H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 89
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The use of the Titan I1 as the launchvehicle for the Gemini spacecraft prompted anextensive program to determine the causes ofpogo oscillations in the various existing vehicles,as well as to develop fixes for these systems.Sufficient knowledge has now been gained sothat the behavior has been explained qualitatively, and simplified models have been developed to study some of the systems over awide range of parameters. Much of the presenteffort is being expended in refining the detailsof these models, so that they will provide abetter quantitative prediction of pogo behavior.Further, the knowledge gained has indicatedthe requirement for dynamic analyses of allfuture systems during the design stage to detectpotential pogo behavior.Considering the current state of investigationof pogo behavior, as well as the variety ofvehicles in which it has occurred to one degreeor another, the present discussion will belimited to only a qualitative description of thetwo types of coupling mentioned above, alongwith a brief description of several c ~ fthe devicesemployed as fixes for the system described.I t must also be emphasized that the intenthere is to recognize the overall role of theliquid fuel systems in the occurrence of thisbehavior, rather than to give a completedescription of pogo behavior in any specificvehicle.The comparatively severe oscillations experienced by Thor-Agena and Titan I1 near theend of first-stage burning have been found tobe the result of an instability in a closed loopcomprised of the propellant system, engine, andstructure.
In the closed-loop system shown infigure 10.22, an oscillating thrust disturbancecauses a longitudinal acceleration response inthe first axial mode; the acceleration in turnacts on the mass of propellants in the tanks andfeed lines, causing pressure fluctuations at theinlel to the propellant pumps; this causes corresponding perturbations in the pump flo~c-rate,aml thus an oscillatory engine thrust, whichcloses the loop. When the closed-loop gain issufficiently large, and the total phase angle issnc.h as to allow positive feedback, the acceleration response will be larger than the acceleration input disturbance; that is, nn unstablesituation exists, and the amplitude of allvariables involved will tend to grow withoutIpZiiiqp1characteristics characteristics- 1(Accelerationfdynamics(ThrustLOX pumpinlet pressureFuel pumpinlet pressurecharacteristics characteristicsvelocityvelocityLOX thrustcontributionFuel thrustcontribution10.8 ENGINE-STRUCTURE COUPLINGSimple ModelI t has been mentioned that the study of pogooscillations has developed to the point a t whichsimplified models, including the appropriatemathematical transfer func'fions of the subsystems involved, can be formulated forthe various vehicles to assist in the study oftheir behavior over wide ranges of parameters.The results of studies of engine-structuralcoupling using such models are given in references 10.22 through 10.25.
Relatively simplemodels can be used to explain the behaviorqualitatively, but more complex approximations are necessary to correlate quantitativelywith the actual vehicle performance. A simpleblock diagram of a model will be used to describethe behavior in the present case.Fuel f e dlinebellows\-tankKll ItankFuel bellowsLOXCavitation bubbles~ombustion9chamber1Ul ~ u e injectorlFIGURE10.22.-Propellanttransfer functions for airnplevehicle representation (ref. 10.23).380THE DYNAMIC BEHAVIOR OF LIQUIDSlimit. The actual limiting and subsequentrestabilizing will occur as a result of changes inthe system.The entire sequence of pogo behavior forengine-structural coupling can be described bymeans of figures 10.23 and 10.24.
For simplicity, only tn-o subsystems in the loop arewed : the structurnl system, whose resontint frequency, w,, increases n.ith burning time, and thepropellant feed line, whose frequency, w,, remains c o n s t a ~ ~ t The.feed line frequency, w,,is the resonant frequency for pressure oscillations in that subsystenl and, hence, depends onthe propellant compressibility chariicteristics,including effects due to gas bubbles from pumpInput,0Structure-- output-Propellantfeed lineFIGURE10.23.-Closed-looppositive feedback(ref. 10.23). 'systenl-Time after liftoff1,t,> t,>t,Structuralgaini \10F e d linegain10phaseFedlinephase--180'WlFIGURE10.24.-Frequency"PFrequency I cps )wIresponse for sample rubsystem(ref. 10.23).cavitation or otherwise, and apparent compressibility effects as a result of the flexibility of thelines themselves.
