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CoA R E P O R T AERO No. 207
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VLtóGTUIGBOUW ICUNDEBIBLIOTHEEK
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THE COLLEGE OF AERONAUTICS
CRANFIELD
THE LATERAL DYNAMIC STABILITY OF A GROUND
E F F E C T WING
by
CoA Report A e r o No. 207 November, 1968.
THE COLLEGE OF AERONAUTICS CRANFIELD
The L a t e r a l Dynamic Stability of a Ground Effect Wing
by
P , E. K u m a r . B, Sc. (Eng), A. C. G. I.
SUMMARY
The equations of l a t e r a l motion of a wing in ground effect have been developed and the rolling r e s p o n s e r e s u l t i n g from a g r o u n d - s h a p e forcing function h a s been obtained. L i n e a r i s e d d e r i v a t i v e s at a fixed height above ground w e r e obtained from p r e v i o u s wind-tunnel t e s t s and substituted into the equations.
It was concluded that the rolling r e s p o n s e of a GEW to a h a r m o n i c forcing function, such a s a b r o a d s i d e on s e a - s t a t e , was of the s a m e frequency a s the function and that the amplitude r a t i o was dependent mainly on the values of 1 and i .
CONTENTS
S u m m a r y
1.0 Introduction
2. O G e n e r a l equations of l a t e r a l nnotion
2. 1 L i n e a r i s e d equations in ground effect
3. 0 Stability c r i t e r i a
3. 1 E x t r a c t i o n of d e r i v a t i v e s 3. 2 The coefficients of the quintic
3. 3 Case of a single wing with endplates
4. 0 Rolling with yawing s u p p r e s s e d
5. 0 Conclusions R e f e r e n c e s
L i s t of m a i n s y m b o l s — m m a s s of c r a f t
U, V, W v e l o c i t i e s in x, y, z, d i r e c t i o n s .
p, q, r angular velocities about x, y, z, wind a x e s . U, p e t c . l i n e a r and angular a c c e l e r a t i o n s r e s p e c t i v e l y . X, Y, Z f o r c e s in x, y. z d i r e c t i o n s
L , M , N m o m e n t s about x, y, z d i r e c t i o n s . y^, Ip e t c . a e r o d y n a m i c d e r i v a t i v e s .
A2, B2 e t c . coefficients of l a t e r a l stability quintic. ^A' ^c ^*^' n o n - d i m e n s i o n a l i n e r t i a s .
initial incidence.
ip yaw angle
^
bank angle
s i d e s l i p angle, (ground shape in section 4. 0).
A T t / t t m/p SU P a i r density. S wing plan a r e a . l a t e r a l r e l a t i v e density p a r a m e t e r .
^ L ' ^ D o v e r a l l lift and drag coefficients r e s p e c t i v e l y . w frequency of forcing function.
1
-1. o Introduction
Ground effect wings, if they should prove to be a p r a c t i c a l proposition, will o p e r a t e over t e r r a i n s the shapes of which will i m p a r t both longitudinal and l a t e r a l a e r o d y n a m i c loads on the craft itself. Such a condition might e a s i l y o c c u r if a GEW w e r e to fly diagonally a c r o s s oncoming waves. Reference 3, c o n s i d e r e d the longitudinal dynamic stability of a ground effect wing. This p r e s e n t note t a k e s a brief look at the l a t e r a l stability. The motions envisaged h e r e might o c c u r if the craft m e t waves b r o a d s i d e on,
2. 0 G e n e r a l equations of l a t e r a l motion.
T h e s e equations can be found in r e f e r e n c e 1, and a r e r e s t a t e d h e r e , r e l a t i v e to wind a x e s , for the s a k e of c o m p l e t e n e s s . The notation u s e d is a s given in r e f e r e n c e s 1 and 2. m { y - r V + <ÏW) = X m(V-pW + rU) = Y m(W- qU+ pV) = Z Ap - (B-C)qrfD(r2-q2) - E(pq+r) + F ( p r - q ) = L Bq - (C-A)pr+E(p2-r2) - F(qr+p) + D(pq-r) = M (1) C r - (A-B)pq+F(q2-p2) . D(pr+q) + E ( q r - p ) = N
Since we a r e only c o n c e r n e d with the l a t e r a l a s y m m e t r i c motion in t h i s p r e s e n t r e p o r t the longitudinal equations of X, Z and M will hitherto be ignored. F u r t h e r , a s s u m i n g that the GEW is s y m m e t r i c a l about its longitudinal a x i s , we can neglect the u, w and q d e r i v a t i v e s appearing in the Y, L and N equations.
