Health monitoring of Bridges based on multifractal theory

Ling Zhao; Jiawei Ding; Haiming Liu

Journal & Issue Details

Journal Details

Format: Journal
First Published: 01 Jan 2016
Publication timeframe: 2 times per year
Languages: English

The multifractal theory is applied in an analysis of bridge disturbance signals with the aim of investigating their nonlinear characteristics, and then the recognisable fault features are extracted from them. By calculating the box dimension and correlation dimension of the bridge disturbance signal, the dimensional characteristics of the disturbance data are analysed to distinguish the health-state of the bridge. Finally, taking the bridge disturbance data as an example, and by using the multifractal spectrum analysis of the disturbance data, it is concluded that the multifractal method can accurately identify the fault state and realise the bridge health monitoring.

Keywords

box dimension
Correlation dimension
Bridge health monitoring
Multifractal spectrum

Article

Introduction

In recent years, with the rapid development of China’s economy, bridges, as the lifeline of transportation, have been developing rapidly. However, faulty construction, design defects, natural disasters and other problems lead to the bridge damage in varying degrees, and sometimes even collapse and other serious consequences, which has brought a great negative impact to economic development. Therefore, domestic and foreign scholars began to pay attention to the health monitoring of bridges. As bridge excitation signals always exist in complex environment, scholars have put forward a variety of mature methods, such as Stochastic subspace identification (SSI) method [1], environmental excitation method (ERA), frequency domain decomposition method (FDD) [2], peak picking method (PP), time domain parameter identification method (ITD), etc. Liu et al. [3] distinguish and identify dense modes by combining the extended analytical mode decomposition (AMD), recursive Hilbert change and variable frequency synchronous compression wavelet transform (ZSWT), which can track the instantaneous characteristics of time-varying structures under environmental excitation. Mao et al. [4] proposed a method for automatic modal parameter identification of long-span bridges and several machine learning methods, such as PCA, K-means clustering and hierarchical clustering, and these were applied to modify data-SSI. Compared with other improved SSI algorithms, the proposed method uses PCA to eliminate the noise component in the criterion vector of modal verification before identifying modal parameters, so as to improve the identification accuracy and save the operation time. In addition, Mao et al. [4] also proposed an algorithm to determine the optimal parameters of cluster analysis in this study, and finally realised continuous automatic identification of modal parameters of Sutong Bridge through this method. Zhu et al. [5] proposed a Bayesian frequency domain algorithm, which has been applied to the vibration test of Jiangyin Bridge. It has good identification effect and can distinguish and identify dense modes. In order to reduce the noise level of bridge health monitoring signals, Yan Peng [6] proposed an improved EMD wavelet correlation denoising algorithm, and compared it with EMD wavelet threshold denoising algorithm and wavelet default threshold denoising algorithm, and proved that the improved EMD wavelet correlation denoising algorithm has better denoising effect in bridge environment. Liu et al. [7] proposed a near-field pulse seismic signal denoising algorithm based on complementary set empirical mode decomposition in view of the nonlinear and non-stationary characteristics of seismic pulse signals, and applied the algorithm to bridge excitation signal denoising with good results. Cheng et al. [8] believed that there was a one-to-one correspondence between the bridge structural forms and real physical modes, which was verified by experiments. Then, combining with the idea of clustering, a real modal screening algorithm was proposed and applied to the bridge health monitoring of a long-span cable-stayed bridge. However, the above methods need to overcome the inability to accurately distinguish and identify the dense modes with similar eigenvalues. The bridge disturbance signals are nonlinear, non-stationary and non-periodic. Multifractals have unique characteristics in revealing the nonlinearity and discontinuity of complex systems, and are especially applicable to various nonlinear phenomena. Fractal dimension, which is an important parameter used to describe the behaviour of nonlinear systems, can be used to qualitatively analyse the motion state of nonlinear systems, especially to quantify a certain process, so that health monitoring of Bridges can be easily realised. At present, many scholars have characterised the irregularity of the signal by using the single component formation, but the single component fractal can only reflect the overall characteristics of the signal, and lacks the ability to characterise the local singularity [9]. Although multifractal cannot reflect the overall irregularity of the signal, it can well represent the local behaviour of the signal. This paper calculated the box dimension and correlation dimension, and has drawn the box dimension and correlation dimension of the double logarithm curve; this analysis shows that a preliminary judge bridge immunity data have typical fractal characteristics. The analysis of multifractal dimension spectrum provides further evidence that the bridge immunity data have fractal features, and can distinguish different bridge health state.

Fractal Theory

Fractal theory has been widely used in the field of nonlinear analysis today. Fractal dimension is a measure of fractal, which describes the regularity of many irregular things in nature through a simple number. Of course, different degrees of complexity have different fractal dimensions. Therefore, the health state characteristics of Bridges can be obtained by calculating the box dimension and correlation dimension of the bridge disturbance data.

