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VOL. 14, NO . 4,AP RIL 1977 J.A I RCRA FT 3 75
Prediction ofJum p Phenomenain Roll-Coupled M aneuvers of AirplanesA. A.Schy*and M . E.HannahtNASA LangleyResearchCenter,Hampton, Va.
An easily computerized analytical method is developedfor identifying critical airplane maneuversin whichnonlinearro tational coupling effects may cause sudden jumps in the response to pilot s control inputs.Fifth-and ninth-degreepo lynomialsfo r predictingmultiplepseudosteady statesof roll-coupled maneuversar ederived.T he program calculates thepseudosteady solutionsand theirstabi l i ty.T he occurrenceof jump-likeresponsesfo rseveral airplanesand a variety of maneuversis shownt o correlatewell with the appearanceof multiple stablesolutions for critical control combinat ions .The analysis isextendedto include aerodynamics nonlinear in angleof attack.
IntroductionPHILLIPS' original analysis of the roll-coupling problemconsidered th e rotational couplingeffects of constant rollrate on the stability of the shor t-per iod longitudinal an dlateral oscillations. Although th econstantrolling constraint isartificial, it is physically realis tic fo r well-behaved airplaneswith good damping in roll . Phil l ips ' analysis predic ted thatdivergence-like motions would be expected at cer tain cr i t icalroll rates, when th e usual linearized stability analysispredic ted perfect ly acceptable behavior . Th is dangerouscoupling effect of rapid rolling, leadingt o large deviations inincidence angles and tail loads, was confirmed in manynonlinear computerized simulations and in flight. In con-nection with these simulation studies, there were many at-tempts to extend Phill ips ' analysis . The main object ive ofthese studies was to obtain a sim plif ied method for predic t ingth epeak motions in roll-coupled maneuvers.P in ske r2 an d Rhoads an d Sc hu l e r3 showed that Phill ips 'critical roll rates also could be obtained as steady-state(autorotational) solutions of the approximate equations ofmotion used byPhillips. Thismethod has theadvantages thatitalso predicts steady-state solutions for the other variable s, ismore readily generalized, and permits the use of co ntrol inputvalues as independent parameters, instead of roll rate.However , attempts to use this method to predict peakdisturbances reliably in coupled maneuver were not suc-cessful. In the present studyw ehave returned to this methodof calculat ing the steady states of the approxim ate equationsof motion, which we shall call th e pseudosteady-state (PSS)method, because it neglects th e effects of vary ing weigh tcomponents in body axes. However , instead of trying topredic t th e magnitudesof response peaks, th emethod willbeused to predict those control input combinations that m aycause sudden "jumps" in the response as the motion is"attracted"to a newstable pseudosteady -state.
Preliminary RemarksA recentpa pe r ,4which presents an excellent review of pastwork on roll coupling, also shows that it is necessary to in-c lude gravity effects to predict peaks in the response. Th eauthors introduce an expansion using g/V as a small
Presented a t the AIAA 3rd Atmospher ic Fl ight Mechanics Con-ference, Arlington, Texas, June 7-9, 1976 (in bound volumes ofConference papers, no paper number); submitted June 25, 1976;revisionreceived Nov. 29, 1976.Indexca tegory: Airc ra f t Handl ing, S tabi l i ty ,and Control .*SupervisoryAerospace Engine er. Associate Fellow AIAA .fAerospaceEngineer.
