## 3 Model Verification on a Bearing Test Rig
### Model Verification with Vibration Data  
A comparison was made between the test and simulation in the time domain and frequency domain. According to the test, the vibration waveforms in the time domain are shown in Figure 6. As can be observed, the amplitudes of vibration signals from C4 bearing were larger than those from CN bearing. To show their differences more clearly, the root mean square (RMS) of the vibration signals was calculated and presented in Figure 7b. As expected, the RMS value of C4 was larger than that of CN. Furthermore, when under larger clearances C4, the RMS value showed an uptrend with the increase in load, which is in line with the numerical simulation in Figure 7a. However, under smaller clearance CN, the RMS value showed different trends in both the simulation and test signals. The RMS value of the simulation data showed a downtrend with loads, while that of the test data was nearly similar among different loads. The downtrend of the simulation data may be due to the influence of resonance caused by load. When under the same clearances, the bearing load distribution varied with the external load on it. When load was relatively small, there were less balls in the contact region, which may result in larger vibrations. The trend of actual measurements may be influenced by background noise or other joint components.  
Except for the time domain analysis, the frequency spectra of the simulation signal and test signal were calculated and compared in Figure 8. Note that \"800-10\" in Figure 8a means that the load was 800 N and the clearance was 10 \(\upmu\)m. Such expression was adopted in the following text. From the frequency spectrum of the simulation signal, it is easy to distinguish the low frequency band and the high resonant frequency band. However, the spectrum of the test signal is quite complicated with multiple unknown resonant frequencies, which may be generated by other components in the test rig  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Bearing Type** & **Bearing Class** & **Clearance (\(\upmu\)m)** \\ \hline \multirow{2}{*}{6206 \(ZZ\)} & CN & 8.33 \\  & C4 & 40.86 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Experiment test bearing clearance values.  
Figure 6: Waveforms under different loads, (**a**) 0 N, (**b**) 800 N, (**c**) 1600 N, (**d**) 2400 N.  
As can be seen from the envelope spectrum in Figure 9, BPFO and its harmonics dominated the entire spectrum for different clearances and load conditions. In Figure 10, the harmonics of the rotating frequency \(f_{\rm s}\) and BPFO can be clearly observed together, though there are a number of unresolved frequency components due to the dynamic effects of motor and coupling. Note that the amplitude of BPFO and its harmonics were much higher than other frequency components. This means that BPFO plays an important role in the modulation process for bearing with clearances. In addition, the amplitude of BPFO from bearing with a larger clearance (red solid line) was much higher than that from bearing with smaller clearances (blue dotted line). All these findings are consistent with the simulation study.  
Based on these findings, it is reasonable to conclude that the proposed dynamic model is competent to describe the bearing vibration under different clearances.