H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 26
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4.11).as shou~11in figure 4.31, so that the variousdata will again be presented in terms of rmsvalues. Figures 4.32 and 4.33 show the effectsof percent perforation and hole size on dampingfactor, while figure 4.34 again shows the effectof baffle width, for solid ring baffles.We hare alreadj discussed in detail thecorrelntion with theory, as shown in figure4.3, and, therefore, shall avoid repetition inthis section.The effects of liquid kinematic viscosity onthe resonant frequency and damping ratiowere not considered in these experiments(ref. 4.11); however, if the trends establishedin cylindrical tanks having perforated partitions (sec.
4.4) can be used as crude criteria,then it mag turn out that the effects of kinematic viscosity are significant for perforatedring baffles.Tests were also conducted wit11 two solidring baffles in the tank to determine theeffects o r 1 damping- one bnHe being placedabove and one below the liquid surface, thezc6.04.02.0Liquid resonant frequency parameter, w e d l gFIGURE4.30.-Effectsof double rings as a function ofbaffle depth (ref.
4.38).results being shown in figure 4.30. Thesedata indicate that the baffle above the surfaceis effective only if its distance from the freesurface is less than about dJR=0.125, whilethe submerged baffle is effective from aboutdJR=O to d,/R=0.375. Tests of this natureare useful to designers in thrtt. they indicatethe baffle spacing corresponding to someminimum acceptable damping ratio.Conical-Section BafflesThe perforated conical ring baffles have alsobeen investigated sufficiently t,o warrant somebrief presentation of data within this section.Figure 4.35 shows the variation in dampingfactor as a function of baffle location, withbaffle width as a paramet,er; one set, of data forthe inverted baffle is also included. Data fortwo perforated baffles (50 percent open) areshown in figure 4.36.
These data would seemt o indicate trends t ~ n dvalues quite similar tothose obtained for the flat rings.127DAMPING OF LIQUID MOTIONS AND LATERAL SLOSHINGmental mode in a cylindrical tank is given by-Symbol---0--X,ld0.001840.004170.00833p=- 2ngelRp cos e ( I )--*---*a--)'-,,+J (el RJl(el)eR,%<r<RRMS(X,ld=0.00184 to0. m33where p is the pressure, c=1.84 is the firstroot of J((e) =0, and d,, ao,r, R, e, 8, are relatedto tank geometry and coordinates as shown infigure 4.39.
I t is seen that the pressure distribution varies with the cosine 8 aroundthe circumference of the baffle and varies onlyslightly with the coordinate r for values ofw/R generally employed in practice. Thetotal moment Mb acting to overtllrn the baffle"00.080.160.240.320.40Baffle depth, dSIRFIGURE4.31.dEffect of excitation amplitude on damping effectivenese of a solid flat ring baffle as a function ofbaffle depth (ref. 4.38).Conical-section perforated-ring baffles werealso studied experimentally in the laboratory,with relation to simulation of a specific fullscale vehicle (ref.
19). (This study also included floating can devices, as discussed insec. 4.3.) Figure 4.37 shows force responsecurves, comparing the full-scale prototypeand model data, while figure 4.38 shows thevariation of damping factor wit,h an equivalentReynolds number parameter, of the typeme~ticr?edpre+;,zus!-~.JI2 t!:k latter figure,data on floating cans (independent of liquiddepth since they are surface devices) are alsoincluded.Loads on Rigid Flat Ring BafflesSome early data on baffle loads were givenby Srmstrong and Kachigan (ref. 4.39).
Thedynamic pressure at any point on the baffleresulting from liquid sloshing in its funda--RMS ( X,ld 0.00184 to 0.00833 )Test liquid, waterd = 36.6 cmgw 1 R = 0.157a20L"00.080.160.240.320.40Baffle depth, dS 1 RFIGURE4.32.wEffect of percent perforation on dampingeffectiveness as a function of baffle depth (ref. 4.11).128THE D ~ A M I CBEHAVIOR OF LIQUIDS1 d = 36.6VcmoTest liguid, water.w 1R Q 1571IIII1a oao. 16o.
24a 320.40Baffle depth, d S l RFIGURE4.33.-Effect of perforation hole size on dampingeffectiveness as a function of baffle depth (ref. 4.11).u00.10.20.3a 4 0.50.6Baffle depth, d S I R0.70.8FIGURE4.34.wVariation in damping factor with bamelocation for eolid baffles (ref. 4.28).-0.4-0.200.20.4a6Baffle depth, d s l R0.81.0FIGURE4.35.-Variation in damping factor with bafflelocation for solid conical-section baffles (ref. 4.28).Baffle depth, d S I RFIGURE4.36.WVariation in damping factor with bafflelocation for perforated conical-section baffles (ref.4.28).129D A M P ~ GOF LIQUID MOTIONS AND LATERAL SLOSHING9Phase angle( Deg.)100Frequency parameter,dw21gFIGURE4.37.-Comparison of model and ABMA full-wale tests of force response in a tank with conical rings (ref.
