H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 8
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This results in a faster convergence ofthe solution. The velocity distributions in thecontainer are obtained by omitting the firstterm in the braces; that is, omitting the termcos 4 for the radial velocity component ofu, and sin 4 for the angular component of u,.General notes pertaining to translationaloscillations:(1) The results for container motion y=yoeiw are given in appendix A (also seefig. 2.3).(2) The following definitions are applicablein appendix A:In natural logarithmI'the gamma functionthe derivative of the logarithm of thegamma function, that is,In this table oo, xo, and 4o are the rotationalamplitudes about the y, z, and z axes. Note:(1) The equations for the pitch-type motions,80 and xo are valid for a coordinatesystem with its origin located midwaybetween the tank bottom and the undisturbed fluid surface, on the vertexaxis of the tank.
(See fig. 2,3, s', y', z'coordinates.)(2) The equations for the roll molions, 40;are valid for a coordinate system withits origin located in the undisturbed freesurface.(3) To obtain the displacements of the freesurface in tank-fixed coordinates, subtract that resulting from container motion from that of the space-fixed system.Thus.(4) The quantitiesCmare defined in appendixA.where y =0.5772 is the Euler constant.(3) I t should be noted that, because ofsingularities that occur in the velocitypotential, the results contained in thissection are not valid for a = % and rue%.For these two cases, special modifications must be made to form admissiblefunctions.Rototionol Forced Oscillat/onsLiquid behavior induced by pitching- androll-type motions of the container is described by the equations given in appendix B.(5) The forces given in appendix B are withrespect to the space-fixed (inertial) coordinate system.(6) I n appendix B (rotational excitation) theterms in the upper portion of the bracketscorrespond to the motion 8=80eina,whilethose in the lower portion correspond t ox=xoeina.Forced Osc~llotionsResultinp From Tank k n d i n oBauer (refs.
2.31 and 2.33) has solved theproblem of liquid sloshing in a ring sectortank resulting from a res scribed bending-typemotion of the tank walls. That is, given amotion of the walls of the form %(z)etw orLATERAL SLOSHING IN MOVINQ CONTAINERSyo(z)eint (see fig. 2.6), he obtained equationsfor the velocity potential, liquid free surfacedisplacements, pressures, forces, moments, andvelocity distribution, as given in the following.Note there is assumed to be no interactionbetween the container and fluid. (Interactions between the moving liquid and the elasticcontainer will be studied in ch. 9.)The velocity potential is (the first termwithin the brackets corresponds to the motionz,,eiQtand the second term to y,,eiQt):9-+cos +1 = e n [{iQxo(z)r(Qyo(z)r sin2,+whereiQabmm~mn(z)=6.(2)i)(tmnFIGURE2.6.--Coodinate system and tank geometry forq2[id (tmn(1-$)bending excitation (ref.
2.31).z)d=-Cl29[IyciQtzO(0)r cos +(0)r sin+The pressure a t depth (-2) isx [ S ~ (tm.f €mng;)+-C O S ~(tmn])k(2.30)dfand primes denote Merentistion with respectto z. All other quantities have been previouslydefhed.The surface displacement of the liquid measw e d from the undisturbed positionThe liquid forces in the z and y directions are21k~mtzm(ktmn)]+~~(tmn))(2.31a)28TEE DYNAMIC BE~AVIOR OF LIQUIDSThe term M*- Q2eiQt z,,(z)dz in front of the double summation in F, represents the inertial force;hS:hi.e., the force that would be produced by an equal volume of solidified liquid.The liquid moments with respect to the point (0, 0, -h/2) are4hM+=MrQ2erQr2nua2(l+k2)yo(-h)-sin 2 t u cos 2 m2mThe first integrals in equations (2.32) represent the contribution of the pressure distribution fromthe circular walls.
The second integral is the contribution of the pressure a t the tank bottom,while the remaining integrals can be identified as the contribution to the moment from the pressuredistribution a t the tank sector walls. The last term in these equations represents the moment ofthe undisturbed liquid about the point (0, 0, -h/2).The velocity distribution is given asiQxO(z)cosO)ur=e'n'[ { i R y o ( z ) ain 4+?9 A,(z) cos (E 0) % Ch. (£in :)],,m-o(2.33a)LATERAL SLOSHINGu,='-e f ~[{'"t( sin)+C2 A ~ ( z )sin (g4)-iSly~(z) cos 4n-o rn-o-=em#29IN Moms CONTAINERSI)]CUza (trnn acos4[{iSlm$(z)s 4) cm/zaiQryA(z) sin 4 } +g2 A&n(z)~ 0 (gn - ~n - oThe velocity distribution in the tank is obtainedby omitting the first term in u, and u,; theseterms represent the tank motion.Solutions for the special cases of bending ofsector, annular, and circular tanks are givenin the following paragraphs of this chapter.(€mn:)I(2.33~)motions of a 45O sector tank, as summarizedbelow.For translation in the x- and y-directions,with excitations of the forms xoerntand y,,eint+(T, 4, z, t)=iSleiQ1Circular Sector Compartmented Tankwo nzbmnJ(d cash (fThe solution of the hydrodynamic problemfor cylindrical containers of sector cross sectionmay be obtained from the more general solutionspresented in the preceding paragraphs byallowing the diameter ratio b/a to approachzero.
