Some notes on the performance of free surface tanks as passive anti-rolling devices

y., REPORT No l'i 9 AUGUST 1964

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_4i1

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SHIPBUILDING LABORATORY

TECHNOLOGICAL UNIVERSITY

-

DELFT

JR J. J. VAN DEN

and IR J. H. VUGTS

. ',*.

i

i)

'

.,, -.

SOME NOTES ON THE PERFORMANCE OF

t;

SOME NOTES ON THE PERFORMANCE OF FREE SURFACE TANKS

AS PASSIVE ANTI-ROLLING DEVICES.

Peoit Nr. 119.

B

¡r. J.J. van den Boech and Ir. J.H. Vut8.

5hi,building Laboratori.

Univereitj of Technology - Deift.

So*. Notes on the Performrnce of FreeSurfoe Tanks

a. Passive Anti...Rollinj D.vic.s.

3y Ir. J.J. van den Boech _{*n4 Ir. J.ff. Vqgt..}

A tre. surface tank _{i. forced to perfora sinusolda]. oseilla} tioris about a fix.d axis of rotation. The _{way, pattern}

_{in the tank}

i. observed and it turns out that

the sueential

_{phennn, which}

creates a moment that counteract. the impos.d _{motion te the genersu.} tien of a bore a8lsiresdy notèdb7áWatte in 1885

_C6.

ßome result. of the. meuur.ments

of the moment

about the axis

of rotation due to the water transfer in

the tank, an, given.

The quantitative influenc, of

_{systematic changea in severa] paraae}

tena is investigated. Por

_{a certain amount of water in}

the tank the

moment is maximum if

_{no restrictions are iapoesd on the fr.. motion}

of the liquid and if vieooue lose.. are prevented as tar as possi-ble.

The data in this report have tä be considered _{as interim}

r.-suite. The investigation of the

performance of the tank eeparat.ly

and in it. connection with the

motions of the ship will b. con.

titiued.

/

Nomen1ature.

a _{Distanc, from axle of rotation (centre of gravity of ship) to} bottom of tank.

b _{Tank breadth (measured acrose the ship).}

h _{Water depth in the tank.}

i _{Tank length (measured forward and aft).}

°, Amplitud, of forced osoillatjon.

L Circular frequency of forced oecillatio.

-

f \f'

theoretical natural frequency for the fundamental wave motion in a stationary rectangular tank.

C, Experimental natural _{frequency, definid as the point where}

L

- -

_{90 degree. when the tank' performs sinusoidal oscillations.}

M _{Am1itud. of th. moment xerted by the water in th. tank on the} ship.

£

Phase angle of this moment with respect to

the

_{rolling m.tion}

(6

positive: moment leads motion).

-2-

-3-1. _qtrodotion.

Rei]. atabjljzatjon has attracted attention sino. almost a ceri. tury. After the development of active itabilising fins passive tanks ar? again considered for their iplicity, low cost and _{action at} low or even zero apead. During the past uveral types of tanks have bein proposed and tried in practice to moro or less satisfaction. It is not the intention of thi. report to compare the different

ay.-tema; it will only be concerned with tanks extending over the full breadth of the vessel and having * free surface.

Theoretical studies on anti-rolling tank. are limited to the U-tubs tank and are r.duod to an equivalent double pendulum _theory, e.g. I _{This principi. ii net applicable to the case of th. fr..}

surface tank, however. The physical phenomenon is completely dif fe. rout sad must b. olaaaed in the gr.up of wave problema in shallow water. Wave phenomen, in rectangular tanks have been diacusied beth in the shipbuilding and aeronautical field, _{see among otbersta) and} [3]. But roll atabiliation was never the objectiv, of these theore-tical investigations and for the purpose in question a linear theory Was suffioient. Since th. main stabilizing action is created by _a boze travelling up and dein the tank's breadth, which is an essen-tially nonlinear phenomenon, it is hard to believe that any linear theory will produc. reliable approximations for the tank se an anti-rolling device.

