an experimental modal analysis technique for large-scale structures

8/8/2019 An Experimental Modal Analysis Technique for Large-scale Structures

1/7

I n g e n i e u r - A r c h i v 6 0 (1 9 8 9 ) 1 1 7 - - 1 2 3 I n g e n i e u r - A r c h i v Springer-Verlag 1989

A n e x p e r im e n t a l m o d a l a n a ly s i s t e c h n iq u e f o r l a r g e - s c a l e s t r u c t u r e sJ. Wallasehek, Darmstadt

S u m m a r y : Using the theory of stationary random vibrations an experimental modal analysis technique,previously suggested and applied by Luz [2--4], is derived. This method does not require an artificialexcitation of the structure under investigation and is based on measurements of the system's velocityonly. I t is well suited for applications to large-scale structures such as high-rise buildings, towers or bridges.E i n V e r f a h r e n z u r e x p e r i m e n t e l l e n M o d a l a n a l y s e b e i g r o B e n S t r u k t u r e n[~bers ieht : Mit I-Iilfe der Theorie stationi~rer Zufallsschwingungen wird die yon Luz [2--4] vorgeschlageneMethode zur experimentellen Modalanalyse erkl/rt. Dabei wird keine k/instliche Erregung der zu unter-suchenden Strnktur ben6tigt und es genfigt, die Schwinggeschwindigkeit des Systems zu messen. I)as Verfah-ren ist speziell ffir grote Banwerke, wie z. B. Hoehhiiuser, Tfirme oder Talbrficken geeignet.

1 I n t r o d u c t i o nToday experimental modal analysis has become an accepted method for determining the dynamiccharacteristics of linear st ructural systems. The most imp ortan t information about linear struc-tures are essentially the eigenfrequencies and the associated mode-shapes. Since these propertiesbelong to a mathema tical model of the str ucture rat her than to the struct ure itself, they cannotbe measured directly. Usually the structure is excited at certain points and its dynamical responseto this excitation is observed. Then frequency response functions arc calculated by means of fastFourier transform techniques and the parameters of an associated mathematical model (eigen-frequencies and mode-shapes) are estimated using curve-fitting: Although this method hasbeen strongly advanced over the years, it features a serious drawback which becomes obviouswhen largeseale structures such as high-rise buildings or bridges are to be analysed: It is theartificial excitation of the structure by means of shaking or impact which is often not appropria tefor large structures. The huge amount of energy which is necessary to induce structural vibrationsmay cause local damage, and the measurement of the exciting forces usually is not a simple task.

Because of this problem especially civil engineers became interested in experimental modal-analysis techniques which require no artificial excitation. These methods use excitations bynatural microtremors, wind or traffic loads which are available everywhere and at any time. Ofcourse, no quantitative information about the exciting forces can be used in this ease. Never-theless the determination of the eigenfrequeneies and mode-shapes of the structure is possiblein most situations, and how this can be done will be described in the following.

2 F o r m u l a t i o n o f t h e p r o b l e mFor illustration we discuss the ease of a one-dimensional continuous system whose deflection isgiven by the scalar function w ( x , t ) of the spatial variable x and time t . The definition of thesystem' s transfer function G ( x , ~; D) is based on the stea dy-st ate response to a single conce ntrat ed


2/7

118 Ingenieur-A rchiv 60 (1989)

Fig. 1 . Defini t ion of the t ransfer funct ionh a r m o n i c e x c i t a t i o n F ~ ( x - - ~ ) e j ~t ( s ee F i g . 1 , 8 : D i r a c d i s t r i b u t i o n ) . T h e t r a n s f e r f u n c t i o n i sd e f i n e d a s t h e r a t i o

G ( x , ~ ; ~ ) = W ( x ) / F (1 )( g e n e r a l l y c o m p l e x - v a l u e d ) b e t w e e n t h e a m p l i t u d e W ( x ) o f t h e r e s p o n s e w ( x , t) = W ( x ) e ~gta n d t h e f o r c e a m p l i t u d e f . T h e t r a n s f e r f u n c t io n d e p e n d s o n t h e l o c a t io n ~ w h e r e t h e fo r c e i sa p p l i e d a n d o n t h e f r e q u e n c y t g .

