PSI - Issue 17

H. Lopes et al. / Procedia Structural Integrity 17 (2019) 971–978 Lopes et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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to 1.26 W to fulfill the dynamic range of the intensity capture by the CCD of the camera. Figure 4 shows the global view of the experimental setup. The measurements were performed on top of an optical table with active vibration isolation supports to isolate it from external perturbations. Also, two very soft foam were mount on the back and next to the laminated plate to avoid rigid body motions during the measurements.

Fig. 4. Global view of the experimental setup for the measurement of modal rotations. A temporal phase modulation with the recording of five intensity distributions and using constant phase step of π/2 was applied to extract the phase of interference 0 ( , ), /2 ( , ), ( , ), 3 /2 ( , ), 2 ( , ) . The phase of the interference in reference state when the plate was at rest Φ ( , ) and in deformed state when was at maximum amplitude of vibration Φ ( , ) are given by the following equations (Kreis (2005)): Φ ( , ) = arctan ( 7 ( ,3 /2 ( , ) − , /2 ( , )) 4 ,0 ( , ) − , /2 ( , ) − 6 , ( , ) − ,3 /2 ( , ) − 4 ,2 ( , ) ) (6) Φ ( , ) = arctan ( 7 ( ,3 /2 ( , ) − , /2 ( , )) 4 ,0 ( , ) − , /2 ( , ) − 6 , ( , ) − ,3 /2 ( , ) − 4 ,2 ( , ) ) (7) The change of the phase between the reference state, Φ ( , ) , and the phase of the deformed state, Φ ( , ) , is called the phase map of the modal rotation ∆ ( , ) and is given by: Δ ( , ) = { Φ ( , ) − Φ ( , ) − if Φ ( , ) ≥ Φ ( , ) Φ ( , ) − Φ ( , ) + if Φ ( , ) < Φ ( , ) (8) The phase in these maps is noisy and is wrapped within [− , ] . To obtain the modal rotation fields, the phase maps are post-processed by first applying a low-pass filter to remove the noise and then by using an unwrapping algorithm to remove the phase discontinuities. The filtered phase maps are obtained by applying ͵Ͳ times the sine/cosine average filter to noisy phase maps, with a windows size of [5x5] (Aebischer and Waldner (1999)). These values are defined according to a heuristic process, that is dependable on the number and spatial distribution of the phase discontinuities and the level of the noise in the phase maps. The continuous phase field is obtained by applying the Goldstein algorithm (Ghiglia and Pritt (1998)) to the filtered phase maps. In the present study, the illumination and the observation vectors are both perpendicular to the plate surface, which means the shearography system is only sensitive to out-of-plane deformation. Therefore, a following relationship can be stablished between the rotation field for the direction, (e ) ( , ) , or for the direction , (e ) ( , ) , and the phase map, Δ ( , ) , for the mode : ( e) ( , ) = 4 Δ Δ ( , ) (9) (e ) ( , ) = 4 Δ Δ ( , ) (10)