October 1962 THE IMPULSE RESPONSE FUNCTION AND SHIP MOTIONS. W.E. Cummins. Report 1661

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3 Wwllfil IiUhl THE IMPULSE RESPONSE FUNCTION AND SHIP MOTIONS by W.E. Cummins This paper was presened a he Sympsium n Ship Thery a he Insiu flir Schiffbau der Universii Hamburg, January 192. Ocber 192 Repr 11

4 , ' AIMM I r Im=YiumII l I AlU1 uinm ABSTRACT Afer a review f he deficiencies f he usual equains f min fr an scillaing ship, w new represenains are given. One makes use f he impulse respnse funcin and depends nly upn he sysem being linear. The respnse is given as a cnvluin inegral ver he pas hisry f he exciing frce wih he impulse respnse funcin appearing as he kernel. The secnd represenain is based upn a hydrdynamic sudy, and new frms fr he equains f min are exhibied. The equains resemble he usual equains, wih he addiin f cnvluin inegrals ver he pas hisry f he velciy. Hwever, he cefficiens in hese new equains are independen f frequency, as are he kernel funcins in he cnvluin inegrals. Bh represenains are quie general and apply ransien mins as well as peridic. The relains beween he w represenains are given. The reamen cnsiders six degrees f freedm, wih linear cupling beween he varius mdes. UC~Y~~ I I I I I III a II-~I~

5 h, The Impulse Respnse Funcin and Ship Mins W. E. Cummins Inrducin Jus ver a decade ag, Weinblum and S. Denis') presened a cmprehensive review f he -sae f knwledge a he end f wha we may call he "classical" perid in research n seakeeping. Sn afer, S. Denis and Piersn 2 ) pened he "mdern" perid (sme wuld prefer call i he "saisical" perid). The sudies f he frmer perid were primarily cncerned wih sinusidal respnses sinusidal waves, bu he inrducin f specral echniques pened he dr fr he discussin f respnses randm waves, bh lng and shr cresed. The cnsrucin f he specral hery n regular wave hery as a fundain delighed us all, as i presened an apparen jusificain fr he admiedly arificial sudies f he "classical" perid. The aciviy during his las decade has been specacular, wih five majr and many minr faciliies fr seakeeping research being pened. Hundreds f mdels have been esed, many full scale rials have been run, and here has even been sme real grwh in ur knwledge f he subjec. In paricular, he specral l has been sharpened and empered by he empiriciss, and he analyss have made impran advances wih he raher frighful bundary value prblem. In fac, we have all been frging ahead s rapidly ha we appear have frgen ha we are wearing a she which desn' quie fi. The ccasinal pain frm a misplaced e is ignred in ur general enhusiasm fr prgress. The "she" which I refer is ur mahemaical mdel, he frced represenain f he ship respnse by a sysem f secnd rder differenial equains. The she is squeezed n, wih n regard fr he shape f he f. The inadequacy f he she is eviden in he disrins i mus ake if i is be wrn a all. I am referring, f curse, he frequency dependen cefficiens which permi he mahemaical mdel fi he physical mdel (if he exciain is purely sinusidal, ha is). Bu wha happens when we dn' have a well defined frequency? The mahemaical mdel becmes alms meaninglegs. True, a Furier analysis f he exciing frce (r encunered wave) permis he mdel be reained, bu physical realiy is alms ls in he infiniy f equains required represen he min. Le us cnsider his mahemaical mdel briefly, and resric urselves a single degree f freedm. T be cmpleely fair, le us cnsider a pure, sinusidal scillain. The frcing funcin (if he sysem is linear) will be sinusidal, and can be brken in w cmpnens, ne in phase wih he displacemen and ne 9 u f phase. We furher divide he in-phase cmpnen in a resring frce, prprinal he displacemen, and a remainder. The laer we call he inerial frce, and rea i as if i were prprinal he insananeus 1) References are lised a he end f he paper accelerain. The u-f-phase cmpnen, which prvides all he damping, we rea as if i were prprinal he insananeus velciy. We can nw wrie an equain, which has he appearance f a differenial equain, relaing hese varius quaniies: a (w) i + b (w) i + c (w) x = F sin ( + E). Bu a differenial equain is suppsed relae he insananeus values f he funcins invlved. If he peridic min cninues, his cndiin is saisfied. Of curse, i culd jus as well be saisfied by he equain b + (c- a ') x = f () r mre generally (a + d) + bi + (c + d) x = f () where d is arbirary. These are all equally valid mdels. One f hem is be preferred nly if i ruly relaes he displacemen and is firs and secnd derivaives he exciain in sme mre general way. Bu suppse f () were be suddenly dubled. Wuld he insananeus accelerain be given by 2f()-b()i-c(c)x a (w) In general, n! Or suppse he ampliude f he scillain be suddenly increased. Wuld he u f phase cmpnen f f (), immediaely afer he change, be equal bi? Again, in general, n. Thus, a bes, b (w) mus be cnsidered as a sr f "apparen" damping cefficien, a (w) as an "apparen" apparen mass, and he physical significance f bh is bscure. When he scillain cnsiss f several cupled mdes, he