W e w i ll a s s u m e t h a t t h e s y s t e m s t o b e i n v e s t i g a t e d a r e s u b j e c t t o s o -c a ll ed p r o p o r t i o n a ld a m p i n g . T h e n G ( x , ~ ; D ) c a n b e w r i t t e n a s

G(x, ~; O) = ~ H~(O)Ck (x ) ~ k (~ ) , ( 2 )k--1a b i l i n e a r f o r m o f th e ( r e a l- v a l u e d ) m o d e - s h a p e s ~ k(x ) w h e r e

1 1H k ( D ) - - ( 3 )m k w ~ - - ~ + 2 jD kD ~ o ki s c a ll e d t h e m o d a l t r a n s f e r f u n c t i o n , m k i s th e m o d a l m a s s , D ~. t h e m o d a l d a m p i n g a n d w k t h ee i g e n f r e q u e n c y a ss o c i a te d w i t h m o d e # .

O u r a i m i s t o e s t i m a t e ~ k (x ) a n d cok u s i n g m e a s u r e m e n t s o f t h e s y s t e m ' s r e s p o n s e w ( x , t ) o n l y .T h e r e s u l t c a n t h e n e a s i l y b e g e n e r a l i z e d t o t h e g e n e r a l c a s e o f t w o - a n d t h r e e - d i m e n s i o n a ls t r u c t u r e s a l l o w i n g f o r d i s p l a c e m e n t s i n a n y d i r e c t i o n .

3 S p e c t r a l t h e o r y o f r a n d o m v i b r a t i o n sW e a r e i n t e r e s t e d i n t h e s t r u c t u r e ' s r e s p o n s e t o r a n d o m e x c i t a t i o n s l i k e w i n d - p r e s s u r e s o r m i c r o -t r e m o r i n d u c e d b a s e - m o t i o n s . T h e s e l o a d i n g s c a n b e c o n s i d e r e d t o b e t h e r e a l is a t i o n s o f s t o c h a s t i c( fi e ld - ) p r o c e s s e s a n d i n t h i s c a s e a l s o t h e d e f l e c t i o n o f t h e s y s t e m h a s r a n d o m c h a r a c t e r . I n t h em a t h e m a t i c a l s e n s e s to c h a s t i c ( fi el d- ) p r o c e s s e s a r e d e s c r i b e d b y p r o b a b i l i t y d i s t r i b u t i o n s d e f i n e do n p r o b a b i l i t y s p a ce s . H o w e v e r , t h e m o s t i m p o r t a n t i n f o r m a t i o n f o r e n g in e e r in g a p p l i c a t io n s c a nb e o b t a i n e d f r o m j u s t t h e f i rs t - a n d s e c o n d - o r d e r m o m e n t s .

L e t { / ( x , t ; l ~ ) l d e n o t e t h e s t o c h a s t i c f i e l d - p r o c e s s o f t h e e x c i t a t i o n i n F i g . 2, w h e r e , u i su s e d t o d i s t i n g u i s h b e t w e e n t h e v a r i o u s r e a l i s a t i o n s / ( x , t ) w h i c h f o r m t h e f i e ld - p r o c es s . T h e f i r s t -a n d s e c o n d - o r d e r m o m e n t s o f { / ( x , t ; # ) } a r e c a l l e d m e a n

m 1 ( x , t ) = E [ / ( x , t)] (4)a n d c o r r e l a t i o n f u n c t i o n

K : : ( x , y ; t , u ) ~ - E [ / ( x , t ) / ( y , u)] (5)w h e r e E [ ] i s t h e e x p e c t a t i o n o p e r a t o r . T h e s e m o m e n t s c a n b e e x p r e s s e d i n t e r m s o f t h e p r o b a -b i l i t y d i s t r i b u t i o n s o f { / ( x , t ; # )} a n d c a n e q u i v a l e n t l y b e d e f in e d b y t i m e - a n d s p a c e - a v e r a g e s if t h ep r o c e s s i s e r g o d i c .