parameter , and calculate approximate osci l latory controldeflection requirements to enforceth e ar t if ic ial constraint ofconstant rollrate. By use of these special control inputs, theyshow that their approximate ex pansion, including f irst-orderweighteffects, fits the peaks of the integrated equations well .Another recentp a p e r5h as shown that , for any arbitrary t ime-varying roll-rate, the other responses can be predicted usingan expansioninp(t). However ,t h edifficulty with these typesof analysis , which depend on specifying the form of roll rate,is that in nonlinear maneuvers there is no apriori way topredic t th e roll-rate response. Simulator studies ofmaneuver ingairplanes have shown that c er tain combinationsof constant control deflections can lead to very sudden andi rregular "jump-like" responses in all variables, includingroll rate.In order to illustrate this, Fig. 1 show s the roll-rateresponses to two rectangular-pulse aileron inputs for thefighter airplane sample flight condition used in Ref . 5. Thedotted lines show ihep(t) responses assumed in this referenceafter th e aileron is centered. Even for the 5 aileron, th ecalculated response is substantially different; whereas for the10 pulse the roll rate does not even return to zero, but"jumps" into an autorotational state. All of thecom plicatedresponse calculations in Ref . 5 for the assumed p ( t ) in thiscritical range therefore ar e useless, since they would neveroccur in an actual rolling maneuver unless the controls weremoved in a complicated m anner not kn ow n to the pilot .In the present s tudy, the control inputs are the independentvariables instead of the roll rate. An analytical solution forthe pseudosteady-state corresponding to any combination ofcontrol inputs is derived, along with the linear perturbationcharacter ist ic polynomial def ining th e stability of thepseudosteady maneuver. The existence of multiple stable
160
120
P , 80c l e g / s e c
40
0-4 0
t . s e cFig . 1 Comparison of calculatedroll-rate response to rectangularaileron pulsean droll rate assumedin Ref. 5 .
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37 6 A A SCHY AND M . E . HANNAH J.AIRCRAFTsolutions at any combination of control deflections isassumed to be a necessary condition for theoccu rrence of aj u m p in theresponse, resulting from the nonlinear rotat ionalcoupling effects. The identification of such critical controlcombinations is the main role of the approximate PSSanalysis . Comparison of the PSS results to integrated t imehistories for a number of examples indicates tha t varioustypes of jum p phenom ena can be predic ted. In order to obtainaccuratepeak responses, the complete nonlinearequationso fmot ioncan be used for the criticalcondit ionsidentifiedby thePS S analysis. Al though results ar e presented only fo r linearaerodynamics, the analysis is extended to includeaerodynamicsnonlinearin a.
AnalysisThe approximate nonlinear equations of motion arepresented in Appendix A.Thesewere used to calculate all timehistories in this paper . They assume tha t speed is constant ,tha t th eincidence angles (a and 0) ar e small as compared toone radian, and that aerodynamics ar e l inear. Principal axesare used. Seven f i rst -order equations resul t . Th e stabili ty ofth e linearized perturbed motion from any equi l ibr ium
solution to these equations is determined by a seventh-ordercharacteris t icequat ion . In order to obtain an equi l ibr ium orsteady-state solution, it is necessary to solve th e nonlinearequations that resultwhe nal l t imederivativesare set equal tozero.By use of mod erncomputer sand algori thm s, thiscan bedone even for the complete equations and nonlinearaerodynamics, wi th no simplify ing assumptions; however, i tis known tha t such solutions must be spiral pa ths about avertical axis, because of the vertical weight fo rce .6 Since thesudden jump phenomena of concern in this study seemunrelatedtosuch vert icalspirals,b utseemratherto be relatedto the rotat ional coupling effects at any or ientat ion, the PSSapproach is to ignore the variation of body-axis componentsof weight , in order to el iminate the constraint tha t th erotat ionalvelocitybe vert ical . The resultingequations for thePS S solutions ar eobtainedby dropping theg / Vterms in theforceequations(whichdepend on < > and 0) and the associated / > an d 6equations, an d setting th e time derivativetermsin theotherfiveeq uations tozero.Solution for Pseudosteady States
Leavingasidethe rollingequation,t h eother four equationsar e linear in q, f, ft and A a, where the bars indicate PSSsolutions.Thesecan bewri t tenas(1)(2)(3)(4)
In vector-matr ix form, defining xT(p)=q,f, f t A a ) , theseequationstaketh eform
and
mqq+p r+maA a=- mbe be
(pI4-M)x =Wher eI4 is the fourth-order identity
(5)
M=0
-mq10
nr ft p0 00 0cosa0 -)
(6)
bT=(np,0,0,-sma0) (8 )The vector ca ndsubscr ipt cindicateth econtrolinput terms.The solution of Eq. (5) is
=(pI4-M)-(c+pb) (9)Theinversematr ix inEq.(9) is in thesameform tha toccursin th e Laplace transform solution of a set of four first-orderdifferential equations,and is a matrix o frational functions ofp. By use of thenumerical valuesf orM , c, and6,th esolutionfo r Eq. (9) is programmed easily f o r ^ p m p u t e r calculat ion,givingsolut ionsq(p), f ( p ) , 0( p ) , andAa( p) inth eform
X i ( p ) = N j ( p ) / Q ( p 2 ) 1 0Here Q(p2 ) is the quartic expression obtained by evaluating\p1 M\,w h i c htu rnsout to be quadrat ici n / ?2 .
