H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 6
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2.3.)It is often convenient to wr;ite the velocitypotential as the sum of the potential of the container motion, &, and the potential of the liquidmoving relative to the container, Q1. That isIf the container is stationary, #,=0, while if thecontainer is in motion +, can be found by integrating the equationV&=V (container)-iand the constant of integration may be takenas zero, since it can be absorbed in 91.The9, is independent of container geometryand has been evaluated for roll, pitch, and translational motion of the container (ref. 2.18). Itshould be noted thatsatisfies the Laplaceequation for pitch and translation, but not forroll motions of the container. In this lattercase, the sum of &+a1 is made to satisfy theLaplace equation by making 4 the solution ofa Poisson equation.A complete discussion of the derivation of theabove equations may be found in references 2.17and 2.19.
Reference 2.17 should also be consulted for an excellent discussion of the boundary conditions which are simply stated inlinearized form below.At the container wallswhere a/&& denotes differentiation in the direction normal to the surface of a rigid body incontact with the fluid, and v, is the commonvelocity of the fluid and boundary surface in thedirection normal to the surface. ImposingBernoulli's law and the condition that the fluidparticles must stay on the surface, and thenlinearizing the results, leads to the equationsdescribing the free surfacea@zSgb=--By eliminating 6 between these two relations, asingle equation for @ is obtained asFor most cases, it is assumed that po=constant,so thatIn the case of pitching motions of an accelerating vehicle, the free surface condition is(ref.
2.20)8 h + p ~ T -&r+ p ~ T ~ t i -be"p ~ T y - = 0be*bt"bz(2.12)btwhere 8, and 8, are the pitch angles about thez and y axes as shown in figure 2.1, A is thetotal acceleration in the z direction, AT is theacceleration in the z direction due to thrust, andA=AT+g. For a ground test, the accelerationof the vehicle due to thrust is zero so - thatAT=O, A=g. For pitching motions duringlaunch, the effect of the gravitational field magbe neglected so that the body force is dueentirely to the thrust and the inertia force dueto gravity and the body force due to gravitycancel each other; therefore, g=O and A=AT.For oscillations given by 8,= levlei*', el= le.le'@',the free surface condition given above may besimplified by the transformationwhere & and & satisfy the free surface condition*S+Aiii=O- .1PPO%O _--at azwhere z=6(x, y, t) is the equation of the freesurface, and po(z,y, t) is the surface pressure.x, X 'FIGURE2.1.-t&,di~.teafor adcbatingY'LATERAL SLOSHING IN MOVING CONTAINERSFor the case of no pitching motion, equation(2.12) reduces to2.2RECTANGULAR TANKNatural FrequenciesThe mode shapes and frequencies in arectangular tank (see fig.
2.2) are given byLamb (ref. 2.13, sec. 257)which is equivalent to equation (2.11b).The foregoing linear boundary value problemis generally solved by employing the classicalmethod of separation of variables, and muchinformation concerning the behavior of liquidsin moving containers has been obtained forseveral practical container geometries, includingthe compartmented cylindrical tank.In addition to the technique of separation ofvariables, several other methods have beenemployed to analyze lateral slosbing of liquidsin containers. Budiansky (ref.
2.21) used asophisticated integral equation technique tofind the natural frequencies and slosh forcesin the cases of a half full, and nearly fullspherical container and a circular canal filledto an arbitrary depth. Lawrence, Wang, andReddy (ref. 2.22) have shown that the problemof finding a velocity potential O wuch satisfiesthe equations (2.7), (2.9), and (2.11b) (w,=O ineq.
(2.9)) is equivalent to finding a velocitypotential which makes an integral an extremum.This conclusion is significant in that not onlyis the governing d8erential equation automatically satisfied, but all necessary boundary conditions are also. The velocity potential maybe obtained approximately by employing theRayleigh-Ritz technique. This technique hasbeen used rather extensively (refs.
2.22 to 2.25).'hoesch (ref. 2.14) employed an inverse technique to find the natural frequencies andobtained results for several interesting geometries; he also gave an interesting discussion of-ALuw--Iwlab--uaup-L:-A:LuL-A----u wUw U W L I-.--A,-.:--bULlU&LLLWA--.,.-Aa=& m-02 A.,n-0cos[y(z+z)]X cosrf(y+i)]oi.=gk tanh (kh)(2.13a)(2.13b)wherem ,and 6 is theelevation of the liquid above the referencelevel z=h/2.
