~,~-~..z~ ' " , , , - ,%

, • ~r: ̧ ; ~ , , J ~ , % ~ ,,'

....................... g,. & N, I~o. ~074

MINISTRY OF A I R C R A F T PRODUCTION

A E R O N A U T I C A L R E S E A R C H C O U N C I L

R E P O R T S A N D M E M O R A N D A

o

Lateral Stability of Tailless A~rcraft By

: A . W . T H o R w , B.A. and M . F. Cm~'ris

Crown Copyright Reserved,

L O N D O N : HIS M A J E S T Y ' S S T A T I O N E R Y O F F I C E

I948

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NATIONAL AERONAUT~C&L ESTAt~LISHMt~T3

L { g R A R.¥ Lateral Stability of Tmlless Aircrafk

By A. W. THORPE, B.A. and M. FI CURTIS

COMMUNICATED BY THE DIRECToR-GENERAL OF SCIENTIFIC RESEARCH,

MINISTRY OF AIRCRAFT PRoDuCTIoN

Reports and Memoranda _No. 2074

June, 1943"

Summary.--Reasons for Enquiry.--Inforrnation was required oll the probable effect on lateral behaviour of a change from conventional to tailless types.

Range of Invesligation.£.--The essential features of a tailless design are represented by large reductions in the absolute values of the derivatives y~., n,, n,. As few tailless models have been studied, a numerical survey of stability boundaries has been made over a range of these parameters which probably covers the limits set by the all-wing design without end fins.

Curves of constant period and constant damping have been drawn in a few cases and from these curves a numerical comparison of the stability characteristics of conventional and tailless aircraft has been made. '

Conclusions.--(1) For the larger values of n~ and y,, considered, oscillatory instability is more likely to occur at low speed than at high, and instability at high speed is unlikely. For the smaller values of n, and y~, oscillatory instability is more likely at high speed than at low speed, and stability at high speed can be attained only with a small value of -- l,.

(2) Spiral instability is probable at a l l speeds, but at high speed the rate of growth.of this motion will be small.

(3) The survey stresses the need for systematic measurements of y~, n,., n, (particularly the last) in the tailless range.

1. I n t r o d u c t i o n . - - L a t e r a l stabi l i ty characterist ics have been invest igated previously (R. & M. 19891, 3) bu t such investigations have been concerned main ly wi th the convent ional type of aircraft , where it was possible wi th fair approximat ion to use s t andard values for most of the ae rodynamic derivatives. In order to examine the lateral behaviour of tailless aircraft , s imi la r calculations have been made using the range of values l ikely to occur in aircraft of this type.

At ten t ion has been paid chiefly to tile case of the t rue flying wing wi th no fins. In this case there is no l inear relat ion between nv and n,. and it is necessary to t rea t these as independent variables. For a wing with end fins there will be a relat ion of the form n~ = a - / bn~., but the quant i t ies a and b can have such a wide range of values tha t explorat ion on these lines would be impractical . If nv and n~ are assumed to be unre la ted then we can take /~nv and --~l~ as independent variables for p lot t ing s tabi l i ty diagrams, and var ia t ion of the b o u n d a r y w i t h / , is then el iminated.

I t is considered tha t in a tailless aircraft wi thout fins, the values of -- n, and n, are unl ikely to exceed 0 .03 and tha t - - y , will be considerably smaller than 0 .2 . The re la t ive-densi ty factor ~ on an all-wing design should not exceed 40. Informat ion at present available on this type of aircraft indicates tha t iA and ic will be in tile region of 0 .09 to 0" 12 and tha t their difference will probably be small. Wi th these facts in mind the following ranges have been invest igated : /~z~ = 0 to 1.4, -- ~,: = 0 to 0.03, -- yv -- 0 to 0 . 2 for several different combina- tions of inertias.

*R.A.E. Report No. Aero. 1826 received 30th July, 1943.

(79722) A

2

2. Notat ion.- -The notat ion is based on tha t of R. & M. 18()P from which m a n y of the following definitions are taken. Many revisions are due to Dr. Mitchell. 4

The x-axis is taken into the direction of the relative wind in the undis tu rbed condit ion and in all this work has been assumed to be horizontal . The y-axis is along the wing, positive in the s tarboard direction. The z-axis is then perpendicular to the x- and y-axes and is positive down- wards. These axes remain fixed in the body in the d is turbed motion.

The forces and velocities along these axes and the couples and angular velocities about t hem are defined below : - -

Force in I Couple Velocity in Angular Axis direction ' about direction of ! velocity

of axis , axis axis I about axis

Ox X L V +- u p Oy Y M v q Oz Z N w r

These m a y be expressed in the following non-dimensional form : - -

Y L N c,. = -~..~,~ ~,-'~-~-°' c, = y ¢ % , c , , -

p V ~ S s '

; ' - - v , f ' = , ; = ~ ,

w h e r e S is the gross w i n g area and s is the semi-span and ff is the relative densi ty pa rame te r m/oSs.

The uni t of t ime chosen is I = ~, /pSV = l,s/V. The symbol d/dr denotes differentiation with respect to this uni t of time.

The moments of iner t ia about the axes Ox and Oz are denoted by A and C respect ively and these a re expressed in coefficient form as follows : - -

A C i4 - - ~ n s 2 , i c - - r t l S ~ •

"Fhe product of inert ia about the axes Ox Oz is denoted by E and the dimensionless coefficient by i,.: =: -- E/ms ~. The ratios E/A and E/C are denoted by ~, and <. respectively.

The ae rodynamic derivat ives can be expressed as

aC,, " aCl aC,,

~C~ ~C,, ~p - ~ ( p s / v ) ' % - ~ ( p s / v ) '

~C~ aC~ l, =: ~,(rs/V)' n, - ~(rs/V)"

3

In this notation y~, l~, Ip, np, n,, are usually negative and n~ and l, are usually positive. In order to facilitate numerical calculation it is convenient to write

lp ~tp ~.4 ¢c

nr

With this notation the equations of motion under no external forces in horizontal flight become

d

and the stability equation becomes

where

+~ --kS=0,] ) ~ - - 12 ~ = 0 ,

~+n~ ; =0, d +N~=o,

A X 4 + BZ 3 + C2 ~ + D~ + E = O,

A = 1 - - sA e c ,

B = l ~ + n 2 + ~ j 2 - - ~ n ~ + y ( 1 - - ~ ) ,

C = (l~n, + l~n~) + :L (l~ + n2 + ~J2 - *An1) + , ~ + w ,

D = .9~ (l~n2 + 12nl) + (,,L~nl + ./U/I) @- k (~...W -+ .APe4) ,

E = k (£°n~ -- XI~).

