J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 46
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As was already stated, no further restrictiveity at the time tassumptions except those needed to linearize the problem are made.Before discussing methods of linearization we interpolate a briefdiscussion of the relation of the theory presented here to that of otherwriters who have discussed the problem of ship motions by means ofthe linear theory of irrotational waves. The subject has a lengthyhistory, beginning with Michell in 1898, and continuing over a longperiod of years in a sequence of notable papers by Havelock, starting1909.
This work is, of course, included as a special case in what ismpresented here. Extensive and up-to-date bibliographies can be foundin the papers by Weinblum [W.3] and Lunde [L.19]. Most of this workconsiders the ship to be held fixed in space while the water streamspast; the question of interest is then the calculation of the wavedependence on the form of the ship. Of particularinterest to us here are papers of Krylov [K.20], St. Denis and Weinblum [S.I], Pierson and St. Denis [P.9] and Haskind [H.4], all ofwhom deal with less restricted types of motion. Krylov seeks themotion of the ship on the assumption that the pressure on its hullis fixed by the prescribed motion of the water without reference tothe back effect on the motion of the water induced by the motionof the ship.
St. Denis and Weinblum, and Pierson and St. Denis,employ a combined theoretical and empirical approach to the problem which involves writing down equations of motion of the shipwith coefficients which should be in part determined by model experiments; it is assumed in addition that there is no coupling between the different degrees of freedom involved in the general motion of the ship.
Haskind attacks the problem in the same degreeof generality, and under the same general assumptions, as are madehere; in the end, however, Haskind derives his theory completely onlyin a certain special case. Haskind 's theory is also not the same as thetheory presented here, and this is caused by a fundamental differencein the procedure used to derive the linear theory from the underlying,resistance initsbasically nonlinear, theory. Haskind develops his theory by assumingthat he knows a priori the relative orders of magnitude of the variousThe problemattacked in this chapter by aformal development with respect to a small parameter (essentially athickness-length ratio of the ship); in doing so every quantity isquantities involved.isTHE MOTION OF A SHIP IN A SEAWAYa formal series (for a similar type of[J.5]). In this way a correct theory should bedeveloped systematicallydiscussion see F.John249inobtained, assuming the convergence of the series and there wouldseem to be no reason to doubt that the series would converge forsufficiently small values of the parameter.
Asidefrom the relativesafety of such a method purchased, it is true, at the price of makingrather bulky calculations it has an additional advantage, i.e., itmakespossible a consistent procedure for determining any desiredordercorrections. It is not easy to compare Haskind's theoryhigherin detail with the theory presented here. However, it can be statedthat certain terms, called damping terms by Haskind, are terms thatwould be of higher order than any of those retained here. A moreprecise statement on this point will be made later.Onebyof the possible procedures for linearizing the problem beginswriting the equation of the hull of the ship relative to the coordinatesystem fixedformin the ship in thez'(9.1.2)-0h(x' y')9z'>0,9witha small dimensionless parameter.* This is the parameter withftrespect to which all quantities will be developed.
In particular, thevelocity potential 0(X, F, Z; /; ft) =. <p(x* y, z; t; ft) is assumed topossess the development(9.1.8) <p(x 9 y, z;Thet; ft)= fafa y, z; t) +free surface elevation r](x z;9lar velocity a>(t;(9.1.4) ri(x, z\ft) (cf. (9.1.1 ))t; ft)(9.1.5)s(t; ft)(9.1.6)o>(f; ft)t; ft)ft*<p 2 (x,&*;<)+and the speeds(t; ft)and angu-are assumed to have the developments= /%(*, z; t) +(x. z;t) +==So (t) + Sl (t) + ...= o> (0 + ftco^t) +ft*r] 2...9,ft....Finally, the vertical displacement y c (t) of the center of gravity andwith respect to thethe angular displacements** 13 of the ship2andassumedzaxesare#, t/,given byrespectively,* It,important to consider other means of linearization, and we shall discusslater.
