H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 44
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The last assumption isnot strictly true, since the sum of mo and the total massof the infinity of sloshing masses ( d l but one of whichare neglected in this model) should equal the liquidmass.THE DYNAMIC BEHAVIOR OF LIQUIDSIn one series of tests with the liquid oxygentank of the Centaur missile (ref. 6.5), the tankmas oscillated at a frequency much less thanthe sloshing natural frequency, and then thetank was quick stopped. Since the tank wasnot rotated (&=O), equations (6.10) and (6.12)can be combined to givewhere F,,, is the magnitude of the peak sloshingforce immediately after quick stopping, and Xois the amplitude of the tank motion; that is,xo=Xo sin wt. From this equation, the pendulum mass, m,, may be calculated (sinceITthe natural frequency,can be measuredby noting the escitation frequency for which asloshing resonance occurs). The reliability ofthis method is demonstrated in figure 6.2,cylindrical section of height, c.) The scatter inthe experimental data was about & 5 percent;for this reason, upper and lower limit curves aredrawn on the figure.
It can be seen that thesloshing mass is very small when the liquidalmost fills the tank, because the free surfacearea continually decreases as h/(b+c) is increased above 0.5, until for h/(b+c) = l . O thereis no free surface; that is, there is no wave motion. I t should be noted that there was acertain amount of internal hardware in thisscale model, such as a thrust barrel, a fill pipe.a vent pipe, and so forth; however, there wereno slosh baffles present.The location of the pendulum hinge pointabove the center of the tank was determined byquick stopping the tank, as was done in the m,measurements, and then the maximum momentand force were recorded. For these conditions,it can be seen from equations (6.10) aIld (6.11)that h4max=h, Fmax,orResults of typical measurenlents for the Centaurtank are shown in figure 6.3.
There is a considerable amount of scatt,er in the data. Theabrupt change in the slope of the curves ath/(b+c> =0.S is probably caused primarily byt,he fact that the free surface area increases forincreasing liquid depths as long as h/(b +c) <0.5,Liquid depth ratio, hllb + c)FIGURE6.2.-Ratio of pendulum mass to liquid masspresent at each depth ratio as function of liquid depthratio (ref. 6.5).v3which shows t l ~ esloslli~lgmass, m,, as a fl~nc- K -0.10 a 1 0.2 a 3 (14 0.5 0.6 0.7 0.8 a 9ti011 of liquid depth, h , and total ttrnk depth,;s0h + c . (The Centaur liquid oxygen tank isLiqu~ddepth ratlo, hllb + c)essentially all oblate spheroid with rt minorrrdinmetor of b, although the t w ~oblate s ~ h e - FIGURE6 . 3 .
- ~ a t i ~of pendulum hbge point location toroidal segnre~ltsare sepnrated by a very smalltank height as a function of liquid depth ratio (ref. 6.5).ANALYTICAL REPRESENTATION OF LATERAL SLOSHIP-Gbut decreases for increasing liquid depths whenh/(b+c) >0.5.The fixed mass, mo, can be calculated by theequationmo=mT-ml(6.15)where m~ is the total fluid mass. The locationof this mass below the center of gravity of theliquid, and its centroidal moment of inertia,. can be determined by pitching the tank (%#O)and recording the sloshing moment and force.If this is not possible or practical, as was thecase in the experiments described in reference6.5, additional assumptions are necessary. Ifone assumes that the mechanical model shouldduplicate the statical properties of the liquid,ho can be calculated by requiring that the staticmoment on the tank, when it is tipped througha small angle, be the same for both the liquidand the mechanical model.
In other wordswhich is identical to requiring that the locationof the center of gravity of the mechanical modelbe the same ns that of the liquid. However,locannot be determined unless it is possible topitch the tank; if this is done, I, can be calculated directly from equations (6.10), (6.1I),and (6.12).The parameters of a mechanical model usinga spring-mass element instead of a pendulumcan be determined in a similar way.Analytical Derivation of Model ParametersExpmssions for the sloshing forces andmoments are available for a number of simpletank shapes, and for these the model parametersmay be determined without recourse to experiments. Because the theory usually assumesthat the liquid is inviscid and the tank is"clean," the inertial parameters are strictlyva.lid only for a, t.ank cnnt.aining no baffles orother internal hardware: even with baffles,however, the parameters do not change significantly, t,he main effect being an increase indamping.
