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

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 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.

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 _{kt}

Step 2: Draw the logarithmic curve of _{kt}_{kt}

Step 3: The least square method is used to fit the slope _{B}

The correlation dimension is based on the probability of the point set falling into each hypercube [10], and can be defined as:

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.

Step 2: Lets pick any vector _{i}_{ij}_{i}

Step 3: Find the distance distribution function

Step 4: The logarithmic curve of the distance distribution function is drawn, and the correlation dimension _{m}

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

Step 2: Divide the deviation into

Step 3: The least square method is used to fit the mean square error ^{2}(

Step 4: For the 2

Step 5: Change the size of the subinterval length s and repeat Step1–4. If the time series _{q}_{q}_{q}_{q}_{q}_{q}_{q}_{min}, _{max}].

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.

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

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:

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.

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.

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.

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 |

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.

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.

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.

#### 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 |

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 Industrial transfer and regional economy coordination based on multiple regression model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Research of cascade averaging control in hydraulic equilibrium regulation of heating pipe network Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Performance evaluation of college laboratories based on fusion of decision tree and BP neural network 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 Analysis of the relationship between industrial agglomeration and regional economic growth based on the multi-objective optimisation model Constraint effect of enterprise productivity based on constrained form variational computing The impact of urban expansion in Beijing and Metropolitan Area urban heat Island from 1999 to 2019 TOPSIS missile target selection method supported by the posterior probability of target recognition Ultrasonic wave promoting ice melt in ice storage tank based on polynomial fitting calculation model The incentive contract of subject librarians in university library under the non-linear task importance Application of Fuzzy Mathematics Calculation in Quantitative Evaluation of Students’ Performance of Basketball Jump Shot Visual error correction of continuous aerobics action images based on graph difference function Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Radioactive source search problem and optimisation model based on meta-heuristic algorithm Research on a method of completeness index based on complex model Fake online review recognition algorithm and optimisation research based on deep learning Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the two-flow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East Corporate social responsibility fulfilment, product-market competition and debt risk: Evidence from China ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Reconstruction of multimodal aesthetic critical discourse analysis framework Image design and interaction technology based on Fourier inverse transform What does students’ experience of e-portfolios suggest Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity Data mining of Chain convenience stores location The optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics A generative design method of building layout generated by path Parameter Id of Metal Hi-pressure State Equation Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control P-Matrix Reasoning and Information Intelligent Mining Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multi-array antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Application of regression function model based on panel data in bank resource allocation financial risk management Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Badminton players’ trajectory under numerical calculation method Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Difference-in-differences test for micro effect of technological finance cooperation pilot in China Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management Exploration on the collaborative relationship between government, industry, and university from the perspective of collaborative innovation The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanK-A New SVM Kernel for Classification Validity and reliability analysis of the Chinese version of planned happenstance career inventory based on mathematical statistics Visual positioning system for marine industrial robot assembly based on complex variable function Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Research on the influencing factors of agricultural product purchase willingness in social e-commerce situation Study of a linear-physical-programming-based approach for web service selection under uncertain service quality A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Fractional Differential Equations in the Exploration of Geological and Mineral Construction Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China The Law of Large Numbers in Children's Education Data structure simulation for the reform of the teaching process of university computer courses Calculating university education model based on finite element fractional differential equations and macro-control analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Good congruences on weakly U-abundant semigroups Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Mathematical simulation analysis of optimal detection of shot-putters’ best path Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Nonlinear Differential Equations in the Teaching Model of Educational Informatisation Fed-UserPro: A user profile construction method based on federated learning Smart Communities to Reduce Earthquake Damage: A Case Study in Xinheyuan, China Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe _{80}Si_{10}Nb_{6}B_{2}Cu_{2}Computer Vision Communication Technology in Mathematical Modeling The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption MCM of Student’s Physical Health Based on Mathematical Cone Attitude control for the rigid spacecraft with the improved extended state observer Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Cognitive Computational Model Using Machine Learning Algorithm in Artificial Intelligence Environment Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Informatisation of educational reform based on fractional differential equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression System dynamics model of output of ball mill Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Design of Morlet wavelet neural network to solve the non-linear influenza disease system Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Motion about equilibrium points in the Jupiter-Europa system with oblateness Fractional Differential Equations in Electronic Information Models Badminton players’ trajectory under numerical calculation method BIM Engineering Management Oriented to Curve Equation Model Optimal preview repetitive control for impulse-free continuous-time descriptor systems Development of main functional modules for MVB and its application in rail transit Study on the impact of forest fire prevention policy on the health of forest resources Mathematical Method to Construct the Linear Programming of Football Training The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching The modelling and implementation of the virtual 3D animation scene based on the geometric centre-of-mass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}The consistency method of linguistic information and other four preference information in group decision-making Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Fractal structure of magnetic island in tokamak plasma Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Nonlinear strategic human resource management based on organisational mathematical model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Fractional Differential Equations in Sports Training in Universities Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model