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"~ETERMINATION OF M 5s
ID
ERODYNAMI
C H RACTERISTICS
OF R O L L I N G BALLISTIC VEHICLES F R O M D Y N A M I C M O T I O N D A T A
U S I N G E Q U A T I O N S OF M O T I O N M E T H O D S 1
-
ROLL E Q U A T I O N
bv
R. F. ROSS
Aerospace Corporation
San Bernardino, California
J
A M A 7th AerosDace Sciences
Meeling
NEW YORK CITY, NEW YORK/JANUARY 20-22, 1969.
L
F i n 1 mblicotion rights resewed by American Initifuto of Aeronautics and Astronautics. 1?po Avenue of lhe Americas. New York, N.Y. 10019.
Abstracts may be published without potmistion if credit i s given to author and I o AIAA. (Pricei A l A A Member $1.00. Nonmember $1501
DETERMINATION O F MASS AND AERODYNAMIC
CHARAC'I'ERISTICS OF ROLLING BALLISTIC VEHICLES
F R O M 1 l Y N A M I C MOTION DATA U S I N G EQUATIONS O F MOTION M E T H O D S
I - R O L L EQUATION
R. F. R o s s
T H E A E R O S P A C E CORPORATION
San B e r n a r d i n o O p e r a t i o n s
San B t r m r d i n o . California
Acknowledgmcnts
M r . R o b e r t Swanson, a c t i n g a s a c o n s u l t a n t
to T h e A e r o s p a c e C o r p o r a t i o n , p e r f o r m c d a n
e x t e n s i v e l i t c r a t u r e s u r v e y which ied to the
s c l e c t i o n of the " e q u a t i o n s of motion" m e t h o d
d i s c u s s c d in t h i s p a p e r .
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A d i r e c t m e t h o d of d e t e r m i n i n g t h e m a s s and
a c r o d y n a m i c c h a r a c t c r i s t i c s of rolling b a l l i s t i c
v c h i c l e s f r o m d y n a m i c motion d a t a i s p r e s e n t e d .
T h e m e t h o d i s of the s o - c a l l e d " c q u a t i o n s of
motion" type, and t h e a p p r o a c h i s t o ' p c r f o r m a
F o u r i e r - t y p e a n a l y s i s on the body-fixed r o l l a x i s
e q u a t i o n o f motion. Using s i m u l a t e d motion d a t a
( o b t a i n e d by n u m e r i c a l i n t e g r a t i o n of the c q u a t i o n s
of m o t i o n ) typical r e s u l t s a r e p r e s e n t e d , a s a r c
' e x a m p l e s of t h e effects of f r r o r s in the d a t a diic
t o d a t a a c q u i s i t i o n and p r o c e s s i n g .
NOMEKCLATURE
Componcnts of incrtial accclcralion along body-fixed 3' and 7.
axes (ft/sccZ)
Roll torque coeffiricnt
Roll damping coefficient
Rcfercncc length (ft)
Vehicle products of inertia about
body-fixcd a x e s (slug ft2)
Vehiclc products of inerlial about
body-fixed axes (flug f t 2 )
L
Resultant moment about hodg,
roll axis (ft Ih)
M
Vchicle mass (slugs)
Q
Dot rcfcrs to timc dcrivative
(' )
I.
Abstract
P. 9. r
Atmas&eric d m s i t y [slugs/ft3 )
P
Body-fixed camponcnts of vehicle
angular velocity (rad/seci
F r c c stream dynamic p r e s s u r e ,
i p V 2 (lb/ft2)
Introduction
T h e a n a l y s i s of d y n a m i c m o t i o n d a t a o b t a i n e d
d u r i n g b a l l i s t i c v e h i c l e flight t e s t s which i s d i r e c t e d t o w a r d the d e t e r m i n a t i o n of the vehicle's
acrodynamic and m a s s characteristics, and
p o s s i b l e c h a n g e s t h e r e o f duc t o a b l a t i o n , c a n p r o c c c d f r o m two b a s i c p o i n t s of view. T h e i n d i r c c t
a p p r o a c h , w h e r e i n the d e s i r e d c h a r a c t c r i s t i c s
a r e d c t c r m i n e d by j u d i c i o u s p c r t u r b a t i o n o f p r e f l i g h t m e a s u r e d o r p r e d i c t e d values until t h c s i r n u l a t c d d y n a m i c m o t i o n ( o b t a i n e d hy i n t c g r a t i o n of
the e q u a t i o n s of m o t i o n ) s a t i s f a c t o r i l y m a t c h e s
that m e a s u r e d , i s m o s t t e d i o u s a n d o f f e r s no
g u a r a n t e e of a u n i q u e solution. T h e s e s h o r t c o m i n g s a l o n e p r o v i d e s u f f i c i e n t i n c e n t i v e to
i n v e s t i g a t c a mort d i r e c t , but sufficiently g e n e r a l ,
a p p r o a c h to the a n a l y s i s of b a l l i s t i c v e h i c l e m o t i o n
d a t a . The p r c s e n t p a p e r d c s c r i b e s s u c h a d i r e c t
a p p r o a c h b a s e d upon t h e s o - c a l l e d " e q u a t i o n s of
motion" m e t h o d s , and p r e s e n t s the r e s u l t s of
a p p l i c a t i o n of the m e t h o d t o thc e q u a t i o n expressi n g c o n s e r v a t i o n of a n g u l a r m o m e n t u m a b o u t t h e
v e h i c l e longitudinal ( r o l l ) a x i s .