I~ - - L if ( x , t )

w(x,t ) Fig . 2 . Beam under s tochast ic exci ta t ion


3/7

J . Wa l lasch ek : An ex p er im en ta l mo d a l an a ly s i s t ech n iq u e fo r large-scale s t r u c tu r es 1 1 9L e t { w ( x , t ;/ ~)} b e t h e b e a m ' s d e f l e c t io n . T h e j o i n t s t a t i s t i c s o f { / ( x , t ; # )} a nd { w ( x , t ; # } c an b e

d e s c r i b ed b y t h e c r o s s -c o r r e la t i o n f u n c t i o nK 1 u ( x , y ; t , u ) = E [ / ( x , t ) w ( y , u ) ] . ( 6 )

T h e s t o c h a s t i c f i e l d - p r o c e s s e s { / ( x , t ; # )} an d { w ( x , t ; # ) } a r e r e l a te d t o e a c h o t h e r b e c a u s e t oe v e r y r c a l i s a t i o n o f { / ( x , t ; / ~ ) } - - s p e c i f i e d b y f i x i n g # - - t h e r e i s e x a c t l y o n e r e a l i s a t i o n o f{ w ( x , t ; # )} w i t h t h e s a m e #.

I f f o r e x a m p l e t h e b e a m o f F i g . 2 i s d e s c r ib e d b y t h e m o d e l o f E u l e r a n d B e r n o u l l i a n d i fv i s c o us e x t e r n a l a n d i n t e r n a l d a m p i n g a r e t a k e n i n t o a c c o u n t , t h e n t h e r e l a t i o n b e t w e e n t h el o a d i n g a n d t h e d e f l e c ti o n i s g i v e n b y t h e e q u a t i o n o f m o t i o n

1 @ d ., - f i t E I --8 x @ d ~ - ~ @ o A - f ff i w ( x , t ) = / ( x , t ) (7 )( E l : b e n d i n g s t i f f n e s s ; A : c r o s s - s e c t i o n a l a r e a ; o : m a s s d e n s i t y ; d , d .,: e x t e r n a l a n d i n t e r n a ld a m p i n g ) a n d t h e b o u n d a r y c o n d i t i o n s

w ( x , t ) l ~ = o = O ,

~w(X,~x t) x=0 = 0 ,

w ( x , t ) x = L = o ,

~ 2 w ( x , t ) I = 0~ X 2 I x = L

( 8 )

w h i c h h o ld fo r e v e r y p a i r o f r e a li s a ti o n s . I t c a n b e a s s u m e d t h a t s i m i l a r m a t h e m a t i c a l r e l at i o n sb e t w e e n e x c i t a t i o n a n d t h e s y s t e m ' s r e s p o n s e e x is t fo r m o s t m e c h a n i c a l s y s t e m s e v e n if t h ee q u a t i o n o f m o t i o n o r t h e b o u n d a r y c o n d i t io n s a r e n o t k n o w n .

T h e d y n a m i c s o f l i n e a r s y s t e m s c a n b e d e s c r i b e d b y m e a n s o f t h e i m p u l s e r e s p o n s e fu n c t i o ng ( x, e ; t ) w h i c h d e s c r i b e s t h e m o t i o n a t l o c a t i o n x d u e t o a u n i t i m p u l s e a p p l i e d a t l o c a t i o n e a tt i m e t = 0 a s s u m i n g t h a t t h e s t r u c t u r e w a s a t r e s t f o r t < O . I t c a n b e s h o w n t h a t t h e i m p u l s er e s p o n s e f u n c t i o n i s t h e i n v e r s e F o u r i e r t r a n s f o r m

+c o' f( x , e ; t ) - ~ a ( x , e; ~9) e j~t dr9 (9)-o o

o f t h e t r a n s f e r f u n c t i o n . B y v i r t u e o f t h e s u p e r p o s i t i o n p r i n c i p l e t h e r e s p o n s e t o t h e e x c i t a t i o n/ ( x , t ) c a n t h e n b e e x p r e s s e d a s a c o n v o l u t i o n in t e g r a lt

w ( x , t ) = f f g ( x , e ; t - r ) / ( e , r ) d e d r (10)0 D

w h e r e t h e i n t e g r a t i o n o v e r e e x t e n d s o n t h e w h o l e d o m a i n D a n d v a n i s h i n g in i t i a l c o n d it i o n s h a v eb e e n a s s u m e d .