If a>b, the lowest frequency ofinterest is obtained by letting m=l, n=Om, n are integers ranging from 0 toCoordinates of pointI a12. b12. h127+-~ G W L Y U V I Jand liquid natural frequencies. It was shown,for example, that if two containers have thesame free surface and if one is completely contained in the other, then the smaller containerpossesses the smaller eigenvalue. Ehrlich (ref.2.26) investigated the finite merence technique applied to obtaining the fundamentalfrequency in axisymmetric fluid-filled containersof arbitrary bottom geometry.FIGURE2.2.-Coordinate sylrtem and tank geometry forrcctangolat tank.18THE DYNAMIC BEHAVIOR OF LIQUIDSapplication of this analysis to low gravityproblems, it will be discussed further inchapter 11.The higher frequencies may be obtained fromequation (2.13b) once the dimensions a and bare known.
If kh is small, equation (2.13b)may be approximated byForced OscillationsThe response of a fluid to simple harmonicmotions of the tank in translation, pitching, andyawing is given in reference 2.28. This paper isa condensation of reference 2.29, which containsa more detailed discussion, in addition to thesolution of arbitrary tank motions through theuse of Laplace transforms. The assumptionslisted in section 2.1 are employed, with theadditional condition that the angle of containerrotation during pitch is small.Eide (ref.
2.27) has presented the results ofan analytical investigation of the problem ofthe variation of pitch motion of a vehicle underlow inertial forces due to fuel sloshing in arectangular tank. He assumed that the forcesdue to thrust, rotation, and capillary actionare of the same order of magnitude, and thatthey are all much greater than the Coriolisforce, and then determined the velocity potential and frequency of the free surface oscillationgenerated by an impulsive torque. The equations of motion derived for the rotation of avehicle containing a large amount of liquidwere then investigated to determine the generalbehavior of the vehicle and the stability of thefree surface motion.
Because of the specificHorizontal Motion Porolkl to x-AxisThe velocity poten tial, the horizontal forceon the container in the z-direction, and themoment acting on the container about the yaxis (positive moments are given by the "righthand" sign convention) for a displacement ofthe container given by z(t)=z,, sin (Ill) are asfollom:sin (2n+l)XLsinw..hj2%-o[x cosh (2n+ 1) - z+cosh [(2n+ l ) r1[ !] '(r, =h/ah(2.14~)a[tanh (2n+ 1)$1f-, n23= (hlg)4(2.148)8 tanh [ ( ~ + 1 ) ~ 1 1 11$(2n+ l)'rlM"rn=?fsin ntwhere nowu:= (2n+ 1)')I}WF=pghab (total weight of liquid) =gMIPitchino About y-AxisThe velocity potential, horizontal force inthe x-direction, and moment acting on the container about the y-axis, for an angular displacement of the tank walls in pitch given byB(t)=B,, sin Ill, are as follom:LATERAL SLOSHING IN MOVING CONTAINERS+5sinh [(2n+ 1) h]31M#m=e0f2~hnt-8 tanh [(2n+l)*t11P(2n+ l ) h1-+I-+2MA*12j'r:-[(2n+ 1)17,]C 8 tanhg(2n+1)~r,tanh [@n+ 1) rl](2n+ 1) rlI8 tanh [(2n+l)?rr1]+-2f: j-+2:d(2n+l)3~1where I' is the effective moment of inertia ofthe fluid about the y-axis (moment of inertiawith the free surface fixed).Yawing About z-AxisThe velocity potential and the moment actingn-oon the container about the z-axis (there are noresultant horizontal forces), for an angular displacement of the tank walls in yaw given by+(t)=+, sin M, are as follows:20!lTlB DYNAMIC BEHAVIOR OF LIQUIDS(2n+ 1 )r3(2n+16ab(b2(2n+1)2-a2(2m+l)'] sechXjyI[sinh (2n+ 1)cosh [(2n+ 1 )g]z]m n + l)2+a'(2m+1)']~~(2m1)'(2n++1)'[b2(2n+1)2+a2(2m+ 1)'lwhere the resonant frequencies w,, are given by&.=g5 4b2(2n+ l)'+a2(2m+ 1)' tanh+b4 tanh [(2n+ 1)9.3CIRCULAR CYLINDRICAL TANKThe upright cylindrical container of circularcross section is obviously of considerable importance for our discussion here because of itsrelationship to the configuration of large boostvehicles.
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