2000_modak_model updating using constrained optimization

8/6/2019 2000_modak_model Updating Using Constrained Optimization

1/9

PergamonMechanics Resea rch Comm unica t ions , Vol . 27 , No. 5 , pp . 543-55 I , 2000Cop yright 2000 Elsevie r S c ience LtdPrin ted in the US A. Al l r ights re se rved0093-6413/O01$-seef ront ma t te r

PII: S0093-6413(00)00128-2

M O D E L U P D A T I N G U S I N G C O N S T R A I N E D O P T I M I Z A T IO N

S . V . M o d a k , T . K . K u n d r a a n d B . C . N a k r aD e p a r t m e n t o f M e c h a n i c a l E n g i n e e r in g , I n d i a n I n s ti tu t e o f T e c h n o l o g y , H a u z K h a s ,Ne w Del h i , 1 1 0 0 1 6 , In d i a

(Received 29 M arch 2000; accepted or print 7 August 2000)

In t ro d u c t i o n

A v a i l a b i l i t y o f a n a c c u r a t e d y n a m i c f i n i t e e l e m e n t m o d e l o f a s t r u c t u r e i s v e r y i m p o r t a n t t od e s i g n e n g i n e e r s a s i t a ll o w s t h e m t o i m p r o v e t h e d y n a m i c d e s i g n o f t h e s t r u c t u re a t c o m p u t e rl e v e l r e s u lt i n g i n a n o p t i m i z e d d e s i g n a p a rt f r o m s a v i n g s i n t e rm s o f m o n e y a n d t im e . B u t t h e r em a y b e s o m e i n a c c u r ac i e s o r u n c e r t a in t i es t h a t m a y b e a s s o c ia t e d w i t h a f i n it e e l e m e n t m o d e l .T h e Di s c re t i s a t i o n e r ro r , a r i s i n g d u e t o ap p ro x i m a t i o n o f a co n t i n u o u s s t ru c t u re b y a f i n i t en u m b er o f i n d i v i d u a l e l em en t s , i s i n h e ren t t o t h e f i n i t e e l em en t t ech n i q u e . W h i l e o t h e ri n a c c u ra c i e s m a y b e d u e t o t h e a s s u m p t i o n s a n d s i m p l if i c a ti o n s m a d e b y t h e a n a l y s t w i t h r e g a rd st o t h e ch o i c e o f e l em en t s , m o d e l l i n g o f b o u n d ary co n d i t i o n s , j o i n t s , e t c . T h i s i s re f l ec t ed i n th ed i f f e r e n c e b e t w e e n f in i te e l e m e n t m o d e l p r e d i c t i o n s a n d t h e d y n a m i c t e s t d a ta . G i v e n t h eav a i l ab i l i t y o f an accu ra t e d a t a acq u i s i t i o n an d m eas u r i n g eq u i p m en t t h e m eas u red t e s t d a t a ,t h o u g h m a y n o t b e p r e c i s e , is g e n e r a l l y c o n s i d e r e d t o b e m o r e a c c u r a t e t h a n a n a l y t ic a l m o d e lp red i c t i o n s . T h e m o d a l t e s t i n g an d m o d a l ex t r ac t i o n m e t h o d s , [ 1 ,2 ] , a r e a l s o we l l d ev e l o p ed fo ro b t a i n i n g a r e l i ab l e e s t i m a t e o f t h e m o d a l d a t a . T h i s h as fo rm ed t h e b as is fo r ad ju s tm en t o rco r r ec t i o n o f a f i n i t e e l em e n t m o d e l , i n t h e l i g h t o f m ea s u red t e s t d a t a , wh i ch i s r e f e r r ed asm o d e l u p d a t i n g.A n u m b e r o f m o d e l u p d a t in g m e t h o d s h a v e b e e n p r o p o s e d i n r e c e n t y e a r s a s s h o w n i n t h es u rv ey s [ 3, 4] an d t h e d e t a i l s o f n u m b e r o f t h em ca n b e fo u n d i n t h e t ex t [ 5 ]. T h e m o d e l u p d a t i n gm et h o d s can b e b ro ad l y c l a s s i f i ed i n t o d i r ec t m e t h o d s , wh i ch a re e s s en t i a l l y n o n - i t e r a t i v e o n es ,an d t h e i t e r a t i v e m e t h o d s . A s i g n i f i can t n u m b er o f m e t h o d s , [ 6 ,7 , 8 ], wh i ch w ere f i r s t t o em erg e ,b e l o n g ed t o t h e d i r ec t ca t eg o ry . T h es e m e t h o d s u p d a t e d i r ec t l y th e e l em en t s o f s t if fn es s an dm a s s m a t r i c e s a n d a r e o n e s t e p p r o c e d u r e s . T h e r e s u l t i n g u p d a t e d m a t r i c e s t h o u g h r e p r o d u c em e a s u r e d m o d a l d a t a e x a c t l y b u t d o n o t g e n e r a l l y m a i n t a in s t ru c t u ra l c o n n e c t i v i ty a n d t h ec o r r e c t io n s s u g g e s t e d a r e n o t p h y s i c a l l y m e a n i n g fu l . I t e r a ti v e m e t h o d s h a v e g e n e r a l l y b e e nb a s e d o n e i t h e r m o d a l d a t a o r f r e q u e n c y r e sp o n s e f u n c t i o n ( F R F ) d a t a . A n a l y t i c a l m o d e lu p d a t i n g u s i n g m o d a l d a t a i n an i t e r a t iv e f r am e wo rk was f i r s t p ro p o s ed i n [ 9 ]. T h e u p d a t i n ge q u a t io n s w e r e b a s e d o n t h e f ir s t o r d e r a p p r o x i m a t i o n f o r t h e e i g e n v a l u e s a n d t h e e i g e n v e c t o r s int e rm s o f u p d a t i n g p a ram et e r s . I n [ 10 ] a m a t r i x p e r t u rb a t i o n t ech n i q u e fo r r eca l cu l a t io n o fe i g en s o l u t i o n an d e v a l u a t i o n o f e i g en d a t a s en s i ti v i ti e s h as b ee n u s ed . T h e e f f ec t o f i n c l u d i n g

