H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 57
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I t can be shownthat with a (conservative) 5-percent dampingfor the sloshing propellant, the vehicle is stablewith respect to the propellant motion. Withthe bending frequencies during flight time varying from 0.8 to 1.2 cps in the first bending modeand from 1.9 to 2.5 cps in the second bendingmode, the system was found to be stable. I n&htpzrticldar CSSP, tha sarnnd nf the hendingmodes exhibited the smallest stability.Wind ResponseAfter the stability of all generalized coordinates has been established, the response of thespace vehicle resulting from some atmosphericdisturbance can be investigated.
In order todetermine the influence of the vehicle flexibilityand propellant sloshing in the tanks, fourdifferent cases are studied:(1) Rigid vehicle mith no sloshing in t,he propellant tanks.(2) Rigid vehicle with propellant sloshing inthe three heavy tanks (No. 1: LOX tank of thebooster stage; No. 2: Fuel tank of the boosterstage; No. 3: LOX tank of the second stage).(3) Elastic vehicle mith no sloshing in thepropellant tanks.(4) Elastic vehicle with propellant sloshingin the three heavy tanks.The input used is a 95-percent probability windbuilding up in about 11%seconds to a value of75 m/sec.
To this wind, start.ing at 58.5seconds flight time, a 9-m/sec gust is addedwhich starts a t 70-seconds flight time, remainsa t 84 m/sec for about 1 second (an altitudeband of about 250 meters) and 'then dropsdown to 75 m/aec (fig. 7.28). This occurs at aflight time of 70 seconds where the product ofangle of attack and dynamic pressure assumesits maximum value. At that flight time, thesloshing frequencies in the tanks were:Fundamental sloshing frequency in tank 1:fl,=0.44 cpsFundamental sloshing frequency in tank 2:f2,=0.445 CPSFundamental sloshing frequency in tank 3:f3,=0.45 Cps00246810Time (sec)FIGURE7.28.-Wind1214buildup and gust.The bending frequencies mere:First bending: fl,= 1.15 cpsSecond bending: f,,=1.92 cps1618------WL-Y-L*LP_---258THE DYNAMIC-'..FIGURE7.30.-Response646668707274.A-Rigid no sloshI62--BEHAVIOR OF LIQUIDS.-.......--60.*-.-I--76Time (sec)FIGURE7.29.-Response analynis: Engine command signal.In figures 7.29 through 7.40, the response ofeach of the various generalized coordinates iscompared to that of the rigid vehicle withoutand mit,h liquid sloshing and the elastic vehicle\vithout sloshing.
Here, t*heinfluence of sloshing in a rigid vehicle as well as the elasticinfluence can be obtained. Figure 7.29 sho\vsthe engine deflection PC, reaching an angle ofabout 1.2' after the gust hits the vehicle. Thevalue for a rigid vehicle without sloshingexhibits a slightly larger magnitude.
After thegust has been applied, the sloshing liquidrequires a little more than 0.1 engine deflection;the engine follo~vsthe propellant motion withit frequency of about 0.45 cps. This oscillationexhibits a damping value of about 6 percent.The elastic vehicle shows a slightly smallermaximum peak value; it requires, however, alittIe Iarger engine deflection in the transientthan does the rigid vehicle. The oscillation ofthe engine deflection is a t a frequency of about2 cps and has an amplitude of about 0.015',showing that the less stable second mode hasbeen excited.A similar.
behavior can be detected in figure7.30 for the rate of the engine deflection, withthe exception that 8, reaches its stop of 5O/sec.The three cases show no appreciable differencein the attitude angle (fig. 7.31). It reaches itspeak of about 4 O shortly after the gust hasdisappeared. The elasticity of the vehicleincreases the angle slightly by a value of aboutIanalysie: Engine command rate.0.1'. Figure 7.32 shows the result for 6 indeg/sec2. The angle of attack, a, is exhibitedin figure 7.33 and presents a maximum valueof about 10' a t the time the gust is applied.Sloshing and elasticity of the vehicle resultonly in very small differences compared withthe rigid vehicle. The same is true for thetranslational displacement, y, of the vehicle(fig. 7.34).
The translational acceleration ofthe vehicle exhibits the fnct that translationalsloshing results in a larger disturbance thandoes sloshing resulting from pitching (fig. 7.35).The vehicle performs translational oscillationswith a frequency of 0.45 cps as a result of thepropellant oscillations in the tanks. Elasticoscillations have only a minor effect. Afterthe gust has hit the rigid rehicle, the propellantin the first tank (LOX tank of the boosterstage) reaches a maximum amplitude of 21centimeters and performs a damped oscillation0606264666870Time Isec)72FIGURE7.31.Vehicle rotation.7476---i-259VEHICLE STABILITY AND CONTROLTime (sec)FIGURE7.32.-Vehicle0 "'60angular acceleration.I6264Time (sec)66FIGURE7.35.-VehicleIIII68707274translatory acceleration.76Time (set)FIGURE7.33.-Angleof attack..--6062646668Time (sec)FIGURE7.36.-Fimt-alosh.-6067646668Time (sec)FIGURE7.34.-Vehicle707274707274767476amplitude.76translation.with about 5 percent damping (fig.