T h i s d e t e r m i n i s t i c r e l a t i o n h o l d s f o r e v e r y p a i r o f r e a l is a t i o n s a n d a f t e r t a k i n g t h e e x p e c t a t i o no n b o t h s i d e s o f ( 1 0 ) w e f i n d

tt ) = f f e ; t - r ) r ) d e d r (11)

0 D

f o r t h e m e a n a n d

K w w ( x , y ; t , u ) - - f f / f . q (x , e ; t - - v ) g ( y , ~ ; u - - o ) K # ( e , ~]; v, a , d~ d~] d~ dr (12)O D O D

f o r t h e c o r r e l a t i o n f u n c t i o n o f t h e s y s t e m ' s r es p o n s e .M o s t s t o c h a s t i c p r o c e s s e s w h i c h a r e e n c o u n t e r e d i n t e c h n i c a l a p p l i c a t i o n s c a n b e c o n s i d e r e d

t o b e s t a t i o n a r y , w h i c h m e a n s t h a t t h e i r f i r s t- a n d s e c o n d - o r d e r m o m e n t s d o n o t d e p e n d o n t h ec h o ic e o f t h e ti m e - o r i g in . A s a c o n s e q u e n c e t h e m e a n d e p e n d s o n l y o n t h e s p a t i a l c o o r d i n a t e ,a n d t h e c o r r e l a t i o n f u n c t i o n d e p e n d s o n l y o n t h e s p a t i a l c o o r d i n a t e s a n d t h e t i m e - l a g t - - u


4/7

120 Ingenieur-Arehiv 60 (1989)b e t w e e n t h e t w o i n s t a n t s w h i c h a r e c o m p a r e d . W i t h o u t l o s s o f g e n e r a l i t y o n l y c e n t e r e d p ro c e ss e s ,w h o s e m e a n v a n i s h e s i d e n t i c a l l y , n e e d t o b e c o n s i d e r e d , si n c e t h i s c as e c a n a l w a y s b e o b t a i n e db y a s i m p l e c o o r d in a t e t r a n s fo r m a t i o n . T h e o n l y i m p o r t a n t i n f o r m a t i o n a b o u t ~ s t o c h a st i cf i el d - p ro c e s s i s t h e n c o n t a i n e d i n t h e c o r r e l a t io n f u n c t i o n .T h e c o r r e l a ti o n f u n c t i o n o f a s t a t i o n a r y s t o c h a s t i c f ie l d -p r o c e ss c a n b e w r i t t e n a s t h e i n v e r s eF o u r i e r t r a n s f o r m

Kf1(x , t ; y , u ) ~ - f ~ oz (x , y ; 17 ) e j9t-~) d/2 (13)o f t h e n o n - n e g a t i v e f u n c t i o n ~ z ( x , y ; 17) w h i c h i s c a l l e d t h e s p e c t r a l d e n s i t y a n d c o n t a i n s i n -f o r m a t i o n a b o u t t h e p o w e r l o c a l i z a t io n o f t h e s t o c h a s t i c f ie l d -p r o c es s {] (x , t ; /~)} , see [1].

I f t h e e x c i t a t i o n p r o c e ss i s s t a t i o n a r y , t h e n a l s o t h e s y s t e m ' s r e s p o n s e w i l l b e s t a t i o n a r y a n dy ,~w(X, y; 17) = f f G (x , $; D) ~os](~,~ ; D) G * ( y , r t ; D) d~ d~ (14)

D Dc a n b e d e r i v e d f r o m ( 1 2 ) f o r t h e s p e c t r a l d e n s i t y o f t h e s y s t e m ' s r e s p o n s e . O n c e t h e s p e c t r a ld e n s i t y ~ o~ (x , y ; D ) i s k n o w n , a l l o t h e r r e l e v a n t i n f o r m a t i o n s a b o u t t h e s y s t e m ' s r e sp o n s e c a n b ec a l c u la t e d , e .g . t he va r i a nc e o f t he d e f l e c t ion is

+e ca2,~(x) = E[w 2(x , t)] = f ~%~(x, x; 17) d/2 . (15)

No te tha ~ a l l t he se f unc t ions c a n be e xp r e s se d in t e r m s o f ~OzI(~ # ; 1?) a nd the t r a ns f e r f un c t ion o ft h e s y s t e m .S o f a r w e h a v e o n l y b e e n c o n c e r n e d w i t h t h e d i s p l a ee m e n ~ o f t h e s t r u c t u r e . W e c a n h o w e v e re a s i l y c a l c u l a t e t h e c o r r e l a t i o n f u n c t i o n s a n d s p e c t r a l d e n s i t i e s o f o t h e r s y s t e m r e s p o n s e s , l i k ee .g . b e n d i n g - s t r a i n o r v e l o c i t y ; i n t h e l a t t e r e a s e w e f i n d

