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A C C E L X R A T I OYS I N L A N D I N G 7iI TH A T R I C Y CLE-TYPX L A N D I N G GEAR By R o b e r t T. J o n e s I n c o n n e c t i o n w i t h t h e a p p l i c a t i o n of s t a b l e t r i - c y c l e - t y p e l a n d i n g g e a r s t o t r a n s p o r t a i r p l a n e s , t h e ques- t i o n a r i s e s a s t o w h e t h e r c e r t a i n of t h e p a s s e n g e r s may n o t e x p e r i e n c e r e l a t i v e l y g r e a t a c c e l e r a t i o n s i n a n emer- gency l a n d i n g . S i n c e t h e main l a n d i n g w h e e l s a r e b e h i n d t o e c e n t e r of g r a v i t y i n t h i s t y p e of g e a r , a h a r d - b r a k e d l a n d i n g . w i l l c a u s e i m m e d i a t e n o s i n g down o f t h e a i r p l a n e a n d , when t h i s motion i s s t o p p e d d u e t o t h e f r o n t wheel s t r i k i n g t h e g r o u n d , t h e r e w i l l be some t e n d e n c y f o r t h e r e a r m o s t p a s s e n g e r s t o b e thrown o u t of t h e i r s e a t s . The f o l l o w i n g r o u g h c a l c u l a t i o n s a r e d e s i g n e d t o show t h e m a g n i t u d e s of t h e v a r i o u s r e a c t i o n s e x p e r i e n c e d i n a s e - v e r e l a n d i n g u n d e r t h e s e c i r c u m s t a n c e s , DEFINITIONS OP SYidBOLS ( s e e f i g u r e 1 )
Axes .- The a i r p l a n e a x e s a r e f i x e d a t t h e c e n t e r of
g r a v i t y a n 4 a r e c h o s e n p a r a l l e l a n 4 p e r p e n f i i c u l a r t o t h e g r o u n d a t t h e i n s t a n t of l a n d i n q a s shown i n f i g u r e 1, 6 , a n g l e of p i t c h of a x e s r e l a t i v e t o t h e i r i n i t i a l p o s i t i o n on la.n<ine: ( p o s i t i v e n h e n a i r p l a n e i s n o s i n e u p ) Uo, f o r w a r d v e l o c i ' t y of a x e s on l a n d i n g (alonrg X a x i s ) u , c h a n g e i n f o r w a r d v e l o c i t y d u r i n ~ l a n d i n g w, v e r t i c a l d e s c e n t v e l o c i t y of a x e s d u r i n g 1a.nding ( a l o n g Z a x i s ) iT
m, mass o f a i r p l a n e i n s l u g s = -
g m k y , moment of i n e r t i a of a i r p l a n e i n p i t c h i n g M , p i t c h i n g moment a b o u t c , g , of a i r p l a n e b y w i n g s p a n .
S , w i n g a r e a c , w i n g c h o r d f , d i s t a n c e , f r o m a i r p l a n e c , q . t o ta.i 1 p a s t X F , W . , d i s t ~ . n c e f r o m a i r p l a n e c . g . t o f r o n t w h e e l X R e p . , d i s t a n c e from a i r p l a n e C . Z . t o r e a r m o s t p a s s e n a e r h , h e i q h t of c . g , a b o v e g r o u n d ( s h o c k a b s o r b e r ex- ten4e.l) , c o e f f i c i e n t of s l i d i n g f r i c t i o n of t i r e s on l a n d - i n g s u r f a c e ASSUiilED L A N D I N G C O N D I TIONS a ) The a i r p l a n e s t r i k e s t h e p r o u n i a t a v e r t i c a l de- s c e n t v e l o c i t y of w f e e t p e r s e c o n ? a n d w i t h a f o r w a r d v e l o c i t y of Uo f e e t p e r s e c o n d . The w h e e l s a r e c o n s i d - e r e d t o b e l o c k e d a n d t h e l a n d i n g a t t i t u d e a s i n f i g u r e 1.
