H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 63
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The dimen-The average wave amplitudeis shown plotted in figure 8.10, for hid= 1.' -4scan be seen, theory and experiment comparefairly well for the smaller values of Yo, but a tlarger amplitudes they deviate severely. Someidea of the reason for this can be obtained fromfigure S.11, which shows the experimentalliquid response over a wide frequency range ina 14.5-centimeter-diameter, rigid cylindricaltank.
As the forcing freqnency is graduallydecreased from a point slightly to the right ofthe stability boundary (see fig. 8.7), the liquidamplitude continually increases as the unstableregion is traversed. After a steady state isreached, the forcing frequency can be furtherdecreased, even to points considerably to theleft of the left-hand stability boundary, that is,into the stable region for small motions, urldthe steady-state nmplitude increases still more.However, a point is finally reached a t which theliquid motion quickly decays to zero, as indicated by the downward pointing dashedarrow for x0=0.65.
On the other hand, if thefrequency is gradually increased from a pointto the left of the rmstable region, the liquidremains quiescent until the stability boundaryis reached. At this point, the liquid amplitudequickly increases to a sizable magnitude, asindicated in the figure by the upward-pointingdashed arrow. Both of these types of responsetire similar to the well-known "jumps" in othernonlinear systems. 9 photograph of the largenmplitude motion is shown in figure 5.12; it isquite con~plex,as can be seen, with breaking-The other possible solution, oil= B cos or, .
. .,can be shown to be an unstable steady state m d consequently is not observed in practice (ref. 8.20).7 Neither the calculations nor the experimentalmeasurements were taken exactly at the wall, T/R= 1.0,in order to minimize viscous and wall effects.j1THJ3 DYNAMIC BEHAVIOR OF LIQUIDS..
. . ,-.. '.,a-,1.;::.:..*a-:,. .-,...- , I..." ..:,..<,.. . , -.. .,..*-.. :...r.:-...tI-;:.:: : ;;;,:::,,:7.2:. .. .;-,.,,;c -.,.: . .+:.;- .. , . ":." ..i*. - .:- . ....' -. .,., . \ . .2.;:.. .f4. : : <,;;.<;. .,*....f.;.;:.>:.:..- ... ...,!-::,-,%,I0 Experimental dataFIGURE8.12.-Large0a%0.98LOO1.02Frequency parameter a = wlw,,amplitude breaking wave (ref. 8.20).1.04FIGURE8.10.-Comparieonof theory and experiment form= 1, n= 1 mode 1/2-subharmonic rmponee (ref. 8.20).Excitation amplitudestheory predict the point a t which the liquid(1jumps down" from a large to a practicallyzero amplitude; in order to do this, a dissipation mechanism would need to be included.(The upward jump is predicted, however.)Hence, it should not be expected that thetheory would remain valid for very largeamplitude liquid motions.Symmetrical sloshing in n cylindrical t,ankcan be st*udiedby the same methods (ref.
8.20).For example, one finds that for h l R 2 2 , thepredominant dimensionless amplitude Pol forthe first symmetric mode is given by04446485.05.2Excitation frequency cps-5.45.6FIGURE8.11.-Experimentally determined liquid responseform= 1, n=l mode l/%subharrnonic reeponne for varioue excitation amplitudcn (ref. 8.20).waves being present.
Of course, the idealizedt'heoxy present.ed here does not take into accountsuch phenomenn as t'hese. Neither does theIt should be noted that equation (8.36) contains second-order terms, in contrast to theprevious analyses. Even so, the secondorder terms alone do not predict the size ofPOI,and the third-order terms must be retained.,."VERTICAL EXCITATIOK OF PROPELLANT TANKSThe second-order term, OO2,i.e., the amplitudeof the second-order symmetrical sloshing mode,is calucated fromThe solution of equations (8.36) and (8.37),correct to the third order, ispol=A sin UT+A:~-A:~ cos 2 ~ 7Boz=A&-A& cos ~ U Twith the A,, calculated as before.
The approximate sloshing amplitude may then be written as2830.7-$.-Em.-u0.4.-w3-m.-5u,0.3;€=( A sin UT+A:~-A:~ cos 207)JO(&~T)By using this equation the wave form at anytime may be calculated. A typical shape,normalized so that t.he maximum amplitude isunity, is shown in figure 8.13. Theory andexperiment agree very well in this case;however, as before, the actual liquid responseamplitude and the theoretically calclllnted0(1940.960.98L 001.02Frequency parameter u = w~w,,1.04FIGURE8.14.-Comparison of theory and experiment form=O, n = l mode 1/2-subharmonic response.(ref.
