H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 11
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2.31) may also be obtained from themore general solution presented earlier in thissection; equations for the velocity potential,free surface displacements, forces, and momentsare therefore given in appendix G.2.4sinh (a. :)-2c&+y)- a'------.------(B0.9 w,'a'- 1 .
1 ~ :a'- 0 . 5 ~ :a'- 0.9 w ta'= 0.8 at:hla = 2 0g/a = la o1/sec8RotationSPHERICAL TANKSpherical tanks, because of their high volumeto-weight ratio and obvious structural advantages, are very often employed in spacevehicles. Unfortunately, however, the theoretical analysis of liquid oscillations in such acontainer is a problem of considerable mathematical complexity so that a number of investigations employing various approximation techniques have been undertaken. Budiansky(ref. 2.21) employed a sophisticated integralequation technique and obtained liquid naturalfrequencies and forces for the cases of theI- - 1 ) L - 1 1UU-~UU&iitt~~ --..-I- #..TI tnnb"--. Theseresults, together with the solution for the nearlyempty tank and the known behavior of thecircular canal, were used to calculate approximately the liquid behavior for arbitrary depth.McCarty and Stephens (ref.
2.44) obtaineduU(UAJ-AIIYFIGURE2.25.-Freeeurface displacement for variousexcitation frequency ratios in. a circular cylindricalcontainer (ref. 2.3).experimental frequencies which agreed wellwith the theory of reference 2.21. (Additionalfrequency data were obtained for the spherein connection with a study of spheroids (ref.2.45), but not compared with theory.) Rileyand Trembath (ref. 2.23) used the variationaltechnique presented in references 2.14 and 2.22tn cdculate frequencies, with good agreementalso being obtained with these theoretical andexperimental results.
h finite difference numerical scheme was also successfully applied to thespherical tank (ref. 2.26). Stofan and Armstead (ref. 2.46) also employed the theory ofFIGURE2.26.-Magnification function of the liquid force in a circular cylindrical container (ref. 2.3).reference 2.21, but with an approximation technique for the kernel function; the frequenciesand slosh forces calculated by this methodwere again in good agreement with experimentaldata, the slosh forces increasing with liquiddepth to a maximum which occurs a t thefundamental frequency for the half-full condition. Chu (ref.
2.47) developed a numericalprocedure to determine the kernel function,which is related to the Neumann function onthe boundary, and then utilized the method ofreference 2.21 to calculate the liquid naturalfrequencies and the constants required toevaluate the force response. The method isgeneral and may be applied to the tanks filledto an arbitrary depth. Extensive experiments,involving measurements of liquid frequencies,wall pressure distributions, slosh forces, etc.,have also been carried out by Abramson, et d.(refs. 2.48 and 2.49).
Comparisons betweenthese various calculated and measured frequencies, for various modes, are given in figure2.28. The experimental data on frequencies fora sphere given in reference 2.45 also agree verywell with dl of the data given in this k u r e .Experimental data for the fundamental liquidfrequency in a spherical tank has also been reported in reference 2.43, together with relevantempirical equations, as shown in figure 2.29.The empirical equations, valid for certain rangesof the liquid depth parameter h/R, are.
....-. -.,'I . . ...,.. . .,..-,.. .. .'... -,.....*-,a-.-.--.A>"<,..:. ,.-4..-.,...A ,FIGURE2.27.-Magnification function of the liquid moment in a circulrv cylindrical container (ref. 2.3).or, mom generally,and the value of Cl is also given in figure 2.29(for very small liquid depths hlRSO.1, theliquid frequency crtn be obtained from the experimental data for a cylindrical tank with aspherical bottom given in fig, 2.22).
In theseequationsRattayya (ref. 2.25) also employed the variational technique of references 2.14 and 2.22 toobtain natural frequencies and slosh forces forliquid in an ellipsoidal tank, with the sphericaltank as a special case. An empirical equationfor the fundamental liquid frequency in thesphericai tank was iheu developed in the f o~52THE DYNAMIC BEH VIOR OF LIQUIDSVarious of these values for the lowest liquidfrequency are compared in table 2.3.Therefore, the fundamental liquid frequencymay be obtained with about equal accuracy-0.0.40.81.21.6hlRFaired experiments,McCarty-Stephens [2. M]+-+Russian experimentsA20[243]Points calculated from Russian equations (259) [242]-;-?,iments,,Mercury experiments Stofan-Armstead [2 461a4000oTheory, Budiansky [2 211Theory, Chu [2 471Theory, Riley-Trembath [2231aa04.LZ1.62024hlRFIGURE2.28.-Liquidnatural frequency variation withdepth in a spherical tank--comparisons of variourtheorier and experimental data.TABLE2.3.-ComparisonFIGURE2.29.-Lowest liquid natural frequency variationwith depth in a npherical tank-mmparillon of empiricalequations and experimental data (ref.
2.43).of Liqrcid N a t d Frequency XIfor a Spherical TankIf:Mikishev (ref. 2.43)Eq. (59a)Eq. (59b)1.091.22Budiansky(ref. 2.21)_-___---_---1.02•1.00- - , - - - - - - - - - - - - 1.10- - - - - -1. 06 -1.8-----------------------.Obtained from fig. 9 of ref. 2.21.Experiment(fig. 2.28)1.011.011.081.101.221.22_ - - _1.22 - - - - - - - - - - - - - _1.25_--,__2.031.831.92 ........................• 2.040.05----------------------------------0.50- - - - - - - - - 1.00- - -,- - - - -,Fig. 2.22Chu(ref. 2.47)Rattayya(ref. 2.25)(eq.
