H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 70
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The resultingI n Miles' analysis (ref. 9.5) of the coupledcoefficients appropriate for coupled bendingliqnid-bending tank system, the equations ofoscilltttions are summarized in the appendix tomotion for the circular cylindrical tank arethis chapter, and examples for a cantilever tankdetermined by tt Lagrangian formulation, takingare given.1the kinetic energy in the form T=- 7,F;mi,yiijHaving determined the kinetic and potential2 1 jenergyexpressions, these are then substituted1and the pnt,~nt,inlP n e r E RS V=F 7 7.k.r(~.~.2 7 7 """ into Lagrange's equations so t ' n a ~ for, bendingand liquid free surface motion only,the following\\*here the *, are generalbedcorreequations are obtained:sponding to translation, rotation, bending, andsloshing motion of the coupled system. For thempurposes of this discussion, only bending(E$+%~.3 ) 4 3 + x E ~s+3(is+3,1motions of the tank will be discussed, so thatthe coordinates that are of concern are shown( E ~ ~3)93+5+ L1 r 3 .
.+dlr+3=0 (9.18)in figure 9.1. I n effect, f(z) is the mode shapeof tile+306THE DYNAMIC BEHAVIOR OF L I Q ~ D SThe frequency equation obtained from equations(9.1), by assuming periodic motion, isThe subscripts 3 and s+3 correspond to thegeneralized displacements, q3(t)f(z) denotingbending displacement and ~ # + ~ ( t ) +e)(denotingr,displacement in the sth sloshing mode of theliquid free surface from a plane normal to thegenerator of the cylinder. (See fig.
9.1.) Inequations (9.1) and (9.2), the following notationis used:03=4iii@:+h3Zc+%3.3-+=%+3.8+3Q3k3.*+3s =m 3, r + 3~I n general, it may be anticipated that a coupledfrequency (root of equation (9.2)) will occurfor each uncoupled bending and liquid sloshingmode included, and the coupled frequencieswill be d i e r e n t from those of the correspondinguncoupled modes. Further discussion of thisequat,ion will be deferred momentarily.An approximate solution to equation (9.2) hasbeen investigated in reference 9.6, for the caseof small, thin circular cylindrical shells, bothwith simply supported and cantilever boundaryconditions.
For the case of the small uniformcylindrical shell, the liquid frequencies are stillcomparatively low, while the coupled bendingfrequency is high; therefore, a considerablysimplified expression for equation (9.2) may beobtained. Havingand Q3, small compared to w, they may be neglected, and equation (9.2) reduces touncoupled bending frequencyuncoupled sloshing frequencieScoupling coefficientsm eand w, are the effective empty tank mass andresonant bending frequency, respectively. Thebars over the coefficients indicate normalizationby the total liquid mnss.The frequency equation, equation (9.2), hasnn infinite number of roots w=w,, the coupledtank bending, liquid sloshing frequencies.
Ashas been mentioned, for an actual vehicle thelo\vest coupled bending frequencies (i.e., w, andw2, corresponding to the first two roots ofequation (9.2)) occur in the neighborhood ofsome of the uncoupled sloshing modes w,+3,\vhich display significant response amplitudes.As n result, in a specific case, equation (9.2)must be solved, using some judgment as to\\-hat terms of the series may be neglected asinsignificant, and at best only numerical solutions to the resulting equation can be obtained.For this case, the difference between the coupledand uncoupled bending frequencies is thusdetermined by the coefficient art,.
The un#+8, andcoupled inertia coefficients, ? f i b E,+3.m , are always positive. The inertia couplingcoefficient Z , may be either positive, negative, or zero, depending upon the ratio bla.When m3,a+3=0, it follows that a3,=0, causinga sharp minimum in coupling for certain tankconfigurations (the coupling need not actuallybecome zero because of the summation overs and the presence of the small potentialcoupling ,+,). This minimum in coupling isparticularly noted in the results for the cantilever tank near b/a=2. This situation isanalogous to the inertially coupled bending andtorsional vibrations of a beam, in which casethe uncoupling occurs when the shear centerof the beam coincides with the centroid. ForE3.i+3#0, t,he factor [l-&y3,]-l is alwaysgreater than unit,^, thus ~ieldingan increase inthe coupled bending frequency due to theliquid sloshing.Equation (9.3) is solved for the particularcases of a cantilever and in-ended tank for theliquid at varying partial depths b/l.
Figures9.2 and 9.3 show the results for these two types-z,307INTERACTION BETWEEN LIQUID PROPEX.JJANTSAND TETE ELASTIC STRUCTURE0002468Tank fineness ratio Nd1012,IiiI3>.'.*...-7, 4A!- ..; .-.,-.. -- J i:- .. :--**.=-- .'43-y- - ,...>2468Tank fineness ratio 4d1012FIGURE9.2.-TheoreticalFIGURE9.3.-Theoreticalof tanks, plotted as the percent increase inbending frequency due to coupling (u-w3)u3X 100 versus tank fineness ratio I/d for varyingpartial depths bll.
