1、REPORT 1098SUMMARY OF METHODS FOR CALCULATING DYNAMICLATERAL STABILITY AND RESPONSE ANDFOR ESTIMATING LATERALSTABILITY DERIVATIVESBy JOHN P. CAMPBELL and MARION O. McKINNEYLangley Aeronautical LaboratoryLangley Field, Va.Provided by IHSNot for ResaleNo reproduction or networking permitted without li
2、cense from IHS-,-,-National Advisory Committee for AeronauticsHeadquarters, 1724 F Street NW, Washington 25, D. C.Created by act of Congress approved March 3, 1915, for tlle supervision and direction of the scientific studyof the problems of flight (U. S. Code, title 50, sec. 151). Its membership wa
3、s increased from 12 Lo 15 by actapproved March 2, 1929, and to 17 by act approved Nay 25, 1948. The members are appointed by the President,and serve as such without compensation.JEROME C. HUNSAKER, SC. D., Massachusetts Institute of Technology, ChairmanALEXANDER _VETMORE, SC. D., Secretary, Smithson
4、ian Institution, V/ce ChairmanALLEN V. ASTIN, PH.D., Director, National Bureau of Standards.DETLEV W. BItONK, Ptl. D., President, Johns Hopkins Univer-sity.THOMAS S. COMBS, Rear Admiral, United States Navy. Chief ofBureau of Aeronautics.LAURENCE C. CRAIGIE, Lieutenant General. United States AirForce
5、, Deputy Chief of Staff (Development).HeN. THOMAS W. S. DAvis, Assistant Secretary of Coinmcrcc.JAMES H. DOOLITTLE, So. I)., Vice President, Shell Oil Co.MATTHIAS B. GARDNER, Vice Admiral, United States Navy,Deputy Chief of Naval Operations (Air).R. M. HAZEN, B. S., Director of Engineering, Allison
6、Division,General Motors Corp.W1LLIAM I_ITTLEWOOD, ._,_. E., Vice President, Engineering,American Airlines, Inc.HeN. DONAI.D W. NVROP, Chairman, Civil Aeronautics Board.DONALD L. PUTT, Major General, United States Air Force,Vice Commander, Air Research and Development Command.ARTHUR E. RAYMOND, SC. D
7、., Vice President, Engineering,Douglas Aircraft Co., Inc.FRANCIS W. I_EICHELDERFER, Sc. D., Chief, Ui_ited StatesWeather Bureau.iON. WALTER G. WHITMAN, Chairman, Research and Develop-ment Board, Department of Defense.TiIEODORE P. WRIGHT, SC. D., Vice President for Researuh,Cornell University.HUGH L.
8、 DRYDEN, Pm D, DirectorJOHN W. CROWLEY, JR., B. S., Associate Director for ResearchJottN F. VICTORY, LL.D., Executive SecretaryE. It. CHAMBERLIN, Executive O_rerHENRY J. E. REID, D. Eng., Director, Langley Aeronautical Laboratory, Langley Field, Va.SMITh J. DEFRANCE, LL.D., Director, Ames Aeronautic
9、al Laboratory, Moffett Field, Calif.EDWARD R. SRARP, Sc. D, Director, Lewis Flight Propulsion Laboratory, Cleveland Airport, Cleveland, OhioLANGLEY AERONAUTICAL LABORADORY_Langley Field, Va.AMES AERONAUTICAL LABORATORY,Moffett Field, Calif.LEWIS FLIGHT PROPULSION LABORATORY,Cleveland Airport, Clevel
10、and, OhioConduct, under unified control, for all agencies, of scientific research on tl_e.fundamental problems of flightIIProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT 1098SUMMARY OF METHODS FOR CALCULATING DYNAMIC LATERAL STABILITY AND RESP
