1、REPORT No. 638THE INFLUENCE OF LATERAL STABILITY ON DISTURBED MOTIONS OF ANAIRPIANE WITH SPECIAL REFERENCE TO THEMOTIONS PRODUCED BY GUSTS, By ROBEET T. JOKESSUMMARYDisturbed lateral m;.0s+.g-.462-.m-.m-.449-.m-.4m-.442-.4X-.4s0-.Ma.al-.44am5!.m.!lm.441:%1.442.m.H1.442:s!.442-aC47-.ms-.MO-.m-.Ws-.m-
2、.cm-.m-.mo00-.la-.177.9m-(L(L-.o14-.on-.W-.UM.=-.M5-.uf4-.m-.Ced-.W4-.m-.UM- U14-9.9.-. m-.W7-.m-.m-.m-.1*-.1o1-.m-.W7-.lw-.m.These coefhients were est.imted from the outwardcharacteridics of the airplane by methods described inreference 4. The derivatives (?V, CW d OrB (CQRC-sponding to the yawing
3、momente in sideslip and inyawing and to the aide force in aidealip) were assumedto be affected by the changes of fin area. Only thederivative Cb (corresponding to the rolling momgpt inaideelip) waa mnuned to be affected by changes ofdihedral. The effect of dihedral on the Mend force inaideelip was n
4、eglected mnuch as it was found that acompensating error was introduced by the absence ofthe side force due to rcdling in the equations of motion.Another omission is the small adverse effect of dihedrd.angle on the weathercock-etability factor UW Tfieffect is small, pmticularly in view of the wide va
5、ria-tion of C.Bassumed. At a lift coeflkknt of 1.8, repre-senting 10w+qwed flight, a full+pan flap was assumed.Teete show that the tiect of .mmh-aflap is to increasethe weathercock-etability factor somewhat for the wingalone, In practice, the flap might interfere with theair flow-over the h so that
6、the increase of Cxdassumedin this condition would not be realized. -Fige 1 shows the positions of the modihd airplaneson the lateral+tability diagrams. These diagrnma arc-tidy similar to those given in reference 6 cxccptthat a simultaneous increase of C,with C%was aewmmdto show directly the efTectof
7、 increasing the fin mea.The value recommended by Diehl (reference G) forCad works out to about 0.03 for tho wing oadingassumed here. Linda mentioned by iiilliktin (refer-ence 7) correspond to 0.08 CMB0.05. hrdy alldesignem are familiar with the limits of LJhT, for satis-factory lateraI stability giv
8、en by Korvin-Kroukovsky(reference 8). l?igure 1 (a) shows these limits in termsof cJ#andcab.It-Aould be mentioned that Korvin-Ikmkovekys formulas are more suited to the empirical-statistical analysis in which they were employed thrmto the determination of abeolute values of GJC16 forthis stability c
9、hart. In most cases, wind-tunnel testsshow values of C.dsmaller thun those prcdictod so thatProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-INFLUENCE OF LATERAL STABILITY ON DED MOTIONS OF AN NE 509“the s-d range, if givenin wind-tumd values,would pr
10、obably fall somewhat lower than indicated inilgllre 1.The value CW=0 does not., of course, correspond toan airplane with M vtical tail surface. Experiencehas shown that the unstable yawing moment of a largeweU-streamlined fuselage may timtirely ofbet the stabi-lizingaction ofafairhence the smallest
11、area likely to be used in a moderndesign (4 percxmt of the wing gea) was chosen to repre-sent the condition.It ie, in general, M3ioult to prediot the values oftither Cv or C+ for a given design- It will be realizedthat the correqondingvahm of k area and dihedral, asrefereed to in this report, apply
12、onIy under certainidedized conditions and are employed primarily es amatter of convenience in fig idme on the problem.Reference 9 giv= a summa ry of teat values of C.B,rncluding a discussion of pertinent factors and drawingeof the modeIa tested. The data included in that papershould aid the dedgner
13、in judging the vreathercockstability.INPLUENCE OF LATERAL STABILITY ON MOTIONS DUETO ABBITRABY DISTURBANCESGBN- DESOEIPTIONOFLATEEALMOTIO.XSThe equations of lateraI stabiIietmWdallp I BIIIUUI YawngCaa CLt.rn % n P“ n: 66mJi!-L o-,sm-, ma-,666-,mo-,4670-a 409-.m-, m40-, ml-,stml-, mm-, m71.7m?$%.Iw.O
