H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 21
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The logarithmic decrement of successive mave heights may be calculated from these measurements and then thedamping ratio is obtained from equation (4.4).(5) Anchor force decay method.-The rate ofdecay of the peak force in the tank anchor ismeasured after the tank motion is suddenlystopped.
The logarithmic decrement is thenobtained from successive peak values and thedamping ratio 7 is obtained from equation (4.4).The relationship between the damping factorscalculated by the above methods cannot bestated easily unless the liquid dynamic behavioris essentially linear. The mechanisms whichproduce energy dissipation are thought to beknown qualitatively, but only a few have beendescribed quantitatively. The damping maybe caused by the free surface boundary layer(see ref. 4.4), fluid viscosity, turbulence, boundarylayer friction, and interchange of energy betweendifferent sloshing modes.
I n practical applications, of course, none of the previously describedmethods accounts for all the energy dissipated.The degree of nonlinearity of the system issignificant in determining the amount of energyneglected by the various methods. If thesystem behaves Iiuetlrly and the damping issmall, then the relationships between the energy,logarithmic decrement, and damping ratio asexpressed by equations (4.1) through (4.7)are valid.Caution is especially required when considering the damping of fluid forces causedby oscillations of a tank of unsymmetric geometry. I t is conceivable that the damping of107these forces may be quite dependent upon thedirection in which the forces are measured andthe technique used to measure the damping.Reduction of force amplitudes in multipleconnected tanks (such as partitioned tanks)may be the result of phasing of the liquidmotions rather than energy dissipation.
Therefore, the designer must specify exactly what heis interested in-damping of fluid motions ordamping of tank forces.I n table 4.1, and in figure 4.2, experimentallydetermined damping values for an annular ringbaffle in a cylindrical tank with a flat bottomare given as a function of free surface waveamplitude. The values as obtained by thefive experimental methods outlined above arecompared, together with theoretical curvescalculated for conditions conforming mostclosely to the assumptions of Miles' ringdamping theory (ref.
4.2). (See sec. 4.5 of thischapter for a detailed discussion of ring dampingfrom the viewpoints of both theory and experiment.) Experimental damping data for asingle-ring baffle are compared in figure 4.3with the damping factor as calculated from thering damping theory (ref. 4.1 1).O n Damping InvestigationsViscous damping alone, with particular reference to liquids in moving tanks, has beeninvestigated from several viewpoints (refs. 4.12through 4.15) demonstrating that the significantvariables are liquid height , liquid kine ma ticviscosity, and tank size. Theoretically basedequations which predict the damping factor asa function of these variables are modified withthe aid of experimental data.
In certain situations, it may be feasible to rely entirely uponviscous damping of the fluid to suppress theliquid motions.rn,r n e damping ezectiveness of movable orfloating devices has been studied on manyoccasions (refs. 4.16 through 4.22), but generallywith the conclusion that while floating ormovable-lid-type devices may damp liquidmotions substantially, they also involve rat,hersignificant weight penalties.Recently, additional interest has developedin positive expulsion bags and diaphragms (refs.108THE DYNAMIC BEHAWOR OF LIQUIDSTABLE4.1.-Damping Ratios for an Annular Ring in a Cylindrical Tank(Ring position (h-d)lR= 2.11 ; ring-width parameter C=0.235; R= 15.1 cm; test liquid-water)[Ref. 4.61-2Method of damping measurementi/RDrivc forcc .
. . . . . . . . . . . . . . . . . . . . . . . . .Wave amplitude response- - - - - - - - - - - - Wnvr nmplitude decay - - - - - - - - - - - - - - - -Anchor forcc clccay - - - . - - - - - . - - - - - - - - -4.16 and 4.17) because of their possible application t o the problem of propellant transfer underweightless conditions. Experiments indicatethat a nlajor factor nffecting liquid damping isthe thickness of the membrane, since the amountof dnmping increases (as does the first naturalfrequency) with increasing thickness.
