H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 59
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J. derospace Sci., Oct. 1961, pp.819-820.7.22. BAUER,H. I?.: Wind Response of the SaturnLOR Vehicle. Marshall Space Flight Center,NASA, MTP-AERO-63-19, 1963.7.23. HAYS, P. J., AND SUMRALL,J. P.: StabilityAnalysis of Saturn SA-5 With Live S-IV Stage.NASA TM-X-53017, 1964.PRINCIPAL NOTATIONSao=gain factor of attitude channela, =coefficient of control dampingAt= output signal of accelerometerA,= shear aFea of airframe cross sectionca=damping coefficient of accelerometercG =damping coefficient of rate gyroscopec,i=damping coefficient of nth sloshingmode in Xth propellant containerD=dissipation functionD,=dissipat,ion function of propellantD,=dissipation function of vehiclestructureEI= flesurnl stiffness of airframeF= F, F*=thrustFl=stationary thrustF2= gimbal thrust+g=-F =longitudinal vehicle acceleramtion! I ,=structural damping factor of theulh bending modeg,=gnin factor of accelerometer channelG= shear modulusH,=Hurwitz determinantI=mass moment of inertia of totalvehicle about mass center ofvehicleI,=moment of inertia of rate gyroscope about its output axisIox=moment of inertia of nonsloshingliquid mass in Xth propellantcontainer (about its center ofmass)IR=moment of inertia of rotor of rategyroscope about spin axisI:=moment of inertia of structureper unit lengthk=radius of gyration of vehicle aboutits mass centerka=spring constant of accelerometerkB= spring constant of swivel compliancek.~=spring constant of nth mode ofslosh model in Xth tankKo=restoring moment of rate gyroscope per unit anglem =mass of vehiclema=maSS of accelerometerm,~(m,)=mass of nth sloshing mode inXth tankmoh=mass of nonsloshing liquid inXth tankm:=mass of structure per unit lengthMB,= generalized mass of vth bendingmodeMR=gyr09~0pi~reaction torque of rategyroscopeM,=bending moment of vth bendingmode~ , pl=phase-lagpcoefficients of controlsystemq, =generalized coordinatesQ,=generalized forces&,=shear force of vth bending modes= u+iw= complex frequencyt =timeT=kinetic energyTp=kinetic energy of propellantT,=kinetic energy of structurev= cross-vel ocityV=potential energy of vehicleVp=potential energy of propellantVsk=potential energy of elastic deformation of structureVII=potential energy of raising centerof mass in equivalent gravitational fieldV,=potential energy of bending vibrationx= abscissa, distance from mass centerxa= coordinate of accelerometerIxCiRl=distance of center of instantaneousrotation to mass center of vehicle267VEHICLE STABILITY AND CONTROLx,=coordinate of swivel pointxo=coordinate of gyroscopexnx(x,)=coordinate of sloshing mass of nthmode in Xth tankX,A= coordinate of nonsloshing mass inhth tankxR=coordinate of rate gyroscopey=lateral translation of rigid vehicleya= displacement.
ofaccelerometermass relat,ive to vehicleynx=displacement of nth sloshing massin Xth container relative tovehicleY,= normalized bending deflectioncurve of vth bending mode&=indicated angular velocity of airframe at the location of the rategyroscopeI-X= egg2 - "gain parameter" of accel-~=ggserometermnx=slosh mass ratio of nthsloshing mode in Xth containerto total vehicle massv,=w,/w,=frequency ratio of accelerometer to control frequencyv,= w,/w,= frequency ratio of propellant to control frequencyccnx=;..-Xaa=-----=locationparameter of aclxct~lcelerometerj3= engine deflection angle againstvehicle center axispc= control deflection-y,= slosh damping factor2'3c=-=ratioof thrust available forFcontrol purpose to total thrustof vehicleca= abbreviation for accelerometer defined on page 242eh =abbreviation for accelerometer defined on page 242&=abbreviation for gyroscope definedon page 239 ,&=abbreviation for rate gyroscope defined on page 239(,=damping factor of accelerometer(,=damping factor of control systemCo= damping factor of rate gyroscope(,,A=damping factor of nth sloshingmode in Xth tankll,=generalized coordinate of vth bending mnde4=rotation angle of rigid vehicle relative to space4 ,=output signal of attitude gyroscope&=angular velocity of airframe at thelocation of the rate gyroscopew,=natural circular frequency of accelerometerw,=natural circular control frequencywo= natural circular frequency of rategyroscopew,~(w,)=natural circular frequency of nthpropellant mode in Xth containerwR=angular velocity of rotor in rategyroscopew.=natural circular frequency of vthbending mode---_UY_-.'.I___I-_-...-----I.*.I-I__AIIi.- -Chapter 8IiVertical Excitation of Propellant TanksFranklin T.
