•1
REPORT No. 77
0
Klusr.
J DEL
TEGPNISCHE HOGESCHOOL
VUEGTUIGBOUWKUNDE ^temaolilraert 10 - DELFTTHE COLLEGE OF A E R O N A U T I C S
CRANFIELD
4
ESTIMATION OF THE EFFECTS OF DISTORTION ON LONGITUDINAL STABILITY
OF SWEPT WING AIRCRAFT AT HIGH SPEEDS (SUB-CRITICAL MACH NUMBERS)
by
B. S. CAMPION. D.C.Ae.
This Report must not be reproduced without the permission of the Principal of (he College of Aeronautfcs.
REPORT NO. 77 J a n u a r y . 1954 T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D E s t i m a t i o n of t h e E f f e c t s of D i s t o r t i o n on t h e L o n g i t u d i n a l S t a b i l i t y of Swept '.Ting A i r c r a f t
a t High Speeds ( S u b - C r i t i c a l I/Lach Numbers)
b y
-B.S. Campion, B. Sc. (Eng. ) • , D.C.Ae.
S U M M A R Y
The e f f e c t s of d i s t o r t i o n on t h e l o n g i t u d i n a l s t a b i l i t y of swept wing a i r c r a f t a t h i g h speeds ( s u b - c r i t i c a l Mach numbers) a r e c o n s i d e r e d on a q u a s i - s t a t i c b a s i s . The method employed i s b a s e d on t h e t h e o r y of G a t e s and Lyon b u t i n v o l v e s some e x t e n s i o n
of t h i s t h e o r y .
The t r e a t m e n t of wing d i s t o r t i o n i s c o n s i d e r e d i n some d e t a i l and t h e e f f e c t s of b u i l t - i n t w i s t and camber and wing w e i g h t
3
a r e i n c l u d e d u s i n g t h e s o - c a l l e d s u p e r p o s i t i o n method . The 2
a p p l i c a t i o n of t h e a n a l y s i s of Lyon and R i p l e y f o r i n v e s t i g a t i n g f t i s e l a g e , t a i l and c o n t r o l c i r c u i t d i s t o r t i o n i s s u g g e s t e d , b u t means of modifying and s i m p l i f y i n g t h i s p r o c e d u r e where d s s i r a b l e a r e put f o r w a r d .
The a n a l y s i s i s i l l u s t r a t e d by means of a s i m p l e exa:nple of a swept wing f i g h t e r a i r c r a f t f o r which v;^ing, f u s e l a g e and t a i l d i s t o r t i o n e f f e c t s a r e c o n s i d e r e d , and t h e r e s v i l t s a r e cUscussed w i t h r e f e r e n c e t o t h e i n d i v i d \ i a l and combined d i s t o r t i o n e f f e c t s a s w e l l a s t h e e f f e c t of c o m p r e s s i b i l i t y ,
f
2 -«
4
LIST OF COrCTEOTS1. Introduction
2. Y/ing distortion
2.1. Solution of the aeroelastic equilibrium problem for a
flexible lifting surface.
2.2. Effects of wing distortion on the wing lift and pitching
moment contributions.
2.2.1. \7ing -vvith zero b\iilt-in tvd.st and camber;
distortions due to wing weight neglected.
2. 2. 2. Y/ing v/ith built-in t m s t and camber;
distortions due to ^ving vreight neglected.
2.2.3. Vfing vri.th built-in twist and camber when
distortions due to vdng weight are included.
2.3. Effects of wing distortion on the downvra.sh at the tail.
3. Fuselage, control circuit and tail distortion.
3.1. Effect of fuselage inertia loading.
3.2. Calculation of A , A^ etc. including distortion effects.
4. Modifications to existing longitudinal stability theory
4.1. Static margin.
4. 2. Manoeuvre margin,
4.3. Dynamic stability - quasi-static theory.
4.4- Application of the theory.
5. Miscellaneo\as refinements.
5.1. Inclusion of effects of change of density with altitude
on stability.
5.2. Incliosion of effects of changes in the form of the ^ving
lift distribution and movement of aerodynamic centre
due to compressibility.
€.
A simple example.
6,1. Calculation of lift and pitching moment coefficients
of the flexible wing and tailplane.
6. 2.
Introduction of fuselage distortion.
6.3.
Trim curves.
6.3.'!. Trim curves-.yith C = 0 .
m
6. 3 2. Trim curves with C. = - —'••—^
"° N/I-M^'
6.4. Stick fixed static margin.
6.5. S t i c k fixed manoeuvre margin.
6.6. Q u a s i - s t a t i c s t a b i l i t y d e r i v a t i v e s .
7. Concluding Remarks.
L i s t of References
L i s t of Symbols
The longitudinal stability of an aircraft is usually considered in terms of 'static stabilL-y' (measured by the static margir-s), 'manoeuvrability' (measured by the manoeuvre margins) and general dynamic stability. If the stability derivatives are very little modified by frequency effects (i.e. they approximate closely to the quasi-static values), then the static and manoeuvre margins as normally defined can be related to the coefficients E^ and C^ in the 'stability qijartic' given by the usual small displacement theory. The values of the static and manoeuvre margins then largely determine the characteristics of the phugoid and short period oscillations.
^Tien structural distortion effects are introduced, it is again necessary to consider whether the motion of the aircraft (in the long and short period oscillations) occiurs under quasi-static conditions, or whether the dynamics of the separate aii'craft com-ponents should be considered, introducing additional degrees of freedom corresponding to oscillations of individual components. This question is discussed in ref. 4 where it is pointed out that if the frequency of the short period oscillation is of the same order as the lavest natttral frequency of any component (e.g. the wing) the simple qiiasi-static approach is suspect. Once the
quasi-static approach is abandoned, however, the treatment of djmamic
stability when distortion effects are included becomes very difficult. The general treatment of the dynamic stability of a flexible aircraft can be similar to that employed in flutter problems, although the difficulties are enhanced by the fact that coupled oscillations of wing, fuselage, tail, etc., are combined with the overall tigid body'motion of the aircraft. An attempt to formulate the equations governing the motion of an aircraft with flexible fuselage and
wings is made in ruf. 5, and the problem is considered briefly in ref. 4.
In this report the 'quasi-static' approach only is considered, as in refs. 1 and 2. By this method the equations of motion for a rigid aircraft are losed but the values of the
aerodynamic derivatives are modified to include distortion effects, The basic theory is in essentials that of refs. 1 and 2 with certain modifications and extensions which it is believed will permit of a more logicaJ treatment of the effects of iving distortion in cases where there is built-in twist or camber, and the treatment also permits the ready inclusion of the effects of aircraft weight. The conditions for the balance of aerodynamic, elastic and inertia forces are obtained by the superpositi-^ method of ref. 3, vriiich
-4-it is considered has many advantages over other methods.
The analysis is illiostrated by means of an example of a high speed swept vdng fighter, and the results of the analysis are discussed in detail. This example pi-esents a n\miber of features of general interest.
For compressibility effects on a rigid aircraft the important parameter is the Idach number so that the variation of true air speed must be conjidered. YTien the aircraft is flexible however, the distortioneproduced by aerodynamic loading are
dependent on the equivalent air speed. In general, therefore, we must consider the two parameters M and q = ^oV . In this report the suffix M indicates that a derivative is taken at
constant Mach number (e.g. (fC,/öa),,), and in such cases it is also implied that q is constant.
The corrections for variations of inertia loading due to normal acceleration introduced in para. 4 are ' q-uasi-static' i.e. it is assumed that the normal accelerations of all parts of the aircraft are the same as that of the C.G. and that the structure is alv/ays in equilibrium under the applied aerodynamic and inertia loading. This assvanption is similar to the assumption that
frequency effects on the aerodynamic derivatives, etc. may be neglected and m i l similarly becane invalid when the short period frequency is high.
Acknowle dgement
The author wishes to thank Professor A.D. Young of the Department of Aerodynamics for his guidance in the preparation
of the original thesis and in its revision for publication.
