Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis

For deflection signal time series x (k), k = 1,2,...,N, MF-DFA calculation steps are as follows:

Step 1: Determine the ‘profile’ (1)Y(i)≡∑K=1i|xk−x¯|, i=1,…,NY\left( i \right) \equiv \sum\limits_{K = 1}^i {\left| {{x_k} - \bar x} \right|,\;\;\;i = 1, \ldots ,N}

Subtraction of the mean x¯{\bar x} is not compulsory, since it would be eliminated by the later de-trending in the third step.

Among solutions of h(q) there is one that is known as the generalised Hurst index, which represents the correlation of the original sequence. When 0.5 < h(q) < 1, it indicates that the signal has a long-range correlation and that h(q) is closer to 1; and so, the correlation is stronger. When h(q) = 0.5, there is no correlation between the time series. When h(q) < 0.5, the signal only has a negative correlation.

The generalised Hurst exponent h(q) is one way to describe time series with multi-fractal properties. Multi-fractal spectrum [10] completely presents the fractal singular probability distribution form of a signal, which improves the degree of the fine description of signal geometry and local scale behaviour. In the de-trending multi-fractal method h(q) is converted into mass exponent τ (q), singularity exponent α₀ and multi-fractal spectrum f (α). The transformation relationship is as follows: (6)τ(q)=qh(q)−1\tau \left( q \right) = qh\left( q \right) - 1(7)α=τ′(q)=h(q)+qh′(q)\alpha = {\tau ^\prime}\left( q \right) = h\left( q \right) + q{h^\prime}\left( q \right)(8)f(α)=qα−τ(q)=q[α−h(q)]+1f\left( \alpha \right) = q\alpha - \tau \left( q \right) = q\left[ {\alpha - h\left( q \right)} \right] + 1

The multi-fractal singularity spectrum obtained by the MF-DFA method is a set of parameters that can accurately describe the multi-fractal dynamic behaviour characteristics of deflection signals.

The singularity exponent α₀ corresponding to the maximum value of the spectral function f (α) describes the irregularity of the deflection signal. The larger value of α₀ represents the more irregular deflection signal. ∆f is the proportion of large and small deflection signal peak; its value is ∆f = f (α_max) − f (α_min). Multi-fractal spectrum width ∆α is bigger, the deflection signal wave is more irregular, and the value of ∆α can be obtained by ∆α = α_max − a_min. When ∆f and ∆α are larger and α₀ is smaller, and the deflection signal fluctuation is more intense, the multi-fractal features are stronger [11]. Therefore, in this paper, we have selected α₀, ∆f and ∆α as the bridge deformation characteristics.

Using DFA for fault prediction, the main operation is a de-trending operation, and the use of different order polynomial fitting means the removal of different types of a trend in the signal. The scale range of choice within the scale index can reflect the intrinsic characteristics of the signal. Only an appropriate scale can obtain reliable statistics, with the time sequence changing fast and with the maximum recommended being n/20 [12]. The DFA method is used to analyse the deflection signal, and the relationship between the scaling exponent τ (q) and the order q of the wave function was obtained, as shown in Figure 2.

Fig. 2

Due to the non-periodic and non-stationary characteristics of the bridge deflection signals, the scaling exponents τ (q) and q have an obvious nonlinear relationship. From Figure 2, we observe that the distribution in the figure is a convex function, which indicates that the bridge deflection signals have multi-fractal characteristics and the time series has a long-range correlation. Therefore, the MF-DFA method can directly distinguish the state of the bridge.

Figure 3 shows the variation relationship between the generalised Hurst index h_q and q of the deflection signals.

Fig. 3

From Figure 3, we observe that the dynamic mechanism of deflection signal generated by the bridge under different deformation is different, which leads to great differences in the generalised Hurst index under different states. Moreover, the generalised Hurst index h_q decreases with the increase of q, which further proves that the bridge deflection signal has multi-fractal characteristics.

Figure 4 shows the variation rule of the multi-fractal spectrum wave function of the bridge deflection signals with the singularity index in different states.

Fig. 4

From Figure 4, it can be seen that the multi-fractal spectrum of the deflection signals of bridges varies with the degree of deformation. For example, the width of the singularity index ∆α is different, and the distribution of the singularity index α₀ is also regular. On selection of the multi-fractal spectrum parameter, we ascertain α₀, ∆f and ∆α as bridge deformation characteristics; the results are shown in Table 1.

Table 1 shows that when the bridge is in a healthy state, the deflection of the multi-fractal spectrum parameter ∆α value is far less than the fault state, and with the increase of bridge deformation, the spectral width ∆α and singularity index α₀ gradually increase. ∆f has obvious variation in high-frequency mechanical signals [13], but it is not suitable for bridge health diagnosis due to the small amplitude and frequency of deflection signals in bridge health detection.

In conclusion, the multi-fractal spectrum parameter α₀ and ∆α can preliminarily distinguish the health status of the bridge, and the spectrum width ∆α can effectively distinguish the deformation degree of the bridge. Therefore, the singularity exponent α₀ and multi-fractal spectrum width ∆α can be used as characteristic parameters for bridge health diagnosis.