H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 69
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This cannot necessarily bevehicle structure must be considered in theaccomplished by a simple superposition of whatoverall analysis of vehicle dynamics. Theisalready available because of nonlinesritiesinternal pressurized liquid and gas column,present.In other words, the developmenttogether with the shell-like tank of a spaceschemeisan example of progressing fromvehicle, represents a system which, in additionthecomparativelysimple to the more complex.to liquid surface motions, can experienceThesolutionsofwhat have been describedbending and breathing motions of the tankhereasindividualsimplifiedproblems appear,walls, breathing of the tank bottom, andinmanycases,togiveaverygood descriptionpressure oscillations in the liquid column, whileoftheresponsesoftheactualoverall systemall of these motions can couple with the mofor a limited range of input conditions.
Bytions of the rest of the structure and vehicleand large, for a given liquid in a tank of givencontrol system. Considering the complexitiessize and geometry, the various forms of liquidof the rigid tank liquid behavior that haveelastic tank behavior that arise are a functionalready been described, along with the comof the type of excitation. For forced oscillaplexities in analyzing elastic responses oftion, in other words, the responses of theshell-like structures, it is obvious that thevarious coupled natural modes of the systemcombined problem is one of extreme difficulty.depend on whether the input (lateral or longiRecognizing this complexity of the coupledbysieiil, ii ~ ~ C O I I Lappare~lt~ Si i ~ a isorrie si111~1i- tudifin! trnnslntion nnd pit~hing) forms ageneralized force capable of exciting a givenfications must be assumed in the overallnatural mode.
I t may happen that liquidproblem, so that at least solutions for restrictedsurfacemodes are excited for a limited rangeranges of input conditions are obtained. Forofexcitationparameters, but significant elasticthis reason, this chapter has been outlined intotankmodesare not; as a result, for thatsections which discuss the analysis of variouslimited range, the analysis of the liquid in aindividual aspects of the overall coupledrigid tank provides a goad description of theproblem.
Further, it will be seen that withinoverall behavior. On the other hand, if foreach section, further simplifications of theproblem have been assumed, particularly insome ranges of frequencies, significant liquid303--Preceding page blank X r304THE DYNAMIC BEHAVIOROF LIQUIDSsurface motion does not occur, while elasticsloshing have become critical factors in theoverall stability of large space vehicles, espeaction of the tank wall does occur, then thecially since the fundamental lateral bendingpresence of the liquid simply causes the addition of a distributed mass on the tank walls,frequency of vehicles with large-diameter proas well as allows longitudinal pressure modes,pellant containers is very low, and often nearand an analysis neglecting the liquid surfacethe propellant and control frequencies.
As inchapter 7, stability boundaries shall be precondition would provide an adequate descripsented which exhibit the influence of the varioustion of the overall behavior in that limitedpropellant parameters, such as sloshing mass,frequency range.propellant frequency, container location, gainFor linear responses, in many cases, couplingvalues of the control system, gyro location, andof liquid and elastic container modes mayphase-lag coefficients, as well as the influence ofbecome significant because of close proximityelastic parameters, such as bending frequency,of their respective uncoupled natural frequengeneralized bending mass, and structuralcies. For example, coupling of low-frequencydamping.liquid surface modes with low-frequency bending modes of the vehicle may be a distinctpossibility.
This follows from the fact that in9.9 COUPLING OF PROPELLANT MOTIONS WITHthe uncoupled cases, that is with sloshing modesELASTIC TANK BENDING VIBRATIONSin a rigid tank, or with bending modes in aliquid-Wed tank having a capped surface, theGeneral Discussionrespective motions occur a t frequencies relatively near to each other. For the two casesThe significant bending modes of largeoccurring simultaneously, i t is obvious thatrockets and space vehicles usually occur a tstrong coupling might result, so that eachrelatively low frequencies (i.e., less than 5 cps).respective coupled mode would occur at aAs has already been pointed out, the mostfrequency different from the respective unimportnnt liquid surface modes in cylindricalcoupled cases.
