## 2 Fault Signal Characteristics of Test Rig
### Acquisition of Abnormal Vibration Fault Signals Based on FFT and DWT
#### 2.2.2 Discrete Wavelet Transform (DWT)  
Although Fourier transform can transform the signal from the time-domain analysis to the frequency-domain analysis, the corresponding time-domain information is lost during the transformation, and hence it cannot provide comprehensive and effective information for fault diagnosis and analysis. Wavelet analysis is an extension of Fourier analysis [23; 24]. DWT is usually used to analyze nonlinear and non-stationary time-frequency signals. Therefore, the DWT is more suitable for practical applications. In this manuscript, vibration analysis is conducted by using DWT.  
In order to better understand discrete wavelet transform, the wavelet basis functions \(\mathrm{\Psi(t)}\) of continuous wavelet transform is introduced first.  
\[\mathrm{\Psi_{a,b}(t)=|a|^{-1/2}\mathrm{\Psi}\bigg{(}\frac{t-b}{a}\bigg{)}}, \tag{4}\]  
where \(a\) is called the scale factor and \(b\) is called the displacement factor.  
Figure 8: Surface morphology of the inner ring.  
Suppose that the initial signal is f(t), the continuous wavelet transform is defined as:  
\[\mathrm{WT_{x}(a,b)}=|a|^{-1/2}\int f(t)\Psi\Big{(}\frac{t-b}{a}\Big{)}dt=\Big{<} f(t),\Psi_{a,b}(t)\Big{>}. \tag{5}\]  
\(\mathrm{WT_{x}(a,b)}\) represents the projection of f(t) on the wavelet basis function, and the DWT is used to discretize a and b; In general, it is taken \(a=2^{-j}\), \(b=2^{-j}k\), \(j\), \(k\in Z\), Suppose that \((W_{\Psi}f)(a,b)=\Big{<}f(t),\Psi_{a,b}(t)\Big{>}\). The DWT can be obtained as follows:  
\[(\mathrm{DW_{\Psi}f})(j,k)=\Big{<}f(t),\Psi_{j,k}(t)\Big{>}, \tag{6}\]  
where \(\Psi_{j,k}(t)=2^{\frac{j}{2}}\Psi\Big{(}2^{j}t-k\Big{)}\); \(j,k\in Z\).  
Several common wavelets are available in DWT, such as Morlet, Mexican wavelet, Meyer, Harr wavelet, et al. In previous studies, wavelet basis function selected is similar to the original signal in principle. Therefore, Daubechies wavelet is selected in this article. The Daubechies (dbN) wavelet system, which is a discrete orthogonal wavelet, is adopted in this manuscript.  
This study adopts the db3 wavelet system to analyze and reconstruct the fault signal of the rolling bearing, with six decomposition layers. The reconstructed curves of wavelet coefficients D1-D6 and scale coefficient A6 are reflected in Figure 9.  
It can be concluded that with different numbers of decomposition layers, by comparing the original vibration curve with the waveforms obtained by the wavelet coefficient D1-D6 and the scale coefficient A6, the signal reconstructed by the scale coefficient A6 best reflects the fault signal characteristics of the bearing. In Figure 9, the starting time of skidding damage is 8.4 s, which are directly obtained by Figure 4. Figure 9 also shows that there is an obvious difference between the starting time of the skidding damage and the maximum vibration time, and the starting time of skidding damage is ahead of the maximum vibration time.  
By identifying and analyzing the vibration acceleration curve of the rolling bearing from the frequency domain and the time-domain, it can be inferred that the rolling bearing starts to skidding damage at 8.4 s. The vibration characteristics of the bearing under the condition of skidding damage can be described more accurately by combining the information of the time-frequency domain. In addition, radial load, temperature, slip rate mutation, and other factors will cause the skidding damage,  
Figure 9: Discrete Wavelet Transformation.  
and the subsequent analysis will focus on these three factors to analyze the reasons for the abnormal vibration of skidding damage in rolling bearings.