The maximum v e r t i c a l a c c e l e r a t i o n e n c o u n t e r e d i s D w m x a n d t h e maximum b r a k i n g a c c e l e r a t i o n i s Dumax - -...
b o t h a p p l i e d a t t h e p e i n t s of c o n t a c t of t h e
c1 (DW - g)max
t h e main l a n d i n g w h e e l s w i t h t h e g r o u n d .
b ) The motion of the a i r p l a n e a f t e r s t r i k i n g t h e g r o u n d w i l l c o n s i s t of combined p i t c h i n z a n d v e r t i c a l movement. The v e r t i c a l movement of t h e c.g. i s c o n s i d e r e d t o b e p r e d e t e r m i n e d by the a c t i o n of the main shock ab- s o r b e r a n d t o c o n s i s t of a p r a c t i c a l l y c o n s t a n t d e c e l e r a - t i o n of t h e v e r t i c a J 4 e s c e n t v e l o c i t y w .
Owing t o t h e c l e f l e c t i o n of t h e t i r e s a n d i m p e r f e c t i o n i n t h e a b s o r b e r d e v i c e t h e r e w i l l , h o v e v e r , be some d e l a y i n t h e a t t a i n m e n t o f t h e f u l l d e c e l e r a t i o n a f t e r l a n d i n g .
T h i s d e l a y i s assumed t o b e r e p r e s e n t e d b y a s i m p l e f o r - m u l a , v i z : / I f t h e a i r p l a n e r e m a i n e 3 i n t n e a t t i t u Z e n e p i c t e d i n f i g u r e 1 t h r o u g h o u t t h e i n t e r v ~ l of a b s o r p t i o n of t h e s h o c k a f t e r lanrl.ine;, e v e r y p a r t of t i l e machine ~01114 ex- p e r i e n c e t h e same 4 e c e l e r a t i o n Ow a s t h e c e n t e r of ~ r a v . i t y . However, s i n c e t h e b r ~ k i n . 7 l.oarl m D i 1 i s a p p l i e d some d i s t a n c e below t h e c . g . , t h e a i r p l a n e w i l l a l s o ex- p e r i e n c e a n a n g u l a r a c c e l e r a t i o n on s t r i k i n e : t h e . e r o u n d e a u a l t o n e e l e c t i n e t h e v a r i a t i o n i n h Aue t o t h e d e f l e c t i o n of t h e s k o c k a b s o r b e r . T h i s a c c e l e r n . t i o n w i l l c a u s e a r o - t a t i o n i n p i t c h , r e s u l t i n e i n p a r t s of t h e a i r p l a n e be- h i n t t h e c . e . b e i n g a c c e l e r a t e d upwar* r e l a t i v e t o t h e c . ~ . a n d c o n s e q u e n t l y r e c e i v i n q a e r s a t e r s h o c k d u r i n g t h e l a n d i n g . I n a 4 2 j . t i o n t o t h i s , t h e p i t c h i n g a n g u l a r v e l o c i t y a c q u i r e d d u r i n g t h e p e r i o d o f a b s o r p t i o n of t h e s h o c k a n 4 b e f o r e t h e f r o n t w h e e l s t r i k e s t h e g r o u n d must b e reiiuced t o z e r o b y a s h o c k a b s o r b e r a t t a c h e 4 t o t h e .
f r o n t w h e e l . T h i s l a t e r d e c e l e r a t i o n i n p i t c h i n g , s u b s e - q u e n t t o t h e main l a n d i n g s h o c k , t e n 4 s t o l i f t t h e r e a r p a s s e n g e r s f r o m t h e i r s e a t s a n d on t h i s a c c o u n t s h o u l d n o t b e a l l o w e d t o e x c e e d 3 2 . 2 f e e t p e r ( ~ e c o n 4 ~ .
I t ill b e n e c e s s a r y t o c a l c u l a t e t h e a n g u l a r v e l o c - i t y and a c c e l e r a t i o n i n p i t c h i n a a c q u i r e d d u r i n ~ t h e pe- r i o d of a b s o r p t i o n of t h e main s h o c k i n o r d e r t o f i n d t h e r e a c t i o n s e x p e r i e n c e 4 by t h e p a s s e n e e r s i n t h e r e a r s e a t s .
