H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 62
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.- .- -,,and equation (8.12) as0rnC C mnBmane+"fsin mx sin nxm-1 n - l=(8.21)Equations (8.20) and (8.21) can be solvedby the method outlined previously (eq. (8.15)and the accompanying text). Penney andPrice (ref. 8.12) used this method to solve a setof equateionsvery similar to equations (8.20) and(8.21) ; details of their analysis are given inThe constant term bo is l~ecessarilyzero, sincechapter 3. By using the same procedure here,the plane q=0 locntes the mean positior~or theand then examining the resulting equations tosurf ace.determine the orders of magnitude of the variousBefore substituting equations (8.16) nndterms, it follows that a, or 0, are of the order of(8.17) into (8.11) and (8.12), it is much moremagnitude (al)" or (6')"; that is, a2and & areconvenient to cast all the parameters intosecond-order terms in comparison to al anddimensionless form.
Becnuse the first mode is; a g and O3 are third-order terms in comparisonassumed to be predominant, the appropriatetoa1 and p,, and so on.Consequently, it islength for use in nondimensionalizing is L / ~ Trelativelystraightforwardtowrite down theand the appropriate time is (L/27rg)'J2. (Con]governingequationstoanyorderof approximapare this with the nondimensionalizing oftion.However,onlyathird-ordertheory iseq. (8.11).) Let $t be the nondimensionalgivenherebecausethealgebraicworkbecomesform of @, [ the nondimensional form of q, a,estrernely laborious for higher order approxima;the nondimensional form of a,, 8, the nontions.
But as will be seen, it is necessary todimensional form of b,, s the nondimensionalretain a t least third-order terms, since a lowerform of t , and B the nondimensional form of. The nondimensional form of the variables order approximation does not yield any quantitative information about the sloshing amplitude.x and 2 will still be 3: and z, but the tankBecause the amplitude of the free surfacedimensions are now -T _< x 5 T. The nondimenmotion, and not the velocity potential, is moresional surface wave frequency is denoted byeasily correlated with experimental data, theu =wl(2 rg/L) It2, where (21rg/L)w2is the naturala, are eliminated from the approximate equafrequency of free small amplitude waves.tions.
Hence, by modifying equations (3.58)The series expansion of the velocity potentialand (3.59) of chapter 3 to conform to thebecomes.279VERTICAL EXCITATION OF PROPELLANT TANIZSnomenclatureof this chapter and by including thechanges necessitated by the factor (1 - 4 2 e cos 2ur),the first-order linear approxima tion to thesloshing amplitude is found to be81-k(1-4'J2e COS 2 ~ ~ ) 8 1 = 0(8.22)it is the lowest order approximation thatpredicts the sloshing amplitude as a function offrequency.It is now assumed that the solution ofequations (8.24) can be written in the form of aFourier series, either as-The secohd-order approximation is found to be&f (1- 4 2 6 cos 2 4 ~ ) 8 ~ = 0&+2 ( 1 - 4 9 e cos 2 a ~ ) & = k.& =CA, sin nu7n-1(8.23)or as&+(1 -4.2.cos 2ur)B1 ( 1 -2p2-;1=2B?/%+BIn=lE)(t k-i3fi)-s18282+2(1-4u2e cos 2ur)f12=8:j3+3(1-4.4.
. 3 .cos 2 ~ r ) & = 3 & & - ~ 818:(8.24)The first-order equation is a Mathieu equation, and is simply the nondimensional form ofequation (8.8) for m=l and N=2. Thesecond-order set consists of a Mathieu equationfor and an equation for B2 which contains thesolution of t h e first equation as a parameter.This set does not result in any improvement ofthe frequency-amplitude relation but onlyrefines the surface wave shape. The third-orderequations are nonlinear and describe largeamplitude waves to a better approximationthan the lower order equations.
Furthermore,(A)2=-fi1=2B, cos nu7The third-order approximation is(8.25)(8.26)These two steady-sta te solutions have beenchosen by analogy with the linear solutionsdescribed previously. I t is found that retainingonly the fist term in the series equations(8.25) or (8.26) gives satisfactory results.By assuming that the time variation of 8, is ofthe form [Bl(t)ln,the third-order approximatesolution corresponding to equation (8.25') maybe written down asBl = A sin UTfl2=Ai$--Ag2 cos 2ursin or-&sin 3ur(8.27)where A,, is of order A. A similar set ofequations may be written for the solution(8.26).By substituting equations (8.27) into equations (8.24) and collecting coefficients ofsin ur, sin 307, cos 2ar, and constants, the A t ,may be determined. The results are1+2u4-302+8e2(1-2e)u6-4e(1+2e)u4-2eqZ?-(A-?2--1 u4-2e2u616 16 22e)280THE DYNAMIC BEHAVIOR OF LIQUIDSThe coefficients C i , are the combinations of uand E as indicated.
