H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 4
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The secondcomposed of sets of simple spring-massdashpottopic relates to liquid spray formed a t the free(or pendulum) elements, which can then besurface because of relatively high frequency10THE DYNAMIC BEHAVIOR OF LIQUIDSexcitation; the third topic relates to certaininteresting aspects of bubble dynamics andcavitation that have been observed in laboratory studies.Chapter 9 now brings into focus some of theeffects rising from the elasticity of the containers. The first point of discussion is thatof the influence of the liquid in modifying thevibration characteristics of the elastic shell,without considering any significant couplingand primarily as a result only of the addedmass.
Both bending- and breathing- typevibrations are considered. Next, the coupledproblem is considered and interesting forms ofliquid response induced by tank wall elasticvibrations (observed in the laboratory) arediscussed. The influence of the elastic bottomof the tank, with regard to the vertical sloshing problems discussed in chapter 8, is thendiscussed. Finally, the effects of body-bending flexibility on the general problem of vehiclestability and control, for which the rigidbody case was presented in chapter 7, are oublined.Three interesting but unrelated topics thatseem to warrant some presentation, althoughslightly apart from the bulk of the materialpresented in this monograph, are collectedtogether in chapter 10.
The problem of domeimpact is presented first, and while such aproblem actually arose during one phase ofdevelopment of the Redstone missile, the importance of this problem in launch-abortsituations and the closely related one of orbitalmaneuvering and docking is not yet fullyestablished. Vortex formation is an interestingphenomenon, but one which can be suppressedquite simply in most cases by baffling the tankoutlets, and is therefore discussed only briefly.The Titan I1 launch vehicle, in the early stagesof its flight test program, exhibited rathersevere longitudinal oscillations of the entiresystem, now often called "pogo" oscillations.The liquid systems and the elastic structurewere quite well coupled and, together with thecombustion system, formed a closed loop.While not solely a problem of liquid dynamicresponse, it was felt that the phenomenon wasof sdlicient importance to warrant a briefpresentation.The various complex and important behaviorof liquids under' conditions of low and zerogravity are discussed in detail in chapter 11,the concluding chapter of this monograph.Means of simulating such behavior in laboratoryexperiments are especially treated in somedetail, as are the various tecbniques for con trolling the location and orientation of liquidswithin tanks subject to the weightless conditionof the space environment.REFERENCES1.1.
SLOANE,MARGARETN.: The Dynamics of theSloshing Phenomenon (a Literature Survey).Rept. GM-60-5111-5, Space Technology Laboratories, Inc., Apr. 1960.1.2. COOPER,R. M.: Dynamics of Liquids in MovingContainers. ARS J., vol. 30, Aug. 1960, pp.725-729.1.3.
ABRAMBON,H. NORMAN:Dynarnic Behavior ofLiquids in Moving Containers. Appl. Mech.Revs., vol. 16, No. 7, July 1963, pp. 501-506.H. NORMAN:Some Current Aspecte1.4. ABRAMBON,of the Dynamic Behavior of Liquids in RocketPropellant Tanka. Applied Mechanics Surveys, Spartan Books, hlc., Washington, D.C.,1966.1.5. LAMB,H.: Hydrodynamics. Sixth ed. DoverPublications, New York, 1945.1.6. G o ~ ~ s ~ o ~ oG.n aR.:n ,The Tidal Oscillations inan Elliptic Baain of Variable Depth.
Proc.Roy.'Soc, .(London), A, 130, 1930, pp. 157-167.1.7. VANONI,V. A.; AND CARE, J. H.: HarborSurging. Proc. First Conf. Coastal Engineering,Long Beach, Calif., 1951, pp. 60-68.1.8. PENNEY, W. G.; AND PRICE, A. T.: SomeGravity Wave Problems in the Motion ofPerfect Fluids.Part 11. Finite PeriodicStationary Gravity Waves in a Perfect Liquid.Phil.
Trans. Roy. Soc. London, A, 244, 1952,p. 254.1.9. WESTERQAARD,H. M.: Water Pressure on DamsDuring Earthquakes. Trans. Am. Soc. CivilEngineers, vol. 98, 1933, pp. 418-472.1.10. J a c o a s E ~ ,L. S.: Impulsive Hydrodynamics ofFluid Inside a Cylindrical Tank and of FluidSurrounding a Cylindrical Pier. Bull. Seiam.Soc. Am., vol. 39, 1949, pp. 189-204.1.11. WERNER,P. W.; AND SUNDQUIBT,K. J.: OnHydrodynamic Earthquake Effects.
Trans.Am. Geophysical Union, vol. 30, no. 5, Oct.1949, pp. 636-057.INTRODUCTION1.12. JACOBSEN,L. S.; AND AYRE,R. S.: HydrodynamicExperiments With Rigid Cylindrical TanksSubjected to Transient Motions. Bull. Seism.Soc. Am., vol. 41, 1951, pp. 313-346.1.13.
