## 5 Application to Rotating Machinery Fault Diagnosis
### Application to Rolling Bearing Fault Diagnosis  
In order to verify the proposed method, the rolling bearing failure experiment was carried out on a rotating machinery fault test bench. The top view of the experimental device is shown in Figure 10. The adjustable speed motor in this system is a DC servo motor with a power of 600 W. The rolling bearings are installed on four bearing chocks, and both driving gear and driven gear are spur gears. The moment of the inertia of the load is 0.03 kg\(\cdot\)m\({}^{2}\). In this experiment, the tested bearing is the 6311-type rolling bearing, which was formed the inner-race and outer-race faults by laser cutting technology, and the slot width and depth are 0.15 mm and 0.13 mm, respectively. The vibration signals are collected by two piezoelectric acceleration transducers, which are horizontally and vertically fixed to the bearing housing. Figure 11 shows the time-domain waveform of vibration acceleration signal of the rolling bearing with outer-race fault. The sampling frequency is 4096 Hz, and the rotating frequency is \(f_{r}=25\) Hz, and by calculating the outer-race fault characteristic frequency is \(f_{o}=76\) Hz. The vibration signal shown in Figure 11 is decomposed by the RS-LOD method. The value of the pole parameter \(p\) will affect decomposition results and decomposition time; therefore, we need to determine the value of \(p\) firstly. Table 2 shows the number of sifts required to identify each MOC. It can be seen that the number of sifts required for each MOC component and the total number of sifts for per decomposition for a range of p values. When \(p\) takes relatively smaller value at a certain range, the total number of sifts required is less, and the number of MOCs obtained is also less. From these two standards, when \(p\) takes the range of 0 to 10, it is possible to get higher decomposition efficiency. Figure 11: Vibration acceleration signal of the rolling bearing with outer-race fault. Figure 10: Top view of rotating machinery failure test bench. Table 3 lists the IO value and decomposition time for different \(p\) values. It can be seen that with the increase of \(p\), the IO value decreases gradually, and the decomposition time increases during the fluctuation. However, when \(p\) is greater than 10, the decrease of IO value is not obvious and undergoes gradual stabilization, but the decomposition time increases obviously. Therefore, consider the decomposition effects and decomposition time to determine the p value is 10 in this decomposition. When \(p=10\), the decomposition results are shown in Figure 12, and the high-frequency components \(\text{MOC}_{1}(t)\) and \(\text{MOC}_{2}(t)\) are shown in Figure 13. It can be seen that seven MOC components and one residue are obtained from high-frequency to low-frequency, and the decomposition effects are ideal. According to the rolling bearing fault vibration mechanism, the fault vibration signal of the rolling bearing is the modulation signal with the natural frequency of a rolling bearing system as the carrier frequency and the fault characteristic frequency as the modulation frequency. Therefore, the working condition of the rolling bearing can be judged by the envelope spectrum of the fault vibration signal. Additionally, the natural frequency of the rolling bearing system usually exists in the high-frequency components. Hence, by applying the Hilbert transform to the \(\text{MOC}_{1}(t)\), and the obtained Hilbert envelope spectrum, as shown in Figure 14, it can be easily seen that, except for the spectral line of rotating frequency (25 Hz), there are also obvious spectral lines of the outer-race fault characteristic frequency (76 Hz) and its double (152 Hz) can be found in the Hilbert envelope spectrum. Therefore, the fault of the outer race of the rolling bearing can be judged, which is consistent with the actual situation, and shows the effectiveness of the proposed method.