H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 81
Текст из файла (страница 81)
L.; AND SKALAK,R.: Free Vibrationsof Fluid Filled Cylindrical Shells. J. of Engr.Mech., ASCE, vol. 88, no. EM3, part I, June1962.9.21. BARON,M. L.; AND BLEICH,H. H.: The DynamicAnalysis of Empty and Partially Full Cylindrical Tanks, Part 11-Analysis of Uplift andStructural Damage. DASA No. 1123B, Defense Atomic Support Agency, Sept. 1959.9.22. BLEICH,H.
H.: Longitudinal Forced Vibrationsof Cylindrical Fuel Tanks. Jet Propulsion,vol. 26, no. 2, Feb. 1956, pp. 109-111.9.23. PALMER,J. H.; AND ASHER,G. W.: Calculationof Axisymmetric Longitudinal Modes forFluid-Elastic Tank-Ullage Gas System andComparison With Model Test Results. Proc.3509.24.9.25.9.26.9.27.9.28.9.29.9.30.9.31.9.32.9.33.THE DYNAMIC BEHAVIOR OF LIQUIDSAIAA Symposium on Structural Dynamics andAeroelasticity, Boston, 1965, pp. 189-193.CHU, W. H.: Breathing Vibrations of a PartiallyFilled Cylindrical Tank-LinearTheory.
J.Appl. Mech., vol. 30, no. 4, Dec. 1963, pp. 532536.CHU, W. H.; A N D GONZALES,R.: Supplement toBreathing Vibrations of a Partially FilledCylindrical Tank-LinearTheory. J. Appl.Mech., vol. 34, no. 4, Dec. 1964, pp. 722-723.Yu, Y. Y.: Frec Vibration of Thin Cylindric~llShells. J. Appl. Mech., vol. 22; Trans. AShlE,vol. 77, 1955, pp. 547-552.FONTENOT,L.
L.; A N D LIANIS, G.: The FreeVibrations of Thin Elastic Pressurized ' Cylindrical Shells Filled With a Perfcct and Incompressible Liquid Having a Frec Surface.International Symposium on Spncr Technology and Science (Tokyo, Japan), Sept.1963.RABINOVICH,B. I.: The Equations of the Transverse Vibrations of Liquid-Filled Shells (Eng.trans.). NASA T T F-216, 1964.NATUSHKIN,V. F.; BND RAKHIMOV,I.
S.: Oscillations of a Cylindrical Shell Partially FilledWith a Fluid (in Russian). AviatsionnniaTckhnika, vol. 17, no. 3, I!lril, 1q). 7 5 - 7 s .(IAA A64-28276)E. A.; AND PAVLOV,B. S.: OscilSAMOILOV,lations of a Hemispherical Shcll Filled With aFluid (in Russian). Aviatsionnaia Tekhnika,vol. 7, no. 3, 1964, pp. 79-86. (I.4A A64-28277)Longitudinal Sloshing ofHWANO,CHINTSUN:Liquid in a Flexible Hemispherical Tank.Paper No. 65-APM-14, Applied hIechnnicsiFluids Engineering Conference, ASLIE (iiTashington, D.C.), June 7-9, 1965.SHMAKOV,V. P.: The Equations of thc AxiallySymmetric Vibrations of a Liquid-FilledCylindrical Shell.
NASA T T F-219, July 1964.M.:BEAL,T. R.; COALE,C. W.; A N D NAOANO,Influence of Shell Inertia and Bending Stiffnesson the Axisymmetric Modes of a PartiallyFilled Cylindrical Tank. AIAA Paper No.65-412, AIAA Annual Meeting, July 1966.U. 9.; AND ABRAMSON,9.34.
KANA,D. D.; LINDHOLY,H. N.: An Experimental Study of LiquidInstability in a Vibrating Elastic Tank.J. Spacecraft Rockets, vol. 3, no. 8, Bug. 1966,pp. 1183-1188.9.35. KANA,D. D.: Longitudinal Forced Vibration ofPartially Filled Tanks. Tech. Rept. No. 6,Contract No. NASw-146, Southwest ResearchInstitute, Feb. 1963.9.36.
