Arched steel tube concrete bridge detection method based on ultrasonic guided wave frequency dispersion characteristic

Abstract

The invention relates to the technical field of ultrasonic guided wave nondestructive testing, and particularly discloses an arched concrete filled steel tube bridge detection method based on ultrasonic guided wave frequency dispersion characteristics, which adopts a theoretical derivation-simulation verification-experimental analysis method, and comprises the following steps: firstly, establishing a fluctuation characteristic equation based on a rotating coordinate system and a semi-analytical finite element method; deriving a guided wave frequency dispersion equation, obtaining a frequency dispersion curve through numerical calculation, and determining an optimal monitoring mode and an excitation frequency range; a three-dimensional finite element model is constructed for wave field numerical simulation, and the reliability of the model is verified through comparison of a theoretical solution and a simulation solution; then carrying out an ultrasonic guided wave detection experiment, collecting a time domain signal and extracting an actually measured frequency dispersion characteristic; and finally establishing theoretical prediction through time-frequency analysis, comparing numerical simulation with experimental data, and verifying the accuracy of a theoretical model in combination with error analysis. The method can accurately detect the defects of the arched concrete-filled steel tube bridge, and improves the reliability and practicability of complex structure damage identification.

Description

Technical Field

[0001] The present invention relates to the technical field of ultrasonic guided wave non-destructive testing, and particularly to a detection method for arch-shaped concrete-filled steel tube bridges based on the dispersion characteristics of ultrasonic guided waves. Background Art

[0002] Due to its high load-bearing efficiency and composite mechanical properties, the arch-shaped concrete-filled steel tube structure has become the core load-bearing system for projects such as long-span and heavy-load bridges. However, in a complex service environment, multi-scale damages are likely to occur: debonding at the interface between the steel tube and the concrete will weaken the composite effect, and the steel tube wall will crack under the influence of the concrete. Traditional detection methods (such as visual inspection, tapping method, or local strain monitoring) have significant limitations: they cannot accurately locate, have insufficient sensitivity, low detection efficiency, and are difficult to meet the requirements of global monitoring of large structures.

[0003] With its unique long-distance propagation ability, multi-modal excitation characteristics, and high sensitivity to micro damages, ultrasonic guided wave technology has shown significant technical advantages in the field of non-destructive testing of composite materials. Research has confirmed that the dispersion characteristics (group velocity, phase velocity, etc.) and propagation attenuation behavior of guided waves are sensitive to changes in the internal elastic parameters of materials. In particular, multi-modal characteristics such as Lamb waves and SH waves can provide multi-dimensional diagnostic information for interface damage identification. However, in engineering practice, the heterogeneous material interface of typical layered composite structures such as concrete-filled steel tubes will induce complex guided wave mode conversion and energy dissipation phenomena, resulting in the fuzzification of detection signal characteristics. To overcome this technical bottleneck, it is proposed to use low-frequency non-dispersive modes for detection, establish a numerical model of the waveguide structure based on the semi-analytical finite element method, and achieve time-frequency joint analysis through methods such as wavelet transform method and Hilbert-Huang transform method. Combining the dispersion characteristic regulation and signal feature decoupling strategy, the problem of the complexity of guided wave propagation characteristics caused by heterogeneous interfaces is systematically solved. Summary of the Invention

[0004] The object of the present invention is to provide a detection method for arch-shaped concrete-filled steel tube bridges based on the dispersion characteristics of ultrasonic guided waves, which solves the problem of insufficient accuracy in damage detection of curved surface structures in the prior art. By constructing a dispersion curve atlas database and combining time-frequency domain joint analysis algorithms, the precise positioning of millimeter-level defects inside the arch rib is achieved.

