On the form of non-linear roll damping of ships

On the Form of Nonlinear Roll Damping of Ships

A Technical Note by Dr. Yoshiho Ikeda, Osaka Prefecture University, A. y. Humboldt Guest Scientist at TUB

Introduction

As well known, the roll damping of a ship shows strong non-linear characteristics even for moderate amplitudes due to viscous effect. In most ship-motion theories used in

prac-tice nonlinear roll danming is replaced by an equivalent linear form, and the results based on that are fairly good as far as the treating of the basic resonance of the roll motion is concerned.

Recently nonlinear ship-motion theories have been developed, and higher order motions, e.g., the subharmonic and ultra-harmonic resonances of roll motion, the nonlinear roil mo-tion in random sea etc. have attracted the attenmo-tion of many reseachers [i] - [5] . Because the form of the equivalent li-near roil damping is thought to be insufficient for such higher order theories the establishment of a more rational and accu-rate model of roll damping is necessary L 3 J

Resent Works on Roil Damping Model

J.F. Dalze].l [6 Jand A. Cardo et al.E 7 Jconsidered several models of nonlinear roll damping.

J.F.

Dalzell investigated two models of roll damping, namely

iV21

Ai2.IØJ

(1)

Ai3,

-AI (2)

and thought that the coefficients of Eqs. (1) and (2) can be obtained from the extinction curves of a free-roll test of a ship. For Eq. (1) the extinction curve can be fitted by the following equation

dY

_aY

-;-

(3)

d"i'

where denotes the decrease in roll amplitude per half cycle or one swing. The coefficients N21 and N22 are obtained

--b

Tr

(j2iWo))

(9)

from a arid b as follows,

w

-

(2a)

N22

3WG

(2b)

(4)

For Eq. (.2): the extinction curve can be expressed as:

-

=c1 +

dY3

(5)

and the relations between the coefficients of roll damping and those of the extinction curve can be obtained,

t'J

J

(2c)

1J3

37rL)3

h

_4W

d)

(6)

He tried to choose the more reasonable model of these models (1) and (2), but found only that both models are quantitative-ly and qualitativequantitative-ly reasonable within the range and scatter of available experimental data.

Note that the relations of Eqs. (4) and (6) for a free-rol]. tet ate valid only when the rol]. damping coefficients

(N21, Ñ, N31, and N33) are independent of roll amplitude. It was found, however, that N22 depends strongly on roll am-olitudë (or Kc number) due to viscous effects even for mode-rate amplitudes, and that N21 depends on roll amplitude, too,

fot large amplitudes f motion (see Appendix 1).

A. Cärdo et ai E 7 J investigated two models as follows,

D0,

,1l02Id1).

ib

D0'3

They deduced the same forms of the extinction curve as Eqs. (3) and (4)and the relations between the coefficients of roll damping and the extinction curves are as fol].ows,

3 (Jo

-- C. _/

S

_{(1 0)} ( ))' where

=T01/

,M

/21

g,. _/1' , D

/1

Since both Eqs. (3) and (5) fit fairly well th

expeimhtal

data, as pointed out by Dalze1l,'thy proposed a new method

for deermining a roll damping model by a forced r-ol] test with á constant amplitude of the exciting moment'. Assuming' the linear restoring moment tz) and the exòiting motnent.

co , the maximum roll 4rnplitude 'in resonance can be

obtained on the basis of Eqs. (7) and (8) repreçtively as

follows,'

,

-e

e.

2Wo

4p3c&i2

They concluded that since the expressio,ns in qs, (11) and

(12) are different in two models regarding the dependen-cy of the excitation intensity and of t1e damping coefficients one can obtain definite conclusions on a more realistic

dam-ping model by the comparison of these forecasts wit'h experi-ntents of the' forced roIling in synchronism. Noté. that the relations deduced by the are also based on the assumption of constant coefficients of roll damping (böi, D11, Db3, D1i:

i' 'D) regarding roll amplitude. Moreover from the' pòint,

öf view of an exoeriment, it séemS to be as difficult asby a free-roll test to obtain definite conclusions on the dam-pin4 model ¿sing their scheme.

