RcoLTS -P 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

Isjieed by the Council

EXPERIMENTAL DETERMINATION OF DAMPING 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 moment of 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 publish 2d rccentl s I,,troi! /1db

1/

Our knowledge of the hydrodynamic forces and moments acting on a ship in waves, based on experiments, 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 function 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 necessary 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 frequency range (ships vibrations), provides less diffic 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. Consequently, 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 measured 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 result itig motion.

in addition to tile comparison of calculated and measured shipmotions, tile knowledge of tile mag-

nitudes of the various coefficients, and the influence of hull line modifications on these coefficients 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 tions with more accuracy.

ii till Ii tie nlodi fica-

This would reduce the number of model tests in waves, when the determination of an optimum performance 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 60series 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 shipmodel 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 been given for these two cases.

4

A SPRING

GUIDE

DOUBLE GUIDES ON BALL BEARINGS

HEAVE POT. METER

PURE HEAVING MOTION B

GUIDE

/

PIVOT

SPRING

PITCH POT. METER

PURE PITCHING MOTION C

PITCH POT.METER

COMBINED PITCHING AND HEAVING MOTION Iig.

I

it rrafl genie,,! oJ Is!

'2a. i9

1,/ 24Q -

oc- o'..?J

5

A sketch of the test arrangement is shown in

just mentioned disadvantages led to the preference

Fig. 1. Fig I a shows the arrangement in the case of a

of the system with a spring exciter (as shown in

pure heaving motion; the model is free to perform heaving oscillations, but is restrained for pitch by a vertically guided rod.

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 electronically 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 recorded 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)

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

Fig. 1).

dynamometcr to study the hydrodynamic forces and moments of a mathematical model performing pure heaving motions. The six component dynamometer was used in this case to measure the moments

lIZ

to excite the model. On each end of the model rotating weight is located; the weights were

cou pled by the driving mechanism, but t heir phase differed 180 degrees to obtain a sinusoidally varying moment. Each of these methods has its own advantages and

ii

C=

'S1

/?

+ k.

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 solution:

= z0

C - kr - cos a zo

phenomena.

The system of the rotating weights, of which 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

,I(oI --

Separating the real and imaginary parts, one finds:

equal, which simplifies the study of non lineair some rnodi fications are know ii, has the ci isad vantage

(1)

mass of model added mass waterplane area

a

Particularly when measuring phase angles, inaccuracies may occur in this way. An advantage of the method is the fact that the amplitudes of model motion and exciter are almost

a

= kr

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

sensitive with respect to vibrations of the towing carriage, which leads to a certain amount of noise on the registration and necessitates hand fairing.

FehV

where:

due to heave as well (coupling terms).

disadvantages. The electronical dynamometers appear to be very

+ hz + cz

(1) 2

(2)

and: b

r = -k sin (I (liZ11

Ihe amplitude of heave zo, the phase lag

(3) be-

size and therefore the power of the motor is limited

tween tile motion of tile eccentric and of the model,

by the dimensions of the model; moreover, the exciting moment is proportional to the period

and the circular frequency II are to be measured,

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

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

6

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

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

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

(I

(,)

-1- gy r /- ( c'0/r) ( --fln-l-C) sin ('i ' -f-

where: E

C = ' K + k 122

r sin

+k1 (' z0/r

in which:

+

1, r

A=

cosfl

k 1, cos ;'

-I--- G

+

r

or

;' - &

arc the phase lags and

" F can be substituted For a cheek F = to show the symmetry in the expressions for c and F, a' and D. Here again, the phase lags ;' and , the amplitudes 'j

()2

', r) B" Sin F

(8) where ' and

i

1Iii

(

('i z111r

and the values of A and B (respectively inertia and damping): C

r)B' 'c cos

'

( y,,/r) ( _A,t -I- C) cos F

I

and z, and the circular frequency ii can he

(5)

measured.

(6)

I a and I b, the coefficients of the coupling terms can be determined.

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

L

sin fi

B

Pitch amplitude i/(J, phase lag / and circular frequency '' 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' +

(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.

,,,

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

+ .6. SW 011/'i)!?

The main oarticulars are as follows: Length (bp) L 2.438m 0.325

B

Displacement Area of waterline plane Mass moment of inertia of the model

nm

L=61.9

kg

So= 0.561

rn2

I

2.25

kgmsec

Moment of inertia of 0.170m1 K waterline plane 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) .07 S were tried Tests at zero snecd and at FR

but these did not give reliable results because of serious tank-wall effects. At each of the four speeds the frequency is varied

3 and oc = 13, and the amplitude of

cm. At low ti-ic eccentric between I and 4 or frequencies tile waves generated b the model motions may travel faster than the model itself.

= z c'"'

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

mogra Hi

The model is of the Todd 60-series parentform .60. with a blockcoefficient C

between 0)

The solution of (7) is given by:

1'

Test

B read t ft

'/'

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

+

,.;:!,/

(I)

M = k. lr mass moment of inertia of the model ii,, added mass moment of inertia moment of inertia of waterplane K pitch angle y The solution will be:

+

.6. cos /

(4)

A = I + ,v

r) b' sin i

(

(z 'r) boi COS F

+

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). ± 13 are not imFrequencies higher than cc

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'.

7

large compared with damping forces and therefore a determination of the damping cannot be accomplished here with reasonable accuracy.

Fig. 3 shows the phase lag and motion amplitude

curves for the combined heaving and pitching motion.

The combined heaving and pitching motion is studied for one speed only, viz.: FR .20.

The coefficients a, b, A and B are calculated, using the data of tables I and 2 and the formulae

This test covered the same frequency and amplitude range as mentioned above.