7.36). Thepropellant in the second tank (fuel tank of thebooster stage) reaches a maximum of 13 centimeters (fig. 7.37). The amplitude of the pro-6062646668Time (sec)FIGURE7.37.-Second-alosh7072amplitude.260THE DYNAMIC BEHAVIOR OF LIQUIDSpellant in the third tank (LOXtank of the S-I1stage) is 4 centimet,ers (fig. 7.38), and remainsconstant for a larger time period. For theelastic vehicle without propellant sloshing, thefirst bending mode has a maximum displacement of the generalized coordinate, ?,, of 1.7centimeters right after the gust has hit thevehicle (fig.
7.39). The second mode exhibitsonly 4% millimeters, but shows sustainedvibrat.ions of one-tenth of a millimeter in thetransient, indicating that this has a frequencycontent to which the second bending mode issusceptible (fig. 7.40).Time (set)FIGURE7.40.-Second60626466 68Time (sec)70727476FIGURE7.38.-Third-slosh amplitude.bending deflection.the time the gust is applied is about 1.2' andslightly less for the elastic vehicle.
Then theengine performs a damped oscillation which isalways forced to react to the sloshing of thepropellant and exhibits a slightly larger d u efor the elastic vehicle. If propellant sloshingis suppressed, the engine performs small oscillations a t the frequency of the second bendingmode. As can be seen in figure 7.42, the stopof 5'/sec for 4, has again been reached. Thereis not much difference indicated in the attitudeangle, 4(=4'), as shon-n in figure 7.43. Figure7.44 shows 6. The angle of attack, a,reachesa maximum of lo0, but the difference betweenthe rigid and elastic vehicle is only minorL0a8a4fTime (set)FIGURE7.39.-First bending deflection.Figures 7.4 1 through 7.52 compare theresponse of the elastic vehicle mith propellantsloshing in its tanks mith that of the elasticvehicle without sloshing and with that of therigid vehicle with sloshing.
In figure 7.41, itcan be seen that the engine deflection a t aboutt oa-a 4-4 8-1.2606264666870Time (set)Frcune 7.41.-Engine7274command ~ignal.76.cVEHICLE STABILITY AND CONTROL6062646668 70Time (set)72747660FIGURE7.42.-Engine command rate.626466 68Time (set)FIGURE7.44.-\-ehicle70727476angular acceleration.06062646668Time (set)FIGURE7.45.-Angle6062646668Time (set)FIGURE7.43.-Vehicle70727470727476of attack.76rotation.(fig.
7.45). The translation of the vehicle isgiven in figure 7.46 and exhibits only veryslight differences be tween the various cases.The translational acceleration, however, indicates immediately the effect of propellantsloshing. The maximum acceleration is 17cm/sec2 (fig. 7.47). The propellant exhibits amap,&.;?., gmy;!it~& cf 21 c~n,Lm_et.~rsin cnntainer No. 1. Elastic and rigid vehicles shownearly the same value, except that in the elasticcase the peaks come a t a slightly later flighttime (fig. 7.48).
In the second tank (fig.7.49),t,he maximum is 13 centimeters for a rigidvehicle and about 1%centimeters higher for theelastic vehicle. This is because the bendingmode a t t.his tank location exhibits a largerTime (secJFIGURE7.46.-Vehicletranelation.deflection. This effect is increased for the thirdtank, which is even closer to the antinode of thebending mode and is also located in the dangerzone between center of mass and the center ofinstantaneous rotatmionwhere more baffling is262THE DYNAMIC BEHAVIOR OF LIQUIDSElastic with slosh.606'6062646668Time (set)7072747662646668Time (set)70727476FIGURE7.50.-'l%ird-slosh amplitude.FIGURE7.47.-Vehicle translatory acceleration..*.required.
In figure 7.50, the maximum sloshing amplitude of a rigid vehicle is 4 centimeters,while for an elastic vehicle the amplitudereaches 5 centimeters. The propellant motionexhibits very little damping. The first bendingmode is represented in figure 7.51, and reachesits peak of 1.7 centimeters right after the gusthas hit the vehicle (the sloshing influences thetransient).
The second bending mode, shownin figure 7.52, has a maximum value of 4.5mm and exhibits larger but (because of thelow frequency sloshing) smoother values in thetransient. In both bending modes, the presence of sloshing propellant influences the decayof the bending motion of the space vehicle.60626466 68Time (sec)70727476FIGURE7.48.-First-slosh amplitude.Time (set)F ~ G U R7.49.-Second-sloshEamplitude.Time (sec)FIGURE7.51.-Firstbending deflection.i263VEHICLE STABILITY AND CONTROL16f,* Bending frequency14f,=Torsional frequencyTime (set)FIGURE7.52.wSecond bending deflection.Time (set)7.5EXAMPLEThe following is a simple example whichemploys the previous results. A Saturn Itype vehicle for which half of the thrust(e=%)is available for control purposes is used.The Saturn I booster is powered by a cluster ofeight engines, each of which mill produce188000 pounds of thrust to give a total of1.5 million pounds.
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