H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 61
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Figure 8.5 illustrates such a stability diagram for a 14.5-centimeter-diametercircular cylindrical tank model, with a liquidA1aepclle y u d to the tank d i a ~ c t e r this;tar& isessentially rigid for the low frequency excitationconsidered here and, thus, the preceding resultsare applicable.
Only a few of the unstable andstable regions corresponding to the mode oflowest natural frequency (smallest j(,)areshown. One obvious point of difference between figures 8.4 and 8.5 is that the instabilityregions are much narrower when plotted dimensionally. Since the effect of damping is to1TEE DYNAMIC BEHAWOR OF LIQTJIDSExcitation frequency- cpsFIGURE8.5.-Partial Mathieu stability chart for rn = 1, n = 1 mode.rnake these regions even narrower, one can seethnt the one-half subharmonic wonld probablybe the only large amplitude wave motionobserved in experiments.I n any given tank there are a large numberof modes, each having its own stability diagram.
Figure 8.6 shows some of the onehalf subharmonic instability regions, and thelocation of several harmonic modes, for thesame tnnk model as figure 8.5. From thischart, one can see that the unstable regions forvarious sloshing modes overlap one another; forexample, the first symmetrical mode (m=O,n = 1, in this chart) nnd the second and thirdantisymmetrical modes (m=2, n = l , nnd m=3,? I = 1) overlap considerably.I n fact, for almosttiny combination of escitation frequency andnmplitude, there is sonlo overlapping of unstable regions, nnd it is apparent that thisoverlnpping is even more congested nt higherfrequencies.
The low-order modes can be fairlywell isolated, however, so that they are con-venient modes to study both experimentallyand analytically. These modes are nlso themost important in practical applications.Large Amplitude Subharmonic MotionAs mentioned previously, the only sizableliquid motion that is usually observed in (Lvertically vibrated tank is the !:-subharmonicresponse. I n order to investigate this responsemore thoroughlj-, it is no\\- assumed that thetime dependence of the mth sloshing mode isadequately represented byn,=A sin wt+ B cos wtIf N = 2 in equntion (S.S), then t,he steady-statemot'ion given by this equntion does indeedrepresent n ji-subharmonic response.By substit.ut,ing the nbore equation intoequation (S.8), setting N = 2 , and collectingterms, one finds that----*------ .- ._--_____*_-I.I_I_L11VERTICAL EXCITATION OF PROPELLANT TANgSIi[-d+i,tanh i m hp 3 + g + 2 ~ ) ]P.A sin yt-+[.
.] sin h t + [ - ~ ~ tanh+ ~ i~ h (f+g- 2 ~ ~ ~B~cos) wt+[.]. .I-($+g)or else A=O, B#O, andw2(l+2xoi. tanh i.h)=smtanh i m h(?-+g)These two relations can be combined into amore convenient. form by introducing thenatural frequency of the mth mode, which isgiven byo : = i tanh L h (%+9)and by letting e=rOjymtanh X,,,h be a dimensionless excitation amplitude. Then according tothe general theory of Mathieu's equation, smallliquid motions will tend to increase withoutlimit in the frequency range given bycos h t = 0Hence, to the first approximation either B=Oand A # 0 , with the ndditional requirement that02(1-2xoL tanh k h ) = i m tanh L h275Equation (8.10) is valid only for the first fewmodes (small values of m) because the higherorder modes are damped quite severely.Figure 8.7 shows an experimentally determined stability boundary for the firstantisymmetrical sloshing mode in a 14.5-centimeter-diameter, rigid cylindrical tank.
As canbe seen from this figure, the theoretical predictions of equation (8.10) and experimentalresults are correlated very well except for aslight decrease in the width of the experimentalboundary which is probably caused by viscouseffects.3It should be recalled that the excitation frequencyis 2w.-Excitation frequency cpsFIGURE8.6.-General liquid behavior at low frequencies.276THE DYNAMIC BEHAVIOR OF LIQUIDSEven with these nonlinear boundary conditions, the potential, @, can still be expandedin a series of orthogonal eigenfunctions ofequation (8.1); that is@ = C a mcosh i,(h+z) Sm(z1 Y) (8.13)oi,sinh LhFrequency parameter,-.WIW,,FIGURE8.7.-Theoretical correlation for rn =1, n =1 mode112-eubhannonic stability boundary (ref.
8.20).Although the linearized theory predicts aninfinitely large sloshing amplitude in the unstable region, experience has shown that evenin this range, only a finite amplitude is obtained,and if it or the fluid pressure is to be determined,a nonlinear analysis must be employed. Sincethe only mathematical source of nonlinearityfor an inviscid liquid in a rigid container is inthe free surface boundary conditions, equations(8.2) and (8.3) should not be linearized entirely.As n~e~ltionedprel-iously, the theory developed in this section can be justified onlyfor the lower order sloshing modes.
Thus,the effect of surface tension may be neglectedfor the relati~elylong wa\-elengths under consideration here.4 The pressure boundary conditions may now be written as- ( g - 4 0 2 ~ 0 COS 20t)Z1*-,=O(8.11)I t shoiild be noted that equation (8.11) isevaluated at z--q instead of a t z=0, the positionof the undisturbed free surface.By resolving the fluid velocity into components normal to the free surface, the kinematic boundary condition may be written asThis conclusion is true only so long M g, or anequivalent steady longitudinal acceleration of the tank,is not too small. If g=O, the surface tension forcenbecome predominant, as discussed in eh.