2.1

The Box Dimension

The box dimension is the simplest and most widely used fractal dimension, which can better reflect the dynamic structural changes of the dynamic system on the whole [10]. The steps for calculating the box dimension are as follows:

Step 1: Computes the count of the grid N_kt(1) $N_{kt} = \frac{1}{k \cdot t} \sum_{i = 1}^{M_{0} / k} | max {T [(i - 1) k + 1 : ik + 1]} - min {T [(i - 1) k + 1 : ik + 1]} | + 1$ {N_{kt}} = {1 \over {k \cdot t}}\sum\limits_{i = 1}^{{M_0}/k} {\left| {\max \left\{ {T\left[ {\left( {i - 1} \right)k + 1:ik + 1} \right]} \right\} - \min \left\{ {T\left[ {\left( {i - 1} \right)k + 1:ik + 1} \right]} \right\}} \right| + 1}

Step 2: Draw the logarithmic curve of ln(N_kt) − ln(kt), A segment of the linearity sign on the curve is determined as the scale-free region of the signal. The linear regression model satisfies the following equation in the scale-free region ln(N_kt) − ln(kt). (2) $\ln (N_{kt}) = - D_{B} \ln (kt) + b$ \ln \left( {{N_{kt}}} \right) = - {D_B}\ln \left( {kt} \right) + b

Step 3: The least square method is used to fit the slope P of the linear regression equation, that is the box dimension D_B. (3) $y = px + b$ y = px + b

2.2

Correlation Dimension

The correlation dimension is based on the probability of the point set falling into each hypercube [10], and can be defined as: (4) ${dim}_{c} (Ω) = lim_{ε \to 0} \frac{1}{\ln (ε)} \ln (\sum_{k = 1}^{N (ε)} p_{k}^{2})$ {\dim _c}\left( \Omega \right) = \mathop {\lim }\limits_{\varepsilon \to 0} {1 \over {\ln \left( \varepsilon \right)}}\ln \left( {\sum\limits_{k = 1}^{N\left( \varepsilon \right)} {p_k^2} } \right)

The correlation dimension is sensitive to the time course behaviour of dynamical system and can well reflect the dynamic process of dynamical system. The steps to calculate the correlation dimension are as follows:

Step 1: One - dimensional time series is reconstructed in phase space by time-delay method. (5) $\begin{array}{l} X_{i} (m, τ) = (T_{i}, T_{i + 1}, T_{i + 2}, \dots, T_{i} + (m - 1) τ) \\ i = 1, 2, 3, \dots, N \end{array}$ \matrix{ {{X_i}\left( {m,\tau } \right) = \left( {{T_i},{T_{i + 1}},{T_{i + 2}}, \cdots ,{T_i} + \left( {m - 1} \right)\tau } \right)} \hfill \cr {i = 1,2,3, \cdots ,N} \hfill \cr }

Step 2: Lets pick any vector x_i, and calculate the distance R_ij from the other N − 1 vectors to x_i. (6) $R_{ij} = {[\sum_{l = 0}^{m - 1} {(T_{i + l τ} - T_{j + l τ})}^{2}]}^{1 / 2}$ {R_{ij}} = {\left[ {\sum\limits_{l = 0}^{m - 1} {{{\left( {{T_{i + l\tau }} - {T_{j + l\tau }}} \right)}^2}} } \right]^{1/2}}

Step 3: Find the distance distribution function C(ε). (7) $C (ε) = \frac{N_{1} (ε)}{N (ε)}$ C\left( \varepsilon \right) = {{{N_1}\left( \varepsilon \right)} \over {N\left( \varepsilon \right)}}

Step 4: The logarithmic curve of the distance distribution function is drawn, and the correlation dimension D_m is obtained. (8) $D_{m} = lim (\frac{ln C (ε)}{\ln (ε)})$ {D_m} = \lim \left( {{{\ln C\left( \varepsilon \right)} \over {\ln \left( \varepsilon \right)}}} \right)

2.3

Multifractal Spectral Algorithm

By comparing the fractal dimension of the normal data and the fault data, the fault and the normal data can be distinguished preliminarily. Then, by analysing the multifractal spectrum of the disturbance signal, the fault characteristics of the disturbance signal can be extracted better, and the fault state can be identified. The steps of the multi-fractal spectrum algorithm are as follows:

Step 1: Calculate the cumulative deviation of the time series of data against the mean value Y (i). (9) $Y (i) = \sum_{k = 1}^{i} [x (k) - \tilde{x}] (i = 1, 2, \dots, N)$ Y\left( i \right) = \sum\limits_{k = 1}^i {\left[ {x\left( k \right) - \tilde x} \right]\left( {i = 1,2, \cdots ,N} \right)} where: $\tilde{x}$ {\tilde x} is the mean value of time series x(k).