These curves may be appreciably afTected by the assumed bending modeshape, which in these analyses involved aparabolic mode shape for the cantilever tankand sinusoidal for the pin-ended tank.We now return to the discussion of equation(9.2), the more exact frequency equation. Thecurves of figures 9.2 and 9.3 are for the specialcase of a uniform shell where w>>u,+~. Inorder to determine the effect of coupling on theresonant frequencies of the system when anciw , + ~ are of the same order of magnitude, theexact frequency equation, equation (9.2), wassolved for a cantilever tank with a tip massadded in order to lower the bending resonance.Analytically, this involved only a slight modification of the empty tank inertia coefficient, Be,but required considerably more effort in solvingfor the eigenvalues of the frequency determinantof the set of equations (9.la) and (9.lb). Inthe analysis, six modes (one bending and fivesloshing) were used.The results of this analysis showed that theaddition of mass to the vibrating system,although lowering the resonant bending frequency, also reduced the effect of the sloshingmass on the response of the entire system.
Forthe cantilever tank, the addition of a tip massof 10 times the empty tank mass reduced themaximum frequency increase of approximately11 percent, as shown in figure 9.2, to less than1.5 percent. Further increases in tip massreduce the iniiuence of ihe iree yurfucu vuresonant bending frequency still further. Itwould appear, therefore, that consideration ofthe free surface boundary condition will only besignificant in those cases where the sloshingmass is an appreciable portion of the totalvibrating mass, as in the bending of relativelylow fineness ratio, uniform shells. For largemissile structures where the total bending masspercent increase in resonantbending frequency due to free surface effects in cantilever tanks hef.
9.6).i0percent increase in resonantbending frequency due to free surface effects in simplysupported tanka (ref. 9.6).308THE DYNAMIC BEHAVIOR OF LIQUIDSis large, the liquid-sloshing mass can probablybe neglected in the calculation of overall bending resonant frequencies without significanterror. The solution for all six roots of thecoupled frequency equations also showed thatthe coupled sloshing frequencies did not changeappreciably from their uncoupled values.
Thus,rigid-tank-sloshing frequencies appear to beadequate for use in analyzing the bending tankin a space vehicle.Experimental Verification of Lagrangian AnalysisIn order to correlat'e with the results predictedby t,he simplified frequency equation, as givenin the preceding section, experiments wereconducted by Lindholm et al. (ref. 9.6) in whichthe coupling effect on the bending frequencymas measured for thin cylindrical shells withcantilever and simply supported ends. Bending frequencies of several shells containingwater at various depths were measured, both0oa2a4with a free liquid surface and a capped liquidsurface.Data were obtained for two cantilever tankshaving l/d (length to diameter) ratios of 2.76and 5.03. Figure 9.4 shows the experimentaland theoretical capped liquid surface (uncoupled) bending frequencies for the tanks as afunction of the fractional depth of the liquid.The influence of the mode shape on the theoretical curves is readily apparent by thedifference in the two curves representing aliiesr mode shape and parabolic mode shape.I t can be seen that the actual data fell somewhere in between for both tanks tested.
Thisis readily explained from the fact that the actualmode shapes, which were obt.ained by measurements on both tanks, also fell between a linearand parabolic shape.The difference in capped and uncappedresonant frequencies for the same two cantilevertanks is plotted in figure 9.5 as the percentagefrequency increase against fractional depth.Here again, the data fell between that for thetwo assumed mode shapes. I t is obvious that00.60.81.0Fractional depth of liquid (blR)FIGURE9.4.-Theoretical and experimental capped rewnant bending frequency variation with liquid depth forcantilever tanks (ref.
9.6).o0.20.40.60.8Fractional depth of liquid Ibll)FIGURE9.5.-Theoretical1.0and experimental percent increase in reeonant bending frequency versus liquid depthin cantilever tanks (ref. 9.6).-'ItI-.\I,:1,IIIIi!ii11'.,.-*THE ELASTIC STRUCTUREINTERACTION BETWEEN LIQUID PROPELLANTSthe extreme sensitivity of the theory to modeshape requires the substitution of the actualmode shape into the theory in order to get thebest possible prediction of coupled frequencyand percent frequency change.Similar experiments were also performed oncircular cylindrical, simply supported tanks, theresults of which are shown in figures 9.6 and9.7. I t can be seen that the agreement between theory and experiment was not as goodas with the cantilever tanks. The deviationwas explained as probably resulting from a lackofepropersatisfaction of the boundary conditiona t the tank bottom, and a considerable distortion in mode shape which occurs a t partiallyfull liquid depths.
The theoretical curves werebased on a half-sine wave, but measurementsindicated that this shape approximated theactual shape only for a c ~ m p l ~ efulll y or emptytank.In general, it was concluded that for the casesstudied, Miles' theorg appeared to give anadequate prediction of the influence of the liquidmotion on the elastic bending frequency, butits use is complicated because of the strong30912210X;18mU.c 6.2=af-4alCz22o00.20.40.60.8Fractional depth of liquid (bU)1.0FIGURE9.7.-Theoreticaland experimental percent increase in resonant bending fretpency versus liquid depthins u ~ ~ o * etanksd(ref.
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