The sideforce Y c o n s i s t s of gravitational and a e r o d y n a m i c components Yg and Ya r e s p e c t i v e l y , given by
Yg = mg(!/)Sine + , ^ o s e ) . . Ya = vYv + pYp + rYj. + ,^Y^ ^ ' Y J i s the contribution to sideforce due to the bank (^ in ground effect,
F r o m (1) and (2)
m(V+rU) = vYy + pYp + rY^ + (/•Y^ + mg( i^SinG + </iCose) Dividing by ft^ S and using the n o n - d i m e n s i o n a l i s e d l a t e r a l stability d e r i v a t i v e s , we get
„, d ,r N ^ ^ Pt" r b ^ , C L 4C1. ^ ( ^ - - y j + r - ^ y p - - g ^ y ^ - ^ y ^ - ^ - 2 — t a n e - 1 . ^ "
}lGn.C6
-ff-fö^v-l^p-^y^-^^^^- " ' ^ - ^ ( ^ ^ ^ = 0 (^)
The rolling m o m e n t L c o n s i s t s of a e r o d y n a m i c components only given by La = vLy + pLp + r L j . + ({L, where L r i s the rolling m o m e n t due to bank in ground effect.
Along s i m i l a r lines a s before but dividing by pU'S_b_ y^e get 2
2
-«^
ÏA dp + r iD - ^ iE + P^ i F " ^^v - pb lp - r b Ir - (fl^ = O JÜ2 dr M^ jT^ iT^ ~Tü 2TJ
S i m i l a r l y the yawing m o m e n t equation i s
ie d r - p i p - r p ijj - dp i-g - ^ny - pb np - r b nj, * n ,
(4; O (5)
2 . 1
^^2 dT ^2 ^2 dT M2 2U
L i n e a r i s e d equations in ground effect
2U
Neglecting second o r d e r t e r m s in equations (3), (4) and (5), i, e. a s s u m e s m a l l amplitude m o t i o n s r e l a t i v e to wind a x e s , we get
d^ - ^Yv - p yp - *y^ - 0 C L dr ^2 2 dp - _ l g _ ^ . ^ ^ 1 ^ . i g d-d'" ^A iA l ^ d T - i E d£_ - np ^ . ^ 2 gnv + d ? IC dT ic i c dT t a n e I r P -iA - % r • i c «^CL + 2 iA ic f ^ = O
o
(6)F u r t h e r m o r e , by definition p =_d^ and r = dip (7) dT dT
Combining (6) and (7) we get the u s u a l 3 x 3 c h a r a c t e r i s t i c stability d e t e r m i n a n t a s below, w h e r e Xs d / d r
1 ^
- ^ n ^ ic -^2 Iv iA (X-y^) !l' ic ic ic ,X -1 1./ ^o — ) - «' _£. lA - >-yp ^ CT ^^2 2 '^ lA X(X-'^r) ic -X, iEX + l r iA iA X ( l - y r ) . tang = 0 (8)(8) i s identical to the s t a n d a r d a i r c r a f t l a t e r a l stability d e t e r m i n a n t except for the n ^ , 1^ and y^ t e r m s . It m u s t be s t r e s s e d that (1, n , y ) , v a r y n o n - l i n e a r l y with height above ground a s do all the o t h e r d e r i v a t i v e s a p p e a r i n g in (8), As mentioned in ref. 3, for the longitudinal c a s e s , we c a n investigate the stability of the l a t e r a l motion by examining the r o o t s of the quintic in X , obtained from (8). using v a l u e s of the d e r i v a t i v e s at specific heights above the ground,
w h e r e
The equation obtained from (8) is of the form A2 X ^ + Bg X'i + C2 X3 + Dg X^ + Eg X + Fg = O A2 = 1 - iE^ (9) lA ic B9, = - y v ( l - i E ) - Ip - n r - iE (np + Ir) lAic lA l A i c
3 -Ir + _M2nv ( 1 - y i l - i l yp)+ ^ 2 1 v , i E ( 1 - y r ) . y ic ^ iA ^2 T A ~ ic ^"2 Ttf" - M2_ ( ^ n ^ + 1^) iA ic D2 = y v ( ^ ( l r n p - l p n r ) . i ^ ( i E - ^ - l . ) . ^ v ^ _l£ + ïAic iA ic ic iA 4 - £ £ l t a n e + i E ( ^ + y , ^ ) + _ i ^ ( i ^ y p . i p y ^ ) j .^^2_l^^(i_ECL,^„e 2 iA 2 i^M2 i ^ ic 2 . j ^ ( i - y r ) + £ l ^ + y ( ^ - £ i j ^ ) + ^ n r 1^ _ ly n^ ic ^ 2 2 ic '^ 2 iA ic ic E2 = ^ ( ^ ; 2 ^ ( l 3 ^ ) l p C L t ^ „ e + ^ ( ^ ^ y ^ ) ) -ic iA ^^2 iA 2 iA 2 ^ I v n p C L . Mgn^ . l - y r . n r , CL^.y, _ P _ t a n e - _±_L'. ll)-^ ( _ + - ' 0 ) + iA ic 2 i c 2 ic 2 + ^ 2 y v ( l r n ^ - n ^ g i A i c 2 _ ^ l i l t a n e ( n v l ^ - l y n ^ ) (10) i A i c 2 3. O Stability c r i t e r i a
