# Wideband Vibro-Acoustic Modulation for Crack Detection in Wind Turbine Blades
## 2 VAM Model for Detecting Cracks in WT Blades
### Comparison between Traditional VAM (Active Vibration) and Proposed VAM (Passive Vibration)
#### 2.3.2 Passive VAM  
For the simulation of the VAM effect to create wideband noise excitation signals, we employed Equations (1) and (3), where \(P(\omega)\) represents the wideband noise spectrum with random amplitude and phase. An example of the spectrum of the blade vibrations for case 1, without additional rotational loads, is shown in Figure 10.  
Figure 8: The VAM Spectra of the 52-m WT blade: (**a**) for the frequency of 2031.6 Hz (normalized frequency \(\zeta=0.01\)); (**b**) for the frequency of 2304.6 Hz (normalized frequency \(\zeta=0.24\)). It is seen that the side band components are higher for the frequency closer to the resonance frequency. Figure 9: The dependencies of MI on the probe frequency for the blades with parameters presented in Table 2: for the 52-m WT (blue line) and the eight-foot section (red line). Variations in the signal spectra due to resonance frequency and attenuation changes during the blade rotation are not typically seen in the signal spectra, and we will consider the modulation of the total signal energy \(E(\tau)\). Here, the slow time variance \(\tau\) is used. In our modelling, the sampling rate for this slow time was 10 Hz, while the sampling rate for the description of the resonance blade vibration was 10,000 Hz. \[E(\tau)=\int_{\omega_{1}}^{\omega_{2}}\left[\frac{P(\omega)}{({\omega_{o}}^{2} -\omega^{2}+i\delta\omega)}\right]^{2}d\omega, \tag{15}\]  
where \(\omega_{o}=\widetilde{\omega}_{o}(1+\alpha cos\Omega_{0}\tau)\), and \(\delta=\delta_{o}(1+\beta cos\Omega_{0}\tau)\). The frequency band for energy calculation was chosen around the blade resonance frequency: \(\omega_{1}=\omega_{0}/2\), \(\omega_{2}=2\omega_{0}\). The calculations of the slow time variation in the signal energy and its spectra form the basis of the DEMON signal processing, which is widely applied in the field of submarine detection [55; 57]. The spectra of the DEMON signal are as follows:  
\[D(\omega)=FFT[E(\tau)], \tag{16}\]  
In the DEMON spectra, a peak will appear at the rotation frequency \(F\) representing a crack or other non-linearities that may occur on the WT blade. This peak can be detected when it is higher than the surrounding spectral noise components. In this wideband VAM model, the signal-to-noise ratio (SNR) is computed as the difference between the frequency component with the rotation frequency and the maximal spectral component outside of the rotation frequency. Figure 11a illustrates the SNR of the DEMON spectra of the rotating blade excited via wideband noise. Figure 10: The frequency response changes for the test with an 8-foot blade section [8]. The blue color lines show experimental data, and the red color lines are interpolated by Equation (3). The SNR is calculated using the following formula:  
\[SNR=A_{F}-A_{max}, \tag{17}\]  
where the \(A_{F}\) is the amplitude of the DEMON component with the rotation frequency, and \(A_{max}\) is the maximum of the noise peak outside of \(A_{F}\). Figure 11 displays the DEMON spectra of the three cases summarized in Table 3. The analysis of the signal is conducted within a time window of 10,000 s.  
The influence of the non-linear parameters describing the resonance frequency shift (a) and amplitude dependable attenuation (\(\beta\)) was investigated by comparing the SNR and MI for cases with and without amplitude-dependable attenuation (\(\beta\) = 0). This comparison is presented in Table 3.  
As can be seen in the table, damping non-linearity is particularly important for wide-band signal modulation. The SNR of the DEMON signal decreases with the absence of damping non-linearity (\(\beta\) = 0). In contrast, damping non-linearity does not significantly influence the MI parameters of the harmonic probe wave. The conducted modelling allows the prediction of the measured SNR of the processing time. Increasing the time window of signal analysis leads to an increase in the SNR. Figure 12 presents the dependence of the SNR on the signal-processing time, which is up to 10,000 s (about 2.8 h), for the 52-m blade (case 2 in Table 2).