On roll damping by free-surface tanks

Paper Ño. 4

The issue of (his _{copy of the paper is on the express understanding that no publication, either of the whole or in abstract,} will be ,nade until after the paper has been read at the Meeting of THE RO YAL INSTiTUTION OF NA VAL ARCHITECTS in the Weir Lecture Hall, 10, Upper Beigrave Street, London, S.W. I, on March 23, 1966. The institution is not, as a body, responsible for the statements niade or the opinions expressed by ihdividual authors or

speakers.

Written Contributions to the Discussion should reach the Secretary, R.J.N.A., before May 3, I 966.

ON ROLL DAMPING BY FREE-SURFACE TANKS

-By Ir. J. 'J. VAN DEÑ BoscH* and Ir. J. H. VUGTS*

Summary

Roll damping by means of water transfer in a rectangular tank with a free surface, as introduced by P. Watts at the end of the last century, is.investigated.

The influencé of a periodically varying moment on the rolling notion is detérmined, starting from certain simplifying assumptions.

The magnitude of the moment which is created by the moving water mass in an oscillating

tank, is détermined eperirnentally. The influence of the tank parameters on the amplitude

and phase of the moment is ascertained. Introduction

Since the rise of steam power for ship propulsion and the consequent disappearance of the sails on seagoing ships, there have been many attempts to replace the steadying effect of the

sails on the rolling motion.

Bilge keels proved effective to a certain degree añd it became also clear that a ship with a small metacentric-height possessed

an easy motion. However, other demands can make it unde-sirable or even impossible to use a small metacentric height. The range of the statical stability or the stability in damaged

conditions are examples. Also there may be purely economical reasons to prefer a vessel with a large beam.

In 1883 Philip Watts read a paper at a Meeting of this Institu-tion concerning a roll-damping tank and in 1885 a second paper

followed on the same subject."2

In the papers and the

discussion thereon Watts and also R. E. Froude described in

detail the mechanism by which a roll damping moment is

created by the wave actiOn of a fluid in a rectangular tank

placed abbard a' ship. Although the tank acted satisfactorily, interest, died away in the years that followed and the work of

Watts seems to have been nearly forgotten.

In 1911 a. roll damping tank was introduced by Frahm.t3 This tank consisted essentially of a U-tube in Which by the oscillating water a moment, counteracting the rolling motion

was created. Although a number of ships were provided with

this type of tank it never succeeded in becoming popular. This

may partly be due to the difficulty to "tune" the tank for

different conditions but also there appeared to exist, as in the days of Watts a läck of confidence in a device incorporating

large quantities of free-moving water. Since the last world war controllable fins have been employed quite frequently and these

are reported to work efficiently. A disadvantage of the fins is

that the effect is largely dependent on the ship's speed.

In recent years a roll damping tank was introduced consisting rif two wingtanks connected by a part with a free fluid surface.

This tank acts partly like a Frahm-tank because pressure

differences are created, by viscous effects, partly like Wàtts' original tank, because the wave action; although hampered, is

still present.

Much hàs been published about the U-tank by Frahm and

others. This knowledge is accessible toeveryone who wants to make usé of it.

* Stiipbuilding Laboratory, Technischó Hogeschoòl, Delft.

1

Although.Wattsgave a very clear description of the phenomena in the free-surface tank, the data were insufficient to be of general use. The idea, however, appeared sound and deserved more attention than it had received hitherto.

For this reasoh an

investigation was startéd with the aim to furnish the data usable for thé design and judgment of a free-surface tank of this type. Although the mechanism of the damping action was quite clear

and some relations could be obtained theoretically, a mainly experimental approach was considered to furnish the desired

data in the shortest time.

List of Symbols

Apart from the customary nomenclature concerning the:

geometry of the ship the following symbols are used:-D, = Depth of tank.

14, = Virtual mass moment of inertia about the rolling axis

K = Rolling momént in general.

- K = Rolling moment due tó the Waves.

Kwa = Amplitude of K.

K, = Rolling moment due to the tank water. Kta = Amplitude of K,

N4, = Damping coefficient of roll. R4, = Restoring coefficient.

S4, (w) = Spectral density of rolling motion.

S (w) = Spectral density of wave amplitude.

(w) = Spectral density of wave slope. T4, = Natural roll period.

T, = Theoretical natural period of water transfer in the tank.

b = Tank breadth (measured across the ship).

c = Velocity of the hydraulic jump.

h = Water depth in the tank at rest k = Wave number.

k4, Radius of gyration for roll.

'i = Tank length (measured forward and aft).

s = Distance from tank bottom to rolling axis.

