# 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)  
#### 2.3.1 Active Method  
This model considers the application of the probe wave with the frequency \(f\).  
\[\sigma=Pcos\omega t, \tag{9}\]  
where \(\omega=2\pi f\).  
In our model, it is assumed that the vibrations in the blades are measured via accelerometer, the electrical signal of which is directly proportional to the acceleration of the blade. This obtained signal is then presented in the following form:  
\[u(t)=A(t)cos(\omega t-\varphi(t)), \tag{10}\]  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Test Data** & \(f_{1}\) & \(f_{2}\) & \(Q_{1}\) & \(Q_{2}\) & \(\alpha\) & \(\beta\) \\ \hline eight-foot section of WT & 335.85 HZ & 337.1 HZ & 81.2 & 111.5 & 0.0004 & 0.0272 \\ blade & 2001.7 HZ & 2024.6 HZ & 5.7 & 5.9 & 0.00113 & 0.003 \\ of 1900 Hz to 2250 Hz. & 2590 HZ & 2602.1 HZ & 14.6 & 14.8 & 0.0005 & 0.00135 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The linear and non-linear parameters of the oscillator model for the considered WT blades.  
Figure 6: Experimental frequency responses observed in [6] for the 52 m blade in various frequency bands. (**a**) The figure represents the frequency change between frequency band 1800 and 2250, (**b**) The figure represents the frequency change between frequency band 2500 and 2700. Blue sold lines represent the experimental data for intact and damage status, while the red dash line represents curves interpolations by Equation (3).  
The amplitude and phase are defined using the following Equations:  
\[A=\frac{\omega^{2}C}{\sqrt{\left(\omega_{o}^{2}(t)-\omega^{2}\right)^{2}+\left( \delta(t)\omega\right)^{2}}}, \tag{11}\]  
\[\tan\varphi=\frac{\delta(t)\omega}{\omega_{o}^{2}(t)-\omega^{2}}, \tag{12}\]  
The constant \(C\) is frequency-independent coefficients, reflecting the amplitudes of the externally applied force and accelerometer sensitivity. The resonance frequency and attenuation undergo slow modulation, as described in Equations (5) and (6). Consequently, the spectrum of the signal \(u(t)\) will contain a main component with an excitation frequency \(f\) and two sideband components with frequencies \(f+F\) and \(f-F\).  
In this paper, we evaluate the modulation of the harmonic probe wave or noise blade vibrations, and results are presented in normalized form. Therefore, the signal spectrum will display the relative level of sideband components.  
In the VAM method using harmonic probe wave, the Modulation Index (MI) is often used as a measure to assess the level of non-linearity [50; 56]  
\[MI=\frac{U_{-}+U_{+}}{2}-U_{o}, \tag{13}\]  
where \(U_{o}\) is the received spectral component of the probe wave, and \(U_{+}\) and \(U_{-}\) are the sideband components in dB.  
Applying Equations (11) and (12) enables the calculation of signal spectra with side components produced via modulation. Figures 7 and 8 illustrate examples of these calculations for a blade rotation frequency of 0.2 Hz and various probe wave frequencies. For the comparison of various blades, in this calculation, we used the normalized frequency \(\zeta\) calculated by subtracting the probe wave frequency \(f\) from the from the resonance frequency \(f_{0}\), before dividing the solution by the resonance frequency \(f_{0}\).  
\[\zeta=(f-f_{0})/f_{0}, \tag{14}\]  
Figure 7: The VAM spectra of 8-foot blade: (**a**) for the frequency of 337.5 Hz (normalized frequency \(\zeta=0.01\)); (**b**) for the frequency of 357.2 Hz (normalized frequency \(\zeta=0.05\)).  
It can be observed that the 8-foot blade section exhibits a higher level of side band components compared to the 52-m blade, which occurs due to the higher Q factor in the 8-foot section of the blade.  
The MI response dependencies on the probe frequency were also calculated, and the results are shown in Figure 9.  
It reveals a significant dependency of the MI on the frequency when using the VAM model, and the case with lower attenuation shows stronger variation in the MI.