H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 42
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A closeup of the adapter fixture and%Lateral restraining cables/I-ThorLateral restraining cables.Gimbal t h r u s t padPneumatic bellows13,700 kg.; shakerSupport standFIGURE5.51.-Suspension system for the Thor-Agenalongitudinal tests.THE DYNAhIIC BEmVIOR OF LIQUIDSFIGURE5.52.-Air-bellowsfixture for the Thor-Agenalongitudinal tests.air bellows is shown in figure 5.52. Servo cont,rol of the bellows pressure maintains theaverage elevation of the vehicle essentiallyfixed for any weight condition, including transient conditions such as associated with highpropellant efflus rates. Water will be used tosimulate the oxidizer and fuel in the Thor,while Freon and Apco-467 will be used in theAgena.The experimental results will yield data onthe transmissibility of vibratory forces throughthe various types of structures used in largelaunch vehicles. Test objectives include acquisition of impedance data relative to the inputforces generated in the engine section and forcestransmitted to the payload area, and the effectsof propellant flow on the dynamic response ofthe launch vehicle structures.
Experimentaldata will be compared with current state-ofthe-art analytical procedures to assist in determining appropriate parametric values for usein advanced analytical methods.REFERENCES5.1. A~URPHY,GLENN: Similitude in Engineering.The Ronald Press Co., 1950.P. IY.: Dimensional Analysis.
Yale5.2. BRIDGMAN,Univ. Press, 1931.5.3. DUNCAN,W. J.: Physical Similarity and Dimensional Analysis. Edward Arnold & Co., 1955.6.4. ROUSE,HUXTER:Fluid hlechanics for HydraulicEngineers. Dovcr Publications, Inc., 1961.5.5. I ~ O U S EHUNTER,,ED.: AdvXlced 3lechanics ofFluids. John Wilcy 8: Sons, Inc., 1959.5.6.
VENNARD,JOHNK.: Elementary Fluid hJcchanics. Fourth ed., John Wilcy $ Sons,Inc., 19G1.5.7. LORDRAYLE~GH:Stability of Flow of Fluids andInvestigations in Capillarity. Phil. hlag., vol.34, no. 59, 1892.E.: hlodel Experiments and the5.8. BUC~CINGHAM,Forms of Empirical Equations. Trans. ASXf E,vol. 37, 1915, p. 263.5.9. SANDORFF,P. E.: Principles of Design of Dynamically Similar Models for Large PropellantTanks.
NASA T N D-99, 1960.5.10. ABRAMSON,H. N.; A N D RANSLEBEN,G. E., JR.:Simulation of Fuel Sloshing Characteristics inMissile Tanks by Use of Small Models. ARSJ., vol. 30, July 1960, pp. 603-613.5.11. EPPERSON,T. B. ; BROWN,R. B.; A N D ABRAMSON,H. N.: Dynamic Loads Resulting From FuelMotion in Missile Tanks. Proceedings of theFourth Symposium on Ballistic Missile andSpace Technology, vol.
11, Pergamon Press,1961, pp. 313-327.5.12. DALZELL,J . F.; A N D GARZA,L. R.: An Exploratory Study of Simulation of Liquid Impact in Space Vehicle and Booster Tanks.Tech. Rept. 9, Contract NAS8-1555, Southwest Research Institute, Sept. 1964.5.13.
ABRAMBON,H. N.; A N D NEVILL, G. E., JR.:Some hlodcrn Developments in the Applicationof Scale Rlodels in Dynamic Testing. ASRlEColloquium on Use of Rlodels and Scaling inShock and Vibration, Nov. 1963.D. G.; LEONARD,H. W.; A N D PERRY,5.14. STEPHENS,T . W., JR.: Investigation of the Dampingof Liquids in Right-Circular CylindricalTanks, Including the Effects of a TimeVarknt Liquid Depth. NASA T N D-1367,1962.5.15. SUYNER,I . E.; STOFAN,A. J.; A N D SHRANO,D. J.: Experimental Sloshing Characteristicsand a Mechanical Analogy of Liquid Sloshingin a Scale Model Centaur Liquid OxygenTank. NASA T M X-999, 1964.JOHNLOCKE;LEONARD,H. W.4YNE;5.16.
