[PPT] - In-operation subspace-based covariance-driven structural PowerPoint Presentation

SLIDE 1

In-operation subspace-based covariance-driven structural identification Steelquake and Z24 bridge benchmarks

Laurent M´ evel, Mich` ele Basseville, Albert Benveniste, IRISA, Rennes, France Maurice Goursat, INRIA, Rocquencourt, France basseville@irisa.fr - http://www.irisa.fr/sigma2/ Toolboxes: LMS CADA-X, and Scilab ftp://ftp.inria.fr/INRIA/Projects/Meta2/Scilab/contrib/MODAL/

1

SLIDE 2

Problem : In-operation modal identification of structures

The excitation is typically:

– natural, not controlled. – not measured: ∗ buildings, bridges, offshore structures, ∗ rotating machinery, ∗ cars, trains, aircrafts. – nonstationary (e.g., turbulent).

How to merge multiple measurements setups

e.g. in case of moving sensors?

2

SLIDE 3

Identification and merging

Output-only eigenstructure identification,
In the presence of nonstationary excitation,
Handling moving sensor pools, with some reference sensors.

Wanted:

Avoid merging identification results from the different pools,
Merge the data instead, and process them globally,
Use a standard subspace algorithm.

3

SLIDE 4

– Modelling – Output-only covariance-based subspace identification – Merging multiple measurements setups – Robustness to nonstationary excitation – Numerical results - Steelquake – Numerical results - Z24 bridge

4

SLIDE 5

Modelling FE model:

                  

M ¨ Z(s) + C ˙ Z(s) + KZ(s) = ν(s) Y (s) = LZ(s) (Mµ2 + Cµ + K)Ψµ = 0 , ψµ = LΨµ

State space:

                  

Xk+1 = F Xk + Vk Yk = HXk F Φλ = λ Φλ , ϕλ

∆

= HΦλ eδµ = λ

modes

, ψµ = ϕλ

mode shapes

5

SLIDE 6

Output-only covariance-based subspace identification

Ri

∆

= E

 Yk Y T

k−i

 

k if stationary !

, H =

              

R0 R1 R2 . . . R1 R2 R3 . . . R2 R3 R4 . . .

. . . . . . ... . . .

              

Ri = H F i G , G ∆ = E

 Xk Y T

k

 

O ∆ =

        

H HF HF 2

. . .

        

, C ∆ =

G F G F 2G . . .
H = O C , H −

→ O − → (H, F ) − → (λ, ϕλ)

6

SLIDE 7

Implementation

ˆ Ri

∆

= 1 N

N

k=1 Yk Y T

k−i

k when nonstationary !

, ˆ H =

            

ˆ R0 ˆ R1 ˆ R2 . . . ˆ R1 ˆ R2 ˆ R3 . . . ˆ R2 ˆ R3 ˆ R4 . . .

. . . . . . ... . . .

            

SVD( ˆ

H) + truncation − → ˆ O − → ( ˆ H, ˆ F ) − → (ˆ λ, ˆ ϕλ) ˆ H = U ∆ W T = U

     ∆1

∆0

     W T ;

ˆ O = U ∆1/2

1 O↑

p(H, F ) = Op(H, F ) F

det(F − λ I) = 0 , F Φλ = λ Φλ, ϕλ = H Φλ

7

SLIDE 8

Merging multiple measurements setups

       

Y (0,1)

k

Y (1)

k

       

Record 1

       

Y (0,2)

k

Y (2)

k

       

Record 2

. . .

       

Y (0,J)

k

Y (J)

k

       

Record J

                                                

X(j)

k+1

= F X(j)

k

+ V (j)

k

Y (0,j)

k

= H0 X(j)

k

(the reference) Y (j)

k

= Hj X(j)

k

(sensor pool noj) R0,j

i ∆

= E Y (0,j)

k

Y (0,j)

k−i T

, Rj

i ∆

= E Y (j)

k

Y (0,j)

k−i T

E Y (j)

k

Y (j)

k−i T

not used,

E Y (j′)

k

Y (j)

k−i T

(j = j′) not available

8

SLIDE 9

Stationary excitation

covV (j)

k

= Q , G ∆ = E X(j)

k

Y (0,j)

k T

R0,j

i

= H0 F i G

∆

= R0

i ,

Rj

i = Hj F i G Rπ

i ∆

=

            

R0

i

R1

i

. . .

RJ

i

            

= H F i G , H ∆ =

           

H0 H1

. . .

HJ

           

Nonstationary excitation

covV (j)

k

= Qj , Gj

∆

= E X(j)

k

Y (0,j)

k T

R0,j

i

= H0 F i Gj , Rj

i = Hj F i Gj

Hint: right renormalization of the covariances.

9

SLIDE 10

Robustness to nonstationary excitation

Time-varying excitation within each record

covV (j)

k

= Qk

Approximate factorization of covariances

ˆ Ri ≈ H F i ˆ G

Consistency :

T −1 ˆ F T → F, ˆ H → H

Combination of:

the key factorization property of the covariances,
the averaging operation underlying covariance computation,

allows to cancel out nonstationarities in the excitation.

10

SLIDE 11

Numerical results - Steelquake

Estimated modes - Classical subspace identification Mode 1 2 3 4 5 6 7 8 Freq.(Hz) 3.1 3.92 6.1 9.68 10.8 12.27 13.0 17.7 Dir. X Y Y Y X Z Z Z

11

SLIDE 12

2 7 12 17 22 27 32 37 5 10 15 20 25

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

2 7 12 17 22 27 32 37 5 10 15 20 25

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

Order Frequency

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

2 7 12 17 22

× ×

37 5 10 15 20 25

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

27

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

32

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

2 7 12 17 22 27 32 37 5 10 15 20 25

♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦

2 7 12 17 22 27 32 37 5 10 15 20 25

◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ⊕ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊

2 7 12 17 22 27 32 37 5 10 15 20 25

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

19.09 17.74 12.3 11.8 11.13 6.13 3.9 13.1 12.98 10.81 ◊ ⊕

Twelve sensors (all except sensors 3, 7, 10) (Q09 − 1).