Whcn e x p l i c i t s o l u t i o n s of thc d y n a m i c a l
cqriation(s) of m o t i o n i n t e r m s of t h e d e s i r e d p h y s i c a l p a r a m e t e r s a r e not a v a i l a b l e , a s i s the c a s e
for t h e p r o b l e m c o n s i d e r e d h e r e i n , t h e a p p r o a c h
u s e d to e x t r a c t t h e s e p a r a m e t e r s f r o m d y n a m i c
m o t i o n d a t a i s g e n e r a l l y of the " e q u a t i o n s of
motion" t y p e . Such m e t h o d s , e . g. , t h e d e r i v a t i v e
method, Laplace transform method and F o u r i e r
t r a n s f o r m method.heve been considered a t length
i n o t h e r p a p e r s " 2, f o r the a n a l y s i s of t r a n s i e n t
rcsponse o f l i n e a r s y s t e m s and have been s u b s u m e d u n d c r a s i n g l e , m o r e g e n e r a l t h e o r y by
S h i n b r o t ' which a l l o w s e x t e n s i o n t o n o n l i n e a r
s y s t e m s . A s w i l l be shown, S h i n b r o t ' s m e t h o d is
i n h c r e n t l y s u i t e d to the p r o b l e m a t h a n d , a n d a v o i d s
some of t h e s h o r t c o m i n g s of F o u r i e r a n a l y s i s .
The example r e s u l t s p r e s e n t e d were obtained using
s i m u l a t e d m o t i o n d a t a r e p r e s e n t a t i v e of a b a l l i s t i c
r o l l i n g v e h i c l e r e e n t c r i n g the e a r t h ' s a t m o s p h e r e .
11. S t a t e m e n t of P r o b l e m
Components of radial ccnter of
p r e s s u r e offsct (ft)
v
2
S
Reference a r e a (ft )
V
Vehicle vclocity (ft/sec)
In f o r m u l a t i n g a m e t h o d t o a n a l y z e d y n a m i c
m o t i o n d a t a , c o n s i d e r a t i o n m u s t n a t u r a l l y be
given a t t h e o a t s e t t o t h e n a t u r e of t h e d a t a which
i s to hc u s e d . U l t i m a t e l y , t h e m c t h o d d e s c r i b e d
i n t h i s p a p e r will be u s e d t o a n a l y z e the d y n a m i c
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m o t i o n d a t a o b t a i n e d on flight t e s t s of b a l l i s t i c
v e h i c l e s , h e n c e i t should be c o m p a t i b l e with the
t y p e s of m e a s u r e m e n t s m a d e . It w i l l be a s s u m e d
in t h e p r e s e n t a n a l y s i s t h a t t h e c o m p o n e n t s of the
vehicle acceleration and angular velocity ( a s
m e a s u r e d by t r i - a x i s i n e r t i a l a c c e l e r o m e t e r s and
r a t e g y r o s , r e s p e c t i v e l y ) along three m u t u a l l y
o r t h o g o n a l body-fixed a x e s a r e known f u n c t i o n s of
t i m o , a n d that t h e v e h i c l e v e l o c i t y a l t i t u d e h i s t o r y
i s known. Initially, i t will a l s o be a s s u m e d t h a t
the a c c e l e r a t i o n s a r e m a d e a t t h e v e h i c l e c e n t e r of
m a s s ; h o w e v e r , t h i s r e s t r i c t i o n will l a t e r be r e laxed.