s-called cupling cefficiens are equally cnfused and cnfusing. If we resric urselves a phenmenlgical invesigain f hw a given ship behaves in a given wave sysem, hese difficulies d n cncern us. We simply measure respnses knwn waves. Ms f he wrk ver he pas decade has been f his naure, and much f i has been excellen. Hwever, sner r laer, we are required cnsider n "wha" bu "why," and a mre analyical echnique is demanded. The phenmenlgical sudy can ell us he effec f a change in ship lading n seakeeping qualiies nly afer we have measured i; here is n basis fr quaniaive predicin given he resuls fr ne gyradius. And he effec f a change in frm is presened as an islaed resul, unrelaed and unrelaable he gemeric parameers invlved. We are driven he use f he mdel discussed abve in an aemp clarify he relain f cause and effec. Bu such a pr mirrr f realiy is f lile value, and in fac can d much harm. I am n he firs raise his issue. The difficulies are well knwn and a number f wriers have discussed hem. In paricular, Tick s ) has vigrusly argued agains ur usual pracice and has prpsed a mdel which is very clse he ne which will be exhibied here. His case is based slely upn he general characerisics f linear sysems, while, we shall ake ad-

6 vanage f he principles f hydrdynamics ie he mdel he phenmena. Mre recenly, Davis') has prpsed a rainal apprach frm he pin f view f saisics. This is suggesive, paricularly since i was he specral hery f saisics which firs gave weigh he invesigain f respnses peridic waves. Briefly, he specific bjecives f his paper are: 1. T exhibi a mdel which permis he represenain f he respnse f a ship (in six degree f freedm) an arbirary frcing funcin (wih exciain in all six mdes). The mdel will n invlve frequency dependen parameers. 2. T separae he varius facrs gverning he respnse in clearly idenifiable unis, he effec f each be separaely deerminable. Thus he effec f gyradius will be separable frm added mass. The added mass will be relaed nly inerial frces and mmens. The naure f he damping frce will be exhibied. The effec f cupling will be derivable and he effec f "uning" upn cupling will be deerminable. In his paper we shall n cnsider he cmplemenary prblem f he relain f he exciing frce he inciden wave sysem. This prblem is equally basic, and when i has been adequaely reaed, we will begin have a saisfacry framewrk fr he inerpreain f ur empirical sudies. The Impulse Respnse Funcin The basic l which will be used in his sudy is an elemenary ne, widely used in her fields and well knwn all engineers: he impulse respnse funcin. I is difficul undersand is neglec in ur field. Perhaps as Tick suggess, i is because waves lk sinusidal. Fr any sable linear sysem, if R (), he respnse a uni impulse, is knwn, hen he respnse f he sysem an arbirary frce f () is x () = JR (-) f () d x () = - R (r) f (- ) d. The nly assumpin required (aside frm cnvergence) is lineariy. In he presen cnex his is, f curse, a very srng assumpin, and he puriss will argue ha i implies a hin ship r he equivalen. Hwever all experimenal daa indicae ha he assumpin is a gd wrking apprximain fr small mderae scillains f real ship frms. We shall hyphesize ha he assumpin hlds absluely. Le xi, (i = 1,..., ) be displacemens in he six mdes f respnse: x, = surge (psiive frward) x2 = sway (psiive pr) x 3 = heave (psiive upward) x4 = rll (psiive, deck sarbard) x 5 = pich (psiive, bw dwnward) x g = yaw (psiive, bw pr) Le Rij () be he respnse in mde j a uni impulse a = in mde i. Ne ha Rij () des n necessarily equal zer, hugh in a damped sysem which is n unsable, i will rdinarily be finie. In mdes wihu a resring frce (sway, surge, and yaw), he impulse respnse will asympically apprach sme value. Fr her mdes, Rij () =. If he (fi()) are an arbirary se fr frcing funcins, he crrespnding respnses are xj () = 1 f Rii (r)fi ( - r) dt. [2] i=1 Thus, he marix (Rij ()) cmpleely characerizes he respnse f he ship an arbirary exciain. Befre we g n, le us cnsider he relain f hese funcins he usual cefficiens. Firs cnsider he case where he mdes are uncupled. Le fi () = Fi cs ( - Ei) [3] where Ei is a phase angle whse value will be assigned laer. where xi () = Fi J Rii (r) cs [w ( - T) + Ei] dr = Fi [cs (w + Ei) f Rii cs wrdt + sin (w + Ei) f Rii sin rdr] = Fi [Riic (w) cs (w + Ei) + Rii s (w) sin (w + Ei)] [4] Riic (w) ; f Rii (r) cs rdr [5a] Rii s (w) = J Rii (T) sin wrdt [5b] are he Furier csine and sine ransfrms f Rii (). We shall call hese ransfrms he frequency respnse funcins. We make he furher reducin Taking xi () = Fi [(Riic cs E i + Rii s sin El) Cs w + (Rii s cs Ei - Rii cs Ei) sin w]. an Ei = Rii / Rii c we have x i () = F i [(Riis) + Riic) ]f ls cs. Als f () = F i (RiiC cs - Rii s sin w) [(Riis)' + (Riic) l'/ Nw cnsider he usual represenain [] [7] aiki + biii + cixi = fi (). Using he x i and fi frm [7] and [8], i is easily seen ha ai = l/" ci- Ri [1a] (Rifc)* + (Riis) Rii s b i = [1b] [(Riic)2 + (Riis) ' ] A mre useful relainship is bained by seing ei = in [4]: x i () -- = Rii c () cs + Rii s (w) sin. [11] Fi Thus Rii c and Rii are he ampliudes f he in-phase and uf-phase cmpnens f he respnse a uni ampliude frcing funcin f frequency w. The impulse respnse funcin is relaed hese funcins by Ri i (r) 2 $ Ric () csr d a - f Rii s (w) sin w d [8] [12] using he Furier inversin frmulas. Ne ha Rii c and Rii are uniquely relaed. If ne is knwn, hen by [12] and [5], he her is deermined. Equain [11] can als be wrien X i () [(Rii) 2 + (Riis)]l a cs [ -- Ei (w)] Fi [13] where Rii s (O) an E = [14] Riic (O) Thus, he respnse fllws he exciain by he phase an- 1 (Rii s / Rii c) and has he ampliude [(Rii's) + (Riic) 2 1 '/ '. " I -I I ~ CYC - -LI--~P*Y

