LONGITUDINAL STABILITY AND CONTROLLABILITY OF AN AIRCRAFT

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-01043R001200230006-3
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 458
• Document Creation Date: December 22, 2016
• Document Release Date: September 21, 2012
• Sequence Number: 6
• Content Type: REPORT

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OSTOS' W IY I tl
IA3~GITUDIN&L STA?3ILITY AND CO!Tf `UARILITY
OF AN ttIRC
Approved b; the a'.ini~tr of '.ighcr iuc.at on i SS
as a Textbook for '.iher n, ucatonal stitutions
of Aviatior
STATE PUBLISHING HOUSE FOR THE DEF SE INDUSTRY
Moecot 1951
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This book is a textbook for higher aviation institutes
for the course on longitudin i stability arid cortrollab1liY
of aircraft. It describes t e methods of analyzinj the
questions of longitudinal stability and controllability of
an aircraft and gives recocecendations for the employment of
these methods in designing an aircraft.
On the basis of the material presented in the book,
the student. will be able to make the necessary calculations
of stability and controllability and rationally select the
basic aerodynamic design parameters determining the longitud-
inal stability and controllability of an aircraft.
The book is written in connection with the syllabus of the
course given at the 1o8cow Aviation Institute, and is intended
for students at higher aviation institutes and for aircraft
engineers.
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Preface
(T'ranslator's Note; The fir8t two pages of trio Preface are rniesing in original)
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Ii
The present work is an attempt at such a unified exposition of all the inti
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the problems of stability and controllability of an aircraft on he nrobi( which
are not connoted among themselves, of 'static stai:i11ty and iysuuaic stai1ity, and
to the conclusion that the whole problem( rust under all circur3tance3 be coordinated.
interwoven queations of 1cri itudtal. stabilit;? and controllability of ar: }a-
crag ~.
The mwater i ~, of the ''d?c ok .:eleactf ~i 'tri irr. M r ,iccor
Cotrof lectures delivered ctur th the i.3 t U ! ew 'it t e ..o'.; W ~w ) '
;Lviat~on Institute irieni t?rzho;'ti -o.
Fart oj' the /material th U's book z ;et r. 'i..~,ll tyc i riie .
readers wno desire to familiarize tre^selveir. re:, r :ietai1 it '.:.e _~.ecticn.
under consideratior:.
In the desire not to overload the book, the a t'r:ors were cornpell ed to a cuss
certain questions w}:ieb, in the; r opinion, are zecf.~:iary r hasty a ::i wrai-y
r , is r or ..r.34a(a e 1 n: 1' et ce of the
r:owever, the t,:ndar~cen~.ai ass ~~.rcVe 3 the
style, p aoition
cotapressibility of the air on the longitudinal omert, the analysis or the disturbed
motion of an aircraft, the connection between r~aneuverbility, controllability and
stability, have been discu3sed in rather great detail. The course has been written
by the authors under the assumption that the readers are familiar with theoretical
and experimental aerodynamics and with aerodynamic cacputation.
The book is intended for students at higher aviation institutes and for engin-
eers of designing offices, and may also be utilized by scientific workers in the
;field of aerodynamics, streas analysis, and problems of stability.
Recognizing the complexity of the task undertaken, the authors will be thankful
to readers for any criticism; such remarks will be taken into consideration by them
in their future work.
The authors express their thanks to professor V.S.Pyshnov, Professor Ya.M.Ku-
1L jritskes, and Professor A.K.Msrtynov, who read the manuscript and who have given a
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S - area of aircraft wings;
1 - span of wings;
b - chord of wing;
b - root chord of wine (in plane of
0
bl - wing tin chord;
- jean reoetric chord of w'';
-
Slap
- mean aerod mimic ord .~.
- aspect rat; O of w.';
- wing taper;
` ~ i CKI; r cs of
- relative nrof'ile t^icknesu (createst re 3L~.Ve
profile);
- relative c?a^tber of prof?-e,
- portion of wing area served by flans;
flap " soaan of
flaps (distance between outer tips of flaps);
b , - mean aerod;mamic chord of o~rt of wig served by f laps;
A lap
- angle of sweet ack of wing (angleietween transverse axis of
aircraft 01 and projection of line of foci of wing onto the
coordinate plane xoz);
- dihedral angle V of wing (angle between transverse axis of air-
craft and plane of wing chord);
L - length of aircraft;
Lfus - length of fuselage;
S - area of rectangle described about horizontal projection of fuse-
fus
SYt.t.
lage;
-area of horizontal tail surface;
bh.t. - chord.of horizontal tail surface;
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of aircraft and ' inye axis of elevator);
Lta.c,
Slev- area of e1evtur;
b., - chord of i tnr;
- relative area of hori;:.ontal tail. surface;
S
Sa $..~_. - relative area of elevator;
I - arrraa of horisontal tail vurface (distance between center o_' /-r:,vity
~.t.
ShA
- rt?. ..1 Yl." ad i1i o. 1'J .. ,Y l'~i L:l '~.:11 ;
relative wtatica1 mo^eY:t of
area of ;~.or:.or:t ~. t ... i.
-f
a
V - teed of ii ht or . F'inall
rLere or.
'~ ~i D ..
surface (;!Aig.0.2e) will be in a state of indifferent ~euilibriurn, since any ositicn
occupied by it, on interruption of the action of the disturbance, will be an e-
librium position.
The examples we have considered above relate to the case when the body is at
rest in its initial pos t.ion= However, the same arguments are'also applicable to
the case when the body is in motion. The differerce is only that, in the case when
the body, before being exposed to the disturbance, occupies a definite position in
space, the criterion of stability is the return of the body to the initial position
on interruption of the action of the disturbance; however, in the case when the
body, before the disturbance, was moving in space in a definite war (along a de~'i-
?nite trajectory with a definite law of variation of velocity with time, etc.), the
criterion of stability will be the return of the body to the initial motion on in-
terruption of the action of the disturbance.
The motion of a ball along the runway of a bowling alley may serve as an ex-
`ample of stable motion. Under the influence of disturbances (for example, rough-
"nee in the walls of the groove of the alley), the ball will be sat deflected
from the original motion imparted to it by the bowler. However, as soon as the
'' action of the disturbance stops, the ball returns to its original motion.
f'
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If we hold onto the seat of a bicycle and attempt to roll it in a straight line,
e will note that it is necessary to intervene continuouely.in the motion of the
bicycle, by correcting its random deflectiorns from a rectiliroa.r trajectory which
zre due to the roughness of the roadway. The bicycle ~rhibitan un~tabl.e motion.
The stability of a moving body is the designation given its property, in the
presence of disturbances, of res Lrn.in.' its orlFinal motion after the .ictior o:' he
?~, r1 ~ iy.
.. .L.i1_t,',
r,.
disturbance has ended. i?as ~into the ?t.udy oC : e o,.,.:r, c.? r i r
; b1e e . iir.t r:, ;o:: c .,: :,
we must e+'~ril cor:wider ticc D3 .,
.
r # i. j~,,... r~:- o': a ~..~
its stability , :: f .; "':~: ;~~. .~ w. .! ... !.
ry '1., .,~
r
possible t'es of iisturarcethat wily d e 'lit it "+?}, t' .': i G2'_lir
On GJI:u ng this e io of the t.a" ii
sta ~.er.'~
bod'r from itw, state - e ibr :z',
the pert'.:rbatio~?:~ hour cd
t
wert;iii;. ..'"d ~.
".,:e.
do , f , E ?b i,i ty
with the second v.~r,~.~i-,.: ,.>.i.l.~' '},
i `r
erulibr{ i '.m ao.s, ; Lion are omal~-~? _n solvir ob. cc- n 0 aircraft ,,ab i.. ity
~i .
in the follow na, consider the disturbances 3nall.
The following forces act on an aircraft .light, the force of gri.v ty, to
aerodynamic forces applied to he wings, fuselage, eiapennae, etc.,a".i the tractir
of the engine installed on the aircraft. It is obvious that, it eS;uilibriuin, the
aura of all forces acting on the aircraft must be zero, and the vector of result r.t
aerodynsznic forces and traction must pass trough the center of gravity of the air-
craft. It follows from this that the equllibriurri of the aircraft during different
states of flight may be attained either by displacement of the vectors of the aero-
dynamic forces and traction in such a way that the vector of their resultant passes
through an invariant center of gravity of the aircraft, or by a corresponding dis-
placement of the center of the aircraft in such a way that the resultant of the
Vectare of all the aerodynamic forces and traction passes through the center of
gravity; in this case, the resultant of the aerodynamic forces, the traction, and
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the farce of gravity must be equal to zero, for which purpose the anp].e of attack of
the wings and the operating conditions of the engine must be varied in an aopropiate
manner.
It is obvious from general considerations that the former method of control of
an aircraft, for the accomplishment of which an empennage is necessary, is the more
perfect. It is precisely this method of control that wa used by the noted inventor
Fig.0.3 - A.F.Mozhayskiy's Airplane Model
of the airplane, the Russian seaman A.F.Mozhayskiy, whose airplane (ri.0.;) about
70 years ago, made the first flight in the world. On this aircraft !:ozha?skiy placed
a "tail, which may be raised and lowered and which serves to vary the direction of
flight upward and downward and, by means of the vertical area, movir it to the
right or left to obtain lateral control of the apparatus" (3ibl.l). In this way,
about 70 years ago, Mozhayskiv gave the correct solution of the problem of control-
ling an aircraft. It was only inertness of the Tsarist Government of Russia that
hindered subsequent development and application of the brilliant creative ideas and
designb of A.F.t4ozhayskiy.
It is interesting to note that, when the first flights were made abroad, several
ecades after the flight of the Mozhayskiy aircraft, a second method of controlling
he aircraft was used. 0.Lilienthal controlled his glider by shifting his own body
w.a
ith rpspectrto?the glider in such a way that the center of gravity of the glider
- ate?dfepl&csd in accordance with the displacement of the point of application of
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the aerodynamic forces Meting on the glider. ouch gliders were called balancing
gliders. On the alreraft of the Wright Brothers; a tw1 at rr t.ha 1 rr. t
leading to the corresponding displacement of the point of application of the aero-
dynamic forces. The birds, who show no apparent special organs of control, such as
the rudder of a modern aircraft, control their flight in about the same way. Thus
both Lilienthal and the Wright brother took the path o.f blind .imitatior; of nature,
although such imitation is far from always preferable. or example, most rn.achincs
Trade by man use the rocking Notion and the wheel, but :either rocking nor ari, ryin
siud ar to the wheel is encountered & ong living beings. It is clear that i ethois
of controlling an aircraft as used on the apparatus o Lilienthal and the Wright
Brothers, which are still suitable at low speeds of flight and at so li aircraft
dimensions when the value of the foment acting on the aircraft is varied by sinpl?
displacement of the pilots body or by twisting of the zings, ^ray becone entirely
unsuitable when the speed of flight and the size of the aircraft are increased. ;'or
this reason, in its subsequent deveioppraeent, aviation proceeded along the path indi-
cated by Mozhayskty. Thus the principle of aircraft control, used in modern designs
throughout the entire world, was first worked out in .tussia.
Most Russian aircraft designers used !'oMhayskiy's principle of control in their
designs. Thus, the ussian inventor A.'V.Shiukov in 1909 built a glider with ailer-
ons and a tail, which was a great step forward by comparison with the balancing
glider of Lilienthal. The elevators and ailerons on this glider were controlled by
a single lever in the same way as is done on modern aircraft, In its test flights,
the Shiukov glider proved to be stable and controllable.
The talented Russian designer and scientist S.S.Nezhdanovskiy, as early as the
1890's, investigated the glider, using an empennage, like A.F.Mozhayskiy, to ensure
Its stability. Thus already at the very beginning of the birth of aviation, the
:progressive scientists of our country, correctly defined the method for ensuring
letability and controllability of the aircraft, although the general knowledge in
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The great Russian scientist, founder of aerodynamic science, the Feather of
Russian Aviation" t.Ye.Zhukovskiy, began in 1909 to deliver a course "Theoretical
Principles of Aeronautics" in which, together with other questions, he considered
the iroblem of aircraft stability. This course was the first systematic course on
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this field at that time was naturally in an et~rvenic state. The insufficient know-
3.edg o;t the laws of stability at that tim* lead to frequent accidents with flying
machines,
lygin had alresdy laid down the rereral pr i ^.{ci2Ies or tr.c t teor, ~ 1. ct :.ve re
sistance bassi precisely o the do~mlriibeyond t;:e 5b::~'s of rl :ircr ,'t.
.. , .~ ,
t of aviation and d ,., its ., scientific f orr.,to , aer_od:-;~.: ,~.,rr,_'
~;, a i ,,..,
The general Jeveiot u-
led to substantial extansion of the theoretical ar.i ax eri. ier.ta~ i.e, e or.
stability allowing at the present tine the cor:structior; of well-or :iii .e ar,
rather complete theory of stability, as well as the ~eveloprent of the enri eer nr
,?thuds or calculating the stabilit;, and controllability of the aircrt. :.r o':t-
standing and honorable merit in this field belongs to the scientist: f e..r cc.,rtry,
who worked out and solved a number of problem: in this field..
Without dwelling on the genera! investigations of the stability of rotior of a
body, conducted by the fas ous Russian scientists Academicians L. 4uler (r749) and
Ye.Koteltnikov (1774), Professor .Ye.:hekovskiy (lei?) A.'``?.i.yanurov (l92), and
.`others, we shall confine ourselves here merely to a short survey of he work on
aircraft stability.
stability calculations1 ; however, by that true the .amp'; ._entit
This is explained by the fact that engineers at that time had only a vague idea
of the law of change in the forces acting on aircraft in flight, and did not under-
stand the immense importance of the mutual positior: of the center of gravity of the
aircraft and the point of application of the resultant aerodynamic force. it that
time, the existence of the downwash in the regior, of the tail, ca,.;sei by the wind
life, was also unknown, althau'h this is a factor of exceptional >iFr:ificance it
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the theory of flight to be g.ive~n in the world. In It :hukovekiy analyzad the mo-
ments of the forces acttg on he abYcrYtr't irr t'lignt, anci the measures necessar;'
to ensure stable flight of the aircraft. Academician :.S.Ch~tplygin inve 3tigated
the moments acting on the aircrit wing, using the concept of the "mctacentric
curve", which is a curve rcr~reu~r:tir. an env'cI` p ..1onc which :ere slide the onJJ
of the. vector of the aero4T:zuzis 1"orcen act1r or he wing.
The w',rk o~, ;tr cr i: :! th rwcr:t ? br_iliant dve1orner.t after the
October Revolution, which relpe:, ~cierce a.,`.i in the "xurir:r ,.,
y~rs of w~0{ per, , C.. .O n rah Vn''r d '. . ?:. t~ r 7~C''k. '.
G':'~* : ~:~^JkiP...~`~. r., . `"?t,ti. a.'.F:~.!:, f?.. . .Y..C', ...~.`. .. ~'.~
7.S. yshn?:)Y', 'a..Yatlveyev, a.~ .1t:Y'rF.u, ;s..d other3 .:1d ?{!,;ch .? the eid of
~ w w r r. t n...1 ,
.. ..~,,.
practical arply _C4tt..o:~ 0.c theory a .,Q the .C'~r!;~.o~.,YC..t . 3: C4'?-p t GC.r. ..:' ieu..ri .. of
inn of c x~er,..tui er.tiw.~ ^,1 ctu,.Ldio or
lating uail.; `'L ~ tbi?, y the inequt10r
?on i
The
lS >0~
'a
~ the aircraft with ,^~.ect to wt,tiC ~..i,i}'rit:r
The condition of neut~r~.i~y of the will be exrressai by the equation
tine and qua;:tit~.-
Coefficients of it d1 Static tabili;y' For ' 1ualita
static stability it is in practice more convenient to deal not
Live evaluation of
with the moment itself but with the dimersianless coefficient of this ;anent. The
coefficient of the longitudinal mraent rn is defined by the rein+tion
b certain arbitrarily selected linear
;where 3 is the zing area of the aircraft; , a. uuantity for which the wing chord
is usual],Y ten; and q ' 4, the velocity
d n g stability of the aircraft in the range where the
head. P1~rther, in cones eri B
is linear, it is in practice ?ore convenient to use the rela-
dependence of Cl on a
ith the coefficient
but w
+ of attack, tion off the aoeiPioienL .~ not with the angle
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Fig.i..9 are constructed.
Fig.l.9 - Curve of the Coefficient of
Longitudinal ?4oment (Pitching
Morcent) nn as a Function of
the Coefficient of Lift Cl.
The curve m * f (C1), given in
Fig. i.9, may be obtained in rind-tunnel
tests of the aircraft or its model and
corresr nds to the constant value of
the 3ach number arid the constant value
of the angle of udder deflection 3 .
From the curve Z - r (C1) we car. esti-
mate the value and sign of the static
longitudinal stability. For this ur-
pose, the derivative cmr = COL is
cL i
used, ten at the noint of balancing
(rr ? c. The derivative mgt is
texed the coefficient of longitudinal
static stability of an aircraft.
It is obvious that, by the same reasoning as with respect to the curve
= f(a), the existence of static stability of the aircraft is deterrdr,ed by the
inequation
m; ' < 0.
0
#A single-valued relationship between C1 art a is also obtained if we neglect
the influence of the compressibility of the air; the influence of compressibility
will be discussed in later Chapters of this book.
ticn,of:C1.
The partial derivative is used, since in the general case, the coefficient
agy.-be a unction or,the.Mach nzaber or the flying speed, besides being a fune-
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t ;lift of the aircraft. C1, which is w iquelye, connectod with the angle of attack
In the graphic representation of thin relation, curves sirailar to those sho~m in
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The absolute value of the derivative m1 chxaracteries the degree of static
etabilitl - or inatebility - of the aircraft.
Stabilit Controllabilit ari Safat of 1i ht
To ensure safety of flight, the pilot must be able to detect the instaat -:$t
which the aircraft goes into angles of attack, overloads, or speeds that are dai er-
ous from the point of view of strength or controllability. An irvoluntarr trar,si-
It- she absence of ,:tutc,ry~tic sign-~.=
Lion to these dangerous states must be avoidtd.
t pilot from bringing the -1ircr-:,ft close to u-::?;eTOUs
on the ~~ircraft, preventing he ;
nula??n is Iar w,ely taken over by the forces
attitudes and situations, the plc o= an and deflection of the control levers for the control surfaces. Ir his case, the
..o, '",,,e rer~Lirements for
forces rather than the deflection; ,lav the lra,cr sy
stem of control developed by flight ractice consist in marring the force r;ecesa ;r
roduci. a destructive overload or for brirdn g the aircraft ir;to high and low
for p
eeds sufficiently noticeable for the pilot.
sp
.
The necessity of sufficiently gre,~t
o es to brin the airplane into such critical states of flight m13~;es certain de- .
~orc 8
wards on both the stability of the aircraft :d the tude of the hinge mcrents
of the control surfaces.
To suucaarize the above statements, we gay note that longitudiru l stability-,
controllabilitYand flight safety are intirn telY correlated. It cannot be said
,
that high stability adversely affects the controllability and safety of an air-
craft. On the other hand a high stability, at proper selection of the size of the
rol surfaces and their aerodynamic-cofltpensatiori, improves the controllability
cont
of the aircraft and the safety of flight. However, these conclusions are not
invariably vslid? A change in the conditions of utilization of the aircraft and
future investigations may necessitate certain modifications in these concepts.
t be remembered that flight With an unstable aircraft is possible; how-
It mus
ever, such fights are unpleasant for the pilot or dangerous , therefore,
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inadmieeible. Umwever, flights with an uncontrollable aircraft are entirely irt-
possible.
In an uncontrollable aircraft it is impossible for the pilot to perform cork-
e,cioue nations, and the behavior of such an aircraft does not depend on the actions
of the pilot. Flight on an uncontrollable ;aircraft must inevitably end in catas-
trophe. For this reason, controllability is the deCi3ive factor for the very
possibility of flight.
For a ? athemxatical analysis of questions connected with stability and control
lability of aircraft, a determination of the forces and the wzents of these forces
acting on the aircraft under various conditions is a rrirne req isite.
In the following Chapters we will discuss the basic rnethcts of dete. nini `Le
moments acting on the aircraft in steady and wisteady f1i ht.
bo
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P~ ATE torns ACTIt~C t3F APB AIRCRAFT '1ITHC4JT TAIL SURFACES IP
STAY RCTTLIhFAR FLIGHT
The moment coefficient of an aircraft without horizontal tail surfaces in
steady rectilinear flight, like the rnonent cofficif:rlt of r r ;y aircraft, can be rst
determined by node/ tests in a wind tunnel. 1r: the absence of such f acili-
reliably
es the coefficient of ^ ,ent may be apps .ir" tely found by c~lcuiatin^ the
moments of the individua ele.^rents: wing, fuselage, engine = < celles, etc. The
methods of such calculations are given below.
sent of 'din. with Constan_ tCh
? to the expression for the longitudinal rrx~nent acting on a wing with
r us 1-'`r'i
hard with respect to an axis lying in the plane of the chord at a certain
constant c
from the leading edge of the wing (Fig?2.1). In this case it is convenient
-distance
use the components of the total aerodYna is force acting on the wing, taken in
to
the syste- of fixed axes, in which the line of the chord is taken as the absci.-sa
and the perper$iculax to the chord, directed upward, is taken as the ordinate. Let
gig.2.l Position of the Axes of '4otnenta
us place the origin of cooMinates on the leading edge of the wing. Let the co-
errodynaniic force be C~, C. We will
efficients of the con'iponente of the total a
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eorider the moment positive if it tonds to increase the angle of attack of the
wine and ne?ative in the opposite case. It is not hard to see that the coeffi-
..r
cients C, CDl are interconnected by a polar relationship analogous to the ordinary
polar of the wing. This polar relationship, i.e., the curve CDl - f(CLI), is termed
a polar of the second kind in contrast to the ordinary polar of the first kind. The
relation between the coefficients of these polars is fiver by fonrula s whose deriva-
tion is elementary:
=c~cosa+c,,sina
= CD cos a -- L sine.
Since in practice, the angles of attack a are sral1, we n y, wit;icut cc ider ble
D
where a is extressed radians.
In this case, eq.(2.l) may be simplified and written in the fo
CL t ~cL + Coot;
Co' , Co---Ch2.
error, take cob a 1 and sin a 1
The value of the derivative C~ is considerably less than Cd For this reason and
for further simplification, we may take
C~~ ^ C1
C 1 z Cx--C~d. (2.1')
It is the latter formulas that are generally used in practice. Sire 0D1
represented in the form of the two terms in a certain combination, it will he found
that CDl < ?, while CD > C is always the case. Figure 2.2 shows a typical polar of
the second kind.
As stated above, on the basis of the theory of similitude the wing rx rent may
be represented in the form
jSbq.
The coefficient a2 is termed the coefficient of longitudinal wing moment.
42
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Since the quantities 3, b, and are arrays known, m must be defined for deterriin-
ina the wing rnr+mer?t::
If x~ is the distance to the center of pressure (the points of an~ iication of
the aerodynamic force) from the leading edge of the wing, related to the chord,
Po! 1 et t'"~ Kind
Pouf o} %1 kina~
? ~0? 20? a'
10
.
p ?
-020 -l6 -012 -0,08.0,04 0 0,04 0,0a 0,12 0.16 0,20 c,,,c,,
?ig,2.2 - Folars of First and Second Kinds in a Srecial Case
then, for the cQeffici.ert of rso ent with respect to the xis selected, lcc;ited ;t
the distance xT = x1b from the leading edge, 'ire will have one of the following
Fig.2.3 - Determination of ding Moment; the Axis of Noments Lies in the
Plane of the Chord
D
'expressions (Fig.2.3):
xp- XT) cy, c_ + C4,zt,
43
(2.2)
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trlrou~n ?t+5 kU La AR uu ,w?
rig... , A
."u LWiu41:b ,v.~~r~,'a~iJ1~11a +b~ VIJ4t7.lilt~U ~ 4.i a,t"cvv
of the roment with respect to the axis selected, while the bitter is obtained if we
first deter.M the moment with respect to the axis~assin ; through the le adir; edge
of the wind and teen, by the er~eral rules of ech niCS, r'' ;ssing through the is
,
.- 1(d), ~Ss ?s acionIy] p,ncrri,
selected. On he linear prt or the curve ul
the
), .d ~ rt n ccc ?n or the . rr'. i>1Jk: b4~ i~ :~i,'~..
w W
,
relation is obtained so that t c~ .{2.2
(2.11), iil yield
=C C
(2.L)
whence
We h'.ve obtained a certain erescior few the center of :rc-3sirC of
from which it will he seen h;.t, for he so-called " i e; to profile of he ;1n' 1:
which cas 0, d1js~~.Ceti .~.Grthe CilCr', '.I;rj, in ...."Ui
1 '"i
the center o.* ;ressure is
0 it approachesx cular, at c ~ , P
On the other had, the second part c f the sane ex"''r essicr. (2.2) i'urnls"es
La. -
xt?x1-"~ (2.6)
that the svm within the brackets of eq.(2.5) vanishes. The coefficient of moment
with respect to the axis passing at the distance xF from the leading edge of the
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tion of the axis
It is obvious that, on the chord of the profile, we can always select such a sosi-
 0
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cal results for this case.
