H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 74
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Their approach to the tankfilled with one flliid is essentially the same asthat of Berry and Reissner (ref. 9.14), justdiscussed, except that tangential inertia termsare kept in the shell equations, so that twondditionnl sets of natural frequencies result foreach modal pattern of the shell. The lower setof these natural frequencies correspond to thoseobt nined by Berry and Reissner, using equation(9.10b) for the apparent mass factor; lowerfrequency resonances corresponding to thoseobtained for the case for which equation (9.10a)applies were not discussed.
For multiple,immiscible fluids, a method of superposition isindicated for the solution of certain modes;ho\ve\-er, no nppreciable nurllerical calculationshave been completed.Finally, n few comments must be added withrcgnrd to the fluid pressure modes considered inthe previous two analyses. Only longitudinaloscillational pressure distributions of tlhe typeq=&(r, 6, t) sin ~ U!IX321INTERACTION BETWEEN LIQUID PROPEIILANTS AND THE ELASTIC STRUCTUREwere considered, in which all multiple pressurewave patterns have zero amplitudes a t both endsof the liquid column.
I t must be emphasizedthat other coupled pressure modes, particularlyaxisymmetric modes which result in a maximumpressure amplitude at the tank bottom (as occurfor longitudinal tank excitation), can be excitedin the compressible fluid-elastic tank combination (or even in an incompressible fluid-elastictank system). Such axisymmetric pressuremodes can occur within the same frequencyrange as the nonsymmetric coupled tank breathing modes of the form shown above, dependingon the shell geometry and the properties of thefluid. Pressure modes of this type in a longitudinally excited cylindrical shell filled with acompressible fluid have been studied by Reissner(ref. 9.16).
The shell is considered strictly tobe a circumferential membrane, and a prescribed excitational pressure is assumed to actat t,he bottom of the fluid column. Thisanalysis shows that pressures a t the tank bottombecome very large a t a coupled longitudinalresonance of the fluid column.The very similar, and more practical, case ofa prescribed oscillatory acceleration at thebottom of an elastic cylindrical tank containinga liquid has already been discussed in chapter 8.Longitudinal pressure responses, of the type towhich we are now referring, can be approximated by equation (8.60) in chapter 8, and thefrequencies at which the resonances occur arethose values where this equation predicts infinitepressure. Thus, for a liquid in an elastic tank,the lowest set of the axisyrnmetric, longitudinalpressure modes, in which the radial pressuredistribution is essentially plane, can be approximated by water-hammer theory, as shown inchapter 8. This approximate analysis is aspecial case of that of Reissner's, mentionedahnvn, n_dit.
h~qcmns kss ~ ~ ? l _ r p _pitt ~!cro~rliquid levels. More accurate analyses of thesecoupled longitudinal modes have become available only recently, and w i l l be discussed brieflya t the end of the next two sections.incompressible Fluids (Free Surface EffectsNeglected)T h e hydrodynamic loading inertia and pressurization effects of fluids on the breathingshell can readily be analyzed using incompressible fluid theory, while ~leglectirlgthe freesurface effects by some suitable assumption.In this section, we discuss several methods bywhich this has been done, along with experimental confirmation of the analyses.Based on ideas presented by Reissner (ref.9.16)) and Berry and Reissner (ref. 9.14), whichhare beer1 discussed in the preceding section,Lindholm et al. (ref.
9.9) developed u frequencyequation for the completely liquid filled tank,similar to equation (9.9), using incompressibletheory for the fluid, in an unpressurizeti circular cylindrical shell. The method is essentially the same as that of Berry arid Reissner(ref. 9.14), except that the Laplace equation istaker1 as the goverr~i~lgequation for tlie liquid)and the radial deflection is taken with m 1 :>w=&,7n*2sin -cos no cos wt1and the pressure distribution is taken as(y)m ~ 2q= C~,,Lsincosn8coswtI t may be noted that this form of pressure distribution requires that the free surface remainessentially plane; hence, the free surface effectsare neglected. The liquid apparent mass isin which Arn=mua/l.
I t may be noted thatthis expression is the same as equation (9.10a)when k,4O (or when co-, oo ), a result anticipatedin reference 9.14 for heavy fluids. In this case.equation (9.9) becomesResults predicted by this eqnation for a thinshell filled with water, ulorlg with experimental.correlation, are shown in figure 9.17.
For comptirison purposes, it may be noted that theseresults were obtained from the same cylinderas that used for the empty shell data shown infigure 9.11. The theoretical curves in figure3221THE DYNAMIC BEWVIOR OF LIQUIDS1-+--h l a a00667L020Axial wave length factor X=,3.0m* a l lFIGURE9.17.-Theoretical and experimental natural frequencies of liquid-filled cylinder in breathing vibration (ref. 9-91.9.11 were obtained from equation (9.12) withm,=O.Further experimental work was performed inreference 9.9 to determine the liquid effects onfrequency at partial liquid depths, as shown infigure 9.18.