(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:

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 ampl 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-

h ./ gL

and

B \/ gL AL

Instead of the quantity a in -Iof ii and u/nz are shown in Fig. 4.

sidered.

-,

the values

TABLE I A iupliliiili' aiiil Jil.)ase for pure bc'a vi u FR = .15

FR =

FR = .20

V.)

a

10

0.519 0.539 0.568 0.624 0.722 0.894 1.226 2.318 3.166

11

1.335

12

0.774

2 3

4 5

6

7 8

9

a

2.0 5.4 10.1 15.3

19.0 22.5 30

49 135 166 177

0.527 0.553 0.592 0.654 0.759 0.937 1.267 2.361 3.198 1.367 0.764

in olwii

2.5

6.4 11.1

15.6 18.6

21.6

FR =

.2

z0/r

0.518 0.548 0.592 0.666 0.770 0.919

2.8

6.6 11.2 15.0 18.4

22.4

29

1.192

30

47

2.150 2.890 1.306 0.753

49

129 166 177

128 166 176

.30

a

a

0.512 0.539 0.576 0.632 0.721 0.902 1.250 2.163 2.946

3.3

7.3 11.7 15.5

18.6 23.0 32 50 132 169 180

1.261

0.705

o is given in degrees

TABLE 2 A iii />li 1ol(' (1/1(1 Jibase for pure J)ite/iiui& motion F (I)

= .15

FR = .20

FR = .25

i1

4

0.241 0.258 0.283

5

0.3 33

3

11

0.409 0.528 0.742 1.138 0.800 0.443

12

0.2 92

6

7 8

9 10

FR

-:

fi

1.7

0.244

1.7

4.8

0.261 0.288 0.333

4.4

9.2 14.5 20.5 29

44 86 140 163 172

0.406 0.544 0.807 1.149 0.773 0.448 0.289

45

0.243 0.260 0.284 0.327 0.398 0.523 0.799

8.2 13.0 18.5

27.3

2.1 6.1

9.9 13.5 18.0

25.9 44

88

1.151

92

140

0.755

141

159 169

0.446 0.285

162 172

is given in dcgrccs per cm; fi in degrees

0.2 So

0.265 0.291 0.330 0.395 0.517 0.856 1.227 0.703 0.414 0.285

2.4 5.3

9.0 12.8 17.4

25.6 44 95

44 62 72

8

35-.

_150°

30-.

2.5_

120°

4

-J

I 4

0.

90° 15_

- 60°

30°

05

0

0

0

2

4

3

I

I

I

I

6

5

B

10

I

I

Is

4

3

2

I

I

I

_I80

PITCH

FR. 020

A

l.2

_150°

PNASE

-r..LQcni

a- 20

U,

UJ.

.- ao

_120°

j

x-4.0

0 4

/ I I

06

-J Ui

0.

90° 4

AMPLITUDE

60°

04. _30° 02_ A

0

I

I

I

2

3

lig.

2.

1

4

x

I

I

5

6

iea.curel Jib,, of

wsc

''''1''''

I

I

I

B

9

0

1

II

T

12

I

13

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

0

I

4

,,,,)IIO,,

5

9

-

T

240°

PITCH0 HEAVE FR, =020

0°

12.

180° U-

3.0

0- 50

-

20°

-

900

-

60°

06.

04_

02

- 30°

0

2

I

3

4

5

6

=sc

0 JO

II

2

13

14

15

800

PITCH AND HEAVE

FR =0.20 - 150°

£ -r -2Ocm

.-ao 0- 50

C)

08

06. - 60° 04

30°

02

0 0

I

2

Fig.

__i,_

-,n__--__

3

3

ten, ii rd

6

'nines

--asc

sf 1 55! 1:/tIm/c and /h,

10

II

for so so bi,,:/ hen: 'ins

2

3

,,,,I pIc'iin

14

owl:,,,

IS

-50 rn

4_n0

10

-IS

HEAVE

1

-40 3-

3-

/vgL

'I' A

I

b

3

2

0

B

1%

-

05

0

0

....20

17

/

I

Vt

-

V/kp_.l5

-

HEAVE

-

4

5

6

7

9

8

(J-_ flC

I0/

10

II

9'tc

-

m

-

--

05

0

20

15

OhllI,l

0

13

2

sL

10

2.0

15

---

.

-

-'-5

HEAVE

HEAVE -

V/

//

- 40_

40-

/

3_

-\ -

- 9

V/v3O

S

-

-

-

-7-

A

6

I23 0

05

67S9 0 10

5

II

o

13

12

o

20

lig. 4

I

hi

hutCh

I/U' I

/ (I?

ii re i/Cl I i#I/

3

2

4

5

6

7

89

F

as PIll//i/I/I

to

IS

'I 0

II

2.0

0 2

3

.0

F

PITCH

F

i.o

PITCH

,/fl

:

I

B

I0

005-.

-3 Q5

I

0 2

3

r

T

I

I

4

5

6

QSLO

7

0

9

8

(aJ - SC -

o

a

.

/

4-.

\\1

9

20

0101

-

\._

N 1' 10

0

r

.

o

o

I

2

3

05

4

6

5

7

0

r

9

B

tO

10

II

12

I

.

c.

I).,,,,J,,,t a,,!

/J .1 ,,,,,, ,,,,,,,, ,,,

S

F

/B

0 I

2

0

2.0

PITCH

0 0

3

20

15

-

I

II

"II'

0

o.

F

_I.O

-

V/11- .25

010

o

3

PITCH

9

Q05

12

20

S

F

I-

II

I,., Jut J,ti I,

3

4

5

(5 J'th 1,1,1