H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 9
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The wave forms of thefree surface at times before and after the first' See eqs. (2.22) for definitionm of a,and bmn.LATERhL SLOSHING IN MOVING CONTAINERS33and second natural frequencies are shown infigure 2.14.la,8040..L=c01,Q9 -40- 80- 120FXCOBB2.12.-Ma@cationfactor of liwd moment in a90° aect'x cylindrical tank (excitation along x axis)(ref.
2.3).FIG-2.13.-Mngni6cation factor of liquid force in a 90'eector cylindrical tank (rotational excitation about y-1 ( m 2.3).~For roll oscillations of the container excited by #=#,,era#, the velocity potential is34THE DYNAMIC BEHAVIOR OF LIQUIDSwhereand the quantities K:, 4, tl:, c, K*, 6, q*, and p are defined in appendix C.
The values 2, arethe roots of the equation J:(;,) =O while the values &,-l,n are solutions of the equation J;,-a(&,-1. ,)=O.Figures 2.15 to 2.17 illustrate the magnification factors for the forces and moments of theliquid, while explicit expressions for free surface displacements, forces, and moments are given inappendix C. The values of Lo, L1, and L associated with roll motions (these are Werent fortranslational motions) areGeneral Sector TankThe case of a sector tank with an arbitraxy vertex angle is discussed in references 2.3 and 2.31.The results for translational, rotational, roll and bending oscillations are given in appendixes Dand E. The following points should be noted in connection with these tables:(1) The quantities a,, b,., and c, are given in equations (2.22) and appendix A.
The termsb,, may be simplified to(2) The determinant A,,=Oreduces to JAlla([) =0, the zeros of which are denoted by em,;the roots may be obtained from table 2.2.(3) The functions C(c) simplify to the Bessel functions~.,.(cni).Liquid natural frequencies for the 60' sector tank have been determined experimentally(ref. 2.8) and therefore are also shown in figure'2.10.LATERAL SLOSHING IN MOVING CONTAINERSSL=5.0(llsec)i2=5.4(11sec)(First eigenfrequency at 5.5 radl set, second eigenfrequencyFIGURE2.14.-Waveform of the liquid fm surface in acircular quarter tank (ref. 2.3).FIGURE2.15.-Mapification factor of liquid force in a 90'sector cylindrical tank (roll excitation about z axis)(ref. 2.3).Ring-Compartmented (Annular) TankPerhaps it is worthwhile to mention here,once again, the essential characteristics ofcompartmented tanks that lead to their study.If one wishes to avoid coupling between variouscomponents of a dynamical system, an effective35FIGURE2.16.-Magnification factor of Liquid moment in a90' sector cylindrical tank (roll excitation about z axis)(ref.
2.3).FIGURE2.17.-Magnification factor of liquid moment in a90' sector cylindrical tank (roll excitation about r axis)(ref. 2.3).procedure to employ is often that of modifyingone component in such a way that its naturalfrequencies are shifted appreciably. In thecase of a liquid-filled tank, of course, compartmentation is the means of shifting the liquidfrequencies, as may be seen in figures 2.5 and2.10.
Figure 2.5, especially, shows that sectorcompartmentation does indeed have the effectof increasing the frequency of the lowest modein the uncompartmented tank, but also introduces additionai mocies'with frequencies thatare relatively closely spaced.
On the otherhand, compartmentation by annular ring hasthe effect of decreasing the frequency of thelowest mode while introducing additionalmodes that are relatively widely spaced.The annular tank has been considered theoretically in references 2.2 and 2.34. Equationsfor the liquid natural frequencies, free surfacedisplacements, forces and moments induced by36THE DYNAMIC BEHAVIOR OF LIQUIDSfree, translational, pitching, and bending typeoscillations have been derived and extensiveplots presented in reference 2.2. The methodsemployed are again similar to those outlined inearlier portions of this chapter, with the resultspresented below.
The velocity potential for aliquid in a ring tank subjected to arbitrarypitching and translational motion has also beenderived in reference 2.34, although the fluidforces and moments were calculated only forthose special cases of a circular cylindrical tank.Free OscillationsThe origin of the coordinate system is locateda t the center of the plane of the undisturbedfree surface with the z-axis pointing out of theliquid (fig. 2.18). For free oscillations, thevelocity potential iswhere En, and Dnmare to be determined fromthe initial conditions, f,, are the positive rootsof equation (2.18d) for the case 2a=1, theCmare defined by equation c2.18~)for(t i)the case 2a=1, and k=b/a is the diameterratio of the inner aud outer tank walls.