Damping due to these det-ices has generally beenfound to bc estreruely high.The rnnjor effort devoted to damping of liquidmotion has been on investignting the dampingdue to various types of fixed bnfffes. This is:-,-= 0.253d.0.1680.019.024.027.023.024.029.025.026.021.027. 18'18.020,015.026. 019.014.030.030.021.019.017.015.017.017.012.026.020.022.015.014.014I0. 15.12.ll-- 0.505Y0.026.027.026....................---------- ---------.15. 12.ll.027.031.031.........................................23.23.052.040,.031.022.017,051.039.026.020.015.040.042.016.014.013.010.0090.012.012.015.011.0087....................1.052.040.031.022.051.039.026.020.015.025,016.014.011.022.015.011,010.010FIR10. 19.17.16.14.13.19.17.16.14.13.21.21.032.023.017.013.034.028.024.021Y0.0085.0071.0075.0064.0061.012.0093.0091.0094.0079.023.024.0055.0052.0052.0044.OW6.004i.0044.0047-------------.-----.........................................032.023.017.013.034.028.024.021 ..0079.0059.0047.0039.0062.0052.0058.0041--..-----------------for n good reason-they provide large dampingfor relatively low weight, without other majordrawbacks.
These fixed baffles may be classified into nonring type and ring type. The nonring type includes partitions and cruciforms(these are physically locnted in the same manneras n stringer), while the ring type includes bothsymmetrical rings and asymmetrical ring segments. I n order to save weight, all of thesevnrious baffles are often perforated. I t hasbeen found that, within certain limits on thepercent of area removed b y perforation andDAMPING OF LIQUID MOTIONS AND LATERAL SLOSHING0.07--o0aAnchor force decayWave - amplitude decayWave - amplitude response-ARing forceDrive forceRing a r e a E ( 0 . 2 3 5 1 ( T R ' JTest liquid, water--a 10Wave amplitude ratio,-a 15TIRFIGURE4.2.-Damping ratios for an annular ring in a cylindrical tank as obtained by various methods (ref.
4.6).their diameter, a significant weight saving canbe realized without loss of damping effectivenessor reduction in the resonant frequency. Theeffectiveness of rigid-type baffles depends largelyupon the location of the baffle with respect t othe liquid free surface and on the baffle geometry.Baffles which have a tendency to transfer thefirst-mode sloshing into a rotary motion havebeen briefly investigated in references 4.23 and4.24. Ring baffles having double spirals withopposite directions of rotation appear to bemore desirable than single spirals, but t,helatter, along with swirl plates, induce theproblem of liquid s~ix-1.~ing in circular cylinde~s(refs.
4.3, 4.4, :uid 4.1.5).The effects on damping of the liquid depth.liquid ~linplitude, lrinematic viscosity, 11nd surface tension were specificallj- investigated hireference 4.15, und an empirical relationship forcalculating the damping coefficierit was obtainedin the form4.2 VISCOUS DAMPING IN TANKS OF VARIOUSGEOMETRYwhere v is the kinematic viscosity," . R is the tankradius, g is the acceleration of gravit,y, h is theliquid depth, and 6 is the logarithmic decrementCircular Cylindrical TankA t least three extensive experimental investigations have been carried out on viscous dampThis phenomenon was mentioned in ch.
2 anddiscussed in more detail in ch. 3.3 It is not clear from the text (ref. 4.15) whether thisequation was obtained from an examination of theexperimental data or was obtained by using experimental data to modify equations which appear in nmentioned but unreferenced report of work byRabinovich.110THE DYhTAMIC BEEUVIOR OF LIQUIDS025X)75Surface tension, T ,dynes l c mFIGURE4.4.-Theeffect of surface tension on viscousdamping in a cylindrical tank (ref. 4.15).0oa080.16a24Baffle depth, d,lRa32a40FIGURE4.3.-Comparison of theory and experiment fordamping provided by a Bat solid-ring baffle a s a functionof baffle depth (ref. 4.11).of the amplitude of the free surfacc oscillations.FOPlarge depths, h B > l .O, equation (4.8a)rnay be approximated byEmploying two cylindrical tanks of differentdiameters (20 centimeters and 51.8 centimeters)and selecting liquids with different surface tensions T (but with identical kinematic viscosityv), the effect of surface tension on the dampingwas investigated.
It was concluded that, a tleast for small tanks, surface tension can significantly affect the damping coefficient, as shownin figure 4.4.Damping in a cylindrical cavity with aspherical bottom was also determined experimentally, leading to an empirical equation forthe damping coefficier~tin the formwhere C2 is given in figure 4.5 as a function ofliquid depth. This equation reduces to equation(4.8b) for large liquid depths (h/R>l.O).I t was also determined from these experiments(ref. 4.15) that liquid amplitudes measured a tthe wall less than 0.1R had no effect on thedamping. This is in agreement with otherstudies (ref. 4.13) where no relation was foundto exist between damping and liquid amplitudefor a range of amplitudes a t least up to 0.05R.Apparently unaware of the earlier Russianwork discussed above (ref. 4.15), a similarlyextensive experimental investigation was carried out by Stephens, e t al.
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