J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 51
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) We have already assumed that the motion of theship is a small oscillation relative to the moving coordinate systemWATER WAVES270seems reasonable, therefore, to develop all our basicquantities (taken as functions of x, y,z;t) in powers of /?, as follows:otc,y, z. It(9.8.2)y(x, y,2;t; /?)(9.8.8)17(0, 2; <;0)(9.8.4)<(<;0)(9.8.5)(<; j8)(9.8.6)0,(*; /?)The^== fa= * + fo + /, += o> + /toj + /Wo, += /30 + /W +.....,...,=1.,1, 2,8and second conditions state that the velocity potential andwave amplitudes, as seen from the moving system, aresmall of order /3.
The speed of the ship, on the other hand, and thefirstthe surfaceangular velocity of the moving coordinate system about the verticalaxis of the fixed coordinate system, are assumed to be of order zero.a not unexpected(It will turn out, however, that cu must vanishresult.) The relations (9.3.6) and (9.3.7) serve to make precise ourassumption that the motion of the ship is a small oscillation relativeto the system ox, t/, z.We must now insert these developments in the conditions derivedin the previous section.
The free surface conditions are treatedAs a preliminary step we observe that(9.3.8)<p x (x, rj, z; t; ft)- P[<pl9 (x, 0, z; t) + rff^(x*, 0, z; t)+ P*[fli<pixv(x>*; ')90, z; t)+ ?i(*+*5first....]0]with similar formulas for other quantities when they are evaluatedfree surface yHere we have used the fact that r\ is smallrj.of order ft and have developed in Taylor series. Consequently, thedynamic free surface condition fory==ri arising from (9.2.9) with=on thep=can be expressed in the form(9.3.9)and++++Oofogfot-(3*r) 2this condition is to...]+tf[(grad ^)2++ *K +#! +be satisfied for y=...]OHM* ++ <P*x) +0.Infact, as.always.]inTHE MOTION OF A SHIP IN A SEAWAY271problems of small oscillations of continuous media, the boundaryconditions are satisfied in general at the equilibrium position of theboundaries.
Upon equating the coefficient of the lowest order termto zerowe obtain(9.3.10)g^+the dynamical free surface condition+ co<p)(p lx(sa)<p(p lz=(p ltfory=0,and it is clear that conditions on the higher order terms could also beobtained if desired. In a similar fashion the kinematic free surfacecondition can be derived from (9.2.12); the lowest order term in ftyields this condition in the form:(9.3.11)q> ly+(S Q+ co^)rj lx -a)^rj lz-i? lt=fory=0.Weturn next to the derivation of the linearized boundary condiship's hull.
In view of (9.3.6) and (9.3.7), the transformation formulas (9.2.14) can be put in the formtionson the(x'=x +06^-9'.) -fin*=ly'(9.3.12)y- fa + ftQ^z - ftB^x(z'=z+p0 2lx-(30 n (y-y'c)when terms involving second and higher powers of ft are rejected.Consequently, the equation (9.2.16) of the ship's hull, up to terms in2can be written as follows:ft,z+ fi62lx- pe u (y -y'c )- fth[x + ftQ^y - y' - ftQ^z,- Pvi + flu* - &**] =yand, upon expanding the functionz(9.3.13)+ ftQ2l- 00 n (y -xc)fe,y'c )the equation becomes- ph(x, y) +the dots representing higher order terms inkinematic boundary condition for the hullft....=0,We can now obtain theby inserting the leftside of (9.3.13) for the function / in (9.2.11); the result is(9314)^=l^i*o,hand00l**<>(02i- *)whenItAthe terms of zero and first order only arc taken into account.clear that these conditions are to be satisfied over the domainof the x, t/-plane that is covered by the projection of the hull on theisin the rest position of equilibrium.
