H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 45
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6.15).This is because liquid motions in a sphericaltank are inherently more nonlinear than intanks with parallel sides (see ch. 3); thus theresults of a linear theory, such as the equivalentmechanical model shown in figure 6.7, cannot beexpected to yield results as accurate for spherical tanks as for other configurations.Experimentally determined parameters for apendulum model in a spherical tank were compared with analytical values by Sumner (ref.of these models. Since such models are usedmostly in connection with waves whose lengthsare relatively long in comparison to the liquiddepth, they will not be investigated further.The available analyses for spherical tanks(refs.
6.6, and 6.15 through 6.17) are essentiallynumerical (see ch. 2); hence, the parametersfor the mechanical model are presented in therameter agreed well, eicept for nearly full ornearly empty tanks (where nonlinear effectspredominate), the pendulum-sloshing mass ratiodid not. The other two parameters, pendulumarm length and hinge-point location, also agreedfairly well.Equivalent mechanical models for arbitrarilyshaped conical tanks are not available.
How-FIGURE6.5.-Simplemechanical model for rectangulartanks.6.17).While the filnda.ment,al freqi1enc.y pti-THE DYNAMIC BEHAVIOR OF LIQUIDSFIGURE6.7.-Simple-pendulum mechanical model for spherical tanks (ref. 6.6).ever, a partially complete model for a 45"conical tank (refs.
6.6 and 6.18) is shown infigure 6.8 and table 6.3. The centroidal moment of inertia, l o , of the rigid mass, m ~ is, notgiven in the references; consequently, themechanical model is valid only for translationalmotion of the tanli, and not for pitchingmotions.I n most instances, the foregoing modelsadequately represent the dynamics of liquidsloshing. The two chief restrictions to theiruse are : (1) the excitation frequency should notbe close to the natural frequencies of any of thehigher order sloshing modes, and (2) the modelis not a valid representation for frequencies inthe very near neighborhood of the naturalfrequency of the fundamental sloshing mode.Tile first restriction can be lifted by using themore complete mechanical models presentedin section 6.3, t ~ n dthe second restriction can berelaxed by including appropriate darrlpingfactors in the model, as will be done in section6.4.
I n some cases, provision must be madeFIGURE6.8.-SimpleTABLE6.3.-Modelmechanical model for 45' conicaltanks.Parameters for 45'Tank[mT=:/Sprhl;10not available]Conbl207ANALYTICAL REPRESENTATION OF LATERAL SLOSHINGfor rotary sloshing. This is discussed in thefollowing paragraphs.Pendulum Analogy for Rotary SloshingIt is an experimental fact that when acylindrical tank is translated periodically at afrequency very near the natural frequency ofsloshing in the fundamental mode, the liquidwave motion ceases to oscillate about a singlenodal diameter. Instead, a rotational wave isobserred to wash around the tank boundary;that is, the nodal diameter rotates. (Thisphenomenon is discussed in greater detail inch.
3.) I t is known also that if the point ofsupport of a conical pendulum is oscillated nearits natural frequency, the pendulum motiondeparts from the plane of excitation, and theplane of the to-and-fro motion of the pendulumrotates (refs. 6.19 and 6.20). Since the fluidmotion away from resonance is representedadequately by a pendulum, it seems possible,therefore, that the rotary motion of the fluidmay also be represented by a pendulum. Amodel of this type has been analyzed and,indeed, it is a fair representation of the essentialcharacteristics of rotary sloshing (refs.
6.21 and6.22).The rotary motion of both the liquid and thependulum analog are caused by nonlineareffects in which energy is transferred betweenvarious modes (or degrees of freedom). Ananalysis shows that in both cases there arethree regimes of motion :(1) Stable planar motion except in a narrowfrequency band centered approximately aroundthe natural frequency.(2) Stable nonplanar motion in a narrowfrequency band immediately above the naturalfrequency.(3) Nonstable motiotl (swirling in which therotation of the nodal diameter changes constantiyj in a narrow frequency band immediately below the frequencies for which motionsof type (2) occur.T h e proof of these statements follows directlyfrom an analysis of the conical pendulumshown in figure 6.9.
