H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 90
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10.26(b)), and by means of thebellows in the bellows device (fig. 10.26(c)). Theaccumulator has bee11 used on the fuel systemand the standripe o11 the oxidizer system inTitan 11. The result of incorporating these devicesistocausethe propellant feed system frequency ,w,, to occur a t a considerably lower frequency.Then, even if it is still above the structuralfrequency, w,, at liftoff, as the two frequenciescoincide early in flight, the structural gain isso low because of the nearly full fuel load,that the net gain never becomes greater thanunity until after the net phase becomes negative. The use of such devices might be avoidedFflcharacteristicsLOX dischargeIflu# ratediagram for complex vehiclerepresentation (ref.
10.23).381T H E DYNAMIC BEHAVIOR OF LIQUIDSre: Center ofgravityre: PipeRigid linkapproximately 5 cps. The analyses indicatethat the engine does not play a significant rolein this case, the unstable coupling occurringthrough the pressure regulation system for boththe fuel and oxidizer tank ullage pressure.Figure 10.27 sho~vsa schematic of the pressure regulation system for either the fuel oroxidizer tank, the only difference being that amixture of helium arid oxygen is used for theoxidizer tank, while only helium is used for thefuel tank ullage. This system nlaintains aconstant gage pressure in the tank ullu,ve asthe tank is drained. A pressure-sensing linetransmits the ullage pressure to 2 1 pneunlaticregulator, which in turn itdjusLz tlie flu\\ rateof helium gas througl~a supply dt1c.t into thetank ullage.At liftoff the transient engine thrrlst andlaunch forces escite the longitltdinal modes ofthe vehicle, and a corresporidixlg oscillationoccurs in the tank ullage volume and pressure.This pressure oscillatio~iresrllts both from theSensing linesComparator valvex,FIGURE10.26.-Schematicsof mechanical devices employed as pogo fixes (ref.
10.22).if the proposed system can be analyzed sufficiently accurately for ~ o t e n t i a pogolbehaviorduring the design stage of xiewly proposedvehicles.10.9 PRESSURE REGULATOR-STRUCTURE COUPLINGAnalyses of pogo-type phenomena in theAtlas vehicle ht~vebeen reported by Rose andHarris (refs. 10.26 and 10.27j. In this vehicle,the oscillations occur for a duration of about20 to 30 seconds immediately after liftoff, andat the longitltdirlal structural frequency ofFIGURE10.27.-S~hematic of Atlas pressure regulationSystem (ref. 10.26).LONGITUDINAL OSCILLATIONSvariation in tank volume and a variation inthe quantity of ullage gas present, resultingfrom tile regulator responding to the ullagepressure oscillation.
This oscillating pressureacts as an effective axial force on the vehiclestructure. Analyses of suit,able models of thesystem have indicated that the closed-loopsystem can be unstable for a period of timewhen the ullage volume is small, immediatelyOF FLIGHT VEHICLES383after liftoff. Subsequently, the gains andphases of the various subsystems change sothat the net gain and phase no longer allow a nunstable system. I t must be emphasized thatthis type of coupling is entirely different fromthe previously described engine-structure coupling, but the net result is the same althoughthe oscillations occur early in flight ratherthan near burnout.REFERENCES10.1.
NOLTING,R. K.: Simulation of Orbital Mooringof Gemini and Agena Vehicles by Means ofDynamically Scaled Models. Proceedings ofSymposium on Aeroelastic and DynamicModeling Technology, U.S. Air Force, RTDTDR-63-4197, 1964.10.2. EIDE, DONALDG.: Preliminary Analysis ofVariation of Pitch Motion of a Vehicle in aSpace Environment Due to Fuel Sloshing in aRectangular Tank. NASA T N D-2336, 1964.10.3. EPPERSON,THOMASB.; A N D BROWN,ROBINSON:Dynamic Loads Due to Fuel Motion in FuelTanks of Missiles.