14、m:K,bB/,47Q:%,4m,467.mn,aso0080-.rmuloIo0000-mm-,m4m: mmm,mm,0016Mo,00m4,-,mm,M4tm,Mwl-0m14z-,mo44-,m14mLBMLad:%g-:%-,641-,Qm,4m.m7,611=4-,mno-,mm-,mo-,mm-,m-,mml-,Oiiulo-,mmo-,KmO-,mo41-,mon4L-.om6-. mm-. rliwll:, p%-,nm-,mm-.9979-,Jm19-,m-amu: 4-,mMOfBJ!f#l,lW-ag-. Om-, wl-,4W-.706-,mo-.4m-,6s8,07
15、4-:E-,4n-,mu-,716n:-t E-1, mo7-. or9m-LWt8-,M!lmJo,m,-.M7nl.ilm.ma.Wn:E-%B476-1.mm-l,eam:Y!J-,1641-.!W6-,mm-,W-,w-,ma-,m-,6476-am,4m-: !$.:w-,146-,MI-:g-:!%-,m-.moL1.maL%O-i=,0129-L 144-;mo0k#4am-o,MIO:w:F4-:E,m7.mm.Olm-:%!-,m,m-Lws-a416-LWI-o.m49.!WIaS16if%?LOW.Os:1:4-LM4-laMa-a4ma644.600.MO-.m-.61
16、6-:!%m-:*-,m-,mm-.040a71ma.4m,4s7-aEm4-. w-, m,040-: E,am-:=#w-: %-,781-,711-m.-= %-144m-ma41770,7mmmd%?m!791,Mllt%-4,lm-Ma646-u&n4O.ma:3-:%-.la-. w-. Om-,187-,m-,M6-,4m-,m4-,ml,mo00000.lmo0000000I.,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-512
17、 REPORT NO. 00S-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICStable HI, were marked off on the ordinate erode. Thetime intervals within which the various Inodea dimhiahor increase by one-half, or by one-tenth in the case ofh (see table II), were then spaced offon the daciasaand points on the curves wer
18、e found by &ninidor increasing the ordinak succ.emively m indicatedby the sign of the root. The oeciUatory mode wasobtained by drawing in the envelope (given by +p,say) and spacing off the quarter periods, beghming atthe point indicated by the phaae angle of this mode.The cosine curve was then simpl
19、y sketched in as ahowm.The final curves were found to give remarkably goodchecks when applied in the original differential equa-tions of motion. Such a check shows the corractneesof both the method of plotting and the analyticaleolutione (equation (1) and table III).If the impreeeed disturbance is g
20、iven as a functionof t by a curve, it will usually be su&ient to approxi-mate this curve by the addition of a number of succes-sive positive and negative steps. The combination ofsteps necemary to reproduce the disturbance leadsdirectly to the addition of the elementary motions forthe resuhant motio
21、n. Otherwise, for example, if thevariation of disturbance is given by L(t), then the re-sultantmotion p(t) at any time t due to L(t) beghmingat t=O may be found by Duhamels theorem, thus(t) s, =tiblt ,oJIim-,zt=flz.() L(O)+- 0pz(t&) L (tJ d, (2)where p(t) and PL(ttl) are obtained from table IILAn ex
22、planation and a graphical method for evaluationof such integrals are given in reference 3.MOTIONS IN SIDE GUSTSThe motion caused by a unit inorement of gustvelocity is found by compounding elementary distur-bances in such a way as to simulate the disturbing actionof the gust. Thus, in a aide gust of
23、 velocity Q, the dis-turbing a.ccekdi.on along Y will be . end angulardisturbances will be u&G,d,.As explained before, the eflecta of any ueual variationof gustiness can be largely foretold from the effect of aunit sharp-edge gust. The variable gust can be budtup from small increment jumps of gust v
24、elocity co-Sponding to Shmp-edge Crom+urrentc and the finalmotion will approaoh that obtained by superposing themotions due to the indivichud elemeuts.The effect of a sharp gust from the aide is similar to,although not exactly the same as, the effect of an initirdangle of aidealip. For the aide gust
25、, it is necessary totake account of the period of penetmtion of the airplaneinto the current. The tit effect wiU be to push thenose of the airplane downwind whereas an instant laterthe current will strike the h, turning the machine intothe gust. The action of dihedral in causing the ma-ohine b roll
26、away from the gust ti also ocwur beforethe fin is affeoted. The effects are, however, of shortduration and do not alter the motion h any great extcutafter the &et fraction of a second, mcept in cases ofsmall weathercock stability where the fuadage con-tibutee a large unstable yawing moment.The compu