Combining E q u a t i o n s ( 1 ) and ( 2 ) , a n d r e a r r a n g i n g s l i g h t l y y i e l d s t h e r o l l e q u a t i o n i n final
f o r m f o r the a n a l y s i s .
QSd
t-Q:$p[&]tA,
xx
~
xx
A ycg
E x p r e s s i n g c o n s e r v a t i o n of l i n e a r and a n g u l a r
m o m e n t u m of a r i g i d b a l l i s t i c v e h i c l e in the bodyf i x e d c o o r d i n a t e d s y s t e m i n which the m o t i o n
m e a s u r e m e n t s a r c m a d e ( a n d which g e n e r a l l y h a s
o n e a x i s a l i g n e d with t h e v e h i c l e l o n g i t u d i n a l , o r
roll a x i s , a s i n F i g u r e l ) , t h e r o l l a x i s r o t a t i o n a l
e q u a t i o n of m o t i o n i s
[?I
xx
-
cg
f.; -
x x 'YY]
[F]
xx
t (rp
-
4
q[#] xx
T h e e q u a t i o n h a s b e e n m u l t i p l i e d by' l/lxx to
a v o i d h o m o g e n e i t y and the q u a n t i t i e s i n b r a c k e t s
( t h e knowns i n a d y n a m i c s i m u l a t i o n ) a r e now the
unknowns i n t h e p r o b l e m . T h e c o e f f i c i e n t s a r e
t i m e - v a r y i n g f u n c t i o n s of m e a s u r e d q u a n t i t i e s a n d
t h e i r d e r i v a t i v e s . I t w i l l b e c o n v e n i e n t to w r i t e
Equation ( 3 ) in a shorthand natation a s follows:
Ai(t) Xi
= B(t)
(4)
"
X.
*
-
FIGURE 1
".
z BOD"-FIXED & X E S
B
*E"LER
T h e approach will be to p e r f o r m a Fourier-type
a n a l y s i s ( a s developed by S h i n b r o t l ) on Equation
( 3 ) to d e t e r m i n e the m a s s and 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 p p e a r i n g t h e r e i n . If the a c c e l e r a t i o n m e a s u r e m e n t s a r e m a d e a t a point o t h e r than
t h e v e h i c l e c e n t e r of g r a v i t y , the a p p r o p r i a t e
e q u a t i o n s t r a n s f o r m i n g the m e a s u r e d a c c e l e r a t i o n
should b e u s e d i n Equation ( 3 ) . T h e only new
quantities introduced a r e the coordinates locating
the a c c e l e r o m e t e r i n the body-fixed c o o r d i n a t e
system.
ANGLES
COORDINATE SYSTEMS AND
EULER ANGLES
111. E q u a t i o n s of Motion Method
T h e l e f t - h a n d s i d e o f E q u a t i o n ( 1 ) is, of c o u r s e ,
t h e c o m p o n e n t of the r e s u l t a n t a e r o d y n a m i c
m o m e n t a c t i n g on t h e v e h i c l e along the r o l l a x i s .
F o r a b a l l i s t i c v e h i c l e ( g e n e r a l l y h a v i n g two p l a n e s
of s y m m e t r y ) , a d m i t t i n g the p o s s i b i l i t y of a p u r e
r o l l t o r q u e , r o l l d a m p i n g a n d a r a d i a l c e n t e r of
p r e s s u r e offset, t h e l e f t - h a n d s i d e of E q u a t i o n (1)
m a y be e x p r e s s e d a s f o l l o w s
L = CeoQSd t Ce
A s pointed o u t by S h i n b r o t l , the d e r i v a t i v e ,
Laplace t r a n s f o r m , and F o u r i e r t r a n s f o r m
methods f o r l i n e a r s y s t e m s have c e r t a i n common
f e a t u r e s when a t t e n t i o n is f o c u s e d on the f o r m a l
p r o c e s s e s w h e r e b y the l e a s t s q u a r e s e q u a t i o n s a r e
o b t a i n e d . T h e s e f e a t u r e s l e a d to the g e n e r a l i z a tion p r o p o s e d by S h i n b r o t , which is to p e r f o r m
t h e following o p e r a t i o n s o n the equation of m o t i o n
under consideration.
p t MR A
- MRzA
(2)
y zcg
ycg
w h e r e p o s s i b l e a n g l e - o f - a t t a c k d e p e n d e n c e of the
r o l l t o r q u e a n d r o l l d a m p i n g t e r m s h a s not b e e n
s h o w n e x p l i c i t l y . T h i s effect m a y be a c c o u n t e d
f o r if n e c e s s a r y s i n c e t h e body-fixed a n g l e - o f a t t a c k c a n be e x t r a c t e d f r o m the m o t i o n d a t a , but
i t is n o t i n c l u d e d in the p r e s e n t a n a l y s i s f o r
s i m p l i c i t y . It i s noted t h a t the a e r o d y n a m i c m o d e l
s h o u l d b c r e p r e s e n t a t i v e of the a c t u a l p h y s i c a l
situation.
