H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 75
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Using Lagrange'smethod, the equations of motion for the coupledsystem are determined, and after the ussumption of simple harmoriic motion, a3(N+ 1) order frequency determinant is obtained.This frequency determinant, similar to that ofLeroy (ref. 9.17), can then be solved numericallyfor any value of N, depending on the accuracydesired. (Because of the complexity of thefrequency determinant that results from theabove procedure, the d e t i i s will not be givenhere, but the original report (ref. 9.18) should beconsulted.)Numerical computntions were performed topredict a number of natural frequencies for twodifferent cylinders subjected to various conditions, and the results were compared with experimental values. Detailed characteristics of the.two cylinders are summarized in table 9.1, p.
330.For the simplo case of s tank pressurized witha light gas and N=O (one-term approxinlationof the displacements), the resulting frequencyequation is identical to equation (9.8), and theexperimental results obtained were similar tothat shown in figures 9.15 and 9.16. For apressurized liquid filled tank and N=O, theresulting frequency equation is similar to equations (9.9) and (9.10) for the case of k,=O,except that additional terms are included whichaccount for the effects of hydrostatic pressureof the liquid and finite cylinder length.
Correlation of predicted results with experimentsfor this case is shown in figure 9.22. (The upperfigure is for cylinder 1, and the lower figureis for cylinder 2.) For partial liquid depths,326THE DYNAMIC BEHAVIOR OF LIQUIDS8 120Uis looFa2-F=Theory (one term)7-8 0 'C-0I024681012141618Number of circumferential waves, n180Internal pressure Experiment160140-?s120loo3-fC280604020O-Theory lone term)-method outlined in the section on breathingvibrations of empty shells to make the endconditions equivalent to freely supported ends.Only m = 1 modes were investigated ; however,the analysis is applicable to higher modes aswell, as long as N is chosen to give sufficientaccuracy.An analysis for p a r t i d y liquid filled cantilever cylindrical shells has been given by Baronand Skdak (ref.
9.20), 11-hich is based on similarwork of Baron and Bleich in references 9.19 and9.21. Although these analyses are for liquids incylindrical shells simply supported a t the bottomand free a t t,he top, the methods of approachcan readily be modified and used with boundaryconditions more applicable to space vehicletanks, as has been done by Mivson and Herr,previously described.The case of longitudinal forced vibration of acylindricnl tnnk filled with an incompressiblefltlid lrns been inrestigated by Bleich (ref.
9.22),~rsinguii tipprosinlute nntilysis. The emphasis ison the nsisym~netric,longitudinal pressure dis tribution in the fluid column, and the effects ofthe \\-all el~st~icityare approximated by considering the tank to be a series of rings, ratherthan using a conlplete shell analysis. Freesurface cffects nrr neglected. The results arecornpur:~blc t c l the findings of Reissner (ref.9.1G),mentioned in the preceding section, aswell as the water-hammer analysis given inchnpter S. That is, for a relatively incompressible liquid in a flesible cylinder, one canneglect the compressibility of the liquid; for thiscnse the nnulysis of Reissner (ref. 9.16), and the\\.titer-hnrnmer analysis of chapter 8, essentiallyreduce t o the analysis of Bleich (ref.
9.22).Althoqh the liquid is considered incompressible, t~ longitudinal asisymmetric pressuremode, having a maximum at the tank bottom,still occurs in a system in which the inertia isassumed to lie solely in the liquid, and theeffective ~ompressibilit~yto be solely in the tank\\.all. I n effect, the resulting modes representn=0 shell displacement modes which can beupproxirnated by-024681012141618Number of circumferential waves, nFIGURE9.22.-Theoretical and experimental natural frequencies for pressurized, water-filled cylinders (ref. 9.18).for both the cases of N = O and of A'= 1 (o11cand two-term npprosimntion to the displaccments), cnlculntions nlso were perfomled iiildcomp~~redto esperirnetltal results.
