Damage Localization from Free Vibration Signals

Transcription

1 Mre inf abut this article: Damage Lcalizatin frm Free Vibratin Signals Yi ZHANG, Dinisi BERNAL 2 Schl f Aerspace Engineering and Applied Mechanics, Tngji University, Shanghai 292, P. R. China.455@tngji.edu.cn 2 Civil and Envirnmental Engineering Department, Nrtheastern University, Center fr Digital Signal Prcessing, Bstn MA 25 bernal@neu.edu Key wrds: Damage lcalizatin vectr, Free vibratin, Transfer functin. Abstract A methd t lcalize damage that uses free vibratin signals frm the damaged state and a mdel f the reference was recently develped[]. This paper summarizes the thery and examines an applicatin where the free vibratin after the passage f vehicles is emplyed t lcalize damage in bridges. The fact that the bridge is made up f multiple parallel beams with a deck that acts cmpsitely is accunted fr in the simulatins. INTRODUCTION This paper reviews a scheme fr damage lcalizatin that perates withut identificatin and which uses a mdel f the reference state and free vibratin respnse frm the damaged state. The methd is derived under the cmmn assumptin f finite dimensinality, linear behavir and sensrs cunt that is n less than the rank f the change in the transfer matrix resulting frm damage. The methd perates by regarding the free vibratin in the damaged state as a sum f a free decay respnse frm the undamaged state plus a frced respnse arising frm crdinates assciated with the damaged elements. The initial state cntributin is remved by prjecting the measurements in the null space f free decay respnse f the undamaged state, leaving nly the frced respnse. The fact that the residual respnse belngs t a space that is related t the damage is utilized fr lcalizatin. 2 DAMAGE LOCALIZATION FROM FREE VIBRATION SIGNALS Accepting the typical assumptins f linearity and finite dimensinality and assuming that damage can be treated as a change in the stiffness matrix K, the respnse t an arbitrary initial cnditin can be written as and, M x t Cx t K K x t x x x x () Frm eq.() ne can write d d d d d + + =, Mx t Cx t Kx t x x x x (2) h h h h h

2 Mx t + Cx t + Kx t = Kx () t x, x (3) p p p d p p where it is clear that we ve taken x t = x t x t (4) d h p Slving fr the respnse in eqs.2 and 3 in the Laplace dmain and adding the superscript t indicate measured crdinates ne has x xh s R() s x (5) and where Rs () C ( Is A), L [ M ] T m 2n, C R and u u x () s R() s L Kx () s (6) p u d Au 2n 2n R are the bservatin 2n 2n and transitin matrices and I Î R is the identity. Since the initial state in eq.(5) is nt a functin f the Laplace variable the expressin can be evaluated at k values f s and stacked. Cmbining the stacked equatin with the definitin intrduced in eq.(4) ne has defining Q k as xd s xp s R( s) x x xd sk xpsk R( sk) T T Rs ( ) Qk Null. (8) Rs ( k ) (7) And pre-multiplying eq.7 by Q k gives k k xd sk xpsk xd s xp s Q Q (9) Substituting eq.(6) int eq.(9) ne can write where Z( S) ( S) x (S) () q d 2

3 Z( S) x ( s ) d xd ( s2) Qk x d ( sk) () ( S) Q q k R( s ) L K R( s2)l K R( sk )LK (2) x d ( s) x d ( s2) xd ( S) x d( sk) (3) and S s s. The cnstraint explited in this apprach, which has been designated as k the free-vibratin damage lcalizatin scheme r FVDL, is t lcalize the damage using the fact that, as eq.() shws, Z(S) is cntained in the span f q when the damage distributin is crrect. Hw well Z( S ) fits in the span f q can be measured by the generalized subspace angle between them [] and we take the discriminating metric as z The prcess can be summarized as fllws: (4) sin( ). Select s-values 2. Fr each pstulated damage distributin cmpute the matrix Q k using a mdel 3. Measure the free vibratin frm damaged structure and calculate Z( S ) using eq.() 4. Calculate q using eq.(2) 5. Calculate z using eq.4 6. The methd anticipates the damaged pattern t be the ne assciated with the highest. z 3. MODELING CRACKS The presence f a crack disrupts elementary beam thery in the near vicinity f the crack. 3

4 One way t mdel the additinal flexibility is t add a rtatinal spring at the lcatin f the crack which, fr a prismatic beam with a rectangular crss sectin can be taken t have a stiffness k r given by [3]. k r EI ( ) 2h 2 (5) where crack depth and h (6) where h is the beam height. Anther way t capture the added flexibility is t prescribe a reduced flexural stiffness EI ver a small length f beam l e. The smeared apprach is useful fr extending the results t nn-rectangular crss sectins and is thus the ne that we use in the numerical sectin. T set the equivalence we assume the distance l e is sufficiently small t treat the mment ver it as cnstant s ne can write Frm where it fllws, after sme simple substitutins that Mle M (7) EI k r e h f ( ) (8) where f ( ) 2 ( ) 2 (9) The functin ( ) f is depicted in fig. fr defined using the value f the reduced EI at the crack lcatin. As can be seen, the effective length (nrmalized t the depth) increases with the depth f the crack t a maximum f arund ne, which ccurs at a nrmalized crack depth f arund.7, and then decreases. 4

