H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 25
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Rings with radialcleamnce, conic sections (perforated and unperforatcd) , and cruciforms in spheroidal tankshnl-e all been investigated (ref. 4.28). Nearlyall of these experiments hare indicnted thesnperioritj- of flat ring baffles iu providing agreat deal of damping for relatively small weightpenalties. We shall therefore devote considerable discussion tc) these within the followingpages.Ring Damping in Cylindrical Tanks-TheoryThe damping of liquid free surface oscillations in a circular cylindrical tank by a flatsolid ring baffle has been predicted theoretically,as we hare already noted, by Miles (ref. 4.2).The theory has been extended by O'Neill(ref.
4.6) and modified by Bauer (ref. 4.37).The liquid, oscillating in its fundamentalmode, produces a ware having its maximumurnplitltde a t the mull. The direction of flowin the vicinity of the wall is essentially vertical,where d,, c, nnd 'i are related to tanks geometryand are shown in figure 4.2. This relationshipis based upon the assumptions: (1) the fluidis oscillating in its fundamental mode andlinearized potential theory accurately describesthe flow except in a small region near the ring;(2) the local flow in the region of the ring isunaffected by the free surface or the tankbottom; (3) the circular frequency is accuratelytipproximated by the potential flow solution forthe undamped circular frequency (see ch. 2 ) ;and (4) the drag coefficient is independent ofthe angular coordinate.
Assumptions (1) and(2) require thatwhile assumption (3) requires that the dampingbe moderat,ely small.An espression for the drag coefficient, basedupon certain experimental data (ref. 4.36), waspresented by Miles nswhere U , is the tirnewise maximum velocity,T the period, and D the plat,e widt,h. Forlarge liquid depths, h/2R>1, and for thedominant sloshing mode in a c,ylindrical tank,the above espression for CD can be rewritten(employing the presen't notation) asSubstitution of equation (4.15) into equation(4.14) yields the damping ratioDAMPING OF LIQUID MOTIONS AND LATERAL SLOSHINGFIGURE4.24.-Dampingdfactor as a function of baffle depth calculated from Miles' theory (after ref. 4.37).(f)"'r=2.83e-4.6--dC;f2(4.16)I t should be noted that the damping decreasesexponentially with increasing dJR.
Calculated curves of damping factor versus baffledepth for various liquid amplitudes and withbaffle width as a ammeter are shown in figure4.24.Two significant contributions relative to--..equation (4.16) h a r e been made in reference4.6. First, the dimensionless force parameterF , ~ ~was~ introdueed10R ~and theof this parameter lvns related to 7 by_"JThis force parameter is based upon the similitudetheory presented in ref.'4.19, and is generally employedit is often easier to measure than is the liquidamplitude.124THE D ~ N A M I C BEHAVIOR OF LIQUIDSemploying linearized potential theory.
Forliquid oscillations in the fundamental mode inn deep tank, the relationship iswhere ng is the acceleration along the tankaxis. Substituting this into equat,ion (4.16)yieldsEither equation (4.16) or (4.17) may be used tocalclilate the dnmping rntio. The advantage inusing equation (4.17) is that no visual obserl-ation of the sloshing liquid itself, but only theforce, is required.
The second significant contribution is the estension of the range of validityof equations (4.16) and (4.17) through experimental observations. I t was indicated thatthese equations may be employed for any valueof 5 orpertinent and reasonnble forRpngR3ring dnmping in propellant tanks, for any ring-(L)submergence % > 0 , and for ring widths correI? spondirig t o C,<0.25. The datn did have considerable scatter, but most of the dampingnitios fell I\-ithin h 3 0 percent of the predictedvalues.It should he noted that Bauer (ref. 4.37) hasdeveloped estensive plots from Miles' theoryshowing the damping ratio for various valuesof baffle depth, baffle width, liquid amplitude,and baffle location. (See fig.
4.24.) He alsocomputed the damping ratio as a function ofliquid depth in a tank having a number of ringbaffles, as shown typically in figure 4.25. Thefigure shows the effects of various baffle spacingand is based on 10 baffles above and 10 belowthe undisturbed liquid free surface.As we discussed earlier, Bauer (ref. 4.37) alsoattempted to account for the condition duringsloshing where part of the baffle emerges fromthe liquid. The effect is shown in figure 4.26and can be compared directly with Miles'result from figure 4.24 (see also fig. 4.3).Ring Damping in Cylindrical Tanks-ExperimentDamping produced by ring baffles in cylindrical tanks has bee11 investigated extensivelyin a number of the references already cited inthis chapter as well as in reference 4.38.
Mostof these studies h a r e indicated that: (1) the flatring baffle is a very effective damping device,and (2) the damping ratio is approximatelyindependent of the mode of excitation (pitchingor tmnslation) .The experiments of O'Neill (ref. 4.6), asdiscussed previously, have extended the ~ralidityof the range of the ring damping theory ofMiles and have compared various methods ofobtaining the damping factor.
Abramson and0Decreasing liquid levelFIGURE4.25.-Damping factor for a series of ring bafflescalculated from Miles' theory (ref. 4.37).FIGURE4.26.-Damping factor as a function of baffledepth calculated from Bauer's theory (ref. 4.37).DAMPING125OF LIQUID MOTIONS AND LATERAL SLOSHINGGarza (ref. 4.11) and Stephens (ref. 4.28) haveinvestigated in detail various geometrical parameters and their effects on liquid resonantfrequency and damping ratio. We shall presentand discuss various of these results in thefollowing paragraphs.Liquid Resonant FrequencyThe presence of a baffle obviously modifiesthe liquid resonant frequency, especially whenit is located near the liquid surface (ref.
4.25).Figure 4.27 shows this effect, with baffle widthas a parameter, for a single flat solid ring baffle.For perforated baffles, the frequency becomesdependent upon both the perforation hole sizeand the percent open (ref. 4 . 1 1 ) , as can be seen4.29. (Solid ring baffle datain figures 4 . 2 ~ ' u n dare also included in these figures, for comparison; the effects of baffle width are given infig. 4.27.) All of these experimental datashow, generally, that the liquid resonant frequency is a maximum for a baffle located a t theliquid free surface, decreases to a minimum a ta baffle depth near d,/R=O.lO, and then increases gradually with increasing d , / R until the2800.080.16Baffle depth,0.24dSIR0.320.40FIGURE4.28.-Effect of percent perforation on liquidreeonant frequencies as a function of baffle depth(ref.
4.11)-first liquid resonant frequency of an unbaffledtank is reached. For baffle depths greater thantl,/R=0.06, it may be observed that for 0.201centimeter-diameter hole perforations the frequency increases with increasing percent of holeperforations; the frequency also increases withperforation hole size for a given percentage perforated area. Since the excitation amplitudesignificantly affects the frequency, measurements were made for various values ranging3then the datafrom 0.00184 L x o / d ~ 0 . 0 0 8 2 andpresented in terms of rms values.For design purposes, the axial spacingbetween each of a series of ring baffles shouldbe less than about d,/R=0.08.
This configuration yields the highest possible resonantfrequency which can be maintained with ugiven baffle system. This effect is shown infigure 4.30 for two ring baffles.Baffle location, dS 1 RFIGURE4.27.wVariation in liquid resonant frequency withbaffle location (ref. 4.28).Liquid DampingThe damping ratio for flat single-ring bafflesis quite dependent upon excitation amplitude,1262.8THE DYNAMIC BEHAVIOR OF LIQUIDSIoa08IIa160.24Baffle depth, dSIR1Ia32a40AFIGURE4.29.-Effect of perforation hole size on liquidresonant frequencies as a function of baffle depth(ref.
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