## 2 Bridge Damage Identification Methods
### Damage Identification Based on Continuous Wavelet Transform
#### 2.2.3 Brief on Variational Modal Decomposition  
As previously mentioned, the acceleration response of the test vehicle contains multiple components of bridge vibrations, adversely affecting the accuracy of damage identification. To improve the identification accuracy, this study employs the variational mode decomposition method to decompose the vibration data of the test vehicle, thereby clearly identifying the characteristic information of each bridge mode for effective damage identification. Variational mode decomposition is capable of decomposing complex signals into multiple intrinsic mode functions, each characterized by distinct frequency ranges and waveform patterns. In variational modal decomposition, the signal is decomposed into several intrinsic mode functions (IMFs). Variational modal decomposition obtains these modal functions by solving a variational problem, unlike other modal decomposition methods. Its main advantage lies in decomposing the signal into modes with stricter bandwidth limitations and gradually reducing residuals. At the same time, variational modal decomposition allows the bandwidth width to be adjusted according to needs during the decomposition process. Compared with empirical mode decomposition (EMD), variational modal decomposition shows significant advantages. First, variational modal decomposition has higher robustness to noise and can effectively separate the intrinsic modes of the signal, while EMD tends to generate spurious modes under high noise conditions. Second, the decomposition results of variational modal decomposition are more stable as it optimizes each mode's center frequency and bandwidth based on the variational principle. In contrast, EMD results are more susceptible to data starting points, leading to endpoint effects and mode mixing. Finally, variational modal decomposition is based on the variational principle and has a more rigorous mathematical derivation process, whereas EMD mainly relies on empirical rules, lacking a solid theoretical foundation. The foundation of variational mode decomposition (VMD) lies in formulating a variational optimization problem. This approach leverages principles from classical Wiener filtering, the Hilbert transform, and frequency mixing to decompose multi-component signals into intrinsic mode functions (IMFs). During the decomposition, it is assumed that the sum of the component signals reconstructs the original signal. The method then iteratively searches for IMFs with limited bandwidths and optimally matched center frequencies to minimize the total bandwidth sum. This approach determines the number of IMFs and can control the number of output IMFs based on actual conditions. Each IMF can be represented as:  
\[U_{k}=A_{k}(t)\cos\Bigl{[}\varphi_{k}(t)\Bigr{]} \tag{7}\]  
where \(A_{k\ (t)}\) is the instantaneous amplitude of \(U_{k}\); \(\varphi_{k\ (t)}\) is the instantaneous phase; and \(t\) is time. The corresponding expression for the constrained optimization problem constructed by variational modal decomposition for the component signals is:  
\[f(t)=\sum_{k}u_{k}(t) \tag{8}\]  
where \(f(t)\) is the original signal, K is the total number of modes, and \(u_{k}(t)\) is the \(k\)-th mode function. \[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum_{k}\|\partial_{t}\left[\left( \delta(t)+\frac{i}{\pi t}\right)^{*}u_{k}(t)\right]e^{-i\omega_{k}t}\right\|_ {2}^{2}\right\} \tag{9}\]  
where \(\{u_{k}\}\) represents the set of decomposed mode functions, \(\{\omega_{k}\}\) is the set of center frequencies for each mode function, \(\delta(t)\) is the Dirac delta function, \({}^{*}\) denotes convolution, and \(i\) is the imaginary unit.