Q(P2)=P4(11)
Positive solutions of Q(p2) =Q define critical values of p,sincet hesolut ionsin Eq. (9) areundefined at thesevalues.A swas originallypointed out in Refs. 2 and 3,thesecritical PSSvalues correspond essentially to the critical roll rates definedby Phi l l ips in R e f . 1. Howev er , these values do not play animpor tan t role in the PSS analysis. The PSS solutions areobtained asfol lows.General ly,when the solutionsfromEq.(9 ) areinsertedintothe rollingequation
The result may be w rit tenIc +lpP+fi (P)/Q(P2 )~/2 ( P ) / Q 2 (P2) = 0 1 2 )
where fc represents the contro l input terms,and//and/2 ar edefined by the numerators of f t r,an dqas 1 3 )
1 4 )Note that th evalues ofp corresponding to Q =0 cannot beused in Eq. 1 2 ) to define corresponding controldeflections,because of the singularity at that point . However, using th econtrol deflections as the independent variables, lc is welldefined,and solutions of Eq. 1 2 ) givethe corresponding PSSvalues for allvariables.Mult ip lying Eq . 1 2 ) by Q2 yields a ninth-degreepolynomialequation for the PSSvalues;).
= ( fc + f p f ) ) Q 2 + f ] Q - f 2 = 0 1 5 )Th e polynomial form is preferable for computer solut ion.Solutionsof Eq. 1 5 ) are valid solutionsof Eq. 1 2 ) if they areno t solutionsof 2 = 0. If theydo occur , for isolated controlvalues,at Q =0solutions, then theymu st be multiplerootso fEq . 1 5 ) to be valid solutions of Eq. 1 2 ) . No such solutionshave occurred in theexamplesconsidered inthispaper .In most roll-coupling studies, th e last term in Eq. 1 2 ) isneglected, assuming th eproduc t qf to be of minor influence.Thefifth-degreepolynomialfor PSS solutionspthen becomes
:T=(nc,-mc,zc,-yc) 7 1 6 )
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37 8 A.A.SCHY ANDM.E.HANNAH J.AIRCRAFT
Table1 Parametersofexampleairplanes
Airplane AAirplaneBAirplaneCAirplane D
Dynamic pressure,l b / f t 2197.0297.386.075.3
K,ft/sec691500269317
I X P > _slug-ft210,9761,70012,00013,476
I Y P > slug-ft257,10012,400
171,00058,966
*ZP-s lug-f t264,97513,600
180,00067,719
M O ,rad/sec2.4172.171.6611.713
*,rad/sec1.834.441.011.65
disappears.Therefore, the PSS analysis predictsthat thisi s acritical range of aileron values,that the roll rate and sideslipmay "jump"tolargevaluesnear stable solutionIII,and thato tmayj u m ptolargenegativevalues.Figure 3 shows corresponding time histories obtained byintegrating th e equations of Appendix A for three constantaileron values. In fact, the jumps predicted for all threevariables are seen to occur, al though the critical aileronmagnitude is near 5, rather than 6. In this case, the peakvaluesarealsopredicted reasonablywell,b ut thisis not to beexpected generally, since theneglectedweighteffects c an havea very importantinfluenceon peak responses.The ticmarks ,showing where various bank angles are reached, are used toindicate the realistic part of the maneuver. In roll-couplingstudies,360 rollsusuallyare assumed to represent a realisticl imiton the maneuver.In this case, the PSS method seems to predict both th ecriticalaileroninput and thej umpmaneuverratherwell.Notetha t th e usualmethod of predicting the critical aileron level,using Phillips' critical roll rate (p =o )^=1.83 rad/sec fromTable1)an dth esimplifiedp/da= (2V/b) (Clda/Cpp), wouldg i v e < 5 a 15 .
