# A Survey on Fault Diagnosis of Rolling Bearings
## 3 Rolling Bearing Fault Detection  
The failure of rolling bearings will break the original energy balance of the system, and the most intuitive performance is abnormal vibration. The bearing fault vibration signal shows an increase or fluctuation in amplitude in the time domain and spectrum lines of fault characteristic frequency with prominent amplitude in the frequency domain. In [15], four empirical formulas were summarized for calculating the theoretical fault frequencies of the inner ring (\(F_{inner}\)), outer ring (\(F_{outer}\)), rolling element (\(F_{ball}\)), and cage (\(F_{cage}\)), as shown in Equation (1).  
\[\begin{cases}F_{inner}=\frac{N_{\text{h}}S_{ab}}{2}(1-\frac{d_{b}}{D_{p}}\cos \varphi)&F_{outer}=\frac{N_{\text{h}}S_{ab}}{2}(1+\frac{d_{b}}{D_{p}}\cos \varphi)\\ F_{ball}=\frac{D_{p}S_{ab}}{2d_{b}}(1-(\frac{d_{b}}{D_{p}}\cos\varphi)^{2})&F_ {cage}=\frac{S_{ab}}{2}(1-\frac{d_{b}}{D_{p}}\cos\varphi)\end{cases} \tag{1}\]  
where \(D_{p}\) is pitch diameter, \(d_{b}\) is rolling element diameter, \(N_{b}\) is rolling element number, \(\varphi\) is contact angle, and \(S_{sh}\) is shaft speed, which are basic parameters. It is possible to detect a bearing fault by observing the fluctuation of the time-domain waveforms or observing spectral lines associated with the fault characteristic frequency. Directly measuring rolling bearing vibrations is impossible in the real world. Generally, the sensor installed on the bearing pedestal is used to collect the signals indirectly, resulting in a significant amount of noise and harmonic interference in the collected vibration signals. The polluted bearing vibration signal is not effective for detecting bearing faults. Therefore, a series of fault detection methods based on vibration analysis was proposed to remove the noise and harmonic interference components in the signals, enhance the fault-related pulses, reduce the difficulty of fault detection, and improve the effectiveness of detection. Based on the difference in signal processing principles, the fault detection methods are mainly divided into four categories: morphological transformation-based methods, filter-based methods, decomposition-based methods, and deconvolution-based methods.  
The common rolling bearing fault detection methods are summarized in Figure 4. The morphological transform-based methods can extract harmonic or impact components of signals by using morphological operators with different structures, whose appropriateness directly influences performance. The filter-based methods can adaptively identify the resonance frequency band that contains rich fault information, where the division of frequency band and the choice of subband are the key factors affecting the results. The decomposition-based methods refer to decomposing the complex signals into simple subband signals, and  
Figure 3: The taxonomy of this survey.  
these methods should address modal aliasing, parameter setting, manually tuning, etc. The deconvolution-based detection methods belong to blind signal processing technology, which recovers fault characteristic signals by designing the appropriate inverse filters and setting the deconvolution period and filter length. In addition to the method based on recursive decomposition, which typically lacks the mathematical model as theoretical support, other methods have the complete mathematical theory.