Prediction of motions of an airplane resulting from abrupt movement of lateral or directional controls

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Material Information

Title:
Prediction of motions of an airplane resulting from abrupt movement of lateral or directional controls
Alternate Title:
NACA wartime reports
Physical Description:
20, 6 p. : ill. ; 28 cm.
Language:
English
Creator:
Wolowicz, Chester H
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Ailerons   ( lcsh )
Yawing (Aerodynamics)   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: A procedure is presented for determining the motions of an airplane resulting from the deflection of the lateral or directional controls for the case of non-linear derivatives. The step-by-step integration on which the procedure is based considers the rolling, the yawing, and the lateral accelerations computed from wind-tunnel data as functions of the sideslip angle. A sample computation table is presented to illustrate the application of the procedure. A comparison is made of different methods for calculating the disturbed motions of an airplane resulting from an abrupt aileron movement. Experimental data, which were obtained from conventional wind-tunnel tests of a model of a recent fighter airplane, are used in the computations for comparing the various methods. The resulting solutions show that, for the case of nonlinear derivatives, the calculated motions are in better agreement with the results obtained from flight tests if the rolling and yawing accelerations computed from static-model tests are condisered as functions of the sideslip angle. The lateral acceleration, which is often assumed to be negligible, should be considered. The variation of the rolling and yawing accelerations resulting from aileron movement probably should also be considered when sufficient data are available.
Bibliography:
Includes bibliographic references (p. 19).
Statement of Responsibility:
Chester H. Wolowicz.
General Note:
"Report no. L-125."
General Note:
"Originally issued May 1945 as Advance Restricted Report L5E02."
General Note:
"Report date May 1945."
General Note:
"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 previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003613074
oclc - 71200896
System ID:
AA00009390:00001


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I(NACA


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WARTIME REPORT
ORIGINALLY ISSUED
May 1945 as
Advance Restricted Report L5E02

PREDICTION OF MOTIONS OF AN AIRPLANE RESULTING FROM
ABRUPT MOVNEMIT OF LATERAL OR DIRECTIONAL COiNROLS
By Chester H. Wolowicz

Langley Memrial Aeronautical Labaratory
Langley Field, Va.








-. .. :." -

N AC) r


..: : .
" ; ,.


.... t .:


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.


L 125


DOCUMENTS DEPARTMENT


L- o. L502
ARR No. L5Z02


J







































Digilized by Ihe Iniernel Archive
in 2011 WillI lunding from
University ol Florida, George A. Snmahers Libraries wiIll sLpport front LYRASIS and Ihe Sloan Foundalion


hlp: www.archive.org details predictionofmnoliO1ang





NACA ARR No. L5E02 2 '

iJATION4L ADVISORY Co TIiITTEE FOR ALEP;TAUTICS


ADVANCE RESTRICTED REPORT

PREDICTION OF O'ciO.Tza OP AiN AlPLATTE REZULTI;tUG FROI .

ABRUPT I!OVEi2.E;T OF I.AT'TRAL OR DIR:CTIO'IAL CONTROLS

By Chester H. 'iolowicz

SU;: AaY

A procedure is presented for determining the motions
of an airplane resulting from the deflection of the
lateral or directional controls for the case of non-
linear derivatives. The step-by-step integration on
which the procedure is based Considers the rolling, the
yawing, and the lateral accelerations computed from wind-
tunnel data as functions of the sideslip angle. A sample
computation table is presented to illustrate the appli-
cation of the procedure.
A comparison is made of different methods for calcu-
lating the disturbed .tiolns of an airplane resulting
from an abrupt aileron movement. Experimental data, which
were obtained front convent z-al wind-tunnel tests of a
model of a recent fighter airplane, are used in the co;n-
putations for comparing the various methods.
The resulting solutions show that, for the case of
nonlinear derivatives, the calculated motions are in
better agreement with the results obtained from flight
tests if the rolling and yawing accelerations computed
from static-model tests are considered as functions of
the sideslip angle. The lateral acceleration, which is
often -essumied to be negligible, should be considered.
The variation of the rolling and yawing acctlerations
resulting from ailuron movement probably should also be
considered when sufficient data are available. The
variation of the dynamic derivatives Lp, N Lr, and Nr
should also be taken into account when sufficient dynamic-
test data are available.
It is shown that the present step-by-step integration
method is reliable for cases in which only the first
quarter-cycle of the motion is required (for example, in
cases in which the maximum value of the sideslip angle is
desired for determining vurtical-tail loads in rolling
pull-outs). For the range past the first quarter-cycle
of the motion curve, the method requires, further refinements







