H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 60
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8.35).When a tank containing a heavy liquid isforced to vibrate vertically, a t a frequency nottoo large in comparison with one of the first fewnatural frequencies of free sloshing discussedin chapter 2, a pattern of standing waves canbe observed on the free surface. Contrary toone's intuition, it has been observed that thewave frequency, that is! the frequency of thesurface oscillations, usually occurs at exactlyone-half that of the container motion, althoughin some cases the liquid frequency is equal tothe forcing frequency, and in other cases,greater.
The frequency of the free surfacevibrations for transverse excitation, of course,corresponds precisely to t'he forcing frequency,as discussed in chapter 2.This apparently anomalous behavior forvertical excitation was first noticed by Faradaydming a Series of investigations of vibratingplates covered by a thin layer of water (ref. 8.1).271VERTICAL EXCITATION OF PROPELLANT TANKSIn his experiments, it appeared that the freesurface waves vibrated with a frequency whichwas only half that of the plate. About 40years later, a similar series of tests was conducted by Mathiessen (refs. 8.2 and 8.3), andon one very important point he recorded anopinion in opposition to that of Faraday:in his experiments, Mathiessen found that theliquid vibrations were synchronous.
Since thetheory of gravity waves had not been developedto a point a t which this contradiction could beresolved analytically, Lord Rayleigh repeatedFaraday's original experiments with improvedequipment. Rayleigh observed, in support ofFaraday's view, that the liquid oscillationswere a one-half subharmonic of the forcingmotion; that is, they occurred at a frequencyequal to half that of the excitation (ref. 8.4).In addition, by generalizing his theory of forcedvibrations which he had originally restrictedto systems having only one degree of freedom,he developed a tentative theory to explain theliquid oscillations (refs.
8.5 and 8.6). Hisanalysis led to a single Mnthieu equation;using this analysis, Rayleigh wns able to showthe existence of subharmonic surface waves.It is now known, as will be shown in thefollowing sections, that vertical vibration causesa quiescent liquid free surface to becomeunstable for much the same reasons as thosediscussed by Taylor (ref. 8.7) in his treatmentof the well-known Rayleigh-Taylor instabilities.The theory of such liquid motions leads to asystem of Mathieu equations.
Because of themore complete development of the theory ofMathieu functions since Rayleigh's time, it isnow possible t o show that Faraday, Mathiessen,and Rayleigh might all have been correct; also,the actual dynamics of the liquid motion, whichare important in missile stability analyses, maynow be calculated.Linearized TheoryAs in most dynamical systems, the equationsof motion for the liquid contained in a verticallyvibrating tank can be developed either from anintegral equation (energy) approach, or from adifferential equation approach. The energymethod, which is especially suitable for approximate calculations or for tank shapes which donot correspond to any standard coordinatesystem, has been developed extensively by-Moiseyev (refs. 8.8 and 8.9) and Bolotin (ref.8.10). However, the more elementary differential equation approach is sufEcient for a largenumber of practical cases, and it is the methodemployed in this chapter.The linearized theory developed here followsclosely the analysis of Benjamin and Ursell(ref.
8.11) and is given with as much generalityas possible. The results are then specializedin later sections when particular examples arediscussed. I n nearly all of this section, theelasticity of the liquid container is neglected;this subject is discussed in detail in chapter 9.Suppose that a rigid cylindrical tank ofnrbitrary cross section, but having a flat bottom,is Wed to a depth, h, with an incompressible,inviscid liquid. The tank is vibrated veiticaliywith an amplitude, x,,, and a frequency, NU,where N is a positive number.
The frequencyof the liquid motion is assumed to be w, so thatspecifying N determines, in effect, whether theliquid response is subharmonic, harmonic, orsuperharmonic.Cartesian axes (x, y, z) are fixed to the tank,with the z-axis pointing upward along thetank axis. The undi~t~urbedfree surface corresponds to z=O and the tank bottom to z=-h.Thus, the axes more with a vertical acceleration iV2w2xocos Nut, and the motion relative tothese axes is the same as if the tank were atrest and the gravitational acceleration wereg-N2w2x0 cos Nwt.
The fluid velocity relativeto the moving axes may be derived from apotential, @, that satisfies Laplace's equationThe Eulerian equations of motion call beintegrated by using @ to yield the unsteady formof Bernoulli's equation-8+E-Lat 2[(??)1+(??)1+(%)]dx-( g - W x ocos Nwt)z=Owhere p is the fluid pressure measured relativeto the ullage pressure and p the fluid density.272TEIE DYNAMIC BEHAVIOR OF LIQUIDSAt the free surface there is an additional pressure restriction of the formwhere v is the surface tension, and R1, R2 theprincipal radii of cnrvature of the surface. I tis acceptable to consider only small motionsand displacements in n linearized theory, sothaton the same bounding curve.
