A method for studying the hunting oscillations of an airplane with a simple type of automatic control

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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


WAlRTIMEll RE PORT


ORIGINALLY ISSUED
May 1944 as
Memorandum Report

A METIOD FOR STUDYING THE HUNTING OSCILLATIONS OF AN
AIRPLANE WITH A SIMPLE TYPE OF AUTOMATIC CONTROL
By Robert T. Jones


Langley Memorial Aeronautical
Langley Field, Va.


Laboratory


NACA


WASHINGTON

NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of
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icdally edited. All have been reproduced without change In-order to expedite general distribution.
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NATTO'AI ADVISORY COMi?';ITT=,O FO AR'O-'AT:IC


F'- RAN DT V' RiP'.FT

f)r the

Arrry .ir Forces, "atrfel C--mr-and

A 'THOPL PO'R STUDY IG T :- FHUTIITG CSCITLATICT~I F 1F

AIRFLA:rl WTTH A T P'.PT.2 TYPE CF AU'TCi.ATIC COC'TROL

3y hobert T. .n-.ne


? U:. '.PY


A method is presented fr.r nrndict;rn the amrplitude
and frequency, under certain sinr.lf.i '..r cor.iti.nrs, of
the huiltina oscillatior.s of an auiLor:a call: ; cznrLrll--.d
aircraft wi't lagI in t!e control ystb orr ir the
rer,onsc of the aircraft to th-. control. f the steer-
lr. device -' actuated by a slimrle ri.,:ht-lft typc of
slnal, the series of clterintingt fixhd-a-.l.itude signals
occurring' during th.e huintln; may .ordinrilY beE re~.prss--nted
by a qi.iare wave. Fo ir.-u]a are giv En c ypr -rinrr the
response to) uch a var'atr1n of signal n'r terms of the
r.-epo:2or to c unit PFlPral. A 'noire complex t-pe of hunt-
inE, which ray involve c'o:.lc reetitior.n of insia ls of
vPryin7r duration, lhas nrit been treated and requiress
further analy-is. SeP-eral exanmples )f application of
the method are included and the. result diScussed.


INT ROD-'CT IC,'


When an airplane or other aircraft is directed by
a simple right-left signal from ar.- aitor&atlc steering
device, the result is u-uallyr a -naintAi.ned huntrinr oscil-
Ip.tion about the Kc'esired path. The a;4'.r! tude of this
o:cillr.tion is influenced by the ar.ount of 1.sckl-i t jr
"dead zpot" in the control system and by th- c'am!'ir.g of
tlh motion of the air-.l.ane. In t!. ['Ulcwir.r an-:.ysi.
the anplitudp and frequency of the.'e orcillations is
investigated in terms if the response cr:arqcteristicc
of the airplane.












A'T' T S IS


The analysis is based on consideration of the
response of the airplane (in terns of angle of :.'rw or
pitch) to a continued (unit) signal (fig. 1). This
response may be calculated by the ordinary theory of
dynamical stability and it will be convenient to repre-
sent it in operational form (references 1 and 2):


Rl(t) = il(D)l(t) (1)


The unit response ordinarily occurs in the form


f(D)
R1 (D) =


from which is obtained

I l t 2t
Rl(t) = C(t) + Cle + C2e + ... (2)


where C(t) is the steady-state motion, C1 and C2
are the constant coefficients of the Heaviside expans1,7n,
and l,' X2, etc., are the nonzero roots of the char-
acteristic equation defining the natural period- of
oscillation and the dampinc of the aircraft without
signal. The function f(D) and the particular solu-
tion C(t) depend on the time variation of cortrol
displacement produced by a signal and on the -tabllity
characteristics of the airplane in the derre.i-s of
frPeedo' in which the control operates. ("er refer-
ence 3.) In the case of a continued signal, the usual
form of the function C(t) is


C(t) = C- + Cot

where Co is the steady rate of turn called f~r by the
signal. (See fig. 1.)












During a hmintinr oFrci 1ati n the automatic steering
device reverses the signal neriodicallv as the airplane
swings through the de'olrede hep.din-'. t y-irc 1 hunr.tinr
oscillation is shcwn in figure 2. .-ere it .s as -umrre
that the reversal of signal is delayed either because of
a "dead spot" in the steering device or becauT-e of back-
lash in the control mechanism or a co.mbination if the
two. As indicated, the oscillation will hav.'e a funds-
mental period 2n/L' but may also involve components of
higher frequency, depending on the natural r:odes of
oscillation of the airplane. Ordinarily the shorter-
7priod components do nrt have sufficient aTplitud:- to
cause a reversal of t-ie siar.a] d.urinP a half cycle. In
these cases the variation of signal with time will be
represented by a Fimrplc "square wave," which may be
expressed as a function of time by


o sin n t(n = 1, 3,, ,, ) (3)
n

or, more conveniently, by the ima,-nar:; part of the
corresponding exponential series; that Is,

I. 4 1 init (
T.P. ._ (A)
n


where t = 0 is taken to ret:resent a time at which the
si!nal beco-es positive.

The response to the alternating signal is obtained
by substituting expresFion (4) for the unit function
l(t) in equation (1). Thus,


R(t) = I.P. 1(D) ,) e at (5)
n

If the airplane is inherently stable, s- that transient
effects following the start of an oscillation di-appear
with time, the rerairinr- steady zscillation will be
represented by
4 \ 1 Tli ei nwx. t
R(t) = I.P. I (inr)eet (6)
n












Equation (6) 1ives the forced oscillation of the
airplane in response to an alternating signal in the form
of a square wave of any frequency w.

>y invest'rating the form of these forced oscilla-
tions at various frequencies it will be possible to
ascertain whether such oscillations, under the condi-
tions of autcmatic control, will give rise to the
assumed alternating s"'.nals of equal duration, and thus
to establish certain ranges of w over which hunting
of this type can occur. It will also be possible to
establish, in these ranges, a correspo:.de;vce between
the frequency of the hunting oscillation and the magni-
tude of the dead spot. With tbn frequency determined,
it is possible also to find the amplitude of the oscil-
lation and the max-munm deviation of the airplane from
its path.

In the simplest cases the required information may
be obtained directly fro7 equation (6). In the case of
more complex motions, further analysis will be required
as follows:

As a first step, separate lR(inw) into its real
and imaginary parts


lI(inw) = A(nw) + i B(nw)


The functions A and 3 may be plotted rsairL-t n) as in
figure 3. T7.ese functions "'ill show peaks near values
of no corresponding to the resonant frequencies of the
airplane. Then, for any particular hunting frequency w,


R(t) T n I(nu) sin n~t + B(ne) cos nwt] (7)
n

At the time of reversal of the ei.nal sin not = 0
and cos not = 1 accordic.ly as the s al is becoming
positive or negative. The a-!Ilitude of the response at
this instant is therefore


4rB(w) + B(3L,) + B( 5) + ..








5


This amir.litude will alac e ta: e e rr.rtitud. ?f the 4,ead
spot. (See fig. C.) / rlt vf


B = ~ ~E(n)
n odd


carn readily be obtainedd fro., the P cur,-e 3f f i'ur-e rc
will show the periods of the hir'Linr c'cillptlti c-rre-
sponding to vari')u width of dead spot.

The slope of the rer"oonse cu':.rve at thi's e a.e I:-stant
-q


-' = ( ) = L-) + A(i ) + .(5 ) + I.
d t r


If the response to a positive signal is necative (as i'..
fig. 1 in order that the rotioc. rerore-ent a p.Derible
hunting osc'llat' :n ('hat is, be con-i tent with the
a _umed variation .?f signal), it i? r.eces.sar:," that


(T) R 6

for a pc.-.tive lead sp.o't, ar.nd that


(II) R' > 0


indicating that the alrplan~ cr:.sres the dead -P.t in
the proper direction. A further- condition s that no
more than one cor-plete crossing of the dead scot occur'
within one-half cycle; that is,


(IT) R(t) > -

(See f'g. 2.) The value of R(t) in the middle of a
half cycle is relatively simple to obtain

R = (A) A 4(w) + A(5) ...











and say be usec as a criterion, though RlA is not neces-
sarily the :axrr'i or 1.int:.umr value of R-t) (see fi`. 4)
and crn."'-tion ITT .: not be satisfied even th'.i4h
RA >-EB'

It should be noted that, in the regions excluded bi
the foregoing conditions, a more complex t-y'e of tl..iting
osill:.tion involving a sequence of signals of different
duratins may occur. In these regions, the RA and RB
curves derived for the square-wave signal no loier aply
to the condition of automatic control. These oscillations
require 3:=..lysis beyond that presented in this report.





Tn order to demonstrate and check the procedure
described, assume a simple response characteristic in
whnih the airDlane i-mediatcly starts turning at a
constant rate, as directed by the signal. With this
response

Co
1l(D) = -o

Co
RI(nri) i
no

and, from equation (7),
4 \, 60
R(t) = oos not
n

which is the Fourier series for a saw-tooth -:ave C"
out of phase with the si.nl. (See fig. 5.) In this
case the response occurs without I-l r-nd the arrrlitude
of tie hunting is exactly ei:il to the dead spot. The
frequency e is TWC/2 divided by the width of the dead
spot.

A simple exanrle re'-rer the oractical case is one
in which the si.:n-l causes a force F to act on a
rss r. In this case the response to a unit g.inal is















and th.s hunt ln. "osillstion its E.-r. to .e


R(t) = 1 n not
n n njw2

Th r exr'ession 1 r ec.r!z-d s e ur'-r zrrie-. fr a
sv.ccesaio on f o '.'r.ol:. se '"-i: I 6). It s'hov.ul b
note- :Th'a Lthere i' no co -.oner-.t c.'t of -h..are wita- thl
si' nal, i.th the rezslt that is "ar-: for all val.ea:
of wo. .3nce the calu)slatlin sh.-"s no ,o-s'.bilit.;, of
hu-nt:-. t:ith a finite deadc s-:.t. Ir. fact, ".t c;an be s.E-n
f'r.Ocr.: er. consli'er..t:' ors that if 1 dead soot ex:.- ted
the .: -c-lat on vnu-.il5 be div reenL.

:tterest' n r. ar ':o1 etinF o :h-s method s .re. furnis:.ec.
b-r cqsg"r in vhich the response to a sijnel shIov.s a la.,
nos~l i" d,'-? to backlash in tha control mechanis,;, in
ac0.i tiT to s de-. sP-ot. A siT.:.l5 ex ;- le of thi. '-ind
S-111 istr3te3d In fi. ure 7. -er:- the r son"c iS lFr
to that in t.he first e:;ar:.l (fi.. 5) exce-)t for th-
t!' ':e 1ig T. 'Usz s '-ade 01 tlh- -. l-ofnov.: n I. o ',erst-.. r

e-TD1. T1us,
-TD
e-T f(t) = f(t T)


A'r-"lyn. thio ooejitor to the res-ons-: in figure 5 one
obtr.in-

R1(D) = -e_
D

R(in co) = (sin r.n: + 5 -o rTr) =A + iB
no.'
and, f nt-al (eq-jion 0 17)),


'- -- n

4 '- o o
r= -o- cos nw(t T)
n nDc












~!ith the 1l~.-t. n2 -esponse, the hunting oscillation
is not confined tc the amplitude of the dead spot and,
in fact, hurting v1ll occur with no dead spot. It is
easily seen by reference to figure 7 that the half
period of the oscillation in this case (no dead spot)
is

7/0 = 2 T



Langley Memorial Aeronautical Laboratory,
national l Advisory Co:mrr'.ttee for Aeronautics,
Langley Field, Va., May 5, 1944.








9





1. Jeffre-,, "-aro..d- Operational cthi- fe i; "ather-at leal
Physcs. Cambrid g Tracts ir. *'.athrE-atic and
"athe-atical Fhysics, I:c. 23, 2d Ed., Car-'ridge
TniT v. Fres-, 1921.

2. JneP., Robcrt T.: A ISimnlife Ann'licat'o:r of the
:!ethod of ?'eratorP to the Calc1'ulati of D'r-
turbed l"ctionr of pn Airplane. 'CA \P. :o.. 530,
193C.

3. Jones, RBbert T. Ca;lulcu?0Ton of the ".ot.ln rf an
A. rplane under the 'n luencc of Trrepular Dir-
turbance Jour. Aero. Sol., v-1. 3., .o. 12,
Oct. 1923, pn. 419-42'.


























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