T h i s p i t c h i n g motion i s a t f i r s t c a u s e 4 by t h e b r a k i n g l o a d z p p l i e d a t t h e w h e e l s b u t , a s t n e motion commences, i s m o d i f i e d by s e v e r a l s e c o n d a r y f a c t o r s 41ie t o t h e mo- t i o n i t s e l f , a s f o l l o w s : a ) A p i t c n i n ~ rnoiaent 4ue t o c h a n g e i n a n e l e of p i t c h a f t e r l a n d i n g a r i s i n g p a r t l y f r o m t h e s t a t i c s t a b i l i t y o f e3 i n p i t c h a n d p a r t l y f r o m t h e d i s p l a c e m e n t o f t h e a i r p l a n e t h e c . ~ , of t h e a i r p l a . n e away f r o m t h e p o i n t o f s u p p o r t .
The p a r t of t h i s s e c o n 4 a r y moment n r j s i n e f r o m t h e a e r o d y - namic e f f e c t i s c a l c u l a t e 4 a s : v h e r e t h e c h a n c e i n a n g l e of a t t a c k a i s assumed t o b e e q u a l t o t h e change i n p i t c h a n g l e b s i n c e t h e a i r p l a n e i s s e n s i b l y c o n s t r a i n e d t o t r a v e l a l o n g t h e g r o u n 4 a f t e r l a n d i n ? . The p a r t of t h e moment a r i s i n q from t h e d i s - p l a c e m e n t o f t h e c . g . i s :
id2 = - m(Dw - g ) h s i n t (3)
The v a r i a t i o n of (DW - g ) w i t h t i m e , e i v e n by e q u a t i o n ( 1 )
w i l l b e a p p r o x i m a t e d i n computing t h i s s e c o n d a r y e f f e c t by s i m p l y t a k i n g 90 p e r c e n t of t h e maximum v a l u e . The d i f f e r e n c e between s i n b a n d b w i l l a l s o b e n e g l e c t e d , F r e s u l t i n g i n A c t u a l l y t h e l e n g t h h w i l l b e d e c r e a s e d somewhat due t o t h e d e f l e c t i o n of t h e s h o c k a b s o r b e r a n 4 t h e a b o v e e s t i - mate of Mb2 may b e c o n s i d e r e d c o n s e r v a t i v e . The t o t a l s e c o n d a r y e f f e c t clue t o r o t a t i o n i n p i t c h i s : b ) A p i t c h i n a moment due t o t h e a e r o d y n a m i c 4amping o f the p i t c h i n q m o t i o n , T h i s moment i s c a l c u l a t e 4 by u s - in,c t h e c u s t o m a r y aer.odysamic f o r m u l a : where t h e s u b s c r i p t s t r e f e r t o t h e h o r i z o n t a . 1 t a i l W p l a n e .
* 7 i t h t h e a , i d o f t h e s e f a c t o r s d e f i n i n e t h e s e c o n d a r y e f - f e c t s d u e t o motion t h e p i t c h i n e m o t i o n due t o a n y t o r q u e a p p l i e d by b r a k i n e : a t t h e w h e e l s may b e c a l c u l a t e d . The e q u a t i o n of m o t i o n i s : i s t h e a p p l i e d t o r q u e a n d i s assumed t o where mky2 X Yo b e known i n t e r m s of t h e t i m e . F o l l o w i n g t h e assumed l a w of v a r i a t i o n of s h o c k - a b s o r b e r r e a c t i o n ( e q u a t i o n ( 1 ) ) a n d c o n s i d e r i n g t h e a p p l i e d t o r q u e a s b e i n g due t o t h e b r e a k - i n g r e a c t i o n a t t h e w h e e l s w e have: GENFRAL SOLUTION OF E Q U A T I O N OF M O T I O N E q u a t i o n (8) may b e i n t e g r a t e d i n g e n e r a l t e r m s and a c t u a l n u m e r i c a l v a l u e s f o r t h e v a r i o u s q u a n t i t i e s may b e s u b s t i t u t e 4 i n t o t h e g e n e r a l s o l u t i o n l a t e r .
