R 0 R 1 R 2 . . . Modeling the temperature effect R - - PowerPoint PPT Presentation

Handling the temperature effect in SHM: combining a subspace-based statistical test and a temperature-adjusted null space

Mich` ele Basseville, Laurent Mevel, Houssein Nasser IRISA (CNRS & INRIA & Univ.), Rennes, France Fr´ ed´ eric Bourquin, Fabien Treyss` ede LCPC, Paris & Nantes, France National Computer & Security project Constructif michele.basseville@irisa.fr -- http://www.irisa.fr/sisthem/

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Introduction

• Usefulness of global vibration-based SHM methods
• Limitations due to temperature effects on the dynamics
• f civil engineering structures
• Wanted: discriminate between changes in modal parameters

due to damages and changes due to temperature effects

• A statistical subspace-based damage detection algorithm:

null space of a matrix built on reference modes/modeshapes at a known temperature

• Proposed solution to temperature handling:

measured temperatures, thermal effect modeling: analytical null space updating

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Content

Parametric subspace-based damage detection Modeling the temperature effect Temperature-adjusted null space detection Experimental results Comparison with a non parametric approach Conclusion

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Parametric subspace-based damage detection

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

Xk+1 = F Xk + Vk F ϕλ = λ ϕλ Yk = H Xk φλ

∆

= H ϕλ Ri

∆

= E

 Yk Y T

k−i

  ,

H

∆

=

              

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

              

Ri = H F i G = ⇒ H = O C

O ∆ =

        

H HF HF 2 . . .

        

, C ∆ =

G

F G F 2G . . .

• G ∆

= E

• Xk Y T

k

• H −

→ O − → (H, F ) − → (λ, φλ)

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Canonical parameter : θ ∆ =

   

Λ vec Φ

   

modes mode shapes

Observability in modal basis : Op+1(θ) =

        

Φ Φ∆ . . . Φ∆p

        

θ0 : reference parameter for safe structure Left null space: ST S = Is, ST Op+1(θ0) = 0 Yk: N-size sample of new measurements Residual for SHM: ζN(θ0) ∆ = vec( ST (θ0) ˆ H )

J (θ0): sensitivity of ζ w.r.t. modal changes; Σ(θ0): covariance

χ2-test: ζT

N Σ−1 J (J T Σ−1 J )−1 J T Σ−1ζN ≥ h

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Modeling the temperature effect

Clamped, planar, axially pre-stressed, Euler-Bernoulli beam Eigen problem

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

EI d4w(x) dx4 − N0 d2w(x) dx2 − ρA ω2 w(x) = 0 w(x)|x=0,L = 0 ; dw(x) dx

• x=0,L

= 0 with ω ∆ = ln |λ| , w = wω ∆ = φλ , w(x) : transversal displacement. E, I, ρ, A: Young’s modulus, cross-section inertia momentum, density and cross-sectional area. N0: quasi-static axial preload in Newton: N0 = EA ε(x) , ε(x) ∆ = ε0(x) − α δT , δT ∆ = T0(x) − Tref ε0: mechanical strain, T0: current temperature, Tref: reference (no stress) temperature, α: thermal expansion coefficient.

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N0 remains spatially constant (no external axial body forces, no axial surface tractions, gravity effects negligible). Only one measure at a given point is necessary: Thermally compensated strain gauges measure ε(x) = N0/EA. Solutions of the eigen problem: 2γ−

n γ+ n (1−cos(γ− n L) cosh(γ+ n L))+(γ+ 2 n

−γ− 2

n

) sin(γ−

n L) sinh(γ+ n L) = 0

with : γ

+

− n =

      

• ρA

EI ω2

n +

   

N0 2EI

   

2 +

− N0 2EI

      

1/2

Solved numerically (no analytical solutions for the clamped case). Analytic expression for mode shapes. Pre-stress effects on mode shapes are negligible (numerical results).