HOUSNER,G. W.: Dynamic Pressures on Accelerated Fluid Containers. Bull. Seism. Soc.Am., vol. 47, 1957, pp. 15-35.1.14. CLOUGH,R. W.: Effecb of Earthquakes onUnderwater Structures. Proc. Second WorldConf. Earthquake Engineering, Japan, 1960,pp. 815-831.1.15. BLAOOVESHCHENSKY,S. N.: Theory of ShipMotions. English ed. Dover Publications,New York, 1962.1.16. VASTA,J.; GIDDINOS,A. J.; TAPLIN, A.; ANDSTILLWELL,J. J.: Roll Stabilization by Meansof Passive Tanks. Trans. SNAME, vol.
69,1961, pp. 411-460.1.17. SEWALL,J. L.: An Experimental and TheoreticalStudy of the Effect of Fuel on PitchingTranslation Flutter. NACA T N 4166, Dec.1957.1.18. GRAHAM,E. W.; AND RODRIGUEZ,A. M.: TheCharacteristics of Fuel Motion Which AffectAirplane Dynamics. J. Appl. Mech., vol. 19,Sept. 1952, pp. 381-388.K.: On the Stability of a Spinning1.19. STEWARTSON,Top Containing Liquid.
J. Fluid Mech.,vol. 5, part 4, 1959.130. S c o q W. E.: The Free Flight Stability of aLiquid Filled, Spinning Shell. BRL Repb.1120, 1135, 1233, Aberdeen Proving Ground,.Md.1-21. MILSS, J. W.: Free Surface Oscillations in aRotating Liquid. Journal of Physics of Fluids,vol. 2, no. 3, May-June 1959, pp. 297-305.111.22. WINCH,D. M.: An Investigation of the LiquidLevel at the Wall of a Spinning Tank. NASAT N D-1536, Aug. 1962.1.23.
BARON,M. L.; AND BLEICH,H.: The DynamicAnalysis of Empty and Partially Full Cylindrical Tanks, Part 11-Analysis of Uplift andStructural Damage. DASA Rept. 1123B,Sept. 1959.1.24. GEISSLER,E. D.: Problems of Attitude Stabilieation of Large Guided Missiles. AerospaceEngineering, vol. 19, Oct. 1960, pp. 24-29,88-71.1.25. Saturn SA-1 Flight Evaluation. Rept. MPRSAT-WF-61-8, NASA MSFC, Dec. 1961.1.26. BISPLINOHOFF,RAYMONDL.; AND MICHEL,DOUGLAS:Some Structural Dynamic Problemsof Space Vehicles. Japan Society of Aeronautical and Space Sciences, Tokyo, Nov. 1963.H. NORMAN:Amazing Motions of1.27. ABRAMSON,Liquid Propellants. Astronautics, vol.
6,Mar. 1961, pp. 35-37.1.28. ABRAMSON,H. NORMAN:Liquid Dynamic Behavior in Rocket Propellant Tanks. Proc.ONRIAIA Symposium on Structural Dynamicsof High Speed Flight (Los Angeles), Apr. 1961,pp. 287-318.1.29. MOISEYEV,N. N.; MYSHKIS,A. D.; AND PETROV,A. A.: On the Problems of Hydrodynamicsin Cosmonautics. XVth International Astronautical Congress (Warsaw), Sept. 1964.1.30. GLUCK,D. F.; AND GILLE, J. P.: Fluid Mechanics of Zero-G Propellant Transfer inSpacecraft Propulsion Systems.
ASME Journalof Engineering for Industry, Feb. 1965, pp. 1-7.Chapter 2Lateral Sloshing in Moving ContainersSandor Silverman and 8. Norman AbramsonelMATHEMATICAL BACKGROUNDWe have discussed in chapter 1 how lateralsloshing of propellants or other liquids may beinduced in a rocket or space vehicle and why itis of importance. We shall present, in thischapter, theoretical and experimental ~esvltspertaining to undamped liquid sloshing behavior in rigid containers. Obviously, any attemptto repeat details of the mathematical developments leading to these results would be far toovoluminous to permit incorporation in a monograph of this type.
Further, while the detailedmanipulations for any specific tank geometryare quite tedious, the principal mathematicalproblem of finding a solution to the boundaryvalue problem for Laplace's equation is straightforward and relatively well known. The naturalfrequencies, mode shapes, potential functions,free surface displacements, and forces andmomenta are given for a variety of containergeometries. In the case of forced motion, theforcing function is assumed to be of the formem'.
Solutions for arbitrary tank motions maybe obtained from harmonic solutions by usingFourier series or Fourier integral techniques,and while some analyses of liquid behavior incontainers subjected to impulsive motion havebeen performed (see ref. 2.1 for specific references), they are not of governing importancefor this monograph.During sloshing in a cylindrical container, thelower part of the liquid performs oscillations asAL-.-Luuuugu it8 rigid Loriy rrud oniy the liquidin the vicinity of the free surface moves independently; as the frequency of oscillation of thefree surface increases, the motion penetratesdeeply into the liquid.
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