1、REPORT No. 560A SIMPLIFIED APPLICATION OF THE METHOD OF OPERATORS TO THECALCULATION OF DISTURBED MOTIONS OF AN AIRPLANEBy ROBERT T. JoNEs) SUMMARYtion of such equations are given by W hence motionsdepend for solution on the integration of simultaneous of the airplane axes relative to both air and ea
2、rthlinear diilerential equations. Methods for the integra- must be considered.313Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-314 REPORT XO. 560 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSThe aerodynamic reactions to the motions arisefrom changed
3、relative air velocities over the dMerentparts of the airplane. The calculation or meamm+ment of these componant aerodynamic reactions leadsto quantities lmown as “resistance derivatives” or%tability derivatives,” which are taken as constantfactors of proportionality between the reactions andthe velo
4、cities or displacements of the motions. For amore detailed exposition of the concept of stabilityderivatives, the reader is referred to standard text-books on aeronautics.On account of the bilateral symmetry of the airplaneit is customary to divide the motions into two inde-pendent groups, the later
5、al and the longitudinal, eachIIdconsisting of thre,degrees of freedom:i w” “/ ,ti - $“”l:.-4 IRolling.,fA) ateral motJo_ Yawing.Sidedipping. /- the deviation of this spiral from a circle Iil-i a3FIrNJM2-Qraphicdmethcdoflwaticgmlumofl()nm.rmm,wherelW).D4-aD+bi-cD+d. DI-SI+fUFRIC%.RI(OX e,+ib OJhowsth
6、e influence of damping on the natural motion)f the airplane.The summation indicated in equation (6) calls forhe plotting of such a logarithmic spiral for each of the:omplex roots. Since the9e roots always occur in con-ugate pairs, the calculation maybe carried out for one)f such a pair and a spiral
7、calculated for the secbndwouldbe exactly conjugate to the first. Thus, it is onlylecessary to perform the foregoing calculations for onenot of each pair, the summations indicated in thequations being carried out in effect by merely doublinghe abscissas of the points of one of the conjugatemls. If A,
8、a+ib and A,=aib, this summationnay be written:The formulas for the integration of terms containingin ni and cos ti may be put into a more convenientorm for the graphical or logarithmic calculations, i. e.,(22)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from
9、IHS-,-,-318 REPORT NO. 560 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSIn these forms the graphical construction of the terms For the velocitie9 of an assumed gust, forms involv-+j(in int proceeds along the- same lines m“ that of the ye “ are useful. Thus, if the gust is considered a ,(in transient”
10、one, disappearing rapidly from an arbi-terms involving complex roots A. Hare the re- trary initial value, the form (A) (fig. 4)suiting diagrams will be circles, divided into equalangles as ni may be (logo+afl)+(eo+bfl) .t=o.iqJ.-iover each of a pair of conjugate roots is accomplishedby doubling the
11、abscissasof the spiral obtained for one.WAYS OF REPRESG GUSTS AND CONTROLMANIPULATIONSGUST DISTURBANIf the disturbances to be considered are due to gusts,the terms YO, ro %noment9, “which will arise from two sourcw: the:eactions due to natural stability and the reactionsproduced by the displaced con
12、trols. The reactionsmising from the motions are found by combining theknown stability derivatives with the angular velocitiesp and r, obtained from the specification equations (27)md (28). The parts of the momenti necessmilympplied by the controls are then obtained by deduct-ing these from the tctal
13、 moments. In the -e of thetieron control, secondm-y moments in yaw result fromtbe application of rolling moment, which modify theunount of rudder control displacement necessary.CONTROL AGAINST GUSTS OR EN- FAILUREIn order to deal with attempted control of a givendistnrbmce it is bportmt to consider
14、that there isnvariably a lag in the pilots reaction in counteringihe motion. In” these cases it is possible to “-rime;hat the disturmce arises instantly, or” nearly sojvhether persistent or not), and that the pilots to speci.iic problems of airplanemotion. The airplane assumed in these calculationsi
15、s a typical 2-passenger machine having the followingcharacteristics:CHAEACPBRL9I1 OF TYPICAL AIBPLANEType: Monoplane; aspect ratio 6; rectangular, roundedtip, Clark Y wing; 3)to the form required for expan-JI(D) fJD)NP)Rm o (34)rhe calculation of the various polynomials in D resultsn:fl(D) =pN,D+ Uo
16、N sin 1.73) (38) rithms of the abscissas. The final points are found toAn additional complex root that is the conjugate of sin 5.60). (45)Since the quotient of these values is to be multipliedinto eifor a series of values of t it will be convenientto use the logarithm of this quotient, simply adding
17、 tcit the various valuw of kt for which the calculation kto be made. This logarithm isIogf ()+fl)o1 = (log 3.99log 40.6) +i(5.235.60)=2 .32O.38 iLLLUSTEATION OF SOLUTION WITH VABIABLE DL9TUBBANCETERMSExample II, Sideslipping during 2-control turnmaneuvar:(a) orP=t=0.z62 sin t+ O.131sin 2t (48)A cons
18、traint of the machine in one of its degrees ofkeedom hwing thus been specified, it is only necessaryto consider the equations for free motion in the remain-ng two degrees. As before, the lateral motion will beissumed to be independent of the longitudinal. Thero:emain only the sideslipping and yawing
19、 motioti toe considered. Their equations are:(49)drI=viV.+pNp+rN,!dthough the equations oontain the rate of rolling and;he angle of bank, these are to be considered as known$ .6 I I-. L%717kc!xw!%olnf-1! / *.,/.$.4 P, 1 30 3 /h/ 2 .2* /j/ / 7- , /: 9“ delip, v/u.$0 =“ v2%-”Jo * “1 2 4 5?Zme,sconds6F
20、IGIJBB7.Eanlt of compntathn; example IL Sldmllp dorlng a Z-mntrolturnmenenver.tom equations (29) and (30) and are, in fact, to beed as the disturbance terms. CallingYo=0Y, 1No (D-ivr)ID 1 II UON, (DN,) If,(D) =.D-Nrj,(D)=1I(D) =DN,D+ UON,(52)(52a)(53)If the airplane is to maintain its altitude while
21、turning, the speed must be adjusted ta give a higherlift than that at an equal lift coefficient in level fhght.At an asaumed lift coefficient of 1 the speed necesmgto maintain altitude while turning at 30 bank isfound to be 95 feet per second. Actually, if thisspeed is held throughout the specified
22、maneuver, thelongitudinal path will be accelerated somewhat; thiscondition will be neglected in the present problem.The necessary stn,bility derivatives calculated for thenew condition are: N,=O.712ufl,=2.40 1 (54)NP=0.323The “disturbance effects” Y. and NOare (see equations(46), (47), and (49)Yn=o.
23、111o.0888 Cos-etc.,performing the indicated integrations results in(DLp)-l(t)=l(t)(t + . . . ) (63)+ etc.If this series is multiplied throughout by L, it becomesidentically the series for M except for the term 1,that isLP(-LP)-+ll (t)=l(t) H, T. IL No. 21,1917; and III, T. R. No. 27, N. A. C. A_, 19
24、18.2. Bryant, L. W., and Williams, D. H.: The Application of theMethod of Operators to the calculation of the DisturbedMotion of an Aeroplane. R. & M. No. 1346, BritishA. R. C., 1931.3. Routh, E. J.: Advanced Rigid DynamicsjVOLII. TheMaaMillanCompany,1905.L Goumat,Edouard:Funotionsofa timplex Variable. Trans-latedby E. 1%HedriokandOttoDunkel,Ginn&Co.,1916.$. Bush, V.: Operational Cirouit Analyaie. John Wiley andSona, Inc., 1929.325Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
