Analytical studies of mast-cable-hoop-membrane type antennas were conducted using a transfer matrix numerical analysis approach. This method, by virtue of its specialization and the inherently easy compartmentalization of the formulation and numerical procedures, can be significantly more efficient in computer time required and in the time needed to review and interpret the results

Gabriele, A.

Loewy, R.

English

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I^
PROGRESS REPORT ON THE
FORMULATION OF A DYNAMIC ANALYSIS
METHOD FOR A GENERIC FAMILY
OF HOOP-MAST ANTENNA
SYSTEMS
(NASA-CR-164981) FORMULATION OF A DYNAMIC
	 582-12300
ANALYSIS HBTHOD FOR A GENERIC FAMILY OF .
HOOP-!BAST ANTENNA SYSTEMS Progress Report
(Rensselaer Polytechnic last., Troy,.ti. Y.)	 Unclas
18 p HC 1102/!!F A01
	 CSCL 205 G3/32 08321
November 2, 1981
A. Gabriele
R. Loewy
INTRODUCTION
We are conducting analytical studies of "mast-cable-hoop-membrane"
type antennae, using a transfer-matrix (Ref. 1) numerical analysis approach.
This methodology has beda chosen as particularly well-suited for handling
a large number of antenna configurations of a generic type. While not
capable of providing, in principle, more information than a proper NASTRAN
formulation, a dedicated transfer matrix analysis, both by virtue of its
specialization and the inherently easy compartment&Z.ization of the formula-
tion and numerical procedures, can be significantly more efficient not only
in computer time required but, more importantly, in the time needed to
review and interpret the results.
'	 -2-
TRANSFEI OF STATE VALMLES •AROW TIE SM
The analysers begins with the formulation of the typical element of
the hoop assembly. Assumptions in this formulation are as follows:
1. The hoop is polygonal,
2. the hoop support cables alternate up and down with each segment
of the hoop,
3. mass effects are approximated by a series of concentrated masses and
mass moments of inertia,
4. stiffness transfer matrices account for the flexibility of hoop
segments, which are modelled as equivalent beam-column-torsion members, including
shear deflections, and
S. initial tensions in the cables and hoop are negligible compared to
spring rates, (these tensions will be represented in subsequent analysis).
The typical hoop element model is conceived 4s beginning and ending in
the middle of an upper cable positionon the hoop, as shown below
jAs	 i	 E_
.X1.	 E
ti
V+A
liote that for hoop element 01 (is i • 1) the right-hand most station is "0"
and the left-hand most station is 11219.
With a transformation of the state variables around the hoop o as follows:
rd
4r
v `s ^- T M7 
m
T	 Sra
	 V21fM 1C^ ^[TI	 m
K C+^
	
