## 6 Simulation and Experiment of Weak Signal Detection Using Double-Coupled Duffing System
### Experimental Study on Weak Fault Signal Detection of Rolling Bearing  
In the previous section, the simulation signal is used to verify the method proposed in this paper. In order to verify the performance of the method in practical engineering applications, it is necessary to test it with the measured signal. In this paper, the bearing failure data of Case Western Reserve University is used. This dataset covers normal data and four types of fault data (inner ring faults, outer ring faults, rolling element faults and retainer faults). The experimental equipment is shown in Figure 14. The data is tested with 6205 bearings produced by SKF. The sampling frequency is 12,000 Hz. The bearing parameters are shown in Table 2.  
In the experiment of Case Western Reserve University, the electrical discharge machining technology was used for single-point fault arrangement. The acceleration sensor is used to collect vibration signals, and the sensor is placed on the motor housing by using a magnetic switchable device. The data selected in this paper are the inner ring fault data and normal data. The method of manufacturing the inner ring fault is to arrange a pit with the size of 0.007 inches on the inner ring. The normal data are not used for any processing for the bearing. When sampling two sets of data, no load is added, and the sensor is arranged \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Rolling Elements \(N\)** & **Ball Diameter \(d\)** & **Contact Angle \(\alpha\)** & **Pitch Diameter \(D\)** \\ \hline
9 & 7.94 mm & 0 & 39.04 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of 6205 bearings. Figure 14: Experimental equipment. Figure 13: Phase plane diagram of Duffing system (without VMD processing). in the six o'clock direction. Calculate the theoretical fault frequency of the bearing, and the calculation equation is shown in Equation (20):  
\[\omega_{inner}=\frac{r}{60}\frac{1}{2}N(1+\frac{d}{Dcos\alpha}) \tag{20}\]  
The spindle speed of the two groups of experimental data is 1797 r/min. According to the calculation, the theoretical fault frequency of the bearing inner ring is 1019 rad/s. The time domain diagram and frequency domain diagram of the signal are shown in Figure 15.  
Figure 15 shows that the frequency component of the signal is very complex and contains a great amount of noise. It is difficult to directly identify the weak fault signal by the traditional FFT method. The signal is normalized by energy and then decomposed by EMD. The two groups of signals are decomposed by EMD, and both obtained nine IMFs. The number of IMFs of the VMD algorithm is set to 9, and the signal is decomposed by VMD. The calculated IMF center frequency is shown in Tables 3 and 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**IMF** & \
\begin{tabular}{c} **Center Frequency** \\ **(rad/s)** \\ \end{tabular} & **IMF** &
\begin{tabular}{c} **Center Frequency** \\ **(rad/s)** \\ \end{tabular} \\ \hline IMF1 & 30,598.45 & IMF6 & 6575.37 \\ IMF2 & 13,208.90 & IMF7 & 4447.15 \\ IMF3 & 13,167.05 & IMF8 & 1996.14 \\ IMF4 & 9827.87 & IMF9 & 522.98 \\ IMF5 & 8262.91 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: VMD decomposition results (normal). Figure 15: Time domain and frequency domain plots of experimental data. (**a**) Time domain diagram (normal); (**b**) frequency domain diagram (normal); (**c**) time domain diagram (inner fault); (**d**) frequency domain diagram (inner fault). The theoretical fault frequency of the bearing is between the center frequency of IMF8 and IMF9, so IMF8 and IMF9 are added as the excitation of the double-coupled Duffing system. According to the method described in Section 4.5, 1000-1040 rad/s is selected as the frequency band to be analyzed. According to the performance parameters of the double-coupled Duffing system, the frequency band is divided into 20 parts, and the center frequency of each sub-band is taken as the built-in driving force frequency of the double-coupled Duffing system.