This work proposes a pump methodology based on both frequency and amplitude modulation to improve the potential of non-linear phased array imaging techniques; while addressing issues of attenuation, energy, and low signal to noise levels. The dual frequency method proposed by Fierro and Meo (2019) where the B-Scan of two frequencies at f₁ and f₂ are used to evaluate the non-linear responses of the structure is expanded by introducing a non-linear modulation parameter and tri-frequency excitation. Fierro and Meo presented a dual frequency modulation technique which used four B-Scan images to determine the modulated response M_R2 (refer to Equation 3). This was done by capturing two B-Scan images at two different frequencies, f₁ and f₂ (refer to Figure 1). Then subsequently two further images were captured by switching the transmitted frequency per phased array element, with the first image (f₁₂) firing f₁ on all odd elements (1, 3, 5…) and f₂ on all even elements (2, 4, 6.), and the reverse being done for f₂₁. Once these four images were captured a final non-linear image was generated by using a subtraction method highlighted in Figure 1 and Equation 3. The method is summarized in Table 1.

A standard broadband 128 element (2.5 MHz−7 MHz—Diagnostic Sonar Probe and System) phased array probe (element pitch = 0.5257 mm and width = 0.5 mm) utilizing a stepped-linear (sequential) firing approach over 32 elements was used for these experiments.

Figure 3 shows the actual delay laws tested for the pump frequency excitation methodology for the tri modulation method. Where: Figures 3A,C refers to the delayed single frequency signal at f₁ and f₃ (identical delay laws), Figure 3B refers to the pump single frequency signal at f₂ and Figures 3D–F refer to the dual modulation signals f₁₂₃, f₂₃₁, and f₃₁₂, respectively. All signals were focused at a depth of 40 mm (just above the machined notch in the tested aluminum samples) with f₁ and f₃ delayed by 540 ns for each element relative to f₂ (please refer to Figure 3). Please note, that the Diagnostic Sonar system was programmed to complete the 6 detailed delay laws sequentially and compile them into a single B-Scan image. With the first six columns relating to beam 1 of f₁, f₂, f₃, f₁₂₃, f₂₃₁, and f₃₁₂, respectively, with beam 2 relating to columns 7–12, and so forth. This data was then post-processed to generate the final B-Scan images. Compiling and processing this data real-time onboard the system is not a large step forward as many systems today already incorporate pulse inversion techniques (requires two B-Scan images with signals of phase 0 and 180°) into the software packages. Figures 3G–L shows the six raw B-Scan images received using the delay laws highlighted in Figures 3A–F for the tri frequency and amplitude modulation method. The images are focused at a depth between 20 and 42 mm and show the notch location (depth-42 mm, width-22), but do not show the presence of a crack.

Equation 4 and Figure 4 shows the subtraction process used to determine the image of the tri frequency modulated non-linear response (M_R3—refer to Equation 4). In the same respect as the dual frequency method, it is known that f₁, f₂, and f₃ do not contain any modulated non-linear content, whereas f₁₂₃, f₂₃₁, and f₃₁₂ contain modulated non-linear content that is related to various combinations between the frequencies used to excite the structure. In particular, sum (f₃ + f₁, f₂ + f₁, and f₃ + f₂) frequencies and different (f₃– f₁, f₂– f₁, and f₃– f₂) frequencies, but due to complex interactions between frequencies it is expected that further intermodulation's between these frequencies may occur. As further frequencies and total energy of the system increases (six B-Scans vs. four B-Scans) it is expected that amplification of the non-linearities produced by defects/damage will occur. Considering, the data implications for the suggested approach, the data captured is 6× that of a single frequency scan and 1.5× that of the dual frequency modulation method, which corresponds to an increase in the time to complete the scan.

Table 2 summarizes the testing strategy of the tri frequency system and highlights the seven steps of the subtraction process. In the tri modulation tests three different frequency test scenarios were evaluated (1) f₁= 3.8 MHz, f₂= 4 MHz, f₃= 4.2 MHz, (2) f₁= 4.8 MHz, f₂= 5 MHz, f₃= 5.2 MHz, and (3) f₁= 5.8 MHz, f₂= 6 MHz, f₃= 6.2 MHz. This was done in order to evaluate the effect of frequency (resonance of the damage) and attenuation of the signal through the thickness of the sample. Note that the purpose of this paper is not to draw definitive conclusions on frequency selection or the attenuation of the signal, but rather to highlight that these are important factors to take into account when evaluating non-linear techniques, this is something to be focused on in future work. In this case the central frequencies (f₂= 4, 5, and 6 MHz) were chosen as they are all close to the central frequency of the probe (5 MHz), with f₁and f₃chosen at f₂± 0.2 MHz. In the scenario of unknown defect resonance, energy into the sample is maximized in order to promote the generation of non-linearities, hence f₁and f₃are near f₂. The importance of the work is to highlight improved detection capabilities of the outlined modulation method, as well as the improved signal to noise levels that can be achieved by increasing the number of modulating frequencies.