	 V
M
' qs
W WERE WE DEOCIAAE
Tl [MIENtil
.
-[JE3 ITS
ly...
I[krr[
y
#COP. Su PPcr-r
. SPRiNGS,
^A 8 i cs  ASE ftPPRo^y^'^ A,d
r
-4-
pT/pN OF "rRANSfER #m-rR lccs
T^P#lA( DER M^-T.elc^S IMPOSE
FoR -J;lf .MOM EAM
U^Pf.^ /4N0 LOWER G4LLC Fokce,3 ON T/+- MME'S ASSuntiNG A
F/X E a ^A^T,
lillak] ar  SC ^ s
1. 0 0- K, caa^^ o 0 o KK-bkpcwp - 0 0 000
0
1
0
0 c
o
o0 00 00 CO	
.
0
o 0 00
O 0 o
0 0 o r o 0 0 o d 0 o0
.
.
p o o k^	 A41.k/ o o - KK,o;,^p 00 0.0
O o 0 O D 1 0 c 00 0
o c o 0 o c o 000 0
O m o 0 o CO 0 dodo0 0 0 0 0 0 0 0 1 00 0p 0 4 O O o 0 O	 ,. 0 l O d
0 0 0 0 0 0 0 0 0'
o d O. O	 .p O O O p d.0 ^	 _
o1
—Kl. Oaw 4^1w S O7'r sw	 SL.wC• — 0 0 0. 0 o 0 , O D0.0 1 O O
°.o o , o 0 0 0 0 0 0 0
'- o. o o -Kcmv
	 cp 1 0 0-1^,ar^^V^ o 0 o 00 0 0 `o O o O 0 o 0 0
0 0 o O O O 1 O O o 0 O
:*; o 0 0 0 O o o i 0 0 0 0
O o co c D O O o D O O
0 0 O p O o 0 O D o 1 O
o p 0 O 0 0 O. D 0 I
S^G.MENr
s	
.
Erl
0
AXES A-OeOUN4 -rllE #60A .Aa lm o r-.
^^.. ®^o o.. o .c o 0 0. o o .o .
O_ CALCL O. p o O ' 8 D .A;" Q O O0 0 0 o O o O o :,e+ o 0
D o: O ^_ p O o O O o O
0 o p CMCL o 0 0 0 0 .c o
o.' .o 0 o O I o 0_ o 0 0 00. - O o o 0 .o ! 0 O c o 0
0 a o 0 0 ..o o CO&CL o 0 00 -A,;%ot 0 O a 0 0 0 =AL . O o O
o ^ o o a D b 0 ^^- .o
0 0 0 A;FD. 0 0 O 0 o conk o
:0
 D. D. O o_ o o ,omd 0 0 0 0* cc
i
1
-s-
TNT CT3 TXAO s FER MAtIR IX /s. AN AX/3 7R4fJCJr*"A7)►oe
te)Af lx ., A «OUP/N b	 C#hA)dE s J14. ORI EAITATroN .oF Hoo/^
A N 0 GuE FURTHf^ OE F/NE
•
 ^
Cc).T ^ f^a Tim 3 A-^ E A s  T^^  W ^ E Tio E
T/i /^ /^1.5 f0^/rII^T/ON ANGLE O TC s ^'Y C^— 'y^ o^
PN Q -
T (` ^ .	 ^If^eE pc = Cat —^ ^
AIN p	 .
3
fl
ORE-GINAL PAGE A
'	 OF POOR QUALM
T#C- /HA AS lrR&J I M MATRICES [M3 .1NCORPat'#'7 . DRAeZMfifArs
FORCE . f^N D /Y10MENT
	
Rcvr_,# 4 rwe *jj-tjAiA"oN .
. O F . ^N ^/NI TES/MAL D /MEAN S /ONS . Of ^^^ GONCICN T1@AT^ /IIIAS3i^4 • ^
1	 O	 O	 cLP M	 O	 O	 O	 o: -: p.. O	 G'.	 .O.
•	 oO	 1 4 M	 p	 o	 O	 O	 O	 O	 O	 O	 O
O O	 p	 I	 O	 O	 O	 O	 o	 O	 O	 00	 0	 0.	 0	 1	 O	 O
	
.0
	 p	 O
_	 O. O.	 o	 O	 p	 I -w'^s^ O	 o	 O	 o	 00 0
	 o	 O	 0.	 0	 1	 MO	 o	 0	 0	 0
^M	 =	 o .0 - .0	 0	 0	 0	 I	 0	 0	 0	 0O	 0	 C	 o	 0	 1 w^rx o	 O
p	 o	 a.	 O	 0	 p	 0	 o	 O	 i	 D	 OO	 O	 O	 D	 o	 D	 O	 o	 O	 O	 1' tsIOM i
o 0 0	 0	 o	 D	 o	 0	 0	 o	 D.
Tµ><	 E cf}s ri c	 rRA Ns r" IVAzxlecs ^E^ ^C EA-r
T^ f1oaP	 SEG mF.^+rs A,S	 MASSLESS EQ ^ ^V Ac..E.vT BE.tM3 /9ND 
11VCL-00 E	 CONStOERArio.,1 O r Axm L-, T'OA• S/CM Ac- AND S&"A
D EF^cT'^oN S .	 ( RE.J^ ^^	 ' .
Eel
11
i
I	 0
s
126M	 EZ
(oEt q b fie=,
O	 o
O	 o
O	 0
D	 o
O	 o
0	 0
0	 0
0	 O
o
- o	 Q	 e..	 0	 0	 0	 ^o
l	 o	 0	 0	 D	 0	 0
I	 O	 a	 O	 o	 00	 0	 1	 p	 o	 o
O	 0	 i	 O	 O	 O
S0 A•	 E	 O	 O
0	 0 ^^_ •/	 E_	 I	 oO	 0	 O	 O	 Q	 f
O	 O	 0	 O	 O. O • G^^T
O	 O	 0	 O	 O O
0	 0	 0	 0	 0	 0	 p
0	 o	 0
O	 0	 0
0.	 o	 .0
O	
.O	 O
O	 D	 p.
0	 0	 O
0	 O	 O
0	 O	 o
0	 0	 c
I	 o	 0
O	 1	 0
0 ^ A	 l
-7-
IHTRODDCING TEE EFltCT OF CASLI FORCIS
The analysis proceeds by accounting for the transverse motions of
the mast as they affect cable forces. Note that axial motions of the mast
are neglected.
v^pEit	 -
	
