase moment of inertia about the rolling axis.

K General moment producing roll.

Roll exciting moment, due to the waves

Kwa Amplitude of K.

Kt Roll exciting moment, due to the tank.

(its phase is such. that it counteracts the rolling motion)

K Amplitude of K

ta t

bamptng coefficient against rolling.

Restoring coefficient.

Spectral den8ty of rolling motion.

ft

_"

_{wave amplitude.}

S " " " wave slope.

27r

Iatural roll period.

24-Theoretical natural period of water transfer in the tank.

b Tank breadth (measured across the ship).

c Velocity of the hydraulic jump.

g Acceleration of gravity.

h Water depth in the tank at rest.

Water depth at the lower side of the hydraulic jump.

h2 _{Water depth at the higher side of the hydraulic jump.}

21c

k -

Wave number.

Transverse radius of gyration.

1 _{Tank length (measured forward and ft.).'}

s _{Distance from tank bottom to axis of rotation Ç positive if}

tank above axis.

xyz _{Coordinate axes; the x-axis is the rolling axis;}

the y-axis is positive to starboard and the z-axis

positive upward.

OÇ

Maximum wave slope at the surface.

Weight of displacement.

V

Volume of displacement.

Phase angle between wave momentK and rolling motion;

positive if K leads '.

Phase angle between tank moment and rolling rnotion;

positive if K leads.

Wave amplitude.

Wave length.

(S

Nondimensional amplitude of tank moment

V

Nondimensional roll damping coefficient.

Mass density of water.

Mass density of tank fluid.

Roll angle. Roll amplitude.

Significant roll amplitude of an irregular rolling motion.

Circular frequency.

Natural roll frequency.

AØ

Theoretical Natural frequency of water transfer in the tank.

Actual natural frequency of water transfer in the tank, defined

List of fj.gures.

Fig. I Illustration of wave phenomena in the tank.

Fig. 2 Principle of experimental set-up.

Fig. 3 Coordinate system and tank parameters.

Fig. 4 Position of bore during one period of rolling at tank

re8Oflafløe

-90 degrees).

Fig,. _a Example of measured-in-phase and quadrature components of tank

moment.

Fig. 5b Example of amplitude of tank moment.

Fig. 5e Example of phase angle of tank moment.

SFie. 6a The actual resonance frequency £Oa«a8 a functton of

Fig. 6b latio &a4)t as a function of h/b.

Fig. 7a Example of for different values of i.

Fig. 7b Example of for different values of

Fig. 7e The nonlinear increase of Kta with at

Fige Ba The dependance of Kta fl h/b.

Fig. 8b " '

4

_{on h/b.}

Fig. 8c The variation of with h/b versus the nondimensionalized

fre-quency

Fig. Bd The nonlinear increase of Kta with h/b at A)

Fig. 9a Nondimeeional amplitude of tank moment versus nondimensional

frequency- for two tank widths.

Fig. 9b Phase angles of tank moment versus nondimensional frequency for

two tank widths.

rig. lOa The dependance of Kta on s/b at 'O

Fig. lOb t, I,

ç

on s/b at 4.)=

Fig. lia Example of Kt for the rectangular tank and two modifications

with the same amount of water in the tank.

Fig. lib Example of for the rectangular tank and two modifications

with the saine amount of water in the tank.

Fig. lic Example of Kta sink for the rectangular tank and two

modifi-cations with the same amount of water in the tank.

Fig. 12 Illustration of the selection of the waterdeptb.

Fig. 13 Example of roll response in regular beam waves with and without

tank.

Fig. lka Results of the calculations for the modal of the sixty series.

Fig. lb

Results of the calcu1ation for the model of the sixty series,

presented as magnification factor.

Fig. 15 Rolling of the model of the Sixty Serias for two different water

deptba.

Fig. 16 Neumann wave slope spectra.

Fig. 17 Comparison of calculated roll spectra for the model of the

Sixty Series.

Fig. 18 Resulte of the calculations for the trawler.

Fig. 19 Comparison of calculated roll spectra foi the trawler.

Fig. 20.1-2O. Nondimensional quadrature component of tank momnt.

Fig. 2i.1.2l.i Nondimenajonsi. in -phase component of tank moment.

Fig. 22.1-22.3 Nondimensional amplitude and phase of tank moment for

s/b = .-0.kO.

Fig. 23.1-23.3 Nondimensional amplitude and phase of tank moment for

s/b = -0.20.

Fig. 2.1-2.3

Nondimensional amplitude and phase of tank moment for

a/b = 0.

Fig. 25.1-25.3 Nondimensional amplitude and phase of tank moment for

Ò

Summary.

The problem of a free surface tank as a rol]. damping

device is

treated.

The wave phenomena in the tank

are

described and it is pointed out that

the roll damping is essentially based on the existence of a hydraulic jump

or bore.

Quantitative information on the counteracting moment, caused by the water transfer in the tank, is provided.

Under the assumption of a pure rolling motion in beam waves calculations are performed to show the reduction in rolling and athwartship's accelerations.

A general procedure to design a free surface tank for this

case is

presented.

The process

is illustrated by an example.

1. Introduction,

Since the disappearance of the sails on oceangoing ahps, with their

stabili-zing wind effect, flava], architects have, been concerned in reducing the

rolling of ships among waves. With bilge keels they performed a first,

success-ful]. attack, but in several casen these did not prove to be sufficient.

A early as

1880

the Britib Admiralty supported the idea of Messrs. P. Vlatts

and R.E. Froude to install 91water chambers" in a ship to counteract the

rol-ling. In two papera [1], [23,read before the Institution of Naval Architects

in London in

1883

and

1885

Mr. Watts published their findings. The idea to

have a eubstantal amount of free water

anti-rolling tank in this most simple design was almost forgotten. It is a remarkable thing that it has lasted so long before the subject was taken up again, fór Lt does not seem doubtful that many ehtp'e officers have used its principle without being aware of it. Ia not it customary for them to sail with slack tanks in stiff ships rolling badly? In most Cases the influence

of slaCk tanks can not be explained by

a mere reduction of

the nietacentric

height. Though each separate double bottom tank in itself is a fairly

inef-fective anti-rolling

device,

the total length of the tanks is so large that

it may have a notable effect on the ship's rolling,

when

the circunistances

are favourable. In section

9

this will be illustrated. It is very remarkable

that Watts and R.E. Froude had such a clear physical insight in the operation

of the tank at that time. During the

present

investigation no really new

things were found out. The systematic measurements only supply the necessary information to understand why the tank performs' as it does and to design tanks for practical applications.

2, Bsc aGeumptiofle.

The general state of motion of a ship in a

confused sea is

a complex problem

to analyse and too little is known about it to use such a theory as a star-.

ting point for the observations. Therefore it is aesumed,that the ship is performing a pure rolling motion about a longitudinal axis through the ship's centre of gravity in regular beam waves, Accepting the system to be linear the roll response to a long-crested irregular bean sea then can be calcula-ted. The influences to be expected from these simplifications will be dis-cussed later.

For the moment it has to be accepted as a hypothetical

condition.

Under the8e circumstances the rolling motion of a vessel is full1 equivalent to the forced oscillation of a damped pendulum suspended from a fixed axis

'S

of rotation.

