Experimental determination of damping added mass and added mass moment of inertia of a ship model

REPORT No. 25S

October 1957

STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE

AFDELING SCHEEPSBOUIV - PROF. MEKELWEG - DELFT

(NETHERLANDS' RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION

(SHiPBUILDING DEPARTMENT - PROF. MEKELW'EG - DELFT.)

*

EXPERIMENTAL DETERMINATION OF

DAMPING, ADDED MASS AND ADDED MASS

MOMENT OF INERTIA OF A SHIPMODEL

by

Jr. J. GERRITSMA

ADDED MASS AND ADDED MASS MOMENT OF. INERTIA

OF A SHIPMODEL

by

Ir. J. GERRITSMA

Publication No. 8, Delfl Shipbuilding Laboratory

Swmpsis

The damping, added mass and mass momentof nertia of a shipmodcl, performing forced heaving or pitching

Oscillations in calm water, are determined cxperimentally.

Th _{influence of model spccd, amplitude and frequency of th motions on these quantities has bcii sttided.} Tb2 coupling terms in the casc of a combined heaving and ptching motion are considcrcd for one model pced.

Some of the experimental results are corn pa red w t Ii calculated valucs, vhic ii hav2 bcen publish2d rccentl s

I,,troi! /1db 1/

Our knowledge of the hydrodynamic forces and moments acting on a ship in waves, based on

ex-periments, is rather small.

Concerning the heaving antI pi telling motions,

which will be discussed in this publication, some experimental data on damping and mass as a func-tion of frequency have been given by Haskind and

Rieman

[1]

and by Golovato [2]. In both cases a

mathematical shipform was

forced to perform

heaving motions in calm water, to obtain the neces-sary information.

Theoretical in vestigations on damping and added

mass have contributed only a few practical results in the low frequency range which is of interest for

ship motions.

1 he calculation of added mass in the high fre-quency range (ships vibrations), provides less dif-fic tilt i es.

Free surface effects arc negligible in this case and

the

resulting boundary condition

facilitates the

calculation of hydrodynamic effects considerably.

Such a boundary condition does not hold in the low

frequency range, which complicates the theory.

The influence of free surface effects on damping

and added mass, prevents an extrapolation of ii gIl

frequency values to the low frequency range.

Con-sequently, the calculation of the coefficients of the differential equations, describing the heaving and

pitching motions, is usually based on simplified ;ind rather coarse assumotions [3, 4].

So! vi ng the thus fou 11(1 differential equations for

amplitudes and phase lags, fairly large differences

Earnetimes are found between calculated and mea-sured ship motions.

To investigate the origin of these differences, it

scems necessary to comoare each of the calculated

coefficients of the equations with measured values,

as each of them contributes to the amplitude and

phase of die resultitig motion.

in addition to tile comparison of calculated and

measured shipmotions, tile knowledge of tile

mag-nitudes of the various coefficients, and the

in-fluence of hull line modifications on these

coeffi-cients seems interesting.

When sufficient knowledge of this influence is available it will perhaps be possible to predict the

effect oii shii,motions of certa in ii till Ii tie nlodi fica-tions with more accuracy.

This would reduce the number of model tests in

waves, when the determination of an optimum per-formance in rough water is the ultimate goal.

Tile data oresented in this article do not hive the pretention to be complete in this respect; they are coil fined to one shinform, namely the Todd

60-series parentform with a blockcoefficient C, .60. Also tile exciting forces and moments due to the waves are not considered. It is intended to determine

the exciting forces 2nd moments of this model in the Ilear future and to complete tile research with

two models of tile same series with blockcoeflicients

C, .70 and C .80.

Measit I/nc ,i,iI/otIs

There are a number of methods to excite a ship-model ill calm water, which have been published

recently.

They will be discussed here briefly.

First, tile work of Haskind en Rieman _{[I ]} must

be mentioned. In their paper a method to obtain

damping and added mass for the forced heaving

motion has been given, as well as tile experimental

results for a mathematical shipform at zero speed. A method to determine the damping and added

mass moment of inertia for the pure pitching motion

and the coefficients of tile so called cross coupling

terms for a combined heaving and pitciling fllOtiOil

is also indicated, but no experinientai results have

DOUBLE GUIDES ON BALL BEARINGS

SPRING

GUIDE

PURE HEAVING MOTION

HEAVE POT. METER

A

PIVOT

/

PITCH POT. METER

PURE PITCHING MOTION

SPRING B

GUIDE

PITCH POT.METER

COMBINED PITCHING AND HEAVING MOTION

C

Iig. I it rrafl genie,,! oJ Is!