For this reason an experimental procedure was followed to ool. loot information about the performance of the tank. It was forced to execute sinusoidal oscillations about a fixed axis while ampli-tude and phase of the moment about this axis were measured.

Applying the results of these measurements to th. ship it must be possible to caloulats approximately for a given ship and a given sea the pure rolling motion in the stabilized case. An exact eolU'. tion of the general problem will probably offer extremely great dif-ficulties because of the nonlinear character of the roll-equation

and of the tank phenomena. Moreover, the coupling effects between rolling

*nd

several other motions then bave to be taksn ittb account.

-4-2. xerimental se.up.

A sketch of the experimental installation is given _{in tig. 1.} An electromotor drtveà via a g.ar.box an oscillator which produces

a sinusoids). translation in the vertical pi-ari.. By Unrolling the

vertical. displacement *3-ong the oirouaførenc. of s wheel thi motj..on

is transformed into _{a eiriusoj4al. rotation about the aie of the}

wheel, An oscillating Cradle is suspended from the

xie by m.aa

of a tube with ball.bearing., so that it is free to move nearly fr.ottonlese3.y about it. At its end the axis is connected to the cradle through a etrain..gauge dynamameter _{aieaeuri.rig the extenial} force and hence the moment required for the oscillation of oradla plus tank. The eu,penejÓn relje,eó the dynmomet,r from the wiight of the cradle and tank and from

_{unwanted transverse}

_forces.

The electronic measuring system produceí the I

_{..phaee and}

ninety degrees outc.ot-pkaaeoomponenti of the ainusoidally varying toro.o This meaeurizg techniqu. le adequetii)t described tu

4

The amplitude or oscillation may be varied _{up to about 30} di.-grees. But a singlo amplitud. of 17, _{dere.e (0,3 radIan) caí the} largest angle used Th. rang. of circular frequrntci.. i. riet ltmi ted from any practical point of view. During the _{t..t. this quanti} ty was generally varied from about c» 0,5 radian per second t.

4 ,5 or radians per second.

The tank is basically of a rectangular form with the following dimensione:

length: 10 n.

breadth: 983mm.

deptb

335 mí.

Some modifications wili. b. described _{separately. rh. upper side of} the tank is open,. but a sheet àf plutic prevents _{the 3-Qe. of water}

during o.oillation.

The amount of water in thetank _{was varied with t.p; of 3kg}

3.

Tb. sipl. rectangul,ar tank.

When the tank is oscillated at Increasing frequencies with a certain amount of water and at a certain amplitude of Motten the physical phenomena change greatly during the test. At low frequen.

aise a very long wave ii prea.nt. Thereupon a train of waves of a very abort wavelength appears which in th. beginning ay interfere with the long wave. After these al]. disturbances suddenly th. bore

arises. The component of the moment, which is 90 degree. out-.of-.phae with the motion now increases rapidly. Over a fairly large rang. the phenomenon then does not chang. significantly, although the water motion beoo*ea mòre Violent an large vortices apaar when the dina'

tion of propagation of the travelling wall of water is reversed.

Next, at a sharply fixed frequency, the bon, pasees into a single and steep wave which rune from on. side of the tank to the other. With a small further increase in frequency the liquid approximates

the frosen state and no water transfer is taking place. After the bore has disappeared the moment exerted by th. water falls down rapidly. Th. above ph.nm.na are illustrated very well by the photos

graph.s in fig. 2. Xt ham to be noted that for these pictures a upe-cial tank of somewhat different dimensions was used.

The results of the measurements and the influsno. of the vari-' oua parameters will now be dicusesd If the problem is considered as two-dimensional there are 5 vaniablee

.the otion is deterntin.d by its amplitude (0)and irequ.nay (c» -'th. tank by its breadth (b) and ita position with respect to the

axis of rotatien (a)

.th. amount of liquid in the tank by the waterdepth (b)

It has to be remarked that the results do not invelve the in-fluence of the topdeok, which will be present in an motual tank.

Tig. 3 chews the coordinate axes and the different parameters.