7 _1_11 ~ ~_ IIU The respnse fr a given frequency, as deermined by he pair f funcins Rii, Ri c, r alernaively, he pair [(R, 1 i)2 + (Riic) 2 ] /, an- (Ris /R i), is a mapping in he frequency dmain f he uni respnse funcin, which is defined in he ime dmain. As equains [4] and [11] permi us pass frm eiher dmain he her, he w represenains are cmpleely equivalen. Viewed in his way, he frequency respnse funcin is a meaningful, useful cncep. I is nly when we ry aribue a deeper meaning i, by imbedding i in a false ime dmain mdel, ha we creae cnfusin. Nw cnsider he mre general, cupled sysem, wih exciains in a single mde f he same frm as given in equain [3]. Then xj () = F, [Rijc cs ( + Ei) + Rij s sin ( + E,)]. [15] If we cnsider he usual represenain X (ajk j + bjk j + Cjk Xj) = fk() j=1 where fk () = fr k + i, we can develp a sysem f equains in he unknwns, ajk, bjk. (The cjk are assumed knwn frm saic measuremens.) All 72 f hese unknwns are presen, in principle, excep where mdes are uncupled. T deermine hem, i is necessary cnsider he respnses exciains in each f he mdes separaely. We hen have enugh equains, if we separae he in-phase and u-f-phase cmpnens, deermine he cefficiens. We have n need fr hem here, s we defer furher discussin unil we face a clsely relaed prblem. I is nly significan ne ha hey can, in principle, be deermined frm he se f impulse respnse funcins, and herefre hey cnain n infrmain which is n derivable frm hese funcins. Seing Ei = in [15], we have he sysem xj () - Rije cs + RijS sin. [1] Thus, Rije and Rij s are he ampliudes f he in-phase and u-f-phase respnses in he jh mde uni ampliude exciain in he ih ' mde. As befre, Rij () = 2 Rij cs dw [17a] + [R(+) - R( f (-) d+x(.) [2] The secnd inegral cnverges, s his expressin prvides a usable definiin f xj (). Nw le fi () = cs. Afer an inegrain by pars, we have xj () = 1 / Ri (r) sin ( -r ) dr + J [Rij ( + r) - Rij (r)] cs r dr + xj (). Our nly cncern is wih he scillary cmpnens f x i. These are easily deermined by cnsidering he asympic frm f he abve expressin as becmes large. Rij ( + )-*Rij (), and he secnd inegral becmes cnsan. If we se hen Xj () = lim [Rii ( + r) - Rij (r)] cs dr --* 1 xj () = (- Rij cs + Rije sin ) [21] where Rij and Rij c are he sine and csine ransfrms f Rij (). We knw ha xj () is sinusidal, wih frequency. Therefre, his expressin hlds n nly fr large bu fr all. If we define Rije = - Rij8/O [22a] Rij s = RijCe/ [22b] hen [1] sill hlds. Ne hwever, ha Rije and Rijf are n lnger ransfrms f Rij because hese d n exis. Neverheless, an inversin is sill pssible. Cnsider f [Rij (r) - Rij ()] cs r dr S [Ri (r) - Rij ()] sin r - Rij(r) sin r dr _ Ri j S / = Rije Tha is, Rij c is he csine ransfrm f [Rij () - Rij ()] and - Rij s sin d) and xj () _ [(Rij c) + (Rijs)],I cs (w - Ej) where F: an Ej = Rij / Rij. [17b] [18] [19] We have passed ver he quesin f cnvergence f he inegrals in equains [2] and [5]. Cnsisen wih ur hyphesis f lineariy, we shall assume Ifi () I is bunded. There will hen be n difficuly unless f JRij (T) I d des n exis. Unfrunae- O ly, in hree mdes here are n resring frces (r else hey are negaive), and evidenly sme care is needed in reaing hese cases. A negaive resring frce implies an unsable sysem, which wuld be beynd he scpe f his analysis. Hwever, he case in which Rij appraches sme nn-zer bu finie limi can be reaed. The divergence f he inegrals can be vercme if we arbirarily assign a value xj (). We frmally wrie x i () = Rij (-r) fi (r) drt Ri (--r) fi (r) d-r + xj () - - r xi () = J Rij (r) fi ( - r) dr Rij () = Rij () + Leing equal zer, - - Rij c cs d 5 2 Rij () Ri (cs - 1) d. When Rij () =, his reduces [17a]. Similarly, 3 [Rij (T) - Rij ()1 sin r dr - Ri j ( ) / + Rij cs r d - [ijc - Ri i ()] / and Rij () = Rij () + Ris - Rij )] sin O d [23] [241

