The Unsteady Aerodynamic Forces on Deforming, Low Aspect Ratio Wings and Slender Wing-Body Combinations Oscillating Harmonically in a Compressible Flow

College Report No. 94

iJiiLfi

.±FÏ " S ^^^^ ^^^^

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE UNSTEADY AERODYNAMIC FORCES ON

DEFORMING, LOW ASPECT RATIO WINGS

AND SLENDER WING-BODY COMBINATIONS

^y

TECHK V L . - ^ . , . . ,-w_ ,:. Kanaalstraat 10 - DELFT • KBK)RT MO. 94 JÜLT. 1955» 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 P I E L D

The Unsteady Aerodynamic F o r c e s on Deforming, Low Aspect R a t i o \7ings and S l e n d e r iring-Body Combinations

O s c i l l a t i n g H a r m o n i c a l l y i n a Gompressible PIOVT

b y

-R. D, l a i n e , B . S c .

S U H II A R Y

A method is presented w-hereby the 'Slender Body Theory' can be applied to the determination of the unsteady aerodynamic forces acting on slender YOJigs and wing-body combinations experiencing harmonic deformations in a com-pressible flov?-. The analysis holds for subsonic

and supersonic speeds, subject to restrictions which are stated and discussed»

A simplification of the method is also introduced which is applicable to many practical cases and calculations are perfonned on this basis v.-hich lead to numerical results for:

1 • 'Equivalent Constant Derivatives' for a deforming slender delta wing using modal functions which are poljmomials of the spanwise parameterj

2, 'Rigid' Force Coefficients for a pitching and plunging, slender, vamg-body combination,

These results are given as closed expressions and in tabular form and some of the results are also shoim in graphical form,

Both the de-ivatives and the 'Rigid' force coeff-icients are defined in such a v/ay as to agree with the usual British Sign Convention,

- 2

TJST OF COI^ii]I\T:S

list of Symbols 1, Introduction

2, The Slender-Body Theory

3, Solution of the Potential Equation

4, Symmetric Flutter Characteristics typicaJ. of a slender delta wing

4«'I, Uncoupled Modes 2|.o2, Coupled liodes

5, Calculation of the Velocity Ibtential and the Generalised Forces for the assumed modes of Paragraph 4,

5.1» Equivalent Constant Flutter Derivatives 6, Representative Results

7, The Pitching and Pl\inging slender body of revolution 8, The Slender Wing-body Combination

9» Discussion References

Appendix I The Cropped Delta - Definitions and Gectietrical Properties

Appendix II Equivalent Constant Derivatives - Definitions Appendix III Results of a calculation using uncoupled nodes

as detailed in iSection . 6,

Appendix IV 'Rigid' Force Coefficients for a slender body of revolution rrith. conical nose,

Appendix V 'Rigid' Force Coefficients for triangular wing on cylindrical body

Appendix VI 'Rigid' Force Coefficients for a slender wing-body combination,

-3-LIST OF SYIvIBOIg

The use of a 'bar' over a syiribol denotes that it is the amplitude of the harmonically oscillating quantity represented by the symbol itself»

Symbol

A,B

F(x)

^s

H(6)

L

Description

portmanteau symbols in eqn, to reference axis

function of II(x) constants in H(8) torsional mode

unsteady lift on rigid wing or body 'Rigid' force coefficients

Defined in Section Appendix I

7

4

4

6

6

n

M

CJO

n

o(

)

^n

%: R(x) %

S

T

constants in series expansion for 0 3 unsteady manent on rigid wing or body 6

'Rigid' force coefficients 6 free stream I'iach number 2 modified ïMthieu function of the third

kind 3 'of order'

constants in series expansion for 0 3

Lagrangian generalised force 4 local radius of body of revolution 7

llaxiraum » i « < > 7 planform area of wing Appendix I factor giving length of vdng-body

combination forebody as proportion

-2^

V

IR

a

b

"'"r'"m

f(x,y) Sr h(6) k freestream velocity

factor giving total length of v/ing-body combination

wing aspect ratio

speed of sound in free stream maximum vi/ing span

local, root and mean chords deformation function

constants in h(8) f lexural mode

vs frequency - Llach No, parameter = -^r—

^a z'rs'

i^rs-(1 ) ,(ZJ

_{^ a'^rs*^ a'} equivalent constant derivatives

rs

B

y

distance of reference section from \ïidng root

length of body of revolution ^ z'rs'^ z'rs ;

(n ) ,(TI.) ;• equivalent constant derivatives

Appendix Appendix

2

Pig, 1 Appendix

3

4

4

3

Appendix Fig. 2 Pig. 3 Appendix

VI

I

I

II

II

•n

P» P

Ó P

>(x)

f a c t o r s giving p o s i t i o n of reference

axis for ï d n g , wing-body caribination

and body r e s p e c t i v e l y

constants in series expansion for 0

local and freestream pressures differential pressure

Lagrangian generalised coordinates polar, radial, coordinate

indices and suffices in modal functions local wing semispan

Appendix VI

3

2 .

3

4

7

4

Pig. 1

-3'

s b / 2

o

se Mathieu function (periodic) 3

t time

u,v,w perturbation velocities 2

w with suffices - various upwash

conditions

x,y,z right-handed Cartesian coordinates Fig, 1

X (y) equation of reference axis 4

z amplitude at reference section in

f lexural mode 4

T'(x) local cross-sectional area of body 7

A factor giving length of conical nose

on body Pig, 4

II = - w T 7

a

amplitude at reference section in

torsional mode 4

/9 functions of \ Appendix III

Y ,b ,c flutter force coefficients 4

'rs' rs' rs

8 non-dimensional spanwise parameter 4

e dimensionless amplitude or thickness 2

^ elliptical coordinatej polar angle 3 snd 7

Ti elliptical coordinate 3

K

= ü = ^ 7

'^ a s '

•\ delta planform factor Fig, 2

V angular frequency 3

p, p _ local and freestream densities 2

<T = — r a t i o of max, body radius t o

-6-0

OO.O) ,CÜ

m

position of reference section = T.b/2

perturbation velocity potential local, root and mean frequency parameters

VG

Ü ) = CO = rrL c r U „^,

etc. Suffix c is used for wing-body combination in place of 'r' Appendix I 2 Appendix V 1, Introduction

In this paper a method is given whereby the aero-dynamic forces can be calculated for slender, lov/ aspect ratio wings deforming harmonically in a compressible flov7,

The method is applied to a slender, cropped, delta Tdng and cei'tain flutter modes are assumed which take the form of polynomials in the spanv-dse parameter. Freedom of the v/ing root is allovred for so that body freedoms can be included. In the latter part of the paper the

aerodyn-amic forces on a pitching and plunging, slender, wing-body combination are evaluated,

The basis of the method is the 'Slender-Body Theory' which has been applied in connection with the (quasi-steady)

stability derivatives for slender, vdngs and vring-body com-binations (refs. 1,2,3,4,5>é),

The application to an oscillating and deforming Tidng has, very recently, been studied by Herbt cind Landahl