As shown in Appendix 1, the coefficients of an,extìnction curve do ñot show a dependence of the energy lo (during a half cycle) due to roll damping on the roll qu1ar velocity but only on thé roll amplitude. Therefore, it seems to b

impossible to determine nonlinear roll damping coefficients depending on róli amplitude from the extinction curves ob-tained by a free-roll test. It should be also noted that since accurate extinction curves can be usually obtained only up to 15 degrees roll amplitude in a free-roll test it is questionable to extend such experimental data to larger amplitudes of roll motion. For example, in Fig. 1

(see Fig. 14 in Dalzell's paper L6J ) we can see a fairly good agreement between experimental and predicted data even in the region of large roll amplitudes. If the coefficients of the cubic and quadratic models were decided on the basis of the data below the 15 deg. region, the predicted values by each model would greatly deviate in the large roll am-plitude region. From the discussions mentioned above we may conclude that a free-roll test is suitable only to obtain an equivalent linear roll damping for ships in moderate amplitudes.

3. Difference between Quadratic and Cubic Models

The nonlinear parts of the quadratic and cubic models of roll damping are expressed as,

bII

(13)

C (14)

Suppose a roll motion

=cosc,t

, then Eq. (13) can be expanded in the form,

bØII:

Eq. (14) becomes to CÇ

--

C4)CS,L.Ut Sin 3t (16) 8 8

-

Si c,jt

--

b Si 3c.

37r0

/7T

o

22

(15) 'o

-5-If we take only the first twO termsof Eq. (15), then both equations are composed Of terms with w*. and 3c . Note that

there is. no term with

2uk

in Eq. (15) though the model is a quadratic one.

As pointed out by Daizéll 6] he experimental res,ts öf

free-roll tests show good agreement with both quadratic ár cubic models in the region of moderate roll amplitudes. Then we may assume that,

2

= (17)

Eqs. (15) and (16) become

b d j

o, 49 A

Sih - o. i A Sii 3* (18)

C.7 A

SiWt -

O;2 A 1. (19)

It can be stated that the differences of the cut-terms bet-ween the: two.models are about 13% and that of the 3'cut-'terrn

about 47%. These differences cause the difference of the motion amplitude in the. basic resonance as. well as in higher

order resonances. It is, yet nQt clear whehQr the efect on th nnlinear charace.stic$ of the roll motion du to these differences is s;ignifiqa.nt Or not. '

4. Determinatin o Nnliné'a'r Roll bathpingby,Experiménts As mentioned in Chapter 3, it ïs difficult tb decompose the measured roll damping into components. experimentally by a

free-roll'test. A forced-roll test is orobably he best way to realize sùci a decomposition. In a forced-roil test, a ship model is forced to rol.1 sinusoïdally with constant f re-quency àn amplitude. The reaction mOment is mäsured and

then,

Substituting Eqs. (21) into Eq. (20) and using the Fourier expansion for the second term in Eq. (20),

Sih(.t

B Ø2i'snwIi/

r4*3c

siv3ct

Suppose that roll damping can be expressed as,

84 4

(20)

and the roll motion itself as

W SP c5t 8

221*

3

544 Ø-Øw L'44

1-3c)54 )Sin

ct

/22

k

--

/ /3 c

B

) sn 3wt

4

-I- ... ,05Tt

Note that the coefficients B44, B4, B depend on 00 as well as ori

j

We can determine the value of each coef fi-cient in Eq. (20) by comparing Eq. (22) with the measured moments for each frequency term by terra. It is necessary to carry out the test systematically for many combinations of frequency and amplitude. Practically it may be not so easy to obtain accurate coefficients by experiments because the roll moment is so small that noise in the measured mo-ment sometimes prevent obtaining accurate data.

5. Requirement for New Model

The most popular expression for the nonlinear roll damping of a ship may be the following one.

(22)

-7-It is necesay,. mörder to check this expression, to

mL

vestiqate each physical cornoonents of the roll damping in detail.

5. 1. Friction cornobnent

The shear stress actil-ig on the surfaçe Of a rolling hull can bè obtained from the second order unsteady boundary

layer theory proposed by Sch1ichtng C8]E9J

f/57 UA

Sn.(wt +--)

U

uA[(T2U24)J

(24)

where denotes the amplitude of the fluid velocity U

(= U7S%wt ) at the outer edge of the boundary layer, and

Ç

is the derivative óf UA with respect to a girth length. The first term of.Eq. (24)can be expressed b the first oné of Eq. (23). The second term of.Eq. (24

consists ofacön-:

stant part and .a

2ct

-part and their magnitude is

propor-tional to dU/dx (where x denotes the cordinate along the girth). Although.the sécond term of Eq.