8 = Rij (c) 1 2 sin s d R,j" sin w d e since 2 Rij" sin w dw sin dw 2 Therefre, [17b] hlds even when Rij (c) #. If Rijc and Rij 8 are knwn, i is n difficul deermine wheher r n Rij () =. Equain [23] gives Rij () in erms f Rij c. Als 2 C Ri () = lim R" sin w dw = I 2 fc sinw d -l)m dxw = ij c () = lim RijB w-+ [25] using a well knwn herem in Furier ransfrm hery (Reference 5, page 12). When he marix f impulse respnse funcins is knwn, ur firs bjecive f finding a represenain f he ship respnse which is free f frequency dependence is achieved. These funcins, which we shall cllecively call he impulse respnse marix, can in principle be deermined experimenally. Equains f Min The ransien respnse f a ship has been cnsidered by Haskind'), wh aemped an explici sluin f he bundary value prblem. This, we shall n ry, as we are cncerned nly wih finding an apprpriae frm fr he equains f min use as a basis fr he inerpreain qf experimenal resuls. We d n agree wih cerain f Haskind's hypheses, and ur resuling equains differ frm his in several impran respecs. Glva 7 ) carried u an experimenal invesigain f he declining scillain min in pich. Hwever, Glva was n aware f he equivalence beween he ransien and seady sae respnses which we have jus discussed, s he aemped nly mach he cefficiens derived frm he ransien experimen, a he frequency f he declining scillain, wih hse frm a frced scillain experimen a his frequency. He was handicapped because f he anmalus behavir f he curve f declining ampliudes. Fr a simple harmnic scillar, his curve is a sraigh line when pled n semi-lgarihmic paper. His curves depared radically frm such a paern. He recgnized ha his implied ha he mahemaical mdel was fauly, and aemped, wih sme success, fi his resuls wih frms based n Haskind's sudy. Mre recenly, Tasai s ) has perfrmed declining scillain experimens in heave, using w dimensinal frms. His resuls are n significanly differen frm hse f Glva. He mached his resuls a he measured frequency wih Ursell's hereical resuls fr frced scillain. The agreemen is quie gd. Case I - N Frward Speed Le he ship be flaing a res in sill waer. We use a sysem f crdinaes (S,, s), fixed in space, wih rigin in he free surface abve he cener f graviy f he ship. A ime =, we suppse he ship be given an impulsive displacemen Axj in he jh mde. The ime hisry f his impulse is n significan, bu fr purpses f visualizain, i may be cnsidered cnsis f a mvemen a a large, unifrm, velciy vj fr a small ime A, wih he min erminaed abruply a he end f his ime inerval. Then Ax = vj A. During he impulse, he flw will have a velciy penial which is prprinal he insananeus impulsive velciy f he ship. I may, herefre, be wrien vjcj, where Vj is a nrmalized penial fr impulsive flw. V, will saisfy he cndiins: where sj n * ij i4 = n S = - Vjj/ n = s, n S =rxn - i s S = surface f he ship j= 1, 2, 3 i =4,5, n = uwardly direced uni nrmal ii = uni vecr in jh direcin r = psiin vecr wih respec c. g. f ship. [2] [27] [28] I is well knwn') ha he abve prblem is equivalen ha bained by reflecing S in s = and aking he surface cndiin ver he reflecin be he negaive f ha ver S. The sluin he Neumann prblem fr he flw uside his cmpsie surface is als he sluin he given prblem in he lwer half-space. Fr nn-pahlgical surfaces, he sluin exiss, and in fac can be cmpued by means f mdern, highspeed equipmen.") During he impulse, he free surface will be elevaed by an amun ATij = -vj A = -- x. [29] a 3 T Afer he impulse, his elevain will dissipae in a radiaing disurbance f he free surface, unil ulimaely he fluid is again a res in he neighbrhd f he ship. Le he velciy penial f his decaying wave min be qj () Axj. I mus saisfy he iniial cndiins cj (&2, 3s, ) = [3] Axj, = ga'j g -Axj n 3 = a axs r a8l)j (,,) a93(j, 2, ) a as Aferward, i saisfies he usual free surface + g a =. a2 a 3 and he bundary cndiin n S -. [311 [32] [33] We may ake his hld n he riginal psiin f S, nly inrducing errrs f higher rder in Axj. This is a classical prblem f he Cauchy-Pissn ype, and here exiss an exensive lieraure n he subjec. Wih cndiin [33], i is mre difficul, by an rder f magniude, han he Neumann prblem. We assume ha i has a sluin. y~li~ I --- I - ' 13 ' lci -- I'-"I -~ - I

9 _ ~ )II/ Nw le he ship underg an arbirary small min in he jh mde, xj (). T he firs rder, he velciy penial f he resuling flw will be simply = J Q isk d + sk 5 d j (- ) (r) dt S= i + (- () d. [341 I is eviden ha he bundary cndiin n S is saisfied n he equilibrium psiin f S, as he firs erm prvides he prper nrmal velciy and aqj / an = n his bundary. Bu als, he value f ae / n n he acual psiin f S will nly differ frm is value n S by erms f secnd and higher rder in xj and is derivaives, s we may cnsider ha [341 hlds n he acual psiin f he hull. T verify ha -he free surface cndiin is saisfied, firs ne ha a8 di. di aj () W + q ( ) + Xr. a 2 d d a a (- ) J () drt. By [2] and [3], n 3 = his reduces Als, a a() j - a j (- ) (T) dt. a' a. Oa az:j _ avi +_ ' " (-r(3(- ) d i3 (T) dt. Ic Subsiuing hese in he free surface cndiin a~ + ae ap () + g a" a 3 \ a r s + ka + g -/ i () dr = [35] by (31] and [32]. Thus, his cndiin is als saisfied, and is he required penial. The frmula [34] is a hydrdynamic analg f [1]. I is quie general, and can, fr insance, be used find he velciy penial due a sinusidal scillain wih arbirary frequency. I is, f curse, necessary knw he funcin cqj (), and his presens unpleasan difficulies. In his sudy we are cnen ha qp, () exiss, and hese difficulies d n cncern us. Of mre imprance han he velciy penial is he frce acing n he bdy. The dynamic pressure in ur linearized mdel is simply ae r