(ref, 8 ) ,

The solution of the 'cross-sectional' problem, for the wing, is analogous to that of a two-dimensional flat plate oscillating in a compressible flow and has been treated by Tiraman (ref, 9) and Reissner (ref, 10),

The use of the 'Slender-Body Theory' allow's the

* Only the regular part of their solution is required in this case,

-7-analysis to apply at subsonic^ , and supersonic

speeds subject to c'=rtain restrictions on Aspect Ratio, Itlach number. Frequency Parameter, Slenderness, Validity at subsonic opcods depends on an approximate satisfaction of the Kutta-Jowkowski condition,

The assumptions of linearised, thin aerofoil theoiy are used, the fluid is perfect, the flow irrotational and harmonic motions are considered throughout,

2, The Slender-Bodv Theory

The coordinate system used is shown in Figure 1 where right-handed rectangular aixes are dravm from an origin, 0, fixed in the vd.ng, vn.th the x-axis parallel to the main

stream and the z-axis upv/ard. It is assumed that the \ïing is a thin, flat plate oscillating about its position of aero incidence in the plane z = 0, but al\7ays lying in the

immediate vicinity of the plane,

The perturbation velocity potential, 0, satisfies the equation

^'^

= 7

(vfe

* hf^

(^-1)

The conditions holding a t the siorface of the wing are specified by a p r e s c r i b e d 'dovrnv/ash' v / ( x , y , 0 , t ) and the s t i p u l a t i o n t h a t the r e l a t i v e normal v e l o c i t y of the a i r and of the wing i s z e r o ,

Applying the ass\miptions of the 'Slender Body-Theory' implies t h a t the x-derivatives i n equation ( 2 , l ) are neglected an.d the tvro-dimensional flov/ a t any c r o s s - s e c t i o n i s then given by the wave equation,

i ^ y y ^ ^ z z = \ - ^ t t (2.2) a

The approximation equation (2.2) is satisfied (ref. 1j5,5) if.

^-4)(^}«'

(2-3)-v;-here U^r-, i s the freestream iviach number and s i s the l o c a l semi-span, or f o r a t r i a n g u l a r v/ing i f

J1 - l 4 : | i R 2 < < l 6 (2.3)b

where M i s the aspect r a t i o .

8

-I f the influence of the ti^ie-derivative term i n (2.2) i s so small ul^t i t can be neglected the equation reduces t o L a p l a c e ' s equation, as for steady flov/, and the difference between the unsteady and steady flow cases mani-f e s t s i t s e l mani-f e n t i r e l y i n the l i n e a r i s e d B e r n o u l l i Pressure equation,

v v

- - D

h M + M.)

(2.4)

This implies that the root frequency parameter must be small (see ref. 1.1 cases 2 and 5 ) .

3, Solution of the Potential Equation

Assuming harmonic notion of angular frequency, v, equation (2.2) beccmes,

2

V

^yy-^^zz-7^ = ° (^-^^

where

^ (x,y,z) e^^ = 0(x,y,z,t) (5.2) The potential 0 is subject to the follov/ing

boundary conditions!

(a) 0 is bounded everyv/here in the flov/ and at

infinity all disturbances should disappear in the proper manner, thus,

i) 0, 0^f 0--^ 0 as v'y + z -> ^-"^ ii) The solution at infinity should i-opresent

v/aves travelling outv/ards frcan the origin, (b) At any point on the wing the prescribed normal velocity must be equal to the normal derivative of 0 at that point.

Let the motion of the point, (x,y), on the v/ing be represented by,

z = f(x,y,t)

= ?(x,y) e^^* (3,3)

f ( x , y ) v / i l l be r e f e r r e d t o as the 'deformation

-9-According to the usual asswiiptions of the

linearised theory the vertical velocity of the point (x,y) is given by,

dz _ ÖZ ^^ d_z ^'^ = dt ~ dt "^ ^CO dx

ivf(x,y) + U^^f^(x,y)j e ^ , or,

w(x,y) = iv f(x,y) + \J^^,t^{x,y) (3.4)

The condition is satisfied over 1he projection of the wing in the plane z = 0 and takes the form,

w(x,y) = ^ (x,y,0) = ivf(x,y) + U.,. f„(x,y) ..(3.5) (c) Outside the -vving and outside the v/ake ^(x,y,z,t)

must be continuous in planes, x = constant, and since it is antisyrmetric in z must satisfy the condition,

^(x,y,0,t) = 0 .

By transforming equation (3.1) to the Elliptical Coordinates, (^,TI), v/here,

y = s cosh 7] cos ^ *)

/ (3.6)

z = s sinh f] sin Z, \

and S ( X ) is the local semi span, Merbt and Landahl (Ref 8) have derived a solution in tenns of Mathieu functions,

Uriinjj che notation of reference 18 the solution takes the form, .,.,

^(x,Ti,S) = 2L.V P ^ Ne^2) (n,k) se (S,k) n=1

v/ith k = I I ..•(3.7)

* The use of a 'bar' over a symbol denotes that it is the amplitude of the harmonically oscillating quantity represented by the symbol itself.

-10-v/here the coefficients, p , are to be determined from

boundary condition (b), equation (3.5) v/hich becomes

in elliptical coordinates,

^ (x,0,^) = s(x) w (x,s cos ^) sin ^

.(3.8)

Differentiating (3.7) vrf-th respect to TI and

putting TI = 0 (on the vdng)

gives,-K,(..o,s) = 2 p j |:;;(He(2>(nA)]

se^(S,k)

—.1 T1=0

- <L^' \

P se (^,k)

n=1

.(3.9)

writing,

V ^ k ) = P n lïï (Ne^'^(^»k)) (3.10)

T1=0

No\7 if w(x,^) is bounded and is a continuous

function of ^, the series representing

0

(x,0,^) in (3.9)

will be uniformly convergent, ^

Multiplying

{j),^)

by se (S,k) and integrating over

the range, 0 to T;, the coefficients P may be detemiined

in an analogous manner to Helf-Range Fourier Coefficients

since an orthogonality relation exists for the ïiathieu

function se (S>k) (see ref, 1 6 ) ,

The P are thus given byj

r;7l

P (k) =

2s

n

w(x,2j) sin S^ se^(S^,k) d^^ (3.11)

•.J o

and finally the p from (3.10)

n

The solution, (3.7)> is nov/ completely determined

and the pressure distribution on the v/ing will be given f ran

(3.7) v/ith Ti = 0,

As discussed in section 2 it is possible, under

certain conditions, to suppress the time- dependent tenn in

equation (3.1) an(ij.,the solution (3.7) then reduces

to,-^ =

HKI

-nTi T • V

s e

L sin n^

1 1

-v/here the L are given by,

- n L^ = P^(x) = ^ I w.sin t,^, sin ng^, d2i^

'''' (3.13)

On the v/ing, ,.,,,

•^Z~/ P»

J =- ^.^ - ^ . s J j i n S (3,14)

"^ n=1 ^

The d i f f e r e n t i a l pressure across the v/ing plane

i s given a s , j

^ï = ^Pc., V J ' ^ x - fe,-// „ (5-^5)

