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 , environmental excitation method (ERA), frequency domain decomposition method (FDD) , peak picking method (PP), time domain parameter identification method (ITD), etc. Liu et al.  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.  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.  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.  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  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.  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.  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 . 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 . The steps for calculating the box dimension are as follows:
Step 1: Computes the count of the grid Step 2: Draw the logarithmic curve of Step 3: The least square method is used to fit the slope
Step 1: Computes the count of the grid
Step 2: Draw the logarithmic curve of
Step 3: The least square method is used to fit the slope
The correlation dimension is based on the probability of the point set falling into each hypercube , 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 Step 3: Find the distance distribution function Step 4: The logarithmic curve of the distance distribution function is drawn, and the correlation dimension
Step 1: One - dimensional time series is reconstructed in phase space by time-delay method.
Step 2: Lets pick any vector
Step 3: Find the distance distribution function
Step 4: The logarithmic curve of the distance distribution function is drawn, and the correlation dimension
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 Step 4: For the 2 Step 5: Change the size of the subinterval length s and repeat Step1–4. If the time series
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
Step 4: For the 2
Step 5: Change the size of the subinterval length s and repeat Step1–4. If the time series
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.
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.
|The box dimension||−1.4486||−1.1203||−1.1230||−1.0980|
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.
|The box dimension||−1.4486||−1.1203||−1.1230||−1.0980|
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Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Analysis of digital humanistic knowledge production based on natural language processing Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering The Construction of Judgment Model for Cultivating Intelligent Accounting Professionals in Universities Based on Logarithmic Periodic Power Law Model Action of Aut( G) on the set of maximal subgroups of p-groups Digital Regeneration and Database Construction of Hunan Embroidery Needlework under the Perspective of Artificial Intelligence Research on the practice of innovation and entrepreneurship education in universities under the environment of big data Application of digital technology in environmental art design Research on loyalty prediction of e-commerce customer based on data mining Research on the evaluation system of tourism management classroom teaching based on multimedia big data analysis The application of ARCS motivation model in college oral English teaching Research on Innovation of Translation Teaching and Translation Strategies for College Students in Multimedia Background A multi-channel convolutional neural network based on attention mechanism fusion for facial expression recognition Exploring the development of cultural and creative tourism under the perspective of ecological theory of tourism destination management Modeling Analysis of User Consumption Data of Internet Culture Industry in the Context of Emerging Technologies Reference: An algorithm for recognizing the main melody of orchestral music based on artificial intelligence of music melody contour Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on 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convolutional neural network models for image inpainting The construction of a practical platform for party building and ideological and political education based on information fusion A study on the influence of the spread of Yangming Studies in Japan on the psychology of the Japanese people based on big data analysis Research on personalized recommendation algorithm for micromechanical sensors based on cloud model Research on the value of modern vernacular literature in contemporary rural construction based on ADDIE model Robust speech recognition based on deep learning for sports game review Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Model Construction of Influencing Factors of Physical Education Core Quality Cultivation Based on Multiple Linear Regression Application of Nonlinear Differential Equation in Electric Automation Control System Research on the operation and management mechanism of cross-border e-commerce shared practice training base Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Strategies for improving Chinese language proficiency based on artificial intelligence technology The success rate model of college students’ new entrepreneurship based on nonlinear differential equations Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Research on the application of marketing strategy of national fitness exercise and dance events in the construction of sports culture based on big data technology Decisions of competing supply chain with altruistic retailer under risk aversion Strategies for constructing mathematical models of nonlinear systems based on multiple linear regression models Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations Research on human resource management performance evaluation based on BP algorithm The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of 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? 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