All the stability diagrams and curves of constant damping have been plotted with ~n~ and --,ul~ as co-ordinates and it is convenient to calculate them first in t e rmsof W and ~ ; for this reason the coefficients which are dependent on these quantities are conveniently written

C = C1 + C~a~ + C3~ ,

D = D1 + D2~ ° + DsJV,

where

(79722)

E = E2~ E~4/',

A = 1 - $A CC

B = (l~ + ~ + ~& - - ~n~) + 9~ (1 - - ~ ~c),

C1 =, ( l ln2 + l~nx) + 9~ (ll + n2 + ~cl2 - - ~An~) , C2 = ~c , C3 = 1 ,

D ~ - - y ~ (l~n2 + l~nl) , D2--= (nl + k ) , D 3 = (l~ + ksA) ,

E2 = kn2, E 2 = kl~.

A '2,

4

3. Stability Boundaries. It has been shown by RouttP that the conditions for stability are that E shall be positive and the expression D(BC -- AD) -- B*E is positive. If E is negative, the motion will contain a divergence, and if D(BC - - A D ) - -B~E is negative the motion will contain a divergent oscillation.

\Ve therefore calculate the points at which E and D ( B C - A D ) - B2E change sign.

3.1. Method of Calculation.--The E ..... 0 boundary is with the assumptions of this work a straight line through the origin and is easily plotted from its equation

E,a2, -- E ~ =: 0 .

The following method of calculating the oscillation boundary is due to Dr. Mitchell.

The oscillation boundary is given by

B E D -- (-22-7t D-IN =- O,

i.e. D ( B C - - A D ) - - B 2 E - R = O .

Let R' = BC -- A D ;

then R' = R ~ ' - - R2 '~ 47 R ( W ,

where RI '=: BC~-- AD~, R ( = A D 2 - - BC~, R~' = B C 3 - - AD,~.

Let the values of ~ on D ~= O, R' =-- 0, E ---- 0 and R =: 0, corresponding to the value ,~f~ be

denoted by - , f , , £fR,, 6e~,, and ~ respectively ; then

E (~, ,4q):= E ~ , - EaJ¢',,

E(2,1. J~/'l) --- E2~e,~: - - E ~ =: O,

s o that E(2,~W~) - - E ~ ( , - ~ f ' l - - ~ f E ) •

Similarly D(~,.,4~) D2(~ 47.£"D) ,

R'(2,,,Arl) = R ( ( ~ , ~ , - 2,,);

therefore R(2,~,/V~) =D(2,~W~) R t ( 2 , 1 d f / ' I ) - - - B~E(~r/V~)

= D2(~1 47 ~,,) R2'(~,~,- 2,,) -- B2E2(2,~ -- 2,,.),

B=E= I I (o(~12 - - [2',e" ~ " --- D=R2' I 2 , t - - + 2',.: . i.e.

The Method employed was to calculate A, B, C~, C2, etc. R ( , R ( , R3', B2E2/D2R(, and hence obtain at the requii~ed values of X; ~ : , ~w , and ~ , and hence cMculate

B2E~ B2E2

and obtain two values of ~ , by solution of the quadratic equation.

5

3.2. Values of Derivatives and other Parameters,--It is considered tha t derivat ives lp, 1,, n~, which are pr incipal ly due to the wing, should be of the same order as for a convent ional design.* For most of the present work therefore, the values assumed in Priest ley's invest igat ion (R. & M. 1989, Pt. P) have been used. These are

At CL = 0 ' 1 At C~ = 1 '0

l,, . . . . - 0 . 4 s - o . 4 o l~ . . . . O" 02 O" 235 np . . . . - -0"03 - -0"05

For CL = 0.1 the product-of- iner t ia t e rm iz.: was taken to be zero and for C~, = 1.0, ij~ was assumed to be -- (ic -- iA) sin ~ cos e, and ~ was assumed to be -- 10 deg.

Boundar ies where R -~ 0 and E = 0 have been calculated for the ext reme values y,, = 0 and y~. = -- 0" 2 wi th each of the values n, = 0 and n, == -- 0 .03 for the following pairs of inertias :--- iA = 0 ' 0 5 , ic-- -0"08; iA = 0 " 0 9 , i c = 0 " 0 9 ; iA=:0"09, i c = 0 " 1 2 ; i 4=0"12 , ic=0"12. All these calculat ions have been made both for CL -- 0 .1 and CL = 1.0. In order to examine the var ia t ion of the boundaries wi th y~ and n; more fully, boundaries have been calculated for all possible combinat ions of the following values of y, and n, : - -

y ~ = 0 , - - 0 . 0 5 , - - 0 . 1 0 , - - 0 . 1 5 , - - 0 - 2 0 ,

n ~ - 0 , - - 0 . 0 1 , - - 0 ' 0 2 , - 0 "03 , j

for it = 0 " 1 2 , i c = 0 " 1 2 , at C L = 0 " I .

Boundar ies have also been calculated to show the effect of vary ing lp, 1,, np from the s t andard values given above in the case y~ -- -- 0.05, n, = - 0.01, i,~ = 0.12, ic ----- 0.12.

Tile numer ica l field surveyed is summar ised in Table 2.

3.3. Variation of Stability Boundaries with the Parameters.--3.31. Variation with y~ and n,,.-- The ' : s p i r a l " boundaries have the equat ion n ~ - l~dr = 0 (for s tabi l i ty the lef t -hand side mus t be positive) or mul t ip ly ing by iA • ic

( - n,) - ( # n v ) l, = 0 ,

so tha t the E bounda ry is a s t ra ight line th rough the origin with slope I,/-- n,.

The oscillation boundar ies are displaced upwards with increase of ei ther -- yo or - . n,. T h e ra te of displacement upwards is greater at CL = 0.1 t han at CL = 1-0 (Figs. 2 and 11). In the one ease in which ,a larger n u m b e r of these ~)arameters was considered (iA -- 0.12, ic == 0.12, CL = 0.1) the var ia t ion wi th y~ and n, was found to be very nearly l inear (Figs. 5-9) so tha t fur ther detai led calculations of this type were considered unnecessary.

3.32. Variation with CL.--I t has been shown elsewhere tha t for the values of the paralneters usual in convent ional designs, the R = 0 boundaries are displaced downwards wi th increase of CL. In the range considered here the R -- 0 boundar ies are displaced downwards wi th increase of CL at the higher values of Yv and n,, but the direction of displacement is reversed at lower values of these parameters . At low values of CL and very low values of y~ and n, tile stable region becomes very small.

