0PREDICTION 0STATE SHIP ROLL DAMPING- OF THE ART

1.INTRDDOCTIONroll motion is one of the most important responses of a ship in waves.The roll motion of a ship can be determined by analyzing various kinds ofmoments acting on thne ship, virtual and actual mass moments of inertia, rolldamping moment,restoring moment, wave excitation and other momumts caused byother modes of ship motion.Among them,the roll damping moment has been con-sidered to be the most important term that should be correctly predicted.Itis needed not only at the initial stage of ship design to secure the safetyof a ship, but also to obtain a better understanding of ship motions in waves.Since the age of W. Froude, a number of theoretical and experimentalworks has been made concerning the predictions of roll damping and rol 1 motion of ships.The recent d.volopment of the "Strip Method" has made itpos-sible to calculate almost all the terms in the equations of ship motions inwaves with practical accuracy, except for the roll damping.The necessityof obtaining the roll damping of ships has been pointed out in the recent*recommendationsof the Seakeeping Committee of the International Towing TankConference(ITTC)Notwithstanding these efforts,itseems that a completesolution of this problem has not yet been rcached.Difficulties in predicting the roll damping of ships arise from itsnonlinear characteristics (due to the effect of fluid viscosity) as well asfrom itsstrong dependence on the forward speed of ship.Moreover,the factthat these various effects have influences on the value of roll damping thatare of the same order of magnitude makes the problem even more complicatedin the absence of bilge keels.After the classical works by Bryan [ 1 ] and Gawn ( 2 1, we can recognize -an epochmaking period a couple of decades ago in the history of researchon roll damping.Experiments on bilge keels by Martin[ 3 ],Tanaka[ 43,3, theoretical works on the vo.rtex flow near bilge keels bySasaji a ( 6 3, consideration of hull-friction damping by Kato [ 7 1, andand Kato ( 5study ifthe surface-tension effect by Ueno [ 8 1,all of these works ap-pear in this period.Furthenior3, we can cite here Hishida's theoretical studies[ 93 on thewavemak.Lng roll damping due to hull and bil.ge keels and flanaoka's mathematicaljL-1-

-2-formulationfor[10]the wave system created by an oscillatory motion ofan immersed flat-plate wing with low aspect ratio.Also, an extensiveseries of free-roll tests at zero ship speed for ordinary ship hull formswas carried out by Watanabe and Inoue [11]Yamanouchi[12].and tests at forward speed bySome of the results of these works have often been usedeven up to the present time or have given background to recent works.Thisfact cannot but remind us again of the difficulty of treating the ship rolldamping problem rigorously.Itcan be said that the recent works started about a decade ago, mainlyassociated with the experimental check of the accuracy of the strip methiod.Much data on radiation forces acting on ship hull,including roll damping,have been accumulated through the forced oscillation tests carried out byVugts(13],et al.Fujii and Takahashi[14],Takaki and Tasai[15],and Takezawa(16].These experiments have clarified that there are stillconsiderable dif-ferences between measured values of roll damping and those predicted by existing methods.In this period much effort has also beei Tuae for obta-iningsnip roll damping,for example, works by Bolton [17],Lugovski et al. [19],Lofft (18), andconcerning the effect of bilge keels, Gersten's studies[20] on the viscous effects, and free-roll experiments by Takaishi et al.(21) and Tanaka et al.[22].Moreover,what should be noted here is the extensive and systematic worksin Japan that have recently been carriedJapan Shipbuilding Research Association,SRl08,succeeded by SR125,out through the cooperation of theespe.uially in the Coomittees of131 and 161 [23].In the prediction method ofship roll damping considered there, damping is divided into several components, for instance, friction, eddy, lift, wave, and bilge-keel components.Then the total damping is obtained by simzing up these component dampings predicted separately.IThis attempt appears to have had a certain success forordinary ship hull forms.The objective of this article isart indamping. . . . .to describe the present state of thethese recent attempts as well as other existing formulas for ship rollFurthermore,for convenience inship design,. .itisintended that

-3-jthe available expressions and formulas should be described in full detail asmuch as possible, so that their values can be calculated promptly once theparticulars of a ship are given.In Chapter 2 the various methods of representation for roll dampingcoefficients and their relationships are stated and rearranged inequivalent linear damping coefficient.damping,Then,for prediction methods of rollsimple methods are introduced in Chapter 3, including the use of datafrom a regression analysis of model experiments.1%terms of antreatment for component dampings iscomponent are fully described there.dicted total damping, which isIn Chapter 4, the neweststated, and available formulas for eachComparisons are made of measured and pre-the sum of the component dampings.Chapter 5 concerns the prediction methods Ror ship roll motion.itisthenot the full present state of the art.The description isHowever,limited toproblem of how to use the formulas of nonlinear roll damping in order toobtain the solution of the roll equation of motion in regular or irregularsea.Finally an example of a FORTRAN statement of a computer program for thecomponent dampings isgiven in the Appendix.