The concept of aerodynamic center is very convenient in the analysis of
stability probless, since the position of the center does not delend* on the angle
of attack nor on coefficient C and depends only on the geometrical shape of the
~,
wing profile.
it will be clear from the following that the compressibility of the air
affect s ...the position of the aerodyrramic center; however, for the time being we will
con*ider the air to be an incoxnpreeaibhe fluid,
45
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wing as will be clear f r the above) grill not depend on -artd, conrrequenti r, , lo
not on the Flo of att~ick
r the t 'Ucf.?icie~:~. ~.ir~eM
hat point on the win; chord with re ect ..o which rimer:
The
penal on the annie o ttzrc~~ rior on 'vT i;r tented the erod t i C tether.
not de
? ed ticr...
line passim; through the profile foci forM~ir.=? the wine is c,ceeds lU"! of `.:e
chord, which fact also explains the remark made ;.bove s to the s3ibi.lity c;f usir.Q
e.(2.7).
The position of the center of sravity of the aircraft with respect to the chcr~
has an extreraely great influence o the wing rent. !J var;rin. xT, the v ue of
the derivative and its sign ca be varied within: side limits. As will be see::
from eq. (2.7), r locating the center of graiity of the aircraft ahead of the r ero-
LI
4
0
a
00
402
0,0f
?a
c
?%0
;o
14
-o
ff
- .
y
t
?
f
y
i
-
-
0,
~
O,Z
0,3
0,4.
0~
Q
0-~
C,,
-
-
-
-
*
Y
r~
-
0
0
-
tig,z?> 7nflucrtce of the Position or the tenter of Gravity in
1evation on the Wing Moment
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 In the preceding ark ~uents, the wing was assu~ ed to be of rect;:n ^iar r' ;:fort:,
with a constant profile along the span. Such win; shares are cet only rarely today,
owing to their runr, tionai utilization of teria (high design weight), especially
in cases of cantilever wings. The wine:.;s used today have a r:onequilatera trapezoid
planforni. The wing profile with respect to the wing span is often taken as vari-
able. iings of such shape, which have a r.urter of aerodynamic advantages, have a
mechanical strength approaching that of a body of eq'al resistance, and are con-
siderably lighter in weight than rectangular wings.
Let us consider how the :nomcnt of a wing of arbitrary shape is expressed with
respect to the center of gravity of the aircraft. We r~nark as a preliminary that,
instead of seeking an expression for the rxoment with respect to the axis passing
-through the center of gravity of the aircraft we may determine the moment with re-
spect to any other axis and then, using the well-known rules of mechanics, define
the axis passing through the center of gravity.
Let us set up the expression.for the.miaent of a wing of arbitrary planforsi
?but of a shape in which the line connecting the foci of the sections (the line of
fool), is perpendicular to the plane of symmetry of the aircraft (Fig.2.6) with
ra3pect to the line of foci. For each elrmantary Wing strip of a width dz and an
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dynamic center, we obtain a negative derivative by placing the center of
L
gravity at the aerodynamic center, we have x ; by placing the center of gravity
behind the aerodynamic center, will be positive.
CCL
In this way, by displacing the center of gravity along the wing chord, we ray
substantially influence the sign and value of the moment of the aerodynamic, forces
on the wing at any sudden change in angle of attack and at constant flying speed.
For this reason, as will ho shown below, the ; sition of the center of gravity or,
to-
as it is also called, , n " of the , r~~, ~ r ir.~ r.i' ~
tar the Itper} ~erin~~s, ;rcraft is an extremely ~,rt..~.
tar influencing the atabillty of the aircraft.
The Moment of a Wing of Arbitr arj Horizontal Contour
L4Q
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,~+h constant spanwise profile, c, will be
In particular, for rlati wuLa >.. -
the same for all wing sections (cam
STAT
Line of Foci
Tr v ezct-
" ;'~ C ~iilj.tin7 ~
area bdz, the motnent with respect to the axis will not, by 1
?
c~ have ','or the of the
t!e w~11 ave ',
a . ?4 of life. Cor9UC itl.;j, h
t?~f u]V1fir~~
where Y? denotes the wing. is wr oth
, d .thin she :.i:..i
In the wing assenblies encoUnterr in ;. r aCticc
acrad~*nc coef ficierts c':er~ only sli~.rl' on the
flow sr~ound the wing, the reason, it rv-~ he
position of the section with respect to the For this hove
m conet in first approximation, on integration. e then
aggtUled + ~ht.t cp,o
since it is =re-
vestigation to the class of trapezoidal wings,
If we confine the in write (~'ig.~?7)
such wings that are most often used in practice, we may
ciselp
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 to 4%3 at ~- ?
(triangular wing).
Thus the moment coefficient of a trapezoidal wing, whose aerodynunic center is
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where
bo
(b0- b1)2 f =b41_(1
b0 1
r ,
is the cocf fici mt of wing t-ar?er or, as it is offer: cr lied, the t . +er,
"xi z z.., t irtrcrir{cir ' } it ?~.(~.2.i ( ) rtd ir;tejr,airn^,,
'~_., 1
win .ire i ~iuzt' t
The cean gctometric dr;g chord is
S
-a t - -~ .
Thus, the product bit Irv he rc; resented in the fcr:
NI?Sb (2'
+1
(
,2.:,-
Cr substitutirzhis expressi n ir, eq. (2. i2 ), we ve
s C.4S+q , ? ?' ?
on representing MZ in the form
.+wsb,,1,
we obtain
Ati __ Ali ?
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M
tit u s I
epending on the taper, the factor for C varies within the limits from I at
(2.17)
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perpendicular to the ptar~e of symmetry of the rcr.ft rE~i tive to the wing F,rk:t
to the meet geo tric ehorc1, is e. ua1 to the moer;t coefficient
always ;renter than to o went coefficient :t C.
ry
S.
of the pro"i c!fhich the
wing is forced. The dogree of eicce ~a ir:ere C. In this case, the contributions of the
i. e., the appearance
takes place no matter
wing to the static stability is reduced. This phenomenon
whether' the aircraft is of the mid-wing, high~Wing or low-wing type.
This unpleasant phenomenon can be controlled by using, in the region of
detac ent on a sweptback wing, special profiles with higher values cf
probably
dis ed by the profile in the rest of the wing, or some other means of
wax than play
the thickening of the boundary layer in these zones of sweptback wings.
preventing
ue that larger values of CL max of the sections, for maple, at the tips
It is abvio
of a wing with positive sweepback, will result from a premature flow separation in
this region.
It must be borne in mind that the above remark, as to the influence of the
number on the slope of the curve mZ ' f(i) at large angles of attacr:, is
Reynolds
completely applicable to sweptback wings.
The calculation of the moa~rents at large angles of attack is not reliable
to the c lexity of the phenomenon. For this region of angles of
enough, owing omp
attack it is better to use the res=ts of wird-tunnel tests on aircraft models.
Influence of the C res b t of r on the Win& yenta
have assumed above that the coefficients of the aerodynandc forces acting
~Ie ha
on a wing of given geometric dimensions are a function only o4' the angles of attack
Fig.2,19 - gchemo&tic gepreeeftation of the Flow irow$ the Wing
69
0
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 of the > a? This assumption ie.susficiently correct only at each uumbcr3,
when the compressibility of air may be neglected in first approximation. As the
Mach nut~-era increase, the assumption that the aerodynamic characteristics are in-
dependent of the Mach nu~ber becomes more and more inaccurate, ard, after the local
velocity at any point of the wing becomes equal to the speed of sound, this assump-
tion sharply contradicts the actual situation.
Before speaking of the influence of the compressibility of the air on the wing
moment, let us briefly recall he physics of the phenomena that take place in he
flow around a wing.
As the Mach number increases, the local velocities and the decompression on
the wing contour also increase, and they increase faster than the each number o:
the oncoming flow. According to this, the pressures acting on the wing contour de-
crease.
If for simplicity, we assume that the configuration of the streamlines around
the wing do not vary with the Mach number, then at some arbitrarily selected but
definite point on the wing contour the condition (Fig.2.19)
PV~--P v
must be satisfied, so that
p
p.
= const.
At the same time, in a coccressible fluid, the density of the air is expressed by
the formula
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where k is the adiabatic index.
The preceding _coMition may thus be written in the fog
$
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It is clear from this expreasion that when the Mach number increases, the ratio
Ya1+ also increases. In reality, wnun tine Mach increases, the coni'iguratioi 01 tre
streamlines does net ruin unchanged, and the phenomenon proves to be far more
codex, but the ratio. V/Vw and, consequently, the decompreasion as well, do in-
crease with increasing Mach. This means that at one and the same angle of attack,
and with increasing Mach number, the coefficients CL and c~ of the profile increase.
In the 193u's an approximate theory of awing in a circulating flow of corn-
pressible fluid was developed. It is possible to use this theoz7, but only hiie
the local velocity at any point of the wing is not yet ?yual to the speed cf sour,..
According to this approximate thecry, at increasing Mach nwnbers, the pressure at
all pointx of the wing varies inversely proportionally to Vi - M` . he Scvie'
scientist S.A.Khristianovich, considering; this problem, came to the conclusicn that
in a more accurate solution, the pressures at different points cc the wing vary in
different ratios with any variation in the ;'Mach number: the decompression in-
creases more strongly the greater the i::itial decompression at the corresponding
points of the profile at M 0, i.e., in an incompressible fluid.
It has been found that, at subcritiral Mc, numbers, the coefficient of lift
of the wing profile increases With M approximately by the law:
(2.3o j
In about this same ratio the moment coefficient of the profile also varies, s that
the aerodynamic center zF ~c~ in the subcritical region of Mach numbers, varies
only slightly. The moment coefficient atCi 0 increases approximately in the
proportion
The variations in C1 and cm? with the l4ach number are rather substantial., Thus,
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wt:er e c'0 is the ~':
.. ~ ~ 1 M'~-~. (2.39)
pp41 +- M') ,. (i oment at acn uw~ers Exceedin the Critical
The iiin ..- ? icai value
if the `~aC;l r+uasber G1 the relayive ,
(M , %p) then the velocities flow i8 h per tnarthe cri
be supersonic a certain regiaof the flow
tiri~l
' be lower than: aw~spaerie
ig' borhaod of the wing, wee the pressures will
rn
the e
x Q.521 . This supersoric zone of flow ; e;,rates
pressure and lower than Pt:`Y ? nozzle. As
the game tray as that observed in a de Laval
a pressure disco^tLty in
the pressure disaar_
the Mach number increases, the supersonic zone broa~iers, ~ M ~ l the
pressure disconti direction of the trailing edge of the profile. At ,
tinuity is shifted in profile.
t is located close to the trailing edge of the p
y
ter foraration of a supersonic gone, the relative
pressures -p t in has the entire forward part of the profile, up to the comp
-
' a in the trail
Po Mach number, whit
conti,miitY vary relatively ?little with increasing
resslon discontinuity, they continue to
part of the profile, beyond the comp
tug
drop ..?_ida ?~ the leading part of the profile the relative de-
sharply. In this ,~,.. .., in n yiTllia u~
"v - - b ~_.vv. -a... as will be seen. from Fig.2.20-
o..-,
cession P " ~ e~~ LO a
trsg part it continues to increase.
~Jf ~'i r: A.L
profile S, :,.
a thtt poin lr the p, at
d increase by afoot 2~ over their values in
the voez"~;i~aCL an ~
an incc,resaible G lu.id.
file contour at which the decompression in an
At M . ~, the point of the pro to the
1 be greatest, the local velocity becomes
irzcoaprossibf1?~,; ~ wool r, mc+ or loss urge zone
of e.ti the ch .,urrber i.icraases further, a re
~pee'd of s~''n~? In this pare, Pressures
u~rso~uc vel~~c~.~ies a~~pears on the profile cor~,aur?
assures o the l:;;i r3 ~ t of
act on the prof ile, i.e. , p~
less ;,ha~x the cr~.ti4al l,~'ess gas cri-
the :
tical ~ ~y ~, .v equal to the speed u, so'
~1r ~1i+V at which u:if.~ ~.. 1f~ .,C~ vVi6i~+i^'~.ur
co nlY LnoWfl, is determned by he expre5si0n
ti~:al t't'eas:a; as is
p 1t IM _ (2.38)
72.
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Thus, as the Mach number increases in the region MH < M< 1, the decompres-
sian zone of the profiiincreasea and gradually shifts toward the trailing edge of
the profile.
This pher~omerion at Ci } 0 is
usually observed first on the upper
wing surface and appears on the under-
side of the wing only at further in-
crease of the Mach r:w^rber. As a
result, the coefficient of wing drag
incr erases sharply, the coefficient o
lilt at first continues tc increase and
then decreases after the ~eco..npression
The coefficient
longitudinal T:oent
increases in ab3o1UyC value, since the
Fig.2.20 - Coefficient of Pressure redistribution of pressure leads to the
versus Mach Number arxi appearance of a moment tending to re-
Fc,atio duce the angle of attack. Al this is
illustrated in .ig.2.21, which gives
experimental data for the trACA 4412 profile.
Obviously, the variations in the aerodynamic coefficient of the wing are sub-
stantial; for this reason and because of the influence of the compressivility of
the air, the wing moment relative to the center of gravity of the aircraft may vary
appreciably. Figure 2.22 gives the curves a1 s f(CL) at various Mach numbers,
correaporxiing to the wing whose characteriatica for a = -O'15' are shown in
Fig.2.23, and to the coordinates of the center of gravity of the aircraft
It 5hou34 be added that large positive pressure gradients occur behirxi the
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4s
Cj
44-
4
a:
CQ
Cm
?
Qf
494
~
ao
ao
Q~ as
a' M
Cm
?a2
Fig.2.21 - Relation of Aerodynamic Coefficients of the Profile and
1ach umber, from eriments
0.! 02
decoapreeaian jump, pomeibly leading to a low separation at the wing eurface and
Urns further co rlie a an airewiy compucc pnenaaenon.
?a1o
04 0.s os 07c
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,0,75
1ig.2.22 - Wing Mont Relative to the Center of Gravity of the
Ajrc t, .:.
,~.~*&~t t YariOus Mach Numbera
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~2-
:
t
:'
o
:L
7T
a,
:
I
:
:
L
d
Q4 0,5
Q
,6
D,7
t1,!
M
2%
:
:
i
:
:
:
:
:
s
c
i
`
~
`-.
1
S
_
:
t
CL' 0
.
,.o
.
?-.~
_._.
- . - .
.w..
I
1
I.
1
.
0,10
4 ors 0,6
-._u._.,__ FiB 2~? -Relation of moo
O.8 M
P to H for Two Profiles
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Figure 2.23a shows the curvea of rip ar~dd Fig.2.23b, the curved x, both as
functions of the Mach number, La will be seen, for different profiles, curves of
different character are obtained, and no universal relationship seem to exist be-
tween these coefficients and the Mach number.
Thus for aircraft flying at superc4tical Mach numbers, the moment coefficient
of the`wing must be taken on the basis of'wind-tunnel tests of a model of the given
aircraft.
~jng Moment at 3zperaoniC Speeds
Let us discuss the determination of the wing moment coefficient in flight at
supersonic speed, i.e., at K 1. In this case, it will be considered that the
local velocities at all points of the wing profile are greater than the speed of
sound. It is well known that the pressure at any point of the profile, at an
assigned K ' 1, is determined approxitrately from the magnitude of the velocity
head of an undisturbed flow an from the magnitude of the angle between the tangent
to the contour of the profile at the given point, as well as from the direction of
the undisturbed flow. In the so-called "linearized" wing theory in supersonic flow,
it is proved that in first approximation the diinensiorless coefficient of pressure
ay put cos
- of intereet for lupersonic flying sp..de? For such profiles we m
file
r
c coafficient of such a pro
~ expressions for the _ ~ e ~. --ody''--sue _
s,3 sin 8 G. The
a4y be presented is the following fora:
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p ~'tl0 is equal to
-2tgA0
..:where a is the angle between the tangent to the profile contour and the direction
- of the undisturbed flow, the sign of this ` angle is ,determined by the rule of signs
for the angle of attack (Fig.2.24)?
Let us confine ourselves to a considration of thin wing profiles, which are
e ..l
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T these fors. PH ann Pg sre
aa...?ei*coefficients c' pressure fcr
r2di rots on the lower ad upper profile surfacee, while }{ and are
corrospo I8 po
the corresponding anJ.o5 between the tangents and the direction of the ~yii5tU1'bedi
flow.
F'ig.2.2L , - Determination of the Arrg1e at Supersonic Flow
Arou 4 the Profile
, ale (~c.l) can be calculated if the geometric characteristics of the
The intrr'
profile are known. Such calculations have been
Y`` ;obtained the following expreesions
c_!( ~21')+ ii(
'i 6
? yam.-,
S.- $,r7P(M)
i,. +I1_!h,1PM~I
(2CF+*,C/)P,(M)
77 .
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made by A.A.Levedev (Bibl.-+) wno
(2?x,2)
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ez~d % is the angle of attack of the profile at Ct ..
MM _ ? ?1 t~N
Cp w is the coefficient of drag due to irition,
k1, k2, k3 are certain rwnerical coefficient3 iependirg orf he shape o' the rrofile.
The values of these coefficients for a few profiles are presented ir: the .fci.-
lowing Table.
Profiles
Rhc oid
Y
4
Formed by two sinusoids
n
2
f
1
i
16
4
16
rc
e
a 4
Forded by two arcs o
3
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is the :relative thickreu of the profile;
I is the camber of the profile:
F (M)-- .. -
!~l !2 J l.4
._:-..t h..oC Fun tiont Fl(M).arid F2(M)
M
8
6
4
-
2
0
6
1
'
Fj
Z
A
is f,6 l,7
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Figure 2.25 gives the functions and ?2(M).
Fora syam tric profile, the expression (2.42) is simplified ar4 takes to 1o1-
lowing form;
.~ ...........a .. _. o + C~ TP'
xp= 051 -k1cF1(M))
cmo =0.
The coefficient. of drag due to friction C{np plays a minor role in stahUity ca cu-
iations and may, in first approximation, be taker. as equal to the coef "icier.t
for subsonic Mach nuubers.
Phese expressions were derived under the assu,.ptior; that the viscosity of the
air is zero. Here, in the leading part of the profile we gbtair p < 0, but in the
trailing part p ' 0, which is responsible for the existence of a wave drag even at
0. As indicated in eq.(2. 0), the quantity p is directly proportional to the
angle G; for this reason, the variation in the angles 6 substantially affects the
aerodynamic characteristics of the profile.
As a resL~lt of the viscosity of the air, a boundary layer is formed near the
wing surface, whose thickness increases in the direction of the trailing edge of the
-file. In this case, the internal flow circulates around a profile having the
charaeterietica of a slightly deformed actual profile, in view of the fact that the
'~ Jthicknese of the actual profile is increased by the so-called thickness of displace-
~aenti of the boundary layer (?ig.2.26). As a result of the viscosity of the air,
of :dieplacament" is considered and applied. -in the
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eo
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the values G of the thickened profile in the trailing part of the profia.a are less
ee of G of the actual profile. The values of @ obtained in the trail-
than the valu
ing part of the profile are also smaller.`
uation (2.i-1) indicates that, as a result of the influence of viscosity, the
gq
value of Cis somewhat reduced at a given angle of attack while CD and rn decrease CL
even mrore.
Fig.2.26 - Influence of the ViBcositJ cA Air cn the Argles 8
The theoretical solution of the prcblen of the flow of a compressible vixcous
gas crow i the wing is verq.complex. For this reason, the calculations car: be
based on eq.(2.43), obtained without allowing for viscosity, and take the above
remarks on the character of the influence of viscosity into consideratior.. As ac
illustration, Fig.2.27 gives the experimental data and co putational curves con-
structed by ?(?!+3 ). As will be seen, the discrepancy between the experir.ental
the calculated values may amannt to 2 - 34 for the moment characteristics,
. data and
J which are of greatest interest to us.
_.<
Let us write the expression for the moment coefficient of a symmetric profile
with respect to the center of gravity locited on the profile chord at a distance of
La before, we have
... mm
Let us _ _.... ,
the position of the aerodsc center, at M > 1. For a syetric
s find
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`rc profile with n relative thickness- f`nr ~ nmr,?a cf" c 0. f1? at t` 2. we have
a, "(j,5 (1- 3 u. C8 x u.732) u.46
As etated previously, at subsonic Mach lumbers, the aerodynamic center ordi-
narily lies within the limits of 0.2 < XF,`< 0.24. Consequently, at supersonic 'ly-
ing speeds, the aerodynamic center shifts considerably in the direction of the
trailing edge of the wing profile. This produces a r oticeable increase in the con-
tributian of the wir~? to the static stability of he aircra"t, if, 'cr exanpie, it
?ig.2.27,- Icperimental and Calculated Data for Profile
xF ^ u.23 at
Jan airplane with the center of gravity lodated at 22~ of the chord
subsonic flying speeds, then on passage fztozn small Hach numbers to ? - 2 for the
example taken, the absolute value of the derivative-_ will increase from the
1 ~~= Ih !+~ Vii .' AI .~. L
-c~uuo --d -v.w. io --- -G.?4. L1LLD faa~,
i(8 1_
_.,...... y... .. ., _ .. . ....... .:.... ...
in i CLilt CtlO f With wiV{1 "nrFa{n other
~Y1W1V ~23V11 VV1 voa?, vw?...
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pe s of the aerodynan%ic characterietics at M > 1; makes it difficult to
culiarities
ceo a are edj" raft raving a satisfactory degree of stability at both 3ubsOnjC ar4
supersonic Kach numbers.
It is clear from eq.{2.w2} that at M'- 1, the expressions for Ci, C0 and rF
become infinite. At Mach numbers close to K ? 1, however, these expressions cease
to be correct, since the velocities obtained on part of the profile contour are sub-
sonic, and the theory is thus inapplicable.
cpa
0 00
Fig.2.22 - Approximate Variation of Aerodynamic Characteristics of a
Profile as a unction of the Mach .;umber
For a region of mixed subsonic arI supersonic flows, as already stated, nc theo-
1iretieal solutions exist for the problem of flow around the body, so that the very
e of
l
op
scanty eacperimental data must be used. Figure 2.2S shows the approximate s
ifG= i uh Mch
t
t (7
e
the curves fcr the. aerodynamic coefficients of the wing, plotted agains
nnmber. The character of the slope, of these curves in tine region W. Loin. a
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features of the profile and is different for different profiles.
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th. rr~Jo~i1 tinns nr the Moment of Swa~tbeck
lines at Lareach N unbers
Swephack wings, especially in recent trues, have begun to be used more and
more, in connection with the fact that, over a certain range of ach numbers, 3weep-
back attenuates the influence of the compxesaibility of air on the aerodynamic char-
acterist,ios of a wing.
Let us imagine a wing cf infinite span and constant chori, around which a
stream of air flows perpendicular to the win: spat. As a result cf the fact that.
the wing profile has a certain thickness, vary-irg alon he chora, the local veiccl-
tiee in the neifhbvrnood o: the chord of the wi?tg will differ '.'rcui the velocity e
the oncomi stream, and the pressure act4.n? on the wing surface will differ f root
the atmospheric pressure.
Fig.2.29 - Flow Around a Wing, Slip-free and with Slip
ti
Let ue now consider the same wing, but with an air stream directed along the
f span, flowing around it. In this case, as a result of the fact that the wirg
i; span is assumed to be tnfinitel7 great, z4 the thickness of the wing is constant
21b each section parallel to the span, the ;local velocities will be equal to the
ores at~oas-
aLtba.oacot- etre+us,..'fhile siU bs equal to the.
-ice
pl%ric:pnssure$._._Tha:esistence of a velocity .oi flow directed along tbs span of an
..
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infinitely long wing of const-nt. chord, 413 not led to the. appearance cf Hero-
*
dynamic forces , .
Let us further imagine this same wing in a circulating flow at a certain slip
angle X (Fig.2.29). By resolving the velocity vector of the flow V into Veos X and
V sin x, we come to the conclusion that the second component, in the analysis of the
pressure forces acting on the wing may be rejected, and that the pressures will be
determined not by the total value of the velocity V, but by its component V cos X ,
which may be termed the "effective" velocity Ve
V.=V cos.
For this reason, in a wing exposed to a circulating flow with slip phencuiena
connected with a shock wave occur at a larger Mach number than in a wing about which
the stream flows perpendicular to the leading edge. This allows the aircraft ie-
signer to advance into the region of rather high Mach numbers, almost without en-
value. For example, if a straight wing of infinite span has M m 0.7, then the
'critical M1ach number at a slip angle X,5?, increases to
countering the unpleaart influence of the compressibility of the air. The differ-
ence in the critical Mach numbers at high slip angles ray reach a considerable
I:
eeEtt:.~.; 3weptback wings may be considered, with a certain. approximation, as wings with-
-out sweepback' but located in a circulating flow at angles of slip equal to the
0
eweepback angle x. In this case, however, the slip effect will be decreased at the
Accordingly, the effective ?4ach number in this case will be
M. =M cos x,
a
.t. for tle.. forces of friction, which do not concern us in this. case.
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center tnd at the tips of a sweptback wing, because of the three-dimensional nature
of the flow in these piacee, which does not permii, tcze Lowe-discussed resolution of
velocity. This is due to the tact that the sweptback wing does not accurately
reproduce the effect of a slipping wing of infinite span. Nevertheless, giving a
wing sweepback is one of the most effective means of penetrating into the region of
high Mach numbers.
Since, in a ;dipping wing of infinite span, the distribution of pressures along
a section per oendicular to the leading cddae is deter pined by the value of the velo-
city Vt V cos X, while, it calculatirg these coefficients, we relate all the
forces and moments to the square of tree total velocity of the oncorr.ing flew, the
coefficients of the forces due to the pressures acting on the wing rust vary in the
ratic
(!a)' ? cap ~.
r
(2.
