## 1 Introduction  
New methods and techniques in technical diagnostics enable improved planning, reduced downtime, and a significant increase in the time between failures. Additionally, understanding the status of rotating machinery helps to boost profits [1]. Rotating machinery, as an important component of mechanical systems, plays a crucial role in power transmission and motion transformation. Due to their harsh working environments and the influence of factors such as vibration, noise, temperature, and humidity, rotating machinery frequently experiences failures during operation. These failures exhibit characteristics such as low magnitude, nonlinearity, complexity, and non-stationarity. Therefore, condition monitoring and fault diagnosis are essential for the normal operation of rotating machinery [2]. Early fault diagnosis techniques relied primarily on manual experience, making it difficult to identify accurately the causes of complex equipment and systems, and requiring significant human and material resources. Later, rule-based expert systems emerged, but they were limited by the formulation of rules and knowledge acquisition. In recent years, data mining-based machine learning has emerged, which can learn complex nonlinear relationships from extracted features and establish diagnostic models for fault diagnosis, demonstrating good performance. However, the application of deep learning, which has gained popularity recently, is still subject to scrutiny in the field of fault diagnosis due to the greediness of data and the lack of model interpretability. Scholars have developed various methods for analyzing and processing rotating machinery vibration signals, such as empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD), and local mean decomposition (LMD) [3; 4; 5; 6; 7]. The EMD method can adaptively decompose signals without the need for any predefined basis functions. However, it lacks a rigorous mathematical foundation, leading to potential instability and inconsistency in the decomposition results. The EEMD method improves the stability of the decomposition results by averaging multiple decompositions. By adding white noise and performing multiple decompositions, it effectively reduces mode mixing. Nevertheless, this approach requires numerous decompositions, resulting in high computational costs and potentially introducing additional noise components that may affect the results. The LMD method yields clearer spectra for each component, offering excellent frequency resolution. However, it faces issues with endpoint effects, the decomposition process is relatively complex, and the computational load is significant. VMD proposed in 2014 [8], effectively avoids mode mixing and is supported by a rigorous mathematical framework. The decomposition process is scientifically grounded and applicable to linear, nonlinear, and non-stationary signals, addressing the aforementioned limitations. Consequently, scholars have extensively employed the VMD method for fault diagnosis in rotating machinery [9; 10]. [PERSON] [11] optimized the decomposition layers and penalty parameters of VMD using improved envelope entropy, selecting components with high correlation to the original signal. By integrating sample entropy, [PERSON] achieved fault diagnosis of bearings. [PERSON] [12] determined the number of VMD decomposition layers by observing the center frequency of VMD components and utilizing fuzzy c-means clustering, achieving effective fault diagnosis. [PERSON] [13] applied the VMD algorithm for adaptive decomposition of vibration signals and used an improved extreme learning machine to diagnose faults in rolling bearings. However, the two key parameters in VMD--the number of modes \(K\) and the penalty factor \(\omega\)--directly affect the VMD decomposition effect and need to be optimized [14]. [PERSON] et al. proposed the center frequency ratio method, which uses the center frequency ratio of adjacent modal components to select \(K\), but \(\omega\) cannot be determined [15].