2, ïïing distortion
YTien considering longitudinal stability it is usual to assume that the ailerons T/ill remain in the neutral position when the wing distorts. Distortion of the main wing structure only is then considered. For most swept wings the loss of incidence due to upward bending is greater than the increase due to twist (referred to an ideal straight flexijral axis). Thus wing distortion produces a net loss of lift curve slope accompanied by forward movement of the aerodynamic centre (see 2.2).
YHien the quasi-static approach is used in obtaining stability criteria and derivatives it is assumed that the structure is al'.vays in equilibrixjm under the applied loading. The problem of treating wing distortion is then basically that of solving for the 'aeroelastic equilibrium' of a flexible lifting surface.
2.1. Solution of the aeroelastic equilibrium problem for a flexible lifting siarface
The problem of calculating the aerodynamic characteristics of a flexible lifting surface is one cf seme difficult^'- due to the fact that distortion under load produces a change of load. Mathe-matically the problem takes the form of the solution of an integral eqiiation.
It is possible to solve the problem (using 'strip' theory) by successive approximation (see ref. 4) or by the method of semi-rigid representation (refs. 4,8). These methods are -^11-known and widely used. They suffer from the disadvantage that since 'strip'
theory m\xst be used induced aerodynamic effects d\je to the distortion itself are neglected; also with the semi-rigid method the accuracy is reduced by the need for approximate representation of the distortion mode. More recently the 'superposition' method has been put forward
(ref. 3) and since it was felt that this method is in seme respects superior to the above it has been adopted here. The following is a brief debcription cf it. The usual assimption of linearity between loading and incidence is fundamental to the method.
At any point on the span of an elastic wing in equilibrium vinder aerodynamic load, the final geometric incidence Oy v/ill be given by:
Op = aj + o^
where a^ = geometric incidence of undistorted wing ('initial' incidence)
Op = change of incidence due to distortion.
/if Op ... X The superposition method has also been developed independently at
-6-If Op is known, the final loading is known and o^ may be calculated, giving a^. Thiis the problem is easily solved
'backwards'. If a number of arbitrary 'final' incidence distrib-utions <Xp. , a__, etc. are chosen and the corresponding 'initial' incidence distributions a_ , a.j.„, etc. are so obtained, then any given initial incidence distributions may be approximately repres-ented by a linear combination of the arbitrary initial distributions. Thus a = A a_. + B^a^ + C„a_, + ...
In practice three or four such terms may be sufficient. The coefficients A„, B , C„ etc. are functions of the parameter qA^, where L^ is the rigid wing lift curve slope allowing for compressibility effects.
But Aga^^ = AgOp^ - AgOg^
^3^X2 = ^ S ^ 2 " ^3*^2 etc.
and t h e r e f o r e
a^ - o,^ = AgOp^ + BgOpg + OgO^^ + . . .
- (^S^l ^ V E 2 -^ °S°E3 "•
-"^
so that Op = AgOp^ + BgOpg + '-'s°F3 •*••••
and hence for the given initial incidence distribution, a_, the final incidence distribution cu, and hence the final aerodynamic characteristics can be obtained. It vidll be seen that by means of this method the aerodynamic and structural problems are separated and may be considered independently.
This method is likely to yield more acc\jrate results for many problems than the other methods mentioned previously, due
mainly to the fact that induced aerodynamic effects due to distortion are readily included and no sweeping aaauraptions need be made about tlie mode of distortion . Once the calculations have been comp]eted for the arbitrary cases chosen, equilibriim conditions for any
combination of q and H are quite easily obtained. As with other methods, some difficulty is encoxmtered if the form of the aerodynamic loading varies appreciably with Mach No. (see 5.2).
moment contributions
2.2.1. Wing v/ith zero built-in twist and caLiber; distortions due to v/ing weight neglected.
If the incidence of the wing (a) is defined as the angle between the chord line of the vring and the direction of flight, measured at the vring root, then we may write
Cj = wing lift coefficient = {/ Law\ —^—) a = Aa ., (l)
^
/ M
where C- = wing lift coefficient of flexible wing due to (root) incidence a.
Law
A = wing lift curve slope (following ref. 1) = • ,
V 0^
/M/aCj. \
It is possible to find ( —r 1 using the superposition method as described above (see al^o ref. 3).
Similarly, taking moments about the rigid vd.ng aerodynamic centre, we may write
" . = f r ) / «
w ^ y M
where C is the pitching mcanent coefficient corresponding to C. and may also be found by the superposition method. For the
'rigid wing' C as defined above is of course zero, but with the flexible wing there is a pitching moment about the rigid wing mean aerodynamic centre (H ,) v/hich is proportional to incidence i.e. there is a movement of the wing aerodynamic centre given by
dc / ac^
giving the mean aerodynaiiiic centre of the f l e x i b l e wing
H = H ^ +
A H.
o oR o
The wing patching moment coefficient about the aircraft C.G is then
. C^^ = Aa (h - H ) (4)
and th3 wing pitching moment coefficient about the new mean aero-dynamic centre (H ) is zero
i.e. (C„ ) = 0 .
m
o wing
2 . 2 . 2 . ¥ i n r with b u i l t - i n twist and camber; d i s t o r t i o n s
due to wing weight neglected,
•"iThen the u n d i s t o r t e d wing has t w i s t or camber i t i s
/r)Of5sible to
-3-possible to consider the effects of incidence, twist and camber
separately. V/e then have, applying the principle of superposition
as in ref. 3, that
( Law I
Cr
= • V'•
I
a +
C T+
CT,
Lw
l
,
^ do. / Lew Ltw
= ^^ - ^Lwo ^5)
where a = root incidence as defined in 2.2.1.
C-. = lift coefficient due to root incidence a on
Law
•wing with zero built-in twist and no built-in
camber.
C^ = lift coefficient on vdng with zero root incidence
and zero built-in twist, but with built-in camber.
C-. = lift coefficient on wing with zero root incidence
Ltw
^
and z e r o b u i l t - i n camber b u t w i t h b u i l t - i n t v d s t .
Lwo ~ Lew Ltnv
- r ~ ^ ) (as b e f o r e )
C_ and 0-., may be found u s i n g t h e s u p e r p o s i t i o n method and a r e fxinctions of speed and Mach No.
'ac.
Thus I - — • ) = I - 3 — ^ ) = A a s i n 2 . 2 . 1 .
Also, the pitching moment coefficient about the rigid wing mean
aerodynamic centre is
C = ( ^ V ^ 1 a + C + C ^
raw \^ aa y mew mtw
where C , C . C , are the pitching moment coefficients
maw ' mew* mtw ^ *
corresponding to C^ , Gj . Gj , and hence
fdc
\
/'ac
\
mw 1 _ / maw ]V aa / j ^ = \^ aa y j^j
ac / acT . A 7T niaw / La^T . r-. n A i . e . ZiH = - — r — / — r — a s m 2 . 2 . 1 . o aa / oa Thenac
C = - = . a + c + ( h - H ^ ) C. raw „ da rawo oR Lw 0 . G, where C = C + C , , mwo mew mtw ' i . e . C = A a , / i n + C + (h - H ,,) (Aa + C^ ) raw- _ o mwo oR Lwo0 . G.
= (C ) + Aa(h - H ) , (6) ^ mC^ . ^ o ' '
wing
where (c ) = C + (h - H „ ) C^. mo' . rawo cR Lwo
vn.ng
and H = H ,, + A H .
o oR o
This expression for C is different from that normally \ised ^C.G.
in that Gj ^ Aa. It vdll be seen, however, that it is conven-ient to express quantities in the forra X + Ya where X and Y are functions of speed and Mach No. but not of incidence.
2.2.3. '''ing ¥dth built-in twist and camber when distortions due to Tdng weight are included
Following the method adopted in 2.2.2 we may vnrite
/ac.