In the calculatior. it is necessary to Lane into consideration the fact that,
together with the actual Mach number, the effective Mach number
MsMco`x,
mist also enter the foc-tula. In addition, it crust be borne in mind that the angle
of attack of a section perpendicular to the leading edge will not be equal to the
angle of attack of the wing measured in the vertical plane containing the velocity
vector of the oncoming flow (cf. Fig.2.29) as well as the fact that the drag like-
i
wise does not lie in the plane.
If all this is borne in mind, it will be found that, for the subsonic region of
~rnrinwt.~on.
'`} ?? Mach numbers, the aerodynamic center of a sweptback wing;
in firAt, nrn p,--
'/does not depend on the sweptback angles, that the moment coefficient at C ? 9 in-
cresee5~5o~ewhat 1esa with increasing Mach numbers than in a straight `wing, and that
he derivative of the coefficient of lift with respect to the angle of attack like-
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atiH~
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wise increases sorewhat 1639 with increasing Mach rwnbers than in a 3trai ;ht wing.
For this region of t eh numbers, in first apprOxia~aation, the expre35ic:U3 are true.
:'or the su, er'5C,:ic r ,ic l; :,slch nUm1 e."3, w~ will
x' A~ x'Mnx
a~x
C/
i. . that ~+'fL that [firN \.~ 4.1 ~ c_f the wing which r'a be ocT:3j e e a3
wing, ~.V.,a., only Li '4: c:.ter ad .C 1t3 t43.
wi'1"5f t.;;ey are liX' `ii...~ 'L~ ~t, ?.;
?S MRa y - cos
r not ^:~U.' ... ....F Jri~t ur
ihe3e fri.-as ar tre e, 3o ic.., a3 a 3hcch wave ?aG~? s
1c X?+CnN
Y' 1
The value and position of the mean aorodyiia~aic chord of a sweptback wing,
be determined, as before, from eqs.(2.26) to (2.2$).
(2.
will
..?ar~ VAq t.n the case of a symmetric
.a1tboa~B~ .teal. derivation of analogous ere33ion8 for an. aetric profilo
? .. ; r45n? undar antA . dif icult es.
t
,,. For simplicity of discussion xe ~;o;uu~o
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C ---
e
!M Z
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Figure 2.30 shows the influence of sweepback cn the aeroiyn,atic coeff iciets of
a rte,:t, 4; r itc span, formed by 4wc arcs f a eir41e with a 1 elUvc i,t~zc boss,
taken along the relative flow, C U. E for M 2.
A3 will be clear, when the angle of sweepback increases, the lift properties ccf
the wing are improved, the aerodynamic center is shifted somewhat forward, while the
rate of such shift increases with ir;creasing sweepbac,. The coeffiiet cl wave
drag, for small values of y, first declir.cs slightly and then increases. It r5'.
be borne tni : id trlatt, at lar;e angles cf :3weerhac, when one croiuc~, .. cos ,
approaches unity, eq.2.:,7 becozae unsuitable, since local subsonic velocity z:nes
appear on the rofile.
It ust be : otod that the change ii. the longitudinal mci e, on ransition
from subsonic to supersonic flyin; spee~fs is less in sweptbacr Wins than in
straight wir.~ s. This fact L y be of advantage in designing, aircraft for svperscnic
flying speeds.
he 'usele :foment
The moment coefficient of the fuselage with respect to the center cf ravi ty
of the aircraft, by anaio3P with the momer.t coefficient of the wing, i;tiay be repre-
sented by the following expression (Yig.2.31):
where G
CL f
;cf
Xf
f.
m:f. (Cmf+.Yff,,
(2.48)
is the moment coefficient of the fuselage relative
to an axis passing through its leading edge;
is the coefficient of lift of the fuselage;
is the dimensionless coordinate of the center of
gravity of the aircraft relative to the nose of
the fuselage;
is the area of the rectangle described about the
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horizontal pro; ecttOf of he fuselage* the co-
efftcients c j ad c~f are referred to this
area,
is the length of the fuselage, with which the
coefficient e~. is related.
i perier:t has sncwr: that t:;e c
the moment characteristics.
k
,I0
uvs.0 2c
4W
Q19
4W
3,
oa
IQ ID SO
S4
2.3C - Influence of 3'reerbacr; o:. tine
Aerody a ic fharacterist,ics
of a'Wingatl4=2.
efficient$ and c +, are atrc.
JeFa:;de::t C t`?e sha e c: he fure:a,-c.
As ar e.caxr e : ia. .3 - rives .::e
values o~ ti dr'.a c cb'.air:ed _.. ex-
.'
t,erin:ents wit:: '.w r c-eels of a fu3e~.ae
cf iifferent fcr, but ~iosely ~esezx-
.i: ~ each ct er. As is clear, ever, a
relatively s,a1 di: erer:ce ir, the
shape c:' t::e .'uselage .ears to a rarken:
difference i:: The aerc r!a,ic _: arac-
teristics. Ycr this rcascn=, i. is ire-
:'erable, ir. ieterrri::i.::^ the ioer cc-
efficient c' The fusel ^e, to use cx-
periierital data, the ?are so since The
interferer.ce of fuselage and i g still
more cotrplicates the deterranatic:: ci
*'fhis area is selected conditionally. It is possible to select instead, for
w
7.e `or ctcn of the fuAAlnaee Then; the coefficients
,.r7~a'apla, the area of ~1~4 4a-~,.o. .,..,..._....
AAd wculd..change in magnitude and the center section of the fuaelage Mould .
enter into the. equation.
j
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On analyzing the results of the teate showri in Fig.2.32, it will be seen that
fuselage No.1, where the flow around thR transverse section is better at small
F'ig.2.31 - i'or Deter ritis:ins the oment o' he ?uselage
ar ies of attack try: in the fuselage Xo.2, has a smaller coefficient of lit than
fuselageNo.2, at one and the same angle of attack. This is natural encugh, since
the poorer the flow around the transverse section of the fuselage, the greater will
be the resistance to the oncoming stream in the presence of an angle of attack, and,
consequently, the greater will be its lift. It is also clear that, at the same co-
efficient of lift, the went coefficient of the fuselage No.l is greater than the
figment coefficient of the fuselage .No.2.
The calculation of the moment coefficients of the fuselage is somewhat :acili-
-tated by the fact that the fuselages of modern aircraft differ only slightly in form
-;
from that of bodies of revolution. For this reason, in first approximation, these
peculiarities of the geometric form of the fuselage may be reflected in the fora of
r
Ane characteristics in parameter for which the aspect ratio X f , ,y be taken
~; G - f b f m8 ,
There bf is the maximum width of the fuselage in planform.
The estimation of the i ment due to the fuselage may be reduced to a deteraina-
~tion of the additional mapent coefficient (additional to
~,~._.,.~ Hof that produced by the
fixing at ?CL ^ 0) ~arniYto a determination of the shift in the aerodynamic center L :
-4ue to the influence of the fuselage. Such an approach assumes the fuselage moment
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to be 1 in comparison with the wing moment ao that the error in estimating the
fuselage moment thus has no subetatial influence on the moment of the whole air-
craft. This essu ption is closer to the actual situation, the smaller the area of
'ig.2.32 - itesults of Tests of Twc :'uselage ;odels
the fuselage pro~ectian is by compaciscr with the wi:.g area. As he area of the
wings decreases, this assumption becomes less and less correct.
The shirt of the aerodynamic center due to the influence of the fuselage Tray
be calculated, if we start from the following considerations.
The moment coefficient acting on the wing and fuselage is obviously equal to
~~1?~lM?~Il" Ap-`x~_Xt~~ moot
this diagram, with xhbeing the chordwise distance from the trailing edge of the
t.
yh?t. where y is the distance in eleva-
root profile to the tail. The quantity h.t.
btoot plotted on the ordinate. In
tion from the, tail to the axial line of the wake, is p1
first approxiartion, yh.t. may be determined by the formula
yht. ' h + xh.t.E1and
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ads'
t.
tirj
0
,,2
42
0,2
A1.082
/
II078
01
'
'
,
I
I
.
Il
i /
s
M?073
-0
08
,
.
'
i
I
/
I
____
1
004
-
0,
i
--
i
8 ?4 ?0,4 0 0,4
A$?iPw * 'A '!..f MAC
Fig.3.31 ? Deceleration of Velocity at Various Mach umbers
uence of the Operating propeller on the Flow Velocity at the Tail Surface
It is known from the theory of the ideal propeller that the velocity in the
. slipstream far front the propeller disk is eqi*l to
(3.21)
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where h is the elevation of the tail above the plane or the wing chord at a 0!
land is the angle of dow sh (in radians) at landing.
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With an operating pros.U.er, wi,tu the tail i.s located entirely within the
slipstream, the effective coefficient of deceleration must theoretically he equal to
k,-k(1+B).
In reality, the velocity distribution along the propeller radius differs somewhat
from that obtained by the theory of the ideal propeller. In adriition, the tail i.a
Fig.3.32 - Supplementary Coefficient of Deceleration
of Velocity During Larding
usually not entirely within the slipstream. For this reason a correction, which
nay be obtained from experimental data, must be applied to the preceding expression.
The final expression for k, takes the form
k,=k(1+k.B).
(3.22)
The coefficient key allowing for the nonuniformity of the velocity distribution
along the tail span should be taken from Fig.3.18.
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ns"luence of Elevator Deflection on the Tail Moment
!y deflecting the elevator upward or downward, the pilot changes the camber of
the profile of the horizontal tail surfaces. The variation of the profile camber
leads to a variation in the tail lift, causing an additional force on the tail,
directed downward or upward, as the case may
Fig.3.33 - he Curves of CL of the
Horizontal Tail Surfaces at
Varicus Angles of Deflec-
tion of the Elevator
CL h,t. a 'h.t. (ah.t. ' no) (3.23)
is the derivative of the curve Cth ? f (a h ? t ?) and n is
xhere ah.t. Bab
so-called mcoefficient of elevator efficiency". To determine the coefficient n,
the
we ~y use the empirical formula
be.
When the elevator is deflected,
the curve C = f ( ah t, ) of the
L h, t. ?
horizontal tail surfaces is displaced,
within the limits of the linear depen-
dence of CL on a, equidistantly to the
right or left (Fig.3.33) by a value
proportional to the angle of deflectiCn
of the elevator. This fact substan-
tially facilitates the analytical cal-
culation of the tail raorrent, allowing
use of the linear relationship
s f(a 6 B). This relation-
CL.t. h.t.,
ship ray be written in the forrr;
S
S- - zO,9 s ,
S.
(3.24)
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For eale, if the elevator area
amounts to 40% of the area of the hori-
zontal tail surfaces, and the area of
co ration equals 20% of the elevator
area, eq? (3.24) we will yield a coef f i-
;, ~I ~ ~ *,-~t~?? equal to:
efficiency
t
or
N'+~a cient of eleva
n= V o,4 4,8 4,51;
Fig.3.34 - For peterflining the
Coefficient of nevator
Efficiency
where SH is the area of he aievator;
:
5h.t. the area of the horizontal tail surfaces ax.c. is the area of the so-called axial compensation,
i,e, , than part. ^f the elevator area located in front of the axis of rotat on
(ig.3.34) ?
n p,9 V'O 4 0,51.
In this wiY, a lO deflection of the
elevator is equal to a change of 0.570
in the angle of attack of the entire tail. the linear dependence
of the horizontal tail surfaces,
In the region of CL mac
of to the down-
ired' and eq.(3.23) is no longer correct. Ogg
on Clh.t. d is ~ at large ogles of
however, the horizontal tail surfaces do not operate
wash,
1 states of flight, snd in practice eq. (3.23 may
be used attack in the pr incipa
cues for the calculation of CL h.t.'
sibilit of Air on the Elevator ficien
lueY-ce of the Co
The above expression for the coefficient of elevator etticieacy, eq?(3'24)' is
r ..t,nck wave on the
accurate for subcritical Mich numbers ? with the appe&TW~ev ~? ? '--u
cf . inf ra ~ Chapter V.
For ri re details on the jmpensation,
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wing, the lif t~~.~~ ~t.t.~r.1(; bens to decline as a
t coelirc.iet~c., ?w ~., ~..~,.~, ..
o ent of the shock on the upper and lower surfaces.
result. of the nonuniform devea
speeds (Pt ' l), the lift coefficient. again in-
increase of M, however, it once more decreases,
ceases. With further
law expreseed by eq.?(2.37) of Chapter U.
the tail, the deflection of
After a
zone of supersonic velocities has fvrtr~ecl on the elevator is nn longer able to ~~ the P~s'~ distribution over the entire
flow caused by the deflected ele~ator are unable
tail, since the disturbances :n the
to penetrate through the front of sonic
velocities. When the elevator is de-
flected in the region of transcritical
n
Mach nwrjbers, the pressure distribu-
on the downstrea~ part
tion varies only
of the tail profile. The distribution
i
M Z M of the pressures along the upstre&
p
part of the profile, however, retrains
Fig.3.35 _ Relation of the Coefficient wachartged? A low sonic velocity is
to reached first at abo.3't the point of the
of Elevator Efficiency
profile contour at which the rarefac-
the Mach C3utrber M
tion was greatest at small each nul-
, he inflowing strewn increase above Mcrit' a region
bey When the pe,ch numbers of t the trailing
rsonic velocities on the profile contour is p~pa8ated toward
sups distribution on deflection of the
edge of the prof tie. Accardin8lJ-, the pressure
Mile. In other words,
varies over an ever seer portion of the tail p
elevator to drop when the Mach number of the tail exceeds the
the elevator efficiency begs
critical value unfavorable tail profile and at atttall
riment shows that, at an
. Expe
angles o to hero or even change
of elevator deflection, the tail moment may decrease
the elevator
the critical Mach numbers? Under such conditions,
its signs beyond
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deflected through a larger angle in order to create a moment of the neces-
vary value at subcritical Mach numbers.
At K > 1, the elevator efficiency is lees than at subcritical Mach numbers.
If we neglect the influence of the fiscosity of air, then theoretically, at M ' 1,
we have the following expression for the coefficient n;
However, at subcritical Mach numbers, the coefficient was approxizrateli pro-
portional to the square root of this ratio.
The relationship between the coefficient of elevator efficiency and the Mach
number is shown schematically in rig.3.3`. As will be seen, one ann the sane ele-
vator is only about half as efficient at supersonic Mach numbers as at subcritical
ones. This fact, in conjunction with the already mentioned shift in position of
the aerodynamic center at supersonic Mach numbers from about 25 to 50% of the chord
require the adoption of special measures to ensure good controllability of the air-
craft at both sell and large Mach numbers.
General prees ion for the Moment of the Horizontal Tail Surfaces
If L denotes the distance from the center of gravity of the aircraft to the
center of pressure of the tail, we have the following expression for the moment of
the horizontal tail surfaces relative to the axis or passing through the center of
gravity of the aircraft;
M h.t. ? yh.t. L
(3.25)
According to the rule of signs adopted, this moment, for an aircraft of con-
ventional design, must be taken with a ninus sign, while for an aircraft of the
duck type, it is taken with a plus sign.
In the general case, when the elevator is deflected, the tail profile can be
coaridered concave since the center of pressure of the horisontal tail surfaces is
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sufficient for practical purposes, we may take
L Lh.t. ? const
lift of the horizontal tail surfaces u y be represented in the form
the
YAe ?gL beShe.' i
its chord on variation in the CL of the tail. For this reason, speak-
shifted along
the arm L is a variable quantity. However, L ? conat may be taken
ing generally,
error. Indeed, in modern aircraft, the tail area an unts to 20-25% of
without great
the wing area, while the distance from the center of gravity of the aircraft to the
is about 2.5 to 3 times as great as the mean wing
axis of the elevator hinges Lh.t.
that the displacement of the center of pressure of the horizon-
chord. If we assume
surfaces relative to the line of elevator hinges, within the limits of the
tat tail
flight angles of attack, mounts to 2t of its chord, then the value of L will vary
2 Q.1)bA. For this reason:, with an accuracy
aoproximataly within the limits (.f"~
(3.26)
is the velocity of the relative airflow at the tail. This velocity is
where ~h.t.
connected with the speed of flight, in the general case when the tail is washed by
the slipstream, by the relation
Vie. uuiV rI?
yield, taking egs.(3.26) and (2.27) into consideration,
(3.27)
from the moment to the moment coefficient of the aileron, eq.(3.25) will
On passing
3h.t. it.t. ke CL h.t.
(3.28)
Msh.t?
'h4~?!& .-representing the static mment of the aileron area related
'~ . actor ~
to the wing are aM the $fl ;e c chord, is denoted by A.
Since the true angles of attack of the tail are Less than the angles of attack
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for a Two-Fin Tail
t. eff
F ?3.37 - For DeterUinir-~
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de ndence t4L:;ces~ 4~.t. and of ~;.t. ~ within the
of the wing, theta is a linear ~'
range of ang].es of attack, so that we may put
limits of the flying
CI,h.t? ah.t.~ h.t.
Thus, the expression for the moment coefficient of the aileron, for the roost general
caaa, takee the following form:
(3.29)
mM h.t? ~ -k e '~h.t.
,) depends on the elongation of the
derivative a of the curve Cyh.t. ' f(?h.t
The h.t.
urfacee. This relation is shaver: in Fig.3.Y~
horizontal tail s
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?n sew airpiR?" ~1e -i.gns, the vertical tail qurfaee are desi. ned in the fcr
of trro plates (f"ig.3.37) located at the ends of the horizontal tail surface. This
eition of the vertical tail surface improves the efficiency of the horizontal tail
surface. The elongation of the horizontal enpenage.for this case must be deter!ii.ned
by the pir1C3l formula
kAah. t.
a4 P Ekp
where t? is the geometric elongation of the horizontal tail surface;
t.
1 is the height of the vertical tail surface (disks);
V. f.,.
t t. is the distance between the fins (cf. Fig.3.37).
r.. with a Jet engine is
When ~ propeller aircraft is gliding, or when an airplane
in flight, the tail is not washed by the slipstream and there is no downwash fron
rnih.t. expressed by the for u1a
the propeller. For there cases,
,~ t (a+~-E - Ef+nh)
~h.t. h? ?
On substituting here the expression for E by eq.(3.15'), we have
where, for smartness, we use the notation
(3.31)
Confining ourselves to the region of linear variation of the curve CL = f ( a) and
noting that c is a constant quantity, we are able to conclude that, in first
t
is a linear function of the CL of the aircraft or of ao. For
approxisrttios-, ~h ~ t ~ suf f i-
c 1cu1sting aircraft of the duck type it y be considered, with an accuracy
cieat for practical purposN, that the angle of downwash at the ailerons is equal to
zero and that there is no deceleration of velocity. For such an aircraft, rZha.
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 LII
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CHAR IV
TOTAL FOMENT of AIRCRAFT IN 8TL DT
RECTILINEAR FLIGHT
Ninent Coefficient Acting on the Aircraft
The total moment coefficient acting on an aircraft in steady rectilinear flight
is the algebraic stir of the moment coefficient of an aircraft without horizontal
tail surface and the moment coefficient of the horizontal tail surface:
mZ + ~zh.t.
In the simplest case, when 0, J = 0, and the effect of the slipstream on
the tail is absent (for example, in an aircraft with jet engines or in an airplane
flying with windmilhing propellers), egs.(2.59) and (3.30), for an aircraft of con-
ventional design, will give
? ~zobh.t. (xFbh.t. - Xdf)CL - kAah.t.(cL + ' E + nb) (4.1)
It has been shown above, in Chapter III, that the angle of downwash a map be con-
sidered a linear function of the coefficient CL, Within the limits of the flight
angles of attack, CL is a linear function of the angle of attacker, and therefore
the wment coefficient m is a linear function of mZ or CL. The relation between rn
and CL ie plotted on the diagraa in the form of a family of parallel lines. The
value of the dihedral of the tail c or the value of the angle of deflection of the
rndder 6 or, finally, the sum e + n6, as may be seen from eq.(4.1), may serve as a
Figure 4.1
paraester of the family of each straight linen for a given aircraft.
shoes roughly the fors of the relation ? f (a)?
With increasing values of CL and its approach to CL max, as indicated in
and on the curve a "spoon", may
Fig?4.1, the linearity of qa ? i (a) is iapaired-
sasastiasi appear, which has al"eady been mentioned in Chapter II. It must be borne
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Fig.k.2 - Zone of Deceleration at Staa11 and Large Values of a
gtiuiralanta of Uea-tor Dsf3ection and Variation
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that in this case, the "spoon" is not formed for the reaaone indicated in
in mind
Cha er II, since here y D? The impairment of linearity in the case of a low
~ T
plane is explained by the fact that, at eufficiently high a ee of attack, the
Fig.4.1 - Approximate Relation m.1 ? f( a)
horizontal tail eurface enters the zone of the most intense velocity deceleration
(Fig .4.2), the deceleration coefficient to drop sharply and the absolute
,
value of the coefficient ma to decrease corre5pondin819'
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150
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* f (c) is affected by the magnitude of the algebraic sum W + nu , which may be
denoted by
value, different variations of the values of and b are required. For exa ple,
the coefficient of the elevator efficiency is n 0.6, then in order to create
~ =~ +nL
With respect to the slope of the curvets m2, it tykes no difference whether We is
varied by varying a or by varying '. That for a variation of ~e by one and the same
* 3? it is necessary either to vary the dihedral of the entire tail assembly by
_ a 3or to deflect the elevator through the angle ??
It follows frorA this that the longitudinal notion of the aircraft may be con-
trolled either by deflecting the elevator or by resetting the stabilizer.
A stabilizer that can be m_~ved during flight is, generally spear, undesir-
able because of design difficulties and because of aerodynamic complications in the
controllability and stability, produced by any deflection of large surfaces. In
so?e cases, however, the use of a fully deflectable tail assembly is expedient.
Tue, if an unskillful selection of the tail profile causes a drop in the efficiency
of the elevator at high Mach numbers, it rosy be more advantageous to vary the angle
of eetting of the entire tail instead of deflecting the elevator.
Aerodynamic Center of Aircraft, Neutral Centerifl
The aerodynamic center of the aircraft, by analogy with the a.c. of the wing,
tern we will uee for designating that point on the wing chord with respect to
is s
which the ~t .;-etfieient of the entire . ~oee not depend on the coeffi-
aient CL.
1b detsroine the position of the aircraft a.c., we will take the derivative of
_ eq.(4;1) with reipaet to C, setting k ? nowt.
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lac .. ?- kM
i
t
where, for brevity, the following notatiof is used:
0
Then the center of gravity coincides with the aerodynamic center of the aircraft,
4
a
the derivative _ must vanish. From this condition we obtain, setting
A, ZPo. r t.
(L.3)
the fact that is al~ys greater than D, the a.c. of an aircraft of
In view of a
design is always located behind the a.c. of an aircraft without hori-
conventlonal 8n
tail surface. On the other hard, in a duck type aircraft, for which the
zontal
c efficient of the horizontal tail surface must be taken with the plus sign,
Foment o
the entire aircraft is located in front of the a.c. of an aircraft with-
. the a.c. of
put horizontal tail surface. The displacement of the a.c. of the entire aircraft,
appreciable value.
relative to the a.c. of an aircraft without tail, reaches a very
=0.06,a~0.07,
Yor example, if - 0.18, then, at k ? 0.9, A 0.5 ah.t.
xPbh.t.
;and D - 8, We Will gave
fo~.
,v.,
N he basis of ege.(4.1) and (4.2), the a~oeent co.L cient f the ai rt. a.. " "-
4~_represented in the folloiwia6 fore
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*ae1e'M4,f h , e(is+t _*1 +) (4.5)
and corresponds to CL 4.
It will be seen from eq.(4.2) that the angle of tail setting and the angle of
elevator deflection 5 do not affect the degree of longitudinal static stability of
the aircraft; they effect only the value, as follows from eq.(4.5).
If the center of gravity of the aircraft coincides with the a.c. of the air-
craft, then, in accordance with the definition of the concept of the aerodynaadc.
center, the 1 nnent coefficient of the aircraft will not depend on GL. In other
words, in this case, the aircraft will be statically neutral.
Now let
1
X, ?rr Xp t~.r + kA~%t. - D)
i
If we introduce this expression in eq.(4.2), we will see that
(~4.3' )
J~.
sac
S
t
`The centering of an aircraft which satisfies the condition (4.3') is called neutral
;centering.
A somewhat different expression mty be obtained for defining neutral centering.
:with this in mind, let us write eq.(4.2), once for actual f"t*_r&!:p and once for
neutral centering
- (Pale.-z)-hMAir
(x/_..z, e) " *I AAe.
H!- subtracting tie first expression tray the second, 1 get
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Knowing the degree of longitudinal static stability of an aircraft at a given
centering, it is easy to determine the ne~Zttral centering from eq.(4.3"). Let, for
a
exam ].e, at x. ? 0.25. m L ? -0.10. Then,
Let us now determine the relationship
between the total longitudinal moment acting
on the aircraft and the velocity head.
Obviously,
m;Sb -q = m~ ~SbAq +
ddn.
-i (dc:). c~Sb.,q?
Fig.4.3 - Approximate Rela- ant in horizontal flight or in flight at
tions 1 ? f(q) low angles 0 of inclination of the flight
path, as is well known,
} r, ~.eo that
4b_:
aur,
~i 1, - m~ ~Sb ~q ?
\U;
(4.6)
; ? ?bs liaMr relation oft an q has bsa, obtained (riga,.)? _ A rarution in the de
o ng i! 1 s t a etaailtL of + n aircraft does _not affect the. angu1 _ ..
