## 2 Fault Signal Characteristics of Test Rig
### Acquisition of Abnormal Vibration Fault Signals Based on FFT and DWT  
#### 2.2.1 Fast Fourier Transform (FFT)  
According to different transform methods and signal properties, amplitude spectrum and power spectrum are commonly used in the spectrum analysis [18; 19; 20; 21]. The amplitude spectrum represents the amplitude corresponding to the harmonic vibration components of each frequency, whereas the height of the spectral line represents the amplitude of the frequency components. The purpose of spectrum analysis is to decompose the various frequency components of the signal to clearly highlight the role of various frequency signal waveforms. FFT is a fast algorithm for calculating discrete Fourier transform (DFT). When using FFT to analyze rolling bearing vibration signal, the DFT of finite length discrete signal x(n) is:  
\[\mathrm{X(k)}=\sum_{n=0}^{N-1}x(n)W_{N}^{nk};k=0,\cdots,N-1;W_{N}=e^{-j\frac{2 \pi}{N}}. \tag{1}\]  
The vibration signal is an important index to measure the skidding damage of rolling bearings. Acceleration, velocity, and displacement is measured as parameters of vibration measurement [22]. This system uses acceleration as the characteristic value of the vibration signal, so that the vibration acceleration signals of the roller and the inner ring are acquired in real-time during the test. The original vibration signal is transformed into a frequency-domain signal by FFT. The time-domain vibration acceleration signal is presented in Figure 3. Figure 4 is the time-domain signal after noise reduction. Figure 5 shows the time-domain signal diagram of the vibration signal after noise reduction in bearing under no-failure conditions. As can be seen from the vibration curve in Figure 3, the original vibration acceleration waveform is unclear and has many blurs. It cannot get faulty signals from the vibration acceleration curve of the original time-domain. Therefore, the noise reduction of the time-domain signal is performed. Figure 4 shows the time-domain signal diagram after noise reduction. The wavelet noise reduction method is wavelet threshold noise reduction, and the specific method is wavelet soft-threshold  
Figure 4: Time-domain signal after noise reduction. Figure 5: Time-domain signal after noise reduction under no-failure conditions. Figure 3: Time-domain signal of vibration acceleration. denoising. The mother wavelet used in this manuscript is wthresh function. Through wavelet noise reduction processing, signal burrs can be well contained. It can be observed in Figure 4 that the vibration signal waveform after noise reduction has greatly improved. By comparing time-domain signal of vibration acceleration under failure conditions in Figure 4 and non-failure conditions in Figure 5, it can be seen from Figure 4 that waveform starts to fluctuate up and down regularly at 8.4 s, so it can be inferred that 8.4 s is the starting time of skidding damage. As can be seen from the spectral diagram of vibration acceleration in Figure 6, the amplitude curve shows a continuous downward trend. A large peak is also seen around 16.44 Hz. In addition, the figure indicates a frequency multiplication value of 16.44, and the amplitude of vibration, such as one frequency multiplication and two frequency multiplications, are relatively large. Abnormal vibration may occur in the rolling bearing, and therefore 16.44 Hz may be the fault frequency threshold of the rolling bearing. Threshold frequency specifically refers to the fault characteristic frequency of skidding damage, and the experimental data in Figure 6 show that the threshold frequency is 16.44 Hz. The roller and inner ring surface are observed after the end of the experiment showed that the bearing surface experienced skidding damage, which is shown in Figure 7. Therefore, it can be preliminary determined that this point is the fault frequency of skidding damage.