# Variational Mode Decomposition Based Time-Varying Force Identification of Stay Cables
## 2 Variational Mode Decomposition Analysis  
The vibration data of a stay cable often contain modal information of each order, that is, components with different frequencies. In the signal processing process, the modal decomposition method [25] is generally used to decompose the original signal into vibration signals of various orders for analysis. Variational mode decomposition [26] (VMD) is a new modal decomposition method that searches for the optimal solution of a constrained variational model established by a specified center frequency to achieve adaptive signal decomposition. For each intrinsic mode function [27; 28; 29] (IMF), the frequency center and bandwidth of the component are continuously and iteratively updated to complete the final adaptive decomposition of the signal in the frequency domain. The IMF component is overwhelmed by noise, and the signal characteristic component is decomposed. VMD algorithm is a completely nonrecursive signal decomposition method that can determine the center frequency and bandwidth of each intrinsic mode function (IMF) by iteratively searching for the optimal solution of the variational model and then adaptively decomposing the original signal into the sum of the individual IMF components. The signal is decomposed into a series of IMFs by VMD processing, and each IMF can be represented as an AM-FM [30] signal \(u_{k}(t)\) as Equation (8) shows:  
\[u_{k}(t)=A_{k}(t)\cos(\phi_{k}(t)) \tag{8}\]  
where \(A_{k}(t)\) is the instantaneous amplitude and \(\phi_{k}(t)\) is the instantaneous phase. The instantaneous frequency is defined as \(\omega_{k}(t)=\phi_{k}^{{}^{\prime}}(t)\). The changes in \(\phi_{k}(t)\) and \(A_{k}(t)\) are relatively slow to \(\omega_{k}(t)\). Within the time interval \([t-\delta,t+\delta]\), \(u_{k}(t)\) can be regarded as the harmonic signal with amplitude \(A_{k}(t)\) and frequency \(\omega_{k}(t)\). Three steps are needed to estimate each IMF: (1) calculating the analytical signal associated with each mode by the Hilbert transform, (2) adjusting the center frequency of each estimate by adding an index term and transforming the spectrum into the baseband, and (3) smoothing the demodulated signal and estimated bandwidth by the H1 Gauss method. Assuming that the signal is decomposed into \(K\) IMFs after VMD processing, the variational constraint model is shown in Equation (9):  
\[ \begin{split}&\min_{\{\mu_{k}\},\{\omega_{k}\}}\left\{\sum_{k=1}^{K}|| \partial_{t}\Big{[}\delta(t)+\frac{j}{\pi t}\otimes\mu_{k}(t)\Big{]}e^{-j\omega _{k}(t)}||_{2}^{2}\right\}\\ & s.t\quad\sum_{k=1}^{K}\mu_{k}(t)=f\end{split}  \tag{9}\]  
where \(\mu_{k}(t)\) is the \(k\)th signal component, \(\omega_{k}(t)\) represents the center frequency of each component, which can be characterized as the instantaneous frequency of each order vibration mode of the stay cable, \(\partial_{t}\) refers to the derivative of time, \(\delta(t)\) represents the unit pulse function, \(\otimes\) is the convolution operation, and \(j\) is the plural sign. The above problem is actually a convex optimization problem. To solve the above optimization problem, the VMD algorithm introduces \(L_{2}\) regularization constraints and Lagrangian multipliers and turns Equation (9) into the augmented Lagrangian function as Equation (10) follows:  
\[ \begin{split} L(\{\mu_{k}\},\{\mu_{k}\},\lambda)=& \alpha\sum_{k=1}^{K}||\partial_{t}\Big{[}\delta(t)+\frac{j}{\pi t }\otimes\mu_{k}(t)\Big{]}e^{-j\omega_{k}(t)}||_{2}^{2}\\ &+||f(t)-\sum_{k=1}^{K}\mu_{k}(t)||_{2}^{2}\\ &+<\lambda(t),f(t)-\sum_{k=1}^{K}\mu_{k}(t)>\end{split}  \tag{10}\]  
where \(f(t)\) is the original signal, \(\lambda(t)\) is the Lagrange function multiplier, \(\alpha\) is the bandwidth parameter, and the meanings of the remaining symbols are the same as in Equation (9). For the above equation-constrained optimal solution of the multivariate function, an alternating direction multiplier algorithm (ADMM) can be used to solve iteratively, and the various order components of the signal can be obtained.