H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 15
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On the other hand, nonlinearities koduced by the introductiod of largedamping into the system, as by baffles or othermechanical devices, are not included within thepresent discussion but will be treated in thefollowing chapter. The nonlinear aspects ofliquid motions resulting from excitation normalto the liquid free surface are, of course, also notincluded within the present discussion but willbe treated in detail in chapter 8.3.2NONLINEAR EFFECTS ARISING FROM TANKGEOMETRYCompartmented Cylindrical TanksThe theory of lateral sloshing in circularcylindrical compartmented tanks, as outlinedin chapter 2, was developed in rather straightforward fashion from the linearized hydrodynamic equations. The earliest experimentalF1c*E3.1.-Large amp1itude brtaking wave dnringlateral eloshing near first-mdc resonance.studies, however, failed to yield very goodagreement between measured and predictedliquid frequencies (ref.
3.1), the measuredfrequencies always being somewhat low. Latermeasurements revealed (refs. 3.2 and 3.3) thestrong dependence of the liquid frequencies onexcitation amplitude and indicated agreementmi& t-he theoretical values for vanishinglysmall excitation amplitudes. This generaleffect is shown in figure 3.2 for 45') 60°, and 90"sector codgurations, the direction of excitationin each case being a bisector of one of thecompartments.Such nonlinear effects, in this instance consisting of a "softening" frequency characteristic,arise essentially from the tank geometry. Thecontinuity of flow toward the center of thetank dictates a higher surface elevation thanthat of an uncompartmented tank because of7980THE DYNAMIC BEHAVIOR OF LIQUIDSExcitation amplitude parameter (xJd)FIGURE3.2.-Effect of excitation amplitude on the lowc~tliquid remnant frequency for 4S0, 60°, and 90' eectortanks (ref.
3.3).Equivalent Reynolds number basedon perforationhole s i n and excitation amplitudep d:Smd( R M =the decreasing area of the cross section (neglecting variations in radial velocity), and hences marked change in liquid frequency.Further, and dramatic, changes in the liquidfrequencies take place when the walls of thesectors are perforated. By plotting the lowerliquid frequency against an equivalent Reynoldsnumber based upon perforation hole size andexcitation amplitude, as in figure 3.3, i t isreadily apparent that two frequencies arepossible for each tank. The higher frequency,corresponding to the lowest values of the equivalent Reynolds number, is essentially asdescribed above for solid sector walls. However,as the equivalent Reynolds number becomeslarger, corresponding essentially to increasingperforation hole size, the liquid frequencydecreases very rapidly to that correspondingto the uncompartmented tank.
The transitionzone depends upon a number of factors, but i t,Flcuar 3.3-Variation in lowest liquid resonant frequencywith equivalent Reynolds number for 4S0, 60'. and 90'perforated sector tanks (ref. 3.3).appears that the percent of perforation may beone of the most important of them. I n anyevent, i t is immediately apparent that the shiftin liquid frequencies provided by compartmentation can easily be negated by overzealousattempts to introduce damping or decreasestructural weight by perforation.The force response in a 90' sector cylindricaltank, as a function of excitation amplitude, isalso of considerable interest and has also beenexplored experimentally (ref.
3.4). The datashown in figure 3.4 were obtained by slowlysweeping frequency with constant excitationamplitude, as is customary when exploring thejump phenomenon in a nonlinear system (refs.3.5 and 3.6). As we know from the theoretical81NONLINEAR EFFECTS IN LATERAL SLOSHING0040455.05.5606.57.07.5804045Frequency parameter wz dlg5.05.56.0657.07.580Frequency parameter ~2 dlgExcitation-hid L OoxIncreasing frequency-Decreasing frequencyFICUBE3.4.-Nonlinear liquid force response in 90' sector tank (ref. 3.4).analysis of chapter 2, sector-compartmentedtanks exhibit liquid resonances in sets, corresponding to the orientation of the varioussectors with respect to the direction to theexcitation.
Thus, the forced response curvesshould also exhibit resonant peaks in corre-82THE DYNAMIC BEHAVIOR OF LIQUIDSsponding sets (two peaks for the 90" sectortank), and the nonlinearities already discussedshould lead to amplitude jumps near theseresonant peaks.
Looking a t the data of figure3.4, however, reveals the interesting featurethat the double jumps occurring a t each resonant peak are both dournward (from higher tolower amplitudes). This arises, of course, fromthe fact that the data represent the total forceresponse of all four sectors, even though onlytwo of these are near resonance, and as aresult the customary response picture is somewhat altered. (The nonlinear force response inan uncompartmented circular cylindrical tankwill be discussed in detail later in this chapter(sec.
3.3.) Figure 3.4 (upper left plate) alsoshows the total force response of an equivalent"frozen" liquid mass; in this instance, thephasing of the forces produced by the varioussectors a t the higher frequencies tends tocancel and produce an even lower total forcethan does the equivalent frozen mass.Spherical TanksNonlinear effects in spherical tanks have notbeen investigated to any appreciable extent.However, even a cursory examination of thegeometry leads to the conclusion that the strongboundary curvatures m i l l appreciably modifylateral sloshing characteristics : for mean liquiddepths less than one radius, the expandingvolume would tend to suppress motions of theliquid free surface, while for liquid depths morethan one radius the contracting volume couldtend to induce breaking waves.
