A method is presented for predicting the amplitude and frequency, under certain simplifying conditions, of the hunting oscillations of an automatically controlled aircraft with lag in the control system or in the response of the aircraft to the controls. If the steering device is actuated by a simple right-left type of signal, the series of alternating fixed-amplified signals occurring during the hunting may ordinarily be represented by a "square wave." Formulas are given expressing the response to such a variations of signal in terms of the response to a unit signal. A more complex type of hunting, which may involve cyclic repetition of signals of varying duration, has not been treated and requires further analysis. Several examples of application of the method are included and the results discussed

Jones, Robert T

English

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REPORT No. 801
A METHOD FOR STUDYING THE HUNTING OSCILLATIONS OF AN AIRPLANE WITH A SIMPLE
TYPE OF AUTOMATIC CONTROL
By ROBERT T. JoNDs
SUMMARY
A method is prewnted for predicting the amplitude and
jrequenq, under certuin timplijying conditti, of the hunting
08dkdhn.s of an auto?nuikdy contro.hd aircrafi with kg in
thecontrol~8tem or in thereaponaeoj the aii-era~to the controk.
Ij the 8teering device ix actuaki by a simple right-lejt type oj
8i9dj th-e8L%23 Oj al.ternaiing~-ampliiu& 8ignak Oeeurring
dwri~ the hunting may ordinarily be repreatmiedby a “square
wave,)‘ .Wmuka are gkn expraming the r~pon8e to such a
rariuti-onoj aig?udin term8 Q{ the reqwnae to a unit m“gnd.
A mare compka type oj hunting, which may involve ycli.c
repetition oj ttigna.kof varying duration, hm not been treated
and requiresjw-ther analysis. S’everalexamplm oj appli.eaiion
of the methodare included and the remdt8di.8cu$8ed.
INTRODUC’ITON
W1)CIIan airplane or otlmr aircraft is directed by a simple
right-loft signal from an automatic steering device, the
result is usually rLmaintained hunting oscillation about the
Wsircd path. The amplitude of this oscillation is influenced
by the amount of backlash or “dead spot” in the control
system rmd by the damping of the motion of the airplane.
In the following tnalysis the amplitude and frequency of
these oscillations is investigated in terms of the response
chmacteristic9 of the nirplane.
ANALYSIS
The analysis is based on consideration of the response of
the airplane (in terms of angle of yaw or pitch) to a eontinucd
(unit) signal (fig. 1). This response may be cxdculatcd by
tlm ordinary theory of dynamical stability and is conveni-
ently represented in operational form (references 1 and 2) as
follows:
R,(t) =E,(D) 1(t) (1)
The unit responso ordinarily occurs in the form
j(D)
~l(D) =F~
from which is obtained
R,(t)=C(t)+ (cl@’+c,@’+ . . .) (~)
where C(t) is the steady-stato motion, G and (?2 are the
constant coefficients of the Heaviside expansion, and xI, Az,
rmd so forth, are the nonzero roots of the characteristic
equation defi~~ the natural periods of oscillation and the
damping of the aircraft without signal. The function j(n)
and the particular solution 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. (See reference 3.)
In the ease of a continued signal, the usual form of the func-
tion C(t) is
c(t)=c-,+c#
where Co is the steady rate of turn called for by the sig+tl
and C-, is a constant. (See fig. 1.)
,..-Si@al
o
I h----e’4’”
..-sieady Sfofe, c(t)= c++ cot
, I
FIGUREl.—TyPIcaIIWPOnSe tomntfmal s@al.
During a hunting oscillation, the automatic steering device
reverses the signal periodically as the airpkme 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 beeause of a “dead spot” in the steering
devioe or because of backlash in the contiol meohanism or a
combination of the two. As indicated, the oscillation will
have a fundamental period 27r/aI (where u is the angular
frequency of the hunting oscillation) but may also involve
487
https://ntrs.nasa.gov/search.jsp?R=19930091878 2020-06-17T02:30:40+00:00Z
REPORTNO. 80 l—NATIONAL ADVISORYCOMMITTEEFOR
components of higher frequency, depending on the natural
modes of oscillation of the airpianc. Ordinarily, the shorter-
period components do not have su.fiicient amplitude to cause
a revemal of the signal during a half cycle. In these cases
the variation of signal with time will be repremnted by a
simple “square wave,” which may be expressed as a function
of time by ‘
~ ~~ sin nut (n=l, 3, 6, . . .) . (3)
or, more convenieutiy, by the imagimq part of the cor-
responding exponential series; that is,
where t= O is taken
becomes positive.
I. P. ;Q:e- (4)
to represent a time at w-hich the signal
-..
...%
-..
/%use sh”ti
I
I
.— . -------- —--- —-.——— ~__---!
Deed apt
FtauBE2–Hundngcuotlldonwith%qwre”signal.
The rwponse to the alternatk~ sknal is obtained
substitntk-g expression (4) for &-e &it function 1(t) &
equation (l). Thus,
lf the airplane is inherently stable, so that transient effects
following the start of rtn oscillation disappear with time, the
remaining steady oscillation will be represented by
Equation (6) gives the forced oscillation of the airplane in
response to an alternating signal in the form of a square wave
of any frequency u.
By investigating the form of these forced oscillations at
various frequencies it will be possible to ascertain whether
such oscillations, under the conditions of automatic conirol,
will give rise to the assumed alternating signals of equal
duration, and thus to establish certain ranges of ~ over which
hunting of this type can occur. It will also be possible to
establish, in these ranges, a correspondence between the fre-
quency of the hunting oscillation rtml the magnitude of the
dead spot. With
AERONAU’ITCS
the frequency determined, it is possible
also to find the amplitude of the oscillation and the maximum
deviation of the airplane from its path.
In the simplest cases the required information may be
obtained directly from equation (6). In the case of more
complex motions, further analysis will be required as follows:
& a first step, separate Zl(ina) into its real and ima@iary
parts
Z,(inti) =A(nu) +iB(mo)
The functions A and B may be plotted against nw as in
fieg.u-e3. These functions will show peaks near values of nu
A
und
.B
.- Resmcmt-..
I
[.-frequencies ‘% 1
‘.s
1
I
I B
I
o
P
I
FIWRE 3.-CnrvMshowbgtn-pkesoandoat.d-phnsecompxmntaofresw.IMto
pei-hdIaslgnnl.
corresponding to the resommt frequencies of the airplane.
Then, for any particular hunting frequency u,
At the time of reversal of the signal sin nut= O and
cos nut=+ 1, the sign depending on whether the signal is
becoming positive or negative. The amplitude of the response
at this instant is therefore
[ 1+; M4+;B(34++B(64+. . . ,
This amplitude w-ill also be we amplitude of the dead spot.
(See @g. 2.) A plot of
(n=l, 3, 5, , . .)
can readily be obtained from the curve of B in figure 3 and
w-ill show the periods of the hunting oscillation corresponding
to various widths of dead spot.
The slope of the re9pon,9e curve at this same instant is
()RIB= a~ ~=~[A(a) +A(3w)+A(5w)+ . . .]
A METHOD FOR STUDYING THE HUNTING OSCULATIONS OF MJ AJRPJ..ANE 489
If
in
the response to a positive signal is negative (as in fig. 1),
order that the motion represent a possible hunting oscilb
tion (that is, be consistent with
signal), it is necexmry that
(I) R.k O
for a positive dead spot, and that
(11) R’,>0
the assumed variation of
mdimting that the airplane crosses the dead spot in the
proper direction. A further condition is that no more than
one complete crossing of the dead spot occurs within one-
half cycle; that is,
(III) R(t) > –R.
(See fig. 2.) The value of R(t) in the middle of a half cycle
is relatively simple to obtain \{,
&R.=4 A(u) –;A(3co) +;A(5u) – . . .1
rind may be used as a criterion, though R~ is not necessarily
, the maximum or minimum value of R(t) (see @. 4) and
condition III may not be satisfied even though R~> —R~.
II
I
“(’’++=3=
IHdf-wcfe of hun+lng motion
PIGIJRE 4.–PlotofRA andRB againstfraqnanoy,ahowbg appradmab @“ens inwhich
huntjngasclllatIorMampccsihleandwidthofdeadspotInthawregfons.
, It should be noted that, in the regions exclu~ed by the
foregoing conditions, a more complex type of hunting
oscillation involving a sequence of signals of diflmmt dura-
tions may occur. In these regions, the curves of l?. and
RD derived for the square-wave signal no longer apply to
the condition of automatic control. These oscill@ions re-
quire analysis beyond that presented in this report.
EXAMPLES
In order to demonstrate and check the procedure dem.ribed,
asume a simple response characteristic in which the air-
plane immediately starts turning at a constant rate, as
directed by the signal. With this response
fil(nui~ =.: i
and, from equation (7),
which is the Fourier series for a %nv-tooth” wa,ve 90° out
of phase with the zignal. (See @. 5.) In this case the
response occurs without lag and the amplitude of the hunt-
ing is exactly equal to the dead spot. The frequency u
is d?O/2 divided by the width of the dead spot.
I tl n n n n
\
7!
-+ h
AC(H-mii~ oscillufiqR(t)=fl~nru cos ruwt
s.—_ ———— _.— —_ —
–_.––___y__v_ 3L_v__ v \
Deed spot
\
.,.-U* reqxm.ge:
Rl(t)=-ct,..
‘“ iftluv.-~
FIGU8E 6.-k3Dlph illWhiCh ~UE4 b hStWltQ103R&
A simple example nearer the practical case is one in, which
the sigmd causes a force F to act on a mass m. b this casa
the response to a unit signal is
R,(D)=–;*
and the hunting oscillation is seen to be
The expression is recognized as the l?ourier series fm a
succession of parabolic segments (&g. 6). It should be
noted that there is no component out of phase with the
signal, with the result that RD is zero for all values of a.
Hence the calculation shows no possibility of hunting with
a finite dead spot. In fact, it can be seen from energy
considerations that if a dead spot existed the oscillation
would be divergent.
. .
490 REPORTNO. 801—NATIONAL ADVISORYCOMMITIXD FORAERONAUTICS
F-=-
..-
YWY7 respmse:
\
FIGURE6.-HontimzlUnii@tinwsnckionbYforca
Interesting applications of the method are furnished by
cases in which the response to a signal shows a lag, possibly
due to backlash in th~ control me-~anism, in addition to a
dead spot. A simple example of this kind is illustrated in
figure 7. Here the response is similar to that in the first
~~ample (fig. 5) except for the time lag ~. Use is made of
the well-hcown lag operakw e=~. Thus,
#’f(t) =y(&–7)
Applying this operator to the response in figure 5 gives
R, (D)= —c+ ~
E,(im)=$ (sin moT+i cosnd=-A+~B
and, finally (equation (7)),
R(t) =; ~$ (fi WT ~ n~~+cos ‘nAOTcos ~t)
=: ~g CosmO(t-T)
With the lagging rcspo~e, the ~~t~g oscillationis not
confhed to the amplitude of the dead spot and, in fact,
hunting will occur with no dead spot. It is easily seen by
,
\
A
Od
B
- ------
o -.—— —-—
nu ——
~/
/
//
Fmom 7.—EmmpIe in whlcbmponW showIns.
figure 7 that the half period of tho oscillationreference to
in this cnse (no dead spot) is
LANGLEY IMEMOFUAL AERONAUTICAL LABORATORY,
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS,
LANGLEY I?IELD,VA., Afay 6, 194-4.
REFERENCES
1. J&roye, Harold: Operational Methods in Mathematical I?hysies.
Cambridge Traota in Mathematim and Mathematioul Physics,
No. 23. Second cd., Carobridge Univ. Press, 1931.
2. Jones, Robert T.: A Simplified Application of the Method of
Operators to the Calculation of Distarbed Motiom of an Airphmo.
A’ACA ROp. hTO. 560, 1936.
3. Jones, Robert T.: Calculation of the Motion of an Airplono under
the Influence of Irregular Disturbances. Jour. Aero. %1.,
vol. 3, no. 12, Oct. 1936, pp. 419-425.
.


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

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