As a aecondorder linear system it is represented by the following equation

of motion;

(I)

in which:

8

Until now there is very little information available on the ship's parameters.

the virtual flasS moment of inertia about the rolling axis, including

hydrodynamic effects;

damping coefficient against rolling;

spring factor or restoring coefficient against a static heel;

b

K _{the active moment, in most general sense, which produces rolling.}

For the ship without roll damping tank K is equal to the wave moment (exci-ting moment) K. If a tank is installed there is an additional moment which

produces rolling, though in a negative sense, that is the moment exerted on

the ship by the water in the tank (tank moment) Kt. So in eneral1 without

an further specification it may be said that

¿<=

/Ç+j<

₍₂₎

In principle all quantities in (i) are known oor can be determined experimen..

tally and the rolling motion can

be solved

from it, if not in an easy

Gener&lly speaking they will be functions of the chip's form, forward speed and

frequency

of motion.

It

_{is often stated that rolling is essentially}

non-linear; then the

aaplitude

_{of the motion comes into play as well. The same}

holds for the wave moment. It may be approximated

uaing the

_{well known}

Froude..Krjloff hypothesia, which, however', ignores any body-wave interaction.

But ofar

the ultimate goal is not an

_{exact prediction of the ship's rolling.} The aim of the investigation is to Judge the effectiveness of a paesive tank

system by comparison of the chip's

rolling With and

without

tank. Therefore

it

is not essential that

_{the ship's parameters and the wave moment are known}

accurately. It suffices to have one or' several combinations of reasonable

estimates

to

_{make a clear comparison, for they are the same in both}

condI-tions. Consequently first of all attention is drawn to the term

'

the

3. The water motion in the tank and its effective moment on the ship.

3.1 Physical henomena.

It is clear that many parameters will influence the fluid motion to a greater or lees extent. There are the geometry and the dimensions of the tank, its position with respect to the axis of rotation, the amount of liquid carried in it and the motion to which the tank is subjected by the shipmotions, both as to amplitude and frequency.

Regarding the type of the installation there may be distinguished between two basically different designs. Firstly there is the free surface tank of

Watts, having one large free surface and secondly the IJ...tube tank of Frahm

which has two small free eurfacee.* The last one has been extensively treated in the literature [3],[fJ ,[s] ,[6j. Not entering into the details of the fluid flow the mass of liquid as a whole could be regarded as behaving like a second pendulum, attached to the pendulum representing the ship, over

most of the frequency range in question. In the free surface tank wave pheno-mena take place, among which is essential the generation of a hydraulic

jump or bore. This fact was already noticed by Watts

f2J in 1885.

So the

only thing the two types have in common is that they use the transfer of

water from one side of the ship to the other with a certain phase g with

respect to the rolling motion of the vessel, as a means to provide a coun-teracting moment.

Considering here only tanks of the free surface type there remain two impor-tant variables for a specified tank installation, for which dimensions and

position are fixed. These ar- the amount of fluid, characterized by the waterdepth h, and the imposed motion, especially with regard to its

frequen-cy.

When the waterdepth is large the wave motion in the tank is a simple oscil-lation of the fluid surface for all frequencies. The fundamental mode of

this standing wave has a wave length ). equal to two times the tank's width b,

measured acroSs the ship.

lo

It has to be noted that it is not meant to say that thereby the reduction in metacentric height of a U-tube tank is considerably smaller.

For less deep water, especially for h/b < 0,10, the picture is entirely

different. At low frequencies the standing wave le present here as well.

Thereupon a train ot progressing waves of a very short wave length appears.

After these small disturbances rather suddenly the bore arises, while the phase lag between the water transfer and the imposed motion rapidly in-creases. Over a large frequency range the phenoineu then doesnot change significantly, although the water motion becomes more violent. Next the bore passes Into a solitary wave, a single and steep wave which runs from

one side of the tank to the other. With a mal1 further increae in

fre-quency the liquid approximates the frozen state if the tank is situated below the axis of rotation, Above the axis this does not occur; the water motion Is rather confused then. Anyway hardly any water transfer is taking place any more. The appearance of the hydraulic jump is illustrated by the photographs in fig, 1.

3.2

Theoretical results.

Al known theoretical studies on anti-roUingtanke are limited to the

U-tube tank

and

based on the

principle of an equivalent double pendulum.

As has been pointed out above this is not, applicable to the case of the free

surface tank. Also wave phenomena in rectangular tanks have been decuased

both in the shipbuilding and aeronautical field; see among othere7J and[8].

But they were always concerned with a linearized theory on the

standiig

wave

pattern, which is not the essential feature of a tree.eurtace tank. The hydraulic jump or bore is an essentially nonlinear phenomenon and only the mathematical representation of this flow pattern can yield reliable

theore-tical results. Until now no known theory exists whjçh accounts by ealcula-tion for the measurements of the moment about the axis of rotaealcula-tion,. exerted by the water transfer in the tank. Nevertheless the theory confirms all

obeer-ved tendencies Completely qualitativelyf

[9j.

To

provide reliable

data for the application of roll damping tanks to ships

and to form a

basis

for a further theoretical analysis a test program was

carried out in which all the parameters involved were changed systematically.

3,3

The experimental eet-up.

The problem was investigated by giving a tank model a forced sinusoidal O8CillatiOfl in rolling. The moment of the free water about the fixed axis of rotation was measured by an electronic strain gauge dynamometer. To the output of the dynamometer a harmonic analysis was applied, producing the two components of the fundamental part of it. A sketch of the installation is

shown in fig. 2, while the measuring technique is adequately described in [io].

The amplitude of oocillation was varied in three steps: 0.0333;

_0.0667

and

0.10 radians (1.9; 3.8 and

5.7

degrees respectively). The frequency of

oscil-lation was changed from well below to well abovethe range in which the

hy-draulic jump was present. The position of the tank bottom varied from kO per-cent of the tank's width below the axis of rotation to 20 perper-cent above it. The waterdepth ranged from 2 percent to 10 percent of the tank's breadth.

k !ult of the measurements of Kt for asiniple rectangular' tank.

4.1 Genera]..

When a sinusoidal motion of amplitude q and frequency A) is imposed on the

tank the measurements show that the resulting moment about the axi8 of rota. tion varies sinueoidal].y, too, with the same frequency and a phase lag with respect to the motion ranging from zero to 180 degrees. Only at low and at high frequencies, ¡o mainly outside the region in which the bore is present, higher harmonics of any importance can be noticed. This waS not only obser-ved from visual judging of the recordings, but could easily be found bymea.. suring the second and third harmonic contributions. Consequently the tank

moment can be represented with sufficient accuracy by

(3)

For various values of the tank and motion parameters the magnitudes of

Kt cos and Kta einst were measured. Kta coB is the amplitude of the

component of which is in phase with the tank motion#, while Kta 61ne

is the amplitude of the out-of-phase or quadrature component which leads / by 90 degrees; as a result it is in phase with the angular velocity.

The experiments were carried out with a tank of the following dimensions:

b = 1,00 ni. i = 0,10 ni. 0,50 ni.