[0005] To achieve the above object, the present invention provides a detection method for arch-shaped concrete-filled steel tube bridges based on the dispersion characteristics of ultrasonic guided waves, including the following steps: S1. Based on the arch-shaped structure of the arch-shaped concrete-filled steel tube, establish a rotating coordinate system; S2. Use the semi-analytical finite element method to establish a wave motion characteristic equation for the arch-shaped concrete-filled steel tube, and obtain a guided wave dispersion characteristic equation for the arch-shaped concrete-filled steel tube; S3. Calculate the theoretical wave numbers, phase velocities, and group velocity dispersion curves of the concrete-filled steel arch, analyze the vibration dispersion characteristics of the guided wave modes at different frequencies, and determine the optimal monitoring mode and excitation frequency range; S4. Construct a three-dimensional finite element model of the concrete-filled steel arch structure based on S1, S2, and S3, and draw the simulation dispersion curve in combination with finite element modeling; S5. Conduct ultrasonic guided wave detection experiments on the concrete-filled steel arch structure according to S3 and S4, and export the experimental data; S6. Process the experimental data exported in S5, and compare and analyze the experimental data with the simulation and theoretical data.

[0006] Preferably, the rotating coordinate system in S1 is as follows: Any point ([[]] x , y , z ) in the Cartesian coordinate system is represented by the quasi-cylindrical coordinate system as shown in formula (1): (1); where x , y , z represent the coordinates of the point in the Cartesian coordinate system, r represents the perpendicular distance from the point to the z axis, i.e., the radial coordinate, represents r and x the angle between the positive direction of the z axis and the q axis, i.e., the azimuth angle, ' is the axial coordinate of the point in the quasi-cylindrical coordinate system,

[0007] Preferably, in S2, the semi-analytical finite element method is used to establish the wave motion characteristic equation of the concrete-filled steel arch, and the guided wave dispersion characteristic equation of the concrete-filled steel arch is obtained, as shown in formula (2): (2); In the formula, is the circular frequency; K 1, K 2, and K 3 are the system stiffness matrices; M is the system mass matrix; is the global nodal displacement vector; is the wave number; i is the imaginary number.

[0008] Preferably, in S3, the theoretical wave number, phase velocity and group velocity dispersion curves of the structure are calculated according to the wave motion characteristic equation of the concrete filled steel tubular arch structure, and the specific solution formulas for the group velocity and phase velocity are as follows: (3); (4) In the formula, is the phase velocity; is the group velocity; is the right eigenvector; is the left eigenvector.

[0009] Preferably, in S3, the low-dispersion bending modes are selected as the optimal monitoring modes according to the vibration dispersion characteristics, and the optimal excitation frequency range is determined to be 0.5 kHz to 2.5 kHz.

[0010] Preferably, in S4, a three-dimensional finite element model of the concrete filled steel tubular arch is constructed according to S1, S2 and S3, and the material properties, boundary conditions and structural interface peeling defects are defined; Among them, the material properties include steel density, steel Young's modulus, steel Poisson's ratio, concrete density, concrete Young's modulus and concrete Poisson's ratio; The simulated wave number, phase velocity and group velocity dispersion curves are extracted and compared with the theoretical solutions in S3 to verify the model accuracy.

[0011] Preferably, in S5, an electromagnetic ultrasonic array sensor is selected in the ultrasonic guided wave detection experiment to collect the time domain signal and extract the measured dispersion characteristics. Ultrasonic guided waves are emitted to the concrete filled steel tubular arch specimen, and the experimental data including the propagation distance, amplitude and time difference are recorded simultaneously.

[0012] Preferably, in S6, damage location is carried out through the reflection and transmission coefficients in time-frequency analysis, and the material damage degree is reflected by the energy attenuation coefficient. The experimental data are compared with the simulation and theoretical data, and the accuracy of the theoretical model is verified by combining error analysis.

[0013] Therefore, the present invention adopts the above-mentioned detection method for concrete filled steel tubular arch bridges based on the dispersion characteristics of ultrasonic guided waves, and the beneficial effects are as follows: (1) The present invention proposes a non-destructive monitoring method for concrete filled steel tubular arch structures. By optimizing the sensing configuration and signal analysis algorithm, while ensuring the operation convenience, the deformation detection accuracy of millimeter level is achieved, which is especially suitable for the long-term health monitoring of large curvature bridge structures.