Final Rept., ContractNo. DA-23-072-ORD-1062,Southwest Rrsearch Institute, June 1957.T. B.; BROWN,R. B.; A N D ABR.\u10.4. EPPERSON,SON, H. N.: Dynamic Loads Resulting FromFuel Jlotion in Missile Tanks. Proceedingsof the Fourth Symposium on Ballistic Missileand Space Technology, vol. 11, pp. 313-327,Pergamon Press, 1961.10.5. STEPHENS,D. G.: Experimental Investigationsof Liquid Impact in a Model PropellantTank. NASA T N D-2913, 1965.JOHNF.; A N D GARZA,LUIS R.: An10.6.
DALZELL,Exploratory Study of Simulation of LiquidImpact in Space Vehicle and Booster Tanks.Tech. Rept. No. 9, Contract No. NAS8-1555,Southwest Research Institute, Sept. 1964.T. J.; ToMassoN~,J. E.;ANDSEIFERTH,10.7. COKONIS,R. W.: Dome Impact Analysis-An Approximate Solution. Martin Co. Rept., TN LV211, &lay 1963.10.8. PINSON,L. D.: Propellant-Dome Impact Analysis. NASA Internal ,l.Iemorandum, Nov.i863.10.9. Handbook of Astronautical Engineering.
Firstedition, McGraw-Hill Book Co., Inc., 1961.10.10. BRADY,W. F.; POPE, M. D.; A N D PODE,L.:Ablestar Experimental Studies. Rept. SGC32R-19, Space-General Corp., Aug. 1962.10.11. ABRAMSON,H. N.; CHU, W. H.; GARZA,L. R.;A N D RANSLEBEN,G. E., JR.: Some Studiesof Liquid Rotation and Vortexing in RocketPropellant Tanks. NASA T N D-1212, Jan.1962.10.12.
ROUSE, H.: On the Role of Eddies in FluidMotion. American Scientist, vol. 51, no. 3,Sept. 1963, pp. 285-314.D.: Report on thc I.U.T..i.lI.10.13. K~~CHEMINN,Symposium on Conccntra:etl l'ortis IIotiol~sin Fluids. J. Fluid lIctch., vol. 21, pt. I ,Jan. 1965, pp. 1-20.L.: Essentialb of Fluid Dynamics.10.14. PRANDTL,Hafner, New York, 1949.10.15. DERGARABEDIAN,P.: The Behavior of Vort,exMotion in an Emptying Container. Proc. ofthe 13th Fleet Transfer :nld Fluid 1IechanicsInstitute, Stanford University, June 1960,pp.
47-61.10.16. WESKE,J. R.: On the Orig~nand Jlect1:tnisrn ofYortex 1Iotiun at :iw 11:lct of 1t::ttkcs PlicedNear a Flat Surf:~ct,. Cniv. of LIibrylandTech. Note BY-1.52, AFOSIt TN-58 863,1958.10.17. BINNIE,A. &I.; A N D H-IRRIS,D. P.: The Application of Boundary-Layer Theory to SwirlingLiquid Flow Throuqh e Nozzle. Quart. J .AIech. Appl. 1I:ith., vol.
3, pt. 1, 1950, pp.89-106.10.18. BINNIE,.A. )I., IIOOKINGS,G. A.; IXD KAUEL,11. Y. 11.: The Flow of Swirling WaterThrough a Convergent-Divergent Nozzle. J.Fluid lIech., vol. 3, 1957, pp. 261-274.10.19. KAMEL,M. Y. >I.: The Effect of Swirl on theFlow of Liquids Through Bottom Outlets.ASME Pnper 64-WA/FE-37, ASME WinterAnnual Meeting, New York, Dec.
1964.10.20. HUTTON,R. E.: Fluid Particle Motion DuringRotary Sloshing. J. Appl. Mech., Trans.ASME, Series E, vol. 31, >far. 1964, pp.123-130.10.21. GLUCK,D. F.; A N D GILLE, J. P.: Fluid Mechanics of Zero-G Propellant Transfer inSpacecraft Propulsion Systems. J. Eng. forIndustry, Trans. ASME, Series B, vol. 87,Feb. 1965, pp. 1-8.10.22. RUBIN,S.: Instability Model of hlissile Longitudinal Oscillation Due to Propulsion Feedback.
