1、REPORT 1042SO.ME EFFECTS OF NONLINEAR VARIATION IN THE DIRECTIONAL-STABILITYAND DAMPING-IN-YAWING DERIVATIVES ON THELATERAL.STABILITY OF AN MRPLANE By LEOXASD SmESPIELDSUMMARYA theoretical investigation has lkwn made to detemnine theqfect of nonlinear 8tability deriratiws ma the lateral stabilityof
2、an airplane. Motions were calculated O? the awumptionthat the directional-stability and the damping-in-yawingderiratires are functions of the angle of sideslip. The applica-tion of the Lapace transform to the cakulaiion of an airglanemotion when certain types o-fnonlinear deriratire are present i8de
3、scribed in detai?. The types of nodinearittis amumedcorrespond fo the condition in which. the ralues of the directionaJ-tability and damp ing+n+awing derirafices are zero for smatlangles f sidesip.The results of the irwestigation indicated that under certaincondition the nonlinear staliility derirat
4、ices a+wum.ed in theanalysis caused a motion which had diferent rates of dampingfor the large and small amplitudes of motion, with cery littedamping at the small amplitudes. In general, the period of theredta nt oscillation hmeaaed with time.NTRODUCTIOBJRecent fli-ght teats of seyeral airphines desi
5、gned for high-speed h.-f the deacl spot, each OM of the cases rcprwnti :Iifferent type of variation of C= with 6 in order to simultitlthe effect of several possible flow conditions on thv side forcecting on the vertical surface. For cases 1 anti 2, (n= 0.28d for case 3, .Cm=0.41. TIN corresponding v
6、lu( of “da, for all three cases is 0.39. 1 should be notml in figuw Jthat for cases 2 and 3, C=0 at 19of 2 and 2, whereasfor ease 1., d= has a finitr value at p of 2 and 2.METHOD OF CALCULATING MOTIONSince the nonlinearities shown in figure 1 can hc 1rmtulas linear derivatives of diffwent values wit
7、hin and outside ofthe dead spot, the airplatw motion is calcuh hd on thr Iw.isof classical linear theory. 1hc equations of motion rind hgeneral method of calculating W motion of tin airphm rrgiven in references 1 and 2. Thr methods of rcfcrcwcs 1and 2 are based on the I aplare trmsformat ion which i
8、n-herently; r takes into account th; inititil ronditicms of thvproblem. Because the I,mplace transformation .conskkmthe initial clisplacements and initiaI wlocit ies of the problem,this mod directly applicalh to tlw rtih:ulation of thvmotion of an airplane which hus nonlinww dtwivalivcssimilar !0 th
9、e .derivativw prcscnLcd in figure 1.The nondimensional linearized latwa (qutitions of mo-tion, referred to the sttibility aws, tire for rolling, for yriw-ing, and for sideslipping, rcsprctively:Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.SOMEEFF
10、ECTS OF X-OHJIXEARY WIt.hhr deadspotW/JS,lbflt 1- ml alb-: - 101.1 IOL1 Slugqft :-. -. -. -. -.-., - -.-: -o.OmsE o.mmv, fuw- 759 ib9CL. -.”,-. . . . . . . . . . . . . . . - - .-. o.ala o.asII,fi. . 2i.i 27.7r, deu. -.- .-. -.- . . . . . - . . . . -. - . . . . . . . . - o 0Kzf _ -. .- . . -. .- . .
11、. . - - - CL05L3 o. 0si2K3 . . - . . . . . . . . . . . - . -. -.- -. O.o+m o. CQmJp, prmdf-.-._.-i.-.-:- -0.402 -a 402C., pmmdLm . . . . _ -0.0155 -o. 01%Ci, prrmdhm. -: -. - . . - . . - . . . . . - o. lx -a lmCF9, per *n - o 0Cr, rwmdhur . . . . . . . . . . . . . . . . . . . . . . . . . . . - .0 i-
12、lq.dw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - -21,0,20 20,0,2.0C, Nr rlldlan - . . . . . _ . - a 302 0C.p (raw I and 2), pm mdtan. . . . . - 0.% oC.fl (caw 3), pcrmdfso . .-.-. - -.-;-. . au o,airphme is either outside of or within the dmcl spot. Fromthe
13、analytical solution of themotion, based ontenmssandae.rodpamic characteristics of the “fi%t column of tilde 1and m initial condition of P=5, the time histoq- of p wascomputed for several values of sb until the due of b forwhich B=2 WM reached. For values of so greater than thebfhichresults in = 2, t
14、his analtica solution is ticomectsince the airplane has no-iventered into the deud spot and thevalues of .C,6 and C% are zero. Thus, a new solution mustbe calculated with the use of the values given fi- the secondf.olumn of table I with new initial conrlitions. The newinitial conditions are determin
15、ed by substituting the value ofsfi at which =2 in the original analytical solutions of ,*, D, D,*, and . Once these”-initial conditions areknown, another sol of analytical solutions are computed ford, 4, $, and their derivatives from equntions (3) and (4).This procedure is followed every time f? cro
16、sses through 2or 2. The final resultant motion in sicleslip is the sum oftill the analytical solutions in whereas the C,B for case 3 is 0.41. The motions prcscntdin figure 2 are for C_P=0.28; however, the motions forC,P= 0.41 would exhibit oscillations of approxima tcly the snmcdamping ancl a slight
17、ly srnaller period.The motions of the airplane in sideslip, showing the cfiucof the nonlinearities illustra tMlin figures 1(a), 1(b), nnd 1(c),are presented in figures 3 to 5, respectively. ln all CW.CS,an initial disturbance in sideslip of 5 wns msumcd. Thopronounced effect of the nonlinearities on
18、 the ltiternl motionis noted by a comparison of figure 2 nnd uitlwr one of figures3, 4, or 5. k all three figures (figs. 3 to 5) the motion for=2, the most stable case, approaches n consttint value.The analytical solution of the motion for the case of q=2in figure 3 indic.stes that, within the dmcl
19、spot, the airphmewill oscillate at a period of 0.56 seconds and T,IS=3.38seconds nndwill eventually approach the vnhw of = 0.00!32.Similar motions would be obtained for the rase of ?=2 illfigures 4 and 5. As q is decreased, tlw dmnping of LIWoscillatory motion depends upon the nonlinearity nssunmlan
20、d the values of q. In figuro 3, tlm motion for ?=OOdamps nt a slow rate at. the large amplitudes until thr oscil-lation reaches ,an amplitude of approximatePly 2.4 whucthe dnmping of the oscillation is zero. MI prriod of 1lWoscillation increases from L5 to 1.85 sccomls. For the cmc ofq= a very light
21、ly dmped oscillation is npparrnt wilhintl.w first few seconds and the airplnne may L)c considww-1tobe neutrally stable at an amplitude of +4,5. In figurrs4 and 5 the motion for q=O” cearly indicfitm thnt thu for T= 2, the oscillatory molionis slightly unstable, neutrally stable oscillation would bee
22、xpected to occur in figures 4 nnd 5 for the combinations ofa -valueof q between 0 md 2 and the dmd spo t wwnlrdin the calculations or for q= 2 and u smaller clcn(l spo.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SOME EFFECTS OF NO&GINEAlt VARI.M?
23、ION IN CndANT) C=r OX LATEIM.L STABIH OF k%” AIRPLANE “ 1013 ,.,.-5t-60 I 1 I I I 4 I I I If 2 3 4 5 & 7 8 9 10t, secFm= 2-Calcn18krl moffon ofrm sIrPIane due to an fnithd distarimnce CusfdesIfP fir swami values of?.Provided by IHS Not for ResaleNo reproduction or networking permitted without licens
24、e from IHS-,-,-4.t-1-4-5-c?REPOHT 104?-NAITOXAL ADVISORY COMMITTEE FOR AERONAUTICS -2.-.-. :I ! I I I 1 I I I/ I I I I2“ 3. “a “.5 “. 6 7 -“-”8 -9 10 II i.? f3t sec -.FICTRE3.The effvct of the nonlhwnr dorivatvca deswibc.d in rlguro l(a) oo the motion of tin alrplma,Provided by IHSNot for ResaleNo r
25、eproduction or networking permitted without license from IHS-,-,-.5!4 -3 -2 -I -b60 -w-t -2 -3 4-4-5-SOME EFFECTS OF A-OXT.JXEAR“ARIATIOSIX CmdAXD CmrOX LATERAL STABILZIT-OF AK AIRPIuASE 10M _.v.:9)- a-2t sec. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from
26、 IHS-,-,-*REPORT 1042NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS-5-.I 1“ I I I I I I I I I I I I I ! I II2 3 “4. &- 7 “.8t,ac 9“ !:. .-:FICCRE&-The eflect Of the nuwr derIvatIws dewrilmd in flguro l(o) on the nK!tIonef an airine.In general, the results indicate that the damping of thelateral oscilla
27、tion calcdated with the use of derivatives con- 1stant with amplitude is a determining factor in the type ofmotion obtained where nonlinear derivatives are present,As the inherent damping of the Iateral oscillation decreases,a smaller dead spot will result in a neually stable oaciktion.Obviously, if
28、 the inherent dRmping is zero, a neutrally stableoscillation alreacly exists with zero dead spot.Some .additiomd mkulations were made for the u-wewherethe airplane is disturbed within the dead spot. The motionsfor an initial condition of #= 1 wero computed for q=2and 0 with the assumption of the non
29、linearity dwcribed illfigure 1Q). The results are presented in figur( G. It shouldbe noted that the only Merence between figures 4 and 6 isthe initial condition assumed in the calculations, ln figure 6,the motion for q= 2 is unstable and gradually approachesProvided by IHSNot for ResaleNo reproducti
30、on or networking permitted without license from IHS-,-,-130m E3?FECTS013IroNLnnMRVARIATTOh-ES C.BANDC=.ONLATEEULSTABILFIYOF AN AEiPI#iIWil . 1017_.the amplitude find period of the motion for the case of q= 2in figure 4. The motion for q=0 in f&me 6 is slightly m-instableand will probably increase un
31、til its amplitude andperiod are in close agreement with the motion for the caseof q=OO in figure 4. Calculations have indicated that theosdlatory motion of the airplane within the dead spot Mdouble amplitude about every 4 seconds for = 2” andabout every 30 seconds for q=OO. If the motion is unstable
32、n-ithin the dead spot, either the airphme motion d be neu-traIly stable with an amplitude equal to or greater tham theamplitude of the dead spot or the motion” mill be unstable.The loss in damping and the increase in period which ap-peared in some of t-helateraI oscillations in figures 3 to s cmbe a
33、ttributed to the type of nonlinearity aastied. Fromclassical dynainic stabilitg theory, it is weIl known that thedamping of the oscillation is a function of d., and the periodof the oscilktion is a function of C*P. If the airplane is con-sidered as a mass-spring dashpot system, C.d is the equiv-alen
34、t spring constant of the systam and C., correspon tothe damping constant contributed by the dashpot. Thus asCapia reduced the period increases, and as C., is reduced thedamping decreases.213637+845CONCL”iNG EEMARKSThe results of the investigation. made to. determjne theeffect of nordinearities assum
35、ed in the ana&sis on the lateralstabihty indicate that under cartain conditions a motion isobtained frhich has dMerent rates of damping for the largeand small amplitudes of motion, with very Iittle damping atthe small amplitudes. In general, the period of the resultantoscillation increases with time
36、.#hWGLEY AERONAUTICAL lkB.ORATORY,NATTCIXS ADVISORY Comcmmm FOR .+?LSROXAUHCS,IJXNGLEY “FIELD, TA., eptember 19, 1980.REFERmcEs1. llokrzYcki, G. .&: Application of the LapIace Transformation tothe Solution of the LateraI andhngitudiual St.abil- uaticns.NACA- 2002, 1950.2. Murray,HarryE., and Grant, FrederickC.: -Methodof Caloulatfngthe Lateral Motionsof AfrcraftBasedon the LaplaceTransform.NACATN 2129, 1950.3. Churo RueIV.: ModernO”rationaIMatheu.wticsinEngineering.MoGraw-HillBook Co., Inc. 194Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
