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JOURNAL OF SHIP RESEARCH.

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22(3) 178-185

1978

Dalzell, J.F.

A NOTE ON THE FORM OF SHIP ROLL DAMPING

0022-4502

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ACADEMY LIBRARYUNIVERSITY OF NSW@ADFA

DATE OF COPYINGJournal of Ship Research, Vol. 22, No.3, Sept. 1978, pp. 178-185

1 9 O C T 2 0 1 1

COpy M AD E IN COMPLIA NC E

A N o te o n th e F o rm o f S h ip R o ll D am p in g W ITH TH E COPYR IG HT ACT

J. F. Oalzell2

The object ive of the present work was to develop an approximat ion to the convent ional mixed linear-plus-

quadrat ic ship rol l damping model so that analyt ical obstacles could be overcome in the applicat ion of the

functional series expansion to nonlinear ship rol ling. A mixed l inear-plus-cubic approximat ion was found

to be reasonable for this purpose. In the course of analyses, there were indications that this model may

be closer to an "equivalent approach" than to an "approximation."

Introduction

IN MOST CASES of practical application of seakeeping theory orexperiment, the problem comes ultimately to that of predictionof the statistics of maxima of ship response in realistic (random orirregular) seas. Methods of estimation of the magnitude of oscil-latory ship motion statistics are reasonably well in hand so long asthe motion being predicted can be assumed tobe a linear functionof regular wave height. The methods outlined by St. Denis andPierson [IP involve, in addition to the linearity assumption, theassumption that the wave process isGaussian. These assumptionsbeing fulfilled, the spectrum of response may be estimated, andfrom this a reasonable estimate of the statistics of maxima andminima can be formed.

In this context, ship rolling continues to pose a problem. Theoften-used single-degree-of-freedom equation for roll (which istraceable to the work of Froude [2]) isnonlinear in damping mo-ment. This same nonlinearity, or the recognition of its possible

occurrence, is to be found in many multi-degree-of-freedomanalyses. In the case of both pitch and heave it has been possibleto develop linear theoretical approaches to damping which con-form reasonably well to observation. This has not been the casewith rolling. There presently appears to be no theoretically basedprediction method for roll which iscompletely free of empiricismwith respect to roll damping.

In current practice the usual approach to prediction is somevariant of an equivalent linearization technique (Kaplan [3],Vassilopoulos [4])for the estimation of the variance and spectrumof roll. In this technique, nonlinear elements in the equation ofmotion are replaced with linear elements chosen soas tominimizein some sense the resulting errors for random excitation.

Other approaches have been investigated. Yamanouchi [5]approached the problem with a perturbation technique for the

solution to the nonlinear differential equation, obtaining solutionsfor the nonlinear roll spectrum in terms of the sum of the linearspectrum and various convolutions. A number of investigationshave been made of the application of the Fokker-Planck equationmethod [6] for the solution of nonlinear vibration problems.Evaluation of roll response statistics according to this approachpromises to be extremely difficult when the spectrum of excitationis not white (flat). Haddara (7) uses a modified Fokker-Planckapproach which results in estimates for roll variance for the casewhen the excitation spectrum is white.

1This note summarizeswork carried out under Naval Sea SystemsCommand General HydromechanicsResearchProgram, administeredby theDavidTaylorNavalShipResearchandDevelopmentCenterunderContract NOOO14-75-C-0278.

2 DavidsonLaboratory,StevensInstituteof Technology,Hoboken,NewJersey.

SNumbersin bracketsdesignateReferencesat end of paper.Manuscript receivedat SNAMEheadquarters May23, 1977;revised

manuscript receivedAugust25,1977.

Yet another approach to the nonlinear prediction problem is thefunctional series model. In this approach there is a serious

drawback with respect to application to the usual nonlinear rollingequation. This was pointed out some time ago by Vassilopoulos[8]. If the functional series and a differential equation are to berelated, it appears that the terms in the equation must be analyticfor small values of the variables. This is not the case for the"quadratic" term ordinarily used to represent the damping non-linearity. Thus the first problem in any attempt to apply thefunctional series approach to rolling isto overcome the quadraticdamping obstacle, and this note (which isan abridgement of [9])summarizes some steps taken in this direction.

An approximation to the quadratic term

The single-degree-of-freedom rolling equation may be con-sidered in the form: .

liP + !l(ip) + ! i (<p) = F(t ) (1 )

where:

<p= roll angleI= roll inertia

li(ip) = damping function! i (<p) = restoring functionF(t ) = excitation

t = time

In the literature there appear a number of forms of restoringfunction. None sofar seen are fundamentally different from anodd series in < { 7 .