(24)

affects the occurance of flow separatiOn, the value of its integral over the hull surface does not affect the roll moment any more

5.. 2. Wave coiponnt

The.wave damping moment èrived from a linear wàve theory can be expressed' by the first terth of Eq. (23).

Recently, sone researchers calculated the second-order hydro-dynamic forces due to second-order osciiiaions in waves, and pointed out the importance: of the second order forces which oscillate with 2u frequency as weil. as some secönd order

Fig. 2 shows examples of measurements of radiated waves for a two-dimensional cylinder rolling in _{a forced-roll} mechanism. The results of the Fourier analysis are shown in Table 1. For 00 = loo _{the higher order terms with 2.ut} and 3ct are. dominant. For _{00 =} 240, _{the terms with 2wt}

and 4wt are more. significant.

Table 1. Ratio of the higher order components to the first order component of the measured radi-ation wave created by a forced-roll cylinder.

The results-suggest that the higher order components of wave damping should be taken into account in a nonlinear roll dam-ping form. Among these higher order components the second and third order ones associated with 2wt and 3t are more im-oortant because of the nonlinearity of the restoring moment which. includes a third order component and because of the cross-coupling effect between heave and roll restoring mo-ment which includes a second order component, as pointed out by Paulling [iiJ

** .3 To express the term with 3t, it is necessary to add B44

₀

to Eq. (23). Since the expression of Eq. (23) does not contain any component with 2wt, as mentioned above, it is also neces-sary to add a new term with 2ujt to Eq. (23). It is perhaps

00=100

_Ø=24°

constant 0-.122 0.244 wt 1.0 1.0

2t

2.35 - 2.02 3ct 2.35 0.547

4t

1.19 2.35 5wt 0.324 1.77

reasonable to add

B.*2

because the sècond order moment ith deduced, from thequadratic terms of the fluid velocityin the Bernoulli'. equation.

5.3. Eddy component

Generally speaking, the eddy component of the roil damping', the bilge.. keel component included, may be 'expressed b' the second term of Eq. (23) SiPcë there is no experimental in-.

fQmation about hïgher.order terms of this component of roll damping we can nOt help but derïve it from other experimental results of viscous drags. ..: . . . '.

According to the Sarpkaya's results [1-3'J , tlie yiSbus

for-ces actiPg ón a sinusoidally oscillating crcùlà cylinér

may be expressed with three terms as follows,

F

?DL

Cr.i

S',t

where -. Cd C5c,jt fros wi/

k

uT)/,

2ff/

C3cos(34-5)

(25)

where the cylinder oscillates with the velocity of Umcost He Pointed out that C3 aiid

5 can be expressed by the drag

and inertia coefficients Cd and Cm and that among the higher order terms those with the third harnprii..c 3wt are the.most important ones.

This résult suggests that the eddy.còrnpöneht ofrol1 damoing seems to be expressible by the seçond term Of E. (23) and

a new term witì '

5.4 new model

On the basis of the discussions in the previous chapters, a new model of' nonlinear roll damping may be deduced, namely

(B4444iJø

B4Eø/f

&44y

(ßa44;)Ø

(26)

linear wave component B44F: friction component B44L: lift component

B: second order wave component BE: eddy component

third order eddy component B**

: third order wave component

44w

In this equation, the third term contains a constant part which does not represent a damping. Suppose

_{0 =}

Ø0coswt,

Eq. (26) can be changed into,

3 p*ç

2

I ,./.2_w

i 2t

-t- (- ¿ _-

s

3

t

(27)

Now we can see the contributions of each component to each harmonic term. The first term of Eq. (27) coincides with the equivalent linear roll damping.

6. Other effects

When we treat the capsizing and extreme roll motions of a ship, it is necessary to take into account additional contributions to roll damping, for example, deck wetness , shitming water effect, breaking wave effect and so on.

In this chapter an experimental result for an immersing deck is introduced. The model used for this experiment is

shown in Fig. 3. To obtain the effect of the deck-edge immersion only an under water circular cylinder hull was chosen, which affects only friction damping. Examples of the

roll moment record during one swing are shown in

Fic. 4.

