H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 79
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This seems to be sufficient since, usually,even in large space vehicles, only three of thetanks will exhibit large sloshing masses. Thesloshing propellant masses of tanks with lightpropellants and tanks of smaller diameter canbe neglected. The characteristic polynomialin s isi\INTERACTION BETWEEN LIQUID PROPELLANTS AND THX ELASTIC STRUCTURE,i341and the coefficient determinant is(9.17)IThe coefficients Bj depend on the parameters:11,1Ip,=m,/mo=ratios of modal mass of liquidover total mass of space vehicle{,=control dampingus= w,/w,= frequency ratio of undampedpropellant frequency to undampedcontrol frequencyy,= damping factor of propellantf,=x,/k,=ratio of the coordinate a t thelocation of the modal mass of thepropellant to the radius of g-vrationof the space vehicleY, Y'=displacement and slope a t the firstbending mode.
(Subscript 1, 2, 3means at location of sloshing masses,subscript B means at looation ofgyroscope, and subscript E meansat swivel point of the engines.)IBIan-,=10BaMB/mo=generalized mass ratiotE=xs/kr=ratio of distance of swivel pointof engines from center of gravity toradius of gyrationf,, &=distance between rear tank and uppertanks, respectivelyao=gain value of attitude control systemg,=struc tural dampingp,, p2= phase-lag coefficientsThe stability boundaries are characterizedby the roots, one of which, the real part, willbe zero, while t,he others are stable roots. Withthe Hurwitz theorem for a stability polynomialof nth degree (ref.
9.42), this isB,=O and H,,-l=Owhere H,-l represents the Hurwitz determinantof the formBs...IB,. ..I(n- 1) lines and columns...IRepresenting the stability boundaries in the(t,,7,)-plane, the Hurwitz determinant HI,=0results inwhere the functions C,((J are polynomials in, The stability boundary for the undampedliquid is Co(€,)=O, representing intersectionpoints with the €,-axis. For all points (5,) ys)above the stability boundary, the system isstable.
Because B,=O, the stability boundaryis limited both at the left and a t the right.From B,,=O, it is recognized that the corresponding stability boundaries to the rightand left are given in the form of straight linesperpendicular to the €,-axis. These boundaries,however, play no practical role. Substitution342~m DYNAMICBEHAVIOR OF LIQUIDSof ts=yr=O into the Hunvitz determinantsdetermines whether the origin is in the stable orunstable region.A necessary and sufficient condition forstability is that (ref. 9.43)(1) The coefficients B,, B,,,, Bn-3. .
.>OA similar result is obtained for increasing control system damping. An increase in thegeneralized mass and bending frequency showsthe opposite effect. The larger the bendingfrequency, the more the system becomes independent of the location of the position gyroscopeand structural damping. A strong effect isif n is evenB1, Bo>Oexperienced by changing the phase-lag coeffiif n is oddcients pl and p2. I n the unstable area, anincrease in p, yields an increase in stability.(2) The Hurwitz determinants H,_,, Hn-a, For positive bending slope, the usual structural.
. .>odamping is sufficient to guarantee stability inThe situthe practical range of O<p,<0.2.if n is evenation is quite different with regard to theif n is oddphase-lag coefficient p2, increase of whichexhibits decreasing stability and limited possiTo obtain the basic results of the influencebilities for improvements in the location of theof the propellant sloshing on the stability of thegyro.
For certain values, the gyro can only beflexible vehicle, the propellant is consideredlocated within a short range of small positivefree to oscillate in only one container. Thisbending slope. For p2>0.006, no gyroscopeis done by removing the fourth and fifth lineslocation can be found that will yield stabilityand column from determinant equation (9.17).within the practical range of structural dampingThis yields the characteristic polynomial in sof the space vehicle.For an elastic space vehicle with a simplecontrol system having no phase-lag coefficierlts(eq.
(7.26) with gi=O and p4=p2=O), itNumerical Resulhcan be seen that the danger zone for instabilityIn the numerical evaluation, the distanceof the vehicle is (compared to a rigid vehicle)of the swivel point from the center of gravityenlarged to both sides of the center of gravity,was chosen as x~=12.5meters and the radiusand the center of instantaneous rotation.
Thisof gyration as k,=12.5 meters. The totalrequires locally more (about three times more)length of the vehicle is 57.5 meters. Furtherdamping than for a rigid space vehicle (ref.more, half of the thrust has been considered9.44). For increasing modal mass, p , moreavailable for control purposes.damping is needed in the danger zone toFor an elastic vehicle without propellantguarantee stability of the vehicle, as can be seensloshing, it can easily be shown that for a gyrofrom figure 9.33. The vehicle is stable if sscope location exhibiting positive bending modecontainer of large modal sloshing mass is locatedslope, the vehicle is essentially stable, if properoutside the danger zone. The influence ofstructural damping is present and the phase-lagdecreasing frequency ratios, v,, of the propellantcoefficient p, is smaller than 0.025.
The variousnatural frequency, us, to the control frequency,parameters have essentially the followingw,, is also exhibited in figure 9.33. For v,>leffects. Increasing the control frequency enand increasing, the stability in a decreasinghances the stability in the domain of positivedangerzone is enhanced. The approach of thebending mode slope at the gyroscope location,controlfrequency toward the propellant freand enlarges the instability in the unstable area;quency,or vice versa, not only increases thei.e., for a gyro location exhibiting negative slopedanger zone but also requires more damping toof the bending mode.