Fig. ₃ shows the coordinate axes and the various parameters.

The influence of each parameter on the tank moment will now be discussed separately. Apart from the frequency of motion this amounts to getting an idea of the influence of each parameter on the strength of the bore (h2-h1)

(see fig. _{), as the moment Kt originates mainly from a shift of the fluid} weight in the tank by the existence of the bore. This is illustrated in fig. 11,

f

showing schematically the position of the bore at different points _{of time}

during one period of rolling at the natural frequency of the water transfer in the tank.

k. 2 Frequency of motion.

As described in section 3.1 the phye±cal phenomena change with frequency.

The measured moments change correspondingly. An example ±8 given in fig. 5a.

From these measurementB the amplitude of the tank moment K.

_and

the angle 8t'

by which the water transfer is lagging behind the rolling motion, can be

calculated separately. They are reproduced in fig. 5band c. The continuous increase in the phase lag mainly takes place during the time that the bore is

present; Kt is nearly constant then.

The natural period of the water motion cafl be approximated by taking acount

of the velocity of the hydraulic jump as

,

in which h is the

undisturbed waterdepth.

Twice the breadth of the tank has to be travelled by the bore in one period of water transfer. Therefore the natural period of the water transfer is given by

.,

A)

'-I-1

ei)

and the natural frequency by

Te

(5)

Of course this is only an approximation for the actual velocity of a bore

is given as

1k

in which h2 denotes the larger and h1 the smaller waterdepth; c is the velo-city of the jump relative to a steady basic flow. In this case it concerns a fixed amount of liquid and the condition of continuity does not permit a

steady flow, nor a constant strength (h2u.hl). On the contrary

_and

h2 vary

during one period and are dependent on the magnitudes of the other

para-meters as well. Therefore it is not surprising that the actual frequency 4) at which

t equals -90 degrees does not correspond exactly with the calcuis

lated The agreement becomes better, however, when decreases or h/b

increases and thus the departure of h1 and 112 from h becomes lese important.

This is ±liuntrated

in

the figures 6a and b.

4

k.3

The amplitude of oscillation.

When the amplitude of oscillation increases the strength of the bore and

thereby Kt increasea,too. As has been explained above this influences the

curve of phase angles as well, An example of the measured Kt and ¿-t for

different values of is given in figures 7a and b. The dependance of Kta

on is not linear but a a first approximation according to the square

root. This tendency also follows from theoretical observations, where it has been shown that the strength of the hydraulic jump is proportional to

fig. 7c.

k.k Water depth.

The water depth is a

particularly

important parameter, because from (k)

it is olear that for a certain tank (b fixed) the only possibility to change the natural period of the water transfer is a change in waterdepth. And it

is equally clear that at or

near

this natural period the water transfer je

largest and circumstances are most favourable for roll

_damping.

The effect of increasing waterdepth is twofold. In the first place the curve

of phase angles versus frequency of motion is shifted to the higher frequen-cy range, in accordance to what has been said above. (see k.2). When they

are plotted versus the non-dimensional tunir factor W/ú). then there is

hardly any difference notable, except for the higher relative frequencies; that is for the region in which the bore transforma into the solitary wave.

In the second place the moment amplitude increases because of the larger

amount of water in the tank. But here again the increase is not itnear, but

approximately. quadratic. Thia has been confirmed by theoretical reasoning, too, in which i.t is shown that the strength of the bore varies proportional

to

The dependance on h/b is illustrated in the figures

_8a

through d.

k.5

Tank lenh.

perpen-dicular to the x-axis. Therefore Kta is directly proportional to the tank

length, while will not be influenced.

'..6 Tank breadth.

The fact that physically the phenomenon in the tank is a wave problem implies that for scaling up, Froude's law has to be followed. Therefore it can be expected that the moment exerted by thé tank fluid is proportional to the fourth power of the modelecale. Or, considering in a two-dimensional problem the moment per unit tank length this will be proportional to the third power of the modeiscale, which clearly will be governed by the tank's breadth b,

To investilate this relation side walls were placed in the tank, so that

b = /1+ b. To create a comparable flow pattern the tanks should be filled aocord&ng to the same ratio h/b. When the measurementeKare plotted in a

nondimensional way, that is ¿ versus (.)/4 and ,M.= t; Versus òJ/A)t i

as in fig.

_9a

and b, it appears that the results fully confirm the

expecta-tions.

From this it will be understood that the tank's breadth with the factor b3

dominates the counteracting moment and that for actual application the tank's breadth should be made as large as possible.

k.7 Position of the tank with, respect to the axis o rotation.

The difference in the two conditions with the tank mounted below or above

the axis of rotation is the diffe:rence in the direction of the centrifugal

force, adding to, respectively subtracting from the force of gravity acting on the water particles. Also the direction of the athwartship's acceleration is reversed.

The measurements reveal that the moment amplitudes become larger and the phase angles slightly smaller the higher the tank is situated. Therefore

the very important quadrature component of the tank moment is subjected to

two opposite influences However, the decrease in the phase lag is of little

1*poxaAce between about 60 and 90 degrees because of the flat top of the sine curve in this area. But the increase in moment amplitude is considerable

the

tank is

placed with respect to the axis of

rotation. An example is shown

in fig. lOa, b and c,

k.8

The irifluenco of the topdeck of

the

tank,

By choosing a sufficiently deep tank the influence of the topdeck on the measurements of the tank moment was eliminated, except perhaps for a single point of extremely violent water motion. It has been found that the influence of the topdeck can be neglected when the depth of the tank ja three times the

undisturbed water depth. When lees,

f8peOial].y at

the tank's sides, the

deve-lopment of the bore may be hampered and the effectiveness of the tank

redu-ced. By

₍₅₎

_{and (21) a preliminary estimation of the tank depth can be given} as 2 GM.

Modifjcatjone of the rectqnu]a tank.,

18

Next to the basic rectangular tank several modifIcations were tried, although this subject was far from exploited fully. This is partly due to the enormous number of possible modifications and the great extension of the number of. involved parameters, whjle any basic information on the characteristics of the tank was still lacking, and partly to the fact that the same tendencies were observed for every modification tried.

The influences whIch the alterations have on the hydraulic jump can be summa-rized as a phase shift to the lower frequencies, whIle the phase curve may

be a bit more flattened in addition, and a considerable decrease in the strength of the bore. As the fluid transfer and the counteracting moment is for the

greatest part determined by this bore-strength any decrease in the latter is principally undesirable, although its effect may be partly compensated for by the shift in the phase lag. As a result the important quadrature component

sin Ç

is lowered in the maximum value it obtains and is shifted to lower

frequencies. So with a modification of the tank Kta sin dt approximately takes

the place of a correoondin6K5 sin for a lower h/b in the Case of the simple rectangular tank; compare section k.. An example of these results is shown in fig. lia, b and c. For some further results te referred to [111.

As indicated above the influence of obstructions to create viscous losses is to hamper the development of the bore. The fact that this will only lead to

a diminished tank

acUon

wa6 already notedby R.E. Fraude in the discussion

on [i] and it bas been confirmed experimentally. Only for the very long

rol-ling periods there may be expected an advantage

in

the use of some

modifi-cation. Then the extremely low water levels Can be avoided, which will suffer

from ttionäl resistance of the tank's bottom and sides and from the fact

that already at a small heel all the water w%.21 run into one corner of the

tank and the effective breadth is greatly reduced. But. i such a condition

6.