Mot1oz amplitude and frøU.noy.

An example of the measured in-phes. and quadrature

componente

is given in fig. 4., for a large frequency range and an amplitude of oscillation o, 0,10 Ç5,7 degrees). Therefrom amplitude and

-6

phase can b. calculated eeparate].yi ace fig. ¿eb and c. The exparis.n-tal resonance point of ths tank water 90 d.greea) doce not oor'

respond with the theoretioal natural frequsnoy. Th. velocity of the hydraulic jump is o - Th. natural frequency results trot the

condition that twic. the breadth of the tank has been travelled in exactly one period. So:

2b

or:

cot'

Actually the path which has been covered is lesa

than 2b

by an amount depending on the saplitude of Motion and the arm a. The firstaa. tien of the influence of the amplitud. in the natural frequency is given by:

ot

bi2%

gh,

âegatl've below the axis of rotation. A second influence of the am plitud. resulte from the grewing strength of the bore, that i. a

lar-ger waterdepth between the top of the wave and the bottom of the tartk No measurements of the water level durigosci11ation are present, however.

A complete picture of the dep.ndance on the amplitude and fre-quency of motion for one breadth, one position of the tank and one water level is given in the figuree

6..

through 6e. The general t.nd.n cies of these curves are equally applicable to other cases. From fig. 64 it appears that the moment amplitude increases approximately

_a000r

ding to the square root of the amplitude of metion.

Water le!el.

From (2) it le clear that for _{certain tank (a fixed) th. only}

possibility to change the natural frequency will be a change in water depth. Fig. 6 shows that th. component of the exerted moment, which is 90 degrees out-of-phase, le largest at or in the immediate

neigh-bourhood of

the resonance point. Since it la this component which wi].l increase the ship's damping against rolling it will be _evident

that the tank natural frequency in prinøtpl. has to be established at

or near the ship's natural frequency

-

-7-The effect of increasing waterd.ptb te twofold. In the first place the curv, of phase angles versus frequency is ehif ted. Second-ly, the aoa.nt amplitud. inør.as.s because of the larger aa.unt of water in th tank; ese fig. 7s and b. When the variation in

ampli-tud. and phsi. is plotted versus th. ratio h/b, ourv.. lik, fig. 7e and d appear. The results for th. smallest watsrdeptb show slightly different tendencies becaus. of viscous influ.no.s. Th.re appears

to b. approximately a quadratic relation between tus moUnt ampli.. tude and th. ratio . Thu. M a

ank br.adtk.

The influsnos of tank breadth was inv.etigat.d by placing side walls in the tank on a distano. of 3/4 of the maximum breadth. To

oz'ate comparable two-dimensional ph.noa.na with two tank br.adths it is n.c.s.ary to have different waterd.pths, according to the same ratio h/b.

Toraula (E) shows that the natural frequencies in these eases will be a factor - 1, 1 differnt. However, the phase curves

of both tanks coincide it the frsiu.noy basii is reduced to the equivalent nondimensienal. ratio 'u; see fig. 8a sdb.

Moment amplitudes are greatl different. from Zig.

_8e

it can b. seen that the tank breadth is a very important quantity for the magnitud. of the atabilliing moment. If th. moment per unit tank length

iS

sad. nondimensional by ptb3 the parameter M/pgb5l.app.ars

to

b. a good representativ, in the frequency range in which a ber. is present. This nondisensional representation is given in fig. 8d.,

o.ition Qfhetnk,.

When the distance from the tank to the axis of rotation 4e.. oreases the moment amplitudes become larger and.the phase angle. slightly smaller. The results of the measurement. are given in fig.

9. The differ.nce in phase angles is of little importance between shout 60 and 90 degrees, because of th. flat top of the sino curve in this area.

-8-H

-8-The increase in moment amplitud, is considerable, however. Therefore the quadrature component is larger as the poaition of the

tank is higher. Untox.tunate].y it was not possible to placa the tank above the axis of rotation until now so that xperinrnntally

this tendency oan only be established fox' positions beneath the axis.