= Phase angle between the wave moment and the rolling - motion.

= Phase angle between the tank moment and the rolling motion.

= Wäve amplitude. A = Wave length.

ILa = Non-dimensional amplitude of tank Ínomênt u4, = Non-dimensional roll damping coefficient.

p = Mass density of water. p, Mass density of tank fluid.

= Roll angle.

a =

Roll amplitude.

Ç6a 1/3 = Significant amplitude of irregular rolling motion. = Static angle of heel.

w = Circular frequency.

= Natural roll frequency.

= Natural roll frequency corrected for

the in-phase

moment of the tank.

w, = Theoretical natural freqüency of water transfer in

the tank.

Simplified Theory of the Roffing Motion Basic Assumptions

-In order to simplify the problem, only the pure rolling motion

of the ship about a longitudinal axis through the centre of

gravity was considered, Linearity was assumed for the ship without the tank.

The equation of motion can be represented i n first approxima-tion by the equaapproxima-tion (1).

l+N+R=K

(I)

wherein:-I is the virtual mass moment of inertia about the rolling axis,

N is the damping coefficient for rolling, R is a stiffness coefficient,

K is the exciting moment in most general sense, producing

rolling, ç' is -the roll angle.

When K is harmònic the resulting motion will, after sufficiently long time, also be harmonic of the-same-frequency but will show a phase lag in relation to the moment K.

In the case under consideration this moment is composed of

the moment which the waves exert on the ship and the moment

which is exerted by the water transfer in the tank. The wave moment in regular beam waves can be assumed to be harmonic. FrOm the first series of experiments with asinusoidally oscillating

tank model it was evident that the tank moment could also be

considered to be harmonic in the frequency range of interest.

If

the motion is expressed by

- _{= Çt sin wt . . . . . (2)}

the moment can be presented by

K = Kwa sin (w t+ EKÇ,) + K1,, sin (wt+ E1) (3) in which

Kw,, is the wave moment amplitude; K,,, is the tank moment amplitude;

EK4, is the phase angle between the wave momçnt and the

motion;

, is the phase angle between .the tank moment and the

motion (e, is negative);

Ç1a is the motion amplitude; and w is the circular frequency.

lt is more customary to introdUce the phase angle EK in the expression for the motion instead of in the exciting moment,

but it proved more convenient to depart from this practice.

The tank moment which is written as an external moment can be brought to the left-hand side of the equation and

con-sidered as a correction on the coefficients.

Substitution of the expressions for the motion and the moment

yields:-ON ROLL DAMPiNG BY FREE-SURFACE TANKS

(R - I w2 - -A R)-sin w

t +

(N w - A N w) cos w t =

K.

_{wasln(w(±Ex)}

_{.. (4)}

in which A R Kgacos e, (5)

and

AN=

K,,, SiflE . . (6)

(V9ia V

Evidently A R, the reduced component of the tank moment

which is in phase with the motion, can be considered as a

correc-tion on R,.

Likewise A N, the reduced component which

lags the motion by 90 degrees, can be considered as an extra

damping(e, is

negative, so A N is positive).

The solUtion of equation (4)

reads:-K..,,

(9)

- i w '- A

+ (N - A N)2. w2

and _K4'=

(N,,AN,,)w

At the natural frequency of the

system:-wt,c = y _l V

the phase angle becomes90 degrees and the motion amplitude is governed largely by the dampingof

the system as at this

frequency the amplitude

becomes:-1(

- (N,, AN,)

w,t,

(IO)

w

stands for the corrected natural frequency; the original

natural frequency of-the ship

is:--wv1ç

V

JR

(9a) In Fig. I a typical example is shown of the in-phase and quadra-ture components of the tank moment versus the frequency.

The effective natural frequency is now defined as the frequéncy

at which the in-phase moment K,,, cos e, is zero. Around this

frequency the quadrature component K,,, sin , reaches its

largest value. As the motion is damped by the latter component of the moment it is.evident that the natural frequency of the tank -has

to be -chosen at or near the ship's natural frequency.

- A R and A N depend strongly on the frequency of-the motion. At

relatively low frequencies A R

is positive like the terni

R - I w2 and - rather large

while A N

is - unimportant.

Consequently ,, is somewhat larger than for the ship without

tank {formula (7)). At high frequencies R - I w2 is negative

and A R also.

Çonsequently in this range f'a is also slightly larger than for the ship without tank.

Only when K,,, is proportional to the amplitude of the motion,

A R-and A N will be independent of q,,. In that case the

system is linear and the solution of the equation of motion is

readily calculated. Ifthe system is not linear, i.e. if K,,, is not proportional to çb,, the solution has to be found by an iterative

process.

