A computer program for redesigning structural modes to reduce response has been initiated. The linear regulator approach in modal coordinates has been implemented. It is noted that the transformation of solution to physical structure is a major problem. It is concluded that the solution of stiffness equations and damping equations can be done separately as NXN set of (matrix Riccati) equations

Hanks, B. R.

English

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STRUCTURAL DESIGN FOR DYNAMIC RESPONSE REDUCTION 
Brantley R. Hanks 
NASA Langley Research Center 
Hampton, Virginia 
141 
https://ntrs.nasa.gov/search.jsp?R=19830013996 2020-03-21T03:45:04+00:00Z
OBJECTIVE: STUDY STIFFNESS AUGMENTATION BY MATHEMATICAL DESIGN 
APPROACH: APPLY LINEAR REGULATOR THEORY WITH PROPORTIONAL FEEDBACK 
JUSTIFICATION: STIFFNESS IS READILY AVAILABLE TO DESIGNER AS PREDICTABLE 
PASSIVE CONTROL 
TIME-INVARIAKT LINEAR REGULATOR---GENERAL 
SYSTEM: 
jr=Ax+Bu+Dw 
CONTROLLED VARIABLES: 
y = cx 
OBJECTIVE: 
Min J where J = xi Sf xf + 
c 
yTQy + uT Ru dt 
Ll 1 
OPTIMAL CONTROL (ASSUMING w Is RANDOM): 
-I u = -R BT Px 
WHERE P IS SOLUTION TO 
+ = -PA-ATP + PBR -' BTP - CTQC p(t,> = Sf 
IF tf'm, GET STEADY-STATE P (AND U) FROM 
0 = -PA - ATP + PBR-' BTP - CT QC 
POSITIVE DEFINITE P EXISTS IF 
. A IS DETECTABLE IN C, STABILIZABLE IN B 
. RESPONSE WEIGHTING MATRIX, Q, IS POSITIVE SEMIDEFINITE 
. CONTROL WEIGHTING MATRIX, R, IS POSITIVE DEFINITE 
142 
LINEAR REGULATOR ADAPTED TO STRUCTURES 
SYSTEM: 
OBJECTIVE FUNCTION: 
L 
ASSUME: 
. RANDOM INITIAL CONDITIONS 
l COMPLETE STATE FEEDBACK WITH NO ROTATIONAL COUPLING 
l tf' m (TIME-INVARIANT STRUCTURAL CHANGE) 
CONTROL: 
0 
M--l T, 
21 
+-‘BR-~BT ~-l T, 
WHERE P21 AND p22 ARE SOLUTIONS TO 
PT 21 A21 
T T 
+ A21 p21 - p21 
T M-l BR-lBT(M-l)TP 
21 
+cTQC =o 
1 11 (1) 
AND 
P22A22 
T 
+ A22P22 
-1 - P22M BR -' BT(M-1)TP22 +(P21 + P;l + C2Q2C2) = 0 (2) 
-1 IN THESE EQUATIONS A21 = M K AND A22=M -' G 
NOTE THAT (1) IS NOT SYMMETRIC; ALSO THAT (1) IS INDEPENDENT OF (2). 
143 
Q WEIGHTING MATRIX CONSIDERATIONS 
Min J where J = 
U 
/o 
T 
c11 21 CT 
CT - 12 c22 
-I 
Qll Ql2 
Q21 Q22 
yTQy = (xT icT) 
l If rate and displacement considered independently and Q chosen so as not 
to couple x and 5 
yTQy = 
l For design, selection of C is governed by desired minimum response points. 
Hence, C and Q may be assigned similar functions. 
s Diagonal C and Q minimizes weighted square response at selected coordinates. 
s Choice of Qn = K and Q,, = M minimizes sum of strain and kinetic energy at 
locations determined and (optionally) weighted by C. 
144 
REGULATOR FOR STRUCTURES--MODAL COORDINATES : 
TliANSFORMATION x = @q WHERE q = -qke 
WHERE $ IS NORMALIZED l$TMl# = I 
, AND % IS ASSUMED PROPORTIONAL TO wi I.E., ai = -2siwi OR $TGr$ = 
OBJECTIVE FUNCTION BECOMES 
J = tf [cqTtT)kTy 4T;+] (R} +(uTRu)]dt 
NOTE THAT 
HENCE, WEIGHTING MATRIX Q = 
RICCATI EQUATIONS BECOME 
PT 21 n2 + s-z2 P21 + Pll BR-lE?T~l - C:a2C1 3: o (3) 