P
T
1.
2.
Multiply by N a r b i t r a r y (but sufficiently
smooth) functions fn(t).
Integrate the resulting N equations bet w e e n two definite l i m i t s , s a y , z e r o and T
C a r r y i n g o u t t h e s e o p e r a t i o n s on the r o l l
equation i n shorthand f o r m r e s u l t s in an o v e r d e t e r m i n e d s e t of l i n e a r e q u a t i o n s i n the u n knowns Xi.
2
.-J
T
j A i ( t ) fn(t) dt
o
Xi = / i ( t )
fn(t) dt
0
n = I, 2,
. .
.
N
( 5)
4
It i s i m p l i e d in Equation (4) that the unknowns, Xi,
a r e i n d e p e n d e n t of t i m e , h o w e v e r , in w r i t i n g
Equation (5) t h i s r e s t r i c t i o n i s r e l a x e d , i . e . , t h e
X i ' s a r e a s s u m e d ' t o he constant over the interval
T . A g a i n , E q u a t i o n (5) m a y be a b b r e v i a t e d a s
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A n , i. Xi. = Bn
n = 1, 2 , .
.
. N
(6)
B e f o r e p r o c e e d i n g t o t h e s o l u t i o n of E q u a t i o n s
(11). the s e l e c t i o n of the s o - c a l l e d m e t h o d f u n c t i o n s
f,(t) a n d t h e i n t e r v a l T, w i l l be d i s c u s s e d . B r i e f l y ,
the m e t h o d f u n c t i o n s a r e c h o s e n so that t h e c o n Stant t e r m s which r e s u l t f r o m i n t e g r a t i o n by p a r t s
v a n i s h . C o n s i d e r , f o r e x a m p l e , An, 6 .
T
A
n, 6
= J(rpfn(t]
t qin(t)) dt - q ( T ] f n ( T ) f q(O,]fn(0)
k,
n = I, 2 , .
. .
N
="x
T
n = 1, 2 , .
. .
N
=
(9)
I m p l e m e n t a t i o n of t h i s m e t h o d , g i v e n t h e
r e q u i r e d m o t i o n d a t a ( u s u a l l y s t o r e d on m a g n e t i c
tape), is v e r y straightforward. The integration
i n t e r v a l , T , a n d the n u m b e r of d i s c r e e t f r e q u e n c i e s , N, a r e s e l e c t e d s o t h a t t h e f r e q u e n c y
s p e c t r u m of t h e a n a l y s i s i n c l u d e s t h e m o t i o n f r e quenc ie s.
n r
2T
O<t<T
n, e v e n
IV.
i/
(12)
The method outlined above h a s been p r o g r a m m e d f o r the d i g i t a l c o m p u t e r w i t h t h e flexibility
of changing t h e a e r o d y n a m i c m o d e l , a s r e p r e s e n t e d by E q u a t i o n ( 2 ) . R e g a r d i n g t h e n u m e r i c a l
m e t h o d s a g e n e r a l i z a t i o n of S i m p s o n ' s r u l e d u e
t o F i l o n 6 i s u s e d to e v a l u a t e t h e i n t e g r a l coeffic i e n t s An, i, a n d a m o d i f i e d G a u s s r e d u c t i o n
s c h e m e due t o C r o u t ' is u s e d t o s o l v e t h e d e t e r m i n a n t s e t r e p r e s e n t e d by E q u a t i o n s ( 1 2 ) .
the d e s i r e d r e s u l t i s obtained. In p r a c t i c e a f i n e r
f r e q u e n c y i n t e r v a l t h a n n l T i s often r e q u i r e d , s o
the above a p p r o a c h is m o d i f i e d slightly. T h e
frequencies and method functions a r e chosen a s
follows:
wn
= A i , n Bn
(8)
where
wn
MOTION FREQUENCIES IN
BODY-FIXED COORDINATES
F o r the p a r t i c u l a r m o d e l c h o s e n t o d e m o n s t r a t e the " e q u a t i o n s of m o t i o n " m e t h o d , E q u a t i o n s
(12) a r e a s e t of e i g h t e q u a t i o n s i n e i g h t u n k n o w n s ;
however, f o r s o m e applications only 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 t r e a t e d a s unknowns.
