H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 10
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. . ... zL.., .:*. . . .. .. 1 .- . ... .,. ..x.@.:Ax>:..,:..-........:,:-;.s.m-..*--<-$,2-*~w-7,pe-y..v* *: ............>.*?, c: ::.. 2. . . . . .. . . .-;. .-4*;:..0-'... ..-...'*,-..., ..;..... .7 ".!L,..,,.: . .. .. .. . .. .. .. . .......*,;-....-..40THE DYN~MICBEHAVIOR OF LIQUIDS-~,....:?. 7.- dTT4...p.7.L-s..;:*< ,.:i.c*.,. . ....-$:?.A.
,. -,a,.,41LATERAL SLOSHING IN MOVING CONTAINERS.u+=iQzOernrsin 4cosh [f (z+h)]($- 1) cosh (C. ;)w = i Q ~ ~ ecos' ~ '4(€.:)}Ql(2.47b). 4. sinh E(z+h)](2-'-o1) cosh([.k)The velocity distribution in the tank is obtained from equations (2.47) leaving out the terms66unity" in the parentheses, since these represent the tank motion.Rotational Forced O~llationsThe velocity potential for a liquid in response to small forced rotational oscillations BoefQtabout an axis through the center of gravity of the undisturbed fluid is@(r,4, z, t)=iQeoernta2cos 4Iac+g" '=o6-1) cosh (c.2)where the coordinates are located at the center of gravity of the liquid (z', y', z' coordinates offig. 2.18).The wave form 6 of the free fluid surface measured in the inertial system isThe free fluid surface displacement in the tank is 6*=6-doeatr cos 4.
The pressure distributionin the tank at a depth z is-x""I($-1)/If*\" 01cosh(&,:). , ,-, ,,.'_ .. ? : . . ...... .,.-..'42.'....-.... . . . . . _. .-...THE DYNAMIC BEHAVIOR OF L I Q m"''.,.. .;'.':.-,...:...i. . . .. .-.The fluid forces and moments are given by.'where the first term is the component due to the gravitational force, andFigure 2.21 also shows, as an example, the forces and moments resulting from rotational excitation,plotted versus frequency for k=0.9.The velocity components areu,=iS2eoeiQtacos 4t'+Z€Ki(L2($-1)CO*(€,,-*2an2u6=iS2eoeiQiasin;);) [(A+; k) cosb [k (z+;)]i) .s ($ :)-;si*[:C-z)I])+\r(2.52a).43LATERAL SLOSHING IN MOVING CONTAINERSt)(3-1)~ = i Q 0 ~ e ' ~cos' a I$,cosh (€,,ail29sinh)(3a) cosh (& D+iwsh [t(;-.)I ](2.524The velocity distribution in the tank is obtained by omitting the terms z/a in u, and u+ and ria in w,since these represent the tank motion.Forced Oscillations Resulting From Tank BendingThe origin of the coordinate system is taken in the center of the plane of the undisturbedfluid surface and the z-axis is pointing out of the liquid.
The tank walls r=a, b are assumed tohave a bending-type motion in the direction of the paxis with an amplitude zo(z)e'Qt. Analyticalresults obtained from the solution to this problem (refs. 2.31 and 2.33) are given as follows.The velocity potential is@(r,4,2, t)=iQemCws Okc1(: F.'),(: tub) (~:-n? [{Q2sinh(:t.2)COS~where 2, is given by equation (2.44) and prime denotes Herentiation with respect to z. Thefree surface displacements measured from the undisturbed surface are6=Qge'n8 cos I$(gla)[; t,g{d(-~,'THE DYNAMIC BEHAVIOR OF LIQUID644The pressure distribution in the tank isKc, (:p=pQ2emfa cos ocosh& oosh[: &,(z+h)]rf ~ )(: €,,h)(4-d'(€1[n2 sinh0')(i[{Q~(i €I&)+:tn)4.g cash(: tat)]4dzLATERAL SLOSHING IN MOVING CONTAnYERS.The moment of the fluid referred to the position of the undisturbed center of gravity of the liquid isf..
-- ..... .... .... ..,...,.....46...-. . ..,.,..7 .,.. ..'THE .DYNAMIC BEHAVIOR OF LIQUID6M.t=O'.'..,,.?,I/.-..(2.56d)The velocity components are+;1 tug cosh (: t n z ) } { d ( - h ) +J'x ~ ( tcash) [: tn(t+h)] 4 )-I[t+ W S ~t n ( ~ + h ) ] { l d l ( t[na)(i k t )+;1 tug cosh (: €.€)I&+au*=iQeiotasin 4t-1-+;ZLa~o(2)"'4a In cashn-O[ { Q ~sinh(: € n h ) ( 4 - ~ 3+:zo=inefotacos +(; en?)b[Q2%(o)-gd(o)llug cosh(: t n z )(: €.€)Idt+ a'(: pnh) (4-n2)[ { 0 2 a h cosh (i €3)COS~. > : .LATERAL SLOSHING IN MOVING CONTAINERSFor %=constant, all of these results transforminto the results of the translational oscillationsin z-direction of the rigid tank.Circular Uncompartmenhd TankGeneral DiscussionWe turn our attention now to the simple caseof the cylindrical tank of circular cross section,uncomGtmented either by annular or radialwalls.
The problem of lateral liquid sloshingin such a tank may be obtained, of course, fromthe general solutions for the ring sector tankgiven esrlier, or from direct analysis. Thelatter is perhaps well exemplified by Kachiganand Schmitt (refs. 2.35 to 2.37) who employedLaplace transform techniques to determine theliquid response induced by arbitrary translational and pitching motions of the tank.