CHU, W. H.; A N D KANA,D. D.: A Theory forNonlinear Transverse Vibrations of a PartiallyFilled Elastic Tank. Final Rept. Part I,Project 02-1748, Southwest Research Institute,Mar. 1966.9.37. BHUTA, P. G.; A N D KOVAL,L. R.: CoupledOscillations of a Liquid With a Free Surface ina Tank Having n Flexible Bottom. J. Appl.Math. and Phys., vol. 15, no.
5, 1964.9.38. BHUTI, P. G.; A N D KOVAL,L. R.: HydroelasticSolution of the Sloshing of a Liquid in a Cylindrical Tank. J. Acoust. Soc. Am., vol. 36,no. 11, Nov. 1964, pp. 2071-2079.9.39. TONG,P.; A N D FUNO,Y. C.: The Effect of WallElasticity and Surface Tension on the ForcedOscillations of a Liquid in a Cylindrical Container (Part I: Analysis). SM-6+40, Grad.dero. Labs., Cal. Inst.
of Tech., Oct. 1964.9.40. BAUER,H. F.: Effects of Interaction of Structure,Control, and Propellant Sloshing Upon theStability of Large Space Vehicles. MTPAERO-61-89, Marshall Space Flight Center,NASA, 1961.9.41. GEISSLER,E. D.: Problems of Attitude Stabilization of Large Guided Miss'les.
AerospaceEng., vol. 19, Oct. 1960, pp. 24-29, 68-71.9.42. HUR\YITZ, A.: Uber die Bedingungen, unterwelchen eine Gleichung nur Wurzeln mitMathenegativen reellen Theilen besitzt.matische Annalen, vol. 46, 1895, pp. 273-284.9.43. FULLER, A. T.: Stability Criteria for LinearSystems and Reliability Criteria for R CNetworks. Proc. Cambridge Phil.
Soc., vol.53, 1957, pp. 878-896.9.44. BAUER,H. F.: Stability Boundaries of LiquidPropellant Elastic Vehicles With Sloshing. J.Spacecraft Rockets, vol. 3, no. 2, Feb. 1966,pp. 240-246.PRINCIPAL NOTATIONSA, B, C=displacement constant coefficientsa= tank radiusao,a,=gain value of attitude control systemb =liquid depthc,=0.3 =equivaJent wavelengthfactor constantc,,=velocity of sound in fluid atreference static pressureD= 12(1-v2)Eha =stiffness coefficientd= tank diameterE=modulus of elasticityj= frequency (cps)INTERACTION BETWEEN LIQUID PRO~~LLANTSAND THE ELASTIC STRUCTUREf,,=natural frequency of m, nthbreathing mode of a circular cylindrical shell(CPS)f ( 2 ) =mode shape function forbending tankg= acceleration of gravity.ge= structural dampingh=shell wall thicknesshB=bottom thicknessJ,,I,=ordinary and modifled nthorder Bessel functionsk,=~=compressibility factorCoof fluid in a breathingtankki.
j=generalized stiffness coefficientk,=radius of gyration of spacevehicle about the masscenterI= tank lengthMB/mo=generalized mass ratiom=axial wave number forbreathing cylindrical tankm,=empty tank massm,=fluid apparent massm,, j =generalized mass coefficientn=circumferential wave number for breathing tank-Z -=-axialinternal static"-2Ehpressure 'parameter-n s =POaz =tangentialinternalstatic pressure parameterp,,=internal static pressurepl, pr=phasslag coefficientsQ ( T ) =unsteadypressure radialmode functionq=unsteodp internal pressureq, =generalized coordinateT , 8, Z=cylindrical coordinates (fig.9.1)T= kinetic energyt= timeu8(t), V,(t), W, (t) =axial, tangential, and radialdisplacement of middlesurface of breathing shellas functions of time351u, v, w=axial, tangential, and radialdisplacement of middlesurface of a circular cylindrical shellV=potential energyz, y, z=cylindrical shell coordinates(fig.