T h i s will b e d i s c u s s e d f u r t h e r i n t h e n e x t s e c t i o n .
T h u s , if the m e t h o d f u n c t i o n s (which t o g e t h e r with
t h e i r f i r s t d e r i v a t i v e s m u s t b c i n t e g r a b l e on[O, T I ) ,
a r e chosen to bc
f n ( t ) = shunt
-
Applying t h e l e a s t s q u a r e s p r i n c i p l e t o t h e
l i n e a r , i n h o m o g e n e o u s , o v e r d e t e r m i n e d s e t of
e q u a t i o n s r e p r e s e n t e d by E q u a t i o n (6) y i e l d s the
following d e t e r m i n a n t s e t
A.~ , Ann, I. Xi
( 7)
0
FIGURE 2
.
D i s c u s s i o n of R e s u l t s
A s mentioned previously, the dynamic motion
data u s e d i n this analysis i s simulated (generated
using a s i x - d e g r e e - o f - f r e e d o m computer p r o g r a m )
a n d t h u s r e p r e s e n t s d a t a of t h e h i g h e s t q u a l i t y f o r
the m o d e l u s e d . T h e v e h i c l e m a s s a n d a e r o dynamic c h a r a c t e r i s t i c s , and r e e n t r y conditions
( t a k e n a t 300 Kit) a r e shown i n T a b l e 1 . A l l of
t h e c h a r a c t e r i s t i c s shown a r e t a k e n to be c o n s t a n t
d u r i n g t h e r e e n t r y flight, h o w e v e r , i t is i m p o r t a n t
only t h a t t h e y be c o n s t a n t o v e r t h e p e r i o d being
a n a l y z e d . An e x t e n s i o n t o a c c o u n t f o r unknowns
changing w i t h t i m e , w h e t h e r t h e c h a n g e be due to
a b l a t i o n o r Mach a n d R e y n o l d s n , > m b e r e f f e c t s ,
will be d i s c u s s e d l a t e r
T h e s e l e c t i o n of a f i n i t e i n t e g r a t i o n i n t e r v a l
r e q u i r e s t h a t t h e r e be c o n t a i n e d i n t h e i n t e r v a l a n
a d e q u a t e r e p r e s e n t a t i o n of the v e h i c l e a n g u l a r
m o t i o n . F i g u r e 2 s h o w s t h e two d o m i n a n t f r e q u e n c i e s ( f r o m l i n e a r i z e d t h e o r y 4 , 5, which a p p e a r
in the acceleration and angular velocity data.
T h e s e c u r v e s s e r v e a s a g u i d e to t h e s e l e c t i o n of
the interval size, however, the frequencies
a s s o c i a t e d w i t h n o n l i n e a r e f f e c t s should h e con;
s i d e r e d if n e c e s s a r y .
3
P
3 C l n l l l l V H9lH 1 V S N O l l l l l O S NMONXNn N3A3S
f-
I
Z 318Vl
8866 ' 0
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6666 ' 0
6666 '0
9666 . O
~ G G G'n
nnno ' I
1JX 282
9866 ' 0
~~
(3) (7)
*
*
(+I
.LTITUi)E
xx
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i.ono
.
n. 9998
n. 9999
1. nnn
1.000
n. 9997
0.9999
n. ~
1.000
n. 9983
0.9999
I . no0
1.001
1. nnn
1.on0
1. nnn
i.onn
n. 9999
i.non
I. nnn
n. 9999
1.004
0 . 9992
1.aon
0.9990
0.9972
n. 9998
0.9978
1.nnn
n. 9998
0.9998
1.001
I. nnn
n. 9987
1.000
1. on0
1,001
1.nnn
I . on2
n. 9870
i.nno
0.9999
1.001
1.005
n. 9984
0.9802
0.9998
1.001
~
9
8
136 Kft
99
73
*Normalized using simulation value.
