## 4 Materials and Methods
### Performance Analysis of the Proposed Algorithm
#### 5.2.2 Fault Characteristics Analysis of Bearing  
Under the condition of a sampling rate of 12 KHz, the effectiveness of the bearing fault diagnosis method based on the energy distribution characteristics of the EEMD eigenfunction is verified for the case of faults in the drive end. Figure 2 presents the waveforms, spectra, signal envelopes, and envelope spectra under the signals of this dataset. Furthermore, Figure 7 successively shows the vibration signal waveform, the envelope signal waveform, the IMF1 to IMF4 components obtained by the EEMD decomposition of the envelope signal, and the energy characteristic vector ED of the normalized IMF component. Figure 8 shows the ED distribution maps when the vibration signal is measured at the driving end with an acceleration vibration sensor at a sampling rate of 12 kHz, and the sample data are selected as 105.mat, 106.mat, 107.mat, 108.mat, and 118.mat. At a sampling rate of 12 kHz, the energy intensity of the vibration signal increases first and then decreases from high frequency to low frequency. When IMFx is within the range of 2.0 to 2.2, the normalized energy of the bearing drive end dataset reaches its maximum value and subsequently exhibits a monotonic trend. Figure 9 illustrates the ED of the vibration signal measured at a 48 kHz sampling rate at the driving end using an acceleration vibration sensor. The sample datasets, including 109.mat, 110.mat, 111.mat, 112.mat, and 122.mat, are specifically selected for analysis. At a sampling rate of 48 kHz, the arrangement of ED is not monotonically decreasing. When IMFx is within the range of 2.1-2.3, the normalized energy of the bearing drive end dataset reaches its maximum value and subsequently exhibits a non-monotonic trend. Specifically, two peaks emerge when IMFx is within the range of 0-10. This distinctive feature can significantly improve the discriminability of such faults. Figure 10 illustrates the measurement of vibration signals at the fan end using an acceleration vibration sensor with a sampling rate of 12 KHz. The selected sample data for ED analysis are stored in files 270.mat, 271.mat, 272.mat, 273.mat, and 274.mat. Similar to Figures 8 and 9, for bearings at the fan end with faults as depicted in Figure 10, the energy intensity of the vibration signal does not gradually decrease from high frequency to low  
Figure 6: Normal, CWRU normal. frequency due to the impact of vibration and shock caused by the faulty components. Instead, it exhibits relatively high values in one or more intermediate frequency bands. When IMFx is within the range of 3.0 to 4.5, the normalized energy of the fan end dataset reaches its maximum value and subsequently exhibits a monotonic trend. This is mainly because when the bearing malfunctions, the energy intensity distribution of the vibration signal changes due to the vibration impact on one or more frequency bands in the middle. Consequently, the presence of a fault in the bearing can be determined based on the ED characteristic vector. This diagnostic method is both straightforward and intuitive. Figure 8: Bearing drive end under a sampling rate of 12 KHz (the monotony following the peak). Figure 7: Envelope signal EEMD decomposition result (105.mat). In addition, the difference in energy distribution (ED) mode between healthy bearings and faulty bearings primarily arises from the following physical mechanisms: (1) In healthy bearings, vibration energy is predominantly concentrated in the low-frequency band due to normal rolling contact and mechanical rigidity. Specifically, IMF1 corresponds to high-frequency large noise (the outermost layer of the envelope), IMF2 to medium-frequency medium noise, IMF3 to low-frequency small noise (the innermost layer of the envelope), and IMF4 to the DC component (approximately zero with a zero mean). (2) In faulty bearings, local damage such as pitting or cracks generates transient impacts during rolling, which significantly increases high-frequency energy and excites the inherent resonant frequency of the bearing structure.