The first order roll moment increases rapidly with the roll

amplitude as shown in Fig. 5. The results of corresponding Fourier analysis are shown in Table 2. The results show that the terms of the roll moment with the third harmonic and the fifth harmonic are the most dominant among the higher order ones.

Table 2. Ratio of the higher order components to the first order component of the measured roll moment in large roll amplitude

=o.389,4=31.1°

=O.796,Ç=31.

¿onstant

O.Oo

0.00

LOt

1.0

1.0

2jt

0.00

0.00

3tut

0.577

0.181

4t

0.00

0.00

5,t

0.229

0.329

7. Conclusions

A rational nonlinear roll damping model can not be de-termined by a free-roll test, although a free-roll test is suitable to obtain the equivalent linear damping in mode-rate roll amplitudes.

A forced-roll test seems to be a suitable tool to deter-mine the coefficients of nonlinear roll damping.

It is necessary for establishing a rational nonlinear roll damping model to investigate each physical compönent of roll damping in detail.

8. Acknowledgement

The work has been done during the stay of the author in the Technical University of Berlin as a research fellow of the Alexander von Humboldt Foundation. Prof. H. Nowacki of the Technical University of Bénin is sincerely

acknow-ledged for his kind help and warm èncouragements. The author should like to. express his gratitude to Dr. A.. Papanikolaou of the Technical University of Berlin for his valuable

- 13

-Referen ces

Li.] Potash, R.L.: Second-Order Theory öf Oscillating 'Cylinders, JSR, Dec. 1971, pp. 295

[2,.] Papanikolaou, A., and Ñowack'i, H.: 'Second-Order Theory of Osaillating Cylinders In a Rgula.St,eep Wave, Proc. of

O.N.R., 1980

[3..] Tas'ai, F., and Koteravarna, W!: Nonlinear Hydrodynamic For-ces Acting on Cylinders Heaving on the Surface of a Fluid, Rent. 'No. 77, Res. Inst. of Appl. Mech., Kyushu Univ., 1976 [4.] Cardo, A., Francescutto, A., and Nabergoj, R.:

U'ltraharmo-ni'cs and SubharmòPics ïh the ollincT Motion of a Ship': SteadyState Solution, ISP, Vol. 28, No. 326, 1981 E .J.Cardo, A., Francescutto, A.1 and Nabèrgoj, R.: On thé

Maximum Amplitude in Nonlinear Rolling, Second Inter-national Conference on Stability of Ships and Ócean Ve-hicles, 1982

[6.]Wr'ight, J.H.G., Marshfield, W.B.: Ship Röll Response and Capsize Behaviour in Beam Seas, Transactions of the Royal Institution of Naval Architects, Vól. 122,1980

'[7.] Dalzell, J.F.; A Note bn the Formof ShipRoll Damping,' JSR,

Vol. .22, No. 3, 1978 ' . '

Cardo,: A., Francesôutto,A., 'and Nabergoj: 'On Damping Models in Free and Forced Rölling Mbtion, Ocean' Engineering,'-Vol. 9,

No. 2, 1982' '

Schlichting, H. Boundary-Layer Theory,' six-edition, McGraw-Hill, 1968, pp. 411

Ikeda',' Y., and Tanaka, N.: On Viscous Dragof Oscillating Bluff Bodies, 12th ScientiFic and Methodological Seminar of Ship Hydrodynamics, 1983

[11'.] Yamashita, S.: Calculations of the Hydrodynämic Forces Ac-tinc upon Thin Cylinders Oscillating Vertically with Large Amplitude, Jour. Soc. Naval Arch'. of' Japan, Vol. 141, 1977 [12.1 Kyozu'ka, Y.: Experinental Study on Secpnd-Order Foröes At-.

tinq on Cylindrical Body in Waves, Proc of 14th Symposium of Naval Hydródinamics, 1983 '

[13.'] Pau'liing, J.R., Roseñherg, R.M.: Unstable Ship Motions ResuJ.ting from Nonlinear Coupling, JSR, June, 1959

Sarpkaya, 'T., .Isaacson, 'M.: Mechanics of Wave Forces on Offshore Suctures, Van Nostrand Reinhold Co., 1981

