H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 16

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Also, all thef n l may be computed from the preceding equation, since all of the bti are known; hence, thevelocity potential can be determined completely.By following the same procedure, the kthapproximation is found to bek+lConsequently, if an/umis not an integer, theunique solution of equation (3.15) havingaof 2 s isf ~ , = O f o r n # m ; f ~ ~ = C c o s (3.16)~where C is an arbitrary constant. It is moreconvenient to write C=ag/urn where a isarbitraxy; thus, the zeroth order potential is&=% qrncos -rurnwhich is just the potential for free linearvibrations in the mth mode.The first approximation (n= 1 in eq. (3.10)) is(g$+ig$)r-0=--hlamag cosY, 0)+a2 sin 2rF?'(x, y)+ a 2-2F Ff)(x, y) sin ir1=2,4, 6,. ..for k even, and(%+Lbr2 a;s)b~-ka,,, ag cos +m(xl y, 01=~ - 0+ a 2 g F?(Z, y) cos i r1=1, 3,5,..for k odd.

All of the F:" are known in termsof lower order approximations, but, as before,it is convenientwrite them asF:"=g ~I:)#,,(x,y, 0)n-1.. .Then, the two equations preceding the F: equation reduce toNONLINEAR EFFECTSITF b:4 {sincos 1 ),n+msin~ + f ~ =hturn-ag- cos r+a2 F bzif2+$f,=a2Forced Vibrations of the LiquidtItIT)where the sine corresponds to the odd and thecosine to the even values of k, and at the sametime the index I takes on even or odd values,respectively.I t is necessary and sufficient that resonancedoes not occur in order to insure, that the twopreceding equations possess periodic solutions;thus,hk=Oforkoddht=-bita9for k evenThe potential 4 for forced vibrations can beconstructed by adding a complementary solution 4, to a particular solution 4, which satisfiesthe Laplace equation and the boundary condition on the wetted walls.

4, satisfies theLaplace equation and the homogeneous boundary conditions in terms of #,.The nonlinearfree surface conditions are satisfied by the sumof 4, and #,, and can be put in the followingforms(3.17)Using these solutions, the velocity potentialand the free surface shape can be easily computed. If a is set equal to unity, the lattertakes the formFor simplicity, instead of a complete Fourierseries in t, assumeu,=: sin (rt)f(t, y)Note that the frequency is a function of amplitude; that is,and that periodic vibrations with an arbitraryamplitude lying inside the circle of convergenceof equation (3.18) are possible.This theory, as Moiseyev pointed out, has anumber of shortcomings.

In particular, theprocedure is possible only when u,/u,,, is not aninteger for n#m. Also, the amplitude of thewave approaches its limiting value very rapidlyand then the wave disintegrates. Thus, itseems that either a linear theory or a theorytaking into account the energy dissipation inwave disintegration is needed. Nonetheless, itis impossible to conduct an andysis of resonancephenomena in a liquid with only a linear theoqso that a nonlinear theory can be justified onthese grounds. In order to determine the liquidresponse near resonance, it is necessary that theliquid be excited at the proper frequency; theforced response of the fluid must then be calculated. This is done in the next subsection.(3.21)where p is a parameter specifying the amplitudeof the excitation.The problem is to find periodic solutions ofthis system having a period of 2r/w.

However,the problem is not unique; that is, there canexist solutions that reduce to the solution ofthe free vibrations as p+O, or that reduce to atrivial solution as p 4 . The latter solutionwas selected by Moiseyev.The vibrations far away from resonance canbe constructed in a manner similar to that usedin the previous subsection; that is, it is assumedthatThe amplitude-dependent period of the forcedvibrations is equal to the given period of theexcitation force; thus, the amplitude should bedetermined from the relationc22are functions of the derivatives of+.,andI, but can be treated as functions of 2, y, t, in the powerseries method.THE DYNAMIC BEHAVIOR OF LIQUIDSFrom equations (3.19) and (3.23), the amplitudet thus is given by3-+I~=A~+~?a t X,By following the same reasoning as before, onefinds, for example, thatsin (ut)(3.33)wherewhere f(x, y) in equation (3.21) has beenwritten as f ( ~ Y),=Z~C~#~(X,,Y).Near resonance, the solution (eq.

(3.25)) cannotbe used. The method used here by Moiseyevis to consider the detuning u2.-4 as small. I nother words, it is assumed that near resonanceandIB1=-- a" *l+tl'a2 a2az2(3.36s)with a a parameter. Thus, from equation (3.9),one finds thatNear resonance the series expansion for Q shouldbe expressed in the parameter palb where nowthe ratio alb may take on other than integralvalues. Since the forcing function is proportional to p, the power method should startfrom fraction a/b<l, i.e., b=nla, n1 being aninteger. However, it can be shown that noperiodic solutions exist unless nl =3.Carrying out the analysis for nl=3, onefinds that ifw4 c - n-1x 4~lpnand(3.28a)From equations (3.32a) and (3.32b), it canbe seen that the first approximation isAs before, it is assumed thatH e ~ c e the, "unique" periodic solution of equation (3.37) isfnl=Oforn#n+jml=M sin (wt) N cos (cut)then the approximate equations are(3.39a)(3.39b)where M and N are constants to be determinedfrom the t.hird pair of equations by requiringthat the resultant of a31 the resonance terms beZero.87NONLINEAR EFFECTS IN LATERAL SLOBHINGLikewise, the second pair of equations resultsinfa=Ml sin (ot)+Nl cos (ut)+second harmonics(3.40)where M, and Nl are constants similarly determined by the periodic requirement of the fourthapproximat.ion, which is the solution of thefourth pair of equations.Thus, the solution near resonance can becomputed to any degree of approximation.However, the details of its computation arevery complex and so are not given in full here;the interested reador is referred to Moiseyev'soriginal paper.Thus the nonlinear boundary conditions remainto be satisfied.By assuming that the waves cannot exceed acertain maximum amplitude, the equation ofthe deformed free surface can be written asThe algebraic labor is reduced by writing allequations in nondimensional form.