< p = 3606a =-10
a) Roll rate
0 2c) Angle ofattack
Fig .3 Calculatedresponseso f airplaneA forseveral aileron inputs.Tics mark where certain bank angles are reached.
Airplane BIn this case, Etkin has considered a small jet airplane andshown that a combined pitch-down maneuver and a smallaileron input can lead to avery different kind of divergence,although roll coupling is still th e dominant effect .7 For thestandard aileron roll, high angle of attack aggravates theproblem,sinceit converts into a largesideslipin the rapid roll.
250-
200-150-100-50-0-
-50 --100--150-
P .d e g / s e c
-200
S T A B L EU N S T A B L EO S C I L L A T I O ND I V E R G E N T
-10a) Roll rate5040 302010
e g 0-10-20-30
6 6 a , de g 4
b) S ideslip302 0
o f d e g 1 0
0
6 a,d eg
U
_L-8 -6 -4 -2A I L E R O N D E F L E C T I O N , 6a , de g-10c)Angle ofattack
Fig. 4 PSS solutions for small je t with pi tch-down elevator forairplaneB,6e=2.
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APRIL1977 JUMP PHENOMENA INROLL-COUPLEDMANEUVERS 379In this example, the pitch down causes negative a, whichconverts to negative / 3 , which, in turn, acts th rough thedihedraleffect tocause rollingfaster thanthe Phillipscriticalvalue. Thereaf ter , a diverges to large positive values; thusconfirming th evalueof Phillips criterion, asnoted byEtk in .However , the previous explanation is useful only after th eproblem has been observed in the time histories,sincet hehighroll rate could no t have been expected for small ailerondeflections. The PSS method on theotherh and, does predictthat, with a 2 elevator input ,a ju m p in the response can beexpected at aileronvaluesabove 4 or 5, as shown inFig.4.Essentially, the same argument as was used in the previouscasef orFig.2withregardto the conf igurat ion ofcurvesI, II,and III can beappliedto Fig. 4.Herej u m p stonegative0a ndlarge positivea.are predicted ,alongwiththe jum p tolargep.
Thetimehistories for 4 and 5 aileron areshowninFig.5,an d they confirm the existence and nature of the jumpphenom ena at the predicted critical control input values.T h etendency to develop negative sideslip and the positivedivergence in angle ofattack after the rollratebuilds up canbe seen.T he criticalrollrateo f Phi l l ipsisuseful inexplainingth e instability leading to the divergence-like mot ions , but thePS S analysis shows a priori which cri t ical control com-binations shou ld be studied using the complete s imulatorequations.In both of these cases, the PSS analysis has predictedcontrolinputs wherej u m p phenomena occur in theresponse.Although themech anism of the couplingwas quitedifferent ,th e critical condit ion corresponds in both cases to a disap-pearance of the real solution of the fifth-degree equationco rresponding to the basic branch of t he / ?v s 5a curves . This
solution combines with another (divergent) real solution toforma complexroota thigheraileron values.AirplaneC
This case is a flight condit ion of another fighter airplane.Firs t , th eeffects of applying aileronalonewillbe considered.For thiscase, th e results of the ninth-degreeequation will becompared with th e fifth-degree results, to show why onlyfifth-degree results are presented in the other cases. Figure 6showsthe PSS solution, an d Fig. 7shows th eassociated timehistories for several constant ailerondeflect ions . For the sakeof brevi ty, only roll-rate resuts will be shown in this and theremainingexamples .In Fig. 6, the fifth-degree solutions are practically coin-cident with th e associated ninth-degree solutions . The fourextra ninth-degree solutions are practically equal to the valuesfo r Q=0 and arevery divergentfor allvalueso faileron. Theyseem to be asymptotes for theother solutions , but otherwiseplayed no important role in the examples considered in thisinvestigation.The fifth-degree solution is single valued at da=Q, in -dicating tha t this case is not autorota t iona l . A new stablefifth-degree solution appears as curve III at < 5 a = -24, sotha tvalue is potentiallycri t ical . The roll rate may jump herefrom near60/secto two or three times tha tvalue.T h ebasiccurve inFig.6 is very nonlinear fo r da valueslarger than5 inmagnitude, indicating that th enonlinear rotational couplingbecomes impor tan t before th e critical range of aileron. Thiscan be seen by a sort of "saturation" effect in p at theseaileron magnitude s.
i i i i 0 2 4 6 8 10b) Sideslip t sec
a. d
3 02 0
eg 1 0
0- i n
p
-5^ = 1 8 0 ^
360 ^
p - 180 (p 360i i i i i i I I c)Angleof attack sec
Fig . Calculated responsesfor aileroninputsin criticalrange of Fig.4.
