17-2
REDUCTION OF SEARCH ORDER IN SURFACE RAY ANALYSIS FOR A
CLASS OF NONDEVELOPABLE SATELLITE LAUNCH VEHICLES
RM J h a , SA Bokhari, V Sudhakar and PR M a h a p a t r a
D e p a r t m e n t of Aerospace Engineering
Indian I n s t i t u t e of Science
Bangalore, 560 012 India
ABSTRACT
A Geodesic C o n s t a n t Method (GCM) developed by t h e a u t h o r s is applied t o t h e high
frequency EM problems for a nondevelopable s a t e l l i t e launch vehicle ( S L V )
modeled by a g e n e r a l paraboloid of revolution a n d a right circular cylinder.
T h e junction transition point is obtained by t h e application of t h e Hertz’s
principle of P a r t i c l e Dynamics for t h e f i r s t t i m e in EM theory. The GCM is also
applicable t o cone-cylinder t y p e of SLV w h e r e t h e junction transition point
c a n n o t b e l o c a t e d analytically in s p i t e of t h e developability of t h e individual
s u r f a c e s . A c t u a l m u t u a l coupling r e s u l t s h a v e been p r e s e n t e d f o r a pair of
arbitrarily placed slot a n t e n n a s on a n SLV.
lntroductioo
In this paper, t h e g e n e r i c c l a s s of hybrid quadric s u r f a c e s of revolution
(h-QUASORs) has been t r e a t e d f o r t h e high frequency EM analysis. I t is possible
t o e f f e c t i v e l y model a very wide c l a s s of rotationally s y m m e t r i c bodies with
t r u n c a t e d finite sections of QUASORs. S a t e l l i t e launch vehicles (SLVs) and
subsonic missiles, for example, c a n o f t e n be a c c u r a t e l y modeled by a combination
of a right circular cylinder and a general paraboloid of revolution (GPOR). An
e f f i c i e n t method based on a m o r e g e n e r a l property of surface-diffracted ray
paths, derived from Hertz’s principle in P a r t i c l e Dynamics, has been presented
in this paper, t o d e t e r m i n e t h e r a y g e o m e t r i c p a r a m e t e r s f o r h-QUASORs in
g e n e r a l for t h e high frequency E M applications [1,2]. The specific e x a m p l e
chosen is t h a t of a n SLV modeled by a non-developable h-QUASOR s t r u c t u r e .
Ray Tracing on a Satellite Launch Vehicle
An SLV is a n e x a m p l e of a n h-QUASOR (Fig. I)which may b e described by t h e
t w o s e t s of hybrid p a r a m e t r i c equations [31
x = au cos
where
a
x=f’cos
where
B
y = au s i n
B
z=
-U’
w i t h Iu(
5 ut
(1)
is t h e shaping p a r a m e t e r of t h e t r u n c a t e d GPOR and
B
p is t h e
y=j‘sin~
z = z
with
-zc
5 z 5 -U:
(2)
radius of t h e cylinder.
T h e main difficulty in t h e application of t h e GTD t o such problems a r i s e s
from t h e fact t h a t t h e h-QUASORs in general a r e n o t developable. Hence i t is
not possible t o m a k e use of t h e developability property along with t h e extended
CH2064-2/80/OOOO-O362 S1.W 0 1 9 8 9 IEEE
352
Fermat's principle t o d e t e r m i n e t h e transition point. Even in those c a s e s w h e r e
t h e individual s u r f a c e s of a n h-QUASOR a r e developable, such as a cone-cylinder
model of a n SLV, t h e e q u a t i o n f o r t h e transition point is nonlinear whose
solution c a n n o t b e obtained in t h e closed form.
For non-developable h-QUASORs, w e invoke a m o r e g e n e r a l property t h a t when
a n e x t r e m a 1 p a t h crosses t h e junction b e t w e e n t w o a d j a c e n t surfaces, t h e a n g l e
with r e s p e c t t o a smooth junction line is preserved.
This property of t h e
geodesic follows from t h e Hertz's principle in P a r t i c l e Dynamics 141. Hence in
Fig. I, t h e angles and t h a t t h e geodesic m a k e s with t h e junction line of t h e t w o
surfaces, a r e equal. Since t h e a n g l e invariance property is a local one, i t d o e s
not impose global requirements such a s t h e developability of t h e surface. A
satisfactory s t a r t i n g point f o r 8, is provided h e r e by applying t h e a r c P P I o n t o
t h e t a n g e n t plane of t h e cylinder a t P I [31
[U,
(a2+4u~)1/2-ut(a2+4u~]1/2 a2
+ __
h e r e PPI =
4
2
ZU~+(~'+~U:)'/~
In
(4)
2ut+(a2+4u:)
Equation (3) leads t o a quick convergence of t h e solution. For those c a s e s
where 5 is away f r o m t h e junction e d g e while P is close t o it, t h e s t a r t i n g
point (3), f o r all p r a c t i c a l purposes, m a y b e used t o r e p l a c e jt,t h e r e b y
obviating t h e need f o r numerical s e a r c h f o r Bt.
Numerical Results and Discussion
The height-to-radius r a t i o Ht/R has been used t o c h a r a c t e r i z e SLVs in t h e
following discussion. A r e a l i s t i c SLV d i a m e t e r of 10.0
is chosen, f o r which
t h e r a y - t h e o r e t i c assumptions a r e valid. The r e p r e s e n t a t i v e Ht/R r a t i o is k e p t
a s 5.0.
Since t h e SLV is described in t w o r o t a t i o n a l (parabolic a n d circularcylinder) c o o r d i n a t e s y s t e m s t h e r e a r e t h r e e geodesic directions viz. U, (on
G P O R ) and z (for circular cylinder) along with t h e c o m m o n geodesic p a r a m e t e r #.
Mutual coupling results h a v e b e e n p r e s e n t e d t o show t h e variation along all
These r e s u l t s
t h e s e t h r e e principal geodesic directions of t h e h-QUASORs.
r e p r e s e n t t h e f i r s t e v e r a t t e m p t a t ray t r a c i n g on a convex nondevelopable
s a t e l l i t e launch vehicle, and d e m o n s t r a t e t h e power of t h e GCM as a g e n e r a l
technique for s u r f a c e ray tracing.
REFERENCES
[ I ] P.H. Pathak and N. Wang, "Ray analysis of m u t u a l coupling b e t w e e n a n t e n n a s
on a convex surface", IEEE Trans. Antennas & Propagat. (USA), vol. AP-29,
no. 6 , pp. 911-922, Nov. 1981.
[Z] S.W. Lee, "A review of GTD calculation of m u t u a l a d m i t t a n c e of Slot
conformal arrays", E l e c t r o m a g n e t i c s (USA), vol. 2, pp. 85-127, Dec. 1982.
353
[31 R.M. J h a , S.A. Bokhari, V. Sudhakar and P.R. Mahapatra, "Ray analysis on a
c l a s s of nondevelopable s a t e l l i t e launch vehicles", Electronics L e t t e r s
(CB), vol. 24, pp. 22-23, J a n . 1988.
[41 L.A. Lyusternik, The Shortest L
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