H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 14
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RATTAYYA,J a s n V.: Sloshing of Liquids inAxisymmetric Ellipsoidal Tanks. AIAA PaperNo. 65-114, presented a t the AIAA 2nd Acrespace Science Meeting, New York, N.Y.,Jan. 25-27, 1965.2.26. EHRLICH,L. W.; RILEY,J. D.; STRANGE,W. G.;AND TROEBCH,B. A.: Finite-Difference Techniques for a Boundary Problem With anEigenvalue in a Boundary Condition.
J.Soc. Industr. Appl. Math., vol. 9, no. 1, Mar.1961, pp. 149-164.2.27. EIDE, DONALDG.: Preliminary Analysis ofVariation of Pitch Motion of a Vehicle in aSpace Environment Due to Fuel Sloshing ina Rectangular Tank. NASA T N -2336,1964.2.28. GRAHAM,E. W.; AND RODRIGUEZ,A. M.: TheCharacteristics of Fuel Motion Which AffectAirplane Dynamics. J. Appl. Mech., vol. 19,no. 3, Sept. 1952, pp.
381-388.2.29. GRAHAM,E. W.; AND RODRIGUEZ,A. M.: TheCharacteristics of Fuel Motion Which AffectAirplane Dynamics. Douglas Aircraft CO.Rept. No. SM-14212, Nov. 27, 1951.J. W. S.: The Theory of Sound. Vol.2.30. RAYLEIQH,11. Dover Publ., 1945.2.31. BAUER,HELMUTF.: Theory of Liquid Sloshingin Compartmented Cylindrical Tanks Due toB e n d i n g Excitstion. MTP-AERO-62-61,NASA-MSFC, July 1962.2.32. BAUER,H. F.: Tables and Graphs of Zeros ofCroas Product Beseel Functions.
MTPAERO-63-50, NASA-MSFC, June 1963.2.33. BAUER,HELMUTF.: Theory of Liquid Sloshingin Compartmented Cylindrical Tanks Due toBending Excitation. AIAA J., vol. 1, no. 7,July 1963, pp. 1590-1596.2.34. OKHOTSIMSKII,D. E.: Theory of the Motionof a Body With Cavities Partly Filled With aLiquid. NASA T T F-33, May 1960. Translated from Prikladnaia Matematika i Mekhanika, vol.
20, no. 1, Jan.-Feb. 1956.2.35. KACHIQAN,K.: Forced Oscillations of a Fluidin a Cylindrical Tank. ZU-7-046 (ContractAF04(645)-4), Convair Astronautics, Oct. 1955.2.36. SCHMITT,ALFREDF.: Forced Oscillations of aFluid in a Cylindrical Tank Undergoing BothTranslation and Rotation. ZU-7-069 (Contract AF04(645)-4), Convair Astronautics,Oct. 1956.2.37.
ARMSTRONQ,G. L.; AND KACHIQAN,K.:Propellant Sloshing. Section 14.14115, PP. 14-LATERAL SLOSHING IN M O W CONTAINERS2.38.2.39.2.40.2.41.2.42.2.43.2.44.2.45.14 to l k 2 7 . In Handbook of AstronauticalEngineering (Koelle, H. H., ed.), McGrawHilI, 1961.W.: On the Sloshing of Liquid inMILES, JOHNa Cylindrical Tank. Rept. No. AM6-5,The Ram-Wooldridge Corp., Guided MissileResearch Div., GM-TR-18, Apr. 1956.B A ~ R HELMUT,F.: Fluid Oscillations in aCircular Cylindrical Tank. Tech.
Rept. No.DA-TR-1-58, Army Ballistic Missile Agency,Apr. 1958. (Available from DDC; AD-160272.)B A ~ R HELMUT,F.: Fluid Oscillatioxq of aCircular Cylindrical Tank Performing Lissajoue-Oscillations. DA T R No. 2-58, ABMA,Apr. 1958.ABIXAMSON,H. NORMAN;AND RANSLEBEN,G m o E., JR: Simulation of Fuel SloshingCharacteristics in Missile Tanks by Use ofSmall Models.