8.20).response do not agree quantitatively a t largeamplitudes, as shown in figure 8.14. Thereasons for this disagreement are similar t othe ones pointed out previously for antisymmetrical sloshing.90' Sector Cylindrical TankLndimensional radial position - rlRFrcuas 8.13.--Comparison of theoretical and experimentalwave shape for m=O, n = l mode 1/2-~iubharmonic(ref.8.20).Such tanks are of much practical interest, ash w been discussed i11 earlier chapters, andtherefore experimental studies of liquid surfacesubharmonic motion have been undertaken byKana (ref. 8.25). The frequency of the fundamental subharmonic mode corresponds to thesector 2-4 mode for lateral translation.
(Seech. 3.) E m e r e r , 5 this ::n,sse, the liquid i:every sector executes the same sort of freesurface motion, although the phasing of theliquid motion relative to the excitation mayvary from sector to sector. The secondmode frequency corresponds to the frequency ofthe fundamental axisymmetrical mode in acircular cylindrical tank, which is sector 1-3resonance for lateral excitation.
The response284THJ~ DYNAMIC BEcurves for both modes are quite similar inappearance to figure 8.11.Spherical TankThe behavior of liquids in spherical tanksunder vertical excitation has also been investigated by Kana (refs. 8.25 and 8.26). Again,the liquid slirface modes appear predominantl-jas one-half subllarmonics of the excitationfrequency and are qualitatively similar inshape to those occurring in a cylindrical tank.Of course, there is a variation of the liqliidresonant frequencies with depth, hut no newaspects of the slibharmonic liquid responseseem to arise.Summary of Subharmonic ResultsThe types of subharmonic sloshing discussed in this section nre the only ones forwhich any substantial nrnount of laboratorydatn are nvailable.
Even in these cases,there are no data reported on fluid pressuresnnd forces.The available t h e o r i e ~ ,while~probably not~iccurateenough to calculate large amplitudesloshing and fluid pressures, are sufficient todetermine the frequency range for which largeamplitude motions should be expected; insome instances this infornlation is all that isrequired. As an additional note; one shouldrecall that the method of solving nonlinearproblems outlined earlier is not the only feasibleone. For example, one might attempt anexpansion in powers of e, the nondimensionalforcing amplitude, instead of an expansion inpowers of 81 (or pl1 or bO1as the case may be).Or one might employ the method developedby Hutton (ref. 8.28) in which the two nonlinear boundary conditions are combined togive one equation involving only @.
Evenwith this procedure, however, the resultingequation is correct only to a certain preassigneddegree in the velocity potential. The essentialpoints of this analysis, and other possiblyfruitful approaches, are given in chnpter 3.Only for rectangular or cylindrical tnnks; no analysishas yet been completed for cither the 90' seotor compartmented cylindrical tank or for the spherical tank.A general thoory for tanks of arbitrary geometry hasbeen formulated (ref. 8.27), but no computations havebeen made.LVIOR OF LIQUIDSLarge Amplitude Harmonic and SuperharmonicMotionsUp to this point, the only liquid motion thathas been discussed in detail is the large amplitude subharmonic response, chiefly because it isthis response that would probably prove to bethe most troublesome and the easiest to excitein actual tanks.
However, other types ofresponses are possible since, according tofigure 8.4 or 8.5, harmonic and superharmonicmotions may be excited for certain forcingfrequency-amplitude combinations. I n contrast to the subharmonic motions, however,there appear to be two entirely different typesof harmonic and superharmonic responses:small amplitude motions which seem to correspond to stable responses, and large amplitudemotions which correspond to unstable solutionsof equation (8.8). Although the excitationfrequency for these responses is higher than forsubharmonic motion, it is still sufficiently lowthat the tank elasticity may be neglected.As mentioned previously, the widths of theunstable regions of the Mathiell stability chartfor harmonic and superhnrmonic responses arequite narrow, even for no damping. Withdamping, 11 rather severe amplitude cutoffexists; tlirtt is, the unstable regions do notextend completely to the line xo=O in figure 8.5.This amplitude cutoff may be circumvented,however, by using a small enough tank or alarge enough excitation; the reason for this isthat the width of the unstable regions increaseswith the nondimensional excitation amplitudeapproximately as a2fN, where N is, as before,the ratio of the forcing frequency to the dominant liquid frequency.
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