## 2 Theory of Method
The essence of 2D-VMD is a non-recursive and completely adaptive decomposition of the original function to obtain a specified number of modal components. Let the expression of the synthesized two-degree-of-freedom directional vibration signal sequence be \(f(x)\), which is equivalent to solving the following optimization problem [47]:  
\[\psi\;=\;\left[\varphi_{1},\varphi_{2}\cdots\varphi_{N}\right]^{T} \tag{2}\]2  
where \(m_{AS,K}\) is the analytical signal obtained from the modal component \(m_{K}\) according to \(\omega_{K}\); \(\beta_{K}\) is the weight coefficient of the mode; \(\|\cdot\|^{2}_{2}\ =\ \int_{\mathbb{R}^{2}}\lvert(X)\rvert^{2}dx\) can be understood as a norm in a two-dimensional case. Lagrange multiplier and quadratic penalty function are introduced to further constrain the above problems:  
\[ \begin{array}{l}L(\{m_{K}\},\{\omega_{K}\},\lambda)\ =\ \sum_{K}\beta_{K}\|\
abla[m_{AS,K}(x)e^{-j \langle\alpha_{K},x\rangle}]\|^{2}\\ +\|f(x)-\sum_{K}m_{K}(x)\|^{2}_{2}+\left\langle\lambda(x),f(x)-\sum_{K}m_{K}( x)\right\rangle^{2}\end{array}  \tag{14}\]  
Among them, \(\lambda:\mathbb{R}^{2}\rightarrow\mathbb{R}\) is a Lagrangian multiplier. The problem is further transformed into an unconstrained saddle point problem by the augmented Lagrangian equation defined above. \[m_{K}:\mathbb{R}^{n}\rightarrow\overset{\min}{\mathbb{R}},\omega_{K}\in \mathbb{R}^{n}\lambda:\overset{\max}{\mathbb{R}}^{n}\rightarrow\mathbb{R}\ \ L(\{m_{K}\},\{\omega_{k}\},\lambda) \tag{15}\]The Lagrange multipliers are combined with the quadratic penalty terms:  
\[ \begin{array}{l}L(\{m_{K}\},\{\omega_{K}\},\lambda)\ =\ \sum\limits_{K}\beta_{K}\| \
abla[m_{AS,K}(x)e^{-j\langle\omega_{K},x\rangle}]\|_{2}^{2}\\ \qquad\qquad+\|f(x)-\sum\limits_{K}m_{K}(x)+\frac{\lambda(x)}{2}\|_{2}^{2}-\| \frac{\lambda(x)}{2}\|_{2}^{2}\end{array}  \tag{16}\]  
In each iteration, the modal components, center frequency and Lagrangian multipliers in the updated state are taken as the saddle points of the augmented Lagrangian equation to solve the optimal data [59; 60; 61]:  
\[m_{K}^{p+1}\leftarrow\ \ \ \ \begin{array}{l}\operatorname*{argmin}\\ m_{K}:\mathbb{R}^{n}\rightarrow\mathbb{R}\end{array} \ L(\{m_{i<K}^{p+1}\},m_{K},\{m_{i>K}^{p}\},\{ \omega_{i}^{p}\},\lambda^{p}) \tag{17}\]  
\[\omega_{K}^{p+1}\leftarrow\ \ \ \begin{array}{l}\operatorname*{argmin}\\ \omega_{K}\in\mathbb{R}^{n}\end{array} \ L(\{m_{i}^{p+1}\},\{m_{i<K}^{p+1}\}, \omega_{K},\{\omega_{i>K}^{p}\},\lambda^{p}) \tag{18}\]  
\[\lambda^{p+1}\leftarrow\lambda^{p}+\tau(f-\sum\limits_{K}m_{K}^{p+1}) \tag{19}\]  
where \(\tau\geq 0\). After eliminating the first variation of the modal component and reaching the optimal condition, solve the modal component \(\overset{\wedge}{m_{K}}\) to obtain the Wiener filter update:  
\[\overset{\wedge}{m_{K}}^{p+1}(\omega)\ =\ (\overset{\wedge}{f(\omega)}-\sum \limits_{i\
eq K}\overset{\wedge}{m_{i}}(\omega)+\frac{\overset{\wedge}{ \lambda}(\omega)}{2})\frac{1}{1+2\beta_{K}|\omega-\omega_{K}|^{2}},\forall \omega\in\Omega_{K} \tag{20}\]  
where \(\Omega_{K}\) is the frequency domain half space. The center frequency \(\omega_{K}\) in the update is:  
\[\omega_{K}^{p+1}\ =\ \operatorname*{argmin}\limits_{\omega_{K}\in\mathbb{R}^{2}} \{\beta_{K}\|\
abla[m_{AS,K}(x)e^{-j\langle\omega_{K},x\rangle}]\|_{2}^{2} \tag{21}\]  
The maximum Lagrange multiplier \(\lambda\) is:  
\[\lambda^{p+1}(x)\ =\ \lambda^{p}(x)+\tau(f(x)-\sum\limits_{K}m_{K}^{p+1}(x)) \tag{22}\]  
The above processing reduces the influence of some external noise on the fault feature information to some extent, but the energy of the fault feature information will inevitably leak due to the filtering processing. However, we can use the IMF component with the smallest energy leakage to complete the fault diagnosis of many typical nonlinear dynamic systems. Therefore, screening the IMF component with the smallest amount of energy leakage is the key to optimizing 2D-VMD technology.