200r-
150-
100 -
deg/sec 50
-50
-100
S T A B L E ,th I_ _ _ _ _ D I V E R G E N T ) 9 Jnth - D I V E R G E N T J 9 in d e g
-30 -25 -20 -15 -106a , d e gFig . 6 Comparison of fifth-degree an d ninth-degree roll-rate PS SsolutionsforairplaneC ,aileron only.
Fig . 7 Calculated roll-rate response of airplane C forseveral aileroninputs,de=0 .
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380 A.A.SCHYANDM.E.HANNAH J.AIRCRAFTReferring to the time histories in Fig. 7, responses areshown for da=-10, -20, and -24. For 6f l =-24, aviolentj u m pin theresponseoccursbetween 3 and 4sec.Rollrate jumps to very large positivevalues, as predicted, and alarge, irregular oscillation appears near 200/sec. Similarj umpsandoscillationsoccurin the a and0responses, whichare not shown. Although the aileron value where th e j umpphenomenon occursis well-predicted by the PS ScurvesinFig.
6, the qualitative nature of the jumps is only crudelypredicted. Thisis not surprising, because of the importanc e ofthe large oscillatory components in the final motion. Asshownby Hacker and Oprisiu,4 theseosc illations arestronglydependent on gravity effects. This can be easily verified byintegrating the equations of motion neglecting the gravityterms.Before proceedingto the next example, afinalcomm ent onthe ninth-degree solutions is necessary. The fact that th eninth-degree solutions provided no significant information inthe examples used inthispaper doesnotnecessarilyimplythatthey will never bepractically useful. Although th e programfo r calculating solutionsc an handlerudder inpu ts, thesehavenot yet been introduced. It is possible that improper rudderinputs could cause large yawing motions, which could maketh e qr term muc h more important. Also, other airplaneconfigurations might have very different maneuver charact-eristics, in which the contribution of qr may be more im-portant. On the basis of these limited results, it would bepremature todismissthe ninth-degreeformulationas useless.The next example is for the same airplane and flightcondition,but shows how a pitch-downelevatorinputaffectsthe response to the aileron. This control combination issimilar to that considered for airplane B. The results ar eshown inFigs.8 and 9.Since there are multiple solutions at 6f l = 0 in Fig. 8, anautorotat ion condition is predicted for the pitch-downmaneu ver; and the criticalaileronis near da =-4,whereth e
100
50P.d e g / s e c 0
S T A B L EU N S T A B L E O S C I L L A T I O N- D I V E R G E N T
OLJT
; _r^~-~--50 -
-100-1 i i i i i I i r~ T0 -8 -6 -4 -26a, deg10
Fig.8 Roll-rateP SS solutionso f airplaneC forsmallaileron inputswith pitch-downelevator,be 3 .
200 r
160
120
P, deg/sec 80
40
-40
S T A B L EUNS TA B L E O S C I L L A T IO ND I V E R G E N T
Fig. 10 Summaryplot of PSS roll-rate solutions forairplaneC withcombined elevator and aileroninputs.
p . d e g / s e c
40
-50 -40 -30 fia>deg -20 -10 0Fig .11 Roll-ratePSS solutions ofairplaneDfor aileroninputs.