-On behalf of the comparison of the motions with and without

the tank in operation, acceptable values for the coefficients of

the equation of motion -have to be determined.

The virtual mass moment of inertia J, is usually estimated

s

k1, is the radius of gyration for rolling; ¿ is the displacement;

y ¡s the volume of the displacement; p is the mass density of the water; g is the acceleration of gravity.

k is usually. given as a fraction of the ship's beam. According to existing data this value

¡s:-O33<--<O45

Katot4 gives an expression for k/B containing certain hull

parameters.

-5

KSINE

KtaCOS C

-Fia. 1.EXAMPLE OF THE MEASURED IN-PHASE AND QUADRATURE COMPONENTS OF THE TANK MOMENT VERSUS FREQUENCY OF MTION

lt is customary to calculate the damping with an estimated value of the noñ-dimensional damping

coefficient:-N,5

V5

= \/l R

v ranges from 008 to O2O.

The stiffness coefficient R is the linearized moment of statical stability for one radian of

heel:-...(12)

The wave excited moment can be calculated by integrating the pressure on the hull in the incident wave For this calculà-tion the Froude-Kriloff hypothesis, which says that the pressure distribution in the wäve is not disturbed by the presence of the ship, is assumed to be valid. The waves.are long with respect to the ship's beam and draught. Reflection effects and the

influence of the relative velocity and acceleration on the

exciting moment are neglected. This results. in the following

expression: .

3

. . (13)

k is the wave number 27T/A; À is the wavelength.;

a ¡s the wave amplitude;

Ze is a certain mean depth below the water surface, which can be approximated by half the draught: Ze = d/2.

The Behaviour of the Vessel in Irregular Beam Seas

The statistical determination of the ship's motions is based on the principle of superposition which implies linearity of the system. _{The tank-action is markedly non-linear, but to all} appearance the non-linear effects can be neglected for small roll amplitudes. Therefore linearity is assumed for this part. of

the paper.

From the earlier mentioned equations the ratio of the roll angle amplitude and the wave slope k 1a can be derived.

e2

j(i _A2_R)2±(v_z)2.A2

in.which-:-A =

the frequency ratio,

w =

the original natural fequency of tke ship.

(14)

(15)

= -/I R

Under the simplified conditions as assumed, çba/k _ais equal to the amplitude response function, IH._k (w)I defined for the

wave slope.. The spectral density for an irregular long-crested. wave pattern can be presented for our purpose by a wave slope spectrum whichisderived from the wave energy spectrum

by:--

Sk(th)dw=S(w).dw

. (16)

The roll spectrum then

follows.from:-S (w) d w = I'H,k (w)12 follows.from:-S

(w) dw ...(17)

The significant roll amplitude for a narrow spectrum can be calculated from the familiar

expression:-Ç6a 1/3 (18)

where m

=rjSc(w)dw

(19)

The efficiency of the roll damping tank can be measured by the comparison of the significant roll amplitude for the ship with and without the tank. This comparison should be made for a realistic wave spectrum.

Model Tests with a Rectangular Tank Experimental Set Up

The experiments were designed with the simplified theoretical approach in mind, which has been set forth so far. The experi-ments consisted of the determination of the relation between the

motion, the tank dimensions and the resulting moment

components.

A skètch of the experimental setup is given in Fig. 2. The

model-tank is fixed to a frame- which is forced to perform an harmonic oscillation about a fixed axis. The exciting moment is transmitted to the frame by a strain gauge dynàmometer.

KtaC OSE ..SINE

Ça k

Motøi' Gear box

ON ROLL DAMPING BY FREE-SURFACE TANKS

EIrctr,ic strain indicator Carrier amplifier Th Modulated carrier Resolver Carijer Scotds yoke WSnes'a

The output of the strain meter is resolved in two components

by a mechanical-electronical device. The values read off from

the dials of the instrument are direct measures for the in-phase and quadrature components of the moment. More detailed

information on the measuring technique is given in Ref. (5).

The Water FIów in the Tank

The moment is produced by the shift of the mass of water in the tank. When a tank containing water is heeled, the water

flows to the lowest side.

When this heeling is occurring

periodically it depends on the frequency whether the mass of

water can keep in phase with the motion.

For very low frequencies the water surface remains of course

horizontal, which means that the phase angle between the

moment and the motion

is zero. The moment amplitude

corresponds in principle with the well-known reduction of M due to the free fluid surface.