The resonances for boththese systems are lightly damped.' Pogo behavior results from the interaction of the tworesonances as the changing structural frequency,w,, sweeps past the propellant feed line frequency, w,.The interaction of the two resonances car1be seen more in detail from figure 10.21. Herethe changing structural gain (acceleration response per unit force inputj and phase areshown nt three different times, and can befeed linecompared to the consttintp i n (pressure response to unit prezil1r.e i n p ~ l t )tlnd phase. The structllrtil gain and resonantfrecl~lencyboth increase with ti~ne,due to thedecreasing propellant mass in the vehicle.Damping in the feed line is some\\-hit greaterthan that of the structure, so that its phasechanges more gradually with frequency tlia~idoes the structural phiise.
It mily be riotedthiit the structural phase is lSOOleading \ISOOlagging) below resonance, passes thro11i.11 90'letidirlp (270' lugging) c ~ t resontince, c~rid becomes zero (:160° lnpginc) above resonance.This is because the observation point is tit theengine section in the 1011-erend of the structrlre,and the accordion-like motion results in such aphase response.
The feed line inlet, also nearthe enrine section, has the customary phaseresponse of 0' below resonance. 90' ltw atresonance, and 180' lag above resonance.The net loop gain for the t\vo systems is the~ r o d u c tof the individual pains,iind the netphase is the sum of the two individual phtlseangles. Instability will occur when the netgain is greater than unity, and the net phaseangle is zero. At t = t , , W, is a t wl, which isbelow w,, and the net gain is less than unity.This corresponds to a stable condition earlyin flight. At n later time t =t2, W, and w,coincide, and the structural gain has increasedto where the net gain is greater than unity, andthe net phase is zero; hence, instability resultsand the oscillations grow.
At n still l ~ t e time,rt = t3, the structural gain has further increasedso that the net gain is high, but the net phaseangle has become negntive so that the system isagain stable.LONGITUDINAL OSCILLATIONS OF FLIGHT VEHICLESMore Complete ModelThe above explanation of pogo oscillationsfor engine-structural coupling has been basedon only two subsystems: the structure and thepropellant feed system. In the completesystem, other subsystems, such as those shownin figure 10.22, influence the basic behavior tosome extent.
Therefore, in order to makequantitative predictions from a model, all ofthe significant subsystems shown must berepresented. In fact, an even more completesystem is shown in figure 10.25. This blockdiagram represents a simplified layout of themodel used to study engine-structural couplingin Titan I1 and Thor-Agena. The complexityof the interaction of the various subsystemscan readily be realized. An important featurein this schematic is that the net thrust iscomposed of the vector addition of engine thrustas well as apparent thrust felt by the enginefrom the (inlet pressure) >( (feed line area),from both the oxidizer and fuel lines.Although the system, as depicted in figure10.25, is considerably more complicated thanI .AccelerationIcharacteristicscoefficientCombustionII PI1 E:&1Fueld i echaracteristicscharacteristicsFuel dikhargeflow rateFIGURE 10.25.-Blockas i t is shown in figure 10.23, the occurrenceof the pogo instability results from the samebasic behavior: the sweeping of the structuralresonant frequency through the constant propellant feed line resonant frequency, a t a timeduring fight when the structural gain is high.As far as is known, resonance does not occurin any of the other subsystems in presentvehicles; these subsystems simply alter thenet loop gain and phase.
Their presence is,of course, very important, because they caneither deter or promote the instability, depending on their contribution to the total loop gainand yl~ase. I t must also be emphasized thatthe entire closed-loop system is composed of theoxidizer and fuel loops in parallel, and a pressureresoqance occurring in either loop can cause theengin 3-structural instability.Fixes for Engine-Structure CouplingSeveral devices have been proposed ns fixesfor engine-structural coupling, two of whichhave been used successfully on Titan TI.Three such device.; are shown in figure 10.26.Basically, all of these devices have but onepurpose: to alter the propellant feed line pressure characteristics so that the changingstructural frequency, w,, can never coincidewith the feed line frequency, up, or a t leastso that the coincidence will occur a t such rt t,imeduring flight that the net gain and phase willnot result in an unstable system.Additional compliance is added to the propellant feed system by means of a mechanicalpiston and spring in the accumulator (fig.10.26(a)), by means of a gas bubble in thestandpipe (fig.
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