The rollins motion iith and without roll damping tank.

6.1 General.

In section 2 it has been found that rolling

i8

determined by the equation of

motion:

J#N=Iç-,.k

(7)

Generally the wave moment is stated as

K = K sin cot

so that the phase angle between moment and motion la part of the solution:

sin(Cot-.! ).

a

Then the tank moment (io) has to be presented

by-Kt = by-Kt sin (wt - e_k + as

t has been determined with respect to $

For the pre8ent purpose it is more convenient, however,

to

use (8), (9) and (io).

In regular beam waves K is of the form

-'-'

(#d)

(8)

in which () is the wave frequenct and is the phase difference between the

exciting moment K and the resulting rolling motion

(q)

Through the experiments, in which a forced harmonic oscillation was given to a tank model, Kt has been determined as

(')

Now considering the tank moment a little closer one can write

KL1R,.$

(I/I

in which

R

and

Therefore the roll equation (7) is reduced

to

3

(Ai-4ìV,)

#(Ç_ilR,395 =kj),ái)

provided that the solution is anyway harmonic, for that

i8

the condition

Under which is determined and can be substituted by the two additional.

tezmB ¿tì'*. and

¿iQ

_.q

in the left hand sido. Now assume that this

in true with enough accuracy, then the problem cafl be solved easily. Both

thin condition and the method

of solution will

be discussed further in

6.3.

6.2 The shii without tank.

ror the ship without tank O and the rolling motion is given by

(7) if

N R and are known. According to section 2 they are not

cone-dered in detail here,

The following may

be said about each of these

quanti-ties.

The mese moment of inertia about the longitudinal axis through the centre

of gravity is generally expreseed as

=

¿2.

i

with the transverse radius of gyration, including hydrodynamic effects,

Existing data mootly give k as a faction of the ship's

beam, say

¿1.J3 8 t O.'1 .

Kato [12] producea a formula, derived

from full

scale meaeurement of the

roll period for many different ships:

20

Cb = block coefficient

are*A

Cu upperdeck coefficient _{= length z brèadth}

d = moulded draught

H effective depth = D

+ L

pp

D = moulded depth

A = lateral

area

above the moulded depth

= length between perpendiculars

B ship's beam

f = a coefficient

f = 0.125 for cargo- and paseengerliner

= 0.133 for tankers

= 0.200 for bonito fishers

= 0,177 for whalers.

N.

The damping is usually estimated by the aid of the nondimensional damping

coefficient defined by

¿"

(/7)

For most ships it

will have a value between 0.07 and 0.20.

R.

The restoring coefficient is the only one which is known accurately

for

a

given ship. The lineartzed moment of statical stability for one radian of heel is g ven by:

=

¿1.

K

Using the Proude-Krilof f hypothesis the exciting moment _{can easily be found}

by integration of the

pressure

in the undisturbed incident wave over the

ship's hull. For waves long with respect to the ship's beam

_and

draught this

22

vr

k = wave number =

= wave amplitude

Ze

a certain mean depth below the water surface; often is approximately

chosen z = 1/2 d.

With the fact that k i the maximum surface wave slope

Ç

and substItuting

(18) this can be written as

1Ç.

_Rççc<w.e

The phase of the exciting moment with respect to the rolling motion now

fol-lows

from the solution of the equation of motion (see footnote with formula (9)).

For the ship without tank the equation of motion la considered to be linear.

When the coefficiente ai-ê detérmined in accordance with (is), (i?) and (18),

by lack of any better information, they are constants and linearity of the mathematical representation la always assured.

6.3

The ship with tank.

By equation (1k) the determination of the rolling motion when the vessel is

equipped wIth a roil damping tank is i-educed to the same problem as above.

The same equation has to be solved, but now with different coefficients. There

is one important difference, however, and that is that ¿tand L1Q depend on

the amplitude of motion ç& and thereby the equation has become nonlinear. In other words: the motion is nót exactly harmonic any more, which is in direct

contradiction with the starting-point for the measurements of This can not

be overcome by applying the method of superposition because thIs principle does not hold as Kt is essentially nonlinear. Fundamentally this reasoning is

correct. However, it may already be felt that no matter how

nonlinear

the tank

in itself is as one element of the shIp, the linearity of the whole aystem is

hardly affected thereby. Even a model equipped with a relatively large tank

did flot show a notable departure from a pure harmonic rolling motion in a

wide frequency range around resonance, neither in waveB nor when oscillated in calm water. Therefore it is a fair approx imation to consider the rolling

motion to be

stIll

harmonic and the condition for equation (11+) l.a fulfilled.

This being established its solving offers no great difficulties, _{although it}

tn which only zS has to be determined. The three values _{= 0.0333;}

= 0.0667 and çS = 0.10, for which the measurements have been carried out,

can now be substituted into equation (1k) with the corresponding and

and where the left band side equals the right hand side the correct value of

a is found. Thie process hae to

be repeated

for every W under consideration.

It can easily be done on a computer and even by hand no extremely lengthy calculations are required. The results of the comparison will be discussed in

7 Design procedure for a passive roll damping tank.

From section 14 lt follows that a tank can beat be placed high with respect

to the axis of rotation and over the largest breadth available. This leads to an upper tween deck compartment in the midshIp part of the ship. Other reasons give preference to this place as well. The general problem has been simplified to that of pure rolling. Until now it is not known how the other motions influence the tank eifect and reversed: what the tank does to the ship motions, especially yawiùg.; All these Influences are least pronounced

in the midship section.

The loading condition for which the tank is meant determines the frequency range in which the bore has to be present to have a favourable rol], damping

effect. Roll resonance appears at the natural frequency given by

w=yi;=

₀

.4

(20

2k

As rolling has a small damping the resonance peak isitarrow and roll angles

outside the frequency range of about 0,7O4) to about 1,25W0are completely

unimportant. Therefore only thie range need be considered for the best action

of the tank. Bt due account has to be given to the fact that R la

Influen-ced by the presence of the tank and that thereby the region in which the lar-gest rolling angles occur is shifted. This will be discussed further below.

Before proceeding towardS the most important poInt, that is the determination of the required water depth, it should be ascertained that the reduction of the statical stability by the free surface of the tank is permissible. This may limit the possible tank length or even prevent the application of any tank.

Thia point alone already indicates that especially relatively stiff ships are suited for roll damping tanks.

The above being done there remains to provide as much roll damping as possible, that le to choose h/b according to a large quadrature component Kta

for the frequency range in question. Information on this quantity is presented

in the figs. 20.1 through 20.14 at the end of the paper, in a form which allows

for easy application to practical problems. When the location of the tank is fIxed ita a/b is known and the right diagram can be selected from the four

standard values s/b -

0,140; -

0,20; 0 and + 0,20. If desIred interpolation

is permitted but this will generally not be necessary. It suffices to choose

the neareet standard value. Then the diagram is entered for the desired fre.

quency range 0.70 W to 1.25 ¡7. The h/b giving the largest

=

'

at4

is

real or estimated. Of course it is important

that

there-('4CQU.