Step 2: Divide the deviation into m-equal length intervals of length s,m = int(N/s). Since N is not necessarily divisible, the process is repeated in the opposite direction of the divergence, resulting in 2m-equal length intervals.

Step 3: The least square method is used to fit the mean square error F²(s, v) of each subinterval. (10) $F^{2} (s, v), v = 1, 2, 3, \dots 2 m .$ {F^2}\left( {s,v} \right),v = 1,2,3, \cdots 2m. When v = 1,2,3, . . . m. (11) $F^{2} (s, v) \equiv \frac{1}{s} \sum_{i = 1}^{s} {[(v - 1) s + i] - y_{v} (i)}^{2}$ {F^2}\left( {s,v} \right) \equiv {1 \over s}\sum\limits_{i = 1}^s {{{\left\{ {\left[ {\left( {v - 1} \right)s + i} \right] - {y_v}\left( i \right)} \right\}}^2}} When v = 1,2,3, . . . 2m. (12) $F^{2} (s, v) \equiv \frac{1}{s} \sum_{i = 1}^{s} {Y [N - (v - m) s + i] - y_{v} (i)}^{2}$ {F^2}\left( {s,v} \right) \equiv {1 \over s}\sum\limits_{i = 1}^s {{{\left\{ {Y\left[ {N - \left( {v - m} \right)s + i} \right] - {y_v}\left( i \right)} \right\}}^2}}

Step 4: For the 2m subinterval, the wave function F(s) of order q is defined. (13) $F_{q} (s) = {\frac{1}{2 m} \sum_{v = 1}^{m} {[F^{2} (s, v)]}^{q / 2}}^{1 / q}$ {F_q}\left( s \right) = {\left\{ {{1 \over {2m}}\sum\limits_{v = 1}^m {{{\left[ {{F^2}\left( {s,v} \right)} \right]}^{q/2}}} } \right\}^{1/q}}

Step 5: Change the size of the subinterval length s and repeat Step1–4. If the time series x(k) has long range correlation, then there is a power-law relationship between the q-order wave function F_q(s) and s. (14) $F_{q} (s) \infty s^{hq}$ {F_q}\left( s \right)\infty {s^{hq}} If the time series x(k) has only the simplex fractal property, the generalised Hurst exponent h_q is constant, and when the time series is uncorrelated or short-range correlated, then h_q = 1/2. When the time series has multifractal characteristics, the relationship between h_q and q is nonlinear. According to the standard partition function, the scaling exponent τ_q can be obtained, and the relation between it and the generalised Hurst exponent h_q is as follows: (15) $τ_{q} = q h_{q} - 1$ {\tau _q} = q{h_q} - 1 According to the Legendre transformation in statistical physics, the singularity exponent α and the spectral function f (α) are defined as follows: (16) $α_{q} = d τ_{q} / dq$ {\alpha _q} = d{\tau _q}/dq(17) $f (a) = q (α_{q} - h_{q}) + 1$ f\left( a \right) = q\left( {{\alpha _q} - {h_q}} \right) + 1 Substitute Eq. (10) into Eqs. (11) and (12) to obtain the relationship between the singular exponent α and spectral function f (α) of the multifractal spectrum and the generalised Hurst exponent h_q. (18) $α_{q} = h_{q} + {qdh}_{a} / dq$ {\alpha _q} = {h_q} + qd{h_a}/dq(19) $f (a) = q (α_{q} - h_{q}) + 1$ f\left( a \right) = q\left( {{\alpha _q} - {h_q}} \right) + 1 According to the spectral function f (α), the fractal characteristics of time series can be judged. When f (α) is constant, the time series has the feature of simplex fractal. If the change rule of f (α) is the curve with positive convexity, then the time series has fractal characteristics, and the interval of f (α) ≥ 0 is denoted as [α_min, α_max].

Data Analysis of Bridge Disturbance

The bridge disturbance database used is the monthly bridge disturbance data from 2017 to 2018 obtained by sensors installed in the two spans of Chongqing XXX Bridge. In order to extract the structural characteristics of the bridge disturbance signal and monitor the health of the bridge, 4 test samples of bridge disturbance data are selected from the bridge database for analysis.

3.1

Algorithm design of bridge health monitoring

The proposed algorithm design of bridge health monitoring used for the complete monitoring is shown in Figure 1.

Flow chart of bridge health monitoring algorithm

3.2

Bridge disturbance signal characteristic analysis based on box dimension and correlation dimension

Figure 1–3 shows the variation curve of the disturbance signal over time in 2 A group of abrupt signals, and three groups of normal signals were selected as sample data, then the global time domain waveform of each group of signals was plotted respectively according to the difference of sampling points, as shown in Figure 1 (a–b). Figure 1 (a) shows the mutation signals selected by comparison with other signals, and (b), (c) and (d) are the normal signals selected from different months. Figure 2 (a–d), and Figure 3 (a–d) is the local amplification of the data.