C^ = A°a + C° +
L w L w o N. wii ^T,,
M
where n-1 = l/g x normal acceleration of aircraft (positive
C^oV^s upwards)
i.e. n = —rz , in flight for which the inclination
of the flight path to the horizontal is small.
C, = lift coefficient on wing with zero root incidence, zero built in twist and no built-in camber, due to deflection of the wing under its effective weight.
The affix ° denotes the condition n = 0, so that the effects of wing weight are entirely contained in the last term.
ac,
As w i t h Gp e t c . , — r ^ may be fovind u s i n g t h e s u p e r p o s i t i o n method.
S i m i l a r l y , talcing moments about t h e r i g i d wing aerodynamic c e n t r e we have
/ac
V
/'^
! _ maw i _ . „ o . _ / mnw 'raw r, now J „ o mnw fc,\ C = 1 — t — j . a + C + n ! — x — / \p) •L-rtw { da /^r mwo \ an '--'M and a b o u t t h e C.G. we have/ ac \ ° / ^C \
C =
! _ ^ )
.a + C° + n(
- f ^ l
/ ac^. \
+ (h - H ^ ) (A°af C° ) + n - ^ (h - H ^ ) ^ cR' mwo' V y F = (C ) ° + (h - H°) A°a + (h - F^)A . n ^ mo' . ' o ' ^ o ' n wing . . . ( 9 ) / w h e r e • . •r where H = o A = n
1
<-In practice the and (C )° mo' . wing-10-/ac / ac^ N °
H^ . AH°, AH^ = - 1^-^/
-^J
f Lnw A TTn 1 mnv/- / Lnw 1L
«"A
'
'^"
"K
'-/
'-'J
«cB
-
^K-affix 0 could be dropped from A°, C° , C° *• ' Lwo' niwo vdthout confusion occurring.
The effect on existing longitudinal stability theory of
i
using these more considered in §
complicated expressions for C, , C is
2.3. Effect of wing distortion on the downwash at the tail
The incidence of any point on the rigid tailplane is given by
^ 1 = a + n^ - e^
- where e = local downwash angle. • ' Hence, following \ we could write
Ac^,
due to tailplane whei^ A p = a = cK
1s ^ = Values superposition me ^ f c k ^ were known.the method employed in dealing vdth vdng distortion,
= ^ a ^^ * ^ T ) - ^ e ^R incidence
^°LT a a
c
root incidence of tailplane with zero wing dov,Tiwash = a + n„
^°LT ^ ^ R
downwash angle at tailplane root.
of A. and A. might then be found using the
la 1e '^ °
thod, provided that the spanwise downwash distribution
^ ^ ^ ^ k V'e coiold then write
^ ^ L T
1
= A, (a + Tij - Sj^)
/due to .,,
du3 to incidence
where ^ = ^R { ^f J
^ la •'
and A. = A.
1 la
In practice, however, the downwash distribution is not usually knoTAi with any accuracy even for the rigid wing, Jxnd since it will c'.iange slightly with speed due to wing distortion it will
not be possible to carry cue a single aeroelastic equilibrium calculation for the tailplane applying to all speeds. In view of these difficulties it seems acceptable to use a mean downwash angle, the mean being weighted in favour of the regions of greatest tailplane lift.
Since the downwash is produced by vdng lift it is logical to use an expression for the mean downwash angle ê having the saue forra as the expression for C, .
Lw
• •
Then ë = A a ê + ë + A . n ë = ë + e + ë ,.(10)
a o n n a o n ' where the lift distribution corresponding to A a produces a mean
downwash angle e , the lift distribution corresponding to C_ produces a mean do-Tnwash angle e , and the lift distribution corresponding to nA prod\ices a mean downwash angle e
The value of e v/ill approximate to the value of e„ defined above, and we va'ite
^ ^ - h i
= ^"oR ^ ""oR ^ ^nR
»so that the use of e does not necessarily imply an approximation, Since the lift distributions A a etc. may be obtained by using the superposition method, it is possible to obtain the do^wiivash distributions corresponding to these lift distributions by
super-position of the downwash distributions corresponding to the arbitrary lift distributions employed. This can, however, be a somewhat
involved process, and in practice it may be acceptable to use
estimated values of ë etc. with semi-empirical corrections foi the a
effects of vdng distortion.
The above expression for ë is more complicated than the usual expressions for dovmwash angle, and the effect of using it in stability theory is considered in §4.
I
-12-3. Fuselage, control circuit and tail distortion
Distortion of the fuselage, tail and control circuits modifies the tail lift and pitching moment contributions, and also the elevator hinge moment. This has been considered in ref. 2, where however compressibility effects have not been explicitly
considered.
2
Lyon and Ripley write
°LT = ^1 °To + ^ 2 ^ -^ H^
"H
= B + B, ^ o "^ ^ 2 ^ * ^-b^I
where a^^ = a + ii^^ - ë
Tj- = tail setting angle at zero windspeed ('built-in' tail setting angle)
Ti, p are control angles eq-uivalent to movements of the pilot's controls ("11,^ of ref. 2) as is Tim if a 'variable incidence' tailplane is used
Alternatively, we may write
^LT - ^1T °!r ^ ^"P "^ ^^3^
Cjj = B^ + B^jO^ + Bg-n + BjP
where a_ = a + TI - ë
r\- = t r u e (root) t a i l - s e t t i n g angle
A - f ^
1T - a 0^ and likewise B= value of A. when fuselage is rigid,
^°H
1T
^Otj,3.1, Effect of fuselage inertia loading
The above expressions may be used to include the effects of all distortion xmder piarely aerodynamic loading, the values of A., A etc. being modified to take account of these effects. If
/the effect ...
je This applies to tailplanes of 'fixed' incidence - wi-öi a 'variable incidence' tailplane A _, is the value of A when both fuselage and incidence control circuit are rigid.
t h e e f f e c t of b e n d i n g of t h e f u s e l a g e under i t s ovm weight i s t o be i n c l u d e d a s l i g h t m o d i f i c a t i o n of the e x p r e s s i o n s f o r C^„ and C„ i s n e c e s s a r y .
Suppose a p p l i c a t i o n of a ( t o t a l ) normal a c c e l e r a t i o n of ng produces a n e t change of n. a a ^ c n i n t h e v a l u e of a^p.
Then G^^ = A^^a^ + nk^^ j - ^ + A^ri + A^(3 (l 1)
where a^ = t r u e r o o t t a i l p l a n e i n c i d e n c e f o r n = 0. C,^ = C,° + nA,^ -r-^ (l 2) T h i s may be w r i t t e n
_ r'
^LT ~ LT '^ '"^IT a rï and s i m i l a r l yo
'^"TCjj = Cjj + nB^Ta^ ^ (^3)
= ^ o ^ ^ 1 ^ T o ^ ^ 1 T r l - ^ ^ 2 ^ ^ V j
If application of ng produces a net change of tail setting angle ^ ri^, we have
A^T /ATI^ qS^ A ^ T = Z^i^Ti
where ATI_. = change of ri_ due to ng at zero windspeed
q =
hY^
load on tail change in r\„
F„ = fuselage bending stiffness =
Then 'tJ'Hrn = A q S . 1 T ^ T
3.a ^ =
''^. ^ f — ^
^ .. .,...(14)
a n n on I
^ \
'
* ^ T — /
/ ^ T , 2a^a a ^ a nj = " ' a M^^a n^ - - an ,. 2 - ^ F ^ ;
V (15) /3.2. ..,TECHNISCHE HOGESCHOOL
VUEGTUIGBOUWKUNDE - 1 4 - Kaaaal»traat 10 - PET ff
3.2. CalctiLation of A , A^ e t c . including d i s t o r t i o n effects
The a n a l y s i s of ref, 2 i s adopted here but with c e r t a i n
modifications and the incltision of compressibility e f f e c t s .
