H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 80
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The change of the gain value, a,,,of the attitude control system exhibits hardlyany influence. Increasing bending frequency,, shifts the danger zone from the front partof the vehicle toward a location between thecenter of gravity and the center of instantaneous rotation, which is the danger zone of a rigidvehicle (fig. 9.37).A very important influence upon the stabilityof a space vehicle is exerted by the location ofthe position gyro. (See fig. 9.38.) Shifting thegyro from the taii oi the vehicie toward the noseshows a continuous decrease in stability andrequires increasing baffling to maintain stability.
The danger zone shifts with increasingmagnitude toward the rear of the vehicle.Location of the gyroscope in the region ofpositive bending mode slope is therefore desirable; however, because of the unavailabilityof a location and the gyro's use for many stages,t1346THE DYNAMIC BEHAVIOR-0.8-3'2 -BendingGvroscoDe0(18OF LIQUIDS-3.2mode-1.6-0.808GyroscopeTAFIGURE9.39.-Stability boundaries for an elastic vehicle-28.. .0FIGURE9.40.-Stabilitywith an idealized control system (negative bendingslope), varying mass ratio, slosh frequency, and gainfactor.boundaries for an elastic vehiclewith an idealized control system (negative bendingslope), varying bending frequency, mode shape, andgeneralized mass ratio.this cannot be accomplished, so that filtersand phase-shaping networks have to providethe necessary lags. The change of the phaselag coefficient pl for decreasing magnitudeshows a decrease in the danger zone and anenhanced stability.
A small change of thephase-lag coefficient p2 (for a gyroscope loc&tion in the rear of the vehicle) does not exhibitany appreciable influence.The results of the second case of a gyro location at a negative bending mode slope is exhibited in figures 9.39 through 9.41. The dangerzone is considerably larger than in the case of agyroscope location at positive slope of thebending mode. I t is even larger than in therigid body case. The danger zone is located aftof the center of instantaneous rotation and extends nearly to the end of the vehicle.
Withincreasing sloshing mass, more damping is re-quired in this zone; for smaller sloshing masses,it also extends toward the nose of the spacevehicle.The change of the frequency ratio, v,, exhibitsthe following behavior: if the control andsloshing frequencies approach each other, thedanger zone increases toward the end of thevehicle and requires more local damping;further increase of the sloshing frequency (ordecrease of the control frequency) enhancesstability, and thus decreases the danger zoneand locally required damping in the propellantcontainer; approaching the vehicle bendingfrequency exhibits a strong increase in therequirement of damping which, for a furtherincrease of v,, is again considerably reduced forv,>8.5, and a reduction of the danger zone isalso noticed.
Maintaining control, sloshing,and bending frequencies well separated yields,INTERACTION BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTUREBending modeFIGURE9.41.-Stabilityboundaries for an elastic vehiclewith an idealized control system (negative bendingslope), varying control system parameter, and structuraldamping.a small baffling requirement to maintainvehicle stability. The change of the gain value,ao, of the attitude control system exhibits nochange of the danger zone and only a slightincrease of local damping requirements forincreasing magnitude.Figure 9.40 shows the influence of increasingbending frequency.
For low bending frequency, of course, the danger zone is large and347nearly covers the vehicle, past the center ofinstantaneous rotation, with large local d.amping requirements. With increasing bendingfrequency the danger zone reduces t o that of therigid body case and diminishes the localbaffling requirements by a factor of about 7.The influence of the location of the positiongyroscope is again shown and exhibits essentially the same character as in the previouscase. Positive slope of the bending mode atthe location of the position gyro is very favorable for the stability of the vehicle.
There isonly a very small danger zone around thecenter of instantaneous rotation where themall friction of the propellant in the containeris already sufficient to maintain stability of thevehicle. For a gyro location exhibiting zeroslope of the bending mode, the baffling requirements are about the same as those of therigid body case, while for a location of thegyro on the vehicle with negative bending modeslope, the danger zone is enlarged from thecenter of instantaneous rotation toward therear of the vehicle.
There is also an additionaldanger zone in front of the center of instantaneous rotation. Both zones require large localdamping. For increasing magnitude, thechange of the generalized mass ratio, MB/mo,shows a slo~vly decreasing danger zone anddecreasing local baffle requirements.Figure 9.41 shows the effects of change of thephase-lag coefficients. The increase of p, atfirst shows a slight decrease in stability butimproves stability at a further increase. Theslight variation of the phase-lag coefficient p2does not exhibit much change in stability.Finally, the increase of structural damping, g23,slightly improves the stability of the spacevehicle with respect to propellant sloshing.APPENDIXThe kinetic and potential energy coefficients,mi, and kg,, for the tank, subject to coupledbending and liquid oscillations as shown infigure 9.1, are given below.
