Automatic modal identification of bridges based on free vibrations and advanced signal decomposition techniques

2.1. Variational mode decomposition

Let $t$ and $K$ be the time variable and the total number of components $f_k\left(t\right)$ to be extracted from the recorded signal $f\left(t\right)$ by means of the VMD technique, respectively. In the time domain, the VMD technique aims at extracting the relevant components $f_k\left(t\right)$ from $f\left(t\right)$, also known as modes or intrinsic mode functions (IMFs), and estimating the corresponding central pulsation ${\omega }_k$ through the following steps:

1) the Hilbert-Huang transform is applied to compute the analytical signal for each 𝑘th IMF.

2) the frequency spectrum of the mode is shifted to baseband by mixing with an exponential function tuned to the respective estimated central frequency.

3) the $H_1$ Gaussian smoothness of the demodulated signal (i.e., the squared L²-norm of the gradient) is performed to estimate the bandwidth for each 𝑘th IMF.

Considering these steps, the signal decomposition according to the VMD technique is thus performed by looking for the solution of the following variational problem:

where ${\partial }_t$ and $‖\bullet ‖$ are the gradient operator and the L²-norm operator, respectively. The variational problem in Eq. (1) can be solved by using a quadratic penalty term and enforcing the constraints by means of the Lagrange multipliers technique. This leads to the following augmented Lagrangian function:

where $\alpha$ is the quadratic penalty factor and $\lambda$ is the Lagrangian multiplier whereas $〈\bullet 〉$ is the L²-inner product. The solution of the problem given in Eq. (1) thus corresponds to the saddle point of $\mathcal{L}$, which is obtained through a sequence of iterative sub-optimizations named alternate direction method of multipliers. From a computational standpoint, however, the complete final algorithm is more efficiently posed in the spectral domain rather than in the time domain. It is noted that the number of IMFs to be extracted $K$ and the penalty factor $\alpha$ rule the performance of the VMD technique. Therefore, their value must be carefully determined for the accurate decomposition of the signal. In the present work, the procedure proposed by Mazzeo et al. [10] has been implemented for the automatic optimal tuning of both control parameters.

2.2. Empirical Fourier decomposition

The EFD technique is an adaptive decomposition method introduced recently by Zhou et al. [9] that consists of two main steps. First, a segmentation procedure produces $N$ frequency partitions of the frequency spectrum of the signal to be analyzed. Next, a zero-phase filter bank is required to perform the actual decomposition.

The segmentation process is carried out within a normalized frequency domain $\left[0,\pi \right]$. Therefore, signal frequency lines must be also normalized. Initially, the boundaries of the $N$ contiguous frequency partitions ${\mathrm{\Lambda }}_n=\left[{\omega }_{n-1},{\omega }_n\right]$ are detected. The frequencies ${\omega }_0$ and ${\omega }_N$ are not required to be equal to $0$ and $\pi$, respectively. The first $N$ frequencies corresponding to the largest maximum magnitudes detected in the signal spectrum are then sorted in descending order and are denoted as $\left\{{\mathrm{\Omega }}_1,\dots ,{\mathrm{\Omega }}_N\right\}$, whereas ${\mathrm{\Omega }}_{0=0}$ and ${\mathrm{\Omega }}_{N+1}=\pi$. For each pair of consecutive frequencies ${\mathrm{\Omega }}_n$ and ${\mathrm{\Omega }}_{n+1}$, the partition is determined by picking the frequency value ${\omega }_n$ at which a global minimum is attained as follows:

where ${\widehat{X}}_n\left(\omega \right)$ is the Fourier spectrum amplitude between ${\mathrm{\Omega }}_n$ and ${\mathrm{\Omega }}_{n+1}$ whereas $\omega$ is the frequency variable. The second step of the procedure is the construction of a filter bank. Zero-phase filters are considered to avoid possible interference due to the transition phase, because it can cause mode-mixing effects. The zero-phase filter bank is based on the frequency partitions obtained after the segmentation (i.e., the boundary frequencies of each partition identify the cut-off frequencies). Therefore, let $\widehat{f}\left(\omega \right)$ and $\widehat{\mu }\left(\omega \right)$ be the Fourier transform of the signal to be analyzed and the zero-phase filter in the frequency domain, respectively. The generic filtered component has the following expression in the frequency domain:

The modal component ${\widehat{f}}_n\left(\omega \right)$ can be finally expressed in the time domain by means of the inverse Fourier Transform ${\mathcal{F}}^{-1}\left[\bullet \right]$ operator, i.e. $f_n\left(\omega \right)={\mathcal{F}}^{-1}\left[{\widehat{f}}_n\left(\omega \right)\right]$. Compared to the VMD technique, only one control parameter is involved into the EMD technique, which is the number of the frequency partitions $N$. In principle, its value can be obtained from the frequency spectrum of the signal. Unfortunately, a simple counting of the peaks in the Fourier Transform of the signal is likely to produce a wrong segmentation because the signal spectrum is always noisy in practical applications. Therefore, an iterative procedure has been proposed to automatically tune the proper value of $N$, in such a way to avoid both mode-mixing and mode-splitting phenomena [11].

2.3. Robust damping identification

Once the $k$th component $f_k\left(t\right)$ has been extracted by means of either the VMD technique or the EMD technique, the associated modal damping ratio is evaluated by means of the area ratio-based method [12]. Let $2Z_k+1$ be the number of zero-crossing points in $f_k\left(t\right)$. It is possible to detect a total of $2Z_k$ regions enclosed between $f_k$ and the time axis $t$ with areas $A_{1,k},A_{2,k},\dots {,A}_{2Z,k}$ (see Fig. 1). So doing, the damping ratio ${\xi }_k$ is estimated as follows [12]:

where $R_k=\ln \left[{\sum }_{i=1}^{Z_k}{A}_{i,k}/{\sum }_{i=Z_k+1}^{{2Z}_k}{A}_{i,k}\right]$.

The application of the area ratio-based damping identification method is recommended since it is more robust against the noise as compared to the conventional logarithmic decrement technique.

Fig. 1Detection of the regions with area Ai,k defined by the 𝑘th signal component for the application of the area ratio-based damping identification technique

2.4. Mode shapes identification

The mode shapes are estimated by processing the free vibration response recorded from all $S$ available sensors after the decomposition accomplished via VMD technique or EMD technique [13].

Fig. 2A schematic diagram of a vibration separator

Particularly, the normalized 𝑘th mode shape vector ${\mathbf{\varphi }}_k$ is expressed as follows (see Fig. 2):

The effect of the noise can be mitigated by averaging the values in Eq. (6) for all peaks.