! i (<p) = L Bj<p j (2 )

j=1.3 •...

In most applications, only the first (linear) term is used. In anyevent, the form of equation (2) presents' no fundamental prob-lem.

The classical damping function may be written in the form

(3 )

The absolute value in the second term of equation (3) is theissue.

The foregoing form for roll damping has also been used bothexplicitly and implicitly in modern multi-degree-of-freedomanalyses. There appear to have been extremely few deviationsfrom the form of equation (3) in the last century. A few recentinstances may be cited. Haddara (10) replaced the second termwith a cubic term in ip for illustrative purposes and possiblybecause

the quadratic form was impossible to handle in his derivation.Lewison [11]also replaced the quadratic term with a cubic term.His reason appears to have been largely to facilitate analog com-

178 0022-4502/78/2203-0178$00.41/0 JOURNAL OF. SHIP RESEARCH



puter setup, though he remarked (without elaboration) that thecubic model better fitted the data. Takaki and Tasai [12] per-formed forced oscillation experiments of a ship model with bilgekeels, and found in their analysis of the roll damping that an im-proved fit of the data was obtained by adding a cubic term in (P

to the standard model, equation (3).The foregoing was sufficient encouragement to consider the

replacement of the quadratic term in equation (3) with an oddseries in ( p . If it is assumed that the coefficients N 21 and N 22 inequation (3) are given, then the problem is to relate these to thecoefficients of an odd expansion. One approach is to make aleast-squares fit of an odd series to I ( P I (P over some assumed rangeof ( p , say ( - ( P c < ( P < ( P c ) . The fitting equation was therefore as-sumed in the following form:

1 ( p 1 ( p = I: C < k ( p k / ( ( P c ) k - 2 (4 )k = I , 3 , . . . _

Itwas further assumed that the odd series istruncated at the termwith exponent K. In this case the (K + 1)/2 normal equations inthe unknown coefficients C < k become

K C < k 1k = I ~ . . . (k + j + 1) - j + 3 (with j= 1,3, ... K) (5)

After solving for the C < k , the result for truncation of the seriesat the cubic term is:

Figure 1 indicates the degree of approximation involved inequation (6). Only the positive half of the quadratic form isshownand the evaluation ismade for normalized I ( P I ( p . The maximumdeviations within the range of the fit (± ( P c ) amount to 3 or 4 per-cent of c I e . Because both the fitted coefficients are positive inequation (6), the general behavior of the approximation beyondthe fitted range is the same as that of the quadratic term.If ~tis assumed that equation (6) is an adequate approximation,

substitution in the damping function, equation (3), results in

N ( ( p ) ' " [ N 2 1 + ~ ( P cN z z l ( P + 35 N _ 2 2 ( p 3 = N ( ( p ) (7 )16 48 ! P c - .

Conceptually, equation (7) might be thought of as a sort ofequivalent non linearization. In the sense of making estimates ofroll variance utilizing the functional series method, the form ofequation (7) issuitable. However, the presence of the parameter( P c means in principle that estimates of both the roll and roll ve-locity variances would have tobe made for an arbitrary choice of( P c , these used to estimate a reasonable solution domain of ( p , andthe process iterated if required.

Theory for analysis of roll extinction data

In order to assessfurther the approximation of equation (7),twothings seemed in order. The first was to find some realisticmagnitudes for N21 and N22, and the second was to consider what

1.0

// ,

0.8

0.2

0.0 0.2 0.4

(6 )< J l / 4 > c

Fig. 1 Two-term odd series fit to I < p I < p

violence to the left-hand side of equation (1) was being done rel-ative to observed behavior of ships.

Asnoted in the Introduction, theoretical estimates ofN21 or N 2 2

appear for practical purposes to be totally lacking. Though the-oretical estimates of N21 are made in the more sophisticated ofmodern ship motions algorithms, these seem invariably to comeout too low, the typical procedure being to add something to ac-count for" viscous"damping effects soas tobring computation intoline with observation. In fact, what are considered realistic esti-mates of N 2 1 and N 2 2 are almost totally empirical. In the vast

majority of studies where some distinction 'ismade between "lin-ear" and "quadratic" components of roll damping, the numericalresults are obtained by analysis of ship or model sallying experi-ments. The data-reduction approach has been basically the samesince the time of Froude [2].