/ The e a s i e s t way t o i n t e g r a t e t h i s equa.tion i s t o make u s e of t h e H e a v i s i d e e x p a n s i o n t h e o r e m . The f i r s t s t e p i s t o o b t a i n a n a l e e b r a i c r e s o l u t i o n f o r 6 , v i z , 1 1
--- iho = --
b . = -
(10) (Dl M o
n2 - M q D - I $
According to equation (9) Mo is given by two, terms. The distribution of the operator over these terms re- sults in: The expansion of the first term is and of the second te~rm h t -nt e 1 e -nt
--- = ---- C --
e + A F (D) F (-n)
(A + n) F t (A)
where the A's are the r o o t s , o f the auxiliary equation P(h) = 0.
In the present problem The substitution of these roots into the expansions (12) and (13) results in: where b, is the anqle of pitch a t any ihstant under the condition of a sud?en impact, that i s , under.the condition t ha t i,i reaches its full value instantaneously, and Omax f r o m 6 , to t 2 r e p r e s e n t s the amount'to be subtracted account for the delay in the full action of the shock ab- sorber: The motion represented by the foregoing equations is to b e continued only so long a s is required for the land- ing gear to absorb the impa.ct .of the landing. If wo is the vertical descent velocity on landing, this time is found from t 1 w O = - $ D w d t .
(18) The total vertical stroke of the shock absorber necessary to absorb the vertical velocity wo with the specified acceleration' is : Silbsequent to the time (tl) of absorption of the vertical descent velocity w o , the loaas at the wheels will b e much reduced a n d a new'equation of motion with 4ifferent values of i d and Mo will be i n force. The angular ve.loci ti,es a n 3 di splacements calculate4 from the original equations at the time i i = t l should be taken a s initial conaitions in the new equation. Presumably, the time of absorption o f the landine: shock will not be sufficient for the front. wheel 'to strike the ground 'and: the conditions un4er the new equAtions of motion will be extended to that time, In the n-~merica.1 calculations that fcllow it was foun4 that the equations for the sec- ond phase of the motion after landing represented a con- tinued rotation in pitch at a practically constant angu- l a r v e l o c i t y , t h e a e r o 4 y n a m i c s t a b i l i t y a n 4 d a m p i n e e f f e c t s ., n e a r l y . c o u n t e r b a l a n c i n c t h e r e d u c e d b r a k i n g d n 4 n o s i n g - o v e r moments. For t h i s r e a s o n i t w a s a s s u s e d t h a t t h e a n g u l a r v e l o c i t y i n p i t c h a t t h e i n s t a n t t h e f r o n t w h e e l s t r u c k t h e \ ground w a s t h e same a s t h a t a c q u i r e d d u r i n g t h e shock-ab- s o r p t i o n p e r i o 4 .
CALCULATIONS FOR AIRPLANE A .l) Assumed s p e c i f i c a t i o n s of a i r p l a n e : w = 1 8 , 0 0 0 l b .
b = 80 f t .
S = 9 3 9 s q . f t .
c = 11,8(i3 f t .
f = 3 6 . 6 f t .
XYJ. = 1 4 f t .
X R . p , = 1 4 f t .
h = 8.4 f t .
ky = 1 1 . 7 5 f t .
S t = 1 5 4 . 6 s q . f t .
2) A t a l i f t c o e f f i c i e n t of a p p r o x i m a . t e l y 2 . 0 t h e l a n d i n q s p e e d w i l l b e 6 0 m i l e s p e r h o u r . .
3) Assume a v e r t i c a l d e s c e n t v e l o c i t y of w o = 1 5 f e e t p e r s e c o n d a t t h e i n s t a n t of l a n ? i n c a n d assume t h a t t h e maximum v e r t i c a l a c c e l e r a t i o n e n c o u n t e r e d a f t e r l a n d - i n g i s 4) Assume t h a t t i l e l a w e x p r e s s i n e t h e b u i l d i n g up of a c c e l e r a t i o n a f t e r t h e i n s t a n t of c o n t a c t w i t h t h e <round i s ( s e e f i g u r e 2) T h i s f o r m a l a i s t a k e n t o r e p r e s e n t r o u g h l y t h e con- d i t i o n s o b t a i n e d w i t h a n o r d i n a r y o l e o l a n d i n g g e a r a n d was d e v i s e d a f t e r e x a m i n a t i o n of some e x p e r i m e n t a l r e c o r d s o b t a i n e d i n d r o p t e s t s of m i l i t a r y a i r p l a n e s .
