H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 46
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d(6.28)4 Terms due to the increase in static fluid pressurewith depth have been neglected in the equationsdeveloped in this chapter.THE DYNAMIC BEHAVIOR OF LIQUIDSvalue of h, for n = l and 2 is shown in figure6.13; it can be seen that for small liquid depths,the sloshing mass, ml1 is below the center ofgravity.The magnitude of I. can be determined bywriting down the equation for the pitchingmoment of t,he mechanical model. I t is0.08001324dhThe factor -gZmnyn arises because of theunbalanced gravit,y moments when the tank ismoved a distance, yo, and the masses displacedan additional yn a t the same time.
By substituting equations (6.21) into (6.29), themoment can be expressed asFIGURE6.12.-Ratio of modal masaee to liquid maes forM=-&,cylindrical tanks (ref. 6.29).For n = 1, these results are the same as those ofthe simple model described in section 6.2.The size of m, rapidly decreases for n > lbecause of the term tn(E2,-1) in the denominator of equation (6.27); this is shown graphically in figure 6.12 which shows m, and nz2 incompt~risonto mT, us ti function of hld. TheC~ ~ + m $ f , + f :mnh:The actual pitching moment caused by the sloshing (cf. ch. 2) is+mT5n-1;;htanh 2En - ( g / ~ ) ~ + g h1-dhwi[l -( ~ / w n ) ~ l€ n 2 (ti-1)[whereIi '+--212 ( d ) 6:tanh 6c 1-- tnhtt(fz-1).-In-1211ANALYTICAL REPRESENTATION OF LATERAL S L O S H I N GFIGURE6.14.bRatio of effective liquid to rigid liquidmoment of inertia for cylindrical tanks (ref.
6.26).hldFIGURE6.13.-Locationand magnitude of modal maseesfor cylindrical tanks (ref. 6.29).A direct comparison of equations (6.30) and(6.31) gives the same expressions for m, and h,as before, and in addition it shows that[AXow mTd2perimental proof of the model's validity willnecessarily be delayed until section 6.4, whendamping is introduced (refs. 6.25, 6.30, and6.33).
For convenience the formulae for thevarious parameters are collected in table 6.4.TABLE6.4.-ModelParameters for CylindricalTank[See ch. 2, table 2.4, for values of En; r n ~ = > & r p @ h ](h/d)~+$] in equation (6.32) isthe moment of inertia of a solidified mass ofalofliquid equal in weight to the a c t ~ ~weightliquid. Hence IF is the actual moment ofinertia of the inviscid fluid about its center ofgravity. A plot of this moment of inertia interms of the moment of inertia of the solidifiedliquid is shown in figure 6.14. As can be seen,the moment of inertia of the liquid can be quite(-))(tanh 21"iy) tanh 21. hd-fnh(~2-1)hn----2tanhfnEl -hd[Asmall in comparison to mTg(:y+i]rbecause not all of the liquid participates in apitching motion of the tank about-the liquidcenter of gravity; part of it is completely a trest, or nearly so.'The foregoing mechanical model is a nearlyexact duplication (it is exact for linear motionsof an inviscid fluid) of fuel sloshing in a cylindrical tank as long as the excitation frequencyis not almost equal to one of the natural frequencies of sloshing.
If this is the case, thendamping devices must be added to the modelso that the force and moment response do notbecome infinitely large for w-w,.Thus, ex-A limited number of results for other typesof cylindrical tanks, such as annular crosssection and quarter-sec tion geometries, areavailable in scattered references; see especiallyreferences 6.27 through 6.32.Rectangular Tank (refs. 6.1 4 and 6.1 3)The parameters of an equivalent mechanicalmodel for fuel sloshing in a rectangular tank212THE DYNAMIC BEHAVIOR OF LIQUIDSFree surface-f112 K,hnf/motion is supposed to be in the plane of thefigure, but a similar model is valid for motion a tright 'angles to this plane.
The formulae forthe various parameters are given in table 6.5and shown graphically in figures 6.16 through6.18.TABLE6.5.-Model+---[ m T = p ~ h = f l u i dmass per unit width](g)IK.=VI.~%=mTFIGURE6.15.-ComplexParaineters for RectangularTank(tanh (2n-I).8W(ra(2n-l)h;)taoh(2n-1)r;hmechanical models for rectangular tanks.L4Tank aspect ratio, hlwFIGURE6.16.-Ratio of sloshing masees to fuel mass versustank aspect ratio for rectangular tanks (ref. 6.13).may be determined in a manner similar to thatused for cylindricnl tanks. A model employingspring masses is shown in figure 6.15. TheTank aspect ratio, hlwFIGURE6.17.-Rationof location of sloshing masser toliquid depth versus tank aspect ratio for rectangulartanks (ref.
