## 4 Experiments on Feature Extraction of Machinery Faults
### Experiment on the Swashplate Axial Piston Pump
#### 4.3.2 Feature Extraction Based on SNTF  
Given the cost of computation, we only select the sampled signal for further analysis, which is segmented at the length of 20,480 points. Figure 22 provides the waveform and spectrum of the original signal during 0-0.08 s.  
Next, we convert the one-dimensional signal to a two-dimensional phase space representation by virtue of PSR method. With the sampling rate of 25,600 Hz and the speed of 3415 Hz, the embedding dimension \(m\) is 2048, and the time delay \(\tau\) is set to 1024. Therefore, a total of 19 phase point vectors are generated, creating a data matrix of \(2048\times 19\).  
On the basis of the Stankovic principle, we empirically choose for the Hanning window, the width of 31 where the gradient of the concentration measure \(H(N)\) turns from steep to flat. Then, STFT is performed on phase point vectors to find the time-frequency distributions, which are arranged in a three-dimensional coordinate system to form the multi-aspect data tensor \(\underline{\mathbf{Y}}_{d}\in\mathbb{R}^{2048\times 2048\times 19}\).  
Based on the HALS-SNTF algorithm, as major peaks in the frequency spectrum are found at 1695 Hz, 4616 Hz, and 8094 Hz, the reduced-dimension index is set to three, and the data tensor is decomposed further into factor matrices, within which we are interested in the frequency matrix \(\mathbf{A}\) and the time matrix \(\mathbf{B}\). By the vector product \(\mathbf{a}_{i}\mathbf{b}_{i}^{T},(s=1,2,3)\), three reconstructed TFDs are obtained, respectively, corresponding to three different feature subspaces, as provided in Figure 23a-c.  
Figure 23: The reconstructed TFDs based on the SNTF factors: (**a**) Reconstructed TFD 1; (**b**) reconstructed TFD 2; (**c**) reconstructed TFD 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Bearing Model** & **Running Speed** & **Sampling Rate** & **Fault Characteristic Frequency** \\ \hline D811 IQ & 3415 RPM & 25,600 Hz & 689.9 Hz \\ \hline \hline \end{tabular}
\end{table}
Table 3: Relevant experimental settings.  
Figure 22: The original signal for further analysis: (**a**) Waveform; (**b**) frequency spectrum.  
Generally, the vibration signal of a faulty ball bearing can be regarded as a compound signal with amplitude modulation. More specifically, it is the periodic impulse (the signal of a certain fault) that modulates the system's resonance signal. From the TFDs in Figure 23, we can see that the periodic feature in the time axis is most prominent in TFD 2, and this feature lies in the presence of the resonance frequency at around 4300 Hz and 8500 Hz. Whereupon, to extract the fault feature, we chose TFD 2 to perform ISTFT and envelope demodulation successively. The results are presented in Figure 24.  
The envelope spectrum in Figure 24b extracts the frequency of 687.5 Hz, which is consistent with the theoretical fault frequency of the thrust bearing. For comparison, the envelope spectra of the reconstructed TFD 1 and TFD 3, are presented in Figure 25. They do contain the fault frequency component, but they are apparently less observable than the spectrum in Figure 24.  
From the above results, the impulse feature of the vibration signal is finally extracted. Thereupon, we can confirm that the feature extraction method based on SNTF is verified efficacious for primary applications in the field of machinery fault diagnosis.  
Figure 24: The results of envelope analysis based on reconstructed TFD 2: (**a**) Envelope waveform; (**b**) envelope spectrum.  
Figure 25: The envelope spectra based on the reconstructed TFD 1 and 3: (**a**) Envelope spectrum based on the reconstructed TFD 1; (**b**) envelope spectrum based on the reconstructed TFD 3.