4.1. FBH Tests
The results of performing laser vibrometry in the FFT mode were the amplitude-frequency spectra of the PMMA plate (
Figure 4) and the images of material surface vibrations in the range from 50 Hz to 11 kHz obtained with the scanning step of 31.25 Hz.
At each resonance frequency, the magnitude of vibration amplification was evaluated by determining the signal-to-noise ratio (
SNR) that is a common practice in comparing the efficiency of various experimental and data processing techniques. The
SNR approach evaluates signals in contrast to background and/or system noise [
25]:
where
and
are the amplitudes of vibration velocity in defect and non-defect areas, and
σnd is the standard deviation of
.
It is seen from
Figure 4 that the frequency spectra by
x,
y,
z coordinates look similar by shape but different by amplitude. For each coordinate, the mean vibration amplitudes were calculated in chosen frequency bands and then compared to exhibit the contribution of each coordinate into the resulting vibration picture. The mean vibration velocity is maximal by the
z coordinate being 300 mm·s
−1 to compare to 130 and 230 mm·s
−1 by the
x and
y coordinates respectively. This demonstrates the maximal contribution of bending deformation in the case of the resonance stimulation of FBH-like defects. It is also worth noting that the resonance frequencies for all three components are identical. In other words, the amplification of vibrations occurred by the
x,
y,
z coordinates at the same frequencies, or frequency bands, and following the same order of resonance frequencies. For example, the
x,
y,
z vibrations of the fundamental resonance in the particular FBH appeared in the frequency range from 2344 to 2750 Hz (
Figure 5). In this range, the maximal amplification of vibrations (
SNR = 35) took place by the
z coordinate (out-of-plane LDR) while, by the
x and
y coordinates (in-plane LDR), the maximal
SNR values were only 12 and 24, respectively.
In spite of the fact that LDR frequencies differ by three coordinates, all three vibration components are equally involved in the process of vibrations. This is also confirmed by the identical distribution of vibration velocities along three coordinates in the amplitude-frequency characteristic (AFC) of the PMMA plate (
Figure 4). By having analyzed the AFC of the plate in the frequency range from 50 Hz to 13 kHz, three resonance frequencies were identified over the rectangular FBH by each Cartesian coordinate. The shapes of resonance vibrations were considered separately and compared at each particular LDR frequency. The results of the analysis are presented in
Table 1.
The SLDV results have demonstrated that the maximal amplification of wave front vibrations from 0.07 to 0.6 mm·s−1, i.e., by nine times, appeared at the defect fundamental resonance frequency (2750 Hz) by the z coordinate (out-of-plane LDR). The vibrations by the x and y coordinates (in-plane LDR) have been amplified only by three and four times, respectively. This confirms the conclusion that out-of-plane vibration components play a decisive role in the build-up of resulting defect signals at fundamental LDR frequencies. For example, when considering the FBH resonance of a higher-order, it was found that the contribution of single vibration components into the total vibration pattern was non-uniform. The maximal vibration velocity by the x coordinate (up to 1.4 mm·s−1) was observed at the frequency of 6094 Hz being higher than vibrations by the z (1.1 mm·s−1) and y (0.9 mm·s−1) coordinates. However, all three components contributed to the total vibration pattern obtained by averaging the components at the frequency of 6094 Hz. It is believed that such an LDR feature may be useful when performing the NDT of materials and parts. For instance, in the test case considered above, the summation of x, y, and z vibrations at particular frequencies made the corresponding defect indication closer to the real defect by size and shape. Furthermore, by analyzing vibrations at the third resonance frequency of 9250 Hz, the maximal contribution was provided by the x-component (in-plane LDR) to compare to the y- and z-components. Therefore, the x, y, z vibration pattern at 9250 Hz resembled the pattern obtained by x and y coordinates, i.e., when the contribution of out-of-plane vibrations was minimal. In general, it seems that, even if individual vibration components can be different at each particular frequency, the magnitude of total vibrations remains comparable to both in-plane and out-of-plane LDR components. The total vibration patterns obtained for each spatial coordinate showed the defect indication which is square-shaped and close in size to the true FBH. It is believed that total vibration patterns obtained by three coordinates in the mode of 3D scanning may supply the most reliable information about FBH parameters, including maximal SNR values, thus enhancing the efficiency of defect detection.
The results of numerical modeling illustrate that calculated FBH vibration shapes averaged by three coordinates repeat those by the z coordinate. This is explained by the predominant contribution of z-vibration amplitudes into the total pattern. Therefore, the comparison of theoretical and experimental results on determining vibration shapes and resonance frequencies has been performed for the z-component.