54 3


2/9

5 44 S . V . M O D A K , T . K . K U N D R A a n d B . C . N A K R A

secon d order e igend ata sens i t iv i t ies was s tud ied in [11]. Rec ent ly in [12] it i s p ropose d to em ployb o th ana ly t i ca l and ex per imen ta l moda l da ta f o r eva lua t ing s ens i t iv i ty coe f f i c i en t s wi th theob j ec t ive o f improv ing convergen ce and widen ing the app l i cab i l ity o f m e thod to cas es wherethe r e i s h ighe r e r ro r magn i tude .There have b ee n a t t emp ts to u s e d i r ec t ly the meas u red FR F da ta f o r iden t i f y ing the s y s temmat r i ces a s done in [1 3 ] . I n [1 4 ] a me thod , nam ed as R es pons e func t ion me thod (R FM ) , has b eendeve loped wh ich u s es m eas u red FR F da ta to upda te an ana ly t i ca l mo de l .P a ram ete r i za tion o f a f in i t e e l emen t m ode l i s one o f the impo r tan t is s ues. G ener ic e l emen t mas sand s t i f fness matr ices hav e been in t rod uced in [15] whi le in [16] th is form ulat ion has beenapp l i ed to the p rob lem o f j o in t iden t i fi ca t ion . A s t r a t egy fo r pa r amete r i za t ion o f a we lded jo in tand a c l am ped end has b een p ropos ed in [17 ] and the pa r amete r s upda ted b y s ens i t iv ity me thod .S imi la r ly in [ 1 8] the m e thods o f pa r amete r s ub s e t s e l ec t ion has b een ex tended to the s e lec t ion o fg roups o f pa ramete r s .M ode l upda t ing u s ing gene t i c a lgo r i thms has b een p ropos ed in [ 1 9 ,20 ] a s i t i s ex pec ted tha t thegene t i c a lgo r i thms wou ld he lp to f ind the g lob a l min imum. An upda t ing app roach u s ing neu ra lne twork s b as ed on FR F -da ta i s p ropos ed in [21 ].I t e r a t ive me thod u s ing moda l da ta , wh ich has a l s o b een r e f e r r ed a s inve r s e e igens ens i t iv i tyme thod ( IES M ) in the l i te r a tu re , has b een amon g the mo re s ucces s fu l me thods o f mode lupda t ing . Th i s i s m os t ly b ecaus e o f the f l ex ib i l i ty i t o f f e rs in the ch o ice o f upda t ing pa ramete rs .I n th i s pape r a new mode l upda t ing me thod , named as mode l upda t ing u s ing cons t r a inedop t imiza t ion , wh ich s eek s to improve the co r r e la t ion b e tween the ana ly t i ca l and the meas u redmoda l da ta i s in t roduced . The p ropos ed me thod invo lves f r aming o f f in i t e e l emen t mode lupda t ing p rob lem as a cons t r a ined non l inea r op t imiza t ion p rob lem wh ich w hen s o lved y ie ld sco r r ec tions to the s e lec ted up da t ing pa r amete r s. An ob j ec t ive func t ion q uan t i f y ing the d i f f e r enceb e tween ana ly t i ca l and m eas u red na tu r a l f r eq uenc ies and m ode s hapes i s min im ized s ub jec ted toineq ua l i ty cons t r a in ts b as ed on b oun ds on u pda t ing pa r amete r s and va lues o f M oda l a s s u rancec r i te r ion (MA C ) [22 ].S ome numer ica l t e s t cas es have b een p r es en ted wh ich were conduc ted to demons t r a t e thee f f ec t ivenes s o f the p ropos ed me thod . Upda t ing u s ing IES M has a l s o b een pe r fo rmed wi th thein ten t ion o f a s s es s ing the pe r fo rmance o f the p ropos ed m e thod .Theory_