11、ONSEAND FOR ESTIMATING LATERAL STABILITY DERIVATIVES 1By JOHN P. CAMPBELl, ltll(t _IARION O_,:|cKINNEYSUMMARYA summary o meth,d._.for makil_g dy_mmlc lateral stabil;tyarm re_.ponse calculatiom_ aml fl_r e,s.timating th.e aerodyl_amicstability deriratit,es required.or axe i1_ these calculatiot_s ix p
12、re-sented. The proee,_.ses o.f pelformi_g ealculatim_s qf the timehistories _( lateral motion,_, qf the period aml dampit_g fff thesemotio_s, aml of the lateral utabillty boumlaries are pre._elded axa series of simple ,_traiyhtorward ,_teps. l(.ristin(.I method.s.forestimati_g the xtabil;ty derhat;r
13、,* ace xummariced amt, il_some cases, simple 1_e : empirical.formuhcs“ are prese_ded, llr:f-eret_ee is also made to reports presel_tlng e.rperimeutal datathat should be use/ul ;r_ maki_tg e,_timates of the deriratim_.Detailed estimatint_ meth,ds are pre,_ented .or Iou-._ub._,_ic-,_peed conditiol_s b
14、at ,Id!/ a brief discu._,_io_ aml a li,_t o. refer-eT_ce,s are giren .or tralt,sol_;c- aml ._uper._onic-._peed condithm,s.INTRODUCTION)ynamic lateral stabilily has not reeeived widespreadattention in the past I)e(ause it has not, generally been aserious prol)lem in the design of airplanes. Considera
15、tionof dynamic lateral stability has recently I)e(ome more im-portant, however, l)ccm_sc current design lren(Is toward theuse of low aspect ratio, sweepback, and higher wing load-ing have, in many cases, hd h) unsatisfactory (lynamic lat-eral stability. Airplane designers are therefore finding itnec
16、essary to make such calculations in connection with thedesign and modification of airplanes. In many cases thesecalculations are difficult, to perform for designers who havehad no previous experimwe in theoretical stability workbecause most: of the pul)lished theoretical analyses are notpresented in
17、 a forIn thai is especially suited to the conq)u-tation of dynamic stal)ility. The estimation of the stabUityderivatives required in (lynamic stability calculations hasalso been found to be difficult in many cases. Althoughtheoretical and experinlcntal data on these derivatives haveappeared in numer
18、ous put)lieations, no single publication haspresented methods for estimating the derivatives for alltypes of airplanes.One approach to a l)rcsentalion of metho(Is of cah_ulalingstability and estimating stability derivatives in a form suit-able for use by designers was made by Zimmcrman inreference 1
19、. Although this report has proved to be of valu-able assistance to designers in making dynamic stabilitycalculations, recent trends in airplane design have caused itsSttl_rsedes NACA TN 24(19, “Summary of Meth.ds for Calctlhding l)ynamic LaterMan4.Marion O. McKinney, 1951.usefulness to be seriously
