H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 20
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Soc. (London), vol.A244, no. 254, 1952.TAYLOR,G. I.: An Experimental Study of Standing Waves. Proc. Roy. Soc. (London), vol.A218, 1952, pp. 44-59.TADJBAKHSH,I.; AND KELLER,J. B.: StandingWaves of Finite Amplitude. J. Fluid Mech.,vol. 8, no. 3, July 1960, pp. 443-451.FULTZ, D.: An Experimental Note on FiniteAmplitude Standing Gravity Waves. J. FluidMech., vol. 13, no.
2, June 1962, pp. 193-213.VERMA,G. R.; A N D KELLER,J. B.: Three-Dimensional Standing Surface Waves of Finite Ampli-,tude. Phys. Fluids, vol. 5, no. 1, Jan. 1962,pp. 52-56.BAUER,H. F.: Nonlinear Propellant Sloshing in aRectangular Container of Infinite Length.h'orth American Aviation, Inc., S&ID Report,SID 64-1593, 1964.GAILLARD,P.: Theoretical and ExperimentalResearch of Nonlinear Oscillations of Liquidsin Containers and Channels of Constant Depth(in French). Publ. Sci.
Tech. Minist. llAir,France, no. 412, 1963.3.18. CHU, W. H.: Liquid Sloshing in a SphericalTank Filled to an Arbitrary Depth. AIAA J.,vol. 2, no. 11, Nov. 1964, pp. 1972-1979.3.19. HUTTON,R. E.: An Investigation of Resonant,Nonlinear Nonplanar Free Surface Oscillationsof a Fluid. NASA T N D-1870, 1963.3.20. ROGGE,T. R.; AND WEISS, H. J.: An Approximate Nonlinear Analysis of the Stability ofSloshing Modes Under Translational andRotational Excitations. NASA CR-220, 1965.3.21.
WEISS, H. J.; AND ROGGE,T. R.: A NonlinearAnalysis for Sloshing Forces and Moments on aCylindrical Tank. NASA CR-221, 1965.3.22. BERLOT,R. R.: Production of Rotation in aConfined Liquid Through Translational Motionof the Boundaries. J. Appl. Mech., vol. 26,no. 4, Dec. 1959, pp. 513-516.3.23. ABRAMSON,H. NORMAN;CHU, WEN-HWA;GARZA,LUIS R.;AND RANSLEBEN,GUIDOE.,JR.: Some Studies of Liquid Rotation andVortexing in Rocket Propellant Tanks. NASAT N D-1212, 1962.3.24. MILES,J. W.: Stability of Forced Oscillations of aSpherical Pendulum. Quart. Appl. Math.,vol.
20, no. 1, Apr. 1962, pp. 21-32.3.25. HUTTON,R. E.: Fluid Particle Motion DuringRotary Sloshing. J. Appl. Mech., vol. 31,no. 1, Mar. 1964, pp. 123-130.PRINCIPAL NOTATIONSa= an arbitrary constant or effectivegravitational acceleration(?,=the nth constant in equation (3.25)d=dianleter of the tankF=force exerted on t,he tank by thefluid.f(x, y ) = a function defined by equation(3.21)f n t = defined in equation (3.13)g=gravitational accelerationhn=the nth coefficient defined by equation (3.12)H= Neumann functionh=maximum depth of the liquid atrestn=outer normal to the boundaries oran integerp,,=naturalfrequency of the mthsloshing modeyo= ullage pressurep =pressurer, 9, z=cylindrical coordinatesq=velocity vector of the fluid+S=the free surface when the liquid isstationarys=the distance measured along thefree surfaceT=t,he period of vibrationst =timeU,, U2=functions of (t, x, and y)U=same as zov,=local normal velocity of the wall2, y, z=rectangular coordinates with positive z upwardxo=amplitude of excitational displacemen tc=a parameter related to the nondimensional amplitude of thesurface elevationfree surface elevationA= characteristic numberA,=nthcharacteristic number=wavelengthp = a parameter related to the amplitude of excitationp = a constant, see equation (3.50)r=-NONLIPU'EAREFFECTS IN LATERAL SLOSHINGtransformed frequency, see equation (3.89)p=density of the fluidX=the wetted wall or the summationsign&=the moving part of the wetted wallu = see equation (3.70)u,=the frequency of the nth freevibration..-.~ = anondimensional time defined byequation (3.12)$J= velocity potential#(x, y, 0) =characteristic function#n(x, y, 0) =nth characteristic functionv=- -103$(z, y, z) =function defined by equation (3.6)w=frequency of tank motionw,=nth natural frequencyDDt--StokesSuperscripts:( )"=thederivativesecond derivative of ( ) withrespect to nondimensional time( )'=derivative of ( ) with respect totime( ) *=nondimensionalized quantities( )(''=steady-statevalue of ( ) unlessotherwise specifiedChapter 4Damping of Liquid Motions and Lateral SloshingSandor Silvernzan and H .
Norman Abramson4.1INTRODUCTIONPropellant sloshing is a potential source ofdisturbance which may be critical to the stability or structural integrity of space vehicles,as large forces and moments may be producedby the propellant oscillating at one of itsfundamental frequencies in u partially filledtank. Since the liquid oscillatory frequencymay nearly coincide with either the fundamentalelastic body bending frequency or the dynamiccontrol frequency of the vehicle at some time ofthe powered phase of the flight', the slosh forcescould interact with the structure or controlsystem. This could cause a failure of structuralcomponents within the vehicle or excessivedeviation from its planned flight path.