Minorsky [13] (pp. 186-188 and pp. 192-196) derives themethod according to the first approximation of Kryloff and Bo-goliuboff. The basic result is the differential equation of theamplitude of response, assuming that damping is small, that ex-citation is zero, and that angles are small. In particular, if theseassumptions are applied to equation (1), there results

(9 )

_______________ --n 1o m en clatu re _

Equation (9) is assumed to be an adequate representation of the

a, b = coefficients in quadratic fitting

equation for rol l ext inction per half

cycle

l 1 ( c p ) = restoring function

B1= linear coefficient of res tor ing func-

tion

c,d = coefficients in cubic fitt ing equation

for ro ll excitation per half cycle

F(t) =excitation

I= roll inertia

N( I p ) = damping function

l : ! ( i p ) = cubic approximat ion to quadratic roll

damping model

! : l , ( i p ) = cubic damping function model

N 21 = coefficient of I~r ~rt of qusdratic

roll damping mo&l

N 22 = coefficient of quadratic part of roll

damping model

N31 =coefficient of linear part of cubic ro ll

damping model

N33 =coefficient of cubic part of cubic roll

damping model

n,m =nondimenslonal time

t =time

T = roll period, period

Y = roll amplitude

Yo > I ! : ini tial rol l ampli tude in ship sallying

experiments

1'1,1'3 =coefficients in cubic fit ting equat ion

for rol l ext inction per cycle

0 1 0 0 2 =coefficients in quadratic fitting

equation for roll extinction per

cycle

c p =roll angle

ip c = assumed range of rol l velocity

w = frequency, roll frequency

S E P T E MI S E R 1978 179

.-..------------



ship during the sallying experiment. The fundamental result fromMinorsky (13) for the differential equation of roll amplitudes is

. 1 5 0 2.. dY = - -- N (Y w cose) cose ~

2'111w 0

wherew = roll frequency =VB;]iY = roll amplitude in the sense that the rolling isrepresented

in the form.

I { - { t ) = Y (t ) s in (w t + constant)

Substituting the quadratic damping function, equation (3), inequation (10):

y = - _1_ , 2 . . [N21Ywcos~ + N22Y2w21 cose ] cose jcose de2'111w J «

= _ N21 Y _ 4N22W y2 (11)21 3 '111

It is convenient for analysis of sallying data to make a change inthe time variable in equation (11). That is, let

t =nT

where T = roll period =2'11"/. Making this change in equation(11)

The typical analysis of the curve of declining angles obtained ina ship sallying experiment amounts to estimating dY / dn (the de-crease in roll amplitude per cycle) as a function of mean amplitudeand then choosing coefficients to make a best fit of the data to anequation of the form of equation (12). In particular

- dY = 01 Y + 02y2 (13)dn _

is the form often used. Having estimates of 01 and 02 from thedata, estimated values of N21 and N22 may be formed

l w B1N21 =-01=-01

'II" 'll"W

31 3B1N22 = -0 2 = -02 (14)

8 8w2

Froude found it convenient to estimate the decrease in roll per halfcycle (d Y / d m), and much of the available data are presented inthis way. In this case the data are reduced according to

- dY =aY + by2 (15)dm

sothat in this system N 21and N 22may be estimated from equation(14) with 01 = 2a and 02 = 2b .

Given equations of the form of (12), (13), or (15), the equationof the curve of declining angles from the sallying experiment maybe found [13]. The result in terms ofthe notation ofequation (15)may be written:

Y = Yoexp [-am] (16)

-b1+ - Yo(1- exp[-am])

a

where the initial conditions of roll amplitude =Yoat time m =0have been assumed.

Turning to the second point mentioned at the outset, once thenumbers in the approximate damping function, equation (7),havebeen chosen, the form of equation (1) changes. For example, forthe zero-excitation small-angle case asassumed for the ship sallyingexperiment, the original equation

lc p + BI';' +N21cP + N221 0 1 cP = 0

1,;11iiUJ~~eQto one,of the form

li p + Bgo + NS1C p + Nsscp =0 (17)

(10)

The question is (apart from validity of empirical constants), willsuch a change imply responses which do not reconcile with ob-servation? Accordingly, it is of interest to make believe thatequation (17) represents the ship in the sallying experiment. In

this case the damping function would be

l i_(0) = NS10 + Nsscp (18)

Substitution in equation (10) results in the differential-equationof rolling amplitude

y = _ NSI Y _ 3Nssw 2 ys (19)21 81

or, making the time variable change as was done to produceequation (12):

dY = _ 'll"NSl Y _ 3'11"Nssws

dn l w 41(20)