As follows from
Table 1, the LDR vibrations at each frequency appeared by all three coordinates being slightly different by
x and
y coordinates. This phenomenon was described elsewhere to illustrate that polarization of different LDR mode shapes with respect to defect orientation may have a considerable effect on defect detectability [
18]. The in-plane LDR takes place due to a vibration mode produced by waves of a higher velocity. It is worth mentioning that frictional heating plays a predominant role in the signal buildup when performing ultrasonic vibrothermography.
By taking into account the above-mentioned phenomena, the first three modes of FBH resonance vibrations were identified in the frequency range from 2750 to 9500 Hz to reveal a good agreement with the theoretical data by shapes and frequencies of resonance harmonics. The first calculated mode at the frequency of 2690 Hz corresponded to the experimental results of the scanning at the frequency of 2750 Hz (the second mode of the LDR frequency was characterized by the theoretical/experimental values of 5030 and 6094 Hz, respectively). It is worth noting that calculated shapes of resonance vibrations in total patterns were different in regard to the experimental
x,
y component values. In fact, the calculated total pattern repeated vibrations by the
z coordinate while the experimental pattern combined vibrations by the
x and
z coordinates. A certain discrepancy between simulated and experimental results (up to 10% by the central frequency) was also observed at the higher frequencies. It is believed that such a discrepancy appeared because of different conditions of inputting ultrasonic waves into the sample and some peculiarities of FBH manufacturing, namely, the presence of rounded corners, high material roughness, etc. Moreover, the laser scanning was performed in the case, where the finite-size piezoelectric transducer was attached to the sample edge (
Figure 2b), but the modeling assumed the ideal condition without taking into account any environmental factors. However, in spite of the above-mentioned discrepancies between the theory and experiment, the obtained results matched reasonably well to confirm the validity of the proposed mathematical model.
4.2. Analyzing Acoustic Stimulation Power
As mentioned above, the resonance acoustic stimulation of defective composite samples causes the essential amplification of vibrations in defect zones. The sample material experiences maximal compression/tensile displacements in these zones. In this study, the maximal stresses appearing in the FBH under resonance acoustic stimulation were evaluated. The PMMA plate (
Figure 1) was stimulated at 1250 Hz, which was the fundamental LDR frequency for the rectangular FBH in this sample. The surface in-plane vibrational response by the
x coordinate was investigated using a 3D scanning laser Doppler vibrometer (
Figure 2), and the magnitude of applied stresses accompanying acoustic resonance stimulation was compared with the PMMA tensile strength.
As a result of laser scanning, the absolute sample displacement of 0.7–1 m occurring during compression/tension of FBHs (length L) was determined by the x coordinate. This value corresponded to the in-plane component of vibrations (normal strain).
Suppose that the stresses, which appear under ultrasonic stimulation, are essentially lower than the corresponding strength limit. In this case, all deformations will appear in the elastic regions, and the magnitude of the stresses accompanying acoustic resonance stimulation should be compared with the PMMA tensile strength. By introducing the relative displacement
ε (
Δl/
L) and the Young modulus
E, the normal stress
σ can be found as
σ =
E ×
ε. Next, one can evaluate vibrations and the relative displacement by
x as follows:
Δl/
L = (0.7 ÷ 1) m/31 mm = (0.023 ÷ 0.032) × 10
−3. For PMMA, the Young’s modulus is 2800–3300 MPa [
23], and the maximal stress appearing in the FBH under ultrasonic stimulation is:
σexp = (2800 ÷ 3300) MPa × (0.023 ÷ 0.032) × 10
−3 = (64.4 ÷ 105.6) kPa. Furthermore, the PMMA tensile strength is
σmax = 65 MPa [
26]. The ratio between the tensile strength of PMMA material and maximal stresses appearing in the FBH under ultrasonic stimulation is
σmax/
σexp = 65 MPa/(64.4 ÷ 105.6) kPa = 615 ÷ 1000. The estimates obtained confirm a nondestructive character of ultrasonic resonance stimulation of such composite materials, which are close to PMMA by their mechanical properties.
4.3. 3D Scanning of Impact Damage in CFRP
It was shown elsewhere that vibration patterns allow the visualization of defect contours and, hence, the evaluation of defect lateral size and shape by applying acoustic excitation at the corresponding resonance frequencies [
26], as well as using sweep excitation in the low- and mid-kilohertz range [
27,
28,
29]. Defect identification can be enhanced by analyzing a total vibration pattern obtained by averaging results at some resonance frequencies. Following this concept, a 1.5 mm-thick CFRP plate containing an 18 J impact damage was inspected (
Figure 6), and the laser vibrometry results were validated by the results of the ultrasonic C-scan test. An ultrasonic inspection was performed by using an Ideal System 3D scanner supplied with a focused transducer (resonance frequency 15 MHz, diameter 16 mm, focusing depth 50.8 mm and scanning step 1 mm). The results of ultrasonic inspection were presented as C-scans (see
Figure 7).
