H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 72
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Under these conditions, a method ofderiving a frequency equation given in reference9.8 can be summarized as follows: First, obtainwhere 9 defines the angular position in a crosssection; n and m/2 are, respectively, the numberof circumferential and axial wavelengths; andA, B, and C are constants. Then, using theseforms, the strain energy is expressed in termsof the displacements and the kinetic energy interms of the time rate of change of the displacements.
Further, both strain energy andkinetic energy are functions of the three dis-312THE DYNAMIC B E ~ V I O ROF LIQUIDSplacement variables u, v, and w, and theLagrange equations are utilized to form threeequations relating the displacements. A frequency equation is then formulated from thesethree displacement equations.Some results obtained from the methoddescribed above are shown in figure 9.9, inwhich frequency is given in the nondimensionalform&=j m n00L020\vherefmnis the natural frequency of the m, nthmode. In the axial wave length factor Am,the1 is t'he shell length, and a is the ratio of shellthickness h t o mean radius a.
I t might bepointed out that these results represent only oneof three natural frequencies that exist for eachnodal pattern, all of which can be obtainedfrom the above theory. However, these are thelowest of the three freauencies, and are therefore considered the mosi significant. From figure 9.9, progressing from figures 9.9(a) to 9.9(d),1040 0L0Axial wave length factor Am = m r a l l203.0FIGURE9.9.-Theoretical frequency curves for cylinders with freely supported ends (ref. 9.8).40iI..I,-- -.----.-.-----A\iIINTERACTION BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTURE313energy in the vibrating shell.
This is shownqualitatively in figure 9.12 where a strain energyfactor as a function of circumferential wavepattern is shown at a constant value of A,= 3.81,and a =0.01. Note that this curve correspondsto the frequencies of a vertical cross-plot offigure 9 . 9 ~at a constant value of Am=3.81. I tis now apparent that Rayleigh's purely extensional vibration theory is a good approsimationonly for small values of n (depending on t,hevalue of A,), the inextensional t,heory is goodonly for larger values of n, and neit,her isaccurate over int,errnediate vrtllies of n.
Funget al. (ref. 9.10) have shown that in the emptycylindrical shell the minimllm frequency isapproximately given byand has a nodal pattern of the nearest integer toO01.020Axial wave length factor. A,,,FIGURE9.10.-Natural-... ..--.' .. f - . !- ...- .. i.-..:=Ii1. .!. ._ _ > -.. .. - ... ,... .. .,- ;........ . .- :.,.:-...:..,v.........,..I-..-7:.],....i,'i",'~~.';,,,.
,--1- I..C. _...,; 4 . ..; -?3.0frequencies of empty cylinder8(ref. 9.8).the effect of wall thickness on the frequency isreadily apparent. An experimental correlationwith figure 9.9(d), also taken from reference 9.8,is shown in figure 9.10, where the agreement isseen to be very good.Figure 9.11, obtained by Lindholm et al.(ref. 9.9), is given for comparison with the abovework of Arnold and Warburton. The theoretical curves in this plot also are for freelysupported circular cylinders, but are based onthe shallow shell theory of Reissner.
Here theordinate scale is plotted to a slightly differentnondiiensional frequency parameter. Thenondimensiond form of all the above resultslends itself i v e l to tise ir deeig, qprox~iations.A very interesting observation can be madefrom figures 9.9 through 9.11. That is, for agiven tank and m-pattern (fixed A,), the lowestnatural frequency does not necessarily have thesimplest circumferential pat tern.
Arnold andWarburton (ref. 9.8) have shown that thisbehavior results from the partitioning of strainenergy between bending energy and stretchingAs long asxm<<mConsidering that in space vehicles (particularly in the first stages where cylindricaltanks are used) the tank walls are usuallyintegral with the shell of the entire structure,the assumption of freely supported ends maynot be a good approximation to the actualtank end conditions.
An approximate methodof estimating the effects of restraining momentsapplied at the ends has been presented byArnold and Warburton (ref. 9.11). Theincrease in frequency for a given mode as aresult of fixing the ends perfectly can beestimated by applying an appropriate correction factor to the results presented in figures9.9 t h r c ~ g k!?.!I, and the:: an c s t i ~ a t eef theactual increase can be made by selecting someintermediate frequency. This procedure is illustrated in figure 9.13.Having selected the curve for a particularmode from figures 9.9 through 9.11, say anm=3 and n=arbitrary curve, it would appearas shown in figure 9.13, and would represent anaxial mode pattern as shown in figure 9.13(a).THE DYNAMIC BEHAVlOR OF LIQUIDSFIGURE9.11.-Theoretical and experimental natural frequencies of empty cylinder8 (ref.
