H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 17
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5 9 ~ reduw ,,t-a,=a,(3.60a)equations for the second harmonics areaa=saa(3.60b)--41 ahe11) + aa,= a, cos (t&+a,)(3.618)where a, and a, are arbitrary constants. Thus,the firs~ordersurfaceelevation becomes1&=a1 (al+iZ a!-aa)+a2(2+2a?)f=xmThe equations for the third harmonic are-a8=k1(ia,+g1 at+ -5 a:+g1384a;a211+p&=a13a4)tuzal +&+d (%+a1a2+i a!)(: a:+33 a2+%33afa2-g3 4-23 a 4 ) - 3 c t s a l - ~c,sin (t$+ u,) cos (sx) (3.61b)81.Consequently, the free surface oscillations willnot be strictly periodic in time unless a,=Oexcept for those cases when s=n2, n an integer.In order to pose a definite physical problem,Penney and Price studied those cscillationsthat reduce to a single harmonic term when theamplitude is made small. In particular, attention was centered on those oscillations whichtend tof=tcos x sin t(3.62)The equations for the fourth harmonic are-a,=bl(a1a8+,3+g4 = ~ 1afl1+4(ha:+ala2+ 2aaas f tends to zero.By using equation (3.62), the nonlinear equactions (3.58) to (3.59) can be solved to higherorder, more exact approximations.The second-order equations give a betterapproximation to the wave shape than do thefirst-order ones, as will be seen.
They are+&9a1+4a(Aa mentioned previoudy, ao= 0.-a,=u,and a,=-sus(s>2)91NONLINEAR EFFECX'S ZN LATERAL SLOSEINGThusand1a,=%(ut)+-641 d cos (3ut)--2112cos (ut) --452048For strictly periodic motion a 2 = O in order toavoid a frequency 8 times the fundamentalfrequency. If this is the case, thenSimilarly,a' , and a, are zero to the second order.Thus, the natural frequency of the waves isunchanged to the second order.The second-order approximation can be substituted into the third-order terms for thethird approximation; thus,t5 cost5cos(501) (3.71a)t5 cos(3ut)--1024l5 t5The corresponding free surface shape is given by3137 t5) sin (ut)al=(r+@ t3-3575+(4ta-m sin ( 3 4l1 ts)+=163 e5 sin (5ut)"-(3.72a)The third-order periodic solution that reducestoal=t~mtisal-• cos (J q t )5+zta cos@dl-i(3.61)&o, aa=O, a a = O to the third order.
It follo~sthat s,aa, i~2are dl of fourth or higher order.Thus, in the third-order approximation thefrequency of the waves depends on theiramplitude.Similarly, to the fourth-ord& approximationa1 is still given by equation (3.67) andTo the fifth ordert3-35g1%=(&-Ge3--(ut)+-ta-- 7168 8) cos (3ut)(32'6 There are no fifth-order terms in o. or a. when s iscven. ,sin (ut)2195t5)--473088'=8"--'26sin (3ut)t5ain(kt)(3.72~)cos ( 2 ~ t )1, C+ cos~ ( 4 4 (3.72d)s='45t5sin (ut)-= 3072 t 5 sin (3,,t)768with u defined by equation (3.70):The fifth-order wave shape can be examinedeasily for d= (n+%)r, where n is an integer,--'---AS l l l l i V LICal=c COBt6)+LA mn + n r :oWOYUIAU~l r - r+:mmULIGCiSaULLLIGu u u wr-v-uuruuu~n ~ r n a m t m & l ~at rest and the wave everywhere reaches itsmaximum amplitude.
The wave shape isTHE DYN~MIC BEHAVIOR OF LIQUIDS92\A closer examination shows that t.here is no~ o i n ton the surface which is always at rest,that is, there are no true nodes; moreover, thefree surface is never perfectly flat. Thefrequency of the wave as a function of itsamplitude is given by equation (3.70)reached, the velocity q+ is zero and the pressuresatisfies the Laplace equation a t the tip of thecrest.
Then(8)*=-(3)*Also, by symmetry-2%(3.74)(&)+=-(axso that it may be seen that a finite amplitudewave in an infinitely deep tank is a nonlinearsoftening dynamic system.Penney and Price also worked out t.he limitingcondition on the maximum amplitude of thewave.If the ullage or the atmospheric pressure isnearly zero, and the surface tension (excludinglow gravity cases) is also nearly zero, then onemay assume both are zero to obtain a simplifiedcriterion of instability. Consider an elementof volume of liquid on the free surface as aparticle, then the only vertical forces are thosedue to vertical pressure gradient, 2 dz dy dr, thea2body force, pg d2 dy dz, and the inertial force,- b / b t .
