1、,.:NATIONAL ADVISORY COMMIITEEFOR AERONAUTICSREPORT 1169MATRIX METHODS FOR DETERMINING THELONGITUDINAL-STABILITY DERIVATIVES OF ANAIRPLANE FROM TRANSIENT FLIGHT DATADONEGAN.m -,.*1954-.-,-For smleby the SrsperIntendentor Documents, U. 9. Gmernment Printing Of(ls Washington Uj D. C. Yenrlr mbemiptk,
2、S1O;ibrclgm $11dngfa copprleamrkeacemdlng to aim . . . . . . . . . Ma 2.santsProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT 1169MATRIX METHODS FOR DETERMINING THELONGITUDINAL-STABILITY DERIVATIVES OF ANAIRPLANE FROM TRANSIENT FLIGHT DATABy JA
3、MES J. DONEGANLangley Aeronautical LaboratoryLangley Field, Va.IProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NationaI Advisory Committee for AeronauticsH.eadquartem, 1612 H Street NW., TJashington%, D. C.Created by act of Congress approved March 3
4、, 1915, for the supervision and direction of the sc.ientifk studyof the problems of flight (U. S. Code, title 50, sec. 151). Its membership was increased from 12 to 15 by wtapproved March 2, 1929, and to 17 by act approved May 25, 1948. The members are appointed by the Prcsidwlt,and serve as such wi
5、thout compensation.JEROME C. HUNSAKER,Sc. D., Massachusetts Institute of Technology, CluzirvnanDEfrLBV W. BROXK, PH. D., President, Rockefeller Inst,itmt e for Medical Research, Ilce Chairman,JOSmPHP. AnAMs, LL. D., member, Civil Aeronautics Board.ALLENV. ASTIN,PH. I)., Director, National Bureau of
6、Standards.PWESTOFJR. BASSETT,M. A., President, Sperry Gyroscope Co.,Inc.I,HONARCARICHEL, PH. D., Secretary, Smithsonian Inst.i-t ution.RALPHS. DAMON,D. Eng., President, Trans World Airlines, Inc.JAMES H. DOOLITTL,Sc. D., Vice President, Shell Oil Co.LI,OYDHARRISON,Rear Admiral, LJnited States Navy,
7、Deputyand Assistau t Chief of the Bureau of Aeronautics.ROFiAL M. HAZEN, B. S., Director of Engineering, AllisonRALPU A. OFSTIE, Vice Admiral, United States Navy, IXputyChief of Naval Operations (Air).DOIVALtIL. PUTT, Lieutenant General, llnited States Air Iurw,Deputy Chief of Staff (Development.DON
8、ALDA. Qumr,w, D. .Eng., .4ssist ant Swretary of Drf,wsc(Research and Development).ARTHUR E. RAYMOND,Sc. D., Vice lresidetI;rlgilccrig,Douglas Aircraft Co., Inc.FEM$CIS W. R.EICH12LrREER,Sc. D., !hief, Vnited St.atdsWeather Bureau.OWVALURYAN, LL. D., member, Civil Aeronautics Board.Division, General
9、Motors Corp. NATHANF. TWININO,of Staff.General, nitecl States Air Force, (!hi!fHUGH L. DRYEN, PH. D., Director JOHN F. VICTORY,LL. D., Executive SecretaryJOHN W. CROWLEY,JR., B. S., Associate Directorjor Research EDWARD H. CHAMBERLAIN, Emcutice OJicerHENRY J. E. REID, D. Eng., Director, Langley Aero
10、nautical Laboratory, Langley Field, Va.SMITH J. DFRANc, D. Eng., Director, Ames Aeronautical Laboratory, Moffett Field, Cnlif.IOIWARDR. SHARP, Sc. D., Director, Lewis Flight Propulsion Laboratory, Cleveland Airport., Cleveland, OhioT.ANGLEY AERONAUTICAL LABORATORY AUESAERONAUTICAL LABORATORY LEWISFL
11、INTPROFULSIOLABOA.WMWLangley Field, Va. Moffett Field, Calif. Cleveland Airport, Cleveland, OhioConduct,under unijied control,for all agencies, of scientificresearchon thejundmnentalproblemsof jZightIiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-R
12、EPORT 1169MATRIX METHODS FOR DETERMINING THE LONGITUDINAL-STABILITYDERIVATIVES OF AN AIRPLANE FROM TRANSIENT FLIGHT DATA By JAMESJ. DOXEGANSUMMARYThree matrix method8 are pregented jor determining the/ongitudinal-Wability derivative from trarmient jlight data.One tnethod, which requiregfour meawmeme
13、nts in time-historyf orm and utilizes the incremental tail load to separate the pitch-,ing-moment derivatives Cti and C,. see appendix EcL rate of change of lift coefllcient with elevatordeflection per radian; see appenclix Ec. see appenclix ECzerate of change of lift codficient with pitchingvelocit
14、y per radian; see appendix B18upereedesNAOA TA 2902,tMatrix Methods for Deterrniniug the LongiturlinfMtability Deriv8tivw of rm Airplane From Transient Flight Data” by JamesJ. Dorkgan, 195S.1Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT 1169
15、NATIONAL ADVISORY COMMITTEE FOR .4ERONAUTICSpitching-moment coeflmient of airplane,i14/Frate of change of pitching-moment coefficientwith angle of a.tt.ack per radian; see ap-pendix Erate of change of pitching-moment coefficienttwith ele.vator deffe.ction per radian; seeappendix Erate of change of p
16、itc.hing-momen coefficientwith pitching velocity per raclian; seeappendix Erate of c.ha.nge of pitrhing-mornent coefficient,with see appendix Epitching-moment coeffkient. of horizontal tailsurface, .llJt/q#SJcaccekration due to graviy, f t/see/seeairplane moment of inertia, slug-ftaairplane mclius o
17、f gyration about pitchingaxis, ftlift, lbuirplane mass, 117/g,slugspitching moment of airplaneairplane load factorp “z lb/sq ftdyna.rnic pressure, ,wing mea, sq fthorizonta.1-tail areatime, sectrue velocity, ft/secairplane weight, lblength from center of gravity of airpla.ne toaerodynamic center of
18、tail (negative forconventional airplanes), ftcoeilic.ients of transfer function relating 6and 6; scc appendix Ewing angle of attack, radianstail angle of attack, radiansflight-path angle, radiansangle of pitch, a+elevator deflection, radiansdownwash angIe, radianstail efficiency factor, q,/gmass den
19、sity of air, slugs/cu ftdummy variable of integrationMatrix notation:II II rectangular matrix1 square matrix column matrixcl integrating matrix (see ta-ble I)IP,II 1111,111Irectangularmatricesdefined in appendix Esubscripts:i denotes row elements in matrixt tailfl,wing-body combinationFor sign conve
20、ntions used, see figure 1.A C1OLover a symbol denotes the first derivative with:i:spcct to time, and two dots over a symbol denote thesecond dmiva tive with respect to time.imposesa restriction on the ge-nertdit Of the llldhOd, it tipp!urs Jbe j ustilled since it rechwes computation time to tihnosto
21、ne-half that required for method :4 and for thp munplespresented herein gave results whioh are in good agreementwith those of method A.The method is outlined by merely stating the mppropriat rcqua.tions, the clevelopmcmb of which is contained in append isB. The procedure is as follows:(1) Compute th
22、e time history of Aa by using equations (5)and (6).(2) Determine a time history of the intermediate qtmntitAt from the expressionProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-METHODS FOR DETERMINING AIRPLANE LONGITUDINAL-STABILITY DERIVATIVES FROM
23、FLIGHT DATA 5(3) Calculate time histories of lAadlAd andstA6dtby using the integrating matrix IICI and the timeh;tories of Aa, Awhereas all three flights are analyzed by method B. Com-putations are shown in the tables for flight 1, but for theother flights only the results are given.Table I contains
24、 the integrating matrix IICII based onSimpsons law (ref. 3) which is used in all three methods,The airplane characteristics and flight cond!tions areshown in table II (a) for all thee flights. Although thegeometric parameters are the same, the parameters such asweight, speed, Mach number, center-of-
25、gravity position,and altitude vary slightly between the three runs.In table II (b) the coefficients of the transfer functionwhich relates pitching velocity to elevator deflection de-fined by equation (8) and computed by the method out-lined in reference 4 arc shoyn. These preliminary constantsarc re
26、quired in methods A and C and the actual computa-tions me shown in a subsequent table.Time histories of measured and derived quantities forflight 1 are shown in table 111, The quantities in columns9, Q, and 0 are measured and th other five quantitiesare derived from the measured quantities. In these
27、 tablesmore decimal places are carried in the measured quantitiesthan are warranted by instrument accuracy in order to as-sure no 10SSin accuracy in rounding off. The measurementsProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 REPORT 1169.NATIONAL
28、ADVISORY COMMITTEE FOR. AERONAUTICSof incremental tail load AL were ma.ikhle only for the timeslisted, and, since these. c.ovcrecl approximately the naturalperiocl of the short-period oscillations of the aircrxft, the.TABLE H.-A1RPLAE WARACTER1ST1(Y5, FLIGHT WX-DITIONS, AND TRANSFER-FUXTIOX COEFFIlE
29、XTS()Airplane cinmicterlstlcsrmcfflight coridlttons 1 Fllghtl I?ligM2 ) Fllght3 :data. were considered sufficient. More of the tinw h iqtnritlsof the other variables were available and were used.Method A,The princip comput titions illust rut ingmethod 4 are presented iu table IV; some of the int erm
30、wl-iate steps outlined in method : are simple computations andare therefore uot inc.lucled in this tabk. Tablp IV (u) isobtained by applying equtit.ion (7) to t,hc data given in tt-ihk111 ancl illustrates stp (3) of method .4.In table IV (b), the computations illustrtiting the d(tr= tmination of e a
31、nd C,is based on equation (15).The refined vahies of C= and are determined in t )hIV (d) by step (13). Two of the rolunnw are taken dirwtlyfrom table 111 and the other column is derived by usc ofequation (18). IFinal results obt.sine.d with” method A for tlw data of fhghts 1 and 2 after three ibmati
32、ons me shown in table6, ft . . _. _ _ . .c, ft . . . . . . . . .t%nter.of-gmvlty posltlon,percentM ,A.C ./r, l$110J0.4971806i;1.175289.363,R0.289301fi;O.001i6714.0!: 14.0%27.82 27.440.062854 0.0+3266625J2; 2;9;o.iQ4 O.i961802i; 132$.661711,175” 1,175289.3 289.3512 51458.050 6%8800.297595 0.293060:8;
33、 w. 50.870.001276 0.001281III.9;,ft2 . ._ . . _ . . r ft/sec.-_. _ . . . .( lb_ . . . -W/qs h. . . . . . . . . . . zg, fl . . . . . . . vi,. -. . . . . . . . . .P!slllEdftJ. - . . . . . .!11(b) Coefficientsof airplane transferfunction ,I Flight 1 Flight 2t . CoMicient 0.18 10.829 0.70A-$, . . . . .
34、. K, . . . . 1 -W j 1.4-10.010 : ;2-16.526 “1ITT (e).TABLE 1.11.TIME HISTORIES OF MEASURED AND DERIVED QUANTITIES FOR FLIGHT 1I - . -. -Derived,t, -Measured.-.Ah.lAt-:-.030886-. 10737s-.177000-.204757. 220102-. 21506G-.201806-.181643-.160433-.133012-.031420;g.c.g.1243733.164326.213857.2WS20: :1.3L 4
35、1.5L ti!:2.12.2:2.526H29$.()313.2U3.Eo.0G9703.055312.072SS0.074625.071414.070698.067923.0624391.067923.064712.043240.032377-.012505-.02302, 0260zJ-. 04720i7-.075428-.071623-. 078J374-.032513-.088003-.095707.037504-.020227-.024950-.026422: :f12a-: :;21722835kl?o0.03792-.12006-.35392-. 62b63-.02272-1,
36、 169.M-1.40936-1.56730- L 74432-1.85203-1.75064-S. 75064-1.45320-: i);0J:C6:-. 000G23-.012526;%J:t-.022 7-.04665-.07003-.092 3-. H3Q97-.127257-. 13$?72!-.14856-.14754-.14500-.125729: c%!12-. 03J41=%.2-IW3 o.031 !5y3.127 ;: h33$o.010591.063067.183905.328389.474922.01G391.724134.812714.875329.038172.9
37、02055.850022.744036.?S6752;003;:-.022336-.250374-.470502-. 67W53. 854892-1.010045-1.124362. -_ - .-.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.FOR DETERMINING AIRPLANE LONGITUDINAL-STABILITY DERIVATIVES FROM FLIGHT DATA 7TABLE IV.COMPUTATIONS ,
38、ILLUSTRATING METHOD .4(a) FM approximation of CLaand CL) by step (3).- AnO. 014640.082351.210433.340384.424566. 3Q1427.300243.53$027.516067.470316.424566.353025.239144.217773.100651.034181.003630-.054901(c) Determination of Cm=and CL, by step (10)t-.0.1.2:.6.6.7.81:1.11.2;:1,61,61.7tab.81:1.1.Au-0.0
39、34981-.032478-.026362-.020274. 014992-.028194: fl:;l;.011195.016361.021140.025542.026734AC t FadIIAd An tqs lab:III,:olumn,(b) Determination of Ci rimd C.3.4.5.6,7.81:1.100.010591.068067. 1835Q5.328389;:;:WSZ;.724134.312714.875629.9081731.2:1.5;!1.31.92.02.12.22.32.4III.,.Cmj API =iAwt9.934264Ci= -1
40、. 57W36Cni = -0.1589C.: Flight 2flight 1.,. I.,.“A -”L.- l_0.113 6.700.105 0.4460.llm4 0.0760.0304 0,0330.003 -0.7110.001 -0.1810,001 -0.0780.003 -0.9685.050.4612.08cLm-_. 7.09clip0.468CL;-0.072cL;- . _ 0.028coma. . . . . . . . . . . . . . . . . . . . . . . -0.622cm;- . -. _ _.-. _. -0.171c,ni. . .
41、. . . . -0.068cm$. .- . .-_ . -. -. -0.914(acL/aa),- 4.84wad. . . . . . 0.425hcL,/h6-2.19,.2!)W37552Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 REPORT 1169AATIONAL ADVISORY COMMITTEE FOR AEROh.4UTICSMethod BiThe principul computations illuedrak
42、ingmt!t,hocl B are presented in table V. Again, some of theintwmeciate steps outlined in method B are simple com-put titions and me. therefore omitted. In table V(a) the com-putation demonstmting the determination of C,c, C,i, andC,* by step (4) by using the relation (25) is shown. Threeof the cohmm
43、s me. obtained by operating on columns , ,tind of table III with the integrating matrix IICl I givenin table 1.Table V (b) illushwtw step (7), the determination of CLeaml CL5 using equation (19). Two of the columns are ob-tained ciire.ctly from twble III and the other column isderived by using equat
44、ion (18).In table V c) find results obtuined with method B forthree sets 01 fright clata me shown.Method C,The principal computations of method C tirepresented in tabIe W. Table VI (a) shows the computa-t i.ou of Kl, K2, KS, and K. from flight 1 data by the methodof reference. 4. The integrals in ta
45、ble VI (a.) were computedby reading the film at 0.05-scconcl intervals and using theintegrating matrix for At= 0.05 seconcl; this interval wasne.ceesary in order to obtain reasonable results for themethod. tse of the time interval At= 0.1 l did notproduce suffkie.ntlly accurate values of KS and that
46、 is, the airplaneshould be operating uncler c.ondit.ions in the linear range of thecoefficients, the maneuver should be of the pull-up or push-down variety where little loss in airspecl oc.mrs durirg themnneuver and where clisplac.ement angles are small, and themaneuver shoulcl start from n level-fl
47、ight trim condition andshould be in the Mach number range in which these assump-tions are wdicl.Since the choice. of the methocl t.o lx! uscxl dcpen(ls primoriljon the number of memuremente which arc availahh, mrt h flIA is recommended when four bnsic mwwurcnwnts nrv tivail-able, method B when three memurwnents urc availal)lc, amlso forth. If, howtwer, ml accurate VUIUCof k is known inadvance, then met,hocl B is recomnlen thc rwort Isshould then be read as carefully M possible. Measured tuil-,Ioacl clata should be corrected for efucts of inertia. TIw,accuracy of the imulys