F o r positive stability we r e q u i r e the coefficients A2 F2 of equation (8) to be positive. This is a n e c e s s a r y but not sufficient condition for stability, The complete s e t of conditions a r e that the t e s t functions T i 5 a s defined in section 4, 0 of ref, 3, s h a l l be positive.
3,1 E x t r a c t i o n of d e r i v a t i v e s
The r o l l and yaw d e r i v a t i v e s , 1 J , UJ,^ y and ly, ny, yy r e s p e c t i v e l y , can be deduced from the r e s u l t s of ref. 4 for specific heights above ground. The r a t e of r o l l and r a t e of yaw d e r i v a t i v e s will for the p r e s e n t , have to be e s t i m a t e d using existing methods. It m a y be possible to obtain the r a t e of yaw d e r i v a t i v e s from t e s t s on the College of A e r o n a u t i c s Whirling A r m at a l a t e r date.
F o r a plain wing with constant sganwise loading, as m a y be expected for a wing with endplates, ref, 1 givesl^, Z ^ C L , and for constant spanwise d r a g I r "" - 3 C Q . The sideforce due to r a t e of yaw is a s s u m e d to be negligible for a p l a n a r wing. The contribution to y^. thus a r i s e s from the endplates. A s s u m i n g
that the lift c u r v e s l o p e s of the wing and the endplates is the s a m e then the o v e r a l l sideforce on endplates i s :
^Y = - PUQ r b Sg aj^ a g w h e r e suffix E r e f e r s to the endplates
, ' . 6Cy = _2 r b S E aq 0 ^ , 2 w h e r e S^ i s wing a r e a 2U^ S^ a ^ y = d( ^ ^ ) . (yr)E 2yr = Sw ^1 4 S E Sw °E ai «^E
and i s a function of height since a-^ is height dependent.
S i m i l a r l y the r a t e of r o l l d e r i v a t i v e s can be e s t i m a t e d and a r e a s follows! Ip -' - 0 . 2 ( a i ) c L ( a i ) c L = 0 _ np • - 0. 2 C L + 10, 'i'^D d a ° yp = 0
F o r a typical c a s e of a single wing with endplates at a height of ho = 2, 0 2 5 " the r e s u l t s of ref, 4 yield
C L = L 4 , a = 8 ° , (ai) _ = 0,213 and (ai) _ •, . = 0,097 p e r d e g r e e ^ L ~ 0 L " ' C D = . 0 1 5 5 , (dCD/doO) ^ 0 . C L tan = . 0 9 8 , S E = 0.04, a g = 2^ 2 SyV I r
¥
Ivh
= = = = 0. -. -. -4, 467 091 135 18 n r = -,00517 np = - , 2 8 ny = +.0665 n , = - . 0 7 9 y r = -.0155 y p = 0 yv = 0 y^, = . 4 9 6 andIt should be noted that a t l a r g e r heights o r lower lift coefficients yy is negative and significajit in the c a l c u l a t i o n s ,
3, 2 The Coefficients of the quintic
In view of s o m e of the d e r i v a t i v e s being v e r y s m a l l , o r z e r o , in ground effect we can pick out the dominant t e r m s and a p p r o x i m a t e the r e l a t i o n s (10) to:
A2 = 1 B2 = -yv - i £
iA
^ 2 = y v l p - i r np + ' ^ n y - 1^ ^^2 and 1^ i s the l a r g e s t t e r m iA i A i c ic iA
5
-°2 =Ay^^<<>"-tlil ^^^^^^
iA ^A (11) E2 = - ^ 2 ^ n y l ^ - l r y y n ^ ^ 2 i A i c ^A ^c F , = - ^ 2 ^ C L ny U—*-^ iiir
AU the coefficients A2 F2 a r e positive.