These conditionsnre typical of an eigenvalue problem, and consequently t>heyshow that and q can each beexpanded in terms of a complete orthogonal setof functions S,(x, y). Hence, from equation(8.3) it can be seen that1:- .-.:...I,..* . .I--**IIin the fluid interior, andwhere z=q(x, y, t ) is the equation of the freesurface. Consequently, the linearized form ofBernoulli's equation that must be satisfied atthe free surface is 2-(g-N2w2a0 cos ~ w tV )] 2-0 =O(8.2)I t must also be required that the velocity ofthe free surface normal to itself must equal thefluid relocity normal to the sorface.
Whenlinearized, us shown in chapter 2, this requirement takes the fol.1111.n(x, Y,t ) =m-05 %(~)S"(X,Y)-+--" a'qm-oS, ( r , y)m-o(8.7)Substituting equations (8.7) into equation (8.2)shows thatThe remaining boundary conditions are thatthe perpendicular velocities at the walls andbottom must be zero; that isnt the walls, andnt tile bottom, z= -h.By combining equations (8.3) and (8.4) itcnn be seen that b ~ / b n = Oon the curve bounding the free surface; hence, from equation (8.2)one can show thatThis equation is strictly valid only if the liquidcontact angle at the tank walls is 90".* I ',f.iion the bounding curves or surfaces.
Xm are theeigenvalues. I t then follows (ref. 8.11) that therequired expansions are5 ~ a ~ ( t ) s , , , (y)s ,y, z , t ) = - 2 km(t) cash im(zfh)i,sinh j;,h"%Ias,-bn -Oad-I-am+Xrntanh L ~ ( ? ~ + ~ - cos~ PN N~ ~ X ~for m=O, 1, 2, . . ., and, of course, m specifiesthe particular mode under considoration. Usingthis equation, the unknown time varying liquidsloshing amplitude call be determined.I t may be seen immediately that one solutionof equation (8.8) is am=O, identically. Thissolution corresponds to no sloshing motion atall; i.e., the liquid column acts as a rigid bodyand vibrates up and down as a slug.
Underideal conditions such a response may be possible,but if the free surface is disturbed in any way,one should expect that the disturbance wouldtend to grow into some more skable liquidi.I<*VERTICAL EXCITATION OF PROPELLANT TANKSImotion.
Consequently, the problem at handis one of the stability of a quiescent free surface,and equation (8.8) actually determines thekinds of excitation for which the free surface isunstable; that is, it determines the conditionsfor which disturbances tend to grow. I n orderto find these conditions as simply as possible,it is best to let the frequency of the liquidmotion be variable and the excitation frequencybe fixed at w, instead of, as heretofore assumed.letting the liquid frequency be fixed and theexcitation frequency vary through the parameter N. (It will soon be seen that theoriginal formulation, however, is more convenient in R nonlinear analysis.) So, lettingN = 1, and introducing the parameters4h; tanh imhPm=141210864(s+- y)FIGURE8.4.wStability chart for the solutions of Mathieu'sequation.allows equation (8.8) to be written rnore simplyas~+(p,,,-2p,,,dl-cos 2T)am=0(8.9)which may be recognized as the standard formof Mathieu's equation.I n the usual treatises on Mathieu functionsit is shown that equation (8.9) caxi have stableor unstable solutions, depending on the valuesof p, and q,.
The regimes of stabilitp andinstability are usually shown in a plot such asfigure 8.4. The shaded areas in this figure correspond to unstable solutions of equation (8.9),the solutions for which a,+as T+-.Onlythe first three unstable regions are shown, andthe predominant frequency of the iiquid. motionfor each area is indicated on the plot. Theunshaded regions correspond to stable solutionsof equation (8.9).When xo and w are given, a point on thestability chart may be determined for each ofthe sequence of eigenvalues i,.To determinewhether or not the free surfnce is unstable, p,and Q, must be calculated for each mode andtheir position on the stability chart observed.'I t can be argued that at least one of the points(p,, q,) for some mode lies in an unstableregion, whatever xo and w. In practical cases,however, where the system is slightly dissipative, the higher order modes tend to besuppressed, and an appreciable disturbance ofthe free surface occurs only when one of thelower order modes is excited.A good deal more information about possibletypes of unstable liquid motion can be obtainedby replotting figure 8.4 with dimensionalparameters.
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