## 1 Introduction
[17] propose an intelligent multi-classification algorithm based on hybrid neural networks and principal component analysis (PCA), which has shown excellent performance in the fault diagnosis of rotating machinery. However, its diagnostic accuracy still needs to be improved under actual extreme variable load conditions. [PERSON] et al. [18] propose a method based on the variable predictive model classification discriminant (VPMCD), which can accurately and effectively distinguish between the normal and fault conditions of rolling bearings, as well as different fault modes under variable speed conditions. However, the performance of this method largely depends on the collection of high-quality data, including accurate speed information and clear fault characteristics. [PERSON] et al. [19] propose a new diagnostic method based on the Teager energy operator demodulation computational order spectrum (TCO spectrum) and stacking auto-encoder (SAE) for the fault diagnosis of rolling bearings under variable speed and load conditions. This method can adaptively learn features from the vibration signals of rolling bearings. However, the SAE model requires a large amount of computational resources for training and inference, which may limit its application in resource-constrained environments. [PERSON] et al. [20] propose a fault diagnosis method based on discrete wavelet transform (DWT) and orthogonal fuzzy neighborhood discriminant analysis (OFNDA) and combined these with dynamic recurrent neural network (DRNN) for fault classification. However, this method has shortcomings in handling high-dimensional data and real-time performance. [PERSON] et al. [21] propose a feature extraction method based on wavelet packet decomposition and motif patterns (WPDMP) for the fault diagnosis of rolling bearings under variable working conditions, but the overall computational process is relatively complex, especially when dealing with large amounts of data. To reduce noise and better observe fault-related harmonic components, various techniques have been proposed, such as cepstrum pre-whitening [22] and Daubechies' wavelets [23], as well as cyclo-stationarity-based methods, like the improved envelope spectrum [24] and Wigner-Ville distribution [25]. While these techniques have demonstrated effectiveness in specific scenarios, their performance under variable load conditions and strong noise interference remains limited. The least mean square (LMS) adaptive filter, based on the steepest descent algorithm, is widely used in various fields, such as system identification, noise cancellation, and spectral line enhancement, due to its simple structure, low computational cost, strong robustness, and ease of implementation [26]. However, the fixed step size of the LMS algorithm leads to a conflict between convergence speed and steady-state error, which restricts its practical applications. Consequently, many researchers have conducted studies and proposed improved algorithms. [PERSON] et al. [27] mention the normalized least mean square (NLMS) filter and compare the gradient noise amplification issues of LMS and NLMS, concluding that NLMS is significantly better than LMS. Additionally, the step size factor \(\upmu\) can be iteratively updated as the filter order increases, so the choice of the two parameters of NLMS--the filter order L and the step size factor \(\upmu\)--has little effect on the noise reduction performance. In summary, the NLMS algorithm exhibits stronger adaptability. Because of the above problems, this paper proposes a fault diagnosis method combining parameter optimization VMD-NLMS to diagnose motor bearings under different load conditions. Symbolic dynamics entropy is used to optimize the VMD mode number and penalty factor to prevent over-decomposition and improve the operation speed. Firstly, the optimal number of modes is selected by calculating the ratio of the difference between the minimum symbolic dynamic entropy of the adjacent modes and the minimum symbolic dynamic entropy of the previous mode.