5) C a . l c u l a t i o n of ivib a n d Mq: a ) ivi6 (See e q u a t E o n s 2 a n d 4 . ) T o t a l M b = 6 . 5 b ) M q ( s e e e q u a t i p n 7 . ) Assume t h a t t h e s l o p e of t h e t a i l - p l a n e l i f t c u r v e w i t h a n e l e of a t t a c k i s 4 . 0 .
-36.G2 X 4 X 1 5 4 . 6 X 0012 X 5 8
= ------.-- --- -
- -1 - 1 3
(23) 9 k 77200 6) If t h e l a n d i n g i s made w i t h t h e b r a k e s l o c k e d a n d t h e c o e f f i c i e n t of s l i d i n g f r i c t i o n of t h e w h e e l s on t h e l a n d i n g s u r f a c e i s o n e - h a l f , t h e a p p l i e d a n g u l a r a c c e l e r a - t i o n due t o t h e b r a k i n g i s :
- 2 3 t
= - .3.92 (1 - e ) ( ~ . e e e q u a t i o n 9.) (24)
( 1 t i s t o be n o t e 4 t h a t t h e v a l u e of t h e c o e f f i c i e n t of f r i c t i o n assumed may b e l e f t u n d e t e r m i n e d up t o t h e p o i n t of t h e f i n a l solution.! .
7 ) The e q u a t i o n o f m o t i o n w i t h t h e foreqoirne: q u a n t i t i e s s u b s t i t u t e d f o r i , i q , M o i s : The r o o t s o f F ( D ) = 0 , 'as r e q n i r e d b y e q u a t i , o n s (15) . a n d , , ( I S ) , a r e
------
- 1.13 1 : J l . l 3 ~ + 4 X 6 . 5 .
= --
8 ) The f i n a , l s o l u t i o n , ' c o n s i ' s t i n e of two p a . r t s a.s i n e q u a t i o n s ( 1 5 ) a n d ( I S ) , i s w h e r e bl i s t h e a n ~ l e of p i t c h a t t a i n e d a t t h e t i m e t if t h e l a n d i n g s h o c k i s i n s t a n t a n e o u s . 6 2 , o r t h e a n g l e t o b e s u b t r a c t e ? f r o m b 1 t o a c c o u n t f o r t h e d e l a y i n t h e f u l l s h o c k - a b s o r b e r a c t i o n i s : The c o m p l e t e s o l u t i o n f o l l o w s as The a n g u l a r v e l o c i t y a n d a c c e l e r a t i o n a t a n y i n s t a n t a r e f o u n 4 b y d i f f e r e n t i a t i n g t h i s e q u a t i o n : A t t h e t i m e t = 0 eKt = 1 , s o t h a t t h e a s s u m e d i n i t i a l c o n d i t i o n s on l a n d i n g s h o u l d b e g i v e n b y t h e sums o f t h e c o e f f i c i e n t s o f e i n t h e a b o v e s o l u t i o n s r . T h u s t .
w h i c h i s a c h e c k o n t i e s o l u t i o n of t h e e a u a t i o n s .
The m o t i o n r e p r e s e n t e r i . b y t h e f o r e e o i n e : e q u a t i o n s i s t o b e c o n t i n u e d o n l y s o l o n e a s i s r e a u i r e d f o r t h e l a n d - i n ? g e a r t o a b s o r b t h e i m p a c t o f t h e a i r p l a n e . T h i s t i m e i s f o u n d f r o m t l = 0 . 2 s e c o n d , v e r y n e a r l y X a v i n g f o u n d t h e t i m e o f a b s o r p t i o n o f t h e s h o c k , t h e t o - t a l v s r t i c a l s t r o k e of t h e s h o c k a b s o r b e r may b e f o u n d b y i n t e g r a t i o n ; t h u s o r , A h = 1 . 7 6 f e e t , a p p r o x i m a t e l y I t w i l l b e ~ ~ e c e s s a r y t o c a l c u l a t e b , q , a n d D q a t 0 . 2 secon-3 f r o m t h e e q u a t i o n s ~ i v e n : ( S e e e ~ u a t i o n s ( 2 9 ) a n d ( 3 0 ) .)