For somewhat larger frequencies the water rises at the sides a little above its pósition in the statical condition. Virtually the flow is characterized by a long low standing Wave the length of which is twice the tank breadth. Because the water is being

swept up the moment amplitude is somewhat larger than with very low frequencies, but the phase angle is still' not notably

different from zero.

With an increase of the frequency the water forms a train of small waves every time the motion is reversed. The moment

amplitude has risen slightly and there appears to be a very small phase lag.

With a further small rise of the frequency the train of waves

is suddenly transformed into a bore, or hydraulic jump, a distinct step in the water surface. As soon as this phenomenon develops

the phase lag starts to rise almost linear with the frequency. Qualitatively this picture does not change much over a rather large frequency range, but the bore gains in height, the motion

Fic. 2.PRINcIPLE OF THE EXPERIMENTAL SET-UP

4 Strom gau5e dynamomnter Resolver

VV

o

VmI

o

km!

An'pUf er DamadiAqtoe

Is phase component Quadrature component

i

becoming wilder. The 'mdment amplitude fluctuates somewhat

but does not change materially.

The phase angle rises to

about 120 deg.

At the end of this frequency range the bore degenerates intO a solitary wave with steep sides and a smaller difference in the water levels fore .and after the wave. Consequently the mass

transport is less and so the moment amplitude drops.

The

phase angle in 'this frequency range has a value of say 120-¡50 deg.

If the tank is situated below the axis of rotation the water

flowstops practically with a further rise of the frequency: Then, of course, the moment amplitude is zero. 1f the tank is situated abovè the axis of rotàtion the Water motion becOmes somewhat confused without producing any appreciable moment.

The most important of these wave phenomena are illUstrated by the photographs of Fig., 3.

The existence of the hydraulic jump is evidently the cause of an alternating moment with such a phase that the motion of the tank is counteracted.

It has been remarked already that the

quadrature component of the moment is the one which is

responsible for the damping action of the tank. As roll damping is most needed at the resonance frequency of the ship, it appears

reasonable to choose the natural frquency of the tank at or near

the natural frequency of the ship.

The resonance of the water motion in the tank is determined

by the condition that the bore has to travel twice the tank

breadth in one period. For a small height the velocity of the hydraulic jump is given by:

c=t/

.

...(20)

in which g is the acceleration of gravity and h is the water depth.

From this expression it follows that the natural period

is:-T, -

2b (21)

Fio. 3.-ILLUSTRATION OF WAVE PHENOMENA IN THE TANK FOR VARIOUS FREQUENCIES OF MOTION

and the natural circular frequency:.

(22)

Influenceofthe Tank Paranieters

Amplitude and phase of the moment are influenced by the

following parameters:

-The frequency of the oscillation w.-The dependence on the frequency as treated in the foregoing pages is illustrated by the Figs. 4, 5 and 6. The moment cOmponents for different ampli-tudes as a function of the frequency are shown in Figs. 4 and 5

1.00 w 0.75 :1. ¡ 0.50 025 -0.25 -0.50

FIG. 4.IN-pi-iAsE COMPONENTS FOR THREE DIFFERENT AMPLITUDES OF

MOTION;s/b = OANDh/b = 0-04 1 0.50 0.25 s/b nO h/b=d114 010 a00667

s/b0

h/b =004 D OE25. 0.50 0.75 1.00

ON ROLL DAMPING BY FREE-SURFACE TANKS

0.25 050 0.75 . 100

FIG. 5.QUADRATURE COMPONENTS FOR THREE DIFFÈRENT AMPLITUDES OF MOTION;s/b =O ANDh/b = 0 '04

and the moment amplitude and phase in Fig. 6. The measured

quantities are given in a non-dimensional form. The non-dimensional moment

is:-1.00 0.75 2 lOXL 050 0.25 O -180 degrees 0.25 025

cr=

Kia

!'ap,g.b3.I.

in which p, is the mass density of the tank fluid. The reduced frequency

is:--¡b Vg s/b=0 h/b =004 a0.10 050 0.75 1.00 050 075 1.00 '-wVE?

FIG. 6.AMPLITUDE AND PHASE ANGLE OF TANK MOMENT FOR THREE

DIFFERENT AMPLITUDES OF MOTION;s/b = OAND h/b = O-04

The amplitude of the motion aRefe is again made to

the Fig. 6. For very low frequencies the moment is as a matter of course practically proportional to the motion. This is not the .case, however, in 'the frequency range around the natural frequency of the tank. Here the moment is approximately

proportional to \/4a, at least for small motions. For really

large roll angles the increase of' the moment is still less. The influence of the motion amplitude on the phase is twofold. In- the first place the phase characteristic becomes appreciably flatter when the motion amplitude increases, in the second place the effective natural frequency, that is the frequency at

which-thephase angle is 90 degrees, increases. These phenomena -are caused by the relation 'between the total 'amount of water, the height of the bore and its translation velocity. The

expres-sion for the velocity of translation (20) is only valid for small

disturbances. For a larger height, the water levels in front of and behind the step, which determine the velocity are strongly interdependent because of the continuity-condition. Moreover, the velocity of translation of a hydraulic jump is measured

relative to the water in front of the jump, which velocity in

relation to the bottom is continually changing.