(However, see discussion followingequation (6.51) .) Thus, the various masses,pendulum lengths or spring constants, momentsof inertia, etc., given in the tables are validalso for tanks which are not perfectly clean.203Simple mechanical models for cylindricaltanks are considered first (refs. 6.6 through6.1 1).
Figure 6.4 shows the two types ofmodels for which analytical models are available, and the equations for calculating theparameters are given in table 6.1. The momentof inertia, Io, is the same for both models.Not all of the models given in the referencesare identical. In some cases, mo is given asthe value i t would have if all the modal masseswere included in the model; thus, since onlyone of the modal masses is actually included,the total fluid mass is slightly larger than themass of the mechanical model. Also, thevalue of I, is sometimes given as the valuenecessary to leave the center of gravity of thefluid unchanged; in table 6.1, the value of I, isthat which would leave the center of gravityunchanged if all the modal masses were present;thus, the center of gravity of this model isslightly displaced from that of the liquid.Experimental evidence (refs.
6.10 and 6.1 1)has shown that these discrepancies are veryminor.Figure 6.5 shows an equivalent mechanicalmodel for rectangular tanks (refs. 6.12 through6.14); this model uses a single spring-masselement to represent the liquid sloshing. Themotion of the tank is in the plane of the drawing,but a similar model is valid for motions a tright angles to this plane. Equivalent modelsfor the sloshing that occurs when the tank isrotated about vertical axis are available (refs.6.12 and 6.13), but are not reproduced heresince this type of motion is of less interest inspace vehicle applications.Table 6.2 lists the model parameters, whichhave been developed by neglecting all the termsinvolving the masses corresponding to highorder modes in the original equations (refs.6.12 and 6.13).A series of compound double-pendulummodels used to simulate sloshing in long, shallowrectangular tanks, such as the passive antirolltanks employed in certain ships, have beendeveloped by Dalzell (unpublished notes).
I nthese models, it is necessary to include explicitlythe fact that the pivot point of the swingingpendulum is in motion. Figure 6.6 shows twoa204THE DYNAMIC BEHAVIOR OF LIQUIDSTABLE6.1.-Model1( n , ~ =""h;~1" for spring-mass= IlParameters for Cylindrical Tank+Ll for pendulum; N=O1JriEid=frfd[Efor spring-mass model; N = 1 for pendulum model;+$I)(a)h 2-Spring-mass analogyPendulum analogy----------dL1(P)1 19hdn ~ , =la^ (-)4 4hhK l = ntT~0th=s83.68 ;iv1,=h,,L, (d)tanh 3.6834.4h11=-(;)'-?Z~+@i+m~(l+NLI=Zrind+m~3.68((tanh 3.68h 2a)htanh 3.68 ahtanh 3.68 211ah - 1.07hsinh 3.68 a1.07 C O S ~3.68[1.99.5-f-m~~~~~~~~REPRESENTATIONFree surface205OF LATERAL SLOSHINGTABLE6.2.-,ModelParameters for RectangularTank[ m ~ = p h w = f l u i d mass per unit width]& = m T (*h)( t a n h 3.14m l = m .
(&h)htanh 3 . 1 4 ;?)'Wm,,=m~-mlh11=1.57 w tanh 1.57 ;h rn~ hlo=-+(-4)2 m o 2m-12T (hz+wZ)4wZ[1-7+-u?+h-1252dhhwZ+h3 tanh 1.57 wFIGURE6.6.--Compound double-pendulum mechanicalmodel8 for long shallow rectangular tanks.form of graphs instead of tables. Figure 6.7gives the necessary information for a pendulumanalogy. The centroidal moment of inertia ofthe fixed mass is not required in the modelsince, for an ideal liquid, rotation about an axisthrough the center of tank does not cause anyliquid motion or sloshing forces and moments.Comparisons of the test results and the predictions of the mechanical model do not agreeas well for the spherical tank as they do forcylindrical and rectangular tanks (ref.
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