RZCCARTY,A N D WALTON,WILLIAMC., JR.: ExperimentalInvestigation of the Natural Frequcncies ofLiquids in Toroidal Tanks. NASA T N D-531,1960.5.17. MCCARTY,JOHNLOCKE;A N D STEPHENS,DAVIDG.: Investigation of the Naturd Frequenciesof Fluids in Spherical and Cylindrical Tanks.NASA T N D-252, 1960.H. WAYNE;AND WALTON,WILLIAM5.18. LEONARD,C., JR.: An Investigation of the NaturalSIMULATION AND EXPERIMENTAL TECHNIQUES5.19.5.20.5.2 1.5.22.5.23.5.24.5.25.5.26.5.27.-5.28.5.29.5.30.Frequencies ancl Ilode Shapes of Liquids inOblate Spheroidal Tanks. NASA T N D-904,1961.STOF.\N, ANDREWJ.; A N D ARMSTEAD,ALFREDL.: Analytical and Experimental Investigationof Forces and Frequencies Resulting FromLiquid Sloshing in a Spherical Tank.
NASAT N D-1281, 1962.STOFAN,ANDREWJ.; A N D P.\vLI, ALBERTJ.:Experimental Damping of Liquid Oscillationsin a Spherical Tank by Positive-ExpulsionBags and Diaphragms. NASA T N D-1311,1962.SCMNER,IRVINGE.: Preliminary Experimental Investigation of Frequencies and Forces ResultingFrom Liquid Sloshing in Toroidal Tanks.NASA T N D-1709, 1963.STOFIN, ANDREWJ.; A N D SUMNER,IRVISGE.:Experimental Investigation of the SloshDamping Effectiveness of PositivcExpulsionBags and Diaphragms in Spherical Tanks.NASA T N D-1712, 1963.J.:SUNNER,IRVINGE.; A N D STOFAN,ANDREWA11 Experimental Investigation of the ViscousDamping of Liquid Sloshing in Spherical Tariks.NASA T N D-1991, 1963.SUIIYER,IRVINGE.: Experimental Investigationof Slosh-Suppression Effectivcncss of AnnularRing Baffles in Spherical Tanks. NASA T ND-2519, 1964.ABRAMSON,H.
NORMAN;CHU, WEN-Hw.4; andKANA,DANIELD.: Somc Studies of NonlincarLateral Sloshing in Rigid Containers. J.Applied Mechanics, vol. 33, no. 4, Dec. 1966.HUTTON,R. E.: An Investigation of Resonant,Nonlinear, Nonplanar Free Surface Oscillationsof a Fluid. NASA T N D-1870.
1963.SILVEIR.~,MILTONA.; II.LGLIERI,DOMENICJ.;AND BROOICS,GEORGE W.: Results of anExpcrimental Investigation of Small ViscousDampers. NACA T N 4257, 1958.SILVEIRI, MILTON A.; STEPHENS,DAVID G.;A N D LEONARD,H. W-AYNE:An ExperimentalInvestigation of the Damping of Liquid Oscillations in Cylindrical Tanks With VariousBaffles. NASA T N D-715, 1961.ANDSTEPHENS,DAVIDG.; LEONARD,H. WAYNE;SILVEIRA,MILTONA.: An Experimental Investigation of the Damping oi Liquid Gsci!!ations in an Oblate Spheroidal Tank With andWithout Baffles.
NASA T N D-808, 1961.HOWELL,E.; A N D EHLER,F. G.: ExperimentalInvestigation of the Influence of MechanicalBaffles on thc Fundamental Sloshing Mode ofWater in a Cylindrical Tank. Rept. No.GW-TR-69, Guided Missile Res. Div., TheRamo-Wooldridge Corp., July 6, 1956.1955.31. HARRIS,C. XI.; A N D CREDE,C. E.: Shock andVibration Handbook. Vol. I, IIcGraw-HillBook Co., Inc., 1961, pp. 2-15.5.32. STEPHENS,DAVID G.: Experimental Investigation of Liquid Impact in a Model PropellantTnnk. NASA T N D-2913, 1965.ULRIC S.; CHU, WEN-HWA;K.\N.I,5.33. LINDHOLM,DANIEL D.; A N D ABR-IMSON,H. NO~M.IN:Bending Vibrations of a Circular CylindricalShell With an Internal Liquid Having a FreeSurface. AIAA J., vol.
