## 1 Introduction
### Variational Mode Decomposition (VMD)  
This article tackles the issue of limited robustness encountered during the extraction of fault feature frequencies within gearbox systems. Utilizing the VMD method, signals indicative of wear or tooth breakage faults within the gearbox are decomposed into several Intrinsic Mode Function (IMF) components. Subsequently, a correlation coefficient analysis is employed to meticulously identify those modal components that are imbued with fault signals. These selected components then undergo envelope spectrum analysis, a process that efficiently extracts and illuminates the fault feature frequencies, offering insights into the intricate dynamics of the gearbox's operational deficiencies. This refined approach enhances the precision and reliability of detecting and analyzing faults, thereby contributing to the optimization of maintenance and repair strategies.  
Variational mode decomposition (VMD) is characterized as a fully non-recursive decomposition model. Introduced by [PERSON] et al. in 2014, the essence of VMD is anchored in the utilization of central frequencies and bandwidths of the extracted modes [16]. The triumph of this decomposition technique is attributed to its approach of considering the solution as a constrained variational problem. Each process is meticulously crafted, ensuring that the extracted signals are both comprehensive and precise, thus bolstering the reliability and applicability of VMD in various analytical and diagnostic applications.  
\[\underset{\{u_{k}\},\{\omega_{k}\}}{min}\{\sum_{k=1}^{K}\parallel\delta_{t}[( \delta(t)+\frac{j}{\pi t})\times u_{k}(t)]e^{-j\omega_{k}t}\parallel_{2}^{2}\} \tag{1}\]  
\[\text{\emph{subject to}}{\sum_{k=1}^{k}u_{k}}=f\]  
where \(u_{k}\) denotes the decomposition mode, \(\delta\) is the Dirac distribution, and \(\times\) denotes the convolution. \(\omega_{k}\) is the corresponding central frequency of the quadratic penalty term and Lagrange multipliers are introduced to solve the Equation (1). The augmented Lagrange quantities are shown below:  
\[\begin{split} L(\{u_{k}\},\{\omega_{k}\},\lambda)=\alpha\sum \limits_{k=1}^{k}\parallel\partial_{t}[(\delta(t)+\frac{j}{\pi t})\ast u_{k}( t)]e^{-j\omega_{k}t}\parallel_{2}^{2}\\ +Vf(t)-\sum\limits_{k=1}^{K}u_{k}(t)\parallel_{2}^{2}+\langle \lambda(t),f(t)-\sum\limits_{k=1}^{K}u_{k}(t)\rangle\end{split} \tag{2}\]  
where \(\lambda\) is the Lagrangian multiplier coefficient. The model number is provided in advance as a priori information \(K\) and the equilibrium reference \(\alpha\). Equation (2) can be solved by the alternating direction method of multipliers (ADMM) [17]. All decomposed modes and the corresponding central frequencies are then updated according to (3) and Equation (4), respectively:  
\[\begin{split}\hat{u}_{k}^{n+1}(t)\leftarrow\frac{\hat{f}( \omega)-\sum_{i<k}\hat{u}_{i}^{n+1}(\omega)-\sum_{i>k}\hat{u}_{i}^{n}(\omega) +\frac{\hat{\lambda}^{n}(\omega)}{2}}{1+2\pi(\omega-\omega_{k}^{n})^{2}}\end{split} \tag{3}\]  
\[\begin{split}\omega_{k}^{n+1}\leftarrow\frac{\int_{0}^{\infty} \omega\parallel\hat{u}_{k}^{n+1}(\omega)\mid^{2}d\omega}{\int_{0}^{\infty} \mid\hat{u}_{k}^{n+1}(\omega)\mid^{2}d\omega}\end{split} \tag{4}\]  
A detailed mathematical derivation and full algorithm for VMD can be found in [17]. VMD has been reported to have better performance in identifying fault features from noisy and complex vibration signals compared to local mean decomposition (LMD), ensemble empirical mode decomposition (EEMD), and conventional EMD [18].