I'X-' z = + ü

4. Symmetric Flutter Characteristics Typical of a Slender Delta Iv'inp;

The simplest, pointed, low aspect ratio \/ing satisfying the assumptions of the 'Slender Body Theory' is the slender delta v/ing,

Accordingly the analysis as developed in section 3 is applied to the vdng shown in Figure 2,

To describe the possible flutter modes of the wing a reference axis, x (y) is used (see Fig, 2) given by the equation,

2c

x^ = nc^ + -rg21 (1 - X) (1 - m) iyl (4,1) The applicability of an axis such as this to delta

wings is discussed by IVoodcoclc (ref, 17). '

For any particular flutter motioti it is then

prescribed that sections parallel to the line of flight v/ill twist about the reference axis according to some modal

function, such sections remaining themselves undistorted, v/hilst the reference axis itself translates according to

another modal function, each degree of freedom so involved being associated v/ith a Lagrangian generalised coordiiiate, qm

12'

The generalised coordinates are defined at reference sections given by

|y| = I ,.... (4.2)

and all motions are measured relative to the mean position of the vong (plane, z = O ) ,

A non-dimensional spanwise parameter, 5, is introduced such that

y = 61 (4,3) and l8f = 1 at the reference sections.

Each degree of freedom vd.ll lead to an equation of motion and a generalised force, Q^, v/hich can be expressed conveniently in terms of force coefficients as (canitting e^^*),

T~n = -*v' I (.~Y w + b i w + c ; , q „ ,3„2 i::.-,.v\ «j^s 13 rs m rs' ^

^ '

(4.4)

on the assmaptions of the linearised theory, v/here, s = r = number of degrees of freedon

and w = mean frequency parameter

vc Note,- c is the mean chord of the m m

^rr. * half-vdng. See Appendix I,

In v/hat follows the suffices + and - vdll indicate whether a function applies only for y > 0 or y -C 0

respectively,

4,1. Uncoupled ilodes

Let there be one uncoupled node in flexure and one uncoupled node in torsion described by the nodal functions, h(8) and H(8) respectively, so defined that,

lh(+l)j = |H(+I)i = 1 (4.5) If ze represents the translation of a point

on the reference axis, measured frcan the mean position, * The generalised coordinates and forces are amplitude functions but the bar notation is not used in their case»

-13-positive upwards, and ae the rotation, -13-positive leading edge dov/n, of the section through that point, then,

z

= z h(8) }

° / (4.6)

and a ~ a H(8) \

T/here z and a are the amplitudes at the reference sections, y = + Z.

Now, if the generalised coordinates are chosen so that,

q.

' ~ ^ (4.7)

^ ^2 = r °o ,

then the deformation function f(x,y) of equation 0.3) takes the form,

? = Zhq^ + (x-xj i- Hq2 (4.8) m

It vdll be convenient, for the aerodynamic problem, to consider the functions, h and H, each to be polyncanials in 8J thus, for symmetrical modesj

^% / (^.9)

and H = -^--v G \b\^ \

s s ' ./

r,s = 0,1,2, ...

Equation (4.8) becanes, for the half-wing for v/hich y >• 0,

'•.. r ; la • s *

.^. i-..(4.10)

ir ^,

forces Q. and Qp,

It is nov/ required to find the tv/o generalised and

Thus, _{t • I} i">

'•• s ^

+

* 'S ' indicates that the area of integration is over the \ving planform for which 0 <. s(x) < y only,

-12^

where j 4 p dx dy j is the total (incremental) aerodynamic

force at the point (x,y) on the v/ing and is given by

equation (3.15) when the velocity potentiel derived for the

assumed defocraiation function is substituted,

Prom (4,10),

(6f,) = I | S g , 6 - ;

Sq^

[ r

\

and the force, Q., is seen to be built up from a sum of

integrals of the form,

I Q-, i = i^ I A P » SV. S ^ dx d8 (4,12)

•*• >' t ' .

s

+

In the sane way the force, Qp, is expressed as a

sum of integrals of the form

,-

-. ,2

fP

I ^ 2 ! = c M (^-^of) <^^P- ^s- ^^- ^ ^^ (^-^^^

•— s m f J ij

It vdll be clear that many of the integrals (4,12)

and (4,13) vdll be identical, apart from constant factors,

4,2, Coupled Modes

The deformation function, f, now takes the forrat

f^ =

^ A M ^ )

+ (x-x ) ^ H ( 8 ) : . q^ ..(4.14)

r

'

,

. m

{

t h e r e now being r degrees of freedaa,

The functions h and H are as defined before i n

equation (4,9) vdth r = s so t h a t equation (4,14) becanes,

• ?^ = I>^ q^ h g^ + (x-x^) f %l ^^ -..(4.15)

r I m l

and Q. i s given by ;

h i •

% - ^% = I ^ P (6f+). <3x dy .

S +

1 5

-P i n a l l y , from ( 4 . 1 5 ) ,

^r = ^^ i i

4 P

(

Sr +

- ~ ^

• &J

^''*

^

«^^

S^ " . . < (4.16)

This is a siim of integrals like (4.12) and (4.13). 5» Calculation of the Velocity Potential and the Generalised

Forces for the Assumed Modes of Para^yaph 4

The coefficients, P , in the series representation of 0 are given by (3.11) and since f(x,y) is expressed in polynomials of fS! , and hence of |y| , integrals of the follovdng type are met \^dthj

(•»

cos^ ^^ sin ?^ se^(^^,k) d^^ (5,l)

j

Such i n t e g r a l s can be v/ritten as t h e sum of i n t e g r a l s of the form

!'

j s i n p S ^ . se^(^^,k) d^^ (5.2) Using the F o u r i e r S e r i e s expansion of se (S,k)

i n t e g r a l s such as (5*2) become} "

^ ^ B^"^ (k) I s i n r S . s i n p ^ . d^, (5,3)

r=1 ^ :j ^ ^ ^

YJhen s is even (equation 5.1)» the limits on (5.3) are 0 to 7;. quite straight forwardly, as indicated by equation (3,1l) and only a finite number of terms is

obtained for (5.1). '\Vhen s is odd, ov/ing to the

assumption of symmetry, the limits on (5.3) reduce to 0 to %/2

(or %/2 to 71:) and an infinite series is obtained for (5.3)> and hence for (5.1). However, only a few terLis need be retained in practice.

The velocity potential on the vdng, %^, is now fully determined and the corresponding loading is given by equation (3.15).

The generalised forces give rise to integrals like (omitting constants),

-16-and

I X

1^^

+

^

i)

6^ , dx d8 (5,5)

s.

