H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 58
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The booster is about6.55 meters in diameter and 25 meters in length.The total length of the vehicle is about 51.8meters. The eight engines are arranged intwo square patterns, of which'the inner ones arerigidly attached, while the outer engines aremounted on gimbals which permit them to beturned through angles of about 10' to providecontrol of the vehicle during first-stage poweredflight.
Nine separate tanks feed the eightengines. Clustered in a circle about a centertank of 267 centimeters in diameter are eightsmall tanks of 178 centimeters each in diameter.The center tank (noted as tank No. 1 j and iourouter containers (noted as tank No. 2) containliquid oxygen, while the remaining four outercontainers (noted as No. 3) contain fuel (RP-1).The S-IV, which is the second stage, is poweredby six 15,000-pound-thrust liquid hydrogen/liquid oxygen engines. This stage has adiameter of about 5.5 meters and is about 12.2meters long.
A look a t the frequency spectrumversus flight time (fig. 7.53) reveals that the229-848 0-47-1sFIGURE7.53.-Frequencyspectrum versus fight time.control feedback stability problem is a quiteinvolved one. The purpose of such an exerciseis to decide early enough in the planning stageas to how the vehicle should be efficientlydesigned.The sloshing frequencies range from about0.5 cps to 1.5 cps. At 140-seconds flight time,the four inboard engines are shut down, whichdecreases the longitudinal accelerations andis exhibited in the graph by the abrupt dropof the propellant frequencies.
The smallernatural frequencies in the 267-centimeterdiameter containers compared to the 178centimeter container are the result of itslarger tank diameter. The same effect, andthe smaller liquid height, contributes to thesmall sloshing frequency in the S-IV LOXtank. The control frequency is in the neighborhood of 0.2-0.3 cps, while the first lateralv a n a n a frnmU -U l l-1:-Ulll~-- - - %bout2 to 2.4cps during booster flight. The location of themass center and the center of instantaneousrotation of the vehicle, as well as the locationof the sloshing masses, for booster flight isshown in figure 7.54.
Since the sloshing propellant masses are of great importance uponthe dynamics of the vehicle, their magnitudeis presented in figure 7.55 as the ratio of themass of the sloshing propellant in a container~ - ~ - ~ ~ ~ ~ - ~LL GyUV1)"J-uum--264ITHE DYNAMIC BEHAVIOR OF LIQUIDS28-4-Tank 4LGimbal station1020 - 4060800100120 0 140Time (set)FIGURE7.54.-Sloshmaee location versus booster flighttime.to the total vehicle mass, versus booster flighttime.
Immediately, it can be seen that theLOX tank of the S-IV stage which exhibits(because of the mass change of the vehicle.with booster flight time) large values millhave a very pronounced effect upon the stability of the vehicle.In t,he following, we shall apply the results ofthis chapter. The frequency ratios, v,=w,/w,, are always larger than unity, thus indicating a favorable situation. The gainvalue is ao= 1.5 and changes, a t a flight timeof 110 seconds, to the magnitude a o = l .
Ifonly one mass were sloshing, say that of theS-IV LOX container, the conclusions Ire coulddraw a t a flight time of 70 seconds, wherep=0.03, {,=0.7, v,=3, and [,=-1.2,fromfigure 7.13 would be the following: for thisflight time no damping would be needed inthis container since it is located outside thedanger zone, i.e., outside the zone betweencenter of mass and center of instantaneousrotation; at a flight time of 140 seconds, where0020FIGURE7.55.-Slosh406080Time (sec)100120140mass ratio for SA-I booster flight.r=0.09, v,=4, and .$,=-0.9, a little dampingin the amount of about -y,=0.005 would beneeded to maintain stability.Combining all sloshing masses of the boosterin one mass and considering the effect of theS-IV stage sloshing masses as negligible yieldsthe follo~ving:first of all, at a flight time of 70seconds, the combined slosh mass ratio p \vouldbe of the magnitude of about p=0.09; the frequency ratio, v,, is about V, e 3, and the locationt,= 0.
For this particular case, a damping ofabout 1 percent should be provided. At aflight time of 140 seconds, p=0.18, v,=4, andES=0.8, no damping is needed since the sloshingmass is locsted outside of the danger zone.From this simple consideration, one can conclude that a stationary baffle arrangement inthe upper part of the booster tanks is sufficientto maintain flight stability during the boostphase.To obtain, however, a more realistic requirement, we have to look a t the sloshing in severalcontainers.
Considering the S-IV LOX slosh---& ^ _ _ _ _ _ _ I_- --e- -- .-- _...- - - -.-..11!VEHICLE S T A B ~ I T YAND CONTROLing mass together with the combined boostersloshing mass, one can use the results presentedin figures 7.25 and 7.26. At 70-seconds flighttime, the sloshing masses are given by p,=0.09and p2=0.03, while vl=v2= 3. The distance 1of the combined booster sloshing mass, p,, tothe fixed sloshing mass, p2, of the second stageis 1--16.25meters; that is, El=-1.3.Thevalue 6, =0, and we conclude that a damping of7, = 0.013 is needed to have a stable flight condition. At a flight time of 140 seconds,p1=0.18, p2=0.09, v1 =v2=4, Er=0.8, and-1.8, and a damping of about 1 percent isneeded.