2 NACA ARR Noe L5E02

such as those provided by the Runge-Kutta su mation
method. The present step-by-step integration method may
be applied to the solution of motions produced by rudder
movements or by a combination of rudder and aileron
movement, as well as to the solution of motions produced
by ailerons alone.

INTRODUCTION

The increasing importance of predicting the flying
qualities and maneuverability of an airplane has empha-
sized the need for a more accurate method of computing
the lateral motion resulting from abrupt control movement.
Increased speed and maneuverability have, in addition,
made it necessary to predict the maximum sideslip angles
in lateral-control maneuvers in order that maximum
vertical-tail loads may be estimated.
Much work has been done on the subject of disturbed
motions (references 1 to 5), but all the solutions deal
with constant lateral-stability derivatives. These
treatments assume that the rolling-moment coefficient CL
and the yawing-moment coefficient CO are linear func-
tions of the sideslip angle P, the rolling velocity p,
and the yawing velocity r. Wind-tunnel tests, however,
indicate that most present-day airplanes do not possess
these linear variations of C, and Cn with p, since
the degree of linearity is affected by such factors as
the geometry of the airplane, the power, the type of pro-
peller, and the blade angle.
Lack of mathematical equations for expressing the
derivatives as functions of the motions makes the method
of references 2 and 4 inapplicable. The procedure for
the solution with nonlinear characteristics presented
herein is a refinement and an expansion of the integration
procedure of reference 1.
with the wind-tunnel data available at the present
time, only the linear and angular accelerations PYp,
PLp, PN 6L6, and 6N6 may be determined as functions
of the sideslip angle p. Lack of model-test data for
effects of the rate of roll p and the rate of yaw r
still makes it necessary to deal with the dynamic deriva-
tives Lp, L1, and Np determined from theoretical







:ACA ARF Po. 1,502


treatments (reference 3). The d-na ,.ic derivative Nr
is determined partly -'rorn wind-tunnel data and partly
fro.-; theoretical considerations (references 6 and 7).

In the present report three previously established
procedures, based upon constant derivatives, for deter-
.ninin; the disturbed motions of an airplane that result
from abrupt aileron movemerit are co.;:rared with a step-by-
step integration procedure that considers accelerations,
computed from wind-tunnel data, as functions of the side-
slip Cnrle p. This step-by-step intrt ration not only
generally provides more accurate solutions for disturbed
motions but also should prove usF.ful in determining the
vertical-tail loads resulting from rolling pull-out
maneuvers as discussed in reference 5.

Unrublihed experimental data (fig. 1) obtained from
conventional wind-tunnel tests of a nmoael of a recent
fighter airplane are used in calculating the motions, and
the results are compared with flight results.


CCCFFICIENTS AND S-TOLS


The coefficients and symbols used herein are referred
to a system of axes in which the Z-axis is in the plane
of syTuletrcy and perpendicular to the relative air stream,
the X-axis is in the -,lane of symnnetry and perpendicular
to the Z-axis, and the Y-axis is perpendicular to the
plane of sym,-!metry. The coCfLicients and symbols are
defined as follows:

CL airplane lift coefficient (Lift
qS )
CL, lift coefficient of -;in-

ACLf increment of lift coeffi-L..ent resulting from
flap deflection

Cdw ;:rofile-drEg coefficient of wing

ACdo increment of orofil.e-drap coefficient caused by
flap deflection

,l rolling-moment coefficient (Rollin.oment-)

n yawingo-moment coefficient (Yawing moment









NACA ARR No. L5EO2


C0 rolling-moment coefficient caused by aileron
deflection

Cn yawing-moment coefficient caused by aileron
a deflection

Cy lateral-force coefficient Lateral force

b wing span, feet

bf flap span, feet

X taper ratio; ratio of tip chord to root chord

A aspect ratio

b distance from center of gravity to rudder hinge
line, feet

6a aileron deflection, derees; used with subscripts
L and R to refer to left and right ailerons,
respectively