These integrations must, in general, be done by a graphical or numerical means, except v/hen the time-dependent term in the potential equation (3*^) is suppressed and the Mathieu functions take on their degenerate forms,

The application of the analysis to antisymmetric flutter modes follows the same general lines as given for symmetric nodes,

5.1. Equivalent Constant Flutter Derivatives

By analogy vdth the flutter derivatives of two-dimensional (strip) theory it is possible to define a set of 'equivalent constant derivatives',

These derivatives are constant over the span of the v/ing and give the correct generalised forces v/hen inter-preted in the conventional sense,

The lift and moment on a strip of linit vddth are defined in terms of derivatives such as

^z» ^2' ^a' ^5

^ z ' "^5' °a' °5 ^''^^' ^^^

v^here the 'stiffness' derivatives include the 'inertia' derivatives, Z» , n., t Kf » Ja*,, .

Equivalent constant derivatives,

(g ' (V '

rs rs etc, (m ) , (m.) ,

rs rs

are defined from the force coefficients of equation (4,4) in Appendix II»

As vdth the force coefficients the first suffix refers to the generalised force and the second to the mode.

-17-Apart from the analogy vdth 'two-diraensional' derivatives the concept of equivalent constant derivatives is useful in that it facilitates direct canparison of sets of derivatives derived for different modes. For example, direct derivatives in one freedom are made independent of the modes in other freedoms. (See Appendix III).

6» Representative Results

The preceding analysis has been applied to the case of a triangular vdng (Fig. 2, X = O) using uncoupled modes,

Equivalent Constant Derivatives have been calculated for the flexural modesj

h ( 6 ) = \b\^ ; r = 0, 1, 2, and torsional modes;

H(8) = |8P ; s = 0, 1.

Modes such as these have been taken in pairs, one in flexure, 16 j''^'' and one in torsion, 181 '' , giving six

'sets' of derivatives,

The accompanying table of numerical results (Table l) shows the order of the derivatives and their signs (for m = -g-) and a set of general expressions for the derivatives is given in Appendix III together vdth the results of a calculation on a cix)pped delta for r. = s. = 0 only,

The 'damping' derivatives Z, , m. and n. are plotted against 'm' in Pigs, 8 $ and 10

By taking r. = s. = 0 and m = 1 the derivatives are obtained for a rigid pitching and plunging v/ing referred to the trailing edge - use of the usual transformation

formulae then refers the derivatives to any otlaer axis»

This has been done for a triangular wdng (\ = O ) , a cropped delta ("K = I/7) and a rectangular v/ing (\ = I) for an axis at 0,500 c and the results are presented in Table II, ^

In Pig. 11 the'cross-damping' derivatives n. , Z. have been plotted against 'm' for these v/ings,

In this case of a pitching and plunging v/ing the generalised forces Q. and Qp have simple interpretations

-18-and do in fact represent the imsteady lift -18-and moment amplitudes on the c>-aplete v/ing, i,e.,

X , (+ve upv/ards) p U^.S p .U^.Z^ " °r

and

M ^2 T Z ^ , , , V

X — s - , (+ve nose dCT.TO.ards; 2 - 2 ^ 2

p Uf So p U _ Z^ c

^'^^ '" ^ ^^' -^ ^ (6.1)

The expressions for lift and moment vdll be in terms of the dimensionless amplitudes,

( — ) andc / o _r a

and the relevant frequency parameter vdll be,

r

;

2

\

CO = zz— = CO ; . ^ J . r U,^,.^ m V 1+>-/

It is convenient in this connection to define a set of force coefficients for rigid motions only since in later paragraphs unsteady lifts and moments on rigid bodies and v/ing-body combinations are considered,

Coefficients L , L, ,., etc, are defined by the expressions} / ^ \ = - (L + ico L.) \ — 1 - (L + iw L.) a

L

and ,^ • z -^ ....(6.2) and these \dll be referred to as 'Rigid' force coefficients.

As for the definitions giving the equivalent constant derivatives (Appendix IlJ these rigid force coeff-icients are signed to agree vdth the normal British flutter sign convention

1 9

-TABIE I

-'. w'

^'\. r

s ^ x

0

1

0

.0296 ,0296

1

,00768 ,00768 1 2 .00374 .00374

m^

(my^

^V. r

S X .

0

1

0 1 1

,00376 ,00264 ,00431 ,00214

2

,00450 ,00200

*

IJ (*)-' " ^ ^ N ^ r

0

1

0

,786 .786

1

.954 .954

2

.981 .981 i

m^

(my^

'^"""-'^

r

0 1

0

.0542 ,0900

1

-,0225 • .0379 •

2

".135 -.0405

\ ^

r

s \

0

1

0

.781 1,00

1

.996 .955

2

1,18 1,00

m^ (^y^

s "^. 0 1

0

.0530 .0334

1

.0530 .0334

2

.0530 .0334 + Zj (7R)~^ s ^^^^ 0 1

0

.735 .915

1

1.02 .905

2

1.30 .817 - mg (^ )"•• --v^ r s -"->^ 0 1

0

.114 .195 1

1

,114 ,195

2

.114 .195

Derivatives f o r m = •§•; (On ~ ^ ' ^

-20-TABLE II • • ••' • Equivalent Constant Derivatives for rigid wings pliinging and

pitcidng about an axis at -^ c

Ecjuivalent

Constant

Derivatives

K

h

I a

h

^z

"^S

"^a

Triangular

ïdng

( ^ 0 )

% 2

(.524)

•" 4

(.785)

•K 2 %

- 12 "^m ^ 4

(.262) (.785)

57t

•*• 1ÏÏ

(.981)

'K 2

l2%

(.262)

(.197)

% 2 %

20 ^m " T ^

(.157) (.197:

\ 7Ï

' (.392)

Cropped

Delta

(X=l/7)

- . 5 8 5 co^'

+.785

-.245w^ +.785

+1*07

.245 oij

_m

- c 45

.I37w^ -.0845

-.386

1

Rectangular

¥ing

fx=i)

% 2

4 m

(.785)

•" 4

(.785)

7C

^ 4

(.785)

"^ 8

(1.18)

0

8

(.392)

7C 2 7C

(.0655)(.392)

71:

(.197)

...

N O T E ; Figures in brackets are decimal equivalents of fractions of Tt»

-21-7» The Pitching and Pltinging Slender B<>3.7 of Revolution

A set of flutter force coefficients for a pointed

slender body of revolution can be calculated in an analogcjus

manner to those of the vdng by adopting polar coordinates

instead of elliptical coordinates v/hen solving the potential

equation and in the specification of the boundary conditions.

Probably the only case of interest is the rigid

pit<3hing and plunging body and accordingly this case vdll

be dealt vdth. The cartesian coordinate system for the

bo<ay is the same as for the vdng and is shovm in Figure 3

In each cross-section, x = const., take polar

coordinates J

y = r cos S )

ƒ (7.1)

z = r sin

Z

i

then the potential equation

(3*^)

transforms toj

r

1 d Z il'! 1 a ^ v^ sf „ /^ o\

^ r dZ Q.

Consider the bocSy movements to consist of a

vertical translation (+ve upv/ards) and pitching about an

axis,

^ = ^o = "^ ^B (7.3)

parallel to the y-axis (nose-down pitching +ve).

