H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 77
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However,for longitudinal excitation, an additional peculiarity in the breathing vibration was observed. I t was found that the longitudinallyexcited cylindrical shell with liquid wouldreadily respond in breathing modes at frequencies that were subharmonic to the excitation, as well as at frequencies the same as thatof the excitation.
These subharmonic responses have yet to be explained, although theyare obviously the result of some unknownnonlinearity. (The subharmonic response oftanks undergoing verticalliquids in +dexcitation was discussed in detail in ch. 8.)Nonlinear CouplingI t has been pointed out earlier that when tankbreathing resonances occur at frequenciesconsiderably higher than those of the significantliquid-sloshing modes, one would not normallyconsider interaction between the two motionsto be very probable. However, where nonlinearity exists, experimental studies (refs.
9.34and 9.35) have shown that this reasoning doesnot prove to be correct.We pointed out in the paragraphs just preceding that a nonlinear shell response occursfor the forced vibration of a cylindrical shellcontaining a liquid to such a depth that thefree surface is near an axial antinode of theshell modal pattern, and some measured response data for this case were shown in figure9.29. This type of response is typical forvirtually all breathing modes, as long as theliquid depth is within the proper range, corresponding to the shell mode being investigated.Within the region indicated "11" in figure 9.29,a most peculiar coupling occurs between theINTERACTION BETWEEN LIQUID PROPELLANTS AND THJC ELASTIC STRUCTUREhigh-frequency shell motion, and lowv-frequencyliquid surface motion.As indicated in figure 9.29, region I1 iscomposed of a number of subregions, withineach of which some symmetric liquid freesurface mode, a t its respective low frequency,can be excited by the high-frequency breathingmotion of the shell.
The form of the firstthree symmetric liquid modes is shown infigure 9.30, and is based on experimentalmeasurements.As an example, within the region marked"mode 1," in the cylindrical shell described inthe previous section, the first symmetric liquidsurface mode is excited at about 5.1 cps bythe shell breathing motion occurring at, say,856 cps (for excitation frequency a t 856 cps),and about 0.025-millimeter shell amplitude.The liquid surface motion was observed insome cases to be as large as 1.5-centimeteramplitude. Simultaneously, the high-frequencyshell mution exhibits an amplitude modulationa t 5.1 cps. Once this type of coupling sta~rts,it appears to stabilize a t some amplitudecombination for the liquid and shell, andcontinues indefinitely as long as the excitationconditions are unaltered.
The coupling canbe excited by either transverse or longitudinalexcitation of the shell.Further experimental data for this behaviorare shown in figure 9.31, where oscilloscopephotographs of the shell mall and liquid surfaceresponses are compared for three differentexcitational conditions and three differentliquid modes. Only the envelope of the-0.8-0.40(L4Normalized radial lccationa8FIGURE9.30.-Experimental symmetric surface modes generated by breathing vibration (ref. 9.34).-335sweepShell (739 cps)Liquid (5.1 cps)Mode 1---4millisec(a)Shell (740 cps)Liquid (7.5 cps)Mode 2-415 0 millisec(b)4I5 0 millisec(c)FIGURE9.31.-Phaseof tank and liquid motions forcoupled vibration (ref.
9.34).modulated high-frequency shell motion can beseen, and the liquid surface response representsthe liquid motion at the center of the tank.These photographs correspond to responsedata obtained near the m = 2 , n = 5 mode in acantilever shell about three-fourths full ofliquid, conditions different from that represented by the response curve of figure 9.29.However, the response photographs are typicalof the coupling behavior, regardless of whichshell mode is excited, and for both cantileverm"Ufree!y acppcrtcd a d s . The liquid modeexcited depends on the excitational and, therefore, shell response conditions, as indicated bythe four subregions corresponding to the firstfour symmetric liquid modes.
I n the 7.62centime ter-diame ter cylinder studied, thesemodes occurred with largest liquid response a t5.1, 7.5, 9.4, and 11.7 cps, respectively.A theoretical analysis of the nonlinearcoupling described above has recently been336THE DYNAMIC BEHAVIOR OF LIQUIDScompleted by Chu and Kana (ref. 9.36).This report was only being written during thefinal preparation of this monograph; therefore,the details could not be presented here. Also,a somewhat similar coupling of liquid surfacemotions with a vibrating rectangular tank hasbeen described in section 8.3 of chapter 8.9.4 COUPLING OF PROPELLANT MOTION ANDELASTIC TANK BOTTOMGeneral DiscussionIn many respects, it appears plausible thatthe interaction between propellants and elasticbottom motion should display the same generalcharacteristics as that between the propellantsand the breathing vibrations of the tank walls.In fact, vibrations of the elastic bottom correspond to breathing vibrations of the tank.Hence, it might be conjectured that all of thegeneral discussion of section 9.3 concerning theinteractions of the propellants and the elasticwalls also apply to interactions of the propellant with an elastic bottom.