Non-linear elastic wave spectroscopy (NEWS) techniques have been shown to be innovative ultrasonic NDE and SHM inspection methods. Some of these methods center on the theory of “clapping/rubbing mechanism” (such as defects and damages), which generate non-linear elastic effects which can be evaluated using the frequency response of a time domain signal (Ciampa and Meo, 2012). Compared to linear ultrasound methods, these techniques have shown higher sensitivity in diagnosing material micro-defects such as porosity, inclusions, and early stage damage in the form of micro-cracks, delaminations and adhesive bond weakening (Cantrell and Yost, 2001; Cantrell, 2004; Boccardi et al., 2015; Fierro et al., 2016).

These non-linear elastic effects give rise to further harmonic responses known as the second, third and fourth harmonics (and so forth). It has also been shown that subharmonic production can also be produced by such mechanisms which also provide higher sensitivity than linear ultrasound techniques (Korshak et al., 2002; Ohara et al., 2006; Jhang, 2009). Non-linear ultrasound uses these extra harmonics to determine the extent of defects in a material.

The proposed non-linear wave modulation technique focuses on additional sidebands (modulation) that are produced when two different frequency signals (f₁ & f₂) pass over a cracked region. This can be further compounded by the introduction of a third frequency (f₃), giving rise to a complex collection of modulations and intermodulation's.

By adding equations 1 and 2 (β_S + β_D), developed by Fierro and Meo (2015), a sum and difference modulation (β_S+D) parameter can be expressed as:

It can be assumed that the subtracted response can be equated to the amplitudes of (A_S and A_D) as follows:

Thus,

And for a tri frequency system (β_TRI) the equation can be adjusted to:

Finally, in order to quantitatively validate the results the intensity at the crack/notch region (I_C/N) is compared to the mean background noise level (I_B), giving a signal to noise ratio (SNR) as follows:

In order to evaluate the dual frequency modulation method three test samples were evaluated; an undamaged (UD) sample with a notch, a small fatigue crack (SFC) samples (crack length-−3.5 mm) and a large fatigue crack (LFC) sample (crack length-−14 mm). Figures 7A,D show the full field B-Scan image for single frequency excitation at f₁ (4.8 MHz) and f₂(5.3 MHz) for the UD sample. Due to near field effects, it is difficult to visualize the notch location, thus the area of interest between 20 and 45 mm depth has been focused on in Figures 7B,E. The zoomed images clearly show the notch location and depth (~42 mm). The subtraction process highlighted in Equation 3 (M_R2) Figure 7C and Equation 8 (β_S+D) Figure 7F are then used to evaluate the non-linear responses of the UD sample. Both the linear and non-linear methods clearly identify the location of the notch. When using the dual frequency modulation parameter (β_S+D), the response area becomes very small with a high relative amplitude and thus the area has been magnified in order to highlight the response (refer to Figure 7F). It should be noted that the non-linear response is likely due to discontinuities near the notch tip.

The SFC sample was then used to evaluate whether non-linearities can be used to assess a vertical fatigue crack and whether the pump methodology provides benefits over a traditional approach where there is no delay between f₁ and f₂. The first observation is that linear methods are unable to distinguish the existence of the fatigue crack (refer to Figures 8A,D) and are only able to determine the notch location. Figures 8B,E show the differences between the delayed (pump method) vs. the traditional (no-delay) subtracted images for M_R2. From these images it is clear that the pump methodology distinguishes the notch and identifies a response from the crack location, with the traditional delay methodology failing to determine the crack existence. Finally, in Figures 8C,F, the modulated parameter (β_S+D) is evaluated, in this instance the delayed methodology identifies the crack (within ~1 mm of actual position) while the non-delayed method identifies the notch. It should be noted the total energy used to excite the sample for the delayed and traditional methods is equal with the only difference being the delay between f₁ and f₂. Thus, the results provide evidence that a “breathing/ringing” crack scenario is required in order to exploit modulated responses in the case of stiff vertical fatigue cracks. The non-linear parameter β_S+D (Figure 8C) has also shown that it can eliminate linear responses from the notch.