These are essentially the same as those given by Miles (ref.9.5). Following Miles, i = 3 corresponds to abending displacement along 8=0 and i=s+ 3corresponds to a displacement of the freeliquid surface from a plane normal to the axisof the tank. The bars over the coefficientsindicate normalization by the total liquid mass,M=*aabpL:THE DYNAMIC BEHAVIOR OF LIQUIDS3482a2 [ j f+8 -F(-I) (I)]-jf2a3+T=F1-lt~ [(-l)'+lf '(g)+f'~b "+--$1-1(-;)IFaBbcsch -( " { [ y 2 (n+ j f (-I)] cosh ($1-2jf (;):,:(:,:-l)The following notation has been used:The mi,, k,,, and assumed bending modeshapes for the cantilever tank are given as follows.*The second term in the brackets includes the effectof rotary.
inertia of the tank walls not included inref. 9.5. M. is the total tank mass.2Cantilever Tank/'(-3)INTERACTION BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTURE349REFERENCES9.1. BAUER,H. F.: Fluid Oscillations in a CircularCylindrical Tank Due to Bending of the TankWall. Rept. No. DA-TR-3-58,ABXIA,Redstone Arsenal, Apr.
1958.9.2. BAUER,H. F.: Damped Fluid Oscillations in aCircular Cylindrical Tank Due to Bending ofthe Tank Wall. Rept. No. DA-TR-9-58,ABMA, Redstone Arsenal, May 1958.9.3. BAUER,H. F.: Theory of Liquid Sloshing inCompartmented Cylindrical Tanks Due toBending Excitation. AIAA J., vol. 1, no. 7,July 1963, pp. 1590-1596.F. I.: Concerning the Equations of9.4. RABINOVICH,Elastic Oscillations of Thin-Walled B a n FilledWith a Liquid Having a Free Surface.
SpaceTechnology Lab, Translation STL-T-RV-19from Akad. Nauk. SSSR Izvestija O.T.N.lfeckhanika i lfachinostrogenic, no. 4, 1959.9.5. MILES, J. W.: On the Sloshing of Liquid in nFlexible Tank. J. Appl. Mech. June 1958,pp. 277-283.9.6. LINDHOLM,U. S.; CHU, W. H.; KANA,D. D.;AND ABRAMSON,H. N.: Bending Vibrations ofa Circular Cylindrical Shell Containing anInternal Liquid With a Free Surface.
AIAAJ., vol. 1, no. 9, Sept. 1963, pp. 2092-2099.9.7. Hu, W. C. L.: A Survey of the Literature on theVibrations of Thin Shells. Technical Rept.No. 1, Contract No. NASr-94(06), SouthwestResearch Institute, June 30, 1964.R. N.; AND WARBURTON,G. B.: Flex9.8. ARNOLD,ural Vibrations of the Walls of Thin CylindricalShells Having Freely Supported Ends. Proc.Roy. Soc. (London), Ser. A, vol.
197, 1949,p. 238.9.9. LINDHOLM,U. S.; KANA,D. D.; AND ABRAMSON,H. N.: Breathing Vibrations of a Circulareyilndrical Shell With an Internal Liquid.J. Aerospace Sci., vol. 29, no. 9, Sept. 1962,pp. 1052-1059.Y. C.; SECHLER,E. E.; AND KAPLAN,A.:9.10. FUNQ,On the Vibration of Thin Cylindrical ShellsUnder Internal Pressure. J. Aeron.
Sci., vol.24, no. 9, Sept. 1957, pp. 650-660.G . B.: The9.11. ARNOLD,R. N.; AND WARBURTON,Flexural Vibrations of Thin Cylinders. J.Proc. Inst. Mech. Eng. (London), vol. 167,1953, pp. 62-74.9.12. TIMOSHENKO,S.: Theory of Plates and Shells.McGraw-Hill Book Co., Inc., 1940.9.13. REISSNER, E.: On Transverse Vibrations ofThin Shallow Elastic Shells.
Quart. Appl.Math., vol. 13, no. 2, July 1955, pp. 169-176.9.14. BERRY,J. G.; AND REISSNER,E.: The Effect ofan Internal Compressible Fluid Column on theBreathing Vibrations of a Thin PressurizedCylindrical Shell. J. Aerospace Sci., vol. 25,1958, pp. 288-294.9.15. SALEME,E.; A N D LIBER,T.: Breathing Vibrationsof Pressurized Partially Filled Tanks.
AIAAJ., vol. 3, no. 1, Jan. 1965, pp. 132-136.9.16. REISSNER,E.: Notes on Forced and Free Vibrations of Pressurized Cylindrical Shells WhichContain a Heavy Liquid With a Free Surface.Rept. No. GM-TR-87, AM No. 6-15, ContractNo. AF 18-(600)-1190, Guided Missile Division, TRW Space Technology Laboratories,Nov. 1956.9.17. LEROY,JEAN:On Breathing Vibrations of ThinCylinders Partially Full of Liquid (in French).Comptes Rendus, vol. 257, no. 18, Oct. 1963,pp. 2607-2609.9.18.
MIXSON,J. S.; AND HERR,R. W.: An Investigation of the Vibration Characteristics of Pressurized Thin-Walled Circular Cylinders Partially Filled With Liquid. NASA TR R-145,1962.9.19. BARON,M. L.; AND BLEICH,H. H.: The DynamicAnalysis of Empty and Partially Full Cylindrical Tanks, Part I-Frequencies and Modes ofFree Vibration and Transient Response byMode Analysis. DASA No. 1123A (ContractDA-29-044-X2-557), Defense Atomic SupportAgency (available from ASTIA), May 1959.9.20. BARON,M.
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