5.19).is obtained by integration of this pressuredistribution over the baffle. Thus+ ( %\LC/YE17,.)l\tllI(4.19a)I t can readily be determined by integration ofthe force per unit lenqth of baffle over thecircumference of the baffle thatMb=xR2Fa-F=2FbR=-21447rE(4.19~)Figure 4.40 shou-s a plot of the left-hand sideas a function of tl,/R forof equation (4.19~~)various values of ao/R, valid for fluid depthsh/R>l.
For lower depths, a correction factorCF is given asd._I-baffle is therefore(4.19b)and that the total force )'acting on one-half theee'ECF=cosh$)+sinh( E ~$)tanh (el $)(4.20a)to be applied such that,Mb (true) =Mh[eq. (4.19a)lX CF (4.20b)THE DYNAMIC BEHAVIOR OF LIQUIDS-'---Rings(Conical ringshld'0.785h l d = 0.5950 hld-0.505hld-0.505)--------~in~s(0hld=a595)a0 hIds0.595hld=0.50500. 10.2FIGURE4.38.-Variation0.40.60.8 124Equivalent Reynolds number,d20406080 100of damping factor with equivalent Reynolds number for conical-section rings and floating cans(ref.
4.19).Figure 4.41 sho~vsa co~npnrisonof theory andexperiment for the force on 11 ring baffle; the:igreemen t is quite good.Another theoretical analysis of baffle loads~vnscarried out by IJiu (ref. 4 . 4 0 ) . He emplo-ed conformal mapping to transform thebnMe and tank wall into a simple plane forwhich n solt~tion to Bernoulli's equation canreadily be obtained, and then having the velocity tlnd pressure distribntion on the baffle inthe transforn~edplane, the inverse trnnsfurmntion yielding the solutiorl in the physical plnne.Tile final result for the pressure on the baffletakes the forni?I= (2pw)Fwp-3.06-2d6 8 1 0d"' 9"',l 1(y/w)'sin wt cos e ( 4 .
2 1 )where y is tlie coordincite in the radial directionrliet~suredinward from the tank wall.Figure 4.42 sliows a comparison of gressilres(111 baffles 11s computed from equation (4.2 1 )1111d ILS measured by Garza (ref. 4 . 4 1 ) , for0=O0 with solid rings. For f?=30° und 60°,Garza found some slight reduction in the pressures and a general but slight flat,tening of thecurves. The experimental data are generallyslightly higher than are the calculated values.Figure 4.43 compares the experimental forcedata with the predictions of both theories.Except for very shallow baffle depths, thetheoretical predictions rippear to bracket themeasured values. Garza also investigated(ref. 4 .
4 1 ) the effects of baffle perforation,employing about the same range of parametersutilized in previous studies of baffle characteristics (ref. 4 . 1 1 ) . Both increasing perforationhole size and percent perforation resulted inreductions in the force acting on the baffle up toas much as 25 to 30 percent.Ring Damping in Spheroidal TanksOblate SpheroidsThe performance of ling baffles in an oblatespheroidal t8ank has been investigated in reference 4.12. The variables considered wereDAIMPING OF LIQUID MOTIONS AND LATERAL SLOSHING131+R+GeometryFIGURE4.40.-Moments acting on an annular ring baflleas a function of baffle depth and width (ref.
4.39).coordinatesFbL.4 ~ o d a llineDistance to center of pressureof Fb (approximately equal to Rfor small w l R )Fb - The value of the force per unitlength of baffle at 8 0/-Pressure distribution over half - baffleFIGURE 4.39.-Geometry,coordinates, and pressuredistribution for ring pressure calculations.the baffle submergence, tank fullness, bafflewidth, ~ ~ m p l i t l ~ofl ethe free s ~ r f s c edisplacements, and liquid kinematic viscosity. Thevariation of the liquid natural frequency mithliquid depth and baffle location was also determined. The tank orientation is as shownin figure 4.44 and the damping factor wasdefined byThe variation of the damping factor withtank fullness for ring baffles of various widthsis shown in figure 4.44.
I t is seen that t,hedamping increases with baffle width, and alsowith tank fullness until a maximum value isreached when the liquid level is approximat.elyat the baffle and then decreases with increasingfullness. The maximum damping also increasesas hg (fig. 4.44) is reduced. As the baffle ismoved up, above the liquid free surface, thedamping factor curve becomes much sharperwith about the same maximum value. Acomparison of damping factor obtained inboth ring baffled and unbaffled tanks is shownin figure 4.45.
The rather strong variation ofdamping with the free surface displacement Z(measured at the wall) is shown in figure 4.46,for a constant baffle width.Soherical TanksRing damping in spherical tanks has beeninvestigated in references 4.29 and 4.42. Thebaffles used in reference 4.29 were perforatedmith 23 percent open area, a hole diameter of0.02 inch, and a width ratio of w/R=0.285; theconfiguration is shown in figure 4.17. (This hasbeen referred to earlier in this section as thecruciform or verticd arrangement; rotation ofTHE DYNAMIC BEHAVIOR OF LIQUIDSMean experimental values measurednder steady sloshing(Values are RMS pressuresat y l w - 0 .
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