H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 12
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The experimental resultsobtained in references 2.51 and 2.52 for eachcontainer orientation are discussed below, andthe frequency parameters investigated for eachorientation are summarized in table 2.4.Horizontal Exciw'onFOEthis orientation, the liquid natural freqnencieq fnr a n e?hittra_~1;qi~gdapth m ~ &yapproximated by analogy with a ring tank inthe formwhere E, are the roots of equation (2.18d) ifm/2a is set equaI to unity and k is replaced byrt/~o,he is the height of liquid in a ring tankrequired to produce a volume of liquid equalto that contained in the toroid, and r, and r, arethe inner and outer radii, respectively, of theliquid surface.
Figure 2.36 indicates the generally good agreement obtained between frequencies calculated from this equation andthose obtained experirnentally. The liquidsurface forms of the contained liquid for thefirst and second frequencies are also shown inthe fi8;ure. In general, it is noted that(a) A decrease in the liquid depth ratio results in a decrease of both the first andsecond natural frequencies.(b) For constant minor radius, an increasein the major radius results in a demasein both the first and second naturalfrequencies.The problem of determining the fluid forcesin terms of tank geometry and liquid depth hasnot been resolved.
I t has been noted, however, from the typical data of figure 2.37, that.:56THE DYNAMIC BEHAVIOR OF LIQUIDSTABLE2.4.-Summwy of Nondimensionul Fmqumcy Parame~ersfor Liquids in T o m W Tanks.RegionOrientation[Ref. 2.511ModeAll -----Horizontal- - - - - - - - - All -,----1&=-$Vertical transverse- -'*=q;AnalogyParameter(5roB -------- AII- - - - - -Annular circulsrcylinderSphere of radius rcircular cylinder ofradius rfi=w,,&First-----A, C----All n>l--ql-ui,/!EdFSimple pendulumVertical longitudinalFirst,---,,/:&Circular-arc tubeel=ulB - - -,- - - Alln>l--Sphere of radiusR+rR +r*n--&Circular cylinder ofradius rqn=a$x0 en-1For convenience, values of en are listed as follows: el= 1.841 ;c,=5.331 ;a=8.536; e,= 11.706; h,experimentallydetermined natural frequency.the maxlmum slosh force corresponds not tothe lowest liquid mode, but rather-to the secondmode, except possibly for small values of theratio of major to minor tank radii.
The maximum slosh force in the first mode appears tooccur a t a liquid-depth ratio of about threequarters, while in the second mode it appearsto occur at a liquiddepth ratio of aboutone-half.Transverse ExcitationLongitudinal Excitationof the container, theFor thisliquid frequency parameter can be defined variously for each region. I n regions A and C,we have (ref. 2.51)for the first frequency(2.63a)andFor this orientation of the container, and inall three regions, A, B, C, the frequency parameter(where r is the minor radiusC T , , = ~ . ~of the toroid, fig.
2.35) can be obtained fromthe experimental results shown in figure 2.38.No d ~ t aare available for liquid forces due tosloshing in this orientation.for higher frequenciesIn region B,the relations areg sin B.(2.63b)for the first frequency(2.63~)LATERAL SLOSHXNG I N MOVING CONTAINERSand+S=W~dn- gfor higher frequencies€8-1+,The values forin the foregoing equationsmay be obtained from the experimental resultss h o p in figure 2.39. It should be noted thatthe approximations(region A) = 1.0 and#n (region B) = 1.0 are reasonable. I t appearsthat: in region A, frequencies higher than thesecond are strongly dependent upon the ratio ofmajor to minor radius; in region B, all frequencies are independent of the ratio of radii;in region C, all frequencies are dependent onthe ratio of the radii and the tank size.
Theeffocts of tank size are, however, isolated.2.6 CONICAL TANKSThe natural frequencies of liquids in conicalcontainers have been investigated from a num--ber of viewpoints (refs. 2.12, 2.14, 2.22, 2.24,2.43, and 2.53).A variational procedure was employed byLawrence et al. (ref. 2.22) to obtain the naturalfrequencies of fluids in tanks ranging in geometry from shallow to deep. Plots of frequencyversus depth for the cases of liquid modes withone and three nodal diameters (s= 1, 3) aregiven in figure 2.40.Frequencies were determined experimentally(ref. 2.43) for cones of very small semivertexangles narrowing both upward and downward.An empirical relationship for frequency determined from this data iswhere C3 is plotted in figure 2.41.