The length of Ehe tank 1.The length is measured in longi-tudinal direction. ' If the relatively small friction of the water against the bulkheads is neglectedthe flow pattern is

two-dimen-sional which implies that the moment is proportional to the length.

The breadth of the tank

b.The breadth is measured across

the ship, The water transfer in the tank is a wave phenomenon. This means that the tank can be scaled up according to Froude's law of similarity. Therefore the moment per unit of length is proportional to b3, if the sameh/b ratio is considered. This was verified experimentally.

The waler heighth.The importance of the water height for

the "tuning" of the tank will be clear.

Changing the water

depth is the only possibility to alter the natural frequency of the tank once the breadthis fixed. Apart from the influence of h or

rather the ratioh/bon the phase relation, there is also an appre ciable influence on the moment amplitude. When increasing the depth the amount of water is increased and therefore the moment amplitude. For a fixed breadth the moment amplitude around the natural frequency is approximately proportional to _,/hfb.

Because of this feature it can be advantageous to choose a water

depth which i

higher than would follow from the desired

natural frequency.

The vertical position of the tank

s.The influence of the

vertical position is twofold.

In the first place the moment

amplitude at resonance is largest in the highest position of the

tank. In the second place the phase versus frequency curve is

considerably flatter the higher the tank

is situated. Both

features are favourable. The position of the tank is defined by the vertical distance between the tank bottom and the axis of rotation. This distance is denoted by s and called positive if the tank is situated above the axis of rotation The vertical

position is given by its ratio to the tank breadth. Resulls of Systernallc Experiments.

With a rectangular vessel having the following dimensions:-b

= I 000m.

-1

0100m.

systematic tests were carried out. The consideredh/b

ratio's

were:-h'b =

0O2; 004; 006; 008 and 010.

The considered s/b

ratio's

were:-s1'b =

O4O; 020; 0; +020.

The motion amplitudes for which the tests were carried out

were:-= 00333; 00667 and 0l0 radian

r

1 .9; 38 and 5 7 deg.

This range covers most of the possible applications

The results of these experiments are fully discussed in Ref. (6).

In the Appendix some considerations and suggestions are

iñcluded in the design procedure,. treating the application of the systematic data. The results of a calculation for a particular

case are compared with the results of a model test,

Conclusion

The original water chamber used by Watts proves to be an

efficient and very simple device to damp the rolling motion of a ship.

The calculation of the effect of the tank and the comparison

with the motion of the unstablized ship are mainly of a qualitative

nature. A more precise prediction of the results to be obtained,

depends on a móre thorough knowledge of the rolling motion of

the ship alone and of the interactions with the other modes of

motion.

7

FREE-SURFACE TANKS

References

(I) WATrS, P.: "On a Method of Reducing the Rolling of Ships at Sea," TRANS. I.N.A., 1883, p. 165.

WATTS, P.: "The use of Water Chambers for reducing the Rolling of Ships at Sea," TRANS. [.N.A., 1885, p. 30

FRAHM, H.: "Neuartige Schlingertanks zur Abdämpfung

von Schiffsrollbewegungen," Jahrbuch der Schijfbautech-nischen Gesellschaft, Vol. 12, 1911, p. 283.

KATO, H.: 60th Anniversary Series, Vol. 6, The Society áf Naval Architects of Japan.

ZUNDERDORP, H. J., and BUITENHEK, M.: Oscillator Tech-niques at the Shipbuilding Laboratory,Shipbuilding

Labora-tory of the Technological

University, Delft, Report

No. Ill, November 1963.

BOSCH, J. J. VAN DEN, and VUGTS J. H.: "Roll Damping

by Free-Surface Tanks," Shipbuilding Laboratory of the

Technological University, Delft, Report No. 134, Decem-ber 1965.

Also Report No. 83S of the Netherlands

Research Centre T.N.O. for Shipbuilding and Navigation. BOSCH, J. J. VAN DEN: "A Frèe-Surface Tank as an Anti-Rolling Device for Fishing Vessels," F.A.O. Third Techni-cal Meeting on Fishing Boats, Gothenburg. Oclober 1965.

