Vehicle scanning-based enhanced modal identification of a bridge using singular value decomposition

Abstract

Modal parameter estimation of a bridge with the vibration responses measured from an instrumented vehicle moving at a controlled speed is an active area of research. It is challenging to determine bridge natural frequencies and their associated mode shapes, as the measured output-only responses of an instrumented vehicle contain the desired dynamic responses of the bridge, along with the other confounding components related to vehicle dynamics, driving frequency component, and dynamics associated with the roughness profile of the road. These bridge responses are often masked by the dynamic responses associated with the vehicle, and road surface profile. Measurement noise components add to the existing problem of separating bridge frequency from various other said components. In this paper, an attempt has been made to extract bridge frequencies and mode shapes through the output-only responses collected from a traversing vehicle and using singular value decomposition (SVD) combined with the Teager-Kaiser energy operator (TKEO). Numerical investigations are made on the proposed SVD-TKEO-based modal identification technique in the presence of measurement noise. Parametric studies are conducted to investigate the influence of vehicle speed and road surface roughness on the quality of the identified bridge modal parameters using the proposed technique Numerical simulations carried out, show that the proposed SVD-TKEO-based algorithm performs well in identifying bridge mode shapes, even with relatively higher vehicle traveling speed, and handles even roughness of the road surface profile reasonably well. Lab-level experimental studies using vehicle bridge interaction setup, are also carried out using the SVD-based modal parameter estimation technique to explore its practical use.

Appendix

1.1 Singular Value Decomposition (SVD)

Given a one-dimensional time history signal s(t), it can be converted into the form of a matrix called Hankel matrix as follows. As we deal with acceleration time history measurements from instrumented moving vehicle, the same measured response is used to form the Hankel matrix.

$$ {\mathbf{H}}_{m,n} = \left[ {\begin{array}{*{20}c} {s(1)} & {s(2)} & \cdots & \cdots \\ {s(2)} & {s(3)} & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots \\ {s(m)} & {s(m + 1)} & \cdots & \cdots \\ \end{array} \begin{array}{*{20}c} {s(n)} \\ {s(n + 1))} \\ \cdots \\ {s(L)} \\ \end{array} } \right] $$

(A.1)

Where L is the data length of the signal, H is the Hankel matrix of size mxn, and L =m+n-1. The matrix can be decomposed employing singular value decomposition (SVD)

$$ {\mathbf{H}} = UDV^{T} = [u_{1} \, u_{2} ....u_{m} ]\left[ {\begin{array}{*{20}c} {\sigma_{1} } & {} & {} & {} & {} & {} \\ {} & {\sigma_{2} } & {} & {} & {} & {} \\ {} & {} & . & {} & {} & {} \\ {} & {} & {} & . & {} & {} \\ {} & {} & {} & {} & . & {} \\ {} & {} & {} & {} & {} & {\sigma_{m} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {v_{1}^{T} } \\ {v_{2}^{T} } \\ . \\ . \\ . \\ {v_{m}^{T} } \\ \end{array} } \right] $$

(A.2)

where $V \in {\mathbb{R}}^{nxn}$ and $V \in {\mathbb{R}}^{nxn}$ are the left and right orthogonal matrices respectively. D is a diagonal matrix with singular values. $u_{i} \in {\mathbb{R}}^{mx1}$, $v_{i} \in {\mathbb{R}}^{nx1}$.

$$ D = \left\{ \begin{gathered} [diag(\sigma_{1} ,\sigma_{2} ,.....\sigma_{l} ),0] \, if \, m < n \hfill \\ [diag(\sigma_{1} ,\sigma_{2} ,.....\sigma_{l} )]{\text{ if}}\,\,m = n \hfill \\ [diag(\sigma_{1} ,\sigma_{2} ,.....\sigma_{l} ),0]^{T} \, if \, m > n \, \hfill \\ \end{gathered} \right\} $$

(A.3)

where l = min(m, n), $\sigma_{1} ,\sigma_{1} ,.....\sigma_{l}$ are the singular values of the matrix H and $\sigma_{1} \ge \sigma_{2} \ge .....\sigma_{l} \ge 0.$