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0, 25 1 0, l0 0,35.
It is obvious that, to secure longitudinal static stability, the center of
gravity of the aircraft must be roved forward with respect to its position at
neutral centering.
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N f ' and changes only the eeg~nt cut off by
C4~
aoaffjaient of the straight line X
tba trai~ht li~ae on the orai-te,
p-
&nojfl f Ltrcrt th ee t to oweflt
the concept of aircraft.lpalancing, Let un now
In Chapter I we already used
? de~i lvi. ~;,~" cf the eleva~
iexpression for the ba ancing angle of
first! the anal~-t
* that the points of equilibrium of the aircraft, at
? .From eq?(~+?l~ we find 4
for 6
which 'z ` 0,
' r a.t,h. Ct _
kA~, to
11
t the balancing angle of deflection ~ of the elevator
Equation (4?7) shores tha
a linear function or C. Wow, bearing in mind that
is
+ DcL
t
ItAokt a
su otherwise .pecifi d, all our reasoning will relate to an
*tar, wnlees tail eur-.
~dni~-,. i?e., to! tircratt with _the..horiso .L ...
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Pigurti 4.4 s-tic~lli showy the r~-lat.ion of to C . For a stable aircraft, b
eca- L
6 increases
es ee increaeee. On the other hand, for an unstable aircraft,
decxese CL .
effi-
t
co
with increasing C. We now present still another expression for the momen
cient of 'an aircraft , which may be easUy obtained from eq.(4.1)? On taking the
?1z
. ___'ti*o..?. o f this a ression with respect to C? and to 6, we may write
partial aori~~~ of ,...__ --T-
lit=flhs0 r. --kAakr.(ao+?--e,)+m= c~ +m~G,
b~-
where
m== --kAah tn.
1
-; (a--e~p_"Ef+i).
n
'die Balancing, Curve
On the basis of eq.(4.1) we may write that still another expressiol for 5:
(4.9)
(1.10)
As already mentioned, the term balancing curve is used for a curve showing the
dependence of a balancing angle of elevator deflection on a parameter characteriz-
the state of flight. The flying speed V or the velocity head q may be taken as
ing
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arch meter. If the deflection of the.eleve*~r'follows the law of the banciflg
curve. then he aircraft wii1 be in equilibrium at all flying speeds. In view of
the fact that, in steady rectilinear flight' differing only slightly from horizontal
flight, the equation
is valid, let us rewrite eq.(4.s) in the following form
! Q 1 ~ i
~.'""'....,..... hAahr m=~ q ' (4.11)
n hAo~t
where
This expression shows that, for an aircraft with longitudinal static stability,
for which > 0, the angle of deflection of the elevator required for balancing,
.:. z
increases with the velocity head q, since the last summand within the brackets of
eq.(4.11) is negative in sign but decreases in absolute value with increasing q.
413.
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.,,.ii ?ibi rd CenteriM
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taticnlly` unstable aircraft, p, while ' decreases with i,ncraas-. l~u' a e,
he. Figure 4?5
depend on the rrolocitr
neutral a-iraraft, 6 dose not
~.~ie, ` As demonstrated below,. the
of the bale lC
shows the appte form :. _.
fora. a ed by the pilot to the control stick determines, to a considerable
, a .i_. _o ~ c,.f the Rt evator. 4b reduce the mag-
~' .. ` ~'
t ~ ."extent, the value ct the angle or deuvcU;~ u
deflection b, the ugle of setter of the tail it a can be
aitude of the elevator it is
increased. If,, at any state of horizontal steady flight
correepoS~'
ve b 0 en the corre$pnndi~g value of a can be obtained from
necessNU9 to ha ,
*3. From these equations we find, by equating their right-hand
.idea to sero
ced state of flight at which b = 0. ~'?TM
.~~where C d corre.pond to the balsa
L qb < d the
it followm that, for a statically stable aircraft in which mz s
the higher will be the angle of tail setting 'P? For a
~-. , higbe!' the velocity head qb
ble siraratt, we obtain the opposite relation; the higher , the
62 statica111 uruta
!, p11ar 'tttt be the reQuirrd angle. Indeed, in etable aircraft, the last term of 66 e4?(4?12) n. tive asd deareua in ab olute value with increasing q This
b
. ~`
eue in the angle of tail ;setting ~? In unstable aircraft, the
~, ls~ tb an in~ 3
cn_ eridt S is rnrei1id.
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ensuring the equilibrium of a stable aircraft at high values of CL.
It. toiowe from eq. (4.8) that the no atire. angle of deflection of the elevator
required for balancing at high CL close to CL , increase in absolute value when
the center of gravity of the aircraft is shifted forward. On determining the value
of xr from eq. (4.7), we will have
- - ~'~~ "`. AID 0 6.~4.ty
Z,?X b.h.t + ~hAo nI+*Aa4.t Ia-a ci (14.13)
If the maximum negative angle of elevator deflection is limited, a certain
maximum permissible centering of the aircraft, ! at which it is still
T form.
possible to balance the aircraft at the assigned angle b. For any position of the
center of gravity forward of xT iena., balancing is no longer possible.
A study of eq.(4.13) leads to the conclusion that the value of xf form. will be
r the greater, the smaller CL, the larger in absolute value the negative quantity
b h.t.' and the smeller the dowanssh e. We know from Chapter II that the maxi-
zi value of m
is obtained when the flaps are deflected at landing. In this
zobh.t.
case the smEllest angle of downwash a is also obtained. Thus the case of c.ilcula-
t .. tiag the wdnnas permissible forward centering will be the case of landing with de-
flected flaps. The value of xT form. determined from this condition will be more
,t)
than sufficient to ensure all remaining attitudes.
., As an example, let us determine the maximum permissible forward centering for
f 2 . the following condition:
64_
Coordinate of a.c. of aircraft without horizontal tail surface, with
deflected flip.: ~lbb.t. ? 0.16;
Coefficieut of velocity deceleration on landing k - 0.9;
Coefficient of static tail mamait A ' 0.5
Deriratia caws of coefiicieut of tail Litt ? h.. - 0.06
't
.aps
1witb0ut-borisosafal tail smrfaca
.~.-iorrpooettiaieat , xithextaxled fl
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LaMing angle of attack -
Landizig value of Ct ?. l.t-t
Angle of dorm ah on landing, alloying for influence of ground, t11and.
Angle of dihedral of tail a -1o;
aible angle of elevator deflection. on landing pax -150;
Coefficient of elevator efficiency n ^ 0.58?
From eq. (4.13 ) we obtain
x~ . i-,lo -- X1),'3 ~,5 wl.U 6 '
w ..1
Thus, for the initial. data taken as an example, the maxirmm permissible forward cen-
tering eric-g is )% of the MAC. At the same initial data, but in flight at min speed
with undetlected flaps, we would have, at O.02, a 90, CL 1.0
? -0.05
xT form.
Mouln be far more for+~ard.
'i.e., the pe~iseible centering
A stabiliser, movable during flight, is a powerful means for extending the per-
centering. In the preceding exmmple, if the pilot had been able to
aiseible tor~ard
wary the angle 4t stabiliser setting on landing, by bringing it down to -5?, then
...;
F
jthe permissible forward. centering would have been shifted from xT form. a 0.20 to
tore. o?lz.
' Liter in Chapter 1, we will return to the case of landing with deflected
"M4 + stability of the aircraft;
l*pr, in connection with the selection o he degree of
4. essibility of the air on
or the present, let us turn to the influence of the oompr
be aoent of ttr aircraft.
~iD ~ .
5 e rcm'aft 1foMQt at }High 1-lvina p! L
i
Of .an
bori*oflt&3 tail. surface, we considered. the
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? , on the
infl.netnce of the compressibility of air, manifested at high flying speeds
ma~aent ctar*cterietics. Let us app the results we obtained there to anal the
influence of the compressibility of air on the u ment coefficient of
question of the
the entire aircraft.
< Me influence of compressibility is negligible and
As we have seen, at F p,
the sage
causes an incraaaa of t hQ 0;ff ists a and a at antrroximately
mainly h.t.
Thus in this region of Hach numbers, a certain small reduction
ratio 1 . 1 -
stability occurs, which is caused by the increase of the term kAah.t.D in
in static
eq.(4.2)?
At M < the wing aerodynamic center is shifted rearward in cost cases, the
coefficient m increases in absolute value, and the derivative a of the curve
0
of the wing and the coefficient D in the expression for the downwash, de-'
CL f (s) C
As a net result, the derivative esi increases in absolute value, and the
crease.
longitudiaal static stability of the aircraft also increases, as shown in eq.(4.2)?
the value of the balancing angle of elevator deflection b required for
Accordingly,
librium at a certain value of Ct also decreases. Tic decrease in b, at an Un-
squi
to
may prove
favorable coehination of the design aerodynamic parameters in aircraft, be so considerable that instead of the increase of 6 with increasing flying speed,
as follows from eq.(4.11), we may obtain a drop in 6, and at high flying speeds we
ht even require negative angles of b. In this case, the pilot would sense a
the
instant
thi
,
s
?~~' ;.?;
__tendency of the aircraft to spontaneously increase its speed; at
~t2 .,_,~so-cull "ling into a dive" on which we shall dwell in more detail below, in
,.P+~
~~.
JChapter IIappears. Figure 4?6 gives the approximate character of the variation
~ ,
f b as a.f~mction of flying apd for this case.
~`g= At supersoniG flying speeds, the wing &.c. is shifted to about 50% of the mean
erodynaatia chord, the dowm*sh at the tal vanishes, the coe!licim-t of elevator
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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
effir-iehey decrea ee and, as the )! numbers increase, the c0ufficiemt5 a and ah.t.
eft there is a distinct change in the level of longitudinal static
decline As e- re
stability of the aircraft and of the balancing curve.
Fig.4.6 - Forn- of the Balancing Curve on
Going into a Dive
In the special case when the aircraft wing is composed of symmetric profiles
and the fuselage is a body of revolution, whose aerodynaaic center coincides with
that of the wing, we have
;.0, moo a.e r ~ 0
so that eq.(4.8) takes the following form
,JFor eubcritical Mach numbers let us take, for a certain medium aircraft,
I
S6 .
1 of - - (Xfbh.t. XT ; 1 - D)C1 (4.14)
!t 0,9 se 0,07
s f ),0?; Aa'0~5; D m 8;
x'z 0,24;
0,06
9
aft 2 ~ 1I 0M _.
2,0,22.
STAT
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c h r1ber3, for exanpie, for M 1.6, we have, for to
n O,36; k ' 1,0;
s f = 1,0; xp 0,5; ahr
M- I.6
~0 o.s1 0.fl + . J 0,0~ : -O61.
~o.5.O.A66 U,124
57,3 YM - 1.
0,056 D=0;
= 014,
S.
s,
j
As a result of the rear-ru shift o e .?
moment arises tending to diminish th. ~a of attack. 1b balance this moment, the
tail suet be given a moment tending to inareUi the angle of attack, i.e., the
dotmward elevator dst]sction must be redueed. Beoause of the decrease or even com
~....,.9..... ,~ ......
plate disappearance of the dorrnr~eh at K > 1, the angle of attack of the tail -
STAT
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a xr-
.;This example gives an idea of the change in the balancing curve on transition from
i{:~.._ a
(sabcritical Mach numbers to supersonic Mach numbers. Obviously, the balancing
gds?~ angle of elevator deflection may even change its sign. Physically, this is explain-
At the same flying height, C, varies in the ratio
ed as follows:.
rnuic center of the dn,'
fit; f th aero~
 0
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nt tending to decrease the
a LC~rif 1ee ct f(r, a
e~?g~.lyva
t~nt
,
d at QQnb
. creaae~ . an e pn the tail., All this taken together, leads
~a of att~-ck during a dive app~-ar- .
levator deflection.
f
e
ngle o
to a substantial change in the balancing a sacs ?
f Hors ants/ Tail'Surface on the~ircraft Lift
~ curves in flight teats of an aircraft gives gives oper-
~te~ination of Po tests of
that differ more or less shaxp1Y from the polar obtained in
sting polars
geometrically similar nndel of the sane aircraft in the mind tunnel at the same
d ) th numbers.
Reymolde an
is explained by the fact that, in r de1 grind-tunnel tests,
This
sitian at all values of CL; usuall:T the polar are deter-
occupy one and the same po
case, equilibrium of moments is not obtained at all values
mined at b ? C. In this balancing value
since each position of the elevator corresponds to a unique
. of CL,
om a series of steady states
f
r
flight
he lar of aircraft in I
deten~-inS t p?
n
-
po
-- L
. various speeds, i.e., at various values of CL each point of the corre
-
".. at be deterrLined from the
s ads to its orn angle of elevator deflection 6, which may
po es in the
curve. The deflection of the elevator introduces certain chang
~: _ _ b-lanc
lift sad drag of the aircraft. a fact that explains the difference in the polars?
he ma tude of the change in lift of the whole aircraft in
~ us determine t B~
aircraft
th
e
the balancing curve coupared to a lift of
{ flight at a displacement by
jvithout horizontal tail surface.
e~uisite for diep1&Ce1t by the balancing curve is
As. shora- above, a pr uent at any value of
on the aircraft. Conseq lY,
F ~ ~equilibrium of the ffgmet~ts aatia8
the condttioa
v~ ? a ~4Ii. Y.
boar~Oflt41.t:1.eurZace meY be represented in the form
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while, for an aircraft with h~arizontal tail surface located behind the wing, this
ew'ic'n must be taken with a minus sign, and for an aircraft of the duck type,
with a plus sign. The ratio Th.t? entering into eq. (4.15) is nothing other than
the additional coefficient of lift, supplementing the lift coefficient of the air-
craft without tail; this additional coefficient, ACL is produced by the horizontal
tail surface, relative to the wing area and to the velocity head determined fror
the flying speed.
For this reason we can replace eq.(4.15), by
whence
~,odr.?f~~,
?C ? 'Lt $6.4t
split.
(4.16)
where the plus sign refers to an aircraft of the usual design and the minus sign to
a duck type craft.
According to eq.(2.59) in Chapter II,
mzbh.t. - mZobh.t. r Fbh.t. xT) CL
on introducing this expraasion in eq.(4.16), obtain
ewe (4.17)
Equition (4.17) contains neither thearet of the horizontal tail surface S
nor the ! of Ilevator deflection b. This is
'3~..~.oX sett 4i
w_ i1aCe under the aodition of ?quilibriua, the tail lift
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 ~ t nth"
the moma4i~ ~? t"jc srcr?~t ,
ui).ibrt' is entirely deteruined by
necieeeary for e9
out tails the horizontal tai] surface.
evaluate the additional lift p~duced by
Let us
With this in mind, let us put
x AIbht, 0P2: xrb44 0,1$.
A . r.
rcraft of the duck type, x~ -C.C; and
For an ` ~
L
C we obtain the following, v$11es of hCL
For Moue values of ~
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C
0,1 0,3
0.6
0.9
1.3
L
0 i -0~
0,00
+t).O06
.0,411
142
0
eei8n
Stan
0S2
0
40.075
+ u.l~
.
,
dei
Duck gn
,
+0,019 +
This of the horizontal tail surface on the lift
-
Table ehows that the influence
For this reason, the influence of CL
is ?m-ll e of etandard desipi.
:.. for an airpl~
., C of the whole aircraft ie neglected in
A
~....; tail surface on the ~
.;of the horizontal
c&lcnlatione of such aicraft?
{ .; c.
r the role of
aerod~ duck. type aircraft,
~ 'table also ehox that, in the ca, of a increiees ~ so that it can
M
.~'tbe lift by the horizontal tail surface markedly
;~~`~~ used
i~ c'oalcu]stion of the aircraft. In princi-
longs? he negleated it-F. the a?rod3- _ .. , since the
:cenal deli r
dig pA?' tnah yuV
the dusk e?~ is re
STAT
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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
of a duck-tiDe aircraft not only ensures stability and controllability of ttt
tail
aircraft but also produce a lift useful for the aircraft.
At suPte'sonic flying speeds (M ' 1), as indicated in eq.(14?17), the horizontal
tail surface will yield a negative lift at all values of Cl for an aircraft of con
.. 1 _....., or
since increase9. In a duck-type aircraft, on the con-
ventional -
trary, ~ b ta.t.
trary, at ' 1, the positive lift created by the horizontal tail surface increases
still pore.
It lust be remarked, however, that in duck type aircraft it is hard to obtain a
proper land.irl riechanization of the wind. In an ordinary type aircraft the deflec
tion of the flaps, yielding an additional moment acting on the aircraft without tai],
is compensated by the additional moment of the horizontal tail surface, produced by
the increase in downwash angle. In an aircraft of the duck type, this compefl$atinp
nrnent of the tail, for all practical purposes, is absent since the doxT1wash in this
duck-
case is egligibly small. For this reason, on deflection of the flaps in a . r.
type aircraft the horizontal tail surface cannot compensate the moment created by
the wing, unless a special mechanization of the tail is provided, which is very
difficult to design.
5a?4
STAT
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CHAPT 2 V
THE EIEVA'IVR 111N(Z t J T AJ4D THE SICK )RCE
The Elevator Hinge went.
M~4,. .,r ~F,s, aor~uivnAt!~1C forces actin; on
.t__ _ r. ~UCSi5 of the .~?.-.
4.~uF
Up to now we have considered the ~.d
the aircraft with respect to an axis passe through the center of gravity of the
aircraft. Besides these momenta, the calculation of the stability and controll-
ability of an aircraft is greatly influenced by the moments of the aerodynamic
forces acting on the control surfaces with respect to their axis of rotation. Such
tents comprise the hinge moments, i.e., the rooients with respect to the axes of
the hinges abcut which the control surfaces rotate.
The value of the hinge moment deter-
mines the value of the force that the
pilot Bust apply to control the aircraft.
The greater the hinge moment the greater
that force.
Let us in~gine that a horizontal tail
surface with a syim etrical profile is set
L., Fig?5.1 - Distribution of Pressure at a zero angle of attack to the relative
along the Tit Profile at air flow (c ? h.t. 0) and that the eleva-
tor is not deflected (b ? 0), as shown in
b'a h.t. ' o.
Fig.5.l. Since the tail profile has a
t
:certain thickness, the pressures acting on this tail will differ from atmospheric
~cLJ? eeeure and will vary along the tail chord and, in particular, along the elevator.
c forces acting on
,, of flow, the serod~naani
.~ ehord. However, because of the symmetry
a~da of_rotatian..._:,The.r.eeul-
,t.~gmeAte._,xit.reepact_,to its
elevator chord, .will .be:equel.-to zero.
J9rc.t....p,rcul.to the
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Since, at s>tr114ch nulebers', the cu> n of the pressures acting on the tail
r ins practically aluchsnded when the velocity of the flow varies, the value of the
STAT
0
wishing to pmduce a certain aerodynamic force on to tail, de-
the pilot,
fleets the elevator (Fig. 5.2), the eyrmetry of flow about they body is dread, and.
Fig.5.2 - Pressure Distribution along Fig.5.3 - Pressure Distribution along
the Tail Profile on Deflec- the Tail Profile with Vary-
Lion of the Control. Surface ing Angles of Attack
the aerodynamic force appearing on the elevator will not, generally speaking, pass
through the axis of rotation of the control surface. The same takes place if, at
Fig.S?b - For Determining the Hinge Moments
of the Elevator
Junchanged position of the elevator (b 0), the angle of attack of the horizontal
but its value
hi
arises
en
,
v
nge ma
'se, a
Will be different in the two cues.
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 Ti]
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where A is the pressure difference acting on an . element _ of elev'atQr surface;
is the velocity head of the relative air flow on the tail;
x is the' arm of an element of the eontrol-surface area, with respect to
its axis of rotation (Fig.5?4);
dS ? bdx is an element of the elevator area;
and the integration is extended over the entire area of the elevator. Since q ;.t.
can be considered constant at all points of the tail, eq.(5?l) can be presenter in
the form
where
and
.5. j . pxdS, (5.2)
r
b$ is the wean elevator chord.
The integral entering into eq.(5?2), as obvious from the above statements, is
dimensionless quantity which, at?small Mach numbers, is a function only of the
le of attack of the horizontal tail surface a , of the angle of deflection of
t
h
an
.
.
g
the elevator b, and in addition also depends on the geometric relations between
elevator an entire horizontal tail surface. This dimensionless coefficient is
called the coefficient of hinge nament of the elevator and is denoted by mh
,,.~ ?
Thu.,
{
? 4i
(5.3)
:_,(5.3~..+~-ghat the condition. oi_geometric and aerodynamic,. similitude
lO9...
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`h 1 ~r 0.5 x 0,2 15.6 (kg-m)
If, while preserving geometric sirilarity of the tail, the elevator area is
increased to S 3 m2, the elevator chord to h = 0.49 k, and the indicator speed
B
toi 200 Iii/sec, then the hinge rxment will become equal to
? Mh
Tig.5..5.. -. Diagram of. dal Aerodyn zic Balance
STAT
0
if the angles of attack of the horizontal tail surface and the
s
deflections at the elevator are kept cor~st-nt); however, at increaa.ing dimensions of
the tail and increasing velocity head the hinge moment increases in absolute value.
For example, if the elevator, in one case has the area 5 ~ 0,5 tri ,the chord
'h2
1 ~; -se
p ?. then
50 m/sec
V
,
i
0.2m, and the indicator speed is
2002
3 ,c 0.49 16 ti 3675 rah
ti
This shows that, at unchanged nh, the hinge moment has increased over 200
times.
Aerodynate Balance of the Elevator
The example just presented shows that, if no special measures are taken to
;reduce the coefficient mh, then any increase in flying speed and aircraft size will
;cause the hinge moment of the elevator, and with it the stick force, to increase
~,..... sharply.
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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
tipn (5?2) shows that placing the ai~ of rotation of the control surface at a ccrtain distance b, to the raar of its leading edge will cause the various parts of
face'area to yield irm+nts of the same sign about the axis of rota-
the control.-sur
be considerably reduced in magnitude.
Lion (Pig?5?5), 9o that the total mo~ent spy
of n aerodyna11i.C control-surface balance, i.e., an arrangement of
This Is the l.~lC~1 of w ,~..,, ~
the control surfaces such that the aerod;liC forces acting on them will yield,
ect to the axis of rotation, a total nor,ent of the desired small value.
with res
p
aerodynA.Uhic balance, wren the axis of rotation of the control surface
Such a type of
is shifted rearward dth respect to the leading edge, is known as axial compensa-
tion.
Horn
compensation
Axis of rotation
Fig.5.6 - 1Diagra1 of Horn Compensation
There exists
is also possible to obtain aerody~rnic balance in other re~ys.
ont
f
i
r
n
jecting
. e 5.6) in the case of elevators with flanges pro ..;the horn balance (Fig.
of rotation and yielding a moment with a sign opposite to that of the
of the axis
tor. There is also a servo-balafce
le
va
t of the e
t ced the main par ,_~.._.moaaertt produ by
". be Produced in tMO variants (Fig.5?7) ? An additional deflecting
~.:..,_ (which may surface..
~
connected with the stabilizer in such a way that
{ kineartically
1ike a amoll :`udder, .
f,... +.
deflection of tha min rudder by a positive angle, the servo tab is deflected by
f
H and irtce versa, may also be installed in the rear part, of the
r,~..a negative angle, a~ }
3
.abe _ preeeur.a. on, tha..eLr+~at r..ara: redist ib t*i i .
k 11 ~} /~ .,.tn,M,thid. cae-e, _ tiia ag
} eA~sre approaches the axis- of rotation of the
~...t~Y...~2~~..the.. center of pre
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.. ,.
elevator, acrid, c~nsegueritly, the hinge >nornent decreases. In the other variant, the
servo tab is in the form of an individaal auxiliary control surface, intlled in
the rear (Fig.5??,2); the mechanism of action of the servo tab in this model remains'
the same as.in the precedinE one.
The servo tab took its name from the servo rudder (auxiliary ruder) which was
nrorasad about 24 years ago? In such an auxiliary control surface the control stick
is not connected with a main control surface free to rotate about its axis, but with,
a servo rudder, so that the pilot controls the servo rudder which, in turn, deflects
the main control surface.
,.:rotation, of
,;control
_..jureace
Fig.5.7 - Diagram of Servo ~.1ance
Servo tab
Control
surface
Servo tab
Of all the abrve f orcr of aerodynamiQ b&lance, the neat widespread is the
;o axial type, in vier of its structural sim licity and aerodynamic pertection. For
ses, the axial compensation has no influence on efficiency of the
2 practical purpo
de vad.ahu .dares...nai:..inGre~-ae..he:.:drag..OLtbe...tail,_ rd~tl.e..a_ dS leCt a ..Qt the. .
=r..,aro~W.t~b...createa aet~ad~nlamic forcea..whicb. are directed to~rd..the.side.ot the
STAT
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0
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'orces on this control.: surface, thus lowrora~ng, the efficiency of this surface. The
servo tab so introduced increases the tail drag.. The horn tab, at high angles of
rudder deflection, leads to a poor flow around the tail, with separation of the
the servo tab and its control, which under certain conditions nay lead to vibrations
~.,~~ .. 0?Pnne~. ~-
flow, and way produce a shock in the horizontal Lail .,rfaee . (wfT1 to the small
structural heights it is difficult to ensure a sufficiently rigid construction of
of the tail and to flutter of the whole aircraft.
On the basis of al] these considerations, axial comper ation is raost often ssedd
at present while the servo tab is leas popular.
