H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 48
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I n the flexible-wall analogy, thepitching moment caused by the sloshi~lg isbroken up into two components: one due to thepressure distribution on the walls and one dueto the pressure distribution on the tank bottom.A pure couple is applied a t the tank bottom tosimulate the pressure distribution on thebottom. Thus, the pitching moment whichthe mechanical elements actually should simulate is reduced to its correct value, and thespring masses or pendulums and rigid mass arelowered from their rigid wall locations. Sucha mechanical model is shown in figure 6.28;only one sloshing mass is shown, but the samemethod could be used with a complete set ofsloshing masses.
Furthermore, the damping isnot included for reasons of simplicity. Thebottom couple, AIB, acts as shown.Carrying out the details of the analysis givesthe model parameters shown in table 6.6.The bottom couple, ME, is specified in terms ofthe displacement of ml relative to the tankwalls, y,, but since this term usually entersonly in a stability analysis, where Laplacetransforms are cus tomnrily employed anyway,i t should present no unusual difficulties. Theparameters are also shown graphically in figures6.29 and 6.30.By examining the nctual sloshing force itndmoment equations (or experimental results) andbreaking the moment into two parts as discussedabove, flexible-wall models for other kinds oftanks can be derived. However, such modelsshould be viewed only as first approximationsto the actual sloshing in flexible tanks, and notas being exact duplications of these processes.FIGURE6.28.-Equivalent mechanical model for cylindrical tank with flexible walls.TABLE6.6.-ModelParameters for Flexible-TallCylindrical Tankml = mTd(-)4 4htanh 3.681+3.68HI = 3.66ahh-hd sinh 3.68 a-coshasinh 3.68 h3.682221ANALYTICAL REPRESENTATION OF LATERAL SLOSHING00L045L52025FIGURE6.30.-Momenthldof inertia of rigid mass for flexibletank (ref.
6.4).FIGURE6.29.-Parameters for spring mass element8 forflexible tank (ref. 6.4).REFERENCES6.1. ZHUKOSKII,N. E.: On the Motion of a RigidBody Having Cavities Filled With a Homogeneous Liquid. Collected Works (in Russian).Vo1. 3. Moscowr, 1936.V. V.: Stability of Motion of Solid6.2. RUMYANTSEV,Bodies With Liquid Filled Cavities by Lyapunov's Methods. Advances in Applied Mechanics, vol. 8. Academic Press, 1964.D. E.: Theory of the Motion of6.3. OKHOTSIMSKII,a Body With Cavities Partially Filled With aLiquid. NASA T T F-33, 1960.6.4. LUKENS,D. R.; SCHMITT,A.
F.; A N D BROUCEK,G. T.: Approximate Transfer Functions forFlexible-Booster-and-Autopilot Analysis. Convair Rept. No. AE 61-0198, Contract No. AF33(616)-7037, Apr. 1961. (Also issued nsWADD TR-61-93.)I. E.; STOFAN,A. J.; A N D SHRAMO,1). J. :6.5. SUMNER,Experimental Sloshing Characteristics and nMechanical Analogy of Liquid Sloshing in nScale-Model Centaur Liquid Oxygen Tank.,m a*-IYO*..I* n o l i A l v l A-YYY,6.9. SCHMITT,A. F.: Forced Oscillations of a Fluid ina Cylindrical Tank Oscillating in a Carried Acceleration Field-A Correction. Convair Rept.ZU-7-074, Feb. 1957.J.
T.: Experi6.10. WARNER,R. W.; AND CALDWELL,mental Evaluation of Analytical Models for theInertia and Natural Frequencies of Fuel Sloshing in Circular Cylindrical Tanks. NASA T ND-856, 1961.6.11. STEPHENS,D. G.; AND LEONARD,H. W.: TheCoupled Dynamic Response of a Tank PartiallyFilled With a Liquid and Undergoing Free andForced Planar Oscillations. NASA T N D1945, 1963.6.12. GRAHAM,E. W.; AND RODRIGUEZ,A. M.: TheCharacteristics of Fuel Motion Which AffectAirplane Dynamics. Douglas Aircraft Co.Rept. No. SM-14212, Nov. 1951.6.13. GRAHAM,E. W.; A N D RODRIGUEZ,A. M.: TheCharacteristics of Fuel Motion Which Affect6.6.