~ o ~ ( z , y ; 1 7 ) = 1 ? ~ w ( X , y ; 1 7 ) , (16)F o r a p h y s i c a l s y s t e m t h e s p e c t r a l d e n s i t y o f th e v e l o c i t y c a n e a s i l y b e d e t e r m i n e d w i t h t h e h e l po f a d i g i t a l s i g n a l a n a l y s a t o r i f t h e v e l o c i t y o f t h e s t r u c t u r e i s m e a s u r e d .4 Determination of eigenfrequeneies and mode-shapesI f ( 14 ) a nd ( 2) a r e i n se r t e d in to ( 16 ), we f ind

v~(x , y; ~) = E E ~% ~(~) H~(~)//~(~) ~(x) ~(y) (17)k=l 1=1

f o r t h e s p e c t r a l d e n s i t y o f t h e v e l o c i t y w h e r eIkt(D) = f f ~0z(x, y; D) ~E(X) ~l(Y) dx dy (18)

D D

c a n b e i n t e r p r e t e d a s t h e c r os s s p e c t r a l d e n s i t y o f g en e r a l iz e d f or c es . R e a r r a n g i n g ( 1 7) w e o b t a i n~ ( x , y ; 2) = ~ ~ 2 I k k ( ~ 2 ) l H k ( 1 ? ) l 2 ~ % ( X ) % ( y )

k = l

+ E E 17%~(~) H~(~) H~(1?)~(~ ) ~(y ) (19)k~l / = 1k=~l

w h e r e t h e c o n t r i b u t i o n o f t h e t e r m s h a s b e e n s e p a r a t e d i n t o t w o p a r t s . S e t t i n g x ~ y a n d i n t e -g r a t ing ( 19 ) ove r D y i e ld s~ o ~ ( D ) = f y J ~ ( x , x; D) dx = ~ 1?eIkk(D) IHk(X2)i2 Jke

D k=l

+ ~ s~%~(17)//k(~) H~(1?)Jk~ (20)k= lk~-i


5/7

J . Wallasehek: An experim ental modal analysis technique for large-scale s t ructures 121R e H k

I1

\ J(d k

Im H k

1

W k

Fig. 8 . Modal t ransfer Iunet ion H e(Dw i t h

Jel = f We(x) ~q(x) dx . (21)D

F o r s t r u c t u r e s w i t h c o n s t a n t m a s s - d i s t r i b u t i o n J e t = 0 h o l d s f o r k 4 = l b e c a u s e o f t h e o r t h o g o n a -l i t y o f t h e e i g e n f u n c t i o n s . A l t h o u g h J k , w i l l n o t v a n i s h i n g e n e r a l , t h e c o n t r i b u t i o n o f t h e m i x e dt e r m s i n ( 20 ) i s c o n s i d e r a b l y s m a l l e r t h a n t h a t o f t h e f i r s t s u m a s l o n g a s th e e i g e n f r e q u e n c i e s o ft h e s y s t e m a r e w e l l s e p a r a t e d . T h i s i s d u e t o t h e f a c t t h a t f o r n o t t o o l a r g e d a m p i n g t h e f u n c t i o nH e ( D ) h a s a p r o n o u n c e d p e a k n e a r t h e e i g e n f r e q u e n c y o k w h e r e a s i t is s m a l l fo r o t h e r f r e q u e n c i e s(see F ig . 3 ) .

F o r b r o a d b a n d e x c i t a t i o n I ez (tg ) i s a s lo w l y v a r y i n g f u n c t i o n o f D , s o t h a t ( 20 ) a t t a i n s i t sr e l a t i v e m a x i m a w h e r e t h e f u n c t i o n s jH e(.c 2)l~ a r e m a x i m a l . T h e s e f r e q u e n c i e s a r e

De = toe V1 -- 2D~ (22)a n d l ie c lo s e t o t h e e i g e n f r e q u e n c i e s of t h e s t r u c t u r e i f t h e d a m p i n g i s s m a l l. N e g l e c t i n g t h ed a m p i n g i n ( 2 2 ), w h i c h s e e m s j u s t i f i e d f o r m o s t c i v i l e n g i n e e r i n g s t r u c t u r e s , w e c a n l o c a t e t h ee i g e n f r e q u e n c i e s we a t t h e f r e q u e n c i e s w h e r e v - p~ (~ ) a t t a i n s i t s r e l a t i v e m a x i m a .