[0014] (2) The curvature change of the arch structure complicates the propagation path of guided waves. The waves may propagate along curves or scatter, increasing the difficulty of signal analysis. At the interface between the steel pipe and concrete, reflection, refraction, and mode conversion (such as longitudinal wave to transverse wave) are likely to occur, resulting in signal mixing. The present invention constructs a multi-physical-field coupling detection model. By combining finite element simulation and experimental verification, the characteristic frequency band of 0.5 - 2.5 kHz is optimally selected as the excitation frequency, and the bending mode is determined as the sensitive detection mode. While solving the above problems, the detection accuracy and efficiency are improved.

[0015] The technical solution of the present invention will be further described in detail below with reference to the drawings and embodiments. Brief Description of the Drawings

[0016] Figure 1 It is the overall flowchart of the embodiment of the detection method for arch-shaped concrete-filled steel tube bridges based on the dispersion characteristics of ultrasonic guided waves of the present invention; Figure 2 It is the three-dimensional model diagram of the arch-shaped concrete-filled steel tube arch structure of the embodiment of the detection method for arch-shaped concrete-filled steel tube bridges based on the dispersion characteristics of ultrasonic guided waves of the present invention; Figure 3 It is the quasi-cylindrical coordinate system diagram of the arch-shaped concrete-filled steel tube arch structure of the embodiment of the detection method for arch-shaped concrete-filled steel tube bridges based on the dispersion characteristics of ultrasonic guided waves of the present invention; Figure 4 It is the phase velocity dispersion curve diagram of the arch-shaped concrete-filled steel tube arch structure of the embodiment of the detection method for arch-shaped concrete-filled steel tube bridges based on the dispersion characteristics of ultrasonic guided waves of the present invention; Figure 5 It is the wave number dispersion curve diagram of the arch-shaped concrete-filled steel tube arch structure of the embodiment of the detection method for arch-shaped concrete-filled steel tube bridges based on the dispersion characteristics of ultrasonic guided waves of the present invention; Figure 6 It is the group velocity dispersion curve diagram of the arch-shaped concrete-filled steel tube arch structure of the embodiment of the detection method for arch-shaped concrete-filled steel tube bridges based on the dispersion characteristics of ultrasonic guided waves of the present invention; Figure 7 It is the schematic diagram of the ultrasonic guided wave detection system of the embodiment of the detection method for arch-shaped concrete-filled steel tube bridges based on the dispersion characteristics of ultrasonic guided waves of the present invention. Specific Embodiments

[0017] The technical solution of the present invention will be further described below with reference to the drawings and embodiments.

[0018] Unless otherwise defined, the technical terms or scientific terms used in the present invention shall have the ordinary meanings understood by those of ordinary skill in the field to which the present invention belongs.

[0019] The present invention is a method for detecting arch concrete-filled steel tubular bridges based on the dispersion characteristics of ultrasonic guided waves. Aiming at the problem of analyzing the propagation characteristics of guided waves caused by the complex geometric configuration and material interface coupling effect of the composite structure, the algorithm is optimized on the basis of the traditional semi-analytical finite element method. By constructing a rotating coordinate system, the accurate parametric representation of the spatial curved surface of the arch rib is realized. Combining the nonlinear perturbation theory with the multi-modal dispersion characteristics of guided waves, a multi-parameter joint evaluation model based on the wave number change, phase velocity change, and group velocity change is finally derived, and a quantitative mapping relationship between the guided wave characteristic parameters and key quality indicators such as structural interface peeling defects is established. This method significantly improves the numerical simulation accuracy of the guided wave propagation path in complex curved panel and shell structures by introducing tensor analysis tools to process the wave equation in the rotating coordinate system, providing a new theoretical framework for the non-destructive testing of key components of arch bridges.

[0020] As Figure 1 shown, the method for detecting arch concrete-filled steel tubular bridges based on the dispersion characteristics of ultrasonic guided waves includes the following steps: S1. Based on the arch structure of the arch concrete-filled steel tube as Figure 2 shown, establish a rotating coordinate system as follows: As Figure 3 shown, represent any point ([[]] x , y , z ) in the Cartesian coordinate system with the quasi-cylindrical coordinate system as shown in formula (1): (1).