Fuselage d i s t o r t i o n (aerodynamic loading)
Formulae are deduced i n ref. 2 for the r a t i o s A / a ,
AVa„ allowing for fuselage d i s t o r t i o n and these expressions may
be applied i n general i f r, , a^, e t c , are i n t e r p r e t e d as the
values of A , A e t c . «iien compressibility and a l l d i s t o r t i o n
e f f e c t s except fuselage d i s t o r t i o n are included.
T a i l p l a n e - e l e v a t o r d i s t o r t i o n
As pointed out i n ref. 2, t a i l p l a n e d i s t o r t i o n and
e l e v a t o r d i s t o r t i o n cannot be considered separately since the tvro
are completely interdependent. For t h i s reason i t i s not s t r i c t l y
P'jssible t o use the superposition method for the estimation of
d i s t o r t i o n e f f e c t s on the t a i l p l a n e - e l e v a t o r combination. In
ref. 3 a method of t r e a t i n g the ' f l a p - d e f l e c t i o n ' case i s given
vrhich \ises the superposition method, but t h i s i s based on the
assumption t h a t the l o c a l f l a p angle i s not changed by d i s t o r t i o n
of the main surface or f l a p , so t h a t the form of the spanwise
load d i s t r i b u t i o n due to the flap d e f l e c t i o n remains ^he saaae a t
a l l speeds. ''/hile t h i s assumption may not l e a d to seriovis e r r o r s
when dealing with the vdng-aileron combination (although i t seems
dubioxis even in t h i s c a s e ) , i t seems l i k e l y t h a t i t vrotild lead t o
appreciable e r r o r s i f used for a t a i l p l a n e and elevator. For
the treatment of the l a t t e r , therefore, the method of ' s e m i - r i g i d
r e p r e s e n t a t i o n ' as given in ref. 2 would seem to be more s u i t a b l e .
I f the e f f e c t s of tab d i s t o r t i o n and elevator skin d i s t o r t i o n can
be neglected (see below), the e f f e c t s of compressibility may be
introduced (by means of l i n e a r i s e d theory corrections) to the
' s t r i p ' d e r i v a t i v e s used. The c o r r e c t i o n s applied should be
those appropriate to the three dimensional t a i l surface, so t h a t
the overall lift and hinge moment coefficients vary in the co^rrect way although changes in the form of the lift distribution due to compressibility are ignored, as m t h the superposition raethod. It is not possible to carry out the calculations in terms of a
'compressibility - distortion' parameter such as c^\^ (used in the supejrposition method) since the aerodynamic coefficients a^, a„. a,, b,, b^, b, are net all modified by conpressibility in
2' 3 1 2' 3
the same way. It is therefore strictly necessary to carry out calculations for each combination of q and M within the required range.
Elevator tab and tab circuit distortion
In ref. 2 it is shown that if C and C are small (as they usually are), A , A^, B and B_ are almost iinaffected by tab distortion and A,, B, and C, are modified by the factor
(K/K-C4) v/here Cl is the value of C, when tab distortion is ignored and K is proportional to (l/q) x tab stiffness. On this basis the elevator angle to trim, and hence the stick fixed static margin K , is not r ppreciably changed v/hile the stick free static margin K' is slightly changed. Similarly the stick
fixed manoeuvre margin H is almost unaffected while the stick * m
free manoeuvre margin H' is slightly changed. If the tab circuit can be designed so that K is large compared vdth C' for all q, tab distortion effects are then obviously negligible. If this cannot be done, tab distortion effects are easily included by appljdng the above factor to A,, B,, C,, on the reasonable assumption that the secondary effects of tab distortion on elevator and tailplane distortion are negligible.
If power operated controls are fitted this type of distortion does not, of course, arise.
Elfvator skin distortion
The effect óf elevator skin distortion is also considered in ref. 2 v/here it is shown that distortion of the elevator skin, caused by pressure differences between the inside and outside of the control surface, can modify A^, A and A slightly and B , B_ and B, to quite a large extent. The treatment is approximate,
since skin distortion of a fabric or metal covered surface is not proportional to load.
The curves in ref. 2 of panel deflection agaii-st load shov/ that for an unstiffeiiod metal skin the rate of change of deflection with load is small once the small initial deflection required to take any load has been exceeded. If the skin is applied with initial tension (as is nori'nal practice) this small initial deflection ccai be miniiaised. In viev;- of this it seems reasonable to ignore the effects of elevator skin distortion for design purposes on the grounds that it should be possible to design the elevator so as to keep the skin distortion and its effects small enough to be negligible.
The main effect of any skin distortion is to decrease the stick free static and manoeuvre margins, so that if powered controls are used the effects are even less serioiis than with manually operated controls.
/Elevator control ,,.
I
-16-Elevator control circuit distortion
The treatment of elevator control circuit distortion given in ref. 2 can be applied in general if a., a„ etc. are interpreted as the values of A , A^ etc. when compressibility and all distortion effects except control circuit distortion are included.
4. Modifications to existing longitudinal stability theory Y/hen the new expressions of § 2 and 3 for CT . C
"^ LV mw e t c . a r e used vre h a v e , i n g e n e r a l . -o ^T \ / -o - - - N CT = A a + C-. + nA + -rr- } A. {a - A e .a - e - A n e + TI_ } L Lwo n S 7 1 ^ a o n n To' C = C „ + (C ) + (h-H°)A°a + (h-H") A . n m mf mo' . ^ o ' o ' n w i n g
- V- - / A, (a - A ë a - ë^ - A^.n ë^ +I AH ^ < A , ( a — A s u . — c. —A ,11 c. "riJm J V / o— TI ) C.G. 1 1 a o n n To'
* ^ 1 T F I * ^ 2 ^ - ^ V ] ^'^^
irfiere C „ s C c o n t r i b u t i o n from f u s e l a g emi m
^C.G.= ^TV^^
t" s d i s t a n c e frcci t a i l p l a n e mean aei-odynamic c e n t r e t o a i r c r a f t C.G.
The o t h e r new symbols a r e d e f i n e d i n §2 and 3 .
4 . 1 . S t a t i c margin I n r e f . 1 / d C N / d C K' = - - ^
^' c^ = 0
2 W where C^oV = c o n s t a n t = r r R 2^i.e. in level flight n = 1 = constant.
F o l l o T n n g r e f . 1 t h e e x p r e s s i o n f o r C can be m o d i f i e d by t h e i n t r o d u c t i o n of a new t a i l - v o l i i a e c o e f f i c i e n t ¥„,. Thus, when n = 1, where and C = C „ + (C ) + (h-H°) (C^ - C,. - A ) + (h-H'^)A m mf mo' . ^ o ' ^ L Lwo n ' ^ o ' n wing
-T^{G^-^)fc.-
C, - A ) - ë - A s + Tin, Lwo n o n n To iaa,
1
'IT a n + A,^ r ~ - + A Ti + A,pi
V ^ T = AC 1 + A, - ^ [ ^ - ê \ 1 S (_^o aj AC 1+F\c = ^c.G. * ^K) r
(18) On d i f f e r e n t i a t i n g e q u a t i o n ( l 8 ) w i t h r e s p e c t t o C_, and making use of t h e f a c t t h a t • ^ = 0, we f i n a l l y o b t a i n , a s i n r e f . 1:-/dn >
\ —V^A^l^dcJ
C a 0 mSimilarly for the stick free case: - (n = 1)
C s C „ + (C ) + (h-H°) (CT - C^ - A ) + (h-H^) A m mf mo' . ^ o' L Lwo n' ^ o' n wing
r i - - ë ^
"
^T ] ^1 ("^ - ^-^ ^°- •
L Lwo n o n n To^^— - ^-^
- ^^ - A^ ^^ + ^T
- S -
h 1
^ ^ T T T ^ V - B ; ^ O [
09)
where V, ACT
^2^ = ^ - Bj ^1
AC 1+P A;,j = A^^ - ^^ B^^ Ap^3 " ^3 B^ ^3 •
This yields similarly,
^n - ^T^3ldC^;
C ^
0
m
1 8
-4 , 2 , J.ïanoeuvre Margin
-I The manoeuvre margin, H , is given by
4i
^
^^^LL8 ^1
The second term is usiially fairly small and decreases with increase of altitude, so that the value of the manoeuvre margin is greatly influenced by the value of (ac /SiGj) X
M IThen n i s v a r i a b l e vre have
C = C „ + (C ) + (h-H°) (C^ - C^. - A . n) + (h-H^^) A . n m mf mo' . o ' L Lwo n ' o n
wing
- ^ T
|AJ/'^-è)
(C_ -L Lwo n o n n To CT - A n) - ë - A ë . n + Ti_D i f f e r e n t i a t i n g (20) w i t h r e s p e c t t o C^, we have w i t h s t i c k f i x e d ( i . e . TI and p c o n s t . )
M M ^ ^ M
- V ^ J A
1 (^^,o 'a) (^ \ -^ac^ I ƒ ^ T a n i^ac^ i
M' " ' M (21) n vr I f C^ = ;, and C^ L '. „ 2 „ Lo tpV S •"'' ^DV S
o. = -.„ -^(IÓ:) = 4 (-)
L/jj U,Y/ith s t i c k f r e e vre have
a
.v-.i.-^\) -^(^)J-.;!(I^Ii
' M / ^ -^'MJJ
(23)
/ T h e t a i l . . .
