H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 73
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T h a tis, this type of analysis is useful for predictingliquid apparent mass, and free surface effects,and can further be used to predict coupledaxisymmetric, longitudinal pressure modes, fori t is found that these modes coincide with thecorresponding axisymmetric (n=O) shell breathing modes, for the case of a relatively incompressible fluid in a flexible tank. A number of thenvailnble analyses consider only the hydrodynnmic. loading effects of the liquid, neglectingthe free surface effects, while some considerboth. The degree to \vhich free surface effectsare neglected, in the former ease, varies in thecliff erent studies tliat have been performed.All of tlie tlnalyses assume potential flow so thntLapluc~e'sequation is assumed to be the governing differential equation for the liquid; theboundary conditions a t the shell \\-all requirethat the liquid normal acceleration be equal tothat of tlie tank and that the liquid pressureequal the hgdrodynamic loading of the tank\\.all in all cases, but different solutions resultfrom the imposition of different boundaryc.nnditions nt the liquid free surface.I t has been pointed out in chapter 2 thatthe basic linearized free surface conditions forfree surface oscillations are the following: Atthe free surface we must havewhere [ is the elevation of the free surfaceabove the undisturbed surface level, and is nfunction r=r(r, 8 ) in cylindrical coordinates,and cp is the velocity potential.
Equation(9.5), called the dynamic condition, requiresthat the dynamic pressure at the free surfacebe constant (i.e., the ullage pressure), whileequation (9.6), called the kinematic condition,requires that no fluid mass cross the free surface boundary. I t has further been pointedout that both conditions can be combined togiveon the free surface. Just how these conditionsare imposed at the free surface is n-llnt determines whether the free surface coupling isincluded in these analyses using incompressibleflow theory.Nonlinear coupling of free surface motionsand tank breathing vibrations has been investigated experimentally, and analytical investigations are in progress, but no specificresults have yet been reported.Each of the following portions of this sectionis devoted to a discussion of various analysesand investigations of the above-mentionedliquid effects on breathing vibrations of shells.Virtually all of the work has concentrated oncircular cylindrical shells having freely supported ends and rigid bottoms.
Some of theanalyses investigate more than one of theindicated effects, but none is capable ofpredicting all effects.Light Compressible FluidsFung et al. (ref. 9.10) have investigated bothanalytically and experimentally the effects ofinternal pressurization on the breathing modesof freely supported cylindrical shells, for thecase where only a gas is present in the fluidcolumn. The pressurizing gas is consideredto have no mass; as a result, it adds potentialenergy to the overall system, but the inertiaof the system is the same as that of an emptytank. Strictly speaking, this is a limiting caseapplication of compressible flow theory, sincethe inertia of the gas is neglected and, as aresult, the wave velocity becomes large.
Theanalysis is applicable to the actual case onlyfor vibrational frequencies well below thecoupled pressure resonances in the compressedfluid column.In contrast with the method of deriving ashell frequency used by Arnold and Warburton,here the Timoshenko shell equations (ref. 9.12),written in terms of the displacements u, v, w ,are utilized, and the loading terms are comprised of the static internal pressurization anddynamic tank inertia terms. Inertia of theinternal fluid is neglected.
Displacements ofthe type in equations (9.4) are assumed and arather complex frequency equation results.For comparison, a simplified frequency equation is further derived using Reissner's shallowshell theory (ref. 9.13), \\-ith the longitudinaland tangential shell inertia terms neglected inthe governing differential equations. This results in the equationpo being the static internal pressure in thecylinder.
I t must be emphasized that thisexpression is most accurate in general for largervalues of n (i.e., say n>2), where shallow shelltheory applies, and no coupling with fluid pressure resonances is allowed. The existence ofthis trend was determined by comparing theresults of this frequency equation with theresults of the previously mentioned, moreexact frequency equation. Equation (9.8) is,of course, also suitable for the empty tank forpo=O. Results for this case are s h o ~ - nto besimilar to figures 9.9 through 9.11 of t,heprevious section.Results calculated from equation (9.8) whenthe internal pressurization is not zero are shownin figures 9.14 through 9.16.