'2a. i9

1,

_{/ 24Q}

A sketch of the test arrangement is shown in

Fig. 1.

Fig I a shows the arrangement in the case of a pure heaving motion; the model is free to perform heaving oscillations, but is restrained for pitch by

a vertically guided rod.

The exciter consists of a spring connected to

the model via the guided rod, whereas on the other end of the spring a harmonic motion is applied by means of an eccentric.

Amplitude of motion, phase lag between the

motion of the eccentric and that of the model, as

well as the frequency of the excitation are measured.

This gives the necessary information to compute

damping and added mass, as will be seen later.

A modification of this method is possible with the use of an electronical dynamometer instead of the spring, as proposed by Sr. Denis { _].

The elccrronical dynamomerer can be regarded as a spring with a high spring constant.

Golovato

[2]

used a six component strain gage

dynamometcr to study the hydrodynamic forces

and moments of a mathematical model performing

pure heaving motions. The six component dynamo-meter was used in this case to measure the moments

due to heave as well (coupling terms).

His tests include a variation in model speed. Finally Grim [ I used eccentric rotating weights

to excite the model. On each end of the model

a

rotating weight

is located; the weights were

cou pled by the driving mechanism, but t heir phase differed 180 degrees to obtain a sinusoidally

vary-ing moment.

Each of these methods has its own advantages and

disadvantages.

The electronical dynamometers appear to be very

sensitive with respect to vibrations of the towing carriage, which leads to a certain amount of noise

on the registration and necessitates hand fairing.

Particularly when measuring phase angles, inaccu-racies may occur in this way.

An advantage of the method is the fact that the

amplitudes of model motion and exciter are almost

equal, which simplifies the study of non lineair

phenomena.

The system of the rotating weights, of which

some rnodi fications are know ii, has the ci isad vantage

that the exciter is driven by a motor which is placed in the model.

Due to the model motions, it is sometimes

diffi-cult to obtain a pure sinusoidal excitation, as the

size and therefore the power of the motor is limited

by the dimensions of the model; moreover, the

exciting moment is proportional to the period

squared and this necessitates relatively large weights

at low frequencies to get reasonable shipmotions. The system using an electronical dynaniometer

as well as the one using rotating weights have been

tried in the Dcl ft Sluiihuilding Lahuraory, hut the

just mentioned disadvantages led to the preference

of the system with a spring exciter (as shown in

Fig. 1).

The test arrangement for pure pitching motion

is analogous to that for jnire heave and is shown in

Fig. Ib; the arrangement for the combined heaving and pitching motion is given ill Fig. ic,

In each case, the motions are measured electron-ically via microfriction potentiometers, whereas the

phase lag of tile model is defined with respect to a

certain position of the eccentric.

Motion, phase and an accurate time base are re-corded on a "Sefrarn" pen recorder, capable of 20

cm per second paper speed.

We will now proceed with the derivation of the

analytical expressions which are used to determine the various coefficients of the differential equations from tile measurements.

For the pure heaving motion, the following well known differential equation is valid (see Fig. Ia)

lIZ

+

hz + cz

FehV (1) where: a

ii

/?

C =

'S1

+

k.

= kr

mass of model added mass waterplane area

specific gravity of fluid spring constant of exciter vertical displacement of model

The unknown coefficients a and b (mass and

damping) are solved after substitution of the

solu-tion:

= z0

,I(oI

--Separating the real and imaginary parts, one finds:

kr

C - -

cos a zo 2 (2) (1) and: b

= - sin (I

(liZ11 k r (3)

Ihe amplitude of heave zo, the phase lag be-tween tile motion of tile eccentric and of the model,

and the circular frequency II are to be measured,

which allows a and I) to be calculated.

A sinlihar procedure holds for tile pitching motion (see Fig. 1 b). Tile model is

free to pitch but is

restrained for heave.