* Note t h a t -t = d i s t a n c e between t a i l and n o - t a i l aerodynanaic c e n t r e s ^ -t^j, ,
/m
The t a i l contribution to i —-* . -f \ i s , approximately,
^ '•'l c '^
Vi
2 P ,
s t i c k fixed
Vi
and - stick free . 2 u^
The wing contribution to m is not necessarily negligible if the wing is swept.
In the manoeuvre margin theory it is Implicitly assuraed that m is not appreciably changed by mcr/ement of the C.G. Th\is in deriving the above expressions for tail contribution to m the tail moment arm used is the distance between the tail and
'no-tail' aerodynamic centres. This is not strictly correct since the angular velocity q is about the C.G.
''Tien the tailplane is swept its centre section and tips may be fairly large distances fojrward and aft of the mean tail aerodynamic centre, so that more accurately
Vi
ra = - V " + m ^ a i l 2,. q^where ra = c o n t r i b u t i o n t o m due t o r o t a t i o n of t a i l p l a n e
qrp q a b o u t i t s m.ean aerodyTia:7iic c e n t r e . I n g e n e r a l , hovrever, m w i l l be s m a l l . qrp4.3. Dynai.iic Stability - quasi-static theory
dynamic stability and its relation to the static and manoeuvre margins is discussed in ref. 1 where it is shown that the slow divergence associated vdth K <C0 is less serious than the rapid divergence or rapid unstable oscillation corresponding to H ^ O . This means that the value of (aC /aCT-) is of
™ nr L j,j
greater importance than the value of dC /dC_ . In this paragraph we shall discuss the effects of distortion on the quasi-static
longitudinal stability derivatives.
If distortion effects are included on a quasi-static basis, the usual small-displacement equations of motion can be used for the flexible aircraft, but the derivatives have to be suitably modified.
YJe have, ignoring the thrust contributions, (see ref. 1)
I
-20-u - ''D 2'^dM/^' '-20-u - "^L" 2^.31!/^ ƒ
The largest contributions to m and m. are normally from the tail, and we have *
A° ^T - ^1 ^T
^^1'"* t - 2 T ï ^ s - ^ ^ a ' \ T - ^ ( Ï : 5 T S- •
^25)
(tail) (tail) These expressions are approximate only and correspond to those given in ret. 1. The wing contribution to m. is normally very small at s\ibsonic subcritical speeds, but the wing contribution to m may be appreciable for a swept wing. An estimate of m (wing) may be obtained by the method that is described in Appendix III.
The derivatives x and z occur in the stability equations divided by \i. and are then normally srjTall enough to be neglected,
If the distortions of the aircraft components are small it seems reasonable to ignore distortion effects on the drag derivatives, so that C^, (ac_yda).. and (aC^aM) may be estimated for the rigid aircraft. It remains to determine
(aC^/da)j^ , (aCj^/aM)^, (acyda)^^^ and (aC^aM)^ in terms of A°, A., H , etc.
1' o*
Differentiating the expression for C^ given at the
beginning of this section (eqviation 1 6).
-(^]
.A°.A
(^)
.!Ï(A
I" -A»! -A
(^\ f
,
(^) , p. ,
A, (fl)
V^a^H IT a n 2 \.aa/'j, /^To\ fd§\ ^
assuming I—r—/ = { ^ J = ^
M ^
'M/an^ /an_\
(^^ _^r^.
(26)
v^i "w^^^r^LoV^^^
M /Therefore ...Therefore AO ^ ^ TA (A K^ '- \ A f^n\ ?
( r è y = ^ - ^ •••(27)
' " °Lc" s °Lo r^""^'^" ^''^ ^ i
"y.th stick fixed
{^)
= O, so that
._° + ^ A • (1 - A°ë .
stick fixed = 2 J S: ^jB)
( ~ i ) stick fixed = §—^J a^
va a/,, . „
M A S / ^
öo^
1 - ^ + P^ 7~- U,A s - A^_ — ^
CT S CT/ 1 n n IT a n
Lo Lo L
\?ith stick free (using
13).-/g) _ ! i ( , .A°;).(|a) f!lA j -!ll ! i )
giving
V^
O'/M A S „A„ ^T 1 / - • - % ^. / . aa^v
^ - cf ^ r crC^Vn - ^T r l ;
Lo Lo V'
S i m i l a r l y
/ac^N ...o ac, / . ^ aA
f L 1 ok Lwo / a n ) , n
\TTi) ^ °'m ^ m "^vaM/ n "^ " an
a a
^T f ^ ^ 1 , o
-+ ^ 4 TT^r ( a - A e a - e - A n e
+ T I )S l a M a o n n To'
• ' S l j ^ t l l ^a-'^ F i y - r M - a - M • ^^n-^niaS/^V
, ^^/an\ , a / S V '°T ^ ^ T , ^ 2 ,
•*• ^iT TH KW^"" ^ ^T TM VTÏÏ/ "" "^ a^* "ST ^ TTi^
( a M / a M '^ \ •
assuming ( - ^ J a O .
a
. ( i ) = ^ ( ^ ) ('")
^ ' a Lo ^ '^a
a
+ A
G^ /...N . /ac.Since n = r—
^ L o
and i n i t i a l l y n s 1.
A^ith s t i c k . . .
-22-IT-Tith s t i c k f i x e d = O, so t h a t f i n a l l y
(
a M/
,.o ac^. aA Sn, faA^
aA Lwo n T i l / , o - - , - \ aM a M S / a M a o n n To
aM
de ÖA ds 1 s t i c k f i x e d = at
, . , , o . ds \ de dA . de / ÖA - . 0 a \ o n - . n^laM^a-^^ - ^ / - T M - T M ^ n - \ a l !