Dodge8.1INTRODUCTIONThe dynamic response of liquids is discussedin this chapter for the case when the containingtank is vibrated vertically. Vertical and longitudinal vibrations in nctual rockets can arisefrom several sources. One cause of suchvibrations is n dynamic coupling between therocket structure and engine thrust duringflight. Severnl large liquid fuel boosters, including Thor-Agena, Atlas-Agena, Titan I, andTitan 11, have been subject to this kind ofvibration, with the frequency usually corresponding to the first longitudinal structuralmode, typical values being between 10 and30 cps.
Another type of longitudinal vibration,apparently originating through a coupling ofthe vehicle structure and the pneumatic tankpressure regulation system, also has beenobserved during the early part of the flight ofsome missiles; as one example, the frequencyof such vibrations is about 5 cps for Atlas-Dboosters. Some of these types of longitudinalvibrations encountered in flight have beencalled "pogo oscillations," and are discussedmore fully in chapter 10.Vertical vibrations may also occur duringthe time preceding actual launch when thevehicle is held down and its engines broughtup to fuii thrust, which, fcr krge ~ O C E ~ P T Smap:be on the order of several seconds. Thevibrations occurring then are very complexbecause of such factors as the rough burningof the fuel, the interaction of the exhaust andt,he exhaust deflection structure, etc.A complete analysis of the origin of thevarious types of vertical vibration is beyondthe scope of this chapter; however, regardlessof the origin, the effect of the vibration on theliquid fuel depends primarily on its frequencyand amplitude.
At low frequencies the liquidsurface may respond in a large amplitudestanding wave; as an example, the first symmetrical mode in a 14.5-centimeter-diametertank model, for an excitation of approximately7.5 cps, is shown in figure 8.1. The wavemotion frequency for a response of this type isexactly one-half that of the excitcttion; consequently, such motions are called one-half subharmonic responses.FIGURE8.1.-Typicalliquid motion for m=O, n = l modef{-~-&harrnnnic(ref. 8.20).At higher frequencies, the amplitude of theliquid motion is usually quite small.
But forcertain combinations of excitation frequencyand amplitude, these small capillary wavesmay disintegrate, form a dense spray, and1 Precisely what is meant by "high" and "low"frequencies will be clarified in later sections of this'chapter.269-Preceding page blank-z&s270THE DYNAMIC BEHAVIOR OF LIQUIDS.-...1;.:,.)"$ ~ , ~ ~ . ;- ; &.g;,w:-..!~.;>........;,?:.:!;-#*i........,.
.- . ...,; .:;:.SA.J,,.L.,,. --.,r,.<.':... ,:.;.*..:.il..:r%I .,,,**j:FIGURE8.2.-Second-mode surface wave maintained byspray (ref. 8.18).thereby generate a large amplitude, muchlower frequency standing wave. This phenomenon is illustrated in figure 8.2, which shows atypical spray-formed wave in a small rectangular test tank.The free surface motion sometimes becomesvery violent at larger input levels of vibration,and small vapor bubbles are entrained in theliquid.
The bubbles can become negativelybuoyant and thus sink to the tank bottom;such bubble motions are, of course, contraryto the usual state of affairs. Once these sinkingbubbles reach the tank bottom they can eitherform a large aggregate bubble, or, as in themodel tank shown in figure 8.3, they can surgecontinuously into the drain pipe. Both ofthese situations may pose serious problems forliquid fuel rockets, although it must be emphasized that such occurrences have not yetbeen evidenced in any way other than-inlaboratory experiments.All of the foregoing types of liquid responseswill be examined in greater detail in thefollowing sections.8.9LIQUID SURFACE RESPONSE TO LOW-FREQUENCY EXCITATIONHistorical SurveyFIGURE8.3.-nubblesentering drain line in verticallyexcited tank (ref.
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