9.8)Y, Y'= displacement and slope a tthe first bending mode.(Subscripts 1, 2, 3 meanat locations of sloshing masses, subscript Gmeans at location ofgyroscope, i n d subscriptE means at sw-ivel pointof the engines.)z= axial coordinate from liquidcent,er of gravity (fig.9.1)a=h/a=shell thickness factorfor breathing tanky,=damping factor of propellantdimensional frequencyparameter for a breathing tankl=liquid surface displacementabove undisturbed level{,=control dampingh = y = s h e l l wavelength factor for breathing tankp.=mr/mo=ratiosof modalmass of liquid over totalmass of space vehiclev =Poisson's ratiou,=~~/w~=frequencyratio ofundamped p r o p e l l a n tfrequency to undampedcont.rol frequency6,&=distance between rear tankand upper tanks, respectively&=zE/k,=ratio of distance ofswivel point of enginesfrom center of gravityto radius of gyration352THE DYNAMIC BEHAVIOR OF LIQUIDS(,=x,/k,=ratio of the coordinate at the location ofthe modal mass of thepropellant to the radiusof gyration of the spacevehiclepL =liquid mass densitypo=fluid mass density at reference static pressurep,=shell mass densit.yr =bottom membrane tension@=liquid velocity potential9(r, 9 ) =liquid surface mode shapefunctionQt,,art ,=couplingcoefficients forsloshing in bending tankw =circular frequencyw, =empty tank resonant bending frequencyw,=resonant frequency for ithcoordiiate, i = l , 2, 3 .
. .w,+,=uncoupled liquid sloshingfrequenciesChapter 10Special TopicsPart I. Liquid Impact on Tank BulkheadsJohn F. Dabell10.1 INTRODUCTORYREMARKSONDOMEIMPACTThere are a number of possible operationalsituations where a sudden thrust reversal onthe rocket or spacecraft may produce a suddenalteration in the apparent relative gravitationalfield of the fluid in the various propellant tanks,and thus cause or tend to cause the fluid toreorient itself at the opposite end of the tankwith such rapidity as to produce impact-typeloading on the far bulkhead. These situationsmay be divided, for present purposes, into twogeneral types :(1) Maneuvering or docking of spacecraft inan essentially low gravity environment.(2) Thrust termination in the atmosphere.The impulsive accelerations imposed upon aspace vehicle during docking are very much afunction of the operational procedures andlatching and shock-absorbing systems involvedfor the particular vehicle.
Typical studies ofthe first-generation docking program (ref. 10.1)neglect the fluid dynamics of contained fluids.As of this writing, quantitat,ive estimates of theof r;;oGozs of cczt&erl, EqlirlS on t,hemotions of spacecraft seem to be confined tothe admittedly preliminary analysis of reference10.2. Methods for estimating the possibility offluid impact in this case, and of the pressuresinvolved, seem not readily available, though theentire field of low-gravity fluid mechanics isunder intensive study at the present time.The extent of practical estimation proceduresis summarized in chapter 11, section 11.5, towhich i%e reader is referred.The second general situation mentionedabove, thrust termination or alteration in theatmosphere, has been of some concern up tothe present and it will be the purpose in theremainder of this section to review what workhas been accomplished in this area.Thrust termination in the atmosphere beforevirtually all of the propellants have been usedcan be intentional (mission abort, or perhapsassociated with short-range ballistic missilefights) or unintentional (engine failure orpremature ignition during staging).
In eitherevent, the possible rupture of a tank bulkheadcould result in a structural breakup or, in thecase of hypergolic fuels, could result in a fireball.I n the cases of interest to manned space flight,either the intentional or unintentional thrusttermination case will probably spoil the missionand indirectly destroy the booster rocket (atleast until such time as practical boosterrecovery techniques are perfected).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