The mass of the pendulumbob, ml, and the length, L,, are the same as forthe corresponding mechanical model for ordinary sloshing (fig. 6.4). The equations ofMotion of4Support ' E L , coswtFIGURE6.9.-Conical-pendulum model for rotary sloshing.motion for the pendulum, correct to third orderin the direction cosines (n/2-a, n/2-P, y), areThe angle y has been eliminated in these twoequations by the relation sin2 y=sin2 a +sin2 P.The stability of the various types of solution ofequations (6.17) has been studied by Freed(ref. 6.19) and Miles (ref. 6.20). The mainresults are given in terms of a dimerisionlessfrequency, v, defined by v =c2"where[q]w:=g/L,; they are:(1) Simple harmonic, planar motions arestable for v< -0.945 or v>0.757.(2) Simple harmonic nonplan ar motions arestable for v>0.154.(3) Simple harmonic motions, either planaror nonplanar, are unstable for-0.945<v<0.154.I t can be seen that for v>0.757, both stableplanar and nonplanar motions are possible.Which one is obtained depends on the initialconditions; that is, if there is some initial rotation present, a rotational motion is obtained,but, if not, a nonrotational motion is obtained.The stability properties of the differentialequations (6.17) also have been examined withthe aid of analog computers, with essentiallythe same conclusions.208THE DYNAMIC B E R A ~ ~ O ROF LIQUIDSin this section are all derived from analyticalresults, since it is usually not ~ r a c t i c a l todetermine the size of more than one sloshingmass from experimental measurements alone.Although such analytical theories are in generalonly applicable to clean, unbaffled tanks, it ispermissible, in the light of many experimentalverifications, to use the same inertial parameters to simulate sloshing in baffled tanks ifrt dampening mechanism is included in themodel.Cylindrical Tanks (refs.
6.95 through 6.33)Ratio of forcing frequency to first sloshing resonant frequencyFIGURE6.10.-Comparisons of experimental and conicalpendulum model regions of rotary sloshing (ref. 6.22).Figure 6.10 shows a comparison between thestnbility boundaries of a conical pendulum andthose of sloshing in a cylindrical tank. Morerecent esperimental evidence (ref. 6.23) has shownthat the right-hand stability boundary forsloshing is a strong function of slight imperfections in the tank geometry or excitation.For more nnd more exnctly lateral translation,roundness of tank, etc., the stability boundaries for sloshing and for the conical vendulumnpproach each other more and more closely.Thus, a conical pendulum appears to be agood mechanical model for rotary sloshing.Another formulation of a model that includespitching as well as translation of the tank hasbeen developed recently by Bauer (ref.
6.24)for this type of sloshing.6.3COMPLEX MECHANICAL MODELSAs indicated in sections 6.1 and 6.2, a complete mechanical model must include a pendulum or spring mass for each of the infinitelymany sloshing modes of the liquid. Physically,each mass in the mechnnicul system correspondsto the effective mass of liquid that oscillates ineach particular slosh mode, and, from its size,it is possible to assess how significant that modeis. The complex mechanical models describedThe method used to compute the modelparameters from analytical results for sloshingwill now be described in detail for the case ofa rigid circular cylindrical tank, with the analysis following closely that given in references6.25, 6.29, and 6.30.
The mechanical model isshown in figure 6.11; the sloshing action isrepresented by a set of spring masses, but asimilar analysis is valid for a pendulum model.po is the rotation of the tank about an axisthrough the center of grnvity of the liquid; yois the lateral displacement of the tank; and?I, is the displacement of the mass, m,, withrespect to the tank walls.Each of the spring constants, K,,are chosensuch that their ratio to the oscillating mass isequal to the square of the na.tura1 frequency ofthat mode; that is,tn are the zeroes of J;([,)=O.I n order topreserve the static equilibrium of the liquid,the sum of all the modal masses and the fixedmass must equal the total liquid massTo keep the vertical location of the center ofgravity unchanged as it should be for smallliquid motions, ho must satisfy the relation209ANALYTICAL REPRESENTATION OF LATERAL SLOSHINGresponding equations for sloshing.
For thispurpose, it is assumed that the pitching andtranslational motions are both simple harmonicin time; that is, proportional to, say, cos wt.Then equation (6.21) can be solved for y,;the results areFree surfacetandBy using these relations, the total force on thetank can be written asNow from t,he results of chapter 2, the actul~lforce due to sloshing is *FIGURE6.11.-Complexmechanical model for cylindricaltanke.However, a dynamic analysis must be usedto calculate the remaining unknown parameters:m,, h,, and Io. To begin with, the equation ofmotion for each of the oscillating masses can bewritten asThe reaction force of each of these masses onso that the totalthe tank is simply -K,y,,force acting on the tank can be expressed ash direct comparison of equations (6.25) and(6.26) results in the following equations form, and h,:I n this e q ~ a t i o n , the factor Zmngqo arisingfrom the sum of equation (6.21) for all modeshas been canceled by a factor mdcp,; that is,the potential energy change due to gravityalone is zero for a pure pitching motion ttboutthe center of gravity.Equations (6.21) and (6.22) must be put inti form suitable for comparison with the cor-andk-[i-- ---h:1'tanh I.
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