Theroots of the equation A,=O for m= 1 are tabulated for various vdues of k in table 2.1 andplotted in figure 2.4. The natural frequenciesare obtained from the relationand are shown in figure 2.5 for tank diameterratios of 0.2, 0.5, and 0.8 as a function of theliquid height. The frequencies of the cylindrical inner tank are also shown for comparison.The tank configuration h ~ v i n ga ratio of kbetween 0.5 and 0.7 would appear to offer theFIGURE2.18.4ircukr cylindrical ring (annular) tank.most promise of phasing the liquid motions ofthe inner and outer tank such that some sloshforce cancellation is obtained.
Such cancellation is promoted by the fact that for this rangeof k, the liquid masses in the two tanks areapproximately equal.The free surface displacements, measuredfrom the undisturbed surface, and the pressureat depth (-2) are=-1 92 unm[Fnmsin Ng m-0n-0'LATERAL SLOSRINQ IN MOVING CONTAINERS5 2 on.{Lsin m++Burncos mg)p=-ipn-0 m-0cosh [$(z+h)]X37[&kcl(ktnl)](1-6i)Crn(turnetanmt-PgzAs a matter of general interest, the three lowestThe moments about the center of gravity of theliquid areM,=-iMFCn-o~nlZnletualt-E.n l Y ;.Q.
* -d-.(1-P).liquid free surface modes are shown in figure2.19 for k=0.3. The displacements are no&alized to unity at the outer tank wall, and nindicates the mode number.-u--1-2,,,=Y-IoLn-31-aRatio d innerlouter tank radius k * 0.32 1-3(2.41b)etuml',(L 2-arwnlE"le'~nl'Mr=-iMF n-oC t u r Y : ( t n l ) (1-P)-FIGURE2.19.-Free narkce modes for h e oncillation in aring tank, kn0.3, n=O to 3 (ref.
2.2).LThe slosh forces and moments acting on thecontainer caused by the free liquid oscillationsmay be obtained by integration of the pressuredistribution. ThusIt should be noted that only the modes m=1, 3, 5, . . contribute to the forces andmoments. The velocity distribution in thetank is.I-axm h[% (z+h)] QA i)cash (4,(turn!)eimmmtyL(tnrn)(2.43a)38THE DYNAMIC BEHAVIOR OF LIQUIDSu6=t2 9 n[Cnmcos ~coshXr~n=O m=Oo - Dsin ~MI~(z+h)] Cm(€,i)k) yk(tnm)cash (tun-ei"-'(2.43b)-w = z &E. c.,~ sin' M+D'-, cos -1wA.""z($[mcash[$ (z+h)]--1)cosh ([,I-;y("I"r)]-pgzwThe free surface displacements for variousvalues of k and h/a have been calculatd andare plotted in reference 2.2.The fluid forces and moments (with respectto the center of gravity of the undisturbedfluid and positive according to the "right-hand"rule) are given by the equationsTransktional Forced Oscillations-An[ $ - k ~ ~ ( k € ~ * ) ]For forced translational oscillations of theform xsfQ',the velocity potential is given by@(r,g, z , t)=iQzoefO'acos 4a1@- 1 ) ( 1 - P )A.
cash [%(1+h ) ]x[:+hn=O($-I )COS~:)wherezn=2AThe wave form of the free fluid surface, measured from the undisturbed fluid level, and thepressure distribution in the tank at depth ( - 2 )are--and forces and moments are plotted versus excitation frequency for k=0.5 and 0.9 in figures 2-20and 2.21. The velocity distributions are given byu,=iQzoeinfcos gAnts COOShf( 5 - 1 ) cosh(z+h)]([.:)>-,.'..---.-<%'-. :*. ,.--<.-,>.>s:>.
.,.. my-. '.-,-*<a<;,%*,"-,::.v. - *.A. . .. . . . .. . . "> . -._....%'. ....L--_.&. ...4-,-.=..,'--. .,.-..i.-=. . . . . .%a-L.L'7,*-.x,--*<..'IA .,. . , : "*,.- ;?=-z.-y. . .:..z . Y * ~ : ~ . ~.,~,~ . r ..~ L -- + - , - ..a ., - *..-. .-,. .- : . . . .. . -.,.., --,.._ .. ..J~..-,..,-*.. .,a.LATERAL SLOSHING IN MOVING CONTAINERSI39MX, pa*eiat-hla .ZOgla 3.0IIIlsclc*,0/k-a5FICVBE2.20.-Liquid force and momcnt in a ring tank (k=0.5) reeulting from tramslation and pitching excitation-(d2.2).2-8-7---4- :. _ -,.>*-: T.rY<.:.:.,._...%-5.y.&"-+?-..-.* .* 7.??---::...?.,.,-.'&T;?<$-i-..
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