J/R(,

vJ

fore it should be checked that it is nowhere lesa than, say, half its value

at If this is not the case a little different h/b must be chosen to

satisfy the demands. ThIs procedure is illustrated in fig. 12. Now knowing a

value for b/b the

/.t

coo ¿ is read from one of the diagrams 21,1-21 .k, for

the same s/b and the same frequency range. Through (la) is calculated

and next a corrected value for the natural frequency

After that the described process te repeated for the corrected frequency

ran-ge 0,70

(4))J/

to 125

corr. Then a new h/b is found,

which must be checked again with /

a cos ¿ '

giving a new (

This itterative method looks rather cumbersome, but in practIce it is very easy. Mostly the second approximation is enough and a very favourable aspect of the installation is that selecting the proper waterdepth is not very

criti-cal at all.

The diagrams used above for the determination of h/b are given for one

ampli-tude of oscillation only, that je = 0.10. The nonlinear character of the

tank moment makea/L5 sin and

a

COG ¿t

in fact dependent on 6a However,

a roll angle of 0.10 radian (5.7 degrees) is considered as a maximum for nor-ma], circumstances and a mean for worse conditions. Therefore it may be used

for design purposes. For smaller roll angiee/(5 sint and/#' coa Lt are

rela-tively larger, for larger roll angles relarela-tively smaller.

When the tank has to cover more than one condition of loading without adjus-ting the waterdepth the frequency range is extended from 0.70 times the lowest

to 1.25 times the highest 4.]. The same technique then can be applied. In many cases indeed it is possible to have one amount of liquid for different

conditions because of the non-critical character of the performance of the tank. But, of course, always some Compromise will have to be accepted.

After the determination of all the tank parameters the roll equation can be solved in the way, described in section 6.3. The ship coefficiente and

exci-ting moment can be taken In accordance with section

6.2e

whilefra and

can be read from the diagrams 22.1-22.3 through 25.1-25.3. They give ,LL and

for four a/b and three _{values; h/b rangea from 0.02 to 0.10, For the}

ac-tual calculations of the ship's rolling with tank interpolation between the standard values of s/b and h/b is recommended.

many cases a reduction in atbwartahip'e accelerations will be more important

than the angle of rol]. itself. Then a shift of the 'roll response curve to

lo-wer frequencies may attribute to the reduction as welle for the acceleiations are proportional to 4$ squared. On the other hand it will generally be undesi-rable to have rolling periode of about 30 seconds or more. It gives a feeling

of unsafety to have very long rolling periods.

An example of a typical. response curve for rolling in regular beam waves with

and without damping tank is given in fig. 13.

It is not sufficient to judge the rol]. reduction by the merita of the,tank slstem in regular waves. In most cases the 8hip will meet an irregular sea,

for which a long-icreated Neumann beam sea is taken as an example. The

signi-ficant roll angle will then be represented by

(oJ

in which k is the maximum surface wave slope of every regular component

and is the spectral density of the surface wave slope. The latter is

related to the spectral density of the wave.heighte of the wave spectrum by

.

=

'1Ç(cA3).

(SN)

ipX

1t

So for irregular seas the area under a certain part of the curves of

is an indication for the amount of reduction and this may differ greatly from the ratio of the peake in the response curves. Therefore roll damping tanks should not be made too email. What is considered as a sufficient reduction of the regular rolling may not lead to satisfactory resulta as to the irregular

rolling.

The information on the tank moment, presented at the end of this paper, allows

for application to ships having a ratio of over B from about 0.03 to about

0.16. Thia can easily be derived by putting (5) approximately equal to (21),

One remark baa to be added here. It ja showfl

that the

tank action becomes

rela-tively lese when rolling angles increase. On the other hand the action of bilge keela becomes much better for larger rolling angles because of the higher trans-verse velocities. Therefore jt is not advised to abandon bilge keels when a tank ia installed. They behave more like a supplement than a substitute of one

another Moreover the ship bas to rely on

ita bilge keela only

when in sorne

emergency situation the tank bas to be emptied (for instance too low. metacentric

Ship

length between perpendiculars moulded breadth

depth to upperdeck

moulded draught

volume of displacement

8. Exam .le of tank deei:n and the xesultin: rollin: motion.

8.1 A an example the waterdepth will be determined for a roll damping tank

in a general cargo ship. The ship is chosen out of the standard Todd sixty se.. ries with a blockcoeffieient of 0.70. The tank is situated amidships in a tween deck compartment. The principal dimensions of the ship and the tank and the loading condition are tabulated below.

Tank

width b 21.60 m

length i = 4.00 m

vertical position a/b = + 0.10

Cond1tion of loading

displacement (fresh water)

L

= 20228

4,t

metacentric height = 1.305 m nondimensional metacentric = 0.06 height b vertical position of CG

_{7.475 m}

nondimensiona]. transverse radius of gyration

f

= 0.38

natural roll period T = 14.52 sec

natural roll frequency = 0.433 sec

nondimensional natural , =

0.642

statical reduction in nieta-centric

height

due to the

tank

¿MG

_{= 0.166 m}

Shin parameters

dimensionless damping coefficient 0.10

spring factor A .G _26.398 c 106 kgm loment of inertia =

_.k

= 1k1.vt3 x

lo

kgniaec L 152.40 m pp

21.77m

13 .54

m

8.71m

= 20228

m3 28

'x'om fig. 20.3, valid for

a/b = 0, is found that the optimal h/b ratio at

j' = 0,61+2

is 0.05; fig. 20.1+ for u/b +0,20 indicates the same. Both

f igurea 8how that the condition that

is not less

than about half

ita

value at

io approximately satisfied in the range between 0,70 IAJ

and

1,25( . Therefore b/b = 0,05 ia a good

initial value fox' u/b

+0.10 as

well.

Fig. 21.3 and 21.k then indicate a 00

of 0.0069 and

0.0089

respective-ly; that is 0.0079 for a/b = +0.10.

/4

Ofc

¿iR,. = IíofovS

fo'

= &i

¿r..

/6.,

L Çi1/1,f

_-=

Y

=.

o,iC",'

/'/i (/6

2

cj0C

J-ee.'

2°

O)6b20'/2.f

J2Çx o,6oz

2.1v, &) X '/o 3

3/0 8

_{x (o}

CL

Starting the process again for () = 0,602 one finds a h/b 0,05

(very nearly 0,0k) in fig. 20.3 and h/b 0,0k in fig. 20.4. For a/b = +0,10

h/b = O,O4 is selected because this curve does not drop so much at 0,70 (4i

that it becomes less than 0,50 of ita magnitude at

At h/b 0,04 the /4. _{COB ¿} resulta in 0,0057 and 0,0071 (fig. 21.3 and 21.4),

so 0,0064 for a/b = +0,10.

=

Jio,/o8c'ox ¿,öt,óc,'

2,çPo

,..foC

4iQ

_.U,.3

_2nPu

,

i7?

4e

)/

-

i ,c-

.=

O, '-i f f

= O,1i'fc

I'1IS

_4fto.

