H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 51
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The controlequation, in which the rate is governed bythe output of a rate gyroscope, and withadditional accelerometer control, is then (seeeq. (7.27))p~8+pl8+8=a~#~,+alO~+g z ~ (7.30)~where7.3INFLUENCE O F PROPELLANT L A T E R A LSLOSHINGThe dynamics of space vehicles are bestdescribed by the two essential investigations ofstability and response. Stability expresses theability of a system to achieve a state of motionand how rapidly this can be done. In theanalysis of space vehicle performance, oneusuauy 1.s saiisSei: t o daterminc just the rcots(boundaries) of stability, rather than solvingthe total system of equations. The shifting ofroots to more optimal positions, if possible, isa major part of the analysis.
If this is accomplished, however, the response of the vehicleis then required to determine whether or notthe design of the system is appropriate forcertain given inputs. I n a space vehicle, thesedesign values are the available engine deflection,11its rate, maximum bending moments, waveheight of the oscillating propellants, etc., as aconsequence of a given wind increase and gustthrough which the vehicle may have to passduring the ascent phase of its flight. Thedynamic characteristics of the vehicle differin complexity depending, of course, upon thecomplexity of the system itself; they are reducedhere to a less complex system by truncatingthe equations of motion in such a fashion thata more lucid presentation can be providedwithout loss of the more general features of thesystem.Stability Techniques and Stability BoundariesSince the response of a system depends onits stability, the roots of the system are themost basic parameters of the system dynamics.These describe the response r e d t i n g from anyexcitation and depend only on the physicalparameters of the system.
We shall thereforediscuss, in this section, the basic techniques forobtaining the characteristic roots of a dynamicsystem and shall study the variation of thesestability roots with changes in the physicalcharacteristics of the vehicle.The optimization of the response behaviorof the vehicle is based upon the possibility ofshifting the stability roots of the system.(See fig. 7.5.) This means that if the locationof the roots can be changed in the complexplane (root locus plane) (fig. 7.5) in such afashion that they exhibit a larger negative realStable regionStable rootrx'I-:IUnstable regions plane-' u n ~ ~ e r c o t +uArrow indicates increasing stability by changingphysical parameters of the system (root migration)FIGURE7.5.-Rootlocus plane.234THE DYNAMIC BEHAVIOR OF LIQUIDSpart, by changing the physical properties ofthe system, the system will show enhancedstability and will more efficiently achieve asteady-state condition and absorb a disturbancemore rapidly.
A very convenient way to studythe migration of stability roots is by plottingthem as continuous functions in the complexplane; this is called a root locus plot andrepresents a curve of all (or those which we aremost interested in) root locations in the complexplane ( a + i w ) corresponding t o the change ofsome of the physical parameters of the spacevehicle. The effect of the change of any otherparameter can also be obtained.
One cantherefore investigate the necessary changes thatshould be introduced into the system to yieldenhanced stability and response features.I n space vehicles, the characteristic equationis of higher order and a direct analytical solution is laborious and sometimes even unnecessary for design purposes. Although the solutionof such a polynomial can be performed by themethod of Sturn1 and Graeffe (ref. 7.10), or itsmodification by Brodetsky and Smeal (ref.7.11) without any difficulties, in many cases itis sufficient to know ~vhetheror not the realparts of the roots are negative, thus indicating n decaying motion with increasing time(stable motion) or whether they are positiveand represent an unstable motion with increasing amplitllde as time increases.
When adesign parameter is to be considered as avariable, it is especially important to hareavailable criteria for the stability in terms ofthe roots. Criteria of this type are given byRouth (ref. 7.12) and Hurwitz (ref. 7.13).Another method for the determination of thelocation of the roots is the root locus methodby Nyquist (ref.
7.14). I n the following, weshall use only the Hurwitz criteria.A necpsary and sufficient condition that theequationwith real and positive coefficients, a,, hareonly roots with negative real parts, is that t)heax and the values of the determinants, H,, allbe positive. The determinant H, is of theform :Ioala3Iandlala3as.../0a,a310a.a,...H,=a,H,-,=/.....................For a quadratic equation in s, these condit,ionsare satisfied if the coefficients ao,a,, and az Wepositive. For a cubic equation, the Hurwitzcondit'ions areH l = a , > 0 , H 2 = i ~ ~ aa.23> ~ ,andITo obt.nin some insight into the degree ofstability, that.
is, the rapidity of decay of themot,ion, i t is sometimes of interest to determineVEHICLE STABILITY AND CONTROLthe magnitude of the smallest negative realpart. For this reason, one shifts the imaginaryaxis toward the left. bywhere Z>O and real. If t.he transformedpolynomial is just still stable, that is, thesmallest root is very close to the shifted imaginary axis, we obtain a measure of the magnit,ude of stability.The stability boundaries are characterizedby the roots, 0f. which a t least one has a zeroreal part while the others are stable roots.For an nth-degree polynomial the stabilityboundaries are presented bya,= 0 and H,-,=0To determine which side of the stabilityboundary represents the stable region, onepoint (not located on the boundary) is chosen,and its stability is investigated with the previous Hurwitz determinants, II,>O (eq.