Thus, the balance of forces yieldsaz)-Addingequation (3.77a) to equation (3.7713) yieldsI n general, the first bracket is nonzero, thendx=f dz ora t the tip. Thus, the angle enclosed by thecrest of the limiting wave form for the stationary wave is 90" in contrast to 120' for theprogressive wave.To check the limiting form of the stationarywaves, Taylor (ref.
3.12) has shown that thefifth-order approximation is in good agreementwith his experiments, except near the tip of thecrest (fig. 3.5).on the free surface particle.Since the value of po is practically zero,bplbz must be nearly negative or zero. Therefore, equation (3.75) leads to the followingapproximate upper bound for stabilitywhich states that the downward accelerationawlat must be less than the gravitational acceleration, g.The slope of the sharp crest of the limitingwave form can also be calculated.
Along- thefree surface (of constant pressure)(2)ax2 ( d x ) ~ + , x( bpz)kdzdz+(3)*(dr)'=O(3.77% b)At the instant that the limiting wave form isFrcua~3.5.-Lareeamplitude stationary wave ahowingagmment with-~mn& and Price theory (ref. 3.11). -"hesame configuration results if a smooth curvewau assumed. I n that case, the symmetry conditiondemands sz0.93NONLINEAR EFFECTS IN LATERAL SLOSHINGIIThe perturbation method has also beenapplied by Tadjbakhsh and Keller (ref. 3.13)to standing surface waves of finite amplitudein rectangular tanks of finite depth. The solution is obtained to the third order of the amplitude of the linearized surface wave motion.An interesting result was observed to the effectthat there is a depth about 0.17 of the wavelength above which the frequency is softeningand below which the frequency is hardening,neglecting effects of the third and higher orderterms.
For brevity, this phenomenon is called"frequency reversal", and has been confirmedexperimentally (ref. 3.14).The critical depth of frequency reversal estimated by Fultz from his experiments is notaccurate enough to modify the theoreticalpredicted value of 0.17.The perturbation method was again employedby Verma and Keller (ref. 3.15) to three-dirnensional waves, the results being similar to thosefor the two-dimensional waves.Bauer (ref. 3.16) has extended the basicPenney and Price theory to forced oscillationsof the rectangular tank. He considers thetwo-dimensional problem (infinite length tank)with a finite depth of fluid, and carries out theequation to the third order.
I n this way,Bauer is able to derive a response that includessuperharmonic terms in the free surfacemotions. Figure 3.6 shows the liquid freesurface height versus time, for excitation frequency close to the harmonic resonance (a=0.9 w l ) . The top part of the figure shows theliquid height a t the left wall of the tank (x=O),the middle part of the figure the liquid heighta t the center of the tank (x=a/2), and thelower part of the figure the liquid height a tthe right wall of the tank (x=a). In eachcase, the dashed curve represents the resultobtained from linear theory (always zerofor r=a/2), while the ctwves 1, 2, 3 representresults for varying excitation amplitudes ( q =0.02 a, q=0.1 a, x,,=0.2 a, respectively). Thenonlinear effects show up strongly with increasing excitation amplitude.
Similar results areshown in figure 3.7 for an excitation frequencyclose to the superharmonic resonance - w).( -fEhcmt~ 3.7.-Liquid free amface displacementa in aninfinite rectangular tauk under forced oscillation near&: yL+=ex~~+-r~~cE -nIi-r z=p+G-A&ctudcs (ref. 3.16).Extended Hutton Theory for Nonlinear LiquidMotions in a Cylindrical Tank (ref. 3.4)FIGURE3.6.-Liquidfree surface displecemmts in aninfinite rectangular tank under forced milhtion n wharmonic rmmance for varions mitation amplimdw(ref. 3.16).For a circular cylindrical tank undergoingtransverse oscillations, Hutton (ref.
3.19) assumed that the velocity potential of the disturbance can be approximated by the followingform :THE DYNAMIC BEHAVIOR OF L I Q m SThe velocity potential is;=~lf~[$~(r,7) cos (wt)+xl(r, 7) sin (wt)]44+cZfY$0(r, 7)+$2(r, 7)++(24where zb=tank displacement. The f orm of thein terms ' of J,(X,.r)disturbance potential+~2(r,7) sin (2wt)l+e[$s(r, 7) cos ( h t )++c-ht =time~ = # t ~ / ~ ( useet ) equation;(3.82)co=tank displacement amplitude1(3.80a)t =weor=2 c2%t and H,=d(l-vca")$1=[ ji(7) cos e tja(7) sin 01Jl(xiir)C0Set.f4(7) sin ~lJi(X11t)[A1l(z+h)l (3.80,,)C O S ~(Xllh)tj0=Constant (ref. 3.10, p. 30)An-1NA+C [A.ms (a)+&.sin (28)lJ~n-1("'"')NAx ~n-1=ChJo(Xl*r)CNcash('+ h, 1 (3 SOe)cosh (huh)cash [ h ( z + h ) ]c,h (A&)A+X[CZ, ms (%9)+62.
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