By analogy vdth equation (4.6) we define z to

be the amplitude of the displacement of the point,

X = X , on the body axis and

a

to be the amplitude of

the inclination of the body axis to the Ox axis. Then

the motions produce at a point, x, on the body axis the total

upward displacement,

(z^ + (x-x^)

aJ e^"^

,

Taking the body length, Z , as a reference

length the vertical velocity (equation 3.4) is,

w(x) = iv j

l ^ l ^ .(^x^ . ^ ) a^

(7.4)

Writing » (x) for the local caross-sectional area

of the body, the potential near the body takes the formj

2 2

-body i s

where

(0) = - i ( x ) ~ i ^ . s i n ^ (7.5)

The press\ire a t ai^y point on the surface of the

TT ( a n IV T-' s i n g

p = p U ( .} T — + TT- . I I > ^

I I ( x ) = - i ( x ) r ( x ) ^ (7.6)

The unsteady lift and moment, L and M follow

by integration of (7.6) along the bo(3y,

The 'rigid' force coefficients of equations (6.2) have been calculated for a cylindrical body, vdth a conical nose as shovnn in Pig. 4. These are based on the Aspect Ratio of the geometrically similar vdng having its root-chord equal to tho length of the body and

b

\

^ = — (see Figure 7)

where R is the maximum (base) radius of the body. The coefficients are given in Appendix IV. By putting cr = 1 and (\ = (1-X) it v/ill be seen that the expressions of Appendix IV are identical vdth those that v/ould be given for the rigid cropped delta vdng using the ecjuivalent constant derivatives of Appendix III with B = 0 £md ecjuations (6.1),

8, The Slender li'ing-Bodv Ccmbination

8,1 The rigid pitching and plunging ccambination A set of 'rigid' force coefficients vdll nov/ be derived for Tlie slender vdng-body combination shov/n in Figure 8.

Tids problem vdll be dealt vdth rather differently from the v/ing and slender bocSy cases in that the velocity potential vdll be f(3und, not directly as a solution of Laplace' s Ecruation, but from the tv/o-dimensional potential for incompressible flow normal to a flat plate,

The required potential vdll not generally satisfy the tv/o-dim.ensional wave equation (2,2) and hence the

solution vdll be subject to similar restrictions as the v/ing solution for k—'•> 0,

-23-Using the Joukowski Transformation (see Pig, 6) the velocity potential for the flov/ around the bo(3y config-uration of Fig, 6 (see ref, 4) due to a motion v/ of any section in a fluid at rest oan easily be found and conven-iently exja:essed in tv/o parts,

'

_J-^

and

^i0) = - w / s^ (^ 1 + ^ j - 4y^ + w,/R2 - y /

/ V. S^/ V y V

(8,1)

..(p,)_./s2^,4\ ,2 /. .R^X

where B(0) is the potential on the body (r=R) and w(^) is the potential on the wing (S = 0 or 7t, y = r ) ,

It will be clear that the force coefficients for the v/ing-body combination of Fig, 5 ca^i ^^ considered to be the addition of two sets of force coefficients; viz,,

(i) the force coefficients for a triangular wing on a cylindrical body, dov/nstrean of the lateral

plane through the v/ing leading edge and body junction, (ii) the force coefficients for a pointed body upstream

of the vdng leading edge,

The coefficients (ii) have been calc3ulated in ocotion 7 (Appendix IV),

The coefficients (i) can be calculated using (8,1) vdth the axes and notation of Figure 7.

For the conbination, the velocity, i/, will take the same form as for the body alone, i.e. ecguation (7.4), thus, , „

( ! fz\ U^,.

w(x) = ±v } I c j — I - X a + -r^' a + ax ' (8,2)

r,

1

The loading d i s t r i b u t i o n i s given by the pressure equation (3.15) £md l i f t and moment b y .

L = } I \ (Ap)B.<3y + I

( ^ P V ^ ! <3X ! and \ cro \ '^ o '-' R r V, s

M = 1 ' ; (/ip)B.dy + I (i^p)^.dy; (x-x^) dx

^•H(;-° '-'^ '!, (8.3)

-2h^

Complete expressions fear lift and moment on the clyindrical body aiA triangular vdng lead to the 'rigid' force coefficients v/hich are given in Appendix V. These coefficients like those of the boc3y are based on the

triangular vdng having root chord, c , and maximum

semispan s = b/2 = 'R/C,

The force coefficients for the \*Lole v/ing-body conbination of Figure 5 are given in Appendix VI and the variation of the 'damping' force coefficients L, , L, , M, , M. vdth cr is shown graphically in Fig. 12,

In adding the appropriate coefficients of

Appendices IV and V the definitions of Figure 5 £J^^ Appendix VI v/ere used and again the triangular v/ing is used as a basis,

9, Discussion

The use of the 'Slender Body Theory' for unsteady flov/ problems leads to a solution for the aerodynamic forces T/hich does not involve long ccanputation and many geometrical

and other parameters can be carried along in the analysis vdthout having to be specified definitely at the outset,

The restrictions of the theory as ciiscrussed in section 2 seem to be somev/hat severe but there is evidence to show (ref, 8) that for a rigid triangular "v/ing of aspect ratio, 1, at a Jiach number of 1,25 and fcr a frequency parameter, co , up to 6, the theory appears to be cjuite valid. Furthermore, results fcïr an aspect ratio of 0,5 shov/ that when the time derivative terms are neglected the results differ from those given by the complete solution only if CO > 2 for a ïlach number range of 0 - 1.25 (rigid triangular

wSng),

Ovdjig to the need to evaluate several terms of the llathieu function series v/hen deriving the full solution it is much longer tlian the sirtplified case (for snail root frequency pcranetcr) and it v/ould alv/ays be worthv/hile to

question whether the full solution is really necessary in any specified case,

With the type of v/ing to v/hich this analysis can be applied, it is very imln kely that the root frequency parameter vdll exceed about 0,5 so that in many cases the

siniplified approach vrould suffice,

The force coefficients Y»^>c of equation (4.4) are dependent on Mach number and frequency only through the parameter, k, in the general solution, consequently, in

-25-the simplified case which implies that k—•?• O, -25-the coeff-icients are independent of frequency and i'Iach number,

It has been found, both experimentally and

theoretically that the variation of flutter force coefficients with frequency decreases as aspect ratio decreases so that this is not a surprising result from a theory v/ldch is correct

for

m

—ftO,

The preceding remarks can be taken to apply equally well, in principle, to the v/ing-bo<ay combination,

It is interesting to note that the analysis used by Lawrence and Gerber (ref, 19) (subsonic) v/hen taken to

the limit M —^» 0 gives results for a rigid vdng whi<3h agree vdth those found here and by Garricjk (ref, 1 4),

In this connection it is also interesting to study their results v/hen plotted against aspect ratio. The slopes of the curves (force coefficients) at zero aspect ratio are correctly those given by 'Slender Body Theory' but, in general, the curves depart from their original tangents extremely rapidly. It might be suggested therefore that force coefficients derived using 'Slender BcxSy Theory', if applied outside their range of reasonable validity, vdll give magnitudes which, in general, vdll be very different from the 'true' values. (See also refs. 2o and 21 for v/hich

M = 3 ) .