It.tentatively may be concluded that an anti-rolling tank in a ship best can be placid in a tween deck comartaent, that i. &. high aa praoticabi..

k. ?je4iiicati.n. of the rectangular' tank.

Next to the basic investigation of the rectangular tank several

modification. were considered. They consisted of symmetrical and asymmetrical contraction, with smooth and sharp edges and of a local elevation of the bottom of the tank. The modifications are illustra. ted n fig. 10, In which all particulars are given as well.

In these special designa the problem has lost it. two-dim.n. clonai oharactar and a direct

comparison with

the rectangular

tank

by parameters may offer some difficulties.

It turned out that whithin the accuracy of the measurement. the symmetrical and the asymmetrical contraction A and B were identical;

the reeulta were fairly much different from the rectangular tank, h waver, The contraction C on the other hand was fr a certain amount of water nearly in full quantitativ, agreement with

the

un-modified tank. But it has to be borne in mind that a certain amount of water implies a larger wat.rdepth for the tank with contraction So from this point of view it should have given larger moment.. Medificatien C will not be discussed any furthar.

The measurement. of modification A showed two tend.noi.a which afterwards were noticed in each oasi soa.thing was changed with respect to the basic rectangular form. First].y, the phase curv, Ls shifted to lower frequencies and may in addition be a bit more flat..

-9-Secondly, the amplitude

curve )as

_{much more}

rønounc.d top and

decreases remarkably beyond this top. It was thought that this amplie tude-deoremee might b. produced by the tact that the free surface was considerably smaller than in the cace of the rectangular tank. There-fore modificatien Ewa

triad in an

_{attempt to get a frequency shift} with.ut amplitud, reduction. But th. resulta were of a similar oharmo tir as the measurements of the other modificatiena.

In fig. lia and b the amplitude and phase angle of the counter-acting moment are compared for three cases: th. rectangular tank, modification A and modification E. In all caces the tank oontais the same amount of water, so that the waterd.pthe an, different. This was considered te be a good bacia for comparison in these _three dimensional problemi.

Sino. both amplitude and phaee are influenced the important quadratur. component Msiins.d not b. reduced. In fig. ib

_Meint

is compared for the above three _{cacee. Then it can b. seen that the}

tendencies mentioned above an, valid here as well: a shift to lower frequencies and a reduced top.

In addition to the experimente with the asymmetrical c.ntraotion and a smooth entrance the same tank was oscillated with sharp edges to create high pressure losses; see fig. IO, modification D. It ap-penned that the noticed influences were only atrengthen.d a further shift and flattening of the curve of phase angles and still more

re-duced amplitudes. The resulte of modification D compared to those of modification B and the rectangular tank are given in Zig. 12a, b and o.

To separat. geometrical and pure viscous influences finally _a

packet of wire n.tting

was installed in the rectangular tank, It

ex-tended

ever the full width and length of the tank and to a point well above the water uval, Co that it can b. soseidered es a very large )*omogensaua resistance. Th. measurements of this partieulèr installation az's shown in fig.

13e. Several

ether resulte to compare with are given in fig. 13b. In general it can be said that th. higher the viecouc loases the stronger the phase shift and the amplitud. reduction 'will be.

10

-5. Oonoltitos.

1.. A fr.. surfac. tank brought into oscillation creates e counter-acting moment. Magnitud. and phaas of this moment are depending on a number of paramitere and on the typ. of the installation

tri general s. free surfac. tank offers asny possibilities to be used as a passiv, anti-rolling device.

Tb. seential physical phenominon is the appearance of a bore, a discontinuity in th. water level, travelling to and fra. Thereby

the counteracting moment is created by gravity forces having a

correct phase Lag with respect to the motion of the tank. This is illustrated in fig. 1k, showing schematically the position of the

bore at different points of time during one period of rolling for

tank resonance, that is L -90 degrees. Viscosity effects

influen-ce both amplitud, and phase of th. stabilizing moment. The vis.-cous resistanc, ha. the effect that the phase

curve i. shifted to

lower fr.qu.ncies. Besides that the d.vslopment of the bor is

hampered, its .tr.ngth is reduced

and

thereby the moment amplitu-d.. are considerably isis, a. already noted by Troudet5).