The pivot is located ill

the models' centre of

gra v t y and the spring of the exciter is connected

to the model at a distance 1 from the pivot. The differential equation for this case is given by:

A'1' + B'1' + C'j'

Mt"t (4) where:

A = I + ,v

C = ' K + k 122

M = k. lr

in which:

I mass moment of inertia of the model

ii,, added mass moment of inertia

K moment of inertia of waterplane

y pitch angle The solution will be:

i

and the values of A and B (respectively inertia and damping): C 1, r cosfl

A =

1Iii₍₎₂ (5) B L sin fi (6)

Pitch amplitude i/(J, phase lag / and circular fre-quency '' are to be measured so that A and B can be determined.

Finally, for the case of combined heaving and

pitching motion, we have the following coupled

differential equations:

cl

+ bz +

CZ + (I'/' + ("I' + '/'

A '' + B; + C', ± Dz + Fz + Gz = Me""'

(7)

a, A, h, B, F and M are already defined,

whereas

ci, D, c' and E are the coefficients of the coupling

terms;

further:

C = ;' S, + I

C = ' l2 S + k 1- + ' K

= G = ;'/, S.,+ k 1.,

where 1, = horizontal distance between the models'

centre of gravity acid the centre of the waterplane area.

The solution of (7) is given by:

= z c'"'

,,,

Substituting these expressions, the coefficients of the coupling terms can be found:

(z1/r) ( acn + c) sin F

(z 'r) boi COS F

1' + .6. SW

+

_011/'i)!?

+

-f-+

_or (8)

where ' and arc the phase lags and

;' - &

For a cheek F = "

F can be substituted

to show the symmetry in the expressions for c and F, a' and D.

Here again, the phase lags ;' and , the amplitudes

'j

and z, and the circular frequency ii can he

measured.

Assuming a, A, I, and B to be the same as in case

I a and I b, the coefficients of the coupling terms

can be determined.

Test mogra Hi

The model is of the Todd 60-series parentform

with a blockcoefficient C .60.

The main oarticulars are as follows:

Length (bp)

L 2.438m

B read t ft B 0.325 nm

Displacement

L=61.9

kg

Area of waterline plane So= 0.561 rn2

Mass moment of inertia of

the model I 2.25 kgmsec

Moment of inertia of

waterline plane K 0.170m1

The pure heaving and pure pitching tests have

been carried out at four model speeds: FR = .15, .20, .25 and .30 (FR. is the dimensionless Froude number)

Tests at zero snecd and at FR .07 S were tried

but these did not give reliable results because of

serious tank-wall effects.

At each of the four speeds the frequency is varied

between 0) 3 and oc = 13, and the amplitude of

ti-ic eccentric between I

and 4 or

cm. At low

frequencies tile waves generated b the model mo-tions may travel faster than the model itself.

This effect combined with that of reflections

from the tank walls gives a large scatter of the

experimental Doints in the range cc < 3 (lower

for high speed and higher for low speeds of advance). Frequencies higher than cc

± 13 are not

im-portant from a practical point of view although

tile)' are of interest for theoretical work.

lit hiis range however the inertia forces ire \cr

Vfl'. (' z0/r

( y,,/r) ( _A,t -I- C) cos F ( ', r) B" Sin F

+

('i z111r

k 1, cos ;' -I--- G r

(z,1/r) ( a"r -I- c) COS F --f- (

r) b' sin i

₊

(,)

.6. cos /

-1- gy r

,.;:!,/

/-- (

c'0/r) ( --fln-l-C) sin ('i '

r)B' 'c cos

(I) r

+ k 1

sin '

(I

large compared with damping forces and therefore a determination of the damping cannot be

accom-plished here with reasonable accuracy.

The combined heaving and pitching motion is

studied for one speed only, viz.: FR .20. This test covered the same frequency and ampli-tude range as mentioned above.

Fest results

Fig. 2 shows the test results for the pure heaving and pure pitching motion at FR. .20.

On a base of frequency w, the phase lags _{and /,}

and the amplitudes of motion divided by the am-pl it ude of the eccentric r : z/r and j '/r, are shown. In the tables I

and 2 the ordinates of the thus

found curves are summarized for all the speeds

con-sidered.

