## 4 Application of the New Method in Fault Diagnosis of Rolling Bearing
### Experimental Verification  
We use the actual measured data to verify the feasibility of the coupled Lorenz system applied to rolling bearing vibration signal processing. These data are measured by the bearing data center of the Case Western Reserve University, and the test platform used is shown in Figure 7.  
In this figure, the test stand consists of a 2 \(\,\mathrm{hp}\) motor (left), a torque transducer/encoder (center), a dynamometer (right), and control electronics (not shown). The test bearings support the motor shaft. Single point faults were introduced to the test bearings using electro-discharge machining. Vibration data were collected using accelerometers that were attached to the housing with magnetic bases, and the sampling rate is 12,000 S/s. The tested bearing is a 6205-2 RS JEM SKF deep groove ball bearing, and its basic parameters are shown in Table 2.  
The number of the rolling element \(z\) is 9, and the contact angle \(\alpha\) is 0. Referring to Equations (11)-(13), the coefficient relationship between the fault characteristic frequency and the rotation rate can be obtained, when the inner ring, outer ring and rolling element are damaged. These coefficients are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Inside Diameter** & **Outside Diameter** & **Thickness** & **Ball Diameter** & **Pitch Diameter** \\ \hline
0.9843 & 2.0472 & 0.5906 & 0.3126 & 1.537 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Bearing specification table (inches). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Inner Ring** & **Outer Ring** & **Rolling Element** \\ \hline
5.4152 & 3.5848 & 4.7135 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Defect frequencies (multiple of running speed in \(\mathrm{Hz}\)). Figure 7: The rolling bearing test platform. We take the vibration data of the inner ring with 0.007 _inch_ of single point damage as the tested signal, and implement it in the coupled Lorenz system established above. Then, the time domain waveform of the vibration signal and the synchronous error and the power spectrum density of these two signals are obtained, as shown in Figure 8.  
Figure 8c is directly calculated from the actual vibration signal in Figure 8a, and Figure 8d is calculated from the synchronous error data in Figure 8b. At this time, the motor speed is 1748 rpm, and the double rotation frequency of bearing is 59 Hz. Combined with the coefficients in Table 1, the characteristic frequency of the inner ring fault is 157.7628 Hz. Compared with the subgraphs Figure 8c,d, it is obvious that the double frequency of bearing rotation, the fault feature frequency of the inner ring damage and multiple harmonics of fault frequency are greatly enlarged after the noise reduction, which can be easily identified. Furthermore, we use the same method and model to process the vibration data of the outer ring with 0.007 _inch_ of single point damage. The obtained waveforms are shown in Figure 9.  
In this figure, the motor speed is 1796 rpm, and the rotation frequency of bearing is 29.9 Hz. Combined with the coefficients in Table 3, the characteristic frequency value of the inner ring fault is 107.3050 Hz. Compared with the two subgraphs, in Figure 9c, the fault characteristic signal in the actual vibration signal has been completely covered by the background noise, and it cannot be found by the peak search. In Figure 9d, the frequency of bearing rotation, the fault characteristic signal and multiple harmonics of fault frequency can be observed by finding the peak of the power spectrum density of the synchronization error data, and it can be used as the basis of the fault diagnosis. Finally, we use the same method and model to process the vibration data of the rolling element with 0.007 _inch_ of single point damage. The obtained waveforms are shown in Figure 10.  
Figure 8: The vibration signal (**a**), the synchronous error (**b**), the power spectral density of inner ring damage using the vibration signal (**c**) and the synchronization error (**d**).