811 thelengths are multiplied by 2r/L and the time by(2rg/L)lk. The nondimensional potential andwave form are then+=qanen' cos nz00-Penney and Rice Theory for Finite StationaryOscillations in a Rectangular Tank (ref. 3.11)Although the theory of fhite traveling waveshas been known since the late-19th century, i thas only been recently that standing waves offinite amplitude have been examined. Perhapsthe first such theory is that due to Femey andFrice (ref. 3.11), who have worked out the waveshape to the fifth power in the predominantamplitude.y Price's main concIusionsOne of P e ~ e andis that the wave frequency is a function of itsamplitude. I n order to see how this result wasderived, and to present an example of Moiseyev'sgeneral theory, Penney and Price's method willbe given here in detail.Only two-dimensional waves are considered,so that the appropriate solution of Laplace'sequation for waves in an infinitely deep tank isn-n-0(The nondimensional form ofwhere a-,=a,.z and z is still z and 2.)It is convenient first to work out simplifiedexpressions for such factors as(fla, and soforth.

Now ifthenso that%?+aQ=qP,,(t)e " cosac=i ~ +a, cosznz-'-2 ,,=-C a,eiYz(3.41)where the z axis is positive upward with itsorigin ai the wdistur'oed free smDace,and theflow is independent of y. The boundary conditions that equation (3.41) satisfies arewherec0S*=n--m(&I&..,(3.43b)Now letThe coefficient Z,,is, o f c o r n , zero, since Z=Olocates the mean level of the free surface, but it bretained here since doing eo greatlg simpUe8 thealgebra.,From equations (3.44) and (3.45), it followsthatrnSN(s)= C ~ S ~ - ~ ( s - m ) (3.46)or, in another form,'="-15 &sN-l(s-m)ci*r1-Fmm-mm--m(3'45)so that by continued applicationIt will be observed that SO(0)=l, So(s)=O for s#O, Sl(s)=a8 for all s and SN(S)=SN(-6).Now one can write-where= ~ b- s,) = z 0~ 6 , s )XNsN(s)Finally, each term in equation (3.41a) can be evaluated for z=r ,=;eit cos ( j z )-1~ ( i~ ,) e f i * + ~J -23)- m~ +E~(i,s)et(*-Jz=Ed, ; ) + 5)cos (sx){Efi, s-;)+E(~,s+;)(3.50)r-1With these preliminary mathematics thecoefficients a, and a, can be conveniently determined from the free surface dynamic andkinematic conditions.

The dynamic free surface condition requires at z=Z that the pressure be zero; that is, it is required that[-+--+--I,bfatb@bfb@b j-zct.r,-s -0or, in dimensionless form,Iwhich in dimensionless form is+[qanent(-n)--2n-I he%)+,1 m-1-n-mnq,,a,,e(m+n)t cos (m-n) x=0n-1onz=~(3.52~~)sin (nz)]cos ( ~ ) J = o (3.52b)(3.51)The kinematic condition requires thatBy using the previously defined expressions forE(X,s), equation (3.51) can be written asNONLINEAR EFFECTS IN LATERAL SLOSHING-214+f: on cosnzf &+g& . { ~ ( n , n ) + pn-In-I-1cos iz[E(n,1-n)+gcos lx[E(m+n, 1-m+n)+E(m+n,--and equation (3.52b) as-21 4 +m-15 + cos m1 ~ +-{E(n, m-n)-E(n,C~ 1Mz-nm-1 n-1s+m-n)]m+n)+2cos lx[E(n, 1-m+n)+E(n, l+m-n)-E(n, 1-m-n)-E(n, l+m+ n)]-$ na,,{E(~, n)+$ cos lx[E(n, 1-n)+E(n, 1+n)]1-1n--1Sines equations (3.53) and (3.54) are true for allz,it follows that the coefficient of each harmonicin the equations should be identically zerq; thus,andwhere Sis a positive integqr.

Also, from equation (3.54)+Eb; s+ m-n)-E(n,-E(n, 84-m-n)This equation can be used to calculata8-m-n)I -5m,,[E(n,n-1This system of infinitely many first orderdifferential equations is to be solved by themethod of finite term approximations to theinfinite series. (As a check on the algebra,the time derivative a. of the constant liquiddepth should be zero in equation (3.56a).)Note that equation (3.55a) can be regarded asan equation to determine oro in terms of the othera,; but a,, corresponds to- the arbitrary timefunction in Bernoulli's equation and is onlyneeded when computing the pressurk.To obtain an analytical solution, the coefficients a, and a, are expressed in terms ofintegral powers of a parameter c. The a, areassumed to be of order G and a, to be of thesame or higher order; this assumption can bejustified by a critical examination of equations(3.55b) and (3.56b).Equations for a, and a, to the fifth order inc are given below. The equation for the constant term in the wave shape iss-n)+E(n, s+n)](3.56b)a 0 oncethe other a, and a n have been computed.The coefficient of the fundamental term inthe amplitude must satisfy two equationaderived from equations (3.55b) and (3.56b) :I90THE DYNAMIC BEHAVIOR OF LIQUIDS(l+g3 a?+f1 a2 +;1 a l a r 5+ ~&)+asl-al=&The equations for the fifth harmonic are1+z4a2+=andg)+bFor very small oscillations, these equations( 3 .

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