32.3.1 Methods and Algorithms Proposed Generally, the lock-in thermography technique is carried out by stimulating the material with a modulated sinusoidal heat source at a fixed frequency. As just stated, this frequency allows us to investigate the material to a given depth as reported in Eq. (32.1). It follows that in order to obtain information about the full depth of a component, different tests have to be performed at different modulated frequencies and consequently, in the case of a large structure, the application of LT technique can involve elevated testing time. As demonstrated in other works [22–24], the thermal response of material contains information about high order frequencies proportional to the main frequency. By decomposing the thermal response in time domain as the sum of a singular sinusoidal wave, we can write: Tm tð Þ¼aþbt þX 1 n¼1 ΔTn sin nωt þφn ð Þ n ¼1,3,5,7; ð32:3Þ where t is the time, a and b are two constants used to model the average temperature growth of the material, ω is the modulated frequency of the main harmonic (first Fourier component), ΔTn and φn are the amplitude and the phase of n-th Fourier component. In this work, all the constants were obtained through a least-square fit method by imposing the model of Eq. (32.3) to the thermal signal of each pixel and by considering the terms up to n ¼5. In this way we can write: Tm tð Þ¼aþbt þΔT1 sin ωt þφ1 ð ÞþΔT3 sin 3ωt þφ3 ð ÞþΔT5 sin 5ωt þφ5 ð Þ ð32:4Þ Equation (32.4) allows us to obtain information about the amplitude and phase signal of high order excitation frequencies with respect to the main by a single lock-in test. 32.4 Results 32.4.1 Qualitative Comparison of Sinusoidal and Square Wave Excitation Figure 32.2 shows a comparison between the phase maps obtained by using sinusoidal wave and the first harmonic of square wave. As was expected, deeper defects appear in correspondence with a higher modulation period (lower excitation frequencies). In Fig. 32.3, the phase maps obtained with a sinusoidal wave at a modulation period of 120 s and 72 s are compared to the third and fifth harmonics obtained from a square excitation at modulated period of 360 s. The comparison of the images seems to clearly indicate that there is no significant difference between sinusoidal and square wave excitation. The comparison of results obtained with sinusoidal heating shape with third and fifth harmonic of the square wave seems to suggest that there is a significant but yet not dramatic loss of quality. 32.4.2 Quantitative Comparison of Sinusoidal and Square Wave Excitation In order to quantitatively compare the results obtained from sinusoidal and square wave heating shape, the behaviour of two parameters was monitored: the standard deviation of a sound area (in some ways representative of the noise) and the contrast of the phase signal between defect and sound areas (representative of the usually assessed parameter in NDE). Figure 32.2 shows the area used for quantitative analysis. In particular, notationSis used for the sound area anddfor the area with a defect (in this case, defect number 2). In detail, the phase contrast can be defined as: Δφ¼mean φd ð Þ mean φS ð Þ ð32:5Þ where mean(φ) represents the average value of the phase signal assessed on d and S areas. 270 D. Palumbo and U. Galietti
RkJQdWJsaXNoZXIy MTMzNzEzMQ==