The p ropos ed mo de l upda t ing m e thod i s f i r s t des c r ib ed in th i s s ec t ion . Th i s i s f o l lowed b y a b r i e fdes c r ip t ion o f the Inve r s e e igens en s i t iv i ty m e thod ( IES M) .a ) P ropos ed me thod o f mod e l upda t ing u s ing cons t r a ined op t imiza t ion :In the p ropos ed me thod the p rob lem o f upda t ing o f a f in i t e e l emen t mode l in the l igh t o fmea s u red da ta i s f r amed as a cons t r a ined op t imiza t ion p rob lem. The d eve lopm en t o f theob j ec t ive func t ion and the cons t r a in t s b as ed on m oda l da ta , nam ely na tu r a l f r eq uenc ies and m odeshapes , i s expla ine d below.Le t { f~} and { fA} b e the vec to r s o f meas u red an d ana ly t i ca l na tu r a l f r eq uenc ies to b e u s ed inupda t ing . Thes e vec to r s can b e e s t ab li s hed b y f i rs t i den t i f y ing co r r e la ted m ode pa i r s (C MP s ),wh ich i s e s s en t i a l ly a l i s t i nd ica t ing co r r e s pondence b e tween meas u red and ana ly t i ca l modes ,u s ing a co r r e la t ion too l l ik e Moda l a s s u rance c r i t e r ion (MAC ) . The Euc l idean no rm-b as ednorm al ized percentage er ror in natura l f requencies is wr i t ten as,


3/9

M O D E L U P D A T I N G U S I N G O P T I M I Z A T I O N 5 45

I I G } - L fA 1 1 0 0 (1 )E q u a t i o n ( 1 ) r e p r e s e n t s e r r o r i n n a tu r a l f r e q u e n c i e s i n a n a v e r a g e s e n s e . T h e t o t a l p e r c e n t a g ee r r o r in m n u m b e r o f n a tu r a l f r e q u e n c ie s w i l l b e ,c , = m x F I (2)T h e n o r m - b a s e d n o r m a l i z e d p e r c e n t a g e e r ro r i n m n u m b e r o f m o d e s h a p e s i s w r i t te n a s ,

(3)

Where {qSx} and {@A} are the i TMs i m u l a t e d e x p e r i m e n t a l a n d a n a l y t i c a l m o d e s h a p e v e c t o rs . T h et e r m i n s i d e s u m m a t i o n s i g n i n e q u a t i o n ( 3 ) r e p re s e n t s a n a v e r a g e e r r o r i n o n e m e a s u r e dc o o r d i n a t e o f iTMm o d e s h a p e v e c to r . M u l t i p l ic a t i o n o f th i s t e r m b y n , t h e n u m b e r o f m e a s u r e dc o o r d i n a te s , g i v e s a m e a s u r e o f e r r o r in a n e n t ir e i h m o d e - s h a p e - v e c t o r . I n t h i s w a y t h e t o t alp e r c e n t a g e e r r o r i n m n u m b e r o f m o d e s h a p e s c a n b e w r i tt e n a s ,~'2 = n x F 2 (4)T h e o b j e c t i v e f u n c t i o n t o b e m i n i m i z e d , b a s e d o n e r r o r i n n a tu r a l f r e q u e n c ie s a n d m o d e s h a p e sg iv e n b y e q u a t i o n s ( 2 ) a n d ( 4 ) r e s p e c t i v e ly , is c o n s t r u c t e d a s ,

(5)(u ) = ~ e, + G e2W l a n d W 2 in e q u a t i o n ( 5 ) a r e t h e w e ig h t s t o b e g iv e n t o t h e n a tu r a l f r e q u e n c i e s - b a s e d a n d t h em o d e s h a p e s - b a s e d e r ro r s .T w o s e t s o f c o n s tr a i n ts a r e i m p o s e d o n t h e o b j e c t iv e f u n c t io n t o b e m i n i m i z e d . T h e f ir s t s e t o fc o n s t r ai n t s a r e th e l o w e r a n d u p p e r b o u n d s o n t h e v e c t o r o f c o rr e c t i o n f a c to r s { u} , w h o s ee l e m e n t s r e p r e s e n t t h e u n k n o w n f r a c t i o n a l c o r r e c t i o n s t o b e m a d e t o t h e c h o s e n u p d a t i n gv a r i a b l e s , a n d i s w r i t t e n a s ,{ u } L s < _ {u}< { u } v s (6)