It is also possible to reduce the hinge roment of the elevator by installing a
stabilizer that ie movable during flight. With such a stabilizer the pilot, at
Fig.5.8 - Effect of Movable Stabilizer
'_&;..attitudee where a considerable deflection of the elevator is necessary for a pro
t the aid of a special device, to vary the angle of
~~longed period .. of tire, is able, by
.}.~
?the stabilizer installation and thereby to put more load on the stabilizer and less
i
.' Jon the elr~tor (Fig. 5.8)? Since it. make no difterence in chtaini n~ the necessary
1. '
-mcameet of the tail with respect to the eeflter of gravity of the aircraft, whether
31ot J- ~ t~ elawratar or ch n S$ tab aAgl? of.. attack. at -tha..eatira tail,
tbs_
w
p
utd.4. at:_.whiCkLthe. hinge moment of the
combiaetian nt b
tth
1
,
e:
itJuibleto selec
t., Ac.,. _. .
 11
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rudder r -ins withizt allowable liAlita, by contrQlling the stabilizer.
The realization of a stabilizer movable, during flight involves a certain com .
plication of design,. and, as shown later, results in certain difficulties at large
Mach nul!tbers. For this reason, 'in most cases of a movable stabilizer, a trim tab is
used to reduce the stick force for sustained attitudes.
The trim tab is an auxiliary control surface installed in the rear part of the
rudder in the same way as the servo tab, except that it is not kinernaticall' con-
nected with the stabilizer, being controlled instead by the pilot by means of a
separate handwheel (Fig.5.9)? When prolonged flight is necessary in some cases in
Fig.5.9 - !affect of 'him Tab Deflection
which a negative elevator deflection ma;; be required, the pilot, by means of the
handwheel, deflects the trim tab in the opposite direction (downward in Fig.59) by
and angle such that the hinge cioment vanishes or is sufficiently sirrall.
If, at a certain elevator deflection the pilot, after setting the trim tab in
some definite position does not again touch the, trim-tab control wheel, then, the
trim tab will rerr in in the same position with respect to the elevator, at all other
deflections of the elevator, and a force other than zero will appear at the control
stick.
The Coefficient of Elevator Hinge }content
Within the range of the angles of attack of the horizontal tail surface and of
the.angles of elevator deflection used in practice, the, coefficient of. hinge moment
of the elevator
STAT
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tk introducing the generally' adopted natation
we obtain the folloiring expression for the coefficient of hinge moment
The work-up of experimental data leads to the following approxixr to expression
for the coefficients wh and m for elevators with axial compensation (A'. j? !'i P 6 )
-O,12 Su 1- 3,65a4.,,;
Shy
SI
(5.5)
(5.6)
Equations (5.5) and (5.6) show that, with increasing degree of axial compensa-
tion, the coefficients m and m. decline (Fig.5.10). At Sa. x- ? 0.26 both coeffi-
, S6
s
M? v -0114 1 _ 6,6 s~..f aA t
h
Here 3 denotes the area of axial compensation.
ax.c.
cients vanish, and with further increase of the compensation they change their sign,
initiating an overcompensation of the elevator..
If, in designing the elevator, a 28% compensation is selected, then, at small indus-
0004; -0.0020
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trial deviations during the construction of the aircraft, overcompensation may
occur. For this reason, an axial compensation over 25-27% is rarely used in prac-
tice. The order of a*gnitude of the coefficients mh and m for the mean ratios used
in practice: 30 . 0.4 ' 0.24 and ah.t. 0.06 is found to be as
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It will be aeon that the coefficient is a few tines the value of m. In the fly-
ing range, the angle of attack of the horizontal tail varies within far narrower
rn ~sM~e___t -T-~ 19.
~wl`T I I i 1 I I I I I( i____L
05
In
-
n~
r
~e t?
I~
s Vet i 0"2 _^~-,S~ I
Fig.5?1d - m and m~ Plotted Against Degree of Axial Compensation
G$
r.
(5?4')
As already indicated, deflection of the trim tab is generally used for reducing
the hinge e~oment. The variation in the hinge moment, produced. by a deflection of
the trim tab can be determined from the formula
(5.7)
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Limits than the angle of elevator deflection; for this reason the hinge xrrnent is
determined primarily by the angle 6 of elevator deflection. Due to this fact, the
suti~e-nd a in the expression for is sometimes neglected in practice and re-
mhah.t.
placed by the following:
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sere is the area of _ the trinm er;
0
is the mean chord of that part of the elevator
110
is located (P'ig.5.1l);
is the mean chord of the entire elevator;
is the trim-tab chord;
is the angle of deflection of the trim tab.
Fig. 5.11 - For Determining the Hinge Moment Due
to trim Tab Deflection
Influence of the Compressibility of Air on the Hinge ~'aDt nt
We mentioned in Chapter II, that, as shown by S.A.Khristianovich, in the sub-
critical region of F'ach numbers the pressure curve varies with increasing ach m-
ber in such a way that the greater the initial ordinate of the curve at N = 0, the
greater will be the relative variation of this ordinate.
The pressure distribution along the profile of the horizontal tail surface at
email Mach numbers has the form schematically shown in Fig. (5.12). It ma, be con-
cluded that, as the Mach number increases, the pressures acting on the forward part
of the elevator increase in a greater ratio than the pressures on the rear part of
the elevator (cf. Fig.5.12). It follows from eq.(5.2) that the coefficient of the
elevator hinge foment also varies in this ease. If the elevator has axial compensa-
tion, thin the relative role of the compensation in the general balance of moments
increases. At fiat, the total hinge noment varies only slightly at relatively
177
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31 t~a-ch.;n -ber! (loWCr tbsl t ) despi e- ?
n..M, .. r
t htot al- hinge monr~nc r~.es. it ~ Y~..ris,., th mare a
incpid rate of increase of the
sli_g,h#, ..~ .,.. at,, ~relativel
varies or b ~ 1
Fig.5.12 - Pressure Distribution
,long the Tail Profile
at law ~Sach Numbers
Fig, 5 t13 Slope of the Curve rn ; f (;)
at tech ~hubers Near the
Critical Values
ntribution of the trim tab to that moment (Fig.5.13)? Then, above a
relative co roaches tree
ertain Mach number, the aerodynamic moment acting on the trim tab app
c
c xr *flt acting on the rest of the elevator, the coefficient
value of the aerod
of hinge moment begins to decline, and with further increase of M, it iray even
change its sign.
Fig. 5.14 - Influence of Compression Shocks on the Coefficient mh
theoretical considerations, we must expect that, when
on the basis of
Thus
,
reletive flow at the tail increases fond the critical Mach
the Mach number:ot.the
178
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r;
with perwak{." i 1
~.~ Lt..~h~ !,~;1 s ~:nrec-+~ff#a:i.eni;, m.. of 'the elevator ~ axial com s n .,~..~..
~
the :fora an the con-
; decrease d that overcotalpei5&ti0f may set in. In this cep
trol stick maY and tinge control of the aircraft will be made mare
` change its sign
Fig, 5.15 - For ~leterminifg the Angles 3 in Supersonic Flow
critical )+ach number on the tail, the phenomenorn be-
difficult. After exceeding the
Cosa ression shocks appear which, at increasing 1'ach run-
comes still more complex. p
ward the trailing edge of the tail and at a certain critical 1ach nun-
ber, shift to
on the elevator. This shift will not be the sane for the upper and
her will impinge
1 surfaces, so that the picture of flow around the elevator is substan-
lower tai
orexample, let a compression shock appear on the upper surface of
tia13,hanged. F
the wing at a positive angle of elevator deflection (Fig.5.14)? In this case, high
will act on the trailing part of the elevator (on the tab). Owing to
rarefactions
the confluent flow created by the depression of the elevator, at this moment, on
surface the compression shock will still be located on the stabilizer,
the loner ,
o nsator small rarefactions will' As a result, overcompensation
and on the c mPe
will take place. At somewhat higher values of h the. compression shocks on
-
both upper and lover elevator surfaces may prove to be located close to the trail-
a of the elevator. In this case the coefficient of hinge moment increases
ing d;
,5hu'3Y , since the tail will now be entirely in a supersonic flow.
p
y
As we already know (cf. Chapter II), the additional pressure at a certain
s be-
~1~.poior the profile contour in a eupersoaic flow is proportions/ to the angle
~l
.,J ts 'the tangent to the profile contour. At that point and the direction of the
.
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where denotes the angle between the elevator contour profile at any point, and
its chord.
The excess pressure on an element of the upper surface of an elevator will be
Let us imagine a horizontal tail surface with an elevator deflected by a cer-
le _ tho ?elevator having a constant chord and being set at a certain angle
tain
of attack; (Fig.5.15)? The ankle of an element of the upper elevator surface with
the direction of the undisturbed flow is equal to
044+
Correspondingly, for an eienent of the lower elevator surface,
S.r
1~/?$eel.
~.
(5.8)
For the corresponding elements of the lower elevator surface, the pressure is equal
:r to
''2 2
V MY--- ~ V M ~
The resultant of the aendie force acting on the element of elevator s
I tans dS ur-
. lcax where ! is the elevator epan, is obtained by subtracting eq.(5.8)
~... ~
-from eq.(5?9) and multiplying the difference by the velocity head and by the sur-
!
:face element
1
r
lea
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1d (a 4 c Ordx. (5.,icy a
dMh
M~: . (a+a) q( dx.
On integrating eq?(5.11) from x '? -b to x b - b, where b is the chord of the
axial compensator, we find he hinCe rx rnent acting on the whole elevator
,~ Snt?C
` SN
(5.12)
On dividing eq?(5.12) by S baq and then taking the partial derivatives obtain-
ed, with respect to a and b, we find
(5.13)
where, for calculating mh, the angles a and b must be taken in degrees.
The expression (5.13) shown that, in a supersonic flow, the coefficients
,~ '
The Lament of this force with respect to the axis of rotation of. the elevator
Fib.5.16 - For Deter the Cetf icient ir- a Superaonic Flow
4 (hn b?)= b;
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that overcompensation sets in at a.... '? > 0.5. We recall t?i-at,
in 'a eubsvnic flow, mh was considerably enaller than n, and that overcocrpeneation
in that case began at ?23.
B
In deriving aq.(5.13) we did not allow for the influence of the vjscosity of
the air, which, particularly for the trailing part of the rudder, rray introduce
considerable corrections in the result obtained (cf. Chapter II). However, the
expression so obtained does give an idea of the order of lragnitude of the coeffi-
cients m and m in a supersonic flow.
For exarrpla, for a tail having he dthension of the previous enpie, let
M 1.5. Let us calculate the coefficients and . From eq.(5.13) we have
m~ mew- 51,3
(1--2.0,24)= --0,0162.
P1
For subsonic velocities, with this example, we had
mk -.0,0004; m~ -- 0,0020.
Thus, the hinge momenta obtained in the supersonic flow are ach$lly consider-
,1.5,17 .- App?oximt!te Course of
% ? f (M)
ably greater than in a subsonic flow. In
addition, in a supersonic flow it is
absolutely impossible to disregard the
dependence of mh on the angle of attack
of the horizontal tail surface, which is
sometimes done in analyzing a subsonic
flow, the more so since, as we know from
Chapter III, in a supersonic flow there
is no dawnwaah at the tail and the angles
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coffer from the an las cif ttack of the wing cn1 r the u r
:~..~tta.: of the tail
titY. ~, ich is the clihed~-] of the tail
m
(M) for an
.
Figure 5.17 schefl ticallY shows the ~1ope of the curve rt f (
elevator.
giving ~... ..:..wf:s-1 tail surface and thus also to the giving arreepback to the entire ,~~a~..___
.. hilted toward considerable higher Mach numbers, in
elevator,, the shock crave can be s
of
satisfactory slope of the curve r f (M) over a wide range
this ray obtainz.na
ach ni bers.
Forces at the Elevator Control Stick
cay connected as
or ntrol 'wheel for the elevator is kinernatill
The stick o. co
he general case, the kiaezmatic chain ray be established in
ahorrn in Fig. 5.16 ? In t ~
. gig.5.18 - Kinematic Diagram of the Elevator Control
s
such a that the a-ngl n of the stick b is not equal to the argyle
e of deflectio p
ed mange moment,
n of the elevator b? It is obvious that, for an assign
x at deflection
int of
fmn the axis of rotation of the stick to the po
where by iar the dfsta-nc
,anon of the force by the pilot.
- P4 R4 ant .
f4
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Aral coutpensation
Horizontal. tail area
ma0
Angle of tail setts,
k? 0.9
Coefficient of deceleration
. ,..... _
-
Leh At tail given by the expression C? s O.S* CL
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the coefficient of elevator transmission (although thin term is
is usually' called
t
e
.
c
incorr
d .15 b is the working height of the stick (from its axis
In egs,(5.1~) an (5 ), P
int of application of force by the pilot) and is the linear
of rotation to the po '
The rule of signs for forces or, the stick is such that
di5p1ce.~ent of the stick
when the pilot m?-:st push the stick away from hip, the force is positive, and when he
pulls it toward him, it is negative.
fearing in mind eqs. (5.4) and (5.7), eq? 5?~ ray no be rewritten in the
folloxing fora
and r: t
By means of eg5.(5?14) and (5.16), if we know the coefficients m, r4,
calculation, the force on he stick ray he calculated for all
from experiment or
states of flight.
For example, given the following data for an aircraft:
G a ~ kg~m~
`ding loading G
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Derivative of the curve CI f
.I11 t.
( ht) of tail` $h.t. ?, 0.06
..
Derivative of the curve CL f (e) of wing
Let the f1ier. bt means of the 'trim tab, ensure zero stick force at the zndi-
a 0.075
a-
0
Coefficient of transfer from elevator to stick km 1.6
Angle of zero wing lift
cated flying speed Vib 150 m/sec Dui &t a rudder deflection angle of b = _10
state of balancing by force). Let us determine the value of the stick force at the
(
indicated speed V. 200 m/sec and at a rudder deflection angle '1??
1
The lift coefficient in the state of balancing by force ' is determined by the
1O.
4,aea
The angle of attack of the wing in the initial state is
$- p- - ?1 .
O,O1S
of attack of the tail at these states of flight is
1ht_*+t10!SOJ*)
1135
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Since the force dur'jn,
;in thin a e is also zero:
... ..., ",
h R
In the new state, the coefficient of hinge moment is equal to
~ ~ ei-r~ afNl~t*~IA'~(e,.-e.)+ (1- L (5.17)
On introducing these numerical values in eq.(5.17), we get
the functional relationships of ah.t.
STAT
186
If we neglected the summand a and made our calculations from eq.(5.4+), we
h.t.
P ^ +3.0 kg
In calculating P we did not need to determine the angle of deflection of the trim
Relation of the Stick Force and the Velocity of Horizontal Flight
r~d.r to atablish the relation between stick force and speed, in the
~. -ll
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,.. N .. , lb For the time being, ire
flying eyed must be substituted in eq.(~ ). .
f the
d ~
, o
. s:n
, consider that the cotapresaibilltY of the air has no i.nfluenCe on the aerodynamic
0 is obviously' valid.
p
equation m
the state of balancing, the
I
n
oefficients?
c
:From egs.(4?8') and (4?10) of Chapter IV, at balancing states, we have
I' r
,m bht: y' ~ .?. ?_:'
i." a at
**
'
4
horizontal gyring ~S Cq~i1, on the basis of eq. {x.19)
The angle of attack of the
'of Chapter III, to
coefficients of hinge moment in states of balanCing in horizontal
the .
light are equal to
a h 1. A1~~ h t.+ A1+ m jlt -
Aj A
- D cL 4'
C
-EL
.4~
. The stick force for this. states of flight is found from eq. (5.16):
w p and with
and
in an aircraft without longitudinal static stability (mz )
''~ ... ~ 'That,
control surfaces., the force on the stick declines with increasing
overcompensated
rea:ty head (Fig.S?~)? For a statically stable aircraft, if the condition
+
....~ l. . ,
ied..tb! force on he stick also incrSsa with the velocity head. If,
condition is not satisfied, then even for a staticalli stable air
f r:,..this
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craft, t,.e forcc .pan tll a4k u~:+cr~~e "wjt1~ increasing veioc1 t,y head
The increase of the stick force with incr+'3ing.vloc ty head '(4 ' 0) is
necessary for normal control of the aircraft. In this case, the pilot must apply
additional force to inerea~e the flying speed; the general i up rescion is that of
z force r
r
h? , f"V\ to o to effort or to .let to increase the flying speed.
ViV ~rfa&ft M the
In the cue of a negative derivative of the curve of stick force versus velocity
head ( 0), the pilot must counteract the tendency of the aircraft to increase
its speed. It is evident that, in the former case, the aircraft is far more easily
Zi - Stick Force Versus Velocity Fig.5.21 Loads and Springs in the
System of Elevator Control
As we will see later, the slope of the a P f (q) or P ? f (V)
4:'... ,'. close]y related to the degree of longitudinal static stability of the stick-free
Influence of dicers a Serrits s on Force on Stick
Scarti~IrN, to is~rove the stability of the stick-free aircraft, special
ht$ (balancers) or ep'ings are introduced into the control systea. Figure 5.21
i
WS
~
lit th
e
"aircraft.
rea
J ~T
shows the d vicee placed on the control Stick itself , although in y
o b-.
irts of the eleratbr' control erts.,
p
located on
s
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produced by these devisee, in contrast to the
Olb~-iour~lr this h~ moMUt, _ . , .
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l?ls '7
aerodynamic hinge rament, which is proportional to the velocity head, does not
,4opsnd on the velocity head tat all and is determined by the mass of the w&ight arid
F
its location, or by the tensile force of the spring. However, the coefficient of
the hinge moment due to a balancer or a spring, does strongly depend on the velocity
head and is inversely proportional to it.
It must be borne in mind.that in flight at constant load factor, both these
devices, the balancer and the spring, act in entirely the same way when the state of
flight changes: the hinge moment due to either device does not change. On the
other harm, in flight with a varying load factor, the action of the spring and that
of the balancer differ substantially. When the load factor varies, the hinge
moment produced by the balancer varies proportionally to the load factor while the
hinge moment produced by the constant tension of a spring, remains constant.
If the pilot releases the control stick whose system includes balancers or
springs, then, in addition to the force due to the aerodynacsc hinge moment, a fixed
force will act on the stick and will be determined by the mass of the load, riulti-
plied by the load factor, or by the tensile force of the spring. Under the influ-
ence of this force, the elevator will be deflected in the sw a way and will produce
a moment with respect to the center of gravity of the aircraft.
Let us determine the degree of longitudinal static stability for this case.
When the state of flight varies, the rotation of the elevator takes place
relatively slowly; for this reason, the influence of the angular velocity of eleva-
tor rotation on its hinge moment will be neglected. In this case, two forces act
on the control stick: the force P b produced by the weights and springs in the
sp
control system and the force Pa created by the aerodynamic hinge moment of the
elevator. At each instant of time, with a free stick, these forces will be in
;equilibrium, so that the equality
Pbs +Pa-O
p
is valid. In the general case, when the flight takes place at a load factor n 1,
STAT
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the load on the stick due to the balancer and springs is equal to
P. sp a -npb Aso
The force on the stick due to the aerody is hinge woment will be:
pa ?' ~'khmhSBbBkq ~ h BBB ~Y-c h.t.
Thua,
Frog this we find the angle of elevator deflection with the stick free:
RI'. + I , w h
4 '"`
i
a
;
,
i~S.tm,v a s
(5.23)
Because of the deflection of the elevator, the moment coefficient acting on the air-
craft varies by the quantity QmZ m6 and the total coefficients of moment of an
aircraft, with the stick free, will be equal to
111 , ?
S.
(5.24)
.~ a(5.25)
:J Noting that the angle of attack of the horisontal tail surface according to
- .q.(3.39) of chapter III, is equal to
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?' 1ST ~ qpe; +m)
In accordance with the definition of the concept of overload:
nG . CL gq
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The degree of longitudinal static stability in the former case, as we already
f know (cf. (Chapter I or Chapter Iv), is evaluated by the partial derivative.
dtnr:. ~ittl. taros Y.
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r
~ ht ..(~ ,. D)t~+fr+ .- 4
and making use of C L.(5.20), we may, after certain transtormations, write eq.(5.24).
for the coefficient of stick-free moment, in the form
a
(5.26)
Let us now consider two special cases of flight. In the first case, let the
flying speed remain constant during a variation in the angle of attack a or of the
lift coefficient CL; consequently, let a certain load factor n 1 appear, since
the lift is no longer equal to the weight of the aircraft. In the second case, let
e
the flying speed vary in such a way with varying a or CL, that the equality X ' G
is satisfied, and, consequently, the load factor remains, as before, n ? l..
The forrser corresponds, for example, to a sharp >ztneuver of the aircraft, when
the speed is unable to vary, or to a flight in a disturbed atmosphere ("bump"),
when, due to vertical wind gusts, the angles of attack vary, while the flying speed
remains practically the sane. An example of the latter case of flight is the
take-off run or the deceleration of the aircraft in horizontal flight, when the
pilot simultaneo~ly moves the elevator and the throttle of the engine.
it would be more accurate in these arguments, to take the total aerodynamic
~t
5 1oren_ g_.it*tNd. of the lift 2; but in practice, at the flight angles of attack, R
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?varying load factor and constant speed.
it ir- the latter
l
with a ZignG
ct to overload, ~phasixing by this term that we are dealing
spe
derifltive is called a nteasure of the static stability with re-
his
.
rder to evaluate the degree of longitudinal static stabx
~' ? _. ~ __ in ~ the ~oerri-
cane it is necessary to bear in mind that, on any variation in -L.
,
a, owing to the variation of the flying sperm V or the velo-
cient * m likewise varie
city evaluated
city head q. In this case the degree of longitUi. inal static sta6il.i.tq
ivative ~ - which ray be represented in the form
by the value of the total der -
the derivative d may be found from the
In the special case of horizontal flight
general equation
By differentiating this equation, we get
whence
dais ed the assure of static etabilitl with respect to velo-
The derivative ~. is
eaphui$S$ the fact that we are dealing with flight at varying
4~ city. This term
speed and 00A5t nt overlaid.
with respect with respect to fling
in Chapter fU?
l below
,
be considfrsd in detIi
wSll
s
195 ? STAT
0
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z
the absence of the influence of conpreseibilitq, . " in L
Taking the j*rtial and total derivative of eq.(5.26) with respect to Ci, we
(5 ? 27)
to speed at n covet 1) in stick-free flight, bearing in mind that in
`respect ( C
obtain the degree of stabLitty with respect to the overload (at q coast) and with
(5.28)
I,,
? " I
1d r~
As mentioned above, and as will be clear from the expressions obtained, the
stability with respect to overload is effected only by the balancers, while the
stability with respect to speed is effected by both balancers and springs. The
stability with respect to load factor increases with the force due to balancers;
the stability with respect to speed increases with the force from the balancers and
from the springs.
It is obvious that, in the general case, when the control system includes both
springs and balancers, the force on the stick will be obtained, in horizontal
steady flight, by adding the forces duo to the spring and balancers to the force
defined by eq.(5?21)? On performing this operation, we have
?
p,
(5.21')
In the state of balancing by force, at q ? %, the force on the stick is equal
to zero. From this, by analogy to the prgc"'v-an~ eq ation ,
j
obtain
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*m.6ht +
/'a.? ??
By substituting this eq'ation in eq.(5.21'), we obtain
. p' m4. " _ Q _... ,n ;- (5.22')
1 }d
On sing eq. ( 5.22,) and eq. (5.28) , we cone to the conch ion that
P 1'' ' 1-
~rt ~~ Ie
(3.29)
may be written instead of eq.(5.22')?
We renind the reader a$pin that eq.(5.29) was derived under the assumption of
horizontal flight and the absence of any influence of the coiipres5ibiitY of air.
A lore general case, allowing for the influence of compressibilitY, as well as for
a load factor n / 1, will be discussed in Chapter IX.
Froart eq. ( 5,29) , we can draw the following conclusions
1. The cMracter of the elope of the curve F ? f (q) ie completely determined
the degree of etick-'free longitudinal static stability of the aircraft.
2. For, an aircraft that is neutral with the otick free, the force on the stick
for all flying speeds is .q1 to zero.
Pot' an aircraft that ie unstable with stick free, the stick force declines
3
with increasing velocity head
It the relation brtwetn stick force and flying speed is known from flight
aircraft?aad the coefficient Fz is determined by Bosse method (by
tests-at t
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nd-t,unnel r d 1 tests or from flight te~-ts) the degree of stick-free stability of
A rrr ft trt he c1RtMrn+ined from ea. ( 5.29) ?
N
e
e0
By using springs or balancers, the characteristic .., which plays a substan-
tial role in evaluating the controllability of an aircraft, can be varied to a con-
siderable extent.
If the value of the force on the stick due to aerodynar:ic forces, as deter-
min by eq.(5.22), is added to te farce on the stick due to the balancers and
springs, then the total stick force in horizontal flight will obviously be equal to
.T~ ... , U.. Nast -
I as
(5.31)
w r -- V eI l- S' - p -Ps.
~R
This expression shows that the value of the velocity head q ? qb, at which the
.
stick force will vanish in the absence of springs and weights will no longer yield
y. a zero stick force when springs and weights are present, since this velocity head
only causes the force due to the aerodynamic hinge moment to vanish.