I tis therefore necessary to consider means ofproviding adequate damping of the liquid motions and slosh forces and to develop methodsfor accounting for such damping- in analysesmade of vehicle performance.Linear damping is usually introduced intothe vehicle dynamic analysis through theresonance terms of the equations governing amechanical model representation of the liquidforces and moments.
This is done under theassumption that the behavior of the liquidoscillating in its fundamental mode is analogousto the behavior of a linear, viscously dampedsingle dagaa-~f-freedomsystcm. (See chapters6 and 7 for the development of such mechanicalmodels.) The analysis then yields the amountof damping required for vehicle stability. Oneof the main problems which remains is topredict the amount of damping present in agiven tank configuration.The results of some of the numerous investigations in the field of liquid damping inrigid containers will be presented and dis-cussed in this chapter. Unfortunately, the problem is essentially nonlinear and therefore fewtheories are available for predicting damping(refs.
4.1 through 4.5). Our principal knowledgeof damping characteristics is the result of extensive experimental studies.OnDampingThe term "damping" is generally employedto describe the fact that some energy djssipation always o c c ~ mduring fluid oscillations.By far t.he most comprehensive discussion ofdamping effects in propellant sloshing is contained in reference 4.6.If there is no energy input to a system that isoscillating in one of its natural modes, theamplitude of successive oscillations decreasesas a result of energy dissipation.
This decreasing amplitude can be described by thelogarithmic decrement, defined as6=lnhlaximum amplitude of any oscillationMaximum amplitude 1 cycle laterFor a linear system, in which the restoringforce is proportional to the amplitude of thedisplacement (measured from rest position), thetotal energy of oscillation at the peak amplitude~f RW cycle is proportional to the square ofthis amplitude. The logarithmic decrementcan therefore also be written as6=lnEnergy of motion of 1 cycleEnergy of motion 1 cycle laterIn terms of the energy decrement per cycleAEand the total energy of motion E, equation(4.2) can be reduced to105Preceding page blankT H E DYNkMIC BEHAVIOR OF LIQUIDSDamping, whatever its nature, can be represented by an equivalent viscous damping inwhich the damping force opposing the motionis equal to a damping factor multiplied by the~ielocit~yof the oscillating component of thesystem.
The ratio of the actual damping, c, tothe critical damping, c,, is denoted as y and itcan be sho~vn(ref. 4.7) that, for small valuesof rThe damping ratio y may also be expressed interms of energy by substituting equation (4.3)into equation (4.4)(2) The driveforce method.-A platform, uponwhich the tank is located, is driven at constantexcitation amplitude and frequency. The forcein the drive link, the amplitude of the platformmotion, and the amplitude of the wave motionare recorded.
The energy input required tomaintain steady-state oscillations is the sameas the energy dissipated. That component ofthe drive force that is in phase with the tankvelocity delivers energy to the system at thesame rate as the dissipation. I t is the amountof this input per cycle compared with the totalenergy of motion, as measured by the waveamplitude, that determines the damping rate.(3) The wave (or force) amplitude responsemethod.-The wave (or force) amplitude as afunction of drive frequency is determined forconstant drive amplitude.
The bandwidthtechnique (see refs. 4.7 and 4.8) is then appliedto these response data to obtain t,he dampingratio y. The relationship thus obtained isThe damping ratio in the form used by Miles(ref. 4.2) in the theory of ring damping in acircular tank isEdty= - 2wEwhere z d t is the mean rate of energy dissipat,ion over a cycle of period 27r/w and o is theangular frequency of oscillation.Several experimental techniques have beenemployed to obtain the damping factor andare outlined on the follo~vingpages.(1) Ring force method.-Adirect measurement of the force required to anchor the ringbaffle to the tank ~vallis made. The waveamplitude is measured so that the total energyof motion and the \\lave velocity can be calculated from classical theory. The component ofthe ring force that is 180' out of phase with thevertical wave velocity at the ring position givesa measure of energy dissipation, mhiie thedissipation per cycle compared with the totalenergy of motion determines the damping rate.Onc of the principnl means of increasing thednmping in n liquid propellant tank of cylindricalgeometry is obvior~slythat of introducing baffles in t h cform of thin rings nttachcd normal t o the tank wall.Thcvc rind othcr types of bnfflcfi will bc discussed intlctt~ilInter.where Q is the excitation frequency and z is thewave (or force) amplitude.
A typical responsecurve is illustrated in figure 4.1. I n practice,the ratio z , , , / ~ ~ = 2 / ~ 1 Tis often used and thusequation (4.7a) simplifies toFIGURE4.1.-Typicalwave amplitude response curve.DAMPING OF LIQUID MOTIONS AND LATERAL S L O S ~ GA comprehensive discussion of the ~vholefieldof resonance testing is contained in references4.9 and 4.10; included is a critical discussion ofthe bandwidth technique. I t should be notedthat this technique gives reliable results only ifthe natural frequencies are not close and thedamping is not large. Also, the speed of thefrequency sweep and the magnitude of theexcitation amplitude may affect the results.(4) The wave amplitude decay method.-Therate of decay of the free surface displacementafter cessation of tank motion is measured.With zero input, the decreasing energy ofmotion as evidenced by the decreasing maveheight is a result of the energy dissipated by thedamping mechanism.
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