Also,the early work of Miles (ref. 2.38) should benoted, especially as he considered the moregeneral formulation of the problem for a slightlyviscous liquid with time-varying depth in acylindrical tank of arbitrary cross section. Ofcourse, nearly all of the work of Bauer describedearlier in this chapter was specialized by him tothis particular case (refs. 2.2,2.3,2.19, and 2.39).The behavior of liquid in a tank subjected toharmonic translational excitation in two orthogonal directions has also been treated by Bauer(d.2.40).Extensive experimental investigations (refs.2.11, 2.12, and 2.41 to 2.44) have confirmedthese various theoretical analyses. Goodcorrelation has been obtained not only forfrequencies and free surface displacement butfor pressure distributions and force and momentresponse.
While theoretical analyses hadshown that the influence of the flat tankbottom is significant only for liquid depthsless than, say, one tank radius (see fig. 2.5),and this also is substantiated by the experimental data, the iduence of a spherical orconical bottom is less well defined (see alsoSM. 2.6 for iiquici behavior iii coiiki2 tiiik.).F i e 2.22 shows experimentally determinedvalues of liquid natural frequency versus liquiddepth in a cylindrical tank with a sphericalbottom (ref. 2.43). An experimental evaluation of the effects of a conical bottom was madein reference 2.12, with the interesting resulthlaFIGURE2.22.-Experimentallydetermined natural frequency of a fluid in a circular cylindrical tank with aspherical bottom (ref.
2.43).that the conical bottom could be replaced byan equivalent flat bottom (based on liquidvolumes) for estimating both liquid frequenciesand forces, and this appears to be valid at leastthrough the second liquid mode, and possiblythe third, for fluid depths as low as h/d=1/4.The effects of ellipticity of the cross sectionwere investigated theoretically by Chu (ref.2.15), who was seeking some insight into theeffect of small out of roundness on the behaviorof liquids in circular tanks. It was determinedthat the liquid natural frequencies were onlyslightly modified by the distortion of the crosssection, but that entirely new modes (andhence frequencies) were introduced that haveno counterpart in the perfectly circular tank.Somewhat similar results were obtained experimentdy in a study of liquid motions inoblate spheroidal tanks (ref.
2.45) (see alsosection 2.7) in that it was found that, forellipsoidal tanks oriented such that the freesurface cross section is elliptical, a normalmode of liquid motion occurs which does notexist when the tank is oriented such that thefree surface cross section is circular.The previous discussion has been confined toupright circular cylinders; however, the behavior of a liquid in a circular canal (see fig.2.23) has also been investigated, both theoretic d y and experimentally (refs.
2.13 (sec. 259),2.21, snd 2-44). Sbca ~ymmetricalmodes ofoscillation are not induced by transverse motionof the canal, attention is restricted to theantisymmetric modes. Rotation of the canalabout its center produces no sloshing nor, bysymmetry, does vertical tank motion producehorizontal slosh forces. All pressures are in&.-..,w-"....
~II~ ~ ; >*,m.~ ,...,.~ F 2r;..~ -. -.+ ...x ..~ :~-. ~.:w::~x-:yF:..,.: + wj .:._ . . .-.,-......... 3.- " ... - ..-, , . .. ... .. ' ,. , . ., .,.,, --. .._. . --.,. .>.......-...,:?..I"L ..,I.'.? . w : f : .,~ .~ ~ -.!.>7:s =?~. f~., I ~ . r ~.. . ,,.z y F >. , ... . '._. . . . - ..*-*.~i-':.__:., .*. ? ;*'. .:'. .-..!..-,.- .- - , , r - 'II:-L,,.-..cl,'".->48-THE DYNAMIC BEHAVIOR OF LIQUIDS .-Circular canalI transverse I-. .. ...-.IqmZ4.r 203XL6FIGURE2.23.-Variationof liquid natural h q u e r ~ c yparameter with depth for transveme moden m a circakrcanal (ref.
2.44).-n Experimentally,determined naturalfrequency@ ( t4,, Z,t)=inz,,etota cos 4Ia-Circular canal( longitudinal Ithe radial direction and, therefore, the sloshforces acting upon the container pass throughthe center of the section. Some numerical andexperimental results are presented in figures2.23 and 2.24 (refer to ref.
2.21 for the generaltheory of sloshing in circular canals). It isseen from figure 2.23 that good agreement isobtained between theory and experiment fortransverse oscillations. The experimental results given in figure 2.24 indicate that thefrequency parameter for the longitudinal modesin a circular canal are essentially independentof tank geometry (there is no theory available for predicting the frequencies for thisorientation).FIGURE 2.24.-Variationof liquid natural frequencyparameter with depth for longitudinal modes in acircular cmal (ref. 2.44).Forwd OscillationsFor translational and rotational oscillationsof the form wtntand Boe'nt, the velocitypotentids given earlier in this section reduceto (see ref. 2.3 for mathematical details)I+(4-l)Jl(..)008h(e.t)(2.58a)($1)....49LATERAL SLOSHING IN MOVING CONTAINERS:) ($)2JI(en@(T,6,Z, t )= - i ~ e , e ~ ~cos' a ~6[(.+$) cosh~~(t:-l)Jl (en) C O S ~-* (g)sinhwhere en are the roots of the equation J i (an)=Oand the natural frequencies wn are given byd=g, tanh (a.a!)A()sinhc&-y)])(2.58b)Translation(2.58~)The first few roots of Ji(cn)=O are as given intable 2.2, and the lowest two natural frequenciesare plotted in figure 2.5 a s a function of h/a.Appendix F gives the equations for free surfacedisplacements, forces, and moments for bothtypes of excitation, and these results are showntypically in figures 2.25 to 2.27.The solution of the problem of sloshing in acircular cylindrical tank resulting from prescribed bending type oscillations of the tankwalls (ref.
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