1、REPORT No. 791A THEORETICAL INVESTIGATION OF LONGITUDINAL STMULITY OF AIRPLANES WITH FREECONTROLS INCLUDING EFFECT OF FRICTION IN CONTROL SYSTEMBy HARRYGEDENBEEQand LEON.ABDSTERNFIELDSUMMARYThe relahn between the eleoaior hinge-momtmt parametw8and tlw control forcesjor changtx in jorward 8peed and i
2、n maneu-wr8 k 8hown jor 8everd v?uesof static stability and e.?.ewtornuuw bahwe.The sttiiiy of the short-period osdi?ati0m3 h 8hown m asm”es of boundaries giving the limit-s of the stable region interms of the elevator hinge+noment parameters. The e pVW pV%. positive for pullCZF.()stick-force gradie
3、nt in maneuvers ()stick-form gradient for level flight acceleration of gravityhinge moment; positive when tends to de-press trailing edgemass moment of elevator about its le;positive when tailheavymass moment of control stick about its pivot;positive when stick tends to move forwardfrictional hinge
4、momentmoment of inertia of elevator aboutwmoment of inertia of control stick aboutpivotrrdhs of gyration of airplane about Y-axisitsitsdistance between airplane eeriter of gravityand elevator hingelength of control stickpitching moment aboutgravitymass of airplaneairplane center: ofnumber of cycles
5、required for oscillation todamp to half amplitudenormal acceleration per g of airplane duo tocurvature of flight path; accelerometer reocl-ing minus component of gravity forceperiod of oscillation, secondsdynamic pressureelevator areatail areawing arendistance in half-chords (2Vt/c)time required for
6、 oscillation to drunp to hnlfamplitude, secondstimeforward velocitychange in forward velocity from trimmed w-dueweight of airplanelongitudinal force; positive forwarddistance of center of gravity from aerodyna-mic center; positive when center of grnvityis ahend of aerodynamic centernormal force; pos
7、itive downwardangle of attackrmgle of attack at taiIdeflection of elevator; positive for downwnrdmotion of trailing edgeamplitude of elevator oscillationangle of dohcontrol gearing (0,/3,)angle of pitch of airplanedeflection of control stick; positive for for-ward motion of stiokcomphm root of stabi
8、li equationreal and imaginary parts, respectively, of Aairplanedensity parameter (m/PSQH;U +H”+HJ the imagimuy part determinesthe period of the oscillations. tilore specXcally, if there isa pair of complex rootsX=:+iqProvided by IHSNot for ResaleNo reproduction or networking permitted without licens
9、e from IHS-,-,-332 REPORT NO. 79 1NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSthe period in seconds is given byp.-JSW-0.1:l, ft- 2sJst-0.55At-_-_-_-_-_-_-:- 45The basic stability derivatives and parameters obtained fromthese airplane characteristics by methods shown in appendix AareCL=- 43 c“m- 15. 3
10、CL, - - 3.8 Cne-_-_-_- pvwql“ dn 2PD81J “% P the effect on F.is not revemed, however, until the center of gravity is wellbehind the aerodynamic center (in this case, about 0.05c atsea level and 0.02c at 0,000 feet). If CA8=0, the stickforces are independent of the position of the airplane centerof g
11、ravity.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I.4.3/Fn “Fu 5(7/-50 N /40 / -40 A/-.33 m “/ / / /-/0 /,/ 71/0 / .273:5 .4 .3 .2 -J o .dResfcrhg terdwmy, CbFmvm l.lMm force F. nod pull-up form F. as hnctfons of hfnge-momont p.rronehs.-.Z.O.O.%
12、 f40fmndsporswamfoo G C.7fe.3t;SefileT0L i+.-atlck form In pandsfor“17-1.O; F.-nttek form in fmunds par normal occelaratfon.4FU%7/-70 .3-60+40 P30Ud.91Rr 6W8.rO fret; C-7 fm sea Iovef. F.-stictfome fn -for -1$ F.=df soelevol. F.=410k form In poundsfor v- 1.0; F.-ntlok form In pounds per o normrd amx
13、lomtlon.4.3/ + .2/ / 4/ +F, 0 / / F. Bo / /-110 / / A .-100 -/ / “-% /20 y 0-m / A -lo - z, -0.06q 40 pmdn w swam foot; u-7 fcati ma“level. Fetlokform In pmmdn for TV-1.0; Jm-ettok form In paunde per g normalacW3retlon.Provided by IHSNot for ResaleNo reproduction or networking permitted without lice
14、nse from IHS-,-,-336 REPORT NO. 79 lNATIONAL ADVISORY COMMITTEE FOR AERONAUTICSIncrease in altitude will either increase or decreme F.,depending on the hinge-moment parameters. The solidline in figure 5 is the locus of values of C,=,and C, for whichF. is independent of altitude. For points to the le
15、ft of thisline, F- decreases with altitude; for points to the right ofthis line, Fe increases with altitude. This line is determinedby the relationwhich, for the case of figure 5, becomesC,=,= I.50C,8Another method of increasing the stick-force gradient inlevel flight Fti consists in applying a cons
16、tant hinge momenttQ the elevator by means of a spring or bungee. The effectof the spring on the gradient F. is due to the derivativeC*Uwhich depends in the same way on the constant hingemoment, whether it is caused by a weight or by a spring. Abungee, which tends to depress the elevator, will theref
17、oreincrease the stick-force gradient in level flight Fu. Theeffect of the bungee on the stick-force gradient in acceleratedflight F. will be zero because its action depends solely onchanges in forward speed. Its effect on the shorhperiodoscillations will be zero for the same reason.DYNAMIC STADILYNO
18、PIUON m CONTROLSYSTEMThe stability of the shorbperiod oscillations withoutfriction is shown in figures 6 to 11, which also show theboundaries for true static stability (divergence boundaries).Figure 6 is an example of a more nearly complete presmtationof the stability data than subsequent figures be
19、cause itshows the variation of damping and period of oscillationwith the hge-mOmt3nt pmeik! chat and Chd for certainfixed values of the other parametm. The damping, whichis proportional to (, increases with the magnitude of Cb6.The period, proportional to $, demeases as Chatincreases.Another way of
20、presenting this additional stability data isshown in figure 7, which gives the number of cycles the oscil-lation performs before it damps to half amplitude. It isclear from figure 7 that the oscillation is very well dampedunles the restoring tendency is close to zero. In this parti-cular case, only
21、one oscillatmy mode exists. Inasmuch asthere are only three roots in this case (because G and theother at the control stick, which gives the elevator a sufE-ciently powerful tailheavy moment that the resultant stickforce is the same as with the single weight. In the particu-lar case represented, the
22、 noseheavy moment due to theweight at the elevator is equal to the tailheavy moment dueto both weights. Moving the single weight from the con-trol stick to the elevator has a large destabilizing dfect,Overbalancing the elevator while the stick” force is keptconstant has a considerable stabilizing ef
23、fect. This methodof preventing instability has the disadvantage, however, ofincreasing the total amount of unbalancing weight required.In the case 3.3D feet 3Z5 50wne we $ o 2. I#2#/0% 71/ /( /, 72fn-m ,/0/ / / / / 0 - .3/AA“ /“Fn=O/ 0 /;5 74 73 :2 + 0 -.4Res ta+g temfertcy, FIauB 5.-Efloot of alttt
24、udo on VOIUIMof stlok-foma srodk+ut. S* A =0.C -40 ncamdsw Wlaro fret; C-7 feat.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.4N* J ,2 .s1 CDI .3i 2hkl , t ., doscillations1%;028Q-.i. -,2-.3-.4 -.3 -.4.Qe;orinq tendm-, CU o.1FICL,BE7.Xuullwr O( cy
25、ti to damp to half amplltudo NIn agahwt bingo.momwitetom. %-0; P-lq A=. -Loo.IIi1.,I1IItItI,1Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.4Incremi-ig osullafbs 7274 73 :2 71 0Resttiq tendmcy, C%FIOWBB 9.Eflwt of ok.vator mntrol-ayelom momont of f
26、nortb on tho boundnry for lnm8M-lng OSaiUntlontl. #-lz.s.4Increoshg osdhtim I .3II 83P5 I- 125 1 ,- 417 iI1 ,/ *II I1 /,1.IStokJe rim d;!ItI ioDi vIIdIJ JIg/ /I F/. $!0“fDi vergen ce/I71111I-2/1/-. . r- - - Jm ergenc e .3-.4 .3RBtiuq l;ttttI I I Ihcreashg 09cilk7ttis _;3I t1 /II/I /I I .2II /, dIII
27、/ II;/$I III I/ RI1/oIII tII /II /-./II /II/I 72/H-D?3-wHtH3.3 -2 -J o “JOUEE 12-EEeot dltimofmm-mdtie demtirwnlp on tbefibllltylnmllti p.lq /,-0; ,.0.,IIProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICAL INVESTIGATION OF LONGITUDINAL STABII
28、JITY OF AIRPLAllTES WITH FREE CONTROLS 341EFFECT OF VISCOUS FEICTION IN CONTROL SYSTEMIn the pyeceding section, n constant value of the elevator-damping parameter O*D*was assumed. This value was dueonly to aerodynamic damping. The effects of viscous fric-tion in the elevator control system are obtai
29、ned by consider-ing OhJas an additional variable. This variable can beintroduced, as in the preceding section, by showing a seriesof boundaries in the oh=o. plane for vaxious value9 ofohd. The general nature of the effect of friction is shownfirst, however, by presenting boundaries ti the Chaoh=$pla
30、ne with u=, constant and some other parameter varied.This method of presenting stability boundariw makes iteasier to show the effects of other parametem such as air-plane center-of-gravity position and density when frictionis introduced.Tho effect of viscous .fkiction on the dynamic stability,for va
31、rious conditions, is shown in figures 13 and 14 forP=12.5 and P=37.5, respectively. Figures 13(a) and 14(a)refer to the mass-balanced elevator control system; figures13(b) and 14(I) refer to the tailheavy elevator controlsystem considered in the preceding section. It is shown that,if the airplane ce
32、nter of gravity is ahead of a certain point,lthe instability caused by the unbalanced elevator can beremoved by adding viscous tiction to the control system.This critical center-of-gravity position is behind the aero-dynwnic center, and its distance from the aerodymuniccenter decreases as the densit
33、y parameter P inoreases.When the center of gravity is behind this critical position,viscous friction has a destabilizing effect. These oppositeeffects of viscous friction are shown in the Chatoh plane infigures 15 and 16. When the center of gravity is slightlyahead of this critical position, the eil
34、ect of viscous frictiondependa on its magnitude and also on the value of CkJ. Theaddition of a small amount of viscous friction is destabi-lizingbut larger amounts are stabilizing. If the aerodynamicbalance is sufficiently high (C,J=0) and the viscous frictionlies in a certain range, increasing osci
35、llations will ooour.In figure 14(b), for example, if z=.= 0.OIC and Ca= 0.05,the oscillations will be unstable when the elevator-dampingparameter is in the range from 2.5 to 76. (?ha ismore negative than 0.086, no amount of elevator damp-ing ean cnuse increasing oscillations. As the center ofgravity
36、 moves forward, the destabilizing ect of elevatordamping becomes less and fially disappeam.The effect of the density parameter P can be seen by com-paring figures 13 and 14. The critical center-of-gravityposition approaohcs the aerodynamic center as p inoreases.IMm thfsrqmt was wdttm,MepdntW hnfonnd
37、 h h where % someta mudthe stick-llxed manouvm pcdnLWhen = 12.5, elevator damping always has a skbilizingeffect provided x=. is positive. When P=37.5, elevatordamping may be destabilizing over a small range of C*Daand ChJeven when a .C. is positive (0.05c).When the center of gravity is slightiy ahea
38、d of the afore-mentioned oritical position (which is behind the aerodynamiccenter), the conditions under which elevator damping maycause dynamic instability may be advantageously repre-sented in the Cfi=Ck8plane. If a series of stability boundariesare drawn in that plane for various values of elevat
39、or damp-ing, they will all be con.i?medto a region bounded by a linethat will be called the boundary of complete damping. Anillustration of two methods of constructing this boundary isgiven in figure 17. H a series of boundaries in the Ch=,cfiaplane are drawn for various values of the damping, the c
40、om-mon tangent of all these curves is the boundary for completedamping. This boundary ean also be drawn by plotting theminimum values of cfidobtained from plots of the type shownin figures 13 and 14 against corresponding values of chat.The region in the Ckat(?kaplane between the boundmies forcomplet
41、e damping and increasing oscillations is the regionwhere the addition of viscous friction to the elevator controlsystem may cause dynamic instabili.That a boundary for complete damping cannot bo shownfor K= 12.5 if the airplane is statically neutral or stable (z=.is zero or positive) maybe seen from
42、 figure 13. It is possiblehowever, to show a boundary for complete damping underthese conditions for P=37.5. Fi.-e 18 shows these bound-aries for x=.= O and for the critioal value X=.c.= 0.017c,for both a mass-bakmced elevator and a mass-unbalancedelevator. The boundaries for increasing oscillations
43、 anddivergence are also shomn. For the case of the mabalancedelevator (A= O), the boundary for complete damping is astraight line through the origin and therefore correspondsto a tied ratio of the floating and restoring tendencies,or floating ratio. Elevator mass unbalance decreases theregion of com
44、plete damping.FECT OFSOLmFRICTIONIN ELEVATORCONTROLSYSTEMThe boundary for mmplete damping has an importmtbearing on the effect of solid friction on dynamic stability.In order to calculate this effect, the solid friction is replacedby an equivalent viscous fiction that would dissipate energyat the sa
45、me rate. This condition gives an equiwdent4 c,Ckm=; -J (4)for a sinusoidal motion of the elevator with amplitude T andangular frequency q.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I./-.3./L-100 -80 -60 d-40 -20 .I I I I I I I I I 06mpinq /l!-(a
46、) h-o.(b) A=1O.l?mura 13.-Effmt of efavator dmmplng on the boondary for Ina-emlng cdflatlons for vadcmsmnter-af-gmvfty fmatlorrs and ele.vatormawbdnn W Wndftfon?. CA-*-O.l;p-.1II1It ob? A-37A.FIIProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Increos
47、lng Oscillations IL/ -1:/IL I /1/iDlv ergenc e74 73 -2 71t/I/o,Qesforing fenobncy, CP=IW, hm10.Increoshg cxdhtiwsL-7;III1/I4Dlv ergence74 :3 72 71es foring tendency, qvihere q and Chm axe the values at point 3. This formulashows that the amplitude is proportional to the amounL ofsolid friction.The foregoing analysis shows that tbe region in the cba,ctplane between the boundary for increasing oscillations rmdthe boundary for complete damping is the region wheresteady oscillations may occur because o
