H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 76
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Hence, for a shell frequencyhigher than those of the first several uncoupledsloshing modes, it appears that the liquid freesurface effects on the shell breathing motion arenegligible.Computations were performed for the m = l ,n=4, and m=2, n=4 modes for a shell withINTERACTION BETWEEN LIQUID PROPELLANTS AND TRE ELASTIC STRUCTUREh/a=0.00606, and all =O.
1609, for several liquiddepths, and compared in reference 9.24 withexperimental results from figure 9.18. Theresults are shown in figures 9.24 through 9.26,along with the results of additional computations performed in reference 9.25, in whichdistortion of the mode shape was accounted for.For these cases, the terms due to free surfaceeffects were negligible.The ratio of natural frequency for partialliquid depth to empty tank frequency agreeswell with experimental data only for the m = l ,n=4 mode when using the undistorted modeshape approximation. The results from reference 9.25 were obtained by including mode shapedistortion, using a Galerkin procedure (10 x 10matrix), and it was found that the second-moderesults then agreed better t.han the first-moderesults, as can be seen in figures 9.24 and 9.25.Since the empty tank frequency was a criticalfactor in the frequency ratio as given, the absolute frequencies were compared in figure 9.26.I t can be seen that the agreement betweentheory and experiment is good except for theempty tank frequencies.
I t was suspected thatthis discrepancy was the result of nonuniformities in the shell.It must be emphasized that the above resultsindicate that the effects of the free surface are331-Theory w a c o s ( n 8 ) sin(mm.2, n - 4Theory l o x 10 matrixm-2, n - 4Experiment (Ref.
9.9 )Am-2, n - 40I0I0.20a400.600.80Liquid height length ratio, bl.4-1.0FIGURE9.25.-Comparieon of calculated and measurednatural frequency ratio versus liquid depth, m=2, n=4(ref. 9.25).+EY20.VIm-1, n - 4Theory l o x 10 matrixm-1, n - 4Experiment I Ref. 9.9 )Am-1, n - 4--oama60amLiquid height length ratio, MLax,-1.00000.200.40Liquid heightFIGURE 9.24.-Comparimn of calculated and measurednatural frequency ratio versus liquid depth, m= 1, n=4(ref. 9.25).0.60- length ratio.0.80bll1.0FIGURE9.26.-Comparimn of calculated and measurednatural frequency versus liquid depth (ref. 9.25).i332THE DYNAMIC BEHAVIOR OF LIQUIDSnegligible only for the case where the shellfrequency is higher than the first severaluncoupled liquid surface frequencies. No investigation was made to determine the effectsfor a case where these conditions did not hold,although it appears that the computations couldbe based on the analytical results of reference9.24, assilming that it is physically possible tohave a cylindrical shell of some geometry thatpossessed breathing frequencies in a low rangenear the corresponding sloshing freq rlencies forthat geometry.Breathing vibrations, of a shell with free surface effects included, hnve been tlnalyzed byseveral additional investigators; however, onlya brief comment on these yt udies will be itll.ludedhere.Fontenot and Lianis (ref.
8.27) have investigated the case of u full, pressurized cylindricalshell using a perturbntion technique. Rabinovich (ref. 9.28) has analyzed the partially fullcylindrical shell, using the Vlusov shell eqilntions arid a Lagrnnge-Ctit~ch intecral todetermine the hydrud?-n:inlic preisrlre o n theshell. The natural frerl~~cnciesund nt~trlrnlmodes of the shell are obtained by expressingthe modes as linenr combinations of the ernptytank modes, similar to the methods used inseveral of the articles previously described.Computed results for n frrll tank compare wellwith experimental data of Lindholm et nl.
(ref.9.0). Natushkin and Rakhinlov (ref. 9.29)have investigated the partially filled cylindricaltank with arbitrary encl conditions, whileSamoilov and Pavlov (ref. 9.30) have investigated ti liquid-filled hemispherical shell. Thisliquid-filled shell configuration has also beeninvestigated by Hwang (ref. 9.31). I t wasfound that only very wenk coupling existsbetween liquid silrface and shell breathingmodes, similar to the findings for the case of acylindrical tank. Shmakov (ref. 9.32) has investigated axially symmetric vibrations of acylindrical shell containing liquid, includingfree surface effects.
Again, these modes corresporld to breathing modes, such that the tankradial displacement is independent of the angular position a t all levels (n =0). As has alreadybeen mentioned, these modes are of most concern for longitudinal excitation. I t is foundthat the lowest shell mode suffers axial distortion for various partial liquid depths, similar tothat observed for the regular breathing modesof the partially filled shell, already described.One final reference should be included inthis discussion of cylindrical shell breathingvibrations including free surface effects. Ben1et al.
(ref. 9.:33), utilizing a numerical analysis,also have investigated the coupled tank b ~ l g i n g(n =0)-liquid pressure modes. This analysisbecame t~vailnble only shortly before thismonograph wns printed. I t ~ v a sfound thatthe coupled modes separated into two groups:the coupled liqliid surface modes and theco~lpledbulging or pressllre modes.
