H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 13
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2.15)), and approximate values forthe first mode (n=l) are given in figure 2.45.By comparing these empirical equations withexperimental data, it was found that they areessentially independent of both tank size andeccentricity, but that the ratio of experimentally determined to calculahd frequency wasdifferent from unity.LATERAL SLOSHING IN MOVING CONTAINERSNoteFaired curves shmnfrequency parameter may be approximated bya single curve for a given modal number.9.8 CONCLUDING REMARKS0I0., , ,I , , , ,45h2rLOa5ILO-.-h-2r2(R-r), , ,,a5..LO--h-2R7FIG^ 2.38.-Variation of liquid'#hquency parameterwith depth for transverse mod- of vertical toroidaltanka (ref. 2.51).Actual experimental data for liquid frequencies in the longitudinal and transverseorientation are given as a function of liquiddepth in figures 2.46 and 2.47.
The curvesshown are, of course, only faired through theJ-LTc , L ~ . . l d ha n , m a f ~ . l l w - n t d that !haurnurn.r u uuvuu uw v u r v - u ~"second" mode for a spherical tank (fig. 2.28)corresponds to the "third" modes of figures2.46 and 2.47 and that the "second" modes ofthese figures have no counterpart in figure 2.28;this arises from the fact that new modes ofoscillation occur when the planform of theliquid free surface is elliptical, as noted previously (see also ref.
2.15). These data indicatethat, for a given eccentricity, the values of theU W W V ~----From the results presented in the foregoingsections of this chapter, it should be evidentthat a great deal of information is readilyavailable concerning the general subject oflateral sloshing of liquids in moving containers.For the simplest case of an upright cylinder ofcircular cross section, a very complete background of theory and experiment exists forvirtually every aspect of the liquid behaviorand its reaction on the tank. This is true evenfor the relatively more complex cases wherethe circular cross section is compartmented intorings or sectors by annular or radial walls.Virtually the same statement can be made forthe rectangular tank, but for other and moredifficult co&guratio~s our detailed knowledgeis considerably diminished.The cylindrical container of rectangular,circular, or elliptical cross section can, of course,be analyzed &ctly on the basis of classicalhydrodynamic theory.
For other geometries,however, recourse has to be made to numericalsolutions (ref. 2.26) or to other approximatetechniques. The variational methods especiallyhave proven to be quite useful in a variety ofcases (refs. 2.14, 2.22, and 2.25), but there isno one method of analysis that is universallyvalid.While we have considered various modes ofexcitation of these different containers andorientations, coupling arising from more thanone excitation has largely been ignored. Oneformulation of rather general applicability hasjust become available (ref. 2.54), involvingtanks of arbitrary shape moving with sixdegrnes nf freedom. Calculations have beencarried out for tanks of spheroidal and toroidalconfiguration (ref.
2.55). One analysis involvingcoupled translational motion in two orthogonalmotions has been mentioned earlier in this chapter (ref. 2.40) ; another study considered coupledtranslation and pitch (ref. 2.56).The significant effects of nonlinearities ofvarious types have been referred to, but thisdetailed consideration deferred to chapter 3.,60. 8"10FIGURE2.39.-VhatioaIIIIJ0.20.40.60.8LORegional fullness ratio ( h -2r )l2(R -r )0o a2 a4 0.6 a 8 L ORegional fullness ratio lh-2R )12r0of Liquid frequency parametas with depth for langitndinal mod- of v d d tomidal tank (nf.
2.51).0.20.40.60.8LO,Regional fullness ratio h12RPeripheral circle,-161412--LATERAL SLOSHING IN MOVING CONTAINERS-------5-3FIGURE 2.41.-Experimentallydetermined frequencyparameter for conical tanka of s m a l l vertex angle (ref.2.43).-/j 3 Asymptote-----------1-j - 2-63-AsymptoteDirection of tank oscillationsLongitudinal orientation(elliptical planform)ha'tTransverse orientation(elliptical planform)0'Spheroid configurationsfrPlan viewVertex semiangle, a 1FIGUBE2.40.-Frequencyvcmm depth for a mid tank(ref. 2.22).Horizontal orientation(circular planform)Numbera,cmb,cm1234567.887.88163216323x433.47.8816.3233.44067895.85812l2 1816.7525.27.8816.323x4FIGURE2.42.--0rientationr and dimenaiona of spheroidaltanka studied in reference 2.45.Similarly, the very important question ofsuppression of lateral sloshing by mechanicdbafiles or other devices will be treated inchapter 4.
And finally, the very importantproblem of simplified representation of lateralsloshing forces and moments under variouscircumstances will be discussed in chapter 6.645-65.2/TEE DYNAMIC BEHAVIOR OF LIQUIDSModeI st2nd3rd------.iFIGURE2.44.--Comparison of liquid frequency parameterwith cylindricpl tank solution in ephaoidal tanka(ref. 2.25).3.40hlbFIG-2.43.-Variation of ebeh frequency with liquiddepth in oblate ephvoidal tanka (ref. 2.25).Eccentricity,FIGURE2.45.-Variation of parametemeccentricity (ref.
2.45).and kr.1 withLATERAL SLOSHING IN MOVING CONTAINERS40-3.6-3.2'2nd1st3rd4th----------00(L 2I0.40.6Fullness ratio, h12aMode1st2nd3rdI-----Faired curvesI-II/ --Faired curvesIII-Fullness ratio, h12aRange i n a is 7.88cm to 33.4cmI65I0.8LOFIG^ 2.47.-Variationof frequency parameter withratio for liquida in spheroids with transverseorientation (after ref.
