# Blade Crack Diagnosis Based on Blade Tip Timing and Convolution Neural Networks
## 2 Method  
### Numerical Model  
The finite element model and the lumped parameter model are frequently used for the mathematical modeling of blades in scientific research or engineering analysis. The finite element model is based on real blade geometry. Therefore, the vibration analysis results are more detailed and accurate, but the computational time and resource consumption are very high. The lumped parameter model can only qualitatively describe the blade vibration characteristics. However, it can be programmed to analyze them quickly and provide a large and effective signal data set for failure vibration analysis. A lumped parameter model for the combined cracked blade fault diagnosis problem is developed in this research, as shown in Figure 2. The whole circle blade model consists of 6 sectors, all of which are composed of 3 degrees of freedom, including the tip, body, and root of the blade, and 1 degree of freedom, including the rotor disk. The center of the rotor disk is connected to the rotor shaft to rotate the disk around the rotor shaft. The blade only vibrates in the direction of rotation. A dry friction-damping structure generates relative friction between the tips of adjacent blades based on their relative motion. The blades are modeled with 3 degrees of freedom in this work, which can demonstrate higher-order complex motion patterns in blade vibration. A total of four sensors are evenly distributed around the perimeter of the blades, with a 15\({}^{\circ}\) angle between them. The lumped parameter model used in this study does not correspond to the experimental model because the lumped parameter model ignores a large number of real blade characteristics. Furthermore, in order to validate the method proposed in the study, the mutual influence factors between adjacent blades are added to the lumped parameter model, which is also different from the free blades used in the experiments. The individual waveforms are separated from the sensor output based on the experimentally determined time-domain voltage signal and normalized to the original amplitude and pulse signal range according to the sensor probe size. As shown in Equation (1), the Fourier variation of the first four orders is retained to fit a waveform function of the relative distance between the blade and the sensor. \[U(x_{i,j})=a_{0}+\sum_{i=1}^{n}a_{i}cos(i\omega x_{i,j})+b_{i}sin(i\omega x_{ i,j}) \tag{1}\]  
Figure 2: The lumped parameter model. where \(x_{i,j}\) is the distance of the blade tip from the sensor center, \(i\) is the order of the harmonic wave, \(b_{i}\) is the angular frequency, \(a_{i}\) is the cosine term coefficient, and \(b_{i}\) is the sine term coefficient. Cracks are assumed to appear at random in the top, middle, and root of blades in the numerical model, and the variation in cracks regarding stiffness values at different sites is defined as crack severity \(\xi\). The stiffness value \(k_{\rm m}\) at the location where the crack appears on the blade can be calculated by subtracting the crack severity \(\xi\) from the harmonic value \(k_{\rm t}\), as expressed by Equation (2):  
\[k_{\rm m}=k_{\rm t}(1-\xi) \tag{2}\]  
The problem of blade fracture combinations is examined in this research, where cracks may be present in more than one blade. Cracks are added randomly to the top, middle, or root of a blade, and this detection task is a classification task. Blade crack severity is a continuous variable in the range [0, 0.5], with 0 indicating no cracks and the corresponding detection task is a regression task. The system parameter deviation \(y_{\rm err}\) and signal noise \(u_{\rm noise}\) are added to the BTT voltage signals in this study to make the signals derived from the numerical model resemble the real blade signals more. The noise is 30 dB of white Gaussian noise. The deviation of the system parameters affects the magnitude of the blade vibration displacement amplitude, and the signal noise affects the extraction of the blade tip arrival time.