H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 56
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the characteristicdeterminanting the last column and line from the abovedeterminant, equation (7.50).The stability boundaries for four importantpropellant, tank configurations are discussedbelo11-:(1) Tuo concentric containers.-Itmay bepossible to remedy the influence of propellantsloshing by choosing a concentric tank arrangement consisting of an inner tank with circularcross section (radius r=b) and an outer tank ofannular cross section with an outer radiusr=a.
B y proper choice of the diameter ratioI1itI:\,f1I:i-,,7.;i-1: -1':t'.IVEHICLE STABILITY AND CONTROLk= b/a, the liquid masses in the inside containerand outside container can be brought into sucha phase relation that the forces and momentsof these individual tanks cancel each other.Figures 7.21, 7.22, and 7.23 show the resultsof this study for diameter ratios k=b/a=0.3,0.5, and 0.7. The results are very similar tothose of the case of a single container, sincethe difference in the location of the sloshing1-44iFIGURE7.21.-Stability boundaries for sloahing in concentric tanks (k=0.3).masses is very small.
The danger zone isincreased somewhat. toward the rear of thevehicle. For increasing control damping, t,,the stability is decreased. A decrease in thecontrol frequency enhances the stability, as inthe preceding section. For a sloshing frequencyof the center tank below the control frequency,more bamng has to be employed over theenlarged danger zone. For increasing sloshingFIGURE7.22.-Stability boundaries for sloshing in concen.tric tanks (k=0.5).THE DYNAMIC BEHAVIOR OF LIQUIDS-3.2-2 4-1.6-0.80a8FIGURE7.23.-Stability boundaries for sloshing in concen.trip tanks (k=0.7).frequency, the stability increases. To obtainmasimum cancellation effects, the sloshinginasses of the center and outer container shouldbe equal and should oscillate in antiphase.Equal sloshing masses can be obtained for adiamet,er ratio of about k=0.77 for which,unfortunately, the phases are not favorable.If the phases are chosen favorably, as in thecase of a diameter ratio k=0.5, the sloshingmasses exhibit a ratio of 1:5.
This shows thatno pronounced benefit can be obtained by nconcentric tank nrrangement. For the diameter ratios k=0.3, 0.5, and 0.7, the dampingrequired for stability in the container is in theratio 12:9:8.(2) Sector tank arrangement.-Asshown inreference 7.7, compartmentation of containersby radial walls exhibits considerably decreasedsloshing masses. I n the case of a quarter-tankarrangement, the first modal mass is only abouta third of the value of a cylindrical containerwith circular cross section.
But other vibration modes are still important, such as thesucceeding sloshing mode of which the massstill represents 43 percent of that of the firstone. This indicates that, in stability investigations, this second mode can no longer beneglected. This tank arrangement has, inaddition to the reduced modal mass, theadvantage that the fundamental frequency isslightly larger than that of a container ofcircular cross section. (See ch. 2.) Also, thetotal sloshing mass is distributed to variousmodes; thus, i t is not all excited at the samefrequency as in clustered tanks.Two slosh masses are again considered in theequations of motion, representing the firstand second sloshing mode of the quarter-tankarrangement; the results are very similar tothe previous ones.
Again, the danger zone islocated between the center of instantaneousrotation and the center of mass. The increaseof the control damping,in the sithcritirnlregion decreases the stability, ~vhilcn n increasein the supercritical region increases the stabilityregion. For increasing control frequency, thedanger zone is enlarged toward the base of the\,chicle and requires larger damping in the tankin order to maintain stabilitv.
The influenceof the simultaneous change' of the sloshingfrequencies shows that for sloshing frecjuenciesbelow the control frequency, more baffling innn enlarged danger zone (ton-ard the rear) isrequired. Increasing sloshing frequencies result in a decrease of the danger zone towardthe one between the center of mass and thecenter of instantaneous rotation and requiresless danlping in the tanks to maintainstability of the vehicle. I,o\v gain values(ao=l) require more bafiing along u largerdanger zone, while an increase of the'gain, ao, reduces the danger zone and therequirement of damping in the tanks (fig. 7.24).(3) Tandem arrangement qf two tanks.-Fora tandem-tank arrangement, the results canr,,VEHICLE STABILITY AND CONTROL255FIGURE7.24.-Stability boundaries for sloshing in quartertank arrangement.Ii1!j.
