H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 32
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I t has been assumed that theacceleration scale, a,, is fixed by the respectiveacceleration fields encountered by model andprototype. I n many problems, practical fabrication problems control the geometric scaleratio, and, in some instances, the impliedcompression or expansion of the model timescale, equation (5.35)) may not be feasible oreconomic.Given that the acceleration scale is importantin any given simulation, and that the time scaledoes not impose really serious limitations onthe experimental designer, appioxirnate simulation plots showing permissible geometricscale ratios, D,, as a function of the requiredacceleration scale ratio, a,, are useful in showingthe types of simulations which can be achieved.The corresponding simulation plot for the analysis to this point is devoid of restrictions, butno material properties have been introducedinto the analysis thus far.
I t is the question ofexactly what construction materials and fluidsshould be used in the model that introduceserious restrictions on what can be done.These questions will be taken up in the nextsections with the implied understanding thatall the nondimensional parameters developedin the preceding sections will hold.Structural Material PropertiesChapter 9 of this monograph may be referredto for a more detailed account of the importanceof dynamic interaction between the sloshingfluid and the elastic deformations of the cont'aining structure. Thus far in the similitudeanalysis, only the density of a geometricallysimilar structure has been considered.
Whenthe elastic properties of the material are described by Young's modulus, El and Poisson'sratio, t,r23 =ElpaDfor stress, Z, and strain,c,we have(5.36)153The preceding r-terms in conjunction withthose in the previous section define a "replica"structural tank model whose dynarnic deflections mill be geometrically similar to thoseof the prototype a t corresponding times definedby equation (5.20). To illustrate this notion,the theoretical frequencies of the natural modesof vibration of plates may be lkritten asWa=pa('X (a nondimensional function ofplate geometry)(5.40)(t, denotes plate thickness). If the ratio ofnatural frequencies of a plate in the replicnmodel to the corresponding plate in the prototype is taken, we haveIf m4 is the same in model and prototype, ?rls issatisfied and if E, is written from equation(5.36)Er= P P ~ D ~(5.42)then(Wr)2=arlDr(5.43)Equation (5.43) is exactly the same as thefrequency ratio required by equation (5.21).The ratio expression (5.42) provides a meansof evaluating what sort of simulation might befeasible for replica structural modeling usingdissimilar structural materials (ref.
5.13) withsimultaneous satisfaction of inertial scaling,and this expression has been partially plottedin figure 5.1 as lines of D. vs. a, for values ofthe composite ratio E,lp,. The ratio Elpfor anymaterial actually used in the structure of abooster rocket or spacecraft is likely to besomewhere near maximum for known materialsbecause of the premium placed on unnecessaryweighi arid, consecizently, the composite ratio,El/p,, will probably not have a value exceeding1.0, whatever the choice of model materials.If the weak, heavy composites now underdevelopment (ref. 5.13) are included, modelmaterials might be available to provide aminimum value of Ellprof about 0.01. Thusthe simulation plot (fig.
5.1) is divided into aband bounded by the lines: EJpr= 1 and 0.01 in154THE DYNAMIC BEHAVIOR OF LIQUIDSproach still involves strict geometric similarityinsofar as the fluid boundaries are concerned,but involves relaxation of structural geometricsimilarity so that the struct,ural portion of themodel is "adequate" for purposes of studyingfluid interaction wvith the gross tank bendingand "breathing" modes of vibration. Thefollowing development is taken largely fromreference 5.9.The tank walls may be constructed of unstiffened plate, stiffened skin, sandwich structure, or thin skin pressurized for stability.Assuming the stiffness properties to be isotropicwvith respect to dimensions in the plane of theshell, the followving load-strain relations expressthe rigidity of the shell:FIGURE 5.1.-Approximatesimulation ranges-replicamodel of elastic tank, inertial scaling of fluid dynamicsonly.which simulat,ion under the stat,ed assumptionsmay be feasible; far outside this band, simulation is highly unlikely.If simulation of the plastic behavior of thematerial is attempted (yield strength, c,, forexample) :nz7 =a,/P ~ D(5.44)Equation (5.44) is of the same form as equation(5.42), and much the same arguments applyto the feasible range of [(a,),lp,] as applied tothat for E,lp,.