3. 3 Case of a single wing with endplates
The c h a r a c t e r i s t i c equation for this c a s e , using the a p p r o x i m a t e values of the coefficients a s given by (11), is
X^ 4 .09lX^ + 4,18>-^ + .1615^2^^ + . 278^2^+ .0273^^2 = 0 (12a)
iA I A ic ic
In view of the m a s s concentration at the wingtips due to the end-plates it is r e a s o n a b l e to a s s u m e that i^. s ic = 10 say, and n 2 = 10 say, then (12a) b e c o m e s
X^ + , 0 0 9 ^ ^ + 4. 18^^ + .1615>-^ + . 2 7 8 ^ . 0 2 7 3 = 0 (12b)
The solutions of polynomial equations of the above form have been investigated in ref. 5, Since [C^ | / ' ^ | B 2 x D 2 | then according to ref, 5 Cg is pivotal and the equation above has as an approximate factor (X^ +oX + ]D) where a = B2, /3 = C2. Defining V5 = F2 = . 0273 = , 00654 ^ 4,18 V4 = E2 - 0V5 = , 278 - , 009 (, 00654) = . 0665 0385 p 4,18 Vg = D2 - 0V4 - V5 = , 1615 - , 009 (. 0665) - . 00654 ^ 4.18 V2 s C2 - aVg - V4 = 4.18 - . 009(. 0385) - . 0665 = ^ gss
^ ~ iTTs
e J = B 2 - a V 2 - V 3 = . 009 - . 009 (, 985) - , 0 3 8 5 = - , 0 3 8 4 So Ï 1-V2 = 1 - , 985 = ,015 . ' . factors of (12b) a r e (X^+, 009X+4.18)(,985X^+. 0 3 8 5 X \ 0665X+. 00654)-.0384X^+. 015X^ = 0 ignore t h e s e for an a p p r o x i m a t e solution,Hence r o o t s of q u a d r a t i c factor a r e
X, „= - , 0 0 4 5 - 2, 045i
i, Z
and the r o o t s of the cubic a r e
X„ = - , 0 6 8 3 , X4 5= +0.0146 t 0,263 i ,
It can be s e e n thus that one p a i r of complex r o o t s c o r r e s p o n d s to a damped oscillation (in r o l l ) whilst the o t h e r p a i r with a positive r e a l p a r t c o r r e s p o n d s to an undamped oscillation (in yaw). The negative r e a l root
c o r r e s p o n d s to positive s p i r a l stability. All motions a r e for s m a l l angles only.
