The bridge is an important component in the traffic network, its quality being crucial to people’s lives. As of 2016, one-quarter of China’s bridges have potential problems [1] according to statistics. These defective bridges need to be repaired regularly, which is an expensive task. If regular repairs are not undertaken, it will seriously affect the safety and stability of bridges. Bridge health inspection is an important measure to prevent and reduce accidents. Bridge deflection refers to the linear displacement of the centre of the bridge cross-section along the vertical direction of the axis when the bridge is stressed or the non-uniform temperature changes. In bridge health inspection, the bridge deflection information is often used to detect the deformation of the bridge.

The deflection signal will change due to the damage or mutation of the bridge [2]. Currently, the time-frequency analysis method is the main process of handling the non-stationary signal. For example, In [3] Wavelet transform is proposed to separate the transient and slow information of deflection signal. Hu et al. [4] verified the practicability of principal component analysis (PCA) in fault diagnosis, but the result also shows that PCA is not very sensitive to small faults. Due to the influence of nonlinear factors (impact load, damping, friction etc.), the deflection information of the bridge is nonlinear, non-stationary and non-periodic. Therefore, the structure state of the bridge cannot be fully reflected by the physical model of the bridge deflection data based on the traditional method [5].

Now, the method that is widely used to describe the object of study is fractal dimension [6]. As the bridge damage is affected by many factors, using the multi-fractal can depict the signal from overall irregularity and expresses the local behaviour finely [7, 8]. Therefore, this paper analyses the bridge deflection time-domain signal, and then uses the multi-fractal de-trend fluctuation analysis method to carry out the multi-fractal spectrum analysis of the bridge deflection signals, and summarises the connection between the parameters of the multi-fractal spectrum and the health state of the bridge, which provides the theoretical basis and the way of implementation for bridge health monitoring.

Bridge deflection signal is a non-stationary signal with small amplitude variation, and its inherent non-stationary characteristics are difficult to estimate. In this paper, the trend component of the bridge deflection signal is eliminated by the de-trended fluctuation analysis, and the deformation characteristics are described by the multi-fractal spectrum.

For deflection signal time series

Step 1: Determine the ‘profile’

Subtraction of the mean

Step 2: Divide the profile _{s}_{s}

Step 3: Calculate the local trend for each of the 2_{s}_{s}_{s}_{s}_{v}

Step 4: Average over all segments to obtain the _{q}_{q}_{q}_{q}

Step 5: Determine the scaling behaviour of the fluctuation functions by analysing log-log plots _{q}_{i}_{q}

Among solutions of

The generalised Hurst exponent _{0} and multi-fractal spectrum

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 _{0} corresponding to the maximum value of the spectral function _{0} represents the more irregular deflection signal. ∆_{max}) − _{min}). Multi-fractal spectrum width ∆_{max} − _{min}. When ∆_{0} is smaller, and the deflection signal fluctuation is more intense, the multi-fractal features are stronger [11]. Therefore, in this paper, we have selected _{0}, ∆

The deflection signals of bridges in a normal state, fatigue deformation and sudden deformation were collected from the detection signals of Chongqing xxx Bridge. The time-domain diagram is shown in Figure 1.

It can be seen from Figure 1, that the frequency and amplitude of bridge deflection signals are very small, which makes it difficult to obtain deformation characteristics by traditional spectrum analysis methods. Therefore, it is difficult to diagnose and predict bridge health state directly by amplitude spectrum.

By the theory of multi-fractal, the characteristics of deflection signals can be effectively described. Using this characteristic, trend analysis of deflection signals can be carried out by DFA to eliminate the non-stationary trend influence of time series multi-fractal features, and then the multi-fractal spectrum can be calculated for feature extraction.

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

Due to the non-periodic and non-stationary characteristics of the bridge deflection signals, the scaling exponents

Figure 3 shows the variation relationship between the generalised Hurst index _{q}

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 _{q}

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.

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 ∆_{0} is also regular. On selection of the multi-fractal spectrum parameter, we ascertain _{0}, ∆

Feature parameters of the bridge deflection signal.