[0021] Among them, x , y , z represent the coordinates of the point in the Cartesian coordinate system, r represents the perpendicular distance from the point to the z axis, i.e., the radial coordinate, represents r and x the included angle between the positive direction of the z axis and the q axis, i.e., the azimuth angle, ' is the axial coordinate of the point in the quasi-cylindrical coordinate system,

[0022] It should be noted that the quasi-cylindrical coordinate system used in the present invention has a curved axis along the curvature of the elbow, rather than the straight z-axis of the cylindrical coordinate.

[0023] S2. Use the semi-analytical finite element method to establish the wave motion characteristic equation of the arch concrete-filled steel tube, and obtain the guided wave dispersion characteristic equation of the arch concrete-filled steel tube as shown in formula (2): (2); Wherein, is the circular frequency; K 1, K 2 and K 3 are the system stiffness matrices; M is the system mass matrix; is the global nodal displacement vector; is the wave number; i is an imaginary number.

[0024] S3. Calculate the theoretical wave number, phase velocity and group velocity dispersion curves of concrete-filled steel tubular arch according to the wave motion characteristic equation of concrete-filled steel tubular arch as Figures 4 - 6 shown. The specific solution formulas for the group velocity and phase velocity are: (3); (4); Wherein, is the phase velocity; is the group velocity; is the right eigenvector; is the left eigenvector.

[0025] Analyze the vibration dispersion characteristics (wave number dispersion curve, phase velocity dispersion curve and group velocity dispersion curve) of the guided wave modes of the concrete-filled steel tubular arch structure at different frequencies, and obtain the corresponding modes of the concrete-filled steel tubular arch structure at different frequencies. Screen the low-dispersion bending mode as the optimal monitoring mode, and determine that the optimal excitation frequency, that is, the best detection frequency range, is 0.5 kHz to 2.5 kHz.

[0026] S4. Construct a three-dimensional finite element model of the concrete-filled steel tubular arch structure according to S1, S2 and S3, and define the material properties, boundary conditions and structural interface peeling defects; among them, the material properties include steel density, steel Young's modulus, steel Poisson's ratio, concrete density, concrete Young's modulus and concrete Poisson's ratio.

[0027] Combine the finite element modeling to draw the simulation dispersion curve diagram, extract the simulation wave number, phase velocity and group velocity dispersion curves and compare them with the theoretical solutions in S3 to verify the model accuracy.

[0028] S5. Conduct an ultrasonic guided wave detection experiment on the concrete-filled steel tubular arch structure according to S3 and S4. Select an electromagnetic ultrasonic array sensor to collect the time-domain signal and extract the measured dispersion characteristics. Transmit ultrasonic guided waves to the concrete-filled steel tubular arch specimen, and record and export the experimental data including the propagation distance, amplitude and time difference at the same time.

[0029] S6. Process the experimental data exported from S5, locate the damage through the reflection and transmission coefficients in time-frequency analysis, and reflect the degree of material damage through the energy attenuation coefficient. Compare the experimental data with the simulation and theoretical data, and verify the accuracy of the theoretical model by combining error analysis.

[0030] Example 1

[0031] The mathematical model of the semi-analytical finite element method is applicable to stress waves propagating in waveguide media in vacuum. This method is applicable to any complex cross-section, considering that the finite element cells are composed of small quadrilaterals on the plane and curved edges in the

[0032] direction. u The relationship between strain and harmonic displacement can be expressed as: In the formula, L is the differential operator, represents r the angle between x and the positive direction of the z axis, that is, the azimuth angle, , , z differential operators in the direction, r represents the derivative of with respect to z and

[0033] represents the derivative of with respect to After sorting, the strain matrix is obtained as follows: , are the strain matrices, N represents the shape function, , represent the shape functions in the r and directions.

[0034] For viscoelastic waveguide media, the matrix C is a complex matrix to describe the damping characteristics of the material. When damping is ignored, the material stiffness matrix C is: (7); In the formula, C is the material stiffness matrix, E is the elastic modulus, v is the Poisson's ratio.

[0035] The material constitutive relation is: (8); Wherein, is the stress; is the strain.