I t furthermore can be seen that damping is needed in the lower part of the boostercontainers. Since the fundamental bendingfrequency is close to the sloshing frequencies,i265the effects of vehicle elast,icity (ch. 9) mustalso be considered. Furthermore, some mindresponse studies will show the adequacy of thebaffles. (See sec. 7.4.)The actual feedback analysis of the flexiblevehicle with all its slosh masses includedindicated that with smooth container wallsa slight instability occurred in the 178-centimeter containers between 40- and 60-secondsflight time, and in the 267-centimeterdiametercontainer an instability occurs between 60- and90-seconds flight time (ref.
7.23). By employing proper baffles, these instabilities couldbe removed. The same procedure can nowbe performed for the second-stage flight, buthas been omitted here for reasons of limitedspace.REFERENCES37.1. BAUER,H. F.; AND RHEINFURTH,M. H.: Flutterand Stability Analysis. Army Ballistic 31issileAgency, Redstone Arsenal, Ala., DA-TM4-60, 1960.7.2. RHEINFGRTH,M. H.: Control Feedback StabilityAnalysis.
Army Ballistic Missile Agency, Redstone Arsenal, Ala., DA-TR-2-60, 1960.7.3. THORP, F. A.; AND H ~ T C H I N ~ OR.N ,C.: TheDynamics of Rocket-Powered Vehicles. Massachusetts Institute of Technology, Rept.R-206, Feb. 1959.7.4. LUKENB,D. R.; SCHMITT, A. F.; AND BROUCEK,G. T.: Approximate Transfer Function forFlexible Booster and Auto - Pilot Analysis.WADD-TR-61-93, 1961.7.5.
BARTON, M.: GeneraIized Miesile DynamicsAnalysis. Ramo-Wooldridge Corp., 1958.7.6. HEIST,E. K.: E q u a t i o ~ lof~ Motion of MissileWith Sloshing. Guided AMissile ResearchDivision, Ramo-Wooldridge Corp. Rept. NO.GM-TM-146, 1956.7.7. BAUER,H. F.: Fluid Oscillations in the Containerof a Space Vehicle and Their Influence UponStability. NASA-TR R-187, 1964.7.8.
BAUEX,H. F.; E y ~ d r?fc Liqlr;d PropellantVehicles. Proceedings of the ONR/AIA Symposium on Structures and Dynamics of HighSpeed Flight, ONR/AIA Symposium, 1961.7.9. BAUER,H. F.: The Effects of Interaction ofStructure, Control and Propellant SloshingUpon the 6tabilitj of Large Space Vehicles.Marshall Space Flight Center, NASA, MTPAERO-61-83, 1961.7.10. Z U R M ~ ~R.:L ,Praktische Mathematik fiir Ingenieure und Physiker. Springer-Verlag, Berlin,1953.---7.11. BRODETSKY,S.; A N D SMEAL,G.: On Graeffe'sMethod for Complex Roots of Algebraic Equations. Proc.
Cambridge Phil. Soc., vol. 22,1924, pp. 83-87.7.12. ROUTH,E. J.: A Treatise on the Stability of nGiven Llotion. Macmillan, London, 1877, pp.74-81.7.13. HURWITZ,A.: Uber die Bedingungen unterwelchen eine Gleichung nur Wurseln mit negative reellen Theilen besitzt. MathematischeAnnalen, vol. 46, 1895, pp. 273-284.7.14. NYQUIST,H. : Regeneration Theory. Bell SystemTechnical Journal, vol. 11, 1932, 126-147.7.15. RHEINFURTH,M. H.: The Iduence of ControlSensors on the Stability of Space Vehicles.Marshall Space Flight Center, NASA, MTP61-65, 1961.7.16. BAUER,H. F.: The Effects of Propellant Sloshingon the Stability of an Accelerometer ControlledRigid Space Vehicle. NASA TN D-1831,1963.7.17.
BAUER,H. F.: Stability Boundaries of LiquidPropelled Space Vehicles With Sloshing.AIAA J., vol. 1, no. 7, 1963.7.18. HEINRICH,K.; AND ~ A U F M A N , F. ii.: SioshingStability Criteria for Vehicles With One FreeFluid. Ramo-Wooldridge Memo GM-45.3-45,1956.7.19. FULLER,A. T.: Stability Criteria for LinearSystems and Reliability Criteria for RC Networks.
Proc. Cambridge Phil. Soc., vol. 53,1957, pp. 87&896.7.20. GEISSLER,E. D.: Problems in Attitude Stabilization of Large Guided Missiles. Aero-SpaceEngineering, Oct. 1960, pp. 24-29.266THE DYNAMIC BEmVIOR OX' LIQUIDS7.21. BAUER,H. F.: Parametric Study of the Influenceof Propellant Sloshing on the Stability of SpaceCrafts.
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