1
' . = 0.96 sec
3 . 3 < w<5n.0 rad/scc
TABLE 3
SEVEN UNKNOWN SOLUTIONS AT INTERMEDIATE ALTITUDES
$q-fq-g
xx
xx
xs
xx
1.015
1. nnn
1.001
1.011
1. nnn
1. nnn
1.000
I. on3
1. non
1. nnn
0 . o ' ~ ~ 1.001
1.005
1.nno
1. nnn
1.002
1. on1
1.025
1, non
i.nno
1.002
0.9990
o.mn
i.onn
1.onn
n. 9070
n. m 9 n
1. n o 2
I. on0
1. nnn
1.010
i.nnn
1.nnn
0.9082
I
0.9993
xx
I
0.9998
I
80 Kft
68
58
*Normalized using simulation value
TABLE 4
-
EIGHT UNKNOWN SOLUTIONS AT INTERMEDIATE ALTITUDES
d y n a m i c c h a r a c t e r i s t i c s of b a l l i s t i c v e h i c l e s i n a n
i d e a l s i t u a t i o n with g r e a t s u c c e s s , i t i s of i n t e r e s t
to e x a m i n e a few m o r e r e a l i s t i c c a s e s . E x a m p l e
c a l c u l a t i o n s h a v e been m a d e f o r c a s e s i n which
1) the a e r o d y n a m i c m o d e l does not m a t c h the
p h y s i c a l s i t u a t i o n , 2 ) the m a s s p r o p e r t i e s a r e
a s s u m e d t o b e known, but a r e slightly in e r r o r , and
3 ) the p r o c e s s e s of d a t a a c q u i s i t i o n and p r o c e s s i n g
h a v e i n t r o d u c e d b i a s e s in the d a t a i t s e l f .
T o i l l u s t r a t c t h e f i r s t case m c n t i o n e d above t h e
m o t i o n d a t a of the s i m u l a t i o n which d i d n o t include
r o l l t o r q u e e f f e c t s was a n a l y z e d u s i n g a n a e r o d y n a m i c model w h i c h included the c o r r e s p o n d i n g
t e r m . T h i s a c t u a l l y i s t a n t a m o u n t t o having o n e
z e r o r o o t i n t h e d e t e r m i n a n t s e t of e q u a t i o n s . T h e
r e s u l t s , a s shown i n T a b l e 5, i n d i c a t e t h a t o v e r e s t i m a t i n g t h e a e r o d y n a m i c e f f e c t s d o e s not d e g r a d e the b a s i c a c c u r a c y of the m e t h o d .
5
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k)
~
1
L T 1TU D E
xx
IXX
XX
1.000
1.005
0.9980
0.9996
1.001
0.9999
0.9936
1.004
0 . 9980
0.9995
1.001
1.000
1.006
0.9980
0.9970
1.001
0.9995
10-8
1.024
0.9998
0.9874
1.000
1.003
0.9996
1.000
lo-'
0.9260
1.000
0. 9951
1.001
0.9975
0.9995
1.000
10-8
0.9895
1.000
0.9943
1.002
1.003
0.9995
1.000
0.9960
1.000
0. 9729
0.9995
0.9965
0.9985
1.000
0.9920
0.9988
1.001
1.001
1.018
1.000
1.000
0.9996
0.9723
1.002
0.9710
0.9985
1.000
T = 0 . 8 sec
130 Kft
W
0.9010
10-8
100
1.008
0.9450
70
'Normalized using simulation value.
3 . 9 <w < 59 rad/sec
TABLE 5
EIGHT UNKNOWN SOLUTIONS WITH ZERO ROOT (C )
1,
A l l of t h e n o n - z e r o r o o t s are d e t e r m i n e d to
w i t h i n a f c w p e r c e n t a n d t h e a c t u a l v a l u e s of Ceo
o b t a i n e d a r e i n d e e d v e r y s m a l l . T h c r e v e r s e of
t h i s c a s e , n a m e l y , w h e n one of t h e unknowns, s a y
Cep, i s a s s u m e d to be z e r o ( w h c n , in f a c t , i t is
not) d o e s not m c e t with s u c h success. In p r a c t i c e
then i t i s a l w a y s b e s t to i n c l u d e a s u f f i c i e n t l y
g e n e r a l a e r o d y n a m i c m o d e l when i t s n a t u r e is n o t
known a p r i o r i . For flight t e s t s of b a l l i s t i c v e h i c l e s ,
good engineering judgments in this r e g a r d can
u s u a l l y be m a d e f r o m a e r o d y n a m i c p r e d i c t i o n s and
q u a l i t a t i v e i n s p e c t i o n of t h e d a t a .
.9
in
I
loo
75
LIlIlUDE
A s i s o b v i o u s to the r e a d e r , n o t a l l of t h e t e r m s
a p p e a r i n g i n t h e r o l l e q u a t i o n n e e d be s o l v e d f o r ;
t h e i r e s t i m a t e d v a l u e s m a y be i n p u t t o t h e a n a l y s i s .