30

This is only

1.2% different from (A)) It is clear that a new estimate

Corr,

starting from (t)

would yield a waterdepth in between h/b = 0.04 and

corr,

0.05, but closely to 0.04. However, it

is

not worth while doing

so

because of

the small differences. Another reason to let it be 0.04 is that a larger

magni-tude of the quadrature cornj,onent of the moment is retained at the lower side

of the frequency range. Generally the value will decrease somewhat during

the voyage and then the lower frequency range becomes of more importance.

Once the waterdepth being fixed at h/b = 0.014 the moment amplitude/4.a and the

phase angle £ are determined from the figures 2225 for a numberof

frequen-cies and three different rolling angles. Graphical interpolation is used to

obtain the correct values for B/b = +0.10. ThentN and A R0 are known and the

rolling motion Cafl be calculated from (ik) when a wave oxctting moment is

accep-ted. Taking the wave slope &Ç= 0.015; 0.02 and 0.03 (0.86; i.15 and 1.72

de-grees respectively) and kZe

_O.kk39øK

is determined by (20). The

2g we

resulta of the calculations are presented in fig. Ike and b. From thee figures

the nonlinear character of the tank is clearly noticed;

the larger

the rolling

curve has been calculated for a waterdepth h 0.05b; it is presented in fig. 15.

The response of the ship to regular beam waves is given in fig. 1k and it can be seen that rolling angles are reduced with about 50 to 65 percent. This is reached with a comparatively very small tank, for the amount of liquid is only

7k,65

ton or 0.37 percent of the displacement. But it has to b seen next what

the ship's behaviour will be in irregular beam waves. Therefore 5 different

Neumann wave spectra, converted into wavealope apectra were calculated; they

correspond to windvelocitee of 1k, 19, 2k, 26 and 30 knots respectively. They are shown in fig. 16. It should be noted here that these wave spectra have been

theoretically derived and that they probably contain too much wave energy to be fully realistic for actual sea conditions. Therefore the resulting rolling mo-tions should only be regarded by comparaon and not in ad absolute way.

The rolling motion of the ship without and with tank have been calculated for

irregular beam seas with a dominant wind velocity of 21i.,26 and 30 knots. In

the two lower wave spectra the

ship

does not roll at all. By definition is

5

[

(w)]

[

-

t4]

.2

is determined from fig. 1k and from fig. 16.

has been plotted in Cg. 17, while

the

aignifcant rolling angles 1/3 are

tabulated below. Prom this table it can be seen that the reductions are much

less than would correspond to the regular waves. Therefore a tank like

ropo-sed here is too small for this size of ship. If the tank had been extended to a length of 6 m it would contain about 1/2 percent of the ship's displacement and the performance would have been much improved.

significant rolling angle with tank

3,32°

k, Li5°

7,300

wind velocity without tank

2k knots 3,914°

26 'P 6,2k°

s

8.2 As another example the roll response has been calculated for a trawler,

a much emaller and stiffer ship which meets relatively much worse wave condi-tions than the large and more tender general cargo ship. This is immediately clear when the frequency range of the roll response of both vessels ta compared

in fig. 16: for the general cargo ship about tO 0.25 eec up to &)= 0.60 sec

for the trawler about ¿) = 0.50 soc up to t) = 1.5 aec. Therefore the trawler will roll much more and only the irregular motions for wind velocities of 1k and 19 knots have been determined. The higher wave spectra give rise to rolling angles of ko and more degrees, showing that the Neumann wave spectra certainly contain too much energy. Moreover the nonlinearly increasing roll damping will prevent this calculation from:being realistic.

I

The determination of the waterdeptl will not be repeated but all the

parti-culars are summarized below.

= 1,165 sec1

= 0,07

The ship's responses in regular beam Waves are presented in fig. 18 in the form

of the dynamic magnification factor f ' showing a reduction of about

80 percent. S is given in fig. 19 and the significant roll angles in the table

below. The reduction in rolling in the considered irregular seas is again leas than in regular waves, although 60 to/65 percent reduction remains. It will be

noticed, however, that the trawler i equipped with a large tank with a fluid

weight of 2,5% of ita displacement.

significant rolling angles

wind velocity without tank with tank

32

Ship Tank

=

23,90 m

width b = 6.18 m

B = 6,18 m length i = 1.3k a

D = 3,1+6 a waterdepth h = 0.08 b = 0.1+91+ m

d = 2,62 m fluid weight 1+.091 tons 0.025 L.

= i61,1+ tons vertical position a/b = -0.125

= 0,81+ m 8 = 0,136 1(G _{= 2,78 m} R = 0,1+0 = 5,39 eec 1k knots 16.13° _{I 5.59°} 19 "

28.13°

I 11.1+0°

9. SlaCk tanks.

In the introduction it has been stated that the influence of slack double bot-tom tanks on the rolling of stiff ships mostl7 can not be explained by a mere reduction of the metacentric height. To give some evidence of this statement

a rough estimation

of the effect of slack tanks as roll damping devices will

be made.

Consider a general

cargo ship

of about 10.000 tone

dead

weight, having a beam

of 20 m. Suppose that an anti-rolling tank is present with b1 = 20 n and l. = 5

n

and let the ship have a high metacentric height so that the waterdeptb in the

tank h1 = 0.08 b1 =

i.6o

in. The exerted tank moment io

=

Suppose next that the ship bas a double bottom w±th a width o 8 n at either

side of the centre girder. For the same natural period of the water transfer

in

this double

bottom tank the water depth followa from

-

___

'Ut» xf't(u)

2s

M.

Aa the double bottom contains many structural elements the water transfer lags the rolling motion much more than in an ordinary rectangular tank, so that the water leve), must be much higher to obtain a correct tuning with the ship's

motion. Suppose therefore h2 to be 0e50 in.

Then the exerted

tank

moment for a length 1., of the double bottom tank is

=

C.

434

Consequently the ratio is given by

k2

_¡4'

/..(2 1

1/

=

£o/14' 4

J,

2,

¿1.0J

A,.

s

3k

If the slack double bottom tanks have a length of 22 a at starboard and port

side then 12 = 144 a and

A:

_/

or in other words: the combined double bottom tanks are half as effective as a

specially designed roll damping

tank.

Now in

fact a

rough estimation like this is dangerous and misleading as to the

actual magnitudes of the moments involved. In section ₅ it has been described

that obstructions cause

a

considerable decrease in the moment amplitude so that

the constant C in K and K

2 will certainly not be the same. Further the

phase

angles are infli'ienced as ¿ail

and

it is Kt5 sin which matters most.

But anyway it has been ahown that under favourable

circumstances a roll damping

10. Conclusions nd reinks.

It has been shown that for the simplified case of pure rolling tn both regular and irregular beam waves a simple free surface tank is an attrac-tive passive means of roll damping. Ot coume the results will never equal those obtained by active stabilisation. On the other hand aelf-.excted oscillations and roll'inoreaae will not occur for the installations

descri-bed herein.

The tank's poseibility of damping doea not increase proportional to the severenesS of the rolling motion. Relatively it is much larger for small angles than for large angles. For this reason it is not advised to out down

tank dimensions as far as poible, nor to abandon bilge keels. Therefore aleo one has to be very careful in judging the effectiveness of a certain installation by taking into account the conditions for which the effective-ness has been evaluated.