(7.35) 1.For a quadratic equation, the stability boundaries are given by a2=0 and a,=O. For ucubic equation, the conditions area3= 0 and Hs=ala2-aoas= 0and for a quartic equation they area4=0 and H3=0, i.e., ala2a3=a:a4+aoaiAnother method for determining the stability\c-ould .
be that of the locus curve. The polynomial of the complex variable, s, represents acomplex value, ww=P(s)we havewhere [n/2] means the smaller integer closestto n/2, or equal 4 2 , if n is even. This indicates that for even n the expression X(wjwill be an even polynomial of nth degreewhile Y(w) is an odd polynomial of (n- 1)stdegree. For an odd value n, the X(w) is aneven function of (n-1)st degree and Y(wj isan odd polynomial of nth degree.
Therefore,X(w) = X ( -a ) and Y(w) = - Y(-W). Thefunction w(w) is represented in the w-planelocus curve as w assumes the values from- m tow . All roots of the polynomialw=P(s)=O are mapped into the origin w=Oof the w-plane. The location of this point,therefore, with respect to the locus curve willbe characteristic for the root, s,, with respectto the imaginary axis. The fact that X(w)is even and Y(w) is odd reveals that onlyw 2 0 has to be considered. The locus curvestarts out for o = O on the X-axis and thew-plane at u7=ao and represents, with w+a,a curve such as shown in figure 7.6.
In orderto answer now the question of stability of theroots, s,, \c-e imagine a point, s=iw, on theimaginary axis of the s-plane and considerthe connecting lines of the vth roots, s,, to+If all roots, s, are stable roots, they must belocated in the left half plane of s, that is, theimaginary axis, s=iw, separates t,he left stablefrom the right unstable region of thes.= a+ iwp:aiie.Ir;troduciZg u.-iw int.n t,hepolynomial w=P(s) and investigating the mapping of this imaginary axis in the w-planeshould yield some criteria for the stability ofthe roots of t,he polynomial.
Withw- planes=zwandIFIGURE7.6.-Locus-curve method.236THEDYNAMICthat point.This is, however, a factorproduct representationof the polynomialw=P(s)=a_H (s--s,)----a,r,BEHAVIORof theeOFLIQUIDSThe polynomialof nth degreeX n+m:-_stableP(s)=0hasrootsand?'/,m mg=-----_-- unstablewherew-=X+iYX= R cosY=R sin 4Fromif thethis, one obtainsR_a,_-I r,¢=)E ,,If none of the roots, s,, is located on theimaginary axis, i.e., r, _ 0, and if a, >0, it followsthat R>0, which means that X and Y havethe same zeros as cos 4 and sin 4, respectively.If all roots, s_, are in the left-handplane(stableroots),then 4, can cover only theangularregionfrom--7r/2 to -4-_'/2 fors=iwmovingon the imaginaryaxis.
Theangle, _, therefore, can only cover the angularregion --nTr/2 to +n_'/2; or, for _=0to _,it will cover the angularregion 0 to n_r/2.The angle, ¢, is the angle of the complex valuew=Re _ of the locus curve, and one now obtainsthe criterion:The equationP(s)=Ohas only rootswith negative real part, if the locus curvew=P(s)=w(o_)(for s=io_)circlestheorigin w=0 in such a fashion, that theangle, 4, covers the angularregion from0 to nTr/2 as o_ changes from 0 to o_ (counting positive toward the left).If one root is in the positive half plane, sayon the real axis, then the angle coverstheregion from _" to 7r/2, in rotating by r/2 in thenegativesense.For X stable and _ unstableroots,the angle,¢, yieldsa rotationof(X--u):r/2=m_r/2in the positivesense, if o_rangesfrom 0 to coWith n as the totalmm_ber of rootsX--_= rrlone concludes:locuscurvefor_0=0rootsto _coverstheangularregion from 0 to (X--g)_r/2in thepositive sense.This methodis particularlyuseful for investigatingthe effects of changes in physicalparameterson the behaviorof a dynamicsystem,especiallyif high-speedcomputingmachines are available.General Criteria for Stability of Vehicle WithPropellant SloshingNoAfter the equationsof motionhave beenformulated,as indicatedin the precedingsections, a dynamicanalysis of the vehicle inflight must be performed.The interactionofthe bendingof the flexible vehicle with therigid body motion,the sloshing of the propellants,the reaction of the swivel engines andtheir compliance,the excitationsprovidedbyaerodynamicforces and wind gusts, and thecouplingof the controlsystem providingthestability of the space vehicle, must all be takeninto account.The equationsof motion havebeen linearizedand can be solved on a highspeed computer.It is evident that a detailedformulationand dynamicanalysisis beyondthe scope of this monograph;however,it ispossibleto discussmoreor less simplifiedsets of equationsof motion which shall serveto illustrategross effects and yet be of fundamentalimportancein theof a space vehicle.In orderto maintainpreliminarystabledesignconditionsthroughoutpowered flight, it is necessarytoavoid adverse feedbackconditions arising frominherentphase lags of the sensing element anderroneoussignalsfrombendingvibrations.Especiallyfor aerodynamicallyunstablevehicles, which are particularlysensitive to atmosphericdisturbances,artificialstabilizationthrough the control system (with accelerometer)VEHICLE STABILITY AND CONTROLcan alleviate the required control deflections ofthe gimbal engines, and reduce the loads whichcan be rt potential hazard to the flight performance of the vehicle.
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