H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 64
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But a is proportionalto xo/d, where d is the significant tank dimension.' Consequently, the width of the unstablesubharmonic region varies as zo/d, the width of)~,unstable harmonic region varies as ( ~ ~ / dthewidth of the 31.2-superharmonic region varies asFor a circular cylindrical tank, d is the tank diameter; for a thin rectangular tank, d is the tank length.Also, it should be mentioned that "amplitude cutoff"does not imply here that a certain minimum responseamplitude is required for the assumed motions to exist,ae i t does in some other kinds of nonlinear dampedsystems. Instead, once the required minimum ezcilalion amplitude is obtained, the response amplitudecan be arbitrarily near to zero.@285VERTICAL EXCITATION OF PROPELLANT TANKS( ~ ~ / dand) ~ so, on.
Hence, by increasing xo ordecreasing d, any of the nnstable motions mayeventually be obtained.Harmonic and 312-superharmonic responseswere observed by Dodge, Kana, and Bbramson(ref. 8.20) in laboratory tests conducted in asmall cylindrical tank. The theory of thesemotions was not reported in detail, but it canbe developed using the methods given earlierin this section by letting N assume appropriatevalues. For example, the results for harmonicmotion in the m = l , n = l mode, which areanalogous to equations (8.33), (8.33), and(8.34), are+Pol Xol tanh Xold(1- a2e cos UT)Pol-~11811(0.121482~oltanh Xolb-0.263074X:l)f &[Xol tanh Xolb(0.070796X':l-0.060741) +0.263074X:l]=O(8.39)+XZ1 tanh Xqlb(1-a2ecos u7)821+8:1[~21t,anh X21b(0.175403-O.O65931X:,) -0.482670X:l]=0tude harmonic and superharmonic responses arepossible for excitation frequency-amplitudecombinations that lie completely outside any ofthe unstable regions shown in figure 8.4.
Thesemotions do not seem to be a t all the same typeas the responses discussed previously.Figure 8.15 shows an experimental curve forthe m=1, n = l harmonic mode in a 14.5centimeter-diameter cylindrical tank. Thiscurve should be compared with the 112-subharmonic response curve for the same mode givenin figure 5.11. The amplitude of the subhnrmonic response is much larger, hut even so, theharmonic response is large enough to be easilyobserved visually. The main point of difference, however, is that the harmonic response(fig.
5.15) gives the appearance of the harmonicresponse of a slightly nonlinear, damped systemin the vicinity of a resonance, while the 112subharnionic response (fig. 8.11) does not. I nfact, the phase angle of the harmonic responserelative to the tank motion shifts slowly fro111zero, for frequencies less than peak response-, to90' a t peak response, to essentially 180' forexcitation frequencies greater than peak response, esactly as would a simple spring-masssystem.This kind of harmonic resppnse occu1.s in thevieinitmyof what is normally an unstable regionfor harmonic rnotions (if the forcing amplitudeis sufficiently large), but actually it is not anunstable motion; that is, the amplitude does not(8.40)T h e approxinlate third-order solution to theseequations is of the same form as equation (8.35) ;but,, of course, in t,his case all of the A,, (and A)have numerical values different from before.The other harmonic modes and the superharmonic motions may be analyzed in a similarfashion.
Very little experimental data existfor comparison purposes, probably bectiiisemotions of this type are not nearly so predoniinant i n . large tanks as would be the %-subharmonic response.Small Amplitude, Stable Harmonic, andSuperharmonic MotionsIt has been obser\-ed (ref. 8-20] that even inrelatively large laboratory tanks, small ampli-1.25LOOAg.-a75g- a 50Sd(1250%63.73.93.840-Tank frequency cpsFIGURE8.15.-Experimental response for m=l, n = l modeharmonic (ref.
8.20).286THE DYNAMIC BEHAVIOR OF LIQUIDSexhibit a sudden increase when the frequency ischanged slightly. Consequently, this responsedoes not correspond to an nnstable solution ofl h t h i e u ' s equation. On the other hand, it doesnot correspond to n stable solution, either, sincethe theoretical period of any possible stablesolution is not exactly equal to the period ofthe forcing motion except for n few discretevalues of u and c. Small amplitude responses,however, h a r e been partially analyzed by Bhutaand Yeh (ref.