I t is notedthat small angles correspond to large values ofho/r,, and thus it is difficult to obtain good correlation with the results presented in figure-Excitation frequency parameter 1 ahc-2.33.-Effectof kinematic +ity57on aloeh-force parameter in a spherical tank (ref. 2.50).58THE DYNAMIC BEHAVIOR OF LIQUID62.40. Equation (2.64) is valid for h/ro>2.75and for liquid oscillation amplitudes (measuredat the wall) of less than about 0.01 ro. Ananalytical formulation of the problem of sloshing in a cone of small semivertex angle narrowing both upward and downward were latergiven in reference 2.24, along with comparisonsbetween the solution of reference 2.22 and theexperimental results given in reference 2.43.Abramson and Ransleben (ref.
2.12) alsoobtained experimental results for liquidsloshing in cylindrical tanks having conicalbottoms. These data were found to be in goodagreement with the theoretical force responsecalculated by assuming the tank to be an"equivalent" flabbottomed cylindrical tank.The equivalence, based upon equal liquidvolumes, appears to be valid down to liquiddepths of about ho/R=0.50, and a t least throughthe second mode (and possibly the third).Good agreement was also obtained for naturalfrequencies, supporting similar conclusions thathad been arrived a t analytically (ref. 2.22).9.7OBLATE SPHEROIDAL TANKThe natural frequencies of liquids in oblatespheroidal tanks have been investigated bothexperimentally (ref.
2.45) and theoretically(ref. 2.25). Empirical equations and experimental results obtained in the investigation ofreference 2.45 will be reviewed in this section,including some of the effects of tank orientation(fig.2.42), ellipticity and size, and fluid height.48.IIIiIIII-TankTank4 4 -Calculated Experimental minor radius, major radius,R, cmr, cmRegionBdid not affect the frequencyC-CLongitudinal excitationCITransverse excitation"Horizontal excitation0.1 0.20.30.4 0.5 0.6 0.7 0.8Liquid-depth ratio, h 12r0.9LOFIGURE 2.35.4k.tCh ehowing orientation oftoroidd tankawl dimendona of teat codigurationa (ref. 2.51).Equivalent annularcylinderHorizontal tordd------It should be noted that the frequency parametars selected am not unique; several parameterswere investigated in reference 2.45, and thoseyielding the best results were ~ l e c t e dfor presentation.
As was the case with toroidal containers discussed in section 2.5, the parametersinvestigated were based upon fluid behavior inanalogous containers and, therefore, the development of a valid frequency parameterindependent of tank size and geometry isdependent upon the accuracy of the analogy.Recently, the natural frequencies and modeshapes of liquids oscillating in horizontallyorianted (see fig. 2.42) oblate spheroidal tankshave been calculated using variational techniques (ref. 2-25). The fluid forces and moments due to the liquid oscillating in one of itsnatural modes am given along with an equivalent mechanical model.
(See ch. 6.) Numerical results obtained for the fundamentalfrequency in the special case of a spherical tankhave already been given in section 2.4 (eq.(2.60)).Direction of excitation-Liquid surface form for excitatiohfrequencyat or near first and second natural modesFIGURE2.36.--Calculated and experimental valuea of theh t two natural frequarciee for horizontal orientationof tomid (after ref. 2.52).60THE DYNAMIC B E ~ V I O ROF LIQUIDS00TankTankradiusR, cmradiusr, cm9.536.350.
40.81.21.620Excitation frequency,28243.23.6cpsFIGURE2.37.-First- and eecand-mode slosh forces for horizontal orientation of ioroid at liquid depth ratio of 0.5 (afterref. 2.52).Horizontal ExcitationTheoretically calculated (ref. 2.25) naturalfrequencies for three dserent tank geometriesin the horizontal orientation, as a function ofliquid depth, are shown in figure 2.43.The experimental study (ref. 2.45) showedthat an empirical equation for the liquid naturalfrequencies of the formwhere he=- h(3b-h) and en is the nth root of3 2b-hJ:(c,J=O, gave quite good results (the poorestresults occurred for the nearly full tank in thelowest liquid mode).
The basis of this equationis that of the natural frequencies of liquidcontained in an upright cylinder having a radiusr equal to that of the liquid surface and avolume equal to the volume of liquid containedin the spheroid.A comparison of these two sets of frequencydata, for the three lowest liquid modes, is madein figure 2.44.Longitudinal and Transverse ExcitationsEmpirical equations for the liquid modes incontainers with longitudinal and transverseorientations were' developed (ref. 2.45) by analogy with liquid frequencies in elliptic cylindersin theLongitudinal:~ , , , , = d ~ k , , ~ t a n h ( $ k , , , , ) (2.65b)Transverse :where r is the radius of the free surface (fig.2.42) andh4a-hhe=a+sh)(tThe parameters kt,,and k,,,,are proportional tothe positive parametric zeros of the firstderivatives of the Mathieu function (thesefunctions appear in the analysis of the ellipticcylinder (ref.
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