Imagining that the results of a sallying experiment might be ana-lyzed in a way similar to that for the quadratic model [equation

(13)], the coefficients 1'1and 'Yswould be chosen soas to best fitthe data to

(21)

(12) Accordingly, the estimates of NSI and Nss from the experimentwould become

l w BlNS1 = - 1'1 = - 1'1

'II" m»

41 4BlNss =-- 'Ys=-- 'Ys

3'11"w 3'11"ws

(22)In Froud's method of fitting with half periods, the fitting equation

analogous to equation (15) might be written

- dY = eY + dYs (23)dm

and, as before, estimates of NS1and NS3could be formed fromequation (22) by letting 1'1=2e and 'Ys= 2d .

Finally, the equation of declining angles from the ship sallyingexperiment in which the damping isassumed to be the cubic modelmay be derived from equations (20), (21), or (23). Assuming, aswith the quadratic model, that the initial roll amplitude is Yoattime (m ) = 0, the result in the notation ofequation (23) comes outto be

Y= Yoexp[ -em] (24)

{ I +~ Y 5 (1 - exp[-2cm])

r2

Accordingly, if an approximation of the nature of equation (7)is reasonable relative to available observations, the differencesbetween the quadratic. extinction model, equation (13), and thecubic extinction model, equation (21), should be small relative tothe range and accuracy of data. Similarly, if the curves of de-clining roll angles are examined, equation (24)for the cubic modelshould not differ radically from equation (16) for the quadraticmodel within the range of observation.

Some analyses of ship-sallying data

A review of recent texts (Lewis [14], Korvin-Kroukovsky [15],Vossers [16], and Blagoveshchensky [17], for example) fairly con-vincingly indicates the originator of the mixed linear-pius-qua-dratic roll damping representation to be Froude [2]. Itappearsthat there was no dispute at the time about the representation ofreal fluid effects upon roll asbeing quadratic, only about the mixed

JOURNAL OF SHIP RESEARCH



9

8

70 E X P E R I M E N T

Q U A DR A T IC M OD E L , E q . 1 6V 6

E q . 2 4.U C UB IC M OD EL ,I.U 1 00::

'-' 5UJ0

8Vl

.: 4I.U

-'UJ 7

0 '":: '-'6.U

0

3 .: 5-'0c::

4

2

0 5 1 0 1 5 2 0 2 5 3

10· 40o. T W O E X P ER I M EN T S

Q U A D R A T I C M O D E L , E q . 1 6

C U B I C M O D E L , E q . 2 4

Vl

UJUJcc'-'UJ0

.: 3-0c::

2

1~ __ ~ -L ~ L___~ _L__ ~

o 5 1 5 2 0 2 50

C O U N T O F H A L F C Y C L E S

Fig.2 Fitofequations(16) and (24) to curveofdecliningangles obtainedbytwoexperimentswithHMS Inconstant in 1871


Fig.3 Fitofequations(16) and (24) to curveofdecliningangles obtained

inan experimentwithHMS Volage in 1871

model. Since some of the experimental data used by Froude indemonstrating his point are currently available [2], it was of interestto start with these. The data (which appeared in Naval Science,1874, pp. 220 of reference [2]) involve the presumably faired pointson the curves of declining angles for five sallying experiments onfour ships, all of which apparently were built with bar keels.

The first point ofthe exercise was tomake a comparison betweenthe fits of equations (16) and (24) to the original data. The ex-perimental results had been plotted to an enlarged scale and it waspossible tomeasure the roll amplitudes observed at each half cycleto within ±O.05 deg from these plots. Because it was not possible

to quickly develop a direct least-squares fitting procedure forequations (16) or (24), an indirect semi-trial-and-error approachwas utilized.

The results are shown in Figs. 2 through 5 for the data in refer-ence [2] . Figure 2 indicates curves of declining angles for twoexperiments on HMS Inconstant. Each experiment was analyzedseparately. (Froude had combined the two graphically in hisanalysis [2]). The logarithmic scale in this figure, as in those tofollow, tends to accentuate reading errors for the lower ranges ofroll amplitude. The fit of equations (16) and (24) to data for HMSVolage is shown in Fig. 3. Figure 4 involves fits to data obtainedby French experimenters on a ship named Elorn. This experimentis notable for the large initial roll amplitude (32 deg), and it is re-gretted that the details of the ship or experiment could not be

