H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 3
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Such control of the liquiddynamic behavior is generally accomplishedeither by the introduction of various arrangements of baffles intended to provide an ad*quate degree of damping in the liquid system(fig. 1.4)) or by modifying the tank geometry insuch a way as to change drastically the liquidfrequencies.In order to examine in further detail thevarious aspects of this quite complex problem,it should be noted first (from table 1.1) thatboth the liquid-slosh frequencies and the elasticbody-bending frequencies decrease with increasing tank diameter. Now, we have h e a d yseen that for large circular tank vehicles such asSaturn I, the slosh frequencies are so low as tocouple with the rigid-body motions; this isseen clearly in the left side of figure 1.3.Successful flight would depend, in this csse,upon the provision of adequate damping in thepropellant tanks, probably by the use of variousarrangements of b d e s .
Compartmentation,or subdivision, of the tank (fig. 1.I) has a very5INTRODUCTIONScalloped tankCylindrical tank15First bending mode0050100Time, sec1500'501001%Time, secFxcrrnlr 13.-Variation of vehicle frcquenciw with flight time (ref. 1.26).marked effect in iwecrdng the propellant sloshfrequencies and, therefore, would appear to bean ideal method of avoiding coupling with therigid-body motions; the right side of figure 1.3clearly shows this effect. Unfortunately, however, this solution is not as simple as it seems,for the compartmented tanks usually havesubstantially reduced elastic bending frequencies (these decretms ~ ~ o u probablyldalso causesome changes in the control system design,so that the effects of such changes are by nomeans isolated uuto themselva but form a loop),as compared with the circular tank, and,therefore, coupling between the propellantsloshing and elastic modes may result in largedynamic responses; again, this is indicated inthe right side of figure 1.3.
There is stillanother complication that enters this discussionwhen sector-compartmented tanks are considered: While the lowest slosh frequency issubstantially raised by compartmentation, inthe sector tank new slosh frequencies areintroduced that are not widely spaced, so thatthe possibility of coupling with the elasticmodes may be greatly enhanced.The question of relative advantages ofckular-versus-compartmented-tank configurations is one that has no simple answer. Asideeven from considerations of slosh characteristicsand elastic bending frequencies discuss;d above,one should recognize that the compartmentedtank would probably be lighter in weight thanits equivalent circular tank (taking into accountthe weight of baflEiing required in the latter),but it would probably be considerably moreexpensive to fabricate.1.3 PROPELLANT MOTIONS IN SPACE VEHICLESThe previous discussion has centered aroundwhat might be considered a typical case ofliquid propellant sloshing in a large cylindricaltank, the liquid motions being described asarising essentially from lateral motions of thetank.
Further considerations of such lateralsloshing are primarily involved with othertank geometries, such as those shown in figure1.5, which represent oniy a very f e w pasibilities. Obviously, it is of interest to knowthe propellant (or other liquid) slosh characteristics in these various kinds of tanks, andin various orientations.The excitations or tank motions that maylead to propellant sloshing are also quitevaried, encompassing a wide system of directions, amplitudes, and frequencies. Thus, theTHE DYNAMIC BEHAVIOR OFTankbaffles (perforatedSimplecylinderModifiedcylinderOblatespheroidsModifiedellipsoidToroids-z ring,stifl'enersSphereFIGURE1.5.-Typical space vehicle tank codgurations.FIGURE1.4.-Tnmcatml-cone-type ring baffles for suppression of lateral sloshing.contained liquids can be expected to respond ina variety of ways and, in fact, do exhibit anamazing number of kinds of motions of varyingcomplexity, none of them really simple and allof them difficult to predict and understand(refs.
1.3, 1.4, and 1.27 through 1.30), althoughnot all are of great importance in actual spacevehicles. Without going into detail a t thistime, perhaps it is worthwhile to mention justa few of these:Lateral sloshing.-This is the type ofliquid motion (essentially antisymmetricmodes) under discussion, occurring primarily in response to translational orpitching motions of the tank (figs. 1.6 and1.7).Vettical sloshing.-Liquid motions (usually symmetric modes) occurring primarilyin response to motions of the tank normalto the equilibrium free surface (fig. 1.8).R o W M sloshing (swirl motion) .Liquid motion exhibiting an apparentswirling of the liquid about a normal axisand arising as an instability of the antisymmetric lateral sloshing mode nearr8SOnanC8.Vortex formation.-Developmentof aventilated vortex in the outlet of a tankwhile under draining conditions (fig.