T h e s e c o n d s e t o f c o n s t ra i n t s s e e ks t o p u t a l o w e r b o u n d o n t h e M A C - v a l u e s o f t h e m o d e s u s e di n u p d a t in g . T h i s s e t o f c o n s t r a in t s a r e i m p o s e d t o e n s u r e t h a t t h e l e v e l o f c o r re l a t io n b e t w e e nm e a s u r e d a n d a n a l y t i c a l m o d e s s h a p e s , r e p r e s e n te d b y M A C - v a l u e , e x i s t i n g b e f o r e u p d a t i n g i s atl e a s t r e t a in e d a f t e r u p d a t i n g . T h e c o n s t r a in t f o r i h CM P i s g iv e n b y f o l l o w in g i n e q u a l i t y ,


4/9

5 4 6 S . V . M O D A K , T . K . K U N D R A a n d B . C . N A K R A

W h ere r i g h t an d l e f t s id e o f t h e i n eq u a l i t y rep res en t s M A C -v a l u e c o r r e s p o n d i n g t o i h C M Pb efo re u p d a t i n g an d a t j th i t e r a ti o n r e s p ec t i v e l y . I f t h e u n co n s t r a i n ed v e r s i o n o f t h e m e t h o d i su s ed , wh ere t h e M AC -v a l u e co n s t r a i n t s a r e n o t i m p o s ed , t h en C M P s wi l l h av e t o b e e s t ab l i s h eda t each i te r a t io n . T h e M AC -v a l u e i s ca lcu l a t ed a s,

( 8 )

T h e o b jec t i v e fu n c t i o n g i v en b y eq u a t i o n ( 5 ) i s m i n i m i zed s u b jec t ed t o i n eq u a l i t y co n s t r a i n t sg i v en b y eq u a t i o n ( 6 ) an d ( 7 ) . T h e an a l y t i ca l n a t u ra l f r eq u en c i es an d m o d e s h ap es ap p ea r i n g i nt h e o b j ec t i v e fu n c t i o n an d co n s t r a i n t s a r e r e l a t ed t o an a l y t i ca l s t if fn es s m a t r i x [ KA] an d a n a l y t i ca lm as s m a t r i x [ M A] b y an e i g en v a l u e p ro b l em , wh i ch fo r i h m o d e i s wr i t t en a s ,

= , V A [ M . ] { O . } ' (9 )

W h e r e ~A=(2"/~fA) . Th e [K A ] and [MA] a re i n t u rn t h e fu n c t i o n s o f t h e s e l ec t ed u p d a t i n gp aram et e r s . T h e o p t i m i za t i o n p ro b l em i s s o l v ed u s i n g ro u t i n e fo r co n s t r a i n ed m i n i m i za t i o n o fn o n l i n ea r fu n c t i o n s av a i l ab l e i n M AT LAB . T h e ro u t i n e i s b as ed o n s eq u en t i a l q u ad ra t i cp ro g ram m i n g i n wh i ch a t ev e ry i t e r a ti o n a q u ad ra t i c p ro g ra m m i n g s u b -p ro b l em i s fo rm u l a t edan d s o l v ed . T h i s r eq u i r e s t h e f i r s t an d s eco n d o rd e r d e r i v a t i v es o f th e o b j ec t i v e fu n c t i o n an d t h eco n s t r a in t s . T h e f i r s t o rd e r d e r i v a t i v es were d e r i v ed an d s u p p l i ed t o t h e ro u t i n e wh i l e t h e Hes s i anm at r i x , t h e m a t r i x o f s eco n d o rd e r d e r i v a t iv es , i s co n s t ru c t ed b y t h e ro u t i n e i t se l f .b ) In v e r s e e i g en s en s i t i v i t y m e t h o d ( IES M ) :T h e u p d a t i n g e q u a t io n s f o r t h is m e t h o d a r e b a s e d o n l in e a r a p p r o x i m a t i o n o f t h e m o d a l d a t au s i n g T ay l o r ' s s e r i e s ex p an s i o n . Fo r r h e ig en v a l u e ( s q u a re o f t h e r h n a t u ra l f r eq u en cy i nrad . /s ec . ) an d r h e i g en v ec t o r (t h e m o d e s h ap e) , l i n ea r i s a ti o n g i v es ,

"" f O X ); o r . = , ~ ~ " / ' p , (10)

{ : } i = { : } ; + Z { : } , . A p , ( i 1 )W h ere t h e e i g en v a l u e an d e i g en v ec t o r d e r iv a t i v es can b e ca l cu l a t ed f ro m t h e r e l a ti o n s h i p sder ived in [23] . A p i i s a smal l change in the i th u p d a t i n g p a ram et e r P l, wh i ch a re n u i n n u m b er s .Eq u a t i o n ( 1 0 ) an d ( 1 1 ) can b e wr i t t en fo r a ll t h e m o d e s t o b e u s ed i n u p d a t i n g . T h e s e eq u a t i o n st o g e t h e r , a f t e r d i v i d i n g an d m u l t i p l y i n g b y P i an d t h e n wr i t i n g u i i n p l ace o f Ap i /p ,, can b e wr i t t eni n th e fo l l o w i n g m a t r i x fo rm ,[ s L + , , . . . . { u L x , = 1 2 )


