H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 24
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This representation seems about as effective as any, inview of the overall complexity of the data.The most important feature of these curves,obviously, is the large decrease in resonantfrequency with increasing perforation hole sizeand percent open area. Clearly, the desiredupward sluft in frequencies to be obtained by7 These data are also compared with the responsecalculated for the cruciform (vertical configuration)baffle case by an equivalent mechanical model theory(ref.
4.29) (see also ch. 6). This theory, beingcompletely linear, does not provide very goodpredictions.8 See ch. 3 for a discussion of significant nonlineareffects in compartmented tanks.9 Recall that reduction in total force response incompartmented tanks may occur as the result ofphasing between secton. (See chs. 2 and 3.)118THE DYNAMIC BEHAVIOR OF LIQUIDS' Uqvld DampingKtb\h.'0010Side view''III0.240.320.40I0.48Tank fullness, h12bFIGURE4.16.-Variationof damping factor with tankfullness for cruciform baffles in an oblate ~pheroid(ref. 4.12).FIGURE4.17.-Vertical (cruciform) baffle arrangement inspherical tank (ref.
4.29).compartmentation can be completely nullifiedby overzealous att,empts at weight saving byperforation. The precise nature and definitionof the transition between the two frequencybranches involves a complex interaction between various of the parameters.The damping at the resonant frequency forthe 90°, 60°, or 45' solid wall tanks is low,averaging approximately 0.04. However, atfrequencies below the resonant value, the liquidsloshing is effectively damped. The experimental data (ref.
4.32) showed that perforatedsectors with less than 10 percent open area willincrease the damping ratio to approximately0.1 while maintaining a liquid resonant frequency corresponding to a solid wall compartmented tank. For partitions with open areasgreater than 10 percent, the damping producedis greater than 0.10, but the correspondingliquid resonant frequencies approach that of anuncompartmented cylindrical tank. The liquidviscosity and excitation amplitude have aslarge an effect on the liquid damping ratios asdo the perforation hole size and percent of openarea, and as much as all of these factors have onthe liquid resonant frequencies.Attempts to present the damping ratiosversus Reynolds number, or other variousparameters which include the factors affectingthe damping, failed to yield an effective pictureof these complex data.
The best that could bedone is something such as that shown in figure4.20, which is a three-dimensional plot of thedamping ratio versus the dimensionless resonantfrequency parameter, 3d/g, and the percentageof open sector area for a 45' sectored tank.The results presented (ref. 4.33) are for threevalues of sold, with water and methylenechloride as the test liquids. Maximum damping is produced for sectors with open areas of16 to 23 percent. However, the excitationamplitude must be quite large to maintain theresonant frequency corresponding to solidsector walls. Additional tests with sectors ofsmaller hole diameter ratios (dn/d=0.00139 and0.00278) and open areas up to 23 percentincreased the damping ratios to an averagevalue of 0.15 while maintaining a frequencycorresponding to the solid wall sector tank.Above 23 percent open area, the results becomeinconsistent in frequency and damping ratio.Test results for 60° and 90' tanks withd n/d =0.0056 are also available (ref.
4.33). Forthe 60-percent sector tank, increased dampingDAMPING O F LIQUID MOTIONS AND LATERAL SLOSHING119loo+Phase angleI Deg. )6Experiment, unbaffled tankExperiment cruciform bafflesExperiment, horizontal bafflesTheory, cruciform bafflesl Mechanical model, see chapter 6 J-.-3--O--5Y ' 0.32 horizontalY ' 0.15 cruciformP-l4p g d (X,ld)3Dimensionlessforceamp1itudeTest liquids -water, methylene chloride2100124567Dimensionless frequency parameter,398W*1011dlgm+Phase angle(Deg.),1000A 8.3565I+.-aExperiment, unbaffled tank--*-Experiment, cruciform baffles--*--Experiment, horizontal baffles_.
Theory, cruciform baffles( Mechanical4Dimensionlessforcearnpiiiucie3model, see chapter 6 )Y = 0.15hld.0.50Test liquids -water. methylene chloride210Dimensionless frequency parameter, wSdlgFIGURE4.18.-Liquid force reaponee in a baffled spherical tank (ref. 4.29).12120THE DYNAMIC BEHAVIOR OF LIQUIDS45' Sector tankPerforation hole dia. dh l d = 0.00%SymbolX,ld0.00187mdh=0.004170.008330.205~111Test liquids; water, methylene chlorideFIGURE4.20.-Variation in damping ratio for 45' sectortank (ref. 4.33).Equivalent Reynolds number based on perforation hole sizeand excitation amplitudeIRdr;g"')YXOFIGURE4..19.-Vnriation in lowest liquid resonant frequency for 90'. 60°, 45' sector tanks with equivalentReynolds number (ref.