I n o r d e r t o f i n d t h e m o d e - s h a p e s ~ k ( x ) w e t a k e i n t o a c c o u n t t h a t d u e t o t h e c h a r a c t e r i s t i cb e h a v i o r o f H k (f 2) t h e s u m ( 19 ) i s l o c a l l y , f o r D ~ ~ oe , d o m i n a t e d b y a s i n g l e t e r m

w ~ , ~( x, y ; P - ~ w e ) ~ ~ [ e e ( ~ e ) I H e ( ~ e ) l ~ ~ e ( x ) ~ e ( Y ) . ( 2 3 )D i v i d i ng t h e n ~ t ~ ( x , y ; c o e ) b y T ~ ( y , y ; o~ e) w e e s t i m a t e t h e m o d e - s h a p e ~ k( x) a s b e i n g p r o p o r -t i o n a l t o

~ ( x , y ; o~e) (24)~] ~( y , y ; o~e) "T h e m e t h o d p r o p o s e d t o d e t e r m i n e ~ oe a n d ~ e (x ) c a n b e a p p l i e d i f t h e v e l o c i t y o f t h e s t r u c t u r e

u n d e r i n v e s t i g a t i o n i s m e a s u r e d . I t i s r e s t r i c t e d to t h e e a s e o f s l i g h t l y d a m p e d s t r u c t u r e s w i t hs e p a r a t e d e i g e n f r e q u e n c i e s a n d s t a t i o n a r y s t o c h a s t i c e x c i t a t i o n s w h i c h h a v e n o n - v a n i s h i n gs p e c t r a l c o n t e n t in t h e f r e q u e n c y - r a n g e i n v e s t i g a t e d . H o w e v e r , t h e s e l i m i t i n g c o n d i t i o n s a r e m e tb y m o s t l a r g e -s c a l e c i v il e n g i n e e r i n g s t r u c t u r e s .

5 Applicat ionsT h e m e t h o d a s d e s c r i b e d in t h i s p a p e r h a s b e e n s u c c e s s f u l l y u s e d i n t h e p a s t t o d e t e r m i n e e i g e n -f r e q u e n c i e s a n d e i g e n m o d e s o f c i v i l e n g i n e e r i n g s t r u c t u r e s [ 2 - - 4 ] . A m o n g t h e s y s t e m s w h i c h h a v eb e e n i n v e s t i g a te d w e r e t h e f r e e w a y b r id g e o v e r t h e K o e h e r - v a l l e y i n s o u t h e r n G e r m a n y ( s p an1 1 0 0 m , h e i g h t 1 80 m ) , a H D I ~ n u c l e a r p o w e r p l a n t a n d a h i g h r is e b u i l d in g a t t h e U n i v e r s i t y o fS t u t t g a r t . I n a l l e a se s t h e r e s u l ts s h o w e d g o o d a g r e e m e n t w i t h d a t a o b t a i n e d b y o t h e r m e t h o d s .T y p i c a l l y a b o u t 1 00 m e a s u r e m e n t p o i n ts h a v e b e e n s e l e c te d a n d t h e v e l o c i t y o f t h e s t r u c t u r eh a s b e e n m e a s u r e d u s i n g W i l l m o r e - t y p e s e i s m o m e t e r s o f h i g h s e n s i t i v i t y .


6/7

122 Ingenieu r-Archiv 60 (1989)

6 S om e ope n p r ob le m sB e s i d e s th e p r i n c i p a l l i m i t a t i o n s d i s c u s s e d a t t h e e n d o f S e ct . 4 a n o t h e r p r o b l e m r e l a t e d t o t h em e t h o d d e s c r i b e d a b o v e m a y a r i s e if t h e m a i n s o u r c e o f e x c i t a t i o n i s m i c r o - t r e m o r i n d u c e d b a s em o t i o n . E s p e c i a l l y f o r s t i f f s t r u c t u r e s t h i s m a y b e t h e m a i n e n e r g y s o u rc e . T h e n t h e b o u n d a r yc o n d i t i o n s o f t h e s y s t e m a r e n o lo n g e r a u t o n o m o u s . S i n c e t h e b o u n d a r y c o n d i t i o n s ar c e s s e n t i a lf o r t h e e i g e n f r e q u e n e i e s a n d e i g e n m o d e s , w e m u s t c l e a r l y s p e c i f y w h i c h b o u n d a r y c o n d i t i o n sa p p l y i n t h e t h e o r e t i c a l i n v e s t i g a t i o n .