*There is most likelihood of variat ion in l~ and n~ since these parameters depend on the moment of inertia of the lift distr ibution about the axis of the aircraft. Tailless designs incorporate a fairly large washout and at low CL the lift at the tip m a y be negative, thus substant ia l ly reducing bo th l,. and %. The effect of var ia t ion of these parameters is deMt with in §3.35 and Figs. 14-19. The effect of changes in l~ is not large, but overest imation of the numerical value of n,, m a y make the conclusions of this report ra ther on the pessimistic side.

6 The " spiral " bounda ry is a s traight line th rough the origin with slope l,/ .... n , i.e. the slope

is roughly proport ional to CL.

3.33. Variation with t~.--The stabil i ty boundaries of this report are plot ted with /,n~ and -- f~,I~, as co-ordinates and in this way are made independent of E*. I t is obvious however t h a t if t hey are plot ted in the usual way against n~, and -- l,, the oscillation b o u n d a r y w i l l be displaced downwards with increase of t~.

I

3.34. Variation with Iner t i a . - -The few variat ions of inert ia coefficients t ha t have been con- sidered are not sufficient to give any clear indicat ion of the manner in which the oscillation boundar ies va ry with inertia. They do, however , serve to i l lustrate the general result t ha t the region of stabili ty tends to decrease when iA and ic increase together.

3.35. Variation with lp, l, and np.--Calculat ions have been made to invest igate the effect on the R ---- 0 boundaries if lp, l, and np deviate from the s t andard values used in the remainder of the work.

An increase of -- lp causes a fairly large increase of the stable region in the case considered both when CL = 0.1 and when CL = 1.0. This is shown in Figs. 14 and 17. The var ia t ion of l, seems un impor t an t at CL = 0" 1, bu t increases in impor tance when CL = 1 . 0 ; in both cases a decrease of lr causes the R = 0 b o u n d a r y to be displaced downwards (Figs. 15 a n d 18). Changes in nt~ seem to have a small effect at CL = 1.0, but a much greater effect at CI. - = 0 . 1 , the bounda ry being displaced downwards with increase of -- n~ (Figs. 16 and 19).

The E = 0 b o u n d a r y is independent of Ip and n~ and has a slope proport ional to -- l~.

4. Curves of Constant Damping . - -Curves of constant damping of the oscil latory and of the spiral motions have been calculated in a few cases to indicate the gradient Of damping across the boundaries. The inertias iA = 0"12, ic = 0"12 have been used and the curves have been calculated for the pairs of v a l u e s y v = 0 , n r = 0 ; y~ --- -- 0"05, n , = - - 0 . 0 1 ; 3'~ ---- -- 0"10, nr = -- 0"02.

4.1. Method of Calculat ion. - -The curves of cons tant damping have been calcula ted by the method described by Brown (R. & tVI. 19056).

If r~ _-4= is~ are two roots of o

A~ ~ + B ~ 3+C~ 2 + D ~ + E = 0 ,

then As~ ~ - - if3(rz) s f - f2(r 3 s f + ifl(r,) s, + f(r~) = O,

where f(r~) = Arz ~ + B r f + Crp + Dr1 + E ,

f~(r,) = 4Arp + 3Brp + 2Cr, + D ,

f2(rz) - 6Arp + 3Br~ + C ,

f.~(r 3 = 4Art + B ,

and equat ing real and imaginary parts

A.h 4 -- f~(rz)sP + f(r~) ---- O, t "

f~(r,ls, 2 --'f~(r~)= O. ]

Now f(r~) can be written

7

f(r,) = f01(r ) f02(r,) ze + f0 (r,)w, where fol(r,) = Ar24 + Br~ 3 + Clrt~.+ D1r~,

fo~(r~) - - C~r~ ~ + D2r~ + E ~ ,

fo3(r,) = C3rl ~ + D 3 r , - E ~ ,

and similarly for fl, f~, f.~.

Now if we write al = As,4 - - f~(r3s~ 2 + fo~(r,) ,

b~ - - f o 2 ( r , ) - f2~(r3s, 2 ,

c~ = fo3(r,) - - f~3(r,)s, ~ ,

a2 = fn(r,) - - f~l(r3s, ~ ,

b., = A s ( r 3 ,

c~ = flz(r,) ,

the equations become

a ~ + b ~ + c ~ X = 0 , "~

as + b~w + c ~ = 0 ,

which can be solved for ~ and ,,¢'.

4.2. Resul t s o f Per iod and D a m p i n g C a l c u l a t i o n s . - - T h e curves of constant damping show that there is little variation of the gradient of damping across the R = 0 boundary with change of y~ and n,.

When C L = O. 1 the gradient of damping across the boundary E = 0 is very small so that no appreciable change in the spiral motion is likely to occur within the practical region of l~ and n~.

5. A i r c ra f t wi th F i n s . - - T h e present work has been undertaken mainly to investigate the characteristics of a pure flying wing with no fins. If there are fins there will be a linear relation between n~ and n, due to the contribution of the fin. The value of l~ to give R ----- 0 or E = 0 may be interpolated at any n,, and n, from the figures given and a stability diagram may be constructed. The stability diagram would in this case have an R boundary of degree 4 and an E boundary of degree 2 as in the case of conventional aircraft. The R and E boundaries will in general have one more intersection in the region of positive n~ and negative l, and there will be two completely Stable regions, one for small values of l~ and no and one for larger values of both parameters. The constant b in the relation n, = a + b n~ will for tailless aircraft be much smaller than for conventional aircraft (since the fin arm will be smaller), hence the two intersections of R = 0 and E = 0 will be farther apart and the region of large l~ and n~ may not be accessible.

\

6. N u m e r i c a l Compar i son wi th Convent ional A i r c r a f l . - - V e r y few data are at present available on the probable values of the derivatives for all-wing aircraft. Recent wind-tunnel tests on a swept-back wing have shown that n~ may vary from about 0.005 at low incidences to 0.01 at high incidences ; l~ can be varied for any design by a change of dihedral, bu t there will be a considerable change in l~ with CL due to the large sweepback which is necessary in tailless aircraft to solve the problems of longitudinal stability and trim. In these recent tests l~ varied from -- 0.04 to -- 0.11. The same tests indicate that y~ will probably be in the region of -- 0.01.

During systematic tests of rolling moment due to sideslip 7 yawing moments were measured on a few swept-back and swept-.forward wings. These results indicate that there is no great change of n~ due to sweepback at zero incidence, but that there is a greater increase of n~ with incidence on those wings with greater sweepback.