4.323.ratios c~tfrequencies corresponding to the solidsector wall exist only for open areas betwee11 Snnd 16 percent open nrea, depending on theexcitation amplitude. Additio~ial testsdh,/d=0.00139 and 0.00278 yielded Jnem damping ratios of t~pproximately0.12 at 16 percentto 23 percent open sector area, dependin:: onexcitntion amplitude. The smaller open areac orresponds to the lower escitation amplitudes.I n the 90' tank, sectors with tlh/d=0.0056 losetheir compnrt~nentationeffect at small excittltion ninplitudes; however, tests with sectorwalls htiving dh/d=0.00139 and 0.00275: and upto 30 percent open wen produced metin dtlmpingrrltios of 0.12 tit frequencies corresponding to tliesolid sector ~ ~ for1 xo/d=0.004117 cuid 0.00833.Tests nt xo/d<0.00417 with tl,,/d=0.00139gave results similar to the solid sector.
Theresults for dh/d=0.00279 were incoliclusive,and n o consistent damping ratios or resonantfrequencies could be obtained.Asymmetrical Plate Segment BafflesBaffles having tlie form of a segment of ncirculnr p l ~ ~ tande , mounted normal to the wnllin a circular cylindrical tank, have been investigated in references 4.23, 4.34, and 4.35. Experiments conducted in reference 4.23 indicatedthat these baffles could possibl~provide largevalues of damping by transferring energy fromthe first mode of liquid oscillation into a highfrequency checkerboard mode.Subsequentexperiments carried out in reference 4.34 indicated that one particular asymmetric bafflearrangement provided Inore damping than didone particular ring baffle. (The areas of thetwo types of baffles were equivalent and theywere equal to 16 percent of the cross-sectionalnrea of the tank.) Somewhat contrastingresults were obtained by Garza (ref.
4 . 3 5 ) , whofound that a particular ring baffle providedgreater damping than did a particular asymmetric baffle a t baffle depths from d,/R=O, totl,/lZ=0.12 and approximately the same amountof danlping a t greater depths. I n the latterexperiments, the bame nrea was 28 percent ofDAMPING OF LIQUID NOTIONS AND LATERAL SLOSHINGthe tank cross-sectional area; the experimentalresults obtained are shown in figure 4.21.These contrasting results are 11ot contradictorybecause of the different baffle areas involved;however, they do indicate clearly the need forfurther experiments before asymmetric bafflescan be recommended for use in design.Ring & asymmetrical bafflesC; 0.286 ( 28.6% baffle areaw 1R 0.
157 Test liquid, water-Solid points indicate multivaluedpossibilities for frequency due toring swments00.080.160.24B f f l e depth, dsl R0.32121been studied experimentally in quite somedetail and, to some extent, theoretically aswell. As we discussed early in this chapter,Miles (ref. 4.2) using data based upon reference4.36 succeeded in obtaining an analytical expression for the free surface damping producedby a solid ring baffle. The range of validity ofMiles' theory was later extended (ref. 4.6).Bauer (ref. 4.37) modified Miles' results toaccount for the fact that part of the baffle mayemerge from the liquid during a slosh cycle.The damping caused by the ring baffle breakingthrough the free liquid surface, though, is notaccounted for (fig.
4.22). Experiments (refs.4.6 and 4.11) have indicated, as we saw infigure 4.3, that for a certain baffle and tankgeometry, this damping can be considerablylarger than that predicted by Miles. Forgreater baffle depths, the theoretical prediction:,are generally quite good.0.40FIGURE4.21.-Liquidresonant frequencies and dampingratios for ring and asymmetrical baffled cylindricaltanks (ref. 4.35)..I=.,-.-Tstersrtinn99between a flat solid-ring baffleand the liquid free surface at shallow liquid depth(ref.
4.38).v---...r.4.5DAMPING BY FIXED BAFFLES: RING TYPEIntroductionSymmetrical ring baffles, often perforn tedto save weight, are c o m m o n l ~used in rocketsand spacecraft to reduce the effects of propellant sloshing. The damping of liquid motions in cylindrical tanks by ring baffles hasA great variety of modifications to the basicflat, solid annular ring baffle can easily beenvisioned, some of which are sketched infigure 4.23.
Generally, it appears that anything that is done to reduce the sharpness of122THE DYNAMIC BEHAVIOR O F LIQUIDSand thus normal to an annular ring located as m d distance below the free surface. Energydissipation results from wave motion being2opposed by the ring pressureCD, where o is%IFlat plateFlat plate45OHatUPLipVanesFingersT withcutoutsThick plateFlat plate45 downthe local wave velocity producing the pressureand CD is the local drag coefficient. I t is thenshown (ref. 4.2) that the damping ratio forring damping in a circular cylindrical tank canbe expressed byLip withcutoutsTFIGURE4.23.-Ring baffle configuration8 (ref. 4.23).the edge plate (lip, snndwich, T, etc.) reducesthe baffle effectiveness.
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