9-91.as in figure 9.13(b). The frequency f2 for thiscondition may be obtained from the curve atan equivalent wavelength factor given bywhere c,=0.3. The frequency f2 is then anestimated upper limit, and the actual frequency I\-ould fall sorne15-here be tween f, andf2. Details of this approximation are givenin reference 9.1 1.Breathing Vibrations of Shells Containing Fluids0261014Number of circumferential waves, nFIGURE9.12.-Strain energy due to bending and stretching(ref. 9.8).Having described in some detail the generalaspects of breathing vibrations of emptycircular cylindricill shells, 71-e now turn to theeffects of including multiple, immiscible, heavyand light fluids inside the shell. For thepresent case of space vehicle applications, weconsider a cylir~drical shell partially filledwith a heavy fluid (liquid) to some arbitrarydepth, such that the liquid has a free surface,and is topped by a light fl11ic1 (gas) at somearbitrary pre~sure.
Among various propertiesthat can be significant in the problem at hand,both fluids possess relatively small viscosity,and both possess finite compressibility anddensity, although these properties are d r n stically different for the liquid and the gas.During different parts of the flight, the relativeamount of liquid and gas inside the tankchanges as fuel is expended. In view of thisrather brief statement of the problem, thissection is devoted to a general discussion ofthe influence that the multiple fluid columnexerts on the vibrating tank, and the methodsof analyzing that influence.Considering the typical fluids, pressures,and geometries used in current vehicle tanks,ml~iiiiiberuf the h u w n snecLsit B S ~ S ~ Bthat~Sof the internal fluid column on the breathingvibrations of the tank can be summarized interms of several basic categories, most of whichdepend on the relative heights of the liquid andgas in the multiple fluid column.
Thesecategories are:(I) The viscosity of typical rocket propellants is small; therefore, inviscid flow theory isusually used in analyses of the problem.- AWavelength factor, A,=mtalRFIGURE9.13.-Effectsof end conditions on breathingfrequencies of empty shells (ref. 9.11).For a given tank with freely supported ends,corresponding to a given XI, one would obtaina given frequency f, at A,. However, for atank of the same geometry but with perfectly$x-ed ends, the mode pattern would appear316,THE DYNAMIC B E W O R OF LIQUID8(2) Internal static pressurization causes astiffening effect in the shell; it adds potentialenergy to the system.(3) The multiple-fluid column possesses bothcompressibility and inertia ; therefore, i t cansustain pressure modes of oscillation a t variousfrequencies, even in a rigid tank. I n the flexibletank, this adds additional degrees of freedom,and linear coupling of these modes with thebreathing modes of the tank is expected.Further, the possibility of the existence of highpressures a t some point in the tank, due to aresonance in such a pressure mode, can be veryimportant in itself.
Thus, the system cansustain coupled pressure modes as well ascoupled wall breathing modes.(4) The inertia of the fluid column causes ahydrodynamic loading on the shell wall which,in effect, results in a nonuniformly distributedapparent mass being added to the wall, so thatthe inertia of the breathing tank is considerablyincreased. This effect r e s d ts predominantlyfrom the liquid pnrt of the fluid column, becauseof its greater density.(5) The liquid free surface phenomena addadditional degrees of freedom to the breathingfluid-tank system, so that linear coupling of thismotion must be considered.(6) I t has been found tliat, umder cbertttinconditions, the free surface motion can causethe tank brenthing vibration to become stronglynonlinear, giving rise to n r i o u s nonlinearcoupling effects.The above listing of the influence of the flulidon the breathing tank is given more or leyh inthe order of increasing complexity of sniilysis.Two basic methods of approach hnve beer^ uhcdto analyze the fluid behavior-compressible flo~vtheory, and incompressible flo~v theory, butneither alone is capable of predicting all effects.The beha\-ior is spread over a wide frequencyrange and, b y and large, each analysis is restricted to some part of this frequency rtinge.Inviscid co~n~ressiblefluid theory, along \\.it11some set of shell equations, has been used toinvestigate effects mentioned in p n r a g r ~ p h s(2), ( 3 , and (4) above.
The wave equation isused as the governing differen tin1 equation forthe fluid, and the interaction \\.it11 the tank wallis specified by equating the normal accelerationsof the fluid and the shell a t the shell wall, alongwith equating the fluid pressure and hydrodynamic loading a t the same point. This typeof analysis is particularly useful for determiningthe coupling effects of pressure modes in thefluid column, and can be used to predict thehydrodynamic inertia effects of both the liquidand the gas, but is not normally used to predictliquid free surface effects.Inviscid incompressible fluid theory, alongwith some set of shell equations, has been usedto predict effects indicated in paragraphs (4)and ( 5 ) , and even those of ~ a r a g r a p h(3).
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