S E P T E M B E R 1978

o E X P E R I M E N T

_ _ _ _ Q U A D R A T I C M O D E L , E q . 1 6

_ _ - - C U B I C M O D E L , E q . 2 42 0

Vl

UJUJcc'-'UJ

o 1 0

:;8

0c::

65

4

30

3 5

1 510


Fig.4 Fitofequations(16) and (24) to curveofdecliningangles obtainedinan experimentwithElorn in 1872

2 0

o E X P ER I M E NT


C U B I C M O D E L , E q . 2 4

2o 5 10

3 0

1 5

3 5

2 0

4 0

2 5

4 5

3 0


Fig.5 Fitofequations (16) and (24) to curveofdecliningangles obtainedinexperimentwithHMS Sultan in 1873

found. Figure 5 involves the declining angle curve for HMSSultan, which, owing to the ship's light damping in roll, is un-usually long, and accordingly the results were broken into two partsfor plotting purposes in Fig. 5.

Gawn [18] presents results of several sallying experiments carriedout by Froude on an ironclad warship, HMS Devastation. These

were in the form of various declining angle curves plotted to rel-atively small scale. One of these results was picked arbitrarily,and the curve was read at each half cycle, excluding data fromamplitudes lessthan 1 deg since the values could not be read muchcloser than ±0.1 deg. The resulting data were subjected to theprocedure outlined in the foregoing, with results given in Fig.

6.On the whole, in Figures 2 through 6,there isnot much to choose

between the abilities of the quadratic model, equation (16), andthat of the cubic model, equation (24), to represent the originaldata. In both cases,effort was made to produce fits having com-parable root-mean-square (rmsl deviations from the data. Withinthe range of data, either would serve as a reasonable interpolator.In none of the cases shown would a purely linear damping function

181



o E X P ER IM E N T


C U B I C M O D E L , E q . 2 4

V

LU

W

a:'-'w 3'"_j'_

0a:

2

o 5 10 1 5 20


Fig. 6 Fit of equations (16) and (24) to curve of declining angles found

by experiment with HMS Devastation in May 1873

result in as good a fit, and it is strongly suspected that neither anassumed purely quadratic nor an assumed purely cubic dampingfunction would yield comparable fits. Relative to the present

objectives, the results suggest that the replacement of the classicaldamping model, equation (3), with the cubic approximation,equation (7), can result in a quite reasonable approximation to thetransient roll response solong as an appropriate choice of coeffi-cients can be made.

More modern results from sallying experiments are presentedas roll extinction curves; that is, the numerical differentiation whichprovides the decrease in roll per cycle or half cycle isperformed

1.0

dY

dm

o F RO M F AI RE D E XP ER IM EN TA L R E SU LT S


C U B I C M O D E L , E q . 2 3

0·5

o 43 5

Y , R O L L A M P L I T U D E , D E G R E E S

Fig. 7 Fit of equations (15) and (23) to roll extinction curve obtained for

HMS King George Vin 1914

182

and the results analyzed or faired by the experimenter prior topresentation in the form of a plot of -dY Idm versus Y.

Gawn [18] presents three such sets of ship data. The first twosets to be discussed are from experiments carried out on a 1912-vintage battleship (HMS King George V) and a 1918 destroyer

(HMS Vivian). In both cases Gawn [18] presents faired linesrepresenting -dY Idm as a function of Y. For present purposes,values of -dY Idm were taken off at equal intervals of Yand theresults treated asdata to which equations (15)and (23)were fittedby the method of least squares. The resulting fitted lines arecompared in Figs. 7 and 8 with points representing the originalcurves. There is again not much to' choose between the fits, asomewhat surprising result in view of the likelihood that theoriginal fairing of the two sets of data was done to the quadraticmodel.

The results presented by Gawn [18] of experiments on a thirdship are of the same form but unfaired. The ship was a destroyerof a 1935 design (HMS Nubian). There were six sallying experi-ments carried out. Values of dYIdm were originally derived fromeach and the results were pooled to form some 50 pairs of dYIdm

and Y. These points were measured from the chart presented byGawn and appear as circles in Fig. 9. Least-square fits of equa-tions (15)and (23) were made to the pooled data and the resultingfitted lines are also shown in Fig. 9. The differences between thequadratic and cubic models are clearly less than the scatter oforiginal data. .