F o r e x a m p l e , if a v e h i c l e w e r e m a s s - b a l a n c e d
b e f o r e a f l i g h t t e s t t h e m o m e n t s a n d p r o d u c t s of
i n e r t i a o b t a i n e d could be u s e d d i r e c t l y . An a d i a n t a g a of t h i s a p p l i c a t i o n of t h e m e t h o d is when
the p r o d u c t s of i n e r t i a , s a y , a r e v a n i s h i n g l y s m a l l
a n d h a v e l i t t l e e f f e c t on t h e d y n a m i c m o t i o n ( m a k i n g t h e i r v a l u e s c o r r e s p o n d i n g l y difficult t o e x t r a c t
in the p r e s e n c e of m u c h l a r g e r e f f e c t s ) . T h e r e s u l t s of s u c h a c a s e a r e p r e s e n t e d on F i g u r e 3.
H e r e the m o m e n t s a n d p r o d u c t s of i n e r t i a a r e
a s s u m e d known, hut a 5% e r r o r in a l l p r o d u c t s of
i n e r t i a h a s been i n t r o d u c e d . T h e r e s u l t s , a s
shown, indicate s m a l l e r r o r s in the p a r a m e t e r s
involving r a d i a l c e n t e r of p r e s s u r e o f f s e t a n d roll
damping.
FIGURE 3
-
. Klt
50
THREE UNKNOWN SOLUTIONS
WITH 5 PERCENT ERROR IN
ALL PRODUCTS OF INERTIA
e x p e c t e d f r o m t h e n a t u r e of the t e r m , with the
s e n s e of t h e error i n C j being o p p o s i t e that of the
c r y o r in d e n s i t y or veloflity.
On F i g u r e s 4 t h r o u g h 7 a r e shown the r e s u l t s
obtained when e r r o r s w e r e introduced in the r a t e
g y r o a n d a c c e l e r a t i o n d a t a . It is not i n t e n d e d that
t h e s i z e of the e r r o r s c h o s e n r e f l e c t t h e e s t i m a t e d
q u a l i t y of r e d u c e d f l i g h t data; o n l y the r e l a t i v e
s e n s i t i v i t y of t h e v a r i o u s p a r a m e t e r s of the s o l u tion to s u c h e r r o r s i s of i n t e r e s t . T h e e r r o r s in
a n g u l a r v e l o c i t y u s e d w e r e 5% of t h e r e e n t r y v a l u e
f o r r o l l r a t e , a n d 5% ( a s a b i a s ) of the peak value
a t 1 0 0 Kit f o r p i t c h and yaw r a t e s . A s i s shown on
F i g u r e s 4, 5 a n d 6, the unknowns m o s t a f f e c t e d
a r e t h o s e w h o s e c o e f f i c i e n t s i n the r o l l e q u a t i o n
e x p l i c i t l y involve t h e c o r r e s p o n d i n g c o m p o n e n t of
a n g u l a r velocity. T h e e r r o r s in t h e r a d i a l c e n t e r
of p r e s s u r e o f f s e t a n d r o l l d a m p i n g p a r a m e t e r s
a r e very small.
F i n a l l y , a s e r i e s of e x a m p l e c a l c u l a t i o n s w a s
m a d e ( a g a i n u s i n g t h e b a s i c , or n o - r o l l t o r q u e
m o t i o n d a t i . ) i n which errors w e r e i n t r o d u c e d i n
t h e d a t a i t s e l f . It w a s found that a 5% e r r o r in
e i t h e r air density o r vehicle velocity i s reflected
d i r e c t l y a s a p p r o x i m a t e l y a 5% error in t h e d e t e r
m i n a t i o n of t h e r o l l d a m p i n g c o e f f i c i e n t , w i t h ail
o t h e r p a r a m e t e r s b e i n g u n a f f e c t e d . T h i s is a s
6
.J
T =
0.64 IDC
4.9
<
Y
<
(+J
74 RAD/SEC
xx
ERROR IN ROLL R 3 T E = 0,05P0
2,
0
8
I20
0
a
0
0
0
I00
0
*Normalized using
0
IIIIIYI.1IM
0
I
80
I
ALTITUDE - Kll
100
ALTITUOI
Downloaded by UNIVERSITY OF ADELAIDE on October 28, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1969-102
d
5'
i;o
- SEVEN
UNKNOWN SOLUTIONS
WITH 0.05 pe ERROR IN
ROLL RATE DATA
IO0
ltTllUOE
FIGURE 5
.Kfl
FIGURE 7
I
80
. Kft
- SEVEN UNKNOWN SOLUTIONS
(products of inertia now shown)
WITH 0.05-G BlAN ERROR
IN YAW ACCELERATION
DATA
A b i a s of 0 . 0 5 g ( a b o u t 5% of t h e t r i m l a t e r a l
a c c e l e r a t i o n a t 50 Kit) w k s i n t r o d u c e d i n the yaw
a c c e l e r a t i o n a n d t h e e r r o r s i n p r o d u c t s of i n e r t i a
w e r e quite l a r g e . The remaining p a r a m e t e r s a r e
shown on F i g u r e 7 . T h e reason t h a t a b i a s i n
a c c e l e r a t i o n d a t a h a s s u c h a p r o n o u n c e d e f f e c t on
t h e r e s u l t s is that i n conjunction with the r a d i a l
c e n t e r of p r e s s u r e offset, it c o r r e s p o n d s t o a n
a p p a r e n t rolling m o m e n t i h a t should h a v e ah e f f e c t
on t h e r o l l r a t e . T h e i n c l u s i o n of the r o l l i n g
m o m e n t t e r m in t h i s c a s e should a l l e v i a t e t h i s
s i t u a t i o n t o s o m e d e g r e e in that a rolling m o m e n t
which e x a c t l y b a l a n c e s the c o m b i n a t i o n of t h e bias
a n d c e n t e r of p r e s s u r e offset would be found.