P = 3E j + q )j ()j aj ( )i (r) d Sa jj + s f a(i (--) a j ( ) dt [3] The ne hydrdynamic frce (r mmen) acing n he hull in he kh mde is hen given by where = -j mjk + n j (T) d a (- )k d a skd -~~ i, ~j -T = ii mjk + Kik ( r mjk 8 ) *j (T) d Sk d [37] [38] Kik () = a ( sk d. [39] Sa We can nw wrie he equains f min f he ship which is subjeced an arbirary se f exciing frces, {fk()}. These will be I [(mj, + mk) mjk Rk + Xj + JKjk (--) xj (T) dt] = fk () j= 1 -- [4] where mj = ineria f he ship in he jh mde Cjk xj = hydrsaic frce in he k h mde, due displacemen xi in he jh mde bjk = Krnecker dela ( b jk = 1 if j = k, = if j = k). Case II - Ship Underway The case f he ship experiencing small scillains abu a reference psiin f mean unifrm velciy is much mre cmplex. A pair f funcins, pqj and qj, n lnger suffices, alhugh he paern f ur analysis will be similar ha fllwed in Case I. We use a fixed reference sysem, wih 3 = n he free surface and wih he c.g. f he ship a l = a ime =. We suppse he ship be mving wih a unifrm velciy V in he, direcin. Cnsider he Cauchy-Pissn prblem defined by [3], [31], [32], and [33], excep ha nw [33] is hld n he mving surface S. This prblem has a sluin qj (&, 2, 3,, V) which is, f curse, idenical wih he cpj f Case I when V =. Using his qcj and he Vj bained in Case I, we may wrie he velciy penial fr seady min, 8 = V [V ( - V, 2, 3 ) f q i r, - r)dl] [41] - where q 1 (T, - ) = ql ( - VT, 2, 3, - r, V). Tha his saisfies he bundary cndiin n S is eviden, as Vp 1 prvides he necessary insananeus nrmal velciies, and al / an = n S fr all r. The free surface cndiin is als saisfied, as may be verified by direc evaluain, as in Case I. The velciy penial fr he flw generaed by he ship mving wih cnsan velciy, afer an impulsive sar a ime zer, is = V [p (i- V, 2 R, 8 ) + J ql (r, -r) dc] [42] The free surface and ship surface cndiins are saisfied as befre. The surface elevain a = is -Fjk S P Sk d a - V + q (,)

10 as required, and he iniial cndiins are me. Therefre, his mus be he saed penial. We shall need he seady min velciy penial fr he case in which he ship is displaced by Axj frm is reference psiin. We culd, f curse, cnsider he displaced ship as a cmpleely new hull and wrie dwn a penial similar [41], wih new funcins Vq and q 1. Insead, we deermine he crrecins he 'V and qp, discussed abve, which are necessary saisfy he new bundary cndiins. We wish a 1jj such ha V1 + Axj 'j = n 3 = [43] which implies ha 91j = n 3 = [43a] Als r Sn (a + Axj 91j) = n i I n S (displaced) Axj n il - n S (displaced) On On [441 [44a] In hree cases, sluins are immediaely available. If j = 1 91 ( - Ax, 2, 93) =, - Axl a ± (Ax l ) a, is a sluin f [43] and [44] since in his case we have simple ranslain. Therefre Similarly [45a] 2 [45b] Fr j = 3, here is n such simple sluin. Ning ha he righ side f [44a] is zer n S (riginal), i is nly necessary find is change when S is displaced. Then Ax Ax 3 32 an anb 3 r a 3 s az r -is -.1 [4] an ana 3 [ If j =, he displacemen is simply a rain in yaw. The ranslain f a yawed bdy is equivalen simulaneus ranslains parallel and perpendicular he bdy axis. Therefre, he sluin [43] and [44] is s V1 (, + 2 Ax, 2 -- A, 3) Ax z (1l +2 Ax, 2-1 Ax, V3) = + Ax ( 1 - ' + 2) a, 3 2 / V1 = 2 W1 l 1 2 If j = 4, r 5, he firs erm f [44a] becmes [n + Axj (i _ 3 xn)] -i. The secnd erm is -- - n " VV1 n S (displaced) an [47] which may be wrien, using values f n and VV evalued n S (riginal), -. -) -* -[n + Axj (i _3xn)] * [V1 + Axi (ij -3 xr * ) VVi] If we drp erms f higher rder in Ax and use [27], cndiin [44a] reduces a - 3 xn* i - [j _3xn 'V + n- (ij _xr- V) V 91] an a314 = an and an = 53 n a v k a,-- 1 a, a3 a anan 3 an[48a] [48a] a 3 -l 2 V [48b] Cndiins [43a] und [44a] are sufficien deermine Vj." Sricly, [44a] hlds n S (displaced), bu we nly inrduce errrs f rder (Axj) 2 if we ake he ship surface cndiin hld n he riginal S. Similarly, [4] and [48] can be applied n he reference psiin f S. T Vj crrespnds a (q 1 j, wih -_- -g 1 -aw fr 3 =, = a a 3 and wih cndiins crrespnding [3], [32], and [33] hlding. Again we ake he ship surface cndiin hld n he reference psiin f S. We need ye ne mre pair f funcins. The nrmal derivaive 3qp / n will differ frm zer n S (displaced) he firs rder in Axj. T crrec i, we define a funcin which saisfies he cndiins Axj a WJ -- a Ax n n- qi (T, - T) dt n S (displaced) and and = n 3 = [5] [51] As we d n inend exhibi sluins fr 9j, we shall n reduce he righ side. We als need a (qj, wih - g and he her apprpriae cndiins als hlding. [52] We nw have all he pieces needed wrie he velciy penial fr he flw abu he ship when displaced by Axj frm is reference psiin. I will be E = V {['V, + f Cql (T, - The erms T) dr] + Axj [V1j + f yplj (T, - ) dr] - + Axj [9j + J qj (, - ) dt]} - V (9i1 Axj,ij) [49] [53] prvide he necessary nrmal velciy in he displaced psiin. The nrmal velciies due V Axj Vj and V J" p (-, -r) dt cancel, and nne f he her erms cnribues nrmal velciies f firs rder in Axj. Therefre, he ship surface cndiin is saisfied in he displaced psiin. Furher, each pair f erms in brackes saisfies he free surface cndiin, as may be verified by direc evaluain.. We als have all he pieces needed assemble he penial fr he flw generaed by a ship experiencing small scillains {xj ()). I will be 8 = V { [1 + J q (T, - ) dr] + 7 [xj 1j + Jf qj (T, -T) xj (T) dt] j=1 - a - a I I I r I a II I I ~r~"