.11,... .1ibe seen in figure 7.25. Tank No. 1 is designatedas the rear tank, and No. 2 is the forward tank.The distance between the tw-o sloshing massesis called l(€,=t/k). The sloshing frequency ofthe liquid in these two tanks is the same. I nthe numerical evaluation, the diameter wastaken to be approximately 100 meters (256inches). I t can be seen that for increasings l ~ s h k g=ass, I loss in stability region is encountered; furthermore, the danger zone isshifted slightly toward the rear of the vehicle.Increasing control damping, f,, increases thestability area. An increase of the controlfrequency, w,, results in an increased bafflingrequirement over an enlarged danger zonetoward the aft of the space vehicle.For sloshing frequencies below the controlfrequency, a large amount of damping isFIGURE7.25.-Stability boundaries for sloshing in twotanks in tandem arrangement.required in the enlarged danger zone in orderto maintain stability.
Increasing the sloshingfrequency decreases the danger zone and theamount of damping required.Changing the gain value, a,, has an effectsimilar to that for a single tank. Small gainvalues require strong baffling over an enlargeddanger zone. An increase in the gain value256!THE DYNAMIC BEHAVIOR OF LIQUIDSenlarges the stability region and shifts thedanger zone slightly toward the center ofinstantaneous rotation.The influence of the difference, [,=l/k, inthe tank location exhibits, for increasing distance between tanks, a shifting of the dangerzone aft on the vehicle, with slightly lessdamping requirements. This indicates that,in a vehicle in which one sloshing mass isstationary during flight and in which the otherliquid mass shifts aft on the vehicle, the entirerear part of the vehicle has to be providedwith appropriate damping to maintain stability(fig.
7.26).k..-a6-a2o0.20. 61.0FIGURE7.27.-Stability boundaries for sloshing in threetanks in tandem arrangement.For satisfactory flight performance of aspace vehicle, its stability is essential. Thisis usually obt,nined by propcr design of thevehicle and proper choice of thc control systemby which the thrust vector will be controlled.In addition to the stability, the response ofthe system to winds nus st be well within thelimits of the control deflections, gimbal rates,and maximunl permissiLle structural loadson tho airframe. I t is therefore essential toinvestignte tJhe response of the space vehicleto atnlospheric disturbnnces.
The reductionof these responses can be accomplished byproper airframe design, such that the structurecan withstand the londs, and by designing theshape of tlle vehicle in such a fashion thataerodynamic forces and moments are minimized, as well as by a properly selected control system.Wind buildup and gusts may require largeengine deflections and engine rates, and mayr! .tI.._i7.4 RESPONSE O F A VEHICLE TO ATMOSPHERICDISTURBANCES(4) T a d m arrangement qf three tanks.In almost all space vehicles, the considerationof sloshing in three propellant containers issufficient for simplified stability boundarydeterminations.
The liquid propellants in anyremaining tanks exhibit either small sloshingmasses (because of their low density or differenttank geometry) or larger natural frequenciesof the propellants. For this case the totaldeterminant, equation (7.50), must be treated.The results are similar as in the tandemarrangement of two tanks. Figure 7.27 exhibitsthe shifting (€2) of the two booster slosh massestoward t,he rear of the space vehicle (as ittakes place during the draining of the first-...jstage containers during first-stage flight). Thedanger zone shifts aft on the vehicle with increasing slosh mass difference; this againindicates that the boost,er must be providedwith appropriate baffles t,o maint.ain stability.For elastic vehicles, the situation changesmore or less, depending on the elastic propertiesof the vehicle (refs.
7.20 and 7.21). Thiswill be discussed further in chapter 9.FIGURE7.26.-Stability boundaries for rigid vehicle withsimple control system and two tandem tanks (distancebetween tanke varying).c* ' ,,1.-257VEHICLE STABILITY AND CONTROLinduce bending vibrations and propellant sloshing. In order to study the response of a vehicle,me limit o~rselvesagain to investigations inonly one plane. This does not seem to be asignificant restriction, since the interactionsbetween pitch, yaw, and roll motions are verysmall for this type of vehicle.
The equationsof motion have been linearized and can besolved for variable coefficients by use of theRunge-Kutta method (ref. 7.22).In the following, the numerical results arepresented for a particularly large space vehicle.The complexity of the problem does not allona detailed analytical evaluation of t,he equationsof motion. The presentation of the results forthis particular vehicle, hon-ever, should exhibitthe basic idea and the valuable conclusionsthat can be drawn from such an investigation.Before one can talk about t'he response of avehicle, the stability of such a system has to beestablished.
For this reason, a root locus phasestudy of the vehicle, with the inclusion of twobending modes and three sloshing masses (onefor each of the heavy propellant tanks) is firstperformed. For control damping, a rate gyroscope a t the engine gimbal station (a veryfavorable position) with a damping factor,&=0.7, and a natural frequency, fo=16 cps,is employed. The control frequency is considered to be 0.2 cps and the structural dampingis chosen to be 1 percent.
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