However, it should be said here that the essential point isthat a linearization can be made for any body having the form of a thin disk:it is not at all essential that the plane of the disk should be assumed to be vertical,as we have done in writing equation (9.1.2).** Since we consideronly small displacements of the ship relative to a uniformtranslation, it is convenient to assume at the outset that the angular displacementcan be given without ambiguity as a vector with the components lf f 8 relativeto the #, t/, 2-coordinate system.issome of them,WATER WAVES250(9.1.7)OMft)= ftO^t) + 0<i(0 +- y' = ftVi(t)y c (t;(9.1.8)ft)-=i>1,2,3,eThese relations imply that the velocity of the water and theelevation of its free surface are small of the same order as the "slenderof the ship.
On the other hand, the speed s(t) of theship is assumed to be of zero order. The other quantities fixing themotion of the ship are assumed to be of first order, except for co(t),but it turns out in the end that co Q (t) vanishes so that a) is also of firstness parameter"order.ftThe quantitydescription of Fig. 9.1.2;it iswas definedwith theto be noted that we have chosen toy'c in (9.1.8)in connectionexpressquantities with respect to the moving coordinate systemin that figure. The formulas for changes of coordiindicated(x, y, z)nates must be used, and they also are to be developed in powers ofallft;for example, the equation of the hull relative to the (x, y, z) cois found to beordinate system2lx-ftOu (y-y'c )- fth(x, y) +...=oand rejecting second and higher order terms in ft.In marine engineering there is an accepted terminology for describing the motion of a ship; we wish to put it into relation with the notation just introduced.
In doing so, the case of small deviations froma straight course is the only one in question. The angular displacements are named as follows: O l is the rolling, 2 + a is the yawing, andafter developingthe pitching oscillation. The quantity ft#i(t) in (9.1.5) is calledthe surge (i.e., it is the small forc-arid-aft motion relative to the finite3 isspeed s (t) of the ship, which turns out to be necessarily a constant),while y cy'c fixes the heave. In addition there is the sidewisc disreferred to as the sway; this quantity, in lowest order,dzplacementcan be calculated in terms of s Q (t) and the angle a defined by (9.1.1)in terms of co(t) as follows:(9.1.9)dz*,,Sf*J^(a> Q (t) turns out to vanish.In one of the problems of most practical interest, i.e. the problemof a ship that has been moving for a long time (so that all transientssincehave disappeared) under a constant propeller thrust (considered to besimply a force of constant magnitude parallel to the keel of the ship)THE MOTION OF A SHIP IN A SEAWAY251seaway consisting of a given system of simple harmonic progresof given frequency, one expects that the displacement comwavessingin general be the sum of two terms, one independent ofwouldponentsthe time and representing the displacements that would arise frommotion with uniform velocity through a calm sea, the other a termsimple harmonic in the time that has its origin in the forces arisingfrom the waves coming from infinity.
On account of the symmetry ofthe hull only two displacements of the first category would differfrom zero: one the vertical displacement, i.e. the heave, the other thepitching angle, i.e. the angle 3 The latter two displacements apparently are referred to as the trim of the ship. In all, then, there would bein this case nine quantities to be fixed as far as the motion of the shipinto a.concerned: the amplitudes of the oscillations in each of the sixdegrees of freedom, the speed s Q9 and the two quantities determiningisthe trim.A procedure to determine allofthemwillnext be outlined.Weproceed to give a summary of the theory obtained when thescries (9.1.2) to (9.1.8) are inserted in all of the equations fixing themotion of the system, which includes both the differential equationsand the boundary conditions, and any functions involving ft are inturn developed in powers of ft. For example, one needs to evaluate (p xon the free surface yin order to express the boundary conditionsrjthere; one calculates it as follows (using (9.1.3) and (9.1.4)):(9.1.10)Vm (x9TI,z; t; ft)=x, 0, z; t)ft[<p l9 (x, 0, a; *)+ ^fotfW*'+ rpp^(x, 0, z; t) +z...]'>Weobserve the important fact to which reference will be madelaterthat the coefficients of the powers of ft are evaluated at y0,i.e.
at the undisturbed equilibrium position of the free surface of thewater. In the same way, it turns out that the boundary conditions=for the hull of the ship arc automatically to be satisfied on the verticallongitudinal mid-section of the hull. The end result of such calculations, carriedftisout in such aas follows:Thewayas to include all terms of first order indifferential equation for(pis,of course, the La-place equation:(9.1.11)in theareaAVix*domain y< 0,i.e.the lower half-space, excluding the planeis the orthogonal projection of theof the x, t/-plane whichWATER WAVES252hull(cf.on the xFig.
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