For present purposes, a selection of the available model exper-imental data was made and similar analyses carried out. Vossers[16] presents a set of unfaired model roll extinction data for oneexperiment in Fig. 66 of his text. Figure 10 shows these data andthe least-square fits of equations (13) and (21) (which are theequations corresponding to the roll extinction per whole cycle).Figure 11indicates a similar result using model roll extinction datafor one experiment presented by Blagoveshchensky [17]on page640 of his text. Figure 12 is a third similar analysis for the pooled

results of a number ofexperiments by Lalangas [19]on a Series60,0.60 block parent model at zero speed. Just as in Fig. 9 for shipdata, the results in Figs. 10through 12for model data indicate thatthe differences between the quadratic and cubic models are lessthan or the same as data scatter.

Of the many model roll extinction experiments carried out by

Martin et al [20], one was abstracted for analysis and this was thecase where their model was bare (without bilge keels or artificial

2

o F R OM F A IR ED E XP ER IM EN TA L R ES UL TS


C U B I C M O D E L , E q . 2 3

dY

- dm

6


Fig. 8 Fit of equations (15) and (23) to roll extinction curve obtained for

HMS Vivian in 1925

JOURNAL OF. SHIP RESEARCH



2

d Y- dn

2

o P O O L E D R E S U L T S O F S I X E X P E R I M E N T S


d Y C U B I C M O D E L , E g . 2 3- T m

o 2 3


Fig.9 Fit of equations (15) and (23) to roll extinction data obtained in experiments

with HMS Nubian

60 E X P E R I t \ E N T A L D A T A

Q U A D R A T I C M O D E L , E q . 1 3dV

E q . 2 1dn C U B I C M O D E L ,

,/cf/

/

3 '"

0

(7

o~~

~ 0 5 1 0 1 5 2 0

,y V, R O L L A M P L I T U D E , D E G R E E S0

Fig. 11 Fit of equations (13) and (21) to model roll extinction data pre-

sented by Blagoveshchensky

o E X P E R I M E N T A L D A T A


C U B I C M O D E L , E q . 2 1

iI

o L ~______J__--,----"--_"___----- 5 · 0

o 2 4 6 8 1 0 1 2

V , R O L L A M P L I T U D E , D E G R E E S

Fig. 10 Fit of equations (13) and (21) to model roll extinction data pre-

sented by Vossers

roughness) and was at zero speed. This extinction result waspresented as a faired line. This line was read off at even incre-

ments of roll amplitude and the values appear as circles in Fig. 13 .The least-square fits ofboth the quadratic and cubic models to thepoints representing the faired line are also shown. In this instancethe cubic fit appears slightly better than the quadratic.

The last example picked was presented asa faired experimentalresult by Vossers [16J (p. 116) and is the equivalent of a roll ex-tinction curve. The result was obtained by Motora et al [21 ] bymeans of a forced-oscillation technique, and is notable for theextremely large roll angles involved. Asin the previous example,the faired data was read off at roughly uniform intervals of rollangle and the result treated as data in least square fits of equations(13) and (21). Figure 14 indicates the results. The cubic modelyields a physically believable result; the quadratic model doesnot.

2 . 5

/;

o P OO LE D E XP ER IM EN TA L D AT A


C U B I C M O D E L , E q . 2 1

o

o

o

o 0

~ 0

_ . - 8 " ' - - ;

o 5 1 0

Fig. 12

V , R OL L A M P L I T U D E , D E G R E E S

Fit of equations (13) and (21) to model roll extinction data pre-

sented by Lalangas

SEPTEMBER 1978 183



2 Table 1 Particulars of some of the ships and models for which

F RO M F AI RE D E XP ER IM EN TA L R E S U L Tcomparisons are made in Figs. 2 through 14

0

/

YQ UA DR AT IC M OD EL , E q . 13

Displace-- T n Figure Length, Beam ment, T

C US IC M OD EL , E q . 21 No. Name/Source ft ft tons GM (jt) s

2 Inconstant 337.0 50.3 5200 2.8 15.973 Volage 270.0 42.0 2990 2.9 11.595 Sultan 325.0 59.0 9205 2.5 17.657 King George V 555.0 89.0 25550 5.53 14.7

,/ff8 Vivian 309.0 29.5 1185 2.3 9.19 Nubian 364.0 36.5 2541 2.96 9.8

11 Blagoveshchensky 11.2 2.22 0.214 0.51 1.3412 Lalangas 5.0 0.67 0.0148 0.018 1.35

if 13 Martin 6.67 0.94 0.0333 0.091 1.51

Metric conversion factors:

oL~d'~1 ft = 0.3048 m.1 ton = 0.9 metric tons .