REPRESENTATIONi2.OF ROLL DAPIPING COEFFICIENTSMany ways of representing roll damping coefficients have been used, de-Ipending on whether roll damping is expressed as a linear or nonlinear form,jwhich form of the non-dimensional expressions is to be used, and by whatexperimental method its value was measured, for instance, forced-roll testr-or free-roll experiment.Some of the expressions most coimonly used areintroduced here and the relationships among them are reviewed and rearrangedin terms of a linearized damping coefficient.2.1Nonlinear Damping Coeffitient sThe equation of roll motion has recently been expressed as a three-degree-of-reedom form, including sway and yaw motions simultaneously."However, inorder to limit the discussion here to the problem of nonlinear roll damping,we can write down the equation of the toll motion of a ship in the followingsimple single-degree-of-freedom form:J4- B4 (In Eq. (2.1),p C4,4M(w t).(2.1)represents the roll angle (with the amplitude4A3'Athevirtual mass moment of inertia along a longitudinal axis through the centerof gravity and Co the coefficient of restoring moment, which is generallyequal to W'GM (W the displacement weight of the ship and GM the metacentricheight).Furthermore,Nstands for the exciting moment due to waves orexternal forces acting on the ship,Finally,B4wu the radian frequency andtthe time.denotes the roll damping moment, which is now considered.Although only the main terms of roll motion have been taken into accountin Eq. 22.1,,coupling terms being neglected,itcan be said that Eq. (2.1)almost corresponds to that of three degrees of freedom when we consider thewave excitation term Mas the Froude form with a coefficient of effectivewave slope.This is because the concept of the effective decrease of waveslope turns out, after Tasai's analysis [24], to correspond to the effect ofthie sway coupling terms.We can express the damping mnme'tandjjinthe formB,as a series expansion of I

-5-B) B2 JP2. (2.2)B1The coefficientswhich is a nonlinear representation.B2 ,,in.In otherEq. (2.2) are considered as constants during the motion concerned.words, these values may possibly depend on the scale and the mode of theAmotion, for instance, on the amplitudewhen theWand the frequencyship is in a steady roll oscillation.Dividing Eq. (2.1) byAwe can obtain another expression per unit,mass moment of inertia: y Y 2t(2.3) m (It),nwhereBaAB2B3AAV(2.4)jWn 2TIn Eq.(2.4),rathe quantitiesTw.AOtn ,mandTnrepresent t-he natural frequencyand the period of roll, respectively.A term of the formEq. (2.2). /i lmight be added to the right-hand side ofThis term corresponds to the effect oflevel of the ship hull.Uenosurface tension at waterf 8 ] investigated this effect and concludedthat the surface tension might cause a considerable error in the values ofthe damping coefficients when a small model is used inoscillation.However,this effect issmall amplitude ofnot considered hereafter,because thesurface tension depends strongly on thn condition of the painted surface ofthe model hull as well as on that of the water surface,be neglected inand because it canthe case of roll amplitude with moderate magnitude for aship model of ordinary size.To obtain the values of these coefficients of nonlinear damping directlythrough a steady-stateforced-roll experiment,in which4Aspecified, we would probably need numerical techniques to fitof the assumed equation to the measured data.andwarethe solutionSuch an attempt does not seem

