# Bearing Fault Diagnosis Using Grad-CAM and Acoustic Emission Signals
## 2 Proposed Method
### Bearing Characteristic Component Analysis  
Bearing failures generate periodic burst signals that are represented as the bearing characteristic frequency harmonics in the spectrum [20]. The outer race way with a crack on its surface (ORCS) emits a periodic pulse each time when the rolling element passes over the cracked surface. Since the outer race is a static component of the bearing and the applied load to cracked surface is always stable, the amplitude of the impulses does not change. The inner race way with a crack on its surface (IRCS) generates a series of impulses when each rolling element hits the crack on the inner race of the bearing. By rotating the inner ring with the shaft, the response of impulses grows up periodically when the inner race passes loaded zone, which is oriented to the direction of gravity. Since this phenomenon modulates the impulses by rotating speed, the sideband of rotating speed appears nearby the characteristic frequency of inner race. The rolling element with a crack on its surface (RECS) generates impulses by hitting inner and outer races. The magnitude of the impulse is affected by whether the contact occurred in the loaded or unloaded zones. Similarly, the sideband of RECS is a fundamental train frequency [20]. Figure 3 illustrates the examples of the ideal signals for ORCS, IRCS, and RECS.  
Figure 2: The flowchart of the envelope analysis. Hilbert-transform is applied to the acoustic emission (AE) signal to calculate the 90-degree phase-shifted signal. Then, the analytical signal is calculated by sum of the original signal and its Hilbert-transform as an imaginary number. Next, the envelope signal is calculated by applying absolute operation to the previously computed analytical signal. Finally, the fast Fourier transform of the envelope signal provides an envelope spectrum.  
Accordingly, the bearing characteristic (or defect) frequencies are categorized into ball pass frequency on the outer race (BPFO), ball pass frequency on the inner race, (BPFI), and ball spin frequency (BSF). BPFO, BPFI, and 2\(\times\)BSF are caused by the bearing failures of the outer race, inner race, and rolling element, respectively. The bearing characteristic frequencies are defined as follows [20]:  
\[BPFO=\frac{Nb}{2}\times S\times\left[1-\left(\frac{Bd}{Pd}\times cos\theta \right)\right], \tag{4}\]  
\[BPFI=\frac{Nb}{2}\times S\times\left[1+\left(\frac{Bd}{Pd}\times cos\theta \right)\right], \tag{5}\]  
\[BSF=\frac{Pd}{2 Bd}\times S\times\left[1-\left(\frac{Bd}{Pd}\times cos\theta \right)^{2}\right], \tag{6}\]  
where \(Nb\) is the number of rolling elements, \(S\) is the shaft speed, \(Bd\) is the diameter of the rolling element, \(Pd\) is the pitch diameter, i.e., the distance between the center of a rolling element and the center of the inner race, and \(\theta\) is a contact angle of the rolling element with respect to the shaft.  
The bearing characteristic components (BCCs) were extracted as an input vector of the CNN. BCCs are defined as follows:  
\[BCC(k)=f(\omega),\;\omega=0,\;\ldots,\;F_{max}, \tag{7}\]  
where \(f\) are the values of the envelope spectrum and \(F_{max}\) is the frequency, which is higher than all the harmonics of bearing characteristic frequencies as below:  
\[F_{max}=max(BPFO,BPFI,BSF)\times n+f_{side}, \tag{8}\]  
where \(n\) is the number of frequency harmonics and \(f_{\text{side}}\) is the sideband of the highest characteristic frequency. Table 1 shows \(f_{\text{side}}\) for each type of bearing characteristic frequency. In this paper, \(F_{max}\) was equal to BPFI, which is the highest among the bearing characteristic frequencies. Figure 4 depicts the extraction process of BCCs.  
Figure 3: Examples of the ideal signals for each type of bearing fault.  
Since the variation of magnitude makes the training CNN unstable, BCCs are min-max normalized to be used for input data of CNN as follows:  
\[NBCC(i)=\left(BCC(i)-min(BCC)\right)/\left(max(BCC)-min(BCC)\right),\ i=0,\ \ldots,\ F_{max}. \tag{9}\]