[0036] Combining the above formula, the control equation of the waveguide structure is obtained: (9); Wherein, is the circular frequency; K 1, K 2 and K 3 are the system stiffness matrices; M is the system mass matrix; is the global nodal displacement vector; is the wave number; i is an imaginary number.

[0037] Convert formula (9) into a first-order wave number characteristic equation as follows: (10); (11).

[0038] For any given frequency , solving equation (10) gives 2M eigenvalues and the corresponding 2M eigenvectors. These eigenvalues include the wave numbers of the forward wave and the backward wave. From equation (10), 2M left eigenvectors and right eigenvectors can be obtained. The phase velocity of the m mode at frequency is expressed as: (12); Wherein, represents the phase velocity, is the wave number in the mode.

[0039] Using the eigenvalues and eigenvectors, the displacement field of the forward wave in the bending region can be obtained as: (13); Wherein, is the upper half of the right eigenvector ; is the line source loading point; is the amplitude in the m mode determined by the geometry and boundary conditions, is the displacement field, z ' is the axial coordinate of the point in the quasi-cylindrical coordinate system, represents the summation from m = 1 to m = M, e is the natural constant, exp represents the power of e, and the following content represents the power of e.

[0040] In guided wave non-destructive testing, the original signal often contains reflection, transmission, and scattering components. Through signal processing (such as time-frequency analysis, modal separation), the propagation characteristics of different modes can be effectively distinguished, the signal distortion characteristics caused by defects can be accurately extracted, and the interference caused by structural curvature and material differences can be avoided.

[0041] The time-domain signal of ultrasonic guided wave propagating in the structure can be expressed as: (14); Where, A represents the amplitude, is the attenuation coefficient, d is the propagation distance, k is the wave number, f is the frequency, t is the time, is the time-domain signal, is the pi, and cos is the cosine function.

[0042] The actual signal is composed of multimodal superposition: (15); Where, m is the guided wave mode, represents the sum of all terms corresponding to m that meet the conditions, represents the amplitude in the m mode, is the attenuation coefficient in the m mode, is the wave number in the m mode.

[0043] S6. Damage location is carried out through the reflection and transmission coefficients in time-domain analysis: (16); Where, Z 1 and Z 2 are the material impedances, R and T are the reflection coefficient and the transmission coefficient.

[0044] Then the time difference : (17); Where, d is the distance to the damage location, is the group velocity.

[0045] The short-time Fourier transform formula is: (18); Among them, is a discrete signal; is a window function, represents the result of the short-time Fourier transform, which is a function of time t and frequency f, is a complex exponential function, represents the signal variable, represents the integral operation of variable from negative infinity to positive infinity.

[0046] Damage location is represented by the group velocity and the time difference: (19); Among them, represents the time difference.

[0047] The degree of material damage is reflected by the energy attenuation coefficient: (20); Among them, is the initial amplitude, A is the received amplitude.

[0048] First, the present invention solves the waveguide equation of the concrete-filled steel tubular arch structure by establishing a numerical model, obtains the dispersion curves of its phase velocity and energy velocity, and systematically analyzes the vibration dispersion characteristics of the structure. Subsequently, based on the finite element modal analysis method, the dominant modes with lower dispersion characteristics are selected, and at the same time, it is ensured that the energy velocity of the selected modes at the corresponding frequencies is significantly different from that of other modes, which is beneficial to the accurate identification and in-depth analysis of the waveform signals in subsequent experiments. Through this comprehensive research method, the effective modes suitable for structural health monitoring and their corresponding excitation frequency ranges can be scientifically determined.

[0049] After completing the selection of the monitoring modes and the preliminary range definition of the excitation frequencies, the present invention further conducts a refined identification of the optimal excitation frequencies. The acoustic elastic scaled boundary finite element method is used to systematically solve the frequency-energy velocity correlation curve and the energy velocity sensitivity distribution curve of the structure, and by identifying the peak positions of the energy velocity sensitivity, the most suitable excitation frequency parameters are accurately determined. This method realizes the coupled optimization of the excitation frequency selection and the structural dynamic characteristic response, and provides a theoretical basis and implementation benchmark for subsequent experimental research.