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Temperature-adjusted null space detection

• θ0 : reference modal parameter for safe structure
• Yk: N-size sample of new measurements;

T recorded

• Update the modal parameter θT :

δT − → ε(x) − → (ωn)n − → (λn, ϕn)n − → θT

• Update the null space S(θT )
• Compute the residual ζN(θT ) ∆

= vec( ST (θT ) ˆ H )

• Compute the χ2-test

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Example - Beam within a climatic chamber

• A laboratory test-case provided by LCPC

Climatic chamber in Nantes

• Vertical clamped beam

subject to decreasing temperatures

• Small local damage: horizontal clamped spring attached

to the beam, with tunable stiffness and height

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Decreasing temperature effect on the first 4 frequencies

20 30 40 50 60 70 80 90 100 110 120 52 53 54 55 56 57 58 59 60 εx − αθ(µm/m) First − Frequency safe − theory safe − identify(cosmad) damaged − identify(cosmad) 20 30 40 50 60 70 80 90 100 110 120 142 144 146 148 150 152 154 εx − αθ(µm/m) Second − Frequency safe − theory safe − identify(cosmad) damaged − identify(cosmad) 20 30 40 50 60 70 80 90 100 110 120 276 278 280 282 284 286 288 εx − αθ(µm/m) Third − Frequency safe − theory safe − identify(cosmad) damaged − identify(cosmad) 20 30 40 50 60 70 80 90 100 110 120 452 454 456 458 460 462 464 466 εx − αθ(µm/m) Fourth − Frequency safe − theory safe − identify(cosmad) damaged − identify(cosmad)

First 4 frequencies vs. thermal constraint. Computed (black) and identified (safe, damaged)

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Implementation issues

• Frequencies : computed = identified

(the clamped boundary condition, obtained with tightening jaws, is not perfect) De-biased temperature-adjusted modal parameter: (∀ε)

• θT (ε) ∆

= θT (ε) +

• θ0(ε0) − θT (ε0)
• Compute the key matrices:

residual sensitivity J and covariance Σ for every realization of each scenario

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Handling the temperature effect

20 30 40 50 60 70 80 90 100 110 120 1 2 3 4 5 6 x 10

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εx − αθ(µm/m) χ2 20 30 40 50 60 70 80 90 100 110 120 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

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εx − αθ(µm/m) χ2

Original χ2-test New χ2-test

20 30 40 50 60 70 80 90 100 110 120 1 2 3 4 5 6 x 10

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εx − αθ(µm/m) E(χ2) safe damaged 20 30 40 50 60 70 80 90 100 110 120 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

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εx − αθ(µm/m) E(χ2) safe damaged

Original - Average New - Average Safe and damaged

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Comparison with a non parametric approach: empirical null space merging data at # temperatures

20 30 40 50 60 70 80 90 100 110 120 1 2 3 4 5 6 x 10

7

εx − αθ(µm/m) χ2 20 30 40 50 60 70 80 90 100 110 120 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

6

εx − αθ(µm/m) χ2 20 30 40 50 60 70 80 90 100 110 120 2 4 6 8 10 12 14 x 10

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εx − αθ(µm/m) χ2

Original χ2-test Analytical updating Empirical merging

20 30 40 50 60 70 80 90 100 110 120 1 2 3 4 5 6 x 10

7

εx − αθ(µm/m) E(χ2) safe damaged 20 30 40 50 60 70 80 90 100 110 120 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

6

εx − αθ(µm/m) E(χ2) safe damaged 20 30 40 50 60 70 80 90 100 110 120 2 4 6 8 10 12 14 x 10

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εx − αθ(µm/m) E(χ2) safe damaged

Average Average Average Safe and damaged

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Conclusion

Temperature effect in vibration-based SHM Statistical parametric model-based approach Statistical subspace-based damage detection algorithm Temperature-adjusted null space Example: clamped beam within climatic chamber Comparison with a non parametric approach (empirical null space, merging data at # temperatures) Ongoing: statistical nuisance rejection Future: in-operation examples, extension to 3D temperature fields, thermal model parameterization

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