The determinant Amna([)=O then reduces to J:,(~)=o,the zems of which aredenoted as ..,,,tThe expansion functions(2.33b)+ {c,,,yo)+K)(1-q2) cosh K4m4](2.34a)For pit&ing and yawing motions about the xand y axes, xoemland 6,,emt& z, t)=iSleinlX@Zcos 4IS0 S e c r ~TankBauer (ref. .2.5) determined the velocitvpotentials 'and the' natural frequencies cornwsponding t o . translation, pitch, yaw, and roll('1CP, 2, t)=i~e(~la2{6>'(?-:)+For roll excitation about the z axis, withexcitation &em':2 cos (8m-4)4(2m-1)[4(2m-1)z-l]r2 [~A-~..-2(2m-1)e~m-~,~lrl*2J~m-r(u*)cash (I*+K*)cos (8m-4)4+;(2m-l)[4(2m- 1)2-1](1-q*~ cosh K*In these equations, various quantities are employed as follows:(2.34~)THE DYNAMIC BEHAVIOR OF LIQUIDSabmnq2Bmn=~mn(1-v2)cosh K [(37-;)sinh (;)-2):(I);(ooshwhere ; are the roots of J i r n 4 0 = 0 andJ(u*)=J8rn-4E(r/a)].
Further, the quantitiesf*, u*, q*, and K*are defined byThe natural frequencies are obtained from theequationwhere the vdues a, are the positive roots ofJL(t) =o. Figure 2.7 shows the natural frequencies( 2) as afrnn=-function of liquiddepth; the roots a, are given in table 2.2.Figures 2.8 and 2.9 show, respectively, theforce in the x-direction corresponding to excitation in that direction and the moment aboutthe z-axis corresponding to roll motion, as afunction of the forcing frequency.Experimental data on force response in 45'sector tanks in translation have been obtainedand compared with these theoretical predictions(refs. 2.6 to 2.8).
Figure 2.8 shows relativelygood correlations in the magnitude of the forceresponse except, of course, in the immediatevicinity of the first mode resonance. On theother hand, there are marked differences in theexperimental and theoretically predicted resonant frequencies, with the former always lessthan the latter. This "softeningJ1characteristicis a consequence of the essentially nonlinearbehavior of compartmented tanks and thereforeis primarily dependent upon excitation amplitude, as can be seen in figure 2.10.
(Such nonlinear effects, essentially dependent upon geometry, will be discussed in more detail in ch. 3.)FIGURE2.7.-Naturalfrequency parameter for 45' -tortank (ref. 2.5).oOo Sector TankBauer (refs. 2.3 and 2.4) has also obtainedtheoretical results for predicting the behaviorof a liquid in a 90' sector (quarter) cylindricaltank both by solving the hydrodynamicequations (ref.
2.4) and by considering thisconfiguration as a special case of the ring sectortank (ref. 2.3). The forces and moments dueto translational, pitching, and roll excitationswill be summarized in the following.The velocity potential for translationalexcitation in the xdirection (wfQ1)is., t)=inz,crn{+~,,,b,,COBcos2m4 cash(K+f)J2rn(u)qa(1e q g ) cash K(2.36a)LATERAL SLOSHING IN MOVING CONTAINERS31where--,2 am=r ((-l)m+lm2-t)%-*em, are the roots of the equationand the7 , K , and r as given in equation (2.21) are valid if tm, is, replaced by em,.RGulE2.9.-MaWifiation&ctor for liquid momat M ,for roll excitation in 45O sector tank (ref. 2.5).The natural frequencies may be obtained fromthe equationFIGURE2.8.-Magnificationfactor for x component ofliquid 'fonx for excitation in x direction in 45' sectortank (ref.
2.5).and are shown graphically in figure 2.5, whilethe roots em, are given in table 2.2.Equations for the free surface displacements,forces and moments [with respect to the point(0, 0, -h/2)] are given in detail in appendix C.The forces and moments are plotted againstexcitation frequency in figures 2.11 and 2.12.In appendix C, the first term in F, correspondsto the inertial force of the liquid (fig.
2.11),the last term in the moment expressions correspond to the static moment, and the termsTABLE2 . 2 . - R ~ of J'm (tmd)=Oza;[Refs. 2.4 and 2.81THE DYNAMIC BEHAVIOR OF LIQUIDS-A-Cylindrical tank, WOsectors--a--Cylindrical-.+.-tank, 60'sectorsCylindrical tank, 4S0sectorsCylindrical tank theoreticalX,_@3@@45' sector 60° sector 90' sectoro0. 0020.0040.Excitation amplitude parametera ooa(o. oiox,l dFIGURE2.10.-Effsct of excitation amplitude on the bwastrcmonant frequmcy for 45'. 60'. and 90' rector tankr(rd.2.8).FIGURE2.11.-MagnScation kctor of liquid force in a 90'wr cylindrical tank (excitation abngx a i r ) (ref.
2.3).LOand L are defined byThe velocity potential for rotational oscillations about the y-axis excited by 8dm' isAs in the case of the 45O sector tank, measuredforce response (refs. 2.6 and 2.8) is in relativelygood agreement with theory, but measuredresonant frequencies are substantially less thanpredicted. The variation of frequency withexcitation amplitude for the 90' sector tank isalso shown in figure 2.10. Note again therelative values of frequencies for sector tanksof various vertex angles (compare with fig. 2.5).In addition, it should be noted that the ordinates (magnification functions) of the figuresin this chapter have the units of 1/8ec2.where A,, and B,, are given in appendix B,providing that appropriate values of a, andb,, are used2 and that Em, is replaced by t,.The free surface displacements, forces andmoments are given in appendix C, while theliquid force is plotted versus excitation frequency in figure 2.13.
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