1、GO_.DOC.1 and _ 1.In this region, the rolling aircraft has two modes of oscillation, bothof which are undamped and have frequencies different from those of theoscillations of the nonrolllng aircraft. If the pitching frequency ofthe nonrolling aircraft a_ equals its yawing frequency ah_, then onemode
2、 of oscillation of the rolling aircraft has a frequency equal to thisfrequency plus the rolling frequency and the other mode of oscillationhas a frequency equal to this frequency minus the rolling frequency. Ingeneral, for a_ not equal to _, one frequency of the roiling aircraftis greater than the h
3、igher frequency of the nonrolling aircraft; and theother frequency is less than the lower frequency of the nonrolling aircraft.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12 NACATN No. 1627Whenone of the frequencies of the non_olling aircraft equ
4、als thefrequency of the steady rolling motion C_ or _= i), the aircraftbecomesneutrally stable in one mode, as shownby the fact that thefrequency of this modeequals zero. This phenomenonmaybe explainedphysically on the basis that the restoring forces acting on the nomrollingaircraft which produce a
5、certain oscillation frequen(_y are Just offset bythe centrifugal forces which attempt to swing the fuselage out of linewith the flight path whenthe aircraft rolls with this frequency. Thiseffect is somewhatanalogous to a rotating shaft operating at its criticalspeed. In fact, if the pitching and yaw
6、ing frequencies of the aircraftare both equal to the rolling frequency, the conditions are exactlysimilar to those encountered when a shaft having equal stiffness in alldirections rotates at its critical speed. Whenthe frequencies of theaircraft in pitch and yaw are different, and only one of these
7、frequenciesequals the rolling frequency, the conditions may be shownto be analogousto those encountered when a shaft of flattened cross section rotates atone of its two critical speeds. It may be of interest to note that thetheory for the behavior of such a shaft is identical with the theorydevelope
8、d in this report for the rolling aircraft.Whenone frequency of the no_rolling aircraft is less than thesteady rolling frequency and the other is greater, the rolling aircraftbecomesstatically unstable in one modeand performs a straight divergenceas measuredby instruments fixed in the aircraft. If bo
9、th frequenciesof the nonrolling aircraft are less that the steady rolling frequency,however, the rolling aircraft is stable, as shownby the small stableregion in the lower left-hand corner of figure 3 for a_ and a_rbetween 0 and i. Here again there are two modesof undampedoscillation.In this region,
10、 whenthe values of _ and a_ are equal, the stabilityis analogous to that of a shaft having equal stiffness in all directionsrotating above its critical speed. When _ and % both approach zero,which meansthat the static longitudinal and directional stabilities bothapproach zero, the two frequencies of
11、 the rolling aircraft both approachthe rolling frequency. Physically, this condition meansthat the rollingaircraft can have its axis tilted from the flight path and, because ofits lack of static stability, will continue to roll about this tiltedaxis. This rolling motion will cause periodic changes i
12、n the angles ofattack and yaw with a frequency equal to the rolling frequency. Theseperiodic changeswould be measuredas constant-amplitude pitching andyawing oscillations by instruments fixed in the aircraft.A small stable region exists where the frequency of one modeofoscillation of the nonrolling
13、aircraft is less than the rolling frequency,and in the other direction the aircraft has a certain degree of staticinstability. This stabilizing effect of the rolling motion maybest bevisualized by considering the motion of the aircraft wlth respect tofixed axes. A fin which provides stability in onl
14、y one direction (say,yaw) will makethe rolling aircraft stable about both axes, providedthe rate of roll is fast enough, because the fin rapidly turns from oneplane to another. This effect only occurs for a relatively limitedProvided by IHSNot for ResaleNo reproduction or networking permitted withou
15、t license from IHS-,-,-NACAT_ No. 1627 13range of parameters, however, and is shown in figure 3 as the stableregion in the range of negative values of a_2 and _/2. A negativevalue of a_2, corresponding to an imaginary value of the frequency,represents an exponential divergence defined by the equatio
16、n t0 = Ae-la_This same equation, of course, represents a sinusoidal oscillation offrequcncy a_ for real values of a_. Figure 3 was plotted in termsof _02 and _ 2 rather than _e and _ in order to include theimaginary values of these frequencies.In the lower left-hand corner of figure 3 there is a reg
17、ion ofincreasing oscillations as measured by instruments fixed in the body.In this region, where the nonrolling aircraft has a large amount ofstatic instability, the longitudinal axis of the rolling aircraftperforms a maneuver approximating straight divergence with respectto fixed axes; but because
18、of the rolling, this motion shows up as anincreasing oscillation with respect to the body axes.The effect of distributing weight along the wings as well as alongthe fuselage on the behavior of the rolling aircraft, again with zerodamping in pitch and yaw (_0 = _ = 0), is shown in figures 4 and 5-Fig
19、ure 4 presents the contour lines of the frequencies of the rollingaircraft on a plot of a_ 2 against _2 for F = -0.666. This valueof F corresponds to the case where the moment of inertia about theX-axis equals 0.2 times the moment of inertia about the Y-axis. Theresults indicated by this figure are
20、similar to those for the case whereall the weight is located in the fuselage. A somewhat smaller value ofthe directional stability is required, however, to avoid divergence inyaw of the rolling aircraft. Figure 5 is a similar plot for F = 0.This value of F corresponds to the case where the moment of
21、 inertiaabout the X-axis equals the moment of inertia about the Y-axis. Inthis case a rolling motion produces no inertia yawing moment on theyawed aircraft. With large stability in pitch, the yawing frequency ofthe rolling aircraft would therefore be expected to be the same as thatof the nonrolling
22、aircraft. The results of figure 5 indicate that thefrequency a_2, whlch represents mainly a yawing motion with largestability in pitch, approaches asymptotically the yawing frequency _as _ becomes large. Furthermore, the divergence boundary in yaw,which occurs at _ = 0 for the nonrolling aircraft, i
23、s unchanged bythe rolling motion.The special case where _ = and IX = 0 may be analyzed moresimply by use of the equation of motion of the body with respect to axesfixed in space. This analysis allows a clearer pl_vslcal interpretationof the motion of the body and serves as a check on the results obt
24、ainedProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 NACATN No. 1627previously by meansof Eulers equations. This special case correspondsto conditions existing along a 45 line through the origin in figure 3.The motion of the system with respect to
25、 axes fixed in space is derivedin a following section of this paper, but first the results alreadyobtained by meansof Eulers equations are stated. It maybe seenfrom figure 3 or derived from formula(12) that the frequencies of therolling aircraft with respect to body axes for this case are givenby th
26、e formulasHere, _l and a_2 are nondlmsnsional frequencies expressed as ratiosto the steady rolling frequency. The pitching frequency a_ of thenonrolling aircraft is equal to the yawing frequency ah_, and eithersymbol might be used. The memberon the right-hand side of the equationfor a_2 indicates th
27、e absolute value of the quantity _0 - 1. If theseformulas are put in terms of actual frequencies, rather than nondimensionalfrequencies, they become_lPo = _Po + Po_2Po = l_ePo - Po IHence, the frequencies of the rolling aircraft are given by the sumandby the absolute value of the difference between
28、the frequency of thenonrolling aircraft and the rolling frequency.The solution for the motion based on the equations of motion withrespect to fixed axes is now considered. The dynamic system is shownin figure 6(a). The restoring forces provided by the fins will usuallybe the samewith respect to fixe
29、d axes as with respect to axes rollingwith the body. The forces would be exactly the same, for example, ifthe body had a fin in the form of a circular cylinder. The assumptionthat the forces are the samewould be a close approximation to theconditions existing with a conventional four-fin tail. Becau
30、se all theweight is located along the X-axis, _ rolling motion of the body aboutthe X-axis has no effect whatever on the motion of the X-axls of the bodywith respect to fixed space. The motion of the X-axis of the body,therefore, is composedof vertical and horizontal oscillations offrequency a_po, e
31、xactly as in the case of the nonrolling body. Themost general motion of the axis is a combination of these two componentswith arbitrary amplitudes and phase difference. This combination inProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN No. 162
32、7 15general causes the axis to swing so that sI4y point on the axis traces anelliptical path, as shownin figure 6(a). In order to see how thisresult corresponds to that obtained previously for the motion withrespect to body axes, the frequencies measuredwith respect to fixedaxes must be converted to
33、 frequencies measuredwith respect to axesrolling with the body. This conversion is a kinematic transformation,with no dynamics involved. Ordinarily, the motion of the axis wouldbe resolved into vertical and horizontal componentsas mentioned previously.If the body rolls when the axis is undergoing a
34、vertical or horizontaloscillation, however, the resulting oscillations with respect to bodyaxes will not have constant amplitude. In order to obtain resultsequivalent to those previously described, it is necessary to break themotion of the axis into componentswhich lead to constant-amplitudeoscillat
35、ions with respect to body axes. Two such motlons are possible:one a clockwise and the other a counterclockwise rotation of a point onthe rear of the body. This point movesin a circular path withfrequency _Po“ These motions are shownin figure 6(b). These circularmotions of the body with frequency e_p
36、o are possible motions becausethey maybe obtained by combining vertical and horizontal oscillationsof equal amplitude with a phase difference of 90. Any possible motionof the aircraft maybe produced by combining these two circular motionswith the correct phase difference and amplitude. Examples of p
37、ossiblecombinations are given in figures 6(c) and 6(d). Figure 6(d) shows thatthe elliptical path, which is the most general type of motion, may beproduced by this combination.The frequencies of the rolling aircraft as seen from body axes maybe derived by considering the angle-of-attack changes as t
38、he body rollswhen its axis is performing one of the two circular motions. The casewhere the axls revolves in a counterclockwise direction with frequency a_powhile the body rolls clockwise with a frequency Po corresponds to theformula_lPo = a_Po + PoThe case where the axis revolves in a clockwise dir
39、ection withfrequency a_po while the body rolls clockwise with a frequencycorresponds to the formula Po2Po: l Po-poiThe results obtained by the analysis based on fixed axes msy thereforebe converted to body axes to give the same result as that obtaineddirectly from the analysis based on body axes.A c
40、ase in which the two solutions might be considered to disagreeis one in which the rolling frequency equals the pitching (and yawing)frequencies. The results plotted in figure 3 show a condition of neutralstability to exist at this point, whereas the stability of the axis ofthe body in the analysis b
41、ased on fixedaxes was stated to be independentProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-16 NACATN No. 1627of the rate of roll. It should be noted, however, that 8xiv slight out-of-trlm pitching momentapplied to the rolling aircraft at this poin
42、twould produce a vertical and horizontal momentvarying sinusoidally withtime at the natural frequency of the axis. A condition of resonancewould therefore exist and the vertical and horizontal amplitudes ofthe undampedsystem would increase indefinitely. In the analysis basedon body axes, this sameou
43、t-of-trim pitching momentapplied to theneutrally stable system would cause the angle of pitch to increaseindefinitely. The two methods of analysis, therefore, lead to thesameresult.If under the conditions where the rate of roll equals the pitching(and yawing) frequencies, the ax/s of the body is dis
44、placed in pitch,then a yawing velocity will be introduced with respect to body axes.Any damping forces proportional to yawing velocity would extract energyfrom the system and prevent the amplitude from building up. The dampingis therefore expected to increase the stability of the system, at leastund
45、er conditions where the pitching and yawing frequencies are close tothe rolling frequency. The effects of damping axenow considered on thebasis of the theory.Case of damped oscillations of nonrolling aircraft.- The rate ofdecrease of amplitude of the oscillations of the nonrolling aircraft isdetermi
46、ned by the damping ratio _. The fraction of the originalamplitude to which the oscillation decays in one cycle is shown as afunction of _ in figure 7. For _ = 0.2, the oscillation dampsto 0.25 of its original amplitude in one cycle. This amount of dampingis greater than that usually found for either
47、 the _itching or yawingoscillation of an aircraft of high density and is used to give anextreme example of the effect of damping on the stability of therolling aircraft.The divergence boundary for the rolling aircraft is determined bysetting the coefficient e of the quartic (equation (i0) equal to z
48、ero.The divergence boundary for the case _e = _ = 0.2 and IX = 0 is givenon a plot of a_92 against a_ 2 in figure 8. This figure also correspondsto any values of _8 and _ satisfying the relation _0_ = 0.04because these quantities enter into the coefficient e only as a product.By comparing the bounda
49、ries of figure $ with those of figure 3, it may beseen that the addition of damping has broadened the stable region in theneighborhood of the point a_ = i, a_ = i, that is, where the frequenciesin pitch and yaw are close to the rolling frequency. In other parts ofthe figure, the boundaries are but little changed. The boundary betweenincreasing and decreasing oscillations is not shown in figure 8.In practicej when the frequen