TABLE I

A iupliliiili' aiiil Jil.)ase for pure bc'a vi u in olwii

o is given in degrees

TABLE 2

A iii />li1ol(' (1/1(1 Jibase for pure J)ite/iiui& motion

is given in dcgrccs per cm; fi in degrees

Fig. 3 shows the phase lag and motion amplitude

curves for the combined heaving and pitching

motion.

The coefficients a, b, A and B are calculated,

using the data of tables I and 2 and the formulae

(2), (3), (5) and (6); the result is shown in the

Figs. 4 and 5.

For an easy conversion to other model scales, the

various diagrams are provided with dimensionless scales; the dimensionless damping coefficients for

heave and pitch are respectively:

h ./ gL

Instead of the quantity a in -I- -, the values

of ii and u/nz are shown in Fig. 4.

and B \/ gL

AL

2.4 5.3 9.0 12.8 17.4 25.6 44 95 44 62 72 FR = .15 FR = .20 FR = .2 a FR = .30 a a z0/r a 0.519 2.0 0.527 2.5 0.518 2.8 0.512 3.3 0.539 5.4 0.553 6.4 0.548 6.6 0.539 7.3 0.568 10.1 0.592 11.1 0.592 11.2 0.576 11.7 0.624 15.3 0.654 15.6 0.666 15.0 0.632 15.5 0.722 19.0 0.759 18.6 0.770 18.4 0.721 18.6 0.894 22.5 0.937 21.6 0.919 22.4 0.902 23.0 1.226 30 1.267 29 1.192 30 _1.250 32 2.318 49 2.361 47 2.150 49 2.163 50 3.166 135 3.198 129 2.890 128 2.946 132 1.335 166 1.367 166 _1.306 166 1.261 169 0.774 177 0.764 177 0.753 176 _0.705 180 = .15 FR = .20 FR = .25 FR -: fi 1.7 0.244 1.7 0.243 2.1 0.2 So 4.8 0.261 4.4 0.260 6.1 0.265 9.2 0.288 8.2 0.284 9.9 0.291 14.5 0.333 13.0 0.327 13.5 0.330 20.5 0.406 18.5 0.398 18.0 0.395 29 0.544 27.3 0.523 25.9 0.517 44 0.807 45 0.799 44 0.856 86 1.149 88 1.151 92 1.227 140 0.773 140 0.755 141 0.703 163 0.448 159 0.446 162 0.414 172 0.289 169 0.285 172 0.285 (I) F i1 0.241 3 0.258 4 0.283 5 0.3 33 6 0.409 7 0.528 8 0.742 9 1.138 10 0.800 11 _0.443 12 0.2 92 V.) 2 3 4 5 6 7 8 9 10 11 12

35-. 30-. 2.5_ 15_ 05 0 0 _150° 120° 4 -J 4

I

0. 90°

-

60° 30° 0 6 B

lig. 2. iea.curel Jib,, of

''''1''''

iii Jihie for Jitive /,l1,i,i au/ Ju,i'e /iiIhiiit ,,,,)IIO,,

l.2 U, UJ. 06 04. 02_ I PITCH FR. 020 -r..LQcni a- 20

.- ao

x-4.0 I I A x I I

j

/

I

I

I I A PNASE AMPLITUDE I I _I80 _150° _120° 0 4 -J Ui 0. 90° 4 60° _30° 0 I 0 I 2 I 3 1 4 I 5 I 6 I I

wsc

B 9 I 0 1 T II 12 I 13 I 4 5 4 Is 3 2 10 5 4 3 2

12. 06. 04_ 02 C) 08 06. 0 4 02 PITCH0 HEAVE FR, =020 U- 3.0 0- 50 0 I 2 3 4 5 6 =sc

PITCH AND HEAVE

FR =0.20 £ -r -2Ocm

.-ao

0- 50 - T JO II 2 13 14 15

Fig. 3 ten, ii rd 'nines sf 1 55! 1:/tIm/c and /h, for so so bi,,:/ hen: 'ins ,,,,I pIc'iin owl:,,,

-

900

-

60° - 30° 800 - 150° 0 - 60° 30° 0 240° 0° 180°

-

20° 0 I 2 3

__i,_

-,n__--__

10 II 2 3 14 IS 6 --asc

3 --40 _'I' Vt ....20 /vgL _B A 1% 0

OhllI,l

0 05 10 15 2.0

3-1 05 HEAVE V/kp_.l5 -I / 17

I

_b 0 0 2 3 4 5 6 7 8 (J-_ flC 9 10 II 2 13 0

I0/

15 20

-sL

.