ARS J., vol. 30, no. 7, July1960, pp. 603-612.A~IXAMSON,H. NORMAN;AND RANSLEBEN,GmDo E., JR.: Liquid Sloshing in RigidCylindrical Tanks Undergoing PitchingMotion. T R No. 11, Contract DA-23-072ORD-1251, Southwest Research Institute,May 1961.MIHBHEV, G. N.; AND DOROZHIUN,N. YA.: AnExperimental Investigation of Free Oscillationsof a Liquid in Containers (in Russian). Isv.ALad. Nauk SSSR, OM. Tekh. Nauk, Mekh.i Maahinostr., No. 4, pp. 48-53, July/Aug.1981.
Trannlatad into English by D. Kana,Southwest Research Institute, June 30, 1963.MCC~BTI, JOHN LOCHE;AND STEPHENS,DAVIDG.: Investigetion of the Natural Frequenciesof Fluids in Spherical an&>Cylindrical Tanks.NASA T N D-252, 1960.LEONARD,H. WAYNE;AND WALTON,WILLIAMC., JR.: An Investigation of the Natural Frequencies and Mode Shapes of Liquids in OblateSpheroidal Tanks. NASA T N D-904, 1961.67ANDREWJ.; AND ARMSTEADALFREDL.:2.46. STOFAN,Analytical and Experimental Investigation ofForces and Frequencies Resulting From LiquidSloshing in a Spherical Tank. NASA T ND-1281, 1962.2.47. Cau, WEN-HWA:Fuel Sloshing in a SphericalTank Filled to an Arbitrary Depth.
AIAA J.,vol. 2, no. 11, Nov. 1964, pp. 1972-1979.2.48. ABRAMSON,H. N.; CHU, W.-H.; AND G A ~ A ,L. R.: Liquid Sloshing in Spherical Tanks.T R No. 2, Contract NASS-1555, SouthwestResearch Institute, Mar. 1962.H. NORMAN;CHU, WEN-HWA;AND2.49. ABRAMSON,GARZA,LUIS R.: Liquid Sloshing in SphericalTanks. AIAA J., vol. 1, no. 2, Feb. 1963, pp.384-389.IRVING E.; AND STOFAN,ANDREWJ.:2.50. SUMNER,An Experimental Investigation of the ViscousDamping of Liquid Sloshing in Spherical Tanks.NASA TN D-1991, 1963.JOHNLOCKE;LEONARD,H.
WAYNE;2.51. MCCARTY,AND WALTON,WILLIAMC., JR.: ExperimentalInvestigation of the Natural Frequencies ofLiquids in Toroidal Tanks. NASA TN -531,1960.E. : Preliminary Experimental2.52. SUMNER,IRVINGInvestigation of Frequencies and Forces Resulting From Liquid Sloshing in ToroidalTanks. NASA T N D-1709, 1963.2.53. LEVIN, E.: Oscillations of a Fluid in a Rectilinear Conical Container.
AIAA J., vol. 1, no.6, June 1963, p. 1447.2.54. LOMEN,D. 0.: Liquid hopellant Sloshing inMobile Tanks of Arbitrary Shape. NASACR-222, 1965.2.55. LOMEN,D. 0.: Digital Analysis of Liquid Propellant Sloshing in Mobile Tanks With RotationalSymmetry. NASA CR-230, 1965.H.