'*' d n öK "^ ^iT a M l a n) '^ a M ^ "^ a M ^j
\ ^T 1 / -
^°T^
(31) W i t h s t i c k f r e e , i t can be shown t h a t ( u s i n g 1 3 ) .-l^aMJ^ ~ *• B^ a M ^ " B^ 1 a M "^ a M ^°' * ^TO ' '^
o -2 ë a - e - A ë ) a o n n de —\ de de aAC
1-^,0 . de —) oe oe d:i . \ d-i - , o g o , n n - \" "^ L ^ ^a "^ "^ a MJ ~ a l l ~ ''n a M "
T M^nj
dcLj-y aB.„ aav„ aB,) ^ / a c ^ \ / A . B . „
°T\ . IT _rT p _ J . ( _ L . f _ L W _ n - _ J T
M
• a n a Mj c^^ Va
M/^^
I B ^n B^
(32)
"^ ^iT a M i a n/ "^ aM
a^y
and hence finally
""aA^ ^°Lwo ^^
•^
ai-^ " l i T •*•
r°L\
[ -T-r:] s t i c k f r e e = V a My aA S ^ dA / . _ ^ T-r^ + TT- \ T-rr I a+Ti„ - A ° ë a - ë - A ë \ a M S / a M\^ To a o n n/C
I aA - .0 a o n - » n j
r , . o . ae ~i de aA . ae \
•°' Ui ^a'^
T M . J"
F M•
T M^n" "^n a M/
•^ a n • aM "^ IT a
MVa n/ a
M^ "^
TW'B^a
MJ A s„ _r ^Lo d%\ C^_ S C ^ . V I n n I T a n y(33)
'Lo V whereaA^ aA^ A^ aB^ a l ^ aA^ aA^
A^ SB^TH ~ TH, " BT T M ' ' ' ^ T ¥ * aM " ai " B ^ ÖM ?/
aA
ITa i
^ S aA^ A^aB^ aA^ ^ ^ a.^
M " B- Ö M ^ a M ' a M
a M ~ a M B^ a
M'" a
M' a
M~ a
M B .a
MaB, a/u
3 : : 2 : r 3 ^ H i
a M •
S i m i l a r l y s t i c k f i x e d ( f f a ) s t i c k f i x e d = ( ^ ) + (h-H°) A° - V^ _ A^ (l - A° ë ) v a a / y \ a a / j ^ ' o^ C.G. 1 a '/ac^-x r „ r • öa^~
+ T T - - r ^ i (h-H ) A + V^ ^ A.A e - A,^ - r - i ,. C^^V Ö a . 4 p o^ n C.G. L 1 n n 1T a nj< (34) / a n d , , . 'Loy'^ki
and /ac \
[rtl
s t i c k f r e e M C.G. 1 e ) a ' '°Lo ^^^41 ° ^
+ V, C.G.1 _' ^v
A_,A e - A^^ ^—• I n n I T a n (35) A l s o ac N •r— s t i c k f i x e d a M; •'a / a c „A a(c ) . / - , o dü' j + aA an nl
o' aM ) f ^A.+ (h-H^:; ^ - A^ ^ - V, , ( T i (o, -A° ë a - ë - A ë +TI^ J o-^ a M n a M C.G. I a M de \ aë aA / OÜ. - . o - a a o n n To' aë / ^ . o . de \ de öA . de ( ü - o _ a ^ o n - _ , _ n l.,aM a a M / a M a M n n a M ^1T ^ > d /'^<V^ ^ p ^ ^ "^ èM a n * 1T a M Va n / ^ a M ^ i^ a M i j f ac. \ ( f . aa-—1)
-
C T I F S I
l^n^K) - ^C.G.LVn^ •*• .^T a"! | >
(36) l a My' \ ^ '^a ; .JJ /ac • a M ( - r ^ i s t i c k f r e e/ac A a(c ) . f ,,o aH°) eA
mf mo'wing /, „o\ dk . o o { /, „HN n - VW/['- a M ^' ^ ^ (^-^^o^ a i - ^ T M > * ^^"^o^a^M
an'"'' _ r a l ^ ^ _ ^ "^
- A -r-ê - V_ „ ) T-r? (a - A° ë a - ë - A ë + TI ) n a M C.G. > a M ^ a o n n To
f /:»AO . ^ aë \ aë aA . aë "1
7 / a A - . . o a i o n - . n + A. - a i - r r r e + A -r-TT j - TTJ ~ T T ? S - -^ T"^ 1 [__ vaM a a uj a n a M n n a i/i_i •*• öM a n IT a M \ a n / a M • "^ "^ è M ^ ~ B2' a M r + T T - I T I ? < AC- va M/ j n o' C.G. (h-H^) + V_ _ Lo a (_ ~ 1 - ö°tP~')
Vn'^ "^ ^1T T^Jh
(37)
The value of P (irdtial value) to be used in the above will be given by equation (l9) with 0 = 0 . The initial value
m
of T) may then be found from (18), again vdth C = 0 . Eqviation (16) then gives a. In the above it is assumed that V_, „ is constant i.e. that i^ is constant. This is not strictly true since in general the mean aerodynamic centre of the tailplane vrill move, but the error involved is probably small.
I
t
-24-4 . -24-4 . Application of the theory
S t a t i c margins
Perhaps the simplest way of determining the s t a t i c margins i s t o use the formiolae
K = - V_ A. l ' ^ ^
^n = - ^i h
(Sr)
^'c = 0
m n V ^ 3 V^T,'' m Cjj = 0The values of TI and (3 t o trim for a given value of C„ = C_ are given by equations (l 8) and (l9) v d t h C. = 0 . These v a l u e s can
lil
then be plotted against C_, (or M ) and values of (dn/dC,,) ^ K K O = U and (dp/dC^)^ _ are then obtained by graphical or
numerical differentiation. This procedure is suggested in ref. 2, and must involve some loss of acc\iracy, although since the trim curves can be obtained by calc\ilation this loss can be minimised by the use of a sufficiently large number of ordinates, ^rlaen the trim curves are plotted against C^. the high speed end of the range becomes 'compressed' and the low speed end elongated, so that it is probably better to plot control angles against Mach number.
Then ^ — = - - r ^ ^ , since C_M = constant. R R
dD
It would be possible to find the value of -577^ from the
relation «^^R
V.a
aL
V.' "" 20^ Va
uJj
" 2 0 , \a
aL
Va
M M a5 n V ^ g ^ M ^ ^^R-'^^-^V ^
^°R • /ac^^^'
^'M
This method, however, suffers from the disadvantage that values
aA aA
of -r-r: , "r-r? etc. must be obtained by graphical differentiation d M d M
(in general) so that the loss of accviracy is likely to be greater than that involved in finding -r^ .
R
In general we have
T
N
s - + C, = C„ Cos YipV^S L R '^e
T, T- C^, = C^ Sin Y.
V s ^ ~ ^
''
where T^ = component of thiTist force normal t o f l i g h t
d i r e c t i o n
Tn, = conponent of t h r u s t force in f l i g h t d i r e c t i o n .
Y = angle between flight path and horizontal. Thus in level flight C^ = C, for T„ = 0. This
K LI IN
r e s u l t i s not, as i s suggested i n ref. 6, dependent upon the drag
being small compared with the l i f t .
Manoeuvre margins
The manoeuvre margins may be obtained by d i r e c t
s u b s t i t u t i o n of values f o r A , A e t c . in the equations of 4 . 2 .
I t i s thus a simpler matter to obtain the manoeuvre margins than
t o obtain the s t a t i c margins due to the absence of d e r i v a t i v e s
with r e s p e c t t o forward speed i n the expressions for the former
margins.
5, Miscellaneous refinements
5 . 1 . Incl\jsion of e f f e c t s of change of density with a l t i t u d e
on s t a b i l i t y
The e f f e c t of density v a r i a t i o n vdth a l t i t u d e on l o n g i
-t u d i n a l s -t a b i l i -t y i s considered by Dr. Neumark i n ref. 18. Only
the l e v e l f l i g h t condition i s there considered, so t h a t 0, = C„,
S t a t i c and manoeuvre margins
Neumark gives;
/ac \ „ /ac \
„ [ m > M j _mn = " v a c , j * 2C^ Vfa
^ M ' ^ '" • ° L M/„'
/ ' rM2\r^m\ f, ï^N-l") M f'^ra)
M " ^L
where N =
g - R K g
and R = gas constant for a i r , K. = lapse r a t e .
-26-The static margin (stick fixed or free) as defined by Gates and Lyon in ref. 1 is K above (for 0 ^ = 0 ^ ) ; K _ is the
'generalised static margin' (stick fixed or free) v/'hich is
proportional to E. the last term in the stability quintic when density variation with height is included. However, the quantity K „ is no longer a measure of the stick movement (or force) to change speed,
/
ac
\
Prcm the above formulae given K and I -r—- ) (which ^ M
is fo\md when calcxilating the manoeuvre margin) K „ can be evaluated.
