1、A THEORETICALREPORT No. 787INVESTIGATION OF THE ROLLING OSCILLATIONS OF AN AIRPLANEWITH AILERONS FREEBy DORISCOHENSUMMARYAn andytnk ix thus) 2=b= span of aileronsZe root-meanquare aileron chordSymbols used in describing motions (all angles are inradians):9P!7v8PvaaeP$46PrYNLBacceleration of gravityd
2、ensity of airdynamic pressure ()+pvsteady-flight speeddistance along flight pathdistmce along fight path traversed during one oscilla-()2Ttion, semispans ;sideslip velocity (positive to right)angle of attack of wingeffective angle of attaok due to flap deflectionangle of sideslip (positive when side
3、slipping to right)angle of yaw (positive when nose turns to right)angle of roll (positive when right wing is down)total angle of aileron deflection (positive with right wingdown)rolling velocity (d#Jyawing velocity (d.#/d4side force ositive to right)yawing momentrolling moment in rolling-moment coef
4、ficient; lift in liftcoefficienthinge momentNondimensional quantities:airplane densiQ pammeterairplane moment of inertia about X-axisairplane momeht of inertia about Z-axis.aileron moment of inertia about hinge axis.mass-moment parameter, hinge axis. Jon-dime.mionfd expression for effect .of inertia
5、of aileron system in causing aileron deflec-tion when airplane is accelerated in roll.) mLporailwOmoneW= a ib/2$, mtio of flap chord to airfoil chord at a given sectionb12 d differential. opefator.D= d =jijmIn particular,% D+=!# ;x root of stability equationa real part of A, proportional to rate of
6、damping of motionsn magnitude of imaginary part of X, proportional to fre-quency of oscillations0. yawing-moment coefficient (!J%L)C, rolling-moment coefficient +b/()C, hinge-moment coefficient aCL lift coefficient L()qx(7C= side-force coefficient gSubscripts attached to moment Coticients indicate t
7、hopartial derivative of the coefficient with respect to thequantity denoted by the subscript. In particular,Cha= $ hinge-moment coefficient due to unit”aileron deflec-tion, or restoring tendency. Restoring tendencyis positive when surface is overbalancedC,= =b hinge-moment coefficient due to unit ch
8、ange inlocal angle of attack, or floating tendency.Floating tendency is positive when surface floatsagainst the relative windcbD8= Wemoment coefficient due to unit rato ofdeflection of ailerons (generally the aerodynamicdamping, but may include viscous friction inthe control system)Cla+ rollimoment
9、due to unit aileron deflection, oreffectiveness of the ailerons in producing roll()ac.ma part of additional lift due to angular velocity of flopcaused by acceleration of potential flow ( T4 ofreference 2)()acL5D5.P art of additional lift due to angular velocity offlap caused by effective increase in
10、 camber(11 f reference 2-Z 0 )c,(vAp art of hinge moment due to angular velocity offlap caused by acceleration of potential flow( 4 11, where T, and Tll are given in reference 24T t; )()ac, =part of hinge moment due to angular velocity offlap caused by effective increase in camber(Tn Tlg1 -where T,l
11、 and T12are given in reference 28n- t, )The variable DO is held constant in taking the partialderivative with respect to 6 or D6, which is equivalent toholding a constant.4.2Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICALINVESTIQAON OF RO
12、LLING OSULLATIONS OF AIRPLANE WITH KHJEROASFREE 257The following symbols are adopted because of commonusage:O,=q aerodynamic damping of the airplane in rollb $?ANALYSISEQUATTONS OF MOTIONThe general equations of lateral motion with ailerons free,coupling the rolling motion of the airplane with the y
13、awingand sidedipping motions and with the movements of theailerons following rLsmall disturbance, are aa follows:(1)m(b+V consequently,$=0. The moment of inertia of the ailerons was also takenequal to zero. (The validity of a comparison made on thebasis of zero moment of inertia will be checked in a
14、 subse-quent section.) The hinge-moment parameters Cl. and C8were retained as the principal variables.TABLE ILAILERON CHARACTERISTICSvalueDe this root passea throughzero rdong a line designated in figure 2 as the spiral divergenceboundary. b the region around the positive dh=-ti theremaining four ro
15、ots form two complex pairs, indicatingthut the motions have two oscillatory components. Alongthe Iongdashed curve one oscillation disintegrates into twoaperiodic modes, divergent or convergent accordingly as theosctiations are stable or unstable; at values of oh= and08 outeide this curve the motion
16、is composed of one oscilla-tory mode, which is almost always stable, and three non-oscillntory components. Inside the curve, the two oscillatorycomponent5 are stable so long 8s (?Bbis negative. As Cbecomes positive, instability sets in, as indicated by theoscillatory stability boundary. In general,
17、only one modebecomca unstable; the same oscillation breaks down intotwo aperiodic modes at a slightly larger value of Cfia.Ina small region (A13 in g. 2) defined by the intersectionof the two bmnchcs of the boundary, both modes are un-stable. This detoil and othem occurring outside the stableI Oscil
18、latory sfability I UrIe stable,c.neun.fkle I.5,4.3.2%,/0-./-.2 -J o J 2Ch6FIOUBE2-ObnrecieI end stabiffty of the mmp-ments of tbe moths fonnd by wlntfcm ofthe equetfons hforo tbo ehfrmtlm of sldedfppfmgand yawhg. (Sb8dfnS fndfuah tbatmstab!a region.) Aileron chord, let abfoll chofi C-O; I.=IZ dfbedm
19、f and% 6%aL-in.4.3.2.1cG-. /.-I I l I I I!.0041/1-.8? -J .1 .2EdFIOWEE3.IMa of dfvemenc%M fndfcatcil by the vafae of tfIOIXSItfved IW of tieetabllky eqwifon. EIOII CilOd, lCL-LO.region, or near the boundary, are not Cofidmed of Ypractical importemce; they me mentioned in order to answerquestions tha
20、t might otherwise be suggested by inspectionof the figure.Rate of divergence, four degrees of freedom,-lksnmchas figure 2 indicates that the motions will be unstable formost combinations of values of 6haandCha,itseem advisableiirst to examine the nature of the divergent instability, whichappears alm
21、ost unavoidable. The condition for neutralstabili (zero root) is that the cmstant term of the stabilityequation vanish; that is,o(e,flo+ Cm) + ch(c.#l- c,#nt-m 1.-0.Nature of the motions, two degrees of freedom.-For thecase defined by tables I and II, the motions are as demribedin figure 4. The stab
22、ility equation is a cubic, and them is%u one real root, which becomes zero at the divergenceboundary. The remaining two roots form a complex pair,indicating an oscillatory mode, inside the region defined bythe longdashed curve. Outside this region cdl three rootsare real and no oscillations occur. T
23、he oscillations becomeunstable at a small positive value of CA8,which is almostindependent of the value of Cna.Comparison of results, two and four degrees of freedom,The results of the two computations can now be tested foragreement. Comparison of figures 2, 3, and 4 suggests thatthe effective diver
24、gence boundary of the cross-coupledmotions (shown by the dotted line in fig. 2) may be assumedto coincide with the true divergence boundary in the shnpli-fied caae. Thus, where the simplified analysis indicates achange km stability to instability, there is actually osudden transition from a slow div
25、ergence to a rapid one.The comparison may be extended into the first quadrant ofthe charts. Here the divergence bound appears, in thomore esact analysis, SE a branch of the boundary betweendamped and undamped oscillations (line OA, fig. 2). Tlmoscillations are; however, on the point of breaking clow
26、ninto aperiodic modes and the instability would in practicebe indistinguishable from uniform divergence. In accord-ance with these observations the line of zero roots obtainedfrom the simplified analysis will be termed the “divergenceboundmy,” with the understanding that such a designationis strictl
27、y true only when the cross-coupling is negligible.Further comparison of es 2 and 4 shows that theoscillatory stability boundary of the simplified trentment,although shifted slightly by the introduction of the additi-onal deggees of freedom, is so little altered that it also moybe retained as part of
28、 the stability boundary. Aloreover,the position of the line enclosing the oscillatory regionremains essentially unchanged and still indicates the valuesof the hinge moments at which one oscillation breaks down.It may therefore be concluded that, except for the presenceeverywhere of an additional mod
29、e of oscillation to be dis-cussed subsequently, the broad aspects of the solution forthe more complex case may be deduced from the results ofthe simplified analysis.Comparison of the roots at a number of poiuta shows thatthe results of the two calculations are in close qwmtitntiveagreement, also, wi
30、th regard to the oscillatory mode com-mon to both analyses. Thus, both the period and the damp-ing of the oscillations of one mode can be obtained from therm.dts of the simplified analysis.The oscillations of the second mode have both dampingand period virtually independent of the hinge moments ofth
31、e ailerons. In the case chosen for illustration the periodis of the order of 30 semispans, or, if the span is 40 feet andthe wing loading 40 pounds per square foot, about 3%seconds, throughout the range of Cfidwith CA*negative; themotion damps to half amplitude in the course of one oscilhwtion. Beca
32、use the aileron chamcteristica are not involvedProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORIHICAL INVESTIGATION OF ROLLING OSCILLATIONS OF AIRPLANE WITH AILERONS FREE 261and because of the magnitudes of the period and damping.this mode rLppe
33、arato be the normal lateral oscillation of theairplane with controls fixed and m such is treated elsewherein the literature. For the assumed airplane this mode doesnot become unstable anywhere within the region indicatedas stable by the simplified analysis.Effeot of aileron moment of inertia on cros
34、s-coupling.-Itseems desirable to check the foregoing conclusion againstresults obtained with the moment of inertia of the aileronsystem retained in the equations. For this purpose, theroots of the stability equations have been calculated atC*a=().16 and C*J=0.02, 0.1, 0.2, and 0.3, with la= O.025.Wi
35、th four degrees of freedom, the stability equation has sixroots, Of these, one root indicates the spiral mode and, inthe unstable region, has the same values as are given byfigure3 for the case with zero moment of inertia. A secondreal root corresponds to the real root of the simplified equa-tion. T
36、he four remaining roots form, in general, two oscil-latory pairs. These roots are compared with those of thesimplified equation in the following table:At Chb=0.02, where the periods are of the same order ofmagnitude, the effect of the cross-coupling is seen. Else-where the period and damping in both
37、 calculations agreewithin 1 percent. It appears reasonable to conclude thatthe statements of the preceding section hold in spite of theomission of the aileron moment of inertia from the calcula-tions.SIMPLIFIED ANALYSISUsipg the reduced form of the stability equation makes itpossible to investigate
38、the effects on the stability of the air-plane of varying the aileron charactitics, and even to givecertain general formulas. Because most modern airplanesme dwigned with ailerons completely mass balanced, thweformulas may be still further simplitled by assuming equalto zero.Aileron-free oscillations
39、,-The oscillations associate dwith freeing the aileron controls can now be investigated inmore detail. If a pair of roots is assumed in the formA= admi, a relation can be derived giving the frequencyn in terms of the coe.flicients of x in equation (7). Thisrelation is too lengthy to be presented in
40、its general form;however, calculations have been made from it and the re-sults will be shown in the form of lines of equal periodP=2r/n on the stability charts.The damping of the oscillations is more readily expressiblethan is the period, particularly if a iixed value of the tie-quency is aasumed. k
41、loreover, calculations of the dampingfor zero frequency and for the highest frequency likely to beencountered in practice showed that the expression could bestill further simplified by omitting the terms containing thefrequency and c,= (sinoe these terms apparently canceledeach other) without any ap
42、preciable loss in acouracy. Thus,with t equal to zero, the damping a is, to a good approximat-ion, the smaller root of the quadraticwhich k independent of (?ha.At the stability boundary, the damping a is zero, and,therefore, c C that is,Figure 4, however, show-s the variation with oh= to beactually
43、quite small.Stiok-force oriterion, The divergence boundary is ob-tained by setting the constant term of the stability equationequal to zero; then,(11)This comj.ition for neutral stability is identical with theequation for zero slope of the hinge-moment curve:(12)and is therefore also identied with t
44、he condition for zerostick force in pure roll or in a rapid rolling maneuver. Inas-much as the stick force per unit deflection of the ailerons isc lines of constant stick force are obtainedproportional to ,by replacing the zero in equation (12) by appropriate con-dD the equationfor constant stick fo
45、rce therefore results in a family ofstraight lines padlel to the divergence boundary of equa-tion (11) and the criterion for light stick force for givenaileron di.metions and effectiveness is the closeness withwhich that boundary is approached. A comparison of oneaileron with another, however, shows
46、 that the stick forcewill also be proportional to the value of E=*b=.Method of investigating the effect of friction.-when theeffect of friction in the control system is considered, it isnecessary to distinguish between two types, viscous frictionand solid friction. Viicous friction, which va.rk with
47、 the843110-5%1SProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-262 REPORT NO. 787NAPIONfi AWISORY COMMJIITEE FOR AFIRONAmCSspeed of the flap deflection, is exactly equivahnt to an in-crease in GDa,heretofore cotidaed be due OY to tieaerodynamic dampi
48、ng of the ailerons. Solid fiction acts ina more complex way but may be approximated by anequivalent viscous damping, the amount varying inverselywith the amplitude of the deflection. (A more detailed dis-cussion of this approximation is given in reference 7.)Thus, in the course of a damped oscillati
49、on, for example,the npparent Cm increases md fie qu=tion of fie ectof the friction reduces to the question of whether an increasein Ch=a is stabilizing or destabilizing.ELANATION OF CHARTSThe stabili charts (figs. 5 to 9) are intended both asillustrations of the application of the preceding formulasand as working charts from which the behavior of a particularset of ailerons on a conventional airplane may be predi