8

T h e r e is e v e n less i n f o r m a t i o n o n ' t h e v a l u e of n, a n d we c a n o n l y f o l l o w R e f e r e n c e 1 a n d a s s m n e t h a t n~ fo r a w i n g a l o n e wil l b e of t h e o r d e r 0 t o - - 0 . 0 1 . F o r the r e m a i n d e r of t h e d e r i v a t i v e s it is a s s u m e d t h a t t h e v a l u e s wi l l a p p r o x i m a t e t o t h o s e g i v e n in R. & M. 1989, P t . P a n d § 3.2 of t h i s r e p o r t . *

I t h a s b e e n s h o w n s t h a t f r o m c o n s t r u c t i o n a l a n d p e r f o r m a n c e c o n s i d e r a t i 0 n s c o m p a r a b l e c o n v e n t i o n a l a n d a l l - w i n g d e s i g n s a r e g i v e n b y t h e f o l l o w i n g T a b l e : - - ,,

Weight (lb.) . . . . . . . . . W' Span (ft.) . . . . . . . . . b Wing loading (lb./sq. ft.) . . . . . .

Conventional

60,000 100 50

All-wing

60,000 100 35

The change of loading f rom conven t iona l to an all-wing design will largely consist of a r emova l of load in the rear fuselage and an increase of load in the wing tips. This will give li t t le var ia t ion in the m o m e n t of iner t ia C, bu t a considerable increase in A so t h a t the difference C -- A becomes small. The values assumed for the two cases are therefore, for the conven t iona l aircraft iA = 0 .0625, is == 0. 1225 and for t he tailless iA = 0 .12 , ic --= 0 .12.

We can now summar i se the i m p o r t a n t p a r a m e t e r s for t he two types of aircraft .

Conventional [ All-wing I

Weight (lb.) . . . . . . . . W 60,000 60,000 . . . . . . b 1 O0 ! 1 O0 Span (ft.) ..

Wing loading (lb./sq. ft.) . . . . w 50 :¢5

Inertia coefficients .. f i a 0- 0625 0.12 . . . . ) i e 0.1225 0.12

3'~ --0- 2 0 and - -0 .05 , , , f a t sea level 13 9

Relative density par,uneter /~ ) a t 40,000 it'. 52 36

['r" ,, , F a t sea level .. I "42 i 1 • 19 J ~ ' = - ° ' l ~ a t 40,000 ft. 2.85 2 '38

Unit of time t " " / (" 1 o f at sealevel -" 4-50 3.77 ( - - L = - ' - , ~ a t 40,000 It. 9.01 7-54

For compar i son , curves of cons t an t d a m p i n g have been d rawn for the conven t iona l aircraft ou t l ined a b o v e at Cc = 0 .1 . As a rough compar i son wi th the curves for the tailless aircraft at C L - - 1 .0 the curves of R. & M. 1989 Pt. I I 2 rnay be used. T h e s e ' c u r v e s are no t s t r ic t ly comparab le since the curves of this repor t are for level flight and those at C~, = 1-0 in R. & M. 1989, P t . I I ~ are for t an 0o =-: -- 0- 1.

To obta in a numer ica l compar i son we shall consider the values for the conven t iona l a i rc ra f t :- -

( a ) n ~ - - 0 . 0 2 , l,, - - - - 0 . 0 2 ,

(b) = 0 . 0 2 , = - 0 . 1 0 ,

(c) n~ = 0 . 1 0 , l~ = -- 0 . 0 2 ,

(d) n ~ - 0 . 1 0 , l~ = - - 0 . 1 0 ,

at C L = 0 " I a n d C L = 1.0.

• But see footnote to §3.2.

1

For the tailless aircraft

(~)

(a)

(V)

(~)

(4

(~)

9

n, = O, y~ - - O, n~ = O,

n, -- O, yo = O, n~ -= O,

n , = - 0 . 0 1 , y ~ - : - - 0 . 0 5 , n ~ - : O ' O 1 ,

~ , ,= - - 0 " 0 1 , y ~ = - - 0 " 0 5 , n~= : ' 0 "01 ,

l~== - - 0 . 0 1 . }

I,, ....... 0 05 ,

L = - - O 01,

Iv- : - - 0 05,

Cl. -~: ()" 1 ,

(o)

n, = O, y~. = O, n~ = 0.01 , l, = -- 0.01 ,

n, = 0 , y~ ----- 0 n~ = 0.01 l~ = . . . . 0 .05 ' ' ' C , . -=: 1 " 0 .

n , = - - 0 . 0 1 , y ~ = - - 0 . 0 5 , n ~ = 0 . 0 2 , I~- . . . . 0 . 0 1 ,

n ~ = . - - 0 . 0 1 , y~'---=--0.05, n ~ = 0 . 0 2 , l v = - - 0 . 0 5 ,

The ' informat ion at present available indicates tha t tailless designs will probably approximate more closely t o cases c~, fl, s, ~, than to the others but will probably have parameters within the above ranges.

The dampings of the motions are compared in Table 1.

It will be seen tha t a tailless aircraft is almost certain to be spirally unstable. At low Cl, the rate of divergence will ,be slow and at high CL the rate of divergence is of the same order as for a conventfonal aircraft.

The oscillation may be unstable in all conditions and in no case will the damping be large. The lateraI oscillations may be less t roublesome on a tailless aircraft because of the rather longer period.

In view of the uncer ta in ty of the derivatives it is difficult to form any definite conclusions about the stabil i ty of tailless aircraft . These calculations show tha t the achievement of stabili ty may be difficult. This difficulty is due to the small values of n,. and n, and at present there is no information on the size of n~. It wil l be seen from the stabil i ty diagrams, Figs. 5-9, tha t this derivat ive is of considerable importance and until more information becomes available it is difficult to obtain with any accuracy an esI/imate of the stabil i ty characteristics of tailless designs.

7 . C o n c l u s i o n s . - - T h e conclusions of this work are compared with the cmresponding conclusions for a convent ional design (from R. & M. 1989, Pt. I1).

Conventional aircraft Tailless aircraft

(1) At high speed, spiral instabil i ty is very unlikely to be met .

(2) The oscillation will usually be stable at high speed provided no > 0.

(3) At low speed, spiral instabil i ty is almost certain.

(4) At low speed, the oscillation will usually be unstable except for low 1,, or n~.

At high speed, spiral instabil i ty is likely to occur unless - - l , is large.

The oscillation is likely to be unstable* a t high speed unless - - l~ is small or - -n , or - - y , have the larger values considered.

At low speed, spiral instabil i ty is even more likely to occur than on a conventional aircraft.

Oscillatory instabil i ty is more likely to occur at low speed than at high for the larger values of - - n, and - - y , considered. For the smaller values o f , y, and - - n, the oscillation is more likely to be unstable at high speed than at low. A tow value of l,. will be required for stability.

*But s e e footnote to §3.2.