Rept. No. TOR-269(4126)-28, Contract No. AF04(695)-269, Aerospace Corp.,Sept. 21, 1964.384THJS DYNAMIC BEHAVIOR OF LIQUIDSDAVIS, W. F.; LYNCH,T. F.; AND MURRAY,T. R.: Thor 20-Cycle Longitudinal OscillationStudy. The Shock and Vibration Bulletin,no. 34, pt. 2, Dec.
1964, pp. 177-196.MCKENNA,K. J . ; WALKER,J. H.; A N D WINJE,R. A.: Engine-Airframe Coupling in LiquidRocket Systems. AIAA J. Spacecraft Rockets, vol. 2, no. 2, .Mar.-Apr. 1965, pp. 254-256.N. A.: Analytical Model for MissileRADOVICH,Axial Oscillation Induced by Engine-StructureCoupling. Proc. of AIAA Unmanned Spacecraft Meeting, Mar.
1965.ROSE, R. G.; A N D HARRIS, R.: DynamicAnalysis of a Coupled Structural/PneumaticSystem Longitudinal Oscillation for AtlasVehicles. Paper No. 64-483, Annual AIAAMeeting, Washington, D.C., June 1964.ROSE, R. G.: Dynamics of the Atlas-5cpsLongitudinal Oscillation Following Launch nsRelated to the Tank Pressure RegulstionSystem.
Vol. 1, Longitudinal Model Development, General Dynsmics/Astronautics, Dcc.31, 1963.PRINCIPAL NOTATIONSA,= projected lateral areaa=radius of line vortexb=distance between initial positionof a particle and the uppertank bulkheadCDa_,=drag coefficient a t zero angle ofattackD=drag forceD,=drag a t t=O, v=vo, u=Ok ,,d=base diameterdo= orifice diameterd ,=tank diameterF,=acceleration force of apparatus(ref.
10.3)F,= normal aerodynamic forceF, =axial aerodynamic forceFro=initial normal aerodynamic forceI.',, =initial axial aerodynamic forceg =local gravitational accelerationH= total headh=tank depthI=mass moment of inertia of vehicleK=discharge coefficient(kl-k2)=inertia coefficientskl, k2, k3=coefficientsL=lift forceI= overall vehicle lengthibl= aerodynamic momentM,=mass of apparatus (ref. 10.3)m =vehicle massm l=mass of a particle of fluidrn,=mass of a particle of fluid inexperimental apparatusn,= acceleration index (ref.
10.2)%=amplitude of half-sine relativeacceleration pulsenl =relative acceleration a t maximumindicated impact force-n =averagerelative accelerationn(t) =acceleration pulseP,,, =maximum impact pressurep =pressurepa= ambien t tank pressurep, =fluid vapor pressureQ=drnining flow rateq =dynamic pressure .R,, R,=forces acting on an isolatedparticler =radiusro= Earth radiusS=stroke of apparatus (ref. 10.3)&=base area of boosterT=engine thrustT,=duration of experimentt =timeta=minimum time for a particle toimpact dome, experimental apparatus (ref.
10.3)t,= time of initial contacttl,,=minimum time required for agiven fluid particle to reachthe tank bulkheadu= normal velocityV=fluid velocityVs=swirling velocity of fluidv= tangential velocityuo= initial velocityW=mgX=initial swirl3, =moving coordinates\LONGITUDINAL OSCILLATIONS OF FLIGHT VEHICLESz,, z,, yl, y,=coordinates of a particlexr, y t=coordinate~?=distance through which a particletravels in time t(y,)o=relative particle acceleration a tt=OZ=distance of center of pressureforward of vehicle center ofgravitya =vehicle angle of attackcro=vehicle angle of attack at t = O/3=angle of the quiescent fluid freesurface with the normal to thetank axisI' =circulationT = peak wave height385~ C o c =adjusted cross-flow drag coefficient.O=inclination of apparatusCc= dynamic viscosityv=flight pat'h anglevf =kinematic viscosity of fluidp=mass density of air (local)pf=mass density of fluidu=surface tensionT =nondimensional timeQ =product of tangential velocityand tank radiusw=vorticityo,=propellant feed line frequencyw,=roclret structure resonant frequenciesChapter 11Liquid Propellant Behavior at Low and Zero gWilliam C .
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