12

SLIDE 13

2 7 12 17 22 27 32 37 5 10 15 20 25

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

2 7 12 17 22 27 32 37 5 10 15 20 25

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

2 7 12 17 22 27 32 37 5 10 15 20 25

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

2 7 12 17 22 27 32 37 5 10 15 20 25

♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦

2 7 12 17 22 27 32 37 5 10 15 20 25

◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊

2 7 12 17 22 27 32 37 5 10 15 20 25

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

Frequency Order

Reference sensors: 6/8; moving sensors: 11/14 (Q09 − 1), 9/15 (Q10 − 2), 1/4 (Q10 − 3), 2/13 (Q09 − 4), 5/12 (Q10 − 5).

13

SLIDE 14

2 7 12 17 22 27 32 37 5 10 15 20 25

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

2 7 12 17 22 27 32 37 5 10 15 20 25

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

2 7 12 17 22 27 32 37 5 10 15 20 25

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

2 7 12 17 22 27 32 37 5 10 15 20 25

♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦

2 7 12 17 22 27 32 37 5 10 15 20 25

◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊

2 7 12 17 22 27 32 37 5 10 15 20 25

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

Frequency Order

Reference sensors: 1/6/7; moving sensors: 4/5 (Q09 − 1), 8/13 (Q10 − 2), 12/15 (Q10 − 3), 11/14 (Q09 − 4), 2/10 (Q09 − 5).

14

SLIDE 15

2 7 12 17 22 27 32 37 5 10 15 20 25

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

2 7 12 17 22 27 32 37 5 10 15 20 25

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

2 7 12 17 22 27 32 37 5 10 15 20 25

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

2 7 12 17 22 27 32 37 5 10 15 20 25

♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦

2 7 12 17 22 27 32 37 5 10 15 20 25

◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊

2 7 12 17 22 27 32 37 5 10 15 20 25

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

Frequency Order

Reference sensors: 4/5/12; moving sensors: 1/9/15 (Q10 − 1), 2/6/8/10/13 (Q10 − 2), 3/7/11/14 (Q10 − 3).

15

SLIDE 16

Bending mode in Y direction at frequency 3.92 Hz.

16

SLIDE 17

Bending mode in X direction at frequency 10.8 Hz.

17

SLIDE 18

Slab torsion in Z direction at frequency 17.7 Hz.

2 12 22 32 42 5 10 15 20

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

In-operation subspace-based covariance-driven structural identification Steelquake and Z24 bridge benchmarks

Laurent M´ evel, Mich` ele Basseville, Albert Benveniste, IRISA, Rennes, France Maurice Goursat, INRIA, Rocquencourt, France basseville@irisa.fr - http://www.irisa.fr/sigma2/ Toolboxes: LMS CADA-X, and Scilab ftp://ftp.inria.fr/INRIA/Projects/Meta2/Scilab/contrib/MODAL/

1

Problem : In-operation modal identification of structures

– natural, not controlled. – not measured: ∗ buildings, bridges, offshore structures, ∗ rotating machinery, ∗ cars, trains, aircrafts. – nonstationary (e.g., turbulent).

e.g. in case of moving sensors?

2

Identification and merging

Wanted:

3

Contents

– Modelling – Output-only covariance-based subspace identification – Merging multiple measurements setups – Robustness to nonstationary excitation – Numerical results - Steelquake – Numerical results - Z24 bridge

4

Modelling FE model:

M ¨ Z(s) + C ˙ Z(s) + KZ(s) = ν(s) Y (s) = LZ(s) (Mµ2 + Cµ + K)Ψµ = 0 , ψµ = LΨµ

State space:

Xk+1 = F Xk + Vk Yk = HXk F Φλ = λ Φλ , ϕλ

∆

= HΦλ eδµ = λ

, ψµ = ϕλ

5

Output-only covariance-based subspace identification

Ri

∆

= E

k−i

, H =

R0 R1 R2 . . . R1 R2 R3 . . . R2 R3 R4 . . .

. . . . . . ... . . .

Ri = H F i G , G ∆ = E

k

O ∆ =

H HF HF 2

. . .

, C ∆ =

→ O − → (H, F ) − → (λ, ϕλ)

6

Implementation

ˆ Ri

∆

= 1 N

N

k−i

, ˆ H =

ˆ R0 ˆ R1 ˆ R2 . . . ˆ R1 ˆ R2 ˆ R3 . . . ˆ R2 ˆ R3 ˆ R4 . . .

. . . . . . ... . . .

SVD( ˆ

H) + truncation − → ˆ O − → ( ˆ H, ˆ F ) − → (ˆ λ, ˆ ϕλ) ˆ H = U ∆ W T = U

∆0

ˆ O = U ∆1/2

1

O↑

p(H, F ) = Op(H, F ) F

det(F − λ I) = 0 , F Φλ = λ Φλ, ϕλ = H Φλ

7

Merging multiple measurements setups

Y (0,1)

k

Y (1)

k

Y (0,2)

k

Y (2)

k

. . .

Y (0,J)

k

Y (J)

k

X(j)

k+1

= F X(j)

k

+ V (j)

k

Y (0,j)

k

= H0 X(j)

k

(the reference) Y (j)