H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 55
Текст из файла (страница 55)
Thiseffect is more pronounced the smaller the eigenfrequency of the accelerometer. From' a damping factor, c,, which is about twice the criticaldamping or larger, one recognizes, in the casewa=12 rad/sec, that a further increase of thedamping factor decreases the danger zoneslightly from the back and only slightly enhances the stability. A very important parameter in the design of a control system of a spacevehicle is the location, c,, of an accelerometerfor control purposes. The influence of thisvalue can be seen in figure 7.17. An accelerometer location aft of the center of gravity of thevehicle must be avoided; shifting the accelerometer toward the nose of the vehicle enhancesthe stability.
An increase in the controlfrequency, w, (fig. 7.18), below the naturalfrequency of the propellant (v,>l)-increases thedanger zone toward the tail of the vehicle. Forw,=55 rad/sec (the larger natural frequency ofthe accelerometer), the required damping in theliquid for maintaining stability of the vehicle isrelatively small (y,=0.005 and less). Theinfluence of the control damping, l,, is given byFIGURE7.16.-Stability boundaries of a rigid vehicle withadditional accelerometer control of various vibrationalcharacteristics.1II250THE Dm-AMIC BEHAVIQR OF LlQmDSFIGURE7.17.-Stability boundariee of a rigid vehicle withadditional accelerometer control (influence of the location of the accelerometer).FIGURE7.19.-Stability boundaries of a rigid vehicle withadditional accelerometer control (influence of the controldamping).to n decreasing danger zone and less requireddamping.
Approaching the nat,ural frequency,w,, of the accelerometer makes the vehicle Inoreunstable and increases the danger zone towardthe end of the vehicle. The gain value, ao,hasonly a small influence on the stability; its gaingrowth increases stability slightly and somewhatdecreases the danger zone. Of importantinfluence upon stability is the gain value gl ofthe accelerometer because it presents thestrength of the accelerometer in the controlsystem. Figure 7.20 exhibits this influence forFIGURE7.18.-Stability boundaries of a rigid vehicle withadditional accelerometer control (influenceof the controlfrequency).the fact that, for increasing subcritical controldamping, l,<1, the danger zone decreases andthe stability increases, while for supercriticaldamping, {,>1, the stability becomes moreunfavorable (fig. 7.19).
The influence of thepropellant frequency being below the controlfrequency indicates again the enlarged dangerzone n.hicli stretches nearly from the center ofinstnntaneous rotation to'yard the end of thevehicle. Increasing propellant frequency leadsFIGURE7.20.-Stabilityboundaries of a rigid vehicle withadditional accelerometer control (influence of the gainvalue of the accelero~neter).VEHICLE STABILITY AND CONTROLtwo accelerometer frequencies. For an eigenfrequency of the accelerometer of w,=55 rad/sec, one recognized similar behavior as in thecase of the ideal accelerometer case, for a valueof A = 1.5. An increase of X=gg, exhibits anincrease in stability and a decrease of the dangerzone between the center of instantaneousrotation and the center of mass of the vehicle.For further increase of A, the danger zone shiftsforward of the center of instantaneous rotation.With increasing A, more damping is required inthe propellant container in this zone to maintainstability.
For an accelerometer with a smalleigenfrequency, the situation is quite different.For increasing gain value g,, the stabilityconstantly decreases. Here, the influence ofthe accelerometer favors instability; it not onlyincreases the danger zone toward the end of thevehicle but it also requires considerably moredamping in the propellant tank. I t evenrequires more damping than in the case withoutaccelerometer control (X=O).From this, onecan again conclude that large accelerometerfrequency is required to stabilize the vehicle\rit!l respect to propellant sloshing.In conclusion, one can state that the dangerzone is located between the center of instantaneous rotation and the center of mass, and that itcan be diminished by an additional controlelement in the form of an accelerometer,recognizing that the natural frequency andlocation have to be properly chosen.
Theseresults are valid only for a rigid vehicle in whichthe sloshing propellant mass in one containeris much larger than those in the other tanks.Furthermore, it has to be mentioned that thebending vibration of the vehicle has an effecton the propellant sloshing as well as on thechoice of the accelerometer characteristics andits location. If the controi frequency and thefirst bending frequency are sufficiently separatedfrom each other, then the location of an accelerometer requires negative displacement ofthe bending modes (if the bending modes arenormalized at the tail of the vehicle).
Thisindicates that, for the control of the first twobending modes, an approximate location of theaccelerometer forward of the center of gravity251is appropriate. This location would also befavorable from the standpoint of propellantsloshing.Effects of Container Geometry and TankArrangementsA question of great importance in the designof a large space vehicle is the choice of the formof the propellant containers.
As was shownin chapter 2, tank geometry establishes themodal masses and the natural frequencies ofthe propellant. Containers with large diameters exhibit small natural frequencies that inmany cases are too close to the control frequency. The magnitude of the modal massconsiderably emphasizes this unfavorable effectupon the stability. Clustering of numeroussmaller containers not only increases the naturalfrequencies of the propellant (because of thesmaller diameters) but also reduces the modalmasses, of which the latter is a much moreimportant effect on the overall dynamics of thevehicle. In addition to the weight saving andthe slight increase of the natural frequencies,subdivision of tanks by sector wall has theadvantage of distributing the modal masses todifferent vibration modes of the liquid.To summarize the previous results, with increasing mass the stability decreases and theinfluence of the eigenfrequency change of thepropellant of fixed modal mass is such that adecrease of the natural frequency increases thedanger zone toward the end of the vehicle andrequires more local damping in the propellant.With increasing natural frequency of the liquid,the influence of the propellant sloshing on thestability of the vehicle diminishes more andmere.
well friction is in many cases sdiicientto maintain stability.Clustered ContainersIn the case of a cluster of tanks with smallerdiameters, the results are very similar. Thenatural frequency is increased because of thesmaller diameters. The natural frequency ratioof the propellants in a single circular cylindricalTHE DYNAMIC BEHAVIOR OF LIQUIDStank of radius a, and p identical circularcylindrical tanks of the same total volume, issloshing mass that is reduced by l/pln.
Thereis, of course, also a small stability enhancingeffect because of the increase of the naturalfrequency.Rigid Vehick W i t h Plopellant Sloshing in Two and T h mTanksThis shows that the frequency increase is proportional only to the slowly increasing value( p ) . The total sloshing mass, however,decreases more rapidly with the inverse valueof the square root of the number of containers.The ratio of the total sloshing mass of p tanksand the sloshing mass of the single container ishm(p)1tanhtn(p)lfl;L=mil) p112e nht anh aThis isgreat advantage lor thebut, from the design and overall performancestandpoint, the clustering of tanks has structural and weight disadvantages.~h~ slosh damping required fortanks is t,herefore approximately that ofHere, p,=m,/m is the ratio of the sloshingmass in the vth container to the total massof t,he vehicle.
For nontrivial solutions, thecoefficient deternlinnnt , equation (7.50), mustvanish, from which one obtains t,he characteristic polynon~ialin sand for which the coefficients B, depend on thepreviously mentioned parameters. A similarresult is obtained for t~vocontainers by remov-In some cases the influence of the propellantin other tanks cannot be neglected, making thedetermination of stability boyndaries for vehicles with more sloshing masses mandatory(ref. 7.17). The equations of motion areobtained by treating equations (7.20), (7.42),(7.43), and (7.45) with r],=O, X=l, 2, 3 and/orn = l , 2.
The propellant mill be treated asbeing free to oscillate in three tanks. Thisseems to be sufficient, since usually, even inlarge vehicles, only three of the tanks millexhibit large sloshing masses. The sloshingpropellant masses of tanks with light propellants and tanks of smaller diameter can beneglected. with the usual sssumption regarding solutions of the form e"ct, where s is thecomplex frequency, s =a+iw, the differentialequations are transformed into homogeneousalgebraic equations, with.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
