## 4 Experimental Verification
### Validation of Measured Datasets
#### 4.2.2 Inner Ring Fault  
Similarly, the inner ring fault data are sampled, the number of sampling points is 8192, and the inner ring fault signals with loads of 500 N and 1500 N are collected, respectively. Among them, the spectrum and envelope spectrum of the loads 500 N and 1500 N are shown in Figure 12a,b. Figure 11: Outer ring reconstructed signal envelope spectrum. Figure 10: Quantitative comparison (outer ring). The results of the SSA-VMD-MCKD method are shown in Figure 13. When the load is 500 N, the number of modes after the SSA optimization of VMD is 7, and the penalty factor is 563; when the load is 1500 N, the mode number is 5, and the penalty factor is 126. When the load is 500 N and 1500 N, most of the fault information is submerged by noise. The parameter optimization VMD-NLMS algorithm proposed in this paper is used for fault diagnosis. The fault signals of the inner ring of the motor bearing with loads of 500 N and 1500 N are decomposed by VMD, respectively. The modulus selection indicator under different loads is shown in Table 7. Respectively, the VMD modal numbers are chosen to be \(K=6\), \(K=7\). The optimized optimal penalty factor is shown in Figure 14. Table 8 shows the correlation coefficient (inner ring). When the load is 500 N, IMF2 and IMF3 are selected for reconstruction. When the load is 1500 N, IMF3 and IMF4 are selected for reconstruction. The selection of the order and step factor of the filter is shown in Table 9. When the load is 500 N, the RMSE of \(L=192\) is the largest. When the load is 1500 N, the RMSE of \(L=224\) is the largest. Figure 12: Inner ring fault signal spectrum and envelope spectrum. Figure 13: SSA-VMD-MCKD (inner ring). In Figure 15, the kurtosis, correlation coefficient, and the SNR of SSA-VMD-MCKD and VMD-NLMS methods are compared. It can be seen from the figure that the proposed method is superior to the comparison method. In Figure 16, the fault characteristics of the outer ring of the motor bearing with a load of 500 N and 1500 N can be clearly observed. Among them, 1-5 times of the load are observed when the load is 500 N, and 1-4 times when the load is 1500 N. So far, the fault diagnosis has been successful. The experimental results show that the parameter-optimized VMD-NLMS method can effectively extract the fault features of motor bearing inner and outer rings under variable load conditions (500 N and 1500 N) and strong noise environments (SNR = \(-\)13 dB), with diagnostic accuracy and robustness surpassing traditional methods (e.g., SSA-optimized VMD-MCKD). By optimizing VMD parameters using symbolic dynamic entropy (SDE), the issue of modal over-decomposition is avoided, while NLMS adaptive filtering significantly suppresses noise interference, verifying its practicality in complex working conditions. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(K\) = 3 & \(K\) = 4 & \(K\) = 5 & \(K\) = 6 & \(K\) = 7 & \(K\) = 8 \\ \hline
500 N & 0.0080 & 0.0028 & 0.0104 & 0.0498 & 0.0067 & 0.0017 \\ \hline
1500 N & 0.0061 & 0.0073 & 0.0122 & 0.0035 & 0.0633 & 0.0055 \\ \hline \end{tabular}
\end{table} 
Table 7: Modulus selection indicator (inner ring). Figure 14: Inner ring fault signal spectrum and envelope spectrum. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**L** & **64** & **96** & **128** & **160** & **192** & **224** & **256** \\ \hline
500 N & 6.14 & 8.47 & 9.88 & 10.91 & 11.44 & 10.95 & 10.94 \\ \hline
1500 N & 5.83 & 8.17 & 9.65 & 10.69 & 10.96 & 11.19 &Additionally, the strategy of selecting IMF components based on the weighted kurtosis index (WKI) further enhances the identification capability of fault features. However, the following issues still need to be addressed:  
1. The current experiments are limited to a specific bearing type (620 SEKA) and a single fault mode (pitting). It is necessary to extend the validation to more fault types (e.g., cracks, wear) and bearing models to verify generalization;
2.