*1r eq.(5.22+), giving the value of the force applied by the pilot to the
etick, the quantity Pb and P must be taken with opposite signs, as we have done,
kp
since-the force applied by the pilot is opposite in sign and equal in magnitude to
~t -- the force due to the loads or springs ; if the spring tends to deflect the stick
r { Waw 7 ro4 the pilots, then he pilot must. apply force in the direction Nto+rard
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From eq. (5.29) , we. get
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The vanishing of the stick force will take place at another velocity
which _ia determined from the condition
head 4 ~ q
whence
l; - i',, O.
to obtain a balancing by force in this state of flight, the pilct rust
In order
make use of the trim tab or riust vary the angle of stabilizer setting, if the air-
craft has an adjustable stabilizer (in Such a way tt*t the resultant additional
stick force is equal to (P * p). The hinge mordent obtained as a result of the
b sp
t S$b~kq or of the variation in the angle of
deflection of the trim tab
stabilizer setting 1 Q&c~S~b$kq and the corresponding stick force, are propor-
to the velocity head q? For this reason, at a velocity head different
tion/
from qb, the additional force on the stick will be equal to
y
?p ~ (Pb ~ psp) qb
the derivative of P with respect to q, in the presence of springs and
Thus
weights, wild- be equal to
a1' . aW .1'
5.32)
If for any cause, a negative derivative is obtained in flight tests, thus
interfering with the normal control of the aircraft, the position msy be corrected
by introducing springs or weights: of a definite tensile force or mu, as the case
tem. This is illustrated in Fig.5.22.
be, in the control she
l
in
let the d, ivatiT. of the force Pith respect to velocity hea
,,thaili,ght est$ of an aircraft be ? -0.0025 with a vehocitl head in the state
..: t aq
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.vieihd cwwl
Fig.5.22 - Change in the Gradient ? P by Mean of Weights
ar~d Springs
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or baian`eidi err force, `l
To obtain on this aircraft a derivative of stick force ..,.'. - t 0,0025, it
introduce a spring or weight in the control syster, of such
would be required to
dimensions that the resultant force on the stick would be
__ dP 1000 (0,0025 + 0,0025) =5 ice.
I'b4-pP' "h dq ^'dq --.
If, for exaaple, the arm of the load with respect to the axis of location of
by
the stick were equal to 0.25, nhile the operating height of the stick ' 0.5 m,
then the load required would be
G,=5us=1QK4..
Together with the weight, a epring with a correepoMing tensile force might also be
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20i
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U ra is Boosters ai4~ utot~-tic ? udder Coryt~al
. r~ speeds, when the influence of the coc~pressi-
t high
~, we have seen, a
considerable varfaticns
ronounced, they stick forces undergo
i
s p
f the air
bilit o ht making control difficult. These difficu]ties can be
sand map reach high levels t ht the stick
udder o ~..e =
h
e r
from t
o-rolled "nonlinear" transmission
a
overcome by using we know, the stick.
tro acing hydrIulic boosters into the control system. As
ar by in
force is equal to
t other conditions being equal, is proportional to the square
while the hinge monen ,
of the flying speed'
ON.
Fig?5.23 - System of 'Irreversible draulic Booster
. a force an the stick at high flying speeds, it
For this reason, to avoid too high
k variable during flight, which decreases at high flying
is desirable to have a h' selecting the proper
s. Such a nonlinear transmission can be obtained by
~ to small angles
_?p?~ nd
tics of rudder control. Since high flying speeds correspo
db
...
t the derivatives ..
gy
lected tha
malice might. be so sep
of ceder deflection, the lone
tive or positive angles of rudder deflection. In this cue,
decrease at small Heys
ame elevator deflection, at high speeds, would correspond to
_~ however, one and the s
?
greater stick d.t1ecti(' ~~?.-- low pdS.
drawbac
th
e
the stick force does not eliminate all
~ 'lhie method. of reducing
a
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entioned, the generation of a force of o posit . ai n is twssihle. A nonlinear
i
transmission cannot correct such a situ-tion.
A far chore effective means of normalising the efforts is to introduce hydraulic
boosters revereibie or irreversible; in the control system. The mechanism of
action of the irreversible hydraulic booster is shown in Fig.5.23.
On pushing ushing the control stick forward, which is not directly connected with the
rudder, the pilot displaces the valve slide of the hydraulic booster in such a way
V'~ + 'IIrow 4 V,Yo
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Fig.5.24 - Diagram of Reversible Hydraulic Booster
that oil under a definite pressure, produced by a pump, begins to enter the
cavity (a) of the cylinder. Under the pressure of the oil, the piston is displaced,
and thus deflects the control surface. On deflecting the stick in the opposite
direction, the oil, enters the cavity (b) of the cylinder, instead, and the rudder
is deflected toward the other side. In this way, the total hinge mordent of the
rudder taken up by the piston of the hydraulic booster, and the pilot needs only
a minor effort to displace the valve. Since these negligible forces will not give
the pilot "the feeling of control", the desired feeling of effort is ordinarily
,
,Ircrested idols epee *prin!m or weights in the control system, whose forces
d.psfld on the velocity heed car the Mich nwber of flight. In this case,
th. pilot , is not at all confronted with variations in the rudder hinge mo>aent due
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nf1uifce of the ca rmressihilit of the air. 5ich a booster is therefore
i,
irreye 5tb1e"
bl V%
i e
The reversible hydraulic booster in Fig?5?2+ differs from the irrevers
that it absorbs only a portion instead of the entire elevator hinge mon~m as the
ft is la
f the aircra
? !1t shich a1rp7e cow teracte the rotation o,
(6.4)
' Accotslii-8 to eq. (6 "2), the additionsl coefficient of lift of the horizontal tail
surface is equal to
borisogtal tail eurfaci
< 0,
locity of rotation w
a
and ' ve
tbat at s ne stave.
pTOVS ,
S
t
$7
Iti.s
O
aows-t it PbsitiYS i.e.., OPV "' the rotation of the aircraft. In a
cated in front of the
l
o
face is
vhiah th.:horisoa t tail sur .a.~.tt. is .
'? ..
moaMat is ne~4tiYe a~ h?
ef.. ~triwr..of:..t~::~t, th. '~ .
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?Tbe . A,,srtsn ~uur~- ~`~"?+r&cts the rotation of the aircraft, but at
the beainninR of a rotation it is unable to restore the aircraft to its original
Fig.6.6 - Damping Moment of Tail in Duck-Type Aircraft
position. As shorn by eq.(6.6), the damping moment changes its sign with any
change in the sign of the angular velocity A ? Thus, if for any reason, whatever,
an incipient rotation from the position of equilibrium is interrupted, and is re-
placed by a rotation torard the original position of equilibrium, then the damping
s ant will counteract this rotation j t as it counteracted the initial rotation
from the equilibrium position.
In the analysis of stability problems, the actual angular velocity.. can be
conveniently replaced by the dimensionless angular velocity
(6.7)
~.? On substituting the diniensionlesa ang1ar velocity $, according to eq. (6.7), in
we get
6 )
. (6
e
,
.
q
_V the expreeeion for the damping moaaent a
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we ll.frequently use the derivative of
In our eubse+ieut analysis, Zh.t.~ with
reaped to the dimensionless angular velocity . If we denote this derivative
by mah.t.' eq. (6.6) U. yield
On couaring q. (6.8) with eq. (3.30) in Chapter III, for the moment ccsff icient
of the horizontal tail surface, we note that if the moment coefficient of the hori-
~'t?~'t',
zontal, tail mzh.t. is Pr?P?rtianal to the coefficient A - representing
$bA
the dimensionless static u melt of the horizontal tail area,. relative to the center
of gravity of the aircraft, then the coefficient of damping of the moment will be
proportional to the ratio ~h?t.Lh.t.~ representing the dimensionless moment of in-
ertia of the horizontal tail area relative to the center of gravity of the. aircraft.
At a given value of A, it makes no difference in obtaining a definite value
the coefficient,cnament of the horizontal tail surface whether we vary the
of
ratio Ior?t? . To obtain a definite coefficient of damping moment, how-
gh.t.
. .... -~--
A
ever, this does make a differences the aircraft in which the dimensionless arm of
.... .
the horizontal tai
a~
w moment at carious values at A.
It at be born, in mind that the danping properties of the horizontal tail
H I
~~ ...:M.~,. ....._. _ :... _
o, at . angles of. attack ey decrease or even vanish entirel At an angle
C ft i.
i .
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Vi* i*tion of thr Flow arot*Y the Wing on Rotation of. the Aircraft
Tbrad4itional a%glS of attack at ax point of the wing chord, due to the
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or ' a ao igthat the true angle of aLtbck , ,,,, , tti1 exceeds the
of nttmcl, o e t~~ ~ i
crease in
n
of attack at ,oh the C of the.Itail'beromes equal to C. , an
L L mt3c
d to
tail surface obtained. on rotation will not lea
..t attack of the horizontal
wae_obtained, on the iinta-r aa~,...r... of
f the
an increase of the GL of the tail, as attack of the
angle o
f(o ). On the other hand, the increment in ;.curve ~1ht
h.t.
and the
)
,
,
decrease in C (in this case ACt.t. will be negative
to a ~,t.
. tail leads
horizontal tail will not damp the rotation but will tend to inten'ify it. However,
at large angles of attack, as may occur in a tail spin, the whole theory of
stability becomes inapplicable. In the present discussion, we confine ourselves to
ma,
the consideration of small angles of attack, when the relation CL 1(A)
considered linear.
Moment of the_Wins
limits of the wing chord located near the center of gravity of the
Within the
angle of attack produced by the rotation of the aircraft
aircraft, the change
be considered car~stant, as was the case in the analysis of the danip-
can no longer
ontal tail surface. In this case (Fig.6.7) the signs of the addi-
b ~ by anal the angle of the
of attack ecw,e out different in the leading and trailing parts
1rig.6.7
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(6.10)
where is the coordinate of the center of gravity of the aircraft relative to the
leading eidge of the wings.
The occurrence of these additional angles of attacks, according to the curva-
ture hypothesis, is equivalent to the corresponding cataber of the profile; the aero-
dynamic characteristics of the fictive profile of such camber will not coincide with
the aerodynamic characteristics of the actual profile. 'As a result of this, an
additional rorannt, damping the rotation, rill act on the profile.
Without going into detail, we will present only the final result obtained for
a wing of finite span and rectangular planform (Bibl.8)
Ip
. (I-2.r '-
(6.11)
Equation (6.11) may be also used in first approxIntion for calculating trapezoidal
*
wings with a slight sweepback .
Figure 6.8 shows the calculation, rasuite when using eq.(6.1.) for various cen-
eringe of the aircraft XT and for varioue aspect ratios of the wings (since
is.a function of the. aspect ratio).
The damping produced by wings without sweepback is considerably lees than the
i. ~_ ? a ?.t.. 11 . 1~11,...iw.. .?ew~4wa el.4n n?
daptng by the hori:onul tail ii~i'l+acv. Lira u6 w-nu {1.IY 1V111Iwa++~ v..... v~..p .-.~... ....
f
It ?MuIt be borne in mss that eqe. (6.9) and (6.11), in calculating the deri-
} E tatirq of and a the angls of attack suet be taken in radi n~ , instead of in
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From eq. (6.9) we have
1
Egoation (6.11) yields
Pt,
pbvi,oualy, the coefficient of damping u anent of the wing was only about a
twelfth the coefficient of daaping Foment of the tail.
6.8 - Xnf1uence of c.nte:ing of Aircraft aryl j3pect Ratio of the Wing
M
ors the. t
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craft to the sent of. elements of a sweptback wing (Fig.6.9) are greater than the
Fig.6.9 - damping of Sweptback and Hon-Sweptback Wings
217
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same distances on a wing of the sate planform t.; without sweepback. For this
reason, the damping properties of the wing increaae. with increasing sweepback.
The approximate value of them in nt coefficient of dapping of a ling' of con-
stant chord. with sweepsngle rsy be determined on the basis of the following con-
sideratione.
The dietance from the axis oz pawing through the center of gravity of the air-
The elementary moment due to rotation about the center of gravity of the 'air-
craft in any wing 'section, It the distance s (Fig.6.1Q) of the plane of symmetry of
the aircraft, will be equal to
dMj=bdiq (sc11b+ tc~xj,
(6.12)
where x ie the distance from the leading edge of the section taken as the axis of
rotation; bcm, 1fICL are the increments of cm and CL due to the rotation of the air-
aratt.
The total moment acting on the whole wing is obtained b7 integrating eq.(6.12)
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I I
*,. + 1
b r 4c, dz+ 5 ocxd: .
.1 1
Introducing the . diriensionless coordinate . L . and taking `x
i
2
this expression in the fore
6414 ' dc,dz + dctxdi
so that the nneent coefficient becomes
~
m, -1~c.di + 4c.xdz .
we gay rewrite
(6.13)
On the basis of the above-centioned curvature hypothesis, the following approxirate
expreeeions can be obtained for tCm and ACL:
(6.14)
where, in detarsairting a - _~_o_, the angle of attack a must be taken in radiate.
da
.6,10 - for D t rinina tb6. DSa$nj 1 UiX*t of a'w ptb ak mina
218
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6.15)
On substituting ega.(6.14) and (6.15) in (6.13), after integration and simpl.i-
ficabona we obtain the following expression for m' of a sweptback wing with con-
start chord:
Bearing in mind that the taper of eweptback wirngs, generally apeaking, differs from
the theory we are considering a wing of constant chord (n 1), and
~ ^ 1, while in
applying a correction to the general approximation of the argu ents, we obtain the
following final expression for the a~ of a sweptback wing
13 es example, let us use eq.(6.17).for ca1cu1*ting the value of mz for a a 4;
0.25, assuming that in one case V ? 0, and in the other case x ' 45??
)
We then have, for wings without sweepback (k ? d
lUL4+($ .-2.o15)'~j
h. borne in mind thtt the gradient of the lif t a of ? sideslipping
-
th n for a rectangular wing of the same aspect
wdnE,.iL.ij*11er by ? factor of cos X
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For this reason, on the basis of the curvature hypothesis, an expression for the
With respect to the fuselage, the same reasoning is applicable as to the wing.
seen from this sX2Unpla, swscPhack ` cons it~r b1y? i4rO C the damping propArt y 0
moment coefficient' of damping of the fuselage uay be obtained. Calculations show
that the quantity m of the fuselage is snail, for practical purposes, this coeffi-
cient may 3ust as well not be separately calculated, if, instead, we take the damp-
ing properties of the fuselage into account by applying some correction factor to
eq.(6.9), which value must be taken from statistical data.
In this way, the resultant moment coefficient of damping of an aircraft inay be
determined by the formula
m;*"_I.2 hf?_ w
a
i ?O944pLStg!
i
1
(6.18)
where the damping of the fuselage is taken into account by the factor 1.2 in the
first swivand.
For modern aircraft with sweptbick wings, the value of m$ usually ranges from
-5.5 to -7? In view of the fact that a number of assumptions were made in deriving
eq.(6.18), it is recommende4 that experimental data, obtained in special installa-
tion. in wind tunnels, be used in calculating m.
Ian of the Do+aa at the Tail
In comparing the results of calculation by the above methods with empirical
data, a wksd diecr pancy betwe.n theory and experiJeent was found. The experi-.
a ntal date were considerably larger (about 1.5 times) than the results obtained by
M - e laulttion, Tb. primery reason for this diecrepanc7 between theory and experiment
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t of the stationary state hthesis to an eva1aation of the
ie the ~p~,~,ccabil:i
d* prop of the horizontai.tall Qurface.
oration of an aircraft. the conditions of flow
It is fouind thkt in unsteady
riaontal.tail surface cannot be evaluated by means of the kinellatic
A'around the ho
n at the given instant of time alone. It i9 also necessary to
meters of mono
cter of motion of the aircraft in the preceding instants of time.
ally for the chary
us turn to a Mare detailed consideration of the condition of
To prove this, let
horizontal tail surfaces during unsteady loon of an aircraft.
flow about
rcraft have a translational velocity and at the sane time rotate
Let the ai le of
axis oz. Here, to each instant of time t corresponds a definite ang
abut the
the S and, consequently, a definite value of the lift coeff i-
attack of th +
cient CL. corre
dormwssh at the wing is created by the circulation of the velocity
tine t. However, in view of the fact that the tail is located at a
9pancif ng to the
sponding
downstream of the wing, a certain time is required for the velo-
certain distance
e to reach the horizontal tail surface. At a flying speed V
city induced by th wing d a distance between the wing and the horizontal tail sur-
fa of the aircraft ~ ~?t? the time interval:
Ce the velocity induced by the wing will. reach the tail after
.. ,
t the tine t, the dowrnra$h in the region of the horizontal tail
For this reason, a ,
nd to the angle of attack of the ring which existed at the
surface will correepo
t1-t-t
differ from the angle of attack at the time t by the
Tfiie angle at attack wi11
 n +hp farm
~where D is the coefficient depending on the law of spanwise.distribution of circula-
tion and on the mutual position of tail. and wings.
the derivative of eq.(6.20) with respect to a, we mr.y determine the
Taking
h at the tail will differ from the quantity which, by the
;amount by which the downwas
stationary state hypothesis, corresponds to the time t
1.
,-t 4P (6.21)
gs < ~a `'v, .
of this lag of the downwaeh will be an additional lift of the horizontal
The result
tail directed upward (at positive )
z
and additional n went of the horizontal tail surface
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tending to diminish the angle of attack.
the dimeniionle" derivative of the angle of attack with re-
Let us introduce
` - a
:oi the.~x~o~'ei ?$h.t~. with respect to ,
!ion obtal~ for
,
 ~~e _.~o~ .tea
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so that
h in calculating
the lag of the das
As will be seen, the error due to neglecting
the moment of the horizontal tail surface in unsteady notir~n would be an appreciable
qu&uatity?
' tntive coneiderntiofh.?
M K incrs$ region M < ~ the derivatives a ? _!p.t
h. t.
turning to eqe. (6.4)
a~
iaor+U, varies only slightly.
coefficient D
. W and: m ~. liI(IdSS
thi- e the coefficients
~ l
t 1 U1e e!~ ~a in. this
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seeib
on the Mpment of
.. c character
the var tion of the Mach number, all the basic aer
. ' . ' with t
} t also vary, while these variations become Particulr~r],y Bru
4 iatics of the aircraft
that at any variation in M, the coefficients m~
. at )( > ~tcr? It is , only natural. ,
and os rill also var~?
: sal data on values of these coeffi-
- :.' At the ~~t tuna ~ have no expel
i's. Per this reason, in the anal s is of the influence of
){ich numbs
dente ?t ham.
ate , we at confine the study to mere]ar qU-
~ ;the ~Ch ~1ftl'f on that coeftici .
and 6.9), we note that the coefficients tr.~ and mzh.t. are
On comparing eqe.(6.24)
connected th each other by the relations
t
meluence of the
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increase, at greater increases. in rn . uatton (6.17) sbowa that rn~ +~M likewise
iu ea. Thud, i.fl thin rcigian of high Noah number., the damping properties of an
aft increa..e with increuing M.
At M < Ir, the coefficient a begins to decrease, the coefficient ah.t.
alora its rate of growth and decreases on further increase of M, while the coeffi-
cient D declines. As a result, the coefficiente zit and m~ decrease,
M
Fi.6.11 - Approximate Character of the Variation of n andz with
In the region of supersonic speed.. at F >1, the mines a and ah continue to
.t.
(decline with increasing !2, and the dcm ah at the tail vanishes (D a 0). Thus, in
this region of Mach nmabera, the coefficient mz continues to decrease monotonously,
while the coefficient m~ vanishes. Figure 6.11 shows the approximate character of
the slopes of the coefficient mw and me.
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~HA F'rIt VU
`i..'t) r e ~' ii Lit 1J1; 1 J 2 a J. r )1I.1~f i.
Rcsolntion of the Aircraft Motion into Lon its: a]. and Ix~teral ~on~ .a,
In the most general case, the otiorr of an aircraft roust be corsidered .s a
motion in space by a body having six degree; of freedom, i.e., as the sii of three
translational motions with respect to three {r.xes of coordinates and three rotary
motions about those axes. Such motion is described by sL: equations, whose solution
in the general case is highly complex. Zh gtud,rinSg the co^trollability and stabil-
ity of the aircraft, the motion of the aircraft in space is customarily resolved
into longitudinal and lateral components, these two motions being usually taken as
independent of each other.
Longitudinal notion is the term applied to the lotion of an aircraft taking
;place in a plane coincidifl with the plane of synnetrv.of the aircraft, that is in
the plane passing through the longitudinal axis of the aircraft and perpendicular
to the transverse axis of he aircraft oz directed along the wing span (Fig.7.1)?
The basis for the resolution of the aircraft motion into longitudinal and
lateral is the fact that, at small deviationa of the aircraft motion from ey1Inetri
cal motion (end it is such deviations that are in' fact considered by the theory in
twat caaea), it may be considered that the forces and moment acting in the longi-
tudinal plane do not vary. In exactly the same way the forces and morsents acting in
~"~ , .._ _ .... .
the two other coordinate planes do not vary at amaU deviations of the aircraft
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motion in it plane of symmetry.
in the longtt.udinal motion of an aircraft, there is variation of only three of
yig.?.1 - Mane of Sy retry of the aircraft,
Fig.7.2 fight Path of the Aircraft and Position of the Aircraft in
Executing Loop, Climb, and Dive
the fix -independent p&zaaetere which in the general case determine its position and
aotion.in space ao a solid body; for examplo, the speed of flight, the angle of
226
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ation of the fli,aht ?,h, nr tr,p cu~.pw~;,:; ,,, ....-.-..
attack and the angle of inclin
ity with respect to the axes of coordinates fixed with respect to the aircraft and
etc? ;fie e plse of longitudinal motion of any aircraft are
the angle of pitching,
~ g.7,2). The longitudinal motion of an aircraft
the loop, the Climb, and the ~.ive t ~ _~~ ?
e uations which simplify the solution of the problem by Corr
i$ described by three q ~
parison with the most gbneral case.
Lateral motion is the term applies to the motion of
K...a Y.M rit,,~_ when the center of ravit7 of the aircraft
plane. In lateral motion, of the six independent parrrCte
as such three independent r raeters, we may take fo:
sideslip, the angle of bank, and he angular velocity with
Examples of lateral motion of an a1rcz Ml s++ v ~..~ __ _.
horizontal plane, sideslip, and free lateral oscillations.
Strictly speaking, only gliders or multi-engine aircraft with an e'ren numb engines in th,ch the engines are syetricallr located and the propellers are c
the counterrotating type can be considered fixed with respect to the xy plane.
however an aircraft is also asyametric due to rotation of the engine
i, sually, ,
propeller parts in a single direction, a3ymietriC setting of the vertical tail
In a strictly atometrical glider or aircraft, all types of longitudinal motions
ed b deflecting only the elevator, with the ailerons and
or maneuvers rosy be affect y r
* le of sideslip is the term appllad to the angle between the projection
The ang
of the vector.of :velocity on the xs pl.ane~ and the x axis; the angle or bank is used
$ the ..~. _ d,. the horizontal plane._ The , angle of tax is
~f ~i...ot. dot&t' oc- of.. the aircraft in he horizontal plane.
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{it the pilot does not intervene in the control} will also be metric, i.e., it
exerts s etrical action, then the follow disturbed motion of the ajrcraf t
rudder held motionless. If, in a strictly ; ,metrical aircraft or glider, a st
will take plane in the nurse vertical plane as before the start of the disturbance.
forming raneuVers in a vertical plane, ;nomer:ts and forces that tend to bring the
flight path out of the vertical plan appear together with the longitudinal mor:ents
and forces acting on the aircraft. Of these asyrrretric rnoents and forces, the most
substantial in magnitude are the forces connected with the gyroscopic effect of he.
or even entirely absent.
Por this reason, the asyetric nbments are considerably
smaller in jet aircraft than in propelleraircraft.
propellers and the rotating part of the engines, and with the twist i ,tamed to the
air flow by the propellers. To enable. he pilot to per forr r:, maneuver, for ex-
ample a loop in a vertical plane, he is copelled, while deflecting the elevator,
to produce longitudinal rnc* eats by deflecting in a definite runner the ailerons and
rudder, in order to counteract the lateral mr ents that are produced on the execu-
tion of the maneuver. In propeller aircraft, these asyTr etric manents are rather
considerable and noticeable to the pilot; from flying practice, the difference is
well known in piloting technique when executing so-called left and right figures,
for example, a left and right turn.
In aircraft with jet engines, the engine parts rotate on relatively short
radii, but the effect of twist of the air flow passing through the engine is small
longitudinal motion of the aircraft' in the plane of symmetry independently of its
Since 'the aircraft is usually` not strictly sy netric, it follows that in per-
As stated 'above, in studying the behavior. of an aircraft in the air with the
'object of obtaining more vazily visualized results, it is expedient to consider the
aeyiztetric lateral motion. Such a separate atridy of the longitudinal motion pre-
m1rudder ?in an ideal manner, and will alwa3r5 be able to maintain the flight path in
supposes that the pilot, if necessary, will be able to actuate the ailerons and
c)
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w1 h _tiw
r
., d r lfl. .i
..n x~. 'tM. .. >; b6,?~t
b+.A 4iit'3 t1 y.i~~ +Cl.l L, r;,L2 ati~ We.
ihu, longitudinal force3 and moments in this case will not depend on the action oi'
the ailerons and rudder. 01' course, a ne ar or maneuvers such as a turn, a complete
combat rotation of the tail wire, etc., require the simultaneous consideration of
longitudinal and lateral motion, ,i.e., the consideration of the general case of he
notion of the body with six degrees of freedom. In this book, owcver, we ill not
take up,this problen.
Forces 3rd`orert ~ctiz~ ar. the i~irczaft
The motion of an aircraft in the vertical plane will be characterized by three
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the projection of forcenit
s equations connected w
tdependent egw~tion: two ~ed ,z it".
one equation connecting the moments actin on the aircraft.
