## 3 Fault Diagnosis Process  
When VMD decomposition is adopted, SDE is used to measure the time series complexity of IMF components obtained by decomposition. According to the characteristics of the VMD algorithm, when VMD decomposes a signal, the signal may contain interference, resulting in a complex time series and a large SDE value. As the VMD mode number increases, the time series complexity of the IMF components gradually decreases, and the corresponding SDE value also decreases. To prevent over-decomposition in VMD, a modulus selection index is proposed to select the optimal number of modes. The specific calculation method is as follows:  
\[\left\{ \begin{array}{l}E_{k}=\text{min}SDE_{k}\\ \theta_{k,k-1}=\frac{E_{k}-E_{k-1}}{k_{k-1}}\end{array} \right. \tag{13}\]where \(K\) is the modal number, \(E_{k}\) is the minimum value of IMF SDE obtained by decomposition under different modal numbers, and \(\theta_{k,k-1}\) is the ratio between the difference of two adjacent modal numbers and the previous one. When the number of VMD modes increases, the time series complexity decreases, leading to progressively smaller SDE values. Based on this principle, the change degree of adjacent minimum SDE values is calculated according to Equation (13). When the optimal number of modes is reached, the minimum SDE value is significantly reduced, and the modulus selection index will be significantly increased. To minimize the minimum dynamic entropy for the optimized \(K\) modal components, when the number of modes reaches the optimal, the SDE value will reach the minimum with the increase in the penalty factor, and the penalty factor at the minimum SDE is taken as the optimal penalty factor. The parameter optimization VMD-NLMS method proposed in this paper extracts the vibration signal characteristics of motor bearings under variable load conditions, and the specific steps of fault diagnosis are as follows:  
Step 1: The optimization of VMD Mode Number \(K\). The fault signal is decomposed using VMD, and the SDE value of the IMF component obtained after each VMD decomposition is calculated. The ratio of the difference between the minimum sign dynamic entropy of the adjacent mode number to the minimum sign dynamic entropy of the previous mode number is calculated, and the optimal mode number is selected, to avoid excessive decomposition. The SDE parameters are set as follows [31]:  
Embedding dimension \(m=3\). A smaller embedding dimension is chosen to balance the computational efficiency and the ability to capture the dynamic behavior of the signal. A value of \(m=3\) is commonly used in symbolic dynamics for its effectiveness in representing state transitions. The number of symbols \(\varepsilon=6\). The amplitude domain of the time series is divided into 6 symbols, which provides a good trade-off between resolution and computational complexity. The range of modal number \(K\) is set to [2; 14] to cover a reasonable number of potential modes without excessive computation. Step 2: The optimization of the VMD penalty factor \(\alpha\). After determining the optimal mode number \(K\), the penalty factor \(\alpha\) is optimized using the same SDE-based approach. The penalty factor controls the bandwidth of the IMF components, and its optimization ensures that the decomposition is neither too coarse nor too fine. The range of \(\alpha\) is set to [100, 5000], with a step size of 100. The optimal \(\alpha\) is selected at the point where the SDE value reaches a minimum, indicating the best balance between mode separation and noise suppression. The process of optimizing the VMD mode number \(K\) and \(\alpha\) using SDE is illustrated in Figure 1. The flowchart shows the iterative process of increasing \(K\), calculating SDE values, and determining the optimal \(K\) based on the modulus selection index. In the figure, the threshold \(\theta_{k,k-1}>0.04\) for the mode selection index is determined based on the theoretical analysis of the SDE values [10]. Step 3: Signal reconstruction.