[0050] In summary, the most suitable excitation frequencies and the corresponding modes can be determined by using the method proposed in the present invention.

[0051] In one embodiment, the placement method of the guided wave sensor is designed according to the vibration mode diagrams of the concrete-filled steel tubular arch structure and the optimal mode as Figure 7 shown.

[0052] Figure 7 The ultrasonic guided wave device in

[0053] is responsible for generating and controlling ultrasonic guided wave signals. It generates electrical signals with specific frequencies and amplitudes through an internal circuit to simulate the excitation and propagation process of ultrasonic guided waves in the specimen.

[0054] The preamplifier amplifies the weak electrical signals output by the ultrasonic guided wave device, enhances the signal intensity, ensures that the subsequent oscilloscope can clearly collect the signals, and avoids detection errors caused by weak signals.

[0055] The oscilloscope observes and records the waveforms of electrical signals in real time, displays the time-domain characteristics (such as amplitude, time difference) of the propagation of ultrasonic guided waves, and provides raw data for analyzing the interaction law between guided waves and defects.

[0056] The arch-shaped concrete-filled steel tube specimen is used as the detection object, which contains interface peeling defects inside, and is used to verify the recognition ability of ultrasonic guided waves for the damage of this structure. A circle of exciters and receivers are respectively arranged at both ends of the arch-shaped concrete-filled steel tube. Among them, the exciter converts the electrical signal of the ultrasonic guided wave device into ultrasonic guided waves and couples them to the surface of the arch-shaped concrete-filled steel tube specimen to simulate the "emission" of guided waves; the receiver captures the guided waves propagated in the specimen, converts them into electrical signals and transmits them back, realizing the "reception" of guided waves and completing the signal closed-loop.

[0057] The logic of the detection process is as follows: Signal generation: The computer issues an instruction → the ultrasonic guided wave device generates an electrical signal → the preamplifier enhances the signal.

[0058] Guided wave excitation: The exciter converts the electrical signal into ultrasonic guided waves and transmits them into the specimen, which propagates along the arch structure.

[0059] Signal reception: The receiver captures the propagated guided waves, converts them back into electrical signals → the oscilloscope displays the waveform → transmits them back to the computer for analysis.

[0060] Data analysis: The computer extracts the dispersion characteristics through time-frequency analysis (such as short-time Fourier transform), and combines with theoretical / simulation data for comparison to judge the damage of the specimen (such as defect location, degree).

[0061] Among them, the material properties of the arch-shaped concrete-filled steel tube arch structure are shown in Table 1 below: Table 1 Material properties of the arch-shaped concrete-filled steel tube arch structure

[0062] Therefore, the present invention adopts the above-mentioned detection method for arch concrete-filled steel tubular bridges based on the dispersion characteristics of ultrasonic guided waves, uses the multi-physical field coupling simulation technology to establish a guided wave propagation model, integrates material anisotropy and interface contact effects, and uses electromagnetic ultrasonic array sensors to improve the signal acquisition efficiency. The deep learning dispersion compensation algorithm is introduced to break through the technical bottlenecks of traditional modal separation and signal decoupling, and form a complete technical system integrating theoretical modeling, simulation optimization and experimental verification. This method provides a high-precision and highly adaptable engineering solution for the structural health monitoring of arch bridges, and significantly improves the reliability and practicability of damage identification for complex curvature structures.

[0063] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify or equivalently replace the technical solutions of the present invention, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A detection method for arch concrete-filled steel tubular bridges based on the dispersion characteristics of ultrasonic guided waves, characterized in that, It includes the following steps: S1. Based on the concrete-filled steel tubular arch structure, establish a rotating coordinate system; S2. Use the semi-analytical finite element method to establish the wave motion characteristic equation of the concrete-filled steel tubular arch, and obtain the dispersion characteristic equation of guided waves in the concrete-filled steel tubular arch; S3. Calculate the theoretical wave number, phase velocity and group velocity dispersion curves of the concrete-filled steel tubular arch, analyze the vibration dispersion characteristics of guided wave modes at different frequencies, and determine the optimal monitoring mode and excitation frequency range; S4. According to S1, S2 and S3, construct a three-dimensional finite element model of the concrete-filled steel tubular arch structure, and draw a simulation dispersion curve by combining finite element modeling; S5. Conduct an ultrasonic guided wave detection experiment on the concrete-filled steel tubular arch structure according to S3 and S4, and export the experimental data; S6. Process the experimental data exported in S5, and compare and analyze the experimental data with the simulation and theoretical data.