Of course,the geometry of the bottom, as well as thepresence of drain lines and manifolds attachedto an actual vehicle bottom, would considerablycomplicate the analysis of the coupled bottomvibrations.Apparently, only few investigations of elasticbottom behavior have been performed, andeven those are rather simple nature, in thatthey consider only a flat bottom in an otherwiserigid tank.
However, the results of thesestudies still give valuable qualitative indicationsof what behavior might be expected for themore practical, more complicated' geometricshapes and, therefore, we shall summarize thembriefly here.circular plate equation governs the bottommotion in the second.The fluid is considered incompressible andideal so that it is governed by Laplace's equation, while the boundary conditions a t the tankwalls are the same as for a completely rigid tank.At the fluid surface, the boundary condition isequation (9.7), while a t the tank bottom thenormal velocity of the liquid must equal that ofthe vibrating bottom.
In either the case ofthe thin membrane bottom or the thin flat platebottom, the solution for the natural vibrationalfrequencies of the coupled boundary valueproblem results in an infinite order determinantfor the coupled natural frequencies.For the membrane bottom (ref. 9.37), numerical solutions were obtained for the first oeveralsymmetric coupled frequencies by trunclttingthe determinant a t the eighth order. Theresulting modes might be called the first severalcoupled liquid surface modes. Depending onthe value of the membrane tension, the coupledmembrane modes would occur at higher frequencies and could not nornlnlly be ohtninetlfrom only an eighth-order trunca tic)11 of thefrequency determinant.
Figure 9.32 s h o t r s thegeneral behavior that is predicted by thenumerical results for the first tw-o symmetricalsloshing modes of the liquid surface a t a rathershallow liquid depth. The examples are forinWith Free SurkceBhuta and Koval have investigated the interaction between the liquid surface oscillations ina rigid cylindrical tank having a thin, flatmembrane bottom (ref. 9.37), and having a thin,flat plate bottom (ref. 9.38). The two analysesare virtually identical except, of course, that thethin circular membrane equationthebottom motion in the first case, while the thinCouplingT ,Tension i n membrane, dyneslcmFIGURE9 . 3 2 . - 1 ~ f i u ~of~ ~bottom~elasticity on surficemod-(ref.
9.37).337INTERACTION BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTURErigid tank, but the effect is rather small. I tcan further be seen that the coupling effectdiminishes for increasing liquid depth, andbecomes negligible for b/d>%. Finally, it maybe noted that an increase in liquid density tendsto raise the coupled sloshing frequencies veryslightly. I t must be remembered that in arigid tank the sloshing frequencies are independent of the liquid density.
Similar conclusions are found for the case of the thin, flatplate bottom in reference 9.38.water in a 2.54-meter-diameter tank. Additional numerical results are shown in tables 9.2and 9.3 in which the natural frequencies inradians per second are given for the first severalsymmetric modes, and comparison with severalexperimental values for water in a 7.62cen timeter-diameter tank is given in table 9.4.From the numerical examples for this case, itcan be concluded that the bottom elasticitytends to lower the liquid surface naturalfrequencies below their respective values in sTABLE9.2.-Natural Frequencies in a Rigid Tank With an Elastic Membrane Bottom[Ref. 9.371[ p ~ = 1.00 gm/cc](a) b/a=O.lONatural frequency, radlsec, forModenumbers, dynelcm-1.75X 10"8.75X 10"1.75X 1074 .
3 7 ~1078.75X 1071.75X 10'2.9385.4437.6323.1405.6587.7433.2075.6937.7573.2545.7137.7643.2715.7177.7683.2795.7187.7711 Rigid1- - - - - - - - - - - 2- - - - - - - - - - - -3------------tank3.2581.7267.770(b) b/a=0.25Natural frequency, radlsec, for T,dynelcm-Modenumber1.75X 10"8.75X 1 P1.75X 1074.37x 1078.75X 1074.1897.0428.79810.1284.5637.1268.80610.1294.6257.1348.80710.1294.6637.1398.80710.1294.6767.1408.80710.1291- - - - - - - - - - - - - - - 2- - - - - - - - - - - - - - - 3- - - - - - - - - - - - - - - 4-- - - - - - - - - - - - - - -lo81.31X4.6817.1418.80710.1291.75X 10'Rigid tank4.6837.1418.80810.1294.6897.1428.80810.129(c) b/a= 0.50---- -IModenumberNatural frequency, radlsec, for T, dynelcm-I8.75X 1 0 V .
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