As wasturns out that o>0, i.e., that the angularoffirstorder, or, as it couldt/-axis must be smallplane when the shipmentioned earlier,velocity about theitis=also be put, the curvature of the ship's coursemust be smallsince theWATER WAVES272speed in the course is finite. The quantity s^t) in (9.3.4) evidentlyyields the oscillation of the ship in the direction of the #-axis (the socalled "surge").It should also be noted thatcorresponding to (9.3.14), that9>i*=*o(0 2 iAThis means thatequation2-axis.isTheweK+must be regardeduse=z'f}h(x', y')+ b u (y -02i)*wefind,y'c ).two sided, and that the lastA which faces the negativeimply that <p may have disconasto be satisfied on the side ofequation and (9.3.14)lasttinuities at the diskThe next+ h x) -ifA.step in the procedureis to substitute thedevelopmentswith respect to /?, (9.3.2) (9.3.7), in the conditions for the ship'shull given by (9.2.18) and (9.2.22). Let us begin with the integral1spn dS which appearsnsurface of the hull,is the pressure onin (9.2.18).
In this integralSisthe immersedthe inward unit normal to this surface andispto be calculated from (9.2.9). Withx y, z coordinate system the last equations of therespect to the osymmetrical halves of the hull areitwhichis,02"whereZ ==im<Qq(9.3.16)Wecannowwritepn dS=inpn t dS 1+Js 1Jspn 2 dS 2Js 2which n x and n 2 are given byH1 + H ~ kWecan also writepn dS=-egJs\yn dS+JsPl n\dSJs=n ds68] VJswhere p l9 from(9.3.17)pl= -(9.2.9),is0[(grad+IJsiPi n i ds i+Js agiven by2<p)-(s+ (oz)(p x + xcoy, + yt ].THE MOTION OF A SHIP IN A SEAWAYIfS273the hull surface below the #, 2-plane, the surface area S QSandinthiseachareaoftheisofftquantities t/,v2isH Hof orderisorderHence oneft.-= -yn dSffinds the following to hold:JsFromyndS + (l+ j)O(^) + kO(^).fJswe havethe divergence theorem-yn dSf= VIJS 9Vwherethe volume bounded byis3accuracy of orderV=20hdA-f,isS Q andthe x s-plane.9With angiven byf ft(y l +0 Bl x)dB=2ft fJBJAA/?V(hdA-2p*JA(JLthe projection of the hull on the vertical plane when thehull is in the equilibrium position, B is the equilibrium water linearea, and L is the projection of the equilibrium water line on theHereIfisWv WS l9 S 2 onf Pl nare the respective projections of the immersed surfaces2the x, j/-plane we have=dSf1Pl (jr, y,(\JwJs+JHI;t)H lr dW,- !Jwl*J<i50# ndH'i -Pi(*> y>i;t)d\\\- fPi(*>Uf(\YI-k(IfJ^JPl (x t y,H2;zf Pi(*> y>J \\- 2HPl (x, y,2;H*t)dir aW nor WW Ais identical with A.
Each of the differencesan area of order ft. From this and thehowever,ll\2fact that each of the quantities p, H lx2X9ly2y is of order ft, itNeitherA,2lis,,H H,Hfollows that(9.3.18)JPl ndS=lf[Pl (^, y, II,; t)II lx- Pl (x, y, II2 ;t)H 2x]dA+O(p*)\jf+j(\JA[Pi(^ y,Hi;t)H lyy,HI;- Pl (cr,i/,// 20-Pite2/W;t)H 2y ]WATER WAVES274Itfromwas pointed out above that <p may be discontinuous on A. Hence(9.8.17), (9.8.2), (9.8.4)=(x,y,Ht) =.(*,!,,(9.3.19)Here the+ff i;andwewrite<)tsuperscripts denote values at the positivenegative sides of the diskAwhose positivesideisandregarded as the sidewhich faces the positive s-axis.