Until now it is not known if and in which way the other ship motions influ-ence the performance of the tank. The investigations in the Deift Shipbuil-ding Laboratory wifl be continued for the case that the axis of rotation

is no longer fixed; that is including sway and heave effects.

+. Too little is known about the general etate of motion of a ship in a short

crested sea to be able to pre&±ct its beh8viour even without tank, but it

is very unlikely that the installation of a roll damping tank will change the p&cture to any reasonable extent, except of course for the rolling charaCteristics.

Regarding the remarks 3 and k it is strongly recommended that a tank will

be located somewhere in the midship area, where the influences of severa].

motions are least pronounced.

The characteristics of the tank itself are markedly nonlinear, but due to the small rolling angles of the ship equipped with a roll danping tank the

principle of superposition to treat ship motions in irregular seaS can be regarded valid. Experimental evidence for this statement will be presented

elsewhere. For the ehip without tank superposition may be questionable when rolling angles become large.

j

s

11.

_{Acknowledgst,}

36

The authors are indebted t. varien. members of the staff of the Deift

Ship-building Laboratory' for their aseictanc.

in running the

_{whole program. Special} thanks ax'. du. to Mr. A. Qoeman and Mr. A.P. _{de Zwaaxi for carrying out the}

major part of th. experim.nts. Last but not least th. preparation of all the graph. by Mr. de Z.aan i. gratefully acknowledged.

s

References.

Watts,

P., "On a method of reducing the rolling of ships at sea",

T.I.N.A.

1883, p. 165.

Watts, P., "The use of waterchambere for reducing the rolling of shipø

at aea", T.I.N.A. 1885; p.30.

Frahm, H, "Neuartige Schlingertank6 zur Abdimpfung von Sohiftarollbewe-gungen und ihre erfolgreiche Anwendung in der Praxis",

Jahrbuch der Schiffbautecb.nischen Gesellschaft 12, 1911; p. 283.

k. Horn, F., "Zur Theorie der Frahmeolien Schlingerdmpfungatanks't, Jahrbuch der Schiffbautechnischen Gesellschaft 12, 1911; p. ¿i-53.

Chadwick, J.R. and Kiotter, K, "On the dynamics of anti-rolll4ng tanks",

Schiffatechnik 8, Febr. 1954;

p. 8,

Stigter,C., "Performance of Utak as a passive antirolling device", report no. BiSof the Netberlande Research Centre T.N.O. for

Shipbuil-ding and Navigation, February 1966.

Binnie., A.M, 'Waves in an open oscillating tank", Engineering 151,

19k1;

p. 22k

Graham, LW., "The forces produced by fuel oscillation in a rectangular tank", Douglas Aircraft Company, report no. 5M-13748, April 195.

Verhagen, J.H.G., and Wijngaarden, L. van, "Non-linear oscillations of fluid in a container", J. Fluid Hoch. 22, 4, 1965; p. 737.

Zunderdorp, H.J. nd Buitenhek, M., "Oscillator techniquea at the

ship-building Laboratory", Shipbuildiflg Laboatory of the Techflplogica].

Univer.ity,D1ft, report no. 111, November 1963.

Bosch, J.J. van den and Vugte, J.R., "Some notos on the performance of free surface tanks as passive anti-rolling1evices',Shipbuilding

Labora-tory of the Technological University, Deift, report no. 119, August 196k.

Kato,E. on seakeeping qualities of bipa in Japan", Society

Modulated carrier Carrier Scotch yoke Strom ge dynarnometer

Fig. 2 Principle of experimental set-up.

o

o

VThI

In phase component Quadrature component Demodulator

bitegrotor

VV Amplifier

Resobier

L

t

b

(Th

(Th

Kt KtaSIN (wt +

4:'

a«t

axis of rotation

j

h1

view

in

the positive

xdirection(Looking

forward)

z

Fig. 3 Coordinate system and tank parameters.

4 h T T I

s

T

è»max.pos.

Ompos.

₀₌₀

mom.max.neg.

mom.neg.

_mom.zero

5 Wt=E

0=0

=mx.neg.

mom.max. pos.

Fig. 4 _{PosItion of bore during one period ofrolling at tank resonance}

= -90 degrees).

9 wt.2fl

0=0

=mx.po%.

1.5 KGM 1.0

-0.5

-1.0

o

- Fig. 5a Example of measured In-phase and quadrature components of tank moment.

s/bO h/bOD6 4O.1O .

.1

- I 1.0 2.0 3.0 4.0 (L) SEC1

O 5/bcO h/b = 0.06 a =0.10 1.0 _{2.0 3.0} (*) SEC1

Fig. 5e _{Example of phase angle of tank moment.}

4.0 h/b=006

a =0.10

O 10 _2.0

30

(A) SEC1

Fig. 5b _{Example of amplitude of tank moment.}

4.0 j-1.0 '<ta OE5

-180

DEGR. Ct

-90

.35

h/bOO5

2.0

1.5

O

Fig. 6b _{Ratio w/t} as a function of h/b.

025 0.50

a75

1.00

104k

..

.

Fig. 6a The actual resonance _{frequency t}

as a function of a01O o

25

Q50 0.75 1.00

lOhlb

20

t

1.5 Wa/Wt 1.0 0.5

Fig. 7a Example of for different values of

W

W

W SEC

t5

1a1 0.5

7

-/

Fig. 7c The nonlinear increase of Kt with 0a at w = w.

100

0.75

025

050

10 4

3.0

-(u

SEC1

F1g 8b

The dependaflc

of E

t°h/b.

-180

DE OREES 2.0 KOM 1.5 K

ta

to

OE5

Fig. 8d The nonlinear increase of Kta with h/b at _{W =}

s/b=O a010

f

htb!U.O8.j ft._

f

I

h/bQO6 h/h=OIM!

j.

/

/

/

s/b=O a ° w =w4 5.252Vi7E

,-V

7

/

/.

/

/

/

/

I

0

05

io

1.5 2O

w/wt

Fig. 8e The variation of E with h/b versus the nondimensionalized

Irequency _WfW.

O 0.25 0.50 0.75

1o0

10 h/b

Ct

1.5 1.0 lOOlia 0.5 s/b-OE2O 010 O h/bO.O6 S h/b=006 o h/b=OE04 u h/b=06 b=1.00m bOE75m b=i.00m b=75m

-Q'

s/b-O.2O

a010

r

I

r

'I.

y,.

--/ a o h/b=O.06

h/b=06

oh/b=D.OI.

.h/b=04

b=1.00m b=O5rn b=1.00m b=75m o 0.5

io

1.5

20

w/wt

Fig. 9a Nondimensional amplitude of tank moment versus nondimensional

frequency for two tank widths.

o 0.5 1O 15

20

w/wt

Fig. 9b Phase angles of tank moment versus nondimensional frequency

for two tank widths.

180

DEGREES

A

Et

2.0 KGH Kta 1.5 1.0

5

180

DEOR.

