H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 38
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5.31).ISloshing ForcesFlight experience with launch vehicles indicates that the principal problem posed bysloshing fluids in launch vehicle tankage isassociated with the control system. As thepropellants slosh, forces and moments aregenerated on the vehicle which must be overcome by the control system in order to maintain the vehicle on the programed flight trajectory. Thus, it is necessary to design thecontrol system to cope with both the frequenciesand the amplitudes of the sloshing fluids, andhence it is necessary to know these quantities.m - - L - : -----C-lauLlulquc;a L v r uuauuLigthe fmn..',nnrPaIAV,,,,v:,,,andfor limiting the forces, are discussed in previoussections, but it is essential to know the forcesinvolved even for baffled tanks to assure adequate margins in control system gains andcontrol forces.Research techniques for measuring the sloshing forces imparted to tanks are cited in the-A,,FIGURE5.23.-Mode shapes for the &st three naturalmodes of liquid in a spheroid.
(a) First mode; (b) second mode; (c) third mode.,..1178THE DYNAMIC BEHAVIOR OF LIQUIDSFIGURE5.25.-Techniques for detecting magnitudes offluid sloshing. (a) Capacitance wires; (b) floats;(c) pressure pickups; (d) load cells.FIGURE5.24.-Mode shape8 for first three natural modefi ofliquid in a horizontal toroid.
(a) First mode; (h) secondmode; (c) third mode.literature. The apparatus shown in figure 5.26was used to measure the forces on sphericaltanks (refs. 5.19, and 5.22 through 5.24) andtoroidal tanks (ref. 5.21). The spheroidal tank(ref. 5.15) was in~estigat~edwith the apparatusshown in figure 5.18(b). The techniques usednre typical in thttt the tank is oscilltited a t theselected frequency and amplitude t o generatethe desired fluid motions, is "quick stopped" titri point of zero velocity in the cycle, rind tileforces imparted by the fluid to the trink duringensuing fluid oscillntions measured by installingFIGURE5.26.-Experimentaltest facility.a calibrated load cell in the driving link.
Typictll slosh-force traces are shown in figure 5.27.The measurement of residual forces after bringing the Appt~rlitus to rest is preferable to therlormtilly used forced-response technique in thati t obviates the need to subtrtlct out the tareSIMIJLATION AND EXPERIMENTAL TECHNIQUESTimeFIGURE 5.27.-Typicalslosh-force trace. First mode:liquid-depth ratio, h/2R=0.40.forces associated with the tank-support system;however, it does not provide an opportunity toresolve the resultant force into its mass-springand damping components.
Resolution of theforces when utilizing the forced-response technique has been accomplished with the apparatusshown in figure 5.18(a) by feeding the load celloutputs into a resolver system phased with thetank displacements (ref. 5.27).Irregular Fluid Motions-SwirlSloshand Dome-ImpactWhen the frequency of excitation of a tankcontaining a fluid is gradually increased beyondthe natural frequency, the fluid wave will beginto swirl. Experimental results show that t'heswirl is a combination of wave motions androtary motions-particles in the fluid move upand down and rotate around the tank. As aresult of the rotations of the fluid, the amplitudeof the wave before the surface breaks may reachamplitudes several times as large as those dueto pure sloshing a t a natural frequency and,consequently, the fluid forces imparted to thetank may be much higher.
Although the swirlis most pronounced for the first sloshing mode,there is experimental evidence that swirlingmotions are also possible for higher mode;.As discussed in chapter 3, this is a nonlinearproblem akin to large amplit'ude motions of apendulum.- Experimental techniques for evaluation of theforced response during normal sloshing of fluids,such as described in previous sections of this1.79chapter, have been successfully applied (ref.5.26) to analysis of fluid swirl and appear to beadequate.The problem of dome impact arises when thedirection of the resultant acceleration of thetank is reversed such as may occur during engineshutdown of a launch vehicle flying through theatmosphere, or if an engine is ignited while thepropellant is located in the upper part of a tank.(These problems are discussed in further detailin ch.
10.) The resultant impulsive pressuresmay be severe and must be considered as afactor in assessing the structural integrity ofthe tank.Two experimental techniques have been usedto analyze the extent of this problem. References 5.11 and 5.12 describe a technique whichutilizes a pressure-activated carriage (shown infig. 5.28) to accelerate a tank, partially filledwith a liquid, downward a t accelerationssubstantially greater than 1 g.
The resultingimpulsive pressures on the tank d,:me aremeasured by an array of pressure cells.Another technique for investigation of domeimpact fuel slosh is shown in figures 5.29 and5.30. As shown in figure 5.29, the tank(enclosed in a cage, fig. 5.30) is acceleratedupward by a cable attached to a drop weightand passing over a system of pulleys. *After thedrop weight comes to rest by dropping in asandpit, the motion of the tank is also broughtto rest by the braking force of another cablesystem. The downward acceleration of thebraking force, greater than 1 g, causes the fluidto impact the dome, whereupon both the fluidpressure and the resultant fluid forces aremeasured.