On the other hand, "breathing"tnnks of typical geometries also occur a t lowor "shell modes" of the walls of a tank containfrequencies; therefore, mutunl interaction being a capped liquid may occur at s o n ~ e ~ - h a t tween the two motions might be anticipated.higher frequencies than the significant bendingKevertheless, comparatively few investigationsmodes in a space vehicle, depending on the typehave been attempted in this area of the overallof stiffeners used on the tank uralls. As aconpled prohlem, and virtually all of the workresult, one might anticipate only weak couplingis restricted to cylindrical tnnks.with the low-frequency liquid surface modec,Bnuer has analyzed the behavior of an inbut might expect strong coupling witth highercompressible, idettl liquid of arbitrary depth infrequency spray phenomena.
I t must heH c'ir(+~llnrcylindrical tank (refs. 9.1 and 9.2)emphasized that the above type of rensoninp isand in n sectored cylindrical tank (ref. 9.3) thatvalid only for linear behavior. When signifiis sr11)jected to n specified bending oscillation.cant nonlinearities are present, coupling canIn other words, attent ion is focused on the liquidoccur between individual responses even thollghbehttrior in n contniner whose walls are forcedthey may be considered remote in frequency.to oscillnte with a prescribed bending shape.This type of coupling can, in fact, occlir for theBauer's analysis is one step beyond that of theabore-mentioned high-freqliency brenthing t tlnkbtudg of liquid behavior due to translntion orand low-frequency liquid surface modes.
Bothpit~hingin 11 rigid tank; however, it does notlinear and l~onlinear coupling of liquid andallow for complete elastic coupling with theelastic tank motion will be discussed in thistank walls. I n effect, this work is a study ofchapter.the influence of forced tank bending on theFinally, the interaction of the elastic strucliquid behavior, with no allowance for completeture, fluid systems, and the control system mustinteraction between liquid and tank bendingbe considered. Vehicle flexing and propellantINTERACTION BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTUREmotion.
(The results of these analyses areincluded in ch. 2.)More general analyses of the completecoupled bending-liquid sloshing problem havebeen given b y Rabinovich (ref. 9.4) and Miles(ref. 9.5). Both assume the existence of potential flow, so that a velocity potential may bedetermined in the bending cylindrical tank.Rabinovich uses an integro-differential approach to the partially filled tank problem.The liquid potential function is derived using aLagrange-Cauchy integral, from which expressions for the hydrodynamic and hydrostaticforces acting on the tank walls are obtained.These expressions are then introduced into thedifferential equations for the oscillations of athin-walled elastic bar.
Characteristic frequencies of the resulting integro-differentialequation can then be solved by Galerkin'smethod. No numerical examples or results aregiven for this analysis.Miles' analysis (ref. 9.5) of the coupledproblem involves the use of the Lagrangianprocedure for potential sloshing in a bendingcylindrical tank, and includes numerical esamples and results which fairly readily can beapplied to a tank of given boundary conditions.Since this work appears t o be the most readilyapplicable for a given specific tank, it \\-ill besummarized in this chapter in some detail. I naddition, experiments (ref.
9.6) that have beencorrelated with this analysis will ttlso hedescribed.305FIGURE9.1.-Coordinatesystem for bending tank containing a liquid.t)encling !\-all, and # ( r , 6 ) is the shape ofthe liquid free sr~rfacein some mode. Detailsof the derivlttion of the kinetic and potentialenergy coefficients, m t j and kij, may be obtaineddirectly from reference 9.5; but, in essence,they are determined by assuming the existenceof potential fluid flow, and simple inextensionalbeurnlike bending of the cylindrical tank walls,ant1 solving the appropriate potential equationLagrangian Analysiscon.,i>tent \\-it11 the boundary conditions ofthe partially full bending tank.
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