ARMSTRONG,G. L.; AND KACHIGAN,K.: PropellantSloshing. Sections 14.14115, Handbook ofAstronautical Engineering (H. H. Koelle, ed.),McGnw--Hill Book Co., 1961.K.: Forced Oscillations of a Fluid in6.7. KACHIGAN,a Cylindrical Tank. Convair Rept. No. ZU7-046, Oct. 1955.6.8. SCHMITT,A. F.: Forced Oscillations of a Fluid inrr Cylindrical Tank Undergoing Both Translation and Rotation. Convair Rept.
No. ZU7-069, Oct. 1956.E, J. Appl. Mech., vol. 19, no. 3, Sept. 1952,pp. 381-388.6.14. HOUSNER,G. W.: Dynamic Pressures on Accelerated Fluid Containers. Bulletin of theSeismological Society of America, vol. 47,Jan. 1957, pp. 15-35.6.15. ABRAMSON,H. N.; CHU, W. H.;AND GARZA,L. R.: Liquid Sloshing in Spherical Tanks.Southwest Research Institute, T R no. 2,Contract no. NASS-1555, Mar.
1962..7m.--An-A-~'=llt."PU.z:n m,,: c.". IL IUI..,.Y , IIOIYIbU.AQXKV.-VI.IY,P-,:I.~"CIICU222THE DYNAMIC BEHAVIOR OF LIQUIDS6.16. SCHT, A. A.: A Theoretical Analysis of theEffects of Fuel Motion on Airplane Dynamics.NACA T N 2280, 1951.6.17. SUMNER,I. E.: Experimentally DeterminedPendulum Analogy of Liquid Sloshing inSpherical and Oblate-SpheroidalTanks.NASA T X D-3737, 1965.6.18.
HARPER,J.: Propellant Sloshing in a ConicalTank Undergoing Arbitrary Forced Translational Motion. Convair Rept. ZU-7-089TN, Jan. 1958.6.19. FREED,L. E.: Stability of Motion of ConicalPendulums.Ramo-Wooldridge Co. Rept.GbI-45, 3-434, Oct. 1957.6.20. MILES,J. W.: Stability of Forced Oscillations ofa Spherical Pendulum. Quart.
Appl. Math.,1.01. 20, no. 1, Apr. 1962, pp. 21-32.6.21. HOWELL,J. Y.: Motion of Conical Pendulum.Ramo-Wooldridge Co. Rept. GhI 42.4-4,Nov. 1957.6.22. BERLOT,R. R.: Production of Rotation in aConfined Liquid Through Translational hlotionof the Boundaries. Trans. of the ASME, SeriesE, J. Appl. Aicch., vol. 26, no 8, Dec. 1959,pp. 513-516.H . X.;CHT,W. H.; A N D KANA,D.
I).:6.23. ABRAMSON,Some Studies of Xonlinear Lateral Sloshing inRigid Containers. J. Appl. Mech., vol. 33.no. 4, Dec. 1966.6.24. BAITER,H. F.; CLARK,C. D.; A N D WOODWARD,J . H.: A~lalytical hlechanical Model for theDescription of the Rotary Propellant SloshingRlotion. Final Rept., Contract NAS8-11159,Engineering Experiment Station, Georgia. Institute of Technology, May 1965.6.25. ABRAMSON,H. N . ; CHU,W.
H.; A N D RANSLEBEN,G. E.,JR.: Representation of Fuel Sloshing inCylindrical Tanks by an Equivalent Mechanicalhlodel. ARS J., vol. 31, Dcc. 1961, pp 16971786.6.26. MILES, J. W.: On the Sloshing of Liquid in aFlexible Tank. Ramc+Wooldridge Rept. GMTR-73, Sept. 1956.6.27. BAUER,H. F.: Fluid Oscillation in a CylindricalTank With Damping.