4, 0 Rolling with yawing s u p p r e s s e d
Since a GEW will in g e n e r a l , be flying o v e r a flat t e r r a i n o r over c a l m w a t e r it would be of i n t e r e s t to e s t a b l i s h its r e s p o n s e to a v a r i a t i o n in the ground (or w a t e r ) s h a p e . In view of the waves o c c u r r i n g on the w a t e r surface we s h a l l take a p e r i o d i c change in the bank angle of the s u r f a c e , r e l a t i v e to the h o r i z o n t a l , a s the forcing function F ( T )
The r o l l equation i s
^ - ^ P = J^ E^*^) (13) dT iA iA
If the bank angle of the wing r e l a t i v e to the horizontal, i s 4. and that of the ground is ^, then
F ( T ) = ( ^ ^ ) 1 ^ (14) Hence (1) b e c o m e s dT iA iA • • • Defining ^ = dp - Ip è dT iA ^ 0 S i n WT - <^1* ^^2^ -^1(^^2 1 A iA we finally get _ d ^ - ^ p - «^1,^ _ ^ = -Aol<^. Sin WT ^ (15) dT I A iA iA T h i s c a n be r e w r i t t e n in t e r m s of 4 a s
V-_lp 'i-J^i^ = - A^ jSo ^Sin WT iA iA iA
G e n e r a l i s i n g
V+ 2k(^*+ n^4 = ^o n^Sin wr (16)
7
-Equation (16) i s the s t a n d a r d equation for a s p r i n g - m a s s s y s t e m , with v i s c o u s damping, subjected to a forcing function, and its solution is well known, conditions w h e r e Putting we get 4 = g n n ^ S i n ( w T -& ) -kT rj(n2-w2)^+ 4 k 2 w 2 ' -1 2 2 / 2 2^ e = tan 2kw/(n -w ) and x = /yn -k
e ( A Cos XT + B Sin XT )
(18) K = n 2 / ^(n^ 2>^, .._2__-2.
w ) + 4k w with the boundary
(i) at T = 0, (^ = 0 (ii) T=0, ' ^ - = 0 A = K^oSinG B = KjSoSine (w +k -x ) 2kx (19) Hence 4 = KSin(wT-6 ) + K S i n 6 e " ^ ( C o s x T + (w^+k - x ^ ) S i n x T ) (20) ^o 2kx
Equation (20) i n d i c a t e s that the motion of the GEW will be always be o s c i l l a t o r y asympoting to 4 = K S i n ( w T - b). Since Ip <^< 1^ the damping is low
" ^ 2 and in the limiting c a s e of Ip ^ 0, k _ ^ 0 and '^ = n
'-0 n -w Resonance o c c u r s when w = n
i. e, when w = ^A
8
-5,0 Conclusions
The full l a t e r a l equations of motion of a GEW have been developed and it h a s been shown that, in g e n e r a l , both rolling and yawing oscillations will o c c u r a s a r e s u l t of a l a t e r a l change of ground shape, o r a l a t e r a l displacement, Whether t h e s e o s c i l l a t i o n s a r e damped o r not depends upon the height and incidence of the craft. Since the a e r o d y n a m i c c h a r a c t e r i s t i c s a r e n o n - l i n e a r with height, only the rolling (or yawing) motion at a fixed height h a s been c o n s i d e r e d in o r d e r to u t i l i s e the l o c a l i s e d v a l u e s of the stability d e r i v a t i v e s obtained from wind-tunnel t e s t s .
The rolling motion of the single wing in ground effect is o s c i l l a t o r y and of the s a m e frequency a s the forcing function (waves over w a t e r e t c . ) and the amplitude r a t i o depends upon the v a l u e s of Ip and 1^ . N e a r the ground Ip <[< l,f and the neglect of the damping t e r m in the r o l l equation yields the amplitude r a t i o a s being dependent on
, 2 l / ( l + w i A )
^ 1,^
In p r a c t i c e a GEW will probably be in the form of a tandem wing and consequently the d e r i v a t i v e s used in the p r e s e n t analysis will differ, r e s u l t i n g in a different r e s p o n s e , A fuller study using an analogue c o m p u t e r and the n o n - l i n e a r a e r o d y n a m i c c h a r a c t e r i s t i c s of the GEW would probably be beneficial once the basic configuration of the craft had been fixed.
in ground effect.
One significant fact e m e r g i n g is the n e c e s s i t y to evaluate Ip s a t i s f a c t o r i l y
R e f e r e n c e s
1. B a b i s t e r ,
Bryant and Gates
A i r c r a f t stability and control, P e r g a m o n P r e s s , 1961
Nomenclature for stability coefficients. A. R. C. R. & M. 1801 1937,
P . E, K u m a r , On the longitudinal dynamic stability of a G. E. W.
CoA r e p o r t a e r c 202, 1968.
4, P . E, K u m a r .
Hopkin
An e x p e r i m e n t a l investigation of the a e r o -dynamic c h a r a c t e r i s t i c s of wings, with and without endplates, in ground effect.
C o A r e p o r t a e r o 201, 1968.
Routine computing methods for stability and r e s p o n s e investigations on l i n e a r s y s t e m s . R. & M, 2392, 1950.
