H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 78
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7 5 X 1074-37X1w~II8.75X 107II1.75Xlo8iRigid tank338THE DYNMCBEHAVIOR OF LIQUIDSTABLE9.3.-Natural Frequencies in a Rigid TankWith Elastic Membrane Bottom for DiferentLiquid Densities (Ref. 9.37)[r=4.37X 1W dyne/cm; b/a=0.50]Modenumber--Natural frequency, rad/sec-p ~ =0.0378I-----2---,-3,----4------gram/cc5.
2377. 3528. 86210. 142p ~ =0.505gramlcc5. 3217.353& 86210. 1429.22). The effect on pressure distribution wasobtained by considering the volume changes ofthe flexible bottom, but the free surface kinematic condition was neglected. The obtainedapproximate results are valid only for frequencies well below the resonance of the tankbottom. Finally, the effects of a flexiblebottom have been included in the investigation of Palmer and Asher (ref.
9.23).9.5EFFECTS OF THE ELASTIC STRUCTURE ONVEHICLE STABILITY AND CONTROL'General DiscussionThe influence of the liquid on the coupledbottom breathing modes was not discussed inreference 9.37 or 9.38, perhaps because thesemodes would occur at higher frequenciesdepending on the bottom flexibility, and ahigher order frequency determinant wouldbe necessary to predict them.
However,judging by the influence of only the apparentmass effects of the liquid on n breathing shell,it is anticipated that the coupled bottom breathing modes would occur a t considerably lowerfrequencies than their respective empty tankvalues. Such modes would have a very important effect on the pressure distributionin the tank.A formulation of the problem of forcedlongitudinal vibration of liquid in a rigidtank with a thin membrane bottom has beenperformed by Tong and Fung (ref. 9.39).This problem differs from those above in thatthe stability of small motions of the liquidsurface must be considered, and any practicalfuel sloshing that results will be g-subharmonicin nature.
The analysis, in effect, investigateshow bottom elasticity affects the onset of theX-subharmonic sloshing that has been discussed in chapter 8. Since solutions andnumerical computations are not yet availablefrom this analysis, no further discussion canbe presented. An approximate analysis offorced longitudinal vibrations of liquid in aflexible tank, including the effects of a flexiblebottom, has been carried out by Bleich (ref.With the increasing length of space vehicles,the significant fundamental bending frequencies of the structure continually become lower.The close grouping of control, propellant, andbending frequencies creates acute problemsbecause of the interaction of these variousmodes with propellant sloshing.
This indicatesthat an additional danger of instability rcsuiting from propellant sloshing due to theelastic behavior of the vehicle is present, which,of course, becomes more critical because ofthe low bending frequencies and the overalllow structural damping. I t is possible toreduce these interactions by various techniques(refs. 9.40 and 9.41), as follows:(1) Proper location of the sensors, whichminimizes the amplitude of the input signal.This means that an attitude gyroscope shouldbe located a t an antinode; however, this canbe accomplished for only one bending mode,and even then, is not exactly possible becauseof the tankage of the vehicle, and the varyingfrequencies and mode shapes during flight.Generally, the fundamental bending mode isthe most critical mode and is usually phasestabilized, while higher modes are stronglyattenuated.(2) The input from the sensors to the controlsystem should have small gains a t the naturalfrequencies.
This can be accomplished byfilters and shaping networks, if the frequencies1Thia section was written by Helmut F. Bauer.,339INTERACTION BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTURE!TABLE9.4.-Comparison of the Fundamental Theoretiiml and &perirnental Frequencies for Rigid Bottomjand Flexible Bottom Tanks (Ref. 9.37)Natural frequency, rad/secRigid tankbla0.50- - - - - - - - - - - - - - - 1.00- - - - - - - - - - - - - - - -Percent change in frequencyFlexible bottom tankTheoryExperimentTheoryExperimentTheory4.935.005.275.434.904.985.235.400.61.40Esperiment110.75.55Iare not too close to the control frequency.At the present time, the most important pointis the provision of appropriate filters.(3) Proper location of heavy propellants,tank geometry, proper gain settings, andbaffles.As has been seen in chapter 7, the stabilitycharacteristics of the vehicle with respect topropellant sloshing is influenced by the tankgeometry, which determines the amount ofsloshing propellant, its location in the tank,and its natural frequencies.
Subdivision oftanks into small compartments by radial wallsoffers great advantage in the reduction ofsloshing masses, as has been discussed inchapter 2; furthermore, this increases theeigenfrequency of the propellant slightly, thusshifting it away from the control frequency but,unfortunately, a little closer to the elasticfrequencies.