5/9

M O D E L U P D A T I N G U S I N G O P T I M I Z A T I O N 5 47The s ens i t iv ity ma t r ix [S ] depends o n the e igenv a lue and e igenvec to r de riva t ives and cu r r en tva lues o f the upda t ing pa r amete r s w h i l e vec to r .{AZ} con ta in s d i f f e r ence o f ex pe r imen ta l andcu r r en t ana ly t ica l mode l na tu r a l f r eq uenc ies and mode s hapes . The o rde r o f r ows in [S]co r r es pond ing to e igenva lues i s no t the s ame as the o rde r o f the rows co r r e s pond ing to modes hapes . To improve the num er ica l cond i t ion , b a lanc ing i s done b y d iv id ing the row s r e l a ting tothe e igenva lues on b o th the s ides in eq ua t ion (1 2 ) b y the co r r e s pond ing e igenva lue . The ab oveb a lanced eq ua t ion can b e s o lved in a l eas t s qua re way . The {u} s o found i s u s ed to upda te vec to ro f phys ica l va r i ab les {p} and then the upda ted v e r s ion o f ana ly t i ca l f in i t e e l emen t mo de l i s b u i l tu s ing thes e new s e t o f phys ica l va ri ab les . Due to the u s e o f l inea r app rox ima t ion , in eq ua t ion(10) and (11) , the updat ing is car r ied out in an i tera t ive way unt i l convergence is ob ta ined . Theaccu racy o f the m eas u red da ta can b e t ak en in to accoun t in the fo rm o f a we igh t ing ma t r ix [W] ,in which case the so lu t ion for {u} is g iven by ,{ u } = ( I s l [ w l s l ) - ' [ s I l { z } ( 1 3 )

Des cr ip t ion o f the cas e s tudyUpd a t ing o f an und am ped f in i t e e l emen t mo de l o f a s imp le b u t r ep res en ta tive f ix ed - f ix ed b eams t ruc tu r e u s ing s imu la ted ex per imen ta l da ta i s cons ide r ed to demo ns t r a t e the e f f ec t ivenes s o f thep ropos ed me thod . The d im ens ions o f the b eam a r e 9 1 00x 50x 5 m m and i t i s m ode led u s ing th i r tyb eam e lemen t s wi th node a t ends f ix ed g iv ing a to ta l o f 29 nodes each ha v ing th r ee deg rees o ff r eedom, ax ia l , l a t e r a l and ro ta t iona l . The s imu la ted ex per imen ta l na tu r a l f r eq uency and modes hape da ta i s ob ta ined b y gene ra t ing a f in i t e e l emen t mode l b y in t roduc ing ce r t a in k nownd is c repanc ies in the th ick nes s o f a l l t he f in i t e e l emen t s wi th r e s pec t to the a na ly t i ca l mod e l thed e t a il s o f w h i c h a r e g i v e n in T A B L E 1.The p ropos ed me thod has b een eva lua ted fo r the cas es o f comple te , i ncomple te and no i s y da tas ince the r ea l l i f e meas u red da ta i s a lways incomple te , a s i t i s no t r ea l i s t i c to meas u re a l l t hecoo rd ina te s s pec i f i ed in the ana ly t i ca l f in i t e e l emen t mode l , and a lways con ta in s s omeme as u rem en t no i s e . Two l eve l s o f incomple tenes s a r e cons ide r ed b y a s s um ing tha t on ly l a te r a ldeg rees o f f r eedom , a t a l l the 29 nodes and then a t 1 5 nodes s e lec ted a l t e rna te ly , ar e meas u red .Th i s has b een r e f e r r ed a s cas es o f 33 .33% and 1 7 .24% incomple te da ta r e s pec t ive ly , thepe rcen tages b e ing ca lcu la ted a s numb er o f meas u red coo rd ina te s d iv ided b y to ta l numb er o fdeg rees o f f r eedom o f the mode l . The no i s e i s s imu la ted b y add ing 2% and 0 .2% random e r ror tom ode s hapes and na tu r a l f r eq uenc ies re s pec t ive ly .T h e p e r f o r m a n c e o f t h e p r o p o se d m e t h o d i s j u d g e d o n t h e b a si s o f t h e a c c u r a c y w i t h w h i c h t h ed i s c r epanc ies b e tween the ana ly t i ca l and the s imu la ted ex per imen ta l mode l a r e p r ed ic ted . Thepercen tage avera ge er ror in the vector of cor rect ion factors {u} (A EC F) is ca lcu la ted as error inthe p r ed ic ted co r r ec t ion f ac to rs a s a pe r cen tage o f the k now n ex ac t co r r ec t ion f ac to r s a s g iven b yfo l lowing ex p res s ion ,