F ig. 9 Calculated roll-rate response of airplaneC for small aileroninputswithpitch-downelevator,be=3 .
basicsolution disappears. Thu s,the pitch-dow n maneuver hasgreatly decreased th e critical aileron magnitude. For criticalvalues of aileron, th eroll rate should jump to largepositivevalues. The responses inFig.9showthatthejump occursford a =-6, rather than at da=-4. During th e time intervalroughlybetween 4 and 6 sec, as theairplanebanksfrom 180to 360, therollratejumpstolargepositive values. Althoughthe results are not show n, /3 takes asharpnegative jum p and atakes a sh arp positive jum p in thissametimeperiod;and thesedirections are consistent with th e corresponding stablesolutions for curves III. As in the case fo r zero elevatordeflection, the precisenature of thej u m pis affected stronglyby the presence of a large oscillation, which is caused by theweight effects. The prediction that th e critical aileron valuebecomes much smaller for the pitch-down maneuver isconfirmed,how ever.Figure 10 shows a summary plot fo r airplane C of psolutionsvsda fo rvariouselevatordeflections.For simplicity,the negative solutions are not shown. Such plots can begenerated very rapidly by the computer program. It clearlyshowsthat6e>0(pitch down)causes multiple solutions andpredicted possible jump phenomena at much lower ailerondeflections, and that elevator deflections as low as 2 or 3may be critical. In fact,autorotationalsolutions first appearfor 5e between 1 and 2. There are threedifferent types ofautorotational solution for pitch-down elevator, but allcaused very irregularresponse to aileron,as inFig. 9. Pitch-up elevator seems to increase the critical aileron value. In
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APRIL1977 JUMPPHENOMENAINROLL-COUPLEDMANEUVERS 381
6 a = -50
F i g . 12 Calculated roll-rate responses of airplane D for severalai leron inputs.
other examples , such as airplane A , pitch-up elevatordecreased thecriticalaileron value.Figures 11 and 12show PS S solutions and correspondingtime histories for airplane D. These results ar e typical ofseveral examples tha t were studied, but which showed no
multiple solutions within th e normal range of ailerondeflection. In thiscase, both the PSS solutions and the timehistories show that, while there are significant nonlinearcoupling effects at large 6fl , no sudden jump in the responseoccurs. Only roll-rate response is shown in Figs. 11 and 12,but all variables approach values consistent with the PSScurves . Multiple PSS solutions actually appear at 6a=56,which is beyond the maximum aileron value. These results,and similar ones tha t were obtained for a number of otherexamples , show that the PSS method also reliably predictswhenj u m p sin theresponsewilln ot occur.Finally, itshouldb enoted thatth eobjectivein the examplesshown was to cover a fairly wide range of practical cases,although anattempt wasmade to choose examples likely toshow significant coupling effects. None of the examples,however , was studied with the thoroughness tha t would benecessary inpractice for a specificairplane. For example, noresults with rudder inputs have been shown. In a practicalstudy , a set of casesinvolvingal l threecontrols and includinga variety of potentially critical flight condit ions would bes tudied. Although this would yield a mass of compute routput, th ec alculations ar e very simple,can be programmedon a small computer, and require very little time. Theavailability of an interactive terminal an d automatic plottingwould be very helpful in rapid evaluation of results an delimination ofunnecessarydata.
Concluding RemarksA pseudosteady-state analysis method has been describedfo r predicting criticalcontrol input combinations ,which m aylead to j u m p phenomena in the roll-pitch response of air-planes. Calculated responses for a variety of examples haveshown tha t j u m p phenomena do occur at the predictedcontrol combinations . Future research will be aimed atgeneralizing the method and applying it to more detailedstudies of particular a irplanes , which ar e kn ow n to havedivergent tendencies in rapid maneuvers at high angles ofattack.Th e effects of rudder inputswill be included, and theimpor tance of the qr termeffects willbeevaluated in a widervarietyof maneuvers .The extension of the meth od to includenonlinear aerodynamic effects (Appendix B) will beprogram med for solution byvariousiterativemethods.Another avenueof researchwillbe an attempt to developacomputerizable cr iter ion for the design of airplanes andcontrol systems that will no t have jump phenomena in theirrequired maneuver envelope. For example, such a cr i ter ionmight be based on mathematical constraints on the coef-ficientsofEqs.(15) or (16), whi chguaranteethe existence of asingle realsolution.