The measurement of the fluid forcesis facilitated by supporting the dome from theremainder of the tank by a system of straingage beams.Coupling of Fluid Motions and StructuraiDeformationsAlthough the major problems concerningsloshing propellants involve the uncoupledfundamental lateral sloshing modes of the fluid.experimental evidence indicates that substantial coupling of fluid and tank motions mayresult in some instances. Reference 5.33, for180THE DYNAMIC BERAVIOR OF LIQUIDSDrop weightBra11ng forceFIGURE5.29.-Schematic of dome-impact slosh apparatus.by an electromagnetic coil and measured uithinductance pickups.Another study of interest has been reported.(ref.
5.34) which involved the breathing vibr*tions of a circular cylindrical shell containing aliquid. The technique and apparatus usedwere similar to those employed for the study ofthe bending vibrations discussed in the previousparagraph.Coupling ofFIGURE5.28.-SouthwestReeearch Institute apparatusto study dome-impact fuel slosh (refs. 5.11 and 5.12).example, treats the bending vibrations of acircular cylindrical shell with an internal liquidhaving a free surface wherein the presence ofthe free surface liquid was shown to increase theresonant bending frequency of the tank ascompared to a similar configuration with thefluid capped.
The apparatus used in this studyis shown in figure 5.31. - The tank, pin ended inthis case, is mounted in a stand designed topermit variations in tank-end restraints andequipped with a capping device to restrict thefluid surface for comparison with free sl~rfaceresponses. Vibrations of the tank were excitedFluid and TankMotionsIn most of the studies of fluid sloshing, theassumption is generally made that the sloshingphenomena are not appreciably affected by thetranslatory and pitching motions of the tank,and that sloshing data, generated in rigidlymounted tanks or tanks subjected to small.translat,ory movements such as are necessaryto excite the fundamental sloshing modes, areapplicable to launch vehicle design. Thisphilosophy was put to the test in a study (ref.5.35) wherein a tank was mounted in a gimbalsupported on long cables so that it couldoscillate in both translation and pitch.
Asshown in figure 5.32, the tank was also fitteduith an air jet and control system whichpermitted close simulation of a launch vehicleunder flight. By varying jet pressures, pitching axes, and the spring and damping constantsSIMULATION AXD EXPERIMENTAL TECHNIQUES---181PinsupportFIGURE5.31.-Schematic of apparatue ehowing cappingdevice (ref. 5.33).predicted by use of data generated during moreconventional sloshing studies.5.9 LIQUID MOTIONS IN LARGE TANKSDamping in CylindersFIGURE5.30.-LangleyResearch Center tank and instrumentation for dome-impact sloeh studies (ref.
5.32).of the system, it was possible to study bothstable and unstable motions. The results ofthis study showed that conventional ringbaffles immersed in the fluid were substantiallymore effective in damping pitching motionsthan lateral motions of the tank, and that byinclusion of the proper damping, the motions andstability of the system could be adequatelyAlthough most of the research on fluid sloshing has been conducted in small tanks asreviewed in section 5.8, several importantinvestigations have involved experiments withlarger tanks.
Reference 5.14 presents theresults of a study of the effect of time-varyingliquid depth on the damping of the fundamentallateral sloshing mode. A Plexiglas tank, 91.5centimeters in diameter and 183 centimetershigh, was partially filled with water, the lateralsloshing mode mas excited with a paddle, andt'hn d ~ m p i n gwas measured with the capacitancewire system shown in figure 5.25(a) as the fluidwas drained from the tank at high rates. Bycomparing the damping at a given fluid levelwhile draining with the damping at the samefluid level without draining, it was shown thatthe normal rate of efflux of the fluid from a tankhas no appreciable effect on the damping of thefirst lateral sloshing mode.182THE DYNAMIC BEHAVIOR OF LIQUIDSConstraining rodtSupport platformfl+/Reciprocating+-+drive and balance system7r::"(Wave damperaxisTranslationFIGURE5.33.-Two-dimensionalFIGURE 5.32.-Schematicof experimental apparatus(ref.
5.35).tank.FIGURE5.34.eFront view of test equipment.Two-Dimensional Damping Effectiveness of BafflesReference 5.36 presents a study of the measured tx-0-dimensionnl damping effectireness offuel-sloshing baffles applied to ring baffles in acylindrical tnnk. In coritrtist to the usual tippronch n-here a tank of a geometric configuration of interest is fitted I\-ith baffles and thedamping of the slosliing fluids measured, in thisstudy numerous types of t\\-o-dimensionalbnffle segments were mounted as cantileverelements on a reciprocating blnde (fig. 5.33)installed in n t\vc)-dimensionnl tnnk as she\\-n infigure 5.34. Tlie tit~ik\\.as 30.5 s 152.5 s 152.5centimeters in size nnd wns filled to n depth of122 centimeters with ~vnter. -4s shown infigure 5.34, the blade which supported the baflesegment was mounted to a pi\-oted beam andinstrumented so as to measure the forces imp ~ r t e db y t,he fluid to the bnffle segment as thefrequency and amplitude of the support bladevibrations were varied.
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