There are no effects of density variations with altitude on the manoeuvre margins,
Effect on stability derivatives
Additional stability derivatives Xp, z„ are introduced which are derived in full in ref. 18. These are, however,
functions of C aC_/'aM , C-, aC,/aM and of quantities unaffected by distortion so that they are readily determined.
5. 2. Inclusion of effects of changes in the form of the vdng lift distribution and movement of aerodynar^iic centre due to compressibility
If distortion effects are to be included it is very difficult to allow correctly for variations in the form of the wing lift distribution or in the aerodynaiTiic centre position of the rigid wing. If they are functions of lïach n\mber it is strictly necessary to perform a complete set of calculations fcr each combination of M and q. To avoid the heavy laboiir of such a procedure it is suggested that if the shift of aerodynamic centre is not very large then the aeroelastic equilibrium cal-culations can be made first ignoring the shift of aerodynamic centre. Then allov/ance for the shift can be made without
correcting for the secondary distortion effects introduced. If the movement of the aerodynamic centre is large, however, it might be advisable to perform calculations with a range of representative positions, an interpolation procedure being
subsequently adopted.
If the tailplane aerodynamic centre shift is also appreciable it may be necessary to modify the tail arm.
6. A simple example
As an example the hypothetical aircraft illustrated in Fig. 1 was considered. It has a effing of 45 sweep and aspect ratio 3.81, and an 'all-moving' tailplane of similar planform to the wing. For simplicity the structural characteristics of the wing and tailplane were assijmed to be similar, so that calculations
of lift cijrve slope etc. carried out for the flexible wing apply also to the tailplane. Compressibility effects were allowed for by applying linearised theory corrections to the vdng and tail
lift curve slopes, and also to an ass-umed value of C introduced
' mo into the calculations as an extra 'fuselage' pitching moment
contribution,-the distorted vdng has zero tvdst at a l l sections and i s of symmetrical section riving zero contribution to 0
•^ -^ - mo
No other fuselage or thrust contributions to pitching moment or lift were included. Tail lift was included in the total 0^., but wing weight effects were neglected.
Distortion of the wing, fuselage and tailplane was taken into account, but not control circidt distortion, and there is no
tab. This is consistent with a system of completely rigid power
operated controls, and hence the 'stick fixed' case only was considered.
6.1. Calculation of lift and pitching moment coefficients for the flexible wing and tailplane
'ac
Lw
j {
LawI
Our first problem is to find values of A =^ ,
o
and A H for a suitable range of values of q(= ^pV ) and Mach number. The wing is an example of the case considered in 2.2.1, and hence the superposition method was used.
The method used was exactly as described in ref, 3, the procedure being as
follows.-i) The method of Kuchemann vra.s employed to give (for incompressible flow) the lift distributions corresponding to
2 3
incidence distributions a = ri^, a = TI^', a = TI^, a = constant = 1 radian on the rigid wing. (TI^ =. y / w 2 ) ' These lift
distributions were integrated graphically to give values of C_p/A^,
^Ll/\' ^ L A ' ^L3Al' ^^^^^
CT-D = rigid wing lift coefficient per radian of incidence LR
C_. = rigid wing C, corresponding to a = "n , with a = 1 radian at the tip
2
Gjr) = rigid wing C, corresponding to a = TI , ivith a = 1 radian at the tip
\
r
I
-28-C-T = rigid wing C- corresponding to a = T]^, vdth
a = 1 radian at the tip.
Aj^ = lift curve slope of rigid wing.
It was found that
C ^ ^ / A ^= 1.0, C^^^Aj^ = 0.435,
^-^2^^ = 0.261+,
Gj^^/A^ =0.184.
ii) In stage (i) the iocus of aerodynamic centres was
obtained and hence the pitching moment distributions corresponding
2
to a = T) , a = T) etc. were plotted and integrated to give the
position of the mean aerodjni'iraic centre of the rigid wing, and
the pitching moment coefficients about that point for the varioiis
incidence distributions considered,
Then
^ - = 0 , — - = - 0 . 0 7 7 5 , T ^ = -0.0730, ^ = -0.0675
ni ns.
^ HR
where 0 ^j, 0 . etc. are the pitching moment coefficients about
the rigid wing mean aerodynamic centre corresponding to CTT,, C , . etc.
lii) _ A relation between Ap and Mach nimiber was obtained
using the method due to Collingbourne,
iv) The lift distributions for the rigid wing were integrated
to give shear force and bending moment distributions, it being
assumed initially that the wing had a straight flexural axis lying
along the 0.45 chord line. Torque distributions about this flexural
axis were also obtained.
v) Twist and slope distributions for the four cases were
obtained using assumed stiffness distributions, and these
distributions were then modified at the root in an attempt to
introduce corrections corresponding roughly to the root constraint
effects on a swept wing.
vi) The elastic incidence changes of (v) were matched as
described in ref. 3 and in 2.1.1. to give the superposition
coefficients A^, B„, C„ for a range of values of qAp.
vii) Using (iii) and (vi) a graph of A and A H against
o
liiach nvimber was produced for the condition q/q = M , where
q = value of q corresponding to maximum allowable E.A.S. (Pig. 2).
In the above it was assumed that compressibility effects
modified the two and three-dimensional lift curve slopes without
appreciably modifying the form of the lift distribution, or the position of the mean aerodynamic centre, of the rigid wing. The relation referred to in (iii) connects two and three-dimensional lift cvrve slopes, so that when compressibility corrections are applied to the aspect ratio, sweep angle and two dimensional slope, the required relation between M and A^ is obtained.
In (iv) and (v) it was first assumed that, with a straight flexural axis at 0.45c, the ratio pending stiffness
torsional stiiiness was constant along the span, and that each stiffness varied as the cube of the local chord. On the basis of information given in R.A.E. 3truct\ires Reports 9 and 58 it v/as decided that a representative value of the above ratio was 4.0, and that a representative root torsional stiffness was given by
TTT ""^""^ ^f
= / /»• where l„ = length of flexural axis. torqiie at root
It is shown in ref. 19 that with a swept wing of
moderate aspect ratio and conventional construction, the concept of an effective root may be used. The wing may be considered to behave like an unswept wing outboard of this effective root, but inboard of this the root restraint effects are predominant. The information given in ref. 19 suggests that for a wing of 45 sweep the effective root might be about 0.2 semi-span out from the root. On this basis the twist and slope distributions were modified as shown in Fig. 11. Since this modified the overall values of tvdst and slope considerably the root stiffness used was decreased from the value previously quoted, giving
''f ^
(GJ)R
=
—JQ-
.
Methods of calciiLating the r i g i d wing l i f t d i s t r i b u t i o n s
and the e l a s t i c d i s t o r t i o n s of swept wings are discussed i n
Appendices I and I I ,
6, 2, Introduction of fuselage d i s t o r t i o n
Following the treatment i n ref- 2, we have
A.
A
-—^
^1
-where J = constant, inversely proportional to the fiiselage bending stiffness.
Using the A.P.970 fuselage bending stiffness criterion, with M = 0. 8 and K = 0.12
J = ' "^ , where ^ corresponds to V^, q
-30-where V^^ = 'maximum allovirable diving speed' Since the wing and tailplane are similar, A = A.
6.3. Trim curves
Trim curves for the example a i r c r a f t are shcvvn i n F i g s . 3 and 4. Two s e t s v/ere p l o t t e d , one corresponding t o 0 = 0
(zero fuselage c o n t r i b u t i o n and zero wing c o n t r i b u t i o n about c o r r e c t e d aerodynamic c e n t r e ) and the o t h e r t o C = - . ' . '^
mo j —x
Jl -M^
(taken as (C )„ , (c ) . remaining zero), mo'fiise.' mo'wing ^ '
•
I t was ass\jmed for s i m p l i c i t y t h a t e = 0,1 = constant, and the r e l a t i o n
0 , = 0.0420 ( i ) = 2 : ^ L V q / jj2
was used, corresponding to V/^ = 50 lb./ft' and li - -^p Y where V = 1000 f.p.s.