Equation (A.2) can be written in the form of individual components as

$$ H = \sum\limits_{k = 1}^{m} {u_{i} \sigma_{i} v_{i}^{T} } = u_{1} \sigma_{1} v_{1}^{T} + u_{2} \sigma_{2} v_{2}^{T} + ...... + u_{m} \sigma_{m} v_{m}^{T} $$

(A.4)

The original signal can be reconstructed as $\widetilde{s}$ by forming a Hankel matrix $\widetilde{H}$, with only the top c singular values of diagonal matrix D, which cumulatively contributes to about 90% to 95% of the total value. .

$$i.e.\;\;\;R_{c} = {{\sum\limits_{{j = 1}}^{c} {\sigma _{j} } } \mathord{\left/ {\vphantom {{\sum\limits_{{j = 1}}^{c} {\sigma _{j} } } {\sum\limits_{{k = 1}}^{m} {\sigma _{k} } }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{{k = 1}}^{m} {\sigma _{k} } }} $$

(A.5)

The original signal $s$ can be reconstructed as $\widetilde{s}$ using the following steps.

• (i) Convert the matrix ${\widetilde{H}}^{T}$ into a row vector b of size 1xmn.

• (ii) A matrix $P \in {\mathbb{R}}^{Lxnm}$ needs to be constructed with a set of unit submatrices each of size nxn as illustrated below by considering m as 3 and n as 4.

  $$ P = \left[ {\begin{array}{*{20}l} 1 \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & 1 \hfill & {} \hfill & {} \hfill & 1 \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & 1 \hfill & {} \hfill & {} \hfill & 1 \hfill & {} \hfill & {} \hfill & 1 \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & 1 \hfill & {} \hfill & {} \hfill & 1 \hfill & {} \hfill & {} \hfill & 1 \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & 1 \hfill & {} \hfill & {} \hfill & 1 \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & 1 \hfill \\ \end{array} } \right]_{{6 \times 12}} $$

  (A.6)

• (iii) The signal $\widetilde{s}$ of size 1XL can be constructed as

  $$\widetilde{s}=x{P}^{+}$$

  (A.7)

where $P^{ + }$ is the Moore-Penrose inverse generally referred to as the pseudo inverse of P. Since the size of the matrix P depends on the data length of the original signal s, it is expected to be large. Further, the matrix P as illustrated in Equation (A-6) is sparse. Therefore, computing pseudo inverse is computationally involved. In view of this, the signal $s$ is usually reconstructed using an alternative less computationally expensive procedure through counter-diagonal averaging of the elements of the matrix $\widetilde{H}$ [46]. It can be easily verified that the counter-diagonal averaging mode of reconstruction of the signal $s$ is consistent with the reconstruction process illustrated in Equation (A.7). The counter-diagonal averaging process of signal reconstruction is illustrated below in Equation (A.8).

$$\widetilde{s}(k)=\left\{\begin{array}{c}\frac{1}{k}{\sum }_{j=1}^{k}\widetilde{H}(j,k-j+1), \, {1}\le {\text{k}}\text{<}{\text{m}}\\ \frac{1}{m}{\sum }_{j=1}^{m}\widetilde{H}(j,k-j+1), \, {\text{m}}\le {\text{k}}\text{<}{\text{n}}\\ \frac{1}{(m+n-k)}{\sum }_{j=(k-n+1)}^{m}\widetilde{H}(j,k-j+1), \, {\text{n}}\le {\text{k}}\text{<}{\text{m}}\text{+}{\text{n}}-{1}\end{array}\right\}$$

(A.8)

The signal $\widetilde{s}(t)$ can be recovered from the matrix $\widetilde{H}$ using Equation (A.8). $\widetilde{H}$ is the approximated Hankel matrix constructed using selected effective singular values as decided by the cut-off criterion stipulated in Equation (A.5).

It is appropriate to point out here that the total number of non-zero singular values generated by the Hankel matrix, H during the SVD of a signal s(t) is solely correlated with the number of feature frequency components contained in the signal s(t). This is irrespective of the frequency $\omega_{i}$, amplitude $A_{i}$, and phase $\Phi_{i} .$ In the singular value distribution chart, only two adjacent singular values are associated with each frequency component contained in the time history signal, s. The magnitude of the feature frequency component contained in the signal s determines the order of nonzero effective singular values. In other words, the frequency component with higher amplitude will have a higher rank pair of singular values with larger magnitude.