Once these 8, coefficientshave been computed, the nondimensional surface displacement may be calculated to thethird order in A((x, t ) = A sinUTcos x+ (c&,-c:, 2u7)A2 cos 22+ (C;, sin UT-C& sin 3ar)A3 cos 32COSI n this equation, A is calculated as a functionof the frequency and amplitude of excitationfrom the first of equations (8.28).The maximunl vertical distance from crestto trough, yo, may be determined from theprevious equation asThis equation is shown in figure 8.9 for thecase ~ = 0 . 0 5 .
A few experimental points arealsu shown. I t may be seen that the agreelnerit between theory and experiment is fairlyxt)od.T!lc amplitude-frequency relation corresponding to the assumed solution equation(8.26) has been shown (ref. 8.18) t o correspondto an unstable steady state so that the steadystate solution equation (8.26) will never beobserved in actual experiments; however, thetheoretical calculations for this solution areshown in figure 8.9 as the slightly curvedvertical line originating a t u=0.95.@ = a o + g ~ a m n ~ , ( i r ncosn rme)rn-o n-Icosh %,,,(z+h)kgCOS~(8.30)is the mth order Bessel function ofthe first kind, and the eigenvalues, L,, aredetermined by the transcendental equationJ,(h,.r)J : ( ~ , , R )= OThe corresponding free surface displacement isgiven byv=g2 bm,Jm(imnr)cos mem-0 n-1(8.31)The first case considered here is the lowestfrequency antisymmetrical (cos 8) sloshingmode, so that all and bll are the predominantamplitudes in the expansions (8.30) and (8.31).The appropriate length for nondimensionalpurposes is (illtanh j(,,h)-I and the appropriatetime istanh );,,h)-'".By substituting the dimensionless forms ofequations (8.30) and (8.31) into the nonlinearboundary conditions, and then keeping terms upthrough the third order in a l l and b l l , as before,Third order-Cylindrical TankAlthough the two-dimensional theory outlinedubore is relatively straightforward, considerablymore experimental and theoretical work hasbeen done for the more practical case of waresin a circular cylindrical tank (refs.
8.20 throughS.24). For that reason, a brief sketch of themain results is given here.For this situation, the tank geometry isdescribed by 0 5 r 5 R, 0 58 _<27r,nnd the waterdepth is again given by z = - h .The npproprinte velocity potelitin1 is- 0 Experimental results-0.050I180. 9L0Surface response frequency ( u )FIGURE8.9.-CompurieonL1of theory and experiment for112-subharmonicresponse (ref.
8.18).VERTICAL EXCITATION OF PROPELLANT TANKS281it can be shown (ref. 8.20) that PI,, the nondimensional equivalent of bll, is determined bysionless sloshing amplitude can then be writtenasjl+(1-4u0e cos ~ Q T ) P I4-I Krl%(~€(T,8, r ) = A sin a r cos 8 ~ ~ ( i ~ ~ ~ )+(Aio-Ai2 cos 2ar) cos 2 0 J ~ ( j ; ~ ~ r )+(&-A&cos 2ur)JO(j;Olr)+kllS:1~1~+0.165118~~~~~1-0.198686~2~~l~fKolB01-K2lB21) +0.034780~181'1~;~1+k01801811-k~1821811=O(8.32)The rather lengthy constants Kij and k,, aregiven in the appendix to this chapter.The second-order terms are Bol and B2,, andthey correspond to the first symmetric modeand the first cos 28 mode. They may becalculated from the following equations, bothcorrect to the third orderPol+Aol tanh b l b (1-4a2e cos 2ar)Bol-~11811(0.121482A,,ltan11 A,,lb-0.263074X:1)+&[A,,, tanh A,,lb(0.070796X~l-0.060741)+0.263074X:1]=0(8.33)andtanh X21b(l-426 cos 2a7)PZ1@21+XZ1+611811(0.350807~2~tanh X21b-0.482670X~1)+8:1[~21tanh X2,b(0.175403-0.065931A:,)-0.482670X:,] =O (8.34)A complete third-order theory would includeIL few other @,,which are about of the magniHowever, these j3, are not neededtude ofto calculate any of Pll, flol, or A1 and hence arenot given here.The assumed solution to equations (8.32),(8.33), and (8.34) iso;,.Bll=A sin a rThe A,, may be calculated as before by srtbstituting equations (8.35) into equations (8.32).(8.33), and (8.34), and collecting the coefficientsof the various sines and cosines.
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