The LFC sample was used to evaluate a larger crack and the ability of linear and non-linear techniques to identify the crack tip. The same process for the UD and SFC samples was used, Figures 9A,B show the linear predicted crack depth at around ~30 mm while Figures 9C,D show the modulated responses predicting a crack depth between ~30 and ~27 mm, respectively. In this case, due to the larger scattering of the incident wave in response to the larger vertical crack, the noise floor is considerably higher. Therefore, due to the higher background noise a 50% threshold was applied to the linear and non-linear responses, refer to Figure 9. It can be seen that f₁, f₂, and M_R2 predict similar crack tip depths (within 2.8 mm of actual position, refer to Figures 9E,F while the modulated parameter β_S+Dprovides a more accurate prediction within 1.2 mm of the actual crack tip position (refer to Figure 9G).

From the previous section it is clear that the pump frequency modulation method provides advantages over traditional linear techniques, and was shown to provide more accurate assessment for small and large vertical fatigue cracks. As the modulation parameter is a function of f₁ and f₂ and the non-linear frequency combinations, it is expected that increasing the number of transmit frequencies can have benefits in defect/damage evaluation. It is expected that further non-linearities will be introduced through frequency mixing and lead to greater non-linear signal to noise levels. Figures 10A,C show the dual frequency modulation results for f₁ (4.8 MHz) and f₂(5.3 MHz) and Figures 10B,D shows the tri frequency modulation results for f₁ (4.8 MHz), f₂(5.0 MHz), and f₃(5.2 MHz). Although these two cases cannot be directly compared, as they have a different number of frequencies, they show similar results and predict the same crack tip location within 1 mm from the actual crack tip location.

Figures 11A,B shows different frequency combinations tested for the tri modulation method. At low excitation frequencies (f₁ = 3.8 MHz, f₂= 4.0 MHz, and f₃= 4.2 MHz, Figures 11A,C), the prediction of the crack tip improves relative to all the other cases tested (within 0.5 mm), while at high excitation frequencies (f₁ = 5.8 MHz, f₂= 6.0 MHz, and f₃= 6.2 MHz, Figures 11B,D) the prediction of the crack tip is inferior. It is also clearly noticeable that the background noise level increases as the frequency increases. These results can be explained by two factors (1) the transmitted amplitude of the signal is much lower at the higher frequencies due to attenuation and the probes capabilities and (2) due to the stiffness of the crack, larger energy is required to produce the “breathing/ringing”' crack scenario. It is expected that crack stiffness increases from the base of the crack toward the crack tip. Thus, in this case, the lower frequencies ability to excite stiffer parts of the crack (closer to the crack tip) is improved as this is directly related to the energy of the wave.

Table 3 summarizes the estimated crack tip location found by the linear and dual/tri frequency methods. For the SFC sample (crack length of ~3.5 mm) the dual frequency method (f₁ −4.8 MHz and f₂−5.3 MHz) and tri frequency method (f₁ −4.8 MHz, f₂−5.0 MHz, and f₃−5.2 MHz) both predicted a crack length of ~2 mm. While the tri frequency method at f₁ = 3.8 MHz, f₂= 4.0 MHz, and f₃= 4.2 MHz, provided the best estimation of the actual crack length ~3 mm (within ~0.5 mm). While the standard linear method was not able to detect the fatigue crack in this instance. For the LFC sample (crack length of ~14 mm) the dual frequency method (f₁ −4.8 MHz and f₂−5.3 MHz) estimated a crack length of ~12.8 mm; 2.8 mm longer than that predicted by the linear method. It is clear from these results that the non-linear modulated results outperform those of the standard linear methods, furthermore the tri frequency method is also capable of providing better results than the dual frequency method, although this is dependent on the correct frequency selection. Frequency characterization and selection optimization does not form part of this work.

To further quantify the ability of the linear vs. the non-linear methods Figure 12 summarizes and compares the linear, dual frequency modulation (Figure 12A) and tri frequency modulation (Figures 12B–D) methods in terms of the signal to noise ratio (SNR, refer to Equation 10). The intensity at the crack location (or notch—linear case) was divided by the mean noise intensity value for each of the zoomed images. Figure 12A shows that for the dual frequency method, there was three orders of magnitude difference between the modulated parameter β_S+D (2,487) when compared to the linear results (SNR for f₁, f₂, f₁₂, f₂₁ <2), furthermore the linear results are referenced to the notch not a crack location. Figures 12B,C show that SNR can be further improved by increasing the number of excitation frequencies, with the results close to four orders of magnitude greater than the linear cases (SNR for f₁, f₂, f₃, f₁₂₃, f₂₃₁, f₃₁₂ ≤ 2). Figure 12D, shows a low non-linear SNR factor which was expected due to the low tri frequency response at high frequencies (f₁ = 5.8 MHz, f₂= 6.0 MHz, and f₃= 6.2 MHz, refer Figures 11B,D).