= - 3 . 2 4 ' ( d e g r e e s )
-4.6 - 0 . 6 3 2 0 . 4 0 6
= + . I 8 2 2 + 0 . 8 7 4 e - 0 . 6 9 4 e
q 0 . 2 ( 3 5 )
= - 3 . 5 4 r a d i a n s / s s c o n d / s e c o n d
On s , u b s t i t u t i n ~ t h e s e i n i t i a l c o n d i t i o n s i n t o t h e now e a i l a t i o n o f m o t i o n i t i s f o u n 3 t h a t t h e a e r o 4 y n a m i c s t a b i l i t y an4 d a m p i n e moment n e a r l y b a l a n c e t h e now much- r e d u c e d b r a k i n e a n d n o s i n g - o v e r m o p e n t s . F o r t h i s r e a s o n i t w i l l b e s u f f i c j e n t l y a c c ~ r r a t e t o n e ~ l e c t a n y c h a n g i n g c o n c t i t i o n s t h e r e a f t e r a n d t o a s s u m o t h a t t h e a b o v e a n q u l a r v e l o c i t y p e r s i s t s u n t i l t h e f r o n t w h e e l s t r i k c s t h e q r o u n d a t t, = - 1 5 ' . T h i s a n c u 1 a . r v e l o c i t y ( o f 0 . 5 7 7 r a , . i i a n / s e c o n 3 ) p r o Z u c e s a.n u p w a r d v a l o c i t y of 0 . 5 7 7 X 1 4 f t . = 8 . 1 f t . / s c c .
( 3 6 ) a t t h e r e a r m o s t p a . s s e n e e r l s s e a t . I f s t o p p e d s u l d e n l y a t t h i s s v e a d t h e s e a t w o n 1 4 c a u s e t h e p a s s e n g e r t o r i s e 1 . 0 2 f c e t . A t c o n s t a n t + e c e l e r n t i o n , t h e f r o n t - w h e e l s h o c k a b s o r b e r m u s t t r a v e l a p p r o x i m a t e l y X F a g . / x E a P , of t h i s l i s t a n c e t o p r e v e n t t h e p a s s e n g e r f r o m l e a v i n g t h e s e a t ; h e n c e , i t i s c o n c l u d e d t h a t t h e f r o n t - w h e e l s h o c k a b s o r b e r s n o u l Z h a v e n t r a v e l of s l i g h t l y o v e r o n e f o o t ( i n c l u d i n g t i r e 3 e f l e c t i o n ) i n o r d e r t o a c c o m o d a t e t h e a b o v e c o n d i t i o n s w i t h a s a f e a l l o m a n c e f o r i n e f f i c i e n c y o f o p e r a t i o n ( t n a t . i s , d e v i a t i o n f r o m c o n s t a n t a c c e l e r a - t i o n ) .
A s s u m i n e t n a t t h e l a w e x p r e s s i n g t h e r i s e of s h o c k - a b s o r b e r l o a d w i t h t i m e i s t h e same f o r t h e f r o n t w h e e l a s f o r t h e r e a r , t h e m o r e a c c u r a t e v a l u e f o r t h e t r a v e l o f t h e f r o n t - w h e e l s y s t e m i s : ( c h o o s i n g a n e n o r i g i n f o r t h e t i m e s c a l e ) The l i m i t o f i n t e g r a t i o n t l i s f o u n 4 f r o m t h e known ve- l o c i t y b y p e r f o r m s n g a n ad4.i t i o p a l i n t e z r a t i o n , t h u s - 2 3 t
- 8 . 1 = f t l - 3 2 . 2 (1 - s (3 5 )
) a * , C !
w h e n c e t = 0 . 2 9 5 s e c o n d , a n d t h e d i s t a n c e a.pproxima. t e l y .