Pi.7.3 Tra
ectory of Motion and Diagram of Forces Applied to
Center of Gravity of Aircraft
t
~a
The foreea applied in flight to the individual parts of the aircraft, in setting
lied to the
of foreee a
t
d t
pp
ems
o aye
_un the rquatione of motion, may be reduce
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(7.1)
fe. f4f~;
,#
General Equation of the Lontitudinal Motion of an Aircraft
r~nr~.r r.~~ ...a+-err r~-+~sr~rwr r
In the general cave of unsteady motion, the forces and moments applied to the
aircraft are unbalanced. As a result of the fact that the forces are unbalanced,
i;eo??owing. to the fact that the resultant of all external forces applied to the
aircraft ?ia not equal to zero, it will stove with a linear acceleration or decelera-
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In the present course, however, we ydU consider the weic,ht abd mass of the aircraft
to be independent of tire.
enter of gravity of the aircraft (Fig.7.3), and to ra onts with respect to the
e aX? of the' aircraft passing through it8 center of gravity.
The external forces applied to the aircraft will be as follows; the. engine
throat F, the lift Y, the drag Q, and the' force of gravity (.
in theoretical studiei and ca1cu1atiori of the stability axed controllability of
aircraft,, with the exception of aircraft equipped with liquid
t engir es, the
weight and mass of the aircraft are usually core dered to be constant.
In working up the re cults of flight ~neasureiients, ar i in d3articu1ar or deter-
mining the charteristics of lorlgitc hri stab ility arU ccntrollability of an air-
craft from flight teat3, it is nece:ar:i in sane cases to allow for the variation
of the weight and amass of the aircraft with time, that is, it is necessary to allow
or the relation
For the general case of unsteady controlled motion of an aircraft i air, the
longitudinal aerodynamic moment may be axpressed by the function
/I
v, ii. i. (7.2)
In allowing for the influence of the copressibility of air on the longitudinal
aerodynamic moment in the functional relation given by eq.(7.2), it is in practice
;more convenient to express the parameter Yin termb of the Mach number.
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I~atMtars the directio of the speed of flight, is
w,wt~rt
di
Fig.7.4 - Relation between the Varia-,
tion in the Angle of Inclination of
A convenient form of the general equations or longitudtal motion is obtained
called the tangential acceieratzan tin s
equal to he derivative dV/dt.
The value of d w f dt may be round by
applyinb the well-known law of echanicw:
the product of the mass of a body by its
acceleration is equal to the acting force.
The force acting in the direction of the
velocity of flight will represent the re-
sultant protection of all forces applied
to the flight path.
Proceeding in this way, we obtain the
=PCos2---Q--Gsin$.
(A)
If the right side, of eq.(A) is poeitive in sign, then the speed of the aircraft
will be increaeing at the moment of time under consideration. With the right side
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wiU not be equal to zero,. the aircr .f"t will also have an angular acceleration or
deceleration. ' This will b.e reflected by the equations of unsteady motion of the
aircraft in' their analytical :foam.
center of ?avity of the aircraft, i.e., in
by finding, equations far the nagnitude of the linear ?ccelerationsalor., the tangent
and along the normal to the f l_i pht path, artd also for- the value of the angular ac-
celeration of rotation of the aircraft with respect to its transverse axis pas~irig
through the center of gravity of the aircraft.
The acceleration of notion in to direction of t e tztnpent to the p th of the
consequence of the tact that the a ilt4~nt. ill Wlt~+lY or the cxterrnu:4 roA c4
 P sin a) > G cos e
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the speed of the aircraft will decrease.
'fha acceleration of. oration, along the-norta1 to the flight path, of the. center
`of gravity of the aircraft is called centripetal; as we know frcam the course on
yr
mechanics, Its absolrte value is determined by the expression , where V is the
velocity of flight and R the thstsntaneou radius of curvature of the flight path.
In analyting the motion of an aircraft, it is raore convenient to use the expression
r the centripetal acceleration related to the velocity of flight and the
velocity of rotation of the flight path, equal to dFl/dt. It is relatively easy to
e
prove (F.7.4) that, at any instant of motion,
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the aircraft, onto the norn5al to the flight path. The projection of the resultant
so that the centripetal acceleration f be represented in the forrrx
v.
'II
The magnitude and sign of the centripetal acceleration will be determined. by
the magnitude and cign of the projection of the resultant of all forces, applied to
force is equal to the sum of the protections of the forces of its components. For
this reason, the second of the general equations of motion of the aircraft will be
represented in the form
fr
nd
d'
P Sltla 4.
cN
(B)
According to this equation, the centripetal acceleration at angular velocity
'of rotation of the flight path dIdt is the sum of the lift Y of the aircraft, and
;the prijection of the engine thrast onto the normal to the flight path will be
~aeater than the projection of the weight onto the normal. to the flight path, i.e.,
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~`inal.ay, the angular accolera'tion r4ation of the aircraft about its transverse
`Wdt. will be cormeate1 r4.tt~ the mateente:. off` the forces with se ct to he
ease &Xi by the equation
if the angular velocity of rotation of the aircraft with respect to its transverse
a IS (the angular velocity of pitching) 'i denoted by w , then we may write
Y
r*
In this way we obtain a system of twee differential equatiors
~t
At (S3 ) O n?
/1
Iw11 .., is- 1 ? Y () f ue M
ill
an ;additftntai~-rdlation or connection between the v riablee to >Qake the problem of-
finding a? solution of thi. 53'ttem matheftatteally realizable,- at least in principle.
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X,,
(7.4)
determining the motion of the aircraft in the vertical plane (plane of sfinaetry).
As the independent variables in these three equations, we may select any three
parameters out of all the parameters on which the forces and mcnents entering into
these equations depend. In the general case, the forces and gents depend on the
parameters
y. ~, ".!, y. ,,
on their derivatives with respect to times and on t.
It ie here aeetuned that the poaition,of the engine control stick (throttle
;control) does not change during the motion of the aircraft un er consideration.
we see, the total number of variables exceeds the nuu er of equations of the sys-
tern (7.4)? For this reason the system of equations (7.4) rnuet be supplemented by
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cations of motion
........
the Mach nu?ber, etc. Aa a result of thia, the initial eq of
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L we consider the dietnr'bed _ motion of the aircraft, without intervention by
t}71~
plot, then the position of the elevator will be either
wiU be deteritifed by the variation in
and t). In both, these
constant (b 1 coast, ), _ or
the remaining parameters of motion (b
cages the two "superfluous" variables may be
elimthated from the rsten of equations (1.4) by tfltrans o:' two auxiliary equations:
sw,
:, functions of the parameters of aircraft radt,ion and of
09.
(7.5)
the controlled rsotion, "e~??(7?) and eq.(7.5) lust be supplemented by
On.considsrin~
d,efi?nir either the variation of the elevator position with ti.^e b (t), or the
variation as a 'unction of tine, of any Qther of the parameter3 o' natior. oy' he
a rcrart, for c, ample, the ar;gle of attack.
The first of sh Ae equation (7?5) i sy to obtain i the velocity o' t're center
s ?
of avity of the aircraft along the vortical is represented, on the one hand, in
the form of the derivative dHf dt, and on to other hand, as the projection of the
along the flifht path onto the vv ticai. The second equation re5Ults frog
'ty ~
velocx
.
the getric relationMich are indicated in ig.7.3.
Inte ation of the " rations of I'otiens
of differential equations of notion of the aircraft .(7?u) cannot be
. The system
ight-hard sides are coplex
the aircraft position in
5pace. The foresa and moments entering onto the right-hand sides cannot, in the
.
sufficiently simple analytical expressions. The
-1,eneral case, be represented by
;s
t.endence of the forces and moments on t ' parameters of notion of thA aircraft in
.n.~
for example plots of the
lots
l
t
IN ~ y sIn d,
a
,
a
p
cases is. determined by means of expo n
angle of attack abd the Mach number,
f the
y
Aand C, as ftuictiona o
.~Loefficients G
~
'
of longitudinal ~aoa-ent mz as a function of the coefficient';
t
i
-.._
en
.
curYSQ of the coeffic
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the cairar'afL 1tr tope general form consist of a complex y Stern of nor linu' di.f fereri-
tip. equations.
To detornaine the motion of the rircraf itself from its equations, we gust
either make 8imp1uying asswcptions, undex which it becomes possible to irtdgrate
C.,
the egtmtions in the genera1 analytical form, or we must use approximate methods o
integrating these differential ec1Uatio~ZS.
Simplifying assimptiong may be made to obtain: analytical solutions of these
equations of motion only in the investigation of a cial problems o." the drriarics,
i
stability and controllability of aircraft. t an ex_^ple of suc' a 31. plif ication.
we might mention the aast nption that the flying speed or the angle DI attach rer.
~~s~"~s? ~.~sutnltior:
unchanged during w^.steady notion. The rrost wdeli y~ ad a~^;i :ri~t;l;~
that the unsteady dlsturb?d notion of an aircraft differ i from the thitial stead:
is
state of flight only by r+inor deviations of the par eters of motion 'ror~ heir
values corresponding to the initial state of li&'ht. 'T'his assuruptior :ores the
basis for th+ method of small disturbances, which will be discussed th r~r en i .v .., 1
oilow:re
43V 2h1Ca t' + u-
d
4 $ t~1
r _ di
where x .w
tab
9
is the tr zs~ior ratio o. the G: 4,: J. i1ot.
On dev4opt"c the Ci d , .:a.. ~?:iCk'er?3tzc aeterm !i. o this
r w the fficie1` of ?Me3 1 ' c? iStlC :,'
( A ~Ji L' ~
1o'a'1r^t. r'~' +f :`. 'ad.dif+i?:. to 1 ~r ,f lli ..en J 1.ir fir.: V .1.t? ... _
Cei.'rd the fC.li~ t '~ ~1 tJ..~ ,.+ .. ?
t~: pl .l i. .e
riot, Aal, Aa2, Aa, LrC i
On regulating the trangmiScion ratio ~g ?rr the automatic pilot to the rudder,
the automatic lot can be used for influencing she eoofficicnt3 o' the character-
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i v 1~~ ,f ro 1~' ~.
t,s y tiro s rr} he
 i stic egUat ion and, consct a rntiy;. al, ~o th =?A1ar?tictt~r?tic of the darnic . t_bi1i.tt
GtY' the =. ircr xt. i'?.oro #et ilc+1 in.forattcdon tii`.^ inf 4ueccJ tj e fowa d, f tTx-
amAple, in he ;3bov ..tentione I books by ~J.~t. +"c trov and V./;.ho+telf nikO (3ib1. '1O).
T n r f~ 3 Airc: f`
Oin t..0Iy l.t"J.l et the 5tbt.1iti of 4 x
A.~. shown by V .i .Ve&ir4>i ( 1.b,A,
irCrct ftSC aI"lt' iT~ 1i'i1t"e ~ h inr'~~.i f'CTGe o: he coI~ rol s' t!^itl, ir:C t.hv; ' hO
of ~ ~, i ~, "
~,.OaiuNd Ltill'' ' th.O T .e
ru:~i ~. r a r. _. y
pQ)r i+ 1f, Ca7i ~ o 1ct~tY:~t:. Tr_ ,i. tkn
ithrhe =1rC~'.. l t:
of the (99?ch fro ?nT ,;Aw wi.l.1. !1!4 (:,. +Si_ ~.~ S?, {..
j ;ctw O" the ?:{ai Cr P ;e r:it o the, v tY ~ i -
.^1`11l . (4 0oi o t 1oc. '! t? y I~T'ty '?; 1?, ~~ .Y , of
.ni.t.a;; .?e 1, E""{v~o?, lS4 i 1
Ui ttt k c z, oi' the i'' iiiar vol scat:, cT er.t ir. o'rer .,o, ,
rf M 1
t ~u ~ na.~.vn^
ink, for the Influence of the "om-
pres$ibility of Air.
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1.
Fig.8.5 - p1e of the Relation between and rn and the Mach Numbers at
. Various Angles of Attack
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,6.t- Famp1e of the Re1atior~ between CD an.1 a at 7arioua ach ubers
 0
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Length of mean aeraiy i.ic chori b~, r., 1.95 r.;
Wing area S 21 .?
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uations
The calculation of the coefficient of the char c'teristic equation is rather
laborious.
As an e wpla we present the calculation of a turbv-if.'t : L'i ht~er Conve"r.ti0r.al
1esign with a rectangular wing, ha,yi.ng the Ioliowin; Jelin iata:
Weight G 6900 kg;
Length of aircraft L 10.6 rr;
Relative area of horizontal tail surfaceh.t. ----- v?1.:,
Aria of horizontal tail surface with respect to center of gravity ?.,
Arm of engine thrust with respect to center o{ gravity (en ire 3X1.3 pa35es
above center of gravity) Yp 0.25 m.
Let the pours of this aircraft arri the curves CL f(e, ) be characteriZei
by the curves shown in Fig5.8.2 ar;i 8.3, and the currre of thrust of the turbo-jet
engine is shown by ?ig.8.1.
The coefficients of. the characteristic equation depefl or the coefficients
and in and on their derivatives relative to the angle of attack and the
z
U ,r
:ach number. For thin reason, the calculations require supplerzienting the graphs of
?i8.8.1 8.2.ar 8.3 by graphs similar to those shown in Figs.8.4,to 8.8. All the
grifplta rivv..r. .
^ear be con& ructed from the re ul t.R n f imylel
testa of the given aircraft in a wind tunnel.
Let us, take for the calculation, three attitudes
L t. iuil %.1& r of yle --
0
.4.
.' 0 . 0,4, 0.1, and 0,05 ar4 one attitude of gliding at zero thrust and C
41 ...
2000 m
.
The altitude of flight is taken as equal to
on perfor
e obtain
b
,
ove, w
$tartir-B from the deeign data even :a
of the ,haracteristi
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F{ n - ?e of the 2e1at~or: betwcer, a, u.
f
tlt? ie of Attack
,4112
Fig.8.7 - r rnple or the Relation between x and a at Conitar.t Elevator
Poeiti.on and Various ! ach Nwrbers
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380
5,6?
9,4?
124
0,37
447
At. tack
4, tips ' cL p, t
+0.75?
0
40,75
00
-48.4?
-48,4?
A,b4
1030
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1
1
1
1
I
r
_.-
:
q,,r
'
...
., ..
:
a
]_1
h
Ij
i
1IL
j
n Gr at
a a
'
as a
6 V o+~ M
3,8?
- 3,81'
. 0,07
124
0,87
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QugN r/ rY
hale 8.2
0.001
1110
3,43.10'
0,110
0,~1
0,3IN
1,$
/,q
O,if
0,01
0
0.37
0
0,03
0,11
0.013
11,2 10~
0,066
0.060
0,111
0,1$'
1,3?
I
2,0
1,01
-0,10
..0,10
1.9
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 TakinJ; the as pct ratio of the bori:.cta1 tail srfac
We find the square of the radius of inertia r2 by the approximate foriiula
L
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allatior;e, the ini t+al values for the paraf eterr~ of motion. that arr?
present in Table E1Y~M
`fable .' contains the results of the calculation of the coeffici.entts Cc,
L , w etc. from 8.(7.17), (7.16) ('7.20), and (7?21).
Lc lc
he coefficient of damping moment of pitching if an~i the r oite ~t coefficient iue
to 1a~: i rl owriwash irc c1 err'ineb " the al. roxirnte ?' yr r i,1as for the case of a:':
atrcr&ft of coi:vectiot ai 'ley iri with a rectar: uiar wt ? ~- (rbl.11
as we know from 'Tatter HI, 'rar es with
''ach nur ber. This asstzr;ption i5 1e itiT(ate it this Case, inasruch a3 she Caicui-
tion examples giver; by u$ are of ar i' ustrat ve nature. We k~ ll Cor s . er that the
Fi11y, the coefficient of velocity eceleraticr, is taken as k 0.9. In
that case, for all four states of fliht taken, we obtain
Of course, in technical, not illustrative, caIckl.atior: of dvr amic stability
and disturbed rnotior of a specific aircraft, it is necessary to use for the deter-
urination of the values of ah t. arid, the more accurate methods and for? as,
which, in particular, were presented in Captors III and VI of this book.
e Csh}t,?a: ri
: s .
Z +? xi ii f; cart 3 i C} ? " this C C ? tant for all thfc' stis
1?
 Table 3
14,.. 11,1%
~.alt.`s:.i'):1 'd: the '.:)erC C
'
11LI N,
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11`. 11 ~.. y
ltd 4ad~ ,.~. P
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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
C aicu Lat .or; of he (cf.fIcie;
y b i e ? A
1.4$ ?.13L 7.33
~31C:1at:O:. of the _oe:'i
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 where L i t}:e total ienkrt4h of the xti~^ccraft, w( h ar our ca3e3
thin azrcrA. t: we obt inel
r+,1 i l:lw`_~ }xse of ~f aircraft re' w{)of w.4
C~;'o.t.~.~.a u
r f .e ..u eac
TI't'.19 r f r ., e l:w_ . i11 say( 'abie
c~erof ` e ..^:arapter 3t: C cat_ot:3 tree:. ..
U ~J.i.
14 , a ~Aw tD~~ Ilya 1. QA 1y.IT~
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d S1*fl9 $p *d sf atfifiude
Ifvam
~1idr at urc .nine thrust
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"' r. fc:r
1-i' .c , tab1iatT ct!.
coat f i t s vil t
STAT
r' for e c .r. .
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
) to indicator sped i.1? exactly coy :vi, #ie , t.tb he see:1 ~bhowf: by he i. -
t
s Jr;u:'.ent.
J3 iriica' e.i `atle ., a char e of a"trtud.e at; rcZx ,
cn the `:clue of tte coeff ce1t of the chat"3:.terlc eql i On a~ . en f we aJ ioir
. w
~. ~r ;r c he coeffice! ; a , , z
.
for the i"ni,it2e::e".t~ of the Cc~,pre`s.~ib.+.l r..t: a.
values othe coefiiciet aq very relatti',ely little.
The gyp coe /y ~ SS y (~ r~.
ffi ienlY aA var a.e i with the Catil .ie to a sole:~3wwhACi'~R greater cxt enV v: Car
does the coefficient a1 . The var: at _or; of a2 s lue pri; sril y to the }ariatior in
moment due to lag of dowrwash at the tail
The coefficients a an! a,, contrast to the coefficients al and a2, vary
3 4
etron I;; r. absolute value ani even change their si ~? The variation of a3 and a,
is mainly connected with the influence of the coh.':pres31bil1tY of a1r on u and on
st
the coefficient of static stability : and !"~L; the value of the generalized coef-
ficient of drag of the aircraft C also exerts a substantial influence.
Approximate Formulas for ~eterdnin .the Coefficients
cif the Charactersticion
Approxirate formulas have been worked out for determining the coefficients of
the characteristic equation. To obtain these approximate formulas, we bust reject
h. term ~f !end-order value in the correspordifC exact formulas of the precefd-
-ice Chapter. The paasibilit"y of considerable simplifications of the formulas, in
particular, is shown by the numerical examples preaented in Tables 8.2 to 8.5. We
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In Table 8.7, for a fuller characteriatiorh of the initial. states of f11ht,
3
;e resent the indicator flying speed. The value c)f the indicator r3 yin aped `1
connected with the velocity heat, to whose variation actuates the conventional ~spee,1
inctiaator ;installed on the aircraft, If wre a&; ume that this instrument has, no in-
struient e rors and that there ? is no influence of the aircraft on the flow around
the velocity pick p,. then, in flight under sea-Level conditions (760 1!` ar i
 row will set up these appro W ate formu.1a ?
Accordit to. 0g8.(7.17) and (7J13) of the prec
expressed by the formula
~t dtl
In the state; of fli h. ?cost usei i,n acua1 o:eratior:s, the 3Tries
-en v 4.h ...fitv . ~d '"';J ? ~ an I ;;p"iluir ..~ilera11 1o ';oa excee 10?~. I or this reas.; we i..a ~a.,tM w.
sw? s?nd sin a it the fornula for ",, , i.e., we assu:e hat.
~.C
9 .
SpV
from the equation of stead rectilinear f1" ;ht
we replace 2-h--by -, and take cos A cos a
cos f
ing approxi.rate forinu a for 'ma c'
As confirted by Table .2, we `ay consider that
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11'
On considering, analogously, the separate significance o.f each awi and in the
`expressions for the coefficients of the characteristic equation, we obtain the
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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21: CIA-RDP81-01043R001200230006-3
f
considering the:z, however,
* As shown below, only the characteristics of dynaric stAbi ty which are con-
great practical importance.
of the characteristic equations a1 and a2 are of
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follo~ u approu to forr.,ula
where the coefficie2`.43
aji sa'eter~rtired by tie ::' rrti.':t"y3~F
~?+` P
ltyP
!N 1,~ o
tMAI
h"1 +-yea -:
~
e*M
*11.1 +......--
hereafter, in writing anaI;'tica . fori ulas ani expressions we will frequently
drop the subscript "c" for the aermarc coefficients,
to be L Kl71t i,ii tue gaiiai al foz n ("generalised").
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
In orde
to evaluate the error due to u3e off" the a roxi atc equ mtiOne ( .3)
we present in Table . a com.risor of the value of. the oefficient of
the characteristic equation ealctaated by the exact forrula and by the approx i to
ones for the four initial states of flight taken above.
( th.e ChJaract;er:.. t, i.c
uat N,,..
It folicm'a from Table $.8 that, ~n the region where the influence of the co-
-preseibility of air on the aerxtynard.cs of the aircraft is relatively ~r:i1, the
iifference between the exact and approximate values of all coefficients of the char-
acteristic equation does not exceed 2. Taking into account the known inaccuracies
.
in the detertrination of the initial data for the calculation of the coefficients
al, a2, a,,, and a4, we may consider this d111~~'rl1~C6 tv be en4ir y nlo.-ab1c.
In diving at high speeds, when the. altitude and, consequently, also the den-
.8ity of the air varq rapidly with tire, the possibility of assuntng constant density
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Table fL
1xact an-1 A.?prcixi+ate fal~.ies
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
where al , X2 , X3 , X are. the required roots of the equation. In the general.
~
case, the four roots differ in absolute value, a.~l the subscripts denote the roots
in decreasin order of absolute value.
For the characteristic equation of longitudinal stability, in practice, the
first two roflts coneiderablT exceed the last two roots in absolute value (as will
be plain from he example presented). For this reason we may write
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of air (~ corst) which is aziopted ?iu . the .s:dtna17 theory of dynamic stability,
bear e$ queatior ble. It is entirely poss b1o that the variati m in Heusivy of the
air with dltit ~tc, on consideration of a steep dive, yield a much greater dir-
ference than the difference beten the calculations by he exact forrsarlas and hose
by the approxthat.e ones.
Fjridinf the Root of the Characteristic Equator
-
.; is rattle' Goiicr~t~3~t aid :.'ibo:got=s to ... the roots of biqua:iratic enua
t.ton b;; th9 a5'i oi' exact al ebra c r':ethozs. Various a, t roXi `ate eth s are there-
fore ordi arii;Y seJ to find the roots, which curs erabiY shame:.S the t r..e re-
u.:C1,',~.3.:"1,, wller. the
quired for fir.; the roots an ieLa j3 adeou RyR? a c...
~.iOr ~ ,l.~1tE~r , r ,0 is 1c terTe hU)iI
w"G;f9 y iO
ter, 4sJ:'s:..1~;~;yI ;'.1~v ?
ci.u1a` i?afe, or t,}yt,1 O'~lt Y',`ot o::. i'~iuy o C)sc iii io w r
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
The f11~1ft co1: do is teed for this cr~lcuI. t car ct;ilk c p,rr tZ3 to the va1.~2es:
,e wiU assume tit
r ri. as l:?eriii:
c~.:1at.ons wa assw.'.e -haw
?iA .: h'a t. ra33c .t the .:nisi...
;e i s urbe.1
of r,otie":
reiv with . vert~.tcav,~.1oci.t:r of 5 r' /r. '. To 3
the aircraft ~ ? ~ T L e 51y er.~.er7 thSa 3 ly~l~C'I+ i
s.44! ti l1sL^'.`~_.. .4 7'~ m '
r ',} k R , 7 ~qu I rc..i f a^or
I :,aiueY r te 4ira.e've~yr$ at t ~' ~' ~- rs
:3 yh~traC e `. ,.S
in;. ,,r the aircraft _,. the cxax:;,..ie ta,o:
the 1aw2
M' ?1,$1e- cos (1,9S5t+ 0,414) --
coi(0,0S68t+ 1,54);
V? O,4.-Lw cos (1,985t+ I,68)+
+ 1,65r us cr'O,011 + 1, ;
aS-1. -" costI,985t+089)-
- 10071t-4a"" cos (0,0868t+0,1IS);
? -0.1 224e' cos(1,985S + 1,560)+
M
+ Oao2Ir`4M cos (0,0868t f 1,514),
1 I a;..~tt:rs
/ '
in eq=(RJ.5) the a1 e of af.tack an of ritchirv are ;even in degrees, the
angular velocity in ra;iians per second, arid the velocity alo: ,the flight path in
meters per second. The argurents of the co91nes are given in radians. Figure .9
dives the gran of the variation in AV, Az, A8 and Awe with time, with the dis-
turbed irotian coneiderod lasting for one thiute, while Fig.E.iO shows this tion
-- STAT
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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
H
Lal i sturbaxlee.