Pig, 10 shov/s that for an axis at the trailing edge of the v/ing, no matter which torsional nodes are chosen, the direct damping derivative m, is zero indicating that an undamped pitching oscillation v/ould be possible - for all ajcis positions 0 <;ra < 1 the derivative gives positive dacrping»

V/hen CO ——^ 0 the 'rigid' force coefficients give the values of lift and moment for the steady case as found by Jones, Spreiter and others (refs, 7 and 4,)

2 6

-REFEREIICES

No, Author

1. Ribner, H.S.

2» anith and Beane

Nonweiler, T.R.F,

4» Spreiter, J.R.

5. Henderson, A.

6» Ribner, H.S.,

llalvestuto, F.S.

7» Jones, R.T,

8 , Ivlerbt, H , , L a n d a h l , M,

9, Tiraman,

Van de Voeren,

Greidanus,

10, Reissner, E ,

Title, etc,

The stability derivatives of

law

aspect ratio triangular v/ings at

subsonic and supersonic speeds,

N,A.C,A. - T.N. 12*23.

Damping in pitch of bodies of

rev-olution at supersonic speecis.

Inst .Aero, Scs, Preprint 311 (Feb, 1951).

Theoretical stability derivatives of

a highly swept delta wing and

slender-bocSy combination,

. C, of A, Report No, 5 0 .

Aerodynamic pi"operties of slender

•v/ing-body combinations at .jubsonic,

transonic and supersordc speeds,

N,A,C,A, - T,N, 1662,

ütching moment C and C at

°

mq ma

supersonic speeds for a slender delta

v/ing and slender body combination and

. approximate solutions for broad delta

v/ing and slender body,

N,A,C,A, - T,N, 2553.

Stabili-ty derivatives of triangular

vdngs at supersonic speeds,

N,A,C,A, - T.R, 908.

Properties of lew aspect ratio wings

at speeds belcjw and above the speed

of sound.

N.A.C.A, - T,N, 1032,

Aerodynamic forces on oscillating low

aspect ratio wings in compressible flcjw,

K,T,H, Aero, T.N, 30,

Aerodynamic coefficients of an

oscill-ating aerofoil in two-dimensional

subsonic flow,

Jcjum,Aero,Scs, December 1951,

On the application of ilathiou functions

in the theory of subsonic con'ipressible

flow past oscillating aerofoils,

27 28

-No.

11

12

13

14.

15

16.

17.

18.

19,

20,

21.

Author Lin, C,C, Reissner, E,, Tsien, H.S. Ward. G.N. Miles, J,¥. Garrick, I.E. Lcmax, H. Byrd, P.P. McLachlan Woodcock, D.L, Templeton, H. Lav/rence, H.R. Gerber, E.H, Woodcock, D.L. Lehrian, D.E, Title etc

On two-dimensional non-steady motion of a slender body in a compressible fluid. Jnl. Maths, and Physics. Vol. 27, 1948

Supersonic flov/ past slender pointed bodies,

Quart Jnl. Mechs, & App, Maths., 2, 1949 On non-steady motion of slender bodies, Aeronautical Quart. 2, November 1950 Some research on high-speed flutter. 3rd. Anglo-Zimerican Aero. Conf. 1951

Theoretical aerodynamic characteristics of a family of slender wing tail body

combinations.

N.A.C.A. Tech. Note 2554

Theory & application of Mathieu functions. Clarendon Press. Oxford, 1947

Symmetric flutter characteristics of a hypothetical delta v/ing.

R.A.E. Report Structiires 68.

The technicjue of flutter calculations. R.AoE, Report Structures 37.

The aercjdynamic forces on Icrw aspect ratio v/ings

Jnl. Aero. Scs, November, 1952 Aerodynamic derivatives for a delta vdng oscillating in elastic modes. R.A.E. Report Structures 132 Aerodynamic coeff:'.cients for an oscillating delta v/ing.

-29-APPËNDIX I

The Cropped Delta

-Definitions and Geometrical Properties

See Figure 2 ;

-Mean chord, c = 4 - c ( l + ' \ ) (l)

-Area, S = b c (2)

,2 ,

Aspect Ratio, Ji = |- = — (3)

m

Local sendspan, s = .x for,

2c (1-X)

"" 0 Cx <(l-\)c^ (4)

Reference section position,

I = ^, —

(5)

Spanvdse parameter, 8 = "f (6)

2c

Local chord, c = ^ . (1 -

T ( 1 - X ) 8 )

(7)

(l+\)

Ratio,

r= ^

(8)

m

2c

Reference Axis, x = mc + - ~ (l-'\)(l-m)

\yl

= A + B l8>

A ^ 2m

°m 1 + \

thus

_f

and

b B , (1-X) /, X

m (1+^)

vc

(9)

(10)

Precruency parameter, w =

r;

— (II)

-30-APEEHTOIX II

Equivalent Constant Derivatives - Definitions

The follov/ing equivalent constant derivatives are appropriate to a system vdth the usual British sign convention i.e. z-axis dov/nward, lift positive upwards, moment and angle of attack positive nose-up.

(a) Uncoupled Modes

i. One mode in flexure, one mode in torsion

(-Yl1%*°1l) = ï^ I h^ . dS

iiA

(-Y.ioW + c . „ ) = Z ^ '12 m 1 2 ' a ^^ h.H d8 c m

r.lA

(-YQ^HT, + ° O J = i-^J 1 ?~ H.h d8 _{'21^m ^ " 2 1} 2 > ° i ! „ m 'Jo

oV'

(-Y22%i + °22 OVI^ 2

Jo ^ ° '

niA

b . . (jj = CO , Z . 11 m m z ~ , h^ d8 c m b . o CO = CO Z, I 12 m m o

fH/'

Jo

V m/

\iA

b r , . CO = CO ( - r a . ) 21 n m ^ z^ \2 '^m = '^- ^-' 1 / ^ ^ H,h d8 d8 ,

(-m.)

/ ^ ^

" o ^ '

V m / •• o ••»1A 3 \ H^ dS .

ii, r modes in flexure, s modes in torsion Derivatives such as (I ) ^ and (m ) j,

-31 s u c h a s , v d l l b e d e f i n e d i n t h e same manner as i n ( i ) b y i n t e g r a l s

j]iA n i A

I c h , h , dS and — H , h , , d8 r e s p e c t i v e l y , I* 3? I C^^ S JC (b) Coupled Modes c U o

Ü-

'"^ (-Y c o + c ) = ( Z ) ^ ' r s m r s ' , ^ z ' Z„ rs , z A1A h , h d8 r s e t c » o

il A

(-Y 0)^ + c ) = (I ) — h H d8 r s ,1 a ' o

i l A

2

( b CO ) = CO ( Z . ) ^ r s m ' , m ^ a ' • r s (—) h H d8 I c ) r s V m /

5

J. Ö i V o e t c , - by a n a l o g y v d t h ( a ) i , 2 A t e r m such a s ( - v co + c ) i s o b t a i n e d from ^ ' r s m r s ' j z

the real part of the term in the total expression for Q whic3h involves both h and h J and (b co ) , . , . .

r s ' ^ rs m' Z, IS obtair fron the imaginary part of the same term: other terms are obtained in a similar manner,

-32-APEEI^IDIX III

Results of a calcula.tion using uncoupled nodes

(a) Equivalent Constant Derivatives for Trianpiular Wing (\=0)

Plexural mode h(6) = |8i^ , r = 0,1,2

Torsional mode H(8) = 1 8 ^ , s = 0,1.

r=0. s=0

I = - -r w M

z

b m

h

= -^ f • ^

\

= - [ ? • (.863-1.36m) 0)2 - j !