//

---HEAVE

V/v3O

-'-5 -- 9 - 40_ / -\ - - - S 3_ -A

-7-6 .0 0 o 2 3 4 5 6 7

89

0 II 2 3 o as toF IS

'I

2.0 9'tc m 40--- -V/ HEAVE

I23

67S9 0

II 12 13 0 05 10 5 20

005-.

:

I

,/fl

B I0

-3

I Q5 0 I I T 2 3 4 5 6 7 (aJ -SC -8 r 9 0 II 12 3

QSLO

S 20 o o o. 0 15 20 010 I-Q05 F 9 a PITCH V/11 -20 _I.O .25 II

"I

I . 4-.

I'

_N

/

10 0 0 o I 2 3 4 5 . 6 7 r B 9 tO r II 12 3 o 05 10 2.0 0101 F 9

-PITCH S F _1.0 -

\\1

/B

1'

\._

0 0 0 I 2 3 4 5 6 7

5CC

9 10 II 12 3 0 (5

_tOtiy

5 20

Ig. 6. CorfJuii'n Is of cou/ilin /er7sls for eon,Io,,rI be, /s a,,1 JiiIehiii ..J k9sc 008- - 10-0.04- 0.6 - D +1 +1

-0 -0

- d - 1CALCULATED

004- 0.6-

_d:D - 1.0-0.08- 1.4 -I I I 5 I I I I

0

I 2 3 4 5 6 7 8 9 10 II 12 3 (a.) SEC' I I I I I [ I I I I I I j I ¶

o

0.5 .0 1.5 2.0

::

0.6-kqszc 'j 6 -0000 \CALCULATED

0

II I 2 T 3 I 4 5 I I ¶ 6 7 8 (A)

-

SeC I 9 I 0 III I 12 I 13

0

I I I J I 0.5 I I j I I 1 I .5 I I I I J I 2.0 I I I .0

Similarly Fig. S shows ii,

_{ud /'d/I instead of}

A = I +

_ii.

As the quotient i/1 depends on the distribution

of weights in the model (I) and not on

hydrody-namic properties, thc value of I in the denominator is taken as I (0.2SL) in _{instead of the actual I}

of the model.

As a dimensionless frequency parameter (I;

where B breadth of model, is taken.

\Vith the data of Fig. 3, the now known

coeffi-cients a, I,, A and B, and formula (8), the coeffi-cients of the coupling terms e, F, d and D can be

d etc rm in ed.

These ire plotted in Fig. 6 on a base of frequency.

Ihe d llleflSiOIl less scales are foti nd by multiplying

g

I and

D with

A and e, F with

Discns.siu,, (If the results

As mentioned above, the results for very low

frequencies are not reliable. In this range, the waves

generated by the models' own motions may travel

faster

than the model

itself; conseq uently, the

model is not oscillating in calm water, but in her

own waves.

This phenomenon causes a large scatter of the

experimental points, particularly the measured phase

lags are very sensitive in this respect. When the

generated waves travel faster than the model, the

relation between speed of the iiioc1e1 .ind period of the oscillation is given by:

2iV

< ¼

where: V model speed

T = period

as shown by Brard 17].

For the model in question we find for the

limit-ing case (see table 3):

TABLE 3

2. V/'T= '/

AL

At a speed and period given in table 3 the

dis-tortions of die fluid surface do not cover the area

.ahead of 1ic model.

This relatioll is verified experimentally and good

agreement is found. Also reflections from the tank

walls are important;

it is

thought that even at

slightly higher speeds than given in

table 3 (or

higher frequencies) some in fluence exists, but a

quantitative knowledge of the effect is not at hand.

Furthermore the phase lag is small at low

fre-quencies and an error of some degrees affects

con-siderably the accuracy of the coefficients h and B, as can be seen from formulae (3) and (6).

In general it seems advisable to disregard the

surements in the range m < 3 or 4.

The influence of speed on added mass and added

mass moment of inertia is shown in the figures 4

and 5.

For t" > 6 a five ocr cent. variation, with respect to the mean value of a

in +

ii. is present.