AppendixAResponse time histories, used fo r comparison with resultsof the PSS analysis, were obtained by numerical integrationof th eequationso fm ot ion,assumingconstantair density andspeed and linear aerodynamics , but inlcuding completerotational coupling and weight terms. By use of principalaxes, these can be written as the following first-order
equations , where th e "hats" on the moment derivatives in -dicate that they are defined with th e inertia ratios factoredout, ass ho wninEqs.(Al) to(A3).b r ) A)e ) A2)fi d rd r) (A3)
cos< /> -cos0 0) A4)/3 =-
,.+ (/V ) c o s 0 s i n c / >tan0sin -I -r tan0cos
0=qcos
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38 2 A.A.S CHY ANDM.E.H A N N A H J.AIRCRAFTMultiplying these through by C(p,a) gives tw o quarticequationsinp.P4sina+yp3+P2[(mqnr-nl3) sina+(m+mqnp)cosa+y (z +np)]+p[(rhqn r-n&)y+n(mqcosct+yp) ] + (zfhq m)(n^cosa+y^nr) =0 B9)
4+fp3+p2[fp(ihqnr-np)-fr(m +mq
(BIO)These equationsmay be solved for the PSS values ofp anda by some iterative search algorithm, or they may be solvedgraphically by choosing a set of a values covering areasonable range and plotting the realp solutions of the twoquarticequations vsa . A nycrossing points aresimultaneoussolutions. These solutions then may be substituted into th eexpressionsforq(pta),r(p,a), an d J3(p,a), whose deriva-tion was described in the preceeding to obtain the complete
PS S solution fo reachspecified set of controlinput values. Acomputer program is being written to solve theseequations ,includingth eq rte rm.
ReferencesPhillips, W. H., Effect of Steady Rolling on Longitudinal and
DirectionalStability, NACA TN1627,1948.2Pinsker, W. J. G., Aileron Control of Small Aspect RatioAircraft, in Particular, Delta Aircraft, Aeronautical ResearchCouncilReports and Me moranda 3188,1953.3Rhoads, D. W. and Schuler, J. M., A Theroretical and Ex-perimental Study of Airplane Dynamics in Large-DisturbanceManeuvers, Journal of Astronautical Sciences, Vol. XXIV, July1957,pp. 507-526, 532.4Hacker, T. andOprisiu, C., A Discussionof theRollCouplingProblem, Progress in Aerospace Sciences,Vol. 15,Pergamon Press,Oxford ,1974.5Haddad, E. K., Study of Stability of Large Maneuvers ofAirplanes, NASACR-2447,1974.6Adams, W. M., Jr., Analytic Prediction of AirplaneEquilibriumSpinCharacteristics, NASA TND-6926,1972.7Etkin, B. , Dynamics of Atmospheric Flight, Wiley, NewYork,1972,p p.443-451.
FromtheAIAAProgressinAstronau ticsandAeronau ticsSeriesAEROACOUSTICS: JET AND COMBUSTION NOISE;DUCT ACOUSTICSv. 37
Edi ted byHenry T.Nagamatsu, GeneralElectric ResearchandD evelopmentCenter Jack V. O'Keefe, Th eBoeingCompany;an d IraR.Schwartz, NASA AmesResearch CenterA companion to Aeroacou stics: Fan,STOL, andBoundary Layer Noise; Sonic Boom; AeroacousticInstrumentation, volume38 in the series.This volume includes twenty-eight papers covering je t noise, combustion and core engine noise, and duct acoustics, withsummaries of panel discussions. Th e papers on jet noise include theory and applications, je t noise formulation, sounddistribution, acou sticradiationrefraction,temperatureeffects,jetsandsuppressorcharacteristics,jetsa sacousticshields,andacousticsofswirlingjets.Paperson combu stion and core-generatednoise coverboththeory and practice, examiningducted combustion, openflames,and someearlyresults of core noise studies.Studies of duct acoustics discusscross section variations and sheared flow, radiation in and from lined shear flow, helicalflow interactions, emissionfrom aircraftducts, planewavepropagationin avariableareaduct, nozzlewavepropagation,meanflow in a lined duct,nonuniform waveguidepropagation, flow noisein turbofans, annular duct phenomena, freestream tur-bulen t acoustics, andvortexshedding incavities.
541 pp.,6x9, illus.$19.00Mem.$30.00ListTOORDER WRITE: Publ i ca t ions Dept ., A I A A ,1290Avenue of the Amer icas , Ne wYork,N. Y.10019