11 II + I [xj Vj + f j (v, - ) xj (T) dv]} + I [ij Vj + J q (T, - r) ij (r) dv] j=1l _( O [54] The ship surface cndiin is saisfied as befre, excep nw he erm Mi, Vj prvides he addiinal cmpnens required fr he scillary velciies. And again, he brackeed pairs f erms saisfy he free surface cndiin. p The dynamic pressure a any pin in he fluid is given V (ij (Vij ae1 SV [ j (T, -T) + 3 ((, - (r) v) dv} - + va) (,-)( + + qj (T,-T) ( - V V - d [55] - a There are w cnvluin inegrals in [55], ne invlving he scillary displacemen and ne invlving he scillary velciy. These may be reduced ne by means f an inegiain by pars. We can g eiher way, bu here is sme advanage in defining J [(Plj (T, -) s ha s ha and f + (P (r, --)] d = Ij ( - ) a I [5] f -- ± = [57] a a a alj (- Oj x ( a aaj r) = a 3 xj () 3 (- - ( - ) ij ( ) dr S Vx J() )j () j (r) dr. [58] f1 a The significance f his funcin Oj can be seen by rewriing he penial fr he unifrm flw wih he bdy defleced (Equain [53]). I becmes V { + J p, (,, -r) dr + Axj [(lpi + Vj)+ ' ()1}. [59] Equain [55] nw reduces P j W j + Xj V ±ij + -x 1 V 2 a [Vlj + pj + 41j ()] a, +S aqj (, -) - (- ) d}r) a ai SV 2 v + V S a, a p (, - ) d. [] We are cncerned wih he scillary value f he hydrdynamic frce, bu n seady cmpnens. The las erm [59] des n invlve he {xj)}. Hwever, when we inegrae he pressure ver S, he fac ha S is changing is psiin in a seady flw field implies ha even his erm cnribues he scillary pressure. These pressures will be funcins f he displacemen nly. Inegraing he pressure ver he surface f he ship, we can wrie he equains f min Y [(m bjk + m j k j ) + bjkx Cjk X j + J Kjk (-v) ij (v) dt] fk () [1] where mj and mjk are as defined in [4] and [38], and bjk = QV + ) j- _ s k d [21 cjkx j = Tal hydrdynamic and hydrsaic frce in he kh mde, due displacemen xj in he jh mde. Kjk(-) = (, -) -V 3(-T) f a a sk d 8 There are symmeries which reduce he number f cefficiens. Fr insance mjk = j Sk d =- f V j d.. [3] 8 If we cnsider he space enclsed by S, he free surface, and an infinie hemisphere, we can apply Green's herem, and we find m'k d = mki [4] Furher, if we cnsider he ransverse symmery f he ship, he marix {mjk) reduces mill m 13 m 15 m22 m 24 O m2 (mjk} _ 31 ma 33 m35 [5] m42 m 4 4 m 4 m51 m 53 m55 m 2 m 4 m -1 Evidenly, he marix (bjk) is f he same frm, excep ha in general bjk * bkj. The marix cjk is even simpler as surge and sway displacemens prvide n resring frces, hydrsaic r hydrdynamic. Therefre (Cjk) = c 31 C 33 c5 [1 C 4 2 O c 4 4 O C 4 c 5 1 C 5 3 C55 O L C 2 C 4 C The marix (Kik ()} is f he same frm as ({bjk}

12 Equains [1], hugh similar in frm hse develped by Haskind, differ frm his in several essenials. Haskind fund n hydrdynamic frce prprinal he displacemen, nr did he find he cmpnens f bi due Vj and ij. He als fund ha bs 3 = b,5 =, and b:, = - b :. The presence f ip, in he definiin f bik makes i unlikely ha such relains hld here. Furher, his kernel in he cnvluin inegral mus differ frm ha fund here. The reasn fr hese differences is ha Haskind negleced erms in saisfying he bundary cndiin n he displaced S which are f firs rder in xj. Wih equain [1], we have advanced a lng way ward he secnd bjecive f his paper. The dynamics f he bdy have been separaed frm he dynamics f he fluid. Furher, he hydrdynamic effecs have been separaed in well defined cmpnens, each f which can be fund (in principle) frm he sluin f a Neumann prblem r a Cauchy-Pissn prblem. Specifically, we draw he cnclusins: 1. The equains f min are universally valid wihin he range f validiy f ur assumpin f lineariy. Tha is, any exciain, peridic r nn-peridic, cninuus r discninuus, is permissible, jus s i resuls in small displacemens frm a cndiin f unifrm frward velciy. The case f min wih a negaive resring frce, r a leas he early hisry f such min, is n excluded. 2. The inerial prperies f he fluid are refleced in he prducs mjk Kj. The cefficiens are independen f frequency and f he pas hisry f he min, s hey are legiimae added masses. Furher, hey are independen f frward velciy. 3. There is an effec prprinal ii which accuns fr sme f he damping. This effec vanishes when he mean frward speed is zer. 4. There is a hydrdynamic "resring" frce (i may be negaive). I is equal he difference beween he hydrdynamic frces acing n he ship due he seady flw in he equilibruim psiin and in he defleced psiin. 