As long as pure rolling is considered the active aom.nt created

by the water

transfer in th. tank never

changea into an .xsiting

moment instead of a

stabilizing moment..

Until now it is unknown in which way other

motions influence the behaviour of th. tank and whether this may give rise to an in.. crease of rolling uñd.r sp.oial condition, or not. Tb. investigai., tion in the Deift Shipbuilding Laboratory will b. continued with

special r.gard to thes.

coupling

effects with away and yaw. k. Moment amplitudes do not increase linearly with rolling angle,

so that th. stabilizing action will certainly become less when rolling increases. This nonlinear tank performance makss

calcul.-tions ot the stabilizing ff.at difficult. Tor harmonio mocalcul.-tions

a staple husar approximation is not permissible and for irr.gu.

ia.r motions the principle of

superposition is riot valid.

Th. experimente. shoW that the Uaz'tatjon Of the moment

_ompl1tude

per u&it tank length atitfiO8 approxirt;'tbe toUowig

r.3.a-t ion.:

M#%b3'Jï

V'

(3)

for one po.iti.nof the tank and for c.

The simple rectangular tank was superior to every modification investigated as to the maximum measured moment. So if such a

tank

can be

tuned sufficiently to the required frequ.ney er

frequencies

by adjusting the wmterd.pth it will probably hav, a greater etabi..

using effect

than any

ether specific tank design. Th. frequency

rangà in whith s

bore is

neticed and in which th. quadrature oes-panent of the moment is

large

is by frit' wide enough to c.ver the narrew resonance peak in ship's rolling for a certain leaded er ballast oendition. Nevertheles, tri lome cases it may be et i*per's tance to widen this range still more to cover a range f ship's

natural frequencies witbeut any attention to b. paid teth. tank.

Tr thi. purpose

special designs may serve.

Phase angles are

reaching 90

degrees above the theoretical

resanas

ce peint. The largest amplitude. are

obtained belew, however, after which the amplitude curve rapidly Zallo to sere.

Therefore

it is not very useful to strive after a phase

angl. of abeut 90

degree., th. more so u the eins of 6k,2 degrees, is already _0,9.

The frequency at which th. tank xduat hay, its maximum effest has to be equal to or.comswhat lower _{thanQt. Thiptmeans that the}

natural period of the tank must be smaller than the

rolling period of the ship.

Tank width i. a very important parameter for the moment'. ampli-tude. Tor th. su. ratte h/b the moment ioeasea with about b3.

Therefore it is preferable to us. the full breadth of the ship. for an anti-r.11ing t;nk and te place it high amidships.

Apart from con.id.ratian. abeut the tank breadth the tank can

beat

b. located high in the ship with r..p.ot to the parateter a; if practicable above the axis of Ietation.

12

-REFERENCES.

i _{JR. Chadwick and K. Kiotter.}

"On the dynamioa of anti-rolling tanks".

Schiffatechnik, february 1954, 8. Heft.

A.M..3jzuije.

"Wawes in an open oscillating tank". Sngine.ring, Vol. 151, 1941, _{p. 224-226.}

E.W. Graham.

"Tb. force. produced by fusi oscillation in a rectangular tank". Doug]ae Aircraft Coapany, report ne. 3M-13748, aprii 1951.

4 H.J. Zund.rdorp and M. Buitenhek..

"Oscillator techniques at the Shipbuilding

Laboratory".

Report no. 111 of the Shipbuilding Laboratory of the Technological University, Duft.

5. P. Watte,

"On a

methed

_{of reducing the rolling of ships at sea".} T.I.N.A. 1883, p. 15.

6 . P. Watts.

"Th. use of water chambr. for reduoing the rolling

of ships

at sea".

Etectrxc atratn indicator Corna, ç(lf er, _Rexotver

1ml

tbtor Gear box

Modulated carrier Carrier Jij Strdu geige dynomometer

PrincipLe of experimentaL set-up.