1, no. 9, Sept. 1963,pp. 2092-2099.5.34. LINDHOLY,ULRIC S.; KANA,DANIELD.; A N DABR.IMSON,H. NORMAN:Breathing Vibrationsof a Circular Cylindrical Shell With an InternalLiquid. J. Aerospace Sci., vol. 29, no. 9, Sept.1962, pp. 105'2-1059.5.35.
STEPHENS,DAVIDG.; A N D LEONARD,H. WAYNE:The Coupled Dynamic Response of a TankPartially Filled With a Liquid and UndergoingFree and Forced Planar Oscillations. NASAT N D-1945, 1963.BRUNOJ.:5.36. COLE,HENRYA., JR.; A N D G.~MBUCCI,lleasured Two-Dimensional Damping Effectiveness of Fuel-Sloshing Baffles Applied toRing Baffles in Cylindrical Tanks. NASAT N D-694, 1961.5.37. MILES, J. W.: Ring Damping of Free SurfaceOscillations in a Circular Tank. J. Appl.Mech., vol. 25, no. 2, June 1958, pp. 27.1-276.5.38. KEULEGAN,GARBISH.; A N D C.IRPENTER,LLOYDH.: Forces on Cylinders and Plates in anOscillating Fluid. National Bureau of Standards Report 4821, Sept.
5, 1956.,S.; A N D HERR, ROBERTW.: An5.39. ~ I I X S O NJOHNInvestigation of the Vibration Characteristicsof Pressurized Thin-Walled Circular CylindersPartly Filled With Liquid. NASA T R R-145,1962.5.40. BROOKS,GEORGEW.: Principles and Practicesfor Simulation of Structural Dynamics ofSpace Vehicles.
Presented a t VPI Conferenceon the Role of Simulation and Space Technology (Blacksburg, Va.), Aug. 17-21, 1964.5.41. HERR, ROBERTW.; AND CARDEN,HUEY D.:Support Systems and Excitation Techniquesfor Dynamic Models of Space Vehicle Structures. Proceedings of the Symposium onAeroelastic and Dynamic Modeling Technology(Dayton, Ohio), Sept.
23-25, 1963.5.42. BBSCZP, GECIEGEW.: Techniques for Simulationand Analysis of Shock and Vibration Environments of Space Flight Systems. ASMEColloquium on Experimental Techniques inShock and Vibration, Nov. 1962.5.43. MIXSON,JOHN S.; C.ITHERINE,JOHN J.; A N DARMAN, ALI: Investigation of the LateralVibration Characteristics of a >&Scale Xodelof Saturn SA-I. NASA T N D-1593, 1963.196THE DYNAMIC BEHAVIOR OF LIQUIDS5.44. Ptlrxsox, JOHNS.; AND CATHERINE,JOHN J.:Experimental Lateral Vibration Characteristicsof a >$-Scale Model of Saturn SA-I With a nEight-Cable Suspension System. NASA T ND-2214, 1964.5.45. ~ I I X S O NJOHN,S.; AND CATHERINE,JOHN J.:Comparisoll of Experimental Vibration Characteristics Obtained From a :&Scale Model andFrom a Full-Scale Saturn SA-1.
NASA T ND-2215, 1964.H. L.;MORG.~N,H. G.;A N D MIXSON,5.46. RUNYAN,J. S.: Role of Dynamic Models in LaunchTchiclc Development. AShiE Colloquium onExperimental Techniques in Shock and Vibration, Nov. 1962.5.47. JASZLICS,IVANJ.; AND MOROSOW,GEORGE:Dynamic Testing of a 20% Scale Model of theTitan 111. Proceedings, AIAA Symposium onStructural Dynamics and Aeroelasticity, Boston,1965, pp. 477-485.S. A.; AND RANEY,J.