WAYNE:2.56. STEPHENS,DAVIDG.; AND LEONARD,The Coupled Dynamic Response of a TankPartially Filled With a Liquid and UndergoingFree and Forced Planar Oscillations. NASAT N D-1945, 1963.\PRINCIPAL NOTATIONSA=acceleration fielda, b= tank dimensionsd=diarnetere=2.71828F= f orceg=gravitational acceleration- h=liquid depthIF= total moment of inertia of liquidi=&J,( ) =Bessel function of first kind oforder vIn=natural logarithmM=momentMF=total mass of liquidN.(End=integral defined in appendix Ap=normal pressureradial coordinate, also radius ofthe free surfaceR= tank radius, spheret=timeU, v, w=rectangular velocity components..68T ~ I EDYNAMIC BEHAVIOR OF LIQUIDSy,u*,w=cylindricd velocity componentsv= velocitye= angular coordinateW,= total weight of liquid2,y, z=rectangular coordinate axesY.( ) =Bessel function of second kind oforder va=vertex angle of sector tank; semivertex angle of conical tank-u = 2 mr=gamma functionb=displacement of liquid free surface from equilibrium positionwith respect to space coordinate6*=displacement with respect to coordinate fixed to tankV= kinematic viscosityp=liquid mass density@=velocity potential'function4= angular coordinateQ= excitation frequencyw =naturalcircular frequencyoperator+=indicates vector quantityv =vec!torSubscripts:n=normalT= thrustH= horizontalLATERAL SLOSBLNQ IN MOVING CONTAINERS37*IID2k0"i2 '"1+mIYa5P-PEI:dYMI>;I{VN-Y'h'-I!VNII-- +=L=-IN108U"FI5-=cE'YYN-t, .:pE sE-1:,$1h10I 0 IV51+Vjs;*+c.P"a,LAPPENDIXB.--4ylindn'cal Tank With Ring br Cross Seaion, Ercitation About y A x i s and Roll.[Ref.
2.31PI*.,,"A-"Ad.1* $ I .~..YDI:;:.n5--{,. .ufl.-akha)a.[I •11-+& (,+l zab{-;(i'J112m.l.n(, . , F + l )I - .[I.1.h i c ~ . i ) . l ( r ) l++ u p ~ ~ :~ :, ~~~ hm2 mnb-,.,n(,l.,V),W.am-1~lm.1ullm.1lV1)- w ~ l m . , , "-Iullll - . - l. h~ll ~* I m . . ]""-'{I-...:...,,.."Nd'm-..,*{&I;s (;Zt.-I)][.41n!:11 . , I.iwiiOLATERAL SLOSHING IN MOVINQ CONTAINER8The ''deubl.
.l'n..'' i...."*'Indlut..that th. "Y' corr..pond.to I , and the"-" tor". l h l h r l l r lor ..pr.sslon.lor Y. sod%.71THE DYNAMIC BEHAVIOR OF LIQUIDSLATERAL SLOSHING IN MOVING CONTAINERSTHE DYNAMIC BEHAVIOR OF LIQUIDSLATERAL SLOSHING IN MOVING CONTAINERSTHE DYNAMIC BEHAWOR OF LIQUIDS0awl;0ew;ew;+NdN * )dI-"NE_ILATERAL SLOS-GIN MOVING CONTAINERSTHE DYNAMIC BEHAVIOR OF LIQUIDS*4%+N IVCflwCI.cg:wrn2+pdN> 3hU'f'+N$1.UI,,O\fgI."I.'h,'I*Y8,,.C213H. Norman Abramson, Wen-Hwa Chu, and Franklin T. Dodge3.1INTRODUCTIONThe preceding chapter presented the basicbackground of technology associated with lateral sloshing in rocket propellant tanks, asderived and obtained from linearized hydrodynamic theory.
It should be readily apparent, however, that in actuality nonlinear effectswill be.present and will even, a t times, governthe character of the liquid motions (fig. 3.1).Such nonlinear effects might be described interms of three classes: (a) those which ariseprimarily as a consequence of the geometry ofthe container and are apparent even for rathersmall amplitudes of excitation and liquid response, (b) those which arise primarily as aconsequence of large amplitude excitation andresponse, and (c) those which involve essentiallydifferent forms of liquid behavior produced bycoupling or instabilities of various of the lateralsloshing: modes.
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