Fig.1.

v..

lnplose cnent O.drdtwe oçerd Amphi len

Demodulator

ILustration of phenoniena for different frequencies.

Fig.2

w .1.00 w.Ug

water hartiontat email wucht,.

w.1.50 w.2.00

begin of the_bere fuRy dev.iop.d..bòr.

w.3.00 w .4_00

successive positions of the bore Figi (continued)

lank

Fig.3

a>O f tank 'bottom Is above the axis of rotation.

a<O if tank bottom is beneath the axis

1O 210 3o

I

kgm/m ₂₀ 15 10 5 Ql deees -180 C -90 au-020m b 09O3m a O10 ('Jot o 020m b0S83m hIb0fl

aio

J 10 2OE W ..c.' W 5IC' 3 OE

Example of moment amp(itude as a function of frequency.

FigJ.b.

Example of phase angles as a function of frequency.

Fig 4c.

LOE

The resonance frequency(E-90°)as a fUnction of the amplitude of motion.

Fig.5

Example of measuring resuLts for different amplitudes of motion

Fig. Sa.

kgny4s 20 15 lo degreeS -180 -90 -020m 0983m tyb.ao1 a,.00333 1. Q.Ó0667 .o a.Oio D a.O5 cia. 0.20 a.03O (i.) ,.c'

Moment amplitudes for different amplitudes, of motion

Fig.6 b.

Lo H

lo 2fJ 3.0

(ii sec"

Phase angles for different amplitudes of motion.

Fig. 6c.

kgmm 15 10 5 0 deees -90 C -.45 020m b 0983m h/b 01

Moment amplitudes as a function of the amplitude of motión forW=.W0t

Fi:g. 6d.

01 0.2 03

a0

Phase angles as a function of the ampLitude of motion for

Fig.6e.

kgm,m 20, 15 lo o a-ûIm bQga3m aO W secl

ExampLe of moment ampLitudes for different water depths Fig7a.

ExampLe of phase angLes for different water depths.

Fig. 7b

10 20 30 'o

kgnMi 15 M 10 5 degroes

180

s-O2Om bOgg3rn

1O

o-Moment ampLitudes as a function of the ratio water depth ¡tank breadth for W=Wot

Fig.7c. a-Qn bO983m aoo10 o hlb

Phase angles as a function of the ratio water depth/tank breadth torWW0t

Fig.7d. 0.04 hlb 0.08 0.12

C 90

t

0.06, 0.08 0.12

d.gr.e 180 E 90 '1 kgr/m 20 M

Ï

151 10 5 o 1.0 lOE 2.0 W ec1

Phase angLes for two tank widths.

Fig.8a.

20

W ,.ci

3.0

30

Moment ampLitudes for two tank widths. Fig8c. 40 ¿O a-O.2Om ao10 o boge3rn tb.O1 b.$xO983 4bOO&1 o bO993m I4bO62 b3jtQ9e3 Wb.0.062 V V

-,,

i

j

/

J

/

f

/

1-/

H a-O2Om aaio o b9B3m b4'tO9B3 o b983in b34xO983 -h/bOOl.1 h/b.001 bOOB2 FO.O62 V

-'r,

«a

-"z

d.gre.s -180 M pg bk o 0.020: 0.015 0.010 0.005

Phase angles for two tank widths on a nondirnensionat frequency basis.

Fig.8b.

05 10

(*3/Wo_t

Moment ampUtudè coefficients for two tank widths on a nondimensionat frequency basis.

Fig.8d. a. -020m 010 o b.O93m b.3/X09e3 o b0983m n b. $ x09B3 I .OE01 tbc0! .G062 4b.0ß52I

H

/

lu

/

e o H a. -020m OElO o b.OE983m b.$K0903 o b.0993m ba4xOSB3 I.0OI.l .O0l.l IIjb.0OG2 i-0O62

-H

,t

\

o

r

H..