A

A , B , C , D , E

C

C1, D1

C2, D~, E2

C3, D3, -- E~

E

fi, A,A,f [ol, fu , f2,, etc.

f o~, f12, etc.

f o3, f l,, etc.

i~, i ,

il.

k

lv

11

JV"

nv

"]'~r

nl

n2

R

R '

RI '

R ( , R~'

10

LIST OF SYMBOLS

Moment of inertia of aircraft in roll.

Coefficients of stability quartic f(Z).

Moment of inertia of aircraft in yaw.

Terms independent of ~ and W in coefficients C and D.

Coefficients of ~ in C, D and E.

Coefficients of Jv in C, D and E.

Product of inertia of aircraft about the axes of roll and yaw.

The stability quartic AZ4 + B~3 + C~ + D~ + E.

The coefficients of the Taylor expansion off.

The terms of f , f l , f~, etc. independent of ~ and ~v.

The coefficients of ~ in f , f l , etc.

The coefficients of W in f , f i , etc.

Inertia coefficients in roll and yaw iA = A l m s ~.

Product of inertia coefficient about the rolling and yawing axes. i/~ . . . . E/ms 2.

- - ~ lv/iA.

Values o f ~ corresponding to ~¢" = W~ on D = 0, R' -= 0, E = 0 and R = 0 respectively.

Coefficient of rolling moment due to sideslip, OCJO ~.

Coefficient of rolling moment due to rolling, bCJO(ps/V).

Coefficient of rolling moment due to yawing OCJb(rs/V).

Idi:,. !,nffic

Coefficient of yawing moment due to sideslip, ~C,/~fl.

Coefficient of yawing moment due to rolling, ~C,/~(ps/V).

Coefficient of yawing moment due to yawing, ~C,J~(rs/V).

- - np/ic.

-- nffi~:.

Dimensionless coefficient of rolling velocity, ~pS/V.

Routh's discriminant, D ( B C -- A D ) -- B*E.

Test function ( BC -- AD) .

Term of R ' independent of ~ and ~".

Coefficients of ~ and jv in R' .

Dimensionless coefficient of yawing velocity, ~ rs /V

fs

S

Sl

l

Y~

8

~A

8 C

2

T

¢

q~, T

11

L I S T O F S Y M B O L S - - c o n l d .

D a m p i n g coefficient of l a te ra l oscil lat ion.

D a m p i n g coefficient of " spiral " mot ion .

Semispan of a i rcraf t , ½b.

F r e q u e n c y coefficient of l a t e ra l oscil lat ion.

Un i t of t ime in d imensionless sys t em, m / p S V .

Dimensionless coefficient of sideslip ve loc i ty .

OCy Coefficient of s ideforce due to sideslip, ½-~-d "

- - yV~

Angle of sideslip for small angles). Angle be tw een pr inc ipa l axis of iner t i a a n d x-axis.

- - E / A = iE/iA.

- - E / C = itffic.

D u m m y var iab le of s t a b i l i t y quar t ic .

A i r c r a f t re la t ive d e n s i t y . p a r a m e t e r , m / p S s .

Time in d imensionless uni ts .

Angle of bank .

Coefficients in q u a d r a t i c equa t i on for ~e

No. Author

1 Priestley . . . . . .

2 Priestley . . . . . .

3 Bryant and Gates . . ..

4 Mitchell . . . . . .

5 Routh . . . . ..

6 Brown . . . . . .

7 Irving, Batson and Warsap

8 Montagnon and Hallowes ..

R E F E R E N C E S

Title, etc.

.. A Further Investigation of Lkteral Stability. R. & M. 1989, Part I. March, 1941.

•. An Investigation of the Periods and Dampings of the Lateral (Asymmetric) Motions. R. & M. 1989, Part n . March, 1941.

.. Nomenclature for Stability Coefficients. R. & M. 1801. October, 1937.

•. A Supplementary Notation for Theoretical Lateral Stability Calculations. R.A.E. Tech. Note No. Aero. 1183. A.R.C. 6797. May, 1943. (To be published).

.. The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies. 6th Ed., Macmillan, 1930, pp. 221-226.

.. A Simple Method of Constructing Stability Diagrams. R. & M. 1905. October, 1942.

.. Model Experiments on the Rolling Moment due to Sideslip of Tapered Wing Monoplanes. R. & M. 2019. July, 1939.

.. Some Aspectsof LaNe Aircraft Design. R.A.E. Report No. Aero. 1800. A.R.C. 6694. May, 1943. (To be published).

T A B L E 1

Numerical Comparison of a Tailless and a Conventional Aircraft

CL = 0" 1 Ground Level

Case

"tl v

- - 1 v

Damping coefficient of oscillation Frequency coefficient of oscillation Damping coefficient of spiral motion

St

J's

Oscillation fPe r iod (secs.) "[.Time to halve amplitude* (secs.)

0"02 0"02

--0.24 1.5

--0.002

5.9 4.1

Conventional

C

0.02 0.10 0.10 0.02

--0.19 --0.57

' d

0"10 0"10

--0" 52

0 0.01

+0" 002

All-wing

0 0.05

+0-025

0"01 0"01

--0-07

1"7 --0.013

5-2 5.2

3"4 --0-0003

2-6 1.7

3"4 --0"01

2"6 1"9

0.25 0

30 --400

0"25 0

30 --30

0.7 +0-001

11 12

0"01 0"05

--0-05 0"8

--0"001

9 16

Spiral motion : Time to halve amplitude* (secs.) 500 76 3,000 t 100 Neutral --800 800

Cz = 0" 1 40,000 ft.

Conventional All-wing

Case

~v - - l v

0"02 0"02

0"02 0"10

Damping coefficient of oscillation rt --0"21 Frequency coefficient of oscillation sz 3-0 Damping coefficient of spiral motion rs , --0.002

fPe r iod (secs.) 5.9 5- 3 Oscillation ).Time to halve amplitude* (sees.) 9.4 40

Spiral motion • time to hah'e amplitude* (secs.) ] 1,000 150

0"05 3 '4

--0"013

0"I0 0"02

--0" 55 i 6"5

--0' 0003 1 i

2"8 3-6

6,500

d

0"10 0"10

- - 0 - 44 6-7

- -0 .0 t

2"7 4-5

200

0 0-01

+0 .028 0.25 0

60 --60

0 0-05

+0 .10 0 . 2 5 0

60 --16

Neutral

y

0"01 0.01

- - 0 " 0 4 5 1"7

+0.001

8-8 37

I-- 1,600

0"01 0"05

+ O- 005 1-9-

--0.001

7-9 --3(/0

1,600

*A negative sign indicates a divergence • tile figure given is then the time to double amplitude.

TABLE 1 (cont.)

CL = 1-0 Ground Level

Case

--l,,

Damping coefficient of oscillation Frequency coefficient of oscillation Damping coeffÉcient of spiral motion

T t

St f~

f P e r i o d (secs.) Oscillation ~_Time to halve amplitude* (secs.)