-6-to have been done.Instead, the usual way that has been taken is to assumesome additional relations concerning, say, energy consumption, linearity of4damping and its independence of2.2rAr which will be described later.Equivalent Linear Damping CoefficientsSince it is difficult to analyze strictly the nonlinear equation statedin the preceding section, the nonlinear damping is usually replaced by acertain kind of linearized damping as follows: Be The coefficientBe(2.5)denotes the equivalent linear damping coefficient.Although the value ofBedepends in general on the amplitude and thefrequency, because the damping is usually nonlinear, we assumeBeisconstant during the specific motion concerned.There are several ways to express the coefficientBin terms of theenonlinear damping coefficients B I, Band so on.The most general wayis tu atjume that th e energy loss due to 2 damping during a half cycle of rollAis the same when nonlinear and linear dampings are used (24].If the motirnissimple harmonicat radian frequencydBe Bl 8 wABFor more general periodic motion,firstEq.32 (2.6)(2.6) can be derived by equating theterms of the Fourier expansions of Eqs.(2.5) and (2.2)[15].For convenience in analyzing the equations of lateral motions,thenondimensional forms of these coefficients are defined as follows:-VpVB2wherep,Vbreadth of ship,sional form:P2gandB' V2-'lfor i 1,2,3(2.7)2gstand for fluid density, displacement volume andrespectively.Then Eq.(2.6)takes the following nondimen-

-7-8 w(2.8)B Corresponding to Eq. (2.3), we can define an equivalent linear dampingBe/2Aa.Akcoefficientper unit mass moment of inertia:4 i4 A i, 40 --Since these coefficientsstilllinear termsaeandhave dimensional values(except for the second,a1In2aee2carj'othersa.case of irregular roll motion,tion of the roll damping expression.[27](23],(2.10)-there isanother approach to lineariza-After the works of Kaplan (26]inandwe assume that the difference of the damping moment betweenlinearized and nonli-near forms can be minimized insquares method.6(2.9).the following dimensionless forms are often used, especially for theF),itsYNeglecting the termthe sense of the leastB 3 for simplicity, we define the discrepancythe form:Thrq we can miniuze(2.11)B1 4; B2 4 (Be;)6E{6 2 },the expectation value of the square of6during the irregular roll motion, assuming that the undulation of the rollangular velocity,cients Be,B*andisB2subject to a Gaussian process and that the coeffiremain constant:(62-2 (-2B--1aBe2E{;e0(2.12)2After some calculations we can reach the formBei Bi 8where the factorItisa.B(2.13),represents the variance of the angular velocityclaimed in recent works [23]roll behavior in irregular seas.that this form isuseful for analyzing

-8-Moreover, as an unusual way of linearization, we can equate the nonlinearexpression to the linear one at the instant when the roll angular velocitytakes its maximum value during steady oscillation:Be B1 W4 AB 2d(2.14)This form seems to correspond to a collocation method in a curve-fittingproblem, whereas Eq. (2.6) corresponds to the Galerkin approach.isSince therea difference of about 15% between the second terms of the right hand sidesof Eqs. (2.6) and (2.14),of roll motion.lationthe latterform may not be "alid for the anialysisBut it may be used as a simple way of analyzing forced-oscil-test data to obtain the values of these coefficients promptly fromthe time history of the measured roll moment.However, the most common way to obtain these nonlinear damping coefficients through forced oscillation tests is, first, to find the equivalentlinear coefficientsystem isBeBein Eq. (2.5) by assuming that the forced-roll-testsubject to a li.near equation, and, second, to fit Eq.(2.6) to thevalues obtained by several test sequences with the amlitudeThen we can obtain the values of these damping coefficients,BI.A,varied.B2andso on, which are independent of the amplitude of roll oscillation.Ittheshould be noted here that this condition,independence ofTherefore we mightamplitude, was not stated when we derived Eq. (2.6).obtain different values of the coefficients from the original ones ifcoefficients, especiallyB,2in the presence of bilge keels.should depend on thetheamplitude, particularlyWe should keep these things in mind when weuse a formula like Eq.(2.6).2.3Extinction CoefficientsA free-roll test is probably the simplestof ship or model.Inway to measure roll dampinga model test, sway and yaw motions are usually restrainedto avoid the effect of the horizontal motions.On the other hand, heaveand pitch motions are often kept free to avoid the error due to the sinkageforce in the presence of forward speed, although itmake the vertical motions as small as possible.taken through the center of gravity of the model,is of course desirable toThe roll axis is usuallythe radius of gyration of