The experimental data first available for relatively smallexcitation amplitudes (ref. 3.7) would tend tosubstantiate such beliefs, although they are notnearly so evident as in the case of the compartrmen ted cylindrical tank.Of course, as excitation amplitude is increased,the nonlinear characteristics of the responserapidly become dominant (ref. 3.4). This willbe discussed in more detail in section 3.3.is increased such effects do appear, much asthey are emphasized in the spherical tank. Asimilar statement could therefore probably bemade regarding almost any other tank configuration. Basically, however, the fact is that formore unusual geometries (ellipsoidal, toroidal,etc.), experimental investigations up till nowhave centered about frequency determinationsfrom free vibrations, so that nonlinearities ofthe type discussed so far in this chapter arenot readily discernible.
It is clear that thesector tank is somewhat unique in this respect.Perhaps passing mention should be made ofone special case associated with the rectangulartank. A rectangular tank of high aspect ratio,pitching about a transverse axis, and filled withliquid to only a shallow depth possesses strongnonlinear response characteristics for even extremely small excitation amplitudes (i.e., smallpitch angles).
This arises, obviously, from"piling up" of the liquid at one end of the longtank. A t larger angles of excitation, and particularly in the presence of baffles, the responsebecomes even more strongly nonlinear; travelingwaves have even been observed under such conditions (ref.
3.8), but these may be more theproduct of interference effects.3.3LARGE AMPLITUDE MOTIONSBasic EquationsSeveral analytical theories have been developed for large amplitude lateral sloshing inrectangular or circular cylindrical containers.(See, for example, refs.
3.9 through 3.17.) Weshall outline various of these in the presentsection, making such comparisons with experimental data as are available, and begin with astatement of relevant boundary conditions.The liquid contained in the rocket tank isassumed to be incompressible and its motionirrotational; thus, there is a velocity potential4 governed by the Laplace equationOther Tank ConfigurationsThe circular cylindrical tank does not appearto exhibit any particularly significant nonlineareffect in lateral sloshing as a consequence ofgeometry.
Of course, as excitation amplitudeand the following well-known boundal-g conditions :(1) The relative normal velocity on the wetted wall, Z, is zero; i.e.83NO,~INEAR EFFECTS IN LATERAL SLOSHING(3.2)where2isbnsink. Alsothe velocity of the liquidnormal to the wall and v, is the localnormal velocity of the wetted boundaries.(2) The free sul'face at z=t(z, Y) is subjectedto a dynamic condition on pressure, p,and a kinematic condition on free surfaceelevationThe dynamic condition isr.(3.6)that the nth characteristic function andnumber can be computed from*(z,Y, 0)y, z)}H(z, Y, 0; z', Y', 0){H(%, y, 2; x', yf, 0)='nlx w , y',(3.3)where thePressure PO often takenas zero for simplicity.
The kinematiccondition, in rectangular coordinates, is(on z=r)I t is noted that the gradient of 4 yields thepositive velocity vector p of the fluid; that is,+Vb=q.+Moiseyev's General Theory (ref.3.9)Fme Vibrutions of the LiquidLet $(z, Y) be a characteristic function (normal mode function). It satisfies the followingconditionsIva#=O(1)(2)3-0on the wetted wallbn-(3)%=A$a?&on the free surface(3.44That is, and # are the characteristic numbersand functions of the integral equation(3.7)Let 4,!"',be the velocity potential andthe free surface elevation of the nth mode offree vibration of infiniteshimallysmall amplitude; thenr;")=#n(z, Y, 0) sin (cnt)(3.8a)The square of the frequency of the nth freevibration isa,= g',(3.9)The mathematical problem formulated in thebeginning of this section has been solved byMoiseyev (ref. 3.9), who used a method of expanparamesion into a power series of anter e; iSe.03 4nen(3.4a)(3.4b)0)ds(3.10)n-rnt=e(3.11)C Lenn=OOne expects that the period of the free vibration will be dected by the wave amplitude whenthe amplitude'is allowed to have a finite magnitude.
Thus, in order to introduce an amplitude3 - - A ----A'a 3 1 t . n m ~ t i p n .,Il~.depeiiueu~Y C ) L I ~ i~~ i.-.t v+hnu u o +hfifimyv v v v r jit is assumed that the dimensionless time can beexpressed asurrrr---.---d1where H is the Neumann function (Green'sfunction of the second kind) containing a unitA method of constructing the Neumann functionnumerically for a spherical bowl harr been demonatrated in ref. 3.18.where the h, are to be determined.
(The liquidmotions are assumed to occur in the vicinity ofthe mth linear mode.)84THE DYNAMIC BEHAVIOR OF LIQUIDSThe free surface boundary conditions are tobe satisfied at z={(x, y) ; but {, from equation(3.11), is still unknown. In order to satisfy thefree surface conditions at z=0, and thus circumvent this difficulty, 4 is expanded in aTaylor's series; that is, the kth term of thepotential is written aswhere jnkare unknown functions of time.By equating the terms of equal powers of cin the dynamic and kinematic free surfaceconditions, a system of equations can beobtained and then solved in succession.
Thezeroth approximation [n=O in equation (3.10)]works out to befor k=O(3.14)It can be shown from equations (3.7) thatThus, by substituting equations (3.13) into(3.14a), one finds thatfor k=l. Fy (x, y) is a function only of thezeroth approximation; it may be rewritten ina more convenient form by expanding it in aseries of the characteristic functions :Then the following system of equations arisesfor the determination of j n l.f~l+f,,,~=-- hlurnag cos r+02bgi sin 27In order to obtain periodic solutions, it isnecessary and sufficient that resonance doesnot occur, and thus hl=O. Hence, it can beseen from equation (3.12) that to the firstapproximation the period of the vibration doesnot depend on the amplitude.
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