Consequently, figure 5.1 willroughly apply to this case also.There appears to be little hope for this typeof simulation in the low gravity case if thelaboratory is a t 1 g (a,>100), a t least not withany reasonable geometric scales. There ispotential in the case that t,he prototype is in arelatively high acceleration field and the modelis a t 1 g, especially if the model can be aslarge as or slightly larger than the prototype.On the whole, replicn modeling of st'ructure isexpensive and sometimes impossible. (See thefurt'her discussion of replic,a modeling in pt.
I1of this chapter.) An alternate approach isdeveloped by Sandorff (ref. 5.9). This ap-The notation is illustrated in figures 5.2 and5.3.'Y*t t t tt N y Shape uf elementI TIillFIGURE5.2.-In-plane loadings and deformations of shellelement: Exteneion and shear (ref. 5.9).SIMULATION AND EXPERIMENTAL TECHNIQUESz-rC--FIGURE5.3.-Transverse loadings and deformations ofshell element: Beam shear, bending, and twisting(ref.
5.9).N,=normal force per unit width on edgesparallel to y-axisN,,=shearing force per unit width, equalon x- and y-edgesM,=bending moment per unit width onedges parallel to y-axis155M,,=twisting moment per unit width, equalon x- and y-edgese,=extensional strain parallel to x-axisy,,=in-plane shearing strain; i.e., shear ofan element parallel to xy planey,,= transverse shearing strain; i.e., shearof an element parallel to xy planew= transverse displacement of neutral planeof shell wall, w(x, y, z)w,,= bending curvature of neutral planeabout y-axis, b2w/bx2w,,= twisting curvature of neutral plane,bZw/bxbyA =extensional stiffness, (A= Et/(l-y2) forhomogeneous plate)p,,= in-plane shearing stiffness, (&= Gtfor homogeneous plate)p, =transverse shearing stiffness, (p,= Gtfor homogeneous plate)Da=bending stiffness, (Db=Et3/12(1-;2)for homogeneous plate)B = torsional stiffness, (B=Da for homogeneous plate);=Poisson's ratio;'= Poisson-type elastic constant indics ting coupling between moment aboutone axis and curvature about theother (;=';for homogeneous plate)E=elastic (Young's) modulus in extensionG=shear modulust =thicknessThe stiffnesses (fl,,, bZ, Da,and B) can bedetermined either analytically or experimentally.
These relations govern the elastic behavior of isotropic sandwich structure andisotropic symmetrically stiffened sheet, andtherefore permit consideration of a wide variation in shell-wall detail to meet overall elasticproperties. Orthotropic stiffened skin structurecan be handled in the same manner, withadditional elastic constants to define anisotropy and cross-coupling of elastic deformationprocesses.The advantage of the above representation ist,hat fine detail such as the actual thickness ofthe wall, dimensions of stiffener detail if astiffened shell is used, attachments, etc., neednot necessarily be duplicated. The overallshell-wall extension and bending are the imme-156THE DYNAMIC BEHAVIOR O F LIQUIDSdiate concern, not extremely local effects suchas material rupture or local buckling.Rupture is a design consideration, however,and, therefore, stress distributions must oftenbe obtained.
Stresses, as such, are not recognized as existing in the shell wall in the aboveload-strain relations which specify only forcesN and moments M. Similitude is to be preserved with respect to N and M, not stress, andto convert from one to the other, the particularsection detail (local thickness, etc.) must beused, which will not necessarily be in scalefrom model to prototype. Similarly, insofaras unit weight is concerned, the significantparameter for the shell wall is patw,mass perunit of surface area, not of volume, where t,is an equivalent distributed thickness measurement for the wall structure.There may be portions of the tank structure,e.g., baffles, which, in deforming elastically,affect the fluid motion in an important manner,and which cannot be treated as thin platesunder plane stress.
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