90

I/bOD6 w =w

Fig. lOa The dependance of Kt .Ofl s/b at w

=

h/b =006

a =010

w

Fig. lOb The dependance of Lt on s/b at w

=

0.40

0.20

s9b

+0.20

OhO

020

O

+20

0.5 a OE10 s/b+O.2O

ìb!DO1

AI

o

io

20

3.0 4.0 W SEC1

100

KGM

uso

Q25

-180

DEGR.

I

E

90

.1O to 2.0

w

SEC1

io

-sib "0.10

r

IF

h/b.55

,h/bbOO5S

h/b =0.04 (1U

sc

10

Fig. libExample of for the rectangular tank and two modifications

with the same amount of water in the tank.

Fig. lia Example f for the rectangular tank and two modifications

0.75

w

0.5O 0.25 1.0 2.0 3.0 (L) SEC1

Fig. lic Example of Kta sin for the rectangular tank and two modifications

2.0

1.5

0.5

o (125

05o

(175

FREO. RINGE FORO.)

FREQRAP4GE FOR

I- w.

--i00____

1.25

wVb/g

Fig. 12 Illustration of the selection of the waterdepth.

s/b=O a=0.10

-ik

AhIb=008

h/b=0fl6

I

r7

IL .1 L,I

20

QREE5

15 10 10 5 lb iiO1O WITHOUt TAN$ 0wmOE015

Fig. 14a Results of the calculations for the model of the sixty series.

O

Fig. 14b Results of the calculations for the model of the sixty series, presented- as magnification factor.

s/b =+O.1O

hfb-=04

- - -SHIP-WITHOUT 4ANK- - -.

,j

a.DD3

SHIP WITH TANK

\\

.- uI_

D

-S

_as0_

0,s

IC

D

025

aso

ais

tao

10

øa 5

a=

002

SHIP WITHOUT TANK

/

,. h/b=0.OL} SHIP h/b=0.05 WITH TANK

\\

o 0.25

050

0.75 1.00 W SEC

WIND VELOCITY 30KNOIS 26 KNOTS 2L KNOTS 19KNOTS 14 KNÓTS

to

1.5

W

SEC1

Fig. 16 Neumann wave slope spectra.

O

0.5

SEC 1.0 s.J) U)

D

D

0.5

U)

D

D

5

Fig. 17 Comparison of calculated roll spectra for the model of the sixty series.

\

WITHOUT TANK 3OKNOTS

WITHOUT TANK 26 KNOTS

-WÌTH TANK 30.KNOTS

WITHOUT TANK 2LKNOTS

/ WITH TANK 26KNOTS

,/ _WIfflTANK2KNOTS

A

io

0.25

0.50

(A) SEC1

0.75

io

0.5

tO

1.5

2.0

W SEC1

Fig. 18 Results of the calculations for the trawler.

WI1HOUT TANK

WITH TANK

WITH TANK I4KNOTS

WITHOUT TANK 19KNOTS

WIThOUT TANK 14KNOTS

WITH TANK 19KNÓÌS

1.0

15

(A) SEC

Fig. 19 Comparison of calculated roll spectra for the trawler.

O

0.5

40

e

U)

Ó

D

20

Fig0

20.1 Nondimensional

quadrature

component of tank moment0

Fig. 20.2 Nondimensional quadrature component of tank moment.

Fig. 20.3 Nondimensional

quadrature

component of tank moment.

Fig0

20.k Nondtmenaional quadrature component of tank moment.

Fig. 211 Nondiuieneional in-phase component of tank moment.

Fig. 21.2 Nondimensional in-phase component of tank moment.

Fig.2.3 Nondimenajonal in-phase component of tank moment0

Fig, 21.+ Nondimonajorial tn-phase component of tank moment.

Fig. 22.1 _{Nondimenaional amplitude and phase of tank moment.}

Fig. 22.2 Nondimenalonal amplitude and phase of tank moment.

.22.3 Nondimensional amplitude and jhaee of tank moment.

. 23.1 Nondimensional amplitude and phase o tank moment.

Fig. 2.2 Nondimensional amplitude and phase of tank moment.

:

23.3

Nondimensional amplitude and phase of tank moment.

2i.1 Nondjmensjonal amplitude and phase of tank moment.

24.2

Nondimenatonal amplitude and phase of tank momènt.

Fg. 2i.3

Nondimensional amplitude and phase of

tank

moment.

rig. 25.1 Nondimensional amplitude and phase of tank moment0

Fig 25.2 Nóndimensional amplitude and phase of tank moment.

-0.0100 -00075 00D50 -00025

S/b

0.40

0.10

S/b 020

4)a010

VMN

____IMII

.iIlIIW___I11SiI

Iff1711

iii I

I

05 lo 1.5 -0.020 0.015 ILa0Et OE010 0.005 o

-0.015 -0.010 -0.005 o

11111

1ir

lällA

___L1iL

ÁZT4TI

'IfMl

to 15

-0.040 -OE020 -OE03 -0.010

a00

-I

-__

-'-1.0 (i)Vi7 1.5 0.5

001 I.LosEt o OE005 wV7 1.5 a0.10

!U!IU!I

-1h10,ofb.

W\J

OE5 _1.0

____WIIL_IIJI

---,---

-I

iUII'i

0.010 0J05

-ÁfM

INII.

.__

i____

k

'i'-L

100 o

4Di0

0.5 1.0 1.5

orno ILosEt 0.020 o -0.010

S/b=

a0i0

O

b. 0.5 lo w 15

0.030 0.020 -180 degrees it -go a010 3.

-

-h10% of

-of b. 05 10 15 0010 D 05 _{lo 15}

0.020 OE015 0,010 0.005 o -180 degrees Et 90 k

fí-.I

O

S/b

+0.20

2

\

0.5 10 is

Ia

0.015 a010 0.005 Et -90 a°0333 O 05 05 1.0 is

E

-Ic -180 degrees

o o

sa.

/b

0

-h10°.of b. _0.10

r

41111

,i

wV7 15

AIAJ

of t. 05 lo 15 05 lo

0.015 010 0.005 Et 90 a00667

\

\

\

wVE1 15

E

A

A'

___

of .

_

u_9

₀₅ 10

wVg

lo 05 -180 degrees

t OE020 015 010 0.005 o -leo

á:

'cid

I

I

v

.1v-il,

05 10 15 05 lo 15 fb

S/b =0

4)a0333

015 0.010 0.005 E. -90 O o

2a3

wVW 15

a10

h10, of b.

ÌII1iI

N

.1111

b

_d1_

_of

0E 10 wV 15 0.5 lo -180 degrees

.4)20 015 010 0.005 i80 degrees Et-90

S/b =O.20

a00667 10 . tb O 10

wv1i

is

4144W

wV7 io

5

i80

Ii

-vv!ArA

_{4h1O0.of b.}

1_ai_ii

ii

0.033 3

__

AUF4

_j

0.5 lo 15

Ia

0.0100 0.0075 0.0050 00025 O t-90 O

..

3

S/b = 0.40

a010 h=1O' of b.

1!

//4

A'ÁtII

r

1O0.of

b. 0.5 lo 1.5 05 lo wV7 15 180 degrees

o o

22

a00667

rirzw

=10 of b.

4Á'A'

f b. 0.5 io _wVW 1.5 o5 lo 15