a. CUPP" BRRa^ce r-r.^
GABtE
I.CWC^
C„ & F
• to Clow&A AOmcoi PT. )
Me 5:r
The azimuthal direction ctr:esponding to station "0" on the hoop is
chosen as the reference direction; and state variables on the mast at the
upper branch point, o, and the lower branch point, o, are referred to
that direction.
A! cable force on the hoop is equal and opposite to that on the mast.
Thus, ae define	 frAz	 C 3
SL
-^	 t Here, subtracting the unit matrix from [V] and[!2k]insures
that in calculating cable forces only transverse displacements come into-play.
ffs
a,sas lrT^^ll^l
Je C04 HOW
-w ti m
	
S r.#m VAffif [•^'S faR THE cEir Eiao of 'n'PK - kooP
' E(^/rl^T
	
W / Ty T^ A^CGoR! ^N1 eE^oGJ
i
FoR -rYPlcA ^ NociP iCLC m 6i 1r N 0, w N"
-9^
hi.
 6oa , TI M 1S Fotr" g R E cmE.d ^ S w-rriN G 	 -
i V3  C^
I
S
	 Aft
+
 f 
sci
a
	
e	 W E G ^'-	 .
At
C	 C
.?N	 C	 Car	 LO
A(4	 all
^ l 	 k IT3
---,,^ ao
A GENeR^^ FORM Fob(	 MIDDLEJO^^ DiaPi MF.Ac^N-r of R
TyP/ C At Noy''' 5 "M a NT wo U I-A T&E e E 3
FTr7+
)] 4
St
S
	
4.	 o	 ci
i	 .
r
rnf
ORIGINAL PAGE
NuMeca of Of POOR QUA
t T Al Tom.4L &oP SI..CMC N7S LI W*UC k/E A^CC44 .
1*#*r RN F_ xmt.N-r .)s DE fooe4 as r*c sewtoN pF Twt NooP 8ew-Wte.3 .
TW O . R• ^ .7A•:ENT V PP" CM 4 S) , S /N[ C T/FE hbNoA Ct o$t s,
W EK NOw T^IA-T . TMC sTAT^ . VAjejo; LE &rI^coP STATioN
-:ME/eo AM- ' 'r*C S&ME #4 TMoSE A r #ooP S r* rloN .ZMw SD
^E cho wR m^ -r" S TArE YAt/.+".s A T^ s7.+7loAj o&Ao r}^
CCV3	 M
o.	 a
Std 8 STl ro ri 1 y	 INTO PREYiou& ExPRE-%Sio A3& Fok
buE c,J^N WR ITE TttE.3e . &6 fuNcnoNS OF
uo	 Lo
[A.4]j
2N	 ^o	 to
t
p0b	 !
a
ISo-^ ^1 N,. I I - T Lllh I II...
TUMSFEA OF STATE VARMLES ' ALOSC TU MAST
The analysis now focuses on the transfer of state variables aloes the
mast. The following assumptions are made for the mast confisuration:
1. The feed assembly has its unique mass,
2. the feed mast has a uniform was distribution.
•	 3. the upper mast has a mass distribution that increases in sections
as we proceed down the mast.
4. the nub has a unique msssg
S. the lower mast has a mass distribution that decreases in sections
as we proceed down the mast,	 • .
6. the bottom mast has a uniform mass distribution. and
7. the mast may be assumed infinitely stiff in the axial direction.
The transfer of state variables from the feed assembly to just before
the upper branch point takes the form:
i
F A.
^y
U:.
To obtain the state variables at
from &11 the upper cables.
x
cobL
 a F6Lc"
IN'rRO ^vt : N G C 44* T/¢^ some 'R S Pf E d/oo3tr
b COO/Al E p '^ 	 E x DE PT T'ME s I G N s ARC REVOOM ^a AIL tU3
W^ ^R t its	 • •	 j"p,
4 - 0)	 ^YT3^j	 a
u	 Nom!
	
