APPENDIX A
APPENDIX A DETERMINATION OF LANDING-GEAR DEFLECTION The v e r t i c a l distances from the ground plane of the center of gravity and Since the r a t e of change of height of the main gear point of attachment with respect t o the ground i s equal t o the rate of gear v e r t i c a l deflection, ~ l t h a t is, I I I 1 ; ; = 6M* I ' the following r e l a t i o n i s valid:
% = io - 1% cos 0 - (d cos cp + S2 s i n cp s i n 8 8
i '
cos cp - d s i n
I The r a t e of v e r t i c a l deflection f o r t h e l e f t main gear is: L -
cos cp - S1 s i n
+ ( S l cos cp + d s i n cp @ cos 0
)
APPENDIX B
APPENDIX B TRIGONOMETRIC RELATIONS FOR THE ANGLE h The angle i n the ground plane between the xo-axis and the f r i c t i o n force on a skid i s denoted by h i where the subscript i indicates the skid being con- The f r i c t i o n force on each skid w i l l be i n a d i r e c t i o n sidered ( s e e f i g . 1).
skid r e l a t i v e t o p a r a l l e l and opposite t o the direction of the velocity of the the ground. If the magnitude of the skid velocity i s known, it i s possible t o determine the trigonometric r e l a t i o n s f o r the h i angles.
a vehicle as measured i n the The r e l a t i v e velocity of any position on ground plane i s expressed as: The ground velocity components VLo and V i o a r e r e l a t e d t o body axes by where
Vxi I = u I + qZi - ryi
I
= v I + r x i - pzi
'Yi The components of the r e l a t i v e velocity of the center of gravity a r e denoted by the primed quantities. I n order t o determine these velocities, it i s f i r s t necessary t o investigate the r e l a t i o n s h i p of the vehicle r e l a t i v e velocity, center-of-gravity r e l a t i v e velocity, and wind velocity, which i s where - V i s the velocity vector of t h e vehicle center of gravity r e l a t i v e t o the airstream -I V i s t h e velocity vector of t h e vehicle center of gravity r e l a t i v e t o t h e ground - i s t h e velocity vector of the wind r e l a t i v e t o t h e ground Vw Assume t h a t a wind of constant velocity i s blowing a t a constant angle Vw Jrw t o t h e i n i t i a l heading X and p a r a l l e l t o t h e ground plane. Then, t h e wind components i n the fixed-earth system a r e :
yqo = vw cos Jrw
vwo = Vw s i n Jrw Transformation of these v e l o c i t i e s t o body coordinates (see fig. 2) yields:
uw = (so cos ~r + vw s i n ~r cos e
0 )
vw = (vwo cos Jr - s i n Jr cos cp + cos Jr + vw s i n Jr s i n 8 s i n cp
%O ) ( 0 0 )
cos Jr + vw s i n Jr s i n 8 cos cp - bwo cos Jr -
ww = (so 0 )
The components of wind velocity given by equations (E6) and t h e components given by equations (8) a r e used i n equation (B4) t o determine t h e primed
of v
v e l o c i t i e s as follows:
ut = [v cos p - vw cos qw cos + + s i n qw s i n ,,l,,s e
(
v' = v s i n p -
vw [(sin qw cos Jr - cos Jrw s i n + cos cp
)
+ cos Jrw cos Jr + s i n Jrw s i n Jr s i n 0 s i n cp
( B 7 )
( )
w' = v cos p s i n e -
vw[(cos qw cos + + s i n JtT s i n 9 s i n e cos cp
)
- ( s i n ~r~ cos ~r - cos qW s i n ~r s i n cp
) I
The v e l o c i t i e s given by equations (Bl) and (B2) are used t o determine the trigonometric r e l a t i o n s for hi, which are: ("0)i cos hi =
("G ) i
(%O)
s i n hi = -
(VG) i
For t h e slideout of t h e vehicle represented by equations (l?), t h e velocity r e l a t i o n s given by equations ( B 2 ) become:
= v cos p - 'iyi - vW(cos $w cos $ + s i n $ w s i n q)
' Pko)i
(B9)
= v s i n B + xi cos e + zi s i n 0) - vw(sin $w cos ~r - cos $w s i n 9)
(v;.O)i I
For t h e slideout of a vehicle i n low p i t c h a t t i t u d e , equations (B2) become:
= v s i n p + hi - vw s i n $w cos $ - cos $w s i n $)
(
)
APPENDIX c
INITIAL CONDITIONS AND CALCULATIONS OF EUIER YAW ANGLE, SLIDEOUT DISTMCE, A N D LATEIIAZ, DISTANCE The f i r s t derivative of V, B, and r with respect t o time can be w r i t t e n i n t h e form of the parabolic-difference equations of reference 14 as where E i s the length of the time increment n i s the time i n t e r v a l being considered By designating t h e touchdown increment by n = 0, the conditions immediately a f t e r touchdown are dependent upon the conditions p r i o r t o and a t touchdown.
The conditions a t touchdown a r e selected a r b i t r a r i l y . It i s assumed for the numerical example t h a t the vehicle has a s i d e s l i p angle due only t o a deviation i n wind d i r e c t i o n from the v e h i c l e ' s i n i t i a l heading and t h a t the vehicle has no i n i t i a l r o t a t i o n a l r a t e s . It i s f u r t h e r assumed t h a t these conditions hold immediately p r i o r t o touchdown. These conditions, together with the v e l o c i t y initial condition, are expressed as: ' 0 = 'touchdown r o = r = O -1 It i s a l s o assumed t h a t the vehicle i s decelerating l i n e a r l y from a time p r i o r t o touchdown t o a t i m e a f t e r touchdown. Then Vml becomes:
v-, = 2vo - v1
The equations f o r the E u l e r yaw angle, slideout distance, and l a t e r a l distance from i n i t i a l heading, obtained by the use of the trapezoidal-difference techniques of reference 1 4 are: The angle f3' i s measured i n the ground plane between the xo-axis and the r e l a t i v e velocity of the center of gravity i n the ground plane. The angle i s determined by where r e f e r r i n g t o equations (12), (E), and (B3), I
V ; = u COS e + wf s i n e
O n I I = v 'YO, and VG i s given by equation (Bl) .
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