CALClJLATIONS FOR A I R P L A I t E B 1) A s s u m e d s p e c i f i c a t i o n s o f a i r p l a n e : w = 5 4 , 0 0 0 l b .
b = 1 3 9 f t .
e t c .
A i r p l a n e B i s s i m i l n r t o a i r p l a n e A e x c e p t t h a t t h e w e i g h t a n d w i n g a r e a a r e i n c r e a s e d by' t h e f a c t o r 3 . The moment o f i n e r t i a i s i n c r e a s e d b y 3 X 3 = 9 , r e s u l t i n g i n t h e f o l - l o w i n g m o d i f i c a , t i o n s t o t h e q u a n t i t i e s o c c u r r i n e i n t h e e q u a t i o n s of m o t i o n : The e q u a t i o n of m o t i o n b e c o m e s : The s o l u t i o n o f t h i s e q u a t i o n , o b t a i n e d i n t h e same way a s f o r a i r ~ l a n e A , i s : - 2 3 t -2 r 58 t 1 . 4 5 t
6 = 0 . 6 0 2 + 0 . 0 0 4 6 e - 0 . 2 4 5 6 e - 0 . 3 6 6
- 2 i 5 8 t 1 . 4 5 t
+ 0 . ~ 3 3 e
q = - 0 . 1 0 6 e - 0 . 5 3 1 e
(42) S i n c e t h e l a n d i n g c o n d i t i o n s a r e t h e same a s b e f o r e , t h e r e q u i r e d s t r o k e o f t h e s h o c k a b s o r b e r i s t h e same a n d t h e t i m e t a k e n t o a b s o r b t h e i m p a c t i s l i k e w i s e 0 . 2 s e c - o n d . The f o l l o w i n g t a b l e l i s t s t h e i m p o r t a n t r e s u l t s o f t h e c a l c u l a t i o n s i n t h e two c a s e s .
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2) iviaximum a n g u l a r v e l o c i t y
[ q ( t = 0 . 2 ) 3
3) ' i~faximum v e r t i c a l r e a c t i o n a t
r e a r s e a t .
4 ) Maximum r i s i n q v e l o c i t y a t r e a r s e a t 5) T r a v e l o f main s h o c k a b s o r b e r . . ( i n c l u r i . i n q t i r e ) 6 ) T r a v e l o f f r o n t s h o c k a b s o r b e r ( i n c l u d i n q t i r e ) n e c e s s a r y t o p r e v e n t r e a r p a s s e n g e r f r o m l e a v i n e s e a t X o t e : The v a l u e g i v e n h e r e i s f o r
(3w - 3 ) + X R . - p . Dq
DISCUSSION A N D CONCLUSIONS The f o r e g o i n ? c a l c u l a t i o n s show t h a t i n l a n d i n g u n d e r t h e assumed c o n d i t i o n s t h e r e a r m o s t p a s s e n e e r r e c e i v e s a somewhat g r e a t e r p r i m a r y l a n d i n g r e a c t i o n t h a n would b e e x p e c t e ' l i n t h e c a s e o f t-he c o n v e n t i o n a l u n s t a b l e t y p e l a n d i n q g e a r u n d e r t h e same c o n d i t i o n s (5.5 a a t t h e r e a r > s e a t compared w i t h 4 g a t t h e c e n t e r of g r a v i t y ) . I n t h e c a l c u l a t i o n s no a c c o u n t was t a k e n of s e a t c u s h i o n i n e , n o r o f a n y f l e x i b i l i t y of t h e s t r u c t u r e of t h e a i r p l a n e a n d , s i n c e t h i s a c c e l e r a t i o n r e a c h e s i t s maximum a n 4 d i s a p p e a r s w i t h i n t w o - t e n t h s of a s e c o n 3 , i t may b e c o n c l u d e d t h a t i t would n o t r e s u l t i n i n j u r y o r g r e a t d i s c o m f o r t t o t h e p a s - s e n g e r . The c o n d i t i o n s assumed c o r r e s p o n a t o a , s e v e r e l a n d i n g t h a t s h o u l d b e e n c o u n t e r e d o n l y i n a n emerqency a n d i t may b e assumed t n a t t h e a d d i t i o n a l a c c e l e r a t i o n due t o t h i s l a n d i n y - g e a r d e s i g n would n o t b e n o t i c e a b l e u n d e r o r d i n a r y c i r c u m s t a n c e s .