I
BO
slmulmlm value
I
120
FIGURE 4
' Normallzed using
YmIUB
- SEVEN
UNKNOWN SOLUTIONS
WITH 0.025 RAD/SEC BIAS
ERROR IN PITCH RATE DATA
A s w a s m e n t i o n e d p r e v i o u s l y , a l l of the u n knowns a r e c o n s t a n t o v e r the i n t e g r a t i o n i n t e r v a l ,
T , i n t h i s a n a l y s i s . Should t i m e - v a r y i n g e f f e c t s
b c i m p o r t a n t , e a c h v a r y i n g unknown m a y be exp a n d e d a b o u t , s a y , t h e m i d d l e of t h e i n t e r v a l a n d
t h e r e q u i r e d n u m b e r of t e r m s r e t a i n e d . C o n s i d e r ing the s i z e of the i n t e g r a t i o n i n t e r v a l s , which
generally a r e s m a l l e r a t lower altitudes for r e e n t r y f l i g h t , c o n s t a n t or l i n e a r a p p r o x i m a t i o n s
should be sufficient.
1.5
s
REFERENCES
-10
i
*
2
*
*Normdlztld urlng
slmulotlon value
5
120
S h i n b r o t , M. "On t h e A n a l y s i s of L i n e a r a n d
Nonlinear Dynamical S y s t e m s f r o m T r a n s i e n t
Response Data,
NACA T N 3288.
2.
B r i g g s , B. R . , a n d Jones, A. L . , " T e c h n i q u e s f o r C a l c u l a t i n g P a r a m e t e r s of Nonlinear Dynamical S y s t e m s f r o m Response
D a t a , " NACA T N 2977, 1953.
3.
S h i n b r o t . M . , "A D e s c r i p t i o n a n d a C o m p a r i s o n of C e r t a i n N o n l i n e a r C u r v e - f i t t i n g T e c h n i q u e s , w i t h A p p l i c a t i o n t o t h e A n a l y s i s of
T r a n s i e n t R e s p o n s e D a t a , " NACA T N 2622,
1952.
80
ALTITUDE . Kft
FIGURE 6
W
1w
1.
- SEVEN
UNKNOWN SOLUTIONS
WITH 0.025 RAD/SEC BIAS
ERROR IN YAW RATE DATA
7
~
4.
5.
N e l s o n , R . , "The Motions of R o l l i n g S y m m e t r i c a l V e h i c l e s R e f e r r e d t o a Body A x i s
S y s t e m , " NACA T N 3737, 1956.
6.
F i l o n , L. N . G . , "On a Q u a d r a t i v e F o r m u l a
f o r T r i g o n o m e t r i c I n t e g r a l s , " P r o c . Roy.
S o c . , E d i n b u r g h , Vol. X L I X , 1928, pp 38-47.
P e t t u s , J. J., " P e r s i s t e n t R e e n t r y V e h i c l e
7.
King, K . S . , " N u m e r i c a l A n a l y s i s , " M c G r a w H i l l B o o k G o . , I n c . , 1 9 5 7 , pp 235-236.
R o l l R e s o n a n c e , " AIAA P a p e r No. 66-49,
J a n u a r y 1966.
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