Spiral motion : Time to halve amplitude* (secs.)

0"02 0"02

--0"45 1"7

-~0-16

17 6.9

--19

Conventional

b C

0-02 O, 10 O. 10 0-02

--0-30 - -0 .79 2.2 3.4 0 +0 -22

13 8.3 10-4 3.9

Neutral --14

0"10 0"10

- - 0 " 68 3"7

+ 0 - 1 0

7"6 4"6

--31

0"01 0"01

- -0 .18 1.1

+ 0 . 2 3

22 14

--11

All-wing

0"01 0"05

--0"02 1-3

+0"12

18 130

--22

0"02 0-01

--0" 23 1"4

.@0-25

i 17 r 11

--10

0"02 0"05

--0"1 1"6

@0"15

15 26

--17

-. C L = I . 0 40,000 ft.

5¢

Case

- - I v

Damping coefficient of oscillation rt Frequency coefficient of oscillation st Damping coefficient of spiral motion G

f P e r i o d (secs.) Oscillation \ T i m e to halve amplitude* (secs.)

Spiral motion " Time to halve amplitude* (secs.) i I

0"02 0"02

--0.40 3-3

+ 0 . 1 7

17 16

--37

Conventional

b c

0"02 O" 10 O-10 0"02

--0" 14 --0" 75 4"3 6"7 0 q O '23

13 8.4 45 i 8.3

Neutral -- 27

d

0"10 0-10

- - 0 - 5 8 7.2

+ 0 - 1 0

7"9 10.8

--62

0"01 0"01

--0"08 2"0

@0.23

24 65

--23

All-wing

0"01 0-05

+ 0 . 3 2.2

+0 -12

22 --17

--44

0"02 0"01

--0-13 2 '7

- -0 ' 25

18 40

--21

0"02 0 '05

t0 .18 3 . 0

+ 0 . 1 5

16 --29

_ 3 5 ¸

*A negative sign indicates a divergence • the figure given is then the time to double amplitude.

14

O" 1

1.0

T A B L E 2

Key to Figures

(1) S tab i l i ty diagrams. S t anda rd values of l w l . np

--yo

0 and 0 . 2

0-05 0"10 0"15 0"20

0 and 0 .2

- - - ~ t r

4) and 0 .03

O, 0 '01 , 0 '02 , 0 ' 0 3

0 and 0 .03

i A .

0"05 0"00 0"09 0"12 0"12

0 ' 0 5 0"09 0"09 0"12

ie

0"08 0"09 0"12 0"12 0 ' 12

0 ' 0 8 0"09 0"12 0 :12

.3 . . . . . . .

Fig. No.

1 2 3 4 5 6 7 8 9

10 11 12 13

(2) S tab i l i ty diagrams. Variat ion of Is,, I,, n,,.

ff Cz = 0.1 Variat ion with l~ Fig. 14 c,. =: 0 .1 Varia t ion with l, Fig. 15

v,,: . . . . 0.05, n~ = - -0 .01 CL = 0.1 Var ia t ion with % Fig. 16 ~,, =: 0"12, ie =: 0 . 1 2 ) C L = 1 . 0 V a r i a t i o n w i t h l , , Fig. 17

Cz 1.0 Var ia t ion with 1~ Fig. 18 = 1.0 Varia t ion with % Fig. 19

(3) Curves of constant period and damping

ia = 0 .12 i v = 0 ' 1 2

C,, , y , ,

0"1 0" 1 0"1 1"0 1 ' 0 1 ' 0

0 0 0"05 0.01 O. 10 0 ' 02 0 0 0"05 0.01 0 .10 0 ' 02

--n~ Fig. No.

20 21 22 23 24 25

(4) Curves of constant period and damping (conventional aircraft)

ia = 0.0625 ie = 0-1225

/, : 13 Fig. 26 ~t = 52 Fig. 27

C1~ = 0 ' 1

4"0 "Z "# ~ ° ' ° ~ / " " " ~ - ,~ V ~0/.*'~

OSCILLATION ~OUMDA~IES

5PIRALOIVERGENEE BOUNDARIES

STASL E SIDE ~ " LJNSTA B L E ,SIOE :~"

/ /

/

<-e

I z.0 0 [i g.

~te

< Q g 13 m M O Z Ul

g

,~ t'Jv ,=,o ~ ' P = 0

~ d 0.~ I.o /.¢ ?Z V

FIG. 1. S t a b i l i t y Boundar ie s C~ = 0"1, i~ = 0 . 0 5 , ie = 0 . 0 8

J.5

- ~ v . "~ '

z . . 0

2.c d

,t,0

FIG. 2.

/

OSCILLATION SOUNDARIES.

SPIRAL DIVERGENCE L~)DUNDARIES.

S'TA {~ L E 5 | D E . . ~

U NS' rA~L F .~ IDE. >

-~ ~-~ 5. °2~

s I

/ /

O.g I.O / ~ qm V

/

S t a b i l i t y Boundar i e s CL = 0 . 1 , i~ = 0-09 , io = 0 . 0 9

I.E

Cn

-~-Z v

FiG. 3.

0

DSCfLLATJON E, OUNOARIES

SPIRAL OIVE, RGENCE. BOUhIDAP, IF--S

STA ~LF-- 51DE.. >) U NSTA~LF-- 51DE. ),

Vv

*qv

I

I ~ ~ 1

p I ~ i

O.E / .O

Stab i l i t y B o u n d a r i e s C,. = 0" 1, i.~ = 0-09 , i o = 0" 12

/ . E

~ u

-/z~v

3"C

ENr'E ~0UNDARIES ....

O/~" 5TAI3 LE 51DF_-. ~i>

UNSTAE~LE 51DE ~"

g ..o 2.0 ul

!o

/

/

/ t

f /

1 " i

~ 0 0.5 J.O ,,4c "77 v

FIG. 4. Stabi l i ty B9undaries C~ = 0- l , i~ ~ 0- 12, ic, = 0 . 1 2

i ' .r

- ,~Zv

FIG. 5.

0 I! f,

3-0 ~f

Z

o tO

Z Ill

>

2.0

OSCILLATION BOUNDARIES

SPIRAL DIVERGENCE BOUNDARIE5 . . . . .

STA E~LE S lOE )>

U N 5TAtSLP-- S IDE . )

/

/

/

.o} / . o o /

~ f

f

i / ~C~'-- l.IJ

/ t . / / . ~ 0 " O ~

t / , / . / "/ ~ 0.5 v o ~ ?Zv i.~;

Stabil i ty Boundaries CL = 0 .1 , ia = 0-12, io = 0 .12 , y , = 0

-,u./~ v

3-0

Z-0

/

! /

/ 1

F I G . 6.