(60))53LATERAL SLOSHLNG IN MOVING CONTAINERS(except for large depths) from equation (2.59)or (2.60) or read directly from figure 2.28.The liquid frequencies corresponding to thesecond and third modes may also be obtainedfrom the data given in figure 2.28. The sloshforces as a function of liquid height may beobtained from the equationsa;iii( ~ , ) + u , J . En= - dtl~ c ~(2.61~)~ ~a8Fa=-M, R )D,' ~ (63 (2.61b)dta (U)- ~ P ( C Sn-1where6,=slosh height at the wall associatedwith the nth modeF,=resulting slosh force acting throughthe centerc2=geometric parameter related to thefluidheightshowninfigure2.28R=radius of the sphereU=transvene displacement of thecontainerD,, E,=c~effi~ient~obtained from @ w e2.30The slosh force parameters D, and En maybe obtained from either reference 2.21 or 2.46with about equal accuracy; the results fromreference 2.21 are presented here (fig. 2.30).Measurements of liquid force response (refs.2.48 and 2.49) have revealed some interestingeffects, especially as regards the influence ofexcitation amplitude in partially filled tanksand the liquid behavior in nearly full tanks.(These effects, which are essentially nonlinearin character, will be discussed in more detailin ch: 3.) Figure 2.3 1shows some experimentaldata on liquid force response (ref.
2.49) compared with theory (ref. 2.21). There is obviously some significant effect of excitationamplitude (even granting some unraliability inthe -experimentaldata), especially as regards anoticeable frequency shift and generally betteroverall agreement for the smaller value ofexcitation amplitude. Figure 2.32 shows thestrong d e c t of geometry in the nearly fullspherical tank by virtually suppressing theliquid force response, with a strong secondaryeffect of excitation amplitude. Some furthercomparisons of force response in a quarter-o0.2a40.6a81.0~21.4~61.8LOhlR- -FIGURE2.30.-Modal parameters D , and Elfor determinh g force response in a sphere (ref. 2.21).full tank are made by Chu (ref.
2.47), againwith reasonable agreement.Sumner and Stofan (ref. 2.50) also carriedout an extensive investigation into the effectsof (a) excitation amplitude, (b) tank size, and(c) liquid kinematic viscosity on the slosh54THE DYNAMIC BEHAVIOR OF LIQUIDS+Phase angle(deg.11,36.836.8--6-Experiment ( X,ld = 0.00414-[Z 211Theory I BudianskyDimensionlessforceamplitude200a5'1.01520253.0Dimensionless frequencyFIGURE2.31.-Liquid40455.05.56.0= A'total force rapme in a half-full rpherical tank (ref.
2.49).force corresponding to the lowest naturalfrequency (first mode) in a half-full tank.Figure 2.33 shows some typical results forconstant excitation amplitude and variableliquid viscosity, while figure 2.34 shows theeffects of varying excitation amplitude. It isseen from these data that, in general, thefirst-mode slosh force parameter increaseswith increasing excitation amplitude and withdecreasing viscosity.9.53.5w ' ( RlgTOROIDAL TANKThe toroidal tank, because of packaging requirements, has been considered for storage ofpropellants and liquids for life-support systems.There is no theoretical analysis available forpredicting the behavior of liquids in such tanks,and only limited experimental investigationshave been reported (refs.
2.51 and 2.52).There are available, however, certain empiricalformulas and experimental results from thesestudies which predict with fair accuracy thenatural frequencies and which have establishedprincipal trends in the liquid behavior. Theeffect of excitation amplitude on the frequenciesand forces, however, was not investigated, norwere the experimental data successful inestablishing the manner in which slosh forcesvary with tank geometry and orientation.(see fig.
2.35.)To simplify the analysis of liquid frequenciesin the various container orientations, thetoroid is "divided" into the regions shown infigure 2.35. This reduces the problem of predicting the liquid frequencies for any fluiddepth to that of predicting the frequencies fordepths in only certain regions, which can bedone in each region by reference to an "analogous" container. Thus, the success in obtain-p y ; : ...c,,r.y;:$;*:?l%,..Tz-~.&w... .. . .
. ;. ,. ... .,.? " . ,:.. .' .I,.,.?...-,,,.:.,.<"-*><-,.~,,. *.,.,:,.... 7.--.*..-->*,....,.;,,*-*.,+ .........- .3:&y?''-. :-:-?.,- ---.. .<~*T..-,~.,T;~*>-~,;-:. . - ..............,--.,-7...?I.<'I..P4!-.-i.;.?,.:.-.,;,7?.(,'..., ...... ,,.- - .,-/......LATERAL SLOSHING IN MOVING CONTAINERS200+Phase angleIrnLiquidH,OHtOIDia ,cm. 3 836.8+.+Experiment(&ld*0.00828)-4-Experiment ( XJd * 0.00414 18pgd { V d 11-"6Dimensionlessforceamplitude/afl0-000.5LO1520253.0X540455.05.560Dimensionless frequency w8 ( Rlg 1 = k1F~cmtr2.32.-Liquid total force response in a nearly tull spherical tank (ref. 2.49).ing a frequency ptnameter independent of tanksize and geometry dependb upon the accuracyof the anaIogy.
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