The c a l c u l a t i o n s f u r t h e r showed t h a t a c e r t a i n m i n i - mum s h o c k - a b s o r b e r t r a v e l f o r t h e f r o n t wheel mas n e c e s - s a r y t o p r e c l l l 4 a t h e p o s s i b i l i t y 0 2 l i f t i n g t h e rea.rmost p a s s e n g e r f r o 3 h i s s e a t . T h i s n e c e s s a r y t r a v e l d e p e n d s on t h e p l a c i n c o f t h e f r o n t a h e e l r e l a t i v e t o t h e C . R .
a n d r e l a t l v e t o t h e b r c k m a r d d i s t a n c e of t h e r e a , r m o s t p a s s e n e e r f r o m t h 3 c . e . , a n 3 i s l e s s e n e l n h e n b o t h d i s - t a n c e s a r e made s h o r t e r . I t may b e c o n c l u d e ' l , h o w e v e r , ' t h a t i n most c a s e s t h e s t r o k e r e q x i r e 4 of t h e f r o n t s h o c k a b s o r b e r m i l l b e l e s s t h a n t h a t of t h e r e a r o n e s .
The p o s s i b i l i t y of t h e r e a r m o s t p a s s e n g e r s b e i n g thromn n p ~ a r c l . when t h e f r o n t w h e e l s t r i k e s t h e a r o u n d de- p e n d s on t h e e x i s t e n c e of a l a r g e b r a k i n q e f f o r t a t t h e main l n n 4 - i n e wheels. T h i s b r a k i n q e f f o r t a l s o t e n d s t o : * c a u s e t n e p a s s e n g e r t o s l i d e f o r w a r d o u t of t h e s e a t .
S u p p o s e 4 l y t h e p a s s e n g e r i s r e s t r a i n e d b o t h f r o m s l i d i n g f o r w a r t a n 3 from r i s i n g upward by a s a f e t y b e l t . I t i s t o be n o t e d t h a t t h e b r a k i n r ? : 4 e c e l e r a t i o n assumed i n t h e p r e s e n t a n a l y s i s mas of s u c h m a g n i t u d e t h a t t h e p a s s e n - g e r s were more l i k e l y t o b e thrown f o r w a r d a.nd j n j u r e d on t h i s a c c o u n t d i r e c t l y t h a n t o b e thrown upmasd a n d i n - j u r e d a s a r e s u l t of t h a s u b s e q u e n t c h e c k i n g of t h e nos- i n g - o v e r m o t i o n s . I t may b e s u p p o s e d t h a t t h i s b r a k i n g d e c e l e r a t i o n w i l l b e p r e s e n t u n d e r s i m i l a r c o n d i t i o n s i\ a w i t h a n y d e s i g n of l a n d i n g g e a r a n d , s i n c e t h e s e c o n d a r y v e r t i c a l r e a c t i o n t h a t a r i s e s when t h e f r o n t w h e e l s t r i k e s t h e ground i n t h e c a s e of t h e s t a b l e - t y p e Kear is o r d i n a r i l y m u c h l e s s t h a n the b r a k i n g r e a c t i o n , it is concluded that there s h o u l d be n o d i f f i c u l t y w i t h the stable t r i c y c l e l a n 4 i n g gear on t h i s a c c o u n t .
, L a n g l e y idemorial A e r o n b u t ical L a b o r a t o r y , N a t i o n a l A d v i s o r y Committee for A e r o n a u t i c s , L a n g l e y F i e l d , Va.
N.A. C , A .
F i g s . 1 ,% Figure 1.- I n i t i a l l a n d i n g c o n d i t i o n s : O e f i n i t i o n s o f symuols and axes.
( 3 o t e : c o n t a c t ~ 4 t h ground made a t t = O ) 0 - 0 4 .08 .12 .16 .20 T i m e , second , F i b u r e 2.- Represgntation o f s h o c k - a ~ s o r b i n g l a g -23t \ u y formula: (Dw-g)--4g (1-e