The data are for water in a freelysupported steel cylinder with a=3.77 centimeters, h=0.229 millimeter, and 1=23.4 centimeters. Further, figures 9.19 anil 9.20 showexperimental data showing the effect of liquiddepth on axial mode pattern for two differentmodes. Distortion of the axial pattern from ssine wavo is especially apparent at liquid depthsother than empty or full.Keeping in mind the above-indicsted distortion of axial mode pattern, Leroy (ref.
9.17)has presented an analysis for natural frequenciesof a partially liquid filled, unpressurized, cylindrical shell. The displacements of the distorted pattern are assumed to be representableas linear combinations of all'the natural modesfor the empty cylinder:x A,,, cosODu=cos neFractional depth of liquid (bU)m(9.13)FIGIJEE9.18.-Experimental variation of natural breathingfrequencies with liquid depth (ref.
9.9).Similar to Lindholm et al. (ref. 9.9), the liquidfree surface is assumed to remain essentiallyplane, but an expression for the compositehydrodynamic pressure loading of the shell wallis determined, having components in eachmode of the linear combinations (eq. (9.13)), andan arbitrary partial depth is allowed. Substituting these expressions into a suitable set ofcylindrical shell equations, an infinite order-.frequency determinant resuits. The naturaifrequencies can then be approximated numerically by assuming a limited number of modesin the composition of the actual distortedmodal patterns. Figure 9.21 shows a comparison of some calculated results of this theorywith the experimental results shown in figure9.18 for two different modes.One of the most complete investigations ofthe effects of both pressurization and liquidsa t partial depths in breathing cylindrical shellswith freely supported ends has been performedby Mixson and Herr (ref.
9.18). An analysiswas carried out, combining a number of themethods already here discussed with thatgiven by Baron and Bleich (ref. 9.19) forpartially filled cantilever shells, and then compared with experimental results. Basically, Lagrange's equations are utilized to obtain theequations of motion of the coupled system. Forthis approach, of course, the kinetic and potential energy of the combined system must be determined. The potential and kinetic energyof the shell is expressed in terms of displacements using a method similar to that used byArnold and Warburton (ref. 9.8), except thatadditional terms are included to allow forinternal pressurization.
Only radial inertiaterms for the cylinder are included, so thatonly the lowest set of natural frequencies isdetermined, and inertia for the pressurizinggas is neglected.ODv=sinn&xm324THE DYNAMIC BEHAVIOR OF LIQUIDS1.000. 75-0.50alE-)r:n0.25,a.u=.-0.--0.25HFractionaldepth of liquidrngZ-0.50bll' 0b l l ' 114bll-112bll-314bll- 1m-1-0.75Symbol--------.n-5-1.0000.250.500.75Normalized axial position, XIFIGURE9.19.-Axiclmode shape distortion for m = l , n=5 mode versus fractional depth of liquid (ref. 9.9).0.25.URE9.20.-Axial0.50Normalized axial position, x l l0.75mode shape distortion for m=2, n=5 mode versus fractional depth of liquid (ref.
9.9).1.00Anticipating axial mode shape distortions atpartial depths, the shell displacement functionsare chosen as linear combinations of a finitenumber of the modes of the empty shell in thefollowing form :2200-2000Experimental curve1800u=cosN74e1600Nv=sin no;1400UnNw=cos n$$1200a2af10008006004002000FIG-00.250.50Depth ratio bK0.751.09.21.--Comparison of Leroy theory with experiments for breathing modes (ref. 9.17).The kinetic and potential energies for thearbitrary depth column of liquid are determined using incompressible potential flowtheory, employing the boundary conditions a tthe shell wall, as have already been discussed,but the free surface effects are neglected in adifferent manner than that of Lindholm et al.(ref.
9.9). Here the boundary condition at thefree surface employs equation (9.5), but g=O isset into this equation. As in reference 9.19, it istti-gued that the signscant liquid free surfacemodes occur a t very low frequencies comparedto those of the breathing tank.
Therefore,their effects can be neglected so that the freesurface is allowed to oscillate, but i t is assumedthat the potential energy associated with thefree surface motion is negligible. I n this case,the kinematic equation, equation (9.6), isstill valid, but it is not necessary in order todetermine the velocity potential, a, in theproblem.71cos -V,(t) sin -S-oLLC U,(t)a =OC W,(t) sin8-0at(9.14)For empty or completely full shells only, wherelittle distortion occurs, the first term of eachexpansion is satisfactory .
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