The results of ultrasonic C-scanning have revealed the known structure of impact damage in a cross-ply CFRP composite including multiple cracks and delaminations of which area increases with depth counted from the front surface. Often one considers the cone-, or tree-, or pyramid-like shapes of impact damage. The minimal delamination with the area of 175 mm
2 was identified at the depth of 0.1 mm, i.e., directly under the point of impact (
Figure 7a), while the largest delamination (area of 1530 mm
2) was detected at the depth of 1.4 mm (
Figure 7b). The
SNR values were 7.6 and 14 in these two cases, respectively.
As mentioned above, the efficiency of laser vibrometry can be enhanced by determining defect resonance frequencies. Performing laser vibrometry in the FFT mode allows for the transforming of signals from the time to phase/frequency domain, thus enabling, first, the analysis of vibrations at individual frequencies and, second, the averaging of the results across the total spectrum. The use of such an approach for performing the spectral analysis of impact damage was described elsewhere [
16]. In our study, a 1.5 mm-thick CFRP plate was investigated by using laser vibrometry in combination with a contact transducer mounted on the sample surface, as shown in
Figure 6. Up to 20 frequencies were identified in the range from 185 Hz to 100 kHz, where various defect sections revealed particular resonances. The maximal mean amplitude of vibration velocity (0.34 mm·s
−1) was observed by the
y coordinate being slightly higher than that by the
x and
z coordinates (0.23 mm·s
−1 in both cases). This demonstrates the domination of compression/tension deformations in the case of resonance stimulation, even if vibration amplitudes are close by all three coordinates.
Figure 8 illustrates the amplification of vibrations in the impact damage zone at some defect resonance frequencies (7.44 kHz, 32.8 kHz and 46.5 kHz), thus proving that resonance vibrations appear in a wide range of frequencies.
Another important factor influencing a picture of vibrations in defects is the contribution of each vibration component into the total signal. Similar to the FBH case, the appearance of resonance at some particular frequencies leads to the amplification of vibrations by
x,
y,
z coordinates. However, different sections of impact damage are characterized by different shapes and amplitudes of vibrations. This statement is illustrated in
Figure 9, where contributions of in-plane and out-of-plane vibration components into the total pattern are shown for the particular LDR frequency (43.6 kHz).
The images presented in
Figure 9 only provide an example of the contribution of the
x,
y,
z components into the resulting signal at a particular resonance frequency, but they allow for the understanding of the basics of resonance phenomena in materials with structural defects. It is seen that, at particular LDR frequencies, the enhancement of vibration amplitudes in defect zones takes place by all Cartesian coordinates. Material deformation occurs by both out-of-plane and in-plane coordinates being governed by the same physical phenomena. Therefore, it can be assumed that LDR frequencies correspond to defect vibrations, which simultaneously appear by all three coordinates and should be considered as a single process.
Figure 10 shows the images of CFRP surface vibrations averaged across the total frequency spectrum for three vibration components. The defect area determined on the sample rear surface by means of 3D laser vibrometry was 1760 mm
2, or 85% in regard to that measured ultrasonically (compare
Figure 7b and
Figure 10). It appears that supplying a frequency-modulated causes vibrations of a greater defect area by activating vibrations at both the basic and higher defect resonance frequencies. Therefore, the determination of defect size and shape will be more accurate by applying a wide-range acoustic stimulation.
Figure 10 illustrates that the amplitude of vibration velocity by the
x,
y coordinates reaches 50 mm·s
−1, while by the
z coordinate it only reaches 15 mm·s
−1. Furthermore, in-plane (
x,
y) components of vibration ensure a greater area of vibrations to compare to the out-of-plane (
z) component. Hence, unlike the case of FBHs with maximal out-of-plane resonance vibrations, in CFRP, the predominant contribution to the averaged vibration pattern of impact damage is provided by deformations of compression/tension, i.e., in-plane components of vibrations. In the case of FBHs, this can be explained by higher vibrations along the coordinate
z, which accompany the removal of some mass of material. In impact damage defects, the predominance of
x,
y vibrations leads to the prevailing tangential interaction of in-plane interfaces and respectively to local heat generation in crack-like defects [
18].