J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 49
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It would not be helpful, though, to try to overcome the difficultyby carrying the development to terms of higher order, for example,even though there would certainly then be damping effects in pitchingand heaving: such damping effects of higher order could evidently notintroduce damping into the motions of lower order. This is fortunatelynot the only way in which the difficulty can be attacked.
One ratherobvious procedure would be to retain the present theory, and simplyadd damping terms with coefficients to be fixed empirically, in somewhat the same fashion as has been proposed by St. Denis and Wein-blum[S.I], for example.There are still other possibilities for the derivation of theories whichwould include damping effects without requiring a semi-empiricaltreatment, but rather a different development with respect to a slenderness parameter.
One such possibility has already been hinted atabove in the course of the discussion of ships of broad mid-sectioncompared with ships of MichelPs type. If the ship is considered to beslender in both draft and beam the waves due to oscillations of theship would be of the same order with respect to all of the degrees offreedom; a theory utilizing this observation is being investigated.Another possibility would be to regard the draft as small while thebeam is finite (thus the ship is thought of as a flat body with aplaning motion over the water), i.e.
to base the perturbation schemeon the following equationandfor the hull (instead of (9.1.2)):development with respect to /?, This theory hasall detail, though it has not yet been published.to carry out thebeen worked outinrespect to damping effects the situation is now in some respectsthereverse of that described above: now it is the oscillations injustthe vertical plane, together with the rolling oscillation, that areWithdampedto lowest order, while theyawing and swayingoscillationsWATER WAVES262areundamped.Itwould seem reasonable therefore to investigate theand make comparisonsresults of such a theory for conventional hullswith model experiments.
This still does not exhaust all of the possibilities with respect to various types of perturbation schemes, particularly if hulls of special shape are introduced. Consider, for example,a hull of the kind used for some types of sailing yachts, and shownschematically in Fig. 9.1.4. Such a hull has the property that its beamFig. 9.1.4. Cross section of hull of a yachtand draft are bothfinite,but the hull cross section consists of twothin disks joined at right angles like a T. In this case an appropriatedevelopment with respect to a slenderness parameter can also bemadein regarding both disks as being slender of the same order.
Theresult is a theory in which all oscillations, except the surge, would bedamped; this theory has been worked out too but not yet published.It would take up an inordinate amount of space in this book to dealwith all of the various types of possible perturbation schemesmentioned above. In addition, only one of them seems so far to permitexplicit solutions even in special cases, and that is the generalizationof the Michell theory which was explained at some length above.
Consequently, only this theory (in fact, only a special case of it) will bedeveloped in detail in the remainder of the chapter. In all othertheories, it seems necessary to solve certain integral equations beforethe motion of the ship can be determined even under the most restrictive hypothesessuch as a motion of pure translation with no oscillain detailEven in the case of the generalizedathecaseof(i.e.ship regarded as a thin disk disposedvertically) an explicit solution of the problem for the lowest orderto the velocity potential in terms of an integralapproximationtions whatever, for example.Michell theory^THE MOTION OF A SHIP IN A SEAWAY268representation, say seems out of the question.
In fact, as soon asrolling or yawing motions occur, explicit solutions are unlikely to beThe best that has been doneso far in such cases has been toformulate an integral equation for the values of 9^ over the verticalprojection A of the ship's hull; this method of attack, which looksfound.possible and somewhat hopeful for numerical purposes since themotion of the ship requires the knowledge of (p l only over the area A,is under investigation. However, if the motion of the ship is confined=6nto a vertical plane, so that co 10> ft i s possible to solve2iThiscanbeseenwithreference to the boundthe problems explicitly.ary conditions (9.1.12) and (9.1.13) which in this case are identicalwith those of the classical theory of Michell and Havelock, and hencepermit an explicit solution for q> which is given later on in section9.4.
After 9?! is determined, it can be inserted in (9.1.21), (9.1.22), and(9.1.23) to find the forward speed $ Q corresponding to the thrust T,the two quantities fixing the trim, and the surge, pitching, and heaving oscillations.* In all, six quantities fixing the motion of the shipOnly this version of the theory will beof the chapter.theremainderpresentedThe theory discussed here is very general, and it therefore could beapplied to the study of a wide variety of different problems. For example, the stability of the oscillations of a ship could be in principlecan be determinedexplicitly.in detail ininvestigated on a rational dynamical basis, rather than as at presentby assuming the water to remain at rest when the ship oscillates. Itwould be possible to investigate theoretically how a ship would movewith a given rudder setting, and find the turning radius, angle of heel,The problem of stabilization of a ship by gyroscopes or other devices could be attacked in a very general way: the dynamical equations for the stabilizers would simply be included in the formulationetc.of the problem together with the forces arising from the interactionsof the water with the hull of the ship.In sec.
9.2 the general formulation of the problem is given; insec. 9.3 the details of the linearization process are carried out for thecase of a ship which is slender in beam (i.e. under the conditionimplied in the classical Michell-Havelock theory); and in sec. 9.4 agiven for the case of motion confined to thevertical plane, including a verification of the fact that the waveresistance is given by the same formula as was found by Michell.solution of the problem* These freeundampedconditions are given.isvibrations are uniquely determined onlywheninitialWATER WAVES2649.2.General formulation of the problemWe derive here a theory for the most general motion of a rigid bodythrough water of infinite depth which is in its turn also in motion inany manner.
As always we assume that a velocity potential exists.Since we deal with a moving rigid body it is convenient to refer themotion to various types of moving coordinate systems as well as to afixed coordinate system. The fixed coordinate system is denoted byO X, F, Z and has the disposition used throughout this book: TheX, Z-plane is in the equilibrium position of the free surface of thewater, and the Y-axis is positive upwards. The first of the two movingcoordinate systems we use (the second will be introduced later) isx y, z and is specified as follows (cf. Fig.
9.2.1):denoted by o9Fig. 9.2.1. Fixedand moving coordinate systemThex, 2-plane coincides with the X, Z-plane (i.e. it lies in the undisturbed free surface), the y-axis is vertically upward and contains thecenter of gravity of the ship. The #-axis has always the direction of thehorizontal component of the velocity of the center of gravity of theship. (Ifwethe path ofdefine the course of the ship as the vertical projection ofcenter of gravity on the X, Z-plane, then our conven-itstion about the a?-axismeans thatship's course.) Thus if R =this axisistaken tangent to theYc(X e C9 Z c ) is the position vector of thecenter of gravity of the ship relative to the fixed coordinate systemand hence c(X C9 c Z c ) is the velocity of the e.g., it follows thatR =y,,the #-axis has the direction of the vector u givenu(9-2.1)withIandKCunit vectors along the X-axis and the Z-axis. Ifwe may writeunit vector along the <r-axis(9.2.2)by=XI+ZKs(t)i=u,i isaTHE MOTION OF A SHIP IN A SEAWAYthus introducing the speedplane.
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