1、REPORT No. 521AN ANALYSIS OF LONGITUDINAL STABILITYWITH CHARTS FOR USE INBy CHARmS H. ZIMMERMANIN POWER-OFF FLIGHTDESIGNSUMMARYThh report preaenh a diwwti of longiiuo?ina.18tabdity in gliding j?ight togetherwith a 8eTie# of churt8wiih which the stalnliiy churactenktti of any airphwmay be readily e+i
2、timated.The$rst portion of thti report is ird (1) Aerodynamic forces and momentscreated by movement of the lifting and control sur-faces relative to the surroumiing air; (2) mass forcesand moments arisingfrom the weight and acceleration,angular as well as linear, of the airplane. The funda-mental ba
3、sis of the discussion presented in this reportis that at all times there exists a stab of equilibriumbetween the mass forces and moments and the aero-dynamic forces and moments.A complete treatment of the stability of airplaneswould be extremely lengthy and very complex. Cer-tain assumptions have th
4、erefore been made. As themotion of an airplane is three dimensional, it is to beexpected that any treatment of the subject will beincomplete if it neglects certain of the components ofthe motion. Fortunately, conventional airplanes aresymmetrical (within limits here applicable) withrespect to the pl
5、ane that includes the fuselage axis andis perpendicular to the span axis. It is obvious that alongitudinal motion having no component of linearveloci perpendicular to that plane or no componentof angular velocity about any axis lyiqg in that planecannot intioduce asymmetric forces or momants.Such mo
6、tion can therefore be treated as an independent .phenomenon.The longitudinal-stability characteristics will neces-sarily be affected by any deflection of the lifting orcontrol surfaces. The influence of wPVSCD=O Iaw Cos +wsc.=ob (1)$vscc.=o. IcOMMH133E FOR A3BONAUTICSwhere (la) refe to forces tangen
7、t to the instantane-ous flight path, (lb) to forces perpendicular to theinstantaneous flight path in the plane of symmetry,and (lc) to moments about an axis through the centerof gravity and perpendicular to the plane of aynmmtry.After displacement from the steady-flight conditionthe equations of equ
8、ilibrium read.1WCOS(y+AY)P(V+ AV)S(G+AOJ= mV$ b (2)d%P(V+AvSc(C+AC.) =mky d cwheredVY acceleration tarigent to flight path., centrifugal acceleration normal to the flightdt path.d% angular acceleration of airplane about thelateral axis.Since the effects of angular velocity and of accelerationupon th
9、e forces are neglected, AoD may be written asd(?.Aad d ACL A=,. AU= may be written as+ Aqd where Aq=q, the angular velocity inpitch, since g is zero in the original condition; andAV, A-f, Aa, and q are small quantities by aesmmption.Terms involving products of two or more small quan-tities will be n
10、eglected.sin (7+ A7)=sin Y COSAY+ccs y SillAy=siIl +A COSandcos (Y+A-y) =COS COSAysin y SillA7=COSYAy sm YThen:Subtracting (1) from (3)dVWA7cos y+ PVSCdV+pV2EAa= m .nb (3)c,8b (4)c,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ANALYSIS OFFrom equat
11、ions (1)w COS”T=;PVWCLLOMMTI%OIM4L STABILITY IN POWER-OFF FLGHT 291w sin y= (C.%CLdaq+c2)1-pm=$=(l (16)For simplicity the biquadratic maybe expressed as:A(x)+B(x)3+ C(A)2+A+E=0 (17)whereA=l=-mQ+*cD+31Unfortunately there is no simple, direct method ofsolving biquadratics. It is possible, however, to
12、factora biquadratic into two quadratics, each of which issusceptible to direct solution.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ANALYSIS OF LONGITUDINAL STABIUTY IN POWER-OFF FLIGHT 293The biquadratic(X)+.B(X)+C(A) + DX+E=Omay be replaced by
13、the expression) +alX+hl(X)+ aJ+b2=0from whichor 1n(18)bIii appearathatB=a,+fiC=ti+b,+bzD=alb,+qb,E=b,b,The general case may be worked out as follows:u,=B-C=a,(BaJ +bl+b,D=a,b,+ (Ba,)b,DTherefore,C=aJBaJ+b,+;ll=a,+(ll-a,)bl In(19)bDropping the subscripts:B=3+ J() c+b+ aand 1(20)bBbDa=bE bThese relati
14、ons can be solved by plotting the curvesof b against a. There are two intersections of thesecurves, in general, corresponding respectively to al,bl, and to a, b.Also from (19)= Cba(Ba)D=aCab-a(B-a) +b(B-a)Ib (21)b=Da0+a2Ba?B2a cSubstituting (21c) in (19a)D aC+aBasC=a(Ba) +( B_2a)(22)oraeas(3B)+a4(3B
15、+2%a3(B+4BQ+az(2B +BD+4?a(BD+BC-4EB) +BCD-D-B =0 (23)By use of the foregoing relationships the coefficients)f the quadratics may be determined with aa high aiegree of accuracy as desired by graphical means ory trial substitutions.At fit glance it appears that a 6th-power equationwch aa (23) would be
16、 harder to solve than a 4th-power?quation such as (17). In equation (17), however, the:omplex roots must be obtained; whereas in (23) it islecessary to solve only for the real values. Equation(23) is useful from a practical standpoint chiefly inbtaining accurate values of G by making trial sub-Jtitu
17、tiom from approximate values obtained from thexcpression for G given on page 6. Because of thevery small value of az it is generally not necessary tohclude the terms in (23) that contain powers of Ghigher than the third.Significance of A.As appem in equations (18),there are possible either 4 real va
18、lues of h, 2 realand 1 pair of complex values, or 2 pairs of complexvalues. The values of B, C, and D in the normalflying range of conventional airplanes are alwayspositive because of the signs and maggtudes of theirconstituent factors. It is obvious that no positive realvalue of X can satisfy the b
19、iquadratic unless E isnegative but that if E is negative there is such asolution for A. A positive real value of 1 signifiesan aperiodic divergence. If pm= i9 positive corre-sponding to static stability, then E is positive and thebiquadratic expression indicates no possibility of anaperiodic diverge
20、nce.The values of B, C, D, and E are, in general, suchthat the solution for X gh%s two pairs of complexvalues. It can be shown by mathematical reasoningnot essential to this treatment that an expression ofKexhere A=F and the period in seconds isp= *(25)under standard conditions.Substituting for a in
21、 equation (23):(2)+ (2r)(3B)+ (2t)4(31P+2C)+ (2)3(+ (2)(wC+BD+4E)+ (2t) (D+B4D)+BPFE=o(26)From equation (21)(w= D+2C+3 +(qCE= Cb,orb,=$andD BE throughout the remainder of thereport and # will be used without the subscriptsto refer to the slightly damped phugcid oscillationpreviously defied by j and
22、,.Derivation of expression describing the sinusoidalThe expression Ay=Aex=Adt can bemotion.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ANALYSIS OF LONGIMJ sTTy IN poE+FF GHT 295replaced by the equivalent expression A7=Ae$tcos (Yt ) where A and 3
23、depend upon the instantfrom whence time is taken as zero. If time is zerowhen AY is at a point of maximum amplitude A= AyOand 6= O so thatAy=A” COS#t (32)It follows thatAa=Aa” COS(+ t,charts may be plotted showing the variations of mewith pm. necessary to secure given vslues of fand 4. A number of t
24、hese charts have been preparedcovering the range of conditions likely ta be encoun-tered in normal flight aud are included in this report.(See figs. 16 to 54.)DISCUSSIONThe mathematical relationships evolved in the pre-ceding paragraphs permit calculation of the probablestability characteristics of
25、a proposed design, but theyoffer little information as to the relative importanceof various factors or as to the reasons for the effectsproduced by chmes in those factors. In the foowingparagraphs the oscillatory motion is fit consideredin detail so that the sequence of flight conditions thatmust ex
26、ist if instability is to tie may be pointed out.Next is given a general discussion of charts (figs. 15to 54) that show the effects on the stability character-dListics of the six fundamental parameters CL,CD,zCD, pm=, and m,. Finally, the effects of various-Z-physical characteristics of the airplane
27、on its stabilityare considered in the light of the earlier discussion.The oscillatory rnotion.-ll%on the terms in AVwere eliminated by simultaneous solution of the equa-tions of equilibrium tangent and normal to the flightpath, respectively, the following equality was found toexist: .The terms to th
28、e left of the equality sign arise out ofthe necessity for equilibrium of forces; the terms to thoright arise out of the necessity for equilibrium ofmoments. If a case be investigated with Pm. = Oor)2+%cD+a+w”%$-c=%$+c2)=0“=-KC”+%“=-icD+%)dSince both D and - and athe system when ) OL CL=CDpd+ CD,=The
29、refore equation (42) may be rewritten as 1CLq(C.iCD,)+sin61+COS61 =0(43)3 CD+ da A70It appeam that instabili of the linear motion forsmall values of Pm= can occur only at nnglea ofattack where CDiCDP. The detrimental effect ofCDf is not as great as at first appears, however, be-cause the damping fac
30、tor 3 CDincreases with CD, also.k the foregoing analysis, the effect of ( has beenneglected and therefore the stdements made cannotbe considered rigorously true when the stability or in-stability is of appreciable magnitude. The analysisdoes, however, point out the more important influencesand the n
31、ature of the interaction of events that bringsthem into play. Equation (43) cannot be applied nenrdo2CLthe stall where ceases to equal .TIh figures 2 to 14, inclusive, and in the discussion ofthe oscillatory motion an arbitrary value of A-ro= 1 hasbeen assumed. No attempt will be made to evnlunteYoi
32、n terms of control movements or gust velocitiesu the actual magnitude is unimportant with respectto stability (assuming that the deviations are not soeat aa to destroy the validity of the basic assump-tions). It seems not out of place, however, to suggest 61,h, etc.,that relative magnitudes of Provi
33、ded by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ANALYSIS OF LONGITUDINAL STABILITY IN PO-OFF FLIGHT 303may have important bearing upon the comfort, ease ofhandling, and design load factor of aircraft and thatfurther work taking these factors into consider
34、ationmight lead to valuable information.Charts of rotational damping factor against statiostability faotor.In the section dealing with applica-tion of mathematical formulas a convenient graphicalmeans of showing the variation of t and # with mnand pm= has been described. This method of pres-entation
35、 was given by Gatea (reference 5) in esse-ntiallythe same fashion although differing in detail.In referen 5 preference is given to charts withcoordinate of tail volume and fore-and-aft locationof the center of gravity. Preference has been givenin this report to charts with coordinate of rotationalda
36、mping factor and static-stability factor because theyme more convenient for tie with data horn wind-tunnel tests, they do not require assumption of arbi-trary fixed values for severil important faders, andthey cover a much wider range for a given number ofcharts. The charts are entirely nondimension
37、al anddC “ Vtiation ofvary ody with o, (?D, and $”da,empirical factors such as the K in ma, q, (SWp. 17), etc., ailect only maor pm=,a9 the case maybe, and their effects upon the stability are readilyapparent.The series of charts presented (figs. 15 to 54) areintended to show somewhat more precisely
38、 and com-pletely than has been done in the preceding discussiondCDthe effects of the paretera C., UKJ, and upon dynamic stability and to provide a convenientgraphical means by which the designer may estimatethe probable stabili characteristics of a proposedairplane and the effects of various changes
39、 withoutrecoume ti extensive calculations. The charts coverthe range of valueE of the parameters that appearlikely to be attained in the near future. The valuesrepresented are summarized in table I.Certain general charactmistics of the charts are im-mediately apparentupon inspection. As pm=increase9
40、from zero, t at first becomes more positive correspond-ing to a decrease in dynamic stability but, at a fairlysmallvalue of pmqchangesits trend andbecomesmorenegative. This tendency is general throughout all thecharts and is apparent whether mais large or small.It will bo remembered that such an tie
41、ct appearedprobable from the discussion of the introduction ofangular momentum into the system during a cycleby m As pointed out at that stage of the report,Acq j sin 84,and )all decrease with increase of pm=;AyO ;the charts of variation of the stability characteristicswith meand pm. show definitely
42、 that the decreaseof the product of these faotom is much more thansticient to nullify the increasw in pm= after acertain value of pm. has been exceeded. Increaseof , hence decrease in period of the oscillations, m“thincrease of pma occurs at practically all values ofmC and pin=.Increase9 of ma give,
43、 in general, more negativevaluea of and consequently more rapid dying out ofthe oscillations. At large values of pm= increasesin mgive more positive values of f but this effectk not of practical importance. It will be noticed thatat fairly large values of pmaincreasing ma has butslight effect but th
44、at mg becomes of increasing im-portance as pm= is made snder. Increasing madecreases + rather gradually, giving oscillations oflonger period. It appeam that if the criterion of sta-bili be taken as the number of oscillations necessaryfor the amplitude to decrease to one-half its originalvalue then t
45、he value of mQis of particuhtrimportance.The charts of figures 23, 27, 28, and 29 show thatincreasing CLwithout changing other factOm increasesquite markedly the tendency to instability. This ob-servation agrees well with the effect to be expectedfrom increasing CLthat might have been predicted from
46、(dCLthe part played by CLin the factor CD)CLdin equation (42).The eilect of increasing CDwithout changing otherfactors appears in figures 29, 30, and 31. It will beseen that increasing ( whereas at large angles of attack a smallchange in pm= or ma may change the dampingcoefficient from a deiin.itene
47、gative value to a definitepositive value. The period of the oscillation decreasesmarkedly with the increase of angle of attack but thedecrease is not so great as would appear to be the casefrom consideration of the increase in #. The period and the increase tiin seconds is proportional to CLis suffi
48、ciently great partly to counteract the increasein +.The effects of various physical characteristics ofthe airplane on the stability.-In the preceding dis-cussion no consideration has been given to the variousdimensional characteristics, aerodynamic interferences,etc., that determine the values of the fundamentalparameters. A large number of factors affect thestability but in many cases the effects are of a minornature. Only the more importmt ones will be dis-cuss
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