OSCILLATION BOUNDARIES SPIRAL- DIVERGENCE BOUNDARIES . . . . .

S T A B L E 5JOE - - UNSTA 8 L E .S IDE >

[I

<e / - Z /

/

/ / / / - . ' - t - ~. o.O~

/

O.E /.o ]4.. ~ v 1.5

Stabil i ty Boundaries C ~ = 0 . 1 , ia=0" 12, io=0" 12, y , = --0" 05

" 4

~-'0

- / X Z v I o,,

1L

z

/ /

FIG. 7.

05Ci LLATI ON 80UNDARIES SP IRAL E)JVERGENCF- E~0UNEDARIES 5 T A S L E S i D E )~.- U NS'1-Ai~LE. S I D E ~-

! ~ . . o . o ' ~ 2 " ~

I o/" . .

, . ~ - ~ - ~ - ' -

i

0"6 #'0 / ~ Tz v 1.5

Stab i l i ty Boundaries Cz = 0" 1, i~ = 0 . 1 2 , {o = 0 - 1 2 , y~ = - - 0 . 1

~ 0

_ ~.7. v

05CILLA'TION BOUNDARIES 51DIRA" 01VERGENCE. BOUNDARIES / S'rA 8 L. F_- 5|DE. .~-~ U NS-FA~LE. SIDE. >

/ II t-

)- t~

t ~

~ 0

/ "o,¢~

j ' S S

/ o.s vo /x "-'t2 v #'$

FIG. 8. S tab i l i ty Boundaries C~ = 0 . 1 ia = 0 . 1 2 , io = 0" 12, y , , = - - 0 . 1 5

O0

4 . 0

-~v

3 , 0

~0

i

/ /

/

OSCILLATION. 60U N DAI~.Y o tl SP IRAL DIVERGENCE 130UNDARIES ~l- S T A B L E 51DE ;>,. / j:-

'" U N 5TA~L~--, 51D F"-., :~ >-

o.°'Y

@ " f f /

i¢ / / 0.0"~1 ~ ~

/ / .-- 1 .0"3~1

J /

/ /

/ /

/ /

/ /

/ / /

1 t

. /

/ i "

/

i I

I

o.~ t .o p~ ~ v J'E

FIG. 9. S t a b i l i t y B o u n d a r i e s C , = 0" 1, ix ----- 0- 12, ic = 0" t 2 , y , = - - 0 . 2

11 ~v~

-,~l v 1

/ ~ o ~ , ~ , o . ~oo~o~,~ --

r / - SEZ ! o . I -

g~ i I

i I

/

' I 0 O-S 1,0

/ ~ 7~-" V i

FIG. 1 0 . . S t a b i l i t y B o u n d a r i e s CL = 1 . 0 , i x : 0 . 0 5 , i o : - 0 . 0 8

1,5

~D

4 0

/.lye0

SPIRAL. 01VE.RGELNCE BOUNOARI E5

( !

/ /

/ /

/ /

/

/ 2"0

I

i

r

1

STABLE- ~IOF_

U N~-('ABLE- 51D~

/

i W =0 f t ~ : ~ O

/ / /

/ !

FIG. 11.

0.5 1.0 1-5 ~ n , v

S t a b i l i t y B o u n d a r i e s Cz = l "0, ia = 0 " 0 9 , ic, = 0 .1 t9

4 0

U

3"0

- ~ . l v

2"0

~P IRAL DtVE.RGENCE- BOUN OA;~lE.c~

''/-~

/ !

! /

I I

I

/ /

/ /

/ /

STABLB. 5 1 D F - ~ >

LIN~7-ABLE. 51 DE

/ / /

I / !

/

FIG. 12.

0 " 5 1"0 /.,L rL V

S t a b i l i t y B o u n d a r i e s CL = 1 • 0, i , = 0 - 0 9 , iv --= O" 12

1.5

3'0

2 ' 0

SPIRAL DIVERGENCE- I~OU N DARI F..S,

/ I-

I

i !

II t ILI

i

I I

I

s t

I

i I !

I STABLE- S IIhE- ))-

UNSTABLE 51DE. }-

I II 1 ~ (

0 I/ Q •

t

/ i t

Z O

J j

6 03 O

0.5 1.0 /..z, rLv 1.5

Fro. 13. S t ab i l i t y Bounda r i e s CL = l .0, i~ = 0-19., ia = 0" 12

-/ZLv

U N S T A B L E

J J

f f l I 7

f S T A B L E

OSOI LLATI ON

OSCILLAT ION

-~Z

o o.5 t.O /.,LyL. V ~..5

FIc . 14. Stab i l i t y Bounda r i e s C~, 0-1, iA = 0.19., ie = 0 .12 , y~ = - - 0 . 0 5 , n, = - - 0 . 0 1 . Varia t ion wi th l~.

3 l/lit/t1 / 0SCILLATION E,0UNDARIES SPIRAL DIVLRCENCE. BOUNOARIE.S . . . . . . . . . . STAI~I,..F- SIDE- }. I, UNST.~BLE SIDE.

V i /

/ / 2 / . . . . . . . "

11 .

(~ / / i / / #! %~, U N S T A B L E . " O S C I L L A T I O N

/ / /

/ / " "~_ ~.~ O. Ol . / ~

,' / 'S2rABLE. OSCILLATION , ,, l:).O~-

, , L~-/

I

" ~ " - . . . OS ~'0 ~ r L V 1.5

FIG. 15. S tab i ] i ty B o u n d a r i e s C~ = 0 .1 , i~ = 0 .12 , i~. = 0 .12 , y~ - - - - 0 . 0 5 , n, = - - 0 . 0 1 . Var ia t ion M t h t ,

bO

-/J-Z v

UNSTABLE 0 S C I L L A T I O N / ~

0SCILLA-I'ION STABLE

O

FiG. 16.

1 . 5 O - S t-o flJ'- ?'~ V

S t a b i l i t y B o u n d a r i e s C , = 0 .1 , iA 0 . 1 2 , ic = 0 . 1 2 , y,. = - - 0 . 0 5 , n , = - - 0 . 0 1 . V a r i a t i o n w i t h % .

UNSTABLE 0.61~ILLATION

/

/ / i - - ' - ~

~ " - " / - - - " STABLE OSCILLATION

FIS. 17.

0"5 1,0 I - 5 / Z T Z v

Stab i l i ty Boundar ies CL - - 1 .0, i~t ~ 0" 12, i~: ~ O" 12, y,, = - - 0 . 0 5 , n, -- - - 0 . 0 1 . Var i a t ion w i t h l~.