5. The effec f pas hisry is embedded in a cnvluin inegral ver i i (). Fr sinusidal mins, his inegral will rdinarily have cmpnens bh in phase wih he min and 9 u f phase. The laer cmpnen cnribues he damping. Hydrdynamics f he Impulse Respnse Funcin We nw have w sysems f relains beween he exciain and he respnse f he ship: he impulse respnse relains, [2], and he equains f min, [1]. The frmer are f greaer value in describing he respnse a given exciain, while he laer are useful in analyzing he naure f he respnse. Bh sysems hld fr small scillary mins, s here are relains beween hem. We shall examine hese. Firs, le us sar wih he equains f min and derive he funcins {Rij ()}. Suppse a ship, mving a cnsan frward velciy, be subjeced a uni impulse in he ih mde a ime =. During he impulse, he equains f min reduce mk Xk + I mjk ij = fi bik j=1 where bik is he Krnecker dela. Suppse he impulse acs during ime A. Then, since jj A = Aij = Rij (+ ) we have mk Rik (+ ) + 1 mjk Aij (+ ) = bik- [71 j=1 As i and j range independenly frm 1, we have 3 equains relaing he w ses, (mij} and (Rij ()}. If he equains f min are knwn, equains [7] fix he iniial cndiins frm which he impulse respnse funcins can be deermined. Cnversely, if he impulse respnse funcins are knwn, hese equains yield he apparen masses. Immediaely afer he impulse, we have xj = () if = ij () + () S= 5j (O) + () F Kij (r) i i ( - ) dr = (). Therefre, cnsidering nly zer rder erms in, he equains f min yield: mk Rik (+ O) + I [mik Rij (+ ) + bjk,j (+ )] = [8] J-1 which relaes he cefficiens {bjk) he accelerains Rij (+ )}. Nw suppse he ship be aced upn by a cnsan uni frce in he ih mde (we assume a psiive resring frce exis in his mde). Then, afer equilibruim is reached, SCjk x = bik J-1 and xj = JRij (r) dr ICjk Rij (T) dt = bik [9] j=1 In mdes wihu a psiive resring frce here is difficuly as here is n guaranee ha all f he cupling cefficiens are necessarily zer. Thus, c2 x, he sway frce due a yaw angle x, will n rdinarily be zer, r even negligible. We shall reurn his pin a lile laer. If we rewrie [1] in he frm j j-1 [7] - I [(mi jk + mjk) Rij () + bjk Rij () + Cjk Rij ()] j=1 we have a se f 3 equains which can eiher be regarded as a se f simulaneus inegral equains fr he kernels {Kjk (T)}, r a se f simulaneus inegr-differenial equains fr he impulse respnse funcins (Rij ()}. We have already seen (equain [1]) ha if fi () = cs hen xj () = Rije cs + Rij s sin Subsiuing hese values in he equains f min, we ge - {[(mj bjk + mik) Rij c - bjk )Rij s - jk Rije j=1 - ) (Rijg Kjkc + Rije Kjik)] cs O + [(mj jk + mjk) ' Rij8 - bjk Rij e - Cik Rijs - a (Rijs Kjks - Rijc Kjk)] sin w = bik cs Fr any given frequency, his is an ideniy, s he ne cefficiens f cs w and sin mus be zer. This gives us 72 equains relaing he ransfrms (Ric, Rijs} wih he ransfrms {Kijc, KjkS}. We have we (Rij s Kik e + Rijc KRjk) [71a] = 5 ik + I 8 [(mj,k + mjk) Rije _ bjk Ris - cjk Rijc] j=1 and -- (Rij c Kk -- Rij Kik s ) Ji-1 = E [(mi jk + mk ) w RijO + bik Rijc - Cjk Rij] - [71b] j=1 ~ ---r - ~I ~I-~I 'rr --- ~ r~ ~- --~~

13 I r, equivalenly I ([(m bjk + mk) W 2 - Cjk - W Kjk s ] Rijc J-1 - (bjk + Kjkc) Rij = - ik [72a] X ((bik + Kikc) w Rijc + [(m, bk + mjk) W 2 - Cik - w Kjk' ] Rij)} = [72b] Thus, insead f he inegral and inegr-differenial equains relaing {R,j} wih {Kjk}, Equain [7], we have sysems f linear equains relaing heir ransfrms. Equains [72] are paricularly revealing. If we were arbirarily se he Kjk e and Kjk" be zer, hese are precisely he equains we wuld ge beween he frequency respnse funcins, Rijc and Rij s and he usual frequency dependen cefficiens. Thus, i is clear hw frequency dependency f he Kjkc and Kijk is frced n hese cefficiens in he cnveninal represenain. The ransfrms f {Rij} als yield useful varians f he relains already given. Fr insance, if we le w =, we have I Cjk Rii () = ik [73] Jia mre general frm f [9]. Als, ning ha ij () w R ij " (W) cs d and i () = - 2 ( Rij (w) cs w dw we have Aj () = 2 w Rij s dw [74a] X J RiJ () = - 2 s WRije dw [74b] Therefre, [7] and [8] may be wrien I [(mi jk + mjk) f Rij s dw] = bik [75] j-1 2 reaed independenly f he hydrdynamics. I is n uncmmn in mdel esing have "incmpaible" parasiic inerias in he differen mdes. Thus, he wing gear may cnribue a differen mass in surge frm ha in heave. By means f he equains f min, he effec f hese inerias upn he mins can be analyzed. Thus, he equains f min prvide a mre pwerful analyic l fr sudying he relainship f he respnse he parameers gverning ha respnse. We can cnclude, hen, ha hese w represenains cmplemen each her; he ne fr respnse calculain, he her fr respnse analysis. In fac, if i is ruly pracicable pass frm ne represenain he her, several pssibiliies presen hemselves: a) Mdel experimens may be designed bain maximum accuracy raher han maximum realism. Hydrdynamic effecs shuld be emphasized in he design since her effecs are separaely deerminable. Thus, ne shuld es a small gyradius in rder ha he effec f he inerial prperies f he bdy iself will be minimized. b) Resrains are permissible if heir characer is fully knwn. Thus, raher han direcly find he impulse respnse marix, in is cmplee generaliy, mre elemenary experimens may be cnduced deermine specific erms in he equains f min. We may resric urselves ne, w, r hree degrees f freedm and bain resuls which are cmpleely valid when