PSI - Issue 5

H. Lopes et al. / Procedia Structural Integrity 5 (2017) 1205–1212 H. Lopes et al. / Structural Integrity Procedia 00 (2017) 000 – 000 3 discontinuities and the phase ambiguities with variations between −π and π , thus allowing the evaluation of noise. A flowchart describing the main steps of this method is shown in Figure 1. The measured unfiltered phase map is first filtered with low-pass filters (Seara et al. (1998) and Aebischer et al. (1999)). The phase discontinuities and the phase transitions in the filtered phase map are identified by defining a level function consisting of multiples of 2 . This identification does not include the information of phase jumps due to the noise. This level function is part of the process which is called continuation, phase unwrapping or demodulation (Kreis (2005)), thus allowing to obtain continuous signals. In this work, the level function is used to unwrap the filtered and unfiltered phase maps. Afterwards, these two continuous phase maps are compared and the phase differences are corrected by adding or subtracting 2π to the continuous unfiltered phase map. By subtracting the corrected continuous unfiltered phase map to the continuous filtered phase map one obtains the noise. This noise represents phase fluctuations, i.e. they characterize the noise in the measured unfiltered phase map. In the case of measurements with shearography, a relation between the noise in the phase and the noise in the rotation can be established, based on the following equation (Steinchen and Yang (2003)): 1207

x w x y 

( , )



x   

( , ) x y





(1)

4



where ∆ϕ (x, y) and w x y x   ( , ) / are the mathematical representations of the measured phase map and the measured rotation field, respectively, being Δx the shearing amount and λ the wavelength of the light source. The noise in the rotation field can be further evaluated by computing the root mean squares (RMS) and the signal-to-noise ratios (SNR).

Measured unfiltered phase map

Continuous filtered phase map

Continuous unfiltered phase map

Corrected continuous unfiltered phase map

Noise

Fig. 1. Flowchart describing the method for the evaluation of noise.

3. Experimental setup

Figure 2 shows the experimental setup used for the measurement of the first four modal rotation fields of an aluminum beam in free-free condition and dimensions 400 mm x 40 mm x 3 mm. This condition was accomplished by suspending the beam at its ends by two rubber bands. The measurements are made using a digital shearography system and by applying the temporal phase modulation. The in-house digital shearography system is based on a Michelson optical interferometer, which creates the interference pattern between two wave fronts laterally sheared,