-~.l. v

i f

~pJ

i l l

!i ~, _-- ~f -= O.aO0

2 ~.t. = 0 , 2 3 5 'd

i t

qJ i j

h [

- , . d v

]

0~CILLATI ON BOUNDARIES

%PlRAL 131VEI~GENCE BOUNI:)/~I~I~.S ......

~-FA BL E 51P~... ~,

u N~.TA BI- g" 511~P- h

UNSTABLE OSCILLATION

STABLE OSCILLATION

0

FIG. 18.

0-5 I.O ~ , rTj V 1"5

Stab i l i ty Boundar i e s CL : 1 "0, i~ 0 .12 , ic = 0- 12, y~ = - - 0 . 0 5 , ,Jz, - - - 0 "01 . "Varia t ion wi th 1,.

I t t

UNSTABLE OSCILLATION

I

STABLE OSCILLATION

o

I~I6. 19.

0-5 t.O / j ~ TL. V 1"5

Stabi l i ty Boundaries CL = ! -0, iA 0" 12, ic = 0" 12, y,, = - - 0 . 0 5 , n , = - - 0 - 0 1 . V a r i a t i o n w i t h n ~ .

t' ,3 b3

~ i " i . ~ Oft ~"

~ cl 01 " ~ - ~

\ _ i ~ ~ = ~ 0 •

I : , 0.~ 0.4" 0'6 ~ "~v 1.0 t. O.S

#

jJ

J - I O

/

r~ and st are the damping and frequency coefficients of the oscillatory motion. r, is the damping coefficient of the " s p i r a l " motion.

FIG. 20. Frequency and Damping Coefficients C~ = 0 .1 , i,~ = 0.12, io -= 0.12, y~ = 0, ~, ---- 0.

k'~

1C

~0 ~

I o , ~.L ~

/ ~ ~'L "-~'s\ / / . ' / / f .

t t" j l ,

t l l l ! 5 ; ?" "'" 5 - - " °~!

o . ~ o . V. ~ , o t ~

FIG. 21.

rz and s, are the damping and frequency coefficients of the oscillatory motion.

r, is the damping coefficient of the " sp i r a l " motion.

Freqhency and Damping Coefficients CL : 0.1, ia = 0-12, ie = 0.12, y, = - - 0 ' 05 n, = - - 0.01.

bO

, /

I / I / I" /

0.~. 0 . 4 0 . 6 0.5 /~n~ V t.O

rz and sz are the damping and f requency coefficients of the osci l la tory motion. r~ is the damping coefficient of the "spiral" motion .

FIG. 22. Frequency and Damping Coefficients C~ = 0 .1 , ia = 0 .12, i , = 0 .12, y,, = --" 0 .10, nr = - - 0" 02.

~ 0

-/~Zv

I,,~

f

I 0

T

!

!

I I

~r

o ~

u l

/ i

t !

I I

J

//

0 I

!

I I

i

i

/ / / /

i t /

/ /

i I

/

~,"

J /

I

/"

/ g

/

J /

f f

/

/ /

/

/

• /

I t

I I

I

f

J

bO

/ x 1/ I I /

l I /

0.6 0.?.

5Z=~.o

0 . 8

I v.o /~r~ v

s/-- ~ .o

1,2

FIG. 23.

r~ and s~ are the damping and frequency coefficients of the oscillatory motion. r, is the damping coefficient of the "spiral" motion.

Frequency and Damping Coefficients CL ----- 1.0, ia = 0 .12 , ic ---- 0 .12 , Yv = 0, n, = 0.

I I !

O! rl i

Y..~i /

I I

I

OI

/ /

/ /

/ .#

/

,#

i

d* / • . ~ . # " "

: I , ~ ,:" I I t , \ j ~ - . - - ~

, .,__I_,,, \ ~,.-- ~

/ llX "t" Z -- - 0 . 1

/ , ,, , . . k

i I I / , \

I / II I t /

X i I /

~z=~o I "sz:25 " k / - . o o . , o. , \ , i o

, , h : ° " , . : , . o I

I ,I

I

I

FIG. 72~.

J . .

r~ and sz are the damping and frequency coefficients of the oscillatory motion.

r, is the damping coefficient of the " s p i r a l " motion.

Frequency and Damping Coefficients CL ~ 1 "0, ia = O. 12, io = O. 12, y~ - - 0 . 0 5 , n, = - -0 .01.

EO "-.1

- ~ 3 . ,

~ o

1.0

# / ' / i l l I

i t

I !

I

/ I

/ I f

I

i I

I

0 I

I I

/ I t

i

I !

/

t /

t i X

/ i /

/ l / /

/ i

~ t

i / I

/

o) /

/ /

/

/

/ . / t

/ I

/

l / / I

/ "

i /

/ ,

I /

xO~ t" # .

f

I

/

f f

h J

/ /

f

V, 7

~ ~ = . - - 0 . ~

# ~

i /

/ /

/ /

/ I /

i

o . , ~ O.G ' 0 . 8 /t.L n. V I-o

FIG. 25.

r~ and s ,are the damping and frequency coefficients of the oscillatory motion. ~; i s the damping coefficient of the "spiral" motion.

Frequency and Damp in g Coefficients Cz = 1.0, i a ~ - 0" 1:2, io ---- 0 .12 , y~ = - - 0-1, nr = - - 0"02.

-0 .OI

tl

" Q . I $ [ , . , , : ,

FIG. 26.

r~ and s are the damping and frequency coefficients of the oscillatory motion.

r, is the damping coefficient of the "spiral" motion.

Frequency and Damping Coefficients (Conventional Aircraft) CL = 0.1, ~ = 13, i~ ~ 0.0625, ic. = 0. 1225

b~ e~

+

,.'7-

0.14.

-'Lv - - 0. I'~

C,,IC~

0 . 0 5

°0.01

"t

Y

/

FIG. 27.

~ I . +s = .o.o_o~

_ ~ -_++__

. . . . T

+°' °°: /+°+" 1 °°+ ° ° + / 11 +° °°" / ° ° ' I °°+ +'° ~+ °"' /

rt and s, are the d~/mping and frequency coefficients of the oscillatory motion. r, is the damping coefficient of the "spiral" motion.

Frequency and Damping Coefficients (Conventional Aircraft) C~ = 0 .1 , v = 52, ia = 0 .0625 , ic = 0-1225.

0 . 1 2 .

Ca0

Ig. & N. ~lo. '.',074 (603~,)

A,R.O. ~eAl~mJ[ g~p~t

Publ icat ions o f the

A e r o n a u t i c a l Research C o m m i t t e e TECHNICAL REPORTS OF THE A}ERONAUT iCAL

RESEARCH COMMITTEE.. - ~934.-35 Vol. I. Aerodynamics. 4os. (4os. 8d.)

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