Simi-larly, a three per cent variation in the value of

A = I + p, is found.

In both cases this variation due to the speed of

advance is less at higher frequencies.

The damping coefficients also show some

depend-ency on soeed, see Fig. 8. To study the combined

effect of speed on damping, mass and mass moment of inertia, the magnification factors and determined

for the pure heaving and pure pitching motions

(see Fig. 7).

In the case of pure heaving motion, the influence

of speed on the magnification factor is fairly large

for " < 7.

It is remarkable that the curves for FR

.15

and FR .30 and those for FR = .20 and FR .25

coincide.

in the case of pure pitching motion, one curve is

valid for all model speeds, except at resonance. In both cases a mean circular frequency: w 7.05 is found at 90 degrees prase lag, corresponding to a

natural period of T 0.89 seconds for pure heave

and pure pitch; the mass moment of inertia of the model is taken as I = in (0.2 5L).

These periods agree with those found from in-cl ii n g tests; it must be mentioned however that these latter are difficult to measure.

The maximum values of the various terms of the

differential equations give an impression of their

relative magnitudes.

For the case ') 7 and FR .20, table 4 gives

the necessary information, assuming unit displace-men t.

(1 meter of heave and I radian of pitch.)

TABLE 4

P) _{actual value of model}

FR 0.075 0.15 0.20 0.25 0.30 o 6.7 3.3 2.5 2.0 1.7 heave pitch

= 565 kg

= 164 kgm')

216 kg

B"i =

51 kgm C. 561 kg C

= 172 kgm

'--16 kg

D-' =

2 kgm ('I')

=-61 kg

Em =

11 kgm

=-32 kg

G

=-32 kgm

Fig. 7. Ala.ii)i',Iion fiIors JHI 1,11,' /,e,, iut i,,? /?urr /ü/iIiin ,,zoiIi,,i 2

0

HEAVE

Fr.I5

//

//

/

/

/

'.. PHASE

N

800 0 50 20° 90°

-

I

a. 3d 0 Fr.2O Fr.25 Fr.3O .

___--

-.-/

/

0

2 3 4 5 6 8 9 U..) _- sec

0

II 12 0 I I 0.5 .0 1.5 2.0

o

z

0

I-(.9

0

PITCH

₁₂ Fr=f2

-

180° I50 _120°

_90°

dD

Fr.30_._

_//

/

. 4.PHASE

//

/

/1/

//

0

2

34

5 U) 6 7 B 9

1011

12

0

0.5I I I j .0 - .5 I I I 2.0

/,f,iiI!JI,II.l UU/ lii l'/' /lP/'/ 1 /I,.J 11l I' I Jo i,tiii,iI llII) I o. I ci 0I so

0

i I I I I i i I LI I II 01 6

'8

Ths,..'

f)

L 9 S V £ 1

0

\.

E 9

-L

- 6

500

- -010

\

0 0 0 F

/

/

OE J 0 d SI-'.d 00 000 000

N

/

1

-

H.LId

013 SI 01 50

0

I I i I i i i j I j LI I II 01 I - 6

'9

I DS C) S I V £ I

0

I 9

_----

L I N

\<\aiv1nD1vD

:E1

0 w $ OS -

-

0Ed

0d

''-'

/

0 -

Sikd

AVH

D

It may be concluded from table 4 that generally,

the coupling terms are not negligible.

Particularly their influence on phase lags may

be important as mentioned by Korvin-Kroukovsky [4].

Haskind derived from theoretical work the

fol-lowing relations between the coefficients of the

coupling terms of a symmetrical shipform (see also

[1]):

cE=d=D0

for V=0

and:

.- ti,

1 for

V / 0

The last mentioned relation is confirmed qualita-tively by our experiments for a' and 1) although the actual model form has some asymmetry (see Fig. 6).

Finally, the linearity of the motions seems well

established for moderate amplitudes of motion (see

Figs. 2 and 3).

In any case, the use of the linear differential

equations may be regarded as a good approximation. The coefficients, found experimentally, are

com-parable with those calculated by

Korvin-Krou-kovsky [4].

In [4] a comprehensive account of his calcula-tions is given for a 5 ft. model of the same lines.