-9-the model isadjusted to the value of the actual ship considered,restoring moment leverGMisand thealso measured through a static inclinationtest.In areleased.free-roll test, the model is roiled to a chosen angle and thenThe subsequent motion is measured. Denote by ln the absolutevalue of roll angle at the time of the n-th extreme value.extinction curve expresses the decrease of*nThe so-calledas a function of mean rollangle. Following Froude and Baker, we fit the extinction curve by a thirddegree polynomial :whe re2 alm bAlAwe -n-i -4'm - c43(deg.)(2.15),n, lni(215a)]/2(2.15b).InThe angles are usually measured in degrees in this process.The coefficients a-,bandc are called extinction coefficients.Therelation between these coefficients and the damping coefficients can bederived by integrating Eq. (2.1) without the external-force term over the timeperiod for a half roll cycle and then equating the energy loss due to dampingto the work done by restoring moment.The result can be expressed in theform4CB8lp"B 4 2lpmB 3(rad).(2.16)Comparing Eq. (2.16) with Eq. (2.15) term by term, we can obtain the relations21804 Wn4(2.17)3180 2-)CoiT3,,ncBv337 0" nY8I.

-3.0-It should be noted here again that the condition for the validity of Eq.(2.17)isthat the coefficientsB1pendent of the roll amplitude.,B2and,ccD2remain valid.varies with roll amplitude.Only the part ofis related to the coefficientbe inde-As we can see in the later chapters, theeffect of bilge keels appears mainly in the termvalue ofB.should,B,2b.B,and, further, the,-In such a case, Eq. (2.17) will notwhich is independent of the amplitudeThe other part ofB2that is inverselyproportional to the amplitude will apparently be transferred tc the coefficienta , and the part proportional to the amplitude will appear inc. Inplace of a term-by-term comparison, therefore, it will probably be reasonableto define an equivalent extinction coefficientthe equivalent linear damping coefficientae aeBeand to compare it withas in the forma bm c-Be(2.18)We are also familiar with Bertin's expression (25],which can be writ-ten in the form(dcg.)NeThe coefficientNcan be taken as a kind of equivalent nonlinear expression,and it has been called an"N-coefficient."NWbThe value ofNand so on,As seen from Eq. (2.15), 0m(deg.)depends strongly on the mean roll angleexpression is always associated with theN20(2.19)whereN10is4rthe value of(2.20)*m ' so that itsvalue, being denoted asN whenOm100, etc.N1 01

3.PREDICTION OF aLL DAMPING:I.SIMPLE METHODWhen the principal dimensions of a ship form are given,the most reli-able way to obtain the roll damping of the ship at present time seems to beSince the scale effect of the damping isto carry oat a model experiment.considered to be associated mainly with the skin friction on the hull, whichmakes a small contribution to total damping,the data from the model testscan easily be transferred to the actual ship case by using an appropriatenondimensionalform of roll damping,for instance,Eq. (2.8).If model-test data are not available, it is necessary to estimate theroll-damping value by using certain kinds of prediction formula.two different ways of estimation at present time.cal,There areOne is to obtain an empiri-experimental formula directly through the analysis of model tests onactual ship forms.The other is to break down roll damping into severalcomponents and then estimate the value by summing up the values of thosecomponents individually predicted.The latter is considered to be more rational, so that it has become therecent trend of approach in Japan.To begin with, however,the former approach are described inthis chapter insome examples oforder to know themag-nitude of ship roll damping easily.3.1Wataiiabe-Inoue-Takahashi FormulaA couple of decades ago,Watanabe and Inoue[11]established a formulafor predicting the roll damping of ordinary ship-hull forms at zero advancespeed in normal-load condition, on the basis of both an extensive series ofmodel tests and some theoretical considerations on the pressure distributionon the hull caused by ship roll motion.modified slightly by them (28]ship forms,Their original formula has beenso as to be applicable to a wider range ofincluding ships w:,ith large values of block coefficient.Takahashi[29] proposed a form of forward-speed modification multiplierto be applied to the value at zero ship speed,speed effect on roll damping.thus expressing the advance-We may call this approach the Watanabe--Inoue-Takahashi formula.-1-

-12-This can be expressed in terms of an ecquivalent linear damping coefficient of the formnBe - Beo [l 0.8{1-exp(l-OFn)}whereBeostands for the value ofbe expressed in terms of the extinction coefficientsavalue canItszero ship speed.atBe(3.1)b , as follows:and2aeABeo 2nA(a

0STATE OF THE ART Professor Yoji Himeno This research was carried out In part under the Naval Sea Systems Command . 44Ap 8 Department of Naval Architecture 4L A and Marine Engineering - College of Engineering The University of Michigan . (I BI/2A (2.4) j 5e Be/2A 4 (2.9) iB B l iA ( 2.4 ) y B3 /A (2.4)Cited by: 216Page Count: 87File Size: 3MB