As can be seen from Figure 1–3, the fluctuation of the curves of the high frequency part and the low frequency part of the overall signal is basically the same except for the difference in the density of the curves. That is to say, the bridge disturbance signals have self-similarity, and it can be preliminarily judged that the bridge disturbance signals have typical fractal characteristics. Based on the principle of box dimension, the fact that immunity signal within a certain range of sampling have fractal features is confirmed. Now, the MATLAB program is used to draw the box dimension and correlation dimension of the double logarithm curve as shown in Figures 4 and 5 and, according to the double logarithm curve, the four data box dimension and correlation dimension, as shown in Table 1, are calculated, respectively. From the analysis of box dimension, correlation dimension double logarithmic curve and the fractal dimension of meter, we may safely draw the following conclusion: (1)

It can be seen from Figure 4 that the box dimension has a certain scale-free region, indicating that the bridge disturbance signal has scale invariance, and further indicating that the bridge disturbance signal has fractal characteristics within a certain range.

(2)

It can be seen from Figure 5 that when the embedding dimension is changed, both the logarithmic curves gradually become larger and finally become stable as the increment, which indicates that the correlation dimension of the bridge disturbance signal converges and there is a saturation value. It is further proved that the signal of bridge disturbance has fractal characteristics within a certain range.

(3)

It can be seen from Table 1 that the fractal dimensions of bridge disturbance signals under the same health state are similar, while the fractal dimensions under different states are different. The difference is obvious, and the different sizes of the fractal dimension describe the different health status of the bridge. Different fractal dimensions have similar rules. Both box dimension and correlation dimension can accurately describe the health state of the bridge.

Table 1

Fractal Dimension.

	Data2	Data5	Data8	Data9
The box dimension	−1.4486	−1.1203	−1.1230	−1.0980
Correlation dimension	0.7705	1.0409	1.0842	1.0448

3.3

Multifractal Spectrum Characterisation

According to the previous single-fold fractal dimension analysis, it is confirmed that the bridge disturbance data has fractal characteristics. In order to better analyse the signal characteristics, based on the principle of the multi-fractal spectrum algorithm, the MATLAB calculation program is written to calculate the multi-fractal spectrum of the bridge disturbance data and draw the singularity index, mass index and multi-fractal spectrum of the bridge disturbance signal, as shown in Figures 6–8.

Logarithmic graph of correlation dimension.

Singularity exponential curve of bridge disturbance data.

Quality index curve of bridge disturbance data.

Multifractal spectrum of bridge disturbance data.

As can be seen from Figures 6–7, both the quality index curve and the HUST index curve of the data are nonlinear, that is, the bridge disturbance data has multi-fractal characteristics. Figure 8 shows the multifractal spectrum of bridge data analysis, and indicates that multiple analysis spectral is positive convexity, and the difference among the data waveform is obvious, verify abnormal waveform data set is mutations signal set, proves that the multifractal spectrum can distinguish good Bridges of different health status.

Conclusion

In this paper, the multifractal method is used to calculate the box dimension and the correlation dimension, then to draw the log-log curve to analyse whether the bridge disturbance data set has multifractal characteristics, and thus preliminarily judge the health state of the bridge. Based on the theoretical proof that the bridge disturbance data has multi-fractal characteristics, the multi-fractal spectrum of the bridge disturbance data set is analysed, and the analysis results show that: (1) The sampling signals of bridge disturbance data collected at a certain time or frequency possess the fractal features. (2) The fractal dimension of vibration signals in the same working state is similar, but the fractal dimension under different working states is obviously different. The health state of each bridge which is described by each dimension, has a similar rule and these two dimensions can accurately represent the characteristics of the health state. (3) The multifractal spectrum can well distinguish the health state from the fault state of the bridge, which shows significant and efficient characteristics in the fault pattern recognition and classification, and meets the needs of engineering applications.

In general, it can be concluded that the use of multifractal theory is feasible in bridge health status identification, and further it is a quantitative, accurate method of bridge health monitoring and diagnosis.

Figures & Tables

Fractal Dimension.

	Data2	Data5	Data8	Data9
The box dimension	−1.4486	−1.1203	−1.1230	−1.0980
Correlation dimension	0.7705	1.0409	1.0842	1.0448

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Applied Mathematics and Nonlinear Sciences

Health monitoring of Bridges based on multifractal theory

Published Online: 19 Oct 2021

Page range: -

Received: 19 Mar 2021

Accepted: 29 May 2021

Keywords

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Fractal Dimension.

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