APPENDIX Design Procedure

General Conments

The data presented in Ref. (6) cover the application for ships with GM/B valùes ranging from about 003 to

about 0 18.

The demands concerning the statical stability of the vessel canimposea restriction of the tank size. The reduction

of

M

due to the free surface has to be acceptable.

Because of the importance of the tank breadth for the moment amplitude it is advisable to extend the tank

over as large a breadth as possible.

The best vertical -position of the tank is as high as possible in theship.

In these test series only the rolling motion was considered.

Because of the uncertainty about the behaviour of the

tank at large athwartship's accelerations

it is not

-

advisable to place the tank far forward or aft,

in

particular with a view on the large yawing motions in

stern seas.

-The minimum depth of the tank should be such as to

allow the water to rise to a height of approximately

three times its height at rest. A preliminary estimation

of the tank depth

is:-D,>214

--Every ship sails with different loading conditions and there-fore the resonance frequency varies; it depends on the ship type

whether it is thought necessary or desirable to adapt the tank

level to the different coiiditions

For certain classes of small ships, trawlers, for iristançe, this

appears not practical, as for every fresh haul M will vary.

Consequently the tank has to be filled to a predescribed level which should account for all possible conditions. It will be clear that then the tuning. of the tank cannot be optimal for

every condition.

For large merchant ships the water-level can be adapted, if desired, to the loading condition, because in most cases

is known.

Concerning the determination -of the tank size, the designer

reduction ofthe rolling angles and accelerations in a given sea state. Critçria in this respect are difficult to establish.

The Determination of the Water Depth

The breadth and the length of the tank being settled, the

water depth can be determined utilizing the diagrams for sin and ¡sa cos e, presented in Ref. (6). The diagrams are made

up forS1'a = o i rad., which seemed a reasonable value to base

the calculation upon. The diàgrams for the nearests/b ratio are selected, or the diagrams for the desireds/b ratio are obtained by interpolation.

The frequency range of interest lies around the natural

frequency w of the ship, from, say, 075 w, to I 25 co, because

outside this range roll amplitudes are relatively unimportant. This implies that only between these two limits extra damping is necessary.

The first step is to determine, by estimation or graphical

interpolation, the h/b ratio which gives the highest quadrature

component at the natural frequency of the ship.

For the

obtainedh/bratio ¿ R (5) and the corrected natural frequency w (9) are calculated, utilizing the in-phase component¡saCOS E,,

which is read from the diagram. The process is repeated for the new natural frequency until the difference between the "input',

and the "output" can be neglected.

This iterative method, although seemingly rather cumbersome, is in practice carried out ma very short time.

If the same water depth is used for different loading conditions,

it is of importance that the entire frequency range, that is from

Ø75 times the lowest corrected natural frequency to I 25 times the highest value, is covered by the quadrature component of

the tank moment.

The Estimationofthe Effect

of

the Tank

A simple approach to ascertain the influence of the tank on the motion is to calculate the ratio of the roll amplitude to the

wave slope (14), assuming that the tank moment is proportional

to the motion amplitude. The roll amplitude-wave slope ratio

can be interpreted as the amplitude response function. The

comparison of the amplitude response functions for the ship

with and without tank already furnishes important information about the efficiency of the tank. 1f the peaks of the curves are compared only the effectofthe tank is somewhat over-estimated, for in irregular waves the, roll response consists ofthe response

to the whole gamut of wave components, and for a part of the

frequency range the reduction in amplitude will be less or non-.existent. The better check will be to calculate the roll spectrum

with a realistic wave spectrum and to compare the significant

roll angles. Which spectrum should be used is difficult to say.

For the calculation a motion amplitude q

= 01 radians is

suggested because rolling angles of 5-6 degrees are in general

quite acceptable. Significant amplitudes below this value do not seem interesting. If the calculation results in significant values

which are materially larger,

it has to be expected that the

performance of the tank is less good than the calculation

indicates.

By an iterative process it is not difficult to compute the motion

amplitude taking into account the non-linearity of the tank

moment, for a predetermined wave slope. lt is questionable if

this is of more than academic interest. The main point seems to be the question if the non-linearity is of such importance that the use of the spectral procedure is not justified any more.

Example

A model of the Series Sixty with a block coefficientCB

= O7o

was supposed to be a I to 50 scale model of a shelter deck ship

with the following main

dimensions:-ON ROLL DAMPING BY FREE-SURFACE TANKS

8

= l524Om.

B = 2l77 m.

D = 1354m.

dmld

= 87l m.

= 20,228 m3.

