# Blade Crack Diagnosis Based on Blade Tip Timing and Convolution Neural Networks
## 3 Results
### Results of Experimental Measurement Data  
#### 3.2.1 Analysis of Blade Fault Vibration Characteristics  
A finite element numerical analysis is adopted to obtain the blade modal and response characteristics under various crack location and crack depth conditions. From the numerical analysis results, the key characteristic parameters for fault identification can be obtained to guide the experimental design. In this study, the numerical model is verified by the variable speed sweep method.  
The Campbell diagram of blade vibration and the modal vibration patterns for the first three orders of vibration are given in Figure 11. The result shows that the first-order resonance of the blade may occur between 500 rpm and 1500 rpm for integer multiples of the speed, with resonance points of 611.4 rpm, 739.9 rpm, 936.9 rpm, and 1292.0 rpm for  
Figure 10: Regression curve of blade crack degree detection results.  
Figure 9: Confusion matrix of blade crack location detection results.  
the line of excitation from three to six, corresponding to vibration frequencies of 61.14 Hz, 61.66 Hz, 62.46 Hz, and 64.60 Hz.  
The test speed is set to rise from 0 rpm to 1600 rpm with a uniform acceleration for 96 s and a maximum speed frequency of 26.67 Hz. As shown in Figure 12, the Campbell diagram of blade vibration is obtained using wavelet analysis based on the measured strain gauge's time-domain signal. It can be seen that the blade resonates at multiple multiples in the speed range.  
Table 2 gives the results of the measured intrinsic frequencies of the strain gauges at each octave, which is very close to the numerical analysis results, with a maximum error of 2.8% for the numerical analysis of the blade vibration frequencies.  
In the numerical model, the natural frequencies at nine uniform depths from 1/30 to 9/30 were analyzed at each crack depth, considering the crack locations at the top of the blade, the middle of the blade, and the root of the blade, respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Frequency Multiplication** & **3** & **4** & **5** & **6** \\ \hline Numerical prediction/Hz & 64.60 & 62.46 & 61.66 & 61.14 \\ \hline Experimental measurement/Hz & 62.84 & 62.49 & 61.04 & 60.83 \\ \hline Numerical error/\% & 2.80 & \(-\)0.05 & 1.02 & 0.51 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimental and numerical analysis of blade vibration frequency results at each octave.  
Figure 11: Campbell diagram of modal frequency of model blade.  
Figure 12: Campbell diagram of variable speed strain signal based on wavelet analysis.  
Figure 13 shows the variation curve of the natural frequency of the cracked blade's first, second, and third-order vibration modes with crack depth. When the crack is in the middle and root positions of the blade, the modal frequency decreases as the crack depth increases. In addition, when the crack is in the top position, the blade modal frequency slightly increases as the crack depth increases. The reason is that the influence of the actual blade crack processing mode reduces the quality of the blade tip position. According to the results, the natural frequency of the blade can be applied as a sensitive parameter to indicate the condition of the blade crack. In the same crack depth case, the blade modal frequency is more sensitive to cracks appearing at the root of the blade.