As for the details the reader may be referred to the paper concerned. The method used for the cal-culation of added mass is based principally on the

well known work of Lewis, whereas the calculation of the velocity dependant terms is based on work by

Holstein and Havelock.

The results are summarized below:

--I/ L

=

l _.c.

sI1

-

111 L + 1/2L u J Sk9Ed - 2/2 L - 1/ L

1D=k1

L'

i

SI.''SI - /2L

S the sectional area at a distance from

mid-ship section;

= the coefficient of accession to inertia

for

this section and

k a coefficient for free surface effects.

The resulting dimensionless parameters are:

Dg

AL

AL

- 0.02!

ii. ni - 0.85,

/1,/i

0.75 and

These values may be compared with the

experi-mental results in Figs. 4, 5 and 6.

The calculated "/n' agrees fairly well with the

average experimental value, except at low

fre-q uencics.

This

is not the case with the calculated "

I

which apparently is too large ,w hercas the calculated coefficients ii

and D are only comparable when

only the order of magnitude of the absolute values is considered. The coefficients of the velocity dependant terms, Ii, B, e and F are calculated with

the following formulae, using the well known strip theory: -I- I/ L - - I: i + / L .1

N()I

- '/ L -4- iI I, e

F =

.1

N()Sd

-

2/2L

where N () is the damping force due to unit

ver-tical

velocity. The expression for N ()

is given

in [4].

Good agreement is found for damping in heave

(sec Fig. 8) except at high frequencies.

The damping in pitch shows significant differ-ences between calculation and experiment (see Fig. 8).

In general a qualitative agreement with

Have-lock's calculation for a submerged spheroid is

found [8].

Havelock determined the damping in heave and pitch for a submerged spheroid with the aid of a two dimensional strip theory and with a three

di-mensional theory.

It was found that the strip theory is valid for

heave in the practical range of frequencies, whereas for pitch a correction factor is needed which is < 1 for low frequencies and > I for higher frequencies.

Finally the calculated values of e and £ arc also given in Fig. 6. Here again the order of magnitude

is correct hut no fu rtller ag recmcii t is fou n ci

Lis/ of niuli, s1',,,hoI

hreadth of model

C,, blockcocfficicnt

jJ( = Froude number = V./ gL = acceleration of gravity

I = miss moment of inertia of model K = inertia of waterline plane

= spring constant

L length of model

in = tiass of model

r amplitude of eccentric

S1, area of waterline plane

where k, = 0.75

where

k, - 1.20

1 period of excitation

V = spccd of model = displacement of heave

amplitude of heave

a, _fi, ,', = phase lags

= displacement of pitch

lash toil, M. I). a rid Riema,,, I. S.: "A met hod of deter ni ii rig

pitching and heaving characteristics of a ship". Bulletin de

lAcadinije des Sciences tIe URSS, Classe des Sciences

tech-niques, 1946, no. 10 (translation Russian-1)utcl, by ir. G.

Vossers)

2 c;olorrilo, P.: "A study of tIre forces and moments on a surface

ship performing heaving oscillations". 1916, 1'.M.B.,

report nr. 1074.

3, lVeinblu in, G. and SI. De,,,,ji, M. "On tIi niot ions of sIr ips at

sea". S.N.A.M.E.. 950.

Korsin - K rook ossky, Il. V.: "In vestigat ion of sIr p iirtt ions in

regular waves". S.N.A.M.K., 1911.

5/. Denii, 'ii. and Piers,,,,, W'. I.: "On t lie iiiotsons of ships in confused seas". S.N.A.M.E., 1913.

Grin,, 0.: " Berec liii u ug de r do rcls Sc hw i rig u ,,gcn ci nes Schi ffs

-korpers crzeugi er I, ydrod yrrr,,iiscIrcn K r.i ftc'', .S.T.G.,

1913.

l3raru!, l(.: 'Introduction i l'tudc th,eorlquc dir t.sngage en

niarche". AIMA, 1949.

llarIork, 7'. 11.: "'YIie dau,,isu,g of Irc.ive and pitch; a

conipiri-son of t wo-d iniensio,i.r I .r rid ti rev-dimensional calculations''.

T.1.N.A., 1916.

Ref ereizees

'Iii amplitude of pitch

H. = added mass

= added mass moment of inertia

= displacement of model

2r

(I) = circular frequency