.J

0 2 4 6 8 10 12 14

Fig. 13


Fit of equat ions (13) and (21) to model rol l ext inct ion data pre-

sented by Martin

50

Ij

t'I

tJ'I

'P

o F A IR ED E XP ER IM EN TA LR E S U L T

o

I/,

40

Q UA DR AT IC M OD EL ,E q . 13

dY 30- dn

C UB IC M OD EL ,E q . 21

20

10


Fig. 14 Fit of equat ions (13) and (21) to model roll ext inction data der ived

from forced osci llation experiments by Motora et al

Table 1 indicates some of the particulars of the ships and models

involved in Figs. 2 through 14. Particulars for two ships and twomodels could not be identified. Asmay be noted in Table 1, thedata analyzed originates from a variety of ships and models.Taking ships and models together, the lengths ranged from 5 to550 ft (1.5 to 167.6 m), The type of appendages ranged fromnothing at all through bilge keels on relatively modern forms tobar keels in warships of a 'century ago. The.initial angle in thesallying experiments ranged from 5.5 to 32 deg. Itappears fromthe results that the cubic approximation is qualitatively correct,and can be made to be quantitatively reasonable within the rangeof validity of the quadratic model.

Using the values of coefficients derived in making the fits ofFigs. 2 through 14, numerical comparisons were made in[9] be-tween the values of damping moments implied by the quadraticmodel, equation (3), and the approximation, equation (7). Itwas

found that the maximum error of the approximation might beexpected to be 2 or 3 percent of the value of the function at themaximum experimental roll velocity, This magnitude of error

184

appears comparable with the magnitude of experimental uncer-tainty, and it therefore appears that the approximation, equation(7), can be justified.

The degree of fit to observable data shown in Figs. 2 through14 for the cubic model suggests another approximation to rolldamping, which isquite simply to assume that it follows the cubicmodel [equation (18)J in the first place. In some of the resultsexhibited in Figs. 2 through 13 the fit ofthe cubic model isslightlysuperior. In fact, for the large-amplitude results of Fig. 14the fitis much more realistic.If there are two analytical models which fit the observable data

with roughly the same magnitude of error, the choice between thetwo must be made on a basis other than the fit itself. In the ab-sence of such other considerations, the models may be consideredequally good within the limitations of observable data; outside therange of data, both are extrapolations. Physically, it is expectedthat the damping function will be odd in roll velocity and positivefor positive roll velocity, Both the mixed quadratic and cubicmodels can be made to fit this criterion through a choice of coef-ficients. The primary consideration in the long history of pref-erence for the quadratic damping representation seems to be thatthe drag on a body in a real fluid isproportional to velocity squaredif the velocity is high enough.

Concluding remarks

The reason for embarking on the present work was that thequadratic time-domain representation for roll damping that hasbeen in use for the past century is a serious analytical obstaclewhich, it appears, must be overcome if improvements in tech-niques for the prediction of nonlinear rolling in random seas aresought. The basis for the acceptance of the mixed linear-plus-quadratic time-domain roll damping model is almost entirelyempirical, asare what are taken to be realistic coefficients in this

model.A mixed linear-plus-cubic model, in which the coefficients are

functions of those of the linear-plus-quadratic model, was proposedas an approximation. The results of analyses indicate that this ap-proximation is both quantitatively and qualitatively reasonablewithin the range and scatter of available experimental data.

The additional result that a linear-plus-cubic rolldamping modelfits experimental data about as well as (sometimes better than) thelinear-plus-quadratic model gives rise to speculation that the cubicmodel might be closer to an "equivalent approach" than to an"approximation." .

Ideally, what would be very useful in the prediction of shiprolling statistics in random seas isa prediction framework, analo-gous to the linear framework of St. Denis and Pierson [IJ, in whichthe following could be accomplished:

(a) At least weak nonlinearities could be accommodated for thegeneral multidegree-of-freedom situation.

(b ) Multidirectional seas could be considered as input.

JOURNAL OF SHIP RESEARCH



(c ) The hydromechanic data required could be produced withconventional techniques.

(d ) For economy, the prediction or a major part of it could becarried out in the frequency domain (as well as to take advantageof the accumulating frequency domain descriptions of real seawaves).

(e ) The statisticsof maxima could be estimated with firmly basedtheory in which the possible effects of nonlinearities are accountedfor.