An anti-rolling tank had to be designed for the following loading

condition:-Leaving

port:-V = 20,228 rn3. GM

=006

B

M= l306ni.

KG = 7475m

k4,IB = OE 38

T= l452sec.

w=04327sec.

= ØØ69 (derived from model experiments).

The breadth which was available for the tank was b =21 60 m

the length 40O m and the vertical position was chosen as high

as possible, with the result that approximately s/b -' 010.

Furthermore it was assumed that during the voyage the

consumption of. 'fuel and stores teñded to lower

KT. The minimum value would be:

1f =

0870 m (ii/B = 004), the

volume of displacement 18,205 ni.3 and the corresponding

natural frequencyw = 03533.

From the existing data-the diagrams for ¡s cos e, and ¡se, 51fl E, fors/b = 0l werè obtained by graphical interpolation. These diagrams are presented in the Figs. 7 and 8.

2L

V

i 12 D 0.8 0.4

/

o 025 0.50 0.75 1.00 1.25

wv_

FIG. 7.DETERMINATION OF WATER DEPTH FOR THE DESIGN CONDITION

FROM THE QUADRATURE COMPONENTS

For the design condition tv = 04327, the reduced natural

frequency is w4,i/b/g =

0642.

According to the diagram for

¡sa sin e,, theoptimal value of h/b lies between 004 and 0O6. Knowing that the natural frequency, corrected for the in-phase

'component ¡sa COSE,, is lower than the original natural frequency, the valueh/b =004 is chosen for a further investigation. The iterative method to find the corrected natural frequency

yields:-= 04l2 and

o=\=06llwith

¡sa sin

g = -

Q9l3 10_2. s/b= 010 =0.10 hib=08

A

h/b =0D6 h/b=004__...1j 0.75 ,T1=O.87m _.

j'

woc w4, i=0.87m

r

!

DES'GNCOND. i=1.3O6m

This last value is very near the optimum. A higher

water-level will' result in a greater shift of the corrected natural fre-quency and according to the diagram the gain will only be very slight increase of the quadrature component. With a view on the decreasing stiffness of the ship towards the end of the voyage the value of h/b = 004 is considered preferable.

As a check the corrected natural frequency for the lowest ii value is calculated,.making use of the curves for h/b= 004.

The result

is w = 0307 and w /b/Rj= 0456.

At this

1.6 , 1.2 w u,

o

j.

(NX 0.8 D 0.4 0.4 0.8 0 0.25 050 0.75 1.00 125 wV7

FIG. 8.DETERMINATION OF IN-PHAsE COMPONENT FOR DESIGN CONDITION

reduced frequency the quadráture component is more than half its value at the design condition while at three-quarters of the corrected natural frequency, ws./b/g = 0342 the quadrature component has still a value of nearly thirty per cent of its value at the design condition. This is considered satisfactory. If the h/b = 005 ratio had been chosen the performance df the tank at the lower frequency range would' have been considerably less, while the gain at the design condition would have been immaterial.

Model Tests

Purpose and- Performance

With the ship model a series of tests was- carried out for the design condition to-compare the rolliñg motion with and without the tank in action. Aroll oscillator'was installed by which the model could be subjected to a rolling moment of constant magnitude having any desired frequency within the range of

interest. A gyroscope served to measure the roll- angles. After the model was ballasted and trimmed to satisfy the design con-dition the following tests were càrried out:

(a) The model was oscillated in roll by subjecting it to a

moment with an amplitude of 6 X 102 kg m corre-sponding to a static heeling angle of O90 deg. for the ship without the tank and 05 degrees for the ship with the tank. The ratio of the roll angle amplitude ç6

9

and the static heeling angle 4 was calculated as a function of the frequency.

The object of this series was to determine the value of ve to use in the càlculation and also to compare the test results with the calculated curve for the stabilized ship. (b) The second series consisted of thö measuremen( and analysis of the rolling motion in irregular beam seas, carried out for two different wave spectra. The tests provided a comparison of the measured and the calcu-lated roll spectra, utilizing the results of the oscillation tests and the measured wave spectra.