5 .8 f ( ) Figure : Nrmalized effective distance f reduced inertia vs nrmalized depth f crack. 4 POSSIBLE BRIDGE APPLICATION A pssible applicatin f the lcalizatin apprach discussed is in bridges since free vibratin can be btained after the passing f vehicles. We begin with a simple examinatin where the bridge is simulated as a simply supprted rectangular cncrete beam. We take the span t be 5 m and the crss sectin.5 by. m. The Yung s mdulus and density are E=.4 Pa and =224kg/m 3 and the mving lad is taken as a 6-tn truck (mdelled as a single lad) that crsses with a velcity f m/s. The beam is discretized using elements and damping is assumed t be % f critical in each mde. Sensrs are spaced unifrmly every 5 meters with the first sensr lcated 3 meters frm the left supprt. There are, therefre, sensrs. We cnsider a nrmalized crack depth f.5 and find, frm fig. that the regin f reduced EI is.8h r.8m. Since the size f each beam element in the mdel is.5m the effective length is nt an even number f elements. Albeit nt equivalent at the level f curvature, the glbal respnse is essentially unaffected if ne adjust the effective EI t accunt fr the reduced length. Namely, ne can simulate the damage using nly ne element and take the reduced EI as the EI at the crack lcatin times.5/.8 =.625 i.e. (.25)(.625).78. The results, depicted in fig.2 fr the ideal case where there is n nise, and fr a nise with a standard deviatin f.m/sec 2 shw that the apprach prvides the crrect lcatin f the crack. 5

6 (a) (b) sensr lcatin Figure 2: Metric f eq.4 in a simply supprted beam with a crack with a depth f half the beam, traversed by a 6 tn lad at v = m/s; a) withut nise, b) including measurement nise with a standard deviatin f. m/s 2. Accunting fr the Deck Real bridges are nt a single beam but (in a very cmmn tplgy) a cllectin f parallel beam with a deck that prvides the radway and that acts cmpsitely with the beams, as shwn in fig.3. Figure 3: Crss-sectin f the bridge cnsidered, the deck is m.25m and each beam is.5m m. Fr lads that crss the bridge withut extreme eccentricity the analysis in this case can still be carried ut using a beam mdel using the effective length f reduced EI derived in sectin 3. One gathers that the presence f the deck will make the effect f a crack n the beams less imprtant and thus harder t lcate. T gain sme quantitative appreciatin fr this we cnsider the case where the beam previusly examined is repeated 4 times and used with a m wide rad way (.25m thick) t frm a bridge deck. Examinatin shws that in this case the mment f inertia at the crack lcatin is reduced t.29ei s the effective ver the same length f a.5m is in this instance (.29)(.625).37. The results btained fr different nise levels depicted in fig.4 shw that the damage is clearly lcated when the nise standard deviatin is.m/s 2 but the apprach fails at. m/s 2. 6

7 5 5 (a) 4 2 (b) 4 2 (c) sensr lcatin Figure 4: Metric f eq.4 in a bridge system, damage is a crack with.5m depth in 4 beams at 5.5 m frm the left supprt, the standard deviatin f the nise level is a) m/s 2, b).m/s 2, c).m/s 2. We nte that ne way t reduce the nise level in the free vibratin is t respnses frm multiple passages. Fr example, if we assume the weight f truck changes between 6-tn and 2-tn and the speed frm m/s and 2 m/s and simulate passages the lcalizatin using the average respnse fr the nise level in part (c) f fig.4 is shwn in fig. 5., which makes evident the imprvement realized sensr lcatin Figure 5: Same as fig.4c but based n the average f simulatins. 7

8 5 CONCLUSIONS The paper examined the ptential merit f a methd fr damage lcalizatin that emplys free decay respnse frm the damaged state and a mdel f undamaged structure t examine the lcalizatin f cracks in beam type bridges. The results btained are nt particularly encuraging since the damage cnsidered in the numerical examinatin is large and the level f nise at which the apprach lst reslutin is relatively small. It is nted, hwever, that imprvements can be realized, if the cnditins allw, by averaging the free vibratin respnses frm multiple vehicle passages. ACKNOWLEDGEMENT This research was carried ut during the stay f the first authr at Nrtheastern University in Bstn where he was hsted by the Structural Dynamics and Identificatin Lab. The supprt f the Natinal Science Fundatin f China under grant N and China Schlarship Cuncil during this stay is gratefully acknwledged. REFERENCES [] Y. Zhang, D. Bernal, Damage Lcalizatin frm Free Vibratin Signals, Junal f Sund and Vibratin, In review (26). [2] H.P. Gavin, Structural Element Stiffness, Mass, and Damping Matrices, Duke University, (24). [3] X. Zhu, S. Law, Wavelet-based crack identificatin f bridge beam frm peratinal deflectin time histry, Internatinal Jurnal f Slids and Structures, 43 (26) [4] D. Bernal, Damage lcalizatin frm the null space f changes in the transfer matrix, AIAA jurnal, 45 (27)