M

Ij = + if (''•82 - 1.15KI)

M

m

f (.288 - »455m)

i/

•

M

n^ = - ^

(,121 - .28am)

M

r'

™a ~ |^|2(»0710 - ,20Qn + ,158m2) w^

- i (,243 - .576m)i ^

^%

= f (-'353 + .703rn - .35^1^) -E

r = 1, s = 0

z 5^ ni

Z, = + -^ , ^

Z 'K

l^

= + 3 M - .139 + .205m) co^ + ,333 (

^

-33-I . = + 3 (.516 - ,349m) M

m^ = 3 (.139 - .205m) co^ , &

m. = - 3 (.182 - »349m) &

m =

_m

2

m,

^ ) {,Mi2 - ,2+OQn + »317m2) w^

+ (- .121 + .288n)/ M

J

f (- .353 + .703ra - .35^2) ^

r = 2, s = 0

5^^

= " 1 ^ •

CO iSR

m

= +

_{16 •}

- 6x ) ^ (.490 - .690ra) c / - .0625S M

+ ^ (.690 - ,49Cto) M

m

m.

_ ^

_{2 (,082 - ,115m) CO &.}

= - - p ^ (,0730 - .123m) JR

m

m.

= | 2 l | 2 ( , 0 7 1 1 - ,200m + ,15am2) co^

- (,121 - .28a!i)3 m.

= - 7c (,178 - ,354n + ,176m'^) R

r = 0. s = 1

'ï 2 _, - 7- ^ ^

6 m

•TC

Z. = + f . ^

3 4

-ï = - 3U(.220 - ,353^) ^l - .333? R

_{a . m ^.} ^- J

Z. = + 3 (,2+42 - , 2 7 ^ ) B.

m = 3 (.110 - ,l76m) w^ , Si

mg = - 3 (.107 - . 2 7 W ^

m = "5^ja (,00359 - ,0100m + ,00800m2) cof

^' - ^ (,0430 - ,0930m)i &

m. = 30 (-,0250 + ,0492m - ,0243n2) S.

r = "'t s = 1 . Z = - ^ ,_{z ^% m} i}? E. Z . = + -^ , JR 2 TC

Z^ = - J4 (,0592 - .0911m) (/ - M &

l. = + I (0,540 - 0,355m)

M

\ = ^ (.0395 - .0612m) co^ , M

m,

= - ^ (•05hO - ,1l8m) M m^ = ^ 1 (,0287 - .0800m + ,064Qm2) co^ a <i 1 ^ ' m

- ^ (.0430 - .0930m).' R

mg = 30 (- .0250 + ,0492ra - ,0243^^) -fi

-3b-r = 2 , s = 1,

^ = - Izö «^ . -E

_{z l b o m}

h

= - ^

•

M

IQ = - I 2 ^•°792 - ,117m) tó^ - 1,00/ £

Z. = + ^ ( , 2 0 6 - ,192im) m. m = - 40 ( - ,00495 + ,00733m) co^ m. z ^ m mj = 30 ( - ,0359 + ,0691m) Si m = ^ { (.0287 - ,0800m + ,064Cbi2) co' - ^ (.0430 - , 0 9 3 t t - i

_{„ m.}

2> mj = 30 (- ,0250 + ,0492m - ,0243ra ) B.

(b) Ecjuivalent Constant Derivatives for Cropped Delta (X, general) r = O, s = O o n l y Z^ = - u /?^ «2 ^

z^ = - [ ' ^ ( / 5 , - - / ? , - - r - / ? p ) « 2 - 7 / ^

_{a ) '•'^3 o 1 '^ic 2 ' m 4 f} / m m 3 , 37i ( l + \ ) 2 / 2 , „ A 2 b B''^ _ O 16 A -^ -.2 l 1 + \ ^^1 c 37ïZc / 1+X+X V m -^ m/ m = 7C (/5^ - — /5, - - ^ 2 - ^ ) üj2 ^ z ^ 3 c '^l TC Z c 2' m m m

-3G-""5 ~ 4 , ^ ^2 a ^ 4c ^ bTcZc / ^ 1+X+X V, m m . ' m a ^ * . ^ ^2 ( \ 2 ^ 8 c ^ 3 TC Z c ^3 1+X+X / !..,v m -^ m -^' V. A_/'/^ ^ p b B ^ >

"—H^3'~^1 -^07^-^2/

m V. m m y TcZc ' / 6 c ^ 2 TcZc • ^ 1 0 / m m V m m / ) co f „ A b B \ + ( "^4 "^ 4c "^ 6TCZC V V n m / B. ,3 f / " >, Tcd+X)-" i / o •/,• A „ b B ^ 1

i

/ 2 r/? A_ 2_t i 1+X "^ ^ 1 " c " 37C» + _{4c i 1+X "*" ^ 1 " C "* 3TCtC /} A ^ _ 2 _ ^ ^ _ A_ _ 2 b B ") m !.^ n -^ ra/ where J /? )0-₂ ^ = 1 + 2X - 6(1+X) 1 + 3X ~ 12(1+X) 1 + 2X -4(1+X)2 1 - X 3(1+X) 1 + 8X -15(1+X)2 3 - 2X + X2

_it

~ 3

4X2

x2

^ "b B ( ^ ^ CR A 3 b B ^ / ^ VKIT • ( TTX "^ ^^2 " ~ " 4TCZC j / m \ m m / (

^4 = T.T:;:r ^ 3 - ^ 5

^6 4(1 + x)^ (1 + 2X - 2X^ + 2/3 X^) ^ 5(1+X)^

TECHNISCHE HOr^ESCHOOL

VLIEG'' ' • .•'vVKUï'iDE Kanc; J - DELFT 3 7

-^ = (1 + 3-^ • 3/2 x-^)

^ 15(1 + x)^

R 1 + 4X '^10 ~ 20(1+X)

Expressions for the above d e r i v a t i v e s v/hen X = l / 7 a r e j

K = - .585 co^ . iR

m

Z = -

( T C ( , 1 4 1

- ,226m) J- - yl M

a . ^ ' m 4 |

Zj = +

,211TC(1»86

- 1,12rii) iE

m„ = Tc(.141 - »226m) co^ M

_{z ^ ' m}

mjj = - ,8457ï (.0944 - .281m) M

n = ,8457c ! (,159 - .384m + .282,m^) co^

_{u '\ m}

- (.0944 - .28lir)j M

J

m. = .6457Ï (- .325 + .650m - ,3250^) M

-38-APPEINDIX IV

'Rigid' Force Coefficients for a Slender Body of Revolution •vdth conical nose

(see Figure 4) Definitions:

Length of body = Z^.