8.21) in their treatment of tankswith n vibrating bottom. The essential ddifference between their analysis and the one presented in the preceding sections of this chapteris that they sat'isfied the boundary conditionson the undisturbed positions of the boundariesin a fixed frame of reference. That is, theirboundary conditions for n cylindrical tank areat the free surface, z=0;&--xow sin wta2tit the tank bot,tom, z= - I t .Hence, this is alinenrized theory, but even if it were not, theirresults are valid only for small motions in anabsolute sense (because of equation (8.43)),whereas in the moving coordinate system employed in the linearized snbharmonic analysispresented preriously, the motion of the tankitself can be finite ns long as the relative motionof the liquid is small.The details of the analysis presented in reference 8.21 nre lengthy and so are not givenIiere, hut the main result is that the marelieight relative to the tank bottom isq(r, t ) = x o cos wt--anCJO(~O*T)n 1 u2- winwhere won is the natural frequency of the nthsymmetrical mode, the only type of sloshing-Tank frequency cpsFIGURE8.16.-Experimental response for m = l , n = l mode2-superharmonic (ref.
8.20).considered. Equation (8.44) is of the correctform to predict the observed harmonic response,but the method of calculating an, according toreference 8.21, seems to depend upon specificinitial conditions and not upon the excitation.Thus, i t is difficult t o compare equation (8.44)directly with experimental results.Bhuta and Koval (refs.
8.23 and 8.24) haveshown analytically the existence of harmonicliquid responses in a series of papers concernedwith tanks having an elastic bottom (but rigidsides). This kind of vibration, of course,excites the liquid in a vertical direction. Theyhave found that the natural frequencies of thefree surface motions are slightly different fromthe case of a rigid bottom. Some details oftheir analyses are given in this monogrt~phinchapter 9, where the internction of the liquidwith the elastic tnnk is considered more fully.I n addition to harmonic responses, smallamplitude superharmonic responses also havebeen observed in laboratory experiments (ref.8.20). I n figure 8.16, a response curve for them = l , n = l double-superharmonic mode in a14.5-centime ter-diame t er cylindrical tank is illustrated.
The peak response for this case isconsiderably sharper than the hurmonic response curve shown in figure 8.15. Once again,this response is not an unstable motion, nordoes it correspond to a stable solution ofMathieu's equation-asa point of fact, nosuitable analysis of this sort of superharmonicmotion has yet been discovered. It appearstb!IiVERTICAL EXCITATION OF PROPELLANT TANKSthat considerably more theoretical work isneeded to put the small amplitude harmonicand superharmonic responses into a satisfactorystate of agreement with experiments, and todetermine their significance in possible applications.Mechanical Model for Vertical Vibrations-4s a complement to the equivalent mechanical model analogies given in chapter 6, itwould be convenient in missile stability analysesto have a similar type of mechnnical model forvertical sloshing.
Developing an equivalentmodel for vertical excitation, however, isconsiderably more difficult than for transverseexcitation because of the tremendous numberof possible liquid motions; e.g., subharmonic,stable and unstable harmonic, stable and unstable superharmonic. But the 112-subharmonicresponse would probablj- be the most importantin applications, so that any equivalent modelshould be directed primarily toward duplicatingthis kind of response.Even for the limited problem of subharmonicresponse, there are no available t~nalyses ofsurfacecos 2wtFIGURE8.17.-Equivalentmechanical model for antieymmetric vertical eloshing.287mechanical models for vertical excitation. -4hypothetical model for the first antisymmetricmode, however, is shown in figure 8.17. Thefixed mass, mo, is supposed to duplicate theinertia effects of the more or less rigid bodyportion of the fluid motion, and the pendulummass, ml, is supposed to duplicate the sloshingmass of the liquid.
The equation of motionfor the penduluik, correct to-the third order inthe pendulum amplitude, 8, isThis equation is approximately of the sameform as the nonlinear equation specifying thedominant amplitude for antisymmetrical vertical sloshing. In fact, it has the same region ofunstable motion, and, consequently, the modelshould duplicate fairly well the actual liquiddynamics, especially if the parameters such as11,m,, and mo are derived from experimentalresults. However, the problem of determiningthe plane of the pendulum's motion remainsto be solved.
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