If we substitute the developments offf 2a! ff 2y and (9.3.19) in (9.3.18), then collect the previousHI*,lyHfpn dS=iJsI491\,,,weresults,findf[(A.-fl n )(t^.-^)JA(9.3.20)![(h y+O u )(sQ<plx -<p lt+)+(h y ^^^^^JAfJAThefp(rintegraly c\)X ndS whichappears in (9.2.22) canJsbe writtenf p(r-t/ cj)xndS=- e g| y(r-y ])xn dScJsJsIfwe use the same procedurepn dS weaswas used abovefind-^^fJAWcft.-^)+-for the expansion ofTHE MOTION OF A SHIP IN A SEAWAY275(9.3.21)f8gpxhdA-*QgpOuJA+e2f/?fJA(y-y'c WA-2Qgpyi ! xhfa-2 QgpoJ x*hdxJLJL+[^(^+e i i)(Wix~9ie) +^(Av-0n)(Wix-^ie)1^JA-Qn[(y-y*)(h*-^i)(*^^JAWe nowassume that the propeller thrustdirected parallel to the #'-axis: thatTisof order fP andisisT = ]WYwherei' isthe unit vector along the #'-axis.We also assume that T isapplied at a point in the longitudinal plane ofsymmetry of the shipThus we have the relationsunits below the center of mass.T=(9.3.22)+P*Ti/30(/J),and(R r(9.3.23)The mass of theship-RXc )of orderis/?.T = IfweTXJJwriteM M^ and ex-pand the left hand side of (9.2.18) in powers offtitbecomesP/^o + MJtit + 003*)] + Jflf^ +0(jJ>)] =(9.3.24)The expansionfpn dSof the left+T-handAf^gJ.side of (9.2.22) gives(9.3.25)where/J/ 31isthemomentof inertia of the ship about the axis whichisperpendicular to the longitudinal plane of symmetry of the ship andwhich passes through the center of mass.Ifwereplace the pressure integralsand thrust termsequations by (9.3.20), (9.3.21), (9.3.22), (9.3.23),coefficients of likepowers offtin (9.3.24)following linearized equations oforder termswefindand(9.3.25)motion of thein the lasttwoand then equate theship.we obtain theFrom the firstWATER WAVES276*(9.8.26)2gg f(9.8.27)=phdA^MJgJA(9.8.28)IxphdA=0JA(9.3.29)+-<p lt ) -(s(fp lx -<p lt )-]dA=0[(s<fl> lxIJA(9.8.80)f [x(s^ lx -(p lt ) + -x(s<p lx-(plt )-]dA^OJA(9.3.31)f [(y-y'e )(s<fp lx -<p lt ) + -(y-y' )(se<p lx-<p lt )-]dA=oJAorby(9.8.29)+(9.3.32)[y(s</plx -(f>lt )|-y(s<jplx -<p lt )-]dA=0.JAFromthe second order termswefind(9.3.33)Mi*i=ef [(hx-0 Z iJA=6[JA(9.3.34)Miy^-lqnJL+Q[(h v+O u )(stf> lx -(p lt ) + +(h y -O n )(s(p lx -(p ltJA= -20g(Vi+*9n)Ma:+QJLiJn =-2pgeSl[y(h v +^i)(^iJA-el(y-yc)(kx -0nJA[h vf (y-y'e )hdA-2 egyi ( xhdx-'2 e g0 31 (JLJJA+6\JA)-]dATHE MOTION OF A SHIP IN A SEAWAY277or by (9.8.30), (9.3.31)r.."- 2^ai^ai?^!IJAr/,(yye^hdAZQgytxhdx2QgQ 3lIJLIx 2 hdx+lTJL(9.8.35)fJAEquation (9.3.26) states that the motion in the ^-direction is asmall oscillation relative to a motion with uniform speed SQconst.=is an expression of Archimedes' law: the restpositionof equilibrium must be such that the weight of the water displacedby the ship just equals the weight of the ship.
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