1 rep6C
to t
_iurin th 3 f) Est th3'ec' (xeC' Ud 3 after tll4(.i
ation (3.15) and `is.b.9 and .1O clear
Bhow the resolutio~i pf t
tulbeu \Dt.ion off' the aircraft into a ah ~rt r.e2 iii an a Lvt~ ;-?l;er O1 iJ .i"r:
^l .111 ... rli: yf
1 Wei.cG[i6t J 1f~i~e~~ e'uM.!y
,7 the (~yA'{ e 1 T . 3?1/
}C~k"ujr~r } ahe T~eL (.od ~" .4 S~ 4.Y
:'~~`~ ~ Yotiol~ '~:,"~h F t
rap"I w1. ti~ilt'aIa te
a:i '~
ar54 l - l.> sec, the swranl3 b
(:or ec Let fr a?H' i he i a: r
lidr:e r-'ots, Fra+, t c.1.! ".
Fig.3.lO - ;variation with Tire of AV, &i 1M), and &a3 d~trir;g
First Stage of Disturbei t?iotion
the rapid decay, the oscillatory character of the short-period motion is :rasked and
it is very sirilar 'to an aperiodic motion. This explains the fact that pilots in
flight do not feel the oscillatory character of he motion. The long-period oscil-
period of T -11bB 68 L 72.3 sec decays slowly. The pilot in
~.at ory ration with a ' o.~
STAT
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Oil 1 are scalc 6a~..
 Declassified in Part Sanitized Copy Approved for Release 2012/09/21: CIA-RDP81-010438001200230006-3 `
ande attack cf the aircraf' rarie3 raps ' 1z, /1e hc
?;e 'r r es rA1w
"iver 31!. 0*~ :`o prove th.3, w , .3 ~u.
~+ .. . + of `' t~1'' .,~
,
C31Cj'?tQ the 'c"~7.,:.`.+e ~1:. the ,1i.u^.+ accieraO.
;:t9 trati5;or3e a 3 as well a3 he vat e of :. i
L
path. PIs Let u3 to e the cry e aircraft a3 reccd- :e
the t" v~ te o horic,ai 1 r? :~h+? be C raC- er.Zti_. ~u 4
th 3 C T V+:,..?~
t?e ara eters;
450 km/hr; C 0.
~ z c~ the aagIQ of attack a ,.3 taken in ra-
(in detet~ri.nir. the derivative, C, nr z
. Let us take the followifE; values for the trass of she aircraft and the rio-
gent of inertia with respect to the lateral rubs, sthich we neat to kncw in cablcu-
1ating the aacelerationet
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i; these b that .er.3 very ii ui. iC U. , , T a ,
~.s shie ~:7 ~~a~r:T*vo these ~ Lf
~"y ea t~CiOc.1LY
}.~ can "4 y recot~ 'iurin flight, for xa.~ipie, }
i~p~
c:.1.la;,~,c~ne cahe ix
recorder, a pitch..aWie recorder, or a. ioad- aCOr recoY"1 '
y Os 1 ~}o ii
!Y Ul.i;(71 w
!'e { o,bp~ 3err P tha t 1 }~t the A~i .q y~e`to1utQr of the 1orcititlw, 1 1sts i Mtn
^ M1 n } ?"- r rn,.
3hortriYi an :e81~1n M',~ i:.t~a.i.. Lena A3 e. liaryra P .e.
those tt.at fffer suhstantaa11; :., flrCtOfl i. T'-'
r' '3(3 ''~?`~ tir1:;Y:.~S~.?~ ,I~!r t;1C ^6t,','~'M~s.?..it':~.+r:
w -..i z 3 .,x..41 V w.l
~~rr `.rp. L~ Li...J1 w4i,'S .1 a'w .~~ ^ _.. ..i, ..
... b: vi 1e ,:h`lV a?
p aid.~eriu '?3 e;i~..~.ar
.ir:ar~' ~y?? ': Sa r' .a
`
! ;:a:,ic f :ices =u-.: ::;~~ ',va , , -
, ;. ~.~~ Y ..,,e aer ,
the ::?i.~'7.r:i,; speed or r;cp
h t e r ? sti bC t~iai strytO o Jf o the G.'- tea.
~cre iJ: .7 C.1Vti ~? W z
 Declassified in
Declassified
. Let tas as 3u e further that the aircraft sadden1y enters the re of a risif
~, . ~t 4 ~l the a.r:-i.e of a ac1.- of the aircraf
air current, at ~ vertical '4~l~toC~. ~u..dl ~'~~
~ as i~.l.(? ~cCQlt~1'awiat`ta of t?a ai.r4:~art a.f;
q ~
j:t t ~a'w
9tW~B7~t71I9~..;P Gelc"'i:1;4~~ 1):! 2 :i~1 C4_LC'ill
'
the ar t of t'1ist\u be.i :o;,iO!; we t Y"
first :~r~s+~ ~, .~
`. iar, ver, t're fo1io1 i_
.14) of the preceui.r"
uiotls ~~re
aa-
)"eb 1 W t hot s s fir.`. o1
rr`a Gt fit: : w. J::3
c . he' ' ` ut it he r.uxe ti
KK r
~)c i) ?C
- Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
onthe basis of the raiues so obtaine1 the co:.c1u5icr: car be obtaie~ that,
first irtstarit o1 disturbed Notion, the rotatio:i of the aircraft will be inco r-
the
ab ir3ore intense than the increase or decrea8e ill flying speecI, as distinctly
visible from Fi ?.8.9 and 8.30. kna1ogou8ly, the notion of the aircraft will be re-
salved into tiro types, even with a sharp cteflectiofl of the elevator.
mien distuxbanaes act on the aircraft, for exw~ple, when the pilot changes the
control eurface8, the motion of the aircraft 'caries in the follow-
..,position of the
r ing libriuw, of aerodynamic moments produces a rotation
sequence. the digtarbed eg1~i
f the a3.ra. aft with gee ct to it8 transverse axis. This rotation of the aircraft
o 1~
a variation in angle of attack and, consequently, also to a change in the
s to
in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP81-01043R001200230006-3
 ''N r.'~Jf ~vlw~rl~'C.i:...n:.:y...:~?!d _.._._.... .. ..e ,..
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"le sC at a: J n ~;.`:' ? :rc"h r. ctarrc3,
1'iftinC f a4 L . A$ a r 3U1t of the rclatiiel; hi ah inertia f the aircraft, the iy-.
uJ speed uric g this i o ....iC, 5
fi1: L3. ~ `e fe ?-1 ...f.1 .ti r: a: ....1.e, c3 a ...i:.,Ae ~.a~h .~i. be J .:.: i;:,rres~. -
t ..
i.~ c .. ... ~,..
aar~Ztq ! he .. r~e f.
.,. , c co rei&. c.~ bep,..1e;ce he ,;Snit be1 an.l try:
?Ma a i~
.~ each { M i.',u 4~ i a r i
?. rteY?.. yy it:.. l+i ... .o,; of the 4 a41 ...o is :.).
R pe .
in t.h.is 4Q3G be rou:i
i.. r dear .f l ' ~.'h lr [cf. eQ. V . J L) O. 1a 3t ?or the a v a ie.la..~er in the l er cal. ; 1ai~.e by .:,.3.crC e v,
c cue an ~..r a~4G V~ci4~pr t~ .,, ...
:ypp ((vv~~~~~~,, }~ ~~yy t?t{nn the /~} @rahr ia ~yt : c y yy q{o/~al with i/ ~t the
t eI or e~~.bfyy~~~~,, YrS:? 1eby he law oG~f 'ir . i o'tier1'#l.c +1. e
e1at.`?,on. be J~reer, c ar'1 ., !]yr be wo~.A1rr.i. jr the eli j...o: S of the forcer f r ~ he un-
9
3 at,lc 3tabil i with iespect to C~rerioad
pol 4 7?Ct MDr~ l.R
Deper"idit on the rneuvers perfor e$ in flight, aria d11 ' ary accor;En
green the variation it C and the var`atior; in a'ach riirber. It wag fourd useful to
segregate the two characteristic extreke caves fro this multiplicity of relations.
The first case arises when the variation in C~ and he f yiri speed, an1, conse-
quently, the each number are correlated with the con.1itiori of constant overload.
s e&J~i riotViVi+ o .hen}..l craft ?
Y +?
 view of aricta,ass pf the stl'a~seqUOrt reason:rF, ~e
`~? F`rom the point of
? s the rrar.~.c i~ori-
tand the 1osa1 factor n to an the ratio of the re uitait or
s b e o f recti~
y 'h9n the thrust
T /C w 1? For any ocher s~a-
:? zoti~al ~,
inear steady f1igJt the load factor n as l kewi.se equal to unit?, since in tli~ s
to 7. In practice the ~ii~dt load factor is often deterLned
case R is always equal my w .,~G)
t7 ~a h~ ratio of the lift to the weight of the aircra~v {n ?
a conventiorAas-
" i
s
bility
tier?a,,ter we mue
t be aware that the term !"static staone, as was pointed out in the introduct~.on to this book.
Declassified in Part - Sanitized Copy
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' ~ uarie~ at co~~sta~(t, s~~orrr~! Cact~ n~l~.ber , due
fl ~ l ~ The second ease ar~,ses when ~~ ~. > ,; ,,-Le.r c~ t,.t~t~ forr
y variation in the resultant loci factor n.s ?tart in up a .
ds to status of steady rect l.thear flight. at ya.ioUs ee.:ts, 'Lh
case corree~r~ ,
to the suaa41en entry of the aircrart into a ris~.rr or fallizt~~
latter case cor, esp+ands
verti 4 ~?i,_~~, r:otic~a6 t,lie ai.rcr~~ft.n e:~~zc~l-
cal air c~lrra.~u, ant l:k~,y.se to the .r~i~ w
., a.aewease ,r., otirerloa,i when t,'.e .~?'=., jeed i.a3
t:t,ti ~?nellVer"~ 1.I":VO~~{'r z9,
,?aGV.,.' iceb9 Cfl, 5 ;de~'~r uUal;u . .. a
y r o,
i 4 coa~4ep'` of lof v.ia w~i4YM l': 1 ?,
:;.e21}.: to 9t1`t1~. ~ ,.r~.~ '.,its r.., '
' ~.. duct, "t ? 3
~ ~'::C s'Tc vli rear
s.ra;, ..~b:
to o~terlcat, a
' -ve ti e ; yic of
' u
ex
}
frog ?l E. ; 7"ate C~
~r+,?~{ 1 y~y ,raj ))) I ..q : ~n :' 4 he a )O
; F he C'E.trT L Y., y.8..} 4i) ? .d' US aJAyfee CO gtati4 47 iICSi 4Y A. lia1y +irder the b1.,rY~A ? of lion or the ajxcr4
?.'.;~ear s.a,: et~.iy
Cif} a~7*, he Y}^t cia.
~
Y ~; ;. bVl:t~ '. lae~r 4` ,;
?Nt e value (.?)~
1,, V Let ? Fj... n 1i the ach rusher 4 Ai;?i` r[y -( t^~? ? it'
~ x3t ; ?- C~it'3t ~ with '..i,C ~lb~C':.~53, vu Z S
? ri~ of ~` y.:lterscc tior= of the cur
point ? ~ ~,~ .? ;o ;e s+~ate e?
a a r
to this sate of fli3w? The (-) s .
flight 1san, a o n ateequal to the r,~ { o he , .
c, ,-,~, the ~,~,C,, ;,3C-aircra, Qr equal 'Y .'at one `r lot
L af_ i3, ~~"{SCl: r
-~R4 k con-
v '1'y~^ the A,..~~;r ~tii r~'1s~:.1.8 j: . a U.. cal
which ~+~ IFwr l
the gi.tt^z illt1 ilraor 1: I,1 ,VCi.`.
 ~
i? , the variation of Ino4nents,
a , atantand on],y the ante of attack variee. According~r?
: clu* to the variation in argyle of attack a, in coefficient C will be characterize
by the curve m ~ at i i2. tie to the variation in CL, the lift and load factor
.., will also vary.
The degree of variation of the coefti.ci.ent mZ with any variation
the eUndaton that iu- conat, will be characterized by the derivative
data ...4.
s already pointed out, in flit such a variation in CL will Correspofd to the
stare of raneuvera at which the load factor of the aircraft varies, while the ve-
r~eie ~t~b~litr rtWtIY! te'/IIUd
Fig.8.12 - curves Qaracterizing the Variation in inx with I x const
, anri n l a const, corresponding to the Concepts of Static Longitudinal
Stability with Respect to Overload and with itespect to Flying Spas
loeities anti 14ech number' re ajn practically the same. ?or this reason, we agreed in
Ohapter V to denote the derivative m used' for the initial state of equiibriwn,
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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/21 : CIA-RDP8
the coefficient of static stability with respect to overload is characterized
by a variation in the coefficient of iongft $inal moment, as a.function of the
p ter C (or angle of attack) alone, under the condition that Y = const'
~~ . L
)t const), while the coerficient of stability with respect to speed is charac-
(
terized as a function of the parameter C1 and the paa'arneter h, the variations in
and M in thin case being rigidly connected with the condition n 1?_
For determining the coefficients of Babil{ty with respect to speed and over-
,
i
ioad:aat other initial' states of flight, the cctrtres mz (cf. Fig.8.U) should be
taken at another position of the elevator and a state of balance eorrehpondingg to
3
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as the coeffl ent of static ton itudifla1 stability With respect to overLa ?
scut that the aircraft ch a frog the init state of st iy ree- '
Let u ~~
transitaoof courne, the speei, angle of attack, and coefficient C will vary,
but the load factor ri will renaiu eonsta t? As already pointed out, the variation
k. in thf s case is found bye solving the equations of steady fU.ght [cf.
in C_
eq, (7.1:1 in Chapter VII ? ktaving the relation between CL and 4t for such states of
rectilinear f1i~t, let us determine from the`diagras7 ire Fig.8.1.2, for the va1uea3
s ,
of t N ,eta; the correstX fldirzg, valuea of CLI, Ci3, C4, enc. Torough the
ints correspond' to each individual pair of values }t , Cam; L3, C13 (cf.
po -
Fi .~.12 let us draw a curve. The load factor ri 1 will correspond to all points
of this curve.
The to'Ca1 derivative of ta,, with respect to C1, determined in the state of air-
craft balancing (% 0) under the condition n # 1 (the heavy line in Fig.8?12),
.
i.e.: at a charge in frying speed and a constant load factor, IS called the coeffi-
,
,_ cientg of static longit?udir~al aircraft atability with respect to velocity, a
 onwi~ .o-t
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0230006-3 j
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Te quantity to represents the coeffici~trt_of static stability Wih respct to
. overly since, as follow frog Chapter Vsl, t a eriva ., 4
cor~ition k = cornet.
x ;
g: In the e~ession for a , ins`yai of.m~ it is easy to introduce the generally
~, 2
the ratio Jo.t ~ aid 4 under cor~tione of rectilinear etead7 flit, after which
the operations dhsor1be4 above must be re tell.
and egq.7 ) which define the decay' ~~ riod of he short-Period oscillation of
the. aircraft, shoes that the coefficient a does not de c d on the degree of static
stablity? wbile the coefficient a,~ does de~racI on it linearly. Accordir;, to the
adopted coefficient of stability with respect to overload. In factt,
Stab1It bf'the ircraft ,
etu 7? of the coefticients al and a~ of the characteristic equation (eq,83
'tie roots of the characteristic equation for snort-period disturbed ~rtions,
` " ` "~ front the C.:~,.e 5i ran .
8lD ~!'t~TIO~It fi'c u$vvruuiiv~..
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/21 :CIA-RDP81-010438001200230006-3
~'~~b:
~a' ar,al,~~ia; Lhe i.~~l~~ezi~~ o~~ the d~ee of atat~.~ atab~.lit~ on: the u~y'na-~~.c
`~~i3ide~e~~.~
'~~~ ~~ rat cage corresr~ls to th? v~1.ua~
In th; s- c~~lee 4 ~;s .fa~lows frog e~~ (~ e3}, t,1e ~,~aria~tiv~ in star is s~abilit~ has
n~, influence on the co~ff~.cient of ~a~:pinti~ of the g~4^t_xari~a:l osciL~.aa,~ons
}~ ~ ~ ~,, but a?~'ects ct~l;~* the ~r:v~ of these oscil.iatior:s, Mich is ~ual.tw
o~ a~,rcra~t- ~.n ~~ prea~+~aa ~f e~~oz~t~ri s~t~.o~, too saes must be
i~~ith ~.r-creasn static stmbilzt~, the coe~fi.ci.et a~ .ncreasesf so that tYse
triad of short-period oaciliaticns dz~ini~hes.
Tie second case corree~r~is to the candition
el=
~ ~~
In this ease, tie short.-reriod disturb~l motion is no loner ose.l.latar~ but
.
consists of . tao ~cattuz~,,p superposed aperiodic dam~l r.~tiona. In the given region,
the variati ona oY st;atic stability a~Lread~ a.f f ec the da~~pin~ of the short-period
motion; the damping o? ona of the par~~ial a}~eriodic notions increases ~.th in-
creas;s.rag static atabilitp while , that. of ~ the second decreases.
'~he~tera~.:apart-p~riod`r~ation is noti an entlrel~ fortunate choice, a3nce this
action is not ~a oacillatoi~. But thin term fa generally aecepta,3 and r~i.ll,
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 ~i
~~~~~- 1.~?ai~0,ci~n tr d 1 !
* rva~:'
:..,
. , ,.:; ~~oa~', atabla ire chart-.ria~ c~ti~n
. ~cat~~,~#i.~,:k~', an airciraf't ~i be
~ f~.n~.te ~t~tic ste~b~.l.it~ witk: ~re~peo~ t~ ~verla'~d; ~1~e
ev~~r~ >~~n th~~e .~~ ~ do
~ta-t~lc i~t~tablli.t~r h~.~~ is d~terr?~i.ned b~ the cond~t~.on
~~
~ ~x ~t ~uu~,d b~ ~s~~.ble to cans~.der -~til~. ar~tk~er cage, caries= d~.n
. mx"i~,, .~ ,
`t0 t~i~ cc~tditiaS
~eta~.la~. a.~?s~.s shows, ha~~r~r, ?~at. ,t ~.,,a11. values of a2, av alrea~,?I-
^ ~, a, ~.; ~ ax so .that the rasolutior~ o? the. :ctia~
~;.pos8iblg to 11~~;~@Gi. the a~"1~L..oa C~; ~,~ r
_a short~r.i~3 ~.~~ a lon~_~ri~., b~coc,~~ inaccurate.
~n~o t~ ~,~tians,
c~ be consic~ere;~ oh~- asp the bat~i~ of th~..total ~aq~:at~or.s o? r`~ition
. ~~s case
at~d of the chara~teristi~ e~~t~on s~f the Swarth des,~ee.
does rat a;nter .~to eq.($.3} far al ar~~ az, th:s as
Since the coefficient
^e that the atr~tic stabi:lit~ ~i.th respect to epees exerts practicaLl.~ :~o ~~
e~.den~
f1r5~nad an the akiort,-~~eriad disturbs ~~tiQr..
Let .us thers anaS~ the: influence of static stabii.ty an the ~.o= ~=-Ferac~u ~,la_
's an sia`~a.ll be perfor~ed by jeans of the approxi.~te fors
turbo ~ti.at~ p "flu. ,
...:. ~ uatiQn defix:~.r~ that ~:otion:
~` uadrat:e charac~erigtic
_.... ~'or the caeS.ficAecdts at the q
~` ~~ e . $,Z~~; this `equuation can, be represented in the fars~
,~cco~zng ~ ~
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 Vi4 A ~ QI'~ ft p I~ V Ir f
~ ~~~~I~~i~.l~~~l~
yi-~ i~wnvv icwcvvvvv-v th ~ J
d~~Yl~y~l43Nid~'E-05~.~~f{ I~~~tui~}i~1'~' ut~~f ~.~m , ^.V
~'~
'~'h~ roots of the chs~aat~ri.st~.c e~,ua~;~on for the 3.on~-ax~~,od ~oti~a~a ,:. be d~,far~+'
~b~ he sei~~x
~or~:~~1?~, the coefficients bi and b~ ~.n this case play the e role as the
cr~ffic~.ents axa~ a in the charac~.eristic e~~ti~n of the short-perivi disturb
~. ~
r~~tion.
A$ mentvns~ti~ above, tho i ~.f4"ore!~cc i~ tkas ~icre~ of ~~~~ir~ of ? o;~;..~rio:i
~i.stur6ecl ration,, ~aravi~e~i teat this ~?>~tion i~ 4nste~b~.~: 4sCJ.t02~;,', HNC@S?t5 no
s~abst~rrtia). ar~+~.~tance vn tho f~ qu~ali~tic~s of the. aircraft. Far this rEasvn, the
coefficient b , definiri this ~ar::~~~,, ~-iII be ~:.sr~~;~.r~e~ :.t; this a~aal;~s~.s. 3'~~
l
'~i~ ao:~e~.;~~r ot~,~;~' the co~i'~"ici ony ~?, dc~finin; the transition ;u.~>;' easar:t for the
.,.
~w.i,oy} to ~~tabe ap~rio+~3 c r::~tion.
Let us ~rther aic:~lify. the ~rablem b~* cons~.der:~z~g on~,~ the sib oz" the coef-
ficient of b ark its car~ect~.ar>< with the static stabi l.ity' of the a:t.rcraft . As ~~
2,
b~ d~r~~strateci abo~~s, .the ~ of the trar~sit.ion of the aircra?'t into the re-
gi.on a~ trnstab2,e aperiatic r;~otio;t a~:.i.? ba ~i~i'fx:~i by the condition
b 0
fiv ~ .the. cs2culation to ~ the ~cti:al' charact;eriatics of tl~e ai,rcraf t,, which, as
~,, ~, .
~y a rulo, ~ibit ~etatic, stability 1~ith'respect to n~verl.oad,, it wz.~ be assumed in ;
the fwrther a-asa that the coefficient a~ zs a pasi.ti~e quantitye Then, +?he
sign of. ~h~ c~tan~itq~ b~ wi.ll be deter~sne~t from the' sign of the coefficient of a~
Tn this . ~caae the staterz~nta u~de .f ar the case oP sr.~ll values of a2 must be taken
;to con~siderat~.on~
~; , -
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 Declassified in Part -Sanitized Copy Approved for Release 2012/09/21 :CIA-RDP81-010438001200230006-3
~-~ ~i~~ct1~ p~c-port~,or~~~, tc~ 'tha
r .. ` , ,,, ~ ~~h~t .the ~~~~~t~' ~
~~ ~ `~~~: ~a~~. ~~'kt~t~~~e~~ ram ~~ ~
~.~~h r~~pect . to ~ '
` ~.e?tt ~a~ ~an~~.t~n ~~~ti~ ~t~b~~ity
c~ef'~~.~a t the ~oef f i-
et~tes~ of rect~~~.n~sar f`~i. ,
for th+a" oo~ti~~~~ of v~~~u~ ~te~ ~ Fi ,~3 ~~ wi~.~. be .
~ of tl'xa t~'~ ahc~~n ~.n
^~ent~ of ~~~ ~~~?~rUt~~-t~1. o~a~ ot~ tote ~l~a~r
~'~.ent G anc~ ~,he corr~s].~t~ fi~~~ch n'~ber r~r
fur~ot~.on of t~v par~tera.t the coeffio L
nwuber..I.~t us" take the ~er~.vat~.?re ~S the
the ale ~f ~,t,t~,ak ~ and the ~,~ch
fun~t~ot~
1e of ~tLack; obvicrue~,Y '
+~th re~t~~t to the ~n
~,
tl~.~ht
~~ of the pxec+~.sn~ ~~ptar, we wr~.te
~ta~rt ira~ from, eq. ~ 1 ~
t
~~~~~
the veloc3.ty' of pro~~?-tion ; of sawn ~ air ?
~-ere a ~s
+:, ~ ~ obtain
~,.,~... , g.31}a
~ ;~~.f i~erenti~t~.n~; ~i' ~ .
~ ~ determ~ne~ ~~~~ the equ~t~on of stea~Y rect:~li~:ear
"a'he ~sr~.vaLi~~ ~ c~.n h
~1M+~o ~'" O EaMl/c
o ,~
~"~ ~ ~'?
:~ .~. _
~~,~~ ~8.31~, ~ hive
u~tion~ 6y aa~, uei.ng the notation
~ dividing both aides of the later ~
STAT
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 n e~.i~.nmtin~ t~a~ quanti,t~ fror~ ~the~e 1~et two =bone, wry obtain
~iN.
na~;~ .the fnl~.owi~ ex~r. e~sion for, ~....
"f'r~e ?ralue of she coeff>cfent of 3tfl~ic st,~blitq ~.th y'?S~Ct to 8~@~ iS cc~rnecte~'
~,th t~Y~ ~i~r~.v~atz~~~ by the relation:
d_
f ,.L' .r wnnM~1rIAM '.}.k~.,_~~UBi1f`.8
..,~~/ {AY MV4j~/yMYi ,
~~3,c a~abil.i,t~ ai~ar~, aircraft, first at states of ~fli~t where the influence of
a~sS
the caefficiante C
~
t
~
i
y o
careeeibili
.~~~~
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: CIA-RDP81 010438001200230006-3 ;
rove or a ease
py pp
w m
Taking e~,(6.3) f'~r the ~oefFicienL of a4, it ie not ~;-.fficult td nbtar=
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r
,'t'he go~ar~t~.t~*,.~ ~rixl be eq~~a~. to the gtity ~ , filch we .l~l mbbrev~at~
~~
~~ C~. ria is yell ~~c~m, ~in the r~~on of anles':of at~i