^	
^ ke
CF,
dr)	
t	 S^ N
'!y	 L0
-19-
G Bs Ti T v T 1 N G /NTo EPA-  Fe k	 true G Er -
UAb
La
t
Co ti'rpN u Al G T/hE rie&J s f 'S ,e . Dow N T7iE tA*z-r,
r
1AI)	 LOWER	 -
^o	 ^	 c,4.8c^ Fa2c Es
We R E	 Is- DEP 1 vy-p I N Tice sAme mrairt
ClioC F 
th R^.
t^
Lo	 ^o
CF
S uas-n Ty	 107b rft EoN FOA	 ) WE caN
uO	 L7cF	 L'
`	 R E A am*j 6E TER ms A•ND	 CxMEsslot1 FoA
/N TERMS o r.
	 ^	 7rAK /N C^ 'l'ffE Tw o Ei
F /Cole
	E	 '
	
SOt {/ FaR	 !N TEfft of
wpm
guc Cdr
p^CX3-^^1-C^^^7-[a7^Cos^-^^7-[a^rvl^ x
b ^^
^P.
flxl- IXI LPJ ^Exl -A CYC
RCxnn^^¢ rNe ABOVE	 -	 '
#N D Cott T/Nuiti1 G -rtt-c 'TTANSPE& m THE &77'OM ac
T1 ^ y1? H`.S T^ G^ E . G E T
DM	 Efl
WNEREr
lylCH]^ LDC^-^J]
. m
YI	 ^
t^
-15-
y	 ^
SUMMn
Note that in free vibration, the forces and moments at the extremes of
the antenna assembly are zero. So those state variable vectors are
O	 •	 I O	 i
I--L X IZ
T0 - 1)^Wol0
o -
O
CL
8•M
Which can be reduced to
o
14
	
c. x 6	 ^^
= DCwo)
i	 o	 s
O
The matrix
	 (Wr.) consists of elements of D(V at the intersections of
its 3rd, 4th, 7th, 8th, 10th b 12th columns and its 1st, 2nd, 5th, 6th, 9th
t
b 11th rows. For a non-trivial solution the determinant of :5 (W M) must
k`	 equal zero. Given values for the geometrical and physical cbaracteristics of
s
the system, we can now determine natural frequencies of the structure, by
trial and error, seeking those values of W4% which make the determinant zero.
4r0
o .
.P-0
a
0
q
FR
P-16-
FINALIZING THE FORMLATION OF THE ANALYSIS
The mast and hoop structure are, in one sense, the skeleton on which the
antenna reflecting surface is hung. It now remains to represent the mass and
stiffness characteristics of the antenna surface in the analysis. We anticipate
doing this by dealing with the antenna reflecting surface as a series of pie-
shaped segments supported at state-variable stations u:+ the hoop and mast.
^"t
17-
REFERENCESr
1. E. C. Pastel and F. A. Leckie, Matrix Methods in Elastomechanics
(McGraw-Hill Book Company, Inc., New York, 1963).
2. Roark, Formulas for Stress 6 Strain (McGraw-Hill Book Company, Inc,
New York, New York)
'	 1


Formulation of a dynamic analysis method for a generic family of hoop-mast antenna systems

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