Vibration-Based Monitoring of Civil Structures with Subspace-Based Damage Detection

Abstract

Automatic vibration-based structural health monitoring has been recognized as a useful alternative or addition to visual inspections or local non-destructive testing performed manually. It is, in particular, suitable for mechanical and aeronautical structures as well as on civil structures, including cultural heritage sites. The main challenge is to provide a robust damage diagnosis from the recorded vibration measurements, for which statistical signal processing methods are required. In this chapter, a damage detection method is presented that compares vibration measurements from the current system to a reference state in a hypothesis test, where data-related uncertainties are taken into account. The computation of the test statistic on new measurements is straightforward and does not require a separate modal identification. The performance of the method is firstly shown on a steel frame structure in a laboratory experiment. Secondly, the application on real measurements on S101 Bridge is shown during a progressive damage test, where damage was successfully detected for different damage scenarios.

Appendix

The damage detection test statistic s is set up using measurements from the reference state of the structure, requiring the estimation of the left null space matrix S, the residual covariance matrix $\Sigma$ and setting up a threshold t.

First, the Hankel matrix $\widehat{\mathscr {H}}_{p+1,q}^0$ is computed from the reference measurements. Then, the left null space matrix S can be estimated from its SVD

$$\begin{aligned} \widehat{\mathscr {H}}_{p+1,q}^0 = \begin{bmatrix}U_1&\ U_0\end{bmatrix} \begin{bmatrix}\Delta _1&0 \\ \ 0&\ \Delta _0\end{bmatrix} \begin{bmatrix}V_1 \\ V_0\end{bmatrix} \end{aligned}$$

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as $\widehat{S}=U_0$, where the SVD is truncated at the desired model order n with $\Delta _1 \in \mathbb {R}^{n\times n}$ and $\Delta _0 \approx 0$.

Second, the residual covariance matrix $\Sigma$ is obtained by separating the available reference dataset into $n_b$ data blocks of length $N_b$, such that the total data length yields $N=n_bN_b$. On each block, the Hankel matrix is computed as

$$ \widehat{R}_i^{(j)} = \frac{1}{N_b} \sum _{k=1+(j-1)N_b}^{jN_b} y_k y_{k-i}^T, \ \ \ \widehat{\mathscr {H}}_{p+1,q}^{(j)} = \mathrm{Hank}(\widehat{R}_i^{(j)} ) $$

Then, $\widehat{\mathscr {H}}_{p+1,q}^0= \frac{1}{n_b}\sum _{j=1}^{n_b} \widehat{\mathscr {H}}_{p+1,q}^{(j)}$ and the covariance estimate of the residual follows from the covariance of the sample mean as

$$ \widehat{\Sigma }= \frac{N_b}{n_b-1} \sum _{j=1}^{n_b} \mathrm{vec }\left( S^T\widehat{\mathscr {H}}_{p+1,q}^{(j)} - S^T\widehat{\mathscr {H}}_{p+1,q}^0\right) \mathrm{vec }\left( S^T\widehat{\mathscr {H}}_{p+1,q}^{(j)} - S^T\widehat{\mathscr {H}}_{p+1,q}^0\right) ^T. $$

Finally, compute the test statistic s for several training datasets of length N from the reference state, and determine the threshold t from the test values for a desired type I error.