n uZ a b s ( u e x A c r - - u e R E o t c r e o ) iA E C F = i ~. x 1O0 (14)E a b s ( u E x A c r ) i

i


6/9


R es u l t s an d d i s cu s s i o nFo r a l l th e t e s t cas es r ep o r t ed h e re t h e l o we r an d u p p er b o u n d v a l u es i n e q u a t i o n ( 6 ) a re t ak en a s-0 . 5 an d 0 . 5 r e s p ec t i v e l y an d s o l v ed b y t ak i n g f i r s t s i x s i m u l a t ed ex p er i m en t a l n a t u ra lf r eq u en c i es an d m o d e s h ap es a s t h e av a i l ab l e m eas u red d a t a u n l es s s t a t ed o t h e rwi s e . T h et h i ck n es s o f each f i n i t e e l em en t is t ak en a s an u p d a t i n g p a ram et e r r e s u l t i n g in t h i r t y n u m b er o fu n k n o w n s . T h e c a s e o f c o m p l e t e d a ta i s c o n s i d e r e d f ir s t i n w h i c h a l l th e d e g r e e s o f f r e e d o ms p ec i f i ed i n t h e f i n i t e e l em en t m o d e l a r e a s s u m ed t o b e k n o wn . T h e d i s c rep an c i es a r e p red i c t edex ac t l y a s s een i n F IG . ( 1 ) s h o w i n g t h e b a r ch a r t fo r th e e s t i m a t ed v a l u es o f t h e co r r ec t i o nfac t o r s , C o n s i d e r i n g t h e cas e o f 3 3 . 3 3 % i n co m p l e t e d a t a t h e d i s c rep an c i es a r e ag a i n p red i c t edex ac t l y . In q u an t i f i ed t e rm s t h i s i s i n d i ca ted b y t h e v a l u e o f t h e i n d ex A EC F w h i ch i s fo u n d t o b e2 e - 0 5 % .Fo r t h e cas e o f 1 7 . 2 4 % i n c o m p l e t e d a t a t h e an a l y t i ca l f i n i te e l em en t m o d e l h as a l s o b een u p d a t edu s i ng t h e b a l a n c e d v e r s i o n o f l E S M . S i n c e th e r e is n o n o i s e th e w e i g h t s W 1 a n d W 2 i n e q u at io n( 5 ) a r e t ak en a s u n i t y an d s i m i l a r l y n o we i g h t i n g i s co n s i d e red fo r IES M as we l l . F IG . ( 2 ) s h o wsc o m p a r i s o n o f t h e p r e d i c te d c o r r e c t i o n fa c t o rs o b t a in e d b y t h e p r o p o s e d m e t h o d a n d t h e I E S M .T h e p r e d i c t i o n s o b t a i n e d b y t h e p r o p o s e d m e t h o d a r e a l m o s t e x a c t , a s c a n b e s e e n w h e nco m p ared t o F IG . ( 1 ), wh i l e t h o s e o b t a i n ed b y IES M are r e l a t i v e l y m o re i n e r ro r . Up d a t i n g i sa l so p e r f o r m e d w i t h 5 a n d 4 n u m b e r o f s i m u l a te d e x p e r i m e n t a l m o d e s u s e d i n u p d a t in g i nad d i t i o n t o th e cas e i n v o l v i n g 6 m o d es . T h i s i s d o n e t o s ee t h e e f f e c t o n t h e p e r fo rm an ce o f t h ea l g o r it h m w h e n f e w e r n u m b e r o f m o d e s a r e a va i la b le . T A B L E 2 g i v e s a c o m p a r i s o n o f v a l u e s o fe r r o r i n d e x A E C F a f t e r u p d a t i n g o b t a i n e d b y t h e p r o p o s e d m e t h o d a n d I E S M w h e n 4 , 5 a n d 6s i m u l a t ed ex p er i m en t a l m o d es a r e u s ed . I t is s een t h a t th e e r ro r i n p red i c t i o n s o b t a i n ed b y IES Ma r e o n m u c h h i g h e r s i d e a s c o m p a r e d t o t h a t o b ta i n e d b y t h e p r o p o s e d m e t h o d . T h e h i g h e r l e v e lo f e r r o r i n c a s e o f I E S M i s p r im a r i l y d u e t o t h e i n s u f f i c ie n t w e i g h t a g e t h e m o d e s h a p es r e c e i v