# Wideband Vibro-Acoustic Modulation for Crack Detection in Wind Turbine Blades
## 2 VAM Model for Detecting Cracks in WT Blades  
### Description of the Proposed Model for WT Blade Crack Detection  
To describe the VAM on the WT blade, we consider a mechanical oscillator model (comprising spring, mass, and damping elements), as illustrated in Figure 3. This model can be applied to various blade resonance modes, each of which has its unique resonance frequency and damping parameters (Q factor). Numerous studies have utilized this mechanical oscillator model to perform WT blade analysis. For example, this model was described using Equation (1) in [8] and Equation (34) in [54]. The rotation of the blade generates internal stress due to its weight, altering the elasticity and damping of any existing cracks. A unique aspect of our model is that it incorporates time variation into the model parameters (resonance frequency and Q factor). This variation is driven by the blade load and gravitational load during its rotation. The mechanical oscillator model is presented in Figure 3.  
The model describes the blade material displacement \(x\) under the action of external force in Equation (1) [8, 54]. \[m\ddot{x}+c\dot{x}+kx=\sigma(t), \tag{1}\]  
Figure 3: The model of the mechanical oscillator. Figure 2: Flowchart of the article and steps performed. where \(m\) is the mass, \(\sigma\) is the applied force, \(k\) is the model stiffness that depends on the blade material and include the non-linear crack stiffness, and \(c\) is the damping (attenuation) coefficient. Thus, Equation (1) can be written in the following form:  
\[\ddot{x}+\delta\dot{x}+\omega_{o}^{2}x=\sigma(t)/m, \tag{2}\]  
where the resonance frequency \(\omega_{0}=\sqrt{\frac{k}{m}}\) and attenuation decrement  
\[\delta=\frac{c}{m}=\frac{\omega_{0}}{Q}\]  
\(Q\) is the quality factor. The classical solution shown in Equation (2) can be presented for use in spectral components of the displacement \(X(\omega)=FFT[x(t)]\). The same solution was presented in Equation (3) in [8] and in Equation (56) in [55]. \[X(\omega)=\frac{P(\omega)}{\left(\omega_{o}{}^{2}-\omega^{2}+i\delta\omega \right)}, \tag{3}\]  
where \(P(\omega)=FFT(\sigma(t)/m)\) is the spectrum of the external applied force. The proposed VAM model is based on the slow modulation of the oscillator parameters by the internal loading generated via the blade rotation. This effect is illustrated in Figure 4, where the crack can be opened to a horizontal blade position when the crack is on the upper side (Figure 4a) and closed when the crack is on the lower side (Figure 4b). The model contemplates a linear oscillator, where elasticity and damping parameters are slowly varied under the gravitational force acting in line with the blade rotation frequency \(F\). This model is applicable if the decay time of the oscillator \(T\) is much shorter than the blade rotation period. \[T=\frac{2\pi Q}{\omega_{o}}\ll\frac{1}{F}, \tag{4}\]  
In the model, two time scales are considered:  
Fast time \(t\) describes high-frequency wideband vibrations in the frequency band 100-5000 Hz. In the simulation, the sampling rate at this time is 10 kHz. Slow time \(\tau\) is used for the description of the blade rotation with the frequency \(F=0.05\)-0.2 Hz. In the simulation, the sampling rate for the slow blade parameter variations was 10 Hz. We assume that two parameters of the model oscillator exhibit minor variations in the rotation frequency and can be described using the following formulas:  
\[\omega_{o}=\overset{\sim}{\omega}_{o}(1+\alpha cos\Omega\tau), \tag{5}\]  
Figure 3.0 Schematic presentation of crack opening and closing under the gravitation force: (**a**) the gravitational force is directed down and opens the crack on the upper side of the blade; (**b**) the blade is rotated at 180 degrees, and the gravitational force closes the crack. Figure 3.1  
where \(\Omega=2\pi F\), non-linear parameters \(\alpha\), \(\beta<<1\), and \(\overset{\sim}{\omega}_{o}\) and \(\delta_{o}\) are the resonance frequency and the damping coefficient without application of the rotation load, respectively.