Parameter | Mutation | Fatigue Signal | Normal_1 | Normal_2 |
---|---|---|---|---|

_{0} | 2.072 | 1.873 | 1.835 | 1.780 |

∆ | 0.0254 | −0.2887 | 0.1633 | 0.04698 |

∆ | 2.55 | 1.31 | 0.775 | 0.576 |

Table 1 shows that when the bridge is in a healthy state, the deflection of the multi-fractal spectrum parameter ∆_{0} gradually increase. ∆

In conclusion, the multi-fractal spectrum parameter _{0} and ∆_{0} and multi-fractal spectrum width ∆

For bridge deflection signal, it is difficult to obtain effective fault characteristic quantity by the traditional spectrum analysis method. Therefore, the MF-DFA method is used to analyse the multi-fractal spectrum of the deflection signals of bridges under different states. The conclusions are as follows:

Deflection signals have obvious multi-fractal characteristics, and the multi-fractal characteristics of bridge deformation state are stronger than those corresponding to the normal state.

The MF-DFA method can effectively identify the fault state of the bridge deflection signals. The multi-fractal spectrum parameter _{0} and ∆

#### Feature parameters of the bridge deflection signal.

Parameter | Mutation | Fatigue Signal | Normal_1 | Normal_2 |
---|---|---|---|---|

_{0} | 2.072 | 1.873 | 1.835 | 1.780 |

∆ | 0.0254 | −0.2887 | 0.1633 | 0.04698 |

∆ | 2.55 | 1.31 | 0.775 | 0.576 |

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vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Badminton players’ trajectory under numerical calculation method Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Innovation of Economic Management Risk Control in Retailer Supply Chain Based on Differential Equation Model Statistical Model of College Ideological and Political Learning Based on Fractional Differential Equations Nonlinear Differential Equation in Anti-aging Test of Polymer Nanomaterials Mechanics of Building Structural Materials Based on Lagrangian Mathematical Model Analysis The Mental Health Education Management of Higher Vocational Students Based on Fractional Differential Equations Application of regression function model based on panel data in financial risk management of bank resource allocation University Ideological and Political Learning Model Based on Statistical Memory Curve Mathematical Equation Research on Resonance Properties of Semantic Wave Fractal Fractals Based on Quantitative Analysis of English Corpus Research on urban landscape big data information processing system based on ordinary differential equations Modeling of fractional differential equation in cloud computing image fusion algorithm Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Self-control Ability Based on the Mathematical Model of Probability Theory Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas Criminal law imputation path for biometric information Research on composite dynamic disaster prevention and control system of mine earthquake and shock in thick and hard rock mines Research on urban energy Internet planning based on fuzzy comprehensive method Research on innovative strategies of college students’ English teaching under the background of artificial intelligence Electromagnetic interference prediction technology of new energy motor drive system Research on the intelligent path of college students’ network ideological and political education based on big data mining technology Research on the application of PLC technology in electrical automation engineering Application of matrix multiplication in signal sensor image perception Analysing the variation of metadiscourse verb patterns in English academic papers from within and between disciplines Impact of COVID-19 policy on stock prices of listed property companies Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence VR-based computer maintenance practical training platform development design and application research Design and implementation of an integrated management system for discipline construction and postgraduate education in universities Research on innovative human capital for China’s economic development based on STI model In-depth analysis of the artistic expression of paper-cut elements in the design of boat space Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction Modeling the pathway of breast cancer in the Middle East Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model Research on the effects of financial literacy on rural household credit constraint Research on educational resource recommendation system based on MRLG Rec Visual analysis of the trend of mental health changes among Chinese college students in the context of the normalisation of the new crown epidemic A study on the phenomenon of anaphoric correction in college students’ English conversation Limit cycles of a generalised Mathieu differential system Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Application of data mining in basketball statistics The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Design of fitness walker for the elderly based on ergonomic SAPAD model Research on the innovation of collaborative education model of scientific research and teaching in colleges and universities based on the theory of process reengineering AtanK-A New SVM Kernel for Classification The innovative path of diversified vocal music education reform in colleges and universities under the advanced educational concepts of China and the West Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Rhino application in ceramic packaging design teaching Study of a linear-physical-programming-based approach for web service selection under uncertain service quality Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Fed-UserPro: A user profile construction method based on federated learning Design and evaluation of intelligent teaching system on basic movements in PE Research on identifying psychological health problems of college students by logistic regression model based on data mining Research on evaluation system of cross-border E-commerce platform based on the combined model A mathematical model of PCNN for image fusion with non-sampled contourlet transform Financial customer classification by combined model Retraction