Army Ballistic MissileAgency Rept. DA-TR-4-58, Apr. 1958.6.28. BAUER, H. F.: The Moment of Inertia of aLiquid in a Circular Cylindrical Tank. ArmyBallistic Missile Agency Rept. DA-TR-5-58,Apr. 1958.6.29. BAUER, H. F.: Mechanical Model of FluidOscillations in Cylindrical Tanks and Introducing of Damping. MSFC Rept. MTP-AERO62-16, Feb. 1962.6.30. BAUER,H. F.: Fluid Oscillations in the Containers of a Space Vehicle and Their InfluenceUpon Stability.
NASA TR-R-187, Feb. 1964.6.31. BAUER,H. F.: Liquid Sloshing in a CjlindricalQuarter Tank. AIAA J., vol. 1, no. 11, NOV.1963, pp. 2601-2606.6.32. BAUER,H. F.: Liquid Sloshing in a 45' SectorCompartmented Tank. AIAA J., vol. 2, no. 4,Apr. 1964, pp. 768-770.6.33. DODGE,F. T.; A N D KANA,D. D.: Moment ofInertin and Damping of Liquids in BaffledCylindrical Tanks. J .
Spacecraft Rockets,vol. 3, no. 1, Jan. 1966, pp. 153-155. Also seediscussion, same journal, vol. 3, no. 6, June1966, pp. 057-959.6.34. RATTAYA,JASTIV.: Sloshing of Liquids inAsisymmetric Ellipsoidal Tanks. AIAA PaperNo. 65-114, 2nd Aerospace Sciences Meeting,Jan. 1965.6.35. LOMEN,D. 0 . : Liquid Propellant Sloshillg i nRlobilc Tanks of Arbitrary Shape. NASARept. CR-222, Apr. 1965.6.36. LOMEN, I.). 0 .
: Digital Analysis of LiquidPropellant Sloshing in Mobile Tanks WithRotational Symmetry. K A S A Rept. CR-230.May 1965.PRINCIPAL NOTATIONSc,= damping coefficient for disk I,c,=dnmping coefficient for nth slosh modet l = cylindrical tank diameterF = transverse slosh in^ force!]=acceleration due to gravity, or equivnlent longitudinal accelerationh = total liquid depthh,=locntion of nth slosh mnss relative t oliquid center of gravity, except seeequation (6.34)Iro=location of fixed mass relative t oliquid center of gravityI,=moment of inertin of disk, see figure6.26Ip=moment of inertia of liquid abouttransverse axis through center ofgravit,yI,=moment of inertia of fixed massI,l,ld =momen t of inertia of solidified liquidabout transverse nsis through centerof gravit,yK , =spring const nnt for nth spring-mnsssystem of modell,=location of n t h slosh mass relative tothe free surfacelo=location of fixed mass relative to thefree surfaceANALYTICAL REPRESEXTATION OF LATERAL SLOSHINGL,=pendulum arm length for nth sloshmodem,=slosh mass of nth modemo=fixed massmT= total mass of liquidM=sloshing moment about liquid centerof gravityt=timex,, yo= amplitude of lateral excitationy,=lttteral displacement of nth mass relative to tank walla, 8, y=directional angles for pendulum modelof rotary sloshing, see figure 6.9a,=dimensionless parameter, see equation(6.34)223?,=damping factor of nth slosh modeV2=Laplacian operatorX,=dimensionless frequency parameter, seefigure 6.20v = dimensionless frequency for rotarysloshingtn= zeroes of J:(5,) =0@=velocity potentialcpo=amplitude of pitching oscillationx=see equation (6.38)$=pendulum angle, or rotational angleof disk relative t,o tankw =frequency of excitationw,=natural frequency of nt.h slosh modeChapter 7Vehicle Stability and ControlHelmut F.
Bauer7.1INTRODUCTIONIn order to analyze properly the performanceof rockets and space vehicles, it is necessaryto consider the general problem of dynamicsand stability of the vehicle under thrust.Since, with the increasing size of space vehiclesand their correspondingly larger tank diameters,the liquid propellant frequencies come closerto the control frequency of the vehicle, therefore the influence of the sloshing propellantcan no longer be neglected. Also, with thisincrease in the diameter of the propellantcontainers, the amount of propellant participating in sloshing and the correspondingliquid forces become rather pronounced andcan idluence the stability of the vehicle considerably.
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