Analysis

of

the Results

The measured curves of Ç6a/Ç6t versus the frequency are shown in Fig. 9, together with the calculated curves. For the

calcula-14 12 cDaI4st 10 8 6 2

FIG. 9.00MpARI50N OF MEASURED AND CALCULATED MAGNIFICATION FACTOR FOR OSCILLATION ¡N STILL WATER; CALCULATION WITH

CONST4NT DAMPING COEFFICIENT

tion use was made of the value y4, = 0069 determined from the measured peak value of Ç6a/Ç6s,. The curve for the rolling

motion with the tank in operation was calculated according to

formula (7). /

The agreement between -the -measured and calculated curves for the stabilized ship does not seem too good, but also there are appreciable -differences for the unstabilized ship, suggesting that the damping coefficient y4, - depends on the frequency, whereas for the calculation a constant value was used. Investi-gating this further the damping was determined as a function of the frequency, making use of the measurements for the unstablized model. Next these results were introduced in the second calculation of the curve for the stabilized- ship. The

results of this calculation are presented in Fig; 10, together again with the measured values. It is evident that with this frequency dependent damping, the agreement between the

s/b=O.10

-41

4mÍIh

'E/b0O4)Th

Ih/b0i8

IIk

-'s.

;

U)"

o SHIP (MEASURED

WTH0UTj.EDI

-I I -MEASURED OSC.TEST CALCULATd VOED69 J SHIP WITH TANK

-/

/1

\\

-/

O 0.25 050 0.75 1.00 wV 1.25

ON ROLL DAMPING BY FREE-SURFACE TANKS

0 0.25 0.50 0.75 1.00 1.25

wV7

calculated and measured characteristics is much better, as well

for the ship without tank as fOr the stabilized ship.

The roll spectra which were measured were compared with the calculated spectra. Utilizing the ç6a/ç6s, values from the oscillation tests multiplied by the. factor e'112 (formula 14)

and the measured wave spectra, the roll spectra were calculated.

The results for the ship without tank are shown in the Figs. Il

lOxSk

1.

20

¶

16

FIG. 12.COMPARISON OF MEASURED AÑD CALCULATED ROLL SPECTRA FOR THE SHIP WITHOUT TANK IN WAVE SPECTRUM 2

28 24 12 8 8 0.25 0.50 0.75 1.00 1.25 SHIP WITHOUT TANK MEASURED OSC.TEST CALCULATED H MEASURED OSC.TEST CALCULATEDWITH SHIP J

iì'

T"1T

SHIP WITHOUT TANK I I _{S CALCULATED} FROM OSC.TEST SMÊASURED

Ì

lOxS SPECTR.2 SHIPWITHOUT TANK

37

-S4CALCULATED FROM OSC.TEST I k' .1 .1 I +S+ MEASURED -- lOxS SPECTR.1

SkIP WITH TANK

4h:

A lOxSk SGMEASURED CÄLCULAT[D -FROM OSC.TEST 0 0.25 0.50 - 0.75 1.00 1.25

î-FIG. 10.COMPARISON OF MEASURED AND CALCULATED MAGNIFICATION FACTOR FOR OSCILLATION IN STILL WATER; CALCULATION WITH

FREQUENCY DEPENDENT DAMPING COEFFICIENT

FIG 11.COMPARISON OF MEASURED AND CALCULATED ROLL SPECTRA FIG 13COMPARISON OF MEASURED AND CALCULATED ROLL SPECTRA FOR THE SHIP WITHOUT TANK IN WAVE SPECTRUM I FOR THE SHIP WITH TANK IN WAVE SPECTRUM I

lOxSk SG 6 t 2 O 0.25 0.50 0.75 1.1 00 1.25 28 24 lOx 20 s4 16 12 8 4 14 12 a/4>st 10 8 6 2

The agreemént for the ship without tànk is not so good in this particular case. Especially for the highest wave speétrurn the measured significant amplitude is considerably lower than the calculated value. This is attributed to the increased damping of the motion with large amplitudes

For the stabilizedship the agreement is much better. Thenon-linearity of the tank moment does not appear toaffect the validity of the spectral procedure. Probably this is partly due to the relatively small r011 angles and partly to the fact that the roll damping of the ship itself increases with increasing roll angles, and so compensates for the decreasing tank contribution. Another example is given in Ref. (7).

14 12 lOxSk

t

10 S. 8 2

Printed In Great Britain by

UNWIN BROTHeRS LIMITED. WOKING AND LONDON

(B5333)

SHIP WITH TANK

l0xS1 SPECTR.2 4 S4MEASUREID Measured -Çt'.. 13 degrees Calculated degrees

Ship exclusive

tank:-Spectrum i 866 919

Spectrum 2....

lO99

l392

Ship with

tank:-Spectrum i 356 369

Spectrum 2 585

57l

025 0.50 0.75 1.00 125 WVP7i

FIG. 14.COMPARISON OF-MEASURED AND CALCULATED ROLL SPECTRA FOR THE SHIP WITH TANK IN WAVE SPECTRUM 2

and 12, for the ship with the tank operative in the Figs. 13 and 14. The significant roll amplitudes are summarized in Table i for all conditions.

TABLE I