Although the latter effect makesthe situation with respect to frequency slightlyless favorable, the main dynamic result isgoverned by the considerably reduced sloshingmass.Stability is also strongly influenced by thetank location; that is, the slosh mass locationwith respect to the vehicle center of mass.pmnn,,"- - 3 CuliursIAp;l a o u u u g b ltuuin the controisystem providing appropriate phases also exhibit a tremendous influence upon the stability.A parametric study is usually made to investigate the possibility of eliminatiig instabilitiesresulting from propellant sloshing and indicatesthe amount of damping necessary in the tanksA.Uyul-'..-to maintain overall vehicle stability.
This isachieved by determining stability boundariesin terms of the amount of damping of thepropellant required in the tank for variouslocations along the vehicle. We shall presentan example of such an analysis.For simplification in the treatment of theequations of motion of a spttce vehicle, anequivalent (analytical) mechanical model describing the motion of the propellant in thetank is used. (See ch. 6.) Only the firstbending mode will be considered and propellantsloshing in only one tank is to be included.
Alleffects due to aerodynamics, inertia, and compliance of the swivel engine are neglected. I tis furthermore assumed that only the amountFs of the total thrust F=Fl+F2 is available forcontrol purposes. Previous investigations of arigid space vehicle with a simple attitude control, as described in chapter 7, showed thepossibility of a significant danger zone whereinstability can occur if the tank is not properlybaffled. This danger zone is essentially between the center of gravity and the center ofinstantaneous rotation. This approximation isonly valid for values l/u,,v:<l,where v,=w,/w,is the ratio of sloshing to the control frequency.For most practical cases, the center of mstantaneous rotation for a rigid vehicle marksapproximately the location where the dangerzone starts.
The other intersection point ofthe stability boundary with the fraxis, however, is more critical to changes in the parameters. If the slosh frequency is below the340TBE DYNAMIC BEHAVIOR OF LIQUIDScontrol frequency (natural frequency of pitchmode), the danger zone increases aft of thecenter of gravity, and more effective bafflingis required for stability. Control dampingincrease in the subcritical region requiresincreased baffling, while control damping increase in the supercritical region requires lessbaffling.Decreasing the gain value, ao, at constantcontrol frequency, increases the danger zoneslightly to the aft of the vehicle and requiresmore effective baffling. A decrease in thebafiiing requirement can be obtained by increasing the slosh frequency by a change oftank geometry, as discussed previously.
Here,it can already be seen that the difficulty inproviding control of forces resulting frompropellant sloshing in the tank of a rigid spacevehicle can be reduced not only by baffling butalso by the proper choice of the tank form, andby proper selection of the control system. This,of course, is even more so if the influence of theelastic behavior of the space vehicle is considered. All the large space vehicles exhibitvery low bending frequencies, which makes theinclision of a t least the first bending modenecessary.Equations of Motion and Control EquationTo obtain some general knowledge of theinfluence of the various parameters and todetermine the amount of damping in the propellant tank necessary to obtain stability, theinteraction of translation, pitching, and bendingmotions and propellant sloshing was investigated (ref. 40).
Only the first sloshing mode isconsidered in the annlysis because the effect ofhigher modes is generally negligible for circularcylindrical t a n h 2The slmhing masa of the second mode is leas than3 percent of the sloshing maaa of the firat mode for nliquid in a cylindrical tank with circular cross section.In a quarter-sector tank arrangement, however, thenext pronounced sloshing mode exhibita a mass ofabout 43 percent of that of the lowest one.
(See ch. 2.)The coordinate system (fig. 7.1) has its originin the center of gravity of the undisturbed spacevehicle; that is, when all generalized coordinatesare zero. The accelerated coordinate system isreplaced by an inertial system such that thevehicle is considered in an equivalent gravitational field. The x-coordinate of this inertialcoordinate system is tangent to the standardflight path. Furthermore, acceleration in thedirection of t.he trajectory, mass, moment ofinertia, etc., are considered constant.The equations of motion have already beenpresented in chapter 7.
Considering translatorymotion, equation (7.16); pitching motion,equation (7.17) ; the first mode of t'he propellantmotion in three containers, equat,ion (7.20), forA= 1, 2, 3 ; n= 1 ; one bending node, equation(7.22), for v = 1; and a simplified control equntion of t,he formone obtains (with the usual ~tssumption forsolution of the form e b e l , where sw, is thecomplex frequency, sw,= u + i w ) homogeneousalgebraic equations of which the coefficientdeterminant must vanish in order t,hat anontrivial solution exists.Stability BoundariesThe main results of the influence of theinteraction of propellant sloshing, bendingvibrations, and the control system upon thestability of the space vehicle can be obtainedif we treat the equations of motion and thepropellant as being free to oscillate in threetanks.
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