This means that moremaintain stability. For v,>5, the wall frictiorlstructural damping would be necessary if thegyroscope is situated at a location on the(y,=0.01) is already sufficient to preventvehicle exhibiting negative bending mode slope.instability.{1INTERACTION BETWEEN LIQUID PROPEIrLANTS AND THE ELASTIC STRUCTUREGyroscopeA343Gyroscope A--%FIGURE9.33.-Stability boundaries for an elastic vehiclewith a simple control system, varying modal mass,control frequency, and control damping.re,The change of the control damping,indicates that, for increasing subcritical damping,r,<l, the stability in the danger zone will bediminished, whereas, for increased supercriticaldamping, {,>l, the stability is enhanced.Furthermore, the danger zone decreases forincreasing damping.The influence of the control factor, a,,, of theattitude control system is considerably morepronounced than in the case of a rigid vehicle.With decreasing gain values, the danger zoneincreases tremendously over almost the complete vehicle length and demands considerabledamping in the tanks to maintain stability(fig.9-34!.The generalized mass of the space vehiclealso has some influence, as shown in figure9.34.
For increasing generalized mass, thetrend exhibits increased stability. This meansthat uniform vehicles (Me/mo=0.25) are morestable than nonuniform ones; it furthermoreindicates that when the propellant is nearlyexhausted, stability will be enhanced since aFIGURE9.34.wStability boundaries for an elastic vehiclewith a simple control system, varying slosh frequency,gain factor, and generalized mass ratio.nonuniform vehicle approaches more and morethe state of a uniform beam as flight timecontinues.One major factor in the stability of elasticspace vehicles is the influence of the bendingfrequency. For large bending frequencies, ofcourse, the stability is enhanced and approachesthe values for a rigid vehicle.
With decreasingbending frequency, the stability decreases andadditional baffling is required (fig. 9.35).The slope of the bending mode at the location of the position gyroscope should preferablybe chosen as positive, that is, the gyro is locatedaft of the bending loop; however, this isprsticnS i?lpcssiE2.Usu~ythe, gJyoscopeis located in front of the bending loop, and t'heslope at its location is negative. I t can beseen that, for increasing negative slope of thebending mode at the gyro location, the dangerzone is increased and the stability is decreased.The further the position gyro is moved fromthe bending loop toward the nose of thevehicle, the more damping is required, and344THE DYNAMIC BEHAVIOR OF LIQUIDSp2=0.00084 second squared.
Two main caseshave been investigated. The f i s t one considersthe position gyroscope in a location where thebending mode exhibits positive slope, whichwould be a favorable location for stability.I n the second case, the gyroscope is located(as is in most space vehicles) in the forward partof the vehicle, an unfavorable location bynecessity where the bending mode slope isnegative.Figures 9.36 through 9.38 show the resultsfor the first case of gyro location at positivebending mode slope. The danger zone isconsiderably decreased, and in most casesshifted toward the vicinity of the center ofinstantaneous rotation.
With increasing sloshing mass, the zone increases slightly, shiftsslowly farther toward the nose of the spacevehicle, and requires more damping.FIGURE9.35.-Stabilityboundaries for an elastic vehiclewith a simple control system, varying bending frequency,and mode shape.the larger the area becomes where b d e shave to be introduced to maintain stability.With twice the bending displacement, thestability decreases, the danger zone increases,and a little more than twice the amount ofbaffling is required to obtain stability.For aneelastic space vehicle having a simplecontrol system with phaselag coefficients pland p2, the situation with respect to propellantsloshing is quite enhanced for proper controlgains?The phase-lag coefficients have been chosenfor this example to be p,=0.05 second anda It may be noted that the simple control systemtreated here is by no means close to those customarilyemployed in actual vehicles, but was selected forreasons of simplicity in this treatment.
Yet, evensuch a simple system already exhibits a tremendousimprovement for the idealized vehicle having no enginecompliances, etc. The proper description of the realcontrol system, taking into account lead network,filters, and actuator system, would be of much higherorder and would make the numerical evaluation veryunwieldy.-8endiqFIGURE9.36.-Stabilitymcde Gyroscope \boundaries for an elastic vehiclewith an idealized control system (positive bendingslope), varying modal mass* control damping, and sloshfrequency.INTERACTION BETWEEN LIQUID PROPELLANTS AND THE ELASTIC STRUCTURE345The change of the control damping, c,, indicates that for increasing subcritical damping,{,<I, the danger zone decreases and thestability in it is intensified.
The influence ofdecreasing the frequency ratio, v,, exhibitsthe following behavior: if the two frequenciesapproach each other, the danger zone increasestoward the end of the vehicle and requires morelocal damping; as the sloshing frequency approaches the bending frequency, the dangerzone appears at the tail of the vehicle and requires more baffling to maintain stability.The result can be summarized by stating thatincreasing control frequency increases thedanger zone, which when near the bendingfrequency will cover the entire vehicle.Increasing slosh frequency, ws<wB, first decreases the danger zone to a small region aboutthe center of instantaneous rotation, and thencreates a new danger zone at the tail of theFIGURE9.38.-Stability boundaries foran elastic vehiclewith an idealized control system (positive bending),varying control system gain, and lag factor.PICURE9.37.-Stabilityboundaries for an elastic vehiclewith an idealized control system (positive bendingslope), varying generalized mass ratio, bending frequency, and mode shape.vehicle as soon as it approaches the bendingfrequency.
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