The lowestnatural frequency and its associated mode shapefor a given empty shell were discussed in theprevious section. From figure 9.14, it can beseen that pressurization alters the mode shapeof the lowest mode to a lower order pattern,while, at the same time, all natural frequencies318THE DYNAMIC B E H A ~ O R OF LIQUIDSequations for the circular cylinder, with loadingterms composed of internal static pressure, shellinertia due t o radial motion only, and fluidinertia from the apparent mass of the pressurizing fluid, the following frequency equation isformulated for the coupled breathing modes2o012345Internal pressure parameter $x67810'FIGURE9.14.-Number of circumferential waves at thelowest frequency versus internal pressure (ref. 9.10).are increased. This is shown in figures 9.15and 9.16, dong with experimental correlationwith equation (9.8).I t is further pointed out (ref.
9.10) that for agiven forced input to a cylindrical shell, theanlpI'itude of the response of the various modesnt resonance is very erratic, and larger responses are by no means confined t o the lowestmodes. Therefore, where breathing responsesin shell-like strlictures are anticipated, a lnrgernnpe of frequencies must be examined.Compressible Fluids With InertiaT o study pressrlre resonunce coupling overu wide runge of parameters in n breathing shell,nlong with the hydrodynamic loading of theshell \vull, it is necessary t o include both thecoinpressibility and inertia of the fluid column,\vllet her it con tnins on1y pressurized gas orliquid, or both pressurized gas and liquid.Such nnnlyses itre applicable at frequencies,where the colnpressibility effects are important,but are xiot readily capable of including theliquid free surface effects.Berry and Reissner (ref. 9.14) have investignted the effects of pressurization with fluidsof significant mass, where only one fluid completely fills the tank.
Using shallow shellwhich is most accurate for circumferential values n 1 2 . I n this equation,D = Ehs/12(1- u s )and mj is the apparent mass of the internalfluid. The fluid apparent mass is derivedassuming small mot.ions of an inviscid, compressible fluid; hence the wave equation issolved with boundary conditions such that theunsteady fluid pressure a t the cylinder wallrepresents the fluid loading on the tank, andthe normal velocity of the fluid .at the tank wallis the same as the jvdl ve1ocit.y. Solutions forthis pressure distribution are taken in the formxxq=ei*' cos no sin 1 Q(r)As a result, the following apparent mass isobtained:mj=poafor k,<X1for k,>X1(9.10b)whereXI=-,pois the reference static fluid density,wa1 kc=--,Cn co being the velocity of sound innathe fluid a t the reference pressure, and J , andI , are the ordinary and modified nth-orderBessel functions, respectively.It can be seen that the solution for the frequency, w, from equations (9.9) and (9.10),is a difficult task.
Tables of solutions areINTERACTION BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTURE02468Internal pressure, p,, dynelcm' x 10''1012FIGURE9.15.-Theoretical and experimental natural frequencies versus internal pressure (ref. 9.10).319320THE DYNAMIC BEHAVIOR OF LIQUIDSgiven in reference 9.14 for some limited valuesof the parameters involved. In general, it isconcluded that the effect of including inertiaand compressibility of the internal fluid columnis to lower the natural frequencies of the breathing vibrations of the shell in comparison with012lnternal pressure, p,3.
dyne / c m a x lo-'4FIGURE9.16.-Theoretical and experimental natural frequencies versus internal pressure (ref. 9.10).their values (i.e., those predicted by eq. (9.8)or for m,=O in eq. (9.9)) when the inertia andcompressibility of the internal fluid are absent.In addition to the coupled shell breathingmode frequencies given by equation (9.9),there also occur higher natural frequencies, eachof which is associated with a particular numberof pressure nodal circles in the oscillating fluidcolumn. That is, the assumed form of pressuredistribution nllo~\~sfor a multiple pressure wavepattern to be established in t,he fluid, in theradial direction.
For heavy liquids, it isestimated that the frequencies of these coupledpressure modes are nearly equal to those of afree boundary column of liquid. In otherI\-ords, the presence of the liquid in the shellhas a significant effect whereby it lorn-ers thetank breathing modes, while the flexible tank\vall has a less significant effect on the frequencies of the assumed form of pressureresonances in the free column of liquid.Saleme and Liber (ref. 9.15) have investigated the effects of a compressible fluid columnon the breathing shell, both for the case of theshell completely filled with one fluid, and alsofor the case of the shell filled with two separate,im~nisciblefluids.
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