For convenience, some of the compared results lireshown in figllres 9.3311 thro~tgli 9 . 2 3 ~ . (Alldatn nre for cyliiider 2, and in= 1 mode*.)Theoretical curves are show11both for the es11c.ttheory and for n simplified tlpprosinlutioli ofthe liquid appnrcnt mass. The experiment nlresults were obtained from cylinders havingfixed ends, but nre conipared to theoreticalresults for freely srtpportecl ends by ~tsinpthewhere At, is determined by the liquid depth b,INTERACTION BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTURE240III0IEmeriment (fixed ends)Ratio of water depth to cylinder length, blLI240 --Ratio of water depth to cylinder length, b U320000.20.40.60.8Ratio of water depth to cylinder length, blLLOFIGURE9.238.-Theoretical and experimental variation of natural frequency of a pressurized cylinder with water depth(ref. 9.18).
pa=3.45X lo4 dyneIcm2.229-648 0-67----22THE DYNAMIC BEHAVIOR OF LIQUIDS400-----Exactvirtual massApproximate virtual mass,,Approximatevirtual mass b o terms00I"00. 2a40.6Q8Ratio of water depth to cylinder length, b l lI01.00 Experiment (fixed ends)Ia20.4. 0.6a8Ratio d water depth to cylinder length, b l l1.0Yrcuar 9.23b.-Theoretical0-0.8(121140.6Ratio of water depth to cylinder length, b l l1.0and experimental variation of natural frequency of a pressurized cylinder with water depth(ref. 9.18).
po= 1.38X l W dyne/crn'.INTERACTION0BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTURE0.2a40.6a8Ratio d water depth to cylinder length, b l l0FIGURE9.23c.-Theoretical329LO(L 2(146a8Ratio of water depth to cylinder length, b l L1.0and experimental variation of natural frequency of a pressurized cylinder with water depth(ref. 9.18).
p0=5.51 X 1W dynelcmz.THE DYNAMIC BEHAVIOR OF LIQUIDSTABLE9.1.-Details of Experimental Cylinders[Ref. 9.181rather than the tank length 1. Obviously thissimplified assumption of displacement violatesthe zero-displacement boundary condition a tthe tank bottom, and a, series is required forexact satisfaction of that boundary condition.A more detailed numerical analysis of theabove type of longitudinal modes has beenperformed by Palmer and Asher (ref.
9.23), andverified experimentally. This work becttnleavailable only shortly before the final print,ingof this monograph so that the details could notbe included here. However, the existence ofthe n=O bulging modes described above isverified, and it is found that these modes areintermixed frequencywise, with the nonaxisyrnmetric breathing modes (n22), described in theprevious sections of this chapter. This reference should be consulted for the details of suchmodes.
One more recent investigation of thesemodes is discussed a t the end of the nextsection.incompressible Fluids (Free Surface Effects Included)Breathing vibrations of partially liquid filledshells, with the free surface effects neglected,were considered in the preceding section. I twas pointed out by Baron and Bleich (ref.9.19), however, that for the case of partiallyfull cylindrical shells simply supported a t thebottom and free a t the top, with free surfaceeffects included, the coupled breathing modeswere still very nearly the same as if free surfaceeffects had not been included, while additionalresonances corresponding to coupled fuel sloshing modes occurred a t very low frequencies, nearthe values they would have in a rigid container.In this present section, similar results will befound for a freely supported cylindrical shell.Chu (ref.
9.24), and Chu and Gonzales(ref. 9.25), have investigated breathin,a vibrations, including free surface effects, for apartially liquid filled, freely supported cylindrical shell. Displacement functions of the formfor a completely empty shell as in equations(9.4) are assumed in the computations for thepartially fulI tank, allowing no distortion of themode shape. The dynamic shell equations ofYu (ref. 9.26) are used, except that tangentialand longitudinal inertia for the shell is neglected.The hydrodynamic loading of the shell wall isdetermined, allowing for both inertia of theliquid and free surface effects. Potential fluidflow is utilized, as has been described previously,except that here the free surface boundarycondition is given by equation (9.7).
I t isfound that, in the hydrodynamic loading, theadditional terms due to the free surface effectsare very small unless the excitation is in thevicinity of a frequency corresponding to anatural liquid mode for a similar rigid cylindricalcontainer. Since in practical cases small damping is always present, it is concluded that theseeffects are negligible except for the very lowestliquid modes.
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