eas co m p ared t o t h e n a t u ra l f r eq u en c i es . T h i s is o n acco u n t o f l a rg e d i f f e r en ce b e t weens en s i t iv i t i e s o f m o d e s h ap es an d n a t u ra l f r eq u en c i es w i t h fo rm er b e i n g o n t h e l o w er s id e . AM o d e s h a p e c o n t a in s a r e l a t iv e l y m o r e d e t a i l e d m o d a l d o m a i n d e s c r i p t i o n o f t h e s t r u c tu r e t h ant h e c o r r e s p o n d i n g n a tu r a l f r e q u e n c y w h i c h b e i n g j u s t a s i n g l e n u m b e r c a n b e s a i d t o b e m o r e o f ag l o b a l p ro p e r t y o f t h e s t ru c t u re . T h e u s e o f m o d e s h ap e i n fo rm a t i o n , it s eem s , i s t h e re fo rei m p o r t an t t o g u i d e t h e a l g o r i t h m t o ward s a s o l u t i o n t h a t i s p h y s i ca l l y c l o s e r t o t h e ac t u a ls t ru c t u re . T h e p ro p o s ed a l g o r i t h m u s es a n o rm al i zed o b jec t i v e fu n c t i o n , wh i ch a l l o ws s u f f i c i en tw e i g h t a g e b e i n g g i v e n t o t h e e r r o r in m o d e s h ap e s . I n p r a c ti c e t h e m o d e s h a p e s o b t a i n e d f r o m ad y n a m i c t e s t a r e l es s accu ra t e t h an t h e n a tu ra l f r eq u en c i es . T h i s m a y b e acco u n t ed fo r b ys u i t ab l y s e l ec t i n g we i g h t s W i an d W 2 in eq u a t i o n ( 5 ) t o r ep res en t m eas u rem en t accu racy .Fo r t h e cas e o f 1 7 . 2 4 % i n co m p l e t e d a t a w i t h 2 % n o i s e r e l a t i v e we i g h t i n g s a r e a l s o i n co rp o ra t eds o a s t o r e f l ec t t h e n o i s e p res en t i n t h e s i m u l a t ed n a t u ra l f r eq u en c i es an d m o d e s h ap es . T h ewei g h t i n g m a t r i x [ W ] i s t ak en a s a d i ag o n a l m a t r i x w i t h d i ag o n a l en t r i e s b e i n g t ak en a sr e c i p r o c a ls o f t h e v a r i a n c e s o f t h e c o r r e s p o n d i n g s i m u l a te d m e a s u r e d q u a n t it y . T h e s i m u la t e dn o i s e o f 0 . 2 % o n n a t u r a l f r e q u e n c i e s a n d 2 % o n m o d e s h a p e s i s ta k e n a s a m e a s u r e o f s ta n d a r dd ev i a t i o n t o e s t i m a t e t h e v a r i an ces . F IG . ( 3 ) s h o ws a c o m p ar i s o n o f p red i c t ed co r r e c t i o n fac t o r so b t a i n ed b y t h e t wo m e t h o d s . I t i s c l ea r , a f t e r co m p ar i n g t h i s f i g u re w i t h F IG . ( 1 ) , t h a t t h ep r e d i c t e d c o r r e c t i o n f ac t o r s o b ta i n e d b y t h e p r o p o s e d m e t h o d a r e c l o s e r to t h e i r e x a c t v a l u e s t h e nt h o s e o b t a i n e d b y t h e I E S M . I n q u a n t i f ie d t e r m s t h e v a l u e o f i n d e x A E C F a f t e r u p d a t in g i s f o u n dt o b e 2 0 . 7 % f o r t h e p r o p o s e d m e t h o d a n d 3 3 . 6 % f o r t h e I E S M .In t e rm s o f t h e co m p u t a t i o n a l e f fo r t r eq u i r ed , i t i s fo u n d t h a t t h e p ro p o s ed m e t h o d i sc o m p u t a t io n a l l y m o r e i n t e n s iv e a n d a l so s l o w e r a s c o m p a r e d t o t h e I E S M . T h i s i s o n a c c o u n t o ft h e f ac t t h a t t h e p ro p o s ed m e t h o d i s b as ed o n n o n l i n ea r o p t i m i za t i o n ag a i n s t IES M t h a t i s b as edo n l i n ea r ap p ro x i m a t i o n .