20、limited. For examl)h, the (,qua-I,ions of reflwencc 1 do not inch_de the I)roduct-: : Kz 2YCh,.ight of me cases, how(ver, (foreXaml)h , f()v qui(k approximation of the efl(,ots of the (.har,wlevisli(, equationby the equal ionsandAx4+lO,:_-_-(X t DXq E-02l-i rT_/.,= log_ 2 0.693- II r _ - II rwhere l
21、_ repr(,s(,nts a real veer X or the real part of ,i eomt)lexroot X=lIi and I represents the imaginary part of acomplex root. Negalive values of T_:, represen! lit(, (linerequired to (loullle amplitude for unstable modes of themotion.The values of the (.oefli(,ients A, B, (, D, and E may beobtained b
22、y the method given in steps 1,2, and :_ of lhe sec-tion on “Calculation of Motions.“ If 1he period and timeto damp are to be eah.ulated for a mmdl(,v of related eases,however, the values of the eoefli(ienls A, H, (7, D, and Emaybe more (,onw,niently (.aleuhtt(,d by a tabular llV(eedure suchas that s
23、hown as table for maMng boundary (,al(,ulations.l_iethods of determining the roots of the liqtmdratiecharacter)st)rs(_c (“c (_that are appropriate to lhe I)arli(ular )rol_lem.Step 2: From the known factors, evaluate tit(, following l)aramelers which are lit( slqbility (leriva.lives in llw form inwhi
24、ch they are used in the cah.ulati()n (f m()/i_ms:le-2ttx2 _n 17-: l)amt checking to dvtvrmim whether the signs of B and D nrethe same.(a! 1Zc(_/ !(c)I!(a) Gust, disturbance.(b) Control manipulatirm./-/!(c) (lust disurbg.nee alld corrective control In:udpulation.F1GuII,: 3. -Illustration of stl_rlmsi
25、l.ion of ntoti0ns t_ Ihterlnin, effect of arbitrarydisturbances.%imc I,. ,d Ihe mosl iml)(rlanl stability _h,rivaliv(,s af-feetn/lal.ral stability are Ill( dirc(./iona slablily derivative(7,a and Ih|, i,lll,l ix(, dihedral deriwtlive (10, t.)u|.huies form,ulr.ll ()y;cilhtloty slablily are usually ca
26、hl|lated as a fun-lion of lh,s., Iwo (erivtlivt,s as ilhtslral|,d in figure 4. Th(,sccahuhtti_ms arc /her|ally (:lrried out It 5 the lncthod slmwnin tah(, I. ll_is labh, colilains a |lu|ncrical eXnml)le nt.Isl(.p-)y-slep inslru(.tirms for using th(, tqbh. The r(,sulis (dhis num(,ric,l ,xulnph, are l
27、)h_lle(t in figure ,1. 11( pro-ce(lt|re illttstrntc, I in laid(, l is litst to assl|nc values o iheilidepen_h,nl vataid ir stul)ilitv |,riwllivt.s except (+z nrt, th|,n rsli-marcel. “lhc v1|luc of t“,_ is _Zemrally assltnwd to hnve I,eenvaried by varying the size o the v|rticd lail and, con-seqltent
28、ly, the lni (.onlrib|lion lo eqch of tle other stabilityderivatives vari.s as (,_ is varied. The values ()f the r.ef-tich.nls .|. /_. (, /), and p. a11d then I, tit(, calculated asfuml ions of/_:_- - 2Nx ( t“(,_ nle ()btaim,d from Ill(, vah|(,s of/_.The values .f /+which satisfy the expressio1_ 1“=
29、0 must hecle_l(I to (h,l(rmin|, whether they salisfy Ill(, other con-diti(m for t.,ulral os(ilhl is a con-_(an( for nny giv(,n value of (,_ and i,_ ahnosl invariablyl)osilive sine(, (h(, (hie(, Iri,(h)minani lPrlllS o J (.onlnin lhe(h,liVntiv(.s (), (, and (% xvhi(, mad(,. For g(,n(,ral ,-+tudi(,s o
