H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 54
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The same behavior(7.49) if one int,roduces u,= w and X =O, whichoccurs if v,=w,/o, decreases. This means that,yields B8=B5=0. The stability boundariesfor a decreasing eigenfrequency of the liquid (ordue t,o B,=O are again straight lines given byincreasing control frequency), damping of theI,= fThe stability boundary, frompropellant must be provided in a container inPthe aft section of the vehicle in order to mainthe Hurwitz determinant Hnhl=O(here H3=0tain stability. Figure 7.13 indicates that theor BlBZB3=B$Z,+RB,), is then given by thedanger zone for instability of the vehicle isexpressionlocated approximately between the center of(K~+KZ€~+K~€~)+~~,(K+K~E~+K~€~) the center of instantaneous rotagravityandtion.Inthis zone, the propellant must be+42(K:+KstI+Kg€:)+ fiy.3=0moreorlessdamped, depending on the magniwheretude of the modal mass of the liquid.
Form.The points of intersection of this stabilityboundary with tshe €,-axis are obtained bysetting r,=O and solving the quadratic equationTlie root's of this quadratic equation yieldFIGURE7.13.-Stability boundariee of rigid vehicle withsimple control system.VEHICLE STABILITY AND CONTROL247increasing modal mass, more damping is neededin the danger zone. This is most unfavorableif the control frequency is below the natural frequency of the propellant; that is, if v,<1.0.For v,>2, the mall friction (y,=0.01) alone issufficient to guarantee stability.The change of the control damping, l,,indicates that, for increasing subcritical damping, c,<l, the stctbility in t.he danger zonemill be diminished while, for increased supercritical damping, P,>l, the stability is enhanced.
This means that less damping isnecessary in the container to maintain stabilityin the case {,>I. No additional baffles arerequired in the danger zone if (for a mass ratio,~=0.1)the control damping {,50.5, or {,>2.0.This indicates that, for these values and theparameters v,=2.5 and ao=3.5, the !\-all frictionin the container is sufficient to maintainstability.Another important question in the design ofu large space vehicle is the choice of the form ofthe propellant containers.
We observed inchapter 2 that tank geometry plays an irnportant role in governing the modal masses andthe natural frequencies of the propellant.Containers with large diameters exhibit smallnatural frequencies, which often are too closeto the control frequency. Of course, themagnitude of the modal mass considerablyemphasizes this unfavorable effect upon thestability. Clustering of numerous smaller containers not only increases the natural frequencies of the propellant (because of thesmaller diameters) but also reduces the modalmasses, which is a much more important effect.In addition to weight saving and the slightincrease of the natural frequencies, subdivisionof tanks by sector walls has the advantage ofdistributing the modal masses to differentvibration modes of the liquid. To summarizethen, we note that with increasing mass, thestability naturally decreases. The influence ofthe eigenfrequency change of the propellantwith fixed modal mass is such that a decrease ofthe natural frequency increases the danger zonetoward the end of the vehicle and requires morelocal damping in the propellant.
With increasing natural frequency of the liquid, the influenceof the propellant sloshing on the stability of thevehicle diminishes more and more. Wallfriction alone is then already sufficient tomaintain stability.The gain value, a,, of the attitude controsystem shows, for decreasing magnitude, adecrease of stability in addition to a smallenlargement of the danger zone toward the endof the vehicle.For these numerical results, a Saturn I-typespace vehicle of a length of about 170 feet wasemployed, as before.and are parallel to the y,-axis. For values ofA=ggn<l, the danger zone is located approximately behind the center of instantaneousrotation and shifts with decreasing gain valueg2 toward a zone between the mass center andthe center of instantaneous rotation.
Thestability decreases, which means more dampingin the tank is necessary for increasing X>1.This indicates that, for a greater influence ofthe accelerometer in the control system, theRigid Space Vehick With Ideal Accelerometer(ref. 7.16)ControlBy introducing an additional control elementinto the control system in the form of an idealaccelerometer (w,>>w,), and properly choosingthe gain value, g2, which determines the influence of the accelerometer in the controlsystem, the danger zone can be minimized considerably.
Because of v,>>l , the coefficientsof the stability polynomial are B6=B5=0, andone obtains again a stability polynomial offourth demee.The same formulas as in t,he"previous case are valid, except that in thevalues k, the appropriate terms with A haveto be considered. The boundaries B4=0 areagain straight lines, given by the equationdanger zone shifts fonvard of the center ofinstantaneous rotation and increases with increasing gain value townrd the nose of thevehicle (fig. 7.14). For propellant containersin this location, strong damping must be employed to obtain stability.