inerpreed by means f he equains f min. c) The recurring difficuly f handling mdes in which he he resring frce is zer r negaive can be easily'vercme. I is clear ha an accurae experimenal invesigain f hese mdes wuld uncver pracical difficulies analgus he hereical nes we have discussed. Hwever, he prblem can easily be slved by impsing knwn resrains (i. e. springs) which will resre psiive sabiliy. The effec f hese resrains is readily includable in he equains f min, i can be remved by calculain, and he crrec impulse respnse, free f resrain, can be deermined. (Vrgeragen am 25. Januar 192) S[(mj bik ± mik) f w 2 Rij d - bjk J'w Ris 1 dw] = 3l1 Cnclusin [7] In he freging, we.have presened w mahemaical mdels fr represening he respnse characerisics f a ship. The equains f min are mre general, as hey apply he iniial sages f an unsable min. Where he w sysems are equally valid, we have relains wlich permi us pass (a leas in principle) frm eiher sysem he her. The impulse respnse funcin is cerainly he beer represenain fr cmpuing respnses. I inegraes all facrs, mechanical, hydrsaic, and hydrdynamic, in he ms efficien manner pssible fr cmpuain. Hwever, fr his very reasn, i is a pr analyical l fr explaining why he ship respnds he way i des r hw he respnse will be affeced if any change in cndiins ccurs. Fr insance, mdels are rdinarily esed wih resrains in cerain mdes. A.resrain in any mde will affec he impulse respnse funcin in any cupled mde. Since he ship is free in all mdes, i is evidenly imprper use hese respnse funcins predic full-scale behavir unless hey are crreced fr he effec f such resrains. The hydrdynamic equains d n suffer frm his disadvanage. Knwn resrains are readily includable and heir effecs deerminable. Or a change in mass disribuin can be References 1.) Weinblum, G., and S. Denis, Manley: "On he Mins f Ships a Sea". Transacins, The Sciey f Naval Archiecs and Marine Engineers, Vl. 58, ) S. Denis, Manley, and Piersn, W. J., Jr.: "On he Mins f Ships in Cnfused Seas". Transacins, The Sciey f Naval Archiecs and Marine Engineers, Vl. 1, ) T i c k, Le J.: "Differenial Equains wih Frequency- Dependen Cefficiens". Jurnal f Ship Research, Vl.3, N. 2, Ocber ) D a v i s, Michael C.: "Analysis and Cnrl f Ship Min in a Randm Seaway". M. S. Thesis, Massachuses Insiue f Technlgy, June ) S n e d d n, Ian N.: "Furier Transfrms". McGraw-Hill Bk Cmpany, Inc., ) H a s k i n d, M. D.: "Oscillain f a Ship n a Calm Sea". Bullein de 'Academie des Sciences de I'URSS, Classe des Sciences Techniques, 194 n. 1, pp ) G 1 v a, P.: "A Sudy f he Transien Piching Oscillains f a Ship". Jurnal f Ship Research, Vl. 2, N. 4, March ) T asa i, Fukuz: "On he Free Heaving f a Cylinder Flaing n he Surface f a Fluid". Reprs f Research Insiue fr Applied Mechanics, Vl. VIII, N. 32, ) W e i n b I u m, G. P.: "On Hydrdynamic Masses". David Taylr Mdel Basin Repr 89, April ) H e ss, Jhn L, and S mi h, A.M.O., "Calculain f Nn-Lifing Penial Flw abu Arbirary Three-Dimensinal Bdies." Duglas Aircraf Cmpany, Inc. Repr ES 422, 15. March

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15 11 INITIAL DISTRIBUTION Cpies 7 CHBUSHIPS 3 Tech Inf Br (Cde 335) 1 Appl Res (Cde 34) 1 Prelim Des Br (Cde 42) 1 Sub Br (Cde 525) 1 Lab Mg (Cde 32) 3 CHBUWEPS 1 Aer & Hydr Br (Cde RAAD-3) 1 Ship Insal & Des (Cde SP-2) 1 Dyn Sub Uni (Cde RAAD-222) 4 CHONR 1 Nay Analysis (Cde 45) 1 Mah Br (Cde 432) 2 Fluid Dyn Br (Cde 438) 1 ONR, New Yrk 1 ONR, Pasadena 1 ONR, Chicag 1 ONR, Bsn 1 ONR, Lndn 1 CDR, USNOL 2 DIR, USNRL (Cde 552) 1 Mr. Faires 1 CDR, USNOTS, China Lake 1 CDR, USNOTS, Pasadena 1 CDR, USNAVMISTESTCEN 1 CDR, ASTIA An: TIPDR 1 DIR, APL, JHUniv 1 DIR, Fluid Mech Lab, Clumbia Univ 1 DIR, Fluid Mech Lab, Univ f Calif, Berkeley 5 DIR, Davidsn Lab, SIT 1 DIR, Expl Nay Tank, Univ f Mich 1 DIR, Ins fr Fluid Dyn & Appl Mah, Univ f Maryland 1 Dr. L.J. Tick, Res Div, New Yrk Univ Cpies 1 DIR, Hydrau Lab, Univ f Clrad 1 DIR, Scripps Ins f Oceangraphy, Univ f Calif 1 DIR, Penn Sae Univ, Universiy Park 1 DIR, Wds Hle Oceangraphic Ins 3 in C, PGSCOL, Webb 1 Prf Ward 1 Prf Lewis 1 DIR, Iwa Ins f Hydrau Research 1 DIR, S Anhny Falls Hydrau Lab 3 Head, NAME, MIT, Cambridge 1 Prf Abkwiz 1 Prf Kerwin 1 Ins f Mahemaical Sciences, NYU, New Yrk 2 Dep f Engin, Nav Archiecure, Univ f Calif 1 Dr. J. Wehausen 1 Dr. Willard J. Piersn, Jr., Cil f Engin, NYU, New Yrk 1 Dr. Finn Michelsen, Dep f Nav Archiecure, Univ f Mich, Ann Arbr 1 Prf Richard MacCamy, Carnegie Tech, Pisburgh 13 1 Dr. T.Y. Wu, Hydr Lab, CIT, Pasadena 1 Dr. Harley Pnd, 4 Cnsiuin Rd, Lexingn 73, Mass 1 Dr. J. Kik, TRG, 2 Aerial Way, Sysse, N.Y. 1 Prf B.V. Krvin-Krukvsky, Eas Randlph, V 1 Prf L.N. Hward, Dep f Mah, MIT, Cambridge 39, Mass 1 Prf M. Landahl, Dep f Aer & Asr, MIT, Cambridge 39, Mass 2 Hydrnauics, Inc., 2 Mnre S., Rckville, Md. 1 Pres, Oceanics, Inc., 114 E 4 S., N.Y. 1 1 Mr. Richard Baraka, Iek, 7 Cmmnwealh Ave., Bsn 15, Mass 1 Dr. M S. Denis, 5252 Sangamre Rd, Glen Ech Heighs, Maryland

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