For pracBtical ra~iges of l a ~ i k parameters, the liquidsurface ruodes were foulid to be essentiallya t the same frequency as in a rigid tank.Further, the symmetric bulging tank modesoccur primarily as an eschange of kineticenergy of the fluid (i.e., all the inertia of thesystem is in the fluid), arid the strain energyof the shell (i.e., a11 the elasticity of the systemis in the shell). The influence of shell bendingstiffness was found to be negligible, and thatof shell inertia secondary, in tanks of engineering interest.Nonlinear EffectsGeneralThe preceding sections of this chapter havebeen devoted to a discussion of various lineareffects of fluids on the breathing vibrationsof cylindrical shells.
Most of the analysesare concerned with the natural free vibrationalfrequencies of the shell and liquid, and smallvibration theory is assumed throughout, forliquid free surface oscillations, pressure oscillations, and shell wall oscillations. I t follow-s,then, to question a t what physical amplitudes,if any, of the various oscillations in the liquidshell system, does the linear theory fail todescribe adequately the overall behavior ofthe system? Further, are the frequenciesinfluenced by the excitation in the case offorced vibration? An experimental investigation along these lines has been performed,and some of the results are summarized inthis section.INTERACTION BETWEEN LIQUID PROPELLANTS AND THE 'ELASTIC STRUCTURE333Kana et al.
(refs. 9.34 and 9.35) have experimentally investigated some of the nonlinearbreathing behavior of several elastic shellscontaining liquids with free surfaces underforced vibrational excitation. Cylindrical shellsof about 0.254-millimeter wall thickness, 22.9centimeter length, and about 3.8 1-centimetermean radius were used with water for most ofthe investigations.
In particular, liquid surfaceresponses and shell wall responses were observedfor both simply supported and cantilevershells subjected to lateral or longitudinalexcitation. Responses were studied for variousliquid depths.Generally, it was concluded that laterallyexcited empty shells display an essentiallylinear response during breathing vibration upto amplitudes as large as one-half the wallthickness for the shells investigated. A typicalempty cylindrical shell response is shown infigure 9.27. Full, simply supported shellsalso display a response that is linear, for the0. 180.020.1600. 14595600605610615620Excitation frequency (cps)- 0.12FIGURE9.28.-Experimental forced response of full simply supported shell (ref.
9.34).EEalux5 0.10most part, up to the same amplitude limit, ascan be seen from figure 9.28 for the same tankand mode. Some slight nonlinear softeningis apparent at the larger amplitudes; a similarresult occurs for a partial liquid depth wherethe liquid surface is in the vicinity of an axial,,A,IlvUn iii the iiiodd patter11 of ihe breathinga5mn3 0.080-arv,0.06shell.0.040.02O lGO14351440 51 4 ~ 14;)Excitation frequency (cps)FIGURE9.27.-Experimental forced response of emptysimply supported shell (ref.
9.34).If the liquid surface is not in the vicinityof an axial node, and particularly if it is in thevicinity of an axial antinode of the shell motion,then a marked nonlinear softening responseoccurs for the shell, as can be seen from figure9.29. Here, the response is linear up to onlyabout one-tenth the wall thickness, and isstrongly nonlinear softening for larger amplitudes. The region marked "I" on the figure334THE DYNAMIC BEHAVIOR OF LIQUIDS+845850855860Excitation frequency (cps)865FIGURE9.29.-Measured forced reeponee of partially fullrimply supported shell (ref. 9.34).is the region of instability of a typical nonlinear softening response, where jump phenomena occur, as indicated by the dashed lines.Region I1 is a second region of instability thatoccurs in this case, and is not a t all typical of anonlinear softening response. The behaviorof the system in this region will be discussedsubsequently.
I t is obvious that "small" vibration for the partially full tank in this caserefers to amplitudes of less than at most onetenth wall thickness. Experimental naturalfrequency data for partial liquid depths, discussed earlier and shown in figure 9.18, weretaken at such small shell amplitudes. Thenumbers on each of the response curves infigures 9.27 through 9.29 give only a qualitative indication of the relative forcing amplitudefor that respective curve.Liquid free surface response in the form ofhigh-frequency ripples is most pronounced forthe case where the mean surface level is a t anantinode of the shell motion. In fact, theexcitation of the high-frequency free surfacemotion may be the source of the nonlinearityof the shell response.
The liquid has a tendency to pile up at the shell wall in the vicinityof the antinodes of the shell motion. For verylow shell amplitudes, with the liquid surfacenear an axial antinode, or for even larger shellamplitudes, with the liquid surface near anaxial node, very little free surface motion isapparent. I t appears, then, that the wallmotion a t the liquid surface is most importantin determining the free surface motion.Similar responses were observed for cantilevershells subjected to either lateral or longitudinalexcitation at various liquid levels.
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