\ ' NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIMK REPORT ORIGINALLY ISSUED May 19J|4 ae Memorandum Report A METHOD FOE STUDYHJG THE HUBTIHG OSC ILLATIONS OF AK AIEPLANE WITH A SIMPLE TYPE OF AUTOMATIC COHTBOL By Robert T. Jones Langley Memorial Aeronautical Laboratory Langley Field, Va. NACA WASHINGTON / NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. ^ - ^2 bOCUMEm'S DEPARTMENT Digitized by tine Internet Arclnive in 2011 witln funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/methodforstudyinOOIang 7' i," ti-VJ NATIONAL ADVISORY CO^'IIIITTST: FOR ALROIIAUTICS M^:i'"RA^DT'J': R3PGRT for the Arrry Air Forces, i'ateriel Comn.and A METHOD ^OR STUDYING TI-J" HUNTING CSC ILLATIONS OF AN AIRPLANE WITH A SIJ.IPIT: TYPE OF AUTOMATIC CONTROL 3y Robert T. Joner? SUIjIMAPY A niethoG is presented frr predict.: ng the amplitude and frequency, uiider certain si:nrlifying conditions, of the hi.uitino; oscilHations of an automatically controlled aircraft with lag in the control system or ir the response of the aircraft to the controls. If the steer- ing device is actuated by a simple right-left type of s'pnal, the series of alternating fixed-amplitude signals occurring during the hunting may ordinarily be represented hy a square v/ave. Forrralas are given expressing the response to such a var'ation of slrnal jn terms of the response to a unit sip-nal. A moi'e conplex t'Toe of hunt- ini£ , which may involve cyclic repetition of signals of vP-ryinf duration, has not been ti'eated and requires further analysis. Several examples of application of the method are included and the results discussed. INTRODT'CTION V/hen an airplane or other aircraft is directed by a simple rieht-left sif;:nal from an autor'atic steering device, the result is usually a naintained huntinr oscil- lation about the desired path. The amplitude of this oncillr.tion is influenced by the anount of backla,'=:h or "dead spot" in the control system and by the damping of the motion of the airplane. Tn the fol.Towing anLn.ysis the amplitude and frequency of these oscillation? is investi£.ated in terms of the response characteristics of the a i r'p lane. ANALYST^. The analysi?- is baped. on consideration of the response of the airplane (in ter^n? of angle of yaw or pitch) to a continued (unit) signal (fig. 1). This response may be calculated by the ordinary theorjr of dynamical stPbility and it will be convenient to repre- sent it in operational form (references 1 and 2): Rl(t) = Ri(D)l(t) (1) The uiiit response ordinarily occurs in the form from which is obtained ! Xnt \ot 1 Rl(t) = C(t) + i_Cie 1 + C^e "^ + , . . J (2) where C(t) i.^ tb.e .-steady-state motion, C^^ and Gg are the constant coefficients of the Heaviside expansion, and A.-,, X.,:,, etc., are the nonzero roots of the char- acteristic equation defining the natural period^ of oscillation and the damping; of the aircraft without signal. The function f(D) and the particular solu- tion C(t) depend on the time variation of control displacement produced by a signal and on the stability characteristics of the airplane in the degrees of freedom, in which the control operates. ('^ee refer- ence 3.) Tn the case of a continued signal, the usual form of the function C(t) is C(t) = C_T + C^t where C^ is the steady rate of turn called for by the signal. (See fig. 1.) During a hiinting oscillation the automatic steering device reverses the sir;nal periodically as. the airplane swings through the desired heading. A typical hunting oscillation Is shown in figure 2, Here it is assumed that the reversal of signal is delayed either because of a "dead spot" in the steering device or because of back- lash in the control mechanism or a combination of the two. As indicated, the oscillation will have a funda- mental period 2Tr/o.i -^^jt may also involve components of higher frequency, depending on the natural m^odes of oscillation of the airplane. Ordinarily the shorter- period com.ponents do not have sufficient am.plitude to cause a reversal of the signa] during a half cycle. In these cases the variation of signal with tim.e will be represented ty a sim.ple "square v/ave," v/hich m.ay be expressed as a fijjiction of time by I 2_ n ^-^ '''^^^'' = 1, 3, 5, ...) (3) n or, m.ore conveniently, by the imaginary part of the corresponding exponential series; that is. rr zl_ incot n n (4) where t = is taken to represent a tim.e at which the signal becomes positive. The response to the alternating signal is obtained by substituting expression (4) for the unit function l(t) in equation (1). Thus, R(t) - I.F. R;^(D)| 2_^ e^^'"^ (5) n If the airplane is inherently stable, so that transient effects following the start of an oscillation disappear with time, the rem.aining steady oscillation will be represented by R(t) = I. P. I ^ ^ ~R3_(lna))e^'^^"^ (6) n Equat'or (6) give? the forced oscillation of the airplane in rssponse to an elterratin^; signal in the forin of a square wave cf any fi'-equency 03. B^r investigatine; the forni of these forced oscilla- tions at vaiiou^ frequencies it vill he nosEible to ascertain "/hether such oscillations, under the condi- tions of automatic control, will give rise to the assuTi&d alternating signals of equal duration, and thus tc establish certain ranges cf go over which hunting of this type can occur. It will al?o be possible to establish, in these ranges, a correspondence betvceezi the frequency of the hunting oscillation and the magni- tude of the dead spot. Vvlth the frequency determined, it is possible also to find the anplitude of the oscil- lation and the maxiT.uin ceviation cf the airplane from its path. In the simplest cases the required inforn;ation inay be obtained directly from equation (6) . In the case of more cor.iplex motions, further analysis will be required as follov;s: As a first step, separate Rn(inw) . into its real and imaginary parts R2(ina}) = A(noo) + i 3(na)) The functions A and 3 may be plotted against vJj^ as in figure 3. These functions will show peaks near values of n-jj corresponding to the resonant frequencies of the airplane. Then, for aziy particular hunting frequency oo. ^'^^) - 4 ^ -Qi(na)) sin nwt + B(nco) cos ncot) (7) n At the time of reversal of the signal sin ncot = and cos nx't = ±1 accordingly as the signal is becoming positive or negative. The amplitude of the response at this instant is therefore 4R (oo) + -i B(3a) + lET 3(50)) + ..'.I This amplitude will alro be the ar:}plltude of the dead spot. {See fig. 2.) A plot cf Rp(u;.) = I ^ ^ B(nco) n odd can readiljr be obtained fror.i the E curve of figure 3 and v.'ill show the periods of the hunting oscillation corre- sponding to various width? of deac' spot. The slope of the response curve at this sane instant «'e = (iX = ^fr'-' + AfiCjL-) + A(5:o) + If the response to a positive signal is negative (as in fig. 1) , in order that the motion represent a possible hunting oscillation (that is, be consistent with the assumed variation of signal), it is necessary that (I) R3 I e for a positive dead spot, and that (II) H'g > indicating that the airplane crosses the dead spot in the proper direction. A further condition is that no TTiore than one coiiiplete crossing of the dead spot occurs v;ithin one -half cycle; that is. (III) R(t) > -Rg (See fig. 2.) The value of R(t) in the middle of a half cycle is relatively simple to obtain: Ra = |rA(^) - i A(3a:) + i A(5aO - ...1 e and "lay "03 usea as a criterion, though Ra is n' sarily tha x"axiiDiim or rrdniraur. value of R(t) (see fig. 4) and condxticn III Tt:ay not be satisfied '■e-^ It should be noted that, in the region.^ excl.uded b^*" the foregoing conditions, a r.ore complex type of hunting oscillation involving a sequence of signals of different durations may occur. In these regions, the R^ and R-, curves derived for the square-wave signal no longer apoly to the condition of automatic control. These oscillations require analysis beyond that presented in this report. EXA'ilPLES In order to demonstrate and check the procedure described, assume a simple response characteristic in T.'hich the aim lane ii-imediately starts turning at a const&nt rate, as directed by the signal. VMth this resnonse Rl(D) = - Co ■" / ^ Cn . Rt (na-.i) = 1 nco and, from equation (7), R(t) = - / ~- cos nojt '■ n^w n ^vhich is the Fourier series for a saw-tooth -.vp.ve 9^^ out of phase with the signal. (See fig. 5. ) In this case the response occurs ?;ithout lag and the amplitude of the hunting is exactly equal to the dead spot. The frequency o) -s ^^0^/2 divided by the width of the dead spot. A simple exairple nearer the pr^'ctlcal case is one in v/hlch the signal causes a force F to act on a mass Ki. In this case the resoonse to a u::l t signal is ~' / -.^ \ PI -«- * ' xn 7)2 and. the hunting oscillstion is seen to be R(t) = I - ,-> —J=— Sin nrot ^r m ^rl ^5 2 n i i u.' The expression is rfcc')^riz3d. as the Fo\ir5er series for a succsssion of oarabolic se^'^'^ei'-ts (flgt 6). It should b3 noted that there is no component out of phase with the signal, Y:ith the result that R-g is zero for all values of (0, Hence the cal-riu] ation sho^s no possibility of h-jjiting v,'ith a finite dead soot. In fact, it can be seen fror.: energy consicer<^tlons that if a dead spot existed the cscillation y;ould be divergent. Interesting apolicstion? o? the method are furnished by cases in which the response to a signal shows a lai^ J A TIM^ ^ •& \ y^ R£SfiO^£:, /^f(^^ \ \ ^ Q \ ^ 5 ^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS /^^aAfS £ TO COA^r/AsfUSJ^ S/OA^AL • •• • • t • 1 -I ■4 £ I I I NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS -PoSS/&^£ MO/^r/AjG Am^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS SHOU//AJG y/^ait/O^ A7£i/^^G//iAJIS/A^ Ui/N/CH /i^ur^r/f^S a^/iLAr/a/ ^s /^R^ pgss/s^^ Ano Wiprn i^^ D^AO SPOT //V THC5£ /jeGiQAJ S Kt) -». Jt-*- ^Z^V-'^^. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS /=lSU/i£ S^ I I I ^ I I I j.A^^ >v^:>^7asfC7 Bcosncot) D /7S(y^£r 7- — NATIONAL ADVISORY CUMMiriEE FOR AERONAUTICS i I UNIVERSITY OF FLORIDA 3 1262 08104 955 2 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 90 MARSTON SCIENCE LIBRARY '.O. BOX 117011 ,^,,.,oA GAINESVILLE, FL 32611-7011 USA •v