H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 71
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9-6).dependence on actual mode shape, thereby indicating the necessity for a numerical formulation of a specific problem.9.3COUPLING OF PROPELLANT MOTIONS WITHELASTIC TANK BREATHING VIBRATIONSGeneral Discussion000.20.40. 60.8Fractional depth of liquid (bU)FIGURE9.6.-Theoretical1.0and experimental capped resonant bending frequency variation with liquid depth forsimply supported tanka (ref. 9.6).Vibrations of vehicle tanks in which the wallmotions are predominantly radial, such thatboth flexure and stretching of the wall occurswhile the longitudinal axis of the tank remainsstraight, are often referred to as breathing~brati$na.
For tha pl_?_rpcsesnf t,hiw chapter:the term "breathing vibrations" will includeshell modes that display both circumferentialand axial wave patterns for rotationally symmetric tanks, with the axially symmetriccircumferential modes being included as onecase. I t is recognized that this type of vibration can be very important in shell-like structures where the walls are thin compared toother dimensions. Although most space ve-310THE DYNAMIC BEHAVIOR OF LIQUIDShicle tanks are not simple shells, since theyusually have internal stiffeners or other structural components of various designs, it is highlydesirable to attempt to predict their breathingvibrational behavior on the basis of simpleshell theory as an approximation to the actualtanks, since analytical predictions are otherwisevirtually impossible.
Of course, the accuracyof the predicted behavior depends on howwell the actual tank conforms to such arepresentation.The purpose of this section is to discussbreathing vibrations of thin shells, as theyapply to rocket vehicle tanks, and to considerthe influence of the liquid propellant andpressurization on this type of vibration of thetank wall. Virtually all of the work in thisarea to date has been concentrated on thecircular cylindrical tank, obviously becauseof its vide use in the main stages ofcurrent vehicles ; theref ore, the present emphasis will also be on that geometry, althoughit must be admitted that breathing vibrationsin shells of other geometry can be equally asimportant.
The intent here is to give, first,a rather general discussion of shell vibration,and to describe in some detail the behavior ofempty circular cylindrical shells to havesome measure of understnrlding of this typeof vibration in purely shell-like structures.Then the variations in this behavior caused byinternal pressurization and the presence offluids will be considered.Breathing Vibrations of Empty ShellsA review of the work done to date on thevibration of empty shells of various geometries,including a general discussion of assumptionsoften used in simplifying the analysis of theproblem, together with an extensive referencelist, has been presentjedby Hu (ref. 9.7). Onlya few comments from this review \\dl berepeated here to introduce our discussion ofempty shell vibrations, and the results of onlya few of the more fundamental andyses wvill bementioned.
I t is pointed out that, because ofthe complexity of the problem, the analyticalas well as the experimental results accumulatedin the technical literature do not present anentirely clear picture of shell vibrations, evenfor the simplest shell geometries. The maindiEculty lies not in the formulation of a setof equations describing the shell vibrations,but rather in the simplification and solution ofthese equations. As a result, various approximate theories are introduced to solve certainclasses of shell vibration problems.RayleighJs theory of inextensional vibrationsassumes that for the fundamental modes, themiddle surface (an imaginary surface situateda t midthickness of the shell wall) of a vibratingshell remains unstretched.
The displacementfunctions of the middle surface can be determined with this condition, and the fundamental frequencies are then found from thepotential and kinetic energies corresponding tothese displacements. On the other hand,Rayleigh's theory of extensional vibrations,wvhich would be more appropriate for certainmodes, assumes that the shell deformationconsists mainly of stretching in the middlesurfnce.Considering the strain energy that is periodically stored in the shell \vall during vibration,the two approximate theories appear to be onopposite extremes. In the theory of inextensional vibrations, the strain energy isassumed to be associated with bending stressesonly, and no membrane streqses exist, since thedisplacements are determined by the inestcnsion condition.
But,, when membrane theoryis used to study estensiond vibrritions of n shell,the strain energy is assumed to be solely associated with membrane stresses and that associated with bending is neglected. These twoapproximations do not contradict each other,but are complementary for the purpose ofdetermining rnrious vibrational modes. Thetendency is to classify shell modes into thesetwo distinct groups.In addition to the above two general approximate theories, there are other approachesproposed for certain specific allell configurations.The Donne11 equations for. circular cylindricalshells and the Reissner equations for shallowshells are exmiples (a shallow shell is one inwhich the ratio of rise to span is much less than1, say one-eighth).
In these theories, thegoverning differential equations are simplifiedby neglecting what are considered to be the311INTERACTION BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTUREunimportant terms, based on geometric arguments for the specific case or on an order ofmagnitude analysis. The accuracy of theresults obtained from any of the approximatet.heories depends entirely on how well the basicassumptions are satisfied in a given case. I naddition, i t must be pointed out that virtuallyall of the existing theories predict the freenatural vibrational modes of a shell; therefore,how TV ell these conform to t,hoseencountered forforced vibration again depends 011 the specificcase investigated.Numerous investigations of circular cylindric d shell vibration have been conducted, oneof the most significant being that of Arnold andWarburton (ref.
9.8). In this work, the emphasis is on vibration of shells wvith freelysupported ends (simply supported ends),whereby the ends of the cylinder are heldcircular but are not held by restraining moments. Under these conditions, the modeshapes of vibration nssunie patterns like thoseillustrated in figure 9.8. For such vibrationalforms, a given mode is completely specified byt 1 ~ 0integer parameters: n, the number of circumferential stationary waves, and rn, thenumber of axial stationary half waves.
Although not sho~vnin the figure, it is understoodthat the n=O, m arbitrary (axially symmetriccircumferential pattern) modes are included inthe present consideration, while n= 1 is not,since it corresponds to a bending motion,already discussed in the previous section ofthis chapter. Theoretically, an infinite numberof such modes is possible in a given shell. I tis recognized that in a space vehicle tank,simply supported end conditions may not beexactly duplicated, and the effects of varyingend conditions will be discussed later.Considering vibration in the above modes,-,-,:+I.,.LLPIUUGLD.-.-l-:-I.,-:----&---:---I--- - - - ~ - - - : - - - lCircumferential vibration forms-n=2,n=4n 3Axial vibration forms-.--(b)---_ANodal arrangementn =3, m .
4(C 1$>j/(Y''Axial node\ CircumferentialnodeFIGURE9.8.-Forms of vibration of thin cylinders withfreely supported ends (ref. 9.8).equations relat.ing the t,hree component displacements u, v, and w in the directions x, y, z(fig. 9.8), of a point in the middle surface ofthe shell. I n the present case, this is mostreadily accomplished by assuming a vibrationform compatible wvith the end conditions; i.e.mmu=A cos - cos n9 cos wtlm m sin n9 cos w tv=B sin Iw=C sinmrxCOS 7t8 COS wtl(9.4)L L ~ J l O l ~JU 1 U G A C P U ~ l U UU~U I tJAMIIJlUU&lassumptions are correct over a wide range ofthe integers m and n, and, therefore, thoseapproximat,ioon cannot be used in this case.For modes of vibration in which both circumferential and axial modes exist, both bendingand stretching of the shell \$-all must be considered.
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