2.45).fulhle08FIGURE2.46.-Variation of frequency parameter withf d n e m ratio for liquid in spheroids with longitudinalorientation (after ref. 2.45).REFERENCES2.1. COOPSR,R. M.: Dynamics of Liquids in MovingContaiiem. ARS J., vol. 30, no. 8, Aug. 1960,pp. 725-729.2.2. BAUER, HELMUT F.: Theory of the FluidOscillations in a Circular Cylindrical RingTank Partially Filled With Liquid. NASAT N D-557, 1960.2.3. BAUER, HELMUTF.: Fluid Oscillations in theContainers of a Space Vehicle and TheirInfiuence Upon Stability. NASA T R R-187,1964.2.4. BAUER, HELMUT F.: Liquid Sloshing in aCylindrical Quarter Tank. AIAA J., vol. 1, no.11, Nov.
1963, pp. 2601-2606.2.5. BAUER,HELMUT F.: Theory of Liquid Sloshingin a 45' Sector Compartmented CylindricalTank. AIAA J., vol. 2, no. 4, Apr. 1964, pp.768-770.2.6. ABRAMSON,H. NORMAN;GARZA,Lms R.; ANDKANA,DANIELD.: Liquid Sloshing in Compartmented Cylindrical Tanks. ARS J., vol.32, no. 6, June 1962, pp. 978-980.66THE DYNAMIC BEHAVIOR OF LIQUIDS2.7. ABRAMSON,H. NORMAN;CEU, WEN-HWA;ANDGARZA,Lms R.: Liquid Sloshing in 45"Sector Compartment Cylindrical Tanks. T RNo.
3, Contr. NAS 8-1555, Southwest ResearchInstitute, Nov. 1962.2.8. ABRAMSON,H. NORMAN;AND GARZA,LUISR.: Some Measurements of Liquid Frequenciesand Damping in Compartmented CylindricalTanks. AIAA J. Spacecraft Rockets, vol. 2,no. 3, May-June 1965, pp. 453-455.2.9. EULITZ, WERNERR.: Analysis and Control ofLiquid Propellant Sloshing During MissileFlight. MTP-P and VE-P-61-22, NASAMSFC, Dec. 1961.2.10. EULITZ,WERNERR.: Practical Consequences ofLiquid Propellant Slosh Characteristics Derivedby Nomographic Methods.
MTP-P and VEP-63-7, NASA-MSFC, Oct. 1963.2.11. ABRAMSON,H. NORMAN;AND RANSLEBEN,G m ~ oE., JR.: Wall Pressure DistributionsDuring Sloshing in Rigid Tanks. ARS J., vol.31, no. 4, Apr. 1961, pp. 545-547.2.12. ABRAMSON,H. NORMAN;AND RANSLEBEN,GUIDOE., Jr.: Some Comparisons of SloshingBehavior in Cylindrical Tanks With Flat andConical Bottoms. ARS J., vol. 31, no. 4,Apr. 1961, pp. 542-544.2.13. LAMB, H.: Hydrodynamics.
Sixth ed., DoverPubl., 1945.2.14. TROESCH,B. ANDREAS:Free Oscillations of aFluid in a Container. In: Boundary Problemsin Differential Equations (Rudolph E. Lmger,ed.), Univ. of Wisconsin Press, 1960.2.15. CHU, WEN-HWA:Sloshing of Liquids in Cylindrical Tanks of Elliptical Cross Section. ARSJ., vol. 30, no. 4, Apr. 1960, pp. 360-363.2.16. A B R A M ~ ~H.N , N.: Dynamic Behavior ofLiquida in Moving Containers. Appl. Mech.Rev., vol. 16, no.
7, July 1963, pp. 501-506.2.17. STOXER, J. J.: Water Waves. IntersciencePublishers, Inc., New York, 1957.2.18. ROBERTS,J. R.; BASURTO,E. R.; AND CHEN,P. Y.: Slash Design Handbook I. NASA CR406, 1966.2.19. Bausa, HELMUTF.: Theory of Fluid Oscillationsin Partially Filled Cylindrical Containers.MTP-AERO-62-1, NASA-MSFC, Jan. 1962.2.20. CHU, WEN-HWA:Free Surface Condition forSloshing Resulting From Pitching and SomeCorrections. ARS J., no. 11, Nov. 1960, pp.1090-1094.2.21.
BUDIANSKY,BERNARD:Sloshing of Liquids inCircular Canals and Spherical Tanks. J.Aerospace Sci., vol. 27, no. 3, Mar. 1960, pp.161-173.2.22. LAWRENCE,H. R.; WANQ,C. J.; AND REDDY,R. B.: Variational Solution of Fuel SloshingModes. Jet Propulsion, vol. 28, no.
11, Nov.1958, pp. 729-736.2.23. RILEY, JAMESD.; AND TREYBATH, N. W.:Sloshing of Liquids in Spherical Tanks. J.Aerospace Sci., vol. 28, no. 3, Mar. 1961,pp. 245-246.2.24. DOKUCHAEV,L. V.: On the Solution of a Boundary Value Problem on the Sloshing of a Liquidin Conical Cavities. PMM, vol. 28, no. 1,1964, pp. 151-154.2.25.
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