'Ikeda, Y., Himeno, Y., Tanaka, 'N.: 'OP Roll Darning Force of Ships - Effect o Friction of HuJI and Normal Porce of Bi]qe Keels - , Jour of the Kansai Soc of Naval Arch , Japan,

Appendix 1.. Free-RoIl Test

A free-roll test is probably the easiest way to obtain roll

damping of shii.ps. i:n this test, the model is heeled to a

chosen. angle (usually about 15-20 degrees) and then released. The model makes damping oscillation at its natural frequency which is approximately constant. The subsequent motion is measured. and an.. extinction curve Is obtained on the basis of the obtained roIl amplitude at each swing. The amount of decrease of the roll amplitude during a swing corresponds to the energy loss due' to roll damping. Note that the value ob-tained by a free-roiling test is' an integral value of the

ener-gy loss during

a swing and coincides with the equivalent li-near damping.

Under sorne limiting conditions, we can determine the coef f i-dents of the nonlinear roil damping by a free-roll test as

folthws Suppose roil damping and corresponding extinction curve are expressed as,

ß44*jØI

_(a-1)

=

(a-2)

where

2

If the coefficients of Eq. (a-1) are independent of the roll amplitude we can get the following relations btween the coefficients of Eq. (a-1) and Eq. (a-2)

t

B

Q =

_j, (a-3)

4

2

j3

_(a-4)

o-

_3WC,T1

However, it was found that B4 is usually a function of the Kc number or roll amplitude and B44 seems to depend on roIl amplitude, too, especially for large roll motions

15

-as concluded from experimental results of heave motion E 3 J.

Let us refer to an example. The drag of a bilge keel can be expressed as,

F

_=fCD

uIu4S

(a-5)

where CD denotes the drag coefficient, Um the velocity of the bilge keel and S the area of the bilge keel.

The extinction coefficients a and b can be obtained as follows,

30f b;k

Y2 CO2

TrVG!

b-

3.2nfbkYw2

-

(a-7)

Note that and were taken in decrree when a and b

were determined by Eq. (a-2). As shown in Fig. A-1, the pre-dicted values of a and b by Eq. (a-7) are in good agreement with experimental results obtained by a free-roll test of an ellipsoid with bilge keels. The result suggests that the second term of Eq. (a-1) contributes to the coefficient a as weil as b, and that the relations expressed by Eq. (a-3)

and Eq. (a-4) are no longer valid in this case.

A prediction formula for CD was proposed by the authors E15],

CD

22.5

2.4

(a-6)

where Kc=(7Tr00)/bBk r the distance between roll axis and bilge keel, bBK the depth of the bilge keel.

di dn 50 30 20 lo o o lo

O

: experimental results by Motora et al. 4O : quadratic model determined by Daizell at cubic model determined by Daizell 20

/

A

'O

/

/

O

3' /

o

40 Y (degrees) roll amplitude

Fig. I Extinction data derived from fbrced roll

experiments in large roi.l amplitude by Motora et al./6/.

(mm) 3 10 o (mm) 17 -T = 0.8 sec T = 0.8 sec

demensions of the model LxBxD = 0.BmxO.237mx0.096m = 0.01775m3 -roll axis = O Ç4 24 deg., T = 0.8sec time Time

Fig. 2 Record of radiation wave due to a rolling cylinder

in large amplitude.

o Series 60, C3=O.6 midship section

30 Omm

30, 20 10 time MR (kg-m) 0.05 - 19 -measured (with ,bu1wak) 31.1 deg. CQCLiÇ = 31.1 deg. 0.796

Fig. 4 Recôrd of roll damping moment dUring öne swing

acting: on rolling cylinder in:large amplitude.

angle for deck edge wetness « =20.ldeg. 20

Q,,

/

/

/

friction component

predicted by Kato formula

30 (deg.)

Fig. 5 First order roll damfling component for two-dimensional cylinder in large amplitude.

0.01

measured (with bulwark)

B4 4 (without bulwark) O. measured 0. 005 o /

/

0.05

Q :measured

- 21

-bilge keel

predicted

(by Eq. (a-7)

naked huil

15

b(mm)

5 O' 15 _b3(mm)

breadth Of b.iIqekeel'

Fig. A-i Extinction coefficients a, and b. due to normal 'force component of

jlgeke1

.