P.: Some5.48. LEADBETTER,Analytical and Experimental Studies of theDynamics of Launch Vehicles. Proceedings,AIAA Symposium on Structural Dynamics andAeroelasticity, Boston, 1965, pp. 523-527.PRINCIPAL NOTATIONSA= any areaA2=any area moment of inertiaa =acceleration (linear)all, =power of jth dimension in kth variablea,=acceleration scale ratio (model topro totype)B= torsional stiffnessb=other linear dimensiorls of a fueltankc, co=\~elocityof sound in liquidcPk=function of the a,kD = chnracteristic tank diameterDb=bending stiff iiessD,=general notation for the jth fundament a1 dimensionD, =geometric scale ratio-ratio of characteristic length in the model tothat in the protot,ypeE= Y oung's modulusEb=bulk modulus, gasEL=bulk lnodulus of fluidF=dinlension of force (or g e n e r a lforce)I*'= (subscript) denoting prototype valueof u parameterG= shear modulusg= graritn tiunal accelerationH=qunntity of heath=depth of liquid in tankI=nrea moment of inertia (struct~ire)i , j , k, ])=indicesL= dimension of lengthA4= dimension of mass (or general mass)M= (subscript) denoting model value ofn parameterM=amomentm =number of fundamental dimensionsm'= masslunit length, structureM,=bending moment per unit width,shell elementMzu=twisting moment per unit width,shell elementn= number of variablesN,=normal force per unit width, shellelementN,,=shearing force per unit width, shellelementY =pressurePC="cavity pressure1'Pd=dynamic pressure in liquidPo=pressure a t liquid free surfacePC=vapor pressureQ= charge (electrical)r =radius of meniscus in an equivalentcapillary tubeT = (subscript) den0 tes ratio of modelvalue of a parameter to the prototype value of a parameterT= dimension of timeT,=critical temperature of liquidTI,T2= temperatures, equation (5.75)t , t, =plate thicknesst,=equivalent distributed model tankwall thicknessW=natural plate vibration frequencyw= transverse displacement of neutralplane of shell wallw,,=bending curvature of neutral plane,shell wallw,,= twisting curilnture of neutral plane,shell wallSLMULATION AND EXPERIMENTAL TECHNIQUESa= Poisson's ratio2 =impressed velocity1"x=impressed acceleration (linear)zk=general exponent of variablezo= tank excitation amplitudeui=general notation ith variable8 =any angular dimensionj4= angular velocity$= angular accelerationpz,= in-plane shearing stiffnessp2= transverse shearing st,8nessy,,= transverse shearing strainy,,= in-plane shearing strainAAsL =changes in interfacial areasAAmA E = total change in interfacial energyA P = Po- P,€=straine,=extensional strainI9= temperature@=contactangleA =extensional stiffnessA= characteristic length ratior =dynamic viscosity, liquidr,=dynamic viscosity, gasv=kinematic viscosity197elastic constantg=fluid free surface elevations, ~ e l o c ities, accelerationssi=ith R-term*:=ratio of the value of the ith n termfor the model to the correspondingvalue in prototype (= 1 for simulation)p=mass density of liquidkas=mass density of gasp,=mass density scale ratio (model toprototype)psolid, ps=mass density of tank structureu= surface tensionurn =solid-gas interfacial energylunit areausL=solid-liquid interfacial energylunitareau,=yield strength of materialF=stresss, .r,=characteristic timessr= time scale ratio (model to prototype)4 =Kinematic Surface Tension-a/ pw, w,=characteristic frequenciesV =any volumeV2=any volume moment of inertiaf,4,'v' =Poisson-typeChapter 6Analytical Representation of Lateral Sloshing by Equivalent Mechanical ModelsFranklin T.Dodge6.1INTRODUCTIONI n the previous chapters, the liquid sloshingcaused by lateral motions of the tank hasbeen described analytically and experimentally,and the forces and torques exerted on the tankby the sloshing pressures have also been given.In addition to insuring that the fuel tankstructure is strong enough to withstand theseloads, the designer must also include thesloshing forces and torques in the stabilityanalysis of the in-flight rocket.
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