11.is still valid and still satisfies the boundary conditions (eqs. (8.4) and (8.5)), which, of course,always remain true as long as the tank flexibilityis neglected. However, it is no longer lgitimate to write q=Za,Sm(x, y). Instead, theunknown am's in equation (8.13) must bedetermined somehow from the boundary conditions (eqs.
(8.11) and (8.12))) and calculatingthese coefficients is the central problem inmost of the remaining discussion. There areseveral methods that might be used to do this;one way mould be to eliminate q from equations(8.11) and (8.12), thus getting one equation in@ only; another way is to extend the linearanalysis by assumingand then determining the am's and bmlssimultaneously.6 The lattor method is convenient,and it is the method used here.The general way of attacking the solution is:Substitute the expansions, equations (8.13)and (8.14), into the boundary conditions,equations (8.11) and (8.12).
Consider theresulting two equations as functions of x and y,and expand them in a series of the orthogonaleigenfunctions Sm(r,y). In this way, twoequations of the formnre obt,ained, where F,,,are, in general, functionsSince the Sm(x,y) areof all the a, and b,.orthogonal, one can readily see that eachF,= 0 separately. Consequently, the problemThe dot over a, indicating time differentiation hasbeen dropped in cq. (8.13), since &, is not a ptiosi equalto b,, as was the case in eq. (8.7)..tI!IVERTICAL EXCITATION OF PROPELLANT TANICfjof solving equations (8.11) and (8.12) has beenreduced to the problem of solving a set ofequations F,=O, m=O, 1,2, .
. ., which involveonly a, and b,. Now assume that the nthsloshing mode is the dominant one, so that a,and b, are much larger than the other a, andb,. By inspecting the equations, the orders ofmagnitude of all a , and b, can be determinedin terms of a, and b,. The equations arethen solved for a, and b, to any degree ofapproximation by neglecting all terms abovethe specified degree.This procedure is well illustrated by theexample of two-dimensional waves. Twodimensional finite amplitude standing waveswere considered first by Penney and Price(ref. 8.12) during a study designed to assistin the engineering design of the Mulberryharbors for the Normandy invasion ofWorld War 11. They worked out the waveform to the fifth power in the dominantamplitude.
Among their results was the factthat the frequency of finite amplitude standingwaves in deep water was always less than thatof an infinitesimal wave, and that the masimnmpossible amplitude wave made u. sharp cornerat it,s crest and enclosed an angle of 90' atthat point.
Taylor (ref. S.13) later made anexperimental study of standing waves, andconfirmed Penney and Price's main conclusions.I n particular, he found that the wave of maximum amplitude did come to a sharp point, asshown in figure 8.8. He also showed that theresponse curve was slightly nonlinear softening,which is a consequence of the decrease in fre-qrlency with amplitude. As might be ex.pected, the maximum amplitude waves tendedto be unstable and break at the crest.
Later,Tadjbaksh and Keller (ref. 8.14) carried out ananalysis of standing waves in water of finitedepth by a method entirely different fromPenney and Price's. They showed that forliquid depths less than about 0.17 times thewavelength, the frequency increases with amplitude; for greater depths, the frequency decreases, as Penney and Price predicted. Thishas been confirmed experimentally by Fultz(ref. 8.15). Three-dimensional standing wavesalso have been analyzed (refs. 8.16 and 8.17);in particular, Mack (ref.
5.17) has shown thatthe maximum amplitude axisymmetric wnvehas a crest enclosing an angle of approximately109.5'.Rectangular TankPenney and Price's general method wasadopted by Yarymovych and Skalak (refs. 8.18and 8.19) to study two-dimensional waves in arertically vibrated tank. The salient pointsof the nonlinear analysis are easily identifiedin this, the simplest of all physically significantvertical sloshing problems, and for this reasontheir theory is given here even though thereare very little experimental data available forcomparison. Only the first symmetrical modeis considered in detail, although it is possibleto produce antisymmetrical sloshing even insuch a seemingly symmetrical situation asvertical vibration, since in reality neither thetank nor the vibration are perfectly verticalxlLFICURS8.8.-Form277of largeat amplitude standing wave (ref.
8.13).278THE DYNAMIC BEE AVIOR OF LIQUIDSand symmetrical. Synimetricnl sloshing doesnot cause any unbalanced forces or momentson the tank structures directly, as does antisymmetrical sloshing, but i t does influence thepressure a t the tank bottom, and thereby couldinfluence the pressure in the combustionchamber and the total rocket thrust.For a two-dimensional wave in an infinitelydeep, thin rectangular tank described byLL-<x<-r h= w , the appropriate velocity2 - -2potential for a wave composed only of symmetric modes is2rm2nm@=ao+% e + '~cos -x (8.16)m-1L2This can he derived from equation (8.14) bynoting that S,(x, yj =cos i,x wit'h i, =?=mil,by letting h+m, and by absorbing 1/~,,,intothe coefficient a,.In line with the general theory outlinedpreviously, it is assumed that the equation ofthe free surface can be writken in the formand the surface displacement isAfter substitution, equation (8.11) may bewritten as(D(1-4u2e cos 2us)E-m -0Lu,e+"'f cos mxi .
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