Length of cordcal nose = A Z^

Reference axis at distance m^ Z^ from nose Frequency Parameter (Ü-, = v ^f/U.x .

The force coefficients are based on the v/ing having a geometrically similar planform to the body, thus;

R^ = -^ , cr and Zg = c^ , / z \ / z \ vc I_ so that I •— 1 = / -^ ! and av ^ , 7C 2 /, 2{S\ 2 ^

Lg = + g o^ . ^

I,=-J^/(£A2B.,_^_d^^2 . J ,

T 7C 2 /„ 2 A \

V - 3 /

TC 2 /^^^'^ , A2^5 2

Z

i'ï- = . 4 ^ 1 - ^ - -^ ^ - ^ - - / < ^ • ^

A \

Hj = - f o^ (3^ - -Bj ^

TC 2 f / l 2A.^ 2 "^^^ 2Am|^

/ 2 A \j

h

=-?^^ (1 - v ' ^

-39-APEEMDIX V

'Rigid' Force Coefficients for Triangular Yfing on C y l i n d r i c a l

Body

(see Figure 7)

~ U

L =

L =

- / I o ^ - c r ^ . ^ o ^ ) «2 ^

4 ^ 3

3

/

0

- m

- (1 - a^f I m

J

L. =

M =

+ f 1 I (2 - 30^ + o^) - ra (1 - a^fi &

_{4 ) 3 c ^ ' ' 1}

I ( i (1 - 20^ + cr^ - i^r^ Zncr)

- m j — - cr^ + c «^3

0 3 . . 9 1

CO

m

M. M =

- I f i (1 - 4 ^ 3 , 3 0 ^ ) _ ^ ( i - a ^ ) 2 | ^

- "2^ (1 - 2o^ + 0-^ - 4o-'^ Zno-)

+ m 2 / I - cr^ + I o^ - 0-^)

CO

M, =

- 2 [ l ( 1 - 2 ^ + 30-^) - ^ (1-0^)2

- f i 1 (1 - 0^) - m fl - 0^ - c^ + 0-^)

2 \ 2

L

c

m 2

_c

2

+ - ^ (1 - 0- ) \M

2 N 2 (

J

-^•fO-APEEI€3IX Yl

' R i g i d ' Force C o e f f i c i e n t s f o r a S l e n d e r '7ing-Body Combination ( s e e F i g u r e 5)

Definitions,-Total length of conbination = Vc (I) Ratio of body length to v/ing root chord = — = T

°^ (2)

But Vc^ = (^B •*• °r'-''""^^ ^ » ^^^ figure 5, so that fron (2), T = V - (1 - cr) . (3) Also ,

and

% = T , ü)^ (5) vc r CO = CO = ï^: c — r U...

It is essential that the position of the reference axis should be unicjue v/hen measured from the apex of the v/ing (m c ) and from the nose of tho body (m^ , Z^)» This recjuires that (see figure 5) J

m^ ZT-, = Z„ - crc + m c • 3 B B r c r (o--m ) i.e. nig = 1 ^ (6) V - 1+m

^ = V - 1 +°o- (7)

Length of conical nose = /\ Z-^ = A To (8) This gives nose-length as a constant proportion of body length

ahead of v/ing root. If a nose length whicdi is a constant proportion of total combination length is stipulated then / > must be replaced by an expression of the fomaj

Z V

- ^ i ^ ^ - ^ ^ j

= const, (9)

Note; In the above definitions: T i 0

4 1

-' R i g i d -' Force Ccsef f i c i e n t s , based on the t r i a n g u l a r

wing, f o r t h e v/ing-body combination of figure 5 a r e then!

h = - 7 - ; — + c r

z 4 ) 3

0 - ^ ) T - ; i j +a3f-cA^ 4 .

&

Lg = + g ( 1 - 0 ^ + 0-^) M

T 2E

a 4

+ c ^

cr'

( m - • § • ) - Tm + o-

4

T fl - ^ ) + !• m

7- - Zncr + m (co

4 c > c

(-^)-^(*-4^49;

- (1 - o^ + A )

iR

^a =^f[(t-"=)^^

^ / l - ^ J T - ( 2 . m ^ )

+ 0^, I- - 0- . m 5 iïl

II_

!(i4)^^

(m^ - i ) - Tm,

(-^>AA^4)]

o3|;(.-^;.|.j

+ c r ' 7- - Zncr + m {co , i R

_{4 c J ( c}

2 4 ^ ( 3 c / ^ c ^3 J

- I- cr3 + cr^ (2 - m ) ( ^

3 ° (

4 2 -M_ = ^ m m c

['

. ^ ' . 0 ^ m ( 1 - m J . I m ^ / l - S | ^ )

3 /

I o

_3y

0 ^

5m:

₁ - 2m

2

AN. „2 /

3 3 - -c^ O - T-';- ^ o ^

2 A

3

2 / + o-

4

-f-"^^(-#-^^

+ 1,1. / 2 j

M

M,

- O l

_{m^) - o-}s2 2 1 + (1 - m )'^ I + 0-^ 1 + (1 - m f

>r

(M

- J

L.-.

RECTANGULAR CO-ORDINATE SYSTEM FIG. I.

CO-ORDINATE SYSTEM FOR SLENDER BODY

FIG. a

SLENDER DELTA WING

FIG. 2.

»,1,

l_

I,

SLENDER BODY OF REVOLUTION FIG. 4.

h-*H"—

REAR R^RT OF WING-BODY COMBINATION FIG. 7 WING-BODY COMBINATION FIG. S oa

<3

^L±

latic -.. L B , -U-B.J X . - j i l a n . • » . • 1 THE TRANSFORMATION X , - X + Fe FIG.& ' C R O S S ' DAMPING DERIVATIVES

POR DEFORMNG WING [Xf-O] FIG. 8.

- 1 - 2

CROSS DAMPING DERIVATIVES FOR DEFORMING WING ^^ FIG. 9. 0 - 8 0 - 4 - 0 - 4 - 0 - 8 - 1 - 6 - 2 0 DAMPING DERIVATIVE Tll^ FOR TRIANGULAR WING

FIG. lO 0 - 4 - 0 - 4 - 1 - 6 TRIANGULAR CROPPED ( ^ = ^ 7 ) - 2 0 ' C R O S S ' DAMPING DERIVATIVES

FOR RIGID WINGS FIG. i L

w 17 n

V V

-REFERANCE AXIS 2 0 0 - 8 0 - 4 - 0 . 4 - 0 - 8 l 2 -4 -6 , BODY RADIUS WING SEMISPAN •8 1 0 A - 1 / 2 V = 2 RELATIVE VALUES OF DAMPING FORCE COEFFICIENTS FOR WING-BODY COMBINATION