30、f stability,h()w(,v(,r, approxintnl(, (),willllory stability l)oundnri(,s tonyI)(, (ahulnt(d nlu(.h more siml)ly by the In(thods shown inr(f(r()i( 24.ks l)()int(l ou( )r(viousy, )nlho(ls of (.nhulaling lines of(.onst.ln( l)(,rio(l and (hm:ping of (lie lllt(,rnl ()s(illalion nrel)r(,+-+(,nl(,(i in r(
31、,f(,ren(,_ 0_ I() 1 l_) lit( sut) (lesirtl.|)e to (,s(nll/te lh(,_(, l)(q“ (_fl(,cts for sore(, of the (h,riva.iiv(,s art, dis(uss(,(l in (hefollowing s(,(,lions wliiei, (!over tit.( eslilna.tion I)lO(!(_(llllPS.In the (asi,s wh(,r(, (h(, R(,ynmhls vutnil)(,r (qr(,(.(s are notdiseusse(|, it can 1)(,
32、 assuni(,(l that any al)r(ll)l varia.tion in):lle derivatives near the stall for low-s(.a.h, data will alsobe I)r(,senl. for lit(, full-s(.ah, airl)ane l)tll will lJrobaby o(urat, a higher lift (oefti(ient be(aus(, of the higher maximumlift, coet_.(.ien( of the airI)hm(,. An indi(a(i()l of the lift-
33、(oeffi(qent range ov(,r which, th(, theory may not be (Xl)(,(tedt.o give reliable vahl(,s of stability (leiivuliv(,s f(). th(, full-s(ah airl)lanw (art lle ol)tain(,(l from hlrge-s(.ah, (hag tiara.The analysis of referen.(, 89 in(licates thal tile variationof the derivativ(,s with lift coeffi(i(,nt
34、is dilr(,r(mt, from tllei.heor(,ti(al vari#tioi), at. lift coelIi(i(,nts a.bov(, that a.(, whichthe (hag (hie to lift in(.r(,i_s(,s abruptly fronl (h(, i(h,al vahi,(:2( L/TrA.TIt( etre(.ts of Math inlmt)er and t)o_v(q a)(, no( lreal(,din th(, s(,(.t:ions on the individual d(,iiva(iv(,s but are dis-(
35、:usse(I briefly ill separate setlions. A d(,laih-(l treat,iiientof th(,se (,tt(,cl.s, inchl(ling design formulas aiid charts, was(:onsi(h, red beyond the SCOl)e of this r(_port.THE SII)ESLIP DERIVATIVES Cr, C%, Ct_No sa(isfa(lorv l)ur(ly (h(,or(,ii(.al m(,lho(s have vet, l)(,(,n(hv(d(ll)(,d for ()(a
36、.ining a(“(Irat(, (,slinilit(,s of tile si(l(,sli I)(h,rivativ(,s ()., (,_, i_li(l (:_ for a ()nil)ht(, li,irl)lan(, l)ri-mltliiv I)(,(.lius(, of laig(, inl(,rfer(,n(.(, (,fr(.cts )(l,W(qi I,Wvaii()(is airl)iime ()nipon,(lils and )(_(.aus(; of l_tge, and ()fit,it,un,t)l(li(llLh(, vli-iiuli(ins ()f l
37、h(, (,rivii-tiv(,s wilh angle ofatta(.k. F(illlllltl_t(lv, these (h,rivaliv(,s (ltil It(, ol)tain(ulfrom eOliv(,nli(iniil win(I-luluie for(,-l(,st (lit-ill-. ,qu(h (,Xl)(,ii-ni,(qi,lal diiiil lu(, (,ss(qiliitl to lho tl(!nla.l(, (,l(_rni,in, alion ofsi(h_sli I) d(,rivILiiv(,s. It is, of (nllS(, high
38、ly (,siral)h, ioJill, V( foi(,-l(,s( (ill,lit for the (xit(l airl)lan(, (h,sign, lln(l(,r()nsi(t(,iliI ion, I)ul l(qlsonat) 3- ti-(;(nrli.l t (,sl iniah,s (tin nsnti-lvl)O lllli-(t( h v (OiT(ting the for(,-test (hi-ta for a g(!n,(qitllvsirnilai“ (it,sign. The nletho(ls of (ori(,ct, ing t h,(, for(,(
39、,-l(,sl,data on It siniii,u. (sign for list in lh(, (liSt, till(it,l“ (onsiil(qltl ionit-r(! (.ov(,riqt ill lh( following s(,(.l,ions. Ill lh( follnllh)_s)i,(S(qi,t,(qt, ih( sut)scril)i wor( “(h,sign“ is use(l to (l(,signil_i(_the (lesign lln(l(,l“ (onsi(leration ali(I the si,il)s(Til)l wor(l“(haiti
40、,“ is llSl,(I It)(hsignate the sini,ilar (h,sign for whi(hfor(.(,-i (,st (ill lit- ill(, availal)le.For(e=t(,sl data should )(, us(,(l to (l(,t, erniine the (!llect onthe si(h,sli I) (hqiva(iv(,s of suHi, airl)an(, (,Oml)on,(n, is ash,a(ling-(,da, high,-lift (h,vi(.t,s, stall-control (h,vi(.(,s, tra
41、iling-e(lge flit_l)S, lilt(l(S, ext(rna slores, cal).ol)i(,s , and dorsal li,ll(tv(mira.l fins. Tim (,tr(,(.l, ()f h,a.dhi,g-(,dg(, h,igh-lifi (lcvi(.es isllsually lil(q+,v to (xt(n(l (o)_ ligi,(,i lift (o(,fli(i(,n( (he SSlllOvariation of ill(, (h,rivli,tiv(, with lift (oeffhi(,nl, as for lh(, I)ah
42、i,wing. Trliilil)g-e(tg(, ttaI)S ofh!n |itlv(, arg(, (,lr(,(,ts on, tilecon,t iit)lllion, s of t)olh the wing an,(l the v(qtical (all to th(;si(hslill (hrivi_tiv(,s (iefer(m(!es 40 lilld 71); li,ll( sin(e li(,seeffe(ts li.l( iiO1 easily (,siini,ai(,(I, it ll,i)l)(,llls tiial in lh(,s(, ()ls(,sns(, o
43、f for(,-t(,sl (hi,lll is ess(mlia. The lld(lilh)n of n,a(elh,santi (,xl(,rnli si()i(,s g(qi(rltIl.v has h(,(,i), foun, d it( (l(,(T(,_ts(. the(lir(,(tional sial)ilit.v fa(.(or (, slighlly. Th(, results of _tlimit(,d li,Iil()lllll of i(,s(,Itl(.l), to (l(,l(,rni,in(, the (,l(qi. Oli llmsi(h,slip d(ri
44、vativ(,s (if the size and shall(, of (.anol)i(,s hltst)(en r(,t)(llt(,(l in i(,f(,l(qie(,s 4(re-seni(d hi r/flwiqlc 79. Exl)trhnental data from otherhlv(,siigaiioli,_ iil, Ve showll tha.t /liff(wcn(es in fliselli.Ti crosssection Ciili utilise very large diltelen(_es in the varililJon of( with lillgO
45、, of l/lt.ll(k. For cxiinl|)l, ill tho case of athll, fiis(qtlTe with Ihc liliijor (woss-s(,cliona| axis horizont.al,till sign of (. hits t)een found to rev(,l_l! at lllO(Icrl/t.c anti)_lfu ,lilzii li.liTi(s of till.liti.;. Force-lest (Iqla are essen(hfl forhill.king oslhnllAes in such (,.ases.+,(3)
46、ViliT-fusehlg( iiltcrf(,r(,n(;(,: I or low-Will7 or high-wingconllgurlilions winlz-fuselagc inti_rferenoc oausos t.llc vahieof (_y_ to tic grlqi.lcr lha.n lt).lit, ot)tahn,(l tiv adding t.hc con-lril)ulions of the wing and fuselage. (_ee refiwences 37 and40.) If the vertical lo(ation of the whig Oil
47、 the lust,lave isglql(,illily sinliar for flu; (wo designs, however, any correct.ionfor it difference ill lifts interference fa.(lor (all hc ncgle(ted.Vertical tail. -,t(cuill/e cstiiiillteS of Cy_t._ l llle IleCoSS,llrybecause this fa(lor is used to estinlaie the tail (ontrit)uiionto severl/l other
48、 derivatives. This fa(.tor is esliecially im-porll_iit il.| low angles of allli.Cc Iwcausi, in lliis (fast, iic tallcoiliril)ulion is oflen Inlieh 7r(,ater than lhe wing-fusel/.n lhe (IS(, () il v(,rti(:al (all loeate(l ()ii (i(_ whig, tl(,l(_ is)in a(|diti()n t() lhe increinentlt| (_tn produced by Lit(; l at|hth,ral force, ilii in(r(,nienlal (_ i)ro(tu(_e(| t)y the in(erfer(,ncoefteta( ()f ih(, v(,li