I n o r d e r t o f i x i d e a s l e t u s c o n s i d e r t h e b e a m o f F i g . 4 w h i c h i s ex c i t e d b y t h e ( m i c r o - t r e m o ri n d u c e d ) m o t i o n o f i t s e n d - p o i n t s a s w e l l a s b y t r a n s v e r s e ( w in d ) l o a d i n g . A s s u m i n g t h e m o d e l o fE u l e r - B e r n o u l l i t o b e a p p r o p r i a t e t o d e s c r ib e t h e s y s t e m ' s d y n a m i c s t h e e q u a t i o n o f m o t i o ni s (7 ) a n d t h e b o u n d a r y c o n d i t i o n s a re

w ( x , t ) x=o = - - y l ( t ) ' w ( x , t ) x = L = - - y . ~ ( t ),

~2w(x,~x t ) x= o = O , ~w(X,~x t) ~=L = O .( 2 5 )

A r e f e re n c e m o d e l w h i c h i s a p p r o p r i a t e f o r t h i s p r o b l e m i s t h e s i m p l y s u p p o r t e d b e a m , a n da p p l y i n g e x p e r i m e n t a l m o d a l a n a l y s i s t o t h i s p r o b l e m w e w o u l d l ik e t o i d e n t i f y t h e e i g e n f re -q u e n e i es a n d e i g e n m o d e s o f a s i m p l y s u p p o r t e d b e a m .

I t i s h o w e v e r n o t p o s s i bl e t o o b t a i n t h e s o l u t i o n o f (7 ), ( 25 ) d i r e c t l y i n t e r m s o f t h e e i g e n m o d e so f t h e s i m p l y s u p p o r t e d b e a m b e c a u s e o f t h e i n h o m o g e n e o u s b o u n d a r y c o n d i t i o n s , [5 ]. W e w i l lt h e r e f o r e i n t r o d u c e a v a r i a b l e

(26)L - - x xv ( x , t ) = w ( x , t ) + ~ y ~ ( t ) + - - /y 2 ( t )

w h i c h d e s c r i b e s t h e e l a s t i c d e f o r m a t i o n o f t h e b e a m w i t h r e s p e c t t o t h e s t r a i g h t l i n e p a s s i n gt h r o u g h t h e b e a m ' s e n d - p o i n t s . U p o n i n s e r t i o n o f ( 26 ) i n ( 7) a n d ( 25 ) w e o b t a i n1 + d2 ~ E I - - + d l + e A v ( x ,t )

= / ( x , t ) + L - - x x[~A~)l(t) + dl~)i(t)] @ -~ [~A~)~(t) + diy~(t)] (27)a n d h o m o g e n e o u s b o u n d a r y c o n d i t io n s . T h e t r a n s f o r m a t i o n (2 6) t h u s r e d u c e d t h e e q u a t i o n o fm o t i o n ( 7) w i t h t i m e - v a r y i n g , in h o m o g e n e o u s b o u n d a r y - c o n d i t i o n s ( 2 6 ) t o a n e q u a t i o n o f m o t i o nw i t h c o n s t a n t , h o m o g e n e o u s b o u n d a r y c o n d i t io n s .

I n o r d e r t o f i n d t h e e i g e n f r e q u e n e i e s a n d e i g e n m o d e s o f t h e s i m p l y s u p p o r t e d b e a m b y t h em e t h o d d e s c r i b e d i n t h i s p a p e r m e a s u r e m e n t s o f t h e ( r e l a ti v e ) v e l o c i t y ~(x , t) h a v e t o b e m a d eb e c a u s e t h e b o u n d a r y c o n d i t i o n s a re o n l y h o m o g e n e o u s i n t h e v a r i a b l e v( x, t ). W i t h t h e h e l p o fs e i s m o m e t e r s , h o w e v e r , o n l y t h e a b s o l u t e v e l o c i t y , b ( x , t ) c a n b e r e g i s t r a te d . I f t h e t r a n s f o r m a t i o n(2 6) is n o t t a k e n i n t o a c c o u n t a n d e i g e n f re q u e n e i e s a n d e i g e n m o d e s a r e c a l c u l a t e d f r o m ( 2 2)a n d ( 24 ), t he e ige n f r e q ue nc ie s w i l l s t i l l be c o r r e c t , bu t t h e e ige nm ode s w i l l be e r r a ne ous .

