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2D full field vibration analysis with multiplexed digital holograms

Pascal Picart, Julien Leval, Jean Claude Pascal, Jean Pierre Boileau, Michel Grill, Jean Marc Breteau, Benjamin Gautier, and Stéphane Gillet

Open Access Open Access

Abstract

opportunities for full field 2D amplitude and phase vibration analysis are presented. It is demonstrated that it is possible to simultaneously encode-decode 2D the amplitude and phase of harmonic mechanical vibrations. The process allows the determination of in plane and out of plane vibration components when the object is under a pure sinusoidal excitation. The principle is based on spatial multiplexing in digital Fresnel holography. Experimental results are presented in the case of an industrial application.

©2005 Optical Society of America

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Supplementary Material (9)

Media 1: MPG (1064 KB)     
Media 2: MPG (911 KB)     
Media 3: MPG (1402 KB)     
Media 4: MPG (1608 KB)     
Media 5: MPG (1586 KB)     
Media 6: MPG (1251 KB)     
Media 7: MPG (1092 KB)     
Media 8: MPG (1321 KB)     
Media 9: MPG (811 KB)     

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Figures (12)
Fig. 1.
Fig. 1. Experimental set-up for simultaneous 2D vibration analysis

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Fig. 2.
Fig. 2. Multiplexed holograms of the car joint piece

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Fig. 3.
Fig. 3. 2D vibration amplitude and phase at a frequency of 680 Hz

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Fig. 4.
Fig. 4. Mean quadratic velocities extracted from the experimental results

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Fig. 5.
Fig. 5. 1064 Ko Movie 1 - Kinetic representation of Fresnel for 2D vibration at 340Hz [Media 9]

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Fig. 6.
Fig. 6. 912 Ko Movie 2 - Kinetic representation of Fresnel for 2D vibration at 430Hz

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Fig. 7.
Fig. 7. 1403 Ko Movie 3 - Kinetic representation of Fresnel for 2D vibration at 460Hz

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Fig. 8.
Fig. 8. 1609 Ko Movie 4 - Kinetic representation of Fresnel for 2D vibration at 680Hz

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Fig. 9.
Fig. 9. 1586 Ko Movie 5 - Kinetic representation of Fresnel for 2D vibration at 730Hz

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Fig. 10.
Fig. 10. 1251 Ko Movie 6 - Kinetic representation of Fresnel for 2D vibration at 880Hz

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Fig. 11.
Fig. 11. 1093 Ko Movie 7 - Kinetic representation of Fresnel for 2D vibration at 900Hz

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Fig. 12.
Fig. 12. 1321 Ko Movie 8 - Kinetic representation of Fresnel for 2D vibration at 930Hz

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Equations (12)

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U ( t ) = u x sin ( ω 0 t + φ x ) i + u y sin ( ω 0 t + φ y ) j + u z sin ( ω 0 t + φ z ) k ,
A ( t ) = A 0 exp ( i ψ 0 ) exp [ 2 iπ S . U ( t ) / λ ] ,
A R ( x , y , − d 0 , t ) = − i exp ( − 2 i π d 0 / λ ) λ d 0 exp [ i π λ d 0 ( x 2 + y 2 ) ]
× ∑ k = 0 k = K − 1 ∑ l = 0 l = L − 1 H ( l p x , k p y , d 0 , t ) exp [ − i π λ d 0 ( l 2 p x 2 + k 2 p y 2 ) ] exp [ 2 i π λ d 0 ( lx p x + ky p y ) ] .
A + 1 R ( x , y , − d 0 , t ) ≅ MN λ 4 d 0 4 R * ( x , y ) exp [ − iπλ d 0 ( u 0 2 + v 0 2 ) ]
× A 0 ( x , y ) exp [ i ψ 0 ( x , y ) ] exp [ 2 iπ S . U ( t ) / λ ] * δ ( x − λ u 0 d 0 , y − λ v 0 d 0 ) .
ψ J i = ψ 0 ± Δ φ x sin ( θ ) sin ( ω 0 t j + φ x ) − Δ φ z [ 1 + cos ( θ ) ] sin ( ω 0 t j + φ z ) ,
Δ φ A = 1 2 [ Δ ψ 13 _ A ] 2 + [ Δ ψ 23 _ A + Δ ψ 21 _ A ] 2 ,
φ A = arctan [ Δ ψ 13 _ A Δ ψ 23 _ A + Δ ψ 21 _ A ] ,
〈 ∣ v A ∣ 2 〉 = ω 0 2 πS ∫ ∫ s ∫ 0 T 0 ∣ v A ( x , y , t ) ∣ 2 dtdxdy ,
v z ( x , y , t ) = − λ ω 0 4 π ( 1 + cos θ ) Δ φ z ( x , y ) cos ( ω 0 + φ z ) ,
v z ( x , y ,t ) = − λ ω 0 4 π sin ( θ ) Δ φ x ( x , y ) cos ( ω 0 t + φ x ) .
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