Curves of T) (tail Retting angle equivalent to move-ment of pilot's control) against C, were prodijced for the rigid aircraft and for the f ollovdng cases of distortion.
-(i) Wing distortion only (ii) Fuselage distortion only (iii) Tail distortion only
(iv) '.'/ing, fuselage and tail distortion.
6.3.1. Trim curves vdth C = 0 mo
The curves are shown in Fig. 3. and it is clear that the distortion and compressibility effects introduced have had little effect on the slopes of the trim curves except at the highest speeds. Let us consider these effects in turn.
The effect of compressibility in the absence of
distortion is to dirplace the trim curve a small amount vrhich is nearly constant for all values of C_, The curve, v/hich is linear and passes through the origin when no compressibility
effects are included, remains very nearly linear down to C, = 0.1. Below this C, the slope of the curve becones slightly more
positive i.e. a stabilising effect occurs. Ref. 6 predicts that for C = 0 the increment of elevator angle to trim due
mu
to compressibility is very nearly constant over the whole speed range the approximation becoming less exact as Mach number increases. This is in agreement with the present restilts.
The d i s t o r t i o n of the wing alone then produces a
n e g l i g i b l e change in the trim curve. The reason for t h i s appears
t o be t h a t the effect of the l o s s of l i f t curve slope due t o
d i s t o r t i o n (the wing t i p s bend upwards) i s offset by the forward
movement of the wing aerodynamic c e n t r e . Thus, for a given C,,
wing d i s t o r t i o n makes i t necessary t o f l y a t a s l i g h t l y higher
incidence so t h a t for a given value of TI the nose down t a i l
c o n t r i b u t i o n i s increased. This e f f e c t i s almost exactly
cancelled by the e x t r a nose-up moment r e s u l t i n g from the forward
movement of the wing aeroc^naiiiic centre, so t h a t the value of
Ti-t o Ti-t r i m i s unchanged.
D i s t o r t i o n of the fuselage alone produces a constant
itncreinent of ii„ t o trim over the 'jvhole range, thus leaving the
slope uncihanged. This i s because with C = 0 and a fixed
C.G. p o s i t i o n the t a i l p l a n e load i s constant over the whole speed
range.
D i s t o r t i o n of the t a i l p l a n e alone a l s o again produces
a constant increment of t a i l angle t o trim and the reason for
t h i s i s again t h a t the t a i l load i s constant a t a l l speeds. Since
the form of the l i f t d i s t r i b u t i o n due to any given twist d i s t r i b u
-t i o n of -the -t a i l p l a n e i s assv-tmed independen-t of Mach number, -the
t v d s t due to a given o v e r a l l load i s the sai^ie a t a l l speeds and
hence the increment of t a i l - s e t t i n g angle to trim a r i s i n g from
t w i s t i s constant.
''iThen a l l the d i s t o r t i o n e f f e c t s are combined the t o t a l
increment of r\ t o trim i s s l i g h t l y .greater than the a l g e b r a i c
strn of the separate increments taken individiially. This i s
because vdng d i s t o r t i o n makes necessary a s l i g h t increase i n
incidence for a given C- which, for a given r],^ , causes an
XJ X O
increase in the tail load. "/hen the tailplane and fuselage are rigid this has little effect on r\ to trim since it is largely cancelled by forward movement of the wing aerodynariiic centre.
V/hen the tailplane and fuselage are flexible, however, the increased tail load causes extra tail and fuselai^e distortion which in turn reqtdre a small additional increment of r\ .
6,3.2. Trim curve with C = - rrrrr
Si2 / p
V1 -M
The cvirves are shovm in Fig. 4 and it will be seen that in this case di!r,tortion and compressibility effects have modified the form of the trim curves considerably.
The effects of compressibility alone are very marted at
-32-the higher Mach. numbers considered. As -32-the lift coefficient
decreases and the Mach number increases, C becanes more and
' mo
more negative - i,e. the tail setting angle to trim out C
° '^
mo
becomes more and more negative, and at high speeds this contribution
to Tin, is large compared with that req-uired to trim out the
other vdng and tail pitching moment contributions. At low speeds
the reverse is the case, so that as speed increases, and the C
' ' mo
contribution becomes more dcminant, the slope of the trim curve
becomes less negative, then zero, and finally positive.
The effect of wing distortion alone, however then makes
no further appreciable difference to the trim c\jrve, as in the
case vrhen C = 0 . Again, the forward movement of the vdng
aerodynamic centre is offset by the increased nose down pitching
racment contribution due to the higher tail incidence as before.
Distortion of the fiiselage alone makes the slope of the
trim curve more positive (destabilising), the effect increasing
with speed. The change in C produces a change in tail load,
which becomes more and more negative (tail down) as the Mach
number increases. As the speed increases the fuselage distortion
due to the increasingly negative tail load produces an increasingly
positive incidence of the tail and therefore an increasingly
negative increment of T|_ is required to trim.
Tail distortion alone has a very similar effect to that
of fuselage distortion. The all-moving tail behaves like a wing,
unlike the usual tailplane-elevator combination. Thus a positive
tail load causes the tips of the tailplane to bend upwards,
producing a positive increment of Ti^n to trim, and con-versely
for a negative tail load. The changes in Ti^p to trim are
there-fore in the sane sense as those caused by fuselage bending.
It will be seen that the resiolt of the combined distortion
effects at the highest speeds is slightly less than the algebraic
svim of the separate effects. This is because, as for C = 0 ,
'^ ' mo '
wing distortion causes a slight positive change in tail load which
in this case at high speeds reduces slightly the magnitude of the
tail load, v/hich is negative being largely determined by C .
Thus the fuselage and tail distortions are slightly reduced and
there is a small reduction in the overall (negative) increment of
n„ due to distortion v/hen the distortion effects are combined.
To
have
6.4. Stick fixed static margin
For an aircraft fitted with an all-moving tailplane we
\ = - ^T ^ (^) ^^- ^L =
^R)
^
-^^= 0
m
/'^"^To^
Values of I —op—j were accordingly obtained from the trim
^ ^ 4 = 0
m
0 01 R curves for the cases C = 0 , C = — ^ — ^
-mo • -mo s/l -M^
6.4.1. 0 = 0 mo
The resulting curves of K against q/q for C = 0
® n ^ ^ mo
are given in Fig. 5. Consider first the effects of compressibility. It will be seen that the "tatic margin increases slightly vdth
increase of Mach number. This is because (drim /^r) is
^° ^ 0 = 0 m
very nearly constant except at the highest speeds where it becomes slightly more negative, whilst A. is increased by compressibility. This result is in agreement with that predicted in ref. 6 for the case C = 0 , '".Tien K is positive in incompressible f love, the
mo n ' restoring tail pitching moment due to a change of speed and
corres-ponding change of incidence (c_V = constant) vdll exceed the destabilising wing contribution. The difference between these contributions will be increased if both wing and tail lift curve slopes are increased by canpressibility in roughly the same ratio, C remaining zero.
mo
Considering now the effects of distortion, vre sea that since vdng distortion alone does not appreciably change the slope of the trim curve or the value of V^, or of A. for the aircraft
T 1
considered, K is almost completely unaffected by wing distortion. The effect of fuselage and tail distortion however is to decrease the value of A (V^ still remains very nearly constant) so that although for C. = 0 the trim curve slopes are not
appreciably modified by these effects, the value of K is decreased progressively as q/q increases compared with the valine for the rigid aircraft.
In this example the loss of static nargin due to fuselage distortion is greater than that due to tail distortion, and the combined effects produce a maximum loss of static margin of the
/order of ..
-34-order of 10 - 15 per cent a t a Mach nijmber of about 0 . 8 .
6.4.2. c _ = - , g f ^
-M^
VI - M