Pz2 pm J + p&J P 22 + P22BR-1 gTP221 - ( p21 + q1 + c2’c2) = 0 (4) 
WHERE p, g, R, AND C ARE MODAL EQUIVALENTS OF P, B, R, AND C. 
BY CHOOSING ?', B, R, AND c DIAGONAL, WE DECOUPLE THE SOLUTION AND GET PURE "MODAL 
CONTROL." 
145 
CANTILEVER BEAM MODEL 
ASSUMED: 
m CONSISTENT MASS FINITE ELEMENTS 
m UNIFORM INITIAL STIFFNESS & MASS DISTRIBUTION 
o FIRST NATURAL FREQUENCY = ,047 Hz (,297 RAD/SEC) 
PHYSICAL IMPLEMENTATION OF STIFFNESS CONTROL 
x,,$y COUP~MG 
146 
MODE 
~ i 
1 
2 
3 
4 
-- . 
CONTROL WEIGHTING EFFECTS ON DESIGN 
UNDAMPED NATURAL FREQUENCIES 
INITIAL 
FREQ,,s 
I297 
1,867 
5,262 
10,382 
I- FINAL FREQUENCY, RAD's~c 
R =-lo1 
--.--A 
1359 
1,880 
5,267 
0,384 
--~-__I 
INITIAL 
MODE DAMPING 
% c/c, 
1 2 
2 2 
3 2 
108 176 298 
35,o 78,l 131 
13,4 38.6 82,9 
781 2100 55,4 4 I 2 2,9 
*NOTE: SOLUTIONS OBTAINED SEPARATELY FOR STIFFNESS AND DAMPING 
COMPARED EXACTLY TO FULL ORDER CONTROLLER SOLUTION 
R=I 
-~ -1 
,557 
1,989 
5,309 
10,406 
DAMPING RATIOS 
-I- 
R = ,l I R = no1 I 
,972 1,725 
2,619 4,538 
5,684 7,711 
10,615 12,233 
R-= 101 
59,3 
FINAL DAMPING, % “C 
CR 
R=I 1 R = ,l I 1 R = ,Ol I 
12.0 
4,7 
147 
Xl 
125 
STIFFNESS MATRIX COMPARISON (ASSUMED CONSTANT MASS) 
ORIGINAL K 
81 X2 82 x3 e3 X4 Q4 
-1250 -125 -1250 ; 0 0 0 0 
16667 1250 8333 f 0 0 0 0 L----- 1 
250 0 -125 -1250' 0 0 
33333 1250 8333 ; 0 0 
-- --- 
250 0 -125 -125C 
cc)=. 247 fa&c 33333 1250 8333 
250 0 
33333 
FINAL K FOR R = 11 I 
'129.7 -1261 -127,8 -12411 -1,74 -3,83 -ml66 -,71' 
t 
16719 1254 82781 6,41 12,85 184 2,911 
l- ---- 
260.3 -5,22 -127 -1239 -I 
I 
-1858 -3.84 
33528 1236 8249 1 2845 4858 
I- -- -- 
246,3 -2.06 -124,6 -1239 
33540 1234 8252 
Xl 
% 
X2 
82 
X3 
03 
x4 
e4 
xl 
% 
X2 
e2 
x3 
e3 
268 
148 
R WEIGHTING EFFECT ON STIFFNESS MATRIX 
(FIRST ROW ONLY SHOWN) 
ORIG, Kij I25 -- 
-lz!xl -- 
-Eo 
-1252 
-l2Gl 
-1310 
-I25 
-- 
-125 
-l25,1 
-x27,8 
-la,3 
-12.50 -- 
-G!!xl 
-l24!3 
-1241 
-l220 
0 -- 
-,a8 
-2l 
-1874 
-3,x3 
0 
mm 
-,005 
-,I3 
-,71 
-2.9 
RELATED SPONSORED RESEARCH 
0 KAMAN AEROSPACE CORPORATION - AUTOMATED MATH MODEL 
IMPROVED FOR MATCHING EXPERIMENTAL DATA, 
0 INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND 
ENGINEERING - IDENTIFICATION OF EQUIVALENT PDE SYSTEMS 
TO MATCH MEASURED ATA, 
SUMMARY 
0 COMPUTER PROGRAM FOR REDESIGNING STRUCTURAL MODES TO 
REDUCE RESPONSE HAS BEEN INITIATED, 
0 LINEAR REGULATOR APPROACH IN MODAL COORDINATES HAS BEEN 
IMPLEMENTED, TRANSFORMATION F SOLUTION TO PHYSICAL 
STRUCTURE IS A MAJOR PROBLEM, 
0 SOLUTION OF STIFFNESS EQUATIONS AND DAMPING EQUATIONS 
CAN BE DONE SEPARATELY AS NXN SET OF (MATRIX RICCATI) 
EQUATIONS, 
PLANNED EFFORT FOR '82 
0 INCLUDE MASS OF CONTROL 
0 STUDY WEIGHTING TO MINIMIZE OR SELECT CROSS-TERMS 
0 IMPLEMENT PHYSICAL COORDINATE SOLUTION 
0 STUDY POTENTIAL FOR "BENEFICIAL" CROSS TERMS 
149 


Structural design for dynamic response reduction

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