1.9).Surjace spray.-Development of a densespray of small droplets a t the liquid-freesurface as the result of high-frequencyrigid-body excitation of the tank, or elasticvibrations (fig. 1.10).D m impact.-Liquid in a p a r t i d y filledtank, under conditions of abrupt thrustcutoff while in the atmosphere, may impactagainst the opposing tank bulkhead.Low gravity phenomena.-Under conditions of very low gravity (orbital or interplanetary fiight), the liquid behavior isgoverned primarily by surface tension (andviscous) forces rather than by inertialforces and may be oriented randomly withinthe tank, depending essentially upon webting characteristics on the tank wall.Docking impact.-Liquid impact upon atank bulkhead arising from docking, orother maneuvers in space flight, when theliquid is initially controlled by low p v i t yconditions.FIGITRE1.7.-FitFIGURE1.6.-First two free surface modes of liquid motionin an upright circular cylindrical tank.The designer must, for each type of liquidmotion, understand its behavior and charactenstics sufliciently well to analyze its possible229-648 0--67---2two free surface mod- of liquid motionin a spherical tank.interactions with other components of thedynamic system and to provide adequateremedial measures.
For each type of tankconfiguration, orientation, and excitation, thedesigner requires, among other things, howl&-erelating to (a) liquid natural frequencies andmode shapes, (b) forced response characteristics,(c) damping with and without baffles or othersuppression devices, and (d) simulation techniques. We shall therefore discuss these varioustopics in some detail in the following chapters.1.4 XOPE OF M E MONOGRAPHThis monograph attempts to present a rathercomprehensive view of the general subject, andTHE DYNAMIC BEHAVIOR OF LIQUIDSFIGURE1.8.-Symmetricfree surface mods of liquidmotion resulting from vertical excitation.while it is doubtful that any single presentationcould be of equal value to students, researchers,analysts, and designers alike, it may be thateach of these will at least find here significantquantities of helpful information.
Each of thefollowing chapters therefore discusses in detailone or more aspects of the general problem wehave outlined in sketchy fashion in thisintroduction.Each chapter attempts to give an integratedview of theoretical and experimental data onthe particular subject at hand, with some concentration on the presentation of results ratherthan on the details of procedures; however,general methods of analysis have been outlinedwhere it was felt that these have reasonablywide application, are relatively little known, ormay be necessary to have at hand for properapplication of the final results of data that arecited. A list of references is also given in eachchapter, although, again, no claim is made forcompleteness. In this connection, it should beremarked that no special effort was made to citetechnicd literature other than that readilyavailable in the English language.The characteristics of lateral sloshing behavior, primarily liquid frequencies, modeshapes, and forced response characteristics, forvarious container configurations are presentedin some detail in chapter 2.
The basic elements of the underlying hydrodynamic theoryare presented, followed by analytical relationships and calculated and experimental data fora wide variety of cases. Only rigid tanks areconsidered.The nonlinear effects rising inherently fromparticular tank geometries (primarily in thecompartmented tank), or as a consequence oflarge amplitudes of excitation, or from instabilities in the large amplitude motions nearresonance (rotational sloshing), are discussedin chapter 3.
The first of these, especially,may have important implications for design.Chapter 4 presents a large quantity of experimental data gained primarily from laboratoryinvestigations of damping characteristics inlateral sloshing in rigid tanks. First, consideration is given to the simple viscous dampingeffects in various tanks and, second, to theeffectiveness of diEFerent types of suppressiondevices. Such devices may take a wide varietyof forms, ranging from floating objects and systems which follow the free surface to manytypes of fixed baffle configurations.Because so much of our presently availableknowledge on this general subject comes as rr,result of extensive laboratory investigations,the subjects of simulation and experimentaltechniques are discussed in detail in chapter 5.The subject of similitude by use of smallmodels based on dimensional analysis is presented in some detail, and in rather generalterms, which can then be specialized to specificcases.
Physical properties of various propellants and other commonly used liquids, as wellas a wide variety of potentially useful modelliquids, are tabulated in the appendix.With the force and moment response of theliquid under various conditions of lateral sloshing in rigid tanks fairly well defined in theseearly chapters, the question then arises of,integrating this knowledge into analyses of theoverall system dynamic behavior. This is usu-I-.INTRODUCTIONFIS-1.9.-Vortexformed during gravity draining of a tankcombined with similar representations for otherdynamic components of the vehicle, and thenthe overall system dynamic behavior can beanalyzed by analog or digital techniques.Various types of equivalent mechanical modelsare derived and discussed in chapter 6, including both very simple and relatively complexones. The general subject of vehicle stabilityand control is then treated in chapter 7, demonstrating how various of the analyses and dataare integrated; elastic body effects, however,are not yet included.The various, and interesting, aspects ofliquid behavior in response to vertical excitations of the tank are discussed in chapter 8.The first topic is that of subharmonic response(i.e., liquid response with ajess thanFIGUREl.lO.-Sarface spray malting from high hVencythat of the excitation) of the liquid surface tovertical excitation.fairly low frequency excitations, with detailedally accomplished by first devising appropriatecomparisons made between theory and the reequivalent (mathematical) mechanical modelssults of laboratory experiments.
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