This goal does not appear to be in hand. The present work was

the beginning of a modest attempt to make a contribution to thismuch larger problem -by investigating the applicability of thefunctional series model to single-degree-of-freedom ship rolling[22]. By means of the linear-plus-cubic damping model, somesuccess was achieved in evaluating the spectrum of roll in a waywhich is relatively straightforward in the context of predictionsof nonlinear random response. The practical significance of thepresent results isthought to be that an additional approach to theideal prediction goal isv.alidated in principle. In practice, thereremains a host of problems, the most important of which is thepossibility that the refinements possible in any of the mathematicalprediction frameworks which have been suggested may overreachthe validity of the assumed physical model for rolling throughoutthe practical range.

References

1 St. Denis, M. and Pierson, W. J., Jr., "On the Motions of Ships inConfused Seas," Trans. SNAME, Vol. 61, 1953.

2 The Papers of William Froude, The Institution of Naval Architects,London, 1955.

3 Kaplan, P., "Lecture Notes on Non-Linear Theory of Ship RollMotion in a Random Sea Way," Trans. ITTC, 1966.

4 Vassilopoulos, L., "Ship Rolling at Zero Speed in Random BeamSeas with Non-Linear Damping and Restoration," JOURNALOF SHIPRESEARCH,Vol. 15, No.4, Dec. 1971.

5 Yamanouchi, Y ., "On the Effects of Non-linearity of Response onCalculation of the Spectrum," Trans. ITTC, 1966.

6 Caughey, T. K., "Derivation and Application of the Fokker-PlanckEquation to Discrete Nonlinear Dynamic Systems Subjected to White

SEPTEMBER 1978

Noise Random Excitation," Journal of the Acoustical Society of America,Vol. 35, No. 11, Nov. 1963.

7 Haddara, M. R, "A Modified Approach for the Application ofFokker-Planck Equation to the Nonlinear Ship Motions in RandomWaves," International Shipbuilding Progress, Vol. 21, No. 242, Oct.1974.

8 Vassilopoulos, L. A., "The Application of Statistical Theory ofNon-Linear Systems to Ship Motion Performance in Random Seas,' In-ternational Shipbuilding Progress, Vol. 14, No. 150,1967.

9 Dalzell, J. F., "A Note on the Form of Ship Roll Damping," SIT-DL-76-1887, Davidson Laboratory, Stevens Institute of Technology, May1976, AD-A031 048/261. '

10 Haddara, M. R, "On Nonlinear Rolling of Ships in Random Seas,"International Shipbuilding Progress, Vol. 20, No. 230, Oct. 1973.

11 Lewison, G. R. G., "Optimum Design of Passive Roll StabilizerTanks," The Naval Architect, RINA, Jan. 1976.

12 Takaki, M. and Tasai, F., "On the Hydrodynamic DerivativeCoefficients for Lateral Motions of Ships," Tmns. West Japan Society ofNaval Architects, No. 46, Aug. 1973.

13 Minorsky, N., Introduction to Non-Linear Mechanics, J . W. Ed-wards, Ann Arbor, Michigan, 1947.

14 Lewis, E. V., "The Motion of Ships in Waves," Chapter IX ofPrinciples of Naval Architecture, John P. Comstock, Ed., SNAME,1967.

15 Korvin-Kroukovsky, B. V., Theory of Seakeeping, SNAME,1961.

16 Vossers, G., Behavior of Ships in Waves, Vol. lIC of ReSistance,Propulsion and Steering of Ships, Technical Publishing Company, H.Starn. N. V., The Netherlands, 1962.

17 Blagoveshchensky, S.N., Theory of Ship Motions, Dover Publi-

cations, New York, 1962.18 Gawn, RW. L., "Rolling Experiments with Ships and Models in

Still Water," The Institution of Naval Architects, Vol. 82,1940.19 Lalangas, P., "Application of Linear Superposition Techniques

to the Roll Response of a Ship Model in Beam Irregular Seas," Report 983,Davidson Laboratory, Stevens Institute of TechnololIT, Oct. 1963.

20 Martin, M ., McLeod, G, and Landweber, L., , Effect of Roughnesson Ship Rolling," Schiffstechnik, Band 7, Heft 36, 1960.

21 Motora, S., Shimizu, H., and Nishikido, T., "On the Measuring ofthe Damping Resistance of Roll Through a Large Angle by a Forced Os-cillation Method," Journal Zosen Kyokai, Tokyo, Vol. 100,1957.

22 Dalzell, J. F., "Estimation of the Spectrum of Nonlinear ShipRolling: The Functional Series Approach," SIT-DL-76-1894, DavidsonLaboratory, Stevens Institute of Technology, May 1976, AD-A031055/761.

185

dalzell j f - a note on the form of ship roll damping

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