2. The inspection method for an arch concrete-filled steel tubular bridge based on the dispersion characteristics of ultrasonic guided waves according to claim 1, wherein The rotating coordinate system in S1 is as follows: Any point ([ x x , y , z ) in the Cartesian coordinate system is represented in the quasi-cylindrical coordinate system [[ as shown in Equation (1): （1）； Among them, x , y , z represent the coordinates of a point in the Cartesian coordinate system, r represents the perpendicular distance from the point to z the axis, i.e., the radial coordinate, represents r the angle between x and the positive direction of the axis, i.e., the azimuth angle, z ' is the axial coordinate of the point in the quasi-cylindrical coordinate system, q is the polar coordinate radius, is the central angle.

3. The inspection method for an arch concrete-filled steel tube bridge based on the dispersion characteristics of ultrasonic guided waves according to claim 2, wherein In S2, use the semi-analytical finite element method to establish the wave motion characteristic equation of the concrete-filled steel tubular arch, and obtain the dispersion characteristic equation of guided waves in the concrete-filled steel tubular arch, as shown in formula (2): （2）； In the formula, is the circular frequency; K 1, K 2, and K 3 are the system stiffness matrices; M is the system mass matrix; is the global nodal displacement vector; is the wave number; i is the imaginary number.

4. The inspection method for an arch concrete-filled steel tube bridge based on the dispersion characteristics of ultrasonic guided waves according to claim 3, wherein In S3, according to the wave motion characteristic equation of the concrete-filled steel tubular arch structure, calculate the theoretical wave number, phase velocity and group velocity dispersion curves of the structure. The specific solution formulas for the group velocity and phase velocity are: （3）； （4）； In the formula, is the phase velocity; is the group velocity; is the right eigenvector; is the left eigenvector.

5. The method for detecting an arch concrete-filled steel tubular bridge based on the dispersion characteristics of ultrasonic guided waves according to claim 4, wherein In S3, select the low-dispersion bending mode as the optimal monitoring mode according to the vibration dispersion characteristics, and determine the optimal excitation frequency range to be 0.5 kHz to 2.5 kHz.

6. The inspection method for an arch concrete-filled steel tubular bridge based on the dispersion characteristics of ultrasonic guided waves according to claim 5, wherein In S4, according to S1, S2 and S3, construct a three-dimensional finite element model of the concrete-filled steel tubular arch, and define the material properties, boundary conditions and structural interface peeling defects; Among them, the material properties include steel density, steel Young's modulus, steel Poisson's ratio, concrete density, concrete Young's modulus and concrete Poisson's ratio; Extract the simulation wave number, phase velocity and group velocity dispersion curves and compare them with the theoretical solutions in S3 to verify the model accuracy.

7. The inspection method for an arch concrete-filled steel tube bridge based on the dispersion characteristics of ultrasonic guided waves according to claim 6, wherein In S5, in the ultrasonic guided wave detection experiment, select an electromagnetic ultrasonic array sensor to collect the time-domain signal and extract the measured dispersion characteristics. Transmit ultrasonic guided waves to the concrete-filled steel tubular arch specimen, and record the experimental data including the propagation distance, amplitude and time difference at the same time.

8. The method for detecting an arch concrete-filled steel tubular bridge based on the dispersion characteristics of ultrasonic guided waves according to claim 7, wherein In S6, locate the damage through the reflection and transmission coefficients in time-frequency analysis, and reflect the material damage degree through the energy attenuation coefficient. Compare the experimental data with the simulation and theoretical data, and verify the accuracy of the theoretical model by combining error analysis.

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