6.13).213ANALYTICAL REPRESENTATION O F LATERAL SLOSHING1.00.80.60.4a2000.5LoL520253.0Rayleigh-Ritz methods. Since some of theresulting equations are too lengthy to bereproduced in a convenient form, these resultsare given here only graphically; for example,figure 6.20 shows the variat'ion of liquid naturalfrequency, w,, in the form of t,he dimensionlessas functions of theparameter, A,=w:R/g,liquid depth ratio, hlL, and height-to-widthrat,io,a= LIR (a= 1.0 corresponds to a sphericalshape). Likewise, the size of the first twosloshing masses is shown in figure 6.21; themagnitude of the fixed muss is equal to m ~ -Zm,.Tank aspect ratio, hlwFIGURE6.18.-Ratioof effective liquid to rigid liquidmoment of inertia for rectangular tanke (ref. 6.13).EL--...-6.?3.-EyiSvz!ezt46mehrninal model for ellipsoidalr nAxisymmetrictank (ref.
6.34).A."Ellipsoidal Tanks (Ref. 6.34)0.6Sloshing in a rigid axisgmmetric ellipsoidaltank is represented by the mechanical-modelshown in figure 6.19. The magnitudes of thevarious model parameters were determined by00.40.81.21.620hlLFlow. 6.20.-Variation of slo~hiig frequencies withLiquid depth for ellipsoidal tanks (ref. 6.34).214TEE DYNAMIC BEHAVIOR OF LIQUIDSwhere a n is given in figure 6.22 for n= 1 and 2.The location of the fixed mass may be computedfromwhere t.he center of gravity location in referenceto the geometric center of the tank isExact methods of calculating the moment ofinertia of the liquid are extremely laborious, b u tan approximate method has been described inreference 6.34. The centroidal moment of inertiaof n rigid body having the same shape and massas the liquid isFIGURE6.21.-Variationof modal masees with liquiddepth for ellipeoidal tanks (ref.
6.34).-1.0044a8L2.l 620hlLFIGURE6.22.-Parameter a, for slosh mass location versueliquid depth for ellipsoidal tanka (ref. 6.34).The location of the oscillating masses, h,trbove the geometric center of the tank, can beobt uined fromh,=Lan-- 9a:(6.34)Since the fluid is not rigid, equation (6.37)overestimates the moment of inertia of thefluid. The proposed method to obtain theactual moment of inertia is to determine theratio of the actual liquid to rigid liquid momentof inertia for a cylindrical tank having anidentical fluid height and fluid mass, say fromfigure 6.14. Then this same ratio is assumed tohold for the ellipsoidal tank; that iswhere x is the ratio (a pure number) obtainedfrom figure 6.14.
Note that this approximatemethod introduces an error only in Io; thus, themoment due to pitching will not be greatlyinfluenced by the errors involved in the approximate procedure.Spherical TanksThere are no readily available treatments ofcomplete mechanical models for spherical tanks.The general study of reference 6.15 contains ananalysis of such a model, but i t contains noThis is not the method proposed in ref. 6.34; however, it is a valid method, since it leaves the center ofgravity of the liquid unchanged.ANALYTICAL REPRESENTATION OF LATERAL SLOSHINGFree surfacenhnKlfh----215I t should be remembered that all of thepreceding models are for ideal liquids with noenergy dissipation.
The models yield goodresults as long as the excitation frequency is nottoo close to any liquid resonance. Near resonance, the sloshing forces and moments arelimited solely by the energy dissipation so thatdamping mechanisms must necessarily beincluded in a valid model. This subject istreated in the next section.6.4 INTRODUCTION OF DAMPING EFFECTSFIGURE6.23.bEquivalent mechanical model with damping included.numerical values or charts giving the size of theinertial parameters. Of course, the mechanicalmodel for ellipsoidal tanks (ref.
6.34) for thecase of a height-to-width ratio, a = 1.0, corresponds to spherical tanks so these results maybe used, provided it is recalled that the totalmoment of inertia of the model is effectivelyzero for such a tank.Conical TanksThere are no analyses of complete models forconical tanks; the only available model is thesingle sloshing mass model of a 45" conical tankdescribed in section 6.2.Loil;aii Oaf. g.35) has i;raseiitad a S-----"I:-..-'GLlGLanalysis in which the equations of motion for atank of arbitrary shape are developed.
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