7/9

M O D E L U P D A T I N G U S I N G O P TI M IZ A TI ON 5 4 9

Conclusion

A new model upda t ing method based on cons t ra ined opt imiza t ion has been proposed. Theprop osed m ethod addresses th e di f f icul ty tha t ma y a r i se d ue to the la rge di f fe rence be tween thesensi tivi ties o f natural frequencies and mo de shapes. The pro po sed method, thoughcomputa tiona lly m ore intens ive , i s wo rking succe ssful ly for the case of com ple te and incompletedata. The method also predicted the correct ions required in the analyt ical model in the case ofnoisy d a ta wi th reasonable accuracy.

Re fe re nc e s1. D.J. Ewins, M od al Test ing : Th eory and Pract ice. R esear ch studies press, Letchworth

(1984)2. N.M . Maia and J.M.M . e Si lva, Theoret ical and experimental m od al analysis. Researchs tudies pres s , John W iley and Sons (1997)3 M . Imregun an d W.J. Visser, The Sho ck and Vibrat ion D igest 23(1), 9(1991)4 J.E. M ottershead and M .I. Friswell, Journal o f So un d and Vibrat ion 167(2), 347(1993)5 M .I. Frisw ell and J.E. M ottershead, Fini te Element M od el U pd ating in Structural Dynam ics.Kluw er Aca dem ic Publ ishers , Dordrecht (1995)6 M . Baru ch and I.Y. Bar-Itzhack, A IA A Journal 16(4), 346(1978)7 A. Berm an and E.J . Nag y, A IA A Journa l 21(8), 927(1983)8 M . Baruch, A IA A Journal 22(4), 561(1984)9 J.D. Coll ins, G.C. Hart , T.K. Ha sselm an and B. Kennedy , AIA A Journal 12(2), 185(1974)10 J.C. Chen and J. Garba, AI A A Journal 18(6), 684(1980)11 K.O. Kim , W .J. And erson and R.E. Sandstorm, A IA A Journal 21(9), 1310 (1983)12 R.M. Lin, M.K. Lim and H. Du, A SM E Journa l o f Vibrat ion and Aco ust ics 117 , 192(1995)13 C .P. Fri tzen, A SM E Journal of Vibrat ion, A coust ics, Stress and Re liabi l i ty in De sign 108,9(1986)14 R.M . Lin and D.J. Ewins, Proceedings o f the 15 ~ Internat ional Seminar o n M oda l Analysis,Belgium, 141(1990)15 G.M .L. Gladwell and H. Ahm adian, Mechanical Sy stem s and Signal Processing 9(6),601(1995)16 M .J.Ratc l ifee and N.A.J . L ieven, M echanica l Sys tem s and Signa l Process ing 14(1) , 3(2000)17 J.E. M ottershead, M .I. Friswell, G .H.T. N g and J.A. Brandon, M echanical S ystem s andSignal P rocessing 10(2), 171 (1996)18 M .I. Friswell, J.E. Mottershead and H. Ahmad ian, Journal o f Vibrat ion and acoustics 120(4),854(1998)19 R.I. Lev in and N.A.J. Lieven, Mech anical Sy stem s and Signal Processing, 12(1), 91(1998)20 D.C.Zimmerm an, K. Ya p and T. Hasselman, M echanical Sy stem s and Signal Processing13(4), 609(1999)21 M .J. Atal la and D.J. Inman, M echanical Sy stem s and Signal Pro cessin g 12(1), 135(1998)22 R.L. A llemang and D.L. B row n, Proceedings o f the 1 t International M oda l Analysisconference, Florida,USA, 110(1982)23 R.L. Fox and M.P. Kapoor , AIA A Journal 12(6) , 2426(1968)


8/9


T A B L E iDi s c rep a n c i es b e t ween t h e f i n it e e l em en t an d th e s i m u l a t ed ex p er i m en t a l m o d e lElement Num ber 3 5 11 16 25 29 [ All other elements

% Dev ia ti on i n T h ick ness + 2 0 % + 4 0 % + 2 5 % + 4 0 % + 3 0 % + 3 0 % ] + 1 0 %

T A B L E 2C o m p a r i s o n o f v a l u e s o f e r ro r in d e x A E C F w h e n 4 , 5 a n d 6 n u m b e r o f s i m u l a te d e x p e r im e n t a l

m o d es a r e u s ed i n u p d a t i n gN o . o f m o d e s u s e d U s i n g t h e p r o p o s e d U s i n g t h e I E S Mm e t h o d

6 0 . 0 9 1 % 2 5 . 85 %5 0 . 0 7 2 % 3 4 . 2 6 %4 3 . 7 6 4 % 3 3 . 8 7 %

1 . 4 - . , .. o

8

0 .4

0 .3

0 .2

0.1

0 1 5 1 0 1 5 2 0 2 5 3 0C o r r e c ti o n F a c t o r N u m b e r

FIG. 1C o r r e c t i o n f a c to r s a f t e r u p d a ti n g f o r t h e c a s e o f c o m p l e t e d a t a p r e d i c t e d b y p r o p o s e d m e t h o d


9/9

MODEL UPDATING USING OPTIMIZATION 551

0. I0 .:OJ0 .1

0

Proposed method IESM

1 5 1 0 1 5 2 0 2 5 3 0C o r r e c t i o nF a c t o rN u m b e r

. !o

1 ~ 1U 13 LU lid ,,~UC o r r e c t i o n F a c t o r N u m b e r

FIG. 2Comparison of predicted correction factors for the case of 17.24% incomplete data

Proposed method IESM0 .4

0. 3

mu0 I 5 10 15 20 25 30

Correct ion Factor Number

0.~r

o , , 0 . : ' " I

0 1 5 1 0 1 5 2 0 2 5 3 0Correction Fa ctor Number

FIG. 3Comparison of predicted correction factors for the case of 17.24% incomplete data with 2%noise

2000_modak_model updating using constrained optimization

Documents