\

05 10 15 20 15 20 C 90 t

M 10 5 o degrei -180

t

.-0J.OE 0 bO983m

-I

s

a003fl + UftO7 O ao.10 o a a02o -030 -020

Moment ampLitudes as a function of tank distance to the. axis of rotation for W:Wt

F 1g 9a

-OElO 0

m...___a

Phase angles as a function of tank distance to the axis of rotation for WW0t

Fig9 b. bG83m h.1biOil a00333 +9Oß567 O aO.1O o aOi5 a a02O -01.0 -0.30 -0.20 -010 o a kgrn/m 20t 15

b..0903 i6aou 0 aOm

a,lô

O a-O2Om aO1O e a-LOm U.1O (i) sec.'

uadrature components as a function of tank distance to the axis of rotation forW

Fig. 9 c. kgm14n - 20 -15 10 MainE 5 iOE 2ß 30 40

e u, b.983 983 modification E

wmwm

a'75 875 Fig. 10 983 u, r-In r-C u, e u, 983 ALL, dimensions in mm u, r-u, I-L J ¿91.5 91.5 modification A modification B 737 ¿19.5 modification C modification D

kgm Z0 degrtes -180 C -90

15

t0 0 a-0.20m b 0983m

tank citits:6kg water

a0.l0 o rectangutartaiik O rnodificatø A U ITwticatan E U- _o o o

\

\

Phase shift for modification A' and E.

Fig.ilb. a-0.20m

b-983m

tank contents:6kg watN a__Ow O rtctan9jtar tank moficatnnA U rTficati E

I

-0/

j

to. 2.OE 30 1e0

w

AmpLitude reduction, for modification A and E

Fig11a. 1.0 20' 30 Lo w, M I 0.5

kgm 20 MsinC is! o aOm b. ogaim tvcoMints:6kg water ;.IO o t mocationA

i

1ktiE

/

/

/

/

I

/

/

I o i OE 2.0 w 30

OEuadrature components for modification A and E compared to the rectanguLar tank. Fig:11C.

20 15 de -180 0' O2Qm bO993m !rd ctents:6kg wt aoO.iO O ,ectarutar tiid A moficitmn:B £ tficati D £ Fig12b. a-O2Om bQ983m táctmts. 6kg wat a.O.IO O tçArtár A rvdlfkitIB' - r,òflcb D

-A

.'--

A-__l._ O r

,-20 30 ¿0 w

AmpLitude reduction for modification B and D. Fig .12 à.

10 20 30'

J

w

Phase. sh!ft for modification 'B and D

't

OEs

E .- 90

- 2ß, MsinE -15 -1.0 -05 aO7Om bO983m tan fikg.wate O rtan9Ax tank A. nficatlanB A mOfICatII,D 10 2.0 O 3OE

Quadrature components for modification B and D compared to the rectanguLar tank.

Fi g.12c.

k 100 ManE kgn -2. MainE -05 o 7.5 25

OEuadrature components for therectangular tank with_{a homogeneous resistance.}

Figl3a.

w,

Qomdrature components for several modifications compared to the rectangular tank.

Figl3b. hIS .010 5.-020.,. 0.0013.,. hIb.0201 O 50th n.t.9A 50th ..ith B, nStn.,D n, .di.,, B n..in.,.

o

lì

r 10 _W w 30 4.0 1.0 2.0 W

p

3, wt.1q2 4'. 0 mom zero 6 wt.51t p.-7ip.. 4'.neg. 'mompo.. 9 wt.2i p.0 4'.mao.pos. morn mao.o.g.

'Position 'of bore during one period of rotting 'at tank resonance(C.-90°);vlew in positive x-direction.

F 1g .16.

wtào 2 _wt.i?4

'pio pO7p

ij.maxpos. b.pos.

mom ma oflag. mom.aeg.

4 wt.31T/4 5 wt.n

P.O

.fleg. 4'. maxneg.

mom pos. mom mdx pos.

7 Wt.31T/2 8 wt'.7tyt

'p. -'p. p. -?po

4' .pot