For values ofA= 1.5, the vehicle is unstable if the tank, evenwith only a 10-percent slosh mass, is located infront of the center of instantaneous rotation,unless additional baffles are provided. Forcontainers behind the center of instantaneousrotation, the vehicle is stable. Furthermore,one recognizes that X 1.0 represents the mostfavorable gain value. I n this case, the dangerzone shrinks to n small region around thecenter of instantaneous rotation, in which casethe mall friction of the propellant is usuallysuflicient to provide a stable flight situation.Changes in the other parameters, such asthe slosh mass ratio, p, the frequency ratio,v,=w,/w,,the control system damping, {,, aswell as the gain value, ao,of the attitude system,exhibit the same influences as in the previouscase.
An enlargement of the danger zonetoward the end of the vehicle occurs for largecontrol frequencies and also for small propellantfrequencies (v,<l), even in the most favorablecase in which A= 1.0.The addition of an accelerometer introducesanother important parameter: its location €,.For X= 1, the most favorable case for an idealaccelerometer, the influence of its locatio~luponstability of the vehicle is unimportant.
Forother values of g,, the location of the accelerometer has considerable influence upon stability.A stronger effect of the accelerometer (say X>-X Variesa,* 3.5P . 0.1V*- 2 55,- 0.7FIGURE7.14.-Stabilityboundaries of rigid vehicle withadditionnl ideal accelerometer control (influence of gainvalue of the accelerometer).1.5) in the control system and a location infront of the center of mass yields large instability if the container is located behind thecenter of instantaneous rotation with the accelerometer being in front of the center of mass.Propellant sloshing in those tanks located forward of the center of instantaneous rotation11-ill make the vehicle unstable if the accelerometer is forward of the center of mass.
Fordecreasing values g2< l/g, the stability behaviorof the vehicle approaches that of a rigid vehicle11-ithout. additional accelerometer control. I tshould, hou-ever, be mentioned here that theseresults are too optimistic, since every accelerometer has its own vibrational characteristicswhich must be considered.Rigid Space Vehicle W i t h Accelerometer Control of NonidealCharacteristicThe dynamic behavior of an accelerometer,its natural frequency, w,, and damping factor,la,have a nonnegligible influence upon theoverall stability of the vehicle. From theresults of equations (7.47), (7.48), and (7.49),i t is recognized that the stability polynomialis of sixth degree; therefore, the stabilitybonndnries are given by HS=O,and B6=0.The main influence arises from the naturalfrequency of the accelerometer.
In the numerical evaluation, two circular frequencies (w,=55and 12 rad/sec) \\-ere considered for the accelerometer. For decreasing accelerometer frequency, with a damping factor, la=$/2, thedanger zone increases from the center ofinstantaneous rotation toward the end of thevehicle (fig. 7.15). The influence of increasingliquid mass has the same effect as previously,with the exception that i t is very much nmplified for small eigenfrequencies of the accelerometer; a large amount of damping is required inthe container in order to obtain stability of thevehicle. For tt natural frequency of theaccelerometer of w,=55 rad/sec, wall friction isin most cases sufficient to maintain stability.For small natural frequencies of the accelerometer, propellant sloshing is excited.
This indicates that the situation is more unfavorablel\ith a "bad" accelerometer than in the case\c.ithout one. The damping required in such nVEHICLESTABILITY AND CONTROLFIGURE7.15.-Stabilityboundariee of rigid vehicle withadditional accelerometer control of various eigenfrequencies.case would be about three to four times as muchas in the case without additional accelerometercontrol. The results of the preceding section,and those presented here, indicate that thenatural frequency of the accelerometer shouldbe chosen as large as possible. In order toemphasize the influence of the accelerometercharacteristics, we consider the effect of thechanges of the undamped natural frequency, w,,the damping factor, c,, and the coordinate oflocation, z,, upon the stability of the vehicle.For increasing natural frequency of the accelerometer w,<w,, an increase of the danger zoneis obtained, and more damping is required inthe container to maintain stability.
Above thenatural frequency of the propellant, a decreaseof the danger zone and enhanced stability canbe observed. This means that less damping isrequired to maintain stability. The larger thethe less damping isfrequency ratio v,/v,=w,/w,,required in the then continuously decreasingdanger zone. The influence of the frequency,w,, and the damping factor, c,, of the accelerometer is exhibited in figure 7.16. The increase ofra enlarges the danger zone and requires more249damping in the propellant container.
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