# Intelligent Diagnosis of Bearing Failures Based on Recurrence Quantification and Energy Difference
## 3 Experimental Implementation
### Hilbert Envelope Analysis  
Hilbert transformations [14] were applied to one set of data from each of the four categories: normal signals, inner race faults, outer race faults, and rolling element faults. A signal has both amplitude and phase information, so it can be determined as follows:  
\[x(t)=A(t)\cos(\omega_{0}t+\theta(t)) \tag{17}\]  
By transforming the real-valued signal into a complex form, information such as the instantaneous amplitude and instantaneous frequency of the signal can be directly extracted from the real-valued signal. \[\widetilde{x}(t)=x(t)+j\widetilde{x}(t) \tag{18}\]  
\[\widetilde{x}(t)=A(t)\cos[\omega_{0}t+\theta(t)]+jA(t)\sin[\omega_{0}t+\theta( t)] \tag{19}\]  
\[\widetilde{x}(t)=A(t)e^{j(\omega_{0}t+\theta(t))} \tag{20}\]  
\[\widetilde{x}(t)=A(t)e^{j\theta(t)}e^{j\omega_{0}t} \tag{21}\]  
where \(e^{j\omega_{0}t}\) is the complex carrier signal and \(A(t)e^{j\theta(t)}\) is the complex envelope. \[|\widetilde{x}(t)|=\left|A(t)e^{j\theta(t)}\right|=|A(t)| \tag{22}\]  
The absolute value of the analytical signal is the envelope signal. The amplitude-frequency plots obtained through fast Fourier transform are illustrated in Figures 5-16. The data within the first 1000 Hz of the envelope signal related to the inner race fault were analyzed. Based on established experience, the characteristic frequency of the inner race fault in the rolling bearing was identified as 161.7 Hz, with the second harmonic located at 323.4 Hz. The data within the first 1000 Hz of the envelope signal related to the outer race fault were analyzed. Based on established experience, the characteristic frequency of the outer race fault in the rolling bearing was identified as 107.6 Hz, with the second harmonic located at 323.4 Hz. Figure 4: Bearing failure experiment at CWRU [18]. Figure 5: Diagram of inner race fault signal envelope. Figure 6: Amplitude-frequency diagram of inner race fault signal envelope. Figure 7: Amplitude–frequency diagram of inner race fault signal envelope (first 1000 Hz). \[\widetilde{x}(t)=x(t)+j\widetilde{x}(t) \tag{18}\]0 Diagram of rolling element fault signal envelope. \[\widetilde{x}(t)=x(t)+j\widetilde{x}(t) \tag{18}\]1 Amplitude–frequency diagram of rolling element fault signal envelope. \[\widetilde{x}(t)=x(t)+j\widetilde{x}(t) \tag{18}\]2 Amplitude–frequency diagram of outer race fault signal envelope (first 1000 Hz). \[\widetilde{x}(t)=x(t)+j\widetilde{x}(t) \tag{18}\]3 Amplitude–frequency diagram of outer race fault signal envelope. \[\widetilde{x}(t)=x(t)+j\widetilde{x}(t) \tag{18}\]4 Amplitude–frequency diagram of normal signal envelope (first 1000 Hz). \[\widetilde{x}(t)=x(t)+j\widetilde{x}(t) \tag{18}\]5 Diagram of normal signal envelope. \[\widetilde{x}(t)=x(t)+j\widetilde{x}(t) \tag{18}\]6 Amplitude–frequency diagram of rolling element fault signal envelope (first 1000 Hz). \[\widetilde{x}(t)=x(t)+j\widetilde{x}(t) \tag{18}\]7 Amplitude–frequency diagram of normal signal envelope. The data within the first 1000 Hz of the envelope signal associated with the rolling element fault were analyzed. Based on established experience, the characteristic frequency of the rolling element fault in the rolling bearing was identified as 107.7 Hz. The data within the first 1000 Hz of the normal signal were analyzed. Based on established experience, the rotational frequency was identified as 59.87 Hz. From the perspective of fault identification accuracy, spectral analysis and Hilbert envelope analysis serve as common diagnostic methods that can reveal certain characteristics of faults. However, these methods heavily rely on existing experience and patterns, which limits their effectiveness. For instance, the spectra of inner race faults and rolling element faults often share many similarities, making their distinguishing features less pronounced and sometimes difficult to discern, potentially leading to diagnostic errors.