27、tations -that follow aro based on the m-resumptionthat the rolling action of the gust begins att=O and that the yawing action begins when the air-phme has traveled far enough to cmry the fin into thecurrent. The case of CW=Owas treated by assuminga yawing couple equal and opposite to that of the 4-p
28、er-cent h applied when t= 0, this couple being neutral-ized at the instant the fin entered the guet.A possible further refinement of the calcultitionswould involve the delay in building up the full lift forceson the varioue surfaces. Mathematical methods fordeahg with various lage or rates of growth
29、 of theaero-dynamic reactions have been developed, but theirdescription is beyond the intended scope of the presentreport. It may be said, however, that, for motions asSIOWas the natural oscillations of a rigid airplnno, thiseffect (judging by the theoretical predictions) is quitenegligible.Figure 3
30、 illustrates the resulti of the calculntiunsbaaed on a 10-footiper-eecond sharp-edge aide gust.The curves shown are for flight at C=l.0 but thesame genertd trends appeared in the calculations madefor other lift coeflkiente.The meet noteworthy dMerence shown is tho cflcctof deficiency of iln area on
31、the banking motion (figs. 3(a) and 3 (b). The airplane with 10 dihedrul andaverage fin was not displaced eQmuch in bank by theaide gust as was the airplane with 6 dihedral nnd asmall fin. The initial rate of rolling, howwmr, wasgrea with the greater dihedral.With a given dihedral an increase of fin
32、men cutsdown the banking motion although, uftcr n certainsize is reached, the gain becomes slight, m is illustratedby ilgure 3 (a). With neutral weathcrcock stability(. 3 (c) the change of headiug on entering the gustis at &st small and, although the motion is staMc, tlwoscillation in azimuth seems
33、to be reinforced for atime by the rolling. This action is to be attributed tothe phase lag between the rolling and the yawing cffectaupon penetrating the ehrwp-edge current. It seemsprobable that an appearance of inherent instabilitymay be reached at a point coneidembly abovo thomathematical limit f
34、or undamped oecihtiona. (Seci 1.) It is known, for instance, thnt unstable oacil-Iationa may remdt from an attempt to hold the wingslevel tith ordinary ailerons unless Cq h a dcfinitapositive value.The side gust is equivalent ti a sudden shift in thewind direction, corresponding to a change in azimu
35、thh M indtiti k figure 3 (c). The normnl airplaneswings about and tends to appronch this heading. Itwill be noted that the nirplane with tho large iin turnsfairly sharply into the wind and, einco t.ho bankingProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IH
36、S-,-,-INFLUENCE OF LATERAL STABILITY ON DISTURBED MOTIONS OF AN AIRPWE 5130 i 2 3 4 5 6Time,9ec(qBSUMOE.(b) ROu motfon.FIGUU 3.-Motbosmuc=l.ari-locd.P.JL.-.-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-514 REPORT NO. 6S8-NATIONAL ADVISORY COMMITTE
37、E FOR AERONAUTI03motion issmall,tends to keep the same t pathrelativeto the earthfora shorttime. Afterabout 0eecmnds,however, the spiraldivqence beginste beapparent and the motion finallyreaultain turningdownwind.An example of ex&me spiral divergence is illustratedby the airplane with no dihedrtd. I
38、n this case, how-ever, the airplane banks and turns directly upwind.The airplane with large dihml illustrates the oppo-site condition and shows the predominance of mcilla-tions that generally characterizes the effect of dihedral.Here the airplane tande back toward its original azi-muth heading, drif
39、ting eidewise with the gust.The airplane with 5 dihedral banked rather sharplyaway from the gust, whereas the airplane with zerodihedral showed an undesirable tendency to bank andslide into the gust. It was therefore a matter ofinterest to try some modiflcatione lying in betweenthese two conditions.
40、 It was realized that the rollingcould not be entirely suppressed by such modificationson account of the phase relationships involved in themotions.It appeared that 1 or 2 of effective dihedral wouldgive about the least banking motion in the side gustand hence this condition wainvestigated. Inasmuch
41、as the airplane might have shown a noticeable spiraldivergwme at low speeds with the normal h area, thisarea was arbitmuily reduced, bringing the weathercock-stability factor Cw down in about the same proportionas the dihedral factor CW The va.luea selected wereC,P=0.025 and C%= 0.086, which corresp
42、onds to2 tieotive dihedral. The position of this airplane onthe lateml-etability chart is Aenoted by the point Ain i3gure 1 (b).The rceulte for airplane A are compared with theothers in figures 8 (a) and 3 (b). It will be noted thatthe bank is somewhat smaller than in the case with 6dihedral and a l
43、arge fi but that the bank persists for alonger time, The difference made by the change from6 dihedral to 2 seems surprisingly small. A eome-what greater difference would be expected if the &had not been reduced. It should be borne in mindthat the yawing disturbance is reduced by cutting downthe fm.T
44、he curve for airpIane B (. 3 (a) shows the resultof attempting to secure spirnl stability (at C=l .0)by cutting down the fm of the airplane with 6 dihedral(Note that airplane A is slightly unstable.) The valueof Cdin this case is about half that assure for themean condition. (* fig. 1”(b).) The bank
45、ing dis-placement seems undesirably large (comparatively) inthis Oase. .OTHER TYPES OF GUSTThe flight velocity of the airplane being normallylarge with reepeot to gust doiee, it ia permimible toconsider the gusts as being stationary in time withrespect to tha fight path. Thus the gusts are con-sider
46、ed to exist se a fixed pattern in the air ahead of thoairplane and not to vary in time within the short spncorequired for the machine to travel ita own length.As mentioned before, when the airplane ontcm across-current in level flight, u gradient of sidcwieevalocity tilong the length of the fuedago
47、will mist.The tiect of this gradient is similar to tho effect of arelative yawing motion supcrpesed on tlw eidc velocity.For a uniform gradient tho additional yawing momentwould be (dz) Xif?r. The calculations involvedthis factor by virtue of the time lag asaumod in applica-tion of the yawing moment
48、 due to the fin, and uponthis basis they should be applicable to any rcasonahlcvariation or gradient of sidewiee veIociy.A somewhat d&rent situation arks when tho air-plane iE climbing or descending though a cross windthat varies tith height, as, for instance, when descend-ing through the earth boun
49、dary layer for n cross-windlanding, for then no perceptible gradient of sidowieevelocity along the length of the airplane will exist.The motions that arise in time cases can ho cmnpoundcdby integration from the motion following nn initifilangle of aideelip. This motion is not greatly differentfrom that caused by entering a sharp moss-current andthe same geneml ooncheions will apply.It appears that a