L Hf ( x , t )

y~ ( t ) , Y2 ( t )

f w ( x , t ) Fig'. 4. Beam with moving supports


7/7

J. Wal laschek: An experim enta l modal analysis technique for large-scale st ructures 123I f w e a s s u m e f o r s i m p l i c i t y , t h a t t h e p r o c e s s e s y l ( t ) a n d y 2 ( t ) a r e s t a t i o n a r y , m u t u a l l y u n -

c o r r e l a t e d a n d t h a t t h e y a r e u n c o r r e l a t e d w i t h / ( x , t ) w e f i n d2 L - x L - s . ( + )~ ( x , y ; o ) e ) ~ ~ % I e e (c o e ) l H e ( o e )l 2 q ) e( x ) q ~ e (Y ) ~ - ~ - - L ( 2S )

w h e r e S n ( D ) a n d $ 22 (~ 2) a r e t h e p o w e r - s p e c t r a l d e n s i t ie s o f y l( t) a n d y 2 ( t ) , r e s p e c t i v e l y a n d i t i se a s i ly s e en t h a t ~ m a y n o t b e u s e d t o d e t e r m i n e t h e m o d e - s h a p e s .

S i n c e it is r e l a t i v e l y e a s y t o t a k e t h e t r a n s f o r m a t i o n (2 6) i n to a c c o u n t , i t is s u g g e s t e d t h a t t h em o t i o n o f t h e s t r u c t u r e ' s b o u n d a r y i s m e a s u r e d s i m u l t a n e o u s l y . } ' o r t w o - a n d t h r e e - d i m e n s i o n a ls t r u c t u r e s , h o w e v e r , t h is w i l l i n c r e a s e th e n e c e s s a r y m e a s u r e m e n t e f f o r t .

7 C o n c l u s i o n sI n t h i s p a p e r a n e x p e r i m e n t a l m o d a l a n a l y s i s t e c h n i q u e f o r l a r g e - sc a l e s t r u c t u r e s w h i c h w a sp r o p o s e d b y L u z [ 2 - - 4 ] h a s b e e n e x p l a i n e d u s i n g t h e t h e o r y o f s t a t i o n a r y r a n d o m v i b r a ti o n s .I t h a s b e e n s h o w n t h a t i t is p o s s i b l e i n m a n y c a s e s t o i d e n t i f y t h e e i g e n f r e q u e n e i e s a n d e i g en -m o d e s o f a s t r u c t u r e b y m e a s u r e m e n t s o f t h e v e l o c i ty o n ly . T h e p r o b l e m o f t i m e v a r y i n gb o u n d a r y c o n d i t i o n s , w h i c h a ri s e s i n b a s e - m o t i o n e x c i t e d s t r u c t u r e s , h a s a l s o b e e n a d r e s -s ed . T h e p r o p o s e d m e t h o d m i g h t b e c o m e i m p o r t a n t i n c o n n e c t io n w i t h v i b r a t io n - m o n i t o r i n g a n dc r a c k i d e n t i f i c a t i o n i n c i v il e n g in e e r i n g s t r u c t u r e s .

References1. Lin, Y. K. : Probabi l i s t ic theory of s t ructu ra l dy namics. M alabar : Krieger-P ubl . Hous e 19672. Luz, E. : Exp erim enta l mo dal analysis of ]argescale st ructures. Pap er presented a t the Conf . on vibra t ionprob lem s in engineering, C hina, Ju ne 19863. Luz, E. : Best immung yon Bauwerksparametern and -zust~nden mit Hi l fe yon Schwingungsmessungenun ter stoch astisch er Anregung. M aterialpr/i~ung 28 (1986) 173--17 74. Luz, E. : Zur experimente l len Modalanalyse yon Bauwerken. Mater ia lprf i fung 28 (1986) 301--3065. Meirovitch, L . : Elem ents of vibra t ion analysis . To kyo : McGraw Hil l 1975.

R e c e iv e d N o v e m b e r 1 0 , 1 9 8 8Dr . - Ing . J . Wa l l a schekIns t i t u t f f ir Mechan ikTeehnische Hoehschu le Da rm stad tHochschulstrai~e 1D - 61 00 D a r m s t a d tFede ra l Repub l i c o f Ge rm any

9*

an experimental modal analysis technique for large-scale structures

Documents