6f flap deflection, degrees

6r rudder deflection, degrees

av angle of attack of vertical tail, degrees

aa absolute angle of attack of wing treasured from
zero-lift line, degrees

S angle of yaw, degrees

(3 sideslip angle, radians except as otherwise
indicated; considered in static wind-tunnel tests
to be equal to -t

Cn6 rate of change of yawing-moment coefficient with
r /C
rudder deflection (6Cn
66r/

6r inverse of rudder effectiveness parameter at

constant lift I 3
Dvj







IIACA iR: No. 1,502 5

7p rate of change of rolling-moment coefficient
with wing-tip helix anfil; -1-I-

Cn rate of change of yawing-mnoment coefficient with
wing-tip helix anglo -

C7, rate of change of rolline-moment coefficient
with rb 1
212
OC, rate of change of yawin--nmoment coefficient
with b /-\
2V rb

Cn, rate of change of yawing-moment coefficient with
angle of yaw \-, )
Lp rate of change of rolling acceleration with rate
of roll b qSb \


r 2V rk 2
Lr rate of change of rollin' acceleration with rate


ofr~ b q Sb
of yaw 'r 2Vr



!p rate of change of yawing acceleration with rate
of roll /Cnr b )
S rate of chan.-e of yawing acceleration with rate


of aw n kz2









MACA ARR Io. L5E02


6L5 rolling acceleration caused by control deflection,
/(C- q.Sb\
radians per second per second ---

(1n1.bFcriots a and r indicate aileron and
rudder, respectively.)

6F16 yawin; acceleration caused :)iT control deflection,
/Cnqqb\
radian? per scconu no-r second 2--

(Subscripts a and r indicate aileron and
rudder, respectively.)

PLP rolling acceleration resulting from sideslip angle,

radians per second per second 7,l --

pDTN yawing acceleration rcisulting from sideslip angle,

radilans per second iJpr second n b\


pY siQdeslippin:3 acceleration resulting from sideslip

angle, fIet per second :-er second (Cyv-)

dp
d- rolling angular acceleration, radians per second
dt per second

di yawing angular acceleration, radians per second
dt per second

do sideslipping velocity, radians per second
dt
dv
-- sideslipping acceleration, feet per second per
dt second

ZLn net induced rolling accelerations at t = n

SNn net induced yawing accelerations at t = n









JACA A": I:o. 15n02 7


p rolling velocity, radians per second except as
otherwise indicated

r yawing velocity, radians per second except as
otherwise indicated

/ angle of roll, radians except as otherwise
indicated

P air density, sl-us per cubic feet

V velocity along X-axis, feet Ie- second

v sideslioping component of velocity, feet per
second

q dynamic pressure, pounds per square foot (V2)

S wing Crea, square feet

r mass of airplane, slugs

kX ralius of yration about X-xYis, feet

k, radius of gyration about Z-axis, feet

t time, seconds

g gravitational acceleration (32.2 ft/sec2)

Ko, ':f, 1, K2, K3 constants used in determinining U

The subscripts n and n 1 denote values corresponding
to the time t and to the immediately preceding time
t At, respectively.


PROCEDU.TE FOR C,'.DT'TITM LATERAL MOTIONS

BY STEP-BY-STEP INTEGRATION

All the procedures considered for determination of
disturbed motions are based upon the following vell-known
dynamic lateral-motion equations.for level flight:

d (
d = 6L + pLo + rL. + FpT (1)









FACA ARR No. L5E02


dr
= 61 + + + rU-! + (Tp (2)
dt

dv =- sin $ rV + ,(3)
dt

-d-- = (4)
dt
V
11 (5)

The individual terms in equations (1) and (2) rep-
resent the values of the instantaneous angular accelora-
tioas produced by the magnitude of the aerodynamic
nmo.nents acting on the airplane at any given instant of
time. The individual terms in equation (3) similarly
represent the instantaneous lateral accelerations produced
by the gravitational and aerodynamic forces. The instan-
taneous accelerations are in;de)enuent of the manner in
which the aerodynamic moments and forces vary and are
dependent only upon the instantrneous magnitudes of the
moments and forces acting at any j.ivon tine.

For the linear case, the acceleration terms such as
pNp and PNp may be expressed as products of an angular
displacement or velocity, as the case may be, and a con-
stant slope representing the acceleration caused by the
disturbance per unit disturbed ;motion. Equations (1)
to ()..) may therefore, be directly in::terated (reference 2).

ror tte nonlinear case, direct integration is seldom
possible. Wr'hen direct integration is not possible, the
accelerations, such as CGN, P'-, and 5L6, determined
from moJel experimental data, may be clotted as functions
of 3; such a olot permits a solution for the nonlinear
case of disturbed motions by the use of step-by-step
inteTration or, when available, a differential analyzer.
No variation of CL6 and 6N3 with p was considered
for the airplane in the present report since no such
experimental data were available.

The appendix presents the data, the references,
the calculations, and the information for curves such
as figure 2 necessary for the formal step-by-step
integration. The expression for Nr, as given in
9*








" A APT- Fo. L5E02


the aprtrndix and used in conjunction with the method of
the pr.eent report, differs sli.itly from the expression
given in reference 6 in that the first term of the equa-
tion for the determination of Cn in reference 6


1 b6 ( taO l on ttail of


which represents the dnnpinr of the vertical tail brd is
suitable for prDopller-off conditions, has been replaced
herein by the expression


-iL.6 Cn 65r
b Cr v

Analysis indicated that the rotation of the propeller
slipstream and sidewash in model tests precluded a reliable
SCn
determination of the verti 'al-tall efTectiveless --

when the expression of reference 6 was used. The expres-
sion givenr in the present report is more general and is
suitable for Fnyl power and piol.ller arranj2ment.

The values of "- and K3 have not been
solveO for in the appendix since they are used for flaps-
deflecte- conditions and the airplane used in the present
report was in the cruising configuration. After the
calculations indicated in the appendix have been made and
after curves such as figure 2 have beei plotted, the
step-by-step integration form shown as taUle I may Oe
used. In using the step-by-step integrstion, it may be
desirable to use time increments of 1/10 second for coni-
putational convenience as well as for brevity of the
solution combined with a fairly good degree of accuracy.

The integration indicated in table I is based upon
the suiruatlon process of solution of equations '1) to (5).
This surr.nat.5on process, as used in table T, may be
expressed as


n = (in) t + Pn-1 (6)
n-n








"TACA ARP No. L5E02


Pn + Pn-1
On = 2 At


(O)
n-i


tC)n-l


tr n dr-
n dt]
r = )n-1


n= (dtnl


at + rn1


At + Pn-l


= L + Pn-Lp + rn-ILr + (p~L



= 6! +n-l P o + rn-1Nr + (P n-1)


= sin n-
V 1


(PY~)n-1
- rn +
n-1 V


The subscripts n and n 1 denote values corresponding
to the time t and to the irrediately preceding time
t At, respect vely.
The first step in using the step-by-step integration
involves the insertion of values for the constant accelera-
tions and derivatives 5 L, 6N6, L,, Lr, Hp, and Nr
in columns (5), (11), (20), (21), (24), and (25) in the
underlined spaces provided in the headings of table I.
The values in radian measure of the initial rate of roll p,
the angle of bank 0, the rate of yaw r, and the angle
of sideslip P should be inserted in columns (5), (8),
(13), and (18) for t = 0. From curves such as figure 2,
the values of pYI, PLr, and pNP should be determined
for the valup of p at t = 0 (P = 0 in the present case).
These values should be inserted in columns (14), (19),
and (23) for t = 0.


where


(7)


(8)


(9)


(10)



(11)


(12)








i!JAA A.,: H!o. L5E02 11


Columns (9), (10), (15), (16), (20).to (22), and (24)
to (26) may now be filled in for t = 0. Column (22) pro-
vides the inicuced rolling accelerations; column (26) pro-
vides the induced yavwing accelerations. The net instan-
taneous rolling and yawinc accelerations may now be
determined for t = 0 by performing the computations
indicated in columns (3) and (11).

By repestin7 the procedure indicated in the headings
of table I and by using. the :ariple values obtained for
t = 0, the values of p, 0, r, and p are obtained
for t = 0.1 secon.:J. After the value of p for
t = 0.1 second has been obtained, corresponding values
of PY, PL, and ip are determined and inserted in.
columns (14), (19), and (25) for t = 0.1 second. The
net induced accelerations ZLn and ZNn for t = 0.1 second
may now be determirned (columns (22) and (26)) a&ii, as a
result, the values in columns (3) and (11) r.sy be deter-
mined for t = 0.1 second. The remainder of table T for
the other values of t may now be solved by repeaiting
the procedure indicated in the hedirn.-s and by using curves
similar to figure 2.

The angle of bank $ was determined by averaging,
the rate of roll p (columns (5) to (7)). This averaging
was not followed thr.-ugh for sin / and for r in the
determination of p, because it was thought desirable
to maintain simplicity in the table and the errors intro-
duce'l. by a disre-C-rd of th.se averages are small and
are within the accuracy of the data used for the c&lcsla-
tions in the a~.-ncxi.

The sten-oy-ste .it :'atioin cr-.sented herein is
not limited t.o the soluition of motions -rod'i.ced by- silerons.
Such i..te6riti)nm rray just as readily be applied to the
solution of cdisturbad motions oroduceod by rudder movements
nor by a combination oi ru'-dder ad. aillron rrovermenrt. For
the c2se of latarel disturbances caused by rudder alone,
6aLa and aN6a would be chjnIed to 5r.LA and 5r5 r.

.'hen the step-by-step intear'ation is applied with
variable derivat ves to flight conditions involvin.r accel-
erations greater than 1 g, the value of the airplane speed
used should be the true airspeed V. The acceleration,
however, must be considered in determining the airplane
lift coefficient. The values of Cn and Ca

(if an aileron movement is concerned) ar.d the derivatives
correspond to the new lift coefficient.









FACA ARR No. L5E02


Cri.,PAk;iT&; OF FROCDFUL.:-' FCF COMP;'PTT';

I4 ATE.AL DISTTJ ..UATJ


"'he charactris-'ti.c curves ohtainF.c by the step-by-
ste- Int -ratio. rre coMo:rEd in fiji-res 3 to 6 with the
res t.i s obtained fro.! ac.t,t' fli_-ht tests, with the
metnorl of diff:rrcntia'L. operators (reference 2), which
is an eyant solution dealinfi :;,ith constant slopes; and
rith "n acs'rov.l imte nielytical solutlmn in which constant
slooes are also used (referencnu !) and which is applicable
only t-o the solution of tLhe sile.slip anrle. '.Vhcn the
maciriu. sUid lsl p an-le r ,r:s cobyl.ted by the approuimate
method of reference ', the co-puted value was found to
be 57.50, ; which -ices nob compare with the 1IL determined
fro.n flight tests. '.'hen the Vdlue of Cn was considered
equal to Ca + ( --, the conmptel value of the maximum
sideslir anr-le was determined to be 1r'.20, which is still
rath-er hih. T1,e prs-ent procedure provides the most
accurate correlation v.'ith flisit test results for all the
mctions consider d.

It should be n.-t:Cd that the refine,:ent used in the
present re,.ort -or te determination of' ,Jr was not used
in tht a:'-.licatirn o; the methods io refrecnces 2 and 4.
It should also tb: noted that v 'V, '" rich is considered
equal to the value of p in radians in all ti- procedures,
is in its rtrictect sense equal to ban p. Thu assumption
TT
that p =- l-ads to much lar-.Cr errors for large values
than I'or fnall values of p. For example, consideration
of thevr two sources f error r'ducev the maximum side-
slip nngle of 20, shown'm f'o- the a,-prnximate procedure of
reference I, to a value of 560. rho I.mproved method in
consideri:w I- accounted for 00, wbhreas the other 270
were A.ccoanted for by th3 fact ;that v/V was considered
equal to *-ar p. In the care of the tep--by-step pro-
cedurc of the prudent report, thr .:anxirmum sid3slip angle
would hlve been equal to about 250 if Nr had been deter-
mined by the method of rafcrc.nco 6. If v/7 had been
considered equal to tan 2, thu maximum sideslip angle
by t!i: step-'- --tcp 'iethod would havL been reduced
about- .
V. ,,.








NACA .?;r No. L5202


For solutions involving the ,.Ecsu!iption of linear
slopes, the slopes u cid in the jrcsent problem were arbi-
traril- measured through *: = 0. If the more usual
practice of selecting the avcra -e sloocs over a wider
ran:r. of yaw angles had been employed, the calculated
results vwo'ld have approached more closely the results
of tho var.iable-slope method. cFr cases in which vertical
tail loads in high-spcod dives are of primary concern,
however, small anglos of sideslip may be critical, and
consideration of average slopes over a wide rangc of yaw
ingles ray be unwise. It a:.!:.'a:s th'rOfore that, although
thc rrcvious proccdurL-s may be recson --,ly reliable in a
number of instances in which the charactcriEtic Ci, C,,
and Cv curves possess ap prcxi.:iatoly linear relation-
ships, nonlinear characteristics occur with sufficient
frLqt'rncy to make the general use of thb nonlinear step-
by-step procedure desirable.

In order to determine the importance of the lateral-
acceleration term 1Y the '-.esent prceecure was
repeated ;:ith = Although the resultin- curves
indicate that the influence of (iY for the subject
airp~ .r.e was not very lar'e, the eff: t of pYg may be
niore significant for cther types of airplane and there-
fore s'."ould not be neglected.

A co.::.-rison of the step-by-stE,- solution using
cons.- !rt s.ycps with t.he -c-tho- l .'f dI ffi.er.ntial operators
(ref cr.ice 2) indicated that vwlie-s obtained by tle step-
by-stm-p solu.tio;' tend-d to Jdeviate a little mor.- from
flight te:-t results tl.an the values obtained by the opera-
tional meeLhod The step-by-stei solution, for this
particular compmxrison, apparently gives a sideslip angle
appro:;.ir-atcly 20 larger than the operational procedure of
reference 2. The tendency of 'hCe step-by-step solution
in tC'^ linear case to deviate a 1:,.ttle more from flight
tests than a direct integration procAdure may reasonably
be )resumed to persist in the a, location of the step-by-
step solution to the ncnlinear ca&e, as in the present
i-eport. Further refinement of the step-by-step procedure
may therefore be expected to provide correspondingly
closer aprecrent with flihr'it,. The r i.ne-Futta su-rmation
method (r-cfere-nces 3 and 0) provides such refinciments of
proce:.ures. The ste---by-step procedure as outlined in
the prnorent report, I vevcer, is believed to provide
sufficient engineering acc:.'.acy wihu;: no more tLan t'ie
i".."'" 1 !'-,.'.. r.- .. i? 1"- i .* r '. '..'"1 ( '," .",.j .'








ITACA ARR Ho. L5E02


Although 6L6 and 56', were considered constants
in the preceding example, further analysis indicated that
the rolling and yawing accelerations resulting from
aileron deflection should also be considered functions
of p for a greater degree of accuracy. It is quite
possible that Lp, Np, Lp, and N are not constant as
ordinarily assumed and as assumed in the present report.
If these parameters are not constant, some of the dis-
crepancy that still exists between flight test results
and the present method would be explained. Until experi-
mental data from dynamic-model tests are available, how-
ever, these values must be presuz-ed constant for lack of
more complete information. Other possible sources of
discrepancy between calculations and flight results are
the assujTirtions of level flight, constant normal accel-
eration, constant seed, and instantaneous control
deflection. For practical purposes, however, it was not
believc.d necessary to tale thjse factors into account.





A rocedurc- base upon step-by-step integration is
presented for drctermining the disturbed motions of an
sir-lanc resulting from the *lfl:cti.or of the lateral or
dirJcticonal controls for th. c:ise of.. nonlinear derivatives.
A comparison of the st-p-'by-st.-': p7roc.du're with other
methods indicated the following conclusions:

.. The calculated disturbed motions of an airplane
resulr.in;, from abrupt control moveeient will be in better
agreement iwith the results obtained from flight tests if
the variation of the experimenantallyJ dcter-'ined rolling,
yawini-, and sideslipping accelerations PLp, PNp, and
pYP with the angle of sideslip p is considered. The
sides lippinr acceleration ,3Y which is often assumed
negligible, should be considered. The variation of the
rolling and yawing accelerations aL6a and 5aNLa
resulting from aileron movement probably should also be
considered when sufficient data are available. The
variation of the dynamic derivatives I, ip, Ly,
and 'Tr should also be taken into account when sufficient
dynamic-test data are available.








ITACA RR ITo. L5S02 15


2. The value of the maximum sideslip angle for use
in the determination of the vertical-tail loads in rolling
pull-out maneuvers should be obtained by using the step-
by-step- integration method.

3. The step-by-step integration may be applied to
the solution of motions produced by rudder movements or
by a combination rudder and aileron movement, as well as
to the solution of motions pro 'uced by ailerons alone
when only tht first' qr,.rt;r-cycle of the motion is desired.


Langley Memorial Aeronautical Laboratory
nationala l A.visory Comnittee for Aeronautics
Langl-y Field, Va.









16 TACA .,RR No. L5E02


APPENDIX


DETER'MINATTON OF DYNAITC LATEPAL P.-OTIONE OF A FT.HTER

ATPPLAITE DUE TO ABRUPT AILEROI MOVEMENT


Data re 'iired.- F)r the fighter airplane used in
the illustration, the data required for the determination
of dynamic lateral motions resulting from abrupt aileron
movement are as follows.


b, ft. ... 2.83 Cd .

bf, percent b 66 ACd .
Of
S. 0.50 Ca .

A . 5 5 C .

1, ft . 20.5 Cn5r .
r a,
5ar, des .. 12.75 V, fps .

5ap, deg ........ -17 q, lb/sq ft

5f, deg .. 0 r, slugs .

a,, deg 17.1 S, sq ft .

CL . 1.42 kX2, sq ft .
CL, ... k sq ft .
ACL .. ......... 0 Z2 sq ft.


Landing gear .


The value of


Procedure.- From


. 0.01

... O

0.04

. .-0.0065
* .-o.oo67

S0.001474

. 1L2.2

S. 24.09

. 358

334

...3352
. 33.52

. 57.9


. .. Retracted


b5 = -2.0 is determined from reference 10.


reference 3, determine

C, = -o.L25

Cnp = -0.0655

Cr =O.5308








:.A A.1 No. L5 02


Then, from reference 6, determine

= -0.5335 l
o 2 + 2X


= -0.2749

From reference 7, determine

K1 = -0.O022


Comrpute the following:


L z b qib
Lp= Lp 2V xP2 2


= -1..354

IT = C r -


b qSb
L b
r r 2V e


= 1.55


!,= = -11.6-Cn 5
!'r : Ii.6- 5 r rav


6 aL a a bk 2"





6' = na mk"


= -0.1


+ Cd, + Kf Ltor
dow f dof


1+ lC 2 6 2 ACLf T + 5 ( L+ ) k q 2
:W -- f


= -o.5108









18 NACA ARR iHo. L'E02


From wind-tunnrel d ata for the configuration con-
sider.ed fig. 1), nlot the following as-ainst P or -W:

j3L, qSO






qs
I)i





The ,alues of 7o, F2, a.nd Y3 hi:ve ILOt been solved
for 2In-e thiy sre L.sed for flis-p-'leflectel conditions
and the airrlane used in the res ent report *-.,s in the
cruising co-il'i ~grati on. F-r leps-deflected conditions,
K,. m:.: t determined fron the following formula from
reference 6:


f = -.*55(-t -----
f \b 2 + 2%


The values of Y2 and YF msy be determined from
reference 7.

Although th,~ valur.- of i, and C in the present
runort lhrve b,.cn d.c tormincd ] :'ly from th'. cur-vJs of
r_ ,"'- -. nc 5, it I:n. bo desir ..ble i so rji cases to include
the ffc-c ts of the vertical tail us' s of thu rmvthod of
rc-fjrJcc 11.








NACA ARR No. L5E02


FVE RE'CES


1. Weick, Fred E., an. Jones, Robert T.: The Effect of
Lateral Controls in Producing '.otion of an Airplane
as Corputed from Wind-Tunnel Data. NACA Pep. Nlo. 570,
1936.

2. Jones, Robrt T.: A Sipolified Application of the
method of Operators to the Calculation of Disturbed
Motions of an Airplane. NACA Pep. IHo. 560, 1936.

5. Pearson, Henry A., and Jones, Fobert T.: Theoretical
Stability an-i Control CnarF-r,teristice of ,'ings with
Varicus Amounts of Taper an. T wist. NACA Pep.
No. 635, 1953.

)4. ;syten, Gerald G.: Analysis of 'Wind-Tunnl3 Stability
and Control Tests in T- l'e s l llying Qualities of
Full-Scale Airclanes. ACA .- F No. 5J22, 1945.

5. Gilruth, Robert T.:: Analysis o' Vertical-T&il Loads
in Polling Pull-7ut Y'.aneuvers. NACA CB INo. L4F14,
19L..

6. Campbell, John P., and Mathews, '.'ard 0.: Experimental
Determination of the Yawing Moment Due to Yawing
Contributed by the Wing, Fuselage, and Vertical
Tail of a :M.idwing Airplane !K:odel. NACA ARP No. 3F28,
1943.

7. Harmon, Sidney ':;.: Determination of the Damping
Mor.ent In Yawing for Tanered 'ings with Partial-Span
Flaps. NACA APR Iro. 3H25, 1945.

8. Lew, F. and Baggott, E. A.: Numerical Studies in
Differential Equations. Vol.T, 'atts & Co. (London),
1954.
9. Scarborough, James 3.: Numerical Mathematical Analysis.
The Johns Hopkins Press (Baltimore), 1950.

10. Ames, Milton B., Jr., and Sears, 5ichard I.: Determina-
tion of Control-Surface Characteristics from NACA
Plain-Flas and Tab Data. NACA Rep. No. 721, 1941.
11. Damber, Millard, J.: Effect of Some Present-Day
Airplane Design Trends -n Requirements for
Lateral Stability. NACA TN Fo. 814, 1941.







NACA ARR No. L5E02 20



a,
a







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r '" 1I I0 0 ^







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64 .
s C ,












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a *o
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NACA ARR No. L5E02


.4








-.-4
0-.4



.04







C)

S-.04











0* 0


-40 -30 -20 -10 0 /0 0
Angle of yaw, V, deg


30 40


FIgure 1. Var/atfon of the drectional and /dera/ coefficienhs
C, Cn,, and Cy, with angle ofyaw 1Y determined from
feit- of a -,ccle model of a fighter airplane the
Zanrley 7-bv/lO-f-'nt funmel. Cruising configurafion;C/.42; ,=:qOf:0 .


Fi g. 1






NACA ARR No. L5E02


-4



-6---- --

-/ !


J5des/lp angle, deg
NATlOIAL ADv1~ORv
COMMITTEE jOh ArhONAUUICS
figure 2.- Curves of the acceleraf/ons 0L, (N ,
and a Y aj functhonS of the sides/p ang/e (3
determined from tefts of a -s6cale model of a
fighter airplane m the Loarey 7-by/O-foo/ funnel 6: -0 6a=0.


1..





Z I?


O
I,








Os
- <.


3 -


Fig. 2





NACA ARR No. L5E02 Fig. 3






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I -9



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sjc




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NACA ARR No. L5E02 Fig. 5






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NACA ARR No. L5E02


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18

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Fig. 6








UNIVERSITY OF FLORIDA

3 1262 08104 996 6

''
*- '


UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE LIBRARY
P.O. BOX 117011
GAINESVILLE, FL 32611-7011 USA














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