H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 19
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3.4).99NONLINEAR EFFECTS IN LATERAL SLOSHINGthis inclined plane about a vertical axis thatwe are attempting to describe. This phenomenon almost invariably occurs in laboratorytesta at frequencies in the immediate neighborhood of one of the reson'ances of the normalsloshing modes, as mentioned above, andoccurs whether the liquid has any initial grossrotation or not; the rotational mode can, however, be initiated at any excitation frequencyby introducing some disturbance which provides s substantial initial rotation to the fluid.Theoretical AnalysesAn attempt at providing an analytical de'scription of this type of liquid motion was b tmade on the basis of the behavior of an equivalent linear conical pendulum (ref.
3.22). Thistheory was able to predict the general characterof the stability boundaries, as shown in figure02426283.0123.4Frequency paramler, d d l g16200hld0.25F I G U 3.10.-AmpiitndeX~of liquid fom rwpanw in 6phui d tu& for v u i o &citation~amplitdas (ref.
3.4).quency (refs. 3.22 and 3.23). The essentialfeatures of this complex liquid motion can bedescribed qualitatively as an apparent "rotation" of the liquid about the vertical axis ofsymmetry of the tank, superimposed on thenormal sloshing motion (ref. 3.23). The motionis even more complicated as a type of "beating"also seems to exist; the first antisymmetricliquid-sloshing mode first begins to transformitself into a rotational motion increasing inangular velocity in, say, the counterclockwisedirection, which reaches a maximum and thendecreases essentially to zero and then reversesand increases in angular velocity in the clock;T&&%d31?,g d er? nn dt~rnately.
Thefrequency of rotation is less than that of thesurface wave motion and therdfore the liquidappears to undergo a verticaI upanddownmotion as it rotates about the tank axis; therotational frequency about this upanddownaxis is about the same as that of .the wavemotion. The liquid free surface, a t least a trelatively low excitation amplitudes, is essent i d y plane, and it is the apparent rotation ofSymbol01Frequenq ratio, wlw,FIG-3.11.--Cornparka of experimental data withtheoretical stability bounduieo for rotational liquidmotion near lint-mode rwonana under transverseexcitation (ref.
3.23).~100THE DYNAMIC BEHAVIOR OF LIQUIDS3.11, although other features of the motioncould not be explained; obviously, the phenomenon involves essentially nonlinear effects.An extension of the theory of the conicalpendulum to include third-order terms in theamplitude soon became available (ref. 3.24))which then led directly to the desired nonlineartheory of liquid motion (refs.
3.19 and 3.25).1°The nonlinear analysis shows three types ofliquid motion to occur: normal planar motion,stable nonlinear (rotary) motion, and unstable(swirling) motion. These are in excellentagreement with the observed behavior (refs.3.4, 3.19, 3.22, and 3.23). The theory alsodemonstrates that the rotational mode of liquidmotion arises as a consequence of a nonlinearcoupling between liquid motion parallel andperpendicular to the plane of excitation, andthat this coupling takes place through the freesurface waves.The essential results from Hutton's theory(ref. 3.19) can be stated fairly simply by reference to the equations (3.83)) which govern thecoefficients of an approximate solution for nonlinear liquid motions near first-mode resonance.These equations are nonlinear and possess anonplanar swirl motion solution as well as theusual planar motion solution.
For s*l motionj l=r,fi=o, j3=0, f:=+FKlr-I(3.88)The amplitude 7 is again determined by a cubicequation; that is,In order thatf, ba real and nonzero, it is necessary thatThe approximate steady-state solution thentakes the form :This is essentially the theory us presented earlierin this chapter. .411 equivalent analysis has r~lsobeendeveloped clscwhcrc. (refs. 3.20 nntl 3.21).G=t113(fl ms 9 ms(ot)+fi sin 9 sin (wt)}J1(Xllr)The swirl motions predicted are not simple rotary motions unless If,\ r lfll and the higherorder terms are negligible.
The instabilityitself then leads to more complex modes ofliquid motion, as described earlier.Experimental DataFigure 3.12 shows some comparison betweentest data and the predicted stable and unstableplanar and unplanar regions. The predictedboundaries of the instability region agreeclosely with experimental data for smallvalues of t. Poorer agreement for the non-- 1.50-51.005 a.8z O.-271.
5001.621.661. 70L 741.781.82Frequency (cps)loFIGURE3.12.-Comparimn of theoretical and experimentalinrtability regions for nonplanar aloloshig (ref. 3.19).NONLINEAR EFFECTS IN LATERAL SLOSHINGplanar motion stability boundary is not surprising because of the difficulty in preciselydetermining this behavior; as the frequency isdecreased, the transition is gradual from thesteady-state harmonic nonplanar motion tothe nonharmonic nonplanar motion existingin the instability region.Some interesting experimental information onswirl motion was also obtained during a study ofnonlinear lateral sloshing a t large excitationamplitudes (ref. 3.4), as shown in figure 3.13.Most of the data shown in figire 3.13 mereobtained by maintaining constant excitationamplitude while varying excitation frequency.At first, data were taken only for liquid motionsarising basically from the first antisymmet,ricslosh mode, with no swirl occurring.
Subsequently, the boundary of the swirl region wasdefined experimentally by maintaining constantexcitation frequency and slowly increasingescita tion amplitude until the swirl motionbegan to appear. The onset of swirl could bedetermined relatively precisely not only byvisual observation but by the appearance of asignificant force normal to the direction ofexcitation on the oscillograph output. Theswirl boundary thus obtained is shown infigure 3.13. I t should be noted that thisboundary is dependent upon both excitationfrequency and amplitude, since in certaininstances it was found that large amplitudebreaking waves could be produced without101Frequency parameter wrdlgFIGURE3.13.-Liquidfree surface response in circularcylhdrical tank showing swirl region (ref. 3.4).I n addition, it should be noted that1within the swirl region the motion of the liquidhas a phase angle of near 90" with respect to theinput displacement.
The data presented hereare intended primarily to sho\v the essentialeffects of increasing excitation amplitudes.REFERENCES3.1. ABR.IMSON,H. N.; GARZ.~,L. R.; AND K.IN.I,D. D.: Some Notes on Liquid Sloshing in Compartmented Cylindrical Tanks. ARS J., vol.32, no. 6, June 1962, pp. 978-980.H.
N.; CHU, W. H.; A N D G.1~z.i~3.2. ABRAYSON,L. R.: Liquid Sloshing in 45" Sector Compartmented Cylindrical Tanks. Tech. Rept. No. 3,Contract NAS8-1555, SwRI, Nov. 1962.3.3. ABRAMSON,H. 3 . ; A N D GARZA,L. R.: Some\T~nsnremezts of Liquid Flaquencies andDamping in Compartmented Cylindrical Tanks.AIAA J.
Spacecraft Rockets, vol. 2, no. 3,May-June 1965, pp. 453-455.3.4. ABRAMSON,H. N.; CHU,W. H.; A N D KANA,D. D.:Some Studies of Nonlinear Lateral Sloshing inRigid Containers. J. -4ppl. Mech., Dec. 1966.3.5. STOKER,J. J. : Nonlinear Vibrations. Interscience Publishers, Inc., New York, 1950 (3dprinting, 1957).H. N.: Nonlinear Vibrations. Ch. 4,3.6. ABRASISON,Handbook of Shock and Vibration Control.McGraw-Hill Book Co., 1961.3.7. ABRAMSON,H. N.; CHU, W. H.; A N D GARZA,L.
R.: Liquid Sloshing in Spherical Tanks.AIAA J., vol. 1, no. 2, Feb. 1963, pp. 38k389.3.8. DALZELL,J. F.; CHU, W. H.;' AND >IODISETTE,J. E.: Studies of Ship Roll Stabilization Tanks.Tech. Rept. no. 1, Contract NONR 3926(0n),Southwest Research Institute, Aug. 1964.N. 3 . : On the Theory of Nonlinear3.9. >IOISEYEV,Vibrations of a Liquid of Finite Volume.Appl. Math. Mech. (PMM), vol. 22, no. 5,1958, pp.
860-870.G. S.: Concerning the Motion of a3.10. NARIMANOV,Container Partially Filled With a Liquid Taking Into Account Large ?ciIotions of the Latter(in Russian). Appl. Math. 31ech. (PPrlM),vol. 21, no. 4, 1957. Also available as a Space1023.11.3.12.3.13.3.14.3.15.3.16.3.17.THE DYNAMIC BEHAVIOR OF LIQUIDSTechnology